Optimization Models for Rail Car Fleet Management [1 ed.] 0128151544, 9780128151549

Table of contents :
Cover
OPTIMIZATION
MODELS FOR RAIL
CAR FLEET
MANAGEMENT
Copyright
Biographies
Acknowledgments
Executive summary
1
Introduction
Problem statement
Main objectives
Structure of the book
2
Review of the models for rail freight car fleet management
Operational models
Empty freight car inventory management problems
Empty freight car distribution
Deterministic approaches
Stochastic approaches
Hybrid approaches
Freight car pooling concept
Combined allocation of empty and loaded cars
Tactical and strategic models for freight car management
Service network design problem
Demand estimation
Models for production and attraction
Distribution models
Models for modal split
Assignment models
Rail freight car fleet sizing models
3
Centralized and decentralized model for empty freight car scheduling
Characteristics of decentralized managerial-control functions in the process of empty freight car allocation
Centralized network model
Solution method of triaxial transportation problem
Basic solution
Feasible solution
Final solution
Problem of degeneration
Decentralization of the main managerial functions in the empty car allocation process
Decentralized terminal model of empty freight car allocation
Validation of the proposed model
4
Fuzzy multiobjective rail freight car fleet composition
The best rail freight car fleet mix problem
Classification of freight cars
Utilization of freight cars according to their characteristics
Proposed problem solution
The selection of relevant criteria
Analytic Hierarchy Process
Theoretical basics of AHP
Mathematical basics of AHP
Fuzzy extension of AHP method
Fuzzy AHP method for the best rail freight car fleet mix problem
The best rail freight car fleet size problem
Theoretical foundations of the problem solution
Fuzzy multiobjective linear programming
Statement and solution of the problem
5
Fuzzy random model for rail freight car fleet management based on optimal control theory
Fuzzy preliminaries
Fuzzy sets
Fuzzy numbers
Triangular fuzzy matrices
The main features of triangular fuzzy matrices
Inverse triangular fuzzy matrices
Necessary and sufficient conditions for the invertibility of fuzzy matrices
Application of Rohn's scheme for determining of the inverse fuzzy matrix
Defuzzification of triangular fuzzy numbers
Fuzzy random variables
Fuzzy stochastic model for rail freight car fleet sizing and allocation
Model parameters
Objective functional
Problem constraints
Fuzzy state vector estimation
Proposed approach for solving the problem based on the fuzzy linear quadratic Gaussian regulator
The components of weighting matrices A, B, and L
Choosing the components of the fuzzy weighting matrix ΓP
Numerical experiments
Comparative analysis of the results of the fuzzy random and random model
6
Stochastic model for heterogeneous rail freight car fleet management based on the model predictive control
Discrete time MPC framework for rail freight car fleet sizing and allocation problem
Design variables
System performance measure and constraints
System dynamics
Nonnegative control constraints
Nonnegative state constraints
Capacity constraints
State vector estimation
Forecasting the state of rail freight cars by ARIMA-Kalman method
Components of weighted matrices A, B, and L
The components of matrix Γ(P)
MPC controller
Optimization problem
Detailed description of the MPC approach
Numerical experiments
7
Distributed and decentralized approaches for rail freight car management
Distributed model predictive rail freight car management
Problem description
Cooperative MPC for freight car flow planning
Decentralized model predictive rail freight car management
Numerical example
8
Conclusions
References
Index
A
B
C
D
E
F
G
H
I
K
L
M
N
O
P
Q
R
S
T
V
Back Cover

Citation preview

OPTIMIZATION MODELS FOR RAIL CAR FLEET MANAGEMENT

OPTIMIZATION MODELS FOR RAIL CAR FLEET MANAGEMENT

MILOS MILENKOVIC  NEBOJŠA BOJOVIC

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States © 2020 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-815154-9 For information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Joe Hayton Acquisition Editor: Brian Romer Editorial Project Manager: Andrae Akeh Production Project Manager: Anitha Sivaraj Cover Designer: Christian J. Bilbow Typeset by SPi Global, India

Biographies Milos Milenkovic is an Assistant Professor at The Faculty of Transport and Traffic Engineering, University of Belgrade, Serbia and Research Fellow at the MIT-Zaragoza International Logistics Program, Zaragoza, Spain. Dr. Milenkovic has a PhD degree of Technical Sciences in the field of Traffic and Transportation with a focus on rail freight car scheduling and fleet sizing problems, Magisterial degree of Technical Sciences with a focus on train dispatching problems and MSc degree based on railway-related intelligent transportation systems from the Faculty of Transport and Traffic Engineering, University of Belgrade, Serbia. Dr. Milenkovic’s main research areas are mathematical optimization, model predictive control theory, time series analysis, project management, and engineering economy with a special focus on transport-related applications. He was engaged as the member of editorial board of a number of scientific conferences and also as a reviewer of a number of prestigious peer-reviewed journals, like “Transportation Research Part C,” “Transportation Research Part E,” “Applied Mathematical Modeling,” “International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems” and others. During his career, Dr. Milenkovic published more than 50 papers in journals as well as on scientific conferences. Dr. Milenkovic has also been employed for 2 years in Public Enterprise “Serbian Railways” as a leading engineer for traffic organization and timetable construction. Nebojsˇa Bojovic is a Full Professor and Dean at the Faculty of Transport and Traffic Engineering, University of Belgrade, Serbia. His professional expertise, research, and teaching interests include Applications of Operations Research, Information Systems, and Control Theory in Transportation Systems with particular emphasis on Rail freight car fleet sizing, Railway vehicle routing and scheduling, Risk management, and Project management. He is the author of 4 monographs, 2 chapters in international monographs, over 30 scientific publications bridge appeared in international journals, as well as over 70 papers presented at international and national conferences. His professional engagement includes membership in Euro Working Group on Transportation, Euro Working Group on Multicriteria Decision Aiding, Academy of Engineering Sciences of Serbia, Board of Directors of the Serbian Tariffs Association, Yugoslav Society of Operational Research, and Yugoslav Association for Project Management.

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Biographies

He was a member of the Organizational and Scientific Committees at five international scientific conferences. He is also a representative of the Faculty of Transport and Traffic Engineering within the international associations European Association for Research in Transportation (hEART) and EUropean Rail Research Network of Excellence (EURNEX). He is a reviewer in a number of high-ranked international journals, such as Transportation Research Part B, Transportation Research Part E, European Journal of Operational Research, Computational and Mathematical Organization Theory, and other.

Acknowledgments This book represents the result of a multiyear research effort dedicated to the field of rail freight car fleet management. In this regard, I would like to express my sincere gratitude to Professor Nebojsˇa Bojovic, the Dean of the Faculty of Transport and Traffic Engineering, who has not only provided me with guidance in my professional career but has also broadened my outlook of life. I would like to express my sincere thankfulness to Professor Susana Val, the Director of the Zaragoza Logistics Center, for her support during my work on this book. The book is supported by the project “Clusters 2.0: Open Network of Hyper-connected Logistics Clusters Towards Physical Internet,” which has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 723265 and project MNTR036022 “Critical infrastructure management for sustainable development in postal, communication, and railway sector of Republic of Serbia.”

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Executive summary Freight transportation represents one of the most important economic activities. A significant portion of the associated operations has been conducted by railways known for their ability to offer cost-effective long-haul transportation. Yet, the share of rail freight is insignificant comparing to road transport. One of the reasons for this lies in the inefficiency of the service provided, especially concerning the duration and reliability of on-time delivery. Railway freight transportation is performed by repositioning of rail freight cars. Considering that the freight cars represent capital assets of high value, the freight car fleet management represents a very important activity for the researchers and transport service providers as well. Research and development of rail freight car fleet management models have been in the focus of theorists and practitioners since the middle of the last century with almost equal intensity. Tasks that appear in the formulation of this problem are usually considered in the literature as simple for description and very hard for solving. The stochastic and dynamic nature of the supply, demand, and traveling time of freight cars, as well as the capacity constraints on a railway network, represent a challenge for every researcher. This book represents the result of multiyear efforts to provide a reader with an insight into one of the most important areas of railway transport management. Mathematical procedures for the effective and efficient utilization of railway freight cars have been developed by taking into account demand satisfaction and minimization of total costs. Based on an assessment of existing approaches, new models for rail freight car fleet management are presented. The developed models are solved by a variety of optimization methods, in case of exact, stochastic, fuzzy, and fuzzy stochastic parameters. Some models also tackle heterogeneity and partial substitutability of freight cars. Different control schemes are considered, centralized where the complete system is modeled and all the control inputs are computed in one optimization problem, as well as the distributed or decentralized control schemes, where local control inputs are computed using the local measurements and reduced order models of the local dynamics.

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CHAPTER 1

Introduction Contents 1.1 Problem statement 1.2 Main objectives 1.3 Structure of the book

1 3 4

Since its foundation, operational research has been used for solving transportation-related problems. Under the influence of the significant competition between different transport modes and the continually increasing pressure from the service users, a need for the development of sophisticated models for the optimization of complex business processes in transportation has been recognized. Deregulation has led to the self-transformation of railway companies from the concept of providing cheap services of low quality toward highly reliable and flexible actor on the transport market. In railway undertakings, freight cars have a significant share in total investments, and the efficient utilization of this capital resource is of essential importance for the survival and improvement of the competitive position of the railway sector on the market.

1.1 Problem statement In the past, when the production of bulk cargo has dominated the world economy, the concept of providing low quality, cheap services was sufficient to enable high profitability of railway companies. However, changes in the economy that were oriented toward incorporating more efficient inventory policies have led to the unsuitability of policy-making based only on the adjustment of business processes with demand increase. Consequently, with the aim of maintaining profitability and competitiveness, railway companies reinvest in advanced technology along with the improvement of their management practices. Railway freight cars represent one of the highest capital investments for most of the railway undertakings. However, this resource was not efficiently utilized in the past. Currently, on most modern railways there is an initiative for changing this trend. Rail freight car fleet Optimization Models for Rail Car Fleet Management https://doi.org/10.1016/B978-0-12-815154-9.00001-0

© 2020 Elsevier Inc. All rights reserved.

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Optimization models for rail car fleet management

management represents an essential factor that will contribute to the realization of this objective. The railway transport system, as well as other transport systems, is characterized by an imbalance of commodity flows. That means that the difference between ingoing and outgoing freight flows is positive in some regions while it is negative in some other regions of a railway network. This leads to a spatial and time imbalance of the supply and demand for empty rail freight cars. The complexity of the problem is additionally weighted by the existence of a number of rail freight car types. In other words, transport demand fulfilling requires the establishment of a balance and therefore it is needed to allocate empty cars with the lowest costs of transportation. Based on the previous, it is clear that an efficient rail freight car management system represents an imperative for all rail transport operators. This system, apart from the need to maximize the fulfillment of transport requests, in a great extent influences the total number of needed cars in the system, as well as the way these cars are used for satisfying transport requests. Consequently, a large rail freight car fleet has higher investments, costs of maintenance and inventories than it is necessary, whereas a shortage of rail freight cars resulting in lower service quality will be present on the contrary. Therefore, there are two main research directions in the field of rail freight car fleet management. The first one is dedicated to rail freight car scheduling. This task considers determining the necessary rail freight car flows. The aim of the second task is optimal rail freight car fleet sizing. The empty rail freight car scheduling process is composed of two phases. The first one is a planning phase whereas the second phase represents its physical allocation with minimum costs and demand satisfaction. Empty freight car transport has a large share in the total transport on a railway and, therefore, an efficient and effective distribution process represents an imperative for the decrease of capital and operational costs. In rail freight car allocation research, the size of the rail freight car fleet is treated as an input parameter, although numerous academic discussions and practical experiences have shown that there is a strong relationship between potential benefits from investments in rail freight car fleet and the operational cost decrease. Determining an investment level in a rail freight car fleet is based on establishing a balance between the costs aimed to provide sufficient capacities and the potential costs of unmet demand. For a long time, models dedicated to this problem have actually solved the task of determining the optimal fleet size for satisfying demand and most frequently with respect to the total cost minimization criteria. Lately, among the researchers appears

Introduction

3

a common consensus about the need for involving the costs of empty freight car allocation in the problem of rail freight car fleet sizing. The aim of these tendencies should be the minimization of a sum of constant capital costs and variable allocation costs. Railway undertakings are faced with the problem of managing a big number of freight cars located in different geographical locations. This problem includes repositioning of these cars in order to satisfy the actual demand (loaded cars) or dispatching (empty cars) toward locations where there is a lack of freight cars according to the known current demand, or toward stations where the demand is expected in subsequent time intervals. The minimization of ownership and utilization costs on the planning horizon represents an imperative of all railway operators.

1.2 Main objectives The three main categories of problems that have to be solved for the sake of offering a competitive transportation service are the provision of sufficient capacities, freight car allocation, and train scheduling. Determining the size of investments in rail freight cars represents the primary task for the first group of problems. The second type of problems considers the determining of empty freight car flows in order to adequately respond to asymmetrical demand between stations. The last category involves the regulation of freight car allocations in order to improve transport and the throughput capability of railways. The focus of this book is the interaction between decisions on freight car fleet sizing and decisions about utilization or freight car distribution, and the combined effects of these decisions on the capacity and efficiency of the railway system. By understanding the direct impact on the level of investments in capital resources, the potential benefits from the improved utilization of freight cars are much higher than it can be estimated solely on the basis of reduced operating costs. For the modeling of the interaction of freight car utilization and rail freight car fleet sizing, two crucial features of the railway system must be recognized: the system is dynamic because the demand is variable during the time, and there is uncertainty in both the system performances and forecasting of the future demand in the system. These two features represent the essence of all the models developed in this book. The main objective of the development of these dynamic optimization models is obtaining the answer to the main questions about defining an

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Optimization models for rail car fleet management

adequate investment policy. This involves the determining of the spaceand-time location of freight cars which are located in stations or if they move in an empty or loaded state. In addition, the second important objective is the improvement of freight car utilization by reducing the empty freight car flows. The third objective is introducing a number of realistic assumptions in the development of models for the simultaneous rail freight car fleet sizing and allocation.

1.3 Structure of the book In this book, the rail freight car fleet management problem is considered from two perspectives, the centralized and decentralized/distributed perspective. The centralized perspective assumes that a complete system is modeled and all the control inputs are computed in one optimized problem. In the decentralized/distributed control scheme, full-scale centralized control problems are decomposed into local controllers that do not cooperate (decentralized setting) or cooperate (distributed setting) by communicating to each other an information set containing local decisions. In all models, different scenarios regarding the uncertainty in demand and traveling time were considered. Chapter 2 provides an overview of the recent solution approaches for an efficient and effective rail freight car fleet management. All three levels of planning are covered. Considering the large size and complexity of the approaches, the paper contains only a textual description. The recent approaches are covered as well as the most important older works. The review is focused on railway transport, even though a lot of approaches made in the field of road and container transportation are certainly important to the context of rail freight car fleet management. Chapter 3 considers the empty freight car allocation problem from a centralized and decentralized perspective. The centralized network model is formulated as a triaxial transportation problem, which apart from supply and demand constraints also considers available capacities of stations and rail sections. The problem is solved by the method of potentials. On the other hand, the practical solution of the decentralized model of terminals is enabled by a decomposition algorithm based on a price mechanism. To conduct this procedure, the triaxial transportation problem is reformulated and Kuhn-Tucker’s theorem is applied. The obtained results demonstrate the possibility of decentralization of the empty freight car allocation process.

Introduction

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In Chapter 4, a two-step approach to the rail freight car fleet composition problem is presented. The first step represents the selection of the most preferable freight car type, series, and subseries from the aspect of the main characteristics of a certain commodity type and it is known as the best rail fleet mix problem. This problem, by its nature, represents a multicriteria decision-making problem. The outputs from this step were used as inputs for the best rail fleet size problem in which the optimal number of the most appropriate car types need to be determined. The first problem is solved using the fuzzy AHP, whereas for the second problem the fuzzy multiobjective linear programming is proposed. Chapter 5 presents a centralized perspective of the freight car fleet management problem in which the demand and traveling times have been considered as fuzzy random and fuzzy variables respectively. This chapter contains the basic premises necessary for solving the considered problem with the fuzzy stochastic optimal control approach. In this part, relevant arithmetic operations with triangular fuzzy numbers and triangular fuzzy matrices are described. The chapter contains a mathematical formulation of the problem in the fuzzy sense and a solution based on the general system theory. Numerical experiments have been given in the last part of this chapter. Model predictive control (MPC) is an increasingly significant and popular control approach because of its ability to handle the constraints on inputs, states, and outputs. Therefore, Chapter 6 presents a general MPC framework for the simultaneous optimization of the size and allocation for the heterogeneous rail freight car fleet, where the partial substitutability among various freight car types is considered. The model includes a specification of the rail freight car demands and traveling times as stochastic variables and considers the state, control, and station capacity constraints while remaining linear and thus generally tractable with quadratic costs. The chapter contains a discrete time MPC framework for the rail freight car fleet sizing and allocation problem with a description of design variables, system performance measures and constraints and the state vector estimation. Then, an MPC controller for the problem is proposed. This chapter finalizes with a numerical example to illustrate the results. Chapter 7 considers a different perspective in comparison to the modeling approaches presented in the previous two chapters. In Chapter 7, the rail freight car allocation problem is solved by applying the distributed and decentralized MPC. Both approaches consider the homogeneous freight car fleet and deterministic demand. The overall optimization problem is

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divided into subproblems, which consider the freight car allocation decisions on their own territory. Each decision-making unit or controller solves its own local problem using the model covering only its part of the railway network. Local problems are solved using MPC. These regional decision-making units or controllers are mutually connected by interconnecting links—railway lines on which the empty and loaded freight car flows between regions are interchanged. The decentralized MPC approach assumes that control agents make control decisions independently of each other. Information exchange between control agents is only allowed before and after the decision-making process. In the distributed MPC setting, local problems are not isolated but depend on the solution of the MPC problem of the surrounding controllers. This means that local MPC problems are solved in a cooperative way by enabling iterated communication between controllers until a compromise solution is reached between different controllers. Chapter 8 contains concluding remarks and future research directions. The presented approaches have been summarized and conclusions described. This chapter also emphasizes the original scientific contributions and potential directions for further research.

CHAPTER 2

Review of the models for rail freight car fleet management Contents 2.1 Operational models 2.1.1 Empty freight car inventory management problems 2.1.2 Empty freight car distribution 2.1.3 Freight car pooling concept 2.1.4 Combined allocation of empty and loaded cars 2.2 Tactical and strategic models for freight car management 2.2.1 Service network design problem 2.2.2 Demand estimation 2.2.3 Rail freight car fleet sizing models

7 9 11 24 27 29 29 33 42

This chapter contains a comprehensive review of relevant contributions to solving the problem of efficient and effective rail freight car management. Dejax and Crainic (1987), Cordeau et al. (1998), and Milenkovi
c et al. (2015a, b) presented a detailed review of the planning models in railway freight transport. The models for the rail freight car management problem can be classified as (Fig. 2.1): •

Operational models that consider short-term problems like the rail freight car inventory management problem, determining of empty freight car flows between a pair of origin-destination stations, and combined management of empty and loaded freight car flows. • Tactical and strategic models for solving the midterm and long-term planning problems. These models include service network design, freight car demand modeling, and rail freight car fleet sizing.

2.1 Operational models The purpose of operational models is the efficient management of a fixed rail freight car fleet, or more precisely, the cost minimization of empty freight car flows under the condition of transport demand satisfaction. In general, problems are defined on a network where there are loaded and empty flows. Optimization Models for Rail Car Fleet Management https://doi.org/10.1016/B978-0-12-815154-9.00002-2

© 2020 Elsevier Inc. All rights reserved.

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8 Optimization models for rail car fleet management

Rail freight car management models

Tactical and strategic

Operational

Rail freight car inventory management

Empty freight car distribution

Combined Freight car allocation of pooling concept empty and loaded freight cars

Hybrid Deterministic Stochastic approaches approaches approaches

Fig. 2.1 Classification of models for rail freight car management problem solving.

Demand estimation

Service network design problems Attraction/ production models

Rail freight car fleet sizing models

Distribution Modal split Assignment models models models

Review of the models for rail freight car fleet management

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In some nodes of the network, such as the railway yards, terminals, or depots, the demand and supply of freight cars are known in advance. The models consider the empty freight car inventory management problems in the stations on the network or the problems of empty freight car allocation between different origin-destination pairs of stations under the constraint of transport demand satisfaction.

2.1.1 Empty freight car inventory management problems Railway freight cars are allocated depending on the existing supply and demand of empty freight cars in the stations on the network. One of the problems considered is also the maintenance of an adequate supply of empty freight cars in the stations on the network. Namely, in everyday realization of planning activities in railway freight transport, car dispatchers have to define the necessary level of cars in inventories and the moment when they have to order freight cars to satisfy the demand for freight transport, to protect from demand variations, and to minimize the expected total cost. Operational costs in railway freight transport can be decomposed into three categories: train costs, station operational costs, and freight car costs. The last component includes the freight car ownership and inventory cost. Total inventory costs represent a sum of all station inventory costs that include: the cost of freight car supply for each station, the holding cost, and the cost of shortage due to the fact that the sufficient number of freight cars for the satisfaction of the actual demand is not available. In every railway company, there is a freight car inventory problem. There are uncertain demand and unreliable deliveries of empty cars in all stations on the network. Therefore, under the assumption that a railway company wishes to completely satisfy the demand for transport service, the need for freight car inventories in every station becomes a necessity. Due to the different demand in different stations and limited capacity of freight cars, a shortage or surplus of freight cars may exist. Different demand rates will generate different replenishment periods and order sizes. Stations with a higher rate of freight car utilization need a higher frequency of inventory replacement or a higher number of cars in orders. Therefore, when the customer comes into the station, his requests may not be satisfied due to an inappropriate inventory car policy which includes the duration of the inventory replenishment cycle, the reordering inventory level, and the number of cars in each order. The remaining part of this section includes a review of the

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Optimization models for rail car fleet management

Table 2.1 Characteristics of models for freight car inventory management Authors

Year

Type of model

Solution approach

Philip and Sussman Mendiratta and Turnquist Milenkovic and Bojovic

1977

Discrete event simulation

Simulation

1982

Two-level decentralized optimization model Fuzzy EOQ model

Dantzig-Wolfe decomposition Differential calculus

2014

most important approaches for freight car inventory management. Table 2.1 contains the most important characteristics of different models. Philip and Sussman (1977) formulated a discrete event simulation model for determining the optimal inventory level for a station as a function of the daily supply and demand variations as well as the cost of holding the cars that are waiting to be loaded, with regard to the cost of car shortage for the transport demand satisfaction. This inventory model represents an alternative to the traditional methodologies of transport optimization for the empty car allocation problems (White and Bomberault, 1969). The applicability of the model on real railway inventory problems was presented. As a conclusion, the authors emphasize the influence that the variability of demand and supply has on the number of freight cars needed in the inventory of a certain station. The authors also recommended integrating the recommended station inventory model into a comprehensive set of models for establishing the balance of freight car inventories on the entire railway network. Mendiratta and Turnquist (1982) presented a successful attempt to define a general model for freight car inventory management on the entire railway network. As a basic reason for problems of empty car allocation, the authors pointed out the problem of insufficient coordination between the decisions made on a central level and the local decisions in individual stations and therefore, they applied railway case methodologies developed from Baumol and Fabian (1964), Lasdon (1970), and Jennergren (1971) for the optimization of decentralized systems. The resulting model includes mutually interacting submodels, the network model, and the terminal model. Each of them represents activities performed on a certain level of the railway system: centralized decision-making on a corporative level that considers flows on the entire railway network, whereas the decisions about inventory size are made in individual stations. Decisions are made on each level (corporative and station) whereas the coordination between two levels is based on the price mechanism. The aim of the system is to maximize the profit of the railway

Review of the models for rail freight car fleet management

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under the constraint of empty freight car demand and supply as well as institutional requests. For the sake of reaching optimality of the entire system, there is an iterative exchange of information between the network and the terminal model. The network model defines internal prices of a transfer (dual variables) for empty freight cars that are the inputs in the terminal model. The terminal model uses these prices for order sizing. These orders are then returned to the network model for the sake of repeating the calculation of transfer prices. Iterations are continued until the results of both models become consistent and the internal transfer prices between the models reflect the opportunity cost of freight cars. The network model is formulated as a linear programming (LP) optimization model solved by the Dantzig-Wolfe decomposition. The terminal model is a formulation inventory management that includes the stochastic demand and traveling times of empty freight cars. It differs from the inventory management models proposed by Philip and Sussman (1977) only in the formulation details. An important aspect of the model is that it functions very well even when the overall shortage of empty freight cars exists in the system. The model can be used as a tool for strategic evaluation and as an operational tool for daily empty freight car distribution. Experiments on given data from the US railways show that the model generates distribution decisions that decrease the empty car kilometers, empty allocations and empty car days without decreasing the percentage of the fulfilled demand. Milenkovic and Bojovic (2014) recommended the fuzzy inventory model for determining of optimal inventory level for each individual station as a function of uncertain daily demand and traveling time, freight car supplying cost, the holding cost, and car shortage cost. Triangular fuzzy numbers were used for representing the daily demand and traveling times between supply and demand stations. All cost components were modeled as exact parameters. The model was tested on a railway network composed of three stations, and cost optimal values of the inventory level and the reordering number of freight cars are calculated.

2.1.2 Empty freight car distribution The freight car flow cycle includes the following components: the time spent under the control of a consignor and consignee, time spent in traveling in a loaded state, and time spent in stations. The analyses conducted on most of the railways show a high percentage of cars spent in an unproductive state, waiting in the node of supply or moving empty between the node of supply

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and node of demand. Even though there are a large number of different types of freight cars in this process, they are grouped in a much lower number of classes that are then used for expressing the demand and supply of freight cars (Bojovi
c, 2007). The local dispatchers have the biggest responsibility for the empty freight car allocation. The railway network is decomposed in a number of geographical regions, in which local dispatchers have the main responsibility of demand fulfillment. Every morning, dispatchers try to fulfill transport requests with the supply of appropriate freight cars. Then, they send the report containing the available freight cars and the demand for the following day to the central system. Due to the distrust of the efficiency of allocation on the central level, local managers frequently do not report the real, but a slightly reduced number of available freight cars, expressing their doubt in the timely delivery of freight cars in the following periods. Distrust of decision making on the central level additionally reduces the number of potential car allocations and even further reduces the efficiency of the freight car allocation system. The problem of empty freight car flow optimization is often described by the classic transportation problem. Freight car allocation costs are usually modeled based on the distance between the supply and demand locations. For the defined nodes of supply and demand and the correspondent costs, the stated problem is solved by LP, network models, or some other optimization methods. Unfortunately, freight car allocation models used in practice include numerous approximations and usually ignore future requests of users or they are based on deterministic forecasts. On real railway networks, complex operations are performed with many stochastic incentives and it becomes imperative that the freight car flow management is performed under the condition of using the model to generate the sources of uncertainty. The demand for freight cars represents one of those sources, but not the most important, considering that users usually order the freight cars a certain period in advance. In addition, many of the most important users are very predictable. Probably the most important source of stochasticity for the railway is the traveling time, which can vary significantly between two nodes. The second most important source is the emergence of new empty cars. In addition to other choices, cars can also be sent to another network. The other railway may return empty cars, but without previous announcement. As a result of these activities, empty cars are coming from other networks in an unpredictable way. There is also a problem of car failures, as well as the opinions of users that the cars are unacceptable (e.g., because they are

Review of the models for rail freight car fleet management

13

not clean enough). During the decision-making process, rail freight car dispatchers strive to include all uncertainties by keeping a certain number of freight cars in reserve so that they may immediately replace alternative cars in case of some unplanned events. However, most of the time they do not have an idea about the financial aspects of keeping the cars in their stations. Consequently, keeping more cars than necessary reduces the level of their productivity. For improving the efficiency of this practice, it is necessary to find the answers to the following questions: how many additional cars and of which type are needed, where these cars need to be stored, and which decisions need to be made in case of excess of freight cars. In addition, there are periods during the year when there are demands above the expected level so that the inventory fleet will not be sufficient for fulfilling the demand. Rail freight car scheduling decisions are usually based on cost minimization or profit maximization criteria. Routing decisions have a direct impact on transport costs or the maximization of profit. Therefore, routing decisions directly affect the freight car scheduling decisions. On the other hand, in the function of empty freight car scheduling to shippers, decisions about empty freight car scheduling may influence decisions about service provision, which are under the influence of routing decisions. In fact, this means that if a sufficient number of freight cars is allocated to a certain destination from one source of empty cars, empty cars together with the demand for loaded cars in that destination can justify the direct train departing. In that sense, empty cars can be used for improving the service of the receiver of loaded cars, who in the opposite case would wait for the accumulation of a sufficient number of freight cars to justify departing toward the final destination of these cars. Decisions about empty freight car distribution influence the departing time of loaded cars. If the orders for empty cars are immediately fulfilled, they are loaded and added to the system in order to be delivered to the final destination as soon as possible. In case the orders for empty cars are not met immediately, the goods for transporting need to wait for available empty cars and therefore, they will enter the network later. Therefore, empty freight car scheduling defines the moment when the loaded shipment appears in the system, which can influence the frequency of train departing. Therefore, if only the loaded freight car departures are considered in the routing decisions, it is possible to reject a route, which will otherwise be provided by the train that is based on empty freight cars. Empty freight car flow management assumes that in every moment the status and location of freight cars on the network are known. Freight car status considers the availability of freight cars, if they are empty or loaded, or if they

14

Optimization models for rail car fleet management

are in good working condition or not. Location refers to the railway station in which the cars are located at a given time. A number of approaches to solving the problem of empty freight car allocation have been proposed in the past. A detailed review of research in this field is given by Dejax and Crainic (1987), Bojovic (2000), and Milenkovi
c et al. (2015a, b). All approaches may be categorized as deterministic, stochastic, or hybrid approaches. The most important studies within each category are given in the following sections. 2.1.2.1 Deterministic approaches The development of LP, network modeling, and algorithmic techniques during the 16 years of the last century has initiated a number of studies that use these techniques for solving the problem of optimal empty freight car scheduling in accordance with the previously defined roles and objectives of allocation. Table 2.2 summarizes the most important characteristics of the presented models. One of the first approaches was proposed by Misra (1972). It minimizes the total cost (in car hours) of imbalance in the number of empty cars of a homogeneous car fleet in the nodes of a railway network for a given time period. The transport algorithm (Ford and Fulkerson, 1974) and simplex algorithm (Dantzig, 1963) are presented as solution techniques in the case of limited capacity routes. The author suggests the transport algorithm as an efficient approach to finding the solution to three types of problems: the allocation of empty cars to shippers, determining of routes for uninterrupted traffic operation, and determining of new space locations (rescheduling locations). The imbalance between the supply and demand of freight cars characterizes many stations and therefore, there is a need for reallocation of freight cars from the nodes where they are in surplus to the nodes where there is a demand for empty cars. The task formulated in this way actually represents a classical transportation problem. The stochastic nature of the empty freight car supply and demand has been also considered and daily optimizations or using the averages are proposed as solutions. The possibility of stochastic modeling of the supply and demand in combination with the sequential application of LP has been also mentioned. Beside the reduction of the necessary number of empty freight cars, this approach defines feasible routes and enables determining of saved train hours. The freight car scheduling model on railways of Louisville and Nashville (Leddon and Wrathall, 1968; Wrathall, 1968; Association of American Railroads, 1982) and the model of optimal flow rules for Southern railways (Baker, 1977) also

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Table 2.2 Characteristics of deterministic models for empty freight car scheduling Authors

Year

Model type

Solution approach

Leddon and Wrathall White and Bomberault Misra

1968

Linear transport model Transshipment problem Linear programming

Simplex algorithm

Ouimet

1972

Herren

1973

Baker

1977

Fernandes Grain and De Sinay Spieckermann and Voß Holmberg et al.

1986 1995

Joborn et al.

2004

Lubbecke and Zimmermann Narisetty et al. Lawley et al.

2005

1969 1972

1996

2008 2008

Transshipment problem Network minimum cost flow problem Linear transport model Linear programming Job-shop scheduling problem Multicommodity network flow problem Multicommodity network flow problem Mixed integer programming Linear programming Mixed integer programming

Out-of-kilter algorithm Transportation/Simplex algorithm Out-of-kilter algorithm Out-of-kilter algorithm Simplex algorithm Simplex algorithm Greedy heuristics Lagrangian heuristics/ Branch& Bound method Tabu heuristics

Branch & Bound method Decomposition approach Sequential heuristics

represent static linear transport models with known supply and demand of empty freight cars and of a homogeneous freight car fleet. Standard simplex codes have been used for solving these models. White and Bomberault (1969) solved the same problem, but they also took into account the time perspective via time-space diagram, which schematically presents different routes that can be used for the sake of accomplishing planned arrival time and location of empty cars. This structure had been used for generating a network for minimizing the total cost of the freight car flow. The freight car fleet is homogeneous in this case, whereas the supply and demand are known. An adapted version of the out-of-kilter algorithm was used for solving this transshipment problem.

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Optimization models for rail car fleet management

Introducing the dynamic aspect represents a significant contribution and the work of these authors has served as a referent for many subsequent researchers and the development of general models for the allocation of empty vehicles. In compliance with the consideration of dynamics of the system for empty freight car distribution, Ouimet (1972) also suggested a model for empty freight car allocation with general characteristics similar to the previously described model. More specifically, the model characterizes a capacitated space-time network, homogenous freight car fleet, and as the objective function the maximized satisfaction of actual and future demands with a minimization of delays, empty car km, and number of empty freight cars in railway yards (through storage arcs included in the network space-time) by out-of-kilter algorithm. Author emphasized the importance of good estimations of the supply and demand and proposed the implementation of the model in every period after forecast updates. This approach has been applied to Canadian pacific railways as a support for empty freight car allocation on the level of the whole system. The allocation model of the Swiss federal railways (Herren, 1973) is also an optimization model whose aim is to maximize meeting of the demand and to minimize the costs of rail yard operations and empty car flow. The model considers the heterogeneous freight car fleet with substitutability options and other specific constraints. This has been achieved by presenting the plan of empty car scheduling as the minimal cost flow model on a time-space network where each node corresponds to a type of freight cars in a certain train departure (or arrival) from (to) a specific railway yard. Arcs on the network represent yard operations, car substitutions, and train flows. Taking into account the size of the Swiss rail network, the model was solved by a modified out-of-kilter algorithm. The results showed a possibility of reducing the freight car fleet with the full customer satisfaction achieved, and significant reducing of costs and traveling kilometers in the empty condition was also achieved. Fernandes Grain and De Sinay (1986) suggested an LP formulation for the static, empty freight car distribution problem of Brazilian railways. The model assumed a homogeneous freight car fleet with known demand and supply. It maximizes the profitability of the system and it was solved by a standard simplex algorithm. Narisetty et al. (2008) presented an optimization model for empty freight car scheduling based on the demand. The problem was formulated as the problem of optimal adjustment between the available freight cars and customer demand. The model considers the heterogeneous freight car fleet. For a given day, the supply of empty cars is composed of the cars returned from other railways, cars that a railway rents from other

Review of the models for rail freight car fleet management

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railways, and cars that changed their status to available. Therefore, the authors assumed that the supply of empty cars that will be available in the current period, as well as the part of empty cars which will become available during the following scheduling periods, is known. The demand for empty freight cars is composed of real and predicted orders. Real orders are the orders already made by the customers, whereas predicted orders correspond to the company’s decisions for sending the cars to their probable destinations based on the historical records. Five main sets of characteristics have an impact on the desirability of a feasible empty freight car schedule: the transport costs, costs of freight car substitutability, penalties due to early/late delivery, customer’s priority, and the efficiency of railway corridor. The objective function is represented as a weighted sum of these costs. The developed optimization model had enabled significant savings in transport costs. In addition, the number of crew members necessary for the demand fulfilling process had been reduced, which resulted in 35% of return on investment. Spieckermann and Voß (1995) formulated the problem of empty freight car distribution as the job-shop scheduling problem, with ships representing railway cars and jobs representing the demand for freight cars. Research was conducted on request of a German company that leases freight cars for users throughout Europe. Transport of freight cars is performed by the national railways that charge fees for empty or loaded freight car flows. The objective is minimization of empty movement costs. The model was solved by a three-stage procedure that is a part of greedy heuristics. At the first stage, a feasible solution was found by applying the rule “earliest due date” (EDD). The second stage tries to improve this solution taking into account the minimization of total delay in order fulfilling. The procedure of the improvement with the aim to reduce transport costs without increasing the delay was applied in the third phase. The algorithm was tested on real data as well as on randomly generated instances. The biggest instance contained 805 requests, 225 railway cars, and 205 stations. The system generated significant cost savings but in some experiments the computational times exceeded several hours. Holmberg et al. (1996) suggested a multicommodity network flow model for the operational allocation of empty freight cars. Every commodity corresponds to one type of freight cars, whereas the connecting constraints limit the total number of empty freight cars that can be a part of a scheduled train. Train flows are represented on a time-space network. The aim of the model was to minimize the costs of transport and car shortage. The value of the car inventory in a station after the planning period was also taken into

18

Optimization models for rail car fleet management

account. The multiperiod planning horizon was considered and the operational model was solved by the rolling horizon approach, in which the decisions corresponding to the initial period are applied, whereas the remaining decisions are updated by solving the model during the following period. The model could also be used on a strategic level for the assessment of the effects of variations in the freight car fleet size. Lagrangian’s heuristic method was compared to the simple branch and bound approach. The results obtained on one real and a set of randomly generated instances revealed the applicability potential of the model. The largest solved instance contained 100 stations and 20 types of freight cars. Substitutability was also considered by extending the main formulation. Lubbecke and Zimmermann (2005) considered the problem of freight car management on industrial railways. Loaded and empty cars enter an industrial area. Appropriate cars are stored and scheduled between terminals, whereas the loaded cars and excess of empty cars leave the industrial area. In the paper, the authors considered the problem of which car to move, where and how. Customers or production terminals make a transport order which includes the track for the delivery of the empty wagon, the type of freight (for unloading terminal) or car type (for loading terminal), the number of needed cars, and the delivery time window. The requested car type can be substituted by similar types. The allocation of individual railway cars on transport requests includes the following operational objectives: minimal classification work, short transportation times, and low cost of freight car renting. The railway network in an industrial area is divided into segments. A group of adjacent tracks often serves the same purpose, as storage areas or tracks reserved for incoming trains. Each track belongs to a certain segment. One block of cars cannot contain cars dispatched or directed to different segments. In other words, a block begins and ends in the same segment or represents a direct connection between two segments. Therefore, the authors decomposed the freight car allocation problem into two subproblems: freight car distribution between segments and allocation of cars within each segment, and they proposed the mixed integer programming formulation for solving these two subproblems. Joborn et al. (2004) considered the problem of empty car distribution in a scheduled railway system. The authors analyzed the cost structure of empty freight car repositioning and concluded that the distribution cost shows the economy of scale behavior. Beside the cost proportional to the number of cars dispatched from an origin to a destination, there is also the cost of car handling in intermediate railway yards, which depends on the number of car groups meant for classification. An optimization model was

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19

suggested which explicitly takes into account the effect of the economy of scale. For the description of possible freight car movements in time and space, a time-dependent network was used, which was transformed into a network with fixed costs of links, which symbolize the flow of cars between a certain pair of stations. The resulting optimization model is a capacitated network in which capacity constraints limit the flow on links. Tabu heuristics was applied for model solving. Lawley et al. (2008) suggested an approach for solving the distribution problem for freight cars used for the transport of bulk commodities (coal, steel) on railways. The analyzed problem was at the same time considered important, from the aspect of the share of bulk commodities in the total transport, and complex, because it uses all available railway resources and information for the minimization of congestion, maximization of capacities, and decreasing of uncertainties of dispatch and arrival times. The benefits of effective scheduling of bulk transports include reducing the demand shortage, crew costs, and delays in loading/ unloading. The scheduling of cyclical deliveries of bulk commodities has a few specificities with respect to the traditional empty freight car distribution problems. First, car allocation is performed only for one commodity type. Therefore, freight cars that are used for transport are homogenous, so there is no need for considering car substitutability in the model. Second, shipment sizes between some origin-destination pairs of nodes are big, which implies that the shipments should be formed only after a fully loaded train can be dispatched from the origin to the destination. Therefore, the demand is modeled on a train-by-train basis instead of the car-by-car basis. Finally, demand is repetitive and predictable. This comes from the fact that the demand for freight cars follows a relatively stable production regime of the users of railway services. In modeling the railway-related problems, other sources of uncertainty may appear, such as the variation of traveling and waiting time, which will increase the necessity for the stochastic modeling approach. In this case, waiting times are calculated by considering the available capacities and time of car handling in stations. Average traveling times are calculated in accordance with the traffic on a rail line and they differ depending on the day and traveling time. Considering that a relatively stable environment is in question, the authors use the deterministic approach for problem solving. The proposed model considers the demand information, railway network characteristics (typology, distance, and time characteristics), loading capacities of shipper/manufacturer and unloading capacities of the receiver/customer, and the characteristics of freight cars (number, availability). The model represents a time-space

20

Optimization models for rail car fleet management

network, encoded as a mixed-integer programming problem in which the physical network interpretation (shippers, receivers, stations, railway lines) has been replicated in every time epoch or planning period along the entire planning horizon. The model can be used for tactical planning as well as for detailed daily operations by selecting an appropriate planning period and network characteristics. The model explicitly considers train routings and operational times in railway yards in order to avoid congestions on a railway network. The solution methodology generates good solutions for big, real networks. 2.1.2.2 Stochastic approaches Freight car management problems in railway transportation are characterized by intensive dynamic information processes and therefore, they are suitable for the application of stochastic optimization techniques. The most frequent form of stochasticity results from the uncertainty related to a certain aspect of demand for rail freight cars (level of demand, location, time), but the other forms can also be present (traveling time, availability of freight cars, sudden failures). After a set of control actions for freight car management was defined, the decision makers have the possibility to consider the output of (some of the) uncertain events, and then to react to these events. The actual dynamics often result in very significant analytical complexities: the initial decisions can be significantly influenced by the capability of the decision maker and the system itself to react on forthcoming random events. From the perspective of the empty freight cars distribution, there are three basic classes of dynamic information: the flow of user requests for empty cars, process of state transitions (cars are unloaded and returned empty by the customer or from another railway company), and traveling times of cars between the stations of supply and demand. On most railways, requests for empty freight cars are made 1 week in advance on average, but a significant dispersion of this average is present, which means that some orders can be made much in advance, whereas other orders are made at the last moment (especially by big and permanent customers). There is not much of the available previous information about empty cars, and the traveling times become known only after the transport is realized. One of the most important sources of uncertainty is found in the traveling times. On a railway, it is not unusual to have a 5–10 days average traveling time of freight cars. This source of noise is particularly complex. In other words, if a certain number of cars are dispatched from the station A for the sake of satisfying the demand of

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Table 2.3 Main characteristics of stochastic models for empty freight car allocation Authors

Year

Model type

Solution approach

Jordan

1982

Nonlinear stochastic model

Jordan and Turnquist Turnquist

1983

Nonlinear stochastic model

1986

Nonlinear stochastic model

Kraft

2002

Powell and Topaloglu

2005

Multicommodity network flow problem Stochastic programming

Frank-Wolfe algorithm Frank-Wolfe algorithm Frank-Wolfe algorithm Subgradient algorithm Piecewise linear approximation

the station B, there is no entirely certain information available about the arrival of empty cars. There are numerous ideas on how to improve the forecast of empty freight car arrivals, which is based on the knowledge about loaded cars currently in operation (its route is known, and if it is possible to estimate the traveling time, then it is possible to estimate when these cars will become available). In Table 2.3, the main characteristics of stochastic models for solving the problem of empty freight car allocation are given. Turnquist and Jordan ( Jordan, 1982; Turnquist and Jordan, 1982; Jordan and Turnquist, 1983) made one of the first attempts in this direction. Their research has resulted in the MOV-EM network optimization model for empty freight car allocation (Turnquist, 1986). This research is innovative in two ways. First, beside the maximization of profit as the main objective, the authors explicitly considered and integrated several cost parameters: the revenue from realized orders, the costs of holding the cars unused in rail yards, the costs of unsatisfied demand, and the costs of moving the cars between stations. The authors based their model on the assumptions about the stochasticity of the supply, demand, and traveling times. The model uses the space-time network whose nodes represent the location of stations in certain time periods, whereas the arcs are the car movements between stations and the car dwell times between time periods in the stations. The model maximizes the expected profitability, which is determined by the known and forecasted supply and demand for empty freight cars (several types in different stations on the network), traveling times, revenues, holding costs, and freight car shortages. The model generates the best decision for the current period and predicts the future allocation decisions based on the

22

Optimization models for rail car fleet management

actual information and forecasts. As time passes, additional information becomes available and the model can be rerun for the new planning period with the available updated information. It is assumed that the expectations and variances of the future demand and supply (in every station for every time period and type of car) as well as the expectations and variances (assumed negative binomial distributions) of traveling times between each pair of stations are known parameters. The authors presented calculation procedures for some of these parameters, as well as the details about stochastic equations derivation that integrate these parameters. The model is a nonlinear problem of optimization with linear decomposable constraints. Authors also applied the Frank-Wolfe algorithm (Frank and Wolfe, 1956) for problem solving. Mendiratta (1981) considered the problem of empty freight car allocation through a decentralized decision-making structure. Two interactive submodels were developed: the network centralized and the terminal decentralized whose coordination mechanism is managed by price schedules. These price schedules are the main carriers of the information about the value of the car supply in stations, and all that through the shadow prices. The objective function of the terminal model takes into account the minimization of transport, storage, and costs of freight car shortage. The network model has the form of a classic transportation problem. The terminal model considers the stochastic nature of traveling times and stochastic features of demand and establishes a certain analogy with inventory systems and optimal control. Kraft (2002) suggested a method for the freight car management process, which includes the reservation system and capacity constraints. The suggested method is partially based on the price bidding approach for the simultaneous analysis of the railway service supply and freight car flows with the aim of profit maximization. The suggested methodology enables the determining of the desirable shipment delivery time, taking into account the needs of shippers and forecasted available train capacities. The shipment routing problem is decomposed into the deterministic process of dynamic car scheduling for shipments already accepted for transport, and the stochastic process of price formation for certain segments of train routes for the prediction of the future demand for which delivery times are still not defined. Both models are formulated as multicommodity network flow problems. Each shipment, composed of one or more freight cars, is considered as a separate commodity. The subgradient algorithm is applied for problem solving. Powell and Topaloglu (2005) used stochastic optimization techniques for solving two-level and multilevel problems for empty

Review of the models for rail freight car fleet management

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freight car allocation. The problem is characterized by a significant level of stochasticity in the supply, demand, and traveling times of empty freight cars between stations. Beside these, the authors highlighted some additional features of empty car distribution, such as the time-shifted information process and the need for discrete solutions. A set of algorithms for two-level problems with a network recursion is considered. The basic characteristics of the problem (integer solution, size of the problem) eliminate scenario methods, stochastic linearization, and Benders decomposition (the method gives fraction solutions). Therefore, for problem solving, the authors suggest methods that approximate the recourse function on the second level with maximal preserving of the basic network structure. The presented techniques were based on the derivation of separable nonlinear approximations of the objective function. In order to obtain integer solutions, the authors use piecewise linear approximations with break points existing only in the integer values of supply. 2.1.2.3 Hybrid approaches Railway companies have a long history of using simulation models for the assessment of the total impact of operational policies and procedures. In general, simulation models represent detailed interpretations of rail yard structures or network structures and transport operations, and therefore, they have a huge credibility in the railway industry. Baker (1977) and Assad (1980a) among others, give a review of the simulation models applied in the railway industry. However, these models are very demanding (from the aspect of computational time and effort). Ratclife et al. (1984) suggested the combined simulation-optimization methodology for solving the task of empty freight car distribution. The applied algorithm was based on the LP (for solving the assignment task for known requests) and stochastic LP (for the allocation of empty cars toward stations in which the orders have been expected). The objective function for these tasks is the minimization of the total traveling time on a network with N stations. The stochastic linear model uses the state of nature, which actually represents a demand vector of the station with a certain probability. As supply and demand are known from statistical records, in simulation experiments these are obtained from known distributions. For the simulation, the computer language SLAM is chosen. The obtained results based on the 36- and 48-h interval between algorithm reruns for the network of Fisco railways show significant reductions in the total traveling time.

24

Optimization models for rail car fleet management

2.1.3 Freight car pooling concept The traditional strategy of empty freight car prepositioning assumes the return flow of unloaded cars in the station of loading. This is a very simple and suitable approach by which a significant part of freight shipments has been allocated from the region of one railway to the region of another railway. For the sake of reducing the cost of empty movement, the freight car pooling concept was introduced. Within the agreement about joint use of railway cars between railways, and customers in some cases, freight cars unloaded in the destination station can be directed to any of the stations for subsequent loading. The car manufacturing industry in the United States, as one of the initiators of this idea, has begun in 1981, with the use of the freight car pooling concept for the transport of automobiles with multilevel auto cars from the manufacturer to the unloading nodes. In 1981, by using this concept, one car manufacturing company made a significant reduction of total traveling kilometers of empty freight cars (64 million car km in the case of 13 production plants and 40 unloaded nodes with 5500 cars in the pool) compared to the traditional method (Association of American Railroads, 1982). The most important contributions in the domain of freight car pooling management are presented in Table 2.4. LP represented the basic methodology for the earliest allocation models (Association of American Railroads, 1982). A number of studies were Table 2.4 Characteristics of the model for freight car pooling management Authors

Year

Model type

Solution approach

Association of American Railroads Kikuchi

1982

Linear programming

Simplex algorithm

1985

Simplex algorithm

Glickman and Sherali

1985

Linear transshipment problem on timespace network Transportation problem

Adamidou et al.

1993

Sherali and Suharko Sherali and Lunday

1998

General Nash equilibrium model Network flow model

2011

Marginal cost analysis

NETFLO algorithm/ Dantzig-Wolfe decomposition Gauss-Seidel algorithm Heuristics decomposition Game theory approach

Review of the models for rail freight car fleet management

25

conducted with the aim of improving this methodology. The most important contributions are given in the remaining part of this chapter. Kikuchi (1985) proposed a model for the central monitoring and management of a homogeneous rail freight car fleet. Reliable forecasts of supply (from the central monitoring system) and demand (from service users) are represented as known. Optimal daily dispatching instructions for every freight car in the system, in the sense of minimized total costs of holding, traveling, and shortage of freight cars, represent the objective of the model. The model is formulated as a linear transshipment problem on a network time-space where arcs are used for the movement and holding of freight cars. Glickman and Sherali (1985) analyzed the same problem but in a heterogeneous case with partial substitutability. Their approach differed in that they developed several models where each model has the aim of searching for the optimal empty freight car allocation from different aspects. The first model of the system minimizes the total cost of allocation and exchange of the joint pool of empty freight cars. The model is formulated as a classic transportation problem based on the problem of the space-time network design. The second model aims for an improved balance of relative benefits between railways according to the defined criteria and with shifting the solution in a certain domain of the optimum of the first model. The third “companywide” model minimizes the total cost with regard to the previous standards about the benefits that are brought to every company by the improved distribution and exchange of freight cars. The comparison of the model solutions enables the estimation of the cost of fulfilling the objectives for each of the railways. The modified code for network optimization NETFLO (network flow) (Kennington and Helgason, 1980) is applied for transportation problem solving. Other models were solved with the algorithm of DantzigWolfe decomposition by using the network code for subproblems and the procedure of heuristic rounding off for obtaining integer solutions. Adamidou et al. (1993) concluded that the problem of finding a global strategy for freight car distribution by maximizing the profit for railways that share freight car fleet could be represented as a generalized model of Nash equilibrium. Their model included pairing variables which connect multicommodity subproblems of individual railways and it was solved by the Gauss-Seidel algorithm which iteratively handles individual subproblems. In solving the individual subproblems, the pairing variables were fixed by using the optimal flows obtained in the previous solving of subproblems for other railways. The approach has been tested on a real case composed of three railways and it was proved that the approach is fast and reliable.

26

Optimization models for rail car fleet management

Different solution strategies were compared as well as the different states of demand. Sherali and Suharko (1998) presented a tactical model for the support in a centralized distribution of empty freight cars for the transport of automobiles. The problem included a group of eight main automobile manufacturers who have formed a fleet of freight cars in order to improve its utilization and to reduce freight kilometers in an empty state. The authors have developed and tested two model formulations. The first model is in essence a transport model of the network flow and it takes into account practical issues based on uncertainties in traveling times, priorities with respect to time and location of demand, a number of objectives corresponding to the minimization of delivery delays and shipment grouping policies. Within this framework, the authors have developed: (a) a decision-making tool based on chance constraints, with the aim of determining whether to include a certain arc on basic time-space network; (b) the mechanism of scaling priority demand locations for determining the equilibrium measure between individual car manufacturers; and (c) a weighted penalty scheme for addressing the probability of varying levels of delay on any route of the time-space network. In the second model, a policy of shipment consolidation was considered by introducing a set of binary variables that control the maximum number of blocks of cars formed in all origin locations during each day. This additional feature resulted in network flow formulations. By reducing the level of flow relaxation by introducing suitable inequalities and suitable phases of preprocessing, it was possible to solve the problem, which has 5000–8000 arcs, in an optimal way. However, for problems of larger size, it was necessary to develop a heuristic method. A scheme that sequentially constructs a solution was proposed. This scheme was based on the use of rules for determining an appropriate collection of demand nodes and the rules for the selection of one of these nodes for the partial fulfilling of its demand. The process was based on the opportunity costs and rules for deciding which source should create a group of cars of a suitable size in order to fulfill the needs of the demand node. By careful consideration of 21 compositions of these three rules on a big number of different test cases, the authors suggested a combination suitable for application. In a comprehensive solving strategy, the first model is used for a fast solving of problems that do not contain shipment grouping. This gives insight into the analysis of the relationship between the benefits of forming a lesser number of trains by forcing the shipment consolidation

Review of the models for rail freight car fleet management

27

policy and higher distribution costs along with the penalties for delay/shortage. Sherali and Lunday (2011) analyzed the problem of equitable appointment of a given rail freight car fleet that is used jointly by a consortium of automobile manufacturers and railways participating in the agreement about the joint use of freight cars. The suggested approach improved the existing practice by the identification of its shortages in relation to the equitable freight car distribution. In addition, four improved alternative schemes for freight car allocation to shippers were suggested. These schemes were based on including the time of waiting in transit for the calculation of proportionality factors for loaded car days, then on two different techniques used for the analysis of marginal costs, and on the game theory approach. The authors gave insight into the interpretation of the game theory of the current scheme for freight car allocation, which is based on Shapley values, and suggested an alternative scheme for car allocation to shippers, through consideration of the total capital costs and operational costs for the proportionality factors derivation. For each combination of the current and suggested schemes of allocation to shippers and operators, the authors tested the relative advantage with respect to the current standard by using a set of real examples based on the data from the TTX company.

2.1.4 Combined allocation of empty and loaded cars The demand for empty cars is a consequence of imbalanced flows of loaded cars. Therefore, it is logical to integrate these two types of flows on the operational level in one model for the allocation of freight cars. Key features of the model for the combined allocation of empty and loaded freight cars are given in Table 2.5. Table 2.5 Characteristics of models for the combined allocation of empty and loaded freight cars Authors

Year

Model type

Solution approach

Gorenstein et al.

1971

Shan

1985

Kornhauser and Adamidou Haghani and Daskin

1986

Multicommodity network flow problem Cost minimum network flow problem Network flow problem

Simplex algorithm Network simplex algorithm Iterative heuristics

Nonlinear mixed integer programming

Heuristics decomposition

1986

28

Optimization models for rail car fleet management

Gorenstein et al. (1971) suggested a computational procedure for the freight car flow control, which represents a part of real-time management systems with the aim of improving the utilization of freight cars and satisfaction of service users. The scheduling system is composed of two models. The first model handles only empty car flows. The second model considers loaded and empty freight car flows and represents an LP multicommodity formulation of the minimal cost flow on a network determined by a yard policy of individual yard terminals and train timetable. The empty freight car allocation model considers only the homogeneous freight car fleet. The authors emphasized the impact that different types of cars and their interrelations may have on the solution and suggested that, in the context of the proposed system, it can be appropriate to evaluate interrelations, and then assess each type of freight cars independently. Shan (1985) developed two models that serve for the improved utilization of freight cars and as a tool for the evaluation of problems that originated from the level of utilization of freight cars used from a number of railways. The first model treats the case of a homogeneous freight car fleet as a transshipment problem on the space-time network. In the second formulation, a few different types of freight cars are considered and the problem is formulated as a minimum cost flow problem on a capacitated network. Both formulations were solved using the network simplex algorithms. Kornhauser and Adamidou (1986) considered the problem of freight car flow management within the context of the United States as a noncooperative game of N persons formulated as a multicommodity flow problem on a network that represents the control options of each railway. These options include holding or dispatching of empty or loaded cars within or outside of the considered railway network. The problem was solved by heuristics that iteratively solves each problem until an equilibrium state has been reached for all pairing variables. The approach is not applicable for real cases due to the sharp increase of the problem’s size and computational time with the number of railways covered. Haghani and Daskin (1986) suggested a combined model for train routing and empty car allocation. The model is based on a spacetime network and takes into account operations in railway yards as well as the train flows between railway yards. The resulting formulation of the nonlinear, mixed integer programming was solved by a heuristic approach by which the problem is first decomposed into time levels, and then the subproblems of the freight car flow and locomotive flow are independently considered.

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29

2.2 Tactical and strategic models for freight car management The empty freight car scheduling problem cannot be efficiently handled only on the operational level. Empty freight car flows represent an important component of tactical and strategic problems such as planning on the national and regional level, the network service design and rail freight car fleet sizing.

2.2.1 Service network design problem Railway freight companies perform their operation on complex networks composed of nodes (which include a number of loaded/unloaded nodes and railway yards for the classification/sorting of freight flows) and physical links (railway lines) which connect these nodes. One of the main aspects of the decision-making process of these companies is tactical activity planning (service network design—routes and service levels, policy of freight flows handling in railway yards and transport routing on the service network) which results in the design of an efficient operational plan on the railway. Some of the most important approaches to the service network design problem are explained in the remaining part of this chapter. Their basic characteristics are given in Table 2.6. Crainic et al. (1984) suggested an optimization model that integrates the relations between the operational policy for train routing, classification and assembly policy in railway yards and the allocation of the classification work between railway yards, on the tactical planning level. The objective is generating economically efficient global operational strategies that enable a good level of service from the aspect of delay and reliability. The nonlinear mixed integer formulation was given, and the heuristic algorithm was tested on real data generated for the case of Canadian national railways. Crainic and Rousseau (1986) presented a general framework for modeling the problem of service network design (which models of transportation to use and which frequency of service to offer) for multimodal freight transport which is based on the network optimization model. The developed approach can be used as a support for improving the process of strategic and tactical planning. The problem was solved by an algorithm based on the principles of decomposition and column generation. Haghani (1989) analyzed the interactions between decisions about train routing and the assembly and empty freight car distribution. The author suggested a model that treats the problem of

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Table 2.6 Characteristics of the model for the service network design problem Authors

Year

Model type

Solution approach

Crainic et al.

1984

Heuristic algorithm

Crainic and Rousseau

1986

Nonlinear mixed integer programming Network optimization model

Haghani

1989

Huntley et al. Holmberg and Hellstrand Kwon et al.

1995 1998

Crainic

2000

Kratica

2000

Kratica et al.

2002

Fukasawa et al.

2002

Campetella et al. Caprara et al.

2006

Zhu et al.

2013

1998

2011

Multicommodity network flow problem Integer programming Mixed integer programming Multicommodity network flow problem Mixed integer programming Mixed integer programming Mixed integer programming Multicommodity network flow problem Multicommodity network flow problem Integer linear programming Mixed integer programming

Algorithm based on decomposition and column generation Decomposition heuristics Simulated annealing Branch and Bound/ Lagrangian heuristics Column generation Branch and Bound Genetic algorithm Genetic algorithm Branch and Bound Tabu search Column generation Math heuristics

tactical routing and train assembly in a dynamic way, which enables the modeling of operational car scheduling together with train routing. The model provides decisions about train routing and assembly similar to the classical static network models (Petersen and Fullerton, 1975; Assad, 1980b). However, due to its dynamic nature, the model can handle the variability of demand and generate decisions about empty freight car scheduling as well as the optimal time interval between subsequent train services on a certain pair of origin destination stations. Heuristic decomposition was developed for problem solving. The solution procedure uses a special problem structure and decomposes it into smaller subproblems. Empirical research was conducted in order to test the performances of the optimization model and solution procedure. Huntley et al. (1995) analyzed the problem of optimizing routes

Review of the models for rail freight car fleet management

31

and scheduling of rail freight cars on case of CSX transportation. Authors developed an optimizer based on a combination of heuristics and integer programming and prove effectiveness of developed algorithm for integrated routing and scheduling. Holmberg and Hellstrand (1998) considered a general problem of network design, which can be applied as the optimization model for empty freight car allocation planning. The authors considered the nondeterministic polynomial (NP)-hard problem of the service network design with one origin-destination pair for each type of commodity on the network. Capacity constraints were not included in the empty car distribution, considering that the problem has a strategic character. However, the possibility of using the version without capacity as a subproblem of the problem with capacities, which could be solved by the relaxation method, was taken into account. For finding an exact optimal solution of the incapacitated network design problem, the Lagrangian heuristics is applied within the branch and bound techniques. The Lagrangian heuristics uses the Lagrangian relaxation as a subproblem, solving the Lagrangian dual with subgradient optimization, in combination with primal heuristics (Benders subproblem) and providing primal feasible solutions in that way. Computational tests made for the problems of different sizes and structures implied that the method is more successful relative to the modern mixed integer programming algorithms from the aspect of the problem’s size and computational time. Kwon et al. (1998) presented a few different ways for improving the current practices of freight car allocation and the dynamic model for the routing and scheduling of freight cars. The developed dynamic model includes heterogeneity and traffic variability and train capacity constraints so that it produces traveling plans that are consistent with the train length limitations and sensitive to the service requests of customers. For the representation of the time and space movement of freight cars on the railway network and determining of the routes and freight car allocation plans during the planning period, a space-time network was used. Car movement along the potential car-block and block-train sequences, car handling activities in stations, and potential car holdings in stations were represented as different types of arcs on the space-time network. Four different types of arcs were used: the traveling arc, handling arc, holding arc, and artificial arc. The traveling arc represents a defined block with a corresponding train schedule. The handling arc represents the activity of handling freight cars in the station. The holding arc represents the holding or storage of freight cars until the next available train dispatching in the station. Artificial arcs were used to represent the delivery time window. The problem was formulated as a multicommodity network flow problem on the time-space network for determining the combined routes and car allocation

32

Optimization models for rail car fleet management

plans for a given planning period. The aim of the model is determining an optimal sequence of shipment flows on the time-space network and the forming of corresponding traveling plans, such as to minimize the total penalty costs (which is equivalent to the maximization of the service standard fulfillment). It was assumed that the train cost, car holding cost, and car time cost are fixed during the planning period. The column generation algorithm was used for solving this multicommodity network flow problem. The model was tested on a railway network based on a subnetwork of one of the main railways in the United States Crainic (2000) made a review of the different approaches to service network design modeling and development of mathematical programming techniques for the service design. In his doctoral dissertation, Kratica (2000) represented a genetic algorithm (GA) for solving the incapacitated service network design problem. Sequential GA implementation contains flexibly realized different variants of the genetic operators of selection, crossover, and mutation. A few fitting functions, different stopover criteria, and policies of the generation replacement were presented as well. The results of research into the given problems showed that sequential GA implementation gives results that are in all cases comparable to or even better than heuristics and other methods known in the literature. Campetella et al. (2006) presented a mathematical model for the design of the service network that represents a set of origin-destination connections. The resulting model includes empty and loaded car movements as well as the costs of the intermediate freight car classification. The model suggests a set of services that should be offered, as well as the number of trains and the number and type of freight cars that should flow on each connection. The service quality, measured through the total traveling time, was determined by minimizing the car waiting time in intermediate yards. The approach leads to a problem of multicommodity network design with concave cost functions of some links on the network. As a solution approach, the authors used the tabu search procedure, which contains the perturbation mechanism for forcing the algorithm toward searching of the larger part of the domain of feasible solutions. Computational results of real examples showed a significant improvement comparing to the actual practice. Fukasawa et al. (2002) presented a method for determining the optimal flow of empty and loaded cars with the aim to maximize profit, revenue, or transported shipment volumes, with the given timetable of freight trains together with their pulling capacities. The authors suggested the integer multicommodity network flow model for the problem whose linear relaxation leads to good upper bounds but with a very large number of variables and constraints. In order for the model

Review of the models for rail freight car fleet management

33

to be implementable in practice, the authors applied a preprocessing phase which reduces the size of the model two to three times. The reduced model can be solved with standard packages for integer programming. Caprara et al. (2011) analyzed the problem of designing a set of profitable freight routes on railway corridors taking into account the level of the service that is requested by various shipments. The profit that is realized by the transport of shipments is the nonlinear function of traveling time. The authors suggested an integer programming model and applied the column generation technique as a solution method. Computational results on a real case of the corridor passing through 11 European countries showed that it is possible to obtain near optimal solutions. Zhu et al. (2013) developed a comprehensive modeling framework for integrated scheduled service network design in rail freight transportation. Authors also proposed an innovative solution methodology to solve large-size mixed integer programming formulation, which integrated exact and metaheuristic principles.

2.2.2 Demand estimation The continuous increase of the population and significant economic activity imply the increasing of the freight flow intensity on a transportation network. Together with the increase of demand for freight transport, the mobility of passengers and freight has been decreasing due to the absence of appropriate capacity adjustments. Increasing the freight transport volume represents a motive for the precise estimation of shipment flows as well as for forecasting of the expected future freight flows. In the context of railway transport, researchers and practitioners agree that the unavailability of an appropriate and reliable information system for monitoring the supply of empty freight cars and the demand of users represents a very important problem. Aside from that, the availability of this information is of key importance for the proper allocation of transport as well as for the needs of the system design. It is necessary to consider three different aspects of this problem: • assessment of the availability of empty freight cars in origin stations which can be obtained from the analysis of previously loaded shipments; • evaluating the need for empty freight cars in the destination stations for the next loading or any other activity; and • forecasting of empty flows between certain pairs of origin-destination stations.

34

Optimization models for rail car fleet management

The first two problems are similar to the problem of forecasting the volume of flow in the passenger or freight segment. The precise estimation of supply and demand for empty freight cars is a necessary input for the models of freight car allocation as well as for the models of tactical and strategic planning. The third problem corresponds to the necessary estimation of origindestination matrices, as independent, or combined with origin-destination matrices of loaded flows, for the national or regional strategic planning in freight transportation. It should be mentioned that it is not easy to estimate the supply and demand of empty freight cars only from loaded flows, and that several additional factors must be considered: maintenance, failures, inclusion of new cars in traffic, the freight car pooling concept, and freight car leasing. Supply and demand are often estimated from the statistical analysis of realized flows of empty and loaded freight cars (Powell, 1987; Dejax et al., 1986). Considering that there is not a lot of research conducted in railway transport, in the remaining part of this section, the most important approaches are explained from the area of demand modeling for the services in freight transportation in other modes of transport. Some reviews of the different types of models in freight transportation are presented in Cambridge Systematics (1997a), the Hensher and Button (2000) book, the paper of Pendyala et al. (2000), in the book about the traveling behavior analysis (Hensher, 2001), as well as in the paper of Willumsen (2001). Many modeling concepts that were applied in the forecasting of freight transport were first developed for passenger traffic. Most of the authors agree that the sequential modeling structure of passenger traffic composed of four steps can be completely applied in freight transport. However, within each of these steps, models for freight transport can be substantially different from the models in passenger transport. The important differences between the market of passengers and freight transport are the diversity of decision makers in freight transport (shippers, transport operators, logistics service providers), a great number of different commodity types, and limited data availability (partially available data because of confidentiality reasons). The four steps in the context of the freight transport modeling system are (De Jong et al., 2004): • Production and attraction. In this step, the size of the shipments that need to be transported from different origin zones, as well as the sizes that need to be transported to different destination zones are determined. Output dimensions represent the tones of commodities. In the intermediate

Review of the models for rail freight car fleet management

35

phases of production and attraction, model monetary units can also represent the dimension. • Distribution. In this step, freight flows between origins and destinations are determined. Dimension is represented by tones. • Modal split. In this step, the allocation of freight flows into modes of transport is determined. • Assignment. After converting the flows in tons in transport units, these can be assigned to networks (in some models, this corresponds to the assignment of road freight flows together with passenger cars on road networks). In the remaining part of this section, a review of the most important approaches within each of these four groups of models is given. Table 2.7 gives a list of papers that will be analyzed. The papers are classified with respect to three criteria: the solution method, the type of problem, and modality. 2.2.2.1 Models for production and attraction In the first step, there are four types of models that are usually applied in practice: • trend models and time series; • system dynamics models; • trip generation zone models; and • input-output and similar models. In trend models, historical trends are extrapolated in the future. Time series of data are used for the development of models of different levels of complexity, from simple models of the increasing factor to complex autoregressive moving average models. Time series models with explanatory variables such as the gross domestic product (GDP), employment, population, and others are also developed and applied in practice. Dougherty (1995) made a review of research that is based on the application of neural networks for the analysis and transport-related problem solving, among which is also the application in the domain of transport demand forecasting. Babcock et al. (1999) analyzed the demand for railway grain transportation. The authors in this paper specified the model for the forecasting of the quarter volume of grain transport on the US railways and they empirically estimate the developed model. The selection of explanatory variables implies that these variables have a theoretical relation with the supply and/or demand for grain transport by railway and that the data for the explanatory variables

36

Authors

Year

Solution approach

Problem type

Modality

Coutu Abdelwahab and Sargious

1978 1992

Demand modeling Demand modeling

Railway sector General

Dougherty Babcock et al. Garrido and Mahmassani Fite et al. Sivakumar and Bhat Wong et al. Nijkamp et al. Train and Wilson Holguin-Veras and Patil Milenkovi
c et al. Milenkovi
c et al. Milenkovi
c and Bojovic

1995 1999 2000 2002 2002 2003 2004 2007 2008 2012a 2012b 2013

Gravitational model Binary probit model Linear regression Neural networks ARIMA Multinomial probit model Multiple regression Assignment model with partial separation Fuzzy linear regression Discrete choice models and neural network models Discrete choice models Estimation method of origins and destinations ANFIS ANFIS, ARIMA Kalman filter

Demand Demand Demand Demand Demand Demand Demand Demand Demand Demand Demand Demand

General Railway sector Road sector Road sector General Railway sector General General Road sector Railway sector Railway sector Railway sector

modeling modeling modeling modeling modeling modeling modeling modeling modeling modeling modeling modeling

Optimization models for rail car fleet management

Table 2.7 Main characteristics of models for modeling the demand for services in freight transport

Review of the models for rail freight car fleet management

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are based on a quarter level. However, there are only a few explanatory variables with a quarter frequency and these have a very low impact on grain transport on the railway. In addition, the economic process that generates quarter grain transports on the railway is very complex and it is very hard to model this process by regression techniques. For these reasons, the authors decided to apply the time series model. The forecasting model is generated by the autoregressive integrated moving average (ARIMA) model developed by Box and Jenkins (1976). The AR (4) model is estimated with a procedure for the estimation based on the maximum likelihood for the period 1987:4–1997:4. The actual volume of transport for this period is compared to the forecast generated by the suggested model. Fite et al. (2002) modeled the demand in road freight transport using the model of stepped multivariate regression through which the volume of transport and a set of national, regional, and local economic indicators are connected. The models were made by using a large set of data from one of the most important transporters in the world. Wong et al. (2003) analyzed the demand for railway freight transport as a process which is continuously dynamic in space and time and which is impacted by a number of qualitative and quantitative factors. For the sake of a more rational planning process in railway freight transport, there is a need for a precise forecast of the demand in a dynamic and uncertain environment. In the analysis of the conventional linear regression, deviations between the realized and estimated values are assumed to be realization errors. The authors in this paper approached the forecasting problem from a different perspective, by considering these deviations as a fuzziness of the system structure. Based on the analysis of the characteristics of the forecasting problem in railway freight transport, the suggested method was applied to a real case. In the Assessment of Transport Strategies (ASTRA) model of system dynamics, changes in transported volumes overtime and feedback to/from the economy, land use, and the environment are explicitly modeled (ASTRA, 2000). In the macroeconomic module, increase of GDP is predicted. Obtained predictions are used in the module of regional economy, which gives demand for transportation from the aspect of flows in tons for each origin-destination pair of nodes. In the transport submodel, these flows are assigned to modes of transport and virtual links. Changes in the transportation demand can influence GDP through transportation costs. Parameters of the system dynamics model are usually not obtainable from statistical estimation, but from the existing literature, by a trial of initial values and checking of the resulting dynamic behavior of the system (trial and error). The system dynamics model could include steps of distribution

38

Optimization models for rail car fleet management

and assignment to different transport modes. However, system dynamics models do not contain enough space and network details for the analysis of interzone flows and transport loads on links. Zone transportation rates for the production and attraction are usually derived from the classification of cross-sectional data about the volume of transportation to/from each zone in the area that is considered in a larger number of homogeneous types of zones (Cambridge Systematics, 1997b). Input-output models, in essence, represent the macroeconomic models based on the input-output tables. These tables describe, in monetary units, what each economy sector delivers to other sectors, including the final demand as well. National input-output tables are developed for many states, usually by the statistical office. A special form of an input-output table, which for many states does not, exists, is a more multiregional or space input-output table. Apart from deliveries between sectors, this table also includes deliveries between regions. The majority of space input-output tables contain only a few big regions within the state. Examples of space input-output models in freight transportation are: • Italian national system for passengers and freight (Cascetta, 1997) which uses 17 sectors and 20 regions and also elasticity coefficients. • REGARD model for Norway, with 28 sectors, which generates the demand used in the Norwegian freight transport model NEMO (EXPEDITE, 2000). • SCENES European system for freight and passenger modeling and its predecessor STREAMS (Leitham et al., 1999) with 33 sectors and more than 200 zones in Europe and with elasticity coefficients (SCENES Consortium, 2001). 2.2.2.2 Distribution models As in the previous step, all distribution models in the literature are based on aggregated data (data grouped according to a defined form). Freight flows between the origin and destination zones are determined by the measures of production and attraction, and the measure of transport resistance, which is expressed as the transport cost or generalized transport cost. The most frequently used method is the gravitational model. In these models, the flow between two zones is a function of the product quotient of the production measure and zone attraction and some measure of the generalized transport cost. Coutu (1978) applied the gravitational model for forecasting the empty freight car flow. In the gravitational model, it is assumed that the flow between a certain pair of origin-destination stations is proportional to the

Review of the models for rail freight car fleet management

39

supply at the origin and the demand at the destination. The proportionality factor depends on the parameters such as the distance and cost of transportation between two nodes. The author described the steps that need to be taken in order to use this methodology on the empty freight car flow forecasting on the Canadian national railways. Canadian national railways use these forecasts for resource allocation planning in order to meet the demand for empty and loaded flows in all nodes on the network. The gravitational model was applied and calibrated in accordance with the techniques suggested by Evans and Kirby (1974). The model produces satisfactory results in the preened case study. Sivakumar and Bhat (2002) suggested an approach that represents the distribution model with a partial separation for the modeling of interregional freight flows. The approach uses the logit model for the estimation of a part of freight flows consumed at a destination area, which are generated in each of the production zones. This idea is aligned with the essence of freight flows—freight flows are generated by the demand for freight at a destination, which is satisfied by its flows from one or more source nodes. The presented empirical application showed that the model gives better results in relation to the gravitational model. Holguin-Veras and Patil (2008) suggested the multicommodity formulation of the origin-destination model, which combines the model, which is based on the type of commodity for the estimation of freight flows, and a complementary model of empty flows. This integration is important having in mind that explicit modeling of empty flows is needed for overcoming the errors in traffic flows estimation. Two cases of the proposed model are analyzed. The first uses the total traffic whereas the second is based on the loaded and empty flows. Results show that the models, which include the submodel of empty flows significantly, overcome the models that do not contain this submodel by their capability to replicate realized freight flows. The comparison of the results between the multicommodity and the single commodity model shows that multicommodity formulation brings significant reductions of the estimation error of freight flows. Milenkovi
c et al. (2012a) applied the adaptive neuro-fuzzy inference system for the modeling of passenger flows in the urban passenger traffic system. Milenkovi
c et al. (2012b) compared the adaptive neuro-fuzzy inference systems and the traditional ARIMA models for the modeling of passenger flows in the urban passenger railway system. Milenkovi
c et al. (2013) modeled the monthly passenger flows on the railway by using the ARIMA model in the state-space form. The identified SARIMA (0,1,1)(0,1,1)12 model is transformed into the state-space form, and afterwards the Kalman recursion is applied for forecasting.

40

Optimization models for rail car fleet management

2.2.2.3 Models for modal split Users of the transport service usually choose the mode of transportation by which they will dispatch their shipment. Models for modal split treat the behavior of users from the aspect of the transport mode selection. The basic criteria of selection mainly correspond to the offered level of service and costs of the available modes of transportation. For the modal split in freight transport, the following models, which are based on aggregated and disaggregated data that can be found in the literature are: • models based on elasticity; • aggregated models of assignment on individual modes of transportation; • neoclassical economic models; • econometric models of direct demand; • disaggregated models of assignment on individual transport modes; • micro-simulation approach; and • multimodal network models. Models based on elasticity reflect the effects of change of one variable (e.g., the cost of a certain mode of transport). Elasticities were derived from other models or from expert knowledge. These models are mainly used for strategic evaluations and/or fast approximation or in cases of limited data. One such example is the PACE-FORWARD model (Carrillo, 1996). Aggregated models of assignment to individual modes of transport are mainly binomial or multinomial logit models estimated from the data about the participation of different types of traffic for a larger number of zones. These models only give a share of different modes, not the absolute volumes of transport (tones) or traffic (vehicles) like the direct demand models. Neoclassical models are based on the economic theory of enterprises. For the cost function, with transport services as one of the inputs, the transport demand function can be derived through Shephard’s lemma (Shephard, 1953). In the direct demand model, the number of trips (or kilometers) by a certain transport mode is directly predicted. The classic example is the model developed by Quandt and Baumol (1966). Models of the assignment that is divided into individual modes of transportation are mainly multinomial logit or logit models, which can be based on the random utility maximization theory. Abdelwahab and Sargious (1992) suggested an approach for demand modeling in freight transportation that is based on a simultaneous decision about the mode of transport and size of shipment. The system of simultaneous switching was applied for the development of the combined demand model, which treats the two mentioned decision-making determinants.

Review of the models for rail freight car fleet management

41

More precisely, the process of transport mode selection was formulated as a binary probit model, whereas two linear regression equations were used for the simulation of the choice of shipment size for the railway and road transport. The authors also considered two alternative evaluation methods. The first method requires the formulation of the maximum likelihood function for the system of simultaneous equations. This method has advantages in the simultaneous one-step estimation of model parameters that enable the analyst to define the constraints based on the estimated parameters. A less computationally intensive method is the two-phased estimation, in which the probit method of maximum likelihood is applied in the first phase for the evaluation of models of the transport mode selection whereas the least square method is applied for the estimation of the shipment size equations in the second phase. This method also known as the method of two-phased least squares is simple, consistent, and more familiar to analysts with limited knowledge of econometrics and mathematical procedures. The suggested model was applied for the transport mode analysis (railway as opposed to road) and shipment size determination for each of the considered transport modes. The application of the suggested model for the analysis of the demand for interurban freight transport by rail and road was demonstrated on real data. Garrido and Mahmassani (2000) applied the multinomial probit model for the space-and-time correlated structure of error for the analysis of the transport demand in road transport. Considering that the resulting model had a great number of alternatives, estimation was conducted by applying the Monte Carlo simulation for the estimation of the model’s likelihood. The model was successfully applied to a set of historical data provided by a road transport international company. Nijkamp et al. (2004) compared the descriptive and predictive power of two types of techniques for the statistical estimation of multimodal transport flows. The models considered were models of discrete choice (logit and probit models) and the neural network model. A comparative analysis was made on a large set of data related to the realized European interregional freight flows for two commodity categories—food and chemicals. The results showed that the neural network models have a predictive potential greater than the discrete choice analysis. Train and Wilson (2007) presented a problem of the discrete choice model estimation based on the limited access to transport markets by shippers. The estimated model was applied for the aggregating of decisions about shipments for the sake of determining a spatially generated transport demand.

42

Optimization models for rail car fleet management

2.2.2.4 Assignment models The final phase of the presented sequential process of freight traffic forecasting refers to the route choice between pairs of regions with regard to the type of transportation and resulting flows. In this phase, trips by road, railway, and internal waterways have been allocated to the routes composed of links of corresponding modal networks. The majority of models do not consider the assignment step, whereas some others include the assignment only for the road transport mode. Assignment on the road network is in some cases conducted together with passenger traffic, while freight traffic represents only a small share of the total traffic. For example, origin-destination matrices for road traffic of the Holland freight model TEM-II (de Jong et al., 2004) were combined with the road passenger traffic in the Holland national model. In order to do this, it is necessary to convert trips in freight into equivalent passenger trips. One additional example of the independent assignment step (instead of the group selection of the transport mode and route in the multimodal network) is the Italian national model, where the choice of the transport mode has been conducted on the disaggregated level (level of individual regions) and the assignment on the level of origindestination pairs of nodes planning.

2.2.3 Rail freight car fleet sizing models Vehicle fleet sizing represents a very important problem for researchers as well as for the transport companies, considering the value of transport assets. Vehicle fleet sizing considers the whole service design (Crainic, 2000) and there are a lot of studies that consider this problem in road traffic (Hall and Racer, 1995; Du and Hall, 1997; Ozdamar and Yazgac, 1999) and express shipments services in air transport (Barnhart and Schneur, 1966) which emphasize the existence of this relationship. Vehicle fleet sizing is also important in systems for material management in production operations (Beamon and Deshpande, 1998; Beamon and Chen, 1998). In a generalized case, vehicle fleet sizing can be defined as a sizing of system of reusable resources. Frantzeskakis and Powell (1990) consider the problem of sizing and scheduling of empty equipment from the aspect of the inventory theory and the developed decentralized policies for the empty equipment inventory control. The authors concentrate on the utilization of a given vehicle fleet, not on the decisions about its size. Jordan (1982), Jordan and Turnquist (1983), Turnquist and Jordan (1986), Turnquist (1985), Powell and Carvallho (1998), Beaujon and Turnquist (1991), and List et al. (2003)

Review of the models for rail freight car fleet management

43

proposed a taxonomy for this category of problems by distinguishing between the deterministic and stochastic models, which they also distinguished depending on whether the cars are fully or partially loaded. Each of these combinations can also be analyzed from the aspect of traffic, as a task from one origin to one destination, from one origin to many destinations, or from many origins to many destinations. As the literature in the domain of vehicle fleet sizing doesn’t specifically address the problem of rail freight car fleet sizing, the remaining part of this chapter contains the most important general approaches as well as the approaches focused on other modes of transportation. All approaches are classified based on three criteria: the solution method, type of problem, and modality (Table 2.8). Some approaches apply exact methods based on the mathematical programming, whereas

Table 2.8 Main characteristics of models for vehicle fleet sizing Authors

Year

Dantzig and Fulkerson Kirby

1954

Parikh

1977

Etezadi and Beasley Saksena and Ramachandran Fu and Ishkhanov

1983

Bojovi
c and Milenkovi
c

2008

Bojovi
c et al.

2010

Zak et al.

2011

Cheon et al.

2012

1959

1986 2004

Solution approach

Linear programming Analytic, statistic method Queuing theory Integer programming Constructive heuristics Heuristics Multicriteria decisionmaking Multiobjective optimization Multiobjective mathematical programming Mixed integer programming

Problem type

Modality

Vehicle fleet sizing Vehicle fleet sizing

Maritime sector Railway sector

Vehicle fleet sizing Vehicle fleet composition Vehicle fleet composition Vehicle fleet composition Rail freight car mix problem Rail freight car size problem Vehicle fleet sizing

Road sector General

Rail freight car fleet sizing

Railway sector

Road sector Road sector Railway sector Railway sector Road sector

44

Optimization models for rail car fleet management

others use heuristics or metaheuristics. The following categorization was made according to the types of problem (Hoff et al., 2010): • problems of vehicle feet sizing where the size of the vehicle fleet is considered; • problems of the vehicle fleet composition where the aim is to determine the composition of vehicles of different types; • problems of the vehicle fleet composition and routing of vehicles in which the composition is combined with vehicle routing; and • problems of routing the heterogeneous vehicle fleet of fixed size where it is necessary to route fixed fleet of vehicles of different types. Table 2.8 contains some of the most important characteristics of models for homogeneous vehicle fleet sizing. The first paper related to vehicle fleet sizing was published by Dantzig and Fulkerson (1954) and considered maritime sector. The authors considered the problem of determining a minimal number of tankers for the realization of a fixed timetable. One of the earliest attempts for solving the problem of vehicle fleet size optimization on the railway was made by Kirby (1959). The author considered the two-sided problem of increasing the level of utilization of owned freight cars in a small railway system and also of decreasing the frequency of renting additional cars. By determining the relative daily cost of owned and rented cars, the author derives an expression for the total expected daily cost. From this expression, the number of additional minimal-cost cars in ownership as well as the rented cars can be determined. Parikh (1977) suggested an efficient approximate approach for solving the fleet size and allocation problem. The method is based on the results of the queuing theory and starts from the assumption that the total number of vehicles of an enterprise can be represented with a few separate fleet parks that have their own customer group. The author also assumed that these parks provide a uniform level of service, so that in the case of arrivals modeled by the Poisson distribution and service times distributed by the Erlang’s law, the model determines, for the already given probability, the optimal investments in fleet parks. Etezadi and Beasley (1983) simultaneously considered the problem of the optimal vehicle fleet mix and fleet size. For problem solving, the mixed integer programming formulation was presented. Saksena and Ramachandran (1986) determined the optimal size of a vehicle fleet (type and number of vehicles) through solving the problem of the transport of workers to certain locations (oil platforms). The proposed methodology enables determining the excess and shortage of each type of vehicle. The approach generates a much higher rate of

Review of the models for rail freight car fleet management

45

vehicle utilization and establishes a long-term strategy of vehicle supplying. Fu and Ishkhanov (2004) considered the problem of freight car fleet composition and suggested a heuristic approach for determining the optimal size and mix of a freight car fleet. Bojovi
c and Milenkovi
c (2008) suggested a solution to the vehicle fleet mix problem. The solution is based on the application of the analytic hierarchy process (AHP) as one of the most popular multicriteria decision-making methods. Bojovi
c et al. (2010) analyzed the problem of optimal freight car fleet composition. The problem was decomposed into two subproblems: the subproblem of optimal mix determining and subproblem of optimal freight car fleet size determining. The first subproblem was viewed as a multicriteria decision-making problem and solved by fuzzy AHP. The solution to the first subproblem was composed of the most suitable types of freight cars that can be used for transport of different groups of commodities. The second subproblem represents the problem of determining the optimal vehicle fleet size based on an already defined structure from the aspect of the freight car type. The fuzzy multiobjective LP was applied for solving the second subproblem. The presented two-phased solution approach was applied in the case of PE “Serbian Railways.” Zak et al. (2011) considered the problem of freight vehicle fleet sizing in the road sector. The mathematical model of the decision-making problem was formulated as a multiobjective mathematical programming problem based on the queuing theory. The solution procedure was composed of two steps. In the first step, a sample of Pareto optimal solutions was generated by the computer program MEGROS. In the second step, this set of solutions was estimated according to the model of the decision-maker preferences. The evaluation of solutions was conducted through the use of the interactive method of multicriteria decision making. In the end, the decision maker selects the best compromise solution. Cheon et al. (2012) analyzed the heterogeneous problem of rail freight car fleet sizing in the chemical industry. The problem statement also includes the inventory policy considering that, for the needs of the chemical industry, freight cars are used not only for transport but also for storing chemical commodities. The authors formulated the problem as a problem of the long-term capacity expansion considering that the freight cars have a long economic life. The proposed formulation also considers a number of economic factors. The mixed integer programming approach was proposed for the modeling and solving of this problem. Compared to the existing approaches, significant improvements were identified.

46

Optimization models for rail car fleet management

The vehicle fleet sizing and routing problem differs from the classical routing problems because it also considers the vehicle fleet composition. Therefore, the criterion function is based on the minimization of the total cost function, which contains fixed costs of the vehicle management and variable routing costs. Table 2.9 contains some of the most important characteristics of models for vehicle fleet sizing and routing. Table 2.9 Main characteristics of models for vehicle fleet sizing and routing Solution approach

Authors

Year

Golden et al.

1984

Gheysens et al.

1986

Salhi et al.

1992

Constructive heuristics

Teodorovic et al.

1995

Gendreau et al.

1999

Constructive heuristics, stochasticity Tabu search

Taillard

1999

Heuristic column generation

Bojovic

2000

Renaud and Boctor

2002

Lima et al.

2004

Optimal control theory Constructive heuristics, set partition Hybrid genetic algorithm

Wu et al.

2005

Integer programming, constructive heuristics Constructive heuristics

Linear programming

Problem type

Modality

Vehicle fleet composition and routing

General

Vehicle fleet composition and routing Vehicle fleet composition and routing Vehicle fleet composition and routing Vehicle fleet composition and routing Vehicle fleet composition and routing, Heterogeneous routing of fixed size vehicle fleet Vehicle fleet size and routing Vehicle fleet composition and routing Vehicle fleet composition and routing Vehicle fleet composition and routing

General

General

General

General

General

Railway sector General

General

Road sector

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47

Table 2.9 Main characteristics of models for vehicle fleet sizing and routing—cont’d Authors

Year

Choi and Tcha

2007

Song and Earl

2008

Sayarshad and Ghoseiri Sayarashad and Moghaddam Sayarshad et al.

2009

Sayarshad and Marler Loxton et al.

2010

Yaghini and Khandaghabadi

2012

Milenkovi
c and Bojovic

2013

2010 2010

2012

Solution approach

Integer programming, Column generation Dynamic programming Simulated annealing Simulated annealing Multiobjective optimization Multiobjective optimization Dynamic programming and golden section method Genetic algorithms and simulated annealing Fuzzy optimal control theory

Problem type

Modality

Vehicle fleet composition and routing

General

Vehicle fleet composition and routing Vehicle fleet size and routing Vehicle fleet size and routing Vehicle fleet size and routing Vehicle fleet size and routing Vehicle fleet size and routing

General

Vehicle fleet size and routing

Railway sector

Vehicle fleet size and routing

Railway sector

Railway sector Railway sector Railway sector Railway sector General

Golden et al. (1984) published one of the first studies in this area. The authors defined the problem and present the formulation in which the unit cost of movement is equal for all types of vehicles and it is considered as a constant parameter with unit value. Therefore, variable costs are independent of the type of vehicle. The authors proposed several heuristics for problem solving. Some are based on the technique of savings for the problem of vehicle routing developed by Clarke and Wright (1964) whereas the others are based on a “giant” traveling salesman problem, which is partitioned into subtrips adjusted to vehicle capacity. The authors also developed a procedure for the computation of the lower bound, which considers the relationship between fixed and routing costs. This lower bound procedure was later applied by Gheysens et al.

48

Optimization models for rail car fleet management

(1986) for creating a special heuristic. The heuristics uses the lower bound approach in the branch and bound technique for creating an optimal fleet mix, and the routing problem is solved by using the vehicle mix as the available fleet. The assignment of users to vehicles is determined by solving a generalized assignment problem according to the method proposed by Fisher and Jaikumar (1981). Salhi et al. (1992) presented the mixed integer programming formulation in which they introduced the variable unit traveling costs. They showed that simple modifications of some well-known methods may contribute to the reducing of variable traveling costs. Teodorovic et al. (1995) considered the problem with the stochastic demand included. The demand of every user has a uniform distribution in the interval [0, b], and the probability of cancellation was calculated on every route where the demand exceeds the available capacity. In the case of route cancellation, before it continues its trip to the subsequent customer, the vehicle must be returned to the depot for unloading. The heuristic approach was applied for creating a large trip, which is divided into smaller routes and assigned to appropriate vehicles by an algorithm for determining the number of the shortest paths on the network. Gendreau et al. (1999) suggested the tabu search method based on the general insertion heuristics, GENIUS (Gendreau et al., 1992) and AMP (Rochat and Taillard, 1995). The method was compared to the results obtained by Taillard (1999) and much better performances were identified for the instances with variable unit movement costs. However, the results were slightly weaker in cases where the routing costs were equal for all types of vehicles. Bojovic (2002) considered the problem of rail freight car fleet optimization along with demand satisfaction by minimizing the total cost. The mathematical model based on the optimal control theory was developed. The proposed optimization model provides all of the information about the railway network, such as the capacity of the railway yard, unsatisfied demand, and the number of empty and loaded cars in any time and in any location. The problem was formulated as a problem of finding an optimal regulator for a linear system, which is under the influence of the Gaussian white noise, with the quadratic performance index and random initial conditions. Renaud and Boctor (2002) described the “sweep” heuristics for solving the problem of vehicle fleet sizing and routing. The algorithm represents an extension of the algorithm for vehicle routing that was proposed by Renaud et al. (1996). Based on generating a large number of routes, the optimal selection of routes and vehicles is solved as a set partitioning problem. The suggested heuristics showed superior performances with respect to the alternative algorithms. Lima et al. (2004) described the

Review of the models for rail freight car fleet management

49

memetic algorithm for problem solving. This algorithm represents a hybrid of the GA and local search based on the GENIUS algorithm. Wu et al. (2005) suggested an approach to solving the problem of vehicle fleet size in the road sector. Operational and tactical decisions for a heterogeneous vehicle fleet were explicitly considered by the LP model with the aim of determining an optimal vehicle fleet size and mix. The demand was assumed as known whereas the time is a stochastic parameter. The solution approach had two phases. In the first phase, the transport demand was allocated to vehicles via Benders decomposition. In the second phase, the initial boundaries and dual variables from the first phase were used, and the solution was improved without increasing the effort for computation memory by using Lagrangian relaxation. Choi and Tcha (2007) presented an approach based on the column generation for problem solving. The authors suggested the integer programming model whose LP relaxation they solve by the column generation method. The dynamic programming schemes developed for the classical vehicle routing problem were modified for the sake of the efficient generation of feasible columns. The suggested algorithm was tested on instances with fixed and variable costs. The results pointed to the dominance of this algorithm regarding the solution quality and the computing time. Song and Earl (2008) presented an integrated model for the problem of determining an optimal policy design by empty vehicle allocation and vehicle fleet size in a system composed of two depots. Arrival times of vehicles and traveling times of empty vehicles were considered as stochastic variables. In this approach, the optimal strategy of vehicle scheduling for a homogeneous fleet is based on the control of the boundary value from the aspect of minimization of the expected discounted cost, which is composed of the cost of maintenance, leasing, and repositioning of vehicles in an empty state. The explicit cost function was formed under the condition of a general boundary value strategy. This function was then used for the derivation of optimal boundary values, optimal schedules of empty vehicles, and the optimal vehicle fleet size. Sayarshad and Ghoseiri (2009) proposed the formulation and solution approach for the optimization of the rail freight car fleet size and allocation in which the demand and traveling times were considered as deterministic parameters. The authors assumed that the unsatisfied demand becomes equal to zero at the end of the planning period, or in other words, that the car requests are completely satisfied during the planning period. The model also provides information about the railway network (the capacity of railway yards, unsatisfied demand, and number of empty and loaded cars in any period and in any station). Computational tests for

50

Optimization models for rail car fleet management

small examples can be solved with the exact approach within the short computational time, whereas in cases of medium and large size that is not possible. Due to this, the authors proposed the simulated annealing algorithm as a tool for model solving. Numerical examples were presented in order to check the efficiency and validity of the proposed algorithm. Sayarashad and Moghaddam (2010) suggested the formulation and solution approach for the rail freight car fleet size and allocation for the case of stochastic demand. The authors proposed a two-phased approach based on the simulated annealing algorithm. Sayarshad et al. (2010) suggested the multiobjective mathematical model and solution procedure for the optimization of freight car fleet planning on the railway. The model simultaneously covers the following objectives: • minimization of the sum of costs which consider the quality of service; • maximization of profit expressed as the difference between revenues generated by fulfilling the customer requests and combined costs of ownership and freight car movement; and • minimization of rail freight car fleet size. The Pareto optimal set is determined and applied for the trade-off analysis between the stated objectives. Sayarshad and Marler (2010) presented the formulation of multiobjective optimization, the solution method, and the analysis of the multiperiod problem of rail freight car fleet sizing. The suggested formulation incorporates the following functionalities into one tool for analysis: the functionality of the simultaneous optimization of a homogeneous freight car fleet and allocation, the functionality of profit optimization and quality, and the functionality of considering the yard capacity constraints. Profit and quality (minimum unfulfilled demand) represent mutually conflicting objectives which are simultaneously maximized. The Pareto optimal set was used for the trade-off analysis. The problem solution contains the optimal freight car fleet size and the optimal strategy of freight car allocation. Loxton et al. (2012) considered the problem of the heterogeneous vehicle fleet composition under the presence of the stochastic demand. The problem is based on determining the number of vehicles, which should be bought for each vehicle type, so as to minimize the total expected vehicle fleet cost. The authors developed an algorithm, which represents a combination of the dynamic programming and golden section method for problem solving. Yaghini and Khandaghabadi (2012) suggested the dynamic and multiperiod model for rail freight car fleet sizing. The demand and traveling times were assumed as deterministic. The hybrid approach based on the combination of GAs and simulated annealing was

Review of the models for rail freight car fleet management

51

used as the solving method. Experimental analysis was conducted on a few test cases. The comparison results of the suggested algorithm and CPLEX reflect the substantial efficiency and effectiveness of the suggested approach. When we consider the problem of routing the vehicles belonging to a fixed heterogeneous fleet, the vehicle fleet size is constant or limited by a maximum number. However, compared to the classical vehicle routing problems, vehicles can be of different types and thus have different fixed and variable costs. With respect to the previous class of problems, the aim is not to design an optimal vehicle fleet, but to use the vehicles of different types in the best way possible. Table 2.10 contains some of the most important characteristics of models for vehicle fleet sizing and routing. Gencer et al. (2006) developed a new intuitive algorithm for solving the problems of heterogeneous vehicle fleet routing. The passenger pickup algorithm (PPA) groups users in suitable clusters, and then finds the best route, which covers all customers in every cluster. PPA differs from the majority of other algorithms in the sense that it takes into account the possibility of vehicle renting in case of a vehicle shortage. By this definition, the problem can be treated as a problem of the fixed vehicle fleet with a variable fleet of rented vehicles. The algorithm also considers the demand splitting possibility. Tarantilis and Kiranoudis (2007) presented the flexible algorithm of adaptive memory. This two-phased constructive heuristics is named generalized route construction algorithm (GEROCA). The first phase determines an appropriate pair of customers and vehicles. In the second phase, the best inserting point is identified and an appropriate customer Table 2.10 Main characteristics of models for heterogeneous routing of vehicle fleet of fixed size Solution approach

Authors

Year

Gencer et al.

2006

Constructive heuristics

Tarantilis and Kiranoudis

2007

TavakkoliMoghaddam et al.

2007

Constructive heuristics, tabu search Simulated annealing

Problem type

Modality

Heterogeneous routing of fixed size vehicle fleet Heterogeneous routing of fixed size vehicle fleet Heterogeneous routing of fixed size vehicle fleet, service split

General

General

General

52

Optimization models for rail car fleet management

or a sequence of customers is inserted. The method is capable of decreasing the requests for vehicle fleet size and costs of movement comparing to the actual practice of vehicle fleet allocation. Tavakkoli-Moghaddam et al. (2007) considered a variant of splitting the delivery for the problem of the routing of vehicles with the fixed-size heterogeneous vehicle fleet. Vehicle fleet cost is dependent on the number of used vehicles and total unused capacity. For finding the solution, the authors developed a hybrid simulated annealing algorithm, which was tested on a couple of examples. The results for the cases of low sizes were compared with the optimal solution obtained by the exact methods, whereas the examples of bigger dimensions were compared with the lower bound obtained by solving the giant problem of a traveling salesman, who visits all customers. The results showed that the suggested heuristics can find a good solution in decent computational time. The logical continuation of the problem of vehicle fleet routing of a fixed size is introducing of time windows for every customer. These time windows define the interval in which the customer serving must begin. Time windows can represent hard constraints or soft constraints. In the case of hard constraints, the solution that does not satisfy these constraints is unfeasible. Soft constraints consider that earlier or later serving doesn’t influence the feasibility of the solution but they are penalized in the objective function. This extension can also be used for both vehicle fleet sizing and routing problems. Table 2.11 contains some of the most important characteristics of the models for vehicle fleet sizing and time windows-based routing of vehicles. Desrosiers et al. (1988) considered the problem of finding the optimal number of homogenous vehicles needed for serving a set of customers under the time window constraints. The approach that uses the augmented Lagrangian method was presented. Two approaches based on the Lagrangian relaxation were analyzed. In the first approach, time constraints were relaxed, whereas, in the second, the relaxed constraints were those which request that each node must be visited. The case of the heterogeneous vehicle fleet was considered by Ferland and Michelon (1988). The authors showed that the exact methods developed for the vehicle scheduling problem with time windows and one type of vehicle can be extended to the multiple vehicles type problem. Three different heuristics and two exact methods were developed. The heuristics are based on discrete approximation, scheduling method, and matching methods, whereas the exact methods use the column generation technique and relaxation of time

Review of the models for rail freight car fleet management

53

Table 2.11 Main characteristics of models for vehicle fleet sizing and routing with time windows Authors

Year

Solution approach

Problem type

Modality

Desrosiers et al. Ferland and Michelon

1988

Lagrangian relaxation

General

1988

Vis et al.

2005

Constructive heuristics, branch and bound, column generation Integer programming

Yepes and Medina

2006

Hybrid local search

Br€aysy et al.

2008

Deterministic annealing

Calvete et al.

2007

Constructive heuristics

Dimenzionisanje voznog parka Heterogeno rutiranje voznog parka fiksne velicine Dimenzinisanje voznog parka Heterogeno rutiranje voznog parka fiksne velicine Kompozicija voznog parka i rutiranje Kompozicija voznog parka i rutiranje

General

Road sector General

General

General

window constraints. Another approach to the vehicle fleet sizing problem under the time window constraints was presented by Vis et al. (2005). The authors described the transport between areas for temporary storage of containers (“buffer” areas) and storage areas in the container terminal. The aim was to minimize the size of the vehicle fleet in such a way that the transport of every container starts within its time window. The problem was formulated as an integer programming model, and the simulation was applied for the estimation of the vehicle fleet size. Yepes and Medina (2006) considered the case with soft time windows in the context of a heterogeneous fleet of fixed size. The authors presented the three-step hybrid local search algorithm based on the probabilistic neighborhood search for problem solving. The first step includes the construction of economic routes based on the greedy randomized adaptive search procedure (GRASP) algorithm (1995) which forms a population of solutions. In the second step, the evolutionary strategy based on the special selection approach is used for the search and selection of best solution in the population, whereas the third step represents the post-optimization method based on threshold algorithms with restarts.

54

Optimization models for rail car fleet management

Br€aysy et al. (2008) presented the deterministic annealing metaheuristic for problem solving. The proposed metaheuristics has three phases. In the first phase, the initial solutions were generated through a savings-based heuristic in combination with search strategies empowered with training mechanisms. In the second phase, an attempt of reducing the routes of the initial solution was made by the local search procedure. In the third phase, the solution from the second phase was further improved through four local search operators, which are the part of the deterministic annealing procedure. Local search operators were used for controlling the solution improvement process. Calvete et al. (2007) developed a mixed integer programming model for the vehicle routing problem with hard and soft time windows (hard time windows differ from the soft in the sense that they don’t allow jeopardizing of defined intervals), a heterogeneous vehicle fleet and a number of objectives. For problem solving, the authors recommended a two-step approach, in which the first phase was dedicated to determining all of the feasible routes and calculating the total penalty for every route due to deviation from the objective. In the second phase, the set partitioning problem was solved for obtaining the best set of feasible routes. In the case of the existence of more depots, the problem becomes more complicated than the basic problem of vehicle fleet composition and routing. The objective of the problem is the determining of users which need to be served by different depots by searching for optimal vehicle fleet composition and the best routes for vehicles. In Table 2.12, some of the most important models for multidepot vehicle fleet sizing and routing are presented. Salhi and Fraser (1996) treated the problem of determining the number and location of depots. They present the iterative approach which combines two different heuristics for the simultaneous solving of the problem of Table 2.12 Main characteristics of models for multidepot vehicle fleet sizing and routing Authors

Year

Salhi and Fraser Salhi and Sari Dondo and Cerda

1996 1997 2007

Solution approach

Problem type

Modality

Constructive heuristics Constructive heuristics Constructive heuristics

Vehicle fleet composition and routing Vehicle fleet composition and routing Vehicle fleet composition and routing

General General General

Review of the models for rail freight car fleet management

55

location, routing, and vehicle fleet composition. Salhi and Sari (1997) considered the problem of simultaneous allocation of customers to depots, the formation of a vehicle fleet, and designing delivery routes. The authors suggested multilevel composite heuristics and design two tests of reduction for the efficiency improvement of heuristics. The heuristics is composed of three levels. On the first level, the initial solution is determined by solving the problem for each depot and its customers. The boundary customers were then added to the existing routes. On the second level, composite heuristics attempts to improve the solutions obtained for every depot. On the third level, composite heuristics which includes all depots was considered. Dondo and Cerda (2007) also introduced time windows. The authors suggested a three-step approach for problem solving. In the first phase, cost-effective clusters were identified, then, in the second phase these clusters were allocated to vehicles, which were then sequenced on routes. The assignment of vehicle arrival times was conducted in the third phase by applying the mixed integer programming model. Network optimization problems differ from classic vehicle routing problems in their structure. Instead of searching for the optimal set of routes by which each customer was visited only once, these problems treat the choice of arcs on the graph for the sake of fulfillment of flow requests and minimizing the total system’s costs. Flow requests are generally expressed in the form of commodities which should be transported from origins to destinations in a given time interval. Table 2.13 contains some of the most important network models for multidepot vehicle fleet sizing and routing. Table 2.13 The main characteristics of network models for vehicle fleet sizing Authors

Year

Sim and Templeton Turnquist

1982

Turnquist and Jordan Beaujon and Turnquist Sherali and Tuncbilek

1986

1985

1991 1997

Solution approach

Statistical method Classification of researches in transportation Statistical method Integer programming Decomposition approach

Problem type

Modality

Vehicle fleet sizing Vehicle fleet sizing

General

Vehicle fleet sizing Vehicle fleet sizing Vehicle fleet sizing

General

General

General Railway sector

56

Optimization models for rail car fleet management

Sim and Templeton (1982) developed an efficient recourse algorithm for the analysis of vehicle behavior from one origin to one destination. Traveling time was under the exponential probability distribution law, whereas the arrivals were assumed to be individual with characteristics of the Poisson distribution. For the definition of the cost function and the rules of vehicle dispatching which are based on the queuing theory optimal vehicle fleet was determined. The task belongs to the category of stochastic approaches, for partially loaded vehicles from one origin to one destination. Turnquist (1985) described a few research possibilities in the area of transport service supply with a focus on service design. The research areas were classified into three main categories: vehicle scheduling, vehicle management, and provision of capacities which includes vehicle fleet sizing. The basic classification of approaches for vehicle fleet sizing problems was defined by three factors: the transport type, shipment size with respect to vehicle size, and deterministic vs stochastic analyses. Turnquist and Jordan (1986) developed a model for sizing the container fleet used for the transport of components from one factory to more locations, for the assembly of these components. The model was formulated based on the following assumptions: • Production of parts is conducted in an arbitrary, deterministic cycle. • The size of the shipment is constant and small in comparison to the number of components produced in one production cycle. • Shipments are formed on a daily level. • Traveling times of containers are deterministic. The last assumption was relaxed in order to assess the impact of the uncertainty of traveling times. Based on the results for the process of the infinity server, the authors derived equations for the simultaneous treatment of the container fleet size and the probability of shortage. The authors illustrated with an example the interdependence between the probability of shortage and container fleet size and show the impact of the uncertainty levels and the number of assembly locations on the container fleet size. With small modifications of the derived equations, it is possible to apply the presented methodology on fleet sizing problems of other types of transportation assets. An important potential effect of improved empty vehicle distribution is reflected in decreasing the fleet size without an impact on reducing the quality of service. Beaujon and Turnquist (1991) formulated a model which includes interactions between the vehicle fleet size, loaded, and empty movements. It is a stochastic optimization model (demand and traveling times are stochastic) on a dynamic network, which was solved by decomposition which interactively treats subproblems of vehicle inventories on

Review of the models for rail freight car fleet management

57

the network and vehicle allocations. In every node, the net number of cars is defined by the Gaussian distribution and associated holding costs and costs of unfulfilled orders. Assuming that the traveling time is deterministic, the problem was reformulated in the network problem with nonlinear costs on some arcs. Allocations of loaded and empty cars were simultaneously determined on a network of relatively small size. The network approximation proved to be appropriate for obtaining a good enough solution from the aspect of vehicle allocation and fleet sizing. The initial solution was obtained by solving the deterministic variant of the problem (variances of the number of cars in stations are equal to zero). This solution will not be optimal, but it gives a set of car movements sufficiently precise for the variance calculation. For the assessed values of variance, the Frank-Wolfe algorithm was applied. Searching for a solution continues by iterative procedure until the change of variance. The authors compared the static deterministic, dynamic deterministic, and dynamic stochastic variant on a hypothetic network composed of five stations on a 6 days planning horizon. The obtained results reflected the importance of simultaneous consideration of vehicle fleet size and vehicle fleet allocation. Sherali and Tuncbilek (1997) suggested the static and dynamic models for freight car fleet sizing in case of freight car pooling. The static model strives to underestimate the necessary size of a freight car fleet due to the fact that it is based on the stationary data. The dynamic model is based on a space-time network which represents the movement of empty cars between origins and destinations during a certain planning period, with the aim to minimize the vehicle fleet size with the satisfaction of the total demand in different time periods. The problem is solved by decomposing the model into a sequence of smaller subproblems with a shorter, mutually overlapping time period. After the problem is solved, decisions for the initial part of the considered period are fixed and the next subproblem is solved with adjusted flows. Examples generated in a random way with realistic assumptions were used for the evaluation of algorithm performances. The models were successfully used by the Association of American Railroads.

CHAPTER 3

Centralized and decentralized model for empty freight car scheduling Contents 3.1 Characteristics of decentralized managerial-control functions in the process of empty freight car allocation 3.2 Centralized network model 3.3 Solution method of triaxial transportation problem 3.3.1 Basic solution 3.3.2 Feasible solution 3.3.3 Final solution 3.3.4 Problem of degeneration 3.4 Decentralization of the main managerial functions in the empty car allocation process 3.5 Decentralized terminal model of empty freight car allocation 3.6 Validation of the proposed model

62 64 67 67 68 68 68 69 72 76

The traditional attitude by which all business decisions of influence on the transportation system should be generated in a centralized way has been seriously shaken by the appearance of new alternative organizational structures and their contribution to the improvement of managing effectiveness. The initial assumption for this consideration has been articulated through a possibility of decentralization of decisions in the use of financial assets, allocation of airplanes, locomotives, and freight cars (Bojovic, 1992). The main barrier for overcoming of organizational issues are insufficiently developed methodological frameworks for decentralized decisionmaking which should lead to higher effectiveness, as opposed to the traditional centralized which are focused on efficiency. Combining these requests generates an extremely complex environment for decision makers. The development of sophisticated information systems in real-time usually does not contribute to increasing the revenues in transport organizations. The main reason is that considering the information system without having in mind the organizational structure does not induce a Optimization Models for Rail Car Fleet Management https://doi.org/10.1016/B978-0-12-815154-9.00003-4

© 2020 Elsevier Inc. All rights reserved.

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Optimization models for rail car fleet management

more influential impact on transport organization. Therefore, for the design and development of business processes in transport, it is necessary to consider separately the influence of three dimensions: the organizational structure, the information system, and the decision-making process. Distribution of these influences and its quantification enables consideration of interrelationships between these dimensions. Formulating a hierarchy in a control problem, which specifies the basic control task (maximization of revenues of the organization) with a dominant individual activity of an organization (resource allocation, actually the allocation of freight cars) is specified, represents a key for connecting the main three dimensions of the task. In the era of emerging innovations, contributions of railways must be expressed by those advantages that do not characterize other modalities. Two main directions through which it is possible to consider these comparative advantages are the economically efficient transport of large volumes and suitability of railway transport for automation and computer control. These advantages, despite their importance, do not lead to a profit unless they do not contribute to a transport service which is attractive for the users. The imperative of the railway should be on the establishment of a commercial philosophy that will have the main aim to enable a client-centric approach for almost every freight car request fulfillment. The implementation of computer-aided support in information handling, as the base for freight car management in a territory, represents an intention to embed an emerging technology in a traditionally based system. The freight car fleet represents the most expensive component of railways. According to operational indicators, this asset spends most of its time in empty condition. Actually, it can be said that the problem of many transportation systems relying on the time or space imbalance of transport requests is resource allocation. Once unloaded vehicles must be allocated or kept in a station for reloading. The high percentage of related operational costs on railways gives solid guarantees for the research and utilization of scientific methods of modern organizational theory and systems control. As it has been already mentioned, empty freight car allocation must be solved through the interactive building of the appropriate organizational structure, information system design, and decision-making process. Movement of loaded cars from origin to destination represents a primary productive function of the railway. Dispatching empty cars from the unloading location toward the next potential consignee comes as the response on the existence of requests for the reuse of freight cars. Processes

Centralized and decentralized model for empty freight car scheduling

61

of intermediation in receiving and delivering of cars are complicated; however, all the empty cars can be in one of two states: • waiting in the supply node (every node on the railway network where empty cars can enter or leave the system); • traveling between the demand and supply nodes. Simultaneously making decisions about empty stored cars, user requests, and available supply practically mean managing thousands of empty cars from one node of the network, and it represents an extremely hard task. The number of cars that are sent or received from a neighboring railway system defines available supply in the system. The state and location of all cars in the system are continuously changing, whereas the user orders vary by days during the week or a season and can be significantly increased in periods of a deficit of freight cars. The existence of different types of freight cars with only partial substitutability between them adds additional complexity to the problem. Decisions about empty flows determine the level of empty stored cars in each supply node taking into account the capacities of the stations or throughput of railway sections as well. In this chapter, the computer-aided decentralized decision-making approach for empty freight car allocation is proposed. The approach includes two models, network and terminal, for an efficient and refined resource allocation procedure. The network model considers revenues and costs of the system also taking into account the available capacities of stations and rail sections according to the existing timetables. The approach is mathematically formulated as a triaxial transportation problem. The solution of this problem is based on a modified method of potentials and it simultaneously enables the computation of the primary solution, the flows of empty cars, and dual solution, shadow prices of supply, demand, and available capacities. The resource allocation mechanism considers price schedules as a linear function of resources. This assumption enables a regional freight car manager in each iteration to “buy” the cars from the central unit by the vector of price so that the profit of terminals is maximized at the same time. In this way, terminals solve subproblems that have a quadratic criteria function and linear constraints and for resolving, the Kuhn-Tucker theorem has been applied. This leads to a vector of demand for a resource in deficit, as a result of individual terminal computations.

62

Optimization models for rail car fleet management

3.1 Characteristics of decentralized managerial-control functions in the process of empty freight car allocation Majority of transportation companies still face the problem of allocation of empty vehicles to those destinations where these units should be reloaded. Therefore, the time-related control decisions about vehicle fleet use have great importance primarily in those companies that contain complete information about the state and location of vehicles. Road haulers, container shipping companies, rent-a-car agencies, and rail freight car leasing companies represent the most obvious examples of problems belonging to this category. Since minor modifications are needed in order to apply the models in different transport systems, here the focus is on empty freight car distribution in the railway domain. The organizational design of transport organizations implicitly assumes having knowledge about main control variables. For a smooth decisionmaking process about the allocation of resources the coordination mechanism, its structure, planning horizon, and decision-making levels need to be defined. A variety of models for empty freight car distribution have been proposed in the past, however, their implementation in practice was not so successful. Due to a very narrow problem definition, the centralization of all management functions, and relying only on observed data in the phase of model formulation. Alternative approaches, which will also be considered in the following chapters related to rail freight car fleet sizing, consider the decentralized decision-making process. The development of the decentralized decision-making process in empty freight car allocation and its interactive use with the centralized approach represents the primary objective in this chapter. The requirement related to the availability of upcoming requests for empty freight cars is considered in this approach and it can be fulfilled by the availability of appropriate information system. The establishment of a coordination mechanism is based on a generalized penalty function with which the components of the price vector for optimal buying or selling of joint resources can be determined. This vector becomes a good guide for the movement of cars on a transportation network and at the same time shows that in some transport networks it could be possible to decentralize the main management functions. In order to demonstrate the proposed procedure, a heterogeneous rail freight car fleet structure has been taken into account together with the planned timetable, total available capacity in trains for empty cars, on all

Centralized and decentralized model for empty freight car scheduling

63

branches of a considered rail network. This leads to the possibility of a more realistic solution of the empty freight car allocation problem. The analytical aspect of the approach can be presented through a modified triaxial transportation problem. Fig. 3.1 illustrates the main components of the freight car allocation model for observed information flows and the adopted control hierarchy. The basic idea of iterative decentralized planning based on the method of price adjustments was proposed by Dantzig and Wolfe (1961). This method Network model Determining of trial shadow prices

Aggregating of initial information

Solution consistency check Coordinating subsystem N

3 2

Terminal model Terminal operation control

1

Design of operative and applicable rules

Technological requests for empty allocation task

Fig. 3.1 The main components of the empty freight car allocation model.

64

Optimization models for rail car fleet management

will be used for the presentation of decentralized managerial activities from an organizational standpoint. The economic interpretation of decentralized resource allocation was described and analytically represented by Jennergren (1973). The vector of the so-called price schedules does not give good results in linear economic systems. The algorithm which will be used for freight car allocation is based on the iterative adjustment of price schedules vector until the optimal vector of price schedules is obtained. In this way, the regions, or terminals, have the opportunity to include their local optimization of the level of production activity in the global optimal combination of a centralized unit. As the support for freight car managers, and according to technological requests of the freight car allocation process, two submodels have been proposed, the centralized network model (includes decision-making from one central place) and terminal model (requests for a deficitary resource are computed on N locations). Although the problem of optimal resource allocation has a nature of multicriteria decision-making problem, the choice of profit maximization criteria implicitly takes into account maximization of revenues, minimization of empty freight car flows, and service quality. In this model, allocations of foreign cars have not been included considering that these cars must be returned to railway companies, which are their owners.

3.2 Centralized network model The centralized network model describes the movement of empty cars toward the demand nodes and from the supply nodes according to the capacities of available links. Let us define N as the number of nodes on a railway network, and P as the number of periods of the planning horizon. It is assumed that the discretization of space and time aspect of the allocation problem is possible. On railways, the decisions about the disposition of freight cars are mostly made daily on the beginning of the period in instances τ0, τ1, τ2, …, τn. Costs and revenues relevant for this centralized model are the costs of empty freight car traveling between network terminals and revenue by loaded cars, respectively. Assumed that in a rail freight car fleet there are V different types of freight cars. Let us introduce the following parameters: rjv: revenue by loaded cars of type v in j station reduced for transportation costs of loaded cars from that station to the given destination; cijv: unit costs of movement of empty freight cars of type v from station i to station j;

Centralized and decentralized model for empty freight car scheduling

65

φijv: flows of empty cars of type v from station i to station j; Djv: net demand for empty freight cars of type v in station j; Svi: net supply of empty cars of type v in station i; Kij: available capacity for empty cars, which can be added to the trains carrying loaded cars (this actually means that they can be allocated by the timetable), in a considered time period along the link ij of the railway network. The task of determining the transportation plan is based on determining the vector φ∗ ¼ {φ∗ijk} by which the total revenue has been maximized, and can be formulated by a triaxial transportation problem in which constraints do not need to be expressed as equalities: N X N X V  Y X  ðφÞ ¼ rjv  cijv φijv

(3.1)

j¼1 i¼1 v¼1 N X

φijv  Djv , for j ¼ 1, 2,…,N and v ¼ 1,2, …,V

(3.2)

i¼1 N X

φijv  Svi , for i ¼ 1,2,…, N and v ¼ 1,2,…, V

(3.3)

φijv  Kij , for i ¼ 1,2, …,N and j ¼ 1,2,…, N

(3.4)

j¼1 V X v¼1

φijv  0, i ¼ 1,2,…, N ; j ¼ 1,2, …,N ; v ¼ 1,2,…, V

(3.5)

It is obvious that coefficients of the objective function and empty car flow form three-dimensional (3D) matrices. The implicit assumption in this formulation is that each terminal represents the supply and demand node at the same time. Based on the form of the criteria function it is evident that the movement of a flow of empty cars, as the response to the requests in demand nodes, will be conducted only if the coefficients of this function are positive. The modified triaxial transportation problem expressed in this way does not consider fulfilling the conditions: N X

Djv ¼

j¼1 V X v¼1

N X

Svi , v ¼ 1,2,…, V

(3.6)

Kij ,i ¼ 1,2,…, N

(3.7)

i¼1

Svi ¼

N X j¼1

66

Optimization models for rail car fleet management

N X

Kij ¼

i¼1 N X V X

V X

Djv , j ¼ 1,2, …,N

(3.8)

v¼1

Djv ¼

j¼1 v¼1

N X V X i¼1 v¼1

Svi ¼

N X N X

Kij

(3.9)

i¼1 j¼1

The method of potentials is usually applied for solving the triaxial transportation problem. This method is based on the duality theory. The dual task of this problem can be expressed in the following way (Raskin and Kiri
cenko, 1982; Taha, 2003): Determine the values of the potentials, λvi, μjv, vij, i 2 I, j 2 J, v 2 V which minimize the function: L ðY Þ ¼

N X V X i¼1 v¼1

λvi Svi +

N X V X j¼1 v¼1

μjv Djv +

N X N X

vij Kij

(3.10)

i¼1 j¼1

and satisfy constraints: λvi + μjv + vij  rjv  cijv ,i ¼ 1,2,…, N; j ¼ 1,2,…,N ; v ¼ 1,2,…, V (3.11) so that {φijv} represents a unique solution of the triaxial problem if and only if n o (3.12) λvi + μjv + vij > rjv  cijv , ði, j, vÞ 2 ði, j, vÞ : φijv ¼ 0 n o λvi + μjv + vij ¼ rjv  cijv , ði, j, vÞ 2 ði, j, vÞ : φijv > 0 (3.13) The economic interpretation of the terminal potentials or dual variables was given by White and Bomberault (1969). The potentials determine the level of change of the maximum value of the objective function with respect to the level of relaxing the constraints. In economic literature, the optimal values of dual variables are usually expressed as the shadow prices or marginal costs of resources of an organization. These terms came from the fact that the optimal value of dual variables determines the contribution of each resource in total profit for a given vector of optimal solutions φijv. In a numerical sense, this value is equal to the increment of profit for the unit increment of resource, under the condition that the resources must be exploited in an optimal way. The costs of holding the cars in each station i of a railway network can be expressed as the value λvi, whereas for the flow of empty cars φijv between a pair of stations i and j it is implicitly assumed that the costs of holding an

Centralized and decentralized model for empty freight car scheduling

67

additional freight car are equivalent to a potential revenue from a car of type v in station j reduced for costs of demand satisfaction or reduced for the number of used positions in a train. Since it has already been emphasized that dual variables, by which the saturation or unsaturation of constraints (3.2), (3.3), and (3.4) is estimated, have important economic meaning, so that λvi represents marginal costs of supplying additional cars of type v to the station i, the positive value μjv > 0 shows that, with the existing allocation of empty cars the demand for cars of type v in station j is satisfied, whereas vij, under the satisfied inequality of (3.4) constraint, indicates that on the link ij there are enough positions for adding the cars in train compositions. The variables considered are also known as the shadow prices. Namely, for a station from which the dispatching has been performed, if λvi is equal to zero, this means that an excess of cars of type v in the station i exists and that it does not have any impact on total revenue. On the contrary, any unit increment or decrement of the number of cars increases or decreases the total revenue for the amount equal to the value of potentials. The same holds for values of μjv, whereas the zero value of vij indicates that the unit increment of the number of cars for dispatching does not influence on the total transportation capacities on a given rail section.

3.3 Solution method of triaxial transportation problem There are a number of computational procedures for solving the 3D transportation problem (Haley, 1962; Raskin and Kiri
cenko, 1982). Considering that there is large computational complexity due to the frequent altering and a big number of parameters, in this chapter a new approach for solving triaxial transportation problem is proposed.

3.3.1 Basic solution Similar to the method of the northwest corner, in this chapter, the method of iterative filling the cells of the 3D matrix φijk whose number of indices i + j + k increases in stages is generalized. The value min(S11, D11, K11) is added to the element φ111. The spent resource cannot be used in the next states. In the next iteration, the cells whose number of indices i + j + k ¼ 4 are filled. Each cell is filled with the value according to the method of the northwest corner. By increasing the sum of the index by one, the procedure is repeated in the same way until the last cell is filled, by each dimension. In these cells, the remaining, unused resources are filled, except in the cells that lay on

68

Optimization models for rail car fleet management

the intersection of the boundary surfaces in which the values that correspond to the constraints are filled (these values can also be negative).

3.3.2 Feasible solution During the cell fulfilling procedure, from φ111 to φN1, N1, V1, the main logic is that all resources are available. In order to stick to this logic we adopt that φijk has the value equal to the least available resource:   (3.14) min Svi , Djv , Kij Only the values of elements φNjv, φiNv, and φijV are forcingly added, so these values may not have physical sense and they can be negative. By a generalization of the redistribution method the spatial chains are made for all negative fields of 3D matrix φijv. Here, the original procedure of spatial chains formation is proposed for transforming the basic solution into a feasible solution by the elimination of negative advantages in matrix φijv.

3.3.3 Final solution The described approach based on the redistribution method serves to eliminate all negative values in matrix φijv. Now, for all cells of zero supplies we form spatial chains. By the method of potentials, with defined prices, we determine the characteristics of spatial chains. By the spatial chain of the highest negative characteristic, we conduct redistribution of supplies. According to the method of potentials, the approach is iteratively applied until all negative characteristics in the cells of zero supplies disappear. In that case, the obtained plan is optimal.

3.3.4 Problem of degeneration The application of the described model may lead to the cases of various degenerations. These cases require special attention and ask for additional steps in the procedure. The most frequent degenerations are: • appearing more than N + N + V + 1 of nonzero supplies in the plane V ¼ 1; • appearing less than N  N  V  (N  1)(N  1)(V  1) of nonzero supplies in the plane V ¼ 1; • appearing of more equal minimum values on “–” places of the spatial chain. These degenerations are treated in the following way. The degeneration in the plane V ¼ 1 is solved by equal treatment of all nonzero elements in that

Centralized and decentralized model for empty freight car scheduling

69

plane whereas the degeneration in the spatial chain is eliminated by introducing a variable ε (in this specific case for ε we adopt a lower value than those in matrix φijv) in an appropriate place.

3.4 Decentralization of the main managerial functions in the empty car allocation process Modeling the process of empty freight car dispatching from the supply nodes represents a complex research task. There is a variety of diverse factors of influence on the complexity of the mathematical model, whereby all these factors have a stochastic character. Since the aim of this chapter is decentralization of empty freight car allocation processes, then the most important is the problem of finding an adequate mechanism of coordination between one central and a number of local decision-making units. The mathematical model of resource allocation in terminals implies response to the received car orders or defining of operational procedures for sending or supplying of freight cars. Decomposition algorithms of mathematical programming may serve for a refined description of decentralization in transportation organizations. The structure of some tasks of linear programming opens a variety of possibilities for analysis of linear economic systems through the task of resource allocation. The organizational model can formally be represented by the executive or coordinate unit, which solves the modified triaxial transportation problem and sends the data to the terminals. These terminals then solve the problem and return the requests to the executive decision-making unit. The coordination mechanism between the executive unit and terminals usually relates to the Dantzig-Wolfe approach. The essence of this approach is that the search for an optimal solution is conducted by decomposition of the initial problem into a series of independent problems of lower dimensions. The advantages of this approach do not only include practical computational aspects but also enable subtle economic interpretation. Since, with this task, the terminals have conducted the redistribution of the common deficit resource, the shadow prices of general constraints are the real carriers of information about the marginal value of cars. Therefore, the solution to the primary problem of the triaxial transportation program does not offer a complete solution, and the dual program needs to be considered. The information about marginal costs, or Lagrangian multipliers of the general constraints of a freight car supply, enables reducing the coefficients of

70

Optimization models for rail car fleet management

profit for terminals for the value of the use of freight cars. Jennergren (1973) proposed an approach with which the central decision-making unit finds the price vector of the common resource by leveling the demand for that resource with the available supply of cars by terminals whereby the price of the resources offered in the amount higher than their demand is equal to zero. Knowing the elements of the distribution prices set, dispatchers are able to test the effects of different strategies for freight car allocation by evaluating the total system’s profit, whereas the terminals may solve their own tasks as quadratic subproblems. In this way, the real assumptions are made for the motivation of the regional manager to actively contribute to increasing the total profit of the whole system. In a mathematical sense, this means that the optimal allocation of resource is not generated by the peaks of the intersecting hyperplanes, but from their internal area. If we assume that the set of additional conditions (3.3) of the triaxial transportation problem corresponds to a general constraint and that the inequalities (3.2), (3.4), and (3.5) are considered as the constraints of subproblems, we may apply the proposed decomposition algorithm. The illustration of the structure of the triaxial transportation problem which can be solved by the decomposition approach is presented in Fig. 3.2. In the procedure of deriving the relations that describe the mechanism of distribution prices, the central premise corresponds to their linear functional connectivity with freight cars. In this way, subproblems of terminals become a quadratic type. Based on the previous derivations we may consider the problem of maximization of the expression (3.15): ½ðr1v  ci1v Þ  φi1v + ðr2v  ci2v Þ  φi2v + ⋯ + ðrNv  ciNv Þ  φiNv 

(3.15)

subject to: G1  φi1v + G2  φi1v + ⋯ + Gj  φijv + ⋯ + GN  φiNv  SO H1  φi1v

(3.16)

 d1  d2

H2  φi2v Hj  φijv

 dj

(3.17)

HN  φiNv  dN φijv  0, j ¼ 1, N , i ¼ 1,N , v ¼ 1,N

(3.18)

71

Centralized and decentralized model for empty freight car scheduling

Net supply in all terminals of the network by all types of rail cars

Common constraint

Terminal

3

1

5

2

4 N

Net demand in all network terminals by all types of cars

1

2

Constraints of Subproblems

3

Available capacities of empty cars on all links

j

N

Fig. 3.2 Decomposition structure of triaxial transportation task.

In this reformulated modified triaxial transportation problem, the introduced matrices and vectors have the following dimensions and components: SO ðS11 , S12 , …, Svi , …, SVN Þ is NV vector 2 3 1 0 0 ⋯ 0 60 1 0 ⋯ 07 6 7 7 Gj ¼ 6 60 0 1 ⋯ 07 4⋮ ⋮ ⋮ ⋮5 0 0 0 ⋯ 1

(3.19)

(3.20)

dimensions of the matrix are NV  NV.   dj D11 , D12 , …, D1v , DNV , K11 , K21 , …, Kij , …, KNN , j ¼ 1,2,…, N (3.21)

72

Optimization models for rail car fleet management

is (N + V) vector. 2 1 1 1 60 0 0 6 V6 60 0 0 60 0 0 Hj ¼ 6 61 0 0 6 N6 60 1 0 40 0 1 : : :

1 0 0 0 0 0 0 :

0 1 0 : 1 0 0 :

0 1 0 : 0 1 0 :

0 1 0 : 0 0 1 :

0 1 0 : 0 0 0 :

: : 1 : 1 0 0 :

: : 1 : : 1 0 :

: : 1 : : : 1 :

: : 1 : : : : :

: : : : : : : :

: : : : : : : :

0 0 0 1 : : : :

3 0 07 7 07 7 17 7, j ¼ 1, 2,…,N :7 7 :7 7 05 : (3.22)

which has the dimensions (N + V)  NV; (rjv  cijv) is N  V vector, j ¼ 1, 2, …, N; φijv is a variable NV vector.

3.5 Decentralized terminal model of empty freight car allocation The implementation of the decentralized organizational structure presupposes the application of the coordination mechanism of a decentralization of business functions of a transport organization. In the context of the empty freight car allocation, these activities are performed through price schedules. By this procedure of buying empty cars, according to the real needs of terminals, follows the price matrix so as to maximize the profit of each terminal. Analytically, this can be expressed as follows: h i  max rjv  cijv φijv  p  Gj  φijv , j ¼ 1,2,…, N n  o (3.23)  φijv 2 φijv Hj  φijv  dj , φijv  0 From Eq. (3.23) it is obvious that profit coefficients for terminals are decreased for the costs of resource use since this resource can be considered as the ownership of all remaining nodes on the network with the use of which the revenue in terminals is generated. Numerical indicators of the use of the shared resources of an organization are usually expressed in literature as a linear function of ordered or delivered freight cars for a given terminal. The set of these values has the form: lvi + k  yijv ¼ lvi + k  φTijv  GjT ðk > 0Þ i ¼ 1, 2,…,N ;v ¼ 1,2,…, V ;j ¼ 1,2,…,N ;

(3.24)

and it represents price schedules. All used indices have the same sense as in the previously stated relations. Dual representation of the number of

Centralized and decentralized model for empty freight car scheduling

73

delivered freight cars, yijv and φijv can make a confusion, however, considering that both the centralized and decentralized model converge to a unique optimal solution in the final number of steps, which is also illustrated by a numerical example, this confusion will be easily eliminated. Therefore, representing the price vector as a linear function of state and freight car requests in terminals, reformulates the stated problem in the quadratic subproblem of profit maximization for each terminal. Its analytical representation is h  i   max rjv  cijv  lvi + k  φTijv  GjT  Gj  φijv (3.25) subject to:

n  o  φijv 2 Fijv ¼ φijv Hj  φijv  dj , φijv  0

From the set of feasible solutions each terminal generates one vector, namely, (φi1v); (φi2v); …; (φiNv), which gives the opportunity for the initialization of the decomposition algorithm of price schedules by selecting a positive, but sufficiently small value for ko, 0 < ko  k, where k is defined as 8 9 > > > > > > > > < = δ  , δ>0 k ¼ min  N (3.26) > > >X >  > > T T > 2  φijv  Gj  Gj > : ; j¼1

The initial values for components of vector l are taken from the feasible solution of the triaxial transportation problem, its dual actually, through the values of potential λvi. In the first step, the central decision-making unit promotes the vector of price schedules l + ko  φTijv  GTj so that each terminal solves a quadratic subproblem: h  i   (3.27) max rjv  cijv  lvit + ko  φTijv  GjT  Gj  φijv subject to: φijv 2 Fijv. By solving these problems for each terminal j the components of the demand vector Gj φijv with the shortage of freight cars can be found. Since this is the quadratic programming problem where the criteria function is of the maximization type, and the set of constraints has a linear form, it is possible to apply the Kuhn-Tucker theorem (Zangwill, 1969). The necessary conditions given in the matrix form represent an explicit application of the constrained gradient method.

74

Optimization models for rail car fleet management

2 3   3 φijv   2ko  GjT  Gj HjT   I jO 6 μv 7  rjv  cijv  lvi  7 4 56 6 7¼ 4 Hj jOjOjI j dj ψ ijv 5 T j ¼ 1,2,…, N ,i ¼ 1,2,…, N, v ¼ 1,2,:…, V 2

(3.28)

with an additional constraint ψ ijv  φijv ¼ 0 ¼ μv  T for all values i , j,v

(3.29)

In matrix equality (3.28), all the elements have the same purpose, I is a unit matrix, O is zero matrix, whereas μv and ψ ijv are the vectors of the Lagrangian multipliers of the corresponding constraints.   (3.30) μv ¼ μ11 , μ12 , …, μjv , …, μNV , v11 , v21 , …, vij , …, vNN j ¼ 1, 2, …, N and μv is (N + V) vector.   ψ ijv ¼ ψ 111 , ψ 211 , …, ψ ijv , …, ψ NNV j ¼ 1,2,…, N ,i ¼ 1,2,…, N ,v ¼ 1,2,:…, V   T ¼ T11 , T12 , …, Tjv , …, TNV , T11 , T21 , …, Tij , …, TNN

(3.31) (3.32)

According to all given derivations, it is clear that the initial quadratic subproblem of terminals can be expressed as the system of linear equations, which fulfill the additional constraining condition (3.29). The next step of the decomposition algorithm includes the computation of a new vector of price schedules. lt+1 + koφTijv  GTj in the following way: n h io lqt + 1 ¼ max 0, lqt + αo ðG1 Þq  φi1v + ðG2 Þq  φi2v + ⋯ + ðGN Þq  φiNv  ðSo Þq t+1 whereas αo represents a positive where lt+1 q , qth component of vector l constant. The numerical efficiency of the considered algorithm is primarily dependent on the selection of positive constants ko and αo. As a good convergence indicator of the algorithm, the following quantity may be of use:   X  N   Gj φijv  So    j¼1 

Namely, if the transition between iterations is not followed by decreasing the value of this amount, the algorithm does not converge so the value αo

Centralized and decentralized model for empty freight car scheduling

75

needs to be decreased. Following the same logic, very slow decreasing of the value of this difference points to the need to increase αo. Based on the previous observations it is clear that the convergence of price schedules vector lo + koφTijv  GTj leads to an optimal solution of the centralized tiraxial transportation problem. From the organizational aspect, this leads to the conclusion that the decentralization of the main productive functions of transport organizations is possible. This has also been analytically proved. In the process of empty car allocation, it is not convenient to perform the information exchange at every step. Knowing that the vector of price schedules converges to the optimal vector so that the amounts of the shared resource added to terminals j converge to the amounts, which are optimal for the entire system, the values in the final iterations become optimal values. Fig. 3.3 presents the block scheme of a computer program for solving the decentralized terminal model. Terminal model Initial values

Construction of a matrix of general and special constraint

Finding the vector of price components

Searching for a solution in dual of feasible solution

Finding an initial demand vector

Searching for the vector from feasible solution of triaxial problem

Determining Lagrangian multipliers Printing the solution

Printing the solution Finding constant component of price vector

Computation and selection of values

Printing the solution

Solving of quadratic problems

Definition of KuhnTucker conditions

Computation of new values of price vector components

Checking the convergence conditions

Solving the system of linear equations

Formation of terminal demand vector Printing the solution

Repeating the procedure with new components of price vector Computation of criteria function values

Printing the solution

Fig. 3.3 Block scheme of computer program for the decentralized empty freight car allocation model.

76

Optimization models for rail car fleet management

3.6 Validation of the proposed model This part contains numerical experiments for the network and terminal model, or in other words, the centralized and decentralized organizational decision-making model. The analysis has the aim to show: • in which conditions and under which limitations it is possible to perform decentralization of the main productive function of railway organizations; • what is the behavior of the convergence rate of quadratic subproblems of terminals; and • what are the computational experiences in the computation of eigenvectors of demands for deficient resources during the solving of the triaxial transportation problems. For this purpose, an aggregated network database is defined based on the real geographical rail network of the Republic of Serbia (Fig. 3.4). Terminals are the regional centers for freight car management, denoted by letters from A to J, and the links connecting these terminals represent the railway lines. Model inputs are • real and tariff distances (km) between stations of the considered railway network; • daily supply and demand of empty cars by all terminals and all freight car types (open, closed, flat, and other); • daily number of available capacities in trains according to the timetable and aligned with the capacities of stations and sections; and • transport costs of empty freight car allocation and revenues from goods transportation. Based on the real and tariff distances between the stations on the network (Table 3.1) transport costs of freight car allocation have been calculated. This data is used for determining the functional dependence between the costs of loaded cars and distances traveled by those cars between the station of loading and station of unloading. In that sense, a computer program is developed and which is based on the method of approximations. From the set of curves, with the highest coefficient of correlation R2 ¼ 0.99, a linear function with coefficients a ¼ 2.458 and b ¼ 1.055 is selected. The price by kilometer, which should be multiplied by this relation is a function of the number of axles and it is given in Table 3.2. The compensation for the transport of full carloads is calculated as the product of the number of axles, the average dynamic weight of loaded cars

Centralized and decentralized model for empty freight car scheduling

1

2

4

77

A

B

3

C

6

F

D

5

7

E

G

8

H

9

10

I

J

Fig. 3.4 Aggregated network interpretation for the model testing purpose.

(8.6 tons) and the amount paid for one order (Table 3.3). Net profit coefficients of the objective function can be calculated as the difference between the costs of transportation of loaded or empty cars and the revenues gained by the transportation service performed. The daily supply and demand of empty cars by types as well as the capacities in train compositions are given in Table 3.4. The net demand for empty

78

Station

A

B

C

D

E

F

G

H

I

J

A B C D E F G H I J

– 110 139 187 200 507 364 395 446 592

104 – 108 77 90 397 254 285 336 482

137 101 – 191 120 427 284 315 366 512

164 60 178 – 71 378 235 266 317 463

187 83 113 65 – 307 164 195 246 392

477 373 403 355 290 – 344 264 119 290

339 235 265 217 152 321 – 80 225 277

377 273 303 255 190 249 72 – 145 197

419 315 345 297 232 112 209 137 – 171

557 453 483 435 370 273 258 186 161 –

Optimization models for rail car fleet management

Table 3.1 Real distances between terminals (upper triangular matrix) and tariff distances (lower triangular matrix)

Centralized and decentralized model for empty freight car scheduling

79

Table 3.2 Transport price per kilometer Price per kma [EUR] Freight car type

Number of axles

Empty

Loaded

Open Closed Flat Other

4 2 3 6

1.01 0.86 0.94 1.87

1.40 1.24 1.32 2.56

a

Estimated values.

cars and available capacities are proportionally aligned according to the net supply of empty cars by terminals. Collected and structured inputs are used for testing the centralized and decentralized decision model. The software is written in the C programming language and developed for the sake of solving the triaxial transportation problem, or in other words, the Kuhn-Tucker’s conditions of the quadratic programming problem. Calculated car flows by each proposed scenario demonstrate the convergence of solutions, which raises the opportunities for application of decentralized organizational structures. These calculations are conducted according to the models proposed in the previous sections. Outputs are presented in Tables 3.5 and 3.6.

80

Station

A

B

C

D

E

F

G

H

I

J

A B C D E F G H I J

–

734 –

852 734 –

967 553 1007 –

1044 671 777 604 –

2199 1790 1949 1713 1472 –

1635 1248 1401 1166 926 1635 –

1790 1401 1554 1327 1044 1327 644 –

1949 1554 1713 1472 1248 777 1166 852 –

2693 2117 2281 2033 1790 1401 1327 1044 967 –

Optimization models for rail car fleet management

Table 3.3 Compensation for transport of full car loads

Centralized and decentralized model for empty freight car scheduling

Table 3.4 Daily supply, demand, and capacities of empty cars

81

82

Optimization models for rail car fleet management

Table 3.5 Centralized freight car allocation model—optimal solution Freight car type: Closed Station

A

B

C

D

E

F

G

H

I

J

A B C D E F G H I J

0 3 3 13 9 0 0 0 1 0

7 2 6 3 20 0 2 0 0 0

2 0 0 15 31 0 0 0 2 0

6 0 10 0 11 0 0 0 4 0

6 7 21 0 35 0 0 0 0 0

8 0 0 20 0 4 0 23 16 13

0 0 8 0 0 7 0 6 12 2

0 0 4 0 0 9 0 0 14 5

0 0 2 0 0 11 0 9 0 12

3 0 1 3 0 8 2 5 2 0

Freight car type: Open Station

A

B

C

D

E

F

G

H

I

J

A B C D E F G H I J

0 0 0 0 0 7 4 6 4 5

0 0 0 0 0 10 0 8 5 9

0 0 0 4 0 12 4 9 4 10

0 0 0 0 0 5 0 9 4 9

21 20 12 32 25 36 24 35 44 0

0 8 14 15 57 0 0 0 0 0

4 4 2 14 11 0 0 0 0 0

0 1 5 9 17 0 0 0 0 0

7 3 8 8 10 0 0 0 0 0

0 3 8 3 2 0 0 0 0 0

Freight car type: Flat Station

A

B

C

D

E

F

G

H

I

J

A B C D E F G H I J

0 0 0 0 0 0 0 5 11 3

0 0 0 0 1 0 2 0 9 0

0 0 0 0 11 0 0 5 9 0

0 0 0 0 0 5 7 1 5 0

2 0 0 22 13 0 0 0 0 0

0 0 0 2 44 0 0 0 0 0

0 0 0 0 0 2 0 3 0 9

3 2 0 1 9 0 0 0 0 0

0 0 2 3 7 0 0 6 0 0

0 0 0 0 1 0 0 0 0 0

83

Centralized and decentralized model for empty freight car scheduling

Table 3.5 Centralized freight car allocation model—optimal solution—cont’d Freight car type: Other Station

A

B

C

D

E

F

G

H

I

J

A B C D E F G H I J

5 0 3 0 0 0 0 0 0 0

0 2 4 13 0 0 0 0 0 0

2 3 9 0 12 0 0 0 0 0

0 4 0 17 21 0 1 0 0 0

0 0 0 0 33 0 0 0 0 0

0 0 0 0 2 18 6 10 15 2

0 0 0 0 19 0 4 0 0 0

0 0 0 0 0 0 4 9 0 5

0 0 0 0 0 0 5 0 20 1

0 0 0 0 0 0 0 0 0 5

Table 3.6 Decentralized freight car allocation model—optimal solution Freight car type: Closed Station

A

B

C

D

E

F

G

H

I

J

A B C D E F G H I J

0 3 3 13 9 0 0 0 1 0

7 2 6 3 19 0 0 0 3 0

2 0 0 19 28 0 0 0 1 0

6 0 10 0 11 0 2 0 2 0

14 7 17 14 17 0 0 0 0 0

1 0 5 5 22 4 0 20 16 11

0 0 4 0 0 7 0 8 12 4

0 0 4 0 0 9 0 0 14 5

0 0 1 0 0 11 0 10 0 12

2 0 5 0 0 8 2 5 2 0

Freight car type: Open Station

A

B

C

D

E

F

G

H

I

J

A B C D E F G H I J

0 0 0 0 0 7 1 7 4 7

0 0 0 0 0 10 2 7 4 9

0 0 0 0 4 12 3 10 6 8

0 0 0 0 0 5 2 8 3 9

13 18 16 28 35 36 24 35 44 0

7 8 9 25 45 0 0 0 0 0

4 4 6 12 9 0 0 0 0 0

2 3 5 7 15 0 0 0 0 0

5 3 9 8 11 0 0 0 0 0

1 3 4 5 3 0 0 0 0 0 Continued

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Optimization models for rail car fleet management

Table 3.6 Decentralized freight car allocation model—optimal solution—cont’d Freight car type: Flat Station

A

B

C

D

E

F

G

H

I

J

A B C D E F G H I J

0 0 0 0 0 0 3 4 11 1

0 0 0 0 2 0 2 1 7 0

0 0 0 0 10 0 1 4 8 2

0 0 0 0 0 5 3 2 8 0

2 2 0 12 21 0 0 0 0 0

0 0 0 7 34 0 0 3 0 2

0 0 0 2 2 2 0 1 0 7

1 0 0 3 11 0 0 0 0 0

2 0 2 3 6 0 0 5 0 0

0 0 0 1 0 0 0 0 0 0

Freight car type: Other Station

A

B

C

D

E

F

G

H

I

J

A B C D E F G H I J

5 0 3 0 0 0 0 0 0 0

0 2 4 13 0 0 0 0 0 0

2 3 9 0 12 0 0 0 0 0

0 4 0 17 21 0 1 0 0 0

0 0 0 0 33 0 0 0 0 0

0 0 0 0 2 18 6 10 5 2

0 0 0 0 19 0 4 0 0 0

0 0 0 0 0 0 4 9 0 5

0 0 0 0 0 0 5 0 20 1

0 0 0 0 0 0 0 0 0 5

CHAPTER 4

Fuzzy multiobjective rail freight car fleet composition Contents 4.1 The best rail freight car fleet mix problem 4.1.1 Classification of freight cars 4.1.2 Utilization of freight cars according to their characteristics 4.1.3 Proposed problem solution 4.1.4 Fuzzy AHP method for the best rail freight car fleet mix problem 4.2 The best rail freight car fleet size problem 4.2.1 Theoretical foundations of the problem solution 4.2.2 Fuzzy multiobjective linear programming 4.2.3 Statement and solution of the problem

86 87 87 91 103 107 110 114 115

The development of the railway freight car fleet structure is under the impact of a variety of technical, exploitation, and economic factors due to the change of transportation requests of the economy as well as the continuous process of replacement and renewal of the freight car fleet (Bojovic and Milenkovic, 2008). Management of the development of the freight car fleet structure represents a complex multicriteria decision-making problem under the impact of the following factors: • the existence of a number of freight car types, series, and subseries; • the total number of freight cars in the fleet; • the value of the freight car fleet; and • the requests of the economy for some series of freight cars which vary due to the change of the volume and structure of production. The users of the service prefer specialized freight cars for each commodity type, which is uneconomical from the aspect of the railway considering that there is a high risk of a shortage or excess of car types appearing due to the fluctuating structure of production. The high level of heterogeneity of freight car fleet negatively affects the costs of freight car maintenance and increases the problems of transport organization and allocation of cars by nodes where they are needed. Therefore, the railway tends to a higher homogeneity of freight car fleet which is in direct opposition with the Optimization Models for Rail Car Fleet Management https://doi.org/10.1016/B978-0-12-815154-9.00004-6

© 2020 Elsevier Inc. All rights reserved.

85

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aspirations of users, so a compromise solution needs to be found, one which will be less costly for the railway but still enough to satisfy all transportation needs in order to avoid opportunity costs. The problem of determining optimal freight car fleet composition is methodologically decomposed into two subproblems: • the best rail freight car fleet mix problem, which considers the selection of type, series, and subseries of freight cars as the primary subproblem; and • the best rail freight car fleet size problem, which considers the determining of the necessary number of railway freight cars as the secondary subproblem. In this chapter, one of the possible ways of optimizing the freight car fleet, its structure, and its size, is proposed. For the first subproblem, analytic hierarchy process (AHP) is used, whereas the second subproblem is solved by using mathematical linear programming. Both methods are applied in a fuzzy environment in the sense that some values of attributes, constraints, or objective functions are not given as strict units but with a level of tolerance allowed.

4.1 The best rail freight car fleet mix problem Selection of the appropriate type, series, and subseries of freight cars represents a primary segment of the fleet composition problem. The development of freight car fleet in the past relied mostly on empirical models based on intuitive predictions of transport volume increment rates as a function of economic development and estimation of trends of main railway business-related qualitative and quantitative indicators. The consequences of this approach to the development of the railway freight car fleet composition were reflected in incompatibility between the structure and the volumes of transport requests on one side, and the structure and size of the rail freight car fleet on the other side. This implied significant unnecessary costs, which led to a decrease of the total accumulative and reproductive capability of railways. Direct railway costs include the costs of amortization, costs of current and investment maintenance, costs of insurance premium and annuities based on the freight car supplying costs. The level of indirect costs is also important if we take into account the total amount of unsatisfied transport orders due to the nonexistence of appropriate freight cars. Therefore, it is necessary for the rail freight car fleet structure to correspond completely to the structure of transport requests of the economy defined through:

Fuzzy multiobjective rail freight car fleet composition

87

•

the type of commodities from the aspect of physical and chemical features and other technology and economy-related requests for transport service; • the physical state of the commodity (solid, liquid, gaseous state of the commodity, bulk and partial load commodity, pallets, and containers); • the law of transport request occurrence (the deterministic or stochastic nature of the generation of transport requests); and • the level of the service (the time for request realization, the waiting interval for taking over the shipment by the railway operator, and reliability).

4.1.1 Classification of freight cars According to the regulations of the International Union of Railways (UIC), the system of unique rail freight car labeling for all cars of this association is introduced. This system enables precise and simple recognition of individual freight cars in the sense of their ownership, the interchange regime, type of freight cars, construction, as well as the correct use in all phases of the transport process. All freight cars can be categorized into four main groups and 13 series: 1. closed freight cars of series: G, H, I, T; 2. open freight cars of series: E, F; 3. flat freight cars of series: K, L, O, R, S; and 4. other freight cars of series: Z, U. Within each group, there are cars of standard and special type. The cars of the standard type have wider use in their own group and out of it, whereas the cars of the special type have been used for transport of specific types of goods. For a wider use, this type of cars is usable mostly without a significant reconstruction or adaptation. Each of the mentioned groups has its own common technical-exploitation characteristics that determine their usability.

4.1.2 Utilization of freight cars according to their characteristics Exploitation of freight cars is determined by a set of regulations (role books, instructions, agreements, and tariffs). According to one classification, the freight cars are used for transport of commodity classified in 659 positions. This classification is defined based on physical-chemical features, state, the level of finalization, purpose, package, and dimensions. The purpose of some types, series, and subseries of freight cars for loading of certain commodity

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types is regulated by the set of instructions. However, in practice, there are certain deviations in situations where the most suitable series and subseries are not available, and in that case, the transportation request must be fulfilled with some other suitable type of freight cars. The problem of determining the optimal type of freight car requires new classification considering the existence of tight technological-economic correspondence between the type of commodity, type, and series of freight cars, transshipment equipment as well as the subsystem of intermodal transport. The current classification of commodities (659 positions according to the tariffs for cargo transport on railways) is not appropriate for the direct application in the model. Historically, technical development of types, series, and subseries of freight cars was under the direct impact of the basic physical-chemical characteristics of commodities: • aggregate state (solid, fluid, gaseous); • physical characteristics of commodities (bulk, partial load, live animals, automobiles, pallets, containers); and • the level of protection (protection from heat, atmospheric influences, vibrations, explosion, fire, and environmental pollution). The initial idea of the methodological approach for the new classification of commodities is based on the following arguments: • The assortment of commodities which appears in railway transport is extremely high if it is measured by the number of positions according to the general classification of commodities. However, all types of commodities considering their manifestation form can be grouped in a few characteristics groups: bulk commodity, partial load commodity, palletized commodity, the commodity in a fluid or gaseous state, live animals, automobiles, machines, long objects, containers, powdery, and grain materials. • Construction solutions of certain series of freight cars came under the influence of different physical-chemical features of certain types of commodities, manifestation form and the type of transshipment facilities, or the way the commodities are loaded or unloaded. In practice, the process of adjusting between transportation requests and series or subseries of freight cars is done directly. In this process, the railway tends to satisfy the requests by offering one universal car type whereas the economy tends to use the special series of freight cars. • There is an important difference between basic technical characteristics of certain series and belonging subseries of freight cars that are in commercial use (the volume, number of axles/doors, and allowed weight).

Fuzzy multiobjective rail freight car fleet composition

89

Having in mind the listed arguments, a new commodity classification in terms of techno-economic requests for types and series of freight cars can be introduced. By the synthesis of the common characteristics of technoeconomic requests, a grouping of all commodities in eight homogeneous groups is proposed. For each group, the common characteristics of commodities, basic types of commodities that belong to a group, types, series, and subseries of freight cars that belong to that type of commodity (Table 4.1). Table 4.1 New commodity classification Group number

Common characteristics of homogeneous groups

I

Mass commodities in bulk state, resistant on atmospheric influences: coal, coke, ore, concentrates of ore, nonmetals, mineral raw materials, construction material, sugar beet, etc. Powder and granular materials in bulk, sensitive on atmospheric influences, without the need for ventilation: cement, grain, lime, plaster cast Packaged commodity on pallets or without pallets sensitive to atmospheric influences: products of chemical industry and related branches, cement, wool, household appliances, cotton Packaged commodity which requests special temperature conditions or ventilation: products from plants and animals,

II

III

IV

Type of freight cars

Car series

Open cars of standard and special type

E, F

Closed cars of standard and special type and the cars of “other types” group

G, T, U

Closed cars of standard and special type

G, H, T

Closed cars of standard type with controlled temperature

G, I

Continued

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Optimization models for rail car fleet management

Table 4.1 New commodity classification—cont’d Group number

V

VI

VII VIII

Common characteristics of homogeneous groups

food products, frozen food products, meat products Commodity from metal complex production: steel construction, reinforced concrete, sheet metal, used iron, other heavy objects, wood, plain sawn Containers, road vehicles, construction and agricultural mechanization, vehicles with a continuous track Long objects, automobiles, and containers Live animals

Type of freight cars

Car series

Flat cars of standard and special type, open cars of standard type

R, S, E

Flat cars of standard and special type

K, R, S

Flat cars of special type

L

Closed cars of standard type

G

In essence, this classification of commodities shows the technoeconomic correspondence between the types of goods and types and series of freight cars. For particular groups of commodities, at least one, two, or three series of freight cars are appropriate. The appearance of more than one freight car series, for certain types of commodities, shows in a way the evolution of constructive solutions for some types of freight cars. The proof for this is the existence of closed cars of the standard type, closed cars of the special type (series of H, I, T) and closed cars of the special type (U series). In this chapter, the procedure is explained in the case of selection of the most appropriate series and subseries of freight cars for the first commodity group. Based on the same approach, results were obtained for other commodity groups (Table 4.10). Cars suitable for the transport of the first commodity group are the cars of series E and F. In the next table, the technical characteristics of these cars and the series/subseries associated with them are given. As it can be noticed from Table 4.2, there are 18 different subseries of cars E and F. All these cars are in commercial use, which means that there are 18 alternatives as possible solutions of the best rail freight car fleet mix problem.

Fuzzy multiobjective rail freight car fleet composition

91

Table 4.2 Freight car series and subseries for the first commodity group Freight car series

E

F

Subseries

Freight car volume

Number of axles

Number of apertures

Allowed weight

Eas1 Eas2 Eas3 Es E El1 El2 Ekkl Ekklo-x Fads Fl Fals Faccs1 Faccs2 Faccs3 Falls Fakk-tz Fakkll

74 70 70 36 36 35 36 21 26 38 38 60 39 34 40 26 23 26

4 4 4 2 2 2 2 2 2 4 2 4 4 4 4 4 4 4

6 6 4 2 2 2 2 2 4 4 4 4 6 4 6 4 2 4

58 58.5 58 29 29 25 29 21 21 56 29.5 57 57 59 56 57 60.5 47

4.1.3 Proposed problem solution The best rail fleet mix problem can be solved by applying the techniques of the multicriteria decision-making based on the following specificities: • The existence of a large number of criteria from the aspect of the railway, the users of the transport service and the society. • The presence of conflicts between criteria. • The existence of different measuring units of certain criteria. • The final number of alternative solutions for selection of a particular car series. The decision model for the problem of determining the optimal series of freight cars can be presented in the form of a m
n matrix: 2 3 C1 X11 X12 ⋯X1n1 Y11 Y12 ⋯Y1n2 Z11 Z12 ⋯Z1n3 7 C2 6 6 X21 X22 ⋯X2n1 Y21 Y22 ⋯Y2n2 Z21 Z22 ⋯Z2n3 7 7 : 6 6 ⋯⋯⋯⋯⋯⋯ ⋯⋯⋯⋯⋯⋯ ⋯⋯⋯⋯⋯⋯ 7 O¼ (4.1) 7 ⋯⋯⋯⋯⋯⋯ ⋯⋯⋯⋯⋯⋯ ⋯⋯⋯⋯⋯⋯ : 6 6 7 : 4 ⋯⋯⋯⋯⋯⋯ ⋯⋯⋯⋯⋯⋯ ⋯⋯⋯⋯⋯⋯ 5 Cm Xm1 Xm2 ⋯Xmn1 Ym1 Ym2 ⋯Ymn2 Zm1 Zm2 ⋯Zmn3 i ¼ 1,…, m; j ¼ 1,…, n1 ; k ¼ 1,…, n2 ; l ¼ 1,…, n3

92

Optimization models for rail car fleet management

where O C1, C2, …, Cm X11, X12, …, Xm1, Xm2, …, Xmn1 Y11, Y12, …, Ym1, Ym2, …, Ymn2 Z11, Z12, …, Zm1,Zm2…, Zmn3

The decision matrix of the freight car series selection Possible combinations of subseries of an individual freight car series The values of railway-related criteria The values of user-related criteria The values of society-related criteria

4.1.3.1 The selection of relevant criteria According to the decision theory, the term decision-making can be defined as the selection of one alternative from a limited set of alternatives. This selection can be done in different ways by: – applying the decision-making techniques which use a set of methods, or procedures in the decision-making process (operations research methods, simulation techniques, and diagnostic techniques); – decision-making roles: previously defined tests for logical judgment; and – decision skills: capability for the efficient use of somebody’s knowledge of problem solving. The decision-making process based on mathematical methods requires defining the set of criteria (attributes). Having in mind that each freight car type has up to 13 subtypes, as well as the existence of techno-economic correspondence between the group of commodities and the types and series of freight cars (up to three series of cars belong to one homogeneous group) it is extremely complex to determine independent sets of criteria, which uniquely reflect the interests of all the involved stakeholders (rail operators, users, and society). However, by a deeper analysis of the nature of these relationships, we may conclude that in most cases the criteria selected by the railway include the interests of users and the society. In this specific problem, for the selection of an optimal type of freight cars for the transport of the first group of commodities (according to the new classification) it is necessary to include the following criteria: 1. Carrying capacity of freight cars. Considering the basic purpose, it is of interest for a railway operator to load in one car as much of the commodity as possible. Therefore, the weight of a train will be higher within the same train length, which gives positive effects on the railway throughput. 2. The volume of freight cars. There is usually a need for the transport of shipments that require the use of entire freight cars regardless of their carrying capacity. The limitation for the loading of these commodities is the

Fuzzy multiobjective rail freight car fleet composition

3.

4.

5.

6. 7.

8.

93

volume of freight cars because the mass of goods is much below the limit of the carrying capacity. Tare weight coefficient. The allowed axle load on railway lines of the Serbian railway network is 22.5 tons per axle. Therefore, the gross mass, of one four-axle car, for example, cannot exceed 90 tons. For the sake of increasing the carrying capacity of cars, it is necessary to decrease the weight (tare) of the freight cars themselves. Considering that the weight of empty cars of different types is different, the most realistic indicator is the relationship between the weight of the empty and loaded cars. This relationship is known as the tare coefficient of freight cars. This coefficient should be as lowest as possible. The buying price of freight cars. This criterion represents one of the most important criteria. The freight car fleet belongs to the main capital assets of a company. Considering the high capital expenses, for the railway, it is important to invest minimal assets for supplying of freight cars. Suitability of freight cars from the aspect of the commodity type. Along with the technical characteristics and buying price, freight cars must satisfy certain technological requirements. Railway operators would prefer to use one universal freight car for the transport of all commodity types. Since that is not possible, it is necessary to structure the freight car fleet in a way that fulfills all transport requirements, or in other words, it should be possible to transport as much of different commodity types as possible with each freight car type. Suitability of freight cars from the aspect of the manifestation form of the commodity. This criterion has an impact on commodities that are sensitive to atmospheric impacts (more or less depending on the package). Suitability of freight cars from the aspect of loading and unloading. Around 30% of the transport time goes on loading and unloading. Loading and unloading of bulk commodities are performed by the users of the transport service. Therefore, the railway operator cannot influence the duration of the loading/unloading operations. On the other side, reducing their duration may result in huge savings, in the transport time as well as in the financial assets. Therefore, the railway operator should prefer to use cars that enable easier and faster loading/unloading. This is particularly important in the case of mechanized loading or unloading, and transshipment from one rail car to another. The number of apertures. Besides the doors, there are other apertures on cars. These apertures usually serve for ventilation. Ventilation is important for the transport of some commodity types.

94

Optimization models for rail car fleet management

9. The protection of commodities during transport. In most cases, railway cars must be classified more than once on their trip between the origin and destination. During each classification, one car knocks against another, whereby the shipment can be damaged. A lower probability of shipment damage leads to a higher level of quality for the transport service. 4.1.3.2 Analytic Hierarchy Process The analytic hierarchy process (AHP) was developed by Tomas Saaty during the 1970s and represents a popular tool in decision analysis. AHP serves as a support to one or a number of decision makers in solving complex problems with many criteria in multiple time periods. The three main concepts of AHP-a are the: analytics, hierarchy, and process. • Analytic: AHP uses numbers. There are many reasons for applying mathematics for the aim of understanding and/or presenting a certain choice to all interested parties. Therefore, it can be concluded that all methods that intend to represent one decision are analytic since they use mathematical/logical reasoning. • Hierarchical: AHP structures the decision-making problem in levels, which are fully aligned to the understanding of the situation from the perspective of decision-making: objectives, criteria, subcriteria, and alternatives. By decomposing the problem into levels, the decision maker can concentrate on smaller sets of decisions. Therefore, it is very important to use the hierarchy during the solving of the complex situation (Fig. 4.1). • Process: As it is well known, decisions of huge importance cannot be made in a single move. Any real problem requires a process of learning, deliberation, and iterative defining of priorities of an individual. AHP provides support and shortening of the decision-making process through insight and understanding that are initialized by this process. 4.1.3.3 Theoretical basics of AHP AHP is based on a set of axioms formulated by Thomas Saaty (1980). This basic set of assumptions provides the theoretical foundations of the method. Axiom 1 For the two given alternatives or criteria i and j on a set of alternatives A, the decision maker can compare pairs aij of these alternatives

Fuzzy multiobjective rail freight car fleet composition

95

Fig. 4.1 Schematic representation of AHP.

under any criteria c from the criteria set C on one scale of relationships, whereby these pairs are reciprocal: aij ¼

1 , 8i, j 2 A aji

(4.2)

Axiom 2 During the comparison of any two alternatives i, j 2 A, the decision maker never assigns relative priority of infinite value to one alternative, under any criteria c 2 C, aij 6¼ ∞, 8 i, j 2 A. In other words, infinite preferences are not allowed. In this situation, there is no selection for the given criteria, since all other alternatives are irrelevant. That means, that individual does not need any tool since for each criterion the answer is known. Axiom 3 Decision problem can be formulated as a hierarchy. Axiom 4 All criteria and alternatives that have an influence on the decision-making problem are represented in a hierarchy. Therefore, all expectations can be presented in the form of criteria and alternatives in structure and the priorities aligned with expectations can be assigned. The last two axioms are subtler than the first two. Axiom 3 emphasizes the potential of expressing the problem as a hierarchy. However, not all problems are appropriate for this representation. If it is possible to define the criteria, subcriteria, alternatives and their mutual relationships in a clear way, then it is suitable to formulate the problem as a hierarchy. Axiom 4 highlights that if the DM is thinking about considering one criterion or alternative, then that criterion or alternative should be accepted. The reason for this assumption is the fact that AHP can show an opposite order. Specifically, the method can produce one order of alternatives if, for example, five

96

Optimization models for rail car fleet management

alternatives are available, and a different order if only one of these five is excluded. These axioms are used for the description of two AHP’s main tasks: formulating and solving the problem as a hierarchy (Axioms 3 and 4) and the formation of the pairing comparison process (Axioms 1 and 2). Determining the priorities for a given set of alternatives A under a given criterion c 2 C includes the completion of one n
n matrix, where n is the number of alternatives that are considered. However, considering that the comparisons are reciprocal, the decision maker should analyze only n(n  1)/2 comparisons in order to fill the evaluations A ¼ (aij). This matrix is reciprocal and positive. The next task is to determine the final order of alternatives based on the pairing comparisons. The first and easiest way to do this represents the normalization of one of the columns. However, the order of the alternatives will depend on the column that is taken into consideration since there are errors during the pairwise evaluations of alternatives. Unlike the other decision methodologies, AHP allows the existence of errors during the provision of preference information. Avoidance of errors implies the need for the decision makers to make one a priori and ad hoc assumption about which alternative should be treated as the basis for the comparison. Through request for defining of n(n  1)/2 pairwise estimations, AHP overcomes this problem. Saaty proposed an approach based on eigenvector for estimation of weights obtained based on comparison matrices A. This approach represents a theoretical and practically proven method for weight estimation. Intuitive interpretation is also included considering that it represents an intersection of all possible ways of thinking on a given set of alternatives. Therefore, the weight estimation of a given set of alternatives is an understandable and easy applicable approach. After the weights has been estimated, it is also necessary to define the consistency measure of the given pairwise comparisons. This consistency ratio (CR) represents the measure that the probability is filled in an arbitrary way. Therefore, CR is a comparison of the actual matrix and a fully random evaluation of the pairwise comparisons. Number 0.1, which is accepted as an upper limit for CR, emphasizes that there is a 10% chance that the decision maker has answered questions in an arbitrary way. The higher consistency implies a lower probability that the matrix will be filled in an arbitrary way and therefore the CR increases. AHP does not require from the decision maker to be consistent; it just enables the inconsistency measure as well as the method for decreasing this measure if it is too high.

Fuzzy multiobjective rail freight car fleet composition

97

The last question which relates to the pairwise comparison of objects treats the measuring scale. Axiom 2 implies limits over the pairwise comparisons aij. In Table 4.3, the upper limit is chosen to be 9. In theory, any number less than infinite can be used as an upper bound. However, extensive practical experience suggests that 9 is an appropriate upper bound. After the generation weighting set ωca for each alternative a 2 A for criteria c 2 C, the principle of the hierarchical composition leads to the calculation of the total priority of alternatives by summing up the priorities for criteria c, ωca, multiplied by the priority of criteria c, υc, or

Table 4.3 Fundamental scale Intensity of importance

Definition

Explanation

1

Equal importance

3

Moderate importance

5

High importance

7

Very evident importance

9

Extreme importance

2,4,6,8

Compromised values

Two activities equally contribute to the objective fulfillment Experience and estimation slightly favor one activity in comparison to another Experience and estimation significantly favor one activity in comparison to another There is very evident favoring of any activity in comparison to the other. Its dominance is demonstrated in practice The proof by which one activity is favored in relation to another is of the highest possible rank of affirmation Sometimes it is necessary to numerically interpolate estimation since there is no adequate word for its estimation Continued

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Optimization models for rail car fleet management

Table 4.3 Fundamental scale—cont’d Intensity of importance

Reciprocal in comparison to the previous

Definition

Explanation

If activity i has one of the above-mentioned nonzero values assigned to it in comparison with j, then activity j has reciprocal value in comparison with i.

The comparison obtained by choosing the smaller element as a unit for the estimation of the bigger element as a multiple value of that unit

The scale of absolute numbers used for assigning numerical values to evaluations made by comparison of two elements, with the lower, which represents the unit, and the higher to which the value from this scale is assigned as the product of that unit.

ωa ¼

X

υc ωca

(4.3)

c2C

4.1.3.4 Mathematical basics of AHP The first and basic task in AHP includes the evaluation of weight for a set of objects (criteria or alternatives) from a pairwise comparison matrix A ¼ (aij), which is positive and reciprocal. Therefore, for a given matrix: 0 1 a1, 1 a1, 2 … a1, n B a2, 1 a2, 2 … a2, n C C (4.4) A¼B @… … … …A an, 1 an, 2 … an, n where aij ¼ 1=aji , 8 i, j ¼ 1, 2,…,n the vector of weights or priorities w ¼ (w1, w2, …, wn) needs to be determined. It should be noted that by using the ratio scale, weights which we evaluate are just units multiplied by a positive constant, for example, w is equivalent to cw where c > 0. Therefore, we will normalize w so that it gives 1, for the sake of simplifying the whole procedure. In the case when the decision maker is fully consistent during the process of matrix fulfilling, the following expression holds: aik akj ¼ aij , 8i, j, k ¼ 1,2,…, n

(4.5)

Fuzzy multiobjective rail freight car fleet composition

99

Then the elements of the matrix A will not contain the error and it can be expressed as aij ¼ wi =wj

(4.6)

or   aik akj ¼ ðwi =wk Þ wk =wj ¼ wi =wj ¼ aij , 8 i, j,k ¼ 1,2,…, n

(4.7)

In this case, it can be possible to normalize any column j of matrix A and obtain the final weights: wi ¼ aij =

n X

akj , 8i ¼ 1,2,…, n

(4.8)

k¼1

However, usually there are errors in evaluation, and therefore, the final results obtained by the normalization of columns will depend on the choice of column. Saaty’s method for weight evaluation computes the weight as eigenvector of matrix A: Aw ¼ λmax w

(4.9)

where λmax is the maximal eigenvector of the matrix, or ωi ¼

n X

! aij ωj Þ=λmax , for each i ¼ 1, 2,…,n

(4.10)

j¼1

This method shows that the final weights w represent an average of all possible ways for comparing alternatives. Therefore, the eigenvector is a natural way of determining the weight. Theoretical experience suggests that this method represents the mean for determining the correct order of a set of alternatives. This method leads to a natural measure for inconsistency. λmax is always lower or equal to n for positive, reciprocal matrices if and only if A is a consistent matrix. Thus, λmax  n provides a useful measure of the inconsistency ratio. Normalizing this measure by the size of the matrix produces a consistency index (CI): CI ¼ ðλmax  nÞ=ðn  1Þ

(4.11)

For each size of matrix n, arbitrary matrices are generated and their average CI, known as the random index is computed. These values are given in Table 4.4.

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Optimization models for rail car fleet management

Table 4.4 The values of random index n

RI

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0 0 0.58 0.9 1.12 1.24 1.32 1.41 1.45 1.49 1.51 1.48 1.56 1.57 1.59

By using these values, the CI can be defined as the relationship between CR and RI. Therefore, CI is a measure of comparison of the given and completely arbitrary matrix in the form of their CRs: CR ¼ CI=RI (4.12) The computation of the matrix eigenvector is derived by raising the matrix A to the power of the increasing exponents k, and then normalizing the resulting system:     (4.13) w ¼ lim Ak e = eT Ak e , e ¼ ð1, 1, …, 1Þ k!∞ There is one general role by which the inconsistency of a matrix is directly related to the error in the matrix, and thus to the convergence time. 4.1.3.5 Fuzzy extension of AHP method Fuzzy logic represents one of the leading methodologies for intelligent technologies, and it has diverse application in industrial and commercial decision-making systems, optimization and control. Fuzzy logic is usually viewed as a probability theory or a so-called disguised probability. Both theories represent the efforts for quantifying and numerical representation of uncertainties. The main difference lies in the fact that probability shows whether an event will happen and with what probability, whereas fuzzy logic is more tolerant and it measures the extent to which some event will

Fuzzy multiobjective rail freight car fleet composition

101

be realized and therefore, it does not make the question about the occurrence of that event. The classic theory of sets clearly defines the boundaries of some set, whereas in the fuzzy theory, these boundaries are not clear, and some attribute values can be added in a fuzzy or imprecise way (in some intervals for example). For each value that belongs to a fuzzy set, there is a membership function, which measures the degree of membership of some number to a given fuzzy set. μA ðxÞ ¼ 1;μA ðyÞ ¼ 1;μA ðzÞ ¼ 0; As it can be seen in Fig. 4.2, number Z belongs to a set A with a membership degree 0, μA(z) ¼ 0. The human brain mainly manipulates the perception of different units, distance, colors, etc. This is the main reason for the suitability of incorporating fuzzy logic in the AHP model. By applying fuzzy logic, the error that the decision maker will certainly make can be decreased since he/she is not forced to be precise and to choose only one value. However, some units will be represented in an exact form since there is no need for assigning fuzzy values. It should be noted that the normalized fuzzy set (when the value of the membership function for at least one member of the set is equal to 1) represents a fuzzy number. Fig. 4.3 illustrates a triangular fuzzy number (TFN), whose name originates from the shape of its membership function, which can also take other shapes, like a trapezoidal shape for example. For the sake of the fuzzy extension in this chapter, the TFN will be used. TFN can be written in the form A ¼ (a1, a2, a3) where: • a1 is the lower (left) boundary of TFN; • a2 is the value with the highest membership values; and • a3 is the upper (right) boundary of TFN. A

z

x y

Fig, 4.2 Comparison of the classic and fuzzy set.

102

Optimization models for rail car fleet management

1.0

0.5

0.0 0

5

10

15

20

25

30

35

Fig. 4.3 Triangular fuzzy number.

The membership function of fuzzy number A can be formulated as 8 0,   x  a1 > > > x  a 1 > > , a1  x  a2 < a2  a1   μA ðxÞ ¼ (4.14) a3  x > > > , a  x  a 2 3 > > : a3  a2 0, x  a3 Since the AHP includes the criterion pairwise matrix and the matrix of attribute comparison with respect to each criterion, the relationships of certain values that belong or do not belong to fuzzy numbers need to be determined. Thus, it is necessary to transform fuzzy numbers in numerical values, then to calculate the left and right score of fuzzy numbers with respect to the maximum and minimum values of all attributes within each criterion (Fig. 4.4). Therefore, during the transformation of fuzzy numbers in standard numerical values it is necessary to first calculate their left and right values as follows: μR ðAÞ ¼ max ½ min ðμA ðxÞ, μmax ðxÞÞ

(4.15)

μL ðAÞ ¼ max ½ min ðμA ðxÞ, μmin ðxÞÞ

(4.16)

μT ðAÞ ¼

μR ðAÞ + 1  μL ðAÞ 2

(4.17)

μR(A)—right score of fuzzy number A; μL(A)—left score of fuzzy number A; μT(A)—final value during the transformation of the fuzzy number in a numerical value.

Fuzzy multiobjective rail freight car fleet composition

Low

Medium

103

High

1

0

0.1 Very Low

0.2

0.3

0.4

0.5

0.6

Medium

Low

0.7

0.8

High

0.9

1.0

Very high

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Fig. 4.4 Fuzzy numbers with respect to defined min and max values.

4.1.4 Fuzzy AHP method for the best rail freight car fleet mix problem To determine the optimal type of freight cars by the AHP method, it is first necessary to define the decision matrix, which for the first commodity group looks like in Table 4.5. For this table the following criteria are used: 1. the carrying capacity of cars (t); 2. the volume of cars (m3); 3. the buying price (103 €); 4. the number of doors/apertures; 5. suitability of cars from the aspect of the commodity type; 6. suitability of cars from the aspect of the commodity manifestation form; 7. suitability of cars from the aspect of loading/unloading; 8. the tare coefficient of cars; 9. protection of commodities during transport. In accordance with the applied AHP method, the problem is decomposed in three levels, where the first level represents the objective of the multicriteria decision-making, the second level consists of the relevant criteria, and on the third level are all possible alternatives. The alternatives, in this case, represent series and subseries of car types. The estimation of the relative weight on the first hierarchical level is performed based on the personal experience and information obtained from field experts. Table 4.6 shows a pairwise comparison of criteria in a fuzzy environment.

104

Criteria Alternatives

1

2

3

4

5

6

7

8

9

Eas1 Eas2 Eas3 Es E El1 El2 Ekkl Ekklo-x Fals Faccs1 Faccs2 Faccs3 Fakkll Falls Fakk-tz Fads Fl

58.10 58.50 58.00 29.00 29.00 29.00 25.00 21.00 21.00 57.00 57.00 59.00 56.00 47.00 57.00 60.50 56.00 29.50

74.00 70.00 70.00 36.00 36.00 36.00 35.00 21.00 26.00 60.00 39.00 34.00 40.00 26.00 26.00 23.00 38.00 38.00

122.00 122.00 122.00 72.00 72.00 72.00 72.00 62.00 62.00 140.00 140.00 140.00 140.00 140.00 140.00 140.00 140.00 82.00

4 4 4 2 2 2 2 2 2 4 6 4 6 4 4 2 4 4

10 10 10 8 8 5 5 8 8 10 10 10 10 9 10 8 9 8

10 10 10 7 7 7 7 7 5 10 10 10 10 10 10 10 10 10

8 8 8 8 8 5 5 5 5 10 10 10 10 10 10 10 10 10

0.38 0.37 0.38 0.39 0.39 0.39 0.44 0.53 0.53 0.40 0.39 0.35 0.43 0.52 0.40 0.32 0.43 0.35

10 10 10 8 8 8 8 8 8 10 10 10 10 10 10 10 10 10

Optimization models for rail car fleet management

Table 4.5 Decision matrix for first commodity group

Table 4.6 Pairing comparisons of criteria (fuzzy environment) Criteria 1

2

3

4

5

6

7

8

9

Eas1 Eas2 Eas3 Es E El1 El2 Ekkl Ekklo-x Fals Faccs1 Faccs2 Faccs3 Fakkll Falls Fakk-tz Fads Fl

58.10 58.50 58.00 29.00 29.00 29.00 25.00 21.00 21.00 57.00 57.00 59.00 56.00 47.00 57.00 60.50 56.00 29.50

74.00 70.00 70.00 36.00 36.00 36.00 35.00 21.00 26.00 60.00 39.00 34.00 40.00 26.00 26.00 23.00 38.00 38.00

110.00–130.00 115.00–130.00 110.00–130.00 69.00–75.00 68.00–77.00 70.00–75.00 70.00–75.00 60.00–65.00 59.00–65.00 130.00–140.00 130.00–140.00 130.00–140.00 130.00–140.00 130.00–140.00 135.00–145.00 130.00–140.00 130.00–140.00 80.00–86.00

4 4 4 2 2 2 2 2 2 4 6 4 6 4 4 2 4 4

8–10 8–10 8–10 7–9 7–9 4–6 4–6 6–9 6–9 8–10 8–10 8–10 8–10 7–9 8–10 6–8 7–10 7–9

8–10 8–10 8–10 6–8 6–8 5–8 5–8 6–8 6–8 8–10 8–10 8–10 8–10 8–10 8–10 7–10 8–10 7–10

6–9 6–9 6–9 6–9 6–9 4–6 4–6 4–6 4–6 8–10 8–10 8–10 8–10 8–10 8–10 8–10 8–10 8–10

0.38 0.37 0.38 0.39 0.39 0.39 0.44 0.53 0.53 0.40 0.39 0.35 0.43 0.52 0.40 0.32 0.43 0.35

10 10 10 8 8 8 8 8 8 10 10 10 10 10 10 10 10 10

Fuzzy multiobjective rail freight car fleet composition

Alternatives

105

106

Optimization models for rail car fleet management

As it can be noticed from Table 4.6, criteria 3, 5, 6, and 7 do not contain precise evaluations of attributes except in the interval form, therefore it is necessary to adapt the existing matrix in order to apply AHP. Fuzzy numbers and their left and right scores for some values are illustrated Figs. 4.5–4.7. Final values for all alternatives within the criteria related to the buying price (3), suitability of freight cars from the aspect of the commodity type

Fig. 4.5 Freight car type Eas1: fuzzy number and left and right scores.

Fig. 4.6 Freight car type Ekklo-x: fuzzy number and left and right scores.

Fig. 4.7 Freight car type Faccs1: fuzzy number and left and right scores.

Fuzzy multiobjective rail freight car fleet composition

107

(5), suitability of cars from the aspect of the commodity manifestation (6), and suitability from the aspect of commodity loading and unloading (7) are given in Table 4.7. Since all values in pairing comparisons of criteria are numerical values, it is possible to make Table 4.8. The final order of criteria obtained by filling and processing the pairwise comparison matrix is given in Fig. 4.8. It can be noticed that the most dominant criterion is the buying price of freight cars. The consistency ratio is relatively low and it has the value 0.084. Based on complete evaluations of all alternatives against each criterion, or more specifically filled and processed pairwise comparison matrices, the synthesis of the best rail freight car fleet mix problem for the case of the first commodity group is given in Table 4.9. Freight cars, which are by all criteria and their weight the most appropriate for transport of the first commodity group, are Eas1 (A1). Therefore, by applying the fuzzy AHP method, the freight cars Eas1 appeared as the most dominant solution, however, the freight cars Eas3, as well as Fals, should also be taken into consideration. The same methodology is applied to all other commodity types. The results are given in Table 4.10. As it has been already mentioned, the second subproblem of the rail freight car fleet composition deals with the best rail freight car fleet size whereby only the car types, series, and subseries selected in the first stage are taken into account.

4.2 The best rail freight car fleet size problem In freight transport railway companies, service users, and society do not have the same technical-technological and economic interests. Each of the actors has specific requests which need to be satisfied. Therefore, this problem can be treated as a multicriteria decision-making problem where it is necessary to define the criteria set and the constraints and make a selection of the solution model (Bojovic et al., 2010). Criteria, in this case, represent objective functions which include the interests of all actors. It is possible to define a number of objective functions, but it is necessary to select the most important ones, such as: • the function of freight car supplying; • the function of freight car fleet immobilization; • the function of freight car fleet cassation; • the function of freight car fleet productivity; • the function of freight car fleet size; • the function of economy’ losses due to freight car shortage.

108

3

5

6

7

Alternatives

μL

μD

norm.

μL

μD

norm.

μL

μD

norm.

μL

μD

norm.

Eas1 Eas2 Eas3 Es E El1 El2 Ekkl Ekklo-x Fals Faccs1 Faccs2 Faccs3 Fakkll Falls Fakk-tz Fads Fl

0.112 0.100 0.112 0.256 0.258 0.255 0.255 0.289 0.290 0.050 0.050 0.050 0.050 0.050 0.033 0.050 0.050 0.221

0.419 0.419 0.419 0.253 0.256 0.253 0.253 0.219 0.219 0.466 0.466 0.466 0.466 0.466 0.483 0.466 0.466 0.288

0.653 0.659 0.653 0.498 0.499 0.499 0.499 0.465 0.464 0.708 0.708 0.708 0.708 0.708 0.725 0.708 0.708 0.533

0.090 0.090 0.090 0.136 0.136 0.222 0.222 0.166 0.166 0.090 0.090 0.090 0.090 0.136 0.090 0.181 0.125 0.136

0.454 0.454 0.454 0.409 0.409 0.272 0.272 0.409 0.409 0.454 0.454 0.454 0.454 0.409 0.454 0.363 0.454 0.409

0.681 0.681 0.681 0.636 0.636 0.525 0.525 0.621 0.621 0.681 0.681 0.681 0.681 0.636 0.681 0.590 0.664 0.636

0.090 0.090 0.090 0.181 0.181 0.208 0.208 0.181 0.181 0.090 0.090 0.090 0.090 0.090 0.099 0.125 0.090 0.125

0.454 0.454 0.454 0.363 0.363 0.363 0.363 0.363 0.363 0.454 0.454 0.454 0.454 0.454 0.454 0.454 0.454 0.454

0.681 0.681 0.681 0.590 0.590 0.577 0.577 0.590 0.590 0.681 0.681 0.681 0.681 0.681 0.681 0.664 0.681 0.664

0.166 0.166 0.166 0.166 0.166 0.272 0.272 0.272 0.272 0.090 0.090 0.090 0.090 0.090 0.090 0.090 0.090 0.090

0.409 0.409 0.409 0.409 0.409 0.272 0.272 0.272 0.272 0.454 0.454 0.454 0.454 0.454 0.454 0.454 0.454 0.454

0.621 0.621 0.621 0.621 0.621 0.5 0.5 0.5 0.5 0.681 0.681 0.681 0.681 0.681 0.681 0.681 0.681 0.681

Optimization models for rail car fleet management

Table 4.7 Values of alternatives for criteria 3, 5, 6, and 7

Table 4.8 Exact values of all alternatives against the criteria set Criteria 1

2

3

4

5

6

7

8

9

Eas1 Eas2 Eas3 Es E El1 El2 Ekkl Ekklo-x Fals Faccs1 Faccs2 Faccs3 Fakkll Falls Fakk-tz Fads Fl

58.10 58.50 58.00 29.00 29.00 29.00 25.00 21.00 21.00 57.00 57.00 59.00 56.00 47.00 57.00 60.50 56.00 29.00

74.00 70.00 70.00 36.00 36.00 36.00 35.00 21.00 26.00 60.00 39.00 34.00 40.00 26.00 26.00 23.00 38.00 38.00

0.653 0.659 0.653 0.498 0.499 0.499 0.499 0.465 0.464 0.708 0.708 0.708 0.708 0.708 0.725 0.708 0.708 0.533

4 4 4 2 2 2 2 2 2 4 6 4 6 4 4 2 4 4

0.681 0.681 0.681 0.636 0.636 0.525 0.525 0.621 0.621 0.681 0.681 0.681 0.681 0.636 0.681 0.590 0.664 0.636

0.681 0.681 0.681 0.590 0.590 0.577 0.577 0.590 0.590 0.681 0.681 0.681 0.681 0.681 0.681 0.664 0.681 0.664

0.621 0.621 0.621 0.621 0.621 0.500 0.500 0.500 0.500 0.681 0.681 0.681 0.681 0.681 0.681 0.681 0.681 0.681

0.38 0.37 0.38 0.39 0.39 0.39 0.44 0.53 0.53 0.40 0.39 0.35 0.43 0.52 0.40 0.32 0.43 0.35

10 10 10 8 8 8 8 8 8 10 10 10 10 10 10 10 10 10

Fuzzy multiobjective rail freight car fleet composition

Alternatives

109

110

Optimization models for rail car fleet management

0.3500 0.3000 0.2500 0.2000 0.1500 0.1000 0.0500 0.0000 K3

K1

K5

K2

K6

K4

K7

K9

K8

Fig. 4.8 Relative priorities of criteria.

All these functions represent linear dependence between desired solutions and variables, which in this case represent the number of cars by series. Constraints which appear in this problem are: • the number of freight cars of a particular car type can only be a positive number; • the available capacity of the freight car fleet must be higher than equal to the volume of the shipment for loading; • the productivity of the freight car fleet by types and series must be higher than the total transport output on a railway network; • the total number of immobilized freight cars by series must be lower than/or equal to the total number of cars which can be immobilized; and • the number of freight cars of a given series must be higher than/or equal to the product of the freight car turnover and the number of cars on loading. The optimal rail freight car fleet size problem in which it is necessary to minimize (maximize) a certain number of linear objective functions under a set of constraints represents a multiobjective linear programming problem.

4.2.1 Theoretical foundations of the problem solution The formulated linear programming task can be expressed in derived form as 8 9 F1 ðxÞ ¼ c11 x1 + c12 x2 + ⋯ + c1n xn > > > > < = F2 ðxÞ ¼ c21 x1 + c22 x2 + ⋯ + c2n xn max ð min Þ (4.18) ⋮ ⋮ ⋮ ⋮ > > > > : ; Fm ðxÞ ¼ cm1 x1 + cm2 x2 + ⋯ + cmn xn

Table 4.9 The final rank of freight car series and subseries for the first commodity group K2

K3

K4

K5

K6

K7

K8

K9

0.015 0.016 0.013 0.002 0.002 0.002 0.002 0.001 0.001 0.012 0.011 0.018 0.010 0.007 0.011 0.020 0.009 0.011

0.018 0.015 0.015 0.004 0.004 0.004 0.003 0.001 0.001 0.012 0.005 0.003 0.006 0.001 0.001 0.001 0.004 0.005

0.010 0.010 0.010 0.027 0.027 0.027 0.027 0.041 0.041 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.021

0.003 0.003 0.003 0.001 0.001 0.001 0.001 0.001 0.001 0.003 0.010 0.003 0.010 0.003 0.003 0.000 0.003 0.003

0.014 0.014 0.014 0.005 0.005 0.001 0.001 0.004 0.004 0.014 0.014 0.014 0.014 0.006 0.014 0.004 0.008 0.005

0.005 0.005 0.005 0.002 0.002 0.001 0.001 0.002 0.002 0.005 0.005 0.005 0.005 0.005 0.005 0.008 0.006 0.008

0.002 0.002 0.002 0.002 0.002 0.001 0.001 0.001 0.001 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004

0.001 0.002 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.002 0.001 0.001 0.001 0.003 0.001 0.002

0.003 0.003 0.003 0.001 0.001 0.001 0.001 0.001 0.001 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003

P

0.073 0.073 0.07 0.047 0.047 0.041 0.040 0.053 0.054 0.062 0.061 0.061 0.060 0.038 0.051 0.052 0.047 0.065

Fuzzy multiobjective rail freight car fleet composition

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A18

K1

111

112

Optimization models for rail car fleet management

Table 4.10 The best rail freight car fleet mix Homogeneous commodity groups

Most suitable freight car type

Volume (m3)

Number of axles

Number of apertures

Allowed capacity (tons/car)

I II III IV V VI VII VIII

Eas1 Gas-z Galms Ibbiis Eas2 Smmp-tz Laekks Gkks

74 92 75.1 57 70 – – 57.8

4 4 4 2 4 4 3 2

6 6 4 – 6 – – 4

58 57.5 57.3 25 58.5 59.5 20 24

subject to: a11 x1 + a12 x2 + ⋯ + a1n xn  b1 ⋮ ⋮ ⋮ ⋮ ak1 x1 + ak2 x2 + ⋯ + akn xn  bk

(4.19)

x1 , x2 , …,xn  0 or in matrix form: max ð min ÞCx

(4.20)

Ax  b

(4.21)

x0

(4.22)

where: C—matrix of objective function coefficients; A—matrix of constraints coefficients; x—vector of variables; and b—vector of right-hand-side constants. The task of multicriteria linear programming does not result in an optimal solution as in the case of single criteria linear programming but in multiple solutions that appear during problem solving, such as: • marginal solutions and ideal values of objective functions; • ideal solution; • compromise solution with compromised values of objective functions; and • efficient solution.

Fuzzy multiobjective rail freight car fleet composition

113

If we consider any of the objective functions and search for its minimum or maximum subject to a set of constraints, we need to solve the following problem: " # n X (4.23) max ð min Þ Fi ðxÞ ¼ cij xj j¼1

Ax  b

(4.24)

x0

(4.25)

The obtained optimal solution x ¼ (x1, x2, …, xn) of the function is known as a marginal solution for a given objective function. By replacing these values in the objective function the optimal value of that function (F0i ¼ Fi(x)), known as the ideal value, is obtained. It is possible to obtain marginal solutions for each function Fi(x), which are mutually different, in the same way. The ideal solution of the multiobjective linear programming problem, x, would be the solution from the set of marginal or other feasible solutions which gives ideal values of all objective functions: Fi ðxÞ ¼ Fi0 ði ¼ 1, 2, …, nÞ

(4.26)

If this solution could exist, then it would mean that we do not consider the multicriteria linear programming problem, and in that case, it would be necessary to choose one solution that can represent a compromise solution. This solution gives to objective functions compromised values with certain deviations from ideal ones, so that the decision maker can select, based on his experience, the preferable solution, which is also known as the solution for realization. It is also possible to define the efficient solution in the following way: some solution is efficient if there is no any other solution xn, which is better under any of the criteria than xe, and by all other criteria it is not worse than xe. There are three basic approaches for solving the problem of multiobjective linear programming: • replacement of all objective functions by one objective function; • selection of solutions considering the priorities between objective functions; and • maximization of objective function vectors, component by component. Depending on the solution approach, a number of methods are proposed, as the solution, such as Sask’s or Tamm’s method, which belongs to the first

114

Optimization models for rail car fleet management

group—the replacement of all functions by a single one, which has a general form F ðxÞ ¼ G½F1 ðxÞ, F2 ðxÞ, …, F1 ðxÞ

(4.27)

4.2.2 Fuzzy multiobjective linear programming In the case of fuzzy multiobjective linear programming, the decision maker does not need to strictly define the values of objective functions, but he can define it within the desired interval. During solving this problem given values can be varied in order to reach an optimal solution. That means that the decision maker actively participates in problem solving. The multiobjective linear programming problem with k objective functions cix, i ¼ 1, …, k can be formulated as min ðc1 x, c2 x, …, ck xÞT

(4.28)

Ax  b

(4.29)

xj  0, j ¼ 1,…, n

(4.30)

where ci ¼ (ci1, …, cin), i ¼ 1, …, k are n-dimensional row vectors, xi ¼ (x1, …, xn)T is the n-dimensional column vector, A ¼ [aij], i ¼ 1, …, m, j ¼ 1, …, n is a m
n matrix of coefficients, b ¼ (b1, …, bm)T is m-dimensional column vector. Since the optimal solution to the problem sometimes does not exist, the Pareto optimal solution is proposed. Therefore, the decision maker may give vague values for each objective function, so that in the interaction with it the following membership functions can be defined: 8 9 0, ci x  z0i > > > > < c x  z0 = i 1 0 i μðci xÞ ¼ (4.31) , z  c x  z i i i > > z1  z0i > > : i ; 1 , ci x  z1i where z0i and z1i denote the values of objective functions whose values of membership functions belong to an interval between 0 and 1. Fig. 4.9 illustrates a possible form of the objective function. One of the possible ways for determining the values z1i and z0i is to calculate the min and max of each objective function under a set of given constraints. The obtained values should be taken into account while determining the desired interval to which the solutions of objective function belong [zmin , zmax ], i ¼ 1, …, k. i i

115

Fuzzy multiobjective rail freight car fleet composition

Fig. 4.9 A possible form of the objective function

When the membership functions of all objective functions are defined, where all decision-maker’s criteria are defined, it is possible to represent the aggregation function: μD ¼ μD ðμ1 ðc1 xÞ, …, μk ðck xÞÞ

(4.32)

The fuzzy multiobjective decision-making problem can be defined as max μD ðxÞ x2X

(4.33)

It can be concluded that the function μD(x) represents the level of satisfaction of fuzzy objectives of the decision maker. If Bellman’s and Zadeh’s method is accepted for the aggregation function, then it follows: min ðμ1 ðc1 xÞ, …, μk ðck xÞÞ (4.34) i¼1, …, k Multiobjective linear programming can be represented as   (4.35) max min fμi ðci xÞg i¼1, …, k Ax  b

(4.36)

x0

(4.37)

The solution to this problem depends on a great extent on the decision maker, whether he/she thinks the minimum operator is appropriate or if it is satisfied with the obtained solutions.

4.2.3 Statement and solution of the problem Determining the optimal number of freight cars based on the described model requires defining the objective function and a set of constraints. After the model is defined, it is necessary to test it, which is in this case conducted based on the statistical data of the PE “Serbian Railways” and experience of railway experts from the field.

116

Optimization models for rail car fleet management

Based on the business report for 2006, 6,608,462 tons of the commodity was loaded, of the planned 7,250,000. It should be mentioned that this loading was made including the cars from other railways as well, but for the sake of simplicity it is assumed that the number of cars used for loading on the Serbian railway network is equal to the number of cars used in other railway networks; that is, the freight car exchanging balance is equal to zero. The daily average number of freight cars in the current or capital maintenance was between 500 and 800 cars. The total number of net ton-kilometers was 4,232,300,000. The average turnover of freight cars was 6.4 days, but it varied for different time series. The data related to the freight car turnover by series was obtained in consultation with freight car managers. Daily productivity was calculated based on the realized (or planned) net-ton kilometers and the daily number of required cars. Productivity by car series was obtained based on the structure of transported cargo and the use of the appropriate series of freight cars. Table 4.11 contains the structure of the transported cargo expressed in percentages. The following functions were selected as objective criteria: • the function of freight car fleet immobilization F1(x); • the function of freight car fleet cassation F2(x); and • the function of freight car fleet size F3(x). The overall synthesis of the problem looks as follows: min ½F1 ðxÞ ¼ 0:06x1 + 0:05x2 + 0:05x3 + 0:04x4 + 0:06x5 + 0:011x6 + 0:027x7 + 0:05x8 

(4.38)

min ½F1 ðxÞ ¼ 0:02x1 + 0:015x2 + 0:015x3 + 0:01x4 + 0:02x5 + 0:02x6 + 0:015x7 + 0:015x8  (4.39)

Table 4.11 Structure of transported cargo Commodity type

Share in total transport (%)

Coal and coke Construction materials Oil and derivates Metallurgy Ores and concentrates Nemetals Other

25.2 8.6 6.9 11.8 12.75 5.4 29.4

Fuzzy multiobjective rail freight car fleet composition

117

min ½F1 ðxÞ ¼ 7:5x1 + 6:8x2 + 6:8x3 + 6:3x4 + 7:4x5 + 6:5x6 + 5:1x7 + 6:8x8 

(4.40)

subject to: 44:66x1 + 44:28x2 + 44:12x3 + 19:25x4 + 44:97x5 + 45:82x6 + 15:4x7 + 18:5x8  32994

(4.41)

1591x1 + 313x2 + 147x3 + 113x4 + 362x5 + 405x6 + 86x7 + 49x8  15526164 (4.42) x1  3083

(4.43)

x2  550

(4.44)

x3  258

(4.45)

x4  158

(4.46)

x5  633

(4.47)

x6  679

(4.48)

x7  130

(4.49)

x8  86

(4.50)

The first constraint (4.41) represents the satisfaction of the need for transport of the total daily quantity of freight. Coefficients with unknowns are carrying capacities of cars by series. Average utilization of freight car carrying capacity of 70% has been considered in the model. In the second constraint (4.42), coefficients with unknowns represent daily productivities of freight cars by series, whereas the right-hand-side constant is the average daily transport productivity on the network. The productivity of freight cars is represented by net-ton kilometers per car-day. The remaining eight constraints (4.43)–(4.50) represent the needed number of freight cars by series. The first step of problem solving is to find the minimum of the objective functions, each individually with respect to the given constraints. The results are given in Table 4.12. The results in the table represent the initial solution, which will be used to determine the membership functions of objective functions, and such a model can be represented as

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Table 4.12 The minimum of objective functions Objective functions

Values

Function of immobilization Function of cassation Function of freight car fleet size

655 229 86,700

max ðλÞ

(4.51)

λ  1  0:0012x1  0:001x2  0:001x3  0:0008x4  0:0012x5  0:00022x6  0:00054x7  0:001x8 + 11 (4.52) λ  1  0:001x1  0:00075x2  0:00075x3  0:0005x4  0:001x5  0:001x6  0:00075x7  0:00075x8 + 11:5 (4.53) λ  1  0:0025x1  0:0023x2  0:0023x3  0:0021x4  0:0025x5  0:0022x6  0:0017x7  0:0023x8 + 29

(4.54)

44:66x1 + 44:28x2 + 44:12x3 + 19:25x4 + 44:97x5 + 45:82x6 + 15:4x7 + 18:5x8  32994 (4.55) 1591x1 + 313x2 + 147x3 + 113x4 + 362x5 + 405x6 + 86x7 + 49x8  15526164

(4.56)

x1  3083

(4.57)

x2  550

(4.58)

x3  258

(4.59)

x4  158

(4.60)

x5  633

(4.61)

x6  679

(4.62)

x7  130

(4.63)

x8  86

(4.64)

The deviations that are allowed in objective functions, by which the system moves to a fuzzy environment, are given in Table 4.13. λ practically represents the satisfaction degree of the model solution. The values of objective function are given in Table 4.14.

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Table 4.13 Allowed deviations Objective functions

Allowed deviations

Function of immobilization Function of cassation Function of freight car fleet size

50 20 3000

Table 4.14 The values of objective functions and satisfaction degree: I iteration Objective functions

Values

λ

Function of immobilization Function of cassation Function of freight car fleet size

652.75 249.135 92.596,275

0.908

In the first iteration, the membership function of the freight car fleet size objective function is omitted in order to analyze the impact of this factor on the model. That is the reason why this function has a significantly higher value than the initial, which is unacceptable. In the next step, the solutions are obtained with all three functions included (Table 4.15). The function of immobilization and cassation remained relatively stable, whereas the value of the freight car fleet size objective function was significantly lower with λ ¼ 0.704 which is acceptable. In the third iteration, a higher tolerance was given relative to the initial value of the freight car fleet value, so that the model gave the following solution (Table 4.16): Table 4.15 The values of objective functions and satisfaction degree: II iteration Objective functions

Values

λ

Function of immobilization Function of cassation Function of freight car fleet size

657.36 232.39 87,644.5

0.704

Table 4.16 The values of objective functions and satisfaction degree: III iteration Objective functions

Values

λ

Function of immobilization Function of cassation Function of freight car fleet size

651.84 251.65 93,574

0.749

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Table 4.17 The values of objective functions and satisfaction degree: IV iteration Objective functions

Values

λ

Function of immobilization Function of cassation Function of freight car fleet size

657.36 232.39 87644.5

0.704

Table 4.18 Optimal freight car fleet size Freight car series

Variable

Number of freight cars

Eas1 Gas-z Galms Ibbiis Eas2 Smmp-tz Laekks Gkks

x1 x2 x3 x4 x5 x6 x7 x8

9232 550 258 185 633 894 130 86

The satisfaction degree was higher relative to the previous solution, but the fleet size function value was significantly higher as well. The number of cars x6 is 2185, which represents a high deviation and for that reason, this solution was rejected. In the fourth iteration the change of the membership function of the cassation function was conducted (allowed deviation of 40 cars), and the obtained solutions are given in Table 4.17. It can be seen that the solutions in the fourth iteration are the same as the solution in the second iteration, which means that the functions of cassation and immobilization have a lower impact on the optimal solution. Therefore, the second iteration gives a satisfactory solution, for the objective function as well as for the number of cars by series, which is close to the initial, and expected also. The optimal number of cars by series and subseries is given in Table 4.18.

CHAPTER 5

Fuzzy random model for rail freight car fleet management based on optimal control theory Contents 5.1 Fuzzy preliminaries 5.1.1 Fuzzy sets 5.1.2 Fuzzy numbers 5.1.3 Triangular fuzzy matrices 5.1.4 The main features of triangular fuzzy matrices 5.1.5 Inverse triangular fuzzy matrices 5.1.6 Defuzzification of triangular fuzzy numbers 5.1.7 Fuzzy random variables 5.2 Fuzzy stochastic model for rail freight car fleet sizing and allocation 5.2.1 Model parameters 5.2.2 Objective functional 5.2.3 Problem constraints 5.2.4 Fuzzy state vector estimation 5.2.5 Proposed approach for solving the problem based on the fuzzy linear quadratic Gaussian regulator 5.2.6 The components of weighting matrices A, B, and L e ðP Þ 5.2.7 Choosing the components of the fuzzy weighting matrix Γ 5.3 Numerical experiments 5.4 Comparative analysis of the results of the fuzzy random and random model

122 122 123 125 126 127 131 134 135 138 142 144 145 147 149 160 164 168

In this chapter fuzzy random theory was applied to capture the uncertainties in some of the key variables of the freight car fleet management. The demand and traveling time of freight cars are considered as fuzzy random and fuzzy variables, respectively. The problem is formulated as the problem of finding a regulator for fuzzy linear system with fuzzy quadratic performance and fuzzy random initial conditions.

Optimization Models for Rail Car Fleet Management https://doi.org/10.1016/B978-0-12-815154-9.00005-8

© 2020 Elsevier Inc. All rights reserved.

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Optimization models for rail car fleet management

5.1 Fuzzy preliminaries Before describing the fuzzy stochastic optimal control for rail freight car fleet sizing and allocation, this chapter summarizes some of the most important characteristics of fuzzy sets, fuzzy numbers, and fuzzy matrices with relevant operations. The main features of fuzzy stochastic are presented at the end of this chapter.

5.1.1 Fuzzy sets e Let X represent the domain for which one generic element is x. A fuzzy set A e over X is defined by a function A : X ! ½0, 1. In the literature this function e is characterized by its memis represented as μe, and therefore, a fuzzy set A A bership function μe : X ! ½0, 1 which assigns to each element xin Xa real A number μeðxÞ in the interval [0, 1]. The value μeðxÞ in x represents the A A e and it has been interpreted as a degree up membership degree of x in A e Therefore, the value of μ ðxÞ closer to 1 implies to which x belongs to A. e A e the higher membership degree of xto a fuzzy set A (Bector and Chandra, 2005). Definition 1 (Exact fuzzy set). The exact subset A of X can also be represented as a fuzzy set in X with the membership function as its own characteristic function, or  1, if x ¼ A μA ðxÞ ¼ (5.1) 0, if x 6¼ A   e in X can be represented as an ordered pair x, μ ðxÞ in The fuzzy set A e A which the elements with zero degree are not taken into account. Therefore, n o e in X can be written as A e ¼ x, μ ðxÞ a fuzzy set A where x 2 X and e A μe : X ! ½0, 1. A

e represent a fuzzy set in X. Definition 2(Normalized fuzzy set). Let A  e of a fuzzy set A e is defined as The height h A   e ¼ sup μ ðxÞ h A (5.2) e x2X A   e ¼ 1, then the fuzzy set A e represents a normalized fuzzy set, othIf h A   e < 1, then the subnormal fuzzy erwise, it is a subnormal fuzzy set. If 0 < h A e can be normalized, or become normal by redefining its membership set A   e , x 2 X. function as μeðxÞ=h A A

Fuzzy random model for rail freight car fleet management based on optimal control theory

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Fig. 5.1 Convex, normal fuzzy set.

Definition 3 (Convex fuzzy set). A convex fuzzy set is characterized by a membership function whose membership values are strictly monotonically increasing or monotonically decreasing. More precisely, if x, y,hand z are in thei e the relation x < y < z implies that μ ðyÞ  min μ ðxÞ, μ ðzÞ fuzzy set A, e e e A A A e is said to be a convex fuzzy set (Fig. 5.1). then A e represents an exact set Aα Definition 4 (α n - cut). α - cut of oa fuzzy set A given by Aα ¼ x 2 X : μeðxÞ  α for α 2 (0, 1]. A

5.1.2 Fuzzy numbers Fuzzy numbers represent a suitable way to represent fuzziness and a lack of precision in data. They are developed based on the fuzzy set theory. A fuzzy number is a convex normalized fuzzy set which has a piecewise membership e function.  The  fuzzy number A is positive (negative), denoted by e>0 A e < 0 if its membership function μ ðxÞ satisfies the condition A e A e is an interval numμeðxÞ ¼ 0for 8x < 0(x > 0). α - cut of a fuzzy number A A ber A, A . Definition 5 (Fuzzy point). The fuzzy set e a defined over R ¼ (∞, ∞) represents a fuzzy point (Fig. 5.2) if the membership function of e ais given by  1, if x ¼ a μea ðxÞ ¼ (5.3) 0, if x 6¼ a

Fig. 5.2 Fuzzy point.

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Optimization models for rail car fleet management

e ¼ ða, b, c Þ, Definition 6 (Triangular fuzzy number). The fuzzy set A where a < b < c defined over R, represents a triangular fuzzy number (Fig. 5.3), if its membership function is given as 8x  a > , axb > >ba < μeðxÞ ¼ c  x , b  x  c (5.4) A > > c  b > : 0, otherwise The remaining part of this section contains a brief description of basic arithmetic operations over triangular fuzzy numbers (Dubois and Prade, e ¼ ða, b, c Þ and B e ¼ ðd, e, f Þ represent two triangular fuzzy num1980). Let A bers. Operations of summation, subtraction, multiplication, and division are defined as follows: e+B e ¼ ða + d, b + e, c + f Þ Summation: A

(5.5)

Scalar multiplication: If k is a scalar, then it holds that e ¼ ðka, kb, kc Þ when k  0 kA

(5.6)

e ¼ ðka,  kb,  kc Þ when k < 0 kA

(5.7)

eB e ¼ ða  f , b  e, c  dÞ A

(5.8)

Subtraction: e  0 and B e  0 if b 0) e  0(A Multiplication: When A eB e ¼ ða, b, c Þðd, e, f Þ ¼ ðbd + eða  bÞ, be, ec + bðf  eÞÞ A

e ¼ ða, b, cÞ. Fig. 5.3 Triangular fuzzy number A

(5.9)

Fuzzy random model for rail freight car fleet management based on optimal control theory

125

e  0 ðb  0Þ and B e 0 When A eB e ¼ ða, b, c Þðd, e, f Þ ¼ ðea + bðf  eÞ, be, ec + bðd  eÞÞ A

(5.10)

e  0 ðb  0Þ and B e 0 When A eB e ¼ ða, b, c Þðd, e, f Þ ¼ ðec + bðd  eÞ, be, ea + bðf  eÞÞ A

(5.11)

e  0 and B e 0 When A eB e ¼ ða, b, c Þðd, e, f Þ ¼ ðec + bðf  eÞ, be, ea + bðd  eÞÞ A

(5.12)

e ¼ ða, b, c Þ, b > 0 is defined as The inversion of a triangular fuzzy number A   (5.13) A1 ¼ ða, b, c Þ1 ¼ b1  ðc  bÞb2 , b1 , b1 + ðb  aÞb2 e and B, e is given by folThe division of two triangular fuzzy numbers, A lowing relation: e A eB e1 ¼A e B

(5.14)

As the inversion and product of triangular fuzzy numbers are approximated values, the division is also an approximated value. e   A eB e1 ¼ ða, b, c Þ  e1  ðf  eÞe2 , e1 , e1 + ðe  d Þe2 Division : ¼ A e B

ea + bðe  f Þ b ec + bðe  d Þ ’ , , e2 e e2 (5.15)

5.1.3 Triangular fuzzy matrices The triangular fuzzy matrix (TFM) and the basic operations with these matrices can be defined in the following way (Shyamal and Pal, 2007; Bhowmik et al., 2008).   e ¼ aij TFM of order m  n is defined as A , where aij ¼ (alij, aij, auij) is mn e aijis the average value of the matrix A, e ijelement of the fuzzy matrix A, l u and aij and aij represent lower and upper bounds, respectively. Same as in thecase the basic operations with TFMs are  of classic matrices,  e ¼ aij and B e ¼ bij represent two TFMs of the same order. defined. Let A The following operations are defined (Shyamal and Pal, 2007):   e+B e ¼ aij + bij (5.16) A   eB e ¼ aij  bij (5.17) A

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      e ¼ aij eB e ¼ bij e ¼ cij ½where A ,B A mp mn np n X and cij ¼ aik :bkj , for i ¼ 1, 2,…,m and j ¼ 1,2,…, p k¼1

ek + 1 ek e A  ¼ A  A  e0 ¼ aji transponed A e A   e ¼ kaij , where k is scalar value kA

(5.18) (5.19) (5.20) (5.21)

Definition 7 (Zero TFM). A matrix of triangular fuzzy numbers is a zero matrix if all its elements are equal to zero. This matrix is denoted as e 0. 2

ð0, 0, 0Þ ð0, 0, 0Þ 6 ð0, 0, 0Þ ð0, 0, 0Þ 6 e : : 0 ¼6 4 : : ð0, 0, 0Þ ð0, 0, 0Þ

3 … ð0, 0, 0Þ … ð0, 0, 0Þ 7 7 : : 7 5 : : … ð0, 0, 0Þ

(5.22)

Definition 8 (Unit TFM). A quadratic TFM is a unit matrix if all its elements on the main diagonal are equal to 1, or aii ¼ (1, 1, 1), whereas all the others are aij ¼ (0, 0, 0), i 6¼ j. This matrix is denoted by Ie. 2

ð1, 1, 1Þ ð0, 0, 0Þ 6 ð0, 0, 0Þ ð1, 1, 1Þ 6 Ie¼ 6   4   ð0, 0, 0Þ ð0, 0, 0Þ

3 ⋯ ð0, 0, 0Þ ⋯ ð0, 0, 0Þ 7 7   7 5   ⋯ ð1, 1, 1Þ

(5.23)

Definition 9 (Symmetrical TFM). The quadratic matrix of triangular e0 , or if aij ¼ aji for each i, j. e¼A fuzzy numbers is symmetrical if A

5.1.4 The main features of triangular fuzzy matrices This section presents some of the main features of the TFMs. The laws of commutativity and associativity of these matrices are valid only in the case of operations of summation. e B, e of order m  n holds: e and C Property 1. For any three TFMs, A, e+B e e¼ B e+ A A

(5.24)

Fuzzy random model for rail freight car fleet management based on optimal control theory

127

    e+ B e ¼ A e+B e e+ C e +C A

(5.25)

e+A e ¼ 2A e A

(5.26)

e+e ee e A 0¼A 0¼A

(5.27)

e and B e represent two TFMs of the same order and let k and l are Property 2. Let A two scalars. Then it holds that   e ¼ ðkl ÞA e k lA (5.28)   e+B e + kB e ¼ kA e k A (5.29) e ¼ kA e + lA e ðk + lÞA (5.30)   eB e  kB e ¼ kA e k A (5.31) e e Property 3. For any two matrices A and B for which the operations of summation e+B eB e and multiplication A e are defined, it holds that A  0 e0 ¼ A e (5.32) A 

 e0 + Be0 e+B e 0¼A A

(5.33)

ðA  BÞ0 ¼ B0  A0

(5.34) e e Corollary 4. Let A and B represent two TFMs and k and l are two scalars, then   e0 e 0 ¼kA kA (5.35)   e0 + l  Be0 e+lB e 0 ¼kA (5.36) kA e Corollary 5. Let A is a quadratic TFM, then it holds: e0  A e0 and A e are both symmetrical eA A

(5.37)

e0 is symmetrical e+A A

(5.38)

5.1.5 Inverse triangular fuzzy matrices For determining an inverse fuzzy matrix, two methods exist in the literature, the scenario-based method and method based on arithmetic (Dehghan et al., e 2009). The first method considers a real matrix (derived from fuzzy matrix A called scenario) and defines a fuzzy inverse with respect to each individual

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e as an scenario. The second method is based on searching for a fuzzy matrix B e e e e inverse of a fuzzy matrix A so that the expression AB ¼ I holds. The first method of the inverse fuzzy matrix will be elaborated in more detail considering that the same is applied for the computational procedure in the case of fuzzy stochastic rail freight car fleet sizing and allocation problem. 5.1.5.1 Necessary and sufficient conditions for the invertibility of fuzzy matrices In this part, necessary assumptions are presented for the evaluation of invertibility of fuzzy matrices. Approximated inversions of uncertain matrices (interval, fuzzy) are given by Ghaoni (2002). Also, singularity and regularity of interval matrices are defined by Rohn (1993a,b), Kreinovich et al. (1998), Pal et al. (2010). ^ (the interval matrix is the Definition 10. The quadratic interval matrix A matrix whose elements are interval numbers) is regular if each matrix A ^ is sinwhich represents one scenario is nonsingular. Also, interval matrix A gular if it contains a singular matrix. Rohn (1993a), Rex and Rohn (1998), and Rump (1998) presented a few necessary conditions for defining singularity and regularity of interval matrices by using spectral radius, eigenvalues, and positive definiteness of some real matrices which are in a certain relationship with the original interval matrix. Regularity and singularity can be generalized in the case of fuzzy matrices.   e¼ e Definition 11. Let there be a fuzzy matrix A aij of n  n dimensions. e is α - regular if each matrix A 2 [A]α is nonsingular. Also, for α < β Matrix A e is (α, β) - regular if every matrix A in [A]α and out of [A]β in interval [0, 1], A is regular.     e In other words, if ½Aα ¼ AðαÞ, AðαÞ and ½Aβ ¼ AðβÞ, AðβÞ , then A will be (α, β) - regular if every matrix A, for which AðαÞ  A  AðβÞ or AðβÞ  A  AðαÞ holds is nonsingular. Additionally, it holds that a fuzzy   e e matrix A is regular if each matrix A 2 supp A is nonsingular. e is singular, if and only if inequality Theorem 1. Quadratic fuzzy matrix A j Acx j  Δ j xj has a nontrivial solution. e represent a singular matrix. In that case there is a singular Proof. Let A   e ∗ matrix A  2 supp A so it  is  possible to choose α 2 (0, 1] such that e e ∗ A 2 A α ¼ AðαÞ, AðαÞ A is singular. According to (Rohn, 0

Fuzzy random model for rail freight car fleet management based on optimal control theory

129

1993a) the system j Acxj  Δ j xj has a nontrivial solution.   Conversely, let the e is singular and the equation jAcx j  Δj x j be satisfied for x 6¼ 0. Thus, A 0 proof is completed. □ 5.1.5.2 Application of Rohn’s scheme for determining of the inverse fuzzy matrix For determining an inverse fuzzy matrix in this chapter the approach (Dehghan et al., 2009) based on Rohn’s scheme is applied (Rohn, 1993a, b). Let AI represent an interval matrix whose elements are closed intervals. AI can be expressed by the matrix of lower bounds A and matrix of upper bounds A:   (5.39) AI ¼ A, A e as the For each regular interval matrix Rohn defined the inverse matrix A 1 least interval matrix which contains all the inverse matrices (AI) ¼ {A1 : A 2 AI}, or more precisely as the interval matrix BI ¼ ½B, B whose boundaries are given by B ¼ min A1 y, z2Yn yz

(5.40)

B ¼ max A1 y, z2Yn yz

(5.41)

and

where the minimum and maximum are determined on the level of individual elements. Ayz is a special type matrix defined in following way. An m  n interval matrix AI is given, such that y 2 Ym, z 2 Yn, where Ym, Yn are sets of all m-dimensional and n-dimensional vectors with components 1 and 1, then    aij , if yi zj ¼ 1, (5.42) Ayz ij ¼ ðAc Þij  yi Δij zj ¼ aij , if yi zj ¼ 1: Afterwards, the same procedure for determining an inverse fuzzy matrix e is applied for the fixed α 2 [0, 1]. It is possible to determine the boundaries A   e by applying the described method as BðαÞ ¼ ½BðαÞ, BðαÞ of inverse A α    and BðαÞ ¼ Bij ðαÞ i, j . The aim is to determine where BðαÞ ¼ Bij ðαÞ i, j

functions Lijand Rij which satisfy the following conditions: Lij ðαÞ ¼ Bij ðαÞ

(5.43)

Rij ðαÞ ¼ Bij ðαÞ

(5.44)

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Optimization models for rail car fleet management

e e for  α 2 [0, 1]. Thus, we can approximate the inverse fuzzy matrix A with B ¼ eij is an LijRij fuzzy number if and only if, for each α1, α2 2 [0, 1] eij where B B α1 < α2, so that the relation ½Bðα2 Þ, Bðα2 Þ ½½Bðα1 Þ, Bðα1 Þ exists. Following theorem ensures the satisfaction of this condition.     e and A e represent two α - cut fuzzy matrices A e so that Theorem 2. Let A α1 α2 α1 < α2. Then it holds that ½Bðα2 Þ, Bðα2 Þ ½½Bðα1 Þ, Bðα1 Þ

(5.45)

        e ¼ Aðα1 Þ, Aðα1 Þ and A e ¼ Aðα2 Þ, Aðα2 Þ . Since α1 < α2, Proof. A α1  α2    it holds that Aðα2 Þ, Aðα2 Þ Aðα1 Þ, Aðα1 Þ so that n

  o n 1   o e e A1 : A 2 A A : A 2 A α2 α1

(5.46)

Thus, from Eqs. (3.4) and (3.5) it is clear that Bðα1 Þ < Bðα2 Þ and Bðα1 Þ < Bðα2 Þ, therefore ½Bðα2 Þ, Bðα2 Þ ½½Bðα1 Þ, Bðα1 Þ exists. □ Considering the fact that only three points exist, because three values for α have been used, 0, 0.5, and 1, Lagrange’s interpolation method can be applied in order to prove that each element of the matrix B(α) is a fuzzy number. In that case the following data table exists: α

α0

α1

α2

B

B0

B1

B2

in which αi represents a scalar in the interval [0, 1] and Bj ðj ¼ 0, 1, 2Þ is an n  n real matrix. By applying the Lagrangian interpolation it is possible to construct a matrix 2 3 L11 ðαÞ L12 ðαÞ ⋯ L1n ðαÞ 6 L21 ðαÞ L22 ðαÞ ⋯ L2n ðαÞ 7 6 7 7 (5.47) L ðαÞ ¼ 6     6 7 4     5 Ln1 ðαÞ Ln2 ðαÞ ⋯ Lnn ðαÞ in which Lij ðαÞ are the values of the Lagrangian interpolation which interpolate the following data table: α

α0

α1

α2

ðBÞij

ðB0 Þij

ðB1 Þij

ðB2 Þij

Fuzzy random model for rail freight car fleet management based on optimal control theory

131

Similarly, it is possible to construct an n  n matrix L ðαÞ ¼ Lij ðαÞ which interpolates the next data table: α

α0

α1

α2

B

B0

B1

B2

  Thus, a matrix L(α) is obtained in which Lij ðαÞ ¼ Lij ðαÞ, Lij ðαÞ approximates the matrix Bij(α).

5.1.6 Defuzzification of triangular fuzzy numbers Different strategies have been proposed for the ranking of fuzzy numbers. Among them are the methods based on the coefficient of variation, the distance between two fuzzy sets, the centroid and original point as well as weighted average. Despite the existence of these methods, none of it can be used for consistent ranking of fuzzy numbers in all possible cases. Imperfections exist in ranking fuzzy numbers by using coefficient of variation, distance between fuzzy sets, centroid and original point and weighted average. In this chapter, the method of signed distance for defuzzification of fuzzy numbers is described (Yao and Wu, 2000; Chang et al., 2004, 2006; Bjork, 2009, 2012) and a comparison of this and the alternative centroid method (Sugeno, 1985) has been performed. Definition 12. The fuzzy set [a, b; α], 0  α  1, a < b which is defined over a set of real numbers R, represents a fuzzy interval of level α if the membership function of the fuzzy set [a, b; α]is given by  α, a  x  b μ½a;b;α ðxÞ ¼ (5.48) 0 , in opposite Let F represent a family of all fuzzy sets defined over R, which satisfy two conditions: e 2 F, α - cut A(α) ¼ [AL(α), AR(α)] fuzzy set A e exists for 8 α 2 [0, 1]; (i) A (ii) AL(α) and AR(α) are continuous functions on the interval 0  α  1. e 2 F, from the theorem of decomposition, A e can be Then for each A expressed as e ¼ [ αIAðαÞ ðxÞ A 0α1

where IA(α)(x) represents a characteristic function A(α) and  α,z 2 AðαÞ μαIAðαÞ ðxÞ ðzÞ ¼ 0,z62AðαÞ

(5.49)

(5.50)

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From Definition 12 and Eq. (5.49) it is clear that x 2 R, αIA(α)(x) ¼ e ¼ [0α1 ½AL ðαÞ, AR ðαÞ; α. μ[AL(α),AR(α);α](x) 8 α 2 [0, 1]. Therefore, A In the remaining part of this section a concept of signed distance for a fuzzy set defined over F is introduced. Previously, the signed distance for a point is defined over R (Lee and Chiang, 2007). Definition 13 (Signed distance). For any a, 0 2 R signed distance from a to 0 has been defined as d0(a, 0) ¼ a. If a > 0 then the distance between aand 0is d0(a, 0) ¼ a. In the case when a < 0 then the distance between a and 0 is  d0(a, 0) ¼  a. Therefore, d0(a, 0) ¼ a represents the signed distance e 2 F, with α - cutA (α) ¼ [AL(α), AR(α)], 0  α  1 between a and 0. For A signed distances for two end points AL(α) and AR(α)measured from 0 are d0(AL(α), 0) ¼ AL(α) and d0(AR(α), 0) ¼ AR(α). Thus, the signed distance of an interval [AL(α), AR(α)] measured from origin 0 can be defined as 1 d0 ð½AL ðαÞ, AR ðαÞ, 0 ¼ ½d0 ðAL ðαÞ, 0Þ + d0 ðAR ðαÞ, 0Þ 2 1 ¼ ½AL ðαÞ + AR ðαÞ 2

(5.51)

according to Definition 13. For every α 2 [0, 1], there is one-to-one mapping between an exact interval [AL(α), AR(α)] and a fuzzy interval of the level α[DL(α), DR(α); α]. ½AL ðαÞ, AR ðαÞ; α $ ½AL ðαÞ, AR ðαÞ

(5.52)

0 can be Therefore, the signed distance from [AL(α), AR(α); α] to e defined as   1 0 ¼ d0 ð½AL ðαÞ, AR ðαÞ, 0Þ ¼ ½AL ðαÞ + AR ðαÞ d0 ½AL ðαÞ, AR ðαÞ; α, e 2 (5.53) e 2 F, AL(α) and AR(α) exist and are integrable for α 2 [0, 1], the Since A following definition holds. e 2 F, the signed distance A e measured from e Definition 14. Let A 0 can be defined as ð1 ð1    1 e e e d A, 0 ¼ d ½AL ðαÞ, AR ðαÞ, 0 dα ¼ ðAL ðαÞ + AR ðαÞÞdα 2 

0

0

(5.54)

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133

e ¼ ða, b, c Þ, α - cut is A(α) ¼ [AL(α), For the triangular fuzzy number A AR(α)], α 2 [0, 1] where AL(α) ¼ a + (b  a)α and AR(α) ¼ c + (c  b)α. From Definition 14, it follows that   1 e e (5.55) d A, 0 ¼ ð2b + a + c Þ 4   e ¼ ða, b, c Þ the relationship between C A e which can Furthermore, for A be derived as ∞ ð

  ∞ e ¼ ∞ C A ð

xμeðxÞdx A

μeðxÞdx

1 ¼ ða + b + c Þ 3

(5.56)

A

∞

  e e and signed distance d A, 0 can be represented in the following way. Let us define M¼ (a+c)/2. Based on the results b d(A,0)¼ (2b a c)/4¼ (b M)/2 it       e e e ¼ ðb  M Þ=6 and C A e  M ¼ ðb  M Þ=3 follows that d A, 0 C A (Figs. 5.4 and 5.5):     e e e < M < c. (a) If M > b,then a < b < d A, 0 b.

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Fig. 5.5 Case M < b.

  e , since the maximum membership degree of A e ¼ ða, b, c Þ is in than C A     e e e . Therefore, point b, which has a higher weight in d A, 0 than in C A in the case of the defuzzification of a triangular fuzzy number, it is better to use the signed distance method than the centroid method. e Ee 2 F, where D e ¼ [0α1 ½DL ðαÞ, DR ðαÞ; α Also, for two fuzzy sets D, and Ee ¼ [0α1 ½EL ðαÞ, ER ðαÞ; α it holds that μ½DL ðαÞ , DR ðαÞ  + ½EL ðαÞ , ER ðαÞ  ðzÞ ¼ μ½DL ðαÞ + EL ðαÞ  + ½DR ðαÞ + ER ðαÞ  ðzÞ α α α α α α α α (5.57)   e + Ee ¼ [ ðDL ðαÞ + EL ðαÞÞα , ðDR ðαÞ + ER ðαÞÞα D (5.58) 0α1



[0α1 ½kDL ðαÞ, kDR ðαÞ; α, if k > 0 [0α1 ½kDR ðαÞ, kDL ðαÞ; α, if k < 0   e  Ee ¼ [ ðDL ðαÞ  ER ðαÞÞα , ðDR ðαÞ  EL ðαÞÞα D e¼ kð  ÞD 0α1

(5.59) (5.60)

If 0  DL(α)  DR(α) and 0 < EL(α) < ER(α) forα 2 [0, 1], then it holds that   e  Ee ¼ [ ðDL ðαÞ  EL ðαÞÞα , ðDR ðαÞ  ER ðαÞÞα (5.61) D 0α1 



DL ðαÞ DR ðαÞ e Ee ¼ [ D (5.62) , 0α1 ER ðαÞ α EL ðαÞ α

5.1.7 Fuzzy random variables Fuzzy random variables are introduced for the modeling and analysis of imprecise measurable functions related to the random experiment sampling space, where the imprecision of the value of these functions has been formalized with fuzzy sets (Gil et al., 2006).

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135

Different approaches to the concept of fuzzy random variables have been developed, and the most important among them is given by Kwakernaak (1978, 1979), Kruse and Meyer (1987), and Puri and Ralescu (1986). The term “fuzzy random variable” has been introduced by Kwakernaak (1978), who defined fuzzy random variables as “random variables whose values are not real but fuzzy numbers” and conceptualizes fuzzy random variables as indefinite perceptions of the exact, but unobservable random variable. Let (Ω, Φ, P) represent a probability space and F(R) is a set of all fuzzy numbers in the set of real numbers R. Formally F(R) represents a type of normal convex fuzzy subsets of Euclidean space R which has bounded α levels for α 2 [0, 1]. This is a type of mapping U : R ! [0, 1] so Uα is a nonempty bounded interval where  fx 2 Rj U ðxÞ  αg if α 2 ð0, 1 Uα ¼ (5.63) clð supp U Þ∗ if α ¼ 0 Therefore, the fuzzy random variable represents the mapping ξ : Ω ! F(R) so that for any α 2 [0, 1] and every ω 2 Ω, real mapping inf ξα : Ω ! R, which satisfies inf ξα ðωÞ ¼ inf ðξðωÞÞα and sup ξα : Ω ! R, which satisfies sup ξα ðωÞ ¼ supðξðωÞÞα represents real random variables (Fig. 5.6). Puri and Ralescu (1986) conceptualized the fuzzy random variable as a fuzzification of a random set. Authors have explored and defined the concept of fuzzy random variables whose values represent fuzzy subsets Rn (Banah’s space) and in that way banded together fuzzy random variables with the already known concept of random sets.

5.2 Fuzzy stochastic model for rail freight car fleet sizing and allocation The mathematical model presented in this chapter simultaneously considers problems of rail freight car fleet sizing and allocation of freight cars in an uncertain environment which is modeled by fuzzy stochastic (Milenkovic and Bojovic, 2013). Investments in rail freight car fleet are made with the aim of satisfying the demand for transport service on a considered railway network. Transport

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Optimization models for rail car fleet management

€ller and Beer, 2004). Fig. 5.6 Model of a fuzzy random variable (Mo

service is performed by dispatching loaded and empty cars to different stations in a considered time period. The use of freight cars represents an important element of decisions about the planning of investments in purchasing new freight cars. Reducing operational costs of freight car utilization represents a permanent objective of all railway companies. Actual research indicates the need for simultaneous decision-making about investments in freight car fleet and allocation of freight cars. The additional complexity of the considered problem lies in the fact that the railway freight car fleet is composed of many different types of freight cars and that a certain type of freight cars cannot be used for the transport of all types of commodities. In practice, unavailability of a certain type of freight cars often implies the need for the use of another type of freight cars. Some series of freight cars usually include a large number of different subseries that are mutually substitutable only up to a certain extent. As it can be noticed from the literature survey, vehicle fleet sizing models can be considered as homogeneous and heterogeneous. The model developed in this chapter assumes a homogeneous freight car fleet. An approach of fuzzy optimal control is applied for problem formulation. Fig. 5.7 summarizes the proposed fuzzy linear quadratic Gaussian (FLQG) controller for rail freight car fleet sizing and allocation. A fuzzy linear dynamic model for homogeneous rail freight car flows (loaded and empty) is developed. An estimation of the

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137

Fig. 5.7 Fuzzy linear quadratic Gaussian controller for rail freight car fleet sizing and allocation.

fuzzy state vector is performed based on fuzzy forecasting of the rail freight car level in stations and the fuzzy dynamic model. Fuzzy forecasts are obtained by the double fuzzy exponential smoothing (DFES). Optimal estimates of the rail freight car system state in the fuzzy sense are derived based on the fuzzy Kalman filter which represents fuzzy linear quadratic estimator (FLQE). The fuzzy stochastic multiperiod EOQ model and the fuzzy single period random newsboy model are proposed for determining the rail freight car inventories by stations. The optimal state estimate and objective functional in the fuzzy sense serve as inputs to fuzzy linear quadratic regulator (FLQR) in order to generate fuzzy optimal control law. Section 5.2.1 includes model parameters. The objective functional in Section 5.2.2 is based on the expected costs of ownership and allocation of empty and loaded cars which are represented as exact parameters. Section 5.2.3 provides the definition of the fuzzy vector state equation for the discrete fuzzy linear system that describes the problem of fleet sizing and allocation of rail freight car fleet. The estimation of the fuzzy state vector based on the forecasted number of freight cars is presented in Section 5.2.4. The solution approach is presented in Section 5.2.5. An approach for detere B, e and Lis e explained in mining the components of weighted matrices A, Section 5.2.6. The Section 5.2.7 presents an approach for determining the components of fuzzy matrix for a terminal part of the objective funce ðP Þ. tional Γ

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Optimization models for rail car fleet management

5.2.1 Model parameters The general system theory was used as a background for modeling the problem in which some of the parameters are treated as fuzzy random. Dynamics of the number of loaded and empty freight cars on a railway network with N stations and on the planning horizon of P days is described by a first-order vector fuzzy difference equation. Let N represent the number of stations on considered railway network. These stations may represent individual station of a real network or a group of neighboring stations with a lower volume of work. This grouping of stations has been performed for the purpose of model testing for several reasons. Frequently it is not possible to provide all the necessary data for model testing on the level which represents a real rail network. Since the main objective of model testing is identifying its main characteristics, analyzing the results, and determining the necessary number of freight cars, using a real rail network would primarily mean favoring the analysis of computational complexity instead of the analysis of obtained results. Control actions about the dispatching of empty and loaded freight cars have been conducted once a day, in discrete time instants 0, 1, …, m, …, n, …, P for each station on the network. The length of the planning horizon cannot be shorter than the longest traveling time of freight cars between the stations on a considered network. Freight cars have been considered as continuous variables. Considering that the model is intended for strategic planning, integer requirements on the flow variables do not need to be taken into account. Demand for freight cars from the station i for station j in period n, Dij(n) is modeled as a fuzzy random  variable  described by a fuzzy Gaussian probability density function N e μij , σ ij . Namely, in uncertain and/or unstable environments instead of representing the demand with only one value E(Dij) ¼ μij, which is very difficult, it is more convenient to represent the demand with an interval (μi  Δ1, μi, μi + Δ2), in which Δ1 and Δ2 are defined by the decision maker. Therefore, the fuzzy expectation of the fuzzy random variable can be represented by a triangular fuzzy number ! X fij ¼ μi ¼ ðμi  Δ1 , μi , μi + Δ2 Þ E D j

Only fully loaded cars have been considered, therefore, the level of demand is measured in units of freight cars. The demand only considers routes between the stations, not the stations itself. The demand stimulates fuzzy flows of loaded freight cars which are represented as Feij ðnÞ.

Fuzzy random model for rail freight car fleet management based on optimal control theory

139

In a general case, a number of cars of different types may be available for satisfying the actual demand, so that the notation could be extended on Feijk ðnÞ for the sake of representing the type k freight car flows. In this chapter, the problem with only one type is considered. Since the demand for loaded freight cars toward any station i may not be the same as the demand from station i to other stations on the network, movement of empty freight cars will be needed for balancing the flows. The fuzzy number of empty cars is denoted as Eeij ðnÞ. Control actions, the set of all departures of empty and loaded freight cars on a planning horizon, are under the influence of freight car arrival time. In many systems, traveling times are uncertain due to the malfunctions of movable assets (locomotives and freight cars) as well as external impacts. Instead of explicit consideration of traveling time, the problem in this chapter (and in the following approaches also) has been formulated through the arrival of freight cars. More precisely, if Feij ðnÞ of freight cars is dispatched from station i in period m, in this approach, a question arises how many of these freight cars have actually arrived at station j in period n. Therefore, parameters e θij ðm, nÞ[e αij ðm, nÞ] are defined as fuzzy proportions of loaded (empty) freight cars dispatched from station i in period m which will arrive in station j in period n. Due to the uncertainty of demand and traveling times, it is desirable to maintain the inventories of freight cars in some station in order to avoid occasional shortages. Let Sei ðnÞ represent a fuzzy number of freight cars in ith station at the end of period n. The time interval between making the decision about the dispatching of freight cars toward the demand stations is usually one day. This means that in the next moment of analysis, after 24 h has passed, the number of cars in ithstation can be described by adding to the state Sei ðnÞ the difference between the fuzzy number of cars arrived during period n + 1 but dispatched in previous time periods and the fuzzy number of cars dispatched during the n + 1 period based on the demand from other stations. The state of the system is defined by a total fuzzy supply of cars in each station for every day during the planning period. Since it is possible that in a station there is an insufficient number of cars to satisfy the demand for the e ij ðnÞ is defined, which represents remaining stations, the fuzzy variable U unsatisfied requests between station i and j in period n. Actually, this variable represents undelivered number of freight cars to station i from station j during period n. It is assumed that unsatisfied orders are transferred to the next time period so that there are no lost demands.

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Optimization models for rail car fleet management

Based on the introduced assumptions and according to the adopted parameters it is possible to establish the following relations between defined quantities: Sei ðn + 1Þ ¼ Sei ðnÞ +

N X  X

 Feji ðmÞ  e θji ðm, n + 1Þ + Eeji ðmÞ  e αji ðm, n + 1Þ 

j¼1 m m, i ¼ 1, …, N; j ¼ 1, …, N; n ¼ 1, …, P, i 6¼ j. e αij ðm, nÞ: the fuzzy proportion of arrivals in the nth period of loaded cars dispatched in the mth period from station i to station j, n > m, i ¼ 1, …, N; j ¼ 1, …, N; n ¼ 1, …, P, i 6¼ j. The demand for freight cars is given as eij ðnÞ: the fuzzy random demand for freight cars between station i and D stationj during the period n, i ¼ 1, …, N; j ¼ 1, …, N; n ¼ 1, …, P, i 6¼ j. This demand can be represented as a sum of random (stochastic) component and fuzzy deterministic component. Since it is assumed that this sequence has the Gauss-Markovian character and taking into account that the Gaussian form is valid in the case of linear transformation, the same sequence can be described with the fuzzy state vector of the fuzzy linear dynamic system with the initial Gaussian fuzzy state vector. This parameter actually represents uncontrollable input in the system. The Gaussian pure random process represents a boundary value of the fuzzy Markovian process with a very large covariance and short correlation time. Since the Fourier transformation of correlation function is constant in time, its spectrum is white. This is the reason why the pure Gaussian random process is usually termed as white noise (Bryson and Ho, 1975). Let us define an auxiliary fuzzy variable deij ðnÞ: eij ðn + 1Þ deij ðnÞ ¼ D

(5.67)

deij ðn + 1Þ ¼ e λij ðnÞe λij ðn  1Þ⋯e λij ðnÞdeij ð0Þ + ωij ðnÞ, n ¼ 0,1, …,P  1 (5.68) e λij ðnÞ ¼

e μij ðn + 2Þ , n ¼ 0,1,…, P  2 e μij ðn + 1Þ   0 E ωij ðnÞ ¼ e

(5.69) (5.70)

where eλij ðnÞ: the fuzzy component of demand for cars between i and j during the period n, i ¼ 1, …, N; j ¼ 1, …, N; n ¼ 0, …, P  1, i 6¼ j.

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Optimization models for rail car fleet management

e ij ðnÞ: the fuzzy random component of demand for freight cars between ω stations i and jduring the period n, i ¼ 1, …, N; j ¼ 1, …, N; n ¼ 0, …, P  1, i 6¼ j. Relations (5.67)–(5.69) enable conversion of any fuzzy random process in the equivalent fuzzy Markovian random process by an appropriate extension of the fuzzy state vector for a fuzzy variable deij ðnÞ. Relation (5.70) represents a fuzzy expectation of the Gaussian fully random process which represents a fuzzy random component of demand. As in the exact case, any fuzzy random vector sequence can be converted into a fuzzy Markovian process by a suitable extension of the state vector and by introducing appropriate substitutions. Let us redefine e θij ðm, nÞ and e αij ðm, nÞ in the following ways: e aji, n + 1m ðnÞ ¼ e θji ðm, n + 1Þ, m < n , n ¼ 0,…, P

(5.71)

αji ðm, n + 1Þ, m < n , n ¼ 0,…, P beji, n + 1m ðnÞ ¼ e

(5.72)

In a similar way, by associating one additional index it is possible to describe variables that represent arrivals of loaded and empty cars from previous time intervals xeji, n + 1m ðnÞ ¼ Feji ðmÞ, m < n , n ¼ 0,…,P

(5.73)

e yji, n + 1m ðnÞ ¼ Eeji ðmÞ, m < n , n ¼ 0,…,P

(5.74)

It is also necessary to introduce additional fuzzy variables feij ðnÞ and eeij ðnÞ which can be defined in the following way: feij ðnÞ ¼ Feij ðn + 1Þ, eeij ðnÞ ¼ Eeij ðn + 1Þ, 8i, j, n

(5.75)

5.2.2 Objective functional Optimal control theory considers the problem of determining the control low for a given system so that a given optimality criterion could be satisfied. The control problem includes the performance criterion as a measure that represents a function of state and control variables. In this approach, the system’s performance measure represents an objective functional of cost minimization that includes expected costs of ownership and allocation of empty and loaded freight cars on a considered railway network. The aim is to determine an optimal policy by which any value of the state could be projected on the control that in the best way satisfies a given objective. The fuzzy linear Gaussian system in this chapter has been considered with the quadratic cost functional. Since this functional includes the cost of allocation and holding

Fuzzy random model for rail freight car fleet management based on optimal control theory

143

of empty and loaded cars, there are no practical constraints to express this functional in the quadratic form. Let the performances of the system in the fuzzy sense be measured by the following fuzzy cost: P1    X  e e e ðnÞ, U e ðnÞ, n J ¼ Δ X ðP Þ + f X

(5.76)

n¼0

Fuzzy cost is composed of two components. The first component represents the fuzzy cost of the rail freight car fleet system in the last period of the planning horizon, which is the function of the fuzzy state vector as a consequence of the control actions during the preceding time periods. This cost is derived based on unit costs of car holding hi, unit costs of car shortages.pij, and unit costs of car ownership during their transit between stations Q. The cost component Q is determined by cars by the period (Beaujon and Turnquist, 1991). The freight car holding cost is not lower than the ownership cost Q, however, in a general case it includes an additional cost of storage and freight handling in the station. In this chapter, hi is defined as a unit cost of freight car holding during a time period in a certain station, where hi Q. The second component in the objective functional represents a fuzzy system cost in P  1 periods eðnÞ of the planning horizon which is a function of the fuzzy state vector X e and fuzzy control vector U ðnÞ and it is derived with respect to unit holding cost hi, unit shortage cost pij, unit ownership or leasing cost during transit Q, and unit costs of traveling of empty eji and loaded lji cars from one station to another in the remaining discrete time periods. All cost components are modeled as exact parameters. In summary, the costs which refer to the freight car fleet system can be represented as follows: hi: unit cost of car holding in ithstation by period. This cost is constant during the planning horizon i ¼ 1, …, N. pij: unit cost of car shortage between station i and station j by period. This cost is constant during the planning horizon i ¼ 1, …, N; j ¼ 1, …, N; i 6¼ j. eij: unit cost of empty freight car traveling from station i to station j, i ¼ 1, …, N; j ¼ 1, …, N; i 6¼ j. lij: unit cost of loaded freight car traveling from station i to station j, i ¼ 1, …, N; j ¼ 1, …, N; i 6¼ j. Q: unit ownership cost of freight car during its transit between station, by period.

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Optimization models for rail car fleet management

5.2.3 Problem constraints The assumed form of the fuzzy difference equation which describes the dynamics of the fuzzy number of loaded and empty freight cars, with specified optimality criteria, represents a problem of linear systems control with fuzzy random parameters. The elements of the vector state matrix are fuzzy random parameters of freight car arrival times in stations. The considered model is described by a time discrete control system whose states vary in accordance with the following fuzzy difference equation: eðnÞX e ðn + 1Þ ¼ Λ e ðnÞ + GU e ðnÞ + V e 1 ðnÞ, n ¼ 0,…,P  1,given Xeð0Þ X (5.77) where index n ¼ 0 corresponds to beginning period. These first-order fuzzy linear dynamic constraints reflect basic relations (5.64)–(5.65) of considered problem. The fuzzy state in the next period represents a function of fuzzy state, fuzzy control, and fuzzy random noise e 1 ðnÞ in the preceding period. X eð0Þ is given and where V  e 1 ðnÞ U e (5.78) U ðnÞ ¼ e U 2 ðnÞ 2N ðN 1Þ h i e 1 ðnÞ ¼ feij ðnÞ U , i ¼ 1,…,N ; j ¼ 1,…, N, i 6¼ j (5.79) N ðN 1Þ

  e 2 ðnÞ ¼ eeij ðnÞ U , i ¼ 1,…, N ; j ¼ 1,…,N , i 6¼ j (5.80) N ðN 1Þ h i e n, e Fuzzy matrix Λ aðnÞ, beðnÞ is a fuzzy matrix of state transitions whose dimensions are ½N ð2N  1Þ + 2N ðN  1ÞðP  1Þ  ½N ð2N  1Þ + 2N ðN  1ÞðP  1Þ e represents an input fuzzy matrix with dimensions Matrix G ½N ð2N  1Þ + 2N ðN  1ÞðP  1Þ  ½2N ðN  1Þ e 1 ðnÞ is a random component vector of the fuzzy random demand for V freight cars during the period n with components ωij(n).   e ,i ¼ 1,…, N e ðnÞ ¼ col e βi ðnÞ e χe δ eε ϕ (5.81) X αi ðnÞ e e ðnÞ is a vector of dimensions [N(2N  1) + 2NFuzzy state vector X (N  1)(P  1)] which describes the current state in the freight car system through the number of loaded and empty freight cars in the period n e kj ðnÞ), and actual (dekj ) demand during the same period (Sek ðnÞ), unsatisfied (U

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145

e and loaded (e and empty (e δ, ϕ) χ , eε) flows which arrive in all stations from preceding time periods. eðnÞ has the following components: X   e kj ðnÞ , k ¼ 1,…,N , j ¼ 1,…, N, k 6¼ j e αi ðnÞ ¼ col Sek ðnÞ, U (5.82)   e βi ðnÞ ¼ col dekj , k ¼ 1, …,N , j ¼ 1,…,N , k 6¼ j (5.83) e χ ¼ colð xe211 ðnÞ ⋯ xeN 11 ðnÞ ⋯ xe21, P1 ðnÞ ⋯ xeN 1, P1 ðnÞ Þ   e y211 ðnÞ ⋯ e yN 11 ðnÞ ⋯ e y21, P1 ðnÞ ⋯ e yN 1, P1 ðnÞ δ ¼ col e

(5.84) (5.85)

eε ¼ colð xe1N 1 ðnÞ ⋯ xeN 1N 1 ðnÞ ⋯ xe1N 1, P1 ðnÞ ⋯ xeN 1N , P1 ðnÞ Þ (5.86)   e ¼ col e y1N 1 ðnÞ ⋯ e yN 1, N 1 ðnÞ ⋯ e y1N 1, P1 ðnÞ ⋯ e yN 1, N , P1 ðnÞ ϕ (5.87)

5.2.4 Fuzzy state vector estimation Fuzzy state vector is estimated by applying the fuzzy forecasted values of the eðnÞ in discrete time intervals and model (5.77). For a number of freight cars Z given statistic time series, for a planning process of freight car inventory levels in stations, it is required to forecast their values in the next periods. These values represent fuzzy state observations, XeðnÞ. Let the analytic form of observation equation be ei ðnÞ, i ¼ 1,2, …,N , n ¼ 0,1, …,P ei ðnÞ ¼ Sei ðnÞ + Δ Z

(5.88)

The first member represents a fuzzy number of freight cars in station i in the time period n, whereas the second is a forecasting fuzzy random error for the same station during the same period. Fuzzy forecasts of the number of freight cars are obtained by a method of double exponential smoothing (Kaufmann and Gupta, 1988). Fuzzy exponential smoothing represents a method of continuous updating of the fuzzy forecast in accordance with the newest observations. Fuzzy exponential smoothing assigns weight to fuzzy observations which exponentially decreases while moving away from the actual period. In other words, more recent fuzzy observations get a higher weight in forecasting than the older fuzzy observations. The DFES can be used when fuzzy data shows a trend. Fuzzy exponential smoothing with trend is similar to the basic exponential smoothing except that two components must be updated for each time

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Optimization models for rail car fleet management

period—level and trend. The level is a smoothed estimation of values of fuzzy data at the end of each period. The trend is a smoothed estimation of average growth at the end of each period. Let a fuzzy time series of the freight car number be by an order sequence of triangular  represented 1 2 e fuzzy numbers, Si ðnÞ ¼ Si ðnÞ, Si ðnÞ, Si3 ðnÞ . For any time period, the smoothed value can be determined based on the calculation of two equations, the first for the level and the second for the trend: Si ðnÞ ¼ αyi ðnÞ + ð1  αÞðSi ðn  1Þ + bi ðn  1ÞÞ

(5.89)

bi ðnÞ ¼ γ ðSi ðnÞ  Si ðn  1ÞÞ + ð1  γ Þbi ðn  1Þ

(5.90)

where yi(n) is an observation of the number of freight cars in station i during the period n, α, and γ represent smoothing constants, and Si(n) and bi(n) are the level and trend components of smoothed time series. The first smoothing equation adjusts Si(n) taking into account the trend in the previous time period bi(n  1), adding this trend to the latest smoothed value Si(n  1). The second smoothing equation then updates the trend, and is expressed as a difference between the last two values. This equation is similar to the basic shape of exponential smoothing, but in this case it is applied for trend updating. There is a number of schemes for determining initial values for Si(n) and bi(n). In this case, it is chosen that Si(1) ¼ yi(1) and bi(1) ¼ y2  y1. Forecast of the number of freight cars for the next period in a certain station is given by the following relation: Zi ðn + 1Þ ¼ Si ðnÞ + bi ðnÞ

(5.91)

Forecast of the number of freight cars for the next m periods in a certain station can be determined as Zi ðn + mÞ ¼ Si ðnÞ + mbi ðnÞ

(5.92)

where Zi(n + m) ¼ [Z1i (n + m), Z2i (n + m), Z3i (n + m)] is a fuzzy forecasted number of freight car in station i. The lower bound Z1i (n + m), middle value Z2i (n + m), and upper bound Z3i (n + m) of smoothed fuzzy values are computed independently by applying the recourse form (5.92). Let us assume now that the fuzzy forecasted value is represented as a linear function of the fuzzy state: eðnÞ ¼ H e XeðnÞ + V e 2 ðnÞ, n ¼ 0,1,…, P Z

(5.93)

e has dimensions N  [N(2N  1) + 2N(N  1)(P  1)]. Fuzzy matrix H e V 2 ðnÞ denotes fuzzy white noise which represents fuzzy forecasting errors.

Fuzzy random model for rail freight car fleet management based on optimal control theory

147

 e 1 ðnÞ, V e 2 ðnÞ be independent fuzzy Gaussian vectors Let Xeð0Þ and V with the following characteristics: h     e 1 ðnÞ ¼ E V e 2 ðnÞ ¼ e xð0Þ;E V 0 (5.94) E xeð0Þ ¼ e x^ð0j 0Þ ¼ e n o  T e ð0Þ cov½xeð0Þ ¼ E xeð0Þ  e ¼M (5.95) xð0Þ xeð0Þ  e xð0Þ   e 1 ðnÞ ¼ R e1 ðnÞ cov V (5.96)   e 2 ðnÞ ¼ R e2 ðnÞ cov V (5.97) where e1 ðnÞ: a symmetrical and positive-semidefinite fuzzy matrix of covariance R of a random component of demand for freight cars, which has dimensions [N(2N  1) + 2N(N  1)(P  1)]  [N(2N  1) + 2N(N  1)(P  1)]. e2 ðnÞ: a symmetrical and positive-semidefinite N  N fuzzy matrix of R covariance of white noise in the forecast.

5.2.5 Proposed approach for solving the problem based on the fuzzy linear quadratic Gaussian regulator Optimal control for a fuzzy linear system with quadratic optimality criteria leads to a linear feedback of state variables (Athans and Falb, 1966). Fuzzy state variables can be estimated by using a filter that represents the system’ model and a signal that is proportional to the difference between the forecasted and estimated fuzzy value. By combining these two structures, the optimal filter with the optimal deterministic regulator for the case of fuzzy input, an optimal fuzzy regulator with embedded feedback, could be obtained (Bryson and Ho, 1975). The analytic solution for the model of the fuzzy linear system will be represented in a discrete form, where the e ðnÞ, fuzzy control U e ðnÞ, and fuzzy output Z eðnÞ are connected fuzzy state X by the following system of fuzzy equations: eðnÞXeðnÞ + GU e ðn + 1Þ ¼ Λ e ðnÞ + V e 1 ðnÞ, n ¼ 0,…, P  1 with given Xeð0Þ X (5.98) eðnÞ ¼ H eX e ðnÞ + V e 2 ðnÞ, n ¼ 0,…,P Z

(5.99)

Therefore, considering the dynamic constraints (5.98), the measurement eð0Þ, Z eð1Þ,…, Z eðnÞ (5.99) and random disturbances, the problem is to error Z e ðnÞ so as to minimize a fuzzy objective determine a fuzzy control vector U functional that has the following form:  P1  X  1 e Te e ðnÞ XeðnÞ e ðP Þ + 1 e e ðnÞ T Ω (5.100) J¼ X XeðnÞ, U ðP Þ Γ ðP ÞX e ðnÞ U 2 2 n¼0

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where

~

∼ ~

~ ~

is a fuzzy positive definite, symmetric matrix, and the remaining matrices are as follows: eðnÞ is [N(2N  1) + 2N(N  1)(P  1)]  [N(2N  1) + 2N(N  1) A (P  1)] a fuzzy positive-definite matrix. e is [2N(N  1)  2N(N  1)], a fuzzy matrix. B e [N(2N  1) + 2N(N  1)(P  1)]  [2N(N  1)], a fuzzy matrix. Lis e Γ ðP Þ is [N(2N  1) + 2N(N  1)(P  1)]  [N(2N  1) + 2N(N  1) (P  1)], a fuzzy positive-definite matrix. The recursive dynamic programming algorithm is applied as a solution approach to this fuzzy stochastic problem of sequential optimization (Bryson and Ho, 1975). It should be noted that fuzzy estimates and controls are independent according to the separation principle. This principle enables the decomposition of the fuzzy optimal control problem into two independent subproblems: the subproblem of fuzzy optimal regulation and the subproblem of fuzzy state estimation. The discrete fuzzy Kalman filter is applied for the fuzzy state estimation. The recurrent fuzzy relations needed for solving the formulated task are described below. These relations are analytically solved starting from the last toward the initial state of the planning horizon. eðnÞ  X e ðnÞ ¼ C eðnÞ U    1 eðnÞ + LeT e ðn + 1ÞG e ðn + 1ÞΛ eðnÞ ¼ G eT Γ e+B eT Γ e C G h i e^ ðnÞ ¼ X e ðnÞ e ðnÞ + K e ðnÞ Z eðnÞ  H e X X 

e^ ðnÞ + G e ðn + 1Þ ¼ Λ eðnÞX eU e ðnÞ X

(5.101) (5.102) (5.103) (5.104)

e ð0Þ is given and jwhere X

 1 eM e ðnÞH eT + R e2 ðnÞ1  M e ðnÞH eT H e2 ðnÞ e ðnÞ ¼ W e ðnÞ  H eT  R K

(5.105)   eT ðnÞΓ eðnÞ  C eðn + 1ÞΛ e ðnÞ ¼ Λ e ðn + 1ÞG e C eðnÞ + A eðnÞ eT Γ eT ðnÞ B e+ G Γ (5.106)

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149

e ðP Þgiven, with Γ     e ðnÞR e2 ðnÞK e T ðnÞ (5.107) e ðnÞ ¼ Ie K e ðnÞ HÞ e M e ðnÞ Ie K e ðnÞH e T +K W e ð0Þgiven, with M eðnÞW eT ðnÞ + R e ðn + 1Þ ¼ Λ e ðnÞΛ e1 ðnÞ M

(5.108)

These equations represent a procedure for solving the formulated fuzzy linear quadratic Gaussian control problem. The fuzzy linear controller is represented by Eq. (5.101). This equation represents the fuzzy optimal control law which minimizes the objective functional (5.100). The fuzzy optimal control law is derived with the dynamic programming algorithm and Belman optimality principle (Speyer and Chung, 2008). Relation (5.102) represents the fuzzy reinforcement control process feedback. The recurrent relations (5.103) and (5.104) represent the estimation of the fuzzy state or the Kalman filter. The Kalman filter is the most popular optimal estimator and the only way to optimize the objective functional is to include this filter in the control system. More precisely, these two equations represent the freight car fleet dynamic system model with fuzzy parameters and a correction term which is proportional to the difference between the forecasted values e ðnÞ. eðnÞand the values estimated based on the fuzzy state function H e X Z e ðnÞ, represents the Kalman gain. The fuzzy proportionality matrix (5.105), K In fact, this matrix represents the relationship between uncertainty in the e ðnÞ, and uncertainty in the fuzzy forecast, R e2 ðnÞ. M e ðnÞ is fuzzy state, W the symmetrical and positive-semidefinite fuzzy matrix of the state vector covariances in period n which has the dimensions ½N ð2N  1Þ + 2N ðN  1ÞðP  1Þ  ½N ð2N  1Þ + 2N ðN  1ÞðP  1Þ Relation (5.107) represents the solution of the feedback fuzzy matrix Riccati equation. The matrix of variances of the estimation error (5.107), e ðnÞ, is given in the form which is suitable for practical computation. W Obtained estimation of the given series of observed fuzzy data is optimal in the sense of the minimized mean quadratic error. Relation (5.108) represents updating of the state estimation covariance before measurement.

5.2.6 The components of weighting matrices A, B, and L The relationship between the elements of weighting matrices in the objective functional and the choice of the adequate values of these matrices are very important and defining them is a very complex task. The influence of these matrices on the quadratic optimality criterion and practical directions for their structuring were given by Athans and Falb (1966).

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Optimization models for rail car fleet management

This section describes the fuzzy stochastic inventory model and its relationship with the state of freight cars in a corresponding railway node. The model is based on a multiperiod economic order quantity (EOQ) model (Taha, 2003). The parameter γ i(n) represents an average level of inventory and shortage of freight cars in station i during the period n of the planning horizon. The parameter ηi(n) is an order of freight cars for the same station and period which is made at the moment when the number of freight cars in the station reaches the level of reordering. The cost minimum composed of the sum of freight car ownership and allocation costs represents the criteria for determining the optimal number of freight cars and the optimal ordering quantity. The total demand for freight cars in station i during the planning horizon is Ti. The ordering cost can be expressed as N   Ti 1 X v  eji + ð1  vÞlji ηi N  1 j¼1

(5.109)

Expression (5.109) represents a period-related cost of freight car supplied from all stations of supply to the station for which the optimal delivery per period has been calculated. Since Ti represents the total demand in station i, the relationship Tη i is an average number of orders per period, whereas i the P average cost of supply   of one freight car order is expressed as N 1 ð Þl v  e + 1  v ji ji . The parameter v represents the coefficient of j¼1 N 1 freight car movement in an empty state in relation to the total number of freight cars. The value of this coefficient has been calculated for all cars on the considered network. The parameters eji and lji represent the unit cost of empty and loaded movement between stations j and i. In order to define the station’s freight car holding costs, the average level of freight cars must be assessed. Under the assumption that the number of freight cars linearly decreases for ηi, from value at the beginning of a planning horizon, on γ i, to the value at the end, the average holding cost of freight cars can be expressed as !! N X γ i + ηi (5.110) hi  E Dij 2 j¼1 The parameter hi represents the unit cost of freight car holding in station i. Dij represents the average daily demand between stations i and j per period which has been calculated in relation to P  1 periods of the considered planning horizon.

Fuzzy random model for rail freight car fleet management based on optimal control theory

151

For defining the costs of unsatisfied car orders the following function has been introduced: 8 X XN   < 0, Dij < γ i Xj X δp γ ¼ D (5.111) ij i j¼1 : D  γ , D  γ ij ij i i j j Therefore, the expected total demand for freight cars which has been transferred to the next period is ! ! ! ð∞ X X X Dij  γ i f Dij d Dij δp ¼ γi j j j ð∞ X ! X ! Dij f Dij  γ i Y ðγ i Þ (5.112) ¼ γi

j

j

where Y(γ i) is a complementary cumulative distribution function. Let us assume that the demands for freight cars between the stations of the considered network are independent random variables subjected to the normal distribution law ( Jordan, 1982). Based on the central limit theorem this sum of the Gaussian type random variables is also approximately subjected to the normal distribution law. Let us denote this sum by νi: νi ¼ Di1 + Di2 + ⋯ + Dij + ⋯ + DiN Since Dij(j ¼ 1, 2, …, N) and N(μij, σ ij) are normally distributed, for νi, νi N(μi, σ i), also holds where μi ¼ μi1 + μi2 + ⋯ + μiN σ 2i ¼ σ 2i1 + σ 2i2 + ⋯ + σ 2iN Therefore, the function of the total expected costs in station i during the planning horizon has the following form: !! N  N X  Ti 1 X ηi + γ i + E ½C ðγ i , ηi Þ ¼ E νeji + ð1  νÞlji + hi Dij ηi N  1 j¼1 2 j¼1 ! "ð # ∞X X Ti XN N + p Df Dij  γ i Y ðγ i Þ j¼1 ij j¼1 ij ηi γi j (5.113) However, during everyday rail traffic operation, the stochastic estimation of demand for freight cars is often biased due to the unstable transport

152

Optimization models for rail car fleet management

demand. For this reason, actual demand for freight cars on railway should be considered in the vicinity of the estimated value with fuzzy characteristics. Therefore, the concept of fuzzy sets is applied in order to extend the presented stochastic freight car inventory model by a fuzzy random demand variable (Chang et al., 2006; Kwakernaak, 1979; Lin, 2008). Since the freight P car  dispatcher cannot estimate the future demand as an exact value E j Dij ¼ μi , it is much easier to estimate it by an interval [μi  Δ1, μi, μi + Δ2]. Consider now a triangular fuzzy number which represents a fuzzy expectation of the fuzzy random demand, ! X fij ¼ μi ¼ ðμi  Δ1 , μi , μi + Δ2 Þ D E j

where Δ1 and Δ2 represent parameters that are determined by the freight car dispatcher. By applying the signed distance method for the defuzzification of P  f E D (Yao and Wu, 2000; Chang et al., 2004, 2006; Bjork, 2009, j ij 2012) we obtain d E

X j

! !

" ! +# ð 1 " " X ! # # X fij fij +E dα D D E 0 j j α α ! X fij + Δ2  Δ1 D ¼E 4 j

fij , 0 ¼ 1 D 2

(5.114) The total demand for freight cars in station PN the period P  1 days P i during e of the planning horizon is denoted as Tei ¼ P1 n¼1 j¼1 Dij ðnÞ, i ¼ 1,2,…, N , so the fuzzy expectation of the total demand can be represented as    tot  tot ei ¼ μtot E T i ¼ μi  Δ3 , μi , μi + Δ4 Using the signed distance fuzzy expectation can be defuzzified as follows: ð h  + io     1 1 n h  i   Δ4  Δ3 e E Tei α + E Tei α dα ¼ E Tei + d E Ti , 0 ¼ 4 2 0 (5.115) Now, it is necessary to estimate fuzzy P expectation of fuzzy stochastic fij as an average fuzzy daily e shortage quantity. Let us define Di ¼ j D demand between station i and all other stations on the railway network. Thus, in accordance with the exact case, where Si ¼ Di  γ i, a fuzzy random ei e variable Sei ¼ D γ i can be defined as

Fuzzy random model for rail freight car fleet management based on optimal control theory

153

ei ðdi ÞðÞeγ i ¼ ðdi  γ i  Δ1 , di  γ i , di  γ i + Δ2 Þ Sei ðdi Þ ¼ D In this case eγ i is a fuzzy point. Therefore, the membership function Sei ðdi Þ is 8 t  ðdi  γ i  Δ1 Þ > > di  γ i  Δ1  t  di  γ i > < Δ1 μe ðtÞ ¼ ðdi  γ i + Δ2 Þ  t (5.116) Si ðdi Þ > , d  γ  t  d  γ + Δ i i 2 > i i > Δ2 : 0, in contrary α  cutSei ðdi Þ is [Si(di)L(α), Si(di)U(α)] ¼ [di  γ i  Δ1 + αΔ1, di  γ i + Δ2  αΔ2] for α 2 [0, 1]. Since Sei ðdi Þ belongs to a family of fuzzy sets over R, the estimation of expectations of the fuzzy stochastic variable Sei in the fuzzy sense is     ei ðÞeγ i ¼ E Sei ¼ 1 E D 2

ð1 ð∞



0 γ + Δ1

 Sei ðdi ÞL ðαÞ + Sei ðdi ÞU ðαÞ fDi ðdi Þd ðdi Þdα

i  ð 1 ∞ 1 2ðdi  γ i Þ + ðΔ2  Δ1 Þ fDi ðdi Þdðdi Þ ¼ 2 γ i + Δ1 2 ð ð∞ 1 ∞ ¼ ðdi  γ i ÞfDi ðdi Þdðdi Þ + ðΔ2  Δ1 ÞfDi ðdi Þd ðdi Þ 4 γ i + Δ1 γ i + Δ1 ð γ i + Δ1 ðdi  γ i ÞfDi ðdi Þd ðdi Þ ¼E ðDi  γ i Þ 

γ

1  ðΔ1  Δ2 Þ 4

ð∞i γ i + Δ1

fDi ðdi Þdðdi Þ (5.117)

Let us consider now Eq. (5.113) and apply the fuzzy stochastic variable for the average daily demand Di and total demand Ti. By changing the crisp ei ðÞe stochastic variable Si ¼ Di  γ i to a fuzzy stochastic variable Sei ¼ D γ i , the total expected cost in the fuzzy sense for station i for the entire planning horizon is obtained as follows:

N   E ðTi Þ Δ4  Δ3 X + νeji + ð1  νÞlji E ½C ðγ i , ηi , Δ1 , Δ2 , Δ3 , Δ4 Þ ¼ 4ηi ηi j¼1

N

η +γ Δ2  Δ1 E ðTi Þ Δ4  Δ3 X + pij + +hi i i  EðDi Þ  2 4 4ηi ηi j¼1 " # ð γ i + Δ1 ð∞ 1 E ðDi  γ i Þ  ðdi  γ i ÞfDi ðdi Þd ðdi Þ  ðΔ1  Δ2 Þ fDi ðdi Þdðdi Þ 4 γ i + Δ1 γi (5.118)

154

Optimization models for rail car fleet management

The demand for freight cars Di has a Gaussian probability density function fDi with finite mean μi and standard deviation σ i and γ i such that



ð∞ γ i  μi γ i  μi + ðμi  γ i ÞΦ E ðDi  γ i Þ ¼ ðdi  γ i ÞfDi ðdi Þd ðdi Þ ¼ σ i ϕ σi σi γi (5.119) 



γ  μi γ + Δ1  μi ϕ i ðdi  γ i ÞfDi ðdi Þd ðdi Þ ¼ σ i ϕ i σ σi γi 



i



γ i  μi γ i + Δ1  μ i γ i + Δ1  μi γ i  μi +μi Φ Φ  γiΦ + γiΦ σi σi σi σi (5.120)

ð∞ γ + Δ1  μi (5.121) fDi ðdi Þd ðdi Þ ¼ 1  Φ i σi γ i + Δ1

ð γ i + Δ1

where fZ(z) is the probability density function of the standard normal stok2 Ð ffi e 2 and Φ(k) ¼ ∞ chastic variable Z, and ϕðkÞ ¼ p1ffiffiffi k ϕ(x)dx are comple2π mentary cumulative distribution function. Thus, Eq. (5.114) now becomes E ½C ðγ i , ηi , Δ1 , Δ2 , Δ3 , Δ4 Þ ¼



 N  E ðTi Þ Δ4  Δ3 1 X + νeji + ð1  νÞlji ηi N  1 j¼1 4ηi

ηi + γ i Δ2  Δ1 E ðTi Þ Δ4  Δ3  E ðDi Þ  + + ηi 2 4 4ηi  



N X γ  μi γ  μi + ðμi  γ i ÞΦ i pij σ i ϕ i σ σi i j¼1

+ hi





γ  μi γ + Δ1  μi  σi ϕ i ϕ i σi σi 



γ  μi γ + Δ1  μi  μi Φ i Φ i σi σi



γ + Δ1  μi γ  μi + γi Φ i Þ  γiΦ i σi σi 

 1 γ + Δ1  μi  ðΔ1  Δ2 Þ 1  Φ i 4 σi (5.122)

The problem now is to find γ i and ηi so as to minimize the function of the total estimated inventory costs in the fuzzy sense. The set of necessary conditions for determining the optimal level of freight cars in the station and the

Fuzzy random model for rail freight car fleet management based on optimal control theory

155

optimal quantity of ordered cars from the remaining stations can be determined by solving the following two equations: !  N  ∂C ðγ i , ηi Þ EðTi Þ Δ4  Δ3 1 X ¼ + νe + ð 1  ν Þl ji ji ∂ηi N  1 j¼1 η2i 4η2i ! 

hi EðTi Þ Δ4  Δ3 XN γ i  μi + p 2γ Φ + + ij i j¼1 2 σi η2i 4η2i



γ + Δ1  μi Δ  Δ2 γ + Δ1  μ i Δ  Δ2  μi + γ i + 1 + 1 Φ i ¼0 σ i ϕ i σi 4 σi 4 (5.123)

N 

∂C ðγ i , ηi Þ hi EðTi Þ Δ4  Δ3 X γ γ  μi ¼ + + pij 2 i ϕ i 2 4ηi σ σi ∂γ i ηi i j¼1



2γ i + Δ1 Δ1  Δ2 γ + Δ1  μi (5.124) ϕ i  + σi 4σ i σi



γ i + Δ1  μi γ i  μi  2Φ ¼0 +Φ σi σi

However, before solving this system of nonlinear equations it is necessary to prove that the function of the total expected costs in a station is strictly convex and therefore has a unique absolute minimum. This function will be convex only if the Hessian matrix H is positive-semidefinite for every γ i and ηi. Let us find the elements of this matrix: 2 2 3 ∂ C ðγ i , ηi Þ ∂2 C ðγ i , ηi Þ 6 ∂γ 2i ∂γ i ∂ηi 7 7 (5.125) H ¼6 4 ∂2 C ðγ , η Þ ∂2 C ðγ , η Þ 5 i i i i ∂ηi ∂γ i ∂η2i

N  2

∂2 C EðTi Þ Δ4  Δ3 X 4σ i  2γ i ðγ i  μi Þ γ i  μi ¼ + pij ϕ 4ηi σi ∂γ 2i ηi σ 3i j¼1 



 (5.126) 8γ i + 5Δ1  Δ2 3 γ i + Δ1  μi ðγ i + Δ1  μi Þ  ϕ + 4σ 3i σi σi ! N   ∂2 C E ðTi Þ Δ4  Δ3 1 X ¼2 + νeji + ð1  νÞlji 2 3 3 N  1 j¼1 ∂ηi ηi 4ηi ! 

N E ðTi Þ Δ4  Δ3 X γ i  μi 2 + p 2γ Φ ij i σi η3 4η3i i

j¼1

γ i + Δ1  μ i Δ1  Δ2 γ i + Δ1  μ i Δ1  Δ2 σ i ϕ  μi + γ i + + Φ σi 4 σi 4 (5.127)

156

Optimization models for rail car fleet management

! 

N ∂2 C E ðTi Þ Δ4  Δ3 X γ i  μi ¼ + p 2Φ ij ∂γ i ∂ηi σi η2i 4η2i j¼1



2γ i + Δ1 Δ1  Δ2 γ + Δ1  μ i + ϕ i + σi 4σ i σi



2γ i γ i  μi γ i + Δ1  μi  ϕ Φ σi σi σi

(5.128)

The determinant of the main minor of the first-order Hessian matrix is given by

N  2

E ðTi Þ Δ4  Δ3 X 4σ i  2γ i ðγ i  μi Þ γ i  μi + pij ϕ 4ηi σi ηi σ 3i j¼1 (5.129) 



 8γ i + 5Δ1  Δ2 3 γ i + Δ1  μ i >0 ðγ i + Δ1  μi Þ  ϕ + 4σ 3i σi σi which is necessary for the convexity requirement. After the rearrangement    of this condition, the same will be true only if ðγ i  μi Þ ϕ γi + Δσi1 μi          γ i + Δ1 μi γ i μi γ i + Δ1 μi 1 i ϕ γi μ Þ + Δ > 0 and  3ϕ ϕ 4ϕ 1 σi σi σi σi σi   γ i + Δ1 μi 5Δ1 Δ2 + 4σ 3 ðγ i + Δ1  μi Þϕ > 0, so by selecting suitable values for σi i

Δ1 and Δ2 this condition could be satisfied. If we analyze the determinant of the principal minor of the second order:

N  2 ∂2 C ∂2 C ∂2 C ∂2 C E ðTi Þ Δ4  Δ3 2 X 4σ i  2γ i ðγ i  μi Þ  ¼ 2 + pij 4ηi ∂γ 2i ∂η2i ∂γ i ∂ηi ∂ηi ∂γ i ηi σ 3i j¼1



γ i  μi ð8γ i + 5Δ1  Δ2 Þðγ i + Δ1  μi Þ  12σ 2i γ i + Δ1  μi ϕ + ϕ σi 4σ 3i σi ( 

N  N  X 1 X γ  μi  νeji + ð1  νÞlji  pij 2γ i Φ i σi N  1 j¼1 j¼1



γ + Δ1  μ i Δ1  Δ2 γ + Δ1  μ i  μi + γ i + Φ i  σiϕ i σi 4 σi

X 

 N Δ1  Δ2 E ðTi Þ Δ4  Δ3 γ i  μi 2γ i + Δ1 + + + pij 2Φ  4 4ηi σi σi ηi j¼1







2 Δ1  Δ2 γ + Δ1  μi 2γ γ  μi γ + Δ1  μ i ϕ i  iϕ i Φ i + 4σ i σi σi σi σi (5.130)

Fuzzy random model for rail freight car fleet management based on optimal control theory

157

Expression (5.130) will be greater than zero when the suitable values for Δ1 and Δ2 are selected, so as to satisfy the following inequalities: 5Δ1 > Δ2

(5.131)

γ i + Δ1 > μi (5.132)



γ  μi γ + Δ1  μi 4ϕ i > 3ϕ i (5.133) σi σi



γ i + Δ1  μ i γ i  μi > ðγ i  μi Þϕ (5.134) ðγ i + Δ1  μi Þϕ σi σi

γ i + Δ1  μ i Δ1  Δ2 σiϕ > (5.135) σi 4





Δ1  Δ2 γ i + Δ1  μi γ i  μi γ i + μi + > 2γ i Φ (5.136) Φ 4 σi σi







2γ i + Δ1 Δ1  Δ2 γ i + Δ1  μi 2γ i γ i  μi γ i  μi + ϕ ϕ  2Φ σi 4σ i σ i

σi

σ i

σ i

2γ i γ i  μi γ i + Δ1  μi 1 2 γ i + Δ1  μi 2 γ i  μi > ϕ Φ Φ + + 4Φ 2 σi σi σi σi σi (5.137)

From the set of programmatically generated values for Δ1 and Δ2 which satisfy conditions (5.131)–(5.137) freight car managers may choose the set of suitable values and estimate the demand for freight cars in a proper way, based on their own experience. Thus, for the given Ti,Dij,pij, eij, lij, σ i, μi, Δ1, Δ2, Δ3, Δ4, the following algorithm can be defined for determining the optimal level of freight car inventories and the optimal reordering level of freight cars in station i during the period n. This algorithm is based on a numerical approach for solving the system of Eqs. (5.123) and (5.124), proposed by Hadley and Whitin (1963). The steps of the algorithm are as follows: Step 0. Find the initial, lowest value of ηi which is ηi∗ and let γ (0) i ¼ 0. Set k ¼ 1 and go to step k: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u XN   u2Ti νe + ð 1  ν Þl ji ji t j¼1 ð1Þ ηi ¼ η∗i ¼ (5.138) ðN  1Þhi (k) (k) (k1) stop, Step k. Based on η(k) i determine γ i from Eq. (5.124). If ηi ηi (k) then the optimal solution is γ i∗ ¼ γ (k) i and ηi∗ ¼ ηi . Otherwise, substitute

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Optimization models for rail car fleet management

(k) η(k) i in Eq. (5.123) and compute γ i . Set k ¼ k + 1 and repeat step k until convergence. Since the point (γ i∗, ηi∗) represents a global minimum of the total cost function, it can be approximated by Taylor’s polynomial by an expansion to the second term:

∂C ∂C 1 ∂2 C ðγ  γ ∗ Þ + ðη  η∗ Þ + ðγ  γ ∗ Þ2 ∂γ ∗ ∂η∗ 2 ∂γ ∗ ∂2 C 1 ∂2 C + ðγ  γ ∗ Þðη  η∗ Þ + ðη  η∗ Þ2 ∗ ∂γ∂η 2 ∂η (5.139)

C ðγ i , ηi Þ C ðγ ∗ , η∗ Þ +

For the selected quadratic optimality criteria in the statement of the basic task, it is required to determine the coefficients of the quadratic members γ i and ηi. Thus, a cost function of the following form exists (Monks, 1982): 1 1 (5.140) C ¼ π s γ 2 + π h γη + π u η2 2 2 Coefficients π s, π h, and π u are determined based on expression (5.140). Therefore, C ðγ i , ηi Þ C ðγ ∗ , η∗ Þ

N ( 2

4σ i  2γ i ðγ i  μi Þ γ i  μi 1 E ðTi Þ Δ4  Δ3 X + pij ϕ + ½ 2 ηi 4ηi σi σ 3i j¼1 " + 8
m, i 6¼ j. e yjim ðnÞ: fuzzy number of empty cars dispatched from station j during the period m, and arrived in station i in period n, i ¼ 1, …, N; j ¼ 1, …, N; n > m, i 6¼ j eðnÞ ¼ A π s1 ,e π s2 , …,e π s2 ,e π sN ,…,e π sN ,e π s1 , …,e π s1 ,…,e π s2 ,…,e π s2 ,…,e π sN ,…,e π sN ,2q diagðe π s1 , …,e P P X X  se a21s ðn + s  1Þ,…, 2q se aN 1s ðn + s  1Þ,…, 2q    

s¼1 P X

s¼1

se a21s ðn + s  P + 1Þ, …, 2q 

s¼P1 P X

P X

se aN 1s ðn + s  P + 1Þ,…, 2q

s¼P1

P X sbe21s ðn + s  1Þ,…, 2q sbeN 1s ðn + s  1Þ,…, 2q

s¼1 P X

s¼1

s¼P1 P X

se a1Ns ðn + s  1Þ,…, 2q

s¼1

P X

sbe21s ðn + s  P + 1Þ, …, 2q P X

sbeN 1s ðn + s  P + 1Þ, …,2q

s¼P1

sbeN 1, Ns ðn + s  P + 1ÞÞ

s¼P1

(5.147)

e is a fuzzy diagonal quadratic matrix whose dimensions Therefore, matrix A are [N(2N  1) + 2N(N  1)(P P 1)]  [N(2N  1) + 2N(N P  1)(P  1)] and which has components e π si , q Ps¼m se aijs ðn + s  mÞ and q Ps¼m sbeijs ðn + s  mÞ. e is [2N(N  1)]  [2N(N  1)] a fuzzy diagonal quadratic matrix which B is positive definite with components e π ui . eðnÞ ¼ pr B  u  diag e π 1 , …, e π u1 , e π u2 , …, e π u2 , …, e π uN , …, e π uN , e π u1 , …, e π u1 , e π u2 , …, e π u2 , …, e π uN , …, e π uN

Fuzzy matrix Le is a matrix of dimensions [N(2N  1) + 2N(N  1) (P  1)]  [2N(2N  1)] with components e π hi .

5.2.7 Choosing the components of the fuzzy e ðPÞ weighting matrix Γ The value of cars at the end of planning may influence the sensitivity of decisions about their sending during the routing within the given time horizon. In case these decisions are not of critical importance for the supply of cars in

Fuzzy random model for rail freight car fleet management based on optimal control theory

161

stations, it is not necessary to take into account this value. In the model, this results from the assumed cyclical nature of demand on a periodical basis. A fuzzy stochastic newsboy model is used for the computation of come ðP Þ of the terminal term. The model that ponents of the weighing matrix Γ includes one variable (system’s state) and a time period will be represented by a second-order gradient method using the quadratic and linear members of state variables. It should be noted that this interval is often longer than one day, which is a normal length between periods 0, 1, …, P  1. In this model, the last period of the planning horizon lasts one day as well, taking into account that the demand is assumed as cyclically repeatable on a periodic base. Based on this model, the optimal level of freight car inventories can be defined for a single period in a certain station under the uncertain demand and with the objective of minimizing the expected total cost. The coefficients of the weighting matrix of the terminal can be found by a single Pterm N period newsboy inventory model. Thus, if D j¼1 ij < γ i then it holds that PN γ i  j¼1 Dij is theP number of cars in the station P during a single period. OthN erwise, a shortage N D  γ will exist in case i j¼1 ij j¼1 Dij > γ i . The expected cost for the last period E[C(γ i)] can be expressed as ð γi XN ð ∞ E ½C ðγ i Þ ¼ hi ðγ i  Di Þf ðDi ÞdðDi Þ + p ðDi  γ i Þf ðDi Þd ðDi Þ j¼1 ij γi

0

(5.148)

PN

where Di ¼ j¼1Dij N(μi, σ i). Assume again, as in the case of the multiperiod inventory model, that there is an average daily demand as a normally distributed fuzzy random variable N ðμei , σ i Þ. The expected cost for the last period is N   X   ei + ei eγ i ¼ hi ðγ i  μi Þ γi  D pij E D E ½C ðγ i Þ ¼ hi E e j¼1

!



(5.149) N X γ i  μi γ i  μi + σiϕ pij ðμi  γ i ÞΦ + hi + σi σi j¼1     ei e ei and E D γ i represent a fuzzy random holding number where E eγ i  D and fuzzy random shortage number of freight cars, respectively, eγ i represents a fuzzy point. The estimate of expectation of the fuzzy random shortage in a fuzzy sense is



  γ + Δ1  μ i γ  μi ei e E D  2γ i Φ i + μi + γ i γi ¼ σiϕ i σi σi

(5.150) Δ1  Δ2 γ i + Δ1  μi Δ1  Δ2  Φ + 4 σi 4

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Optimization models for rail car fleet management

By analyzing the relationship between the expected fuzzy random number of freight cars on holding and the expected fuzzy random number of cars in shortage, it can be noticed that     e i ¼ γ i  μi + E D ei e E e γi  D γi The estimate of the total expected cost in station i during the planning horizon in the fuzzy sense is !

N X γ i + Δ1  μi E ½C ðγ i , Δ1 , Δ2 Þ ¼ hi ðγ i  μi Þ + hi + pij σ i ϕ σi j¼1



γ  μi Δ1  Δ2 γ + Δ1  μi Δ1  Δ2 + μi + γ i +  Φ i  2γ i Φ i σi 4 σi 4 (5.151) The necessary condition for determining the optimal number in the station is ∂E½C ðγ i , Δ1 , Δ2 Þ ¼0 ∂γ i !



N X 2γ i γ i  μi γ i + Δ1  μi +Φ pij ϕ hi + hi + σi σi σi j¼1



γ i  μi 8γ i + 5Δ1  Δ2 γ i + Δ1  μi  ¼0 ϕ 2Φ σi 4σ i σi

(5.152)

where Φ(k) represents a complementary cumulative distribution function. The second derivation of this function is ! N X ∂2 E ½C ðγ i , Δ1 , Δ2 Þ ð8γ i + 5Δ1  Δ2 Þðγ i + Δ1  μi Þ  12σ 2i ¼ h + p i ij 4σ 3i ∂γ 2i

j¼1 2

γ i + Δ1  μi 4σ i  2γ i ðγ i  μi Þ γ i  μi ϕ + ϕ σi σi σ 3i (5.153) Expression (5.149) will be greater than zero if the following inequalities are satisfied: ð8γ i + 5Δ1  Δ2 Þðγ i + Δ1  μi Þ > 12σ 2i

(5.154)

1 σ 2i > γ i ðγ i  μi Þ 2

(5.155)

Fuzzy random model for rail freight car fleet management based on optimal control theory

163

By satisfying these conditions, the function of the total costs is strictly concave and its absolute minimum is unique. The optimal freight car inventory level γ ∗i for the last period of the planning horizon can be determined by solving Eq. (5.152). In the vicinity of the minimum point, the total cost function can be approximated by Taylor’s polynomial by an expansion to the second term: C ðγ i Þ C ðγ ∗i Þ +

∂C 1 ∂2 C | ðγ i  γ ∗i Þ + ðγ i  γ ∗i Þ 2 | ðγ i  γ ∗i Þ ∂γ i γ∗ 2 ∂γ i γ∗ i

(5.156)

i

Since there is only a need for the coefficient of the quadratic term: ! ∗

N X ð8γ ∗i + 5Δ1  Δ2 Þðγ ∗i + Δ1  μi Þ  12σ 2i γ i + Δ1  μ i pij ϕ gi ¼ h i + 4σ 3i σi j¼1

4σ 2i  2γ ∗i ðγ ∗i  μi Þ γ ∗i  μi , i ¼ 1,2,…, N + ϕ σi σ 3i (5.157) Coefficients of the fuzzy state vector at the end of the planning horizon P describe the costs of cars in the station and cars in transit. The first category of these costs has been estimated by the preceding estimation whereas the costs of freight cars in transit are assumed to be a linear function of the fuzzy number of cars on a considered route. e ðP Þ is a [N(2N  1) + 2N(N  1)(P  1)]  [N(2N  1) + 2N(N  1) Γ (P  1)] fuzzy positive-semidefinite matrix in the quadratic term of the state vector for the last period of the planning horizon: P X e ðP Þ ¼ diagðe Γ g1 ,…,e g1 ,…,e gN ,…,e gN ,e g1 ,e g2 , …,e gN ,2q se a21s ðP + s  1Þ,  ,2q s¼1

P P P X X X se aN 1s ðP + s  1Þ,2q se a21s ðs + 1Þ,  ,2q se aN 1s ðs + 1Þ,2q  s¼1

s¼1

s¼P1

P P P X X X  sbe21s ðP + s  1Þ,  ,2q sbeN 1s ðP + s  1Þ,2q sbeN 1s ðs + 1Þ,  ,2q s¼1

s¼1

s¼1

s¼P1

P P X X se a1Ns ðP + s  1Þ,2q sbeN 1, Ns ðs + 1ÞÞ: 

s¼P1

(5.158)

The whole procedure for rail freight car fleet sizing and allocation can be summarized as follows. After the estimation of the current state of the rail

164

Optimization models for rail car fleet management

freight car flows, the fuzzy mean and variance of the rail freight car demand can be estimated. Then, there is a need to make a fuzzy forecast of the number of rail freight cars in each station. For this purpose, a fuzzy double exponential smoothing method is applied. The components of fuzzy weighted e B, e and Le are then determined, and the dynamic programming matrices A, recursive algorithm is applied for solving the stated fuzzy stochastic sequential optimization problem. At the end, the defuzzification is performed to find the crisp results which best comprehend the information contained in the fuzzy outputs.

5.3 Numerical experiments In this chapter, the fuzzy stochastic model is applied for solving the problem of freight car fleet sizing and allocation. Experiments are conducted on a set of hypothetical instances. The computational approach for all operations with fuzzy numbers is performed in the Matlab programming package (MathWorks Inc., 2012) using Intlaba (INTerval LABoratory) programming package for the computation of inverse fuzzy matrices. At the end of the section, a comparative analysis of the fuzzy random and random approach for different network configurations and time horizons is given. Let us consider a network composed of four stations on a planning horizon of 4 days. Each day represents an independent decision-making period, so that P ¼ 4. This fuzzy stochastic model includes one type of freight cars. Table 5.1 presents input data about unit cost of movement of freight cars in an empty state, unit cost of loaded freight car movement, and unit car shortage costs. The fuzzy proportions of loaded and empty car arrivals are provided in Table 5.2. The coefficient of empty movement is 0.35, whereas the quadratic term in unit ownership costs is 10 monetary units (m.u). Transport demand is described with a fuzzy normal probability distribution for all periods during the planning horizon and all combinations of origin-destination stations (Table 5.3). SD represents a standard deviation of the fuzzy number of freight cars. Table 5.4 presents the values of unit holding costs and fuzzy estimation of the initial number of cars in stations on the considered rail network. In Table 5.5 unsatisfied demand is provided from previous periods by origin-destination pairs of station. a

Prof. Dr. Siegfried M. Rump, Institute for Reliable Computing, Hamburg University of Technology, Schwarzenbergstr. 95, 21,071 Hamburg, Germany

Fuzzy random model for rail freight car fleet management based on optimal control theory

165

Table 5.1 Cost parameters of fuzzy stochastic freight car fleet sizing and the allocation problem.

Origin

Destination

Unit cost of empty trip (m.u./car)

1 1 1 2 2 2 3 3 3 4 4 4

2 3 4 1 3 4 1 2 4 1 2 3

500 500 600 600 650 700 650 700 750 700 650 650

Unit cost of loaded trip (m.u./car)

Unit shortage cost (m.u./car)

250 250 350 350 350 400 400 450 500 450 500 450

130 130 130 100 100 100 100 100 100 100 100 100

Table 5.2 Fuzzy proportions of arrivals of loaded and empty freight cars during the period. n

Route 1–4; Route 2–3; Route 2–4;

Route 1–2; Route 4–3; Route 1–3;

0 1 2 3

(0, 0, 0) (0.60, 0.61, 0.62) (0.35, 0.36, 0.37) (0.02, 0.03, 0.04)

(0, 0, 0) (0.50, 0.51, 0.52) (0.28, 0.29, 0.30) (0.19, 0.20, 0.21)

The fuzzy number of loaded and empty freight cars which were in transit at the beginning of the planning horizon is provided in Tables 5.6 and 5.7. Elements of fuzzy weighting matrices A(0), A(1), A(2), A(3), B, and L in the objective functional are determined based on the relations of the fuzzy stochastic inventory model. For all stations, the number of freight cars in inventory and the number of ordered cars is determined after three iterations following the algorithm described in Section 5.2.6 (γ ∗1 ¼ 74,η∗1 ¼ 94;γ ∗2 ¼ 58,η∗2 ¼ 88,γ ∗3 ¼ 58,η∗3 ¼ 68;γ ∗4 ¼ 49,η∗4 ¼ 74). Based on the fuzzy random newsboy model, the number of the freight cars in inventories for the last period of the planning horizon (γ ∗1 ¼ 75,γ ∗2 ¼ 58,γ ∗3 ¼ 58,γ ∗4 ¼ 49) are determined. The fuzzy forecast of the actual number of freight cars per stations is presented in Table 5.8.

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Optimization models for rail car fleet management

Table 5.3 Daily demand between stations on a considered part of the railway network. Average fuzzy stochastic daily demand (SD) Origin

Destination

1

2

3

4

1

2

1

3

1

4

2

1

2

3

2

4

3

1

3

2

3

4

4

1

4

2

4

3

(14, 15, 20) (12) (18, 19, 23) (13) (19, 20, 24) (13) (18, 19, 22) (5) (15, 16, 18) (5) (21, 23, 25) (6) (18, 19, 22) (6) (15, 16, 18) (6) (21, 23, 25) (6) (15, 16, 19) (6) (10, 11, 15) (3) (18, 19, 22) (5)

(20, 21, 26) (12) (15, 16, 22) (12) (20, 21, 26) (11) (16, 17, 22) (5) (24, 25, 28) (6) (15, 16, 18) (5) (16, 17, 22) (5) (24, 25, 28) (6) (15, 16, 18) (5) (21, 22, 24) (5) (10, 14, 17) (4) (13, 15, 19) (5)

(19, 20, 26) (12) (25, 26, 32) (13) (14, 15, 20) (11) (14, 16, 19) (4) (13, 16, 22) (4) (26, 27, 31) (6) (14, 16, 19) (4) (13, 16, 22) (4) (26, 27, 31) (6) (12, 14, 20) (4) (15, 16, 20) (4) (21, 22, 25) (6)

(28, 30, 32) (6) (14, 15, 16) (3) (16, 17, 18) (4) (10, 11, 12) (2) (11, 12, 13) (3) (14, 15, 16) (2) (15, 17, 18) (5) (14, 15, 16) (3) (16, 17, 18) (4) (9, 10, 11) (2) (11, 12, 13) (3) (15, 16, 17) (5)

Table 5.4 Unit cost of holding and initial number of freight cars by stations. Station

Holding cost (m.u./car/period)

Initial number of freight cars (SD)

1 2 3 4

15 20 40 30

(15, 16, 17) (4) (10, 13, 15) (5) (9, 10, 11) (4) (10, 12, 13) (3)

The recursive solving approach (5.101)–(5.108) results is applied in an optimal control law for the given rail freight car fleet sizing and allocation system for four stations on a 4-day planning horizon. Obtained results are graphically illustrated in Figs. 5.8 and 5.9.

Fuzzy random model for rail freight car fleet management based on optimal control theory

167

Table 5.5 Fuzzy unsatisfied demand from the previous period. Origin

Destination

Unsatisfied demand

1 1 1 2 2 2 3 3 3

2 3 4 1 3 4 1 2 4

(2, (3, (3, (1, (2, (1, (2, (2, (3,

3, 4, 5, 2, 3, 3, 3, 4, 5,

4) 5) 6) 3) 4) 5) 4) 5) 6)

Table 5.6 Number of loaded cars dispatched before the beginning of the cycle. Origin

Demand

One period

Two periods

1 1 1 2 2 2 3 3 3 4 4 4

2 3 4 1 3 4 1 2 4 1 2 3

(15, 16, 17) (10, 11, 12) (10, 11, 12) (10, 11, 12) (11, 12, 13) (8, 9, 10) (9, 10, 11) (11, 12, 13) (9, 10, 11) (7, 8, 9) (9, 10, 11) (13, 14, 15)

(8, 9, 10) (15, 16, 17) (15, 16, 17) (11, 12, 13) (5, 6, 7) (10, 11, 12) (3, 4, 5) (10, 11, 12) (14, 15, 16) (4, 5, 6) (15, 16, 17) (8, 9, 10)

For the proper functioning of the defined system, the required size of freight car fleet (FfS) in the fuzzy sense is (155.8, 204.4, 218.3). The fuzzy optimal value of total costs which corresponds to the required size of the freight car fleet is determined by the substitution of values for the fuzzy control variables in relation (5.55) and it is e J ¼ ð274:645, 306:045, 314:300Þ m.u. With the method of signed distance for defuzzification, crisp value of 195.7 cars has been obtained with a corresponding cost value of 300,259 m.u. For the stochastic linear Gaussian regulator which has been obtained by eliminating fuzziness in the model, the optimal value of the freight car fleet is 232.13 cars with the cost value of 382.467 m.u.

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Optimization models for rail car fleet management

Table 5.7 Fuzzy number of freight cars dispatched before the beginning of the cycle. Origin

Destination

One period

Two periods

1 1 1 2 2 2 3 3 3 4 4 4

2 3 4 1 3 4 1 2 4 1 2 3

(5, 6, 7) (2, 3, 4) (3, 4, 5) (3, 4, 5) (4, 5, 6) (2, 3, 4) (4, 5, 6) (11, 12, 13) (4, 5, 6) (5, 6, 7) (3, 4, 5) (3, 4, 5)

(3, 4, 5) (2, 3, 4) (1, 2, 3) (7, 8, 9) (2, 3, 4) (5, 6, 7) (5, 6, 7) (10, 11, 12) (3, 4, 5) (7, 8, 9) (5, 6, 7) (4, 5, 6)

Table 5.8 Fuzzy forecasted number of freight cars in stations (SD) by period. Period Station

1

1 2 3 4

(23, (23, (19, (15,

2

24, 25, 20, 16,

25) 26) 21) 17)

(3) (3) (2) (5)

(22, (25, (17, (17,

3

23, 36, 18, 18,

24) 28) 19) 19)

(3) (2) (3) (4)

(25, (17, (15, (18,

26, 18, 16, 19,

27) 21) 17) 20)

(2) (4) (4) (3)

5.4 Comparative analysis of the results of the fuzzy random and random model In this section, the results of fuzzy random and random approach for various numerical examples are compared (Table 5.9). Results of fuzzy LQG approach have been converted to corresponding crisp values. From Table 5.9 it can be noticed that in both cases, stochastic and fuzzy stochastic, with increase in the number of stations and number of periods for a constant number of stations, a higher value of freight car fleet is necessary that results in a higher level of costs required for the freight car fleet system functioning. In comparison with the stochastic approach, the fuzzy stochastic analysis gives a higher required fleet size which leads to higher costs. This is a logical consequence of including future uncertainties of freight car requests. Therefore, the difference in the size of freight car fleet and corresponding system’s cost increases with a higher uncertainty level. CPU time for the conducted tests is insignificant.

Fuzzy random model for rail freight car fleet management based on optimal control theory

169

Fig. 5.8 Problem solution: fuzzy control actions.

170 Optimization models for rail car fleet management

Fig. 5.9 Problem solution: fuzzy unmet demand and fuzzy number of cars by stations and periods.

Fuzzy LQG

LQG

Problem

Number of stations

Number of periods P

FS

J (m.u.)

CPU (s)

FS

J (m.u.)

CPU (s)

1 2 3 4 5 6 7

2 2 3 4 4 5 6

4 7 7 4 7 7 7

101.3 139.7 170.3 195.7 253.8 318.6 379.2

210,227 360,789 530,911 752,062 1,145,519 1,545,293 1,796,256

2 2 2 3 3 4 4

83.2 111.8 135.9 172.8 220.5 289.3 350.6

92,000 235,298 361,695 591,944 827,154 1,160,892 1,390,896

1 1 1 2 2 3 3

Fuzzy random model for rail freight car fleet management based on optimal control theory

Table 5.9 Alternative problems and comparison with the stochastic approach.

171

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Optimization models for rail car fleet management

Benefits of the suggested approach in comparison with the modeling based on the stochastic optimal control are based on simultaneous consideration of the fuzziness and randomness of the demand for freight cars which leads to the outcome less sensitive to the variabilities in inputs and which is capable of including imprecision and vagueness in the decision-making process.

CHAPTER 6

Stochastic model for heterogeneous rail freight car fleet management based on the model predictive control Contents 6.1 Discrete time MPC framework for rail freight car fleet sizing and allocation problem 6.2 Design variables 6.3 System performance measure and constraints 6.3.1 System dynamics 6.3.2 Nonnegative control constraints 6.3.3 Nonnegative state constraints 6.3.4 Capacity constraints 6.4 State vector estimation 6.5 Forecasting the state of rail freight cars by ARIMA-Kalman method 6.5.1 Components of weighted matrices A, B, and L 6.5.2 The components of matrix Γ(P) 6.6 MPC controller 6.6.1 Optimization problem 6.6.2 Detailed description of the MPC approach 6.6.3 Numerical experiments

174 175 181 182 183 183 183 184 185 188 196 198 198 199 204

From the literature review, it can be seen that the previous contributions were mainly concentrated on a homogeneous rail freight car fleet and/or empty flows in a static deterministic environment. Namely, the approaches were based on a multiperiod integer linear programming or mixed integer linear programming formulation that maximize the profit or minimize the cost of the rail freight car fleet system and these approaches are valid only if the demands, forecasts, disturbances, and parameters are constant overtime. This is never the case, so it is necessary to find an optimal policy in a dynamic environment. For that purpose, in this chapter, a model predictive control (MPC) framework was developed for obtaining an optimal set of control actions that involve state, control, and capacity constraints. The demand Optimization Models for Rail Car Fleet Management https://doi.org/10.1016/B978-0-12-815154-9.00006-X

© 2020 Elsevier Inc. All rights reserved.

173

174

Optimization models for rail car fleet management

as a stochastic variable and heterogeneous rail freight car fleet with a partial substitutability among various freight car types are considered.

6.1 Discrete time MPC framework for rail freight car fleet sizing and allocation problem MPC represents a multivariable control method that is usually used to optimally control complex systems while explicitly considering constraints. The system dynamics is represented by a discrete time predictive model where the next state is a function of the current state, current demands (disturbance), and the current control vector with constraints. In each time instance, the optimal control problem is solved online based on the current state (at time n ¼ n0) and the predictive demands over a N step finite horizon (at time n ¼ n0, …, n0 + N  1). The result of the optimization is a sequence of control vectors overtime n ¼ n0, …, n0 + N  1 but only the first control vector (at time n ¼ n0) is applied to the system. In the next state (at time n ¼ n0 + 1), the optimal control problem is solved again for the time horizon n ¼ n0, …, n0 + N then only the control vector at time n ¼ n0 + 1 is applied and so forth. In this chapter, a general MPC framework for the simultaneous rail freight car fleet size and allocation problem is considered (Milenkovi
c et al., 2015a, b). The proposed methodology is graphically illustrated in Fig. 6.1. A linear dynamic model of the system considers loaded and empty freight car flows of different types. Partial substitutability between freight car types

Fig. 6.1 Rail freight car fleet size and allocation approach.

Stochastic model for heterogeneous rail freight car fleet management

175

has also been covered by the model. State vector estimation is based on the forecasts of the number of freight cars of each type in each station and the dynamic model of the system. Forecasts are obtained by the autoregressive integrated moving average (ARIMA)-Kalman forecasting method. The Kalman filter method is used for deriving the optimal system state over the prediction horizon. For the evaluation of rail freight car inventories by stations, the stochastic multiperiod economic order quantity (EOQ) model and single period random newsboy model are applied. Based on the optimal state estimate, the objective functional, and constraints, the MPC controller is designed and used for solving the resulting rail freight car fleet sizing and allocation problem. Presented approach demonstrates the applicability of the receding horizon control for determining the size of rail freight car fleet based on establishing a balance between exact cost parameters, the cost of unmet demand on one side, and the sum of car ownership and utilization costs on the other. All these costs are given as input parameters for each freight car type. The mathematical model enables through the vector difference state equation, an adequate presentation of the dynamic nature of freight car demand. The proposed optimal MPC controller will be applied for deriving the optimal rail freight car fleet investment decisions.

6.2 Design variables As in the case of the fuzzy random modeling approach, let’s consider a network composed of a set of stations N. The planning horizon P has been divided into discrete time intervals 0, 1, …, m, …, n, …, P. This model takes into account the possibility of substitution of the desired freight car type by an appropriate alternative. Namely, in some cases, a transport of specific commodities can be performed by more than one type of freight cars. In that case, the demand may be specified for an aggregate car type. The aggregate car type is only used in distribution planning, there are no cars of aggregate car type. The demand for an aggregate car type is fulfilled only by original car types. The aggregation of car types is prespecified, that is, which original car types can be used to satisfy the demand for an aggregate car type ( Joborn, 2001). The total number of freight car types is denoted as Tσ , composed of the original types T(T  Tσ ) and aggregate car types Ta(Ta  Tσ ), respectively. For each, original and aggregate car type, t (t 2 Tσ ) requests for rail transportation service exist between stations

176

Optimization models for rail car fleet management

i and j in period n, Dijt(n). Dijt(n) represents the demand for freight car of type t from station i to station j in period n, and it is assumed to have a stochastic character. The demand generates loaded Fijt(n) as well as empty freight car flows Eijt(n) on the network. Control actions which represent the set of all loaded and empty freight car departures are under the influence of freight car arrival times. As in the case of the fuzzy random model, rather than to model the traveling times directly, the problem is formulated in terms of freight car arrivals. More precisely, given that Fijt(m) freight cars of type t (t 2 T) were dispatched from station i in the period m, in this approach the question is how many of these cars actually arrive at station j in the period n. Therefore, θijt(m, n)[αijt(m, n)] are defined as the proportions of loaded [empty] freight cars of type t dispatched from i to j in the period m which actually arrive in the period n. These proportions are random variables since the information about freight car arrival is not known in advance. In order to provide an adequate response regarding new freight car requests in a very uncertain environment, it is necessary to maintain a pool of freight cars in the stations. In this approach, variable Sit(n) represents the number of freight cars of type t (t 2 T ) present in the station at the end of period n. The system state represents the total number of a certain type t (t 2 T ) of freight cars offered at each station for each day over the planning horizon. In case if insufficient freight cars of type t are available at the station i in period n to meet all demand, some demand will be either backordered or lost from the system. Variable Vijt(n) represents the number of units of demand, or freight car loads of type t (t 2 Tσ ) that remains unmet at the end of the period n. The notation for the model formulation is given in Table 6.1. The basic formulations between the defined quantities can be stated as follows: Sit ðn + 1Þ ¼ Sit ðnÞ +

N X X  X

Fjitt0 ðmÞ  θjit ðm, n + 1Þ + Ejitt0 ðmÞ  αjit ðm, n + 1Þ



j¼1 t0 2T t m m, i, j 2 N; t 2 T; n ¼ 0, …, P  1, i 6¼ j Continued

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Optimization models for rail car fleet management

Table 6.1 Notation for model formulation—cont’d Notation

Definition

αijt(m, n)

Portion of empty cars of type t arrived during the period n, and dispatched during the period from station i to station j, n > m, i, j 2 N; t 2 T; n ¼ 0, …, P  1, i 6¼ j Number of loaded cars of type t dispatched from station i during the period m and arrived at station j during the period n, i, j 2 N ;t 2 T ;t 0 2 T t ;n > m,i 6¼ j Number of empty cars dispatched from station i during the period m and arrived at station j during the period n, i, j 2 N ;t 2 T ;t 0 2 T t ;n > m,i 6¼ j Forecasting error of the number of cars of type t at station i during the period n, i 2 N; t 2 T; n ¼ 0, …, P Ratio of the number of empty cars of type t to the total number of freight cars Random variable describing travelling time for freight cars of type t from station i to station j i, j 2 N; t 2 T; i 6¼ j

xijtt0 m(n)

yijtt0 m(n) Δit(n) ϑt τijt

Objective functional & decision variables

J ϕ1(X(P)) ϕ2[X(n), U(n), n]

Sit(n) Vijt(n) Eijtt0 (n)

Fijtt’(n) γ it

System performance measure, which represents the optimality criterion for dispatching empty and loaded cars. In this paper it is also denoted as the cost functional Function of state vector in the last period of the planning horizon derived from the unit holding cost, unit shortage cost and car ownership cost during travel Function of state and control vectors during the period n; it is derived from the unit holding cost, unit shortage cost, car ownership cost during travel and unit cost of moving empty and loaded cars from station to station Number of empty and loaded cars of type t at station i at the end of the period n, i 2 N; t 2 T; n ¼ 0, …, P Unmet demand for freight car loads of type t between stations i and j over the period n, i, j 2 N; t 2 Tσ ; n ¼ 1, …, P, i 6¼ j Number of empty car flows of type t used to fulfil demand for car type t0 between station i and station j over the period n, i, j 2 N ;t 2 T ;t 0 2 T t ;n ¼ 1, …,P, i 6¼ j Number of loaded car flows of type t used to fulfil the demand for car type t0 between station i and station j over the period n, i, j 2 N ;t 2 T ;t0 2 T t ;n ¼ 1, …,P, i 6¼ j; The average freight car of type t inventory status and shortage at the station i over the planning horizon per period considered, i 2 N; t 2 T;

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179

Table 6.1 Notation for model formulation—cont’d Notation

Definition

ηit

The average number of (loaded and empty) cars dispatched from the station i over the planning horizon per period considered i 2 N; t 2 T;

Vectors and matrices

X(n) U(n) d(n) Z(n) ν(n) Λ(n) G Φ M(0)

R1(n) R2(n) H A(n) B L Γ(P)

State vector with components Sit(n), Vit(n), Dijt(n), xijtt0 m(n), yijtt0 m(n) Control vector with components Eijtt0 (n), Fijtt0 (n) Vector of disturbances Vector of the forecasted number of freight cars by station during the period n ¼ 0, …, P Vector of error in the forecasted number of freight cars by station during the period n ¼ 0, …, P State transition matrix represented in terms of unit matrices, zero matrices and matrices Λ1, Λ2(n), Λ3(n), and Λ4 System control matrix System disturbance matrix Symmetric, positive-semidefinite matrix of covariances of the state vector at the initial moment and of the preceding state estimation for the same period Symmetric, positive-semidefinite matrix of covariances of the random component of demand Symmetric, positive-semidefinite matrix of covariances of white noise in the forecast Matrix in the linear term of the state vector in the expression for the forecasted number of freight cars by stations Positive-definite matrix with components π sit and P P qt Ps¼m saijts ðn + s  mÞ and qt Ps¼m sbijts ðn + s  mÞ, t 2 T Positive-definite matrix with components π uit, t 2 T Matrix with components π hit, and rijtt0 t 2 T, t0 2 Ta Positive-semidefinite matrix in the quadratic term of the state vector in the last period of the planning horizon

Relation (6.1) represents conservation of flow constraints for each type of freight cars that include the effects of uncertain travel times for car movements through random θ and α terms. The number of freight cars of type t in the next period is equivalent to the number from the last period plus loaded and empty freight car flows which will arrive during the n + 1 period minus the dispatched loaded and empty freight car flows of type t used to fulfil the demand for type t as well as the types t0 which can be substituted by cars of type t. Relation (6.2) exists for all original and aggregate car types.

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Optimization models for rail car fleet management

The demand for t0 can be fulfilled by any car types t 2 T t0 that can substitute for the type t0 . Unmet demand in the period n + 1 is equal to the unmet demand from the previous period plus the new demand minus the cars that are used to fulfil demand in the time period n + 1. The last relation reflects the natural assumption of nonnegativity of freight car flows, states of cars in every station, and unmet demands. The demand Dijt(n) (t ¼ 1, …, Tσ ) is modeled as a sum of random (stochastic) and deterministic components. This sequence is assumed to be of Gauss-Markovian character. Considering that the Gaussian form is retained in the case of linear transformation, it can be described by a state vector of a linear dynamic system excited by a purely Gaussian random sequence (uncontrollable system input), with initial Gaussian state vector. A Gaussian pure random process represents the boundary value of a Gauss-Markovian process with a very large covariance and very short correlation time. As the Fourier transformation of the correlation function with respect to time is a constant, its spectrum is white (Bryson and Ho, 1975). That is why a pure Gaussian random process is often referred to as white noise. Let us introduce an auxiliary variable dijt(n) for all car types for which the following holds: dijt ðnÞ ¼ Dijt ðn + 1Þ

(6.4)

dijt ðn + 1Þ ¼ λijt ðnÞλijt ðn  1Þ⋯λijt ð0Þdijt ð0Þ + ωijt ðnÞ, n ¼ 0,1,…, P  1 (6.5) λijt ðnÞ ¼

μijt ðn + 2Þ , n ¼ 0,1, …,P  2 μijt ðn + 1Þ   E ωijt ðnÞ ¼ 0

(6.6) (6.7)

Relations (6.4)–(6.6) enable converting any random process to an equivalent Markov random process by a proper enlargement of the state vector for variable dijt(n). Relation (6.7) represents the expectation of the Gaussian purely random process, which is the random component of demand in this case. In order to determine random variables θijt(m, n) and αijt(m, n), we applied Liou’s numerical procedure for evaluating a transient system response (Liou, 1966). More precisely, a certain period of time must elapse between the dispatch of a freight car from one station and arrival at another and therefore, the system’s response to a specified control action is delayed in time. As the dynamics of car arrivals is a stochastic function of the delayed system response to a specified demand, it is necessary to know its probability density distribution. A number of distributions may be used for modeling uncertainty in rail freight car traveling times (Spiegelman et al., 2010). For the

Stochastic model for heterogeneous rail freight car fleet management

181

purpose of this study, travel times of freight cars are assumed to follow a negative binomial distribution ( Jordan and Turnquist, 1983; Bojovic, 2002). In that context, this distribution can be interpreted for describing the delay time that has to elapse before a car dispatched from station i appears at station j. Let us now reformulate θijt(m, n) and αijt(m, n) as follows: ajit, n + 1m ðnÞ ¼ θjit ðm, n + 1Þ, m < n , n ¼ 0,…, P,t 2 T bjitt0 , n + 1m ðnÞ ¼ αjit ðm, n + 1Þ, m < n , n ¼ 0, …,P,t 2 T

(6.8) (6.9)

By associating another index, the variables that denote the arrivals of loaded and empty cars from preceding time intervals can also be described: xjitt0 , n + 1m ðnÞ ¼ Fjitt0 ðmÞ, m < n , n ¼ 0,…, P,t 2 T, t 0 2 T t yjitt0 , n + 1m ðnÞ ¼ Ejitt0 ðmÞ, m < n , n ¼ 0,…, P,t 2 T, t 0 2 T t

(6.10) (6.11)

A pair of additional variables that are defined as follows: fijtt0 ðnÞ ¼ Fijtt0 ðn + 1Þ, eijtt0 ðnÞ ¼ Eijtt0 ðn + 1Þ, 8i , j, t,t 0 , n

(6.12)

has also been introduced.

6.3 System performance measure and constraints The performance measure in case of stochastic rail freight car fleet sizing and allocation model has the same structure as in the “fuzzy” case, except that it contains only crisp values and it contains cost terms related to different types of freight cars, as well as to substitution possibilities. Therefore, the crisp form of the objective functional looks as follows: J ¼ ϕ1 ½X ðP Þ +

P1 X

ϕ2 ½X ðnÞ, U ðnÞ, n

(6.13)

n¼0

The first term is a cost of the rail freight car fleet system in the last period over the planning horizon which is the function (ϕ1) of the state vector as a result of control actions during the previous periods. It is derived based on the same cost parameters but defined for each type of freight cars independently: the unit cost of holding a freight car of type t for one period in a station i, hit, the unit shortage cost or penalty cost per period for one freight car of unmet demand of type t from station i to station j, pijt, and the daily cost of ownership or leasing a freight car of type t while traveling on route, Qt. The cost component Qt is determined per car per period (Beaujon and Turnquist, 1991). The holding cost for a non-traveling freight car of type t is at least the ownership cost Qt, but in general it would include additional cost associated with storage and management of freight cars in a station.

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Optimization models for rail car fleet management

In this paper, hit is defined as the unit cost of holding a freight car of type t for one period at a location, where hit Qt. The second term represents a cost of rail freight car fleet system in P  1 periods over the planning horizon which is the function (ϕ2) of the state vector X(n) and control vector U(n) and it is derived from the unit holding cost, hit, the unit shortage cost, pijt, the cost of substituting a car of type t0 by a car of type t, rijtt’ (t 2 T ,t 0 2 T t ) and the unit ownership or leasing cost during travel, Qt, and the unit costs of moving empty, ejit, and loaded, ljit, cars from one station to another at the remaining discrete instants. MPC-based formulation contains a set of constraints that restrict the feasibility region.

6.3.1 System dynamics The model is described by a time discrete control system whose states vary according to the following difference equation: X ðn + 1Þ ¼ Λ½n, aðnÞ, bðnÞX ðnÞ + GU ðnÞ + ΦdðnÞ, n ¼ 0,1,…, P  1 (6.14) Therefore, the state in the next period is a function of the state, control, and vector of disturbances d(n). Control vector U(n) contains control actions of empty and loaded freight cars dispatched from station i in the period n for all original and aggregate car types.   fijtt0 ðnÞ U ðnÞ ¼ X , i, j 2 N, t 2 T , t 0 2 T t (6.15) eijtt0 ðnÞ 2NT ðN 1Þ + 2N ðN 1Þ T t0 t’2T a

Matrix Λ[n, a(n), b(n)] is a state transition matrix whose dimensions are  X  ½TN ð2N  1Þ + 2N ðN  1Þ T a + ðP  1ÞðT + T tÞ  a t2T  X  T tÞ   ½TN ð2N  1Þ + 2N ðN  1Þ T a + ðP  1ÞðT + t2T a

Matrix "

G

matrix with dimensions ! !# X a  ½2NT NT ð2N  1Þ + 2N ðN  1Þ T + ðP  1Þ T + Tt t2T a X ðN  1Þ + 2N ðN  1Þ T t Φ is the disturbance matrix defined as t2T a " a

unit

represents

matrix

of

an

input

dimensions

NT ð2N  1Þ + 2N ðN  1Þ

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Stochastic model for heterogeneous rail freight car fleet management

T a + ðP  1Þ T +

X

!!# Tt

 ½NT ð2N  1Þ + 2N ðN  1ÞðT a +

t2T a

ðP  1Þ T +

X

!!# d(n) is the vector of disturbances which represent

Tt

t2T a

a random component of demand for cars during the period n with components ωijt(n). The state vector X(n) is of dimensions ½NT ð2N  1Þ + 2N ðN  1Þ !!# X T a + ðP  1Þ T + and describes the current state in the rail Tt t2T a

freight car system through the number of loaded and empty freight cars in the period n, the empty and loaded flows arriving in all stations from the preceding time periods and unmet demand over the same period for the original and aggregate car types.

6.3.2 Nonnegative control constraints For each period n ¼ 0, 1, …, P  1 and each car type t 2 T, the number of loaded and empty freight car flows is a nonnegative number: U ðnÞ  O2 where O2 represents a 2NT ðN  1Þ + 2N ðN  1Þ

X

(6.16) T t zero matrix.

t2T a

6.3.3 Nonnegative state constraints For each period n ¼ 0, 1, …, P  1 and car type t 2 T, the current state in the rail freight car system must be a nonnegative number: X ðnÞ  O3

(6.17)

"

where O3 represents a

NT ð2N  1Þ + 2N ðN  1Þ T a + ðP  1Þ

T+

X

!!# Tt

t2T a

zero matrix.

6.3.4 Capacity constraints We also impose the constraints in which the number of rail freight cars in each station will be limited by the capacities of stations: DX ðnÞ  K

(6.18)

where D is a matrix ½N  ðNT ð2N  1Þ + 2N ðN  1Þ  a P of dimensions T + ðP  1Þ T + t2T a T t  that contains the length of rail freight cars

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Optimization models for rail car fleet management

of certain types. K represents a column vector of dimensions [N  1] with station capacities or maximum numbers of freight cars on the state in a station during every period of the planning horizon.

6.4 State vector estimation Two components are crucial for the estimation of the state vector—the dynamic model of the freight car fleet system (6.14) as well as the forecasted number of cars of each type t, Zt(n), at discrete time instants. Planning of the freight car inventory level in every station on the network requires forecasting of their levels in future time periods. These values represent specific state observations, X(n). The observation equation in analytical form looks like the following: Zit ðnÞ ¼ Sit ðnÞ + Δit ðnÞ, i 2 N , t 2 T , n ¼ 0, 1,…,P

(6.19)

The first term Sit(n) denotes the forecasted number of freight cars of type t at the station i in the time period n whereas the second term Δit(n) is a random error of the forecasts for the same station and period. Various methods can be applied for deriving the forecasts of the freight car levels. In this chapter, the ARIMA-Kalman forecasting model is developed and applied. The details of this technique follow after this section. The forecasting error is determined as the mean value of the differences between the actual and forecasted value of the number of freight cars by stations. The forecasted number of cars in the initial period is assumed to be equal to the mean value of the initial state vector which is calculated by averaging over all periods for which historical data are available. Having in mind that the process of variation in the status of freight cars by periods is relatively stable, this estimation enables an appropriate initial value. A forecasted value or an observation can be represented as a linear function of the state: Z ðnÞ ¼ HX ðnÞ + vðnÞ, n ¼ 0,1,…, P

(6.20)

where follows that 2

Z ðnÞ ¼ ½Z1 ðnÞ, …, ZT ðnÞT

H ¼ 4 diag½Ht O

 2N ðN 1Þ T a + ðP1Þ

 5, t 2 T

X t2T

T a

vðnÞ ¼ ½v1 ðnÞ, …, vT ðnÞT

3

(6.21) (6.22)

t

(6.23)

Stochastic model for heterogeneous rail freight car fleet management

185

Ht is a matrix of dimensions N  [N(2N  1) + 2N(N  1)(P  1)] and vt(n) is a N-component vector of the error in the forecasted number of freight cars by station during period n. vt(n) is a N-component vector of the error in the forecasted number of freight cars by station during period n. Vector vt(n) is the white noise, which represents the forecasting errors for cars of type t. X(0) and {d(n), v(n)} are independent Gauss’s vectors with statistics: E ½xð0Þ ¼ x^ð0j 0Þ ¼ xð0Þ;E ½dðnÞ ¼ E ½vðnÞ ¼ 0

 cov½xð0Þ ¼ E ½xð0Þ  xð0Þ½xð0Þ  xð0ÞT ¼ M ð0Þ

(6.24) (6.25)

cov½d ðnÞ ¼ R1 ðnÞ

(6.26)

cov½vðnÞ ¼ R2 ðnÞ

(6.27)

E[x(0)], E[d(n)], and E[v(n)] are mathematical expectations of the initial state, state, and measure disturbances, respectively. cov[x(0)], cov[d(n)], and cov[v(n)] are covariances of the initial state, state, and measure disturbances, respectively.  R1 ðnÞ ¼ diag R1t ðnÞ ON ðN 1ÞT a N ðN 1ÞT a R1t0 ðnÞ X  T t , t 2 T , t0 2 T a O2N ðN 1ÞðP1Þ t2T a

is a symmetric, positive-semidefinite matrix of covariances of the random component of demand for the original car types t and aggregate car     P types t0 with NT ð2N  1Þ + 2N ðN  1Þ T a + ðP  1Þ T + t2T a T t     P dimensions.  NT ð2N  1Þ + 2N ðN  1Þ T a + ðP  1Þ T + t2T a T t R2(n) ¼ diag [R2t(n)], t 2 T is a symmetric, positive-semidefinite matrix of covariances of white noise in the forecast with dimensions TN  TN.

6.5 Forecasting the state of rail freight cars by ARIMA-Kalman method The forecasts of the number of rail freight cars on state in each station are obtained by a sophisticated mathematical model based on a combination of the ARIMA forecasting technique and Kalman recursions. More precisely, based on the available historical data ARIMA processes for modeling the state of freight cars have been proposed. The estimated ARIMA model is incorporated into the state-space framework and classical Kalman recursions were applied for forecasting the future values.

186

Optimization models for rail car fleet management

The ARIMA method was introduced by Box and Jenkins (1976). The method represents one of the most frequently used univariate time series modeling tools. ARIMA models are based on the autoregressive (AR) model, moving average (MA), and the combination of AR and MA models (Milenkovi
c et al., 2016). The AR model involves lagged terms of the time series itself, MA model includes lagged terms on the noise or residuals. The first requirement for ARIMA modeling is that the time series data to be modeled is stationary or can be transformed into stationary. The letter “I” (integrated) means that the first-order difference is applied in order to stationarize given time series. The generalized form of an ARIMA(p, d, q) model for a series yl can be written as χ p ðωÞð1  ωÞd yl ¼ ψ q ðωÞεl

(6.28)

where εl is a white-noise sequence χ p ðωÞ ¼ 1  χ 1 ω  χ 2 ω2  ⋯  χ p ωp

(6.29)

is the AR polynomial term of order p ψ q ðωÞ ¼ 1  ψ 1 ω  ψ 2 ω2  ⋯  ψ q ωq

(6.30)

is the MA part of order q. (1  ω)d is the differencing operator of order d used to eliminate polynomial trends. ω is the backshift operator, whose effect on a time series yl can be summarized as ωdyl ¼ yld. The Box-Jenkins method includes three steps for fitting the ARIMA models. These are the identification, estimation, and validation of the model (Box et al., 2008). The first step is generally based on an analysis of the autocorrelation function (ACF) and partial autocorrelation function (PACF) and their comparison with theoretical profiles of these functions in AR, MA, and ARMA processes. The output of the identification step is determined as an appropriate (p, d, q) model structure. The model structure selected in the previous step has to be fitted to the time series and its parameters have to be estimated. This is the essence of the second step and it is done by using the conditional sum of squares or maximum likelihood method. Validation of the selected model is performed within the diagnostic check with the analysis of stationarity, invertibility as well as the presence of redundancy in model parameters. If a selected model fails in a diagnostic check, it is necessary to repeat the whole procedure again. After an appropriate model is found, it can be used for the forecasting purpose (Box et al., 2008).

Stochastic model for heterogeneous rail freight car fleet management

187

Subjectivity is mostly present in the model identification step. The reason for this lies in the fact that this step is essentially based on graphical interpretations of ACF/PACF estimates. To cope with this subjectivity and also to improve the determination the final orders of the ARMA processes, there are a lot of model selection criteria proposed in the literature (De Gooijer et al., 1985). The most frequently used are information criteria, such as Akaike information criterion (AIC), Bayesian information criterion (BIC), and the normalized version of the BIC. These information criteria consist of the natural log of the mean square error (MSE), plus a penalty for the number of parameters being estimated (Yaffee and McGee, 1999): AIC ¼ T ln ðMSEÞ + 2k

(6.31)

BIC ¼ T ln ðMSEÞ + k ln ðT Þ

(6.32)

lnðT Þ (6.33) T where T is the number of observations, k is the number of parameters in the model k ¼ p + q + P + Q + 1, and MSE is the mean squared error. The common procedure often involves estimating the largest model which is assumed to correctly capture the dynamics of a time series and then decreasing its size (dropping the lags) until the minimum value of AIC, BIC, or the normalized BIC is reached. The state-space form provides a unified representation of a wide range of linear Gaussian time series models including ARMA and unobserved component (UC) models (Hindrayanto et al., 2010). Statespace representations of ARMA/ARIMA models are presented in Box et al. (2008) and Brockwell and Davis (2002). The Gaussian state-space model for the Po-dimensional observation sequence y1, …, yPo is given as follows (Durbin and Koopman, 2001): Normalized BIC ¼ ln ðMSEÞ + k

ρl + 1 ¼ Yl ρl + I l ηl , ηl N ð0, M l Þ,l ¼ 1,…,Po

(6.34)

yl ¼ Sl ρl + εl , εl N ð0, V l Þ,ρ1 N ða1 , ΠÞ

(6.35)

where ρl is the state vector, ηl and εl are disturbance vectors, and the system matrices Y l , I l , Sl , M l , and V l are fixed and known but a selection of elements may depend on an unknown parameter vector. Eq. (6.34) is the state transition equation, while Eq. (6.35) represents the observation or measurement equation. The transition matrix Y l determines the dynamic evolution of the state vector. I l represents the state disturbance transform matrix. The

188

Optimization models for rail car fleet management

disturbance vector ηl for the state vector update has the zero mean and variance matrix M l . Matrix Sl links the observation vector yl with the unobservable state vector ρl. The observation disturbance vector εl has the zero mean and variance matrix V l . The observation and state disturbances, ηl and εl, are assumed to be serially independent and independent of each other in all time instances. The Kalman filtering (Kalman, 1960) allows a unified approach to prediction and estimation for all processes that can be given by state-space representation. Therefore, the state-space model can be treated by standard time series methods based on the Kalman filter (Durbin and Koopman, 2001; Anderson and Moore, 1979). The variance matrix Π of the initial state vector ρ1 may contain diffuse elements when nonstationary components are included in ρo. In this case, diffuse initialization methods for the Kalman filter exist to evaluate the exact or diffuse likelihood function (De Jong, 1991; Koopman, 1997). After the initialization of transition and measurement equations, the Kalman model can be used for forecasting the number of rail freight cars by applying the Kalman recursive prediction (KRP). Recursive steps of KRP are presented in Durbin and Kopman (2001).

6.5.1 Components of weighted matrices A, B, and L This section describes the procedure for determining the elements of weighted matrices as well as their proper structuring. The importance of this process for the quadratic optimality criterion and recommendations for their structuring were presented in Athans and Falb (1966). Comparing to the fuzzy rail freight car inventory model, in this case, the basic stochastic version is, however, extended to include each freight car type independently. Therefore, for components of matrices A, B, and L, the multiperiod EOQ model (Taha, 2003) was applied. Parameter γ it(n) represents the average freight car of type t (t 2 T) inventory level and shortage in station i for the period n of the planning horizon. Parameter ηit(n) represents the number of cars ordered for a station and a period created whenever the inventory of rail freight cars of type t drops to some reorder level. Determining the optimal number of freight cars and the optimal ordering quantity is based on a cost minimum consisted from a sum of fleet ownership and allocation costs. Dit represents the total demand for freight cars of type t in station i. Dit includes the demand specified in terms of the original car type t extended

Stochastic model for heterogeneous rail freight car fleet management

189

for the demand for the aggregate car type t0 (Dit0 ) which will be fulfilled by the original type t (t 2 T t0 , t0 2 Ta ). This extension is based on an estimation of the past share of type t freight cars in substitutions for satisfying the demand for the aggregate car type t0 . The setup cost can be expressed as N  Dit 1 X ϑt  ejit + ð1  ϑt Þljit , t 2 T ηit N  1 j¼1

(6.36)

Setup cost represents the cost of the delivery of freight cars of type t per period, from stations where there is an excess of freight cars to the station for which the optimal delivery has been determined. The relationship Dη it is the approximate number of orders per period. it The unit average cost of supplying an order of freight cars is PN  1 j¼1 ϑt  ejit + ð1  ϑt Þljit , t 2 T, where the parameter ϑt denotes the ratio N 1 of the number of empty cars of type t and the total number of freight cars. This coefficient is determined for all cars of the original type. Unit costs of empty and loaded movement, for each type t of freight cars between all stations j and station i, are represented by ejit and ljit, respectively. In order to determine the average fleet holding cost, we assumed that the number of cars of type t will vary linearly from ηit, the value at the beginning of the planning period, to γ it, the value at the end. Therefore, the average fleet holding cost can be expressed as !! N X γ it + ηit hit  ,t 2 T (6.37) Dijt E 2 j¼1 Parameter hit is the unit holding cost for a freight car of type t in a station i. Dijt is the average daily demand between stations i and j per period defined on P  1 periods of the planning horizon. In order to formulate the cost of shortage for a rail freight car of type t, we introduced the function: δpt

XN j¼1

Dijt , γ i



8 X < 0, D < γ it j ijt X ¼ X , : D  γ , D  γ ijt ijt it it j j

t2T

(6.38)

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Optimization models for rail car fleet management

The expected total demand for cars of type t, backordered or transferred to a future period is

X X ð∞ X δpt ¼ Dijt  γ it f Dijt d Dijt γ it j j j ð∞ X X Dijt f Dijt  γ it Y ðγ it Þ,t 2 T (6.39) ¼ γ it

j

j

Y(γ it) is a complementary cumulative distribution function. The demand for cars of type t between stations on the network comply with the normal distribution law ( Jordan, 1982). Considering the central limit theorem, the sum of two Gaussian random variables also follows the normal distribution law. Let this sum be denoted by κit: κit ¼ Di1t + Di2t + ⋯ + Dijt + ⋯ + DiNt , t 2 T

(6.40)

Considering that Dijt( j 2 N, t 2 T ), and with N(μijt, σ ijt) normally distributed, there is κ it with N(μit, σ it), where μit ¼ μi1t + μi2t + ⋯ + μiNt

(6.41)

σ 2it ¼ σ 2i1t + σ 2i2t + ⋯ + σ 2iNt

(6.42)

The total expected cost for a freight car of type t for a station i over the planning period can be formulated as N  N X Dit 1 X η + γ it E½C ðγ it , ηit Þ ¼ ϑt ejit + ð1  ϑt Þljit + hit it Dijt E ηit N  1 j¼1 2 j¼1 N Tit X + pijt ηit j¼1

"ð

N ∞X

X

γ i j¼1

j

Dijt f

!

!!

#

Dijt  γ it Y ðγ it Þ , t 2 T (6.43)

in order to find γ it and ηit which minimize the function of total estimated inventory costs for each freight car type t. The set of necessary conditions is determined based on the following two conditions: N ∂E ½C ðγ it , ηit Þ Dit 1 X hit ¼ 2 ϑt ejit + ð1  ϑt Þljit Þ + ηit N  1 j¼1 2 ∂ηit 



 N Dit X γ it  μit γ it  μit  ðμit  γ it ÞΦ ¼0 pijt σ it ϕ  2 ηit j¼1 σ it σ it

(6.44)

Stochastic model for heterogeneous rail freight car fleet management



N ∂E ½C ðγ it , ηit Þ hit Dit X γ it  μit ¼0 ¼  pijt Φ 2 ηit j¼1 σ it ∂γ it where

191

(6.45)

1 2 ϕðγ it Þ ¼ pffiffiffiffiffi eγ =2 2π

and

∞ ð

Φðγ Þ ¼

ϕðηÞdη γ

is the complementary cumulative distribution of the demand density function. Now, for a given Dit, Dijt, pijt, eijt, lijt, σ it, μit, the optimal rail freight car inventory level and the optimal reordering number of freight cars of type t in a station i during the period n can be calculated by the following algorithm: Step 0. Find the initial, smallest value of ηit which is η∗it, and let γ (0) it ¼ 0. Set k ¼ 1 and go to stepk. ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XN  u u2Dit ϑ e + ð1  ϑt Þljit t j¼1 t jit ð1Þ ηit ¼ η∗it ¼ (6.46) ðN  1Þhit (k) (k) (k1) Step k. On the base of η(k) it calculate γ it from Eq. (6.45). If ηit ηit (k) (k) ∗ stop, the optimal solution is γ ∗it ¼ γ (k) it and ηi ¼ ηi . Otherwise, use ηit in (k) Eq. (6.46) to compute γ it . Set k ¼ k + 1, and repeat step k until convergence. Since the point (γ ∗it, η∗it) represents the global minimum of the total cost function, it can be approximated by Taylor’s polynomial by an expansion to the second term:

C ðγ it , ηit Þ C ðγ it ∗ , ηit ∗ Þ + +

∂C ∂C 1 ∂2 C ðγ it  γ it ∗ Þ + ðηit  ηit ∗ Þ + ðγ  γ it ∗ Þ2 ∂γ t ∗ ∂ηt ∗ 2 ∂γ t ∗ it

∂2 C 1 ∂2 C ðγ it  γ it ∗ Þðηit  ηit ∗ Þ + ðη  ηit ∗ Þ2 ∂γ t ∂ηt 2 ∂ηt ∗ it (6.47)

The coefficients of the quadratic terms γ it and ηit can be determined based on the cost function of the following form (Monks, 1982): 1 1 Ct ¼ π st γ 2 + π ht γη + π ut η2 2 2

(6.48)

192

Optimization models for rail car fleet management

Coefficients π s, π h, and π u will be derived from expression (6.47). By analyzing the quadratic terms, the unknown coefficients for all car types are ∗

N Dit X γ it  μit s (6.49) π it ¼ ∗ pijt ϕ σ it ηit σ it j¼1 π hit

¼

∗

γ it  μit pijt Φ σ it j¼1

N Dit X

η∗it

2

(6.50)

N  N 1 X Dit X + 2 ϑ e + ð 1  ϑ Þl pijt t jit t jit 3 3 η∗it N  1 j¼1 η∗it j¼1  ∗

∗

 γ  μit γ  μit  σ it ϕ it  ðμit  γ ∗it ÞΦ it σ it σ it

π uit ¼ 2

Dit

(6.51)

The formulated procedure for finding the unknown coefficients π st, π ht , and π ut has to be applied for each station and each original freight car type. What remains to be done, is to define the quadratic cost coefficients for variables that describe the freight car flows. Namely, the state vector, beside the number of freight cars by types, undispatched car flows, and demand requests, also contains the variables that symbolize the cars in transit during the particular planning period. The quadratic cost coefficient for these variables is assumed to be the linear function of the number of cars on the route considered: ! P X xjitt0 m ðnÞQ ¼ xjitt0 m ðnÞ qt0 + qt xjitt0 m ðnÞ sajits ðn + s  mÞ , t 2 T, t 0 2 T t s¼m

yjitt0 m ðnÞQ ¼ yjitt0 m ðnÞ qt0 + qt yjitt0 m ðnÞ

P X

!

(6.52)

sbjits ðn + s  mÞ , t 2 T, t 0 2 T t

s¼m

(6.53) It is now possible to determine the coefficients in matrices A(n), B, and L of the generalized quadratic performance index.

Stochastic model for heterogeneous rail freight car fleet management

193

Therefore, matrix A(n) has the structure as follows: 2 3 A1 ðnÞ O T ðN ð2N  1Þ + 2N ðN  1ÞðP  1ÞÞ X 2N ðN  1ÞðT a + ðP  1Þ Tt Þ 7 6 6 7 t2T AðnÞ ¼ 6 7 4 O ð2N ðN  1ÞðT a + ðP  1Þ X Tt Þ 5 A2 ðnÞ a

t2T a

T ðN ð2N  1Þ + 2N ðN  1ÞðP  1ÞÞ

(6.54) A1 ðnÞ ¼ diagðAt ðnÞÞ, t 2 T At ðnÞ ¼ diagðπ s1t ,…,π s1t ,π s2t ,…,π s2t , π sNt , …, π sNt ,

π s1t ,…,π s1t ,…,π s2t ,…,π s2t ,…,π sNt ,…,π sNt ,2qt P P X X  sa21ts ðn + s  1Þ,…,2qt saN 1ts ðn + s  1Þ,…, 2qt s¼1 P X



s¼1

sa21ts ðn + s  P + 1Þ,…,2qt

s¼P1 P X

saN 1ts ðn + s  P + 1Þ,…,2qt

P X

sb21ts ðn + s  1Þ,…,2qt s¼P1 s¼1 P X sbN 1ts ðn + s  1Þ,…,2qt s¼1 P P X X  sb21ts ðn + s  P + 1Þ,…,2qt sbN 1ts ðn + s  P + 1Þ,…,2qt s¼P1 s¼P1 P P X X  sa1Nts ðn + s  1Þ,…,2qt sbN 1, Nts ðn + s  P + 1ÞÞ, t 2 T s¼1 s¼P1   A2 ðnÞ ¼ diag At 0 ðnÞ , t0 2 T a

At 0 ðnÞ ¼ diagð2qt



s¼1

sa21ts ðn + s  P + 1Þ, …, 2qt Þ

s¼P1 P X

saN 1ts ðn + s  P + 1Þ, …, 2qt

s¼P1 P X

P X sb21ts ðn + s  1Þ,…,2qt s¼1

(6.57)

sbN 1ts ðn + s  1Þ,…,2qt

 

s¼1 P X

sb21ts ðn + s  P + 1Þ, …, 2qt

s¼P1 P X

sa1Nts ðn + s  1Þ…2qt

s¼1

(6.56)

P P X X sa21ts ðn + s  1Þ,…, 2qt saN 1ts ðn + s  1Þ, …,2qt s¼1

P X

(6.55)

P X

P X

sbN 1ts ðn + s  P + 1Þ, …,2qt

s¼P1

sbN 1, Nts ðn + s  P + 1ÞÞ, t 0 2 T a , t 2 T t’

s¼P1

194

Optimization models for rail car fleet management

matrix A(n) is a ½NT ð2N  1Þ + 2N ðN  1ÞðT a + ðP  1Þ    P T + t2T a T t Þ T + t2T a T t Þ  ½NT ð2N  1Þ + 2N ðN  1ÞðT a + ðP  1Þ P diagonal square matrix and has the components π sit, qt Ps¼m saijts ðn + s  mÞ, PP and qt s¼m sbijts ðn + s  mÞ. B is a ½2NT ðN  1Þ + h i P P 2N ðN  1Þ t0 2T a T t0   2NT ðN  1Þ + 2N ðN  1Þ t’2T a T t’ diagonal square Thus, 

P

the

matrix which is a positive-definite matrix with components π uit and π uit’, t 2 T, t 0 2 Ta. B ¼ diag(B1, …, Bt, …, BT) where

  Bt ¼ diag π u1t , …, π u1t , π u2t , …, π u2t , …, π uNt , …, π uNt , π u1t , …, π u1t , π u2t , …, π u2t , …, π uNt , …, π uNt (6.58)

P a Matrix L is a ½NT ð2N  1Þ + 2N  PðN  1ÞðT + ðP  1ÞðT + t2T a T t ÞÞ h T ð2N ðN  1ÞÞ + 2N ðN  1Þ t0 2T a T t0 matrix with components π i and rijtt0 2

L1

X t 3 T 7 t2T a 7 7 L2 X X t 7 5 O 2N ðN 1ÞðP1Þ Tt  2N ðN  1Þ T

O

6 6 L¼6 ðN 1Þ 6O2N ðN1ÞTa 2NT X 4O 2N ðN 1ÞðP1Þ Tt  2NT ðN  1Þ t2T a

T ðN ð2N 1Þ + 2N ðN 1ÞðP1ÞÞ2N ðN1Þ

t2T a

L1 ¼ diag (Lt), t 2 T where 2 h M1 6 6 O1 6 6⋯ 6 6 6 O1 6 h Lt ¼ 6 6 N1 6 6 O2 6 6⋯ 6 6 4 O2

t2T a

(6.59) O1 O1 M1h O1 O1

3

7 M2h O1 O1 M2h O1 7 7 ⋯ ⋯ ⋯ ⋯ ⋯7 7 h h 7 O1 MN O1 O1 MN 7 7 O2 O2 N1h O2 O2 7 7 7 N2h O2 O2 N2h O2 7 7 ⋯ ⋯ ⋯ ⋯ ⋯7 7 7 O2 NNh O2 O2 NNh 5

O3 O3 O3 O3 O3 O3 Mh1 ¼ [π h1t]NN1 Nh1 ¼ [π h1t]N1N1 Mh2 ¼ [π h2t]NN1 Nh2 ¼ [π h2t]N1N1 MhN ¼ [π hNt]NN1 NhN ¼ [π hNt]N1N1 O1 is a N  N  1 zero matrix O2 is a N  1  N  1 zero matrix O3 is a [2N(N  1)(P  1)]  [(N  1)] zero matrix

(6.60)

6 6 6 6 6 6 6 L2 ¼ 6 6 6 6 6 6 4

diagðrij1t10 Þ

⋯

0

  diag rijT t10 t10

diagðrijt1 t10 Þ

⋯

diagðrij1t20 Þ

⋯

diagðrijt2 t20 Þ

⋯

⋮

⋮

⋮

⋮

0

0

diagðrij1T a Þ

⋯

⋮  diag rijt3 T a

⋯

⋮   diag rijT T a T a

⋮

⋮

⋮

⋮

⋮

⋮

⋮

0

0

0

0

0

0

0

i, j 2 N , t1 ¼ 1,…, T t10 ,t2 ¼ 1,…, T t20 , t3 ¼ 1, …,T T a , t10 , t20 2 T a

0

0

  diag rijT t20 t20

0

⋮

t10

3 7 7 7 7 7 7 7 7 7 7 7 7 7 5

(6.61)

Stochastic model for heterogeneous rail freight car fleet management

2

195

196

Optimization models for rail car fleet management

6.5.2 The components of matrix Γ(P) The number of cars in the last period of the planning horizon influences the sensitivity of decisions about their routing within the given planning horizon. If these decisions are not of critical importance for the supply of freight cars by stations, it is not necessary to take this value into consideration. In the case of models proposed in this book, this results from the assumed cyclic nature of demand on the period basis. For determining the components for the last term, Γ(P) a random newsboy model is applied. It includes a single variable (system state) and a single time period and it is presented by a second-order gradient method through quadratic and linear terms in state variables. The last period of the planning horizon lasts one day. This model determines the optimal level of freight cars for a single period in a particular station in case of stochastic demand and when the objective is to minimize the expected total cost. Based on a single period newsboy inventory model for each freight car type, it is possible to determine the coefficients of the weighted matrix of the P terminal P member. Therefore, in case if N D j¼1 ijt < γ it holds, then the quanN tity γ  D is held during the period. Otherwise, a shortage amount j¼1 ijt PN it PN j¼1 Dijt  γ it will result if j¼1 Dijt > γ it .The expected cost for the period, E[C(γ it)], is expressed as ! ð γit N X γ it  Dijt f ðDit Þd ðDit Þ E ½C ðγ it Þ ¼ hit 0 j¼1 ! (6.62) ð∞ X N N X Dijt  γ it f ðDit ÞdðDit Þ + pijt j¼1

γi

j¼1

P where N j¼1Dijt N(μit, σ it). A necessary condition for determining the optimal number of cars in a station is ∂E½C ðγ it Þ ¼0 ∂γ it ∗

γ it  μit hit  ¼ Φ XN σ it hit + p j¼1 ijt

(6.63) (6.64)

where Φ(k) is the complementary cumulative distribution function. Considering that the function of total costs is strictly concave, its absolute minimum is unique. The second derivative of this function is

Stochastic model for heterogeneous rail freight car fleet management N X ∂2 E½C ðγ it Þ ¼ h + pijt it ∂γ 2it j¼1

!

 γ it  μit ϕ >0 σ it

197

(6.65)

Solving Eq. (6.64) will result in the optimal rail freight car inventory level γ ∗it for the last day of the planning horizon. In the vicinity of the minimum this function can be approximated by the Taylor’s polynomial by expansion to the second term: C ðγ it Þ C ðγ ∗it Þ +

∂C 1 ∂2 C | ðγ it  γ ∗it Þ + ðγ it  γ ∗it Þ 2 | ðγ t  γ ∗t Þ ∂γ it γ ∗ 2 ∂γ t γ ∗t

(6.66)

it

The coefficient of the quadratic term is   γ ∗  μ XN it ,i 2 N ,t 2 T p ϕ it git ¼ hit + j¼1 ijt σ it

(6.67)

    P Γ(P) is a NT ð2N  1Þ + 2N ðN  1Þ T a + ðP  1Þ T + t2T a T t      P NT ð2N  1Þ + 2N ðN  1Þ T a + ðP  1Þ T + t2T a T t positivesemidefinite matrix in the quadratic term of the state vector for the last period of the planning horizon. 2 6 6 Γ ðP Þ ¼ 6 4O

Γ 1 ðP Þ

O T ðN ð2N  1Þ + 2N ðN  1ÞðP  1ÞÞ 2N ðN  1ÞðT a + ðP  1Þ

ð2N ðN  1ÞðT a + ðP  1Þ

X

Γ2 ðP Þ

Tt Þ

X

t2T a

Tt Þ

3 7 7 7 5

(6.68)

t2T a

T ðN ð2N  1Þ + 2N ðN  1ÞðP  1ÞÞ

Γ 1(P) ¼ diag(Γ t(P)), t 2 T where Γ t ðP Þ ¼ diagðg1t , …, g1t , …, gNt ,…, gNt , g1t , g2t , …,gNt ,2qt

P X

sa21ts ðP + s  1Þ,  , 2qt

s¼1



P P P X X X saN 1ts ðP + s  1Þ,2qt sa21ts ðs + 1Þ,  , 2qt saN 1ts ðs + 1Þ, 2qt s¼1

s¼1

s¼P1

P P P X X X  sb21ts ðP + s  1Þ,  , 2qt sbN 1ts ðP + s  1Þ, 2qt sbN 1ts ðs + 1Þ,  ,2qt s¼1

s¼1

s¼1

s¼P1

s¼P1

P P X X  sa1Nts ðP + s  1Þ,2qt sbN 1, Nts ðs + 1ÞÞ, t 2 T

(6.69) 0

Γ 2 ðP Þ ¼ diagðΓ t0 ðP ÞÞ, t 2 T

a

(6.70)

198

Optimization models for rail car fleet management P P X X sa21ts ðP + s  1Þ,  ,2qt saN 1ts ðP + s  1Þ,2qt sa21ts ðs + 1Þ,2qt s¼1 s¼1 s¼1 P P P X X X  saN 1ts ðs + 1Þ,2qt  sb21ts ðP + s  1Þ,  ,2qt sbN 1ts ðP + s  1Þ,2qt s¼P1 s¼1 s¼1 P P P X X X  sbN 1ts ðs + 1Þ,  ,2qt sa1Nts ðP + s  1Þ,2qt sbN 1, Nts ðs + 1ÞÞ,t 0 2 T a ,t 2 T t0 s¼P1 s¼1 s¼P1 (6.71)

Γ t 0 ðP Þ ¼ diagð2qt

P X

6.6 MPC controller MPC is a multivariable control method usually applied for the optimal control of complex systems with an explicit consideration of constraints. The dynamics of the system is represented by a discrete time predictive model by which the state of the system in the next period is a function of the actual state, actual demands (disturbance), and the actual control vector with constraints. In each period, the optimal control problem is solved online based on the actual state (at time n ¼ n0) and the predictive demands over a N step finite horizon (at time n ¼ n0, …, n0 + N  1). The result of the optimization is a sequence of control vectors overtime n ¼ n0, …, n0 + N  1 but only the first control vector (at time n ¼ n0) is applied to the system. In the next state (at time n ¼ n0 + 1), the optimal control problem is solved again for the time horizon n ¼ n0, …, n0 + N, then only the control vector at time n ¼ n0 + 1 is applied and so forth.

6.6.1 Optimization problem The stochastic rail freight car fleet sizing and allocation model is represented by a discrete linear model in which the state X(n), control U(n), and disturbance are related in accordance to the following relation: X ðn + 1Þ ¼ Λ½n, aðnÞ, bðnÞX ðnÞ + GU ðnÞ + ΦdðnÞ, n ¼ 0,1,…,P  1 (6.72) with X(0) given and subject to the following system constraints: U ðnÞ  O2

(6.73)

X ðnÞ  O3

(6.74)

DX ðnÞ  K

(6.75)

Based on the system dynamics (6.72) and constraints (6.73)–(6.75), the problem is to determine the set of control actions U(n) in order to minimize the objective functional of the form:

199

Stochastic model for heterogeneous rail freight car fleet management

ð6:76Þ In order to apply the MPC approach, the objective functional has been transformed into the standard form (Anderson and Moore, 2007): P1   1 1X X ðnÞT Q1 ðnÞX ðnÞ + Ue ðnÞT BU e ðnÞ J ¼ X ðP ÞT ΓðP ÞX ðP Þ + 2 2 n¼0

(6.77) where Q1(n) ¼ A(n)  LB1LT and Ue(n) ¼ U(n) + B1LTX(n). The system dynamics constraint (6.72) now takes the form X ðn + 1Þ ¼ N ðnÞX ðnÞ + GU e ðnÞ + ΦdðnÞ, n ¼ 0,1,…, P  1

(6.78)

where N(n) ¼ Λ[n, a(n), b(n)]  GB1LT. Consequently, the problem of the minimization of the criterion (6.77) subject to dynamics (6.78) and the set of constraints (6.73)–(6.75) can be solved as an MPC problem.

6.6.2 Detailed description of the MPC approach In this part, the MPC approach is clarified. The proposed controller is parameterized by a set of discrete decision periods 0,1, …, m, …, n, …, P, a positive-definite matrix Ψ of [T(2N(N  1) + 2N(N  1)(P  1))]  [T(2N(N  1) + 2N(N  1)(P  1))] dimensions and a vector Ω of dimension [T(2N(N  1) + 2N(N  1)(P  1))]. At any time n, the controller applies the optimal solution of a quadratic programming (QP) problem in which the decision variables represent the empty and loaded flow control actions over the time horizon n, n + 1, …, n + P  1 for each type of freight car. The following state-space model represents the considered rail freight car fleet system: X ðn + 1Þ ¼ N ðnÞX ðnÞ + GU e ðnÞ + Φd ðnÞ, n ¼ 0,1,…, P  1 Z ðnÞ ¼ HX ðnÞ + vðnÞ, n ¼ 0,1,…, P

(6.79) (6.80)

The state noise d(n) and measurement noise v(n) are assumed to be Gaussian distributed with zero mean and respective covariances of R1(n) and R2(n) with cross-covariance R3(n):       R1 ðnÞ R3 ðnÞ dðnÞ 0 (6.81) N , R3T ðnÞ R2 ðnÞ vðnÞ 0

200

Optimization models for rail car fleet management

The defined model is used for predictions about the rail freight car system in the following periods of the prediction horizon, using information (measurements of inputs and outputs) up to and including the current period n (Wills, 2004). Now it is possible to make optimal (from the aspect of minimum variance) predictions of state and output by applying the Kalman filter (Athans and Falb, 1966; Kalman, 1960). Accordingly, let X ðij jÞ represents an estimate of the state at time i given information up to and including time j, where j  i. Then, X ðn + 1j nÞ ¼ N ðnÞX ðnj n  1Þ + GU e ðnÞ + K ðnÞ½Z ðnÞ  Z ðnj n  1Þ (6.82) Z ðnj n  1Þ ¼ HX ðnj n  1Þ  1 K ðnÞ ¼ N ðnÞP ðnÞH T + R3 ðnÞ HP ðnÞH T + R2 ðnÞ 

P ðnÞ ¼ R1 ðnÞ + N ðnÞP ðnÞΛT ðnÞ  ðN ðnÞP ðnÞH T + R3 ðnÞÞ 1  ðHP ðnÞH T + R2 ðnÞÞ R3T ðnÞ + HP ðnÞN T ðnÞ

(6.83) (6.84) (6.85)

The Kalman filter is represented by recurrent relations (6.82) and (6.83). In fact, these two relations are the model of the system extended for a correction term, which is proportional to the difference between the values of Z(n) and the values estimated on the basis of the fuzzy state function H  X ðnÞ. Matrix (6.84), K(n), is the Kalman filter gain. The discrete-time algebraic Riccati equation (DARE) for P(n) is given by Eq. (6.85). To apply the MPC approach, the estimates of the state over the whole prediction horizon need to be determined. These predictions can be made only based on the information up to and including the current time n. Eqs. (6.82)–(6.85) can be used to obtain X ðn + 1j nÞ, and optimal estimates over the prediction horizon can be obtained as follows: X ðn + i + 1j nÞ ¼ N ðnÞX ðn + ij nÞ + GU e ðn + ij nÞ

(6.86)

Z ðn + ij nÞ ¼ HX ðn + ij nÞ

(6.87)

Here, i ¼ n, n + 1, …, n + P  1 and notation Ue(n + ij n) is applied to distinguish the actual input at time t + i from the input used for prediction purposes, namely Ue(n + i j n). Based on the estimated current state X ðnÞ and the controls Ue(i), i ¼ n, n + 1, …, n + P  1, the state predictions are generated and used for the

Stochastic model for heterogeneous rail freight car fleet management

201

formulation of the objective and constraints of the QP problem (Le et al., 2013). The detailed description of the QP formulation is given below. The main components of the prediction, optimization, and receding horizon implementation constitute the MPC law. The dynamic model (6.79) is used for the prediction of the future response of the controlled system. For the given predicted input sequence, the corresponding sequence is generated by simulating the model forward over the prediction horizon of P  1 periods. These predicted sequences are places in vectors U, X defined by 2 3 2 3 Ue ðnj nÞ X ðn + 1j nÞ 6 Ue ðn + 1j nÞ 7 6 7 7,X ðnÞ ¼ 6 X ðn + 2j nÞ 7 Ue ðnÞ ¼ 6 (6.88) 4 5 4 5 ⋮ ⋮ Ue ðn + P  1j nÞ X ðn + Pj nÞ Ue(n + ij n) and X ðn + ij nÞ are the input and state vectors at time n + i which are predicted at time n. Then, X ðn + ij nÞ evolves according to the prediction model: X ðn + i + 1Þ ¼ N ðnÞX ðn + ij nÞ + GU e ðn + ij nÞ + Φd ðnÞ, i ¼ 0,1, …,P  1 (6.89) The initial condition at the beginning of the prediction horizon is defined as X ðnj nÞ ¼ X ðnÞ In compact form, the predicted state sequence generated by Eq. (6.89) looks as follows: X ðn + ij nÞ ¼

0 Y

N ðn + jÞX ðnÞ + Ci ðnÞUe ðnÞ + Ei ðnÞdðnÞ,i ¼ 0,…,P  1

j¼i1

(6.90) or X ðnÞ ¼ N X ðnÞ + CU e ðnÞ + Ed ðnÞ where 2

3 N ðnÞ 6 7 N ðn + 1ÞN ðnÞ 7 N ¼6 4 5 ⋮ N ðP  1ÞN ðP  2Þ⋯N ðnÞ

(6.91)

202

Optimization models for rail car fleet management

C is the (convolution) matrix with rows Ci defined by 2 3 G 0 ⋯ 0 6 7 6 7 N ð n + 1 ÞG G ⋯ 0 6 7 C¼6 7 6 7 ⋮ ⋮ ⋮ ⋮ 4 5 N ðP  1Þ⋯N ðnÞG N ðP  2Þ⋯N ðnÞG ⋯ G C0 ¼ 0, Ci ¼ ith block row of C 2 Φ 6 N ðn + 1ÞG 6 E¼6 4 ⋮

0 Φ ⋮

3 ⋯ 0 ⋯ 07 7 7 ⋮ ⋮5

(6.92)

(6.93)

N ðP  1Þ⋯N ðnÞΦ N ðP  2Þ⋯N ðnÞΦ ⋯ Φ E0 ¼ 0, Ei ¼ ith block row of E We obtain the performance criteria as a quadratic function of Ue(n) by a substitution for X ðn + ij nÞ in Eq. (4-77): 1 J ðnÞ ¼ Ue ðnÞT Ψ Ue ðnÞ + ΩUe ðnÞ + Υ 2 e1C + B e Ψ ¼ CT Q

(6.94) (6.95)

e 1 C + d ðnÞT E T Q e1C Ω ¼ X ðnÞT N T Q (6.96)  1 e X ðnÞ + 1 X ðnÞT N T Q e1N + Γ e 1 EdðnÞ Υ ¼ X ðnÞT N T Q 2 2 (6.97) 1 1 T T e T T e + dðnÞ E Q1 N X ðnÞ + dðnÞ E Q1 EdðnÞ 2 2 Matrix Ψ is a constant positive-definite matrix, Ω is a vector of corresponding dimensions, and Υ is a scalar which depends on X ðnÞ and d(n) with 2

3 2 3 2 3 Q1 0 … 0 B 0 … 0 Γ 0 … 0 6 0 ⋱ 6 ⋮ 7 0 ⋱ ⋮7 ⋮7 7 e 6 7 ,and Γ 7 e¼60 ⋱ e1 ¼ 6 Q 6 7,B ¼ 6 4 5 4 ⋮ B 0 ⋮ Γ 05 4 ⋮ Q1 0 5 0 ⋯ 0 B 0 ⋯ 0 Γ 0 ⋯ 0 Q1 Quadratic cost is now defined as a function of input predictions. The next step is to transform the set of constraints (6.73)–(6.75) in the form AUe  B which can be included in solving the problem of optimizing Eq. (6.94).

Stochastic model for heterogeneous rail freight car fleet management

203

Input constraints (6.73) can be expressed as Ue(n)  O2. Applied to predictions, Ue(n + ij n), i ¼ 0, …, P  1 these constraints can be expressed in terms of Ue(n) as IU e ðnÞ  O2

(6.98)

where I represents a unit matrix. In the same way, the state constraints (6.74) applied to predictions X ðn + ij nÞ, i ¼ 1, …,P are equivalent to Ci Ue ðnÞ  O3 + Ni X ðnÞ + Ei dðnÞ, i ¼ 0,…, P  1

(6.99)

Capacity constraints (4-75) are formulated as follows: DC i Ue ðnÞ  DRi  DSi + K, i ¼ 0,…,P  1 where

2 6 6 R¼6 4

N ðnÞ

(6.100)

3

7 N ðn + 1ÞN ðnÞ 7 7X ðnÞ ¼ N X ðnÞ 5 ⋮ N ðP  1ÞN ðP  2Þ⋯N ðnÞ

Ri is the ith block row of matrix R 2 ΦðnÞdðnÞ 6 N ðn + 1ÞΦðnÞdðnÞ + Φðn + 1Þd ðn + 1Þ 6 S¼6 4 ⋮

(6.101)

3 7 7 7 5

N ðP  1Þ⋯N ðn + 1ÞΦðnÞdðnÞ + ⋯ + Φðn + P  1Þd ðn + P  1Þ (6.102) Si is the ith block row of matrix S, n ¼ 0, …, P  1. In summary, the MPC formulation for the stochastic rail freight car fleet sizing and allocation problem can be stated as the minimization of a quadratic objective over Ue(n) subject to a set of linear constraints:

where

1 minimize Ue ðnÞT Ψ Ue ðnÞ + ΩUe ðnÞ 2 Ue ðnÞ

(6.103)

subject to Ae ðnÞUe ðnÞ  Be ðnÞ

(6.104)

2 3 3 I O2 Ae ðnÞ ¼ 4 Ci 5, Be ðnÞ ¼ 4 O3 + Ni X ðnÞ + Ei dðnÞ 5, i ¼ 0,…, P  1 DRi  DSi + K DC i (6.105) 2

204

Optimization models for rail car fleet management

The stated problem is a QP problem. Ψ matrix is a positive-definite matrix and the constraints are linear. Therefore, this problem is a convex problem which is solved by the QP solver (quadprog) in the Matlab (MathWorks Inc., 2012) for each period n. As it has been stated, only the first element of the optimal predicted input sequence is selected as an input to the rail freight car fleet system and therefore, for updating the state X(n) and the right side of the inequalities, Be(n) in each time period within the planning horizon.

6.6.3 Numerical experiments In this section, a set of numerical experiments has been carried out in order to demonstrate the validity of the developed MPC approach. The approach is explained through an example based on a network of four stations (N ¼ 4). The results of other test instances are presented at the end of the section. All test cases are based on the real rail freight car operation practice of Serbian railways. Four original car types (T ¼ 1, 2, 3, 4) are used for freight transport on the considered rail network. These types also satisfy the demand for two aggregate car types (Ta ¼ 5, 6). This assumption is based on the existence of four basic types (open, closed, flat, and other) on most railways and the common interchangeability between open and flat as well as closed and other car types. Fig. 6.2 illustrates possible substitution possibilities. As it can be noticed, the demand for the aggregate car type 5 will be fulfilled by either an original car of type 1 (open) or 3 (flat), while the demand for cars of type 1 or 3 can be fulfilled only by a corresponding type. The same situation holds for the aggregate car type 6 and original car types 2 (closed) and 4 (other). The planning horizon is composed of P ¼ 4 days where each day is one decision period. The length of 4 days is considered having in mind the longest rail freight car traveling time on Serbian railways.

Fig. 6.2 Substitution possibilities between car types.

Stochastic model for heterogeneous rail freight car fleet management

205

Table 6.2 contains the input data on unit costs of empty trips (ejit), unit costs of loaded trips (ljit), and unit car shortage costs (pijt) in monetary units per car for all routes and rail freight car types. Unit costs of empty and loaded trips are estimated and expressed in national currency. Expected travel times and variances for loaded and empty freight cars are given in Tables 6.3 and 6.4. Proportions of arrivals for loaded rail freight car flows together with the quadratic term in the unit ownership cost, as well as the proportions of arrivals for empty freight car flows of different types are given in Tables 6.5 and 6.6. The coefficient of empty trips is 0.35 for all freight car types. (See Table 6.2.) The demand is assumed to have a stochastic nature. Table 6.7 contains the mean value and standard deviation of daily demand for all periods, types of freight cars, and all o-d pairs of nodes. Unit holding costs (hit) per car per period, the estimate of initial number of cars (Sit(0)) per type per station, available capacities for the storage of freight cars by types (CAPi), and the lengths of rail cars are presented in Table 6.8. Table 6.9 contains unmet freight car requests from the preceding period. The numbers of loaded and empty cars that were initially in transit at the beginning of the horizon are given in Tables 6.10 and 6.11, respectively. Table 6.12 contains the cost of substitution which represents a penalty for using one of the original car types for satisfying the demand for aggregate car types 5 and 6. Random inventory models have been used for filling in the matrices A(0), A(1), A(2), A(3), B, and L. Optimal values for the freight car inventory level (γ it) and the ordering quantity (ηit) for each type have been determined after three to five iterations by using the approach described in Section 6.5.1 (Table 6.13). The random newsboy inventory model (Taha, 2003) has been used for determining the values of the car inventory level (γ Pit ) by stations in the last period of the planning horizon. Based on these values, the matrix Γ(3) has been defined (Table 6.12). For the estimation of the state vector, the ARIMA-Kalman method is applied to forecast the number of rail freight cars in each station. The method is applied for each car type t and station i. Forecasting results are given for the first station and first rail freight car type. A data sample of 100 days before the beginning of the planning horizon was used. The time series representing the state of rail freight cars in the station during the previous 100 days is given in Fig. 6.3. We used the first 85 days to fit the ARIMA models and the last 15 days as a holdout period to evaluate the forecasting performance.

206

Freight car type I

II

III

IV

Route

ejit

ljit

pijt

ejit

ljit

pijt

ejit

ljit

pijt

ejit

ljit

pijt

A-B A-C A-D B-A B-C B-D C-A C-B C-D D-A D-B D-C

300 200 200 100 125 150 150 200 100 200 150 125

100 70 70 85 100 120 100 120 70 120 100 85

300 200 200 200 250 300 300 400 200 400 300 250

300 200 200 100 70 200 100 100 150 150 150 150

100 70 70 70 70 100 70 70 100 100 100 100

300 200 200 200 200 300 200 200 300 300 300 300

200 200 300 200 100 125 150 200 100 150 125 200

70 70 100 120 70 85 100 120 70 100 85 120

200 200 300 400 200 250 300 400 200 300 250 400

200 200 300 150 150 125 200 200 150 200 200 300

70 70 100 100 100 85 120 120 100 140 140 200

200 200 300 300 300 250 400 400 300 400 400 500

Optimization models for rail car fleet management

Table 6.2 Cost parameters for rail freight car fleet sizing and the allocation problem

Table 6.3 Expected travel times and variances for loaded freight cars II

III

IV

Route

Expected travel time (days)

Travel time variance (days2)

Expected travel time (days)

Travel time variance (days2)

Expected travel time (days)

Travel time variance (days2)

Expected travel time (days)

Travel time variance (days2)

A-B A-C A-D B-A B-C B-D C-A C-B C-D D-A D-B D-C

1.3 1.2 2.0 1.0 1.5 1.3 1.5 1.5 1.0 2.0 2.0 1.0

0.30 0.40 0.50 0.50 0.30 0.30 0.30 0.30 0.50 0.50 0.50 0.60

1.2 1.0 1.0 1.4 1.5 1.5 1.0 1.5 1.0 1.0 2.0 1.5

0.50 0.40 0.50 0.40 0.50 0.50 0.40 0.50 0.40 0.40 0.60 0.50

1.0 1.5 2.5 1.0 2.0 1.0 1.0 2.0 1.4 2.5 1.0 1.0

0.55 0.50 0.50 0.55 0.65 0.55 0.55 0.65 0.45 0.50 0.55 0.55

1.0 1.5 1.5 1.0 2.0 1.5 1.0 2.0 1.0 2.0 1.5 1.0

0.50 0.50 0.50 0.50 0.65 0.50 0.50 0.65 0.50 0.65 0.50 0.50

Stochastic model for heterogeneous rail freight car fleet management

I

207

208

I

II

III

IV

Route

Expected travel time (days)

Travel time variance (days2)

Expected travel time (days)

Travel time variance (days2)

Expected travel time (days)

Travel time variance (days2)

Expected travel time (days)

Travel time variance (days2)

A-B A-C A-D B-A B-C B-D C-A C-B C-D D-A D-B D-C

1.3 1.2 2.0 1.0 1.5 1.3 1.5 1.5 1.0 2.0 2.0 1.0

0.30 0.40 0.50 0.50 0.30 0.30 0.30 0.30 0.50 0.50 0.50 0.60

1.2 1.0 1.0 1.4 1.5 1.5 1.0 1.5 1.0 1.0 2.0 1.5

0.50 0.40 0.50 0.40 0.50 0.50 0.40 0.50 0.40 0.40 0.60 0.50

1.0 1.5 2.5 1.0 2.0 1.0 1.0 2.0 1.4 2.5 1.0 1.0

0.55 0.50 0.50 0.55 0.65 0.55 0.55 0.65 0.45 0.50 0.55 0.55

1.0 1.5 1.5 1.0 2.0 1.5 1.0 2.0 1.0 2.0 1.5 1.0

0.50 0.50 0.50 0.50 0.65 0.50 0.50 0.65 0.50 0.65 0.50 0.50

Optimization models for rail car fleet management

Table 6.4 Expected travel times and variances for empty freight cars

Table 6.5 Proportion of loaded cars arrived during period n and unit car ownership costs Qt Freight car type I C-A

D-A

A-B

C-B

D-B

A-C

B-C

D-C

A-D

B-D

C-D

0 1 2 3

0 0.5695 0.3516 0.0353

0 0.5267 0.2805 0.0781

0 0.1733 0.4040 0.3175

0 0.5267 0.2805 0.0781

0 0.1733 0.4040 0.3175

0 0.5695 0.3516 0.0353

0 0.5695 0.3516 0.0353

0 0.1733 0.4040 0.3175

0 0.5267 0.2805 0.0781

0 0.1436 0.3821 0.2945

0 0.5267 0.2805 0.0781

0 0.5695 0.3516 0.0353

0 0.5070 0.2836 0.0900

0 0.1554 0.3750 0.3163

0 0.5070 0.2836 0.0900

0 0.1589 0.3808 0.3168

0 0.5070 0.2836 0.0900

0 0.5070 0.2836 0.0900

0 0.6422 0.3507

0 0.1613 0.4130 0.2928

0 0.6422 0.3507

0 0.1613 0.4130 0.2928

0 0.6422 0.3507

0 0.5267 0.2805 0.0781

0 0.5695 0.3516 0.0353

0 0.1444 0.3543 0.3168 15

0 0.5267 0.2805 0.0781

0 0.1436 0.3821 0.2945

0 0.5267 0.2805 0.0781 10

0 0.5267 0.2805 0.0781

II

0 1 2 3

0 0.5362 0.3484 0.0572

0 0.5362 0.3484 0.0572

0 0.1589 0.3808 0.3168

0 0.5362 0.3484 0.0572

0 0.1589 0.3808 0.3168

0 0.5362 0.3484 0.0572 III

0 1 2 3

0 0.6422 0.3507

0 0.5267 0.2805 0.0781

0 0.1512 0.3661 0.3189

0 0.6422 0.3507

0 0.1613 0.4130 0.2928

0 0.6422 0.3507 IV

0 1 2 3 Qt

0 0.5267 0.2805 0.0781

0 0.1436 0.3821 0.2945 10

0 0.1436 0.3821 0.2945

0 0.5695 0.3516 0.0353

0 0.1444 0.3543 0.3168 15

0 0.5695 0.3516 0.0353

209

B-A

Stochastic model for heterogeneous rail freight car fleet management

n

210

Table 6.6 Proportion of empty cars arrived during period n Freight car type

n

B-A

C-A

D-A

A-B

C-B

D-B

A-C

B-C

D-C

A-D

B-D

C-D

0 1 2 3

0 0.5695 0.3516 0.0353

0 0.5457 0.2761 0.0659

0 0.1733 0.4040 0.3175

0 0.5363 0.2784 0.0720

0 0.5457 0.2761 0.0659

0 0.5457 0.2761 0.0659

0 0.6345 0.3513

0 0.5457 0.2761 0.0659

0 0.5362 0.3484 0.0572

0 0.1733 0.4040 0.3175

0 0.5363 0.2784 0.0720

0 0.5362 0.3484 0.0572

0 0.6072 0.3525 0.0085

0 0.6072 0.3525 0.0085

0 0.6345 0.3513

0 0.5070 0.2836 0.090

0 0.1344 0.3364 0.3125

0 0.6072 0.3525 0.0085

0 0.5070 0.2836 0.090

0 0.5070 0.2836 0.090

0 0.5695 0.3516 0.0353

0 0.5070 0.2836 0.090

0 0.6072 0.3525 0.0085

0 0.6345 0.3513

0 0.1583 0.3787 0.3200

0 0.6345 0.3513

0 0.1312 0.3304 0.3108

0 0.6345 0.3513

0 0.5070 0.2836 0.090

0 0.1312 0.3304 0.3108

0 0.6345 0.3513

0 0.1588 0.3787 0.3200

0 0.6345 0.3513

0 0.5095 0.2833 0.0885

0 0.5695 0.3516 0.0353

0 0.1312 0.3304 0.3108

0 0.5695 0.3516 0.0353

0 0.1312 0.3304 0.3108

0 0.5070 0.2836 0.090

0 0.5070 0.2836 0.090

0 0.1312 0.3304 0.3108

0 0.5695 0.3516 0.0353

0 0.5070 0.2836 0.090

0 0.5070 0.2836 0.090

0 0.5695 0.3516 0.0353

II

0 1 2 3

0 0.5169 0.2822 0.0842 III

0 1 2 3

0 0.6345 0.3513 IV

0 1 2 3

0 0.5695 0.3516 0.0353

Optimization models for rail car fleet management

I

Table 6.7 Daily demand between stations on the considered part of the rail network Daily demand (SD)

Daily demand (SD)

1

2

3

4

Route

1

2

3

4

A-B

I II III IV V VI I II III IV V VI I II III IV V VI I II III IV V VI

10 (2) 16 (4) 5 (1) 4 (2) 9 (1) 10 (2) 8 (2) 13 (3) 8 (3) 7 (3) 10 (2) 9 (1) 12 (3) 17 (4) 6 (2) 5 (1) 10 (2) 10 (2) 3 (1) 7 (2) 11 (1) 6 (1) 7 (1) 9 (2)

5 (1) 9 (1) 11 (3) 5 (1) 8 (2) 11 (2) 7 (2) 11 (2) 10 (2) 3 (1) 12 (3) 11 (2) 9 (1) 15 (3) 7 (1) 6 (2) 11 (2) 13 (2) 6 (1) 11 (3) 12 (2) 3 (1) 5 (1) 6 (1)

4 (1) 10 (3) 2 (1) 3 (1) 12 (2) 13 (3) 5 (1) 6 (2) 3 (1) 5 (2) 13 (4) 12 (3) 7 (1) 3 (1) 1 (1) 3 (1) 7 (1) 15 (2) 11 (2) 8 (2) 3 (1) 6 (1) 6 (1) 10 (2)

7 (2) 12 (3) 5 (1) 7 (2) 11 (1) 12 (2) 9 (2) 5 (1) 9 (2) 6 (1) 11 (2) 10 (2) 11 (3) 7 (2) 13 (3) 9 (2) 9 (2) 9 (1) 12 (2) 7 (1) 9 (2) 9 (2) 11 (2) 9 (1)

C-A

7 (1) 17 (3) 12 (3) 6 (2) 11 (1) 13 (2) 8 (2) 9 (2) 18 (3) 9 (3) 10 (2) 8 (1) 11 (3) 5 (1) 12 (2) 4 (1) 10 (2) 11 (2) 4 (1) 9 (2) 5 (1) 7 (2) 6 (1) 7 (1)

13 (2) 5 (1) 14 (3) 6 (2) 9 (1) 10 (2) 12 (2) 7 (2) 19 (3) 11 (3) 11 (2) 9 (1) 6 (1) 9 (2) 11 (2) 18 (4) 12 (2) 13 (2) 6 (1) 11 (3) 19 (3) 5 (1) 10 (1) 12 (2)

9 (1) 6 (2) 19 (4) 4 (1) 7 (1) 6 (1) 13 (2) 8 (2) 7 (1) 8 (2) 13 (1) 6 (1) 21 (4) 15 (3) 12 (3) 5 (1) 9 (1) 8 (1) 3 (1) 15 (3) 17 (3) 13 (2) 11 (2) 9 (1)

18 (4) 11 (3) 10 (2) 16 (4) 8 (1) 9 (1) 22 (5) 6 (2) 12 (3) 11 (2) 12 (2) 7 (1) 29 (5) 9 (2) 8 (3) 6 (1) 9 (1) 6 (1) 9 (2) 14 (3) 12 (3) 12 (3) 12 (2) 9 (1)

A-C

A-D

B-A

C-B

C-D

D-A

Continued

211

Freight car type

Stochastic model for heterogeneous rail freight car fleet management

Route

212

Daily demand (SD)

Daily demand (SD)

Route

Freight car type

1

2

3

4

Route

1

2

3

4

B-C

I II III IV V VI I II III IV V VI

6 (2) 9 (2) 13 (2) 8 (2) 8 (1) 7 (2) 2 (1) 12 (3) 8 (2) 4 (1) 10 (1) 11 (2)

9 (2) 15 (3) 6 (1) 9 (2) 9 (1) 8 (1) 13 (3) 6 (1) 15 (3) 12 (3) 13 (2) 9 (1)

15 (3) 3 (1) 5 (1) 7 (2) 10 (2) 9 (1) 8 (2) 5 (1) 4 (1) 9 (2) 11 (2) 10 (1)

14 (4) 9 (2) 13 (4) 11 (3) 10 (2) 11 (2) 11 (2) 13 (3) 12 (3) 15 (4) 12 (2) 11 (2)

D-B

7 (2) 13 (3) 9 (2) 3 (1) 5 (1) 4 (1) 5 (1) 5 (1) 15 (3) 10 (3) 6 (1) 9 (1)

9 (3) 5 (2) 27 (4) 7 (2) 10 (1) 12 (2) 5 (2) 4 (1) 5 (1) 3 (1) 8 (1) 9 (1)

3 (1) 6 (1) 5 (1) 18 (3) 11 (1) 10 (1) 4 (1) 7 (1) 11 (2) 5 (1) 11 (2) 5 (1)

11 (3) 11 (3) 14 (4) 19 (5) 13 (2) 12 (2) 8 (2) 16 (4) 11 (4) 9 (3) 14 (2) 10 (2)

B-D

D-C

Optimization models for rail car fleet management

Table 6.7 Daily demand between stations on the considered part of the rail network—cont’d

Station A

B

C

D

Freight car type

hit

S0(SD)

hit

S0(SD)

hit

S0(SD)

hit

S0(SD)

Lt(m)

I II III IV

20 15 15 20

30 39 38 36

15 15 15 15

28 30 32 35

20 15 15 20

26 37 33 32

15 20 15 20

38 28 24 23

14 15 15 16

CAPi(m)

(3.00) (4.00) (2.00) (4.00)

2000

(4.00) (3.00) (2.00) (3.00)

1200

(3.00) (2.00) (3.00) (2.00)

1800

(5.00) (3.00) (4.00) (4.00)

1200

Stochastic model for heterogeneous rail freight car fleet management

Table 6.8 Unit holding cost (hit), initial number of cars in stations with standard disturbance (S0(SD)), maximum station’s capacities (CAPi(m)) and lengths of freight cars (Lt(m))

213

214

Optimization models for rail car fleet management

Table 6.9 Unmet demand from the preceding period Route

Freight car type

A-B

A-C

A-D

B-A

B-C

B-D

I II III IV V VI

4 1 2 3 2 5

3 5 3 4 4 4

9 10 4 7 3 2

3 2 11 5 5 5

5 3 3 4 7 6

7 11 6 8 6 1

C-A

C-B

C-D

D-A

D-B

D-C

12 13 2 6 1 1

4 5 3 7 2 3

5 4 5 9 1 2

3 3 4 11 3 4

10 4 6 6 4 3

2 5 2 5 5 3

I II III IV V VI

Table 6.10 Number of loaded cars dispatched before the beginning of the cycle Route

A-B

A-C

A-D

B-A

B-C

B-D

Freight car type

One period

Two periods

I II III IV I II III IV I II III IV I II III IV I II III IV I II III IV

12 12 14 14 18 21 19 20 21 12 14 14 10 11 13 14 11 12 22 22 23 14 11 13

16 21 8 9 15 9 14 11 10 11 24 26 10 5 15 11 13 15 6 19 13 9 13 14

Route

C-A

C-B

C-D

D-A

D-B

D-C

One period

Two periods

8 19 10 25 14 14 15 25 14 15 5 25 14 13 23 13 25 15 6 11 13 5 8 13

9 13 7 16 21 19 9 21 9 7 19 11 17 14 16 7 13 17 5 23 11 16 12 26

Stochastic model for heterogeneous rail freight car fleet management

215

Table 6.11 Number of empty cars dispatched before the beginning of the cycle Route

A-B

A-C

A-D

B-A

B-C

B-D

Freight car type

One period

Two periods

I II III IV I II III IV I II III IV I II III IV I II III IV I II III IV

3 4 2 6 3 3 3 5 3 2 3 3 3 6 3 3 4 4 4 4 2 6 9 9

4 4 5 4 3 2 5 6 5 4 5 6 2 5 5 9 5 4 4 7 6 5 7 6

Route

C-A

C-B

C-D

D-A

D-B

D-C

One period

Two periods

4 4 6 2 5 5 6 3 4 9 4 2 2 3 11 4 2 3 2 6 6 5 2 6

3 7 9 15 3 5 7 4 4 3 7 11 4 5 3 17 4 6 5 9 2 6 4 7

The best-fitting model was found based on the assessment of the alternative ARIMA models. For the first rail car type in the first station, the most appropriate model was found with the Box-Jenkins approach and AIC criteria. This approach involved (1) the selection of the candidate model set, (2) the estimation of the model and determination of AIC, and (3) a diagnostic check. According to the Box-Jenkins approach (Section 6.5), selection of the candidate models first starts with the evaluation of sample estimates of the ACF and the PACF, in order to determine three major orders of the ARIMA models. As it can be noticed from Fig. 6.4, there is a gradual decrease in autocorrelation values which indicates a long-term trend. Also, there is one very large spike on the PACF plot. This reflects the need for

216

Optimization models for rail car fleet management

Table 6.12 Substitution cost rijtt0 Route

t 5 1, t0 5 5

t 5 3, t0 5 5

t 5 2, t0 5 6

t 5 4, t0 5 6

A-B A-C A-D B-A B-C B-D C-A C-B C-D D-A D-B D-C

10 20 15 15 20 25 15 25 10 10 20 15

15 25 20 20 30 35 20 35 15 15 30 10

10 15 30 15 20 10 10 25 15 25 20 10

15 20 35 20 25 15 15 20 20 35 25 15

including a first-order difference term in the model structure (d ¼ 1). Therefore, the basic structure of the alternative ARIMA models has the form ARIMA(p, 1, q). Now, by evaluating the different AR and MA orders (p and q), ARIMA(2, 1, 14) was selected as the most appropriate model with the lowest AIC (AIC ¼5.3562) with R-squared equal to 92.6%. Model equation has the following form:  1 + 0:577ω + 0:284ω2 ð1  ωÞyl ¼ð1  0:798ω  0:264ω2 + 0:087ω3  0:002ω4 + 0:136ω5 + 0:187ω6 + 0:008ω7  0:108ω8 + 0:088ω9 + 0:006ω10  0:013ω11  0:679ω12  0:401ω13  0:145ω14 Þεl (6.106)

Diagnostic checking proves the stationarity and invertibility of the selected model without redundant parameters. The absence of autocorrelation between residuals in different lags (Fig. 6.5) indicated that the residuals were white noise (Ljung-Box Q ¼ 6.295, P-value >0.05). Forecasting was performed by incorporating the identified ARIMA model in a state-space form. As explicit equation of the identified ARIMA(2,1,14) model was obtained as follows. First, the state vector ρl of ARIMA(2,1,14) model was defined as ρl ¼ ðyl1 , y∗l ,  0:577y∗l + 0:798εl + 0:264εl1 + ⋯ + 0:145εl13 , 0:264εl  0:087εl1 + ⋯ + 0:145εl12 , …,0:401εl + 0:145εl1 , 0:145εl Þ (6.107)

γ it

γ Pit

γ it

Station 1

Freight car type

1 2 3 4

ηit

γ 11 ¼ 71.70 γ 12 ¼ 94.84 γ 13 ¼ 70.03 γ 14 ¼ 62.72

η11 ¼ 50.69 η12 ¼ 64.11 η13 ¼ 38.97 η14 ¼ 40.24

γ 31 ¼ 75.05 γ 32 ¼ 69.79 γ 33 ¼ 89.03 γ 43 ¼ 70.44

η31 ¼ 63.49 η32 ¼ 50.97 η33 ¼ 66.92 η43 ¼ 49.95

γ Pit

Station 2

γ 311 ¼ 49.70 γ 312 ¼ 63.01 γ 313 ¼ 38.57 γ 314 ¼ 39.55

γ 21 ¼ 74.39 γ 22 ¼ 69.41 γ 23 ¼ 67.99 γ 24 ¼ 68.63

Station 3

1 2 3 4

ηit

η21 ¼ 50.80 η22 ¼ 44.99 η23 ¼ 48.97 η24 ¼ 44.05

γ 321 ¼ 49.92 γ 322 ¼ 44.34 γ 323 ¼ 47.99 γ 324 ¼ 43.35

Station 4

γ 331 ¼ 62.06 γ 332 ¼ 50.01 γ 333 ¼ 65.66 γ 334 ¼ 49.02

γ 41 ¼ 67.69 γ 42 ¼ 64.16 γ 43 ¼ 84.32 γ 44 ¼ 78.75

η41 ¼ 39.60 η42 ¼ 47.97 η43 ¼ 59.14 η44 ¼ 46.00

γ 341 ¼ 39.16 γ 342 ¼ 46.97 γ 343 ¼ 58.20 γ 344 ¼ 45.49

Stochastic model for heterogeneous rail freight car fleet management

Table 6.13 Rail freight car inventory level and ordering quantity

217

218

Optimization models for rail car fleet management

Number of rail freight cars on state

100.00

80.00

60.00

40.00

100 97 94 91 88 85 82 79 76 73 70 67 64 61 58 55 52 49 46 43 40 37 34 31 28 25 22 19 16 13 10 7 4 1

Time (days)

Fig. 6.3 Time series plot: daily state of rail freight cars of type 1 in station 1.

Fig. 6.4 Sample autocorrelation function (ACF) and partial autocorrelation function (PACF) of the time series data.

With the corresponding disturbance vector I l ηl ¼ðO11 , εl + 1 , 0:798εl + 1 ,0:264εl + 1 ,  0:087εl + 1 , 0:002εl + 1 ,  0:136εl + 1 ,  0:187εl + 1 ,  0:008εl + 1 ,0:108εl + 1 ,  0:089εl + 1 ,  0:006εl + 1 ,0:013εl + 1 , 0:679εl + 1 ,0:401εl + 1 ,0:145εl + 1 Þ0 (6.108)

Stochastic model for heterogeneous rail freight car fleet management

Lag

Residual ACF

219

Residual PACF

20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

–1.0

–0.5

0.0

0.5

1.0

–1.0

–0.5

0.0

0.5

1.0

Residual

Fig. 6.5 Autocorrelation function (ACF) and partial autocorrelation function (PACF) of residuals.

The transition matrix Y l is, therefore, 16  16 and Y l and Sl are given by

ð6:109Þ Y d ¼ ½I14 , Y c ¼ ½0:5769  0:2842 O141 0 , Y e ¼ ½O115 , Sl ¼ ½1 1 O114  Then the variance-covariance matrix of the state disturbances can be defined as   O11 O115 0 (6.110) I lI l ¼ 0 O151 I ∗l I ∗l 0

The 15  15 stationary part of this variance matrix is given by I ∗l I ∗l , Variance matrices M l and V l have dimensions (11), values 9.54 and 0, respectively. The mean of the initial state vector is given by a ¼ E(α1) ¼ O161 and the corresponding variance matrix is given by   kI 1 O115 with k ! ∞ (6.111) Π¼ O151 Π ∗1515 Matrix Π ∗1515 represents the unconditional variance matrix for the stationary part of the state vector.

2

15:22 6 10:67 6 6 3:46 6 6 21:37 6 6 0:22 6 6 21:06 6 6 22:15 6 ∗ Π 1515 ¼ 6 6 21:07 6 0:58 6 6 21:21 6 6 21:12 6 6 0:90 6 6 7:12 6 4 4:13 1:38

10:67 12:92 5:58 0:14 20:41 20:40 21:01 20:99 20:66 21:15 21:01 20:54 4:89 3:02 1:10

3:46 5:58 7:57 3:18 0:87 20:73 0:01 0:11 20:76 21:82 20:79 20:71 1:38 0:89 0:36

21:37 0:14 3:18 6:91 3:4 0:86 20:39 0:48 0:13 21:03 21:60 20:77 20:74 20:33 20:12

0:22 20:41 0:87 3:4 6:84 3:40 0:75 20:55 0:47 0:22 21:11 21:60 20:76 20:18 0:03

21:06 20:40 20:73 0:86 3:40 6:84 3:40 0:75 20:54 0:47 0:22 21:11 21:60 20:78 20:18

22:15 21:01 0:01 20:39 0:75 3:40 6:66 3:16 0:74 20:40 0:36 0:21 21:09 20:72 20:25

21:07 20:99 0:11 0:48 20:54 0:75 3:16 6:33 3:14 0:94 20:56 0:34 0:29 0:11 20:01

0:58 20:66 20:76 0:13 0:47 20:54 0:74 3:14 6:32 3:15 0:93 20:56 0:35 0:28 0:14

21:21 21:14 21:82 21:03 0:22 0:47 20:40 0:94 3:15 6:21 3:24 0:94 20:58 20:34 20:12

21:12 21:01 20:79 21:60 21:11 0:22 0:36 20:56 0:93 3:24 6:14 3:24 0:95 20:00 20:00

0:90 20:54 20:71 20:77 21:60 21:11 0:21 0:34 20:56 0:94 3:24 6:14 3:24 0:99 0:01

7:12 4:89 1:38 20:74 20:76 21:60 21:09 0:23 0:35 20:58 0:95 3:24 6:14 3:15 0:94

4:13 3:02 0:89 20:33 20:18 20:78 20:72 0:11 0:28 20:34 20:00 0:99 3:15 1:73 0:55

3 1:38 1:10 7 7 0:36 7 7 20:12 7 7 0:00 7 7 20:18 7 7 20:25 7 7 20:01 7 7 0:14 7 7 20:12 7 7 20:00 7 7 0:01 7 7 0:94 7 7 0:55 5 0:20

Stochastic model for heterogeneous rail freight car fleet management

221

The obtained ARIMA-Kalman hybrid forecasting model was incorporated in the Matlab (MathWorks Inc., 2012). For determining the most suitable ARIMA model IBM SPSS Statistics 19 (trial version) was used. Conversion into the state-space model and forecasting were performed with the SSM Toolbox (Peng and Aston, 2006) and SSMMATLAB (Gomez, 2012). The last 15 observations were used for the assessment of the model ARIMA-Kalman model performances. This was performed by comparison of the hybrid ARIMA-Kalman model with the ARIMA model, and evaluating the daily forecast errors based on the following accuracy measures: root mean square error (RMSE), mean error (ME), and mean absolute percentage error (MAPE). According to the results of these accuracy measures, the ARIMAKalman model reports better performances in comparison to ARIMA. It results in a 50.6 reduction in RMSE, 57.1 reduction in MAE, and 55.5% reduction in MAPE. Fig. 6.6 presents the fitting and forecasting results. The same procedure was repeated for other freight car types and all stations. Model equations and short-term forecasts are summarized in Table 6.14. The proposed solution procedure was applied for determining the optimal control law for rail the freight car fleet size and allocation system. The computer algorithm for performing all computations was coded in the Matlab software (MathWorks Inc., 2012). Experiments were carried out on the Intel Core i3-2350M (2.30 GHz) personal computer. Figs. 6.7–6.10 summarize the obtained results.

Fig. 6.6 ARIMA-Kalman: fitting and forecasting results.

Table 6.14 Models and forecasts of cars at stations (SD) per day Freight car type

Time period Forecasting model

1

2

3

61.76 (4.64)

64.04 (5.28)

65.36 (7.96)

55.06 (8.9) 46.72 (9.81)

55.18 (6.2) 46.19 (10.25)

52.65 (8.95) 47.23 (10.31)

40.26 (4.35)

40.96 (4.62)

41.34 (5.96)

Station A

I

II III IV

ð1 + 0:577B + 0:284B2 Þð1  BÞyn ¼ ð1  0:798B  0:264B2 + 0:087B3  0:002B4 + 0:136B5 + 0:187B6 + 0:008B7  0:108B8 + 0:088B9 + 0:006B10  0:013B11  0:679B12  0:401B13  0:145B14 Þεn (1  0.100B  0.517B2)(1  B)yn ¼ (1  329B  0.491B2 + 0.169B3)εn ð1 + 0:035B + 0:202B2 + 0:845B3 + 0:154B4 Þð1  BÞyn ¼ ð1  0:193B + 0:263B2 + 0:769B3 Þεn ð1 + 0:598BÞð1  BÞyn ¼ ð1 + 0:078B + 0:078B2 + 0:022B3  0:091B4 + 0:160B5 + 0:097B6  0:0053B7  0:219B8 0:140B9 + 0:216B10  0:056B11 + 0:555B12 Þεn Station B

I II III

IV

(1  1.079B + 0.355B )(1  B)yn ¼ (1  1.000B)εn (1  0.739B + 0.186B2)(1  B)yn ¼ (1  0.542B  0.064B2)εn ð1  BÞyn ¼ ð1 + 0:164B  0:180B2 +0:118B3  0:177B4 + 0:151B5  0:388B6 + 0:005B7 + 0:039B8 0:444B9  0:144B10 + 0:422B11 + 0:522B12 Þεn (1 + 0.935B + 0.126B2)(1  B)yn ¼ (1 + 0.997B)εn 2

39.77 (3.13) 41.66 (5.91) 46.38 (7.19)

40.62 (2.41) 41.58 (6.25) 47.10 (8.21)

40.90 (4.72) 41.41 (6.11) 49.76 (8.50)

61.37 (9.12)

60.65 (8.20)

61.28 (7.91)

Station C

I II III IV

(1  0.597B + 0.039B + 0.214B  0.059B )(1  B)yn ¼ (1  0.807B)εn ð1 + 0:843B  0:227B2  0:378B3 + 0:288B4 Þð1  BÞyn ¼ ð1 + 0:684B  0:698B2  0:968B3  0:025B4 Þεn ð1 + 0:918B + 0:107B2 + 0:161B3 + 0:568B4 Þð1  BÞyn ¼ ð1 + 0:674B  0:518B2  0:643B3  0:001B4  0:204B5 Þεn ð1 + 0:184B  0:092B2 + 0:130B3 + 0:324B4  0:423B5 Þð1  BÞyn ¼ ð1  0:212B  0:530B2  0:008B3 + 0:065B4 0:820B5 + 0:299B6 + 0:734B7 Þεn 2

3

4

42.42 (5.21) 55.66 (4.90)

45.13 (6.35) 52.41 (3.26)

46.29 (5.01) 53.12 (4.05)

52.20 (5.01)

52.11 (3.92)

57.42 (4.65)

49.58 (7.29)

48.02 (7.35)

45.31 (7.85)

47.43 (3.23)

48.02 (3.56)

49.93 (3.12)

59.43 (4.51)

59.16 (6.20)

57.95 (5.36)

50.57 (6.36)

51.46 (4.89)

52.92 (7.20)

60.00 (5.25)

60.46 (3.92)

61.59 (6.00)

Station D

I

II

III IV

ð1 + 0:270B  0:410B2 + 0:415B3 + 0:478B4  0:319B5 Þð1  BÞyn ¼ ð1  0:059B  0:597B2 + 0:486B3 + 0:212B4 0:163B5  0:023B6  0:546B7 Þεn ð1  0:931B + 0:688B2  0:551B3 Þð1  BÞyn ¼ ð1  0:560B  0:321B2  0:356B3 0:076B4 + 0:471B5  0:158B6 Þεn ð1  0:091B + 0:657B2 + 0:538B3 Þð1  BÞyn ¼ ð1  0:078B + 0:486B2 + 0:646B3  0:186B4 Þεn ð1  1:123B + 0:530B2 + 0:276B3  0:326B4 + 0:159B5 Þð1  BÞyn ¼ ð1  0:945B  0:176B2 + 1:175B3 0:372B4  0:746B5 + 0:741B6 Þεn

224 Optimization models for rail car fleet management

Fig. 6.7 Control actions (loaded cars).

E

E

E E E

E E

E E

Stochastic model for heterogeneous rail freight car fleet management

Control action

E

E

E

225

Fig. 6.8 Control actions (empty cars).

E

E E

E

E E E

E E

E E E

E E

E

E E

E E

E E

E E

E

226

Unmet demand

V

V

V

Fig. 6.9 Unmet demands.

V

V

V

V

V

V

V

V

V

V

V

V V

V

V

V

V

V

V

V

V

V

V

V

V

V V

V V

V V

Optimization models for rail car fleet management

V

V

Number of cars

Stochastic model for heterogeneous rail freight car fleet management

Fig. 6.10 Number of cars in stations by periods.

227

228

Optimization models for rail car fleet management

The model provides rail network information related to the number of cars (Sit(n)) in all stations by type and period, unmet orders (Uijt(n)) and empty and loaded freight car movements for every period, and between all pairs of stations (Xijtt’(n), Yijtt’(n)). Now, it is possible to determine the fleet size required for the proper functioning of the rail freight car system. The fleet size actually represents the average number of rail cars available within the planning period. To get this result, the following expression is applied: FS ¼ where Að1Þ ¼

X XX

P 1X AðnÞ P n¼1

(6.112)

ðEijtt0 ð1Þαijt ð0, 1Þ + Fijtt0 ð1Þθijt ð0, 1Þ + Eijtt0 ð2Þαijt ð0, 3Þ

i, j2N t2T t0 2T t

+ Fijtt0 ð2Þθijt ð0, 3Þ + Eijtt0 ð3Þαijt ð0, 2Þ + Fijtt0 ð3Þθijt ð0, 2ÞÞ +

XX

Sit ð1Þ

i2N t2T

Að2Þ ¼

X XX

(6.113) ðEijtt0 ð2Þαijt ð0, 1Þ + Fijtt0 ð2Þθijt ð0, 1Þ + Eijtt0 ð3Þαijt ð0, 3Þ

i, j2N t2T t0 2T t

+ Fijtt0 ð3Þθijt ð0, 3Þ + Eijtt0 ð1Þαijt ð0, 2Þ + Fijtt0 ð1Þθijt ð0, 2ÞÞ +

XX

Sit ð2Þ

i2N t2T

Að3Þ ¼

X XX

(6.114) ðEijtt0 ð3Þαijt ð0, 1Þ + Fijtt0 ð3Þθijt ð0, 1Þ + Eijtt0 ð1Þαijt ð0, 3Þ

i, j2N t2T t0 2T t

+ Fijtt0 ð1Þθijt ð0, 3Þ + Eijtt0 ð2Þαijt ð0, 2Þ + Fijtt0 ð2Þθijt ð0, 2ÞÞ +

XX

Sit ð3Þ

i2N t2T

(6.115) The optimal fleet size (FS) is 6120 freight cars. This fleet generates the total cost of J ¼ 7.46  106 monetary units. Table 6.15 summarizes the results for other tests that involve an increased number of railway stations. The CPU time for the performed tests is insignificant.

Stochastic model for heterogeneous rail freight car fleet management

229

Table 6.15 Alternate example problems MPC

Problem

Number of stations

Number of periods P

Rail freight car type

FS

I II III IV I II III IV I II III IV I II III IV I II III IV

272.6 298.9 181.2 210.2 1555.1 1599.1 1695.5 1270.6 1925.3 1258.6 1100.9 1736.3 2929.2 2027.1 1645.6 1472.3 3834.2 3439.9 3952.3 3179.1

1

2

4

2

4

4

3

5

4

4

10

4

5

20

4

J (106 m.u.)

CPU time (s)

2.49

00:00:01

7.46

00:00:02

8.95

00:00:04

18.3

00:00:12

38.2

00:00:21

CHAPTER 7

Distributed and decentralized approaches for rail freight car management Contents 7.1 Distributed model predictive rail freight car management 7.1.1 Problem description 7.1.2 Cooperative MPC for freight car flow planning 7.2 Decentralized model predictive rail freight car management 7.3 Numerical example

232 233 235 239 240

Previous chapters treat the problem of rail freight car fleet management in a centralized way. However, the rail freight car fleet system represents a very large-scale system which involves a number of interacting subsystems. The railway network is usually composed of regions and each region can be further divided into districts. The flows of freight cars are controlled by local agents (controllers) responsible for efficient freight car management. Centralized control of these very complex subsystems can be very difficult despite the development of high-speed computers and fast algorithms. Also, there are issues related to the reliability of information sharing between subsystems as well as communication limitations. On the other hand, in case of a completely decentralized freight car fleet system, there is a problem related to control and a lack of interaction, which generates suboptimal freight car control actions. In this case, a cooperative distributed model predictive control (DMPC) may represent the solution. Local agents or controllers, having some knowledge on the flows of freight cars from other subnetworks, will be charged with freight car flow optimization on their subnetworks. Fig. 7.1 illustrates centralized (a), decentralized (b), and distributed (c) MPC architecture for a rail freight car fleet management system comprised from two subsystems. The aim of this chapter is to design a DMPC scheme and to compare it with decentralized and centralized schemes on a common benchmark process related to freight car fleet size and allocation. Optimization Models for Rail Car Fleet Management https://doi.org/10.1016/B978-0-12-815154-9.00007-1

© 2020 Elsevier Inc. All rights reserved.

231

232

Optimization models for rail car fleet management

MPC

MPC U1(n)

X1(n)

Freight car fleet subsystem 1

Freight car fleet subsystem 2

Rail freight car fleet system

U2(n)

U1(n)

U2(n)

Freight car fleet subsystem 1

MPC

X1(n)

X2(n)

(A)

Rail freight car fleet system

Freight car fleet subsystem 2 X2(n)

(B) MPC

MPC

U1(n)

U2(n)

Freight car fleet subsystem 1 X1(n)

Rail freight car fleet system

Freight car fleet subsystem 2 X2(n)

(C) Fig. 7.1 Centralized (A), decentralized (B), and distributed MPC (C) architecture for rail freight car fleet management.

7.1 Distributed model predictive rail freight car management Cooperative DMPC (CDMPC) approach is also aligned with recently proposed cooperative strategies for improving the rail freight car fleet utilization as well as the position of single freight car management on the EU railway market (Smart-Rail, 2016). DMPC is a control approach that copes with control problems of large- scale systems in which there are interorganizational connections between different subsystems involved in a joint activity, limited measurement capability and control access of different subsystems and also different potentially conflicting objective criteria of individual subsystems (Li et al., 2016). The general concept, industrial application, and future research directions of DMPC are presented in the books of Camponogara et al. (2002), Maestre and Negenborn (2014), Li and Zheng (2015), and Olaru et al. (2015). In this chapter, CDMPC approach is proposed for solving the rail freight car fleet management problem. The overall optimization problem is decomposed into subproblems, which consider the freight car allocation decisions in related subsystems. Individual decisionmaking units or controllers solve their own, local problems using the model

Distributed and decentralized approaches for rail freight car management

233

covering only their part of the railway network. Model predictive control (MPC) is applied as a solution approach for every local problem. All regional controllers are mutually connected by interconnecting links, in this case, railway lines on which the empty and loaded freight car flows between regions are interchanged. More precisely, these local problems are not considered as isolated but dependent on the solution of the MPC problem of the surrounding controllers. In order to solve the CDMPC problem, a parallel augmented Lagrangian relaxation approach (Negenborn et al., 2008) is applied. In this case, input parameters are taken as exact values and only one type of freight cars was considered.

7.1.1 Problem description Let’s consider a railway transport network M(N, L) composed of a set of nodes N ¼ [s¼1, …, Nsub Ns and railway lines L ¼ [s¼1, …, Nsub Ls . Cooperative freight car management can be established between a set of railway operators Nsub in the M(N, L) network where every railway operator manages freight flows on their own subnetwork which does not include common nodes and links with any other subnetwork Ns \ Nk ¼ ∅, Ls \ Lk ¼ ∅, 8 s, k ¼ 1, …, Nsub. Subnetworks mutually interact through interconnecting lines. Freight car manager s plans the flows of freight cars on subnetwork Ns by solving the optimal freight car allocation problem over P periods of the planning horizon

ð7:1Þ subject to system dynamics and planning constraints Xs ðn + 1Þ ¼ Fs ðnÞXs ðnÞ + Gs Us ðnÞ + Φs ds ðnÞ, n ¼ 0,1,…, P  1

(7.2)

Ds Xs ðnÞ  Ks

(7.3)

Xs ðnÞ, Us ðnÞ  O

(7.4)

The stated objective criteria and constraints are based on the linear discretetime rail freight car fleet sizing and allocation model presented in the previous chapter. Therefore, the objective function (7.1) minimizes the quadratic costs of owning and distributing empty and loaded freight cars but only on a considered subnetwork. The first term is the cost of the rail freight car

234

Optimization models for rail car fleet management

subsystem Ns in the last period of the planning horizon, which is the function of the state vector Xs(n) resulting from control actions Us(n) implemented during the previous periods. The second term (7.2) represents the sum of costs over the P  1 periods of the planning horizon and it is the function of the state vector Xs(n) and control vector Us(n) in a given subnetwork Ns. Matrices in objective functional are given as follows: As(n): Composed of rail freight car supplying costs π si for each station i on a subnetwork s, and quadratic cost coefficients of freight cars in transit, Q. Supplying costs are determined based on the following formula: π si ¼

N   1 X χ  eji + ð1  χ Þlji , t ¼ 1,…, T N  1 j¼1

(7.5)

eji and lji are the unit costs of empty and loaded trips whereas χ represents the empty freight car-running coefficient (ratio of the number of empty freight cars to the total number of freight cars). Traveling costs are given as a linear function of the number of cars on the considered route: xjim ðnÞQ ¼ xjim ðnÞðq0 + qxjim ðnÞ

P X

sajis ðn + s  mÞ

(7.6)

sbjis ðn + s  mÞ

(7.7)

s¼m

yjim ðnÞQ ¼ yjim ðnÞðq0 + qyjitm ðnÞ

P X s¼m

Bs: matrix with costs of unmet demand π ui . Ls: matrix with freight car holding costs π hi . Γ s(P): matrix that contains rail freight car supplying costs π si and the costs of cars in transit which are considered as a linear function of the number of cars on a considered section s. The dynamic model of the subsystem s is represented by the constraint (7.2). The state in the next period Xs(n + 1) is a function of the state in the previous period, Xs(n), the control actions Us(n) and the vector of demand rates, d(n) which are considered as disturbances in the subsystem. Matrices in the dynamic model are defined as follows: Fs(n): state transition matrix; Gs: system control matrix; Φs: system disturbance matrix.

Distributed and decentralized approaches for rail freight car management

235

The dynamic model (5.2) incorporates the following state and unmet demand relations: Sis ðn + 1Þ ¼ Sis ðnÞ +

N X  X  Fjis ðmÞ  θjis ðm, n + 1Þ + Ejis ðmÞ  αjis ðm, n + 1Þ j¼1 m