Problem Solving In Operation Management [1st ed.] 9783030500887, 9783030500894

This volume examines problem solving and applied systems aimed at improving performance and management of organizations.

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Problem Solving In Operation Management [1st ed.]
 9783030500887, 9783030500894

Table of contents :
Front Matter ....Pages i-xix
Front Matter ....Pages 1-1
Theoretical-Methodological Basis for Complex Organization Diagnosis (Felipe de Jesús Lara-Rosano)....Pages 3-16
Methodology for Building Trend Scenarios (Gabriel de las Nieves Sánchez-Guerrero)....Pages 17-45
Consulting as a Systemic Intervention Process (Benito Sánchez-Lara, Oscar Everardo Flores-Choperena)....Pages 47-64
The Role of Technological, Economic, and Usage Ruptures in the Innovation Process (Cozumel A. Monroy-León)....Pages 65-79
Digraphs in the Analysis of Systems’ Representation of Mathematical Knowledge (Patricia Esperanza Balderas-Cañas)....Pages 81-102
Front Matter ....Pages 103-104
Decision-Making with Multicriteria Optimization and GIS for Park Locations (Mayra Elizondo-Cortés, Adela Jiménez-Montero)....Pages 105-115
A Service Location Model in a Bi-level Structure (Zaida E. Alarcón-Bernal, Ricardo Aceves-García)....Pages 117-133
Determining the Demand in Inventory Policies for Mexican Companies Using Fuzzy Sets (Ricardo Aceves-García, Zaida E. Alarcón-Bernal)....Pages 135-157
Back Matter ....Pages 159-162

Citation preview

Patricia Esperanza Balderas-Cañas  Gabriel de las Nieves Sánchez-Guerrero  Editors

Problem Solving In Operation Management

Problem Solving In Operation Management

Patricia Esperanza Balderas-Cañas Gabriel de las Nieves Sánchez-Guerrero Editors

Problem Solving In Operation Management

Editors Patricia Esperanza Balderas-Cañas Faculty of Engineering National Autonomous University of Mexico Mexico City, Mexico

Gabriel de las Nieves Sánchez-Guerrero Faculty of Engineering National Autonomous University of Mexico Mexico City, Mexico

ISBN 978-3-030-50088-7    ISBN 978-3-030-50089-4 (eBook) https://doi.org/10.1007/978-3-030-50089-4 © Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Prologue

Currently, Systems Thinking is a holistic, transdisciplinary, dynamic, and constructivist approach that allows explaining realities through their understanding and knowledge, considering the context and purpose of the systems that give it meaning, structures and functions that make it up, and processes, procedures, and actors that involved create a complex dynamic. Since its inception, the metadiscipline of Systems Thinking has linked thought and action, and has traveled between the fields of scientific research and problem solving. It is through plural participation that its action is focused primarily on complex systems, either from the analysis of problematic situations, planning, design, optimization and even implementation. The book shows a brief look at the potential of Systems Thinking in Problem Solving for intervention, management, and planning in organizations. It presents eight research works that are integrated in two parts: methodologies and techniques. The contributions with a methodological orientation are shown from the first to the fifth chapter of the book, and from the sixth to the eighth chapter, the contributions are directed towards the techniques. It is increasingly common to find the use of Systems Thinking in the solution of problems of private, governmental, and social sector organizations. Systems Thinking is a conceptual framework that allows understanding reality and with it addressing the problems of organizations making use of mathematics, dynamic systems, optimization methods, planning, economics, social sciences, and behavior sciences among others. It combines theory and practice, quantitative and qualitative aspects, both necessary and complementary worlds for the solution of problems. The country needs to plan its future. The lack of planning in our country for solution of problems has led us to react more than to prevent or design our future. For this reason, faced with the complexity we find the problems once we have them, although we must accept that more and more we are looking for an objective image of the country we want. Perhaps if we had had this image in the past, we would not have lost half of our territory or we would not have made the same mistakes year after year. Although the General Law of Planning of the Republic has been enacted since 1930, it was not until 1980 that the first Global Development Plan 1980–1982 was drawn up. We can say that a culture of planning in the country is beginning to be developed. v

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Prologue

In order to act and transform reality, it is also necessary to have a set of methods and techniques for solving problems, and in this sense also in the country there has been a technical acculturation in the public, private, and social sectors. Mathematics is increasingly used. It is no longer seen as that cold, abstract part full of symbols. Today mathematics is immersed in planning, optimization methods, and simulation having an innumerable set of applications. In tune with the above, the book is emblematic. In its first part, Chaps. 1, 2, 3, 4, and 5, it presents a series of chapters aimed at developing theoretical-­methodological aspects. The second part, Chaps. 6, 7, and 8, focuses on the use and development of models at the technical level, which allow us to approach the knowledge of reality and the solution of specific problems. In order to improve the functioning and management of the organizations, the book proposes in its first chapter the theoretical basis for the diagnosis of organizations that is constructed from the point of view of the complexity sciences, the conceptual principles for the elaboration of a diagnosis and a procedure to carry it out, from the point of view of the sciences of complexity, and new social theories. A relevant stage in this procedure is the dynamic analysis of the organization. In this stage, various elements that define its complexity are identified, such as its attractors, branches, chaotic states, strange attractors, situations on the edge of chaos, and its process or auto organization attempts, which can serve as the basis for building models of computational simulation. Starting out with the theoretical support of systems thinking, the methodological basis of interactive planning and the necessary and complementary use of quantitative and qualitative techniques, the second chapter offers a participative process for building trend scenarios. Such process is comprised of five phases: (1) definition of the system and explanation of current situation, (2) forecasts integration, (3) incorporation of the predictions, (4) future image creation, and (5) description of the connection between the visualized present and future. Using this process, the case study Valle de Toluca Aquifer Scenario by 2020 is presented. Chapter 3 analyzes consultancy as a systemic intervention process. Three explanation lines are defined about the ineffectiveness of such activity: (a) the conditioning of consultancy to client’s preferences, (b) the conditioning of consultancy to the consultant’s practices and knowledge, and (c) the dominating factors in the consultancy scenario which may be certain techniques, practices, tools, methods, and methodologies at time of implementation. Furthermore, on the basis of Midgley’s systemic intervention notion, systemic theoretical methodological elements are identified and found in the consultancy process, which establish favorable conditions to the process. The fourth chapter analyzes the innovation process, seeking to maintain or achieve a competitive advantage in the organization. The author establishes that the innovation process that occurs in organizations has no lineal pathway nor is manifested in quiet organizational conditions. It is postulated that three ruptures occur during the process: the rupture of use—associated to a need of the selected market, the technological rupture—associated to the technological requirements, and the economic rupture—associated to the viable strategic price. The process focuses on defining at least three technical objects to be assessed (which can be products, processes, procedures, services, or methods). The objects are evaluated using different tools, which value its potential in all three mentioned ruptures.

Prologue

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Part one of the book concludes with Chap. 5, to show the application of digraphs in the analysis of mathematical knowledge representation systems in the field of secondary education. The methodology is presented and discussed to analyze, under a systemic approach, the visual reasoning procedures that are given with the use of mathematical representations in the learning of differential calculus topics, in the upper intermediate level. The methodology helps in knowing how the one who learns acquires and utilizes some mathematical representation systems and how he/ she organizes them to produce acceptable answers in the school setting. The systems are modeled using digraphs, and through an experience with high school students, the robustness of the proposed methodology is shown. In the second part of the book, the sixth chapter develops a modeling process which allows for park location out of the selection of green areas in an urban area. The process begins with the structuring of the problem and ends with the use of a procedure that interacts between a Geographical Information System (GIS) and a discrete location multi-objective optimization model. The incorporation of GIS facilitated the visual representation of map information as well as data that the zones have. Also, a study case carried out in Delegación Cuauhtémoc of Mexico City is described. The seventh chapter proposes a model for locating bi-level services using a drug distribution network in the State of Mexico, which has been originally presented as one with a sole objective. The strategy to solve multi-objective problems has been useful in situations where there is more than one objective, which in many cases may contradict themselves, but this approach does not consider the possible inter-­ dependency among them, a condition that takes multilevel programming into account. The proposed model was applied to a distribution of medication networks in the State of Mexico, for which it offers the best locations for warehouses. Finally, the book concludes with the eighth chapter that offers an alternative for the determination of the demand in the control of inventories using fuzzy sets. It deals with the need that many Mexican enterprises have of having an alternative to determine the demand in inventory control, such being considered an additional or unnecessary cost, it’s carried out with the basis of experience and subjective judgments by the administration. To take advantage of this need, the use of fuzzy sets was used to determine the demand and its behavior in inventories control for the MRP models and EOQ. Considering the demand as a fuzzy number, its calculation is executed under uncertainty conditions, and in this way, it incorporates the subjective knowledge and the administrative experience for its determination. Finally, the thought of systems in the solution of problems, from the methods and techniques of optimization, planning, and simulation, will have an increasing development in the measure that they are directed to obtain the best possible result, and in this way raise the quality of life of society. These tools will be very powerful in the twenty-first century, and this book shows it. Mexico City, Mexico  

Manuel Ordorica Mellado Patricia Esperanza Balderas-Cañas Gabriel de las Nieves Sánchez-Guerrero

Prologue

viii WATER SCENARIO BUILDING TREND SCENARIOS

Consulting process

CONSULTING CHAP 3 mental framework

INTERVENTION VISUAL THINKING SCHOOL LEARNING

MATH REPRESENTATIONS

NETWORKS

BI-LEVEL PROGRAMMING CHAP 7

DISTRIBUTION

PROBLEM SOLVING FUZZY OPTIMIZATION

ORGANIZATIONS

INVENTORY AND FUZZY DEMAND

PLANNING FUZZY OPTIMIZA

MRP FUZZY

THINKING OF APPLIED SYSTEMS EOQ FUZZY

COMPLEXITY

DIAGNOSIS DYNAMICS

DIFFERENTIATION MANAGEMENT

INNOVATION PROCESS CHAP 4 RUPTURES

MULTICRITERIA LOCATION

Structure of Part I

MODELING PROCESS LOCALIZATION

Introduction

This book is about problem solving in management of systems engineering cases. The content is presented in two parts, one for methodologies and the second for techniques. In the first chapter, a methodology for the diagnosis of the organization dynamics is discussed from the point of view of the Complexity Approach in order to improve and solve problems related with its operation and management. The author of the second chapter reasons about a process for building trend in planning scenarios for problem solving in public and private organizations by five phases: (1) system definition and explanation of the current situation, (2) integration of forecasts, (3) integration of the predictions, (4) the construction of the future image, and (5) a description of the connection between the present and the future. The third chapter is dedicated to explaining the organizational consulting ineffectiveness where three reasons are presented. Two of those reasons are associated to the actors involved in the consulting process; the third reason is associated to the intervention process in problem solving. In the fourth chapter, the objective of the author is identify the technological, economic, and usage ruptures, for the purpose of showing their importance in the stabilization of a process, aimed at reaching a transformation, named the innovation process. In problem solving, the innovation will be considered as “the introduction of a new or significantly improved product or process of a new marketing or organizational method in the company’s internal practices or external relations.” The author of the fifth chapter presents and discusses a methodology to analyze, under a systemic approach, the visual reasoning processes given with the use of mathematical representations when learning Differential Calculus at a high school level. Her interest knows how those who learn, acquire, and use some of the systems of mathematical representation organize them to produce acceptable responses in the school environment.

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Contents

Part I Methodologies 1 Theoretical-Methodological Basis for Complex Organization Diagnosis ����������������������������������������������������    3 Felipe de Jesús Lara-Rosano 1.1 The Dynamic Diagnostic of a Complex Organization����������������������    3 1.2 Organizations as a Complex System������������������������������������������������    4 1.3 Analysis of Complex Organization��������������������������������������������������    6 1.3.1 Analysis of Complex Organization Dynamics����������������������    6 1.4 Diagnostic of Complex Organizational Dynamics ��������������������������   11 1.5 Conclusions��������������������������������������������������������������������������������������   15 References and Bibliography��������������������������������������������������������������������   15 2 Methodology for Building Trend Scenarios������������������������������������������   17 Gabriel de las Nieves Sánchez-Guerrero 2.1 Background ��������������������������������������������������������������������������������������   17 2.1.1 Scenarios in Interactive Planning������������������������������������������   18 2.1.2 Trend Scenarios��������������������������������������������������������������������   21 2.2 Proposed Procedure��������������������������������������������������������������������������   26 2.2.1 System Analysis and Current Situation Explanation������������   27 2.2.2 Forecasts Integration ������������������������������������������������������������   28 2.2.3 Predictions Integration����������������������������������������������������������   29 2.2.4 Construction of Future Image ����������������������������������������������   29 2.2.5 Connection Between Present and Future: Scenario Writing ������������������������������������������������������������������   29 2.3 Conclusions��������������������������������������������������������������������������������������   42 References��������������������������������������������������������������������������������������������������   43 3 Consulting as a Systemic Intervention Process ������������������������������������   47 Benito Sánchez-Lara and Oscar Everardo Flores-Choperena 3.1 Introduction��������������������������������������������������������������������������������������   47 3.2 Consultancy Problems����������������������������������������������������������������������   48 xi

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3.3 Consulting Process����������������������������������������������������������������������������   50 3.3.1 Kubr’s Consulting Process����������������������������������������������������   51 3.3.2 Morfín’s Consulting Process������������������������������������������������   51 3.3.3 Block’s Consulting Process��������������������������������������������������   52 3.3.4 Systems Method of Ochoa-Rosso����������������������������������������   52 3.3.5 Summarizing the Consulting Processes��������������������������������   53 3.4 Systemic Intervention ����������������������������������������������������������������������   54 3.4.1 Consultancy as a Systemic Intervention ������������������������������   54 3.4.2 Critical Thinking in Consultancy������������������������������������������   55 3.4.3 Systemic Intervention Conditions in Consulting Practice ����������������������������������������������������������   60 3.5 Conclusions��������������������������������������������������������������������������������������   62 References��������������������������������������������������������������������������������������������������   63 4 The Role of Technological, Economic, and Usage Ruptures in the Innovation Process��������������������������������������������������������   65 Cozumel A. Monroy-León 4.1 The Context of Innovation����������������������������������������������������������������   65 4.2 Problem to Be Addressed������������������������������������������������������������������   69 4.3 Innovative Process����������������������������������������������������������������������������   71 4.4 Conclusions��������������������������������������������������������������������������������������   78 References��������������������������������������������������������������������������������������������������   78 5 Digraphs in the Analysis of Systems’ Representation of Mathematical Knowledge ������������������������������������������������������������������   81 Patricia Esperanza Balderas-Cañas 5.1 Representation of Mathematical Knowledge������������������������������������   81 5.2 Visual Reasoning������������������������������������������������������������������������������   82 5.3 Representation Systems of Mathematical Knowledge and Visual Reasoning������������������������������������������������������������������������   84 5.3.1 Visual Reasoning and Visualization��������������������������������������   86 5.4 Analysis of Systems of Representation of Math Knowledge ����������   89 5.4.1 Categories ����������������������������������������������������������������������������   91 5.5 Conclusions��������������������������������������������������������������������������������������   95 5.5.1 Pedagogic Implications and Recommendations ������������������   96 Annex 1: Extract of the Teaching Guide��������������������������������������������������    97 Path of Water Flow Coming Out from a Hose ������������������������������������    97 Annex 2����������������������������������������������������������������������������������������������������    98 References and Bibliography��������������������������������������������������������������������  101 Part II Techniques 6 Decision-Making with Multicriteria Optimization and GIS for Park Locations��������������������������������������������������������������������  105 Mayra Elizondo-Cortés and Adela Jiménez-Montero 6.1 The Problem of Park Locations in Mexico City ������������������������������  105

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6.2 Modeling Process for Park Locations ����������������������������������������������  106 6.3 Structuring the Problem of Park Location and Mathematic Modeling Methodology������������������������������������������  107 6.4 Application for Delegación Cuauthémoc in Mexico City����������������  112 6.5 Conclusions��������������������������������������������������������������������������������������  114 References��������������������������������������������������������������������������������������������������  115 7 A Service Location Model in a Bi-level Structure��������������������������������  117 Zaida E. Alarcón-Bernal and Ricardo Aceves-García 7.1 Introduction��������������������������������������������������������������������������������������  117 7.2 Bi-level Programming Models����������������������������������������������������������  118 7.3 Model Approach��������������������������������������������������������������������������������  119 7.3.1 P-Median Location Model����������������������������������������������������  119 7.3.2 General Model����������������������������������������������������������������������  119 7.3.3 Formulation��������������������������������������������������������������������������  120 7.3.4 Bi-level Programming Problems������������������������������������������  121 7.3.5 Definitions����������������������������������������������������������������������������  121 7.3.6 Discrete Bi-level Problem����������������������������������������������������  122 7.4 Formulation of Bi-level Search Services Model������������������������������  123 7.4.1 General Model, Problem (4) ������������������������������������������������  123 7.4.2 Solution Method ������������������������������������������������������������������  125 7.4.3 Algorithm������������������������������������������������������������������������������  125 7.5 Model Application����������������������������������������������������������������������������  127 7.6 Conclusions��������������������������������������������������������������������������������������  131 References and Bibliography��������������������������������������������������������������������  132 8 Determining the Demand in Inventory Policies for Mexican Companies Using Fuzzy Sets��������������������������������������������  135 Ricardo Aceves-García and Zaida E. Alarcón-Bernal 8.1 Introduction��������������������������������������������������������������������������������������  135 8.1.1 Techniques Known and Used for Controlling Inventories in Mexico ����������������������������������������������������������  136 8.1.2 Data from Inventory Records, Problems with Uncertainty and EOQ and MRP Models����������������������  137 8.2 Economic Order Quantity (EOQ) Model with Fuzzy Demand, Without Production or Deficit������������������������  139 8.2.1 Analysis of Results ��������������������������������������������������������������  146 8.3 MRP Model Considering the Demand of a Fuzzy Number��������������  147 8.3.1 Demand as a Fuzzy Number ������������������������������������������������  154 8.4 Conclusions��������������������������������������������������������������������������������������  156 References��������������������������������������������������������������������������������������������������  156 Index������������������������������������������������������������������������������������������������������������������  159

About the Authors

Ricardo  Aceves-García  studied Chemical Engineering at the Autonomous University of Puebla, Mexico, and obtained a master’s in sciences and a PhD in Operational Research from the National Autonomous University of Mexico (UNAM). He has been working in various projects in the field of Transportation, Operational Research, and Optimization for both public and private organizations. At present, he is a Full-Time Professor and Researcher at the Engineering Graduated School of the UNAM, and his main lines of research are process optimization, network transportation, location services, and services operations. Zaida E. Alarcón-Bernal  obtained a doctoral degree in Engineering in the area of operations research, a master’s degree in Engineering (Hons.), and a degree in Actuarial Science, all from the National Autonomous University of Mexico. She is a Full-Time Professor at the Engineering Faculty of the UNAM in the Biomedical Systems Engineering Department. She belongs to the research group on Optimization in Service Companies in the Systems Engineering Department of the Engineering Faculty at the UNAM. She has participated in several consulting projects in public and private institutions. Her lines of research include hospital logistics, bi-level programming, and stochastic programming. Patricia Esperanza Balderas-Cañas  holds a PhD (Hons.) in Pedagogy, a master’s degree in Mathematics Education, and a degree in Mathematics, all of them from UNAM.  She is Full-Time and Definitive Professor in the Systems Engineering Department, UNAM-FI.  Her research lines are on operations research (inventory theory, combinatorial optimization, and systems modeling) and structural model for learning research. Felipe  de Jesús  Lara-Rosano  holds a Doctoral degree in Systems Engineering from the National Autonomous University of Mexico (UNAM). He was Senior Researcher in the Institute for Applied Sciences and Technology, where he coordinated the Group of Cybernetics and Complex Systems and now coordinates the Academic Program of Social Complexity at the Center for Complexity Sciences of xv

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About the Authors

the UNAM.  He is Life Senior Member of the IEEE and his current interests are around the analysis, modeling, and simulation of complex social systems. Mayra  Elizondo-Cortés  has a Doctoral and Master’s degrees in Operations Research from the School of Engineering of the UNAM and a degree in Applied Mathematics and Computation. She is a Full-Time Professor in the Systems Engineering Department, UNAM- FI. Her main lines of research are optimization, simulation, and multicriteria analysis applied essentially to logistics and supply chain processes. She has published notes on Computational Complexity and Simulation and articles in the Journal of Applied Research and Technology, International Journal of Automotive and Mechanical Engineering, and Journal of Engineering Research and Technology. Oscar  Everardo  Flores-Choperena  has a master’s degree in Engineering from UNAM, also studied Industrial and Systems Engineering at UNITEC. He participated in the TREPCAMP Entrepreneurship Advanced Program at UC Berkeley. Nowadays he is pursuing master´s degree in Technology Management at UNAM. He has been a business consultant for based technology enterprises, he applied lean methodologies for rapid business model validation, and supports the creation of 10 startups; he has applied consulting as a systemic intervention process for the university’s high-­technology business incubator at the National Autonomous University of Mexico. Adela Jiménez-Montero  has a degree in Applied Mathematics and Computation and a master’s degree in Operations Research from the School of Engineering at UNAM. She has 9 years of experience in risk for the financial sector, having worked with two of the largest banking institutions in Mexico. Her professional career began in BBVA, working as a risk adviser for RBB and mortgages products. She is currently working for HSBC as Risk Manager of retail customers and has collaborated in the process of defining original strategies for credit cards and customer view analysis, topics related to the pre-approval process of credits, costs and processes optimization, customer's segmentation, fraud monitoring and control, income inference, development of growth strategies, and risk control, taking care of both customer and bank benefits. Cozumel A. Monroy-León  obtained her PhD in Industrial Engineering from the National Polytechnic Institute of Grenoble, France. For 2 years she worked in the Health Policy Coordination of the Medical Benefits Directorate of the IMSS, designing projects that improve the quality of care of the beneficiaries. From 2006 to 2017, she was a subject Professor at the UNAM Engineering graduate where she taught courses on Technological Innovation, Technology Management, Organizational Change Management, and Technological Development for New Products.

About the Authors

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Manuel Ordorica-Mellado  is an actuary, demographer specialized in mathematical demography, a doctor of Operations Research, and a Mexican academic from El Colegio de México (COLMEX). He was President of the Mexican Demographic Society. He studied the actuary career at the Faculty of Sciences of the National Autonomous University of Mexico (UNAM). Later, he obtained the demography master's degree from COLMEX. He obtained his doctorate with an honorable mention in Engineering with a specialty in Operations Research at UNAM, with his doctorate thesis “The Kalman Filter in Population Planning.” He was Head of the Department of Demographic Evaluation and Analysis in the General Directorate of Statistics at the National Institute of Statistics, Geography and Informatics (INEGI). From 1977 to 1987, he was Director of Population Studies of the National Population Council (CONAPO), as well as consultant in population education for the United Nations Educational, Scientific and Cultural Organization (UNESCO). In the area of ​​teaching and academia, he is Professor-Researcher in Demography and the Doctorate in Population Studies at COLMEX.  He is a Member of the Editorial Board of Population magazine (INED, Paris). Manuel Ordorica es actuario, demógrafo especializado en demografía matemática, doctor en investigación de operaciones y académico mexicano de El Colegio de México. Fue presidente de la Sociedad Mexicana de Demografía. Estudió la carrera de actuario en la Facultad de Ciencias de la Universidad Nacional Autónoma de México. Posteriormente cursó la maestría de Demografía por El Colegio de México; se doctoró con mención honorífica en Ingeniería con especialidad en Investigación de Operaciones en la UNAM, con su tesis de doctorado El filtro de Kalman en la planeación demográfica. Fue jefe del departamento de evaluación y análisis demográfico en la Dirección General de Estadística en el Instituto Nacional de Estadística Geografía e Informática (INEGI). De 1977 a 1987 fue director de Estudios de Población del Consejo Nacional de Población (CONAPO), así como consultor en educación en población de la Organización de las Naciones Unidas para la Educación, la Ciencia y la Cultura (UNESCO). En el área de docencia y de la academia es coordinador de la Maestría en Demografía y del Doctorado en Estudios de Población en COLMEX. Fungió como director del Centro de Estudios Demográficos y de Desarrollo Urbano de El Colegio de México. Forma parte del Consejo Editorial de la revista Population (INED, París). Gabriel  de  las  Nieves  Sánchez-Guerrero  holds a Doctoral degree (Hons.) in Systems Engineering from the Engineering Graduate School at UNAM. Nowadays he is Full-Time and Definitive Professor in the Systems Engineering Department of the Faculty of Engineering (UNAM-FI). His research interests include the heuristic techniques for participatory planning and systems interactive evaluation. His current research is heuristic systemic evaluation. Benito Sánchez-Lara  holds a Doctoral and master’s degrees in the Engineering Program from the UNAM and the bachelor’s degree in chemical engineering. Nowadays is Full-Time Professor in the Systems Engineering Department of the UNAM-FI, where is involved in Organizational and Transportation Systems Planning. His research interests include logistics and supply chain, tactical planning, resilience, and viable systems analysis.

Contributors

Ricardo Aceves-García  Department of Systems, Faculty of Engineering, National Autonomous University of Mexico, Mexico City, Mexico Zaida  E.  Alarcón-Bernal  Department of Biomedical Systems Engineering, Faculty of Engineering, National Autonomous University of Mexico, Mexico City, Mexico Patricia  Esperanza  Balderas-Cañas  Department of Systems, Faculty of Engineering, National Autonomous University of Mexico, Mexico City, Mexico Felipe de Jesús Lara-Rosano  Complexity Sciences Center, National Autonomous University of Mexico, Mexico City, Mexico Mayra Elizondo-Cortés  Department of Systems, Faculty of Engineering, National Autonomous University of Mexico, Mexico City, Mexico Oscar  Everardo  Flores-Choperena  Materials Research Institute, National Autonomous University of Mexico, Chalco, Mexican State, Mexico Adela  Jiménez-Montero  Department of Systems, Faculty of Engineering, National Autonomous University of Mexico, Mexico City, Mexico Cozumel  A.  Monroy-León  Department of Systems, Faculty of Engineering, National Autonomous University of Mexico, Mexico City, Mexico Gabriel  de  las  Nieves  Sánchez-Guerrero  Department of Systems, Faculty of Engineering, National Autonomous, University of Mexico, Mexico City, Mexico Benito Sánchez-Lara  Department of Systems, Faculty of Engineering, National Autonomous University of Mexico, Mexico City, Mexico

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Part I

Methodologies

Introduction This book is about problem-solving in management of systems engineering cases. The content is presented in two parts, one for methodologies and the second for techniques. In the first chapter, a methodology for the diagnosis of the organization dynamics is discussed from the point of view of the complexity approach in order to improve and solve problems related to its operation and management. The author of the second chapter reasons about a process for building trend in planning scenarios for problem-solving in public and private organizations by five phases: (1) system definition and explanation of the current situation, (2) integration of forecasts, (3) integration of the predictions, (4) the construction of the future image, and (5) a description of the connection between the present and the future. The third chapter is dedicated to explaining the organizational consulting ineffectiveness where three reasons are presented. Two of those reasons are associated with the actors involved in the consulting process; the third reason is associated with the intervention process in problem-solving. In the fourth chapter, the objective of the author is to identify the technological, economic, and usage ruptures, for the purpose of showing their importance in the stabilization of a process, aimed at reaching a transformation, named the innovation process. In problem-solving, the innovation will be considered as “the introduction of a new or significantly improved product or process, of a new marketing or organizational method in the company’s internal practices or external relations.” The author of the fifth chapter presents and discusses a methodology to analyze, under a systemic approach, the visual reasoning processes given with the use of mathematical representations when learning differential calculus, at a high school level. The interest of the research focuses on recognizing how the students acquire and use some systems of mathematical representation and how they organize them to produce acceptable answers in the school environment.

Chapter 1

Theoretical-Methodological Basis for Complex Organization Diagnosis Felipe de Jesús Lara-Rosano

1.1  The Dynamic Diagnostic of a Complex Organization In the present context of rapid change and turbulence, it is needed to transform the organizations to give them greater viability, adaptability, efficiency, and dynamism (McMillan 2008). This implies a challenge that is neither a minor nor cosmetic: it is necessary to develop new strategies and methods to improve organizations. This leads to the design of new management practices and the development of different forms of interaction between the organizational system elements (Stacey 2001) and formulates operational processes more flexible and suitable to this circumstance while attaining standards of quality and excellence. To do this, it has been taken into account that from the 1980s onward, a new science approach was developed, which has revolutionized physics, chemistry, and biology (Nicolis and Prigogine 1994). It is the complex systems approach, which comprises, among others, the self-organized systems theory (Holland 1995), the complex adaptive systems theory, the dynamics of social networks theory (Newman et al. 2006), chaos theory (Eve 1997), and fractal geometry (Mandelbrot 1987). In the social sciences, innovative approaches to social theory have appeared also, based on the interaction of individual agents (Epstein 2006; Epstein and Axtell 1996), self-organization, and social emergency (Sawyer 2005). Among these approaches, the ones that stand out are social constructivism (Giddens 1991, 1998), symbolic interactionism (Blumer 1969; Hewitt 1976), complex responsive processes in organizations (Stacey 2001), society and social systems theory (Luhmann 1984), sociocybernetics (Geyer and van der Zouwen 1992; Lara-Rosano 2002), computational sociology (Gilbert 2008), sociomatics (Castañeda 2009), sociophysics (Galam 2012), and communities of practice (Wenger 1998). F. de Jesús Lara-Rosano (*) Complexity Sciences Center, National Autonomous University of Mexico, Mexico City, Mexico © Springer Nature Switzerland AG 2021 P. E. Balderas-Cañas, G. de las N. Sánchez-Guerrero (eds.), Problem Solving In Operation Management, https://doi.org/10.1007/978-3-030-50089-4_1

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This research proposes principles for the dynamic diagnosis of an organization from the viewpoint of complexity sciences and the new social theories, in order to improve the organizational operation and management.

1.2  Organizations as a Complex System Knowledge is building, by the knowing subject, a conceptual representation of the real object or process to be known, in a way that this representation is an appropriate reflection of reality with a vision to solve a problem. This process of building a conceptual model of reality based on perceptual experiences constitutes an epistemological process. This epistemological process must be based on a theoretical conceptual frame, which is the paradigm (Kuhn 1970), which lets you decipher this reality according to a worldview. A paradigm consists of a set of basic concepts or categories, structured in a system of relationships that gives life to a theory about reality. The conceptual model of the focused reality, which is the result of a construction through paradigm, constitutes the study object where problems and specific solutions can already be defined. Conceptual models of reality, through relevant modeling methodologies, can lead to simulation computational models, which allow for the explanation, reproduction, and prediction of researched elements’ behavior (see Fig. 1.1). One of the developed paradigms in recent years, to assist in defining the object of study, is the paradigm of complex systems. Through it, a focused portion of reality under study is conceptualized as a complex system, while the rest of reality that influences or is influenced by the system is defined as the system environment. Then, from the complexity sciences, an interpretation process of the complex system’s dynamics is developed, in which parts of this are functionally structured in an explanatory model of its behavior. A complex system is made up of hierarchies by interrelated subsystems, each of which in turn contains its own subsystems and so on, until reaching certain basic elemental components of the complex system that depend on the problem to be solved. In each of its levels, the system presents interactions and feedback among its elements, which are nonlinear and dynamic. As a result of such interrelations, properties in the superior level emerge which none of its components of inferior level present. For example, the living human body is made up of hierarchies by different body systems, each of which is made up of organs, where there are nonlinear and dynamic interrelations and feedback that originate emergent properties in the immediate higher level, which none of the members present. Thus, the digestive system can process food, decomposing it into its fundamental elements and absorbing them into the blood, but none of its isolated organs can do it. Then, the study of a complex system implies the introduction of totality notions, hierarchy, self-organization, and emergency, and analyzing the phenomena therein as products of properties which arise as a whole but are not manifested.

1  Theoretical-Methodological Basis for Complex Organization Diagnosis Fig. 1.1 Methodological outline for creating the subject of study (Source: Prepared by the authors)

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Computational model or simulation

Conceptal model or subject of study

Relevant Theoretical Paradigms

Paradigm of Complex Systems

Organizational Reality to Discover

Organization is an adaptive complex system of social nature, comprised of human agents as basic members, which interrelate by communication between them and also they carry out the different functions on an appropriate structure of the division of labor, to fulfill the mission and objectives they have, both the organization and its members. Organization has properties, whose values depend on its history and define its present state, as well as the way to transform a set of inputs and stimuli in responses and behaviors. These properties constitute its state variables and are interconnected in a complex network structure as a result of the systemic composition. Organizations are open systems, because they require and are in active interaction with a complex environment constituted by the natural environment, the artificial infrastructure, and the social environment in which they act. In this interaction process, an organization changes state with time, and the states it takes are the product of two factors: (a) the internal dynamics of the organization, which, as time passes by, transforms the present state in a new different state, and (b) the intervention of exogenous actions and events, which come from its environment, altering, predictably or unpredictably, the organization’s state. The internal dynamics of an organization does not emerge spontaneously in the present but instead is always the result of a historical process. Consequently, in analyzing such dynamic, the organization’s evolution should always be considered. In the complex systems that include teleological members who pursue their own

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agendas, as it happens in an organization, it is not enough to apply a casual approach, but complemented with a teleological approach (Rosenblueth et al. 1950; Ackoff and Emery 1972; Beautement and Broenner 2011). Indeed, to analyze an organization, a teleological or intentional point of view should be taken into account (Ackoff and Emery 1972; Lara-Rosano 2014) to consider both the influence of casual mechanisms and teleological and anticipatory mechanisms in the system and member’s behavior, which allows for the consideration of the organization motivated by its objectives and, at the same time, to visualize it with a casual approach, considering the historical and structural factors which have taken the organization to its current situation. Therefore, the internal dynamics of an organization is manifested in all its complexity, when trying to analyze it to identify causes and reasons that have led it to present a discrepancy between its current state and its desirable state. This analysis and identification process of causes and reasons of this discrepancy, between its current state and desirable state, is the organizational diagnostic. In this analysis, the interactions among different hierarchical levels cannot be ignored, nor such interactions can be isolated, decontextualizing them of their natural, artificial, and social environment, but both interactions and the different aspects of the environment are an inseparable part of such dynamic. This turns the diagnostic process into a complex process in which, generally, various areas of knowledge should intervene in a transdisciplinary way.

1.3  Analysis of Complex Organization 1.3.1  Analysis of Complex Organization Dynamics The analysis of organizational dynamics aims to examine the temporary evolution of the organization from an initial time, in order to understand the decisive factors of this temporary evolution. When analyzing the organization from the dynamic point of view, the first step is to define its variables, which include three different types (Lara-Rosano 1990): State variables, whose values determine its internal state in a given moment and wherein the history of the organization is included. When the computational model of the organization is built, the number of state variables should be necessary and sufficient to understand the fundamental dynamics of the organization. Input variables that are of two types: exogenous inputs and control parameters. Inputs make up the different types of matter, energy, or information that the organization receives from its supra-system or its environment to process them and obtain the outputs, products, or results given to such environment. The control parameters are those variables susceptible to be directly manipulated ­intentionally, by the organization’s management, the supra-system, or the environment in order

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to modify the organization dynamics. The manipulation of these parameters always involves a decision and a goal, for example, the acquisition of loans. Output/response variables which, generated in the organization, are projected to the environment as products or services and are the result of the control variable actions on the organization and depend on the state of such. The response variables are also called output variables, and their key features are to be observable and serve as a base to the organization’s performance evaluation. The definition of significant input, state, and response variables in a given organization is one of the critical stages of its analysis, because from it depends, to a large extent, the success of the next stages, as well as their utility and reliability. The state variables of the organization define a multidimensional space called the state space or phase space of the organization (Boccara 2004). In any practical problems, the state variables or functions of these are constrained to take values within a certain range, defining a space of feasible states. The determination of these restrictions involves a careful study of the organization and the interactions with its environment. The determination of space of feasible states of the organization is essential because it defines the limits and ranges of the feasible solutions and, thus, the strategies and heuristics appropriate for the efficient search of operational solutions to the raised problems. Furthermore, it allows for the identification of critical restrictions on which could be acted to modify them and thus broaden the problem-solving possibilities of the organization. An organization can be on a regular or chaotic state, depending on groups’ behavior of potential trajectories. An organization is in a regular state if close trajectories in an initial time would remain close as they evolve. An organization is in chaotic state if initially close trajectories in an initial time are separated with time to an exponential rate, without any intervention of external factors. In the long run, this exponential divergence turns processes into unpredictable processes, because by amplifying small errors in the initial conditions, it is impossible to predict in the long run the state that the organization will have. For example, weather is a chaotic process because even though its state can be predictable in the short term in the next hours or days, it is totally unpredictable in the long term. If the trajectories of a system that come from different initial states converge to a limited region of the space of states, it is called an attractor, and the initial states of the convergent trajectories form the attraction basin. Systems that reach an attractor remain in it, unless fluctuations of the environment intervene. A dynamic system can have two or more attractors, whose attraction basins are separated by borders called separatrixes. The identification of attractors is based on an analysis of past stages of the organization where it enjoyed stability, without notable changes by the environment and identifying the state variables values in such circumstances. The regions of the state’s space covered by these state variables’ values constitute the attractors of the organization.

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When the organization has great periods of stability in upcoming states, the group of such states constitutes an inertial attractor. This type of inertial attractor has three characteristics: 1. It’s efficient, when it comes to the organization not having to carry out new searches of new stable states with new costs. 2. Since there are no new states, the organization does not have learning costs to be able to handle itself in such states. 3. The organization keeps its already established relations with its environment and enjoys the comfort of stable status quo. However, even though sometimes inertia guarantees the survival and growth of the organization in a stable environment, the same inertia can lead to a lack of adaptation against untimely environmental changes and to a deterioration of the organization’s ability to survive in such environment. Furthermore, inertia impedes it to search for better states, which allow it to better fulfill its objectives. The bifurcations are periods in time where there is a great instability and there are several development alternatives of the organization, for example, a corporate merger or a bankruptcy situation. The course to be taken depends on decisions in the organization’s setting, including not taking action. The bifurcation detection in the past of the organization allows for the identification of situations in the past, where opportunities and threats of the environment played a definite role in decision-­ making and the organization’s behavior, and it allows to be aware in future bifurcations, to optimally handle the organization’s dynamics. The regions of chaos are those where the organization’s behavior is totally irregular and unpredictable, having great uncertainty in the states that the organization will assume in the future. Chaos refers to the organizations or some variables of such and not its members. The regions of chaos can be identified in the records and history of the organization and allow for the detection of turbulence and change stages for which the organization had to go through in its birth and consolidation process. Chaos, however, is not something negative necessarily as unpredictability, diversity, and variety create innovation, and innovation is the author of a new order, in giving new solutions to old complex problems (Kauffman 1993). The attractors in which the dynamic systems behave chaotically are called strange attractors (Ruelle and Takens 1971). In an organization, a strange attractor can have two or more relative equilibrium regions around defined values. In such cases, the trajectories in the space of states are attracted to these equilibrium regions, orbiting around them, but without falling in its centers, but escaping successively toward the basins of the remaining equilibrium regions, in an irregular and non-­ repetitive way. An example of two or more strange attractors in an organization would be power distribution in different cliques of the organization, where power revolves around a clique during a certain period of time, to move to another clique after. Having strange attractors is an emergent property of the chaotic complexity. Strange attractors are found, also, from the history and records of the organization. An organization at the edge of chaos is more stable, predictable, and controllable than the chaotic one. The balance between order and chaos allows the organization

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to have the ability to evolve in innovative ways. The soft systems participatory planning approach (Checkland and Poulter 2006) would be a procedure at the edge of chaos that allows the organization to change but in an orderly manner and self-­ organizing without external intervention. Therefore, this participatory planning scheme of an organization, being at the edge of chaos, with its unpredictable trajectories, is a source of diversity, variety, and creativity in human and social dynamics. Thus, the unpredictable, the diverse, and the variety create novelty, and novelty is the author of a new order, granting new solutions to old complex problems. In contrast, on a regular organization, its operation is predictable due to the fact that it does not stray from the established and always gives the same solutions, whether they work or not, affecting its adaptability to changes in its environment. The stability at the edge of chaos is not equilibrium or lack of change: it is a drift toward change that brings a greater ability to meet the objectives. In fact, complex and adaptive organizations navigate between moderately unstructured states of slow changes. These transformations provide conditions for the survival of the adaptive complex organization and the connection with the past required for learning, analysis, and reproduction. This navigation allows for random movements to stimulate creativity and innovation. The regions of states on the edge of chaos in an organization are also detected for the analysis of its history and its records. The potential of the organization can be defined as the ability to reach new functions and objectives, successfully dealing with the fluctuation of its environment. Fitness in a given environment is the probability of success of an organization to survive and reach goals in that given environment. Fitness then depends on the state of the organization as well as the state of each one of the systems of the environment it interacts with. For each environment state, we can then define fitness as a density function of probability of success, a function which also receives the name of fitness landscape, because in a two-dimensional space of states, it will have a sort of rugged shape such as the face of the earth, with peaks that mean a greater fitness and dips that mean a lesser fitness, along with the values of the variables of the given environment state (see Fig. 1.2).

Fig. 1.2  Fitness landscape for a given environment (Source: Own elaboration)

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The ruggedness of the fitness landscape depends on the number of members of a system and the degree of interaction. In a steady environment, the organization would have a single fitness landscape, the problem of estimating the points of greatest fitness would be defined once, and the system’s management problem would be reduced to taking it to states of maximum fitness. Nevertheless, in a changing environment, there are as many fitness landscapes as there are environment states, and the system’s management problem to maximize its fitness becomes a complex problem. On the other hand, the peaks in a fitness landscape not only vary in height but also in acuity that in probability theory are called kurtosis (Marion 1995). Some of them are so sharp that if the organization falls in one of them, the least of changes in its state, considerably changes the organization fitness, which can lead to become an inept organization in the same environment. In other case when the fitness landscape is smooth and flat, there may be considerable changes in the variables of the organization’s state, without altering its ability by much: the organization is strong or resilient in that region of states. The ruggedness of the fitness landscape depends on the interaction between variables of state of a system through feedback, in such a way that a change in one variable of state may compensate for the changes in the other. The potential of an organization is related to the obstacles, strengths, and weaknesses. Obstacles are situations that make it difficult to meet goals and, therefore, affect the potential of the organization and may be internal, which are weaknesses with a genesis inside the organization, and external, which originate in its environment. There are three types of internal obstacles: 1. The lack of congruency between norms and practices refers to the discrepancies between what the system claims as its norms, objectives, and values and what the system actually practices in light of the empirical evidence. For example, there are political regimes that call themselves democratic, announcing the pursuit of a government of and for the people, when in reality the political power lies in the hands of a person or a clique that decides everything in its favor. 2. Other type of internal obstacles is the conflicts between objectives of the system or its members. Conflicts are debilitating to a system, in the sense that attention, energy, and resources are consumed when they should be applied toward meeting their own goals of the system. 3. “Bottlenecks” are defined as those characteristics which diminish the communication capacity, the information processing, and or the implementation of solutions to system’s problems and that, as a consequence, limit its performance. Strengths are those qualities or elements of an organization that in the past have allowed it to perform at a high level of fitness and reach its goals and which constitute the source of its comparative advantages in relation to other similar systems, and are the primary determinants of its successes. For example, the strength of an organization may consist of its well-prepared human resources.

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Weaknesses are the elements and structural and functional characteristics of the system that may impede the meeting of functions and objectives and therefore lower the system’s fitness.

1.4  Diagnostic of Complex Organizational Dynamics The diagnostic of complex organizational dynamics is a research-action process to present and solve specific problems of the organization, where members and investigators participate with its perceptions and organizational activities in the development of solutions to said problems, through a social learning process (Checkland and Poulter 2006). According to the previous discussion, the diagnostic of organizational dynamics must meet the following conditions: 1 . Must be the instrument used to identify and solve real problems of the organization 2. Must be ideal in terms of direct, indirect, opportunity, and long-term environmental costs, for the people in the organization as well as for those affected by it 3. Must take into account and use the resources and social capital of the organization To begin a process of complex organizational dynamic analysis, those responsible for the organization’s management must have a subjective perception of the context and the problem. Indeed, they must recognize that it’s not about a situation of routine and that a new focus that reveals all opportunities possible is necessary, but do not know how to change or what are the costs, risks, and benefits of change (Beautement and Broenner 2011). The methodology for the diagnostic of organizational dynamics consists of the following stages: 1. Analysis of the organization and its environment, including problems, resources, values, and restrictions, from different points of view. This requires participatory activities and research (problems, resources, values, and restrictions, from different points of view through participatory workshops with employees of the organization). A synthetic microanalysis is used to integrate the different elements of the organization in an object of study (Auyang 1988). In synthetic microanalysis, the object of study is understood as a whole. This approach allows explaining of organizational behaviors at a holistic level, through a macro-description and a macroanalysis. Macro-description characterizes concisely the great characteristics of the organization in terms of global concepts, e.g., capital, passive, active, marketing participation, number of employees, etc. It condenses the universe of micro-­ configurations in a few informative macro-variables. Its focus is synthetic analytical. Macro-descriptions use macro-concepts related to systems such as wholes and do not refer to its members.

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When the organization has been defined as a whole as well as its composite nature, then you begin to search for underlying micro-mechanisms, which constitute microexplanations. Micro-descriptions characterize the organization in terms of the states of most relevant members, such as the state of sales per product, client and supplier portfolio, number of factories, etc. The amount of information required is large, because the number of possible states grows exponentially with the number of members, by which a strict selection must be made of the most determining elements to solve the problem we’re dealing with. As a result of the micro-description in the context of the large macro-­ description, a systemic model of the organization may be built, using some of the best tested techniques, which would allow for interactive simulation, such as systems’ dynamics. On the other hand, systems affect and are affected by the dynamic nature of the environment that may be placid with familiar manifestations; changing but foreseeable, confirmed by established processes and apt to be discovered; defying, with innovative emerging patterns that can be explored looking for adaptation; or turbulent, messy, apparently hazardous, and chaotic, with transitory events full of opportunities and apt to experiment and learn (Beautement and Broenner 2011). In the analysis of the environment, the following should be identified: • The environmental structure, meaning, the systems’ relations that they establish among themselves the various elements of such environment, as well as their nature. • The environmental threats, which are potential environmental behaviors that may negatively affect the organization. For example, the approval of new fiscal regulations may constitute a threat for some importing or exporting sectors. • The environmental opportunities, which are potential environmental behaviors which may positively affect the organization. For example, the incorporation of Mexico to the Pacific Alliance may represent an opportunity for manufacturing enterprises with cheap but high-quality products. Next, we analyze the structure of the organization, specifying characteristics, such as (Ackoff 1999): • The type of organization: public, private, or both • The labor process, including types of accessories, operations, and transformations they execute, type of intermediate and final products, and partial and total efficiencies • Past and present performance evaluations including the evaluation of decisions made • Formal power structure, including the formal organization chart, the distribution of decision-making power, and the participation of the various positions in the organization’s responsibilities • Informal power structure of the organization, parallel decision routes, and influential groups; informal organization chart

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• General policies of the organization and organizational culture, rules of the game of the organization, and fostered, tolerated, and fought values • Employment policies: selection, admission, formation, promotion, and replacement of personnel • Information flow charts inside and outside of the organization, information emitters, transmitters, receptors and processors, efficiency, and quality 2. Definition of a desired scenario by the organizational community, through a participatory prospective approach for the definition of short-, mid- and long-range goals. This activity also requires participatory workshops. As a result, the organizational team produces a strategic plan to guide the development process. 3. Assignment of priorities to objectives, restrictions, and resources by the organizational community and a hierarchy of problems to be resolved in the development process, to schedule actions to be taken. For that one must consider the intentional components of the organization with their own objectives. These own objectives do not have the same priorities or may even be mutually counter-­ imposed; therefore, an essential task is not only to identify the objectives but also to estimate its priority for the members. Two types of objectives stand out: 1. Own objectives, which are the objectives that are self-assigned by each intentional member. The definition of own objectives indicates a certain member autonomy, since such objectives are a reflection of one’s own interests. 2. The functions, which are the objectives that a supra-system imposes upon its systems components, with the intention of collaborating for the supra-system to meet its own objectives. Functions are defined by the supra-system and assumed by the system component, which implies a normative dependency or heteronomy. The intentional members must then define themselves, in terms of own objectives, as well as functions, specifying the following elements (Gelman and Negroe 1982): • A member’s own objectives • The functions or role the member will carry out in his/her supra-system • The relationships the member establishes, along with other members at the same level which are part of the same supra-system • The functions to perform in the element each one of the subsystems that make it up • The collaborative relationships, conflict, and exploitation created between the various teleological members at all levels 1. Identification of appropriate resources to solve the defined problems, including executive and operational activities, sources of information, financial resources, and identification of alternative information and communication technologies. 2. Programming, implementation, and follow-up of concrete actions to solve the problems defined in the framework of the sciences of complexity (Beautement and Broenner 2011). This requires evaluating strategies against a space of

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p­ ossibilities, so that the intention can be modified and social practices can be adapted. What could happen? Are strategies valid? Next the intervention must be prepared, shaping and adapting social practices, deciding whether certain behaviors or interventions are tailored to the situation. Should we change? Finally, we must elaborate appropriate options to carry out the intervention, given the dynamic context. It is necessary to create a balance of what is desired with what is possible and establish a list of options that could be appropriate under certain circumstances. The management subsystem must develop talents such as executive flexibility; capacities like multi-scale capacity, ability to organize groups, to become adaptable to different contexts; have a participatory attitude, foster transparency and trust, have a transdisciplinary and inclusive focus, and be a promoter of evaluation. We must detect the drivers of change and adjust the degree of adequacy of abilities to the context, to correspond to the nature of the gaps and tensions of change. Are options relevant for the context? Is there a need to expand or reduce activities as the context dynamics changes, for which we must use a fractal approach independent from the scale and type of context? For periodic monitoring of strategies, it is necessary to evaluate its direct effects on the state variables. The evaluation of the complex organizational dynamics has three phases: Determining the initial values of the state variables relevant to the problem, before the intervention Defining desirable future values of the state variables at the end of the process of change Measuring the actual values of these variables in the present moment in which the evaluation is done, to value the changes reached in regard to the objectives in order to assess the changes made with respect to the objectives The diagnostic of the organization indicates the current state and discrepancies between it and its objectives, as well as the causes and motivations of these discrepancies. This way, in this diagnostic, they must be shown, likewise, the obstacles, weaknesses, and threats the organization suffers to reach its objectives, as well as the strengths, potential, and opportunities it joins, including its attractors with its attraction basins, its bifurcations, chaotic states, strange attractors, situations at the edge of chaos, emergent transformations and processes or self-organizing attempts and ability to estimate its future behavior possibilities, considering the possibility of stability at the edge of chaos, as a complex adaptive organization operating in diverse environment scenarios. The diagnosis is, given its explanatory nature, an essential element in any attempt to better the organization’s dynamics. Regardless of the specific diagnostic of the organization that investigates its structure, functions, state, environment, potential, and dynamics, it is better to

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compare the organization in the study with similar organizations, with the purpose of evaluating its performance. This comparative process of diagnostic is called benchmarking, and to carry it out, it is necessary to adopt a common evaluation framework, made up of common analysis categories and observable and measurable indicators.

1.5  Conclusions This work explained the principles for the dynamical diagnostic of complex organizations, from the point of view of sciences of complexity, with the purpose of improving the design and management of the organization, and relates this concept with the processes of definition and solution of problems in an organization. For that, a procedure is shown for the systemic analysis of the organization, in which it defines its elements, environment, dynamics, variables, space of states, structure, and potential. A diagnostic of itself is included with it. In the organization’s dynamics, various elements are identified which define its complexity, such as its attractors, bifurcations, chaotic states, strange attractors, situations at the edge of chaos, and processes or self-organization attempts, which may be used as a basis for simulation models to explain the organization dynamics and better its management processes. Under this scheme, a methodology was proposed to evaluate the follow-up of the dynamics of complex organization through the definition of state variables of the organization. The evaluation of the complex organization dynamics has three stages: 1. The determination of initial values of the state variables, before interven tion begins 2. The definition of desired future values of the state variables, when finalizing the process of change 3. The measuring of real values of such variables in the present moment when the evaluation is made, to value the changes reached in regard to the objectives

References and Bibliography R.L. Ackoff, Re-creating the Corporation: A Design of Organizations for the 21st Century (Oxford University Press, Oxford, 1999) R.L. Ackoff, F.E. Emery, On Purposeful Systems (Aldine Atherton, Chicago, 1972) H.E. Aldrich, Organizations and Environment (Prentice Hall, Englewood Cliffs, 1979) S.Y. Auyang, Foundations of Complex System Theories in Economics, Evolutionary Biology and Statistical Physics (Cambridge University Press, Cambridge, 1988) P. Beautement, C. Broenner, Complexity Demystified: A Guide for Practitioners (Triarchy Press, Devon, 2011) H. Blumer, Symbolic Interactionism (Prentice Hall, Englewood Cliffs, 1969) N. Boccara, Modeling Complex Systems (Springer, Berlin, 2004)

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G. Castañeda, Sociomática: El estudio de los sistemas adaptables complejos en el entorno socioeconómico. El Trimestre Económico LXXVI(1), No. 301 enero - marzo, 5–64 (2009) P.  Checkland, J.  Poulter, Learning for Action: A Short Definitive Account of Soft Systems Methodology and its Use for Practitioners, Teachers and Students (Wiley, Chichester, 2006) J.M. Epstein, Generative Social Science (Princeton University Press, Princeton, 2006) J.M. Epstein, R. Axtell, Growing Artificial Societies: Social Science from the Bottom Up (MIT Press, Cambridge, 1996) R.A.  Eve, Chaos, Complexity, and Sociology: Myths, Models, and Theories (Sage Publications Inc., Thousand Oaks, 1997) S.  Galam, Socio-Physics: A Physicist’s Modeling of Psycho-Political Phenomena (Springer, New York, 2012) O. Gelman, G. Negroe, La planeación como un proceso de conducción. Revista de la Academia Nacional de Ingeniería, México. 1(4), 253–270 (1982) F. Geyer, J. van der Zouwen, Socio-cybernetics, in Handbook of Cybernetics, ed. by C. V. Negoita, (Marcel Dekker, New York, 1992), pp. 95–124 A. Giddens, Modernity and Self-Identity. Self and Society in the Late Modern Age (Polity Press, Cambridge, 1991) A. Giddens, The Third Way. The Renewal of Social Democracy (Polity Press, Cambridge, 1998) N. Gilbert, Agent-Based Models (Sage, Los Angeles, 2008) C. Gros, Complex and Adaptive Dynamical Systems (Springer, Berlin, 2008) J.P. Hewitt, Self and Society (Allyn and Bacon, Boston, 1976) J.H.  Holland, Hidden Order: How Adaptation Builds Complexity (Helix Books (Paperback), New York, 1995) S.A. Kauffman, The Origins of Order (Oxford University Press, Oxford, 1993) T.S. Kuhn, The Structure of Scientific Revolutions (University of Chicago Press, Chicago, 1970) F. Lara-Rosano, Metodología para la planeación de sistemas: un enfoque prospectivo (Dirección General de Planeación, Evaluación y Proyectos Académicos, UNAM, México, 1990) F. Lara-Rosano, Cibernética y Sistemas Cognitivos, in Ingeniería de Sistemas: un enfoque interdisciplinario, ed. by J. Acosta Flores, (Alfaomega Grupo Editor, México, 2002), pp. 44–70 F. Lara-Rosano, Complejidad en las Organizaciones, in Encuentros con la Complejidad, ed. by J. Flores, M. Mekler, (Siglo XXI y UNAM, México, 2011) F.  Lara-Rosano, Petri models of purposeful complex dynamic systems, in ISCS 2014 Interdisciplinary Symposium on Complex Systems, ed. by A. Sanayei, O. E. Rössler, I. Zelinka, (Springer, Heidelberg, Alemania, 2014), pp. 183–191 N. Luhmann, Social Systems (Stanford University Press, Stanford, 1984) B. Mandelbrot, Los Objetos Fractales (Tusquets Editores, Barcelona, 1987) R. Marion, The Edge of Organization: Chaos and Complexity Theories of Formal Social Systems (Sage Publications Inc., Thousand Oaks, 1995) E. McMillan, Complexity, Management and the Dynamics of Change (Routledge, London, 2008) M. Mitchell, Complexity: A Guided Tour (Oxford University Press, Oxford, 2009) M.  Newman, A.-L.  Barabási, D.J.  Watts, The Structure and Dynamics of Networks (Princeton University Press, Princeton, 2006) G. Nicolis, I. Prigogine, La Estructura de lo Complejo (Alianza Editorial, Madrid, 1994) D. Ruelle, F. Takens, On the nature of turbulence. Commun. Math. Phys. 20, 167–192 (1971) R.K.  Sawyer, Social Emergence: Societies as Complex Systems (Cambridge University Press, Cambridge, 2005) Rosenblueth, A. et al. (1950) Purposeful and Non-Purposeful Behavior. Philosophy of Science, Volume 17, Number 4, pp. 318–330 R.D. Stacey, Complex Responsive Processes in Organizations (Routledge, London, 2001) E. Wenger, Communities of Practice (Cambridge University Press, Cambridge, 1998)

Chapter 2

Methodology for Building Trend Scenarios Gabriel de las Nieves Sánchez-Guerrero

2.1  Background History of those who have dedicated to announce future situations, or to look or write about what is coming in the future, is integrated with a lot of names, for example, the Hebrew prophets announcing the future, I Ching book from ancient China, Dodona and Delphic oracles in ancient Greece, writers Jules Verne and Herbert George Wells, famous biochemist Isaac Asimov, or futurist Alvin Toffler. Society has learned that it is convenient to look to the future from experience and science, because any person or nation has tomorrow assured, natural, financial, and material resources are limited, and people feel and think in different ways. Besides, human being has also learned that thinking about the future can make human difficulties decrease and have the hope of a better tomorrow. Thinking in the future to assure having or being has been and is a practice in all cultures living in Earth. From ancient times, mankind has developed many ways to visualize future, from magic rites, religious acts, oracles, or astrology to modern methods like strategic planning, prospective, or futurology, that integrate in the so-called future studies (Slaughter 2002, pp. 350). At the same time, instruments and techniques have been designed to support their realization, like grimoires and magic formulas, meditation, intoxicating substances, astrological charts, Delphi expert consultancy technique, Box-Jenkins analysis of time series, multiple simulation models, scenarios, etc. Many decades ago, Ackoff expressed concisely that the gap between scientific and technologic development [email protected] its assimilation in humanity was growing, so learning and adapting capacity of society was fragile, and in consequence, moving forward in time to look into the future was more difficult (Ackoff 1981, pp. 3–6). This situation is observed in other complex and uncertain situations that make G. de las N. Sánchez-Guerrero (*) Department of Systems, Faculty of Engineering, National Autonomous University of Mexico, Mexico City, Mexico e-mail: [email protected] © Springer Nature Switzerland AG 2021 P. E. Balderas-Cañas, G. de las N. Sánchez-Guerrero (eds.), Problem Solving In Operation Management, https://doi.org/10.1007/978-3-030-50089-4_2

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difficult to do forecasts @and propose to “look” to the future with another approach, like the many international financial system crisis, the increasing collective violence, or weather’s changes. This has helped future studies to reappear and, particularly, in scenarios building as a good choice to deal with turbulence of different social contexts. For instance, the variable weather conditions have caused unexpected floods and long droughts, producing complex situations with high uncertainty and risk, hydrometeorological models are increasingly less reliable, and decision-makers try to complement them with other proposals that gave them more reliability. Scenarios are an option to fulfill this purpose. UNESCO’s World Water Assessment Programme suggests using scenarios as a practice in environment future studies (WWAP 2012, pp. 244). Also, this proposal has become clearer in many articles about scenarios published in specialized magazines, such as Technological Forecasting and Social Change, Futures, Foresight, Agricultural Systems, Ecological Economics, European Journal of Operational Research, Long Range Planning, etc.

2.1.1  Scenarios in Interactive Planning Planning Planning assumed as the anticipated rational decision-making (Ackoff 1970, p. 2) is studied as a scientific discipline since second half of the twentieth century. Then, it has evolved through the years, being applied to many objects and situations, according to certain needs and interests. Being from transdisciplinary nature, planning has taken a lot from other scientific, technic, and, also, artistic disciplines, resulting in multiple types and approaches and integrating in a knowledge field that becomes difficult to delimitate its limits. In this chapter, we delimit scenarios inside interactive planning, which sustain that, if future doesn’t exist, it is possible to build it from will (free will) @and accepting context restrictions and makers self-restrictions. These limitations can be attended, depending on the attitude to the future that is adopted, being inactive, reactive, preactive, or proactive (interactive) (Ackoff 1981, pp. 51–65). Taking systems’ thinking as background, Ackoff proposes interactive planning’s concept and states that the interactive planner has a creative thinking, promotes people involved participation, and seeks systems’ learning and adaption (Ackoff 1981, p. 63). Interactive planning process is integrated by five phases: formulation of the mess, ends planning, means planning, resource planning, and design of implementation and control (Ackoff 1981, p. 74). In the first phase, reference scenarios’ preparation is located; and in the second, idealized design development occurs (frequently called as normative scenario). Then, inside the second phase, reference scenarios and idealized design are compared to determine the gaps of divergences that planning will pursue to “fill.” In third, fourth, and fifth phases, Ackoff doesn’t consider scenarios explicitly.

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Scenarios Frequently, the word scenario, when read or listened to, is associated with the idea of theatre, visualizing like a space where a scene is taking place, where is an ambient and circumstances surrounding the actors. [email protected] sceneries’ rebuilt past, present, or future, possible scenes can be represented, but finally scenarios’ aim is to communicate the interpretation of a specific situation. From 1950, scenario planning as a concept was introduced from the army to industrial and governmental sectors. In first place, scenarios were used to describe future situations with the purpose to design military plans and take decisions. Beginnings of the 1960s from the twentieth century, Herman Kahn spread @scenario as a concept in future studies, and then, Theodore J. Gordon, H. Hayward, Olaf Helmer, and Norman Dalkey, among others, continued its development as a technique. In France, at the end of World War II, Gaston Berger developed futurist studies known as prospective, which later were continued by Bertrand de Jouvenel, Futuribles International promoter. Nowadays, it is possible to distinguish two main trends: the logical intuitive school and prospective school (O’Brien 2004, p. 711; Bradfield et al. 2005, p. 800). There are several proposals to classify scenarios: Börjeson et al. (2006, p. 725) integrate them in predictable, exploratory, and normative factors. Van Notten (2003, p. 426) gathers them in simple, complex, decision support, intuitive, exploratory, and formal factors. Julien et al. (1974, p. 255) group them in exploratory scenarios (trend and framework) and anticipated scenarios (normative and in contrast). In these classifications, it seems that the focus is in the future and, in general, scenario theme has been developed in this way, although visualized futures are supported by past and present situations. This article takes the idea that four states exist to describe scenarios: a current, past, and future (trend or desired) state, as shown in Fig. 2.1. De León (2010, p. 20) reviews “scenario” as a term, finding that there are a big number of definitions to conceptualize it, depending on the scientific discipline in which it is developed (Theron et al. 2009, p. 620), that is necessary to indicate that there are many scenario approaches like different types of future studies, which have several objectives (Höjer et al. 2011, p. 3); in other words, there is no a unique orientation in scenarios (Durance and Godet 2010, p. 1488); however, scenarios are in the middle of debate (Wright 2003, p. 7). Some definitions analyzed are: • It is a story that leagues historic and present events with hypothetical future episodes; scenarios are mechanisms to produce relevant information to take any decision (Van der Heijden 2005, p. 15). • It is an imaginary sequence of events, @in which is taken into account complex elements, from which is formed a coherent, systematic, comprehensive, and possible story (Coates 2000, p. 116). • They are resources to order @alternate future perceptions and surroundings where decisions can be taken (Chermack 2005, p. 61).

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Events

Future Desired Scenario NBA

National Selection

Current Scenario

Major League Team

Future Trend Scenario

Selection of my Neighborhood

Past Scenario

t Fig. 2.1  Four states in scenario description (Source: Sánchez-Guerrero 2016)

• They are alternate creations of future representations built from people participation: experts, strategist, and administrators (Varum and Melo 2010, p. 356). • It is a description (usually from a possible future) that assumes intervention of many key events or conditions that can occur between original situation time and an established time after (Durance and Godet 2010, p.  1489). These authors emphasize that building has to have a scientific basis, particularly, using formal analytic models. Considering the aforementioned, on this chapter, two basic concepts are used in the definition of scenarios: • The sequence or combination of analytically structured events that conduces to build a possible future • The description of a possible future with logical arguments based in experience In this way, we consider that scenario is conceptualized as a system, that is part of interactive planning process, and is the result of dialogue and reflection of people involved, that use reliable information (statistical and/or opinions) to describe a future state, starting from present and a historical review. It uses a conceptual, methodological, and analytic technical background, integrates heuristic planning, and describes (builds), in a comprehensive, coherent, simple, possible, and credible way, how current situation connects with the visualized future. In other words, scenarios are stories elaborated by actors involved in a system that want to represent scenes that show hypothetical future situations where a dynamic of actors is recreated,@ the events and trends dynamics in a determined historic context.

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Because the present is the only time that exists, from here and now, it can be rebuild any version of history and also look future from different ways. There is no unique answer, so we have to avoid keeping with fixated or prejudged ideas. It is important to remind that anything is possible and when human being imagines something, the idea takes the germ of its realization. Scenario, as a systemic construction, demands the active participation of stakeholders. An open participation (of assembly) with stakeholders of the system is functionally difficult, because gathering all frequently is not possible and the dynamics in these forums is easily contaminated. Leading the elaboration process with all will demand a lot of work and requires a special place, and the cost will increase. Beside, dynamic handling would have major difficulties. It is convenient to try people participate in a structured way, being this with a work plan, in small groups and with organizational levels. A good option is designing similar groups as suggested by Ackoff (1981, p. 163–168) in circular organization.

2.1.2  Trend Scenarios Marco Polo and Christopher Columbus are names that identify great explorers, and they were people that took risks inquiring the unknown and opened new paths to mankind. In this way, trend scenarios are stories that pretend to explore and stay ahead to the future, being virtual incursions from explorers that narrate possible future situations in which can be found obstacles and opportunities. Relating Ackoff’s planning typology (1981, Chap. 3) and Fig. 2.1, the current and past scenarios are the product of facts’ structuration and adopt, respectively, an inactive “doing nothing” or reactive “of nostalgia” attitude. The future trend scenario adopts the preactive attitude of “going ahead of future,” based in significant variable projection, supposing a following of trends. Meanwhile, desired future scenario limits to “redesign future” with a proactive attitude. Here, reference is exclusively made to trend scenarios. Ackoff (1981, p. 75) puts trend scenarios in the first phase of planning interaction cycle, which he calls particularly reference scenarios. Özbekhan (1974, p. 75) calls them logic future, and Bright (1978, p. 41) refers them as forecasts. Ackoff (1981, p. 59) distinguishes forecasts from reference projections. He mentions that a reference projection is an extrapolation of an acting feature that has had a system since recent past and probably will have in the future, supposing that will not have a significant change in its behavior (that is a common idea of a prognostic); meanwhile, he associates forecast with prediction. He mentions that forecasts are white-lab-coat-dressed oracles helped by computers. In a different way, Wheelwright and Makridakis (1976, p. 4–5) consider very important both (traditional) quantitative and (based on predictions) qualitative forecasts. In this chapter, it is considered that a forecast is a value anticipated estimation of a variable that is part of a future event based in objective current or past information. For making a forecast, it is used as an analytical model and focused in a time

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horizon. Forecast supposes that analyzed phenomenon and its context do not change inside the time horizon considered. It is necessary to develop a trend scenario use of forecasts but is not the only thing. With expert opinion and its formal use, underpinnings of prediction, scenarios are complemented. We understand prediction as a structured expert opinion, both inside or outside of organization, being this opinion valuable and useful to comprehension and knowledge of studied system’s future. Expert opinion is very important when there is a significant level of risk and/or uncertainty and has no continuous, reliable, and appropriate information about studied phenomenon; but having reliable and appropriate information and a low risk and uncertainty level, expert opinion always is important and recommended; if an expert is wrong, experts frequently look for facts, mostly imperceptibles, from what we have to focus. Guiding prediction efforts of a group expert when they are participating in a trend scenario development, it is considered that they have system analysis and “logic route” that offers a forecast as supply; it can take too long to distinguish if they do not have it, if an event is more probable than the other. This can be in detrimental for some experts, arguing that they are being leaded or that they have a restrained creativity. However, in practice we have observed many times that when an expert opinion is wanted in a planning reunion to visualize a future image, they themselves wonder if any of the participants has measured or predicted a variable or parameter behavior: SCT-IMT-UNAM (1998), IMTA (2006), and CONAGUA-­ IMTA (2013),1 among other experiences. According to Ackoff (1981, p. 80–84), system analysis implies as follows: location of study system object in time and space, identification and contrast of relations between systems with each subsystem, and relations between system with its supra-­ system. In other words, study object delimitation as a system; spatial, temporal, and realized productive activity location; system aims: vision, objectives, goals, mission; validate the organization in the flowchart reality; in situ verification of materials, money, command lines and information flow; in situ verification of policies, practices, strategies, and tactics vs. written in documents; administration style; graphic review of past and current results; review of various actors involved (stakeholders analysis); internal and government regulations that concern the company and how to apply them. Future study using trend scenarios does not pretend that results obtained are really going to happen; it is pretended to focus to the future. Trend scenarios look to explore and describe viable and possible future situations and explain phenomenon’s relevant variables and its dynamic relations, considering what would happen in the future according to its trends. They pretend to identify what possible events can detonate current trends’ push, which will inhibit or maintain them. Likewise, looks to identify what significant breaks or discontinuities can occur in the future, also relating possible effects of other decisions and some external random events.  Secretaría de Comunicaciones y Transporte (SCT), Instituto Mexicano del Transporte (IMT), Universidad Nacional Autónoma de México (UNAM), Instituto Mexicano de Tecnología del Agua (IMTA), Comisión Nacional del Agua (CONAGUA) 1

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During its development, it is important to identify if some events can occur several times and cause cyclical variations or if they occur occasionally. Cyclic or extraordinary phenomena analysis helps, not only to interpret better the studied phenomenon, it also allows to distinguish that it is possible to produce discontinuities intentionally, because, also, these have occurred before. Consequently, a trend scenario is the description of a possible future, and it is the result of integrating forecasts and predictions for a determined phenomenon or system. In this chapter, a process for its development is proposed. Different processes have been suggested for elaborating trend scenarios. Here are described some shortly: Schoemaker (1995, p. 31) suggests seven phases: scope definition, stakeholders’ identification, basic trends’ identification, key uncertainties’ identification, original scenario themes’ identification, consistence and plausibility verification, and learning scenarios’ development. These seven phases can be grouped into four: system analysis, forecasts’ elaboration and analysis, scenario description, and finally a scenario evaluation. Godet (2000, p. 10) considers, inside scenario planning process, five steps for elaboration: problem formulation-system exam, diagnosis, key variable identification, firm dynamic, and scenario composition. In this proposal, five steps can be gathered in three big stages: system analysis, trend analysis, and scenario composition. Jouvenel (2000, p. 42) establishes five phases: problem definition and horizon selection, system elaboration and variable identification, data collection and hypothesis development, possible future states’ exploration, and strategic options’ sketch. In the same way, five phases can be grouped into three: system analysis, forecasts elaboration, and scenario composition. Fink and Schlake (2000, p. 39) suggest four stages: key factors’ detection, future projections’ prospective, scenario calculation and formulation, scenario analysis, and mapping and interpretation. In this proposal, four stages are known as system analysis, forecasts elaboration, scenario composition, and evaluation. Unfortunately, the previous authors describe their processes without being more explicit in conceptual or methodological frameworks that sustain them and, also, in definition of some terms. From aforementioned processes, it is inferred that they part from a system analysis, then forecasts are made, then scenario is composed and validated. The need of complement forecasts with predictions has been expressed. In essence, what is suggested is to reunite both schools: intuitive logic and prospective school. There is an international and multiple disciplinary opinion about to erase boundaries between quantitative and qualitative, among different scientific, social, and humanistic disciplines, and even between schools of thought associated to many regions of the world. Systemic thought, transdisciplinary search, and back to human essence are a big tsunami.

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Conceptual Development Now, previous thoughts are integrated with Chermack, Gharajedaghi, and Checkland ideas, aiming to create a procedure for trend scenarios’ development. We consider that it is important to pick up who presents theoretical-methodological basis about system thinking and planning, and, in Chermack’s case, because he proposes very important approaches for scenario elaboration and evaluation based in system thinking and planning. Chermack (2005, p. 65, 2011, p. 66) suggests that scenario development is part of planning itself. Likewise, he forms the scenario conceptualization as a system and proposes, with a systemic approach, a five-phase process: project preparation, scenario exploration, scenario development, scenario implementation, and project evaluation. Also, Chermack gives scenario a process meaning and highlights context relevance of scenario planning. Chermack is not more precise on how to do the scenario exploration and building. In order to complete Chermack’s suggestion, it can be picked up Gharajedaghi’s system concept (2011, p. 93). He considers four relevant aspects for a system definition: structure, functions, process, and context. If scenario is conceptualized like a system, then, based in Gharajedaghi, the aspects he proposes have to be part of a scenario conception. Also, Chermack’s system and process idea are added to Checkland’s methodology (1981, p. 169, 286) used for elaborating a conceptual model. This is used for identifying relevant aspects or activities that integrate scenario elaboration process. Checkland’s methodology consists in these steps: 1. Study object is conceived as a productive system, where people interact (executive, employees, workers’ union, suppliers, customers, machinery and equipment, money, raw materials, information, inputs, processes, products, etc.) with a determined purpose in a specific context. 2. There are different system elaborations (pertinent systems) that depend on the role people or groups involved play and goals they want in the system. The aim is to get a unique representative elaboration, given its dynamic nature. 3. System construction is made answering two questions: What is it? and What does the system do? In order to answer the first question, Checkland proposes to write root system definition, which consists in describing concisely six significant elements for the system performance: world vision or system’s raison d ́etre (Weltanschauung); transformation process in which inputs are transformed in products (transformation); benefitted or affected people from system activities (customer); actors or people that make principal system activities (actor); system owner that can decide if system no longer exists (owner); context or environmental restrictions that have to be taken as given (environmental). Based in previous description, the second question can be answered in three steps: a list is generated with the minimum number of verbs (in infinitive form) that describe necessary activities or functions required to fulfill with previous description; verbs are

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connected to lines, like a process, according with a logical sequence; with arrowheads over the lines is indicated the lookalike essential flow. 4. Necessary resources’ flows are identified for system accomplish with its reason d́etre, according with established transformation process, representing them differently from logical dependencies. 5. Verbs’ number is watched seeking that does not increase as much as half a dozen that a logical connection exists between them and have same detail level. Finally, to prove that elaboration expresses what is and does the system. 6. Should it be necessary, make elaborations of different desegregation levels, depending on needs. Checkland associates an external evaluation being to all system elaboration that realizes monitoring and control duties, measures system performance, and allows its learning and adaption. Using Chermack, Gharajedaghi, and Checkland system ideas, the steps or phases of trend scenario elaboration process are deduced. Scenario, conceptualized as a system, it is defined with next expression: Scenario Si as a system

Si = F ( s,f ,p,c,a )t

i



Where s: structure, f: functions, p: processes, c: context, a: actors, t: time, i: possible models Indeed • s: Structure (system elements, its relations and nature) • f: Functions (element’s dynamic relations and its properties to fulfill its goals) • p: Processes (structure and functions interaction; transformation rules of inputs to products) • c: Context (system that includes study system object) • a: Actors (benefitted and affected parties: owner, employees, customers, suppliers, competitors, close society and other actors) There will be (i) different elaborations, depending on relevant system actors, considered hypothesis, and time dynamic (t). Considering our definition or trend scenario previously expressed, its root definition is: A system for owners (of scenarios) that helps decision-making inside planning process and turbulent organization context before it. Receives as input the structured problem, needs, demands and organization interests and produces viable norms for decision-making. Process shows connection between present and identified viable futures, initializing with system analysis and using forecasts and predictions. It is elaborated in a participative and plural way by internal and external organization experts; and it is a useful instrument to evaluate alternatives based on visualized futures, besides being an organizational learning tool that allows to establish a dialogue between organization’s stakeholders. The aforementioned reflects “what it is”.

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Then, conceptual model is elaborated, where previous aspects are integrated using a transformation process. Context: Is the environment where scenario is developed and “lives” current system. It is a turbulent environment, full of contingencies and multiple dynamic relations, which originates its complexity. Structured is constituted with: system analysis, analytical estimation of forecasts, expert consultancy, future image elaboration, and document that connects current situation with future image. The functions (verbs) are to analyze, forecast, predict, build, and write, functions that are needed to fulfill with goal in decision making and organizational learning, see Fig. 2.2. Number of system’s significant verbs or elements, 7 ± 2, has been studied by different authors. For further information, Sánchez-Guerrero (2016, Chap. 3) can be reviewed. Previously, these functions were explained concisely, however, in the next section are described in detail. Actors are who elaborates scenario, users of it, the decision maker and among other participants. Scenario elaboration process is the interaction of previous aspects, organized as a transformation process. It receives as an input the structured problem and generates scenarios as a product to aim the goal.

2.2  Proposed Procedure Three people groups interact in a continuous and narrow way during the trend scenarios elaboration: customer, planning group and experts. In the beginning, customer and planning group establish reference terms in which scenario or scenarios will be elaborated. Conditions will be established about objectives and expected results, as well as costs, time and technical specifications: information, planning horizon, suppositions, and basic values to support scenario; potential experts and communication process continue between them. Planning group elaborates forecasts using statistical information, formal analytical tools and simulation techniques. Besides, captures experts’ opinion to integrate the predictions through applying heuristic participative planning techniques and with them build scenarios. During the elaboration always are adjusts in interaction with customer. It is understood by customer or the organization that will use scenario, particularly, who will contact planning group. Customer can be the decision maker, but not necessarily. In this chapter it is considered as an expert the individual which opinion has great value and usefulness to predict and evaluate intuitively the relative occurrence and importance of different factors, referring to a determined object or system. These experts can be internal and/or external of the organization and its participation’s dynamics can be structured, using organization leveled small groups, similar to Ackoff’s proposal in circular organization.

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Expert’s role is similar to Delphic oracle. Having a great reliability, should the Oracle’s Pythia had predicted only misfortune, it would emerge a “Goodwill Oracle”; then, exactly the opposite had occurred whether all answers would be flattering. It is difficult that an expert always accept any opinion, accepting all would deny him his analysis capability and decrease his reliability; on the other hand, rejecting always all kind of opinion also will put his capacity as an expert in doubt (Sánchez-Guerrero 2003, p. 127). Based in Fig. 2.2, steps for trend scenario elaboration are presented in Fig. 2.3, and then will be explained.

2.2.1  System Analysis and Current Situation Explanation In this step, phenomenon is conceptualized as a system to study and placed in its special and temporal dimensions. Problem that motivates the scenario elaboration is formulated. Also the analysis level and planning horizon is settled. Objectives and expected results are established and, based on the system analysis; significant variables are identified to be studied. Likewise, basic hypothesis or suppositions in which variables will be projected are specified, and finally the based current situation is represented. Identifying relevant variables for scenario elaboration is crucial. It can be used a conceptual diagram for identifying them. It is desirable that only needed variables be (in small number 7 ± 2), that will be at the same hierarchical level, and, if it is possible, being mutual exclusive (Sánchez-Guerrero 2003). Also, it is recommended to use in this step other heuristic participative techniques, as TKJ, SOWT, NGT, among others.

Monitor and control

Predict Problematic needs, demands, interests

Analyze

Build

Write

Scenarios: Viable guidelines for decisionmaking

Forecast

Fig. 2.2  Trend scenario elaboration as a system. (Source: Translated from Balderas and SánchezGuerrero 2015)

28

G. de las N. Sánchez-Guerrero Problematic: needs, demands, interests 1. System analysis and current situation explanation

2. Forecasts integration

3. Predictions integration

4. Construction of future image

Feedback

Evaluation: monitoring and control

5. Connection between present and future: scenario writing (Possible actions)

Scenarios: viable guidelines for action

Fig. 2.3  Procedure for develop trend scenarios. (Source: Translated from Balderas and Sánchez-­ Guerrero 2015)

2.2.2  Forecasts Integration Variables are projected to the future using mathematics tools and supposing a continuous trend: Decomposition Method, using Time Series, Multiple Regression Analysis, Box  – Jenkins, among others. Using these techniques supposes that a series of historic observations representing an underlying path of analyzed phenomenon are available. And they have randomness, being this lacking in purpose, cause or order appearance. In other words, it is well known as a forecast. It is important having information, when possible: reliable, necessary and timely. It requires a consistent theoretical, methodological, and technical background. It is convenient to acknowledge that in Mexico are multiple difficulties for getting information, some are: (a) It is poor and scattered, (b) There are some official sources with different

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versions on the same point, (c) The capture formats of the information are changed frequently, (d) They are published with different format and with different purposes, (e) There is discontinuity in the gathering of information, (f) The information presents distortions from who are collecting it, etc.

2.2.3  Predictions Integration In this step the forecast results are received and complement gathering structured expert judgement. A group of experts is reunited and based on their expertise, they identify some relevant variable and determine the possible trends in the analyzed phenomenon. In this step it is necessary to use a theoretical, methodological and technically consistent frame. Some heuristic participative techniques can be used such as: Nominal Group Technique (NGT), Delphi, S-curve, Cross Impact Analysis, Morphological Analysis, among others (Sánchez-Guerrero 2016). It is desirable to combine techniques, for example, from Delphi analysis a Morphological dynamic analysis (Ritchey 2011). Later on, a dynamic of trends can be analyzed using Cross-Impact.

2.2.4  Construction of Future Image As shown in Fig. 2.3, Information from previous phases is integrated in this step. It may be that people involved in previous steps be the same that elaborate future image, however, it can be a different group or, also, like other occasions, the planning group. It is a hard task in which future image is elaborated using participant’s creative visualization and rationally analyzed information. Using hermeneutic procedure (Beuchot 2004) or using a dialectic debate (Mason and Mitroff 1981) can produce good results.

2.2.5  C  onnection Between Present and Future: Scenario Writing Finally, scenario is written using narrations, with a logical structure that allows establishing the relation between present and future. It is desirable that consequent scenario admits to detect tensions, sequences, discontinuities, cycles, threats, and opportunities, as well as be a document: global, coherent, concise, simple, credible, and possible. It is important that scenario writer or writers get involved and be reflected on it, that follow narrations in a linked way and allow observing themselves and thinking about probable impacts and reactions in their situation, so

30

G. de las N. Sánchez-Guerrero

system learning can be promoted. While this reading is seen in group and participative way, scenarios turn into dialogue and thinking resources. Talking about a specific procedure for writing is very complicated, because each scenario is written as a play; is unique. However, next some important and desirable aspects for its writing are shown. Location, problem definition, suppositions, objectives, and values. System is places in space and time, in a historic process (for example, Toluca Valley aquifer in the state of Mexico in the last three federal administrations and in a planning horizon of 10 years). Problem definition. Herman Khan said that most important during scenario elaboration was simply to think about the problem (for example, the “silent” destruction of Toluca Valley Aquifer, CONAGUA-GTZ 2008). Suppositions are axioms or truths that does not require to be proved, they are assumed and lead, greatly, information capture, process, analysis, and interpretation (for example, un-­ conditional support from state and federal government is assured to hydric sector). Objectives indicate the way scenario will take (for example, global rescue of Toluca Valley Aquifer environment). Values express ethic way for elaborate it and to reach the objectives (for example, in a plural participation and respect plan in which users can organize in associations). Variables  They consist in elaborate a list with relevant internal aspects that typify studied system (for example, water extraction), relevant external aspects that typify general surrounding of the system (for example, water supply for Mexico City), specific aspects on which it will try to affect (for example, liquid pressure level of wells) and concerning aspects about policies related to studied phenomenon (for example, current regulations). Variables and its values identification is a hard and thoughtful task, that looks to differentiate irrelevant factors and relations from which are really needed. As aforementioned, relevant variable number is a small one, therefore it is recommended to use conceptual diagrams, NGT, or other techniques for its identification, besides using any simulation model for a better system comprehension and knowledge. Actors and events  Actors are people or organizations that take a big role in system study and are linked to identified variables (for example, COTAS, Basin Councils, Operative Organisms, CONAGUA). In the other hand, events are events or situations with occurrence probability 1 or 0, in other words, can happen or not, and that are associated to identify relevant variables. It is important to consider those events that impulse or inhibit system trends (for example, urban growth in Lerma municipality will increase 30% in the next 10 years. Another example, water volumes estimation for its bill and later charge is affected by the poor control of wells without authorized granting in Toluca Valley industrial zone). Invariants, important and heavy trends  Invariants are supposed permanent phenomena since studied time horizon (for example, cracks in Toluca Valley Aquifer region). An important trend is the one related to a relevant variable (for example,

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there is a solid trend about great urban growth in Toluca, Lerma, and Metepec municipalities). Heavy trends are movements affecting a long-lasting phenomenon and are difficult to modify (for example, the poor distribution of income or low educational level). Carriers’ facts of future  All normal trend or emerging processes show symptoms that regularly are not visible for almost all people. If a holistic and historic perspective is used for past and current events, they can detect these elements that have the potential to modify existing trend significantly, or “trigger” new trends, sometimes, difficult to revert (for example, silent invasion of people to federal lands over two decades, even some legally occupied, in borders or also inside wetlands that are in the interior of Toluca Valley Aquifer, CONAGUA-DLEM 2008-2010). Thirty-five years ago, sporadic struggles happened in little communities of the country (for example, Irrigation District 03, Tula city, Hidalgo state or Jojutla town vs. Tequesquitengo town in Morelos state), sometimes lightly armed, for the right of water use. Today these conflicts are more frequent and less controllable (for example, the defense Yaqui people from Vicam, Sonora state); is expected that in some years armed social movements occur where water is in dispute. Mixed with other forces that currently are distorting social order in our country, it is likely that terrorist attacks would be more regular and then have water infrastructure as a target and cause social chaos. For more information, Becerra et al. (2006) can be consulted. In Mexico, we have had events that, if they had been detected and faced in time, could have change history. For that, it is desirable not to continue the Mexican trend “Here nothing happens”. The armed uprising phenomenon of January 1st from 1994, headed by EZLN (Zapatist Army for National Liberation, for its acronym in Spanish), did not “appear from nothing”, or one person’s idea. In this way, low government’s action to 1985’s earthquake had an important impact for creating new social structures and triggered merging processes that transformed correlation between political forces. Carrying facts of future represent frequently breaks in trends and must be on focus. Finally, developed scenario will answer to needs of: explanation, experimentation, quantification, foresight, and learning needs. A general procedure for scenario elaboration, adding important elements for the writing, is shown in Fig. 2.4. Here is the Toluca Valley Aquifer (AVT) scenario. It is based in many documents, among the most representative are: SACMEX (2007), CONAGUA-GTZ (2008), CONAGUA-REPDA (2009), Diario Oficial de la Federación (Official Gazette of the Federation, DOF, August 28, 2009), CONAGUA-DLEM (2008-2010), Alcamo and Gallopín (2009), Gallopín (2011), De León (2010), De León and Sánchez-Guerrero (2011). Planning horizon focused on was in 10 years. This scenario integrates quantitative aspects of the forecasts obtained from analysis of trends in historical statistical data and subjective occurrence probabilities simulations from relevant qualitative

32

G. de las N. Sánchez-Guerrero Problematic: needs, demands, interest

1. System analysis and current situation explanation

2. Forecasts integration

Feedback

Evaluation: monitoring and control

5.5 Carriers facts of future

5.4 Invariants, important and heavy trends

3. Predictions integration

4. Construction of future image

5. Connection between present and future: scenario writing (Possible actions)

5.3 Actors and events

Scenarios: viable guidelines for action

5.1 Location, problem definition, suppositions, objectives and values

5.2 Variables

Fig. 2.4  General procedure to develop trend scenarios. (Source: Translated from Balderas and Sánchez-Guerrero 2015)

events for system, derived from the experts opinion on which Cross Impact Analysis technique and KSIM Impact-99 software were used (Niccolas 2000) The events, occurrence probabilities, and impacts, and relations between them were defined using Delphi expert consultancy technique. Scenario elaboration lasted around 7 months because a one-year-project about aquifer and region diagnosis was taken as background. Process of one of the consequent scenarios is shown next.

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System Analysis and Current Situation Explanation Regarding macro location of AVT, it is located in the state of Mexico, it is one of the eight aquifers in this state, and one of the thirty-seven in all Lerma-Chapala basin. Specifically, is located inside High basin of Lerma River, to the South of Mexican Highlands and bordered to the north by Atlacomulco – Ixtlahuaca aquifer, to the South by Tenango hill, to the South West by Nevado de Toluca volcano, and to the East by Sierra de las Cruces and Monte Alto, respectively. It covers a total area of 2,738 km2. It takes up 12.17% of the state of Mexico and has a population between 2 and 2.5 million inhabitants. Regarding micro location, municipalities inside aquifer area are: Almoloya de Juarez, Almoloya del Río, Atizapán, Calimaya, Capulhuac, Chapultepec, Xalatlaco, Joquicingo, Lerma, Metepec, Mexicaltzingo, Ocoyoacac, Otzolotepec, Rayón, San Antonio la Isla, San Mateo Atenco, Temoaya, Tenango del Valle (partially), Texcalyacac, Tianguistenco, Toluca, Xonacatlán (partially), and Zinacantepec. It is necessary for conceptualizing the aquifer as a system to know the functions that realizes and are performed on it. Then, entrances and exits are established as a black box. Aquifer supplies water to the region and Mexico City. The main natural functions of the Aquifer are: • Water storage • Capturing rainwater and surface runoffs System entrances are: 1. Natural recharge: belongs basically to infiltrated volumes by rainwater (177.8 hm3) 2. Underground entrances from Nevado de Toluca and Sierra de las Cruces belongs to horizontal water entrances form recharge zones (157.7 hm3) 3. Induced natural recharge: it is water volume annually returning to the aquifer in consequence of irrigation (1.3 hm3) System exits are: 1. Evapotranspiration: it is water loss from a surface by direct evaporation with water loss by plant transpiration 2. Granted groundwater volume: There are annual volumes of extraction (435.66 hm3) 3. Compromised natural discharge: There are water volumes that come from springs or base volume of water from rivers fed by the aquifer, likewise underground exits are considered, that should be continuous, so next hydrogeological units do not be affected. Brief problematic  Accelerated urban and industrial growth of Toluca Valley in the last decades has contributed mainly with environmental modifications. Most evident has been the high contamination or Lerma River, a sample of it is the constant presence of all sort of garbage in all the riverbed, bad odor, aquatic fauna disappearing, chemical analysis that show heavy metals waste, siltation and the poor or null

34

G. de las N. Sánchez-Guerrero

riverbed slope, and health issues of inhabitants. From its source, Lerma River is contaminated with agricultural, industrial, and urban waste, particularly from Tenango del Valle, Joquicingo, Rayón, San Antonio la Isla, and Texcalyacac municipalities. Along its riverbed, untreated flushes that, in some cases, are treated previously and even though fulfill CAN 01.001 norm regarding sewage, they not accomplish established parameters in Lerma River’s classification published in DOF on April 1st, 1996. A conceptual diagram regarding AVT problematic is shown in the next Fig. 2.5. Problem was established analyzing geographic and economic situation, hydric situation, including water available, quality, uses, management tools aspects, political and institutional aspects, water, health, and environment aspects, and extreme phenomena (floods, cracks, and others). And finally, Watershed Sustainability indicator was used, which is employed by many organizations and universities for basins and other water resources evaluation. This line reflects a low sustainability state comparing with other water resources in the world. Planning horizon was settled for 10 years because limitations in data. Significant variables for study were classified in main themes: (a) economy and demography, (b) investments (technology and environment), (c) social, (d) extreme phenomena, (e) intellectual training and capacity, (f) informed social participation, and (g) government. Actors: There are about 58 actors involved in AVT management and use, like the National Water Commission (Comisión Nacional del Agua, CONAGUA) Central Offices, Local Office of CONAGUA in state of Mexico (EDOMEX), Basin Council, Underground Water Technical Council (Consejo Técnico de Aguas Subterráneas, COTAS), municipal government, many NGOs, Academic Institutes, among others. Some hypotheses were: federal and state unconditional support to hydric sector is supposed. It is supposed that experts interviewed have no political turn. Variables grouped in subject matters were: demographic, economic, technologic, water matters investments, social, government, environmental, extreme phenomena, intellectual capital training, informed social participation. Some carrying future events are: 1 . Silent invasion to federal lands 2. Water denationalization 3. People massive evacuation by floods Forecasts Integration Once relevant variables established, data generation starts for planning horizon; in this case, trends extrapolation and Brown technique for prognostics were used. Trends extrapolation is based in continuity and “naïve” concept, in other words, it is assumed that what happened in past will occur in the future. The available information regarding hydric sector, and particularly of AVT, is deficient in many ways, being some mentioned, it is out of date, wrong data from

Lagoon Dryness

Subsidences

Several discharges Pollutied water discharges in superficial water corps

GDP Recharge zone

Deforestation

Floods

Treatment plants malfunctioning

Urban development

Infrastructure risks

Regional geology, local conditions & land dynamics

Fig. 2.5  Toluca Valley Aquifer problematic. (Source: Own elaboration based on De León 2010)

Risks

Health

Land invation

Change of Lerma river slope

Cracking & subsidence

Recharge volume to the aquifer

Social welfare

Aquifer of Toluca Valley, Mexico City and Metropolitan area

Polluted water volume

Physical & sanitary risks to the population

Water & environment quality

Piezometric levels

Treated water volume

Floods and deforestation

Extracted volumes for local consume and for several uses in the D.F.

Clandestine water trade

Extraction

Loss of biodiversity

Rainfall water catchment

Rain

2  Methodology for Building Trend Scenarios 35

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G. de las N. Sánchez-Guerrero

source, differences in measurements, no continuity in facts, different results despite equal variables measurement, among others, that cause limitations in prognostics development. Some relevant variables forecasts are shown next. According with CONAPO (2010) data, municipal population of AVT could be between 2,500,000 and 3,000,000 inhabitants. On the other hand, following elaborated projections in the study of water available in Toluca Valley (CONAGUA 2003), it is expected that up to 2020 exist about 17,000,000 inhabitants in the 23 municipalities inside the valley. If two projections aforementioned are compared, it shows that official data is lower than considered in the last projection. For year 2005, census indicates that there are 2,500,000 inhabitants in the aquifer; meanwhile, for the same year, the other study indicated approximately 13,000,000 inhabitants. In 2007 GDP was $ 40,875.00 MXN and prognostics reflected a growth trend that vary from $ 65,000.00 to $ 80,000.00 million MXN (1993 prices) for 2020, indicating an average growth rate from 2.72% to 3.66%. On the other hand, according to official estimations, the national average economic growth of GDP is calculated not exceeding 3% for 2020. According to forecasts concerning potable water range official data, coverage of 97.27% is estimated. Water availability for 2020 regarding prognostics can be about 185–481 hm3 range. For year 2020, granted total volume will be about 339.94 Mm3, agricultural volume about 148.80  Mm3, public supply of 125.43  Mm3, and industrial supply of 65.71 Mm3. It is estimated that water extraction for Mexico City in 2020 would be in 24.25–95.66 m3 millions range per year. In year 1968 piezometric levels were in 16.8 m averages, for year 2020 are estimated, on average, in 38.26 m. Predictions Integration Delphic and Cross Impact techniques were used. This stage had as an objective to define events, estimate their occurrence probabilities, and analyze their trends that might happen in AVT. Both trend aspects and no trend aspects based in experts’ opinion were considered. Experts selected belong to many organizations, as: CONAGUA Central Offices, Local Office of CONAGUA in EDOMEX, UNAM and UAEM-CIRA, IMTA, UAM-A, Civil Defense (Protección Civil) from EDOMEX. Once participant experts were selected and confirmed their committed participation, Delphic rounds were generated: 1. First round. Its objective was to determine trend events (events that might happen with a high occurrence probability) 2. Second round. Most important trend events selection and classification 3. Third round. No trend events were established (events that, if happen, might change trends drastically)

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4. Fourth round. Its objective was to assign occurrence probabilities to trend events and no trend events by experts, and impact matrix fill in. For impacts record, experts filled in two matrixes: in first one trend events with other trend events were compared, and in second one trend events with no trend events. The identification of events, their description, and both initial and final occurrence probability are shown in Table 2.1. Same experts estimated initial probabilities and final probabilities were Cross Impact analysis result. Likewise, results obtained by applying KSIM-IMPACT 99 Cross Impact software (Niccolas 2000) are shown in Fig.  2.6. This representation is interesting because a final probability considering events’ interaction was calculated. Events having a major occurrence probability were A- C- B and D Construction of Future Image Future image was elaborated began in the middle of the year 2010. It was realized integrating the forecasts and predictions, maybe the experts’ contribution was the most important thing in this part that, in their commentaries, when were interviewed, showed their visions. Future image elaboration was realized partially in a participatory way, and may not meet all directly, because of existing conditions, and other constraints. These were the steps to elaborate future image: 1. Grouping events according to probabilities: events were classified following high, medium, and low occurrence probabilities. 2. Elaborating relations: Once events grouped in high, medium, and low probability, links between them were found. Conceptual diagram elaborated for system analysis was reused; likewise links with carrying facts of future were established Scenario Writing Current trends regarding aquifer deterioration process seem to be the same: aquifer over-exploitation, urban and productive zone growth, and lower water quality will be the three key events of this process. The high population growth (calculated in 2,500,000 and the 3,000,000 inhabitants of aquifer’s municipalities) and high economic growth (estimated in a GDP range about MXN $ 65,000,000  – MXN $ 80,000,000 for 2020) trends will be reflected in a water demand growth for industrial and domestic use (for year 2020, the total granted volume according to prognostics is situated in 339.94  Mm3, 148.80  Mm3 assigned for agricultural uses, 125.43  Mm3 for public supply, and 65.71  Mm3 for industrial supply), demand that will exceed availability, in other words, aquifer would not have the capacity for supplying (water availability for year 2020, following prognostics, can vary in a range from 185 to 481 hm3) population for the 23 municipalities, and far from it, supplying Mexico City.

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Table 2.1  Identified events by experts Event Description A Aquifer over-draining trend will increase, basically because its inefficient use, and will result in a significant loss in water availability for all purposes. B Lack of institutional and normativity capacity for aquifer handling regulation will continue. C Swamp waters will be reduced because natural and human draining, decreasing zone’s biodiversity. D Underground water quality will be reduced because greater depths extraction and contact with highly contaminated superficial waters. E State of Mexico’s Development Plan, and particularly municipalities inside Toluca Valley, will promote a strong urban-productive growth and informal market, extending Valley’s urban zone gradually. F Social conflicts will increase because water shortage and citizenship distrust to poor government actions. G Continuous appearance of cracks with direction mostly to East-West and in some cases to North-South, resulting on high vulnerability situations. H No government cooperation will increase and more indigenous-peasant actions for natural resources defense, located in Toluca Valley Highlands (mazahua, otomí, and other people), will occur. I Lerma’s two swamps will be amusement parks and Chinampas to promote sustainable agricultural use will be built. J Install a public transportation system in the region (a suburban train) will increase immigration to the zone. K Weather change will produce temperature fall, rainfall levels will increase in AVT and reduction of hydric production in Cutzamala system. L Enrique Peña Nieto’s election as president will be favorable for institutional arrangements between Mexico City and state of Mexico governments mainly related to water importation policies. M Mexico City’s water treatment plant building and operation (in Atotonilco de Tula town) will increase water regulation to Mexico City and will drop water extraction in Aquifer. N Water denationalization will modify supply and demand.

Initial probability 0.87

Final probability 0.9936

0.67

0.9567

0.77

0.97071

0.70

0.9364

0.72

0.95569

0.67

0.66146

0.68

0.84578

0.53

0.90883

0.42

0.00892

0.53

0.96907

0.5

0.81967

0.38

0.56728

0.45

0.06542

0.27

0.69781

Source: De León and Sánchez-Guerrero (2011)

It is detected that maybe agricultural use cannot cause major impact, because number of cultivated hectares has a decreasing trend (93.15  million hectares for 2020, which represents 45% less than year 2007). This shows clear signs about most possible future behavior of water use: required volume decreasing for agricultural use as a consequence of population growth and agricultural soil changing to

39

2  Methodology for Building Trend Scenarios

Final Probability 0.993 0.971 0.957 0.969 0.956 0.936 0.909 0.846 0.820

Initial Probability

A C 0.870 0.770 0.720 0.700 0.680 0.670 0.67 0.53 0.530 0.5 0.45 0.42

B

D G

C H J

0.698 0.661

F

K 0.567

N

L Software output KSIM-IMPACT ‘99

0.38

0.22

M

0.06

I

0.001

Fig. 2.6  Cross impact analysis results. (Source: De León and Sánchez-Guerrero 2011)

industrial and urban, probably the major water use for 2010–2020 range will be domestic. The area urbanization (produced by economic development of the zone and its nearness to Mexico City) will cause a major coverage in potable water and sewage systems as a need that, according to trends, will be higher than forecasted national average for 2020, in other words, population will have restricted access to potable water infrastructure. Strong immigration from Mexico City to the valley, due to real estate and consumer goods “accessible” prizes in one of the 23 municipalities part of AVT, will reinforce urbanization trend, which will cause marshes damage, impacting adversely on the landscape, biodiversity, and superficial water volumes because human draining phenomena. It is estimated that for year 2020, only 66% of sewage will be treated. This will heighten contamination, which will affect health from inhabitants to environment. Besides, it is highly probable that underground water will be contaminated as a consequence of sewage and solid waste leaking, which will provoke that extracted water can only be used for agricultural and industrial purposes and not for domestic consumption. Experts indicate that underground water contamination will have a 93% of occurrence probability.

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Municipal inhabitants will continue in risk of unhealthiness because the presence of diseases related to aquifer contamination, this could cause gastrointestinal, respiratory, and cutaneous epidemics. Another infection focus that will last is sewage floods from Lerma River burst. Actually, it is considered to pipe the river to avoid these situations, however, it would take the risk that contamination emerge from the underground because current cracks in the zone could affect this infrastructure, causing sewage infiltration from Lerma River to the aquifer. It is possible to build water treatment plants for municipalities to counteract superficial contamination, which still be operated by private sector companies without government strict regulation. In a period of 5 years, these companies will follow their usual practices to reduce costs: they will not use their total capacity or give the necessary treatment for complex contamination that will exist in the area. It will be from year 2015, after general problem of the aquifer be intensified, that will be viable to have a better government and social control upon these companies. The building of suburban train scheduled to start in 2015, that will connect Observatorio station of the subway system from Mexico City with Toluca City, Lerma, and Metepec, will have as a consequence a major immigration to the Valley, due to mobilization facilities. In urban development plans it is considered that Toluca, Lerma, and Metepec will be the municipalities with greatest population growth in the region for the next 10 years. The State of Mexico Development Plan is structured in three supports: social, economic, and public security, where environmental is inside economic area. The aforementioned shows that environmental issues are –and will be– treated from an economic perspective, at least in the next years. The current institutional and normative capacity is a decisive fact that increases the probability of a violent and organized social crisis occurs between 2015–2020, as a consequence of questionable institutional functionality and efficiency for responding to water availability and area contamination issues; as a result, institutions will be compelled to change their structures and ways of operation, with the purpose to achieve a major functionality and efficiency in solving problems related to this aquifer. This structural change will force to improve social participation mechanisms in the community. In 2010–2020 period will be two federal and two state elections. All indicates that during electoral period of 2012, the probable two strongest president candidates will take water issue only as a political tool, because both represent opposed interests regarding water, and main objective will be again to win elections before resolving aquifer problems. If State of Mexico’s candidate wins the president election, institutional arrangements between Mexico City’s government and State of Mexico’s government could be favored (as always as same political party wins the election in Mexico City), mainly in relation with aquifer sanity policies, although a significant advance cannot be produced because economic interests from industries and companies that will promote mostly economic and political policies than environmental. In the same way, if left-wing party candidate wins the election, maybe agreements that benefit Mexico City will be promoted, and aquifer sanitation advances could be lightly

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greater than the other candidate because environmental administration trend given in Mexico City, but it would not be significant. The problem with other administrations has been dealt generally adapting political models from international contexts. In other words, neither next federal nor state elections are promising to rescue this aquifer. Maybe until future elections (2017 and 2018) political class will be forced to act in an effective way because future problems and social pressure. Investments regarding water (governmental and private) will show annual significant grow (71% from 2010 to 2020). Likewise, international loan will also get higher with the purpose to resolve overexploitation and contamination situation. This will cause a major indebtedness in State of Mexico and the country. Besides, denationalization mechanisms applied as a government policy come from a worldwide trend; strongly supported by International Organisms (IMF and IDB mainly) and transnational enterprises, especially French (Suez and Vivendi), American (Bechtel-United Utilities), and Spanish (principally Aguas de Barcelona). As a consequence of denationalization, rates and control systems can be modernized and also can be installed “prepaid water” meter. Direct consequence of denationalization is water price increase for customers. A major municipal opening for license to supply drinking water and sewage treatment to private companies will be promoted. In the future, new licenses will be made by government transaction offices, which will create a kind of oligopoly. These government offices, one by one will be managed by private entities. Another way of denationalization is the millionaire license transfers in black market, for year 2015 a great part of licenses will be from real estate and industries and a minor percentage to agricultural use. Consume habits will change, especially in less privileged sectors, mainly in Tenango del Valle, Xonacatlán, and Otzolotepec municipalities, that are characterized by having lowest human development rate and a low active economic population. If water is denationalized, social conflicts will appear because water will be a lacking resource in private hands with unachievable rates for low-income level people and situated in poorest municipalities. Regardless of water denationalization, social conflicts will also emerge, because water will be a low availability resource and highly competitive, and example of this situation will be strong mobilizations of indigenous peasant communities that are located in Highlands of Toluca Valley (mazahua, otomí people, and others) defending natural resources in their territory, likewise, no cooperative attitude with government will be strongly perceptible. In relation with extreme phenomena in the area it will presented two: floods and cracking. Weather change will reinforce floods, because rainfall levels in the zone will increase, additionally, floods will still affect and putting people at risk, and construction on lagoon areas will continue due to high demand on real estate. It will be massive evacuations of people by floods because Valley has a high vulnerability in this matter, which can modify urban and population trends. Floods will not only affect infrastructure, also cracks that will appear continuously. The suburban train projects can be affected by these cracks, as in installation as its operation. Weather change also can promote aquifer recharge and different uses

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of rainfall water. Although, serious and long drought seasons can occur, which could cause a decrease of superficial and underground waterbody volumes in studied area and in others of external character, like Cutzamala system dam (which can cause that Mexico City increases its extraction volumes in Toluca Valley aquifer, generating a lower water supply for its municipalities). Solutions that will be introduced over the course of 2010–2015 will be: the artificial aquifer recharge, application of rainfall water collecting policies and technologies, and sewage treatment in domestic, industrial, and agricultural uses. Another project, although not directly related to aquifer, is that when big plant from Atotonilco for more than 60% of Metropolitan Area of Mexico City sewage treatment will be installed, waste water will be reused, which can contribute to reduce extracted water by Mexico City. It is proposed that turn swamps on to recreational parks with the purpose to protect place biodiversity, generate tourism income, and create pleasant urban environments. This project will still be inoperable, due to represents a different alternative that contradicts traditional solutions (that consist in building big infrastructure constructions). Regarding future intellectual capital training, it will be necessary to rain human resources in matters of recharge, administrative management of water, rainfall technology, and environmental risk management. AVT future is clouded without organized social participation, its future is not in hands of science and technology or academic world, and it is in hands of political agents, of their will and correlation of forces that result of informed and organized social participation.

2.3  Conclusions Methodology proposed, despite of others, has as an advantage that is supported by system thinking, assigns trend scenario elaboration inside planning process and considers quantitative and qualitative techniques are necessary and complementary. It proposes five steps for its construction, particularly, five important aspects for its writing. The process of building the scenario allowed to “Toluca Valley Aquifer’s” stakeholders to recognize aquifer as a system where many actors, problematics, objectives, decision-making, interests and demands interact. It is normal that every stakeholder do their activities and think that what they are doing is right, although is not normal that they recognize that their actions are impacting whole system. A trend scenario saturated by numbers and tables is a cold document that does not lead to decision-making awareness. A trend scenario written only in an artistic way, without objective information and analytic method resulted basis, will be a weak tool in decision-making moments. A complete trend scenario has to be a product that conjugates our analytic and synthetic thinking, in other words, it should offers studied phenomenon awareness and simultaneously rational arguments that lead to decision-making and organizational learning. With both approaches the

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plausibility increases. Trend scenario elaboration demands will, time, money, information, knowledge, and experience. From 6–18  months is the time estimated to fulfill it. As long as we do plan a continuous language in our organization and scenarios a frequent tool to visualize future, preparing time will reduce. One of the most difficult tasks is transforming data into narration. For that, it is highly recommended to use diagrams or matrixes that interrelate the different events and by them write the scenario.

References R.L. Ackoff, A Concept of Corporate Planning (Wiley, New York, 1970) R.L. Ackoff, Creating the Corporate Future (Wiley, New York, 1981) J. Alcamo, G. Gallopín, Building a 2nd Generation of World Water Scenarios. UNESCO-WWAP (United Nations World Water Assessment Programme, Paris, 2009), p. 13 P. Balderas, G. Sánchez, Ingeniería de sistemas, metodologías y técnicas. México: Plaza y Valdes (2015) M. Becerra, J. Sainz, C. Munoz., Los conflictos por el agua en Mexico. CIDE, Gestión y Política Pública XV(1), 111–143 (2006) M. Beuchot, Hermenéutica, analogía y símbolo (Herder, Barcelona, 2004) L. Börjeson, K. Höjer, T. Ekvall, G. Finnveden, Scenario types and techniques: Toward a user’s guide. Futures 38, 723–739 (2006) R. Bradfield et al., The origins and evolution of scenario techniques in long range business planning. Futures 37(8), 795–812 (2005) J. Bright, Practical Technological Forecasting (The Industrial Management Center, Austin, 1978) P. Checkland, Systems Thinking, Systems Practice (Wiley, Chichester, 1981) T.J.  Chermack, Studying scenario planning: Theory, Research, suggestions, and hypotheses. Technol. Forecast. Soc. Chang. 72, 59–73 (2005) T.J. Chermack, Scenario Planning in Organizations: How to Create, Use, and Assess Scenarios (Berrett-Koehler Publishers, Inc., San Francisco, 2011) J.F. Coates, Scenario planning. Technol. Forecast. Soc. Chang. 65, 115–123 (2000) CONAGUA  – GTZ.  Comisión Nacional del Agua  – Deutsche Gesellschaft für Technische Zusammenarbeit, Informe final: Plan de manejo de la Cuenca del Río Lerma en el Valle de Toluca (CONAGUA, México City, 2008) CONAGUA. Comisión Nacional del Agua, Zonas de reseva de agua potable para la ciudad de Toluca 2030 (CONAGUA, México City, 2003) CONAGUA. Comisión Nacional del Agua, Registro público de derechos del agua (REPDA). En continua actualizacioón (CONAGUA, México City, 2009) CONAGUA-DLEM, Plan de manejo de la cuenca del Acuifero Valle de Toluca (CONAGUA, México City, 2008-2010) CONAGUA-IMTA.  Comisión Nacional del Agua-Instituto Mexicano de Tecnología del Agua, Foros de consulta del Programa Nacional Hídrico 2013-2018 (CONAGUA-IMTA, México City, 2013) CONAPO. Consejo Nacional de la Población, Proyecciones de la población 2010-2050. SEGOB (CONAPO, México City, 2010) R. De León, La construcción de escenarios: Escenarios al 2020 para el Acuífero Valle de Toluca. Tesis de maestría, UNAM, México, 2010 R. De León, G. Sánchez-Guerrero, Construcción de escenarios: Escenario del Acuífero Valle de Toluca, in Memorias XIV Congreso Internacional de Investigación en Ciencias Administrativas, (ACACIA, Boca del Río, 2011), pp. 50–65

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DOF. Diario Oficial de la Federación, Declaratoria de clasificación del río Lerma que establece su capacidad de asimilación y dilución (Gobierno de la República, México City, 1996) DOF. Diario Oficial de la Federación, Actualización del balance hídrico, 28 de agosto (Gobierno de la República, México City, 2009) O. Durance, M. Godet, Scenario building: Uses and abuses. Technol. Forecast. Soc. Chang. 77, 1488–1492 (2010) A. Fink, O. Schlake, Scenario management: An approach for strategic foresight. Compet. Intell. Rev. 11(1), 37–45 (2000) G. Gallopín, Global Water Futures 2050: Five Stylized Scenarios (United Nations World Water Assessment Programme, Paris, 2011), p. 20 J.  Gharajedaghi, Systems Thinking: Managing Chaos and Complexity (Elsevier, Inc., Morgan Kaufmann, New York, 2011) M. Godet, The art of scenarios and strategic planning: Tools and pitfalls. Technol. Forecast. Soc. Chang. 65, 3–22 (2000) M. Höjer, K. Dreborg, R. Engström, U. Gunnarsson, A. Svenfelt, Experiences of the development and use of scenarios for evaluating Swedish national environmental objectives. Futuras 43, 1–15 (2011) IMTA. Instituto Mexicano de Tecnología del Agua, La investigación científica, de- sarrollo tecnológico y formación de Recursos humanos del sector hídrico en México. Memoria de la Reunión de Planeación Participativa (IMTA, Jiutepec, Morelos, 2006) H. Jouvenel, A brief methodological guide to scenario building. Technol. Forecast. Soc. Chang. 65, 37–48 (2000) P.  Julien, P.  Lamonde, D.  Latouche, La méthode des scénarios en prospective. L’ Actualité Économique 51, 253–281 (1974) R. Mason, I. Mitroff, Challenging Strategic Planning Assumptions, Theory, Cases and Techniques (Wiley, New York, 1981) H.  Niccolas, Aplicación de la matriz de impactos cruzados en procesos electorales del estado de Hidalgo: Desarrollo de la herramienta de software KSIM Impact-99. Tesis de Maestría, UNAM, México, 2000 F. O’Brien, Scenario planning-lessons for practice from teaching and learning. Eur. J. Oper. Res. 152, 709–722 (2004) H. Özbekhan, The emerging methodology of planning. Fields Fields 10, 63–80 (1974) T. Ritchey, Wicked Problems – Social Messes: Decision Support Modelling with Morphological Analysis (Springer, New York, 2011) SACMEX. Sistema de Aguas de la Ciudad de México, Estudio de los niveles estáticos y dinámicos de los acuíferos del Valle de Toluca para la interpretación del abatimiento que han presentado los acuíferos del Valle de Toluca, Ixtlahuaca-Atlacomulco en los últimos 50 años (SACMEX, México City, 2007) G. Sánchez-Guerrero, Técnicas heurísticas participativas para la planeación: procesos breves de intervención (Plaza y Valdes, México City, 2016) G. Sánchez, Técnicas participativas para la planeación. México: Fundación ICA (2003) P.  Schoemaker, Scenario planning: A tool for strategic thinking. Sloan Manag. Rev. 36(2), 25–20 (1995) SCT-IMT-UNAM.  Secretaría de Comunicaciones y Transporte  - Instituto Mexicano del Transporte  - Universidad Nacional Autónoma de México, Seguridad y competitividad del sistema en el transporte carretero de cara al siglo XXI. Memoria de la Reunión de Planeación Participativa (SCT-IMT-UNAM, Querétaro, 1998) R.A. Slaughter, Futures studies as a civilization catalyst. Futures 34(2), 349–363 (2002) O. Theron et al., Methodology to translate policy assessment problems into scenarios: The example of the SEAMLESS integrated framework. Environ. Sci. Policy 12, 619–630 (2009) K. Van der Heijden, Scenarios: The Art of Strategic Conversation (Wiley, Chichester, 2005) P. Van Notten, J. Rotmans, M. van Asselt, D. Rothman, An updated scenario typology. Futures 35, 423–443 (2003)

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C.A. Varum, C. Melo, Directions in scenario planning literature: A review of the past decades. Futures 42, 355–369 (2010) S.C. Wheelwright, S. Makridakis, Forecasting Methods for Management (Wiley, New York, 1976) A. Wright, Using Scenarios to Challenge and Change Management Thinking: , Working Paper Series (University of Wolverhampton, Business School, Telford, 2003) WWAP. World Water Assessment Program, The United Nations World Water Development Report 4: Managing Water under Uncertainty and Risk (UNESCO, Paris, 2012)

Chapter 3

Consulting as a Systemic Intervention Process Benito Sánchez-Lara and Oscar Everardo Flores-Choperena

3.1  Introduction Companies rely on consultancy experts or firms to solve organizational problems, which frequently involve operations, volume, and creation. Consulting failures have been widely reported in the literature, along with the identification of a variety of causes and proposals to solve them. Three categories of consulting failures have been identified, which are related to consultancy actors and consulting intervention processes: 1) consultancy responses to customer preferences; 2) consultant practices and knowledge; and 3) the techniques, practices, tools, methods, and predominant methodologies used in the consultancy. Systemic theoretical and methodological elements have also been identified in consultancy processes, which are integrated in different stages and establish favorable conditions for processes. Identified elements are based on systemic intervention concepts from Midgley (2000), who defined them as proposed actions realized by an agent to create a change that considers system limits. When these elements are included in the consulting process, it is improved by creating conditions that favor positive results. Furthermore, the chance of changing a typically heuristic process into a rational process increases from the additional elements. This chapter is structured into five sections. The first section discusses consultancy and its issues, whereas the second section describes some consulting processes. The next section features systemic intervention characteristics. In the fourth

B. Sánchez-Lara (*) Department of Systems, Faculty of Engineering, National Autonomous University of Mexico, Mexico City, Mexico e-mail: [email protected] O. E. Flores-Choperena Materials Research Institute, National Autonomous University of Mexico, Mexico City, Mexico © Springer Nature Switzerland AG 2021 P. E. Balderas-Cañas, G. de las N. Sánchez-Guerrero (eds.), Problem Solving In Operation Management, https://doi.org/10.1007/978-3-030-50089-4_3

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section, systemic-theoretical-methodological elements are proposed for the consulting process. Finally, conclusions of the work are presented.

3.2  Consultancy Problems Consultancy has been defined by authors such as Appelbaum and Steed (2005), Block (2009), Kubr (2009), Morfin (1993), Graubner (2006), and Schein (1988) as an external professional service provided by specialized and qualified consultants. The offered services aim to solve problems, discover and evaluate opportunities, improve organizational learning, and establish solutions. Companies that resort to consultancy expect beneficial results and changes; however, in many cases, interventions provided by consultants are ineffective. Surveys by Mohe and Seidl (2007) indicate that consultancy customers only consider these interventions to be moderately effective. Besides indicating improvement areas, little empirical research has focused on the conceptualization and operationalization of what makes a consulting intervention successful from a customer’s perspective (Bronnenmayer et  al. 2014). Zackrison and Freedman (2003) defined ineffective consultancy in terms of incomplete achievements, time expended, and resources and information used. In a Mexican context, formal research on consultancy effectiveness is lacking, but some authors have associated benefits with it. El Colegio de Mexico (2012) reported that there is a relationship between low competitiveness of small and medium enterprises and the insufficient number of consultants who support them. Fondo PyME is a government support program aimed at small and medium-sized enterprises. The PyME Fund has resources for consultancy services. Some of these resources are designated for instructors and consultants’ formation, whereas other resources are allocated to elaborate methodologies, contents, and support materials for training and consultancy. The results of a Fondo PyME Evaluation showed that this fund has no influence on employment or wages in the short term; statistical evidence for the effects on sales is not consistent. It should be noted that evaluation results are integrated and do not define results by categories. The Ministry of Economy (la Secretaría de Economía 2010), in their bulletin Nuestras empresas, Secretaría de Economía (2010) reported that the General Consulting Program, with resources from the SME Fund, had supported 39 projects in 2010 for an amount of MXN $71.9  million, benefitted 3421 enterprises, and retained 14,505 employees. The National Consultancy Program PyME-JICA supported 121 enterprises for an amount of MXN $3,662,384 and retained 2803 employees. González-Sánchez (2012) indicated that consultant activity is strategic for enterprises; for example, increase of people employed and income growth in this industry are 3.8% and 16.5%, respectively National Institute of Statistics and Geography of Mexico (INEGI). In an interview with Ortiz-Ruiz (2009), Toru Moriguchi,

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Chairman of Consulting Service Division, said that young consultants in Mexico are not properly recognized; this is in contrast to Japan, where consultants are employed as entrepreneurs or engineers for at least for 30 years before being consultants. This nonconformity has emerged as a result of the Economic Association Strengthening Agreement between Japan and Mexico. Furthermore, despite the relevance of consultancy, the attention paid to the subject in academic research is surprisingly low, resulting in insufficient empirical data on consultancy practice (Bronnenmayer et al. 2014). However, a literature review by Flores-Choperena (2011) and Flores-Choperena and Sánchez-Lara (2011) identified three reasons for ineffective consultancy, based on the experience of some experts in the consultancy field. These reasons are associated with consulting actors and intervention strategies. The first reasons for ineffectiveness occurs when the consulting activity is based on customer preferences (Block 2009). Therefore, a company (considered to be the client) may assume that the answer to its needs is a consulting intervention; thus, it turns to a consultant, which is considered to be an agent of change that can solve problems reported by the company. In some way, the client has already diagnosed and identified some issues that cause dissatisfaction, discontent, and inefficacy in activities, chores, processes, functions, etc. In this situation, the consultant focuses entirely on what the customer requires. The idea of satisfying customer needs, as a result of the intervention process, distorts the purpose of consultancy because the customer has an expected result in mind, even though the role of any consultancy is to increase the capacity of its client to innovate (Lessem et  al. 2010). Therefore, the client will not accept suggested solutions that are not of interest. Furthermore, if the customer wants to achieve some change during the consulting process, the consultant will be conditioned to perform actions associated with what the customer expects; this conditioning could lead to consultancy being well done but incorrect. In the second reason, the consultancy is limited by the practices and knowledge of the consultant (Midgley 2000). The customer may not realize the origin of his problems; however, he has a feeling that something is going wrong. Therefore, the customer puts his trust in the consultant to take part in his company; he expects that this intervention will improve the current condition of the company because of the knowledge base of a highly qualified consultant (Lessem et al. 2010). The consultant actor must identify and express problems to solve, formulate or select solutions, and implement solutions. In these circumstances, theoretical-methodological limitations and the experience of the consultant actor can distort the way the intervention takes place and its results; in other words, the consultant has a gap between his or her education and practical experience (Lessem et al. 2010). In other cases, the consultant may assume a critical or reflexive attitude in the diagnosis and solution formulation processes. As a result, the consultant selects and apply methods or techniques in which he is an expert; in other words, the consultant tries to solve all problems with the same tool. The third line of explanation is related to the second. In the consulting field, successful tools are transformed into popular panaceas (Ackoff 1995, 2000), which often can monopolize consultancy practice because they are frequently requested by

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customers and offered by consultants. The benefits of these tools, which are associated with globally recognized enterprises, are reported in the literature—most often by consultants who studied in executive programs. Examples include management by objectives (MBO), business process re-engineering (BPR), and the Balance Scorecard (BSC), which were developed by consultant “gurus” such as Peter Drucker (MBO), Michael Hammer and James Champy (BPR), and Robert Kaplan and David Norton (BSC) (Lessem et al. 2010). That a company implants some of the tools considered popular panaceas becomes a requirement to access a market and an evaluation criterion used by regulatory institutions and companies that subcontract services. The implementation of tools considered to be panaceas does not consider the needs or conditions of the company; it is not diagnosed in order to determine its properties and potential related to the problems that want to be solved or improved by the consultant. The solution is implanted without a critical thought about the limitations and scope of each tool, using them only as fashion or for accomplishing a market requirement. The aforementioned ideas associated with consultancy ineffectiveness explain in some way the next set of problems: • At the end of consulting interventions, the customer is not satisfied with the results, changes made in the organization are not complete, or the problem of interest is not solved. • Through consultancy, the customer only wants to obtain a certificate or seal that fulfills regulations or requirements. • The customer and consultant have assumed incorrectly that the company is prepared or has the proper conditions for the tool they want to implement. • The customer and consultant do not know the theoretical-methodological information on which many tools are based on and which demonstrates their limitations and scope. • The decision to implement a certain tool is influenced by the perception that the customer or consultant have of its effectiveness, for many are panaceas to solve all problems. • The consultant’s professional profile is broad; as a result, there are marked differences in intervention approaches, as well as in promised and achieved results. • The success of enterprises in a specific market is defined by whether a specific tool has been implemented. Therefore, in a consultancy relationship, the customer asks for that specific tool and then a consultant implements it without investigating the company’s needs and conditions.

3.3  Consulting Process There are different consulting models and processes described in the literature (Kubr 2009; Kurpius et al. 1993; Lippitt 1977; Lorenzo et al. 2007; Morfin 1993; Schein 1986); in general, they are focused on the customer’s satisfaction as the last

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objective. This section describes four consulting processes developed by Kubr (2009), Morfin (1993), Block (2009), and Ochoa-Rosso (1985) in order to illustrate and explain what a consulting process is. An important finding is that the list of consulting process proposals is large; however, there are similarities in the steps proposed and the intervention methods.

3.3.1  Kubr’s Consulting Process Kubr (2009) indicated that the consulting process is based on a strong customer– consultant relationship, where the latter is considered to be a professional in his or her field. Kurb’s consulting process stages are initiation, diagnosis, planning the means, application, and conclusion. In the initiation stage, the consultant works on the customer relationship, trying to obtain information in order to create a general overview of the current situation, desire for change, and situations that cause the customer’s dissatisfaction. At the end of this stage, the consultant has to make an initial analysis of the situation and preliminary action plan. If the proposal is interesting to the customer, a consultancy contract is established. During the diagnosis stage, the consultant and customer work together at establishing the company’s current situation, identifying the problem, expressing the objectives for intervention, and proposing and evaluating alternative solutions. When planning the means, the problem’s solution is identified. This has to be evaluated by the customer and satisfy the planned objective. The selected solution selection should be participative, with an aim to address all stakeholder expectations. In addition, an action plan and strategy to realize changes are developed, in addition to considering eventualities during implementation. In the application stage, the solution is implemented. If new problems or obstacles arise, it becomes necessary to review the solution and action plan. Finally, for the conclusion stage, the consultant presents the results to be evaluated by the customer, gives the process report, establishes commitments, and, possibly, creates a future collaborative relationship.

3.3.2  Morfín’s Consulting Process Morfín’s (1993) consulting process features three phases: diagnosis, operation, and conclusion. The diagnosis phase is the most complex because it includes the first conversations with the customer, problem delimitation and expression, generation of solutions, and selection and elaboration of intervention strategies. The operation phase is the intervention, during which the customer–consultant relationship is strengthened; the consultant obtains permission to perform the intervention, so he or she must be able to execute it and establish the best intervention conditions. The

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concluding phase includes monitoring and control of intervention results, as well as furnishing of the documents and commitments agreed upon with customer.

3.3.3  Block’s Consulting Process Block (2009) established five phases in the consultancy process: access and contract, data collection and evaluation, information exchange and planning, execution, and extension or conclusion. In the access and contract phase, the customer evaluates whether the consultant is the appropriate person to perform the intervention; it includes meetings, presentations, and questions between the customer and consultant; in addition, the customer indicates the expected results from the intervention. The second phase is performed by the consultant, who evaluates the situation identified by the customer and attempts to structure the problem, identifies people involved in the change, and collects data; the problem to solve is formulated in this phase. The next phase is the customer–consultant information exchange. In this phase, the information obtained in the previous phase is presented to the customer; the customer’s collaboration in the analysis is important but not required. It is important to emphasize change resistance caused by the consulting intervention. For some people, this is the planning phase. Then, the plan is executed in the fourth phase, based on the customer–consultant agreement. This phase is executed by one or both of them. Finally, an intervention report is presented and, if necessary, the intervention scope is extended and the customer–consultant relationship continues.

3.3.4  Systems Method of Ochoa-Rosso Using a Mexican context, Ochoa-Rosso (1985) suggested two intervention procedures. Although they are not specifically referred to as consulting processes, both are focused on real productive systems. The first is directed to productive systems on which corrections or improvements to functioning have to be made, whereas the second is directed to systems for which growth or reductions must be planned. Both procedures have marked similarities, including system identification, system and context analysis, pre-test of alternatives, selecting alternatives, and control after implantation. Important differences include the type of analysis that is performed before presenting solutions or design alternatives; the first analysis is restricted only to the system and the other also considers the context. Another is the performance diagnosis that is applied before making corrections or improvements on productive systems. Unlike the consulting processes described previously, the Ochoa-Rosso (1985) method does not indicate a procedure for establishing some sort of customer–consultant relationship. Rather, the method focuses more on the process than on transactive relationships between customer and consultant.

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3.3.5  Summarizing the Consulting Processes The consulting processes documented in this section share some common elements and highlight the customer–consultant relationship as a decisive factor. They are useful as guides or models. However, they do not consider the conditions on which consultancy is developed, such as complexity, plurality, and context. If we consider consultancy as an activities system, according to Dodder and Sussman (2002), complexity can be found in three dimensions: 1. Internal, which is given by the number of activities involved and their interconnections. 2. Behavioral, which is given by emerging behavior with its origin in the group of activities and interactions; these dimensions emphasize the difficulty in systematizing consultancy stages and uncertainty regarding response times for changes made, as well as possible counterintuitive behaviors that emerge. 3. Evaluative, which is given by competence between decision-makers and stakeholder perspectives; different points of view about performance, process, or consultancy results can occur. Regarding plurality, consultancy is shown as a customer–consultant relationship in which the customer establishes his or her requirements and perspectives, and the consultant offers his or her proposed solutions and interventions strategies. With this definition, it can be assumed that the customer and consultant are people that act individually; however, they can be groups that together are considered to be the customer or the consultant. In the same way, the importance of people involved is dissolved. Additionally, according to Flood and Jackson (1991), depending on the stakeholder’s profile, interests, and objectives, the relationships may be singular, plural, or coercive. In singular relationships, stakeholders share interests, values, and beliefs that are highly compatible; they agree on means and ends, participate in decisions-making, and act based on agreed-upon objectives. In plural relationships, participants have certain compatibility of interests, values, and beliefs; however, they differ on some points. They do not necessarily agree on means and ends, but a commitment is possible; they participate in decision-making and act based on agreed-upon objectives. The extreme relationships are coercive, in which participants do not share interests. Values and beliefs are in conflict, they disagree on means and ends, and a real commitment is not possible; some stakeholders limit others in decision-making, and there is no agreement about objectives. Consulting as a change process confronts the problem that the context of the company also changes. Emery and Trist (1965) said that the context can range from placid to turbulent. The placid context is considered the simplest, in which an enterprise’s purposes, as well as its problems, remain practically unchanged; problems do not emerge systematically but randomly. On the other extreme is the turbulent context, in which dynamic processes and systemic problems emerge. Other authors expanded on the number of contexts; for example, Baburoglu (1988) proposes a vortex context and Roggema (2008) discussed the swarm context.

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3.4  Systemic Intervention Midgley (2000) suggested that an intervention methodology is systemic if three characteristics are explicit. The first involves critical thinking about limits of the system on which the intervention is made, consequences, and possible outcomes of decision making. Then, attempts should be made to identify the system, its boundaries, and its context—in other words, include, exclude, and marginalize properly. This identification is related to the principles and values of the consultant, as through them the limits and scope of the consultant’s own intervention can be evaluated. This ethical element, guiding the actions toward the objectives, becomes important because it considers their consequences on an intervention. Considering that consultancy takes place in the company, Midgley’s (2000) approach involves a tactical planning method to identify critical areas where the intervention has to be made. These can be functions, processes, or physical areas of the same company. The identification of limits includes defining the problematic and the scope of the consulting work. The second characteristic is judgment, which has to be included in the methods selection for guiding the intervention. It is important to perform a selection exercise in which the largest number of methods and tools are analyzed, taking into consideration multiple criteria to define the properness of their use. In this way, complexity, plurality, and context conditions can be considered. In this analysis, the largest possible number of methods should be explored; this would assure the appropriate selection according to specific criteria. This activity aims to ensure efficiency in Ackoff’s (1971) approach, with the best selection and achievement of expected results. Action is the third characteristic. Actions in consultancy are determined in a local context, suitable to a temporal horizon, considering an improvement idea and its objective. A systemic intervention takes into account that any action has to be in order to improve a critical area. Considering the aforementioned characteristics, Midgley (2000) defined systemic intervention as an action with purpose, performed by an agent to create a spatial and temporal change. Also included in this definition can be the limitations of change through visualizing the consequences and possible outcomes, and method selection by exploration. In this way, each intervention is unique and temporal, and has judgments and critical principles that guide its actions.

3.4.1  Consultancy as a Systemic Intervention The consulting processes reviewed previously do not explicitly describe stages characterized by critical thinking and analysis, with the exception of action. Therefore, the second objective of this chapter is to identify and place theoretical-­ methodological elements of systemic intervention in the consultancy process which, when integrated in all of its stages, establish favorable conditions for the process

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itself. It is assumed that the consulting process will be a systemic intervention. The theoretical-methodological elements are presented in the following sections and placed in the consultancy process.

3.4.2  Critical Thinking in Consultancy Critique should be a reflexive process in which the consultant and customer are involved, in addition to other stakeholders in the consultancy, to delimitate and contextualize the enterprise, function, or area on which the intervention will be performed through the consultancy. Boundary definition is an arbitrary activity but not trivial. Therefore, the consultant or customer can decide for how long or where to direct the consultancy; however, this decision requires one to know what is not being considered. The predicament of boundary definition involves deciding which problems will not be resolved or which are not in the scope of solution, which is to exclude and even marginalize them. The condition of critical thinking in the consultancy should determine the purpose of the intervention and the tasks of the consultant. The aforementioned conditions assume that the consultant has theoretical-methodological knowledge and experience to mentally structure the intervention, which then leads to delimitation. A mental structure is elaborated with knowledge and experience and can be modified through time according to a consultant’s skills and the abstraction methods that he or she uses to gather information in the environment. Following Ramaprasad and Mitroff (1984), the mental structures can guide a consultant to determining the purpose of an intervention and the means to modify the status quo through them in three ways. The first way is the application of a preexisting mental structure. This implies an arbitrary selection process, not necessarily reflexive, from which the intervention method is selected, considered suitable, and executed. The selection is led by judgment standards, based in knowledge and experience; the standards are sided with the purpose of the intervention. The definition of the standards is very important; dichotomous standards are not recommended. The use of this mental structure is associated with the second explanation of consultancy ineffectiveness, in which the consultant determines the intervention approach. Also associated is a deductive perspective that considers the situation as a well-defined problem, as Bishop (1967) stated. The second way is basic abstraction. Here, using observations of the problem and data collected from it, the consultant determines what should be included, excluded, and marginalized. The observation skills of the consultant should be developed broadly. The use of this structure can be associated with the definition of what Bryson (1988) called strategic aspects. The third way is reflective abstraction. This is only possible if a consultant’s experience and knowledge, which form his mental structure, are combined and a reflective exercise is performed to make changes to the structure. The consultant’s

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aims to evolve his or her mental structure. In general, this is the recommended approach. In summary, the essential component of the critique is the elaboration of a mental structure, with which the purpose of the intervention can be determined and its means can be selected. This elaboration must be the product of a participative process between the consultant and customer, in which ideas are discussed and the relationships between reality and functioning perspectives can be identified. The same process has to show the assumptions of the consultant and customer, as well as generate a rich but not absolute worldview. Bishop (1967) identified three consultancy stakeholders: the customer, the consultant, and a third party. Their interactions can generate multiple approaches to the problem and possible solutions that may be in conflict. Therefore, it is necessary to discover hidden assumptions and interests of each one, so that other points of view can be understood. The elaboration of a mental structure includes the design or definition of inclusion, exclusion, or marginalization criteria. These can produce standards that have to be shared by all consultancy stakeholders in order to have a shared improvement idea and to choose between the alternatives. In addition, consultancy can be placed in time and space in order to identify changes (if they occurred) after the intervention ends. A consultancy’s mental structure, in which the individual observations of the stakeholders prevail, has to be shared and elaborated using individual mental structures. To explore this idea, Churchman (1971) and his inquiry systems for the elaboration of mental models are used. These systems are described next. Leibnizian System  According to this inquiry system, it is possible to synthesize reality to a rational or logical representation. A reality structure is properly logical and can be expressed in the same terms. It is based on a formulated representation with a theoretical framework before collecting data and facts; reality cannot be understood unless a formal theory of it exists. This form is the perfect example of deductive thinking. Lockean System  This is the opposite of the Leibniz system; however, a fact, conclusion, or proposition is not considered objective unless two or more “experts” agree and share their observations and conclusions. This system is the best choice for well-structured problems, where there is a consensus about nature’s problem and its solution. This system is an example of inductive thinking. It is necessary to point out that agreement between stakeholders is essential because as it adds value to what is considered as reality, allowing refuting or sustaining of the elaborated assumptions. These assumptions about elaborated reality are maintained unless another stronger and more detailed account emerges, which unavoidably causes a change; this is reflective abstraction (Popper 2002; Ramaprasad and Mitroff 1984).

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Kantian System  This system uses both aforementioned inquiry systems. It is recommended for poorly structured problems that lack consensus. The idea is to obtain different points of view and potential aspects of the problem. Hegelian System  While the Kantian system considers Leibnizian and Lockean as complementary systems, the Hegelian system considers them to be thesis and antithesis (depending on what occurred first), with an opportunity to integrate them. The Hegelian system is dialectic; the conflict is the principle involved in order to show assumptions behind each world representation. In this way, reality is created through conflict and confrontation of opposing points of view. Singerian System  The Singerian system proposes a systemic visualization of the solution. Activities are interrelated, and reality is identified by changes or solutions. It is considered to be a perfect example of interdisciplinary integration and also antireductionism. Figure 3.1 illustrates this kind of inquiry system. This method is adapted to a desired purpose. It can begin at any stage of the solution process, or it can be fragmented and grouped depending on the case. The four stages of Fig. 3.1 guide to the solution, and there are 3555 arrangements to tackle it. For more details, see Mitroff et al. (1974) and Suarez Rocha (1995).

Fig. 3.1  A vision of systems for problem-solving. (Source: Mitroff et al. 1974)

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Judgment in Consultancy The term judgment implies the capacity of a decision-maker to make correct decisions, create valuable opinions, or make reasonable conclusions after a process (CUP 2011; OUP 2011). For the purposes of this chapter, the term is associated with the decision made as a consequence of the decision-maker’s capacity—in this case, the consultant. In consultancy, judgment is a method to solve a problem based on the consultant’s knowledge about a set of justified theoretical elements of the subject. Under this idea, it is accepted that methodological pluralism is based on theoretical pluralism; this means that first theory and then method. The method selection is done by a critical examination of stakeholders’ standards—including, excluding, and/or marginalizing theoretical elements considering the standards. When the standards are defined and accepted, the measure of the consultancy results makes evident the changes. Midgley (2000) proposed that accepting the idea of theory first and then method implies an evaluation of the utility of theory in terms of it being considered more or less useful for the expected purpose. In addition, accepting theoretical pluralism (in other words, accepting more than one theory for intervention) leads both consultant and customer to a selection that, when both have a small group of theoretical and methodological elements and experience, slants intervention and produces the following three problems: • The consultant cannot understand and synthetize the different ideas of the stakeholders, particularly of the customer. • Questions about the selected method can emerge along the consultancy, which decreases the action’s vitality and efforts. • The consultant and customer have their own interpretations. The expert consultants associate problems to their research field, work, or experience area. For example, a quality specialist perceives quality problems. To guarantee a suitable method selection in terms of the solution, it is recommended to use at least two inquiry systems. Therefore, the judgment will be the result of a critical thought, even about the scope of the solution. Given the selection, the mental structure is modified across time, so it is necessary to prompt changes through refinement and perfection. Figure 3.2, proposed by Ramaprasad and Mitroff (1984), illustrates three stages in a mental structure development; at the farthest point, the refinement for intuition and interpretation is reached. The dimensions in the graph are reflective abstraction and basic abstraction. On the ordinate axis are two forms of judgment, emotional and logical; the first one is subjective (with feelings), whereas the second one is objective and emphasizes logical reasoning. On the abscissa axis are two forms of abstraction or perception, sensitive and intuitive; the first emphasizes acquiring information through the senses, whereas the second collects information through the unconscious.

Refined / Thought Non refined/Emotional

Judgmental Abstraction/Judgment

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Primitive Structure

Ideal development of mental structure

Refined Structure Unrefined/Sensitive

Basic Abstraction/Perception Fig. 3.2  Development of a mental structure. (Source: Ramaprasad and Mitroff 1984)

In addition to the recommendation of using at least two inquiry systems in the method selection, it is suggested to combine abstraction methods to strengthen and refine the mental structure that is used to formulate the problems and standards to identify improvements. From the combination of abstraction forms, Ramaprasad and Mitroff (1984) proposed four personalities of the participants in the consultancy: synthetic linker, analyzer, data observer, and technical processer. In critique and judgment, the synthetic personality is the most convenient, whereas the technical processer is the least convenient. Figure 3.3 shows a proposed cycle for elaboration and refinement of a mental structure with the elements described in the two first sections. In Fig. 3.3, the process starts in the real plane, then goes to perception and judgment, reaches refinement through inquiry systems, and then turns to action, finishing again in the real world. This cycle, despite its differences, follows Checkland’s (1991) suggestion about soft systems methodology. Action in Consultancy The consulting process is oriented to a problem’s solution and its actions to achieve improvements. Therefore, it is considered that the action characteristic is implied in the consulting practice. In processes such as concept design, the action characteristic is not implied.

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REFLECTIVE ABSTRACTION

MENTAL STRUCTURE (A)

JUDGEMENT EMOTIONAL

DEDUCTIVE

INDUCTIVE

MENTAL STRUCTURE (B)

REASONED PERCEPTION

BASIC ABSTRACTION

INQUIRING SYSTEM

DEDUCTIVE INDUCTIVE

DIALECTIC SENSITIVE PRAGMATIC

ACTION

INTUITIVE

REALITY Fig. 3.3  Development of a mental structure. (Source: Flores-Choperena 2011)

3.4.3  Systemic Intervention Conditions in Consulting Practice Systemic intervention can be summarized as a unique actions system performed in a local and sectorial environment in a given time period. It is critically and thoughtfully selected by considering the experience and theoretical-methodological elements of the stakeholders, with the elaboration of the intervention favoring improvements. This definition allows one to consider consulting as a systemic intervention, as well as all conditions implied be fulfilled: reflexive thinking and critiques about the predicament and problem of interest, its limits, environment, scope, and possible effects, in addition to stakeholders’ creativity, participation, and legitimation. Keeping in mind the aforementioned definition and description of systemic intervention conditions developed in the previous section, an attempt to place systemic intervention conditions in the consultancy process is shown next. It is considered that the critique condition has to be placed in the detection and problem definition phase. The customer and consultant should identify or formulate the problem to solve through consultancy. The judgment condition is placed along with the formulation or elaboration of a solution, which is proposed, formulated, and selected by the customer and consultant. The action condition has to be placed in the solution implementation and results evaluation phases, where customer and consultant execute actions and value their

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results. Placing each systemic intervention condition in the consulting process is an action that opens the possibility to define methodological instruments that allow these conditions to operate in the process—in other words, instruments that promote reflective critique and judgment between stakeholders in consultancy, particularly in the problem and solutions formulation, respectively. Considering that critique and judgment are conditions that are associated with the consulting process, but knowing that these are favorable attitudes that consulting stakeholders may or may not develop, it is recommended to include not only analytic participative techniques, but also heuristic ones. Fig. 3.4 is an attempt to summarize the aforementioned by grouping the consulting and systemic intervention phases and including theoretical-methodological elements. Additionally, in a dynamic systems approach, the influence of consulting conditions is shown.

CONSULTING PROCESS

CRITICAL REFLECTION (Transitive Process) JUDGMENT (Decision Process) ACTION (Acting Process)

SYSTEMIC INTERVENTION

START/LOCATION/ ACCESS

PLURALITY

DIAGNOSTIC/ INFORMATION

GENERATNG/ FORMULATING OF CHANGES OR SOLUTIONS

INTERVENTION/ EXECUTION

Inquiring system and mental structure. Inclusion criteria, exclusion and margining. Evaluation standards.

CLOSE/TERM/ CONTROL

COMPLEXITY

Refining of mental structuring and solution selection

CONTEXTUAL CHANGES

Actions

Fig. 3.4 Location of systemic methodological-theoretical elements in consulting. (Source: Author’s own elaboration)

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3.5  Conclusions Three reasons for consulting ineffectiveness were identified in this chapter, which are associated with consultancy actors and the nature of an intervention process. Succinctly, these lines of explanation are as follows: • The consulting practice is conditioned by customer preferences. • The consulting practice is conditioned by the consultant’s experience and knowledge. • Some techniques, practices, instruments, methods, and methodologies dominate the consulting practice scenario. Consulting as a systemic intervention can be defined as a unique actions system that is performed in a local and sectorial environment over a specific time period, selected in a critical and thoughtful way, and designed to consider customer and consultant experience and theoretical-methodological judgment elements, in order to make improvements. The definition of consulting as a systemic intervention process highlights the purpose and customer, in which the consultant must be interested, keeping in mind what is not considered, included, excluded, and marginalized. It also takes into consideration the intervention’s purpose and consultant’s role. Furthermore, it emphasizes selection of a suitable solution. The theoretical elements for making consulting a systemic intervention are as follows: (1) systemic delimitation and contextualization, (2) mental structure refinement by the customer and consultant, and (3) classification of theory over method to define proper solutions. Methodological elements for making consulting a systemic intervention can be summarized in the inclusion of instruments that encourage reflective critique and judgment between consulting stakeholders, specifically in the problem and solutions formulation. These instruments can turn into analytic, participative, and heuristic techniques. It is recommended that techniques fulfill not just one characteristic, but all three as much as possible. Making consulting a systemic intervention, with critique, judgment, and action conditions, includes reducing conditionings and biases in the consultancy process, as promoted by the customer and consultant. Effectiveness in consulting practice—measured in terms of scope fulfillment, expended time, employed resources, and available information—can increase, making it a systemic intervention. Scope, time, and resources can be improved by spending a small amount of time and resources practicing critical thought and utilizing consultancy stakeholders’ knowledge and experience. Reflective actions and decisions are always preferred.

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Chapter 4

The Role of Technological, Economic, and Usage Ruptures in the Innovation Process Cozumel A. Monroy-León

4.1  The Context of Innovation As pointed out in the special report of The Economist magazine (2007, 1–32), innovation has ceased to be exclusively part of the developed countries. The organizations of developing countries (Organization for Economic Cooperation and Development, 2005) also invest part of their budget in research and development of new products, processes, work methods, or services, which allow them to be competitive in this globalized world. To this end, the countries or organizations  – in developing countries – create areas devoted to this purpose, such as research and development laboratories or market research departments. Generally, the term innovation is related to the generation of new technologies (Carayannis et al. 2015, 8), this is to say, with the creation of technological innovations, such as the Mac computer, cell phone, Internet, and LCD screen. However, technological innovations are not the only ones; there are other types of innovations (Prax, Buisson and Silberzahn 2005, 49; Organization for Economic Cooperation and Development, 2005) such as (1) innovation of the service that McDonalds came up with, by introducing the concept of self-service in fast food (The Economist 2007, 3); (2) the organizational innovation produced by a new way of working, by introducing a quality control circle (Cole 2001, 8–10); (3) innovation of a process where there are significant changes in techniques, equipment, or information systems used (Lager 2010, 286); and (4) product innovation: such is the case of the sustainable gasket developed by UNAM’s Engineering Institute for a Mexican company specializing in gaskets (Benitez 2012) or the financial innovation exemplified with the development of the banking system by the Medicis or the invention of the agricultural credit by McCormick (Prax, Buisson and Silberzahn 2005, 49).

C. A. Monroy-León (*) Department of Systems, Faculty of Engineering, National Autonomous University of Mexico, Mexico City, Mexico © Springer Nature Switzerland AG 2021 P. E. Balderas-Cañas, G. de las N. Sánchez-Guerrero (eds.), Problem Solving In Operation Management, https://doi.org/10.1007/978-3-030-50089-4_4

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There are different definitions of innovation (Schumpeter 1934; Christensen 1997; Carayannis et al. 2015, 7), in this work innovation will be defined as “[…] the introduction of a new or significantly improved product (good or service), of a process, of a new marketing method or of a new organizational method in the company’s internal practices, the organization of the workplace or external relations […] A common characteristic of all types of innovation is that they should have been introduced”. (Organization for Economic Cooperation and Development, 2005), this is to say, that such innovation has had to be launched and accepted in a distinct market or that has been used in the company’s operations where such innovation is intended to be carried out. Similarly, that innovation is not only considered “[...] to be effective should be simple and focused; it should create new users and new markets; it should be conducted in a specific and clear way“[...] (Drucker 2002) but also it should meet a need with the purpose to be integrated into a market. This last condition will be considered as an essential characteristic in all the different types of innovation mentioned in the previous page. However, as stated by M. Jung, Y. Lee, and H. Lee, in order to achieve innovation, the success or failure of technological marketing considers as critical factors the insufficient capital, market conditions deterioration, and insufficient marketing capabilities (2015, 1–22). Authors such as Henderson and Clark (1990, 12) or Markides and Geroski (2005, 12) classify innovations into four areas, according to the impact that each one has regarding the dimensions that each one of these authors establishes. The first ones – Henderson and Clark – categorize innovations into two dimensions (Fig. 4.1). The horizontal dimension impacts the product, service, or components of the process, while the vertical captures the impact of the relationship between the tangible and intangible components, which support its creation. In this way, the radical and incremental innovations are obtained, on both ends. Radical innovation establishes a new dominant design and new essential concept designs represented in components that, together, constitute a new architecture; it breaks the existent paradigms

Without change Relationship between concepts and essential components With change

INCREMENTAL

Modular

Arquitectural

Radical

Innovation

Relnforced

Destroyed

Essential concepts Fig. 4.1  Classification of innovation. (Source: Henderson and Clark (1990))

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and creates new markets (Colombo et al. 2015, 1–7). On the other hand, incremental innovation redefines and expands the established design through the improvement of an individual component; there are no changes in the relationships of the essentially designed concepts. The architectural innovation reconfigures an established system and component relationships in a different way. Finally, modular innovation changes the essential concept designs, such as is the case of the replacement of the analog and digital phones. Markides and Geroski (2005, 12), on the other hand, consider the following two dimensions to classify the different types of innovations: the effects of innovation on consumer’s habits and behaviors and the impact generated by innovation to the tangible and intangible assets of organization (Fig. 4.2). When classifying the type of innovation to be carried out, the characteristics of the target market are known and there is a clearer idea of how much transformation there will be in the assets of organization. As noted in Markides and Geroski’s matrix, radical innovation completely destabilizes the intangible (knowledge and experience) and tangible assets (material and human resources), and it produces a great change in the consumer’s habits. Strategic innovation destroys the assets of organization, but it does not have an important impact in the consumer’s habits. On the other hand, development innovation has little impact on the assets of organization and the consumer’s habits and behaviors. Finally, greater innovation has no important changes in the assets of organization, but it does have a greater impact on the consumer’s habits and behaviors. To this end, the cell phone and automobile at the moment of creation and introduction in the intended market are radical innovations. Over time, companies that produce both of these products seek to maintain their market making technological changes, without modifying the consumer’s usage, generating strategic innovations. Greater innovations aim to expand their segment on the market by impacting the

Greater GREATER

Impact on consumer’s habits and behaviors

DEVELOPMENT

RADICAL

STRATEGIC

Innovation

Lower

Improve

Destroy

Impact of innovation on the tangible and intangible assets of organization Fig. 4.2  Classification of innovation. (Source: Markides and Geroski (2005))

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consumer’s behavior; such is the case of a dairy farm which introduces milk of 250 milliliters in the market, a presentation that they didn’t have before. With this introduction the dairy farm seeks to integrate in its market people that, due to their daily activities, need containers which allows them to transport their milk in a secure and practical way. On the other hand, development innovation aims to maintain its market by carrying out a significant improvement which will keep their customer’s attention, such is the case of a kid’s cereal and when a children’s movie comes out, they change the color of the box with the purpose of having it in context in regards to what their children’s market is living. The creation of an innovation can bring advantages, but it also brings challenges. Within the advantages, we can name that an innovation can make a great impact in organization, when this is developed with the aim to generate a competitive advantage (Porter 1998a, b, 165). In such a way that innovation produces important effects in the company’s structure, such as its growth or allowing for market’s expansion. In the case of strategic innovations, the competitive advantage can generate – when finding the new components that will stabilize the assets of organization – a reduction of time or in production costs. Radical innovations  – where intangible and tangible assets are destroyed and it has a strong impact on the consumer’s behavior – will allow knowing new technical characteristics and new market possibilities. ! It is important to know that innovation can be achieved – to obtain or maintain a competitive advantage – through two types of relationships: “business to business” (BtoB) and “business to consumer” (BtoC) (Prax, Buisson and Silberzahn 2005, 52). The advantage of the first relationship is that you have better control of the costs when achieving the bid of the desired service, as the development time of a product is reduced when the provider specializes in the requested field. The disadvantage is that when a relationship with a specific provider exits, the expansion to other markets is limited. On the other hand, in regard to “business to customer,” the consumer is the fundamental element in the social acceptance of an innovation. In a way that the innovator needs to offer new utilities of the product, service, or process to innovate and find new market alternatives in order to assure its acceptance. Some of the challenges faced by an individual or company interested in making an innovation are (Porter 1998a, b, 220) difficulty in (1) finding new market opportunities or new characteristics of the product, service, or process that one wants to innovate; (2) balancing the new market opportunities – not defined – with the existent tangible and intangible resources, which are defined and limited; (3) “controlling” the risks linked to unpredictable and mysterious behavior of the target market; and (4) having access to the new tangible and intangible assets in order to generate innovation. As introduced, the existence of four types of innovation and their classification according to (1) the relationship between the essential concepts and components or (2) the impact caused to the consumer’s behavior and/or the assets of organization makes us question the following two points: Which elements and tools allow us to achieve an innovation? Which steps must be followed to generate an innovation that meets the needs of the established market?

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4.2  Problem to Be Addressed Cases such as the one of 3M with “Post-it” (Von Hippel et al. 2003, 39–64) demonstrate how an innovation does not follow a linear path and how it faces many changes before achieving the differentiation, which allows for the introduction into a market and to meet a need. The creation of an innovation does not emerge in a continuous way (Cole 2001, 19); to think that a lonely genius invents a spontaneous and different idea which will be introduced in a market that meets a need immediately is erroneous. The definition of the characteristics which achieve the differentiation of the new product, service, process, or economic model in order to achieve the innovation to be accepted in the market is carried out through a series of transformation (Hargadon and Sutton 2003, 66). The difficulty to achieve an innovation in a simple and easier way questions not only about the stages to obtain it but also about the indicators which will allow for its viability measurement (Alcaide-Marzal and Tortajada-Esparza 2007, 33–57) in order to analyze the capability that an organization has to innovate (Zabala-­ Iturriagagoitia et al. 2007, 85–106). Hargadon and Sutton (2003, 65–93) propose four stages to achieve innovation. These consist of (1) collecting key ideas coming from different places; (2) developing and defining ideas obtained discussing them with other members in order to understand how and why the idea functions, in such a way that the positive and negative aspects are known; (3) imagining other application possibilities of the ideas found; and (4) moving from promising concepts to a reality of service, products, processes, and economic models. In contrast, Peter F.  Drucker considers that in order to achieve innovation, it should (2001, 272–275): 1. Analyze the opportunities through one of the following sources: an unexpected event, an incongruence in what is desired, an answer to a concrete problem, a change in the industry or market’s structure, a demographic change, a change in perception, and acquiring new knowledge 2. Go out, question, and listen to know what the market is searching for and wants 3. Be simple and focus in a definite market 4. Begin with focused innovations to meet a need in particular 5. Have leadership in order to achieve differentiation with the competition Similarly, Drucker proposes three points that should be avoided (2001, 275–276): (1) being too smart not to see the details for defining the market and identifying the need of such; (2) branching out and losing sight of the real need that the selected market wants; and (3) innovating for the future, as it is difficult to evaluate the impact that will have in future years. Due to the existing difficulty to make an innovation, to define a specific market and identify its need in order to differentiate itself in regard to the competition, this work proposes an “innovation process” composed of seven stages that will allow us to achieve an innovation.

70 Fig. 4.3 Innovation process – nonlinear. (Source: Own elaboration)

C. A. Monroy-León

IDEA formulation Definition of technical objects Technological analysis Usage analysis Economic analysis Evaluation of technical objects Definition of the INNOVATIVE PROPOSAL

As seen in Fig. 4.3, the innovation process proposed in this research begins with the idea of what we want to innovate. The first transformation is carried out when this idea gives rise to what we will name “technical object” (Gaillard 2000, 40). It is suggested to obtain three “technical objects” from an idea in order to evaluate which one of them we can obtain a product, service, or process that complies with the characteristics of an innovation: to meet a need and to have a definite market to achieve differentiation from the competition. The subsequent transformations to identifying the three technical objects will be carried out through three types of analysis: technological, usage, and economic. Each analysis will allow us to obtain the characteristic – technological, of usage, and economic – wherewith we will differentiate the product, service, or process in order to obtain an innovation. The obtained characteristics will be called technological rupture, usage rupture, and economic rupture, respectively. Finally, the innovation process will end with the selection of one of the three technical objects; to do this, the different ruptures will be evaluated for each technical object, choosing the one that best meets the established need, in such a way that we will obtain a definite innovative offer. The objective of this work is to show the importance of technological and economic usage ruptures in the stabilization of the innovator process. Because, as we mentioned earlier, the trajectory of an innovation is not lineal. The comprehension of said process is carried out with the presentation of each stage independently, giving the impression of a linear sequence. However, each stage provides an element that will support having ruptures; this is why there is a dependency between the various stages of the innovation process. Innovation is achieved through a series of transformations that allow for the defining of its characteristics to differentiate it from competition. This

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differentiation can be achieved through market selection and the identification of a need in such a way that once the process of innovation has been stabilized – stabilization is achieved when ruptures have been identified  – the interested person or organization will be able to obtain an innovative product, process, or service.

4.3  Innovative Process Figure 4.3 shows that the innovation process presented in this document is made up of seven stages. Each one of them is described in this section. The technological and economic analysis of usage, will be explained though the use of tools that will allow for the technological, economic or usage rupture, depending on the case. As stated, we will call rupture to a characteristic that will allow the product to separate the product, service or process from the competition. The first stage to generate an innovation corresponds to the concretization of an idea. A formalized idea is the result of the convergence of a problem with a means that can be understood by the people who will be the innovation’s market. We consider that (1) idea is the mental representation of something material or immaterial and (2) the formalization of the idea is created through one of the following two situations: the possibility to exploit an existing means or the possibility to solve an identified problem (Gaillard 2000, 36–39). To exemplify the formalization of an idea, we will present the case of a marathon runner where the problem he faces during a competition is the inefficient hydration caused by the loss of liquid in the container handed to him by the event organizers at the hydration stations. Remember that the hydration of a marathon runner takes place during the run. It is such that the runner, in order to solve the problem he is having, resorts to the mental representation of the means that will allow him to have an efficient hydration. That way, his idea in this situation will be a container that will stop the excessive loss of hydrating liquid during movement. The formalization of an idea does not involve the way nor the means by which an identified problem will be resolved. In order for it, the idea can be transformed in a commercial offer. For this we consider the second stage of the proposed innovation process, the definition of what we will call “technical object.” The technical object is the tangible form of the idea: meaning, the immaterial form of the offer which will be concretized in a prototype or a concept (Gaillard 2000, 40–44). The materialization of the technical object in a product, service, or process will be achieved once the technological and economic usage characteristics are identified  – ruptures – that will allow for the evaluation if the technical object is capable of covering the needs of the established market. The second stage in the innovation process considers the creation of three technical objects for a defined idea. If we consider again the example of the marathon runner, the three technical objects for the presented idea being proposed are: (1) a light container with a small liquid output. (2) A ball which can be swallowed with a body similar to that of a hot dog with the hydrating liquid inside of it and (3) An

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edible sponge which could absorb enough liquid, but when the runner drinks does not squeeze. Even if the materialization of something that could become an innovative product, service, or process has been initiated, the identification of technical objects does not define the elements that each one of them must have in order to meet the needs of the established market. For example, in our second technical object, the type of liquid to be used is not specified nor the material of the ball, which may not dissolve with the liquid used, neither the size of the circumference that is to be swallowed. The elements for each one of the technical objects are defined during the following three stages: usage analysis, technological analysis, and economic analysis. The proposed sequence to get ruptures will have, as the first step, the technological analysis, followed by the usage analysis, and will end with the economic analysis. We must point out that the three stages are mutually dependent. In other words, we may not obtain the usage rupture (UR) without considering the technological rupture (TR) and the economic rupture (ER). The transformation of a technical object into an innovative offer takes place through a back and forth of structuring which may consider the needs of an established market and the technical economic viability of the immaterial innovative offer, meaning that, it is necessary to identify the quadrant where the technical object is located within the Markides and Geroski Matrix (2005, 12). Upon learning the impact the technical object will have over: (1) The consumer’s patterns and behavior and (2) The tangible and intangible characteristics needed to achieve the materialization of the technical object, comprised of the weight that each of the ruptures have in the type of innovation to generate (Fig. 4.4) without losing sight that in order to achieve any time of innovation it is necessary to consider the three analyses: usage, technological and economic. As shown in Fig. 4.4, those innovations where both variables would impact are considered zones of uncertainty, GREATER Innovation Uncertainty Zone

Innovation impact on consumer patterns and behaviors

RADICAL Innovation Uncertainty Zone

UR / ER

UR / TR / ER

EB

TR / EB

Innovation of DEVELOPMENT Comfort Zone

STRATEGIC Innovation Uncertainty Zone

Innovation impact on the organization’s tangible and intangible actives Fig. 4.4  Position of ruptures according to innovation type. (Source: Own elaboration)

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because in order to generate the innovation we must focus on two or three ruptures, creating a detailed usage and/or technological analysis to achieve the differentiation regarding the competition. In the case of a technical object generating an innovation of development, the person or organization wanting to generate an innovation, must consider that their proposal is located in a comfort zone. When working with a technical object that leads to an innovation of development only attempts to resist the invasion of a new product, service, or process, as mentioned earlier, are sought to be held on to the market segment (Fig. 4.5). This happens through an improvement attempt to the existing product, service, or product where the current technology is at the end of its life cycle: there’s no progress and there is no market increase (Prax, Buisson and Silberzahn 2005, 57), in such a way that conducting an improvement is less costly by not having to: (1) Make important changes to the intangible and tangible characteristics of the organization and (2) Define a new market and a new need. The danger of generating innovations of development as a means to reach a competitor’s advantage is that, within a certain time, this innovation stops being dominant when faced against a new technology that increases its competitiveness. Because in order to reach a radical innovation, special attention must be given in the three ruptures – meaning that, the technological, economic, and usage ruptures, carry the same weight – we work with a technical object that generates a radical innovation in order to explain the way the presented tools in the technological, usage, and economic analyses let us obtain the characteristics that make it possible to differentiate an innovative offer. As mentioned, these three analyses are interdependent but the proposed sequence in this innovation process is: (1) Technological analysis, (2) Usage analysis and (3) Economic analysis. The technological analysis considers that a technology is comprised of four parts: the theoretical knowledge, experience, raw materials, and the material and human resources (Gaillard 2000, 182–189). The “Technical Elements Table” is the Competitiveness

Improvement tentative to resist the invasion of the new product

New Technology

TIME

Fig. 4.5  Innovation dynamics. (Source: Own elaboration)

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Technical

Economic

Key

Sub-key

Element

Characteristics

Characteristics

Elements

Elements

Providers

Theoretical Knowledge

Experience

Raw Materials

Material and Human Resources

Fig. 4.6  Technical elements table. (Source: Own elaboration)

proposed tool to obtain the technical breakdown (Fig. 4.6); in it are embedded all the necessary elements to allow the technical object to materialize into an innovative product, service, or process. This tool allows not only to get the first global visual of all the components – technical elements – but also allows for the understanding of the degree of dominance the person or innovative organization have regarding the creation of the innovative offer. Investment into tangible and intangible resources is needed in order to achieve the materialization of the technical object. With the identification of the key elements, the innovative person or organization will learn the components that will allow them to reach a competitive advantage or maintain the segment of the desired market, since these will allow to differentiate from the competition. This means, the key elements will create technological ruptures. These elements must not be exchanged during the transformations made along the innovation process, due to the fact that they are the core to achieve the desired innovative offer. The sub-key elements are those with which we will be able to make the characteristics equal – ruptures – offered by the competition. In the case of technical object, the hydrating ball for marathon runners, where a runner seeks to generate an innovation that meets the need previously mentioned must: 1. There are no technical elements in the category of experience, given that the runner does not have knowledge of chemistry, materials, or of product development. 2. The materialization of the idea may be achieved through a relation BtoC or BtoB. In the first, the economic and material resources needed must be found in order to develop the innovative project. In the second, the technical parameters will transfer and the necessary intangible and tangible resources in order for the company interested in the innovation form the technical object. 3. The acquisition of theoretical knowledge that allows for the development of the hydrating ball will take too long, that is the reason why having elements in this

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section are not taken into consideration and a decision is made to hire a specialized engineer in this field, which will be a technical element in the area of human resources. 4. The technical element regarding raw materials is “polymer xy” The technical characteristics desired are: edible, not dissolvable when in contact with water or a current energizing drink in the market, not rigid, and not allowing for liquid dripping out. 5. The economic characteristic that each technical element is the market price, at the time of purchase or contract of services of such technical element. 6. The key and sub-key technical elements are identified with the usage analysis. The analysis of use allows us to get the usage rupture, this happens in two parts: the internal analysis and the external analysis. For the internal analysis, a “pie chart” (Fig. 4.7) is used as a tool having as the objective to show in a clear and organized way the various market segments that may be interested (Gaillard 2000, 202–203) in obtaining the Technical Objects (TO) created. In the case of the hydrating ball, it was mentioned that the materialization of the technical object could come about through a relation BtoB or BtoC, for the latter we will consider the marathons as the first segment and, triathlons as the second segment. Those considered as market one will be the Mexico City Marathon, the marathon, and the Mike Run to mention three

Segment2 OT3

OT3

Market2

OT2

Segment1 Segment2

Market1 Segment1

BtoB OT1

BtoC OT2

Market1 Market2

Idea Market3 OT1

Market2

Segment3

Market1 Market1

Fig. 4.7  Pie chart. (Source: Own elaboration)

Market1

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examples. The markets for the second segment will be the Great Triathlon Pacifico Mazatlán and the Los Cabos Ironman. Once the segments are identified in the Pie Chart and the possible interested markets, it must be analyzed whether the technological rupture really meets the need of each of the selected markets. In such way that when conducting the internal analysis, the innovator will discover the slice of pie – segment, market – where the defined need is covered. In the second part to obtain the usage rupture, external analysis, the competition is the primordial component. This analysis is carried out through two tools: the buyer’s usefulness matrix (Kim and Mauborgne 2003a, b, 97–100) (Fig. 4.8) and the technical object value chart regarding the competition (Kim and Mauborgne 2003a, b, 2–35) (Fig. 4.9). In this analysis we must know the usage rupture of the innovative product or service of the competition and the behavior of it with regards to the key elements and our technical objects. In such a way that, when comparing both ruptures the technical elements which produce differentiation will be defined: the key elements that generate not only the usage rupture, but also the technological rupture. In the same way, being able to compete with existing products or services involve the consideration of the key elements of our competition, with some sub-­ key elements of our innovation. With the Buyer Usefulness Matrix (Fig. 4.8) you can identify – as the name indicates- the usefulness the customer gets of our technical object or of the competition’s products or services. We must point out that it refers to a technical object because up to this stage of the process our product or service has not been materialized, and will select the products or services from the competition that meet a similar meet to the one defined with our own technical objects. The location of each one of the technical objects created and the services or products of the competition in the Buyer Usefulness Matrix, takes place when the

Purchase

Delivery

Use

Compliments

Maintenance

Recycling Customer Productivity Simplicity Comfort Risk Pleasure and image Environmental Awareness

Fig. 4.8  Buyer usefulness matrix. (Source: Own elaboration)

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PRODUCT

High

Competition 2

Dominion Level

Competition 1

Low price

easy to use

options

quickness precision

Key and sub-key elements of the technical object Fig. 4.9  Technical Object Value Chart regarding the competition. (Source: Own elaboration)

way the customer receives the need is identified. Kim and Mauborgne (2003a, b), 100–104) consider that the buyer’s usefulness cycle is made up of six stages: purchase, delivery, use, compliments, maintenance, and recycling. These stages are located in the horizontal field of the matrix and cannot be modified; in the vertical field the person or organization wishing to create an innovation will locate, for each technical object and servicer or product from the competition, the key characteristic which provides the usage rupture. If when locating the key characteristic of our technical object in the matrix is found that it is located in the same quadrant as that of the competition, then, a rupture is not being created. The reader must remember that ruptures allow for the differentiation from the competition. Part two of the usage analysis, has the technical object value chart in regard to the competition (Fig. 4.9) (Kim and Mauborgne 2003a, b, 10–11). In contrast with the previous tool, this time the behavior of the competition in function with the key and sub-key characteristics of our technical object. This chart is a qualitative tool where the innovator will provide value, considered according to the technological analysis and the first stage of the external analysis. We must point out that the highest point in the chart corresponds to the key characteristic which causes the usage rupture. As with the previous matrix (Fig. 4.8) the peak does not correspond to what the innovator considers a usage rupture; however, it’s necessary to return to the key elements in the Technical Elements Table (Fig. 4.6) in order to restructure the technical elements of the technical object and achieve usage ruptures with them. Once: (1) All the technical elements are identified with which the technical objects are created and, (2) The need of the selected market is defined, we can begin with the economic analysis in order to obtain, on one side, the strategic price for the market product or service and, on the other, the economic indicators – Net Current Value and the Returned Investment – that allow the profitability of the innovative

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offer. The strategic price not only is a function of the cost of production, but also the price the competition offers its product or service (Kim and Mauborgne 2003a, b, 106–117). The person or business interested in creating an innovation must consider that the selected price will be that which will allow them to be competitive and assure investment return on the innovative project. In case that a technical object innovates a process within an organization, the economic analysis is quantified with indicators that allow to compare the situation with project (benefits to obtain with the technical object) of a situation without project (current situation within the organization). In a way that, the economic evaluation will allow them to know the feasibility of the innovative project in the established – usage, technological – conditions in order to assure economic feasibility of the innovative offer.

4.4  Conclusions When using various proposed tools, it becomes validated that the process of innovation does not have a linear trajectory. An innovation is acquired through a constant restructuring of characteristics and technical elements in order to cover the needs of the selected market. The identification of technical characteristics that allow for technological, usage, and economic ruptures permit the stabilization of the innovation process. The identification of the need that the technical object must meet is crucial in this process, this is what lets us know the key characteristics that the innovation will offer. With the market segment defined it is possible to identify the existing competition and, this specify sub-key characteristics of our technical object. In the same way it is important to underscore that identifying the type of innovation to generate, allows for the positioning of ruptures that must happen. The stabilization of the innovation process can be achieved through technological, usage, and economic ruptures. In order for this to take place, the following must be defined: the technological requirements (technological rupture), the key characteristics that meet the selected need of the market (usage rupture), and the strategic price by which our project will be financially feasible (economic rupture).

References J. Alcaide-Marzal, E. Tortajada-Esparza, Innovation assessment in traditional industries. A proposal of aesthetic innovation indicators. Scientometrics 72(1), 33–57 (2007) T. Benitez, “Technological innovation process for a biodegradable resealable packaging”, Digital theses, National Autonomous University of Mexico, (2012) E.  Carayannis, E.  Samara, Y.  Bakouros, Innovation and Entrepreneurship. Theory, Policy and Practice (Springer International Publishing Switzerland, Switzerland, 2015), p. 232 C.M. Christensen, The Innovator’s Dilemma (Harvard University Press, Cambridge, MA, 1997)

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R. Cole, From continuous improvement to continuous innovation. Qual. Manag. J. 8(4), 7–21 (2001) M.  Colombo, C.  Franzoni, R.  Veugelers, Going radical: Producing and transferring disruptive innovation. J. Technol. Transf. 40, 663–669 (2015) P. Drucker, The Essential Drucker (Harper Collins Publishers, United Kingdom, UK, (2001), 349 P. Drucker, “The Discipline of Innovation”, New York, Harvard Business Review, 2013, 143–156 (2002) J.M. Gaillard “Marketing et Gestion dans la Recherche et Developpement”, Ed. Economica, 2da. edicion, 374 (2000) A.  Hargadon, R.  Sutton, “Creer un laboratoire d innovation” en Les meilleures Articles de la Harvard Business Review sur l innovation, Editions d’ Organisation, 65–93 (2003) R.  Henderson, K.  Clark, Architectural innovation: The reconfiguration or existing. Adm. Sci. Q. 35, 9–30 (1990) M. Jung, Y. Lee, H. Lee, Classifying and prioritizing the success and failure factors of technology commercialization of public R&D in South Korea: Using classification tree analysis. J Technol Transfer 40, 877–898 (2015) C.  Kim, R.  Mauborgne, “Creer un nouvel espace de marche”, en Les meilleures Articles de la Harvard Business Review sur l innovation, Editions d’ Organisation, 2–35 (2003a) C. Kim, R. Mauborgne, “Sachez reconnaitre une idee gagnante quand vous la rencontrez”, en Les meilleures Articles de la Harvard Business Review sur l innovation Editions d’ Organization, 96–125 (2003b) T.  Lager, Developing a process innovation work process: The LKAB experience. Int. J.  Innov. Manag. 14(2), 285–306 (2010) C. Markides, P. Geroski, Fast Second: How Smart Companies Bypass Radical Innovation to Enter and Dominate New Markets (Wiley, 2005), pp. 208 Organization for Economic Cooperation and Development, “Oslo Manual: A Guide to Collecting and Interpreting Data on Innovation”, Grupo Tragsa, (2005), pp. 188 B. Prax, Silberzahn 2005, 49; Manual de Oslo Organización para la Cooperación y el Desarrollo Económico (2005) M.  Porter, Competitive Advantage. Creating and Sustaining Superior Performance (The Free Press, New York, 1998a) M. Porter, Competitive Strategy. Techniques for Analyzing Industries and Competitors (The Free Press, New York, 1998b) J. A. Schumpeter, “The Theory of Economic Development: An Inquiry into Profits, Capital, Credits, Interest, and Business Cycle”, Transaction Publishers, (1934), pp. 255 E. Von Hippel, S. Thomke, M. Sonnack, “L innovation chez 3M”, en Les meilleures Articles de la Harvard Business Review sur l innovation, Editions d’ Organization, 39–64 (2003) J.M.  Zabala-Iturriagagoitia, F.  Jimenez-Saez, E.  Castro-Martinez, A.  Gutierrez-Gracia, What indicators do (or do not) tell us about Regional Innovation Systems. Scientometrics 70(1), 85–106 (2007)

Chapter 5

Digraphs in the Analysis of Systems’ Representation of Mathematical Knowledge Patricia Esperanza Balderas-Cañas

5.1  Representation of Mathematical Knowledge Knowing how learners represent their mathematical knowledge is a problem that can be addressed by a systemic perspective (Ackof and Sasieni 1984), in which the elements of the system studied are mathematical concepts, the relationships are between said concepts, and one of the main objectives of the system is to answer and create arguments for mathematical questions. In this way, the system of representation of mathematical knowledge can also be seen as a structural model (Harary et al. 1965). The structural models are represented as digraphs in which the nodes correspond to the mathematical concepts, and the edges represent the relationships between the concepts. These types of digraphs were used in Balderas (1998) to study the learning of basic concepts in an introductory course of differential calculus at a senior high school level. It was documented based on the mental process of students when the teaching and learning were supported by representations of the mathematical concepts generated by media such as notebooks, boards, and printed materials (books and teaching guides), as well as some technological resources (e.g., advanced calculators, personal computers) by the student and the teacher. This approach formed cognitive processes related to visual reasoning (Zazkis et al. 1996; section 2, p. 13). The term visual reasoning describes the mathematical thinking aspects based on or expressed in visual images (Zimmermann 1991, p.  127). The study of visual reasoning began as a result of cognitive integration (Goldin and Kaput 1996) by two or more representations of concepts. Therefore, the visual reasoning associated with mathematical representations (symbolic or algebraic, charts, numeric, tabular) and discursive (natural language) were analyzed. Hereafter, this is simply referred to as P. E. Balderas-Cañas (*) Department of Systems, Faculty of Engineering, National Autonomous University of Mexico, Mexico City, Mexico e-mail: [email protected] © Springer Nature Switzerland AG 2021 P. E. Balderas-Cañas, G. de las N. Sánchez-Guerrero (eds.), Problem Solving In Operation Management, https://doi.org/10.1007/978-3-030-50089-4_5

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study representations. The process of visualization is strongly related to visual reasoning due to the fact that the first supports the second when a learner creates or uses diagrams, figures, or charts to answer, whether it is created on paper, computer, or mentally. All this information places us in the representation of the problem— concepts which are discussed in a particular context of the learning and teaching of mathematics (for teaching targeted for the learner, see Light and Cox 2001). We speak of a representation as the effect of a mapping between the real world and the mental model (Palmer 1978, p. 276). To document part of the mental processes of students, a focal question is knowing what type of conceptual images they possess in regard to the associated representations. It is assumed that learning is built by the double process between the actions of the learner and his or her conceptualizations. In other words, the mental actions model the conceptual images (conceptualizations) and, inversely, the images limit the actions (Piaget, quoted by Thompson 1994, p. 230). This chapter discusses the methodology, using a systemic approach, for the study of visual reasoning when learning mathematical concepts at the senior high school level. The following questions are specifically addressed: How do students use study representations? How do they organize these representations to produce an answer in school situations?

5.2  Visual Reasoning The study of the questions presented above, under natural conditions of development (school learning), requires a holistic perspective of inquiry (Lincoln and Guba 1985; Keeves 1999) for the systemic building of a school reality, where the following influences are recognized: • The principles of the researcher and the selected paradigm in the presenting of research questions and hypothesis of the work. • The theoretical framework used in the recollection and analysis of data, research, and interpretation of findings. • The context, meaning, and students as elements of a specific social group or in a study program, the teacher, and the material resources in the classroom, etc. Regarding the reliability of qualitative studies is has generally agreed to illustrate the claims and assertions by episodes’ samples (Atkinson, Delamont and Hammersley 1988, Taylor and Boydan 1984, quoted by Cobb and Whitenack 1996, p. 225). In addition, the following considerations contribute to the reasoning and justification of the analysis: • The set of information is systematically analyzed, proving temporary assumptions based on written responses at a primary analytical level. From that analysis, protocols are designed to carry out interviews with the participants, which then

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produce interpretations and initial assumptions; this instance accounts for an empirical construction. • An extended stay of the researcher with the study participants (Lincoln and Guba 1985) elevates the level of credibility of the analysis because the experience of observing and interacting with the participants provides a first-hand source, which allows a critical internalizing when trying to explain the activity (the physical exchange between representations). • Critical input by researchers who are close to the study but not familiar with the participants offers alternative interpretations of the assumptions. The participants in this study were 16- to 18-year-old high school students in the area of physics-mathematics. To identify the central methodological aspects, some written answers are illustrated for the activities included in the teaching guide of Annex 1. The following four points include the main objectives of the teaching materials: (a) Guide the learning of derivatives based on discursive representations (D), charts (G), numeric (N), symbolic or algebraic (S), and tabular (T) materials, where the graphic and tabular representations were primarily generated by advanced calculators. (b) Document the physical links between the individual study representations, without interaction between participants, to analyze the individual representations. (c) Inquire about the cognitive integration between two or more internal representations of the concepts, starting with the links that the student established within the external representations. In this specific way, the interest focused on seeing how participants used their resources to represent mathematical ideas. Questions are presented in discursive form, but in reference to another or other study representations; this situation demands an understanding of the referred concepts and is represented in various ways by the student (potential demand of representation, DR). Some questions ask for explanations without specifying or conditioning the type of representation to be used in order for the participant to decide which one or ones to use to answer. See Balderas (1998, p. 78) for more details on the teaching guide structure. An interpretative element consists of pointing out that the use of a certain representation does not mean they do not possess other representations of a specific concept. In fact, a hypothesis of work says that “the student possesses a mental representation of each concept, which is externally expressed by one or more of the socially acquired representations.” Variables of the study were the connections between the discursive representations, graphics, numeric, symbolic, and tabular materials that students showed when participating in learning activities, to bring an answer closer to the question regarding how representations are used and how they are organized, and to come up with an answer in the school environment. Thereafter, it was necessary to study the

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conceptual organization of the student (regarding concepts and the relationships that connect them). Lastly, the main source of information was written answers. The validation was made by a triangulation process through interviews, with protocols designed to prove the interpretations that created the construction of categories presented in Sect. 3.

5.3  R  epresentation Systems of Mathematical Knowledge and Visual Reasoning In learning and teaching mathematics, the use of representations forces us to reflect about the difficulties of creating them and interpreting them, both for the learner as well as the teacher. From a cognitive perspective, the problem of defining a representation is very complex. Targeted discussions may more closely find the root of the problem of cognitive representation (Palmer 1978) to the problems discussed by various researchers that focus on educational problems (Janvier 1987; Janvier et al. 1993; Dubinsky 1991, etc.). The discussion of the concept of representations can be located in the context specific to the classroom. The human mind, although limited for processing and for working with memory capacity, is very effective at handling ideas and extremely complex processes. This power is based on the interaction between two sources of experience organizations: the structures inherent to long-term knowledge and the ability to exploit physical means of organizing the experience—in the case of mathematics, the notation systems. To make sense of the processes that involve mental structures and physical processes, a language is needed that includes separate “records” for each experience, as well as for the interactions between them. Kaput pointed out two types of operations: those that are mental and hypothetic in nature, and the physical ones that are frequently noticeable. Between both types of operations, cyclical processes occur. Two events stand out from physical and mental operations: one that establishes a deliberate and active interpretation (“reading”) and the other that is less active, less consistently controlled, and less seriously organized, which consists of having mental phenomena evoked by physical matter (“evocations”). On the reverse side, two processes take place: the act of projecting the mental structure of the existing material and the act of producing new structures, called “writing,” which includes the physical creation of existing structures (Thompson, 1994, p. 230) at a mental level. Those mental operations produce images or mental models. Piaget distinguished three types of mental images; the distinctions that he outlined were based on how, according to the image, the reasoning actions were associated with it. The research analyzed here mentioned a type that is formed by

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the actions and reasoning of quantitative relationships (Piaget 1967, quoted by Thompson, ib., p. 230). The categorizing of Piaget originally was created to take into account the stay of the object, but there may also be “insight” (internalization) in the creation of mathematical objects by a person (Dubinsky 1991; Sfard 1991; Thompson 1994). When the development of the imagination of a person stops at this beginning level, it may lead to mathematical understandings that are internalized patterns of actions (Boyd 1992, quoted by Thompson 1994, p. 230). Due to the fact that mental images come from various sources, they tend to be highly idiosyncratic. Therefore, the interpretations associated with each external representation of the same concept differ from one person to the next, which is a distinctive feature of the thinking style and the cognitive structure. According to Palmer (1978), a representation is “a configuration of a class which (entirely or partially) corresponds to, or is associated by reference with, symbolizes or otherwise represents something.” The representation does not occur in isolation; rather, it usually belongs to highly structured systems, personally, idiosyncratically, culturally, and conventionally. This is due, inevitably, to the interpreting act involved between the representation and the represented—a fundamental consideration to distinguish when the student’s answers depend more on the representation than on the concept. For example, the charts that students created on personal computers with a chart builder included identifying extreme values of one more function than of the concept itself (Balderas 1992, pp. 22 and 88). An assumption made by experts claims that internal representations may connect, but that the connections between internal representations may be stimulated by the building of connections between the corresponding external representations (Hiebert and Carpenter 1992, p. 66). A conceptual organization of a student analyzed through conceptual maps accounts for the relationships between concepts. These relationships are key elements for the understanding of visual reasoning because they link or connect the various representations associated with the present concepts in conceptual organization, which is, in turn, a product of reasoning. The conceptualization of that derivative as a product of the historic development of its physical, numeric, graphic, and symbolic representations has its genesis in the instrumental use that pushes forth the formation of several abstract notations for math concepts, as well as in the reflection about the meaning and transcendence of the same. For Piaget and Garcia (1989; quoted by Kaput 1994a, p. 85), a state of consciousness happens after a more or less long process, which allows a particular notion, such as in the case of the derivative, to be made up in a fundamental concept. The passage from the instrumental to the state of consciousness of Piaget regarding the derivative shows four time periods that may be analyzed. The first corresponds to the use of a notion to determine tangent and quadrature, before the work of Newton and Leibniz. The second is when Newton and Leibniz built it, whereas the third was a time of exploration and development in the eighteenth century. The last period is when it was defined in the nineteenth century (Kaput 1994a, p. 86). The first two time periods are of great importance, as representations were created and evolved, as considered in this study.

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The present research builds a representation of the system that a student possesses, which is made up of concepts and the relationships between them, with the objective of coming up with an answer. It examines how some electronic media may be used to represent ideas and mathematical processes (Kaput 1992, p. 516) and how to evaluate efficiency or inefficiency of the same. Specifically, this includes the quick and sequenced production of graphics, the deployment of tabular arrangements organized according to the scales used in the corresponding axis, the possibilities of processing data statistically associated with phenomena of variation, and the programming ability for the systemic and productive reproduction of algorithms. It is also important to point out the interactive ability1 of advanced calculators, in contrast to inherent media such as blackboards, because they produce physical reactions to the actions embarked on. For example, pressing the ZOOM key produces effects over the graphic view; pressing the TRACE key moves the chart from one function and shows additional information, such as the coordinates of the point where the cursor is located (intermittent asterisk), which can move the chart according to the indications of the user (user actions).

5.3.1  Visual Reasoning and Visualization The processing of visual information in itself is a perception through the sense of sight, considered to be a component of cognitive activity that organizes sensory data originating from the exterior (Campos and Gaspar 1995a, p.  3). The appropriate knowledge is the result of the interaction between the cognitive activity of an individual and reality (Piaget 1970, quoted by Campos and Gaspar, id.). With the study of visual processing, we want to answer questions regarding how a student uses information to interpret, identify, and compare visual information included in teaching materials or observed on the screens of advanced calculators, in the school setting, in order to produce answers to questions presented. The systems of visual processing transform initial visual stimulus into neuronal impulses, which allow us to detect and remember some acts about the information received. In addition, we can reduce information; in other words, we do not remember in detail all experiences coming from our senses. This reduction is probably a necessary function for the system of visual processing. Nevertheless, it also produces information when completing details about the base of the information stored in the memory. In addition to storing information and recovering it, it is also possible to hold onto that information, remember experiences and things, and use what is known when needed. The processing of propositions is more complicated. After the identification of individual segments, we must remember letters in a certain order to determine

 Kaput distinguished between inert and interactive media by the capacity of the latter to respond physically to the “inputs” (1994b, p. 380). 1

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words, then combine the meanings of words and use them in grammatical rules that are saved in the memory to later understand what a proposition means. For example, in “the problem” and assignments 16 and 17 (Annex 1), propositional information is included and the student is asked to create a graphic that represents such information. It is possible that the student proceeds, in a sequential form, by first recognizing the information included in the problem sentence, then by recognizing the propositional information. Simultaneously, the student may perform a comparison and complete the previously learnt information (relationships between coordinates of points and scales in coordinated axis) to produce answers to the actions: represent and point out. The visual reasoning seen as a cognitive process produces a certain conceptual structure. Its organization makes the subject express or solve a problem a certain way. Particularly, the written or verbal answer reflects in itself a part of the knowledge of a subject with regard to one or many concepts. The term visual reasoning describes the aspects of mathematical thinking that are based or may be expressed with visual images (Zimmermann 1991, p. 127). However, if these visual images are a product of geometrical reasoning, as idealized mental entities, then they are completely subordinate to axiomatic limitations (Fischbein 1994, p.  242). In the same way, a geometric shape is a mental object that cannot only be reduced to usual concepts or images; neither is only a concept because they are also spatial representations. The presence of graphic and tabular representations, whether external or internal, confirm the visual reasoning and conceptual of the student. Knowledge acquires some organization as a result of cognitive activity (Campos and Gaspar 1995a, p. 6), not taking into account the level, range, or focus of the activity. The other organizations may be understood as: “complex constellations of information units (significant theme items, or concepts) together with a variety of logical links which connect concepts in a particular way” (id.). A conceptual organization, at a certain point, may be studied according to the analytical method proposed by Campos and Gaspar (id., p. 8, and transl.), which they called the Model of Propositional Analysis (MAP). This model was built to: “identify main ideas in a conceptual organization and the organization of the same, according with their conceptual and logical contents” (id., translation).2 In this study, we analyzed conceptual organization in relation to the use of representations generated in advanced calculators, related to a type of mental process denominated visualization. For Moses (1982), visualization is how to “understand the problem, plan an attack [strategy] and carry out that plan, and finally look back and take in all the knowledge gained” (p. 61, translation). The behavior of the student when solving problems is very individualized: “Each student focuses on a given problem, with his/her resources, cognitive structure, inclinations and cognitive styles.” (Moses, id.). From there, one can also question if

 For a more comprehensive review of the propositional analysis model, see Campos and Gaspar (1995a, b). 2

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there is a general technique, or a way of thinking, that brings forth a better performance when solving problems, which encourages students to not feel frustrated and to continue. Moses proposed visualization as a way of thinking “and not only as a problem-solving strategy” (id., p. 63, transl.), through which students may better understand a problem once they get an image of the situation, when drawing the situation or building a concrete model. In any case, they will have translated the problem through a visual means to their own perspective. In visual reasoning, a variety of actions are involved, such as feel, imagine, and draw, as well as “dream, draw diagrams, sculpt, manipulate concrete objects, and manipulate objects mentally with our eyes closed.” (id., transl.). Any person creates visual thoughts to a certain degree for which “creative performance of the solution of problems may become better encouraging the visual thought in classroom activities.” (id., transl.) The role that diagrams, schemes, graphs or any other representation plays during problem solving may be explained, regarding math visualization, as follows. [I]n mathematical visualization, [the interest lies] precisely in the student’s ability to draw an appropriate diagram (with pencil and paper or in some cases, with a computer) to represent a concept or math problem; and use the diagram to reach the understanding, and as an assistance in problem resolution. In math, visualization is not an end in itself, but rather a means to an end, which is the understanding. Notice that, typically, one does not speak about visualizing a diagram, but about visualizing a concept or a problem. Visualizing a diagram simply means to form a mental image of the diagram, but visualizing a problem means understanding the problem in terms of a diagram or visual image. Mathematical visualization is a process of forming images (mental, or with paper and pencil, or with the help of technology) and use such images effectively for the discovery and math understanding. (Zimmermann and Cunningham 1991, p. 3)

In mathematics, the process of visualizing is very frequent because it is a discipline that refers to the abstract objectification and representation of facts from reality. Many of them come to a greater or lesser extent of visual experiences, which range from primitive (e.g., hand movement) to the conceptualization of a tangent to a circle or a curve. Three examples of research related to visualization are mentioned next. Presmeg (1986) found five types of visual imagery in his research: concrete or pictured, of patterns, relative to formulas, kinesthetic, and dynamic. The first one refers to mental paintings, the second to relations contained in a scheme with visual-­ spatial references, the third to relationship evocations, the fourth includes the imagination of body movements, and the fifth to movement (imagining the trajectory of water coming out of a hose). It also establishes that a visual method of solution is one that involves visual imagination, with or without the presence of a diagram, as an essential part of the solution method or of algebraic (or analytical) methods. The grade of visualization is in charge of the number of visual procedures present in the solution of the problem (Moses 1977, quoted by Bishop 1989, p.  11). Krutestkii (1976) referred to the student’s “geometric type.” He felt the need to visually interpret the expression in an abstract mathematical relationship. In this idea, the preference for a student’s way of thinking stands out. Suwarsono (1982; quoted by Bishop 1989, p. 10) pointed out that the grade of visualization is high if the correct answer

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is obtained and the reasoning is based on a diagram drawn by the student, or by a visual image built by the student. The particular quality of visualization, addressed in the research, is the ability of the student to draw a diagram (with paper and pencil or on a computer), to represent a mathematical concept or a problem, and then use it to acquire understanding or help to solve a problem (Zimmermann 1991, p. 3). Mathematical visualization then becomes a means and an end toward learning the solution of a problem. It is said that a problem is visualized when the problem is understood in terms of a diagram or present visual image, whether physically or mentally visual. Lastly, it is agreed that a student carries out a visualization act when, for example, the student labels points in the Cartesian plane or finds points in the same, physically or mentally. The decision to choose certain points is an analytical act in the sense expressed by Zazkis et al. (1996).

5.4  A  nalysis of Systems of Representation of Math Knowledge The study of visual reasoning in question was formed based on the conceptual organization of the student associated with a teaching process, in the case of learning the derivative conducted by learning material and designed to solve assignments with an advanced calculator (Annex 1). In this study, discursive representations (text), graphics, numeric and symbolic (algebraic) information, and tables (arranged with double entrance) were used for the concepts of rate of change, instant rate of change, velocity, secant, tangent, and curve direction, among others. Such representations were presented in printed materials (handouts) by the researcher as well as by the student. In advanced calculators, the student generated numeric, graphical, and tabular results to provide answers to various questions formulated in teaching materials (Annex 1) and in interviews (Annex 2), which were designed to inquire about the connections between the mentioned representations. A working hypothesis that guided the current research can be summarized in the following concept: “Never forget that a particular action is a personification or an illustration of a general relation, only for those who cognitively [speaking] already have that relation. For those who are not like that, it is merely another action.” (Kaput 1994b, p. 394). Another objective of the study consisted of affirming that the advanced calculator allows the student to have experiences to build conceptual and procedural knowledge, in regards to the derivative. It was also taken into account that: “for any given topic, anybody has an idea, some description, even, an explanation by itself, obtained through cultural processes and social interaction.” (Campos and Gaspar 1995a, p. 8). When analyzing the data, the following was considered: The mediation mechanisms are better defined as cognitive structures that are coded in language only partially and, frequently, work at the level of tacit knowledge ... It is important

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P. E. Balderas-Cañas to highlight that when we think of a generalization in this way, the diversity of the school environments becomes a gain, not a loss: when diversity is drastic, the subject faces all types of novelties which stimulate its adaptation; as a consequence, the cognitive structures of the subject integrate more and become more different. Once the novelty is faced and adapted, the subject has a richer perception, and supposedly, acts more intelligently. (Donmoyer 1992, p. 8)

An additional hypothesis of the work assumed that the students have vast scheme repertoires, and the integration between conceptual images of two or more representations is encouraged. Therefore, the cognitive integration (Goldin and Kaput 1996) was analyzed by two or more mental representations of the concepts, in a differentiated manner, beginning with the verification of their presence. An enriched conception of cognitive integration (IC) allowed us to obtain an understanding of that process in such a way that it was possible to discover different aspects or different angles of the same process. Therefore, in agreement with Donmoyer (opacity. p. 81), questions were presented about the kind of IC. Studying the process of IC required medially natural data, meaning descriptions with low levels of inferences about the performance of the student and extracts of interview transcripts. The observation of the physical correspondence between two or more representations of the involved concepts in the study, in which students applied and interpreted, took us to the analysis of IC by written and oral answers. Regarding the written answers and in the first level of analysis, the conceptual organization was studied through MAP; here, the main ideas were identified in each conceptual organization, the organization by itself, and its conceptual and logical content (id, p. 10). The presence and organization of concepts and relationships included in the answers according to MAP were detected. First of all, the linguistic components3 of the written answers were identified; those that corresponded to concepts and relations were preserved. Upon a first reading of the answers, protocols for the interviews were set in place; in their transcripts, the concepts and relations were also identified. Secondly, the physical correspondences were analyzed between representations used by the participants in the item answers of the materials and protocols, with the purpose of providing an account of the cognitive integration between the corresponding mental images. The graphic components of the representations and cognitive processes in math learning, in addition to the semantic and syntactic analysis of the common and symbolic language, also required an analysis of visual reasoning. This last analysis took place through written manifestations (text, graphic, symbolic, and tabular) that occurred at an external level, in written answers as well as in interviews. In the third level of the analysis, the conceptual content of the answer was compared with the conceptual expectations for each item, with the purpose of contextualizing the answer. Tables 5.1 and 5.2 show a summary of the levels and stages of the analysis.  Concepts and relationships as nouns and verbal forms, principally.

3

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Table 5.1  Data organization from written answers Level Conceptual

Number Stage 1. First analytical reading of the written answers 2. Codification in concepts, relationships, modifiers, connectives, and other components of the answer Organization 3. Categorization of proposals, concepts, and relationships (see columns 2, 3, and 4 of Chart 1, Annex 2) 4. Propositional mapping (see Fig. 5.2) Representation 5. Codification of representations that the participant uses for concepts and connections between them (see columns 5 and 6) 6. Configuration of each answer relative to the representations and connections being studied (Fig. 5.1) 7. Comparison between the potential demand of representations of the item and the representations used by the participant Content 8. Conceptual content of the answer

Source: Own elaboration Table 5.2  Data organization from interviews Level Conceptual Organization

Number 1. 2. 3. 4.

5. Representation 6.

7. 8. Content

9.

Stage Transcription of interview (see Chart 2, Annex 2) Selection of participant interventions Codification of concepts, relationships, modifiers, connectives, and other components of the answer Categorization of proposals, concepts, and relationships (see columns 2, 3, and 4 of Chart 3, Annex 2) Propositional mapping Codification of representations that the participant uses for concepts and connections between them (see columns 5 and 6, Chart 3, Annex 2) Configuration of each answer relative to the representations and connections being studied (corresponding digraph) Comparison between the potential demand of representations of the item and the representations used by the participant Conceptual content of the answer

Source: Own elaboration

5.4.1  Categories To document the physical correspondence between the representations being studied, the representations of the concepts that each participant used were coded for one of the categories: D (common language or text), G (Cartesian chart), N (in terms of arithmetic quantities and operations), S (in terms of algebraic language), and T (numeric arrangements of two or more entries). In this stage, the building of the

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category4 was given a concept represented by X, which alludes to the representation Y”, denoted by X → Y. This is a bidimensional connection in the sense that it is established for two representations5 X and Y, and two of the five representations D, G, N, S or T, but different; therefore, theoretically, there are 20 possible bidimensional connections. Code X → Y was used for the representations of concepts according to the type of representation X that was used by the student for his or her answer to the item and for the indirect reference made to the other representation Y. The context that allowed the determination of the type of representation and the reference to another was the teaching material and the participant’s actions with such material and with the calculator. The code X → Y by itself accounts for the connection between two representations and can be used as a base for the coding of connections. The set R = {D, G, N, S, T} of representations being studied is considered to be a set of points. The set of bidimensional connections is a set of “edges.” or “lines” that go from one point to the next, the configurations obtained for each item’s answer can be represented as digraphs.6 Such configuration is a digraph, which is defined as two finite sets (representations and connections) for which there are two functions in the set of connections. The range is included in R, which defines the starting point X and the ending point Y in connection X → Y. The common notation (X, Y) is not used for the bidimensional connection because X → Y is a closer notation to the empirical reference. The axioms that meet the digraphs are as follows: A1: Set R is finite and not empty. A2: The bidimensional connections set is finite. A3: There are no different bidimensional connections with the same starting and ending points; there are no parallel lines. A4: There are no connections within the same point; in other words, there are no loops. Certain features of digraphs allow the description of the representative structure of the participants. Three particularly interesting features are complete configurations, entirely disconnected and transitive (Harary et al. 1965, p. 12). The first one means there is at least one bidimensional connection between any pair of representations. The second indicates that the participant’s answer does not include bidimensional connections. The third provides information about the jumps between bidimensional connections, when these are analyzed in relation to the sequence in the answer. A rationale for the use of digraphs to represent configurations between bidimensional connections is as follows: the patterns of relationships between pairs of representations are studied; in that sense, they “may serve as a mathematical model of

 This sense of dimensional term does not refer to the mathematical connotation.  Suggested term by G. P’olya (Harary et al. 1965, p. 2) 6  Suggested term by G. P`olya (Harary, et. al. 1965, p.2) 4 5

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the structural properties of [an] empirical system about relations between element pairs.” (id. P. 2). For example, with the adjacency matrixes of digraphs, line spacing is generated and can show which one is the base connection that each participant showed in his or her answers. An interpretation of digraphs generated by the configurations of bidimensional connections lies in the coordination between empirical elements, “concepts represented in one form” and “relations or references between two representations”, and their points R = {D, G, N, S, T} and the directed edges (arrows). The third category is made up of tridimensional connections represented by an arrangement of three different letters, from among D, G, N, S, and T, in the shaded regions of Fig. 5.1. The order is not significant and denotes the presence of a relationship r between two concepts represented in any of the following possibilities: X → Y r X → Z or X → Y r Y → Z or X → Y r Z → Y



Ten tridimensional connections were documented in Balderas (1998): DGN, DGS, DGT, DNS, DNT, DST, GNS, GNT, GST, and NST. Both bidimensional as well as tridimensional connections allow for indirect inquiry about the cognitive integration that the participant made between the studied representations. In each written answer, a propositional map was created, such as in the case of Fig.  5.2, in which concepts were identified as relationships and propositions. Concepts belonging to two or more propositions, called nuclear concepts (Campos and Gaspar 1995a), were identified in shaded areas of the corresponding figure. Associated with each propositional map, the sequence of concepts that generated a sequence of representation was identified, which informed how much of the potential demand of representation was reflected, in which proportion each representation was used, and which types of bidimensional connections were identified in such answer (Fig. 5.3).

a

b

D G

T S

N

Tridimensional connections: DSG, DSN, SNT

MA

D

G

N

S

T

D

0

1

0

1

0

G

0

0

0

0

0

N

0

0

0

1

1

S

0

1

0

0

1

T

0

0

0

0

0

Adjacency matrix MA

Fig. 5.1  Digraph for a configuration of bidimensional and tridimensional connections and the adjacency matrix (MA) for bidimensional connections. (a) Tridimensional connections of DSG, DSN, SNT. (b) Adjacency matrix (MA). (Sources: Balderas 1998 and prepared by the author)

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Fig. 5.2  Propositional map of Rocky’s written answer to item 2.4.1. (Source: Balderas 1998)

Representations T S N G D

1

2

3

4

5

6

7

8

9

10 11

12

13

14

15

16 17 18

Sequence Fig. 5.3  Sequence of representations in one answer to item 2.4.1. (Source: Balderas 1998)

The sequencing charts of representations and bidimensional connections for each item and participant facilitated the analysis of the three previous questions. In the horizontal axis, the sequence of concepts included in the written answers according to the order of appearance was recorded. In the vertical axis, the representations being studied were recorded, in alphabetical order, to facilitate the comparison of the sequence of the participants. Noted with a small square was the representation X that the student used for a concept. A circle indicates the representation Y that alluded to another or the same concept, but was represented differently. In this way, we were able to identify a sequence of bidimensional connections X → Y, in each answer.

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Fig. 5.4 Grid

The data organization was made according to the stages listed in Tables 5.1 and 5.2. At the end of the first four stages, matrix arrangements were prepared, as shown in the first four columns of Chart 2 (see Annex 2) and the corresponding maps. Finally, the student interventions were selected from the interview transcriptions (excerpts) and the written answers were similarly analyzed. For example, the interview’s excerpt was codified as illustrated in Fig. 5.4. Then, the corresponding matrix arrangement was built, resulting from the selection of a participant’s interventions (codified with letter A), and the first four stages of the first analysis level were developed (Table 5.2). The data thus obtained was used to compare the written answers, becoming an important interpretation framework for the study (see Annex 2).

5.5  Conclusions This chapter presented an analysis method for the study of the representation systems of the mathematical knowledge acquired through problem-solving activities, based on visual reasoning and use of calculators with advanced capabilities by high school students. This method allowed us to show that the variations in the use of representations D, G, N, S, and T were determined by the relationships between the concepts even more so than by the concepts themselves. Participants responded mainly with discursive representations. In the case of algorithms, they proceeded correctly with symbolic representations, which connected to discursive and numeric representations. They scarcely connected the symbolic with the graphic and the symbolic with the tabular representations, on both the

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written answers and answers obtained during the interviews. Connections were not found between the graphic and tabular representations. In general terms, participants met the potential demand of representations (DR), which indicated they interpreted the information included in the items and produced answers based on the representations included in them. Such fact was an indication that the studied representations D, G, N, S, and T were available in the participants—this is to say, they were part of their resources, even though they did not always form representation systems that are strongly connected, which made the passing from one representation to the other more difficult and the communication of ideas turned ambiguous. Therefore, a discrepancy among the used representations in the instruments and the ones the student used to answer was not considered. The representation systems used by the participants were modeled through digraphs, which turned out to be very complete, entirely disconnected, and transitive, which indicated solid representation systems that were weak and idiosyncratic. In fact, the participant’s answers to problem 2.2 showed their decision to label points, which was considered a visualization act; their decision to choose certain points when explaining their answers was an analysis act in the sense of Zazkis (opacity. p. 22). The matter presented in Annex 1 was initially interpreted as a graphic situation by all participants with some details; so, in this item, representation G was available as part of the tacit knowledge (Campos and Gaspar 1995a, p. 3) and was associated with the concepts that the sentence has. However, only one participant discussed the situation in speed terms and two made reference to gravity. Coincidences in the forms of representation of water flow thorough a hose were not part of group interaction, as reported in Balderas (2011); instead, it was explained from cultural aspects of students. By comparing the line spacing of each adjacency matrix, it is inferred that Diana and Mirell had a base with more connections than Viridiana, regarding item 2.4.1, whereas Omega, Fernando, and Rocky had a better base than Rafael.

5.5.1  Pedagogic Implications and Recommendations The results of the analyzed investigation indicate the need to use symbolic representations in learning materials, with links among the discursive and numeric representations, to help students incorporate the use of symbolic representations to express ideas. The learning material by itself should use symbolic representations related to the discursive and numeric representations of the same concepts. The didactic discourse included in the learning materials should promote relationship building among the concepts represented in a wide variety for the students to have solid representation systems.

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Annex 1: Extract of the Teaching Guide Path of Water Flow Coming Out from a Hose7 Background In different inhabited places, it is quite common to see people use hoses to water plants or trees in parks and gardens. The constant and free water flow (the same quantity and without obstructing with fingers or valve) forms a jet of water. Henceforward, through activities and questionnaires, you will be studying the path that the water follows once it comes out of a hose—in other words, the form of the jet of water. It is very important that you explain both procedures and answers thoroughly. You are free to use drawings, graphs, and words in your explanations. Assignment: 15. On the back of this paper, make a drawing that represents the previous matter. Path Direction The Problem Which path should the water take when coming out of the hose so the jet gets to the base of a tree that is 2 meters away from where the water is coming out (measured by the floor), which is 1 meter high from the floor? Assignments: 16. Illustrate the conditions of the problem in a diagram (to do this, use the back of the page). 17. Now, represent the conditions of the problem on a Cartesian Plane; to do this, use the following grid. Also mark: (a) The coordinates of points S and A, corresponding to where the water is coming out (S) and the tree’s base (A), and (b) The direction of the water when coming out of the hose. Answer the three following questions on the back of this page. 2 .2.1 Which direction does the jet of water follow when coming out freely? 2.2.2 What is its mathematical model? 2.2.3 Is there only one possible path for the jet of water? 18. If your answer to question 2.2.3 is NO, on your graphic screen of your calculator create three different path examples passing through points S and A, and answer questions 2.2.4, 2.2.5, and 2.2.6.

 Part 2, taken from Balderas, P. (1998).

7

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2 .2.4 What is the mathematical model of each path? Answer: 2.2.5 What range vision (or window) did you use in assignment 18? Answer: 2.2.6 What direction does each path have when coming out? Answer: 2.2.7 Therefore, what is your answer to the problem stated on page 23? Answer: 2.3 Only one path Write the answers to questions 2.3.1 and 2.3.2 on the back of this page. 2.3.1 What restriction or condition to the situation do you propose so the jet of water follows a particular path? 2.3.2 What direction does the water have on point A (base of tree) with regard to that particular path? 2.4 Water speed Answer questions 2.4.1 and 2.4.2 in the space below on this page. 2.4.1 What speed does the water have when getting to the base of the tree in the particular path of question 2.3.1? 2.4.2 What relationship does the path direction and the water speed have on point A?

Annex 2 (Figs. 5.5, 5.6, 5.7, 5.8, and 5.9) Proposal

Item

2.4.1

P1

concepts C1: speed C2: water C3: increase C1: speed C4: greater

P2

P3 P4 P5 P6

C2: water C5: highest point C6: parabola C7: 1 C8: gravity C1: speed C3: growth (increase) C9: point C7: 1 C5: highest point C10: 0 C7: 1 C1: speed C11: lapse

Relationships R1: of the R2: goes R3: this means R4: goes R5: to be

R6: gets R7: of R8: is R9: falls R10: goes R11: is R12: is R13: is R14: increase

Two-dimensional Connections 1. 2. 3. 4.

D→S D→G D→S D→S

5. D→G 6. D→G 7. D→G 8. N→S 9. D→S 10. D→S 11. D→S 12. D→G 13. N→S 14. D→G 15. N→S

Tridimensional Connections

Total Connections per Item

D DSG DSG

T

G S

N

DNS

16. N→S 17. D→S 18. D→G

DSG

Fig. 5.5   Chart 1. Rocky's codified answer in proposals, concepts, relationships, and connections. (Source: Balderas, P. (1998))

5  Digraphs in the Analysis of Systems’ Representation of Mathematical Knowledge

Cod.

Part.

Content

189Ro01

/

1.8.9, what is the answer you propose for 1.8.9?

189Ro02

A

The slope gets closer to 8...

189Ro03

/

Uh-huh! Here’s the conclusion, how is it that you know? How do you get it?

189Ro04

A

Mmm

189Ro05

/

You go back to your page 19

189Ro06

A

it says that the slope gets closer to 8

189Ro07

/

189Ro08

A

189Ro09

/

How do you know that that’s 8?

189Ro10

A

iAh!, because by looking at it we can see that going up the slope is 8.04 and going down is 7.96

189Ro11

/

uh-huh! and then...

189Ro12

A

189Ro13

/

Then this.. on that point between those two is 8 Ok, you propose the 8 very good, this... in the item.., it them tells you on 1.8.10 ..., what is your answer?

Yes, how do you know?, are you looking at your table 4? It is assumed that here is point 1, 12 right? oh no, its here, right?, here is zero x is 1 and here is 2, then in that point the slope will be 8

Simultaneous Intervention

99

Annotations

[read his/her written answer]

[from the guide, story]

A al 8 / uh-huh! / yes

/ yes . / mmhu. / mmhu

Fig. 5.6  Chart 2. Excerpt from interviewing Rocky corresponding to item 1.8.9. (Source: Balderas, P. (1998))

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P. E. Balderas-Cañas

Reference Code Prop. 189Ro02

P1

189Ro08

P2

189Ro08

189Ro10

P3

P4

Concepts C1: slope C2: 8 C3: here [x2,y2] C4: point 1,12 C5: here [h] C6: zero C7: x C8: 1 C9: here [x] C10: 2 C4: point [1,12] C1: slope C2: 8 C11: by looking C12: up C1: slope C13: 8.04 C14: down C15: 7.96 C4: point [1,12] C16: two [values] C2: 8

Relationships R1: gets close to R2: it is assumed R3: is R4: is R5: is R6: y (and) R7: is R8: then R9: in R10: is going to be R11: we see R12: is R13: y (and) R14: is R15: then R16: in R17: between R18: is

Two-dimensional Connections

Tridimensional Connections

1. D→S 2. N→T, N→S

DNS

3. D→T 4. S→T

DST

5. S→T 6. N→T 7. S 8. N→S 9. D→S 10. N→T, N→S

NST

11. D→T 12. D→S 13. N→T, N→S 14. D 15. D→T 16. D→S 17. N→T, N→S 18. D→T 19. N→T, N→S 20. D→T 21. D→S 22. N→T, N→S

DNS DNS

DNS

DNS DNT DNT DNT DST DNS

Fig. 5.7  Chart 3. Rocky’s answer emitted during the codified interview in proposals, concepts, relationships, and connections. (Source: Balderas, P. (1998))

Fig. 5.8  Chart 4. Graphs by group of the sequences of the representations showed on the written answers. (Source: Balderas, P. (1998) and prepared by the author)

5  Digraphs in the Analysis of Systems’ Representation of Mathematical Knowledge ITEM

Diana

Viridiana

D 2.4.1

G

T

N

G S

N

DNS, DSG

DNS, DSG

Tri

NSG

NSG

D→S

D→S

MA

N→S

MA D D 0 G 0 N 0 S 0 T 0

G 1 0 1 1 0

N 0 0 0 0 0

G S

T 0 0 0 0 0

MA D D 0 G 0 N 0 S 0 T 0

G 0 0 0 0 0

N 0 0 0 0 0

S 1 0 1 0 0

T 0 0 0 0 0

MA D D 0 G 0 N 0 S 0 T 0

G 1 0 1 1 0

T

N

G S

Rocky

D

D

T

N

G S

DNS, DSG

T

N

N 0 0 0 0 0

MA D D 0 G 0 N 0 S 0 T 0

G 1 0 0 0 0

G S

NSG

N

DNS, DSG

D→S

N→S

S 1 0 1 0 0

Rafael

D

T

N

Co.

(NR)

Fernando

D

T

G S

Omega

D

D

T S

Mirell

101

D→S

N 0 0 0 0 0

S 1 0 1 0 0

MA D D 0 G 0 N 0 S 0 T 0

G 0 0 0 1 0

N 0 0 0 0 0

S 0 0 1 0 0

MA D D 0 G 0 N 0 S 0 T 0

G 0 0 1 0 0

N 0 0 0 0 0

S 0 0 1 0 0

T 0 0 0 0 0

MA D D 0 G 0 N 0 S 0 T 0

G 1 0 0 0 0

N 0 0 0 0 0

S 1 0 1 0 0

Fig. 5.9  Chart 5. Two-dimensional and tridimensional connections (Co. Tri.), found in the participants’ written answers. (Source: Balderas, P. (1998) and prepared by the author)

References and Bibliography R. Ackof, M. Sasieni, Fundamentos de Investigación de Operationes (LIMUSA, México, 1984)., 6a P. Balderas, Adquisición de conceptos de cálculo con apoyo de la graficación en microcomputadora Tesis maestría. México, UACPYP-CCH UNAM (1992) P. Balderas, La representatión y el razonamiento visual en la enseñanza de la matemática. Tesis doctoral, Facultad de Filosofía y Letras, Universidad Nacional Autónoma de México (1998) P.  Balderas, Modelación de las representaciones matemáticas generadas por la interacción en pequeños grupos, En Ingeniería de Sistemas. Investigación e Intervención, P.  Balderas, y G. Sánchez, (coords.) (Facultad de Ingeniería – Plaza y Valdés, México), pp. 67–92. ISBN: 978-607-402-394-7 (Plaza y Valdés), 978-607- 02-2408-9 (UNAM) (2011) A. Bishop, Review of research on visualization in mathematics. Focus on learning problems in mathematics, (11), 1, Winter, pp. 7–16 (1989) M.A.  Campos, S.  Gaspar, The propositional analysis model: A concept- link approach to text-­ based knowledge organization analysis. Reportes de investigación. México, IIMAS-UNAM, (5), 46, junio (1995a) M.A. Campos, S. Gaspar, The propositional analysis model: Semantic analysis of correspondence in knowledge construction. Reportes de investigación. México, IIMAS-UNAM, (5), 49, octubre (1995b) P. Cobb, J. Whitenack, A method for conducting longitudinal analyses of classroom videorecordings and transcripts, educational studies in mathematics, (30), 3, abril (1996) R. Donmoyer, Argumentos para la investigacióon de estudios de caso: redefinición de conceptos de validez interna y externa, En Investigación Etnográfica en Educación, eds. by M. Rueda, M.A. Campos. UNAM, pp. 65–86 (1992) E. Dubinsky, Reflective Abstraction in Advanced Mathematical Thinking. En D. Tall (ed.) Advanced Mathematic Thinking (Dordrecht, Kluwer Academic Publishers, 1991) p. 95-123. Fischbein, The Interaction between the Formal, the Algorithmic, and the Intuitive Components in a Mathematical Activity. En R.  Biehler, R.  W. Scholz, R.  Sträber, B.  Winkelmann (eds.) Didactics of Mathematics as a Scientific Discipline, Dordrecht, Kluwer Academic Publishers, p. 231–245 (1994) G. Goldin, J. Kaput, A joint perspective on the idea of representation in learning and doing mathematics, in Theories of Mathematical Learning, ed. by L.  Steffe et  al., (Lawrence Erlbaum Associates, Hillsdale, 1996), pp. 397–430

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F. Harary, R. Norman, D. Cartwright, Structural Models: An Introduction to the Theory of Directed Graphs (Wiley, New York, 1965), p. 415 J.  Hiebert, T.  Carpenter, Learning and teaching with understanding, in Handbook of Research on Mathematics Teaching and Learning, ed. by D.  A. Grouws, (NCTM, New  York, 1992), pp. 65–97 C. Janvier, Translation processes in mathematics education, in Problems of Representation in the Teaching and Learning of Mathematics, ed. by C.  Janvier, (Lawrence Erlbaum Associates, Publishers, Hillsdale, 1987), pp. 27–32 C. Janvier, C. Girardon, J. Morand, Mathematical symbols and representations, in Research Ideas for the Classroom: High School Mathematics, ed. by P.  S. Wilson, (Macmillan, New  York, 1993), pp. 79–102 J.  Kaput, Technology and mathematics education, in Handbook of Research on Mathematics Teaching and Learning, ed. by D. A. Grouws, (NCTM, New York, 1992), pp. 515–556 J. Kaput, Democratizing access to calculus: New routes to old roots, in Mathematical Thinking and Problem Solving, ed. by A. Schoenfeld, (Lawrence Erbaum Associates, Publishers, Hillsdale, 1994a), pp. 77–156 J. Kaput, The representational roles of technology in connecting mathematics with authentic experience, in Didactics of Mathematics as a Scientific Discipline, ed. by R. Biehler, R. W. Scholz, R. Sträβer, B. Winkelmann, (Kluwer Academic Publishers, Dordrecht, 1994b), pp. 379–397 J. Keeves, Overview of issues in educational research, in Issues in Educational Research, ed. by J. Keevs, G. Lakomski, (Pergamon, The Netherlands, 1999) V.A. Krutestkii, in The Psychology of Mathematical Abilities in Schoolchildren, eds. by T. Joan Teller (trad.), J. Kilpatrick, Y. Wirszup (The University of Chicago Press, Chicago, 1976) G.  Light, R.  Cox, Learning and Teaching in Higher Education (Paul Chapman Publishing, London, 2001) Y. Lincoln, E. Guba, Naturalistic inquiry (Sage Publications, Newbury Park, 1985), p. 416 B.  Moses, Visualization: A problem-solving approach. Math Monograph, 7, April, pp. 61–66 (1982). S.E. Palmer, Fundamental aspects of cognitive representation, in Cognition and Categorization, ed. by E.  Rosch, B.  Lloyd, (Lawrence Erlbaum Associates, Publishers, Hillsdale, 1978), pp. 259–303 N. Presmeg, Visualization and mathematical giftedness. Educ. Stud. Math. 17(3), 297–311 (1986) A. Sfard, On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educ. Stud. Math. 22, 1 (1991)., Springer. P. Thompson, Images of rate and operational understanding of the fundamental theorem of calculus. Educ. Stud. Math. 26, 2–3 (1994) R.  Zazkis, E.  Dubinsky, J.  Dautermann, Coordinating visual and analytic strategies: a study of students’ understanding of the group D4. J. Res. Math. Educ. 27(4), 435–457 (1996) W.  Zimmermann, Visual thinking in calculus, in Visualization in Teaching and Learning of Mathematics, ed. by W.  Zimmermann, S.  Cunningham, (Washington, The Mathematical Association Of America, 1991), pp. 127–137 W. Zimmermann, S. Cunningham (eds.) Visualization in Teaching and Learning Mathematics, Washington: The Mathematical Association of America (1991)

Part II

Techniques

Introduction In the sixth chapter a modeling process is presented that allows us to make decisions using optimization multi-criteria and the location of parks from the selection of a set of green areas conducive, in urban areas such as Mexico City. This modeling process consists of steps ranging from the structuring of the problem to the use of a procedure that interfaces between a Geographic information system (GIS) and multi-criteria optimization model of discrete location. By using this procedure, results are obtained for locating parks. The principal aim of the seventh chapter is to show a network location services model for a specific problem, which has originally been formulated with just one objective. The multi-objective strategy has been useful in situations where there is more than one objective and where in many cases they may be contradictory. Such approach does not consider interdependence among each other. Multi-level programming, on the other hand, does take it into consideration, which allows for a hierarchical organization of the objectives and the consideration of relationships among them. Finally, in the eighth chapter, an alternative to determinate the demand for the inventory control by ussing fuzzy sets for its calculation under uncertainty is shown, and in this way, the subjective knowledge and administrative experience is incorporated in its determination.

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Part II  Techniques

Chapter 6

Decision-Making with Multicriteria Optimization and GIS for Park Locations Mayra Elizondo-Cortés and Adela Jiménez-Montero

6.1  The Problem of Park Locations in Mexico City Urban planning and renovation have a very important role in increasing the quality of life of citizens. Green places, such as parks, represent an essential element for quality of life in urban zones. It has been demonstrated that parks generate great social, environmental, economic, and health benefits for urban populations (Sorensen et al. 1998; Nowak et al. 1997; Chiesura 2004; Grace et al. 2008). In Mexico City (also known as the Federal District in Mexico), overpopulation and limited open space complicate the construction of parks to meet international standards of 9 m2 of green area per inhabitant, as a sustainability and welfare parameter of urban spaces. Additionally, the World Health Organization (WHO) suggests that parks are located near housing, so residents can walk there within 15 minutes. Fortunately, Mexico City’s government and the Environment Department (Secretaría del Medio Ambiente, SMA) have shown interest in preserving and increasing spaces that contribute to society’s welfare. Therefore, the Environment Law of the Federal District, published in the Official Gazette of the Federal District, has set regulations that regulate and promote urban parks. Decision-making processes for urban planning and renovation are complex because of the number and types of factors that are involved. The relationships between these factors are generally not linear. The location of public spaces, such as parks, is a complex problem because several criteria are used to determine whether a space is suitable to become a park (Salazar and Garćıa 2005, 299–300), even when it is planned for very developed cities, such as Mexico City. Deciding where to locate a park is not a simple task because there are many conflicts of interest. Unfortunately, this decision may be based on an analyst’s subjective perception of M. Elizondo-Cortés (*) · A. Jiménez-Montero Department of Systems, Faculty of Engineering, National Autonomous University of Mexico, Mexico City, Mexico e-mail: [email protected] © Springer Nature Switzerland AG 2021 P. E. Balderas-Cañas, G. de las N. Sánchez-Guerrero (eds.), Problem Solving In Operation Management, https://doi.org/10.1007/978-3-030-50089-4_6

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certain variables, giving importance to the required investment. There are some criteria that, if considered together, could strike a balance between a necessary investment and benefits, but this is not commonly done; it is rarely considered that a proper park location determines whether or not a population will receive benefits proportionated by these spaces. A poor analysis of a park location can result in inhabitants not visiting it because of a lack of accessibility or security, thus missing out on the resources. Therefore, it is important to perform a careful analysis for locating parks as a strategic planning activity, considering the existence of an a priori assumption, the cost of providing a service to inhabitants when locating and then building the park to fulfill international standards, and the costs resulting from a bad location decision. It is necessary to provide acceptable solutions and support authorities in making decisions on park locations. A systemic view of economic, cultural, social, and environmental factors could create successful policies to improve the quality of life in urban areas, but this situation makes such processes very complex (Baycan-Levent et al. 2009, 219–239; Neema and Ohgai 2010).

6.2  Modeling Process for Park Locations In real life, there are often circumstances that add complexity to a problem; for example, objectives may not be clear, definitions for different elements of the problem may be ambiguous, assumptions that have to be made can be inaccurate, the necessary logic structure may not be well-understood, there may be a great scope of action on what it is possible to do, or it may not be known if a solution really exists, among others. When these circumstances arise, the problem is poorly structured. Strategic problems, as well as service location problems, frequently are characterized by these circumstances, as well as great uncertainty in results. It is considered very risky if things go wrong; furthermore, the circumstance may occur in a highly dynamic environment (Pidd 1996). Modeling is the process of developing and using models to understand, change, administrate, and control a part of reality. These models can be qualitative, mathematic, or both (Grace et al. 2008). Before deciding what models to use, it is necessary to perform a problem structuration stage, in which the modeler takes a poorly defined and implicit view of reality and molds it to be well defined, well understood, and rational for other people. Important differences exist between reality and the model. For example, reality is complex, ambiguous, and poorly defined, whereas the model is simple, concrete, and well-defined. As for the simple model, the simple adjective can be understood as a model that has few elements and relationships, and for that reason it can be not very representative. However, in the context of model building, simple should mean being fully explicit and capable of being tested and evaluated by other people. At this point, it is useful to distinguish between trying to resolve a well-structured problem, which is solving, and trying to obtain knowledge on a poorly structured problem, which is modeling (Fig. 6.1). In this chapter, a mathematic modeling methodology is proposed, designing a procedure that allows an objective to be achieved from a structured problem—which

6  Decision-Making with Multicriteria Optimization and GIS for Park Locations Fig. 6.1 Important differences between reality and models. (Source: Own elaboration based on Pidd 1996)

Reality

Model

Complex

Simple

Ambiguous

Concrete

Poor-defined

Well-defined

107

in this case is to determine the best places for building parks accounting for restrictions and different objectives.

6.3  S  tructuring the Problem of Park Location and Mathematic Modeling Methodology The first phase for solving the problem of park location is its structuration. In a well-­ structured problem, the analysis objectives are clear, assumptions that have to be made are obvious, all necessary data are available and ready to use, and the logic structure behind the analysis is well understood. All of these items can be obtained by measuring the consequences clarification, decision clarification, uncertainties clarification; defining the framework; formulating the hypothesis; developing the model’s sequence; and formulating tentative conclusions. These activities in a problem’s structuration process have to be undertaken by an inquisitive mind that takes many research questions with a spirit of discovery (Pidd 1996). A general process used to structure a problem is shown in Fig. 6.2. In this structuration process, one of the first questions is: What makes a specific place a great prospect for being a park? Majid et al. (1983) evaluated park locations based on the utility they generate, defined by users’ activities and nearby homeowners’ willingness to pay more for the services generated by parks. Erkip (1997) focused on defining factors that should be taken into account in these decisions: service factors such as distance, accessibility, travel time, comfort measurement, security, physical appeal, maintenance; and user factors, such as nearby population characteristics, density, homogeneity, age, gender, income level, and education level. In other research done in Valencia, Spain, Salazar and Garćıa (2005) found that park proximity is an essential element that influences people to pay high costs in order to obtain the benefits of having a park close to them. Including this and other information in the structuring process, it can be concluded that proximity is a critical aspect. However, a number of factors must be considered, which have been thus far limited to economic aspects that are apparently simple to quantify. In addition, not all criteria can be modeled mathematically, nor is possible to obtain all necessary information for the model. Another question is whether optimization models have been used for public place locations. It was observed that are many uses of discrete and continuous localization models, as well as solution techniques for this purpose (ReVelle et al. 2007). However, current models use one or two evaluation criteria, emphasizing the

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Fig. 6.2 Problem structuration as a constant exploration. (Source: Own elaboration)

To reflect

To answer

To ask

distance criterion. A study on urban park locations by Molano and Sarmiento (2007) was used as a framework to develop the multi-criteria location model used in this chapter. A third question that is important in the problem structuration process is whether the use of a Geographic Information System (GIS) is favorable. Many studies on public service locations (Zhou et  al. 2013), particularly parks (Lee and Graefe 2004), have used GIS. GIS has tools for multi-criteria analysis; however, this analysis only allows one to evaluate spatially represented criteria—that is, it cannot consider restrictions related to costs and available resources. Analyzing available information and following a problem structuration problem leads to three essential points. First, when it concerns to public service locations, objectives about costs and benefits are pursuit. In this context of location of public services, pursuit means that they are sought, that is, it is necessary to achieve both cost and benefit objectives, that is, minimize costs and at the same time maximize social benefits, which can be intangible and difficult to achieve or calculate monetarily. Benefits are difficult criteria to quantify. In this case, it is related to having a park to promote its use; that is, the park should be near users or be accessible, assuming that a conveniently located park is more likely to have visitors. Many selection criteria about suitable areas for parks can be considered simultaneously with the help of mathematic models. However, it is very important to analyze and delimitate the problem, as well as define the scope and predicted results. Therefore, for factors that were to be evaluated but had concerns about information availability and model suitability, the following criteria were considered: (a) Investment (park building costs) (b) Geographic coverage (number of blocks covered) (c) Population coverage (number of people who benefitted) (d) Accessibility (traffic density) Other propositions for models that follow the same line of work, for cities like Mexico City, could also incorporate important criteria such as poverty decrease, natural habitat restoration, biodiversity, and culture protection (Wang et al. 2012). Goal or multi-criteria programming (GP) has been used in decision-making based on the achievement of different criteria or objectives that we want to reach. These models include diverse objective-functions in the same model, which contributes to ensuring that the solutions found correspond to the needs of the people involved in the

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109

decision-making process. The idea of GP is to establish an acceptance level for each criterion. Under this multi-criteria approach, there are other methods for decisionmaking that combine qualitative and quantitative techniques; this is referred to as the analytic hierarchy process. An example of this, applied to validation of proposals for urban renovation in Hong Kong, is shown in the work of Grace et al. (2008). Second, based on the obtained information and considering the aforementioned criteria, a multi-criteria mathematic optimization model was developed—specifically, a discrete location model. This tool was used because service location theory allows one to determine the ideal location for a service, considering several restrictions. To develop the model, which interacts with GIS, the mathematic modeling methodology shown in Fig. 6.3 was used. The proposed model is multi-objective because it has different objective-functions to reach; this aspect is important because it increases problem view and is intended to develop solutions that achieve, as much as possible, all planned objectives. Third, as previously mentioned, GIS was used for collecting data, analysis, and results presentation. GIS has been defined as a computer system (PLC) for capturing, storing, analyzing, and visual presentation of spatial data (Clarke 1995). A GIS is integrated essentially by a set of spatial or cartographic information and a database that contains the qualitative and quantitative characteristics of this information (Villa et al. 1996; Wong and Jusuf 2008; Oh and Jeong 2007; Zhou et al. 2013). When solving this problem for the selected case study, it was intended to develop a group of available urban areas that are suitable for becoming parks. These results can be used to make decisions according to population and government needs, as well as ensure that the investment would be located in the best place. OBJECT/SYSTEM Why? What we are looking for? Finding? What we want to know? MODEL VARIABLES AND PARAMETERS Known? What we know? Hypothesis? What can we assume? Predictions? What does our model Predict?

How? How can we observe the model? Improvement? How can we improve our model?

MODEL

PROOFS

Validation? Are our predictions valid? MODEL PREDICTIONS Verification? Are our predictions good?

VALIDATED AND ACCCEPTED PREDICTIONS Use? How can we use the model?

Fig. 6.3  Mathematic modeling methodology. (Source: Dym 2004)

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The model applied is an adaption of Molano and Sarmiento (2007), which also deals with park locations using a multi-objective model. Some modifications were made to adjust it to Mexico City’s needs. The multi-objective model used in this chapter is shown as follows: max f1 = ∑z j j∈J



(6.1)

max f2 = ∑ pi yi



(6.2)

i∈I

max f3 = ∑vi yi



i∈I

min f4 = ∑ci yi



i∈I

(6.3) (6.4)

Subject to

∑y

zj ≤ Ij

zj ≥





∑y ,

∀i ∈ I

i

i∈W j



(6.5)

i

i∈W j

(6.6)

amin ≤ t ∑ai yi + ∑xi i∈I I

i∈I s

amax ≤ t ∑ai yi + ∑xi i∈I I

i∈I s

(6.7) (6.8)



xi ≥ tmin ymin , ∀i ∈ I s

(6.9)



xi ≤ ai yi , ∀i ∈ I s

(6.10)

zj ≤

∑y , i



i∈W j



i∈W j



i∈W j

(6.11)

∑y

i

=1

(6.12)

∑y

i



∀j ∈ J

≥2

(6.13)

z j ∈ {0,1} , ∀j ∈ J



(6.14)

6  Decision-Making with Multicriteria Optimization and GIS for Park Locations



yi ∈ {0,1} , ∀i ∈ I



xi ≥ 0, ∀i ∈ I S



111

(6.15) (6.16)

Expressions (6.1), (6.2), (6.3), and (6.4) represent model objectives; (6.5), (6.6), (6.7), (6.8), (6.9), (6.10), (6.11), (6.12), and (6.13) express model restrictions; and (6.14), (6.15), and (6.16) define the solution space of decision variables. Specifically, expression (6.1) maximizes block coverage and achieves the geographic coverage criterion; (6.2) maximizes the benefitted population and achieves the population coverage criterion; (6.3) maximizes the sum of traffic density indicators (with this objective-function, the accessibility criterion is achieved); and (6.4) minimizes the investment. Expressions (6.5) determine that each block j only be covered (zj = 1) if some prospect area i is selected and has block j in its coverage area; otherwise, if not a single prospect area that can cover block j is selected, then the right side of restriction (6.5) forces to zj = 0. Conversely, restrictions (6.6) force all blocks j that are covered by prospect area i to be activated (zj  =  1) if prospect area i is selected (yi = 1). Expressions (6.7) and (6.8) ensure that the sum of the destined areas for the park from prospect areas are between amin and amax. Expressions (6.9) and (6.10) ensure that if a prospect area i, where i ∈ IS, is selected, then the area in m2 of this, which will be destined for the park, should be between tmin and ai. Dealing with the multi-criteria model implies the use of an appropriate solution technique. For this reason, programming methods with lexicographical goals are used because this solution method defines relevant goals for the problem and assigns priorities to them. The proposed solution diagram consists of two stages. In stage 1, four mono-objective models are solved; these are formed by one of the equations (6.1), (6.2), (6.3), and (6.4) (which define pursued objective) and restrictions (6.5), (6.6), (6.7), (6.8), (6.9), (6.10), (6.11), (6.12), (6.13), and (6.14) (represented by Omega), with the purpose of finding the ideal solution for each of the sub-problems. In stage 2, we used the programming sequential methods by lexicographical goals. Stage 1 gives the ideal solutions for each sub-problem; the results can be used in the second stage to define minimum acceptable levels, which are calculated using the alpha value, which indicates the minimum acceptable fraction of the obtained ideal. For this second stage, it is necessary to order lexicographically objectives; that is, in a sequential way, the solution for one objective is found so that defined goals can be reached. In this process, value is incorporated, which represents the minimum acceptable fraction of the ideal. The priority order used in this case follows the order in which objectives in the model were shown; the purpose is to find a solution that guarantees good results in indicators with a low cost. Adding GIS to this research favored the visual representation of information in maps, as well as data contained in the zones that form Mexico City and Delegación Cuauhtémoc, in addition to characteristics that they have (e.g., blocks, population, transportation infrastructure, roads, routes). Thus, through this tool, a favorable visual representation of results and scenarios was achieved. Additionally, with GIS

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integration using archives in the shape format of transportation infrastructure and blocks, as well as a database of population per block, traffic density, benefitted population, and blocks covered, parameters were obtained for every coverage area developed for parks of Delegación Cuauhtémoc.

6.4  Application for Delegación Cuauthémoc in Mexico City As previously mentioned, Mexico City has “delegaciones” that do not fulfill the international standards of 9 m2 of parks per inhabitant. Our case study focused on one of these areas with the lowest percentages in Mexico City: Delegación Cuauhtémoc. For model functionality demonstration purposes, a catalogue of 19 parks was defined, as suggested by the urban Development Unit (Unidad de Desarrollo Urbano) of Delegación Cuauhtémoc. These spaces are damaged and do not meet with necessary conditions for population use. However, they are capable of becoming parks for the Delegación under study. Our model will select those that are suitable for an intervention. The WHO suggested that parks be designed so that every resident lives near an open space, no more than 15  minutes walking distance. Therefore, analysis was performed for 5, 10, and 15 minutes, aiming to have a wide view of the impact of travelling time on the population coverage of a park. Therefore, for each one of the parks, three coverage ratio polygons were generated, with parameters for traffic density, total population, and number of blocks close to each park. Once parameters were defined, multi-criteria optimization models were formulated for each one of the used scenarios; three coverage scopes were considered to perform an impact analysis in the coverage polygons generated by parks. Based on the parks available, both the optimization model and GIS were used; the latter provided information about the parameters required for the model to obtain a solution. The model solution and results analysis were obtained using the following strategy: (a) According to WHO, parks should be less than 15  minutes walking distance from a citizen’s house. Therefore, for each area, three scenarios were developed for coverage areas: (a) 5 minutes, (b) 10 minutes, and (c) 15 minutes. (b) For each scenario, GIS provided traffic density, total population, and block coverage parameters, data used in the multi-criteria optimization model. It was solved by a lexicographic method, using LINGO 10 software. Results concern the park numbers for which investments should be made, benefitted population, and traffic density indicator. See Table 6.1. (c) Validation is an important part of any model development. In this case, the sensitivity analysis regarding the generated results of the model in relation to its answer was performed, in the presence of parameter changes. With this analysis, the same problem was solved for different parameter values. The model is considered valid in terms of its high robustness and its response to extreme values, since the results obtained when the behavior of the model with extreme values was evaluated were as expected.

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Table 6.1  Model solution given by LINGO 10 1 2 3 4

Objective function Blocks covered Benefitted population Traffic density Investment

5 minutes 51 10,855 0.0661 533,197,971

Coverage ratio 10 minutes 161 38,018 0.05194 517,172,638

15 minutes 253 45,152 0.05509 533,197,971

Source: Own elaboration

a) 5 min

b) 10 min

c) 15 min

Fig. 6.4  Scenarios: (a) 5 minutes, (b) 10 minutes, (c) 15 minutes. (Source: Own elaboration)

GIS

LINGO

Correction of necessary files for the analysis.

Formulation of the model

Generation of coverage areas

Integrate parameter

Obtaining parameters: • Road density • Benefited population • Coveraged blocks

Problem solution

Results

Validation of results Presentation and analysis of the results

Fig. 6.5  Workflow in GIS and LINGO. (Source: Own elaboration)

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Fig. 6.4 shows a work diagram in which activities performed in GIS and LINGO 10 are summarized. The interaction between both technologies can be seen in Fig. 6.5. Results were obtained by a sensitivity analysis, which indicated that investments should be directed to three parks from the first catalogue of 19: Jardín de las Artes Gráficas, Jardín Ignacio Chávez, and Jardín Alexander Pushkin. Using this solution (considering 15 minutes’ coverage, following WHO standards), 253 blocks will be reached (representing 20.9% of the optimal value) and 45,152 people would benefit (reaching 67.7% of the optimal value). With this selection, the largest population would benefit, block coverage and traffic density indicators would be maximized, and cost would be minimized.

6.5  Conclusions Research was conducted by modeling methodologies for a problem structuration framework, as well as mathematic modeling .They were oriented to the development itself. Their correct use allowed the proper techniques for the stated problem and also contributed to the research field. In this chapter, an interactive procedure was developed following a multi-criteria mathematic optimization model to determine a group of locations suitable to becoming parks. In addition, this procedure was applied to develop a subgroup of parks where investments should be made inside Delegación Cuauhtémoc. This information will be useful as support for decision makers. The proposed model was improved because Molano and Sarmiento (2007) model claimed to change variables into binary numbers; we adjusted a restriction from model analysis and validation so that binary variables were not necessary, which required less computing effort for the model using LINGO 10 resources. Joining different objective-functions in a service location model is favorable in terms of satisfaction concerning intended results. At this point, the mathematic tool is useful because results considering different points of view are obtained, which gives a more participative and realistic view compared with mono-objective optimization and this becomes a more useful support for decision-­making. Using GIS was relevant for application development; in addition to providing spatial information and necessary parameter values for optimization models that were otherwise nearly impossible to obtain, results analysis and presentation were easier. Thus, a clearer idea about the generated impact could be obtained, incorporating different values analyzing diverse scenarios. Using tools such as GIS allows one to perform spatial analysis of public service accessibility—in this case, parks. Future studies should analyze different groups to consider special needs of the population (Zhou et al. 2013) Thus, it will be important to highlight the interaction’s originality and creativity between optimization tools and GIS. Indeed, the proper use of operations research methodology provides verified and validated results for real-life problems.

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References T. Baycan-Levent, R. Vreeker, P. Nijkamp, A multi-criteria evaluation of green spaces in European cities. Eur. Urban Reg. Stud. 16(2), 219–239 (2009) A.  Chiesura, The role of urban parks for the sustainable city. Landsc. Urban Plan. 1(68), 129–138 (2004) K. Clarke, Analytical and Computer Cartography (Prentice Hall, Upper Saddle River, 1995) C. Dym, Principles of Mathematical Modeling (Elsevier Academic Press, New York, 2004) F. Erkip, The distribution of urban public services: The case of parks and recreational services in Ankara. Cities 14(6), 353–361 (1997) K. Grace, L. Lee, H. Edwin, W. Chan, The Analytic Hierarchy Process (AHP) approach for assessment of urban renewal proposals. Soc. Indic. Res., Springer 89(1), 55–168 (2008) B. Lee, A. Graefe, GIS: A tool to locate new park and recreations services. Park. Recreat. 30(10), 34–41 (2004) I. Majid, J. Sinden, A. Randall, Benefit evaluation of increments to existing system of public facilities. Land Econ. 59(4), 1–16 (1983) A. Molano, O. Sarmiento, Modelo de localización de áreas urbanas para construcción de nuevos parques vecinales en Bogotá, Tesis de maestría, Universidad de los Andes, Bogotá, 2007 M.  Neema, A.  Ohgai, Multi-objective location modeling of urban parks and open spaces: Continuous optimization. Comput. Environ. Urban. Syst. 1(34), 359–376 (2010) D. Nowak, J. Dwyer, G. Childs, Beneficios y costos de manejo de áreas verdes urbanas. Manuscrito para publicación en Annales del Seminario sobre Áreas Verdes Urbanas (1997, diciembre) K. Oh, S. Jeong, Assessing the spatial distribution of urban parks using GIS. Landsc. Urban Plan. 82, 25–32 (2007) M. Pidd, Tools for Thinking: Modeling in Management Science (Wiley, Chichester, 1996) C. ReVelle, H. Eiselt, M. Daskin, A bibliography for some fundamental problema categories in discrete location science. Eur. J. Oper. Res. 184(3), 817–848 (2007) S. Salazar, L. Garćıa, Estimating the non-market benefits of an urban park: Does proximity matter? Land Use Policy 24(1), 296–305 (2005) M. Sorensen, V. Barzetti, K. Keipi, J. Williams, Manejo de áreas verdes, Documento de buenas prácticas (Mayo, Washington, D.C., 1998) F. Villa, M. Ceroni, A. Mazza, A GIS-based method for multi-objective evaluation of park vegetation. Landsc. Urban Plan. 35(4), 203–212 (1996) G.  Wang, J.L.  Innes, S.W.  Wu, J.  Krzyzanowski, Y.  Yin, S.  Dai, X.  Zhang, S.  Liu, National Park Development in China: Conservation or Commercialization? Ambio, Springer 41(3), 247–261 (2012) N. H. Wong, S. K. Jusuf. GIS-based greenery evaluation on campus master plan. Landsc. Urban Plan., 84(2), 166–182 (2008) S. Zhou, Y. Cheng, M. Xiao, X. Bao, Assessing the location of public and community facilities for the elderly in Beijing, China. Geo J., Springer 78(3), 539–551 (2013)

Chapter 7

A Service Location Model in a Bi-level Structure Zaida E. Alarcón-Bernal and Ricardo Aceves-García

7.1  Introduction Many operational systems may be modeled as networks. In some cases, the problem involves the location of some service to meet the demand of one or various clients; in a search model, the objective is to locate a set of services which supply a group of clients scattered in a region. However, most of the work found regarding location theory focuses on problem-­ solving emphasizing a single criterion, which generally refers to distance minimization (costs, travel time, physical distance, etc.). When including multicriteria, it shows great progress in regard to traditional modeling, due to the fact that a large number of alternatives can be incorporated in decision-making. The multicriteria programming problems are useful to solve situations where more than one objective, attribute, or goal has to be accomplished but does not incorporate interdependency among them. This is why multilevel programming arises. The multilevel problems include multiple objectives and consider the relationships between them according to a hierarchy. These problems incorporate some characteristics of the multicriteria problems and include the game theory approach. The simplest model of multilevel programming is the bi-level programming. The bi-level programming model deals with hierarchical optimization problems, which have a second optimization problem as part of their restrictions. In all cases, the higher-level problem or leader is used to reflect the objective to accomplish a

Z. E. Alarcón-Bernal (*) Department of Biomedical Systems Engineering, Faculty of Engineering, National Autonomous University of Mexico, Mexico City, Mexico R. Aceves-García Department of Systems, Faculty of Engineering, National Autonomous University of Mexico, Mexico City, Mexico © Springer Nature Switzerland AG 2021 P. E. Balderas-Cañas, G. de las N. Sánchez-Guerrero (eds.), Problem Solving In Operation Management, https://doi.org/10.1007/978-3-030-50089-4_7

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certain goal, which cannot be carried out without considering the reaction of the follower’s problem (Dempe 2002). In this work, the aim is to use a network services search model with two objectives, considering that a hierarchical relation is present among the same and that one depends on the solution of the other. The model contemplates two main objectives: minimal investment and demand satisfaction.

7.2  Bi-level Programming Models Multilevel optimization problems are a logic extension of mathematical programming, which consider the inclusion of multiple objectives and the relationships between them, in addition to being a generalization to Stackelberg’s problem (Stackelberg 1934) for non-cooperative games. The simplest case of multi-level programming is the bi-level model (BLPP), which deal with hierarchical problems in two levels (leader and follower), meaning optimization problems which have a second optimization problem as part of their restrictions. In all these cases, the leader problem is used to reflect the objective of accomplishing a certain goal, and this cannot be accomplished without considering the reaction of the sub-alternate part of the decisions (follower’s problem). The bi-level problem’s approach arises out of Stackelberg’s model (Stackelberg 1934), which is a strategic game in economics, in which the first player makes a movement and the following player reacts rationally to the first’s election. In this game, the gain of one player does not imply the loss of other; the players make decisions in a specific order, the second player react logically to the decision of the first, and both players have perfect information about strategies and payments. The approach as a mathematical model initially appeared in the works of Braken and McGill (1973) who presented it as a mathematical programming model with optimization problems in the restrictions. However, as the bi-level and multilevel programming problems, they were introduced by Candler and Norton (1977). Based in Stackelberg’s game, various authors have contributed to the development of the bi-level and multilevel programming. The main ones are Aiyoshi and Shimizu (1981); Bard and Falk (1982); Bialas, Karwan, and Shaw (1980); Candler and Norton (1977); Wen (1981); and Benson (1989). In regard to the properties, Bard (1984) proposed an equivalency with the single level mathematical problems. Subsequently, some authors such as Ishisuka (1988); Savard and Gauvin (1990); Chen and Florian (1991); Bi and Calami (1991); Dempe (1992); Ye and Zhu (1993); Outrata (1993); and Vicente and Calamai (1994) have proposed necessary and sufficient optimality conditions, in addition to the ones proposed by Bard. With regard to complexity, Jeroslow (1985) showed that the bi-level problem is NP-hard. Years after, Bard (1991) and Ben-Ayed and Blair (1990) confirmed this result through shortest tests. Hansen et al. (1992), established that the lineal BLPP is strongly NP-hard.

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Considering the different characteristics of the problems, the solution strategies proposed to solve these types of problems can be divided into six main types: • Extreme point algorithms used for continuous problems. • Branch and bound algorithms have been applied to convex bi-level problems and, even though they are associated with great computational efforts, are capable of finding the global optimal solution. • Complementary pivot algorithms, combining ideas of extreme point and branch and bound. • Descending methods, which incorporate descending directions that look for optimal local points. • Method of penalty functions, which include exact penalty functions and are limited to local optimal calculations. • Heuristic methods, as genetic algorithms and neural networks. However, problems with whole restrictions have not been paid attention to, and only a few algorithms have been proposed, among which stand out the ones from Bard and Moore (1992) which use as strategy branch and bound. In this work, a whole bi-level programming model is presented which will be solved with the mentioned algorithm.

7.3  Model Approach 7.3.1  P-Median Location Model Many systems can be modeled as networks. In the network search models, the most common criteria consist of minimizing the cost function in relation to travel time, distances, and possibly other aspects of the trip. These types of criteria are known as medians, where a given set of location enters which covers each demand center in the system and an optimal set minimizes the negative effect of the trip. Locations that optimize a criterion like such refer to medians in networks.

7.3.2  General Model Suppose there is a given network, for which a set of nodes V = {v1, · · ·, vn}. Each arch (i,j) of the network has a weight associated c(i,j). In most median location problems, the networks are built in such a way that the demand centers are located on the nodes, for which each node vi has an associated demand g(i). If it is considered that the supply center is located in point x ∈ G, then the average time of the supply center to the random request in the network is:

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∑ g ( i ) t ( x, i )



n



where t(x, i) refers to the shortest travel time of x ∈ G to vi ∈ G. An absolute median of a given network is the location x∗ which minimizes the average travel time. If we have p supply centers to be located, p optimal locations should be determined. This problem is known as p − median problem.

7.3.3  Formulation Let N  =  {1, · · ·, n} the set of indexes for potential location of the medians and A = {1,…,m} the set of indexes for the demand centers. For each i ∈ N, j ∈ A where ci,j is the client assigned cost j to the median searched in location i. For the problem, the following decision variables are defined:

{

yi =

{

xij =

1, if the median is located in location i 0, in another case



1, If the demand center j is assigned to the median located in i 0, in another case



i N , j A



The problem (1) can be formulated as:

min ∑ ∑ cij xij i∈N j∈ A

(7.1.1)



Subject to:

∑ xij = 1, ∀j ∈ A

i∈N

∑ yi = p

i∈N

(7.1.2)



(7.1.3)





xij ≤ yi , ∀j ∈ A, i ∈ N



xij ∈ {0,1} , yi ∈ {0,1} , ∀j ∈ A, i ∈ N

(7.1.4)



(7.1.5)

Restrictions (7.1.2) guarantee that each client is assigned to a distribution center. Restriction (7.1.3) makes sure that eight locations are exactly selected for the medians. Restrictions (7.1.4) make sure that the clients are assigned to a median only if

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such as been selected. Restrictions (7.1.5) specify that the decision variables are binary.

7.3.4  Bi-level Programming Problems Bi-level programming problems (BLPP) are mathematical programming problems where the set of all variables is broken into two vectors x and y; x will be chosen as an optimal solution of the second problem parameterized in y. In all cases, the superior-level problem is used to reflect the objective of accomplishing a certain goal, and such cannot be accomplished without considering the reaction of the inferior part of the decisions. An important structure of this problem is the hierarchical relation between two types of decision-makers. A bi-level programming problem (2) can be formulated as follows: min F ( x,y ) = c1 x + d1 y



x∈X



Subject to : A1 x + B1 y ≤ b1

(7.2.2)

min f ( x,y ) = c2 x + d2 y

(7.2.3)

y∈Y



(7.2.1)





Subject to : A2 x + B2 y ≤ b2



(7.2.4)

Where: n m p q pn pm qn qm c1 , c2 ∈  , d1 , d2 ∈  , b1 ∈  , b2 ∈  , A1 ∈  , B1 ∈  , A2 ∈  , B2 ∈ 



For x ∈ X ⊂  n , and y ∈ Y ⊂  m , F : X × Y → R and f : X × Y → 

The sets X and Y add restrictions as higher or lower fees or integrality requisites. Once the problem leader selects an x, the first term on the follower’s problem in the objective function turns into a constant and can be taken out of the problem; in such a case, f(x, y) turns into f(y). The decision sequence means that y can be seen as a function of x, that is, y = f(x).

7.3.5  Definitions (a) Set of restrictions of BLPP:

S  {( x,y ) : x ∈ X , y ∈ Y , A1 x + B1 y ≤ b1 , A2 x + B2 y ≤ b2 }



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(b) Feasible region of the follower’s problem for each fixed x ∈ X: S ( x )  { y ∈ Y : B2 y ≤ b2 − A2 x}





(c) Rational reaction of the inferior problem for y ∈ S(x):

{

}

P ( x )  y ∈ Y : y ∈ argmin  f ( x, y ) : y ∈ S ( x )  t



(d) Inducible region:

Q  { ( x,Y ) : ( x,y ) ∈ S , y ∈ P ( x )}



To prove the aforementioned, it is assumed that S is not empty and compact. And for all decisions taken by the leader, the lower problem will have a set of answers, this is to say, P ( x) ≠ φ





Set P(x) defines the reaction, while region S represents the set on which the leader will achieve the optimal.

7.3.6  Discrete Bi-level Problem The problem (3) will be considered with the following formulation:

min F ( x,y ) = c1 x + d1 y



x∈X

(7.3.1)

Subject to : A1 x + B1 y ≤ b1

(7.3.2)

min f ( y ) = d2 y

(7.3.3)

y∈Y



Subject to : A2 x + B2 y ≤ b2

(7.3.4)

where:

c1 ∈  n , d1 , d2 ∈  m , b1 ∈  p , b2 ∈  q , A1 ∈  pn , B2 ∈  pm , B2 ∈  qm ,



X ⊂  n and Y ⊂  m

where Sl(y) = {x ∈ X : A2x ≤ b2 − B2y} for every y ∈ Y and Su = {(x, y) : A1x + B1y ≤ b1} for each x ∈ X. It should be assumed that optimal solution to the lower-level problem is unique. ℝ

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Considering that X = ℝ n and Y = ℝ m in the linear bi-level problem (L-BLPP), we have the following models: 1 . Discrete linear bi-level problem where X = Zn and Y = Zm 2. Discrete-continuous linear bi-level problem where X = Z and Y = Rm 3. Continuous-discrete linear bi-level problem where X = Rn and Y = Zm

7.4  Formulation of Bi-level Search Services Model The proposed model describes a network with production centers, distribution ­centers, and client zones considering only one product. The network is structured in such a way that production centers only supply distributors and these, in turn, only service their clients. The plant will take the role as qualified, not the warehouses. These are the parameters: Set of distribution plants Set of candidate zones for warehouses Set of clients Number of warehouses to be searched Fixed operations cost if a warehouse is found in j Capacity of plant i Demand of client k Transportation cost per product unit from plant i to warehouse j Transportation cost per product unit from warehouse located in j to client Delivery time from warehouse located in j to client k

i ∈ P j ∈ A k ∈ F M oj Ci dk c ij

sjk k. jk

t

Decision variables: xij Amount of product to be transported from plant i to distribution center located in j.

{ {

y jk =

zj =

1, If wharehouse is located in j 0, in another case

1, If wharehouse is located in j 0, in another case





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7.4.1  General Model, Problem (4) Minimize

z = ∑ ∑ xij cij + ∑ ∑ dk s jk y jk + ∑ o j z j i∈P j∈ A

k∈F j∈ A

j∈ A



(7.4.1)

Subject to: ∑ xij ≤ cij , ∀i ∈ P

j∈ A



(7.4.2)



∑ xij ≥ ∑ dk y jk , ∀j ∈ A

i∈P

k∈F

∑ zj ≤ M

j∈ A



xij ∈ Z +





(7.4.3) (7.4.4)



(7.4.5)



Minimize ∑ ∑ t jk y jk

k∈F j∈ A



(7.4.6)



Subject to:

∑ y jk = 1, ∀k ∈ F

j∈ A

(7.4.7)





y jk ≤ z j , ∀j ∈ A, ∀k ∈ F



y jk ∈ {0,1} , z j ∈ {0,1}





(7.4.8) (7.4.9)

In the objective function of the leader problem, we seek to minimize the transportation costs, from the plants to the warehouses in the first result. In the following term, we seek to minimize the transportation costs from the warehouses to clients. And the last term refers to the warehouse operation costs, if located in j. With restriction (7.4.2), we guarantee that the number of products sent does not become higher than the capacity of the plant. In (7.4.3), we guarantee that what leaves the warehouses is not greater to what arrives there. With the inequality (7.4.4), we guarantee that the number of established warehouses is less or equal to the number of requested warehouses (median). And (7.4.5) establishes the integrality for x. The objective function of the follower problem is (7.4.6), and the intent is to minimize delivery time, from warehouses to clients. With (7.4.7), we guarantee that only one warehouse delivers to each client. In (7.4.8), we guarantee that a warehouse j can deliver to a client k if the warehouse was located j. With (7.4.9), variables yjk,zj are restricted to be binary.

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7.4.2  Solution Method The algorithm shown next was developed by Bard (1998) to solve whole bi-level problems with binary restrictions in the variables. The key of the algorithm is the acknowledgment of each solution to the binary problem it must have a solution to: in the set of rational reactions. When the search is limited in this way, it is possible to easily cover the inducible region points, formulating and solving the following parameterized program (5) (Roodman 1972): Minimize

f ( y ) = d2 y

(7.4.10)



Subject to:

A1 x + B1 y ≤ b1

(7.4.11)



A2 x + B2 y ≤ b2

(7.4.12)



F ( x,y ) = c1 x + d1 y ≤ α j =1



∑ xj ≥ β n



x ∈ X, y ∈ Y



(7.4.13) (7.4.14) (7.4.15)

where the parameters α and β initially take the values of ∞ and 0, respectively. Restriction (7.4.13) forces compensation between both objective functions. The inequality (7.4.14) restricts the sum of the variables controlled by the leader and in this way selects ramification variables.

7.4.3  Algorithm An implied numbering centered in the leader’s variables of decision and applied to problem (4) is used to resolve the binary problem. The algorithm examines the points that satisfy the constraints (7.4.11, 7.4.12, 7.4.13, 7.4.14, and 7.4.15), set x and the corresponding values, then, solve again to get a new point in the inducible region. Adjusting α and β in each iteration, the algorithm continually diminishes the value of the lead objective function objective until the problem is no longer feasible. Being W = {1,. . .,n}, in the k-nth algorithm iteration. The set Wk of variables assigned during iteration k. A vector path Pk of longitude l = |Wk| corresponding to a designation of xj = 0, xj = 1 for j ∈ Wk. Vector Pk identifies a partial solution for the

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variables controlled by the leader in the l-nth of the search tree; it also indicates the order in which the variables were established. Being:

k + = { j : j ∈ Wk and x j = 1}





k − = { j : j ∈ Wk and x j = 0}





k 0 = { j : j ∈ Wk }



After iterations k, you get ℝk, which is the set of covered points in the inducible region, and you obtain F as a greater dimension associated with the function object of the leader:

{

F = min F ( x,y ) : ( x,y ) ∈ e k



}

where F = ∞ , at the beginning of algorithm. To solve the model, we used the following algorithm (Fig. 7.1):

Actualizar conjuntos Sk+ , Sk-- ,Sk0 y Pk --F == min{F , F ( x k , y^ k ){ -, a == F 1 β == 1 + |Sk+ | Sí Inicialización Hay alguna solución factible?



Etiquetar el nodo como explorado

No

Terminar

β == 0

No

El punto está en la región inducible?

No



Existe algún nodo no etiquetado?

?

?

Sk0 == {1, ... , n{ -a == ¥ , β == 0, F == ¥

?

k == 0 , Sk+ == ∅, Sk-- ,== ∅

Fig. 7.1  Solution algorithm for bi-level binary programs. (Source: Own elaboration)

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127

7.5  Model Application The model presented in the previous sections was used to solve a medication distribution problem by a pharmaceutical enterprise in some clinical sites of the State Health Department of the State of Mexico (ISEM). To meet the duty, the enterprise must have, at least, a warehouse that meets the characteristics determined by ISEM. Additionally, the contract specifies the inventory that each pharmacy must keep (demand center), as well as the delivery time in case an emergency occurs, which happens when the demand center makes an unusual request of a product and the product is not available. Clinics are classified according to its dimension, the services it offers, and the care it provides. The attributes and size of each clinic determine the number of SKUs needed for medications and medical material for urgent care. The SKUs include the most requested products. With this information we’re able to determine the delivery frequency. The average daily SKU requests in the client zones are shown in Table 7.1 For the general delivery, it takes place through a monthly distribution planning for the distribution and storing of products, in order to meet with the inventory regulations; nevertheless, because the demand for medication for each health center is random, daily adjustments take place. The systems used by the enterprise become inefficient the moment an emergency occurs, because the inventory of the rest of the pharmacies is destabilized, potentially creating another emergency, and thus, the initial planning becomes obsolete. A new programming and distribution will be required which will generate new expenses. Because of this previous information, pharmaceuticals propose the option of placing three micro-warehouses (at the most) near the health clinics, which will take care of emergency situations, with the intent of reducing costs that lack of supply generates and the attention times in case of an emergency. The micro-warehouses must be placed in the bigger health clinics, since these have more available space for placing; these are shown in Table 7.2. Table 7.1  Daily average demand in zones of clients Outpatient visit clinic Acolman Axapusco Nopaltepec Otumba San Martín de las Pirámides Temascalapa Teotihuacán

Average demand/day 110 110 70 60 80 120 110

Source: Prepared by the authors based on information provided by the health clinics

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Table 7.2:  Health clinic candidates for establishing micro-warehouses Health clinic Dr Gustavo Baz Hospital Municipal Bicentenario Psiquiátrico Dr. Adolfo Nieto Carlos María de Bustamante

Type Specialized hospital General hospital Psychiatric hospital CEAPS

Location Tepexpan Otumba Tepexpan Nopaltepec

Source: Prepared by the authors based on the information provided by the Health Department

The main objective is to minimize shipping costs, from the warehouse to the micro-warehouses and from these to the pharmacies, as well as the building and operational costs, considering the restriction the distributions must do at minimal time, as established in the contract with ISEM. To solve the problem, the following information is taken into account: • There is only one distribution plant (general warehouse) with an additional shipping capacity of 700 units of medication. • Candidate locations to place a micro-warehouse must have the characteristics on Table 7.3. • It is required to locate at least three micro-warehouses in the area. • The transportation costs of the general warehouse to each candidate are shown in Table 7.4. • The daily demand, given as the average historical demands by type of health clinic, by municipality, is point out in Table 7.5. • Times and delivery costs are shown in Table 7.6. The model for the pharmaceutical problem (6) is formulated as: 4



7

4

4

Min z = ∑ x j c j + ∑ ∑ dk s jk y jk + ∑ o j x j j =1

k =1 j =1

j =1



(7.6.1)

Subject to: 4

∑ xij ≤ 700 j =1



(7.6.2)



7



x j − ∑ dk y jk ≥ 0, j = 1,…, 4 k =1



(7.6.3)

4



∑ zj ≤ 3 j =1

xj ∈ Z+



(7.6.4) (7.6.5)

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Table 7.3  Operational expenses of candidate locations Health clinic Dr Gustavo Baz Hospital Municipal Bicentenario Psiquiátrico Dr. Adolfo Nieto Carlos María Bustamante

Monthly operational costs 583.74 583.74 1652.43 1752.43

Source: Prepared by the authors based on the information provided by the pharmaceuticals

Table 7.4  Shipping expenses from the main warehouse to the candidate locations Origin San Martín Obispo San Martín Obispo San Martín Obispo San Martín Obispo

Monthly shipping cost 585.47 673.46 582.72 709.99

Destination Dr. Gustavo Baz Hospital Municipal Bicentenario Psiquiátrico Dr. Adolfo Nieto Carlos María de Bustamante

Source: Prepared by the authors based on the information provided by the pharmaceuticals

Table 7.5  Daily average demand by client area Outpatient visit clinic Acolman Axapusco Nopaltepec Otumba San Martín de las Pirámides Temascalapa Teotihuacan

Demand 110 110 70 60 80 120 110

Source: Prepared by the authors based on the information provided by the pharmaceuticals

4

7

Min ∑ ∑ t jk y jk j =1 k =1



(7.6.6)



Subject to: 4

∑ y jk , k = 1,…, 7 j =1



y jk ≤ z j ,

(7.6.7)



j = 1,…, 4; k = 1,…, 7

y jk ∈ {0,1} ,

z j ∈ {0,1





(7.6.8) (7.6.9)

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Table 7.6  Trip time from micro-warehouses to demand centers Origin Dr. Gustavo Baz Dr. Gustavo Baz Dr. Gustavo Baz Dr. Gustavo Baz Dr. Gustavo Baz Dr. Gustavo Baz Dr. Gustavo Baz Hospital Municipal Bicentenario Hospital Municipal Bicentenario Hospital Municipal Bicentenario Hospital Municipal Bicentenario Hospital Municipal Bicentenario Hospital Municipal Bicentenario Hospital Municipal Bicentenario Psiquiátrico Dr. Adolfo Nieto Psiquiátrico Dr. Adolfo Nieto Psiquiátrico Dr. Adolfo Nieto Psiquiátrico Dr. Adolfo Nieto Psiquiátrico Dr. Adolfo Nieto Psiquiátrico Dr. Adolfo Nieto Psiquiátrico Dr. Adolfo Nieto Carlos María de Bustamante Carlos María de Bustamante Carlos María de Bustamante Carlos María de Bustamante Carlos María de Bustamante Carlos María de Bustamante Carlos María de Bustamante

Destination Acolman Axapusco Nopaltepec Otumba San Martín de las Pirámides Temascalapa Teotihuacán Acolman Axapusco Nopaltepec Otumba San Martín de las Pirámides Temascalapa Teotihuacán Acolman Axapusco Nopaltepec Otumba San Martín de las Pirámides Temascalapa Teotihuacán Acolman Axapusco Nopaltepec Otumba San Martín de las Pirámides Temascalapa Teotihuacán

Time (min) 96 231 82 68 84 267 122 211 117 40 27 44 256 121 142 242 95 81 100 241 154 303 104 12 57 87 266 188

Source: Prepared by the authors

where: j ∈ A Set of candidate zones for warehouses k ∈ F Set of clients fj Set cost of operation if a warehouse is located in j dk Demand from client k cij Transportation cost by product unit from plant i to warehouse j tjk Delivery time from warehouse located in j to client k Decision variables: xij Amount of SKUs to be delivered from the plant to the distribution center located in j

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Table 7.7 Results Health clinic Dr. Gustavo Baz Hospital Municipal Bicentenario Psiquiátrico Dr. Adolfo Nieto Carlos María de Bustamante

Will establish a warehouse Yes Yes No No

SKUs sent from the plant 300 400 0 0

Source: Prepared by the authors based on the model application results and its solution

For the formulation proposed above, the locations of the candidates and the client areas were taken into account in the order presented in the tables. Solving the proposed model using the packages LINGO 11 and Maple as tools, the following results were obtained: The ideal solution to the bi-level problem is reached in points: x = ( 300,400 ) y1 = (1,0,0,0,1,0,1)





With:

z = (11 , ,0,0 ) , y2 = ( 0,111 , , ,0,1,0 ) , F = 2223, y3 = ( 0,0,0,0,0,0,0 ) , f = 742,



y4 = ( 0,0,0,0,0,0,0 )





With the results obtained, it is intended to optimize the number of micro-warehouses, as well as the way they should be supplied by establishing which pharmacies should be supported. The distribution centers must be located: one at Hospital Dr. Gustavo Baz in Tepexpan and another one in Hospital Municipal Bicentenario in Otumba (candidates 1 and 2; see Table 7.7). The first warehouse will serve the hospitals located in Acolman, San Martín de las Pirámides, and Teotihuacán and the second Axapusco, Nopaltepec, Otumba, and Temascalapa, all this at a cost of $ 2,223.00 (Mexican pesos).

7.6  Conclusions In this work, a model of search in a network was constructed in such a way that more than one objective can be considered, ordering them according to their importance, using as a base the bi-level programming. A bi-level model was developed, considering a problem restricted by a lower level one. The particular structure of bi-level and multilevel programs allows for the formulation of a great number of practical problems, which involve a hierarchical decision process.

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First, it involves more than one objective, which allows the consideration of at least two goals for each problem; also, consider that the answer to the objectives is subject to the reaction of another of greater importance. Proposing a problem with multiple objectives linked to a hierarchical relation brings the math programming models closer to a reality, which is the intended goal to emulate. The main solution algorithms for discreet problems are comprehensive, separation, and relaxing numbering methods. The algorithm proposed and used for solving bi-level searching problems such as the one presented is one of separation and relaxation. Separation consists of writing the bi-level problem as one, of only one level, manipulating the problem structure to add the objective function of the higher level as a boundary restriction, and with its relaxation verifying that each solution is feasible. Regarding computer experience, when using MAPLE 13 to solve the case study, the maximum time to run the algorithm was 2 minutes with 40 variables, 12 of the leader function (broken in binaries) and 28 of the follower function. Nevertheless, with a more appropriate language, both solution times may be improved. In general, the algorithm finds good solutions quickly when branching out several times by iteration; this allows highlighting non-feasible solutions quickly, since branching out one variable at a time increases significantly the computer effort. A case study was used to show the application of the model developed and the solution through a proposed algorithm. In the case study, the pharmaceutical enterprise, in charge of planning medication distribution to a series of health clinics, considers as a priority to minimize the delivery costs of a product item, while still considering that delivery time must be minimum, which is the requirement of the Health Institute. The cost function’s priority is a key point in the development of the solution; therefore, the math model proposal as bi-level allowed the solution to be closer to the reality.

References and Bibliography E. Aiyoshi, Shimizu, Hierarchical decentralized systems and its new solution by barrier method. IEEE Trans. Syst. Man Cybern. 11, 444–448 (1981) G. Savard, J. Gauvin, The steepest descent direction for the nonlinear bilevel programming problem. Technical Report G-90-37, Groupe d’ Études et de Recherche en Analyse des Décisions (1990) G. M. Roodman, Postoptimality analysis in zero-one programming by implicit enumeration. Naval Res. Logist. Quarterly. 19(3), 435–447 (1972) H. Benson, On the structure and properties of a linear multilevel programming problem. J. Optim. Theory Appl. 60, 353–373 (1989) H. Stackelberg, Market Structure and Equilibrium (Springer-Verlag Wien, New York, 1934) J.  Bard, Optimality conditions for the bilevel programming problem. Naval Research Logistics Quarterly 31, 13–26 (1984) J.  Bard, Some properties of the bilevel programming problem. J.  Optim. Theory Appl. 68, Technical note, 371–378 (1991)

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J.  Bard, Practical Bilevel Optimization. Algorithms and Applications (Kluwer Academic Publishers, Boston, 1998) J. F. Bard, J.E. Falk, An explicit solution to the multi-level programming problem. Comput. Oper. Res. 9(1), 77–100 (1982) J. Bard, J. Moore, An algorithm for the discrete bilevel programming problem. Nav. Res. Logist. 39, 419–435 (1992) J. Braken, J. McGill, Mathematical programs with optimization problems in the constraints. Oper. Res. 21, 21–37 (1973) J. Outrata, Necessary optimality conditions for Stackelberg problems. J. Optim. Theory Appl. 76, 305–320 (1993) J.  Ye, D.  Zhu, Optimality conditions for bi-level programming problems. Technical Report DMS-618-IR, Department of Mathematics and Statistics, University of Victoria (1993) L. Vicente, P. Calamai, Geometry and local optimality conditions for bi-level programs with quadratic strictly convex lower level. Technical Report #198-O-150294, Department of Systems Design Engineering, University of Waterloo (1994) O. Ben-Ayed, C. Blair, Computational difficulties of bilevel linear programming. Oper. Res. 38, 556–560 (1990) P. Hansen, B. Jaumard, G. Savard, New branch and bound rules for linear bilevel programming. SIAM J. Sci. Stat. Comput. 13, 1194–1217 (1992) R. Jeroslow, The polynomial hierarchy and simple model for competitive analysis for competitive analysis. Math. Program. 32, 146–164 (1985) S. Dempe, A necessary and sufficient optimality condition for bilevel programming prob- lems. Optimization 25, 341–354 (1992) S. Dempe, Foundations of Bilevel Programming (Kluwer Academic Publishers, United States of America, 2002) U.  Wen, Mathematical methods for multilevel linear programming. PhD thesis, Department of Industrial Engineering, State University of New York at Buffalo (1981) W. Bialas, M. Karwan, J. Shaw, “A parametric complementary pivot approach for two-level linear programming.” State University of New York at Buffalo. 57 (1980) W. Candler, R. Norton, Multilevel programming. Technical Report 20, World Bank Development Research Center, Washington D.C. (1977) Y. Chen, M. Florian, The nonlinear bilevel programming problem: A general formulation and optimality conditions. Technical Report CRT-794, Centre de Recherché sur les Transports (1991) Y. Ishisuka, Optimality conditions for quasi-differentiable programs with applications to two-level optimization. SIAM J. Control. Optim. 26, 1388–1398 (1988) Z. Bi, P. Calami, Optimality conditions for a class of bilevel programming problems. Technical Report #191-O-191291, Department of Systems Design Engineering, University of Waterloo (1991)

Chapter 8

Determining the Demand in Inventory Policies for Mexican Companies Using Fuzzy Sets Ricardo Aceves-García and Zaida E. Alarcón-Bernal

8.1  Introduction The management of inventories is one of the most basic links in the supply chain, and the optimization forms part of a greater planning process within the chain. While this may not be the only one, it is the most important because the rest of the processes (distribution, limits, production, and materials) mostly depend on the inventory strategy chosen. If we consider the main components of a system of inventories to be demand pattern, supply pattern, operation restrictions, request policies, and total inventory cost, it is possible to establish that due to these components, inventories are used as buffers between supply processes and demand. The main differences between these two processes are internal factors such as customer service, scale economies, and easiness of operation, which depend on the decisions taken by the administrators or inventory, production, and sales managers, and external factors such as demand, supply process, and delivery time, which generally are buried under uncertainty. The easiest way to avoid uncertainty of these processes, which has been for the demand, is to keep more units than those anticipated in inventory (security inventory). For restocking supply, keeping a security inventory may justify minimizing risk. As for the delivery time, that is, the time lapse between issuing an order and This chapter used the research of Flores Brito (2010). R. Aceves-García (*) Department of Systems, Faculty of Engineering, National Autonomous University of Mexico, Mexico City, Mexico e-mail: [email protected] Z. E. Alarcón-Bernal Department of Biomedical Systems, Faculty of Engineering, National Autonomous University of Mexico, Mexico City, Mexico © Springer Nature Switzerland AG 2021 P. E. Balderas-Cañas, G. de las N. Sánchez-Guerrero (eds.), Problem Solving In Operation Management, https://doi.org/10.1007/978-3-030-50089-4_8

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receiving the product, also keeping a security inventory may guarantee minimizing uncertainty. Considering the importance of determining demand in inventory control, since its behavior is little known, it is possible to establish that uncertainty originates in the lack of information or historical data about the behavior of the same, as well as there is difficulty because of this lack to estimate a distribution of possibilities function, which represents it. As a result, the effective determination of demand is one of the main problems of enterprises, which generally is specified based on the experience and judgments subject by the administration, linguistically described as “the approximate demand is of b units” (Behret and Kahraman 2011).

8.1.1  T  echniques Known and Used for Controlling Inventories in Mexico In 2007 the corporation Corporate Resources Management (CRM) conducted a study on the Mexican industry about techniques known and used for controlling the system of inventories. The results were published in Campos (2010), where corporations of various economic sectors and different sizes participated (39% large, 37% medium, and the remaining small and micro), obtaining the following results. Regarding the best known techniques, the results were Point of Reorder 92.7%, Economic Order (EOQ) 87.2%, Material Requirements Planning (MRP) 78.0%, Maximum and Minimum 75.6%, Periodical Reviews 67.5%, and System Kanban 50%. In regard to the most used, the study concluded that 90.2% preferred Point of Reorder for the planning of inventories, a very simple technique which is better for materials that show a constant demand, a very unusual characteristic in today’s markets, and its result generates excess and lack of materials; next was the technique of Economic Order (EOQ) with 76.9%, then Maximum and Minimum with 65.9%, and, lastly, Periodical Reviews with 37.5%. On the other hand, only 58.5% has used the MRP technique, which requires a certain control over the management of planning variables (forecast, delivery time, and lot size), as well as applying a historical statistic of data, to be updating information. According to this study, the best known and used in Mexico are Point of Reorder and Economic Order, simple determining techniques of low maintenance with characteristics that are rarely found in the real world, which do not guarantee good service to the corporation. The low use of MRP (Material Requirements Planning) technique continues to draw attention, even though the corporations that use it, generally the variables that supply the said system (such as the case of inventory of security, forecast, or lotting), are not properly calculated by a total lack of knowledge of the various models

8  Determining the Demand in Inventory Policies for Mexican Companies Using Fuzzy… 137

and techniques of inventories administration; likewise, demand behaviors are not properly analyzed, and it is assumed as known and constant. Inventory levels are also not updated regularly. Other important aspect discussed in these article is the little or nonexistent frequency with which it provides maintenance and updates to the dates and variables of the inventories system; for this action to be considered, there is an additional cost in Mexican companies, a delicate aspect in the administration of the same. Consequently, the safety inventories and the quantity of materials to be ordered are generally not adequate. Besides, in this study (Campos 2010), the frequency of use was also determined; and about this concept, two perspectives were analyzed: the first one refers to the percentage of the controlled parts for the technique and second, the percentage of total sum of all inventories. The first perspective, Point of Reorder, controls the largest volume of materials, while the second, Periodical Reviews, controls the least amount of parts. As far as the sum of inventories is concerned, the study showed that MRP controls from 10% to 25% of the units, which in total value equals 40–60%, which indicates that MRP is used to control products that most impact costs. Consequently, the inventories models which will be used in this job are MRP and EOQ, being the most utilized in the Mexican industry. The former is applied to control products that impact costs the most, and the second to control the largest volume of materials. The strategy of using fuzzy sets to determine demand was also proposed, which is one of the main problems that Mexican companies have when managing their systems of inventories, which generally are estimated based on subjective experience and judgments, since it is considered an additional cost to provide maintenance and updates to the data and system variables. So these models will be presented and resolved considering the demand as a fuzzy number, allowing for the incorporation the experience and empirical knowledge that we have to determine estimation and behavior.

8.1.2  D  ata from Inventory Records, Problems with Uncertainty and EOQ and MRP Models Inventory control is an important field in the supply chain. An adequate regulation of the same may significantly improve company profit. In 1913, the Economic Order Quantity (EOQ) equation was introduced by Harris (1990). Since then, a great number of academic works have been published which describe numerous basic model variables EOQ. For a review, see Brahimi et al. (2006). The EOQ model tends to have few parameters, and all data entries to the model are assumed to be known, so the amount of the order which minimizes the total cost function may easily be determined. This model also assumes that demand characteristics and delivery time are known with certainty, which renders having a simple

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and direct mathematical structure to model demand during delivery time. Nevertheless, the lack of culture in place to gather data and maintain history of the same makes operation of this model difficult. In the same fashion as in traditional literature about inventories, the fuzzy Economical Order Quantity (EOQ) model has provided a genesis in the evolution of fuzzy models of inventories. According to Guiffrida, Alfred L. Kent State (2010), six works have discussed the basic EOQ model, under different parameter schemes. The following table shows this review, where C indicates the model with non-fuzzy parameter/attribute and F denotes the model with fuzzy parameter/attribute. The model MRP has received much attention, and there is plenty of literature about it (Orlicky 1975). Nevertheless, the classic solution procedures that have been used do not optimize production decisions. Having as the objective to count with optimum solutions to minimize costs, the MRP problem has been studied through mathematical programming, considering the problem with a determinist structure, which allows for having a more manageable model. MRP has been used primarily when there is a dependency between items’ demands; the use of this model of control of inventories allows for the reduction of their levels and may predict the material requirements for the horizon of planning (Table 8.1). However, its implementation requires precise and complete information about the demand of materials and production planning. Having this information implies that companies manage control and registry of activities, which is the reason why few Mexican companies use this model. For the fuzzy model of Material Requirements Planning (MRP), only a few research works have been accomplished. In Lee and Wu (2006), the application of Fuzzy Sets Theory is to the problem of lot size in an MRP system of one stage. In Mula et al. (2008), is developed a new lineal programming model, labeled NNRPD, for short-term production planning in a MRP manufacturing environment with capacity restrictions, multi-product, multi-level, and multi-period. In Vasant (2004), a curve is used as a belonging function for the selection of a mixture of products in a chocolate factory, where the information available is imprecise or fuzzy. In Mula et al. (2007), a model of optimization is formulated which takes into account the uncertainty that exists in the demand in the market, as well as pending orders Table 8.1  Fuzzy models for EOQ Model Park (1987) Vujosevic (1996) Lee and Yao (1999) Yao et al.(2000) Yao and Chiang (2003) Wang et al. (2007)

Parameter of entry quantity to order C C F F C

Order cost

Maintenance cost

C

Source: Alfred L. Guiffrida, Kent State University. Cap 8. Fuzzy, Inventory Models

Annual demand C C C F F C

8  Determining the Demand in Inventory Policies for Mexican Companies Using Fuzzy… 139

(delayed), the concept of possibility programming is used. Such focus allows for the modeling of ambiguity of the market’s demand, information of costs, etc., which could be present in the production planning systems. In Pendharkar (1997), the fuzzy dynamic programming is used to solve a problem of inventories with production programming, where the linguistic states, such as “stock must be zero at the end of the planning horizon” and “reduce, as much as possible, production capacity,” are used to describe administratively the fuzzy aspirations for the inventory and the reduction in production capacity, of a possible market pullout. Then, in both models, it is required to have several parameters for its performance; in this work, demand is considered as the primary parameter to be known. Therefore, it is necessary to have with information to determine it or with a strategy by which it can be estimated. A strategy to consider for the inventories control systems with this problem is the theory of fuzzy sets, which was introduced by Zadeh (1975) and can be applied to model the demand behavior more realistically and use empirical and subjective knowledge of the administration.

8.2  E  conomic Order Quantity (EOQ) Model with Fuzzy Demand, Without Production or Deficit The Economic Order Quantity models, as already mentioned, represent in Mexico a considerable percentage among the most widely used, still present in companies, particularly micro, small, and medium. The basic EOQ calculates the reposition of order size for an inventory system; for a determined item, see Soodong and Noble (2000). The average CP(Q) cost depends on the amount of Q orders that are done to cover the demand and the number of units stored. In the following figure, we can observe how CP(Q) is convex function. And in Q* the values of costs for maintaining inventory and for ordering are the same (Fig. 8.1). Assumed in the model are the following: The demand for items is a known and positive constant. The delivery time is zero, meaning that any time frame done is received immediately. And the involved costs are also known. The behavior of the policy of inventories for an Economic Order Quantity model, without production or deficit, is presented in the following figure (Fig. 8.2). The diagram presented in Fig. 8.3 was the result of the analysis conducted with the solution of several examples when using different EOQ models and different fuzzy numbers; in the determination of demand, see Flores Brito (2010). The application of proposal stages, for the incorporation of demand, as a fuzzy number in the Economic Order, allows for the solution systematically, bringing the results in a simple way and facilitating its analysis. Stages of the solving process: (a) According to the data that you have in the problem and at the behavior of the storage level, the model to be applied is decided.

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Storage Cost

#

Average

Ordering

2370 2133 1896 1659 1422 1185 948 711 474 237 0 95

285

475

665

855

1045

1235

1425

1615

1805

1995



Fig. 8.1  Graphic of the average cost in an EOQ model. (Source: Hillier and Lieberman. Mac Graw Hill. Quantitative methods for administration, Cap. 11)

Storage Level

Q

Time T

T

T

Fig. 8.2  Model behavior with no production or deficit. (Source: Hillier and Lieberman. Mac Graw Hill. Quantitative methods for administration, Cap. 11)

(b) The definition of demand as a fuzzy number implies that knowledge is within certain information values. For example, if demand is defined as a triangular fuzzy number, it means its value can be found within three data (minimum, the most likely, and the maximum). The definition of a fuzzy number can be determined, through the membership function, or for its graphic representation. It is necessary to get values for the α-cuts; at this point, it is necessary to determine the “step length,” which means the increase that α − cuts will have, where / can be found between 0 and 1. The value of the increase will imply the precision of the calculations. (c) It is recommended to do the calculations separately, this is to say, for the equations which determine the optimum policy, first making the calculations that do not include the part of the fuzziness; once these values are obtained, the ­fuzziness is incorporated. For this point we will have many values for each vari-

8  Determining the Demand in Inventory Policies for Mexican Companies Using Fuzzy… 141

Select the model of Economic Order

Define demad as a fuzzy number

Add to the other calculations the fuzziness of the demand

Deviations analysis

End

Using the α-cuts

Calculate the values for the variables to be determined as fuzzy numbers

Calculate the approximations for each one of the other variables

Using the α-cuts

Fig. 8.3  Proposed stages to solve the model of EOQ, with the con demand as a fuzzy number. (Source: Prepared by the authors)

able, according to the step length of the α − cuts that have been decided on; this is due to the fact that calculations have been made for each value of the demand related to the α − cuts. (d) When calculating each variable, a fuzzy number will be obtained, which originated from the incorporation of the demand as a fuzzy number; in this way if the demand has been defined as a triangular fuzzy number (TFN), the results for the other variables will originate a TFN as well. It is recommended to make graphs for each variable, as this allows for results verification. (e) Once the calculation has been done, using the demand as a fuzzy number (FN), it is necessary to carry out an approximation to the obtained FN; the difference between the calculations and approximations is that the calculations are obtained using the equations and the demand as a FN, while the approximations are obtained using only the values which resulted from the calculations and which define the FN, calculating the α-cuts; that is to say, now each variable is considered as an FN. The aforementioned allows for a definite result for certain values; if the demand is a trapezoidal fuzzy number (TrFN), then they will be obtained for the other variables TrFN. Differences between the calculated values and the obtained approximations can be observed in the graphs.

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(f) When we have the graphs where we observe the calculated values and the approximations for each one of the variables, we can get to the end of the process if the differences between one curve and another one are not representatives; otherwise, it is recommended to make an analysis of the deviations’ curves. Following is an example of the Economic Order model without production and without deficit, considering the unknown demand, which is solved as a diffuse EOQ model, following the previous stages. Be the following data: d=?, units/year; k = 25 um/order; and C = 6.25 um/unit. Where h is 20% of the purchase cost as per the company’s policy, therefore h = (20) (6.25) = 1.25 um*unit/year. To determine the optimum amount per order that should be used to minimize the costs, establish what is the average cost associated with the optimum order amount, specifying how many orders would the company making per year. For this case it is considered that the demand is not clearly known, which happens in various real situations. It is admitted that the demand is a triangular fuzzy number (stage 2) in the following manner:

˜



d = ( a,b,c )



˜ d = ( 5800,8500,13700 )



Its membership function is defined as:



0, if x < 5800   x − 5800  , if 5800 ≤ x ≤ 8500  2700 µd ( x ) =   − x − 13700 , if 8500 ≤ x ≤ 13700  5200  0, if x > 13700 

(8.1)

And can be written for the α − cuts

d∝ = 5800 + ( 8500 − 5800 ) α ,13700 − (13700 − 8500 ) α  , α ∈ [ 0,1] (8.2)

Now the calculations for the α − cuts are carried out, with an increment of 0.1, from 0 to 1 (Table 8.2). The way to do the calculations consists in solving the part that is not fuzzy and incorporating the fuzzy part after (stage 3); it is important to mention that at the end a new fuzzy number for each variable will be obtained. For example, to determine the required amount (Q), the area which involves the ordering cost will be solved (k) and the cost to maintain in inventory (h). On the other hand, the fuzzy number d will be calculated, this last calculation will be

8  Determining the Demand in Inventory Policies for Mexican Companies Using Fuzzy… 143 Table 8.2  Calculations of the α − cuts

α 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

dα 5800 6070 6340 6610 6880 7150 7420 7690 7960 8230 8500

13700 13180 12660 12140 11620 11100 10580 10060 9540 9020 8500

carried out for the α − cuts, and precision will depend on the person doing the calculations. In this case, the increment of α is 0.1. For each determined variable (stage 4), the graph from the obtained fuzzy number is presented, considering the demand as TFN. Subsequently an approximation is presented, which means taking into account the three real numbers that define the number and make the ∝ − cuts, as opposed to obtaining the values by calculating with fuzzy number d. Determining the amount of the order is obtained by incorporating the fuzzy part and the non-fuzzy part.



Q=

2k ⋅ 5800 + ( 2700 ) α ,13700 − ( 5200 ) α h

(8.3)

In Table 8.3, the obtained results are shown for the order size. From Table 8.3, it can be observed, in the first column, the values for α with a step length of 0.1; in columns 2 and 3, the results of the application of the formula Q are presented, considering the demand as a fuzzy number; and columns 4 and 5 show the results of the triangular approximations (using the limits obtained from the calculations of columns 2 and 3); this can be represented as follows. The fuzzy number is obtained, whose triangular approximation (stage 5) is: ˜



Q = ( 482,583,740 )



In the graphs, it is observed that the difference between the Q value obtained from the calculations, using the TFN of the demand, and the value of the triangle approximation is minimum, as shown in the following figure. In the same way that Q was obtained, the value for the following variables is calculated (continuation from stage 4); next the results from the calculations are presented and the corresponding graphs (Fig. 8.4).

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Table 8.3  Triangular calculations and approximates of Q ˜

α 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Q  482 + (101) α , 740 − (157 ) α 

Q 482 493 504 514 525 535 545 555 564 574 583

740 726 712 697 682 666 651 634 618 601 583

482 492 502 512 522 532 543 553 563 573 583

740 725 709 693 677 662 646 630 615 599 583

m(Q) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0

0

76

0

74

0

72

0

70

0

68

0

66

0

64

0

62

0

60

0

58

56

0

0

54

0

52

0

50

48

46

0

0

Fig. 8.4  Graphic representation of Q. (Source: Prepared by the authors with data from Table 8.3)

The length of period (T) and the number of requests (N) are variables which are demand-driven; however, the value of N is obtained, using the value of Q previously calculated. The obtained results are presented for the N variable in Table 8.4. ˜

˜

The N value has a triangular approximation same as N = ( 2,15,19 ) . Whereof, the graphic representation is Fig. 8.5.

8  Determining the Demand in Inventory Policies for Mexican Companies Using Fuzzy… 145

˜

Table 8.4  Calculation and triangular approximations of N N= α 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

dα Q

˜

N = 12 + ( 3 ) α ,19 − ( 4 ) α 

12 12 13 13 13 13 14 14 14 14 15

19 18 18 17 17 17 16 16 15 15 15

12 12 13 13 13 13 14 14 14 14 15

19 18 18 17 17 17 16 16 15 15 15

m(N) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

10

12

14

16

18

20

Fig. 8.5  Graphic representation of number of requests (N). (Source: Prepared by the authors with data from Table 8.4)

In the same way for T, it was necessary to calculate it without the fuzzy part and do the calculations after, including the fuzzy part. The results from the calculations for α − cuts are represented in the following table, followed by the results of the approximations of T as a fuzzy number. The results are represented graphically in Fig. 8.5. Finally, the average cost of the inventory, which also depends on the demand, is defined as follows as TFN. The graphic representation is shown in the following figure (Fig. 8.6).

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R. Aceves-García and Z. E. Alarcón-Bernal T

m('I') 1

Triangular Approximation

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0 08

50

0 0.

0

00 08 0.

0

50 07 0.

0

00 07 0.

0

50 06 0.

0

00 06 0.

50 05 0.

0.

05

00

0

0

Fig. 8.6  Graphic representation of the length period (N). (Source: Prepared by the authors with data from Table 8.5)

8.2.1  Analysis of Results The comparison between the data obtained and the fuzzy numbers determined by calculating approximations using α  −  cuts has been presented in the graphs and tables shown. It can be observed that the approximation is acceptable; therefore, it is considered necessary on this context to calculate general expressions for the deviation between the real curves and its approximations which, on the other hand, give rise to more complicated Eqs. (8.14); for this example, the result can be written as follows:

ˆ ( Q ) = ( 602,729,925 ) Q = ( 482,583,740 )CP





 T = ( 0.054,0.0686,0.083 ) N = (12,15,19 )



(8.4) (8.5)

From the previous results, it can be said that the most probable value for Q is 583 units, while the smallest and greater values for the size of the request are 482 and 749, respectively. For CP (Q), T, and N, a similar description is given, as the calculations depend on the same triangular fuzzy number (TFN) (Fig. 8.7).

8  Determining the Demand in Inventory Policies for Mexican Companies Using Fuzzy… 147 Table 8.5  Calculations and triangular approximations of T T= α 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

2k 1 ⋅ h dα

0.08305 0.08118 0.07943 0.07779 0.07625 0.07480 0.07342 0.07212 0.07089 0.06972 0.06860

˜

T = 0.054 + ( 0.014 ) α , 0.083 − ( 0.015 ) α  0.05403 0.05509 0.05621 0.05740 0.05867 0.06003 0.06149 0.06306 0.06475 0.06659 0.06860

CP

0.054 0.055 0.057 0.058 0.060 0.061 0.063 0.064 0.066 0.067 0.069

0.083 0.082 0.080 0.079 0.077 0.076 0.074 0.073 0.071 0.070 0.069

Triangular Approximation of CP

m(CP) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 580

630

680

730

780

830

880

930

CP(Q)

Fig. 8.7  Graphic representation of CP, which depends on Q. (Source: Prepared by the authors with data from Table 8.6)

8.3  M  RP Model Considering the Demand of a Fuzzy Number In 1947 George Dantzig developed the simplex algorithm to solve linear programming (LP) problems. This technique is applied to a variety of problems in the fields of industry, health, economy, transportation, etc. For this reason, linear programming is a well-studied area and one of the most used tools by companies.

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Table 8.6  Calculations and triangular approximations of CP

α 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

 36852 + (17002 ) α , CP ( Q ) =   86550 − ( 32696 ) α 

˜ CP  Q  = 2hk ⋅ dα + cdα   36852 38553 40254 41955 43656 45356 47056 48756 50455 52155 53854

86550 83283 80015 76746 73477 70208 66938 63668 60397 57126 53854

36852 38552 40252 41953 43653 45353 47053 48753 50454 52154 53854

86550 83281 80011 76741 73472 70202 66932 63663 60393 57124 53854

The application of fuzzy sets in mathematical programming for the most part consists of transforming the classic theories in equivalent fuzzy models (see Kaufmann and Gil 1987). In practical situations, for a typical linear programming problem, it is not reasonable to demand that the objective function or the restrictions be specified in a precise way; in such situations, some type of fuzzy linear programming should be used. Fuzzy or flexible linear programming (FLP) can be applied in different cases, for example, when the right side of the restrictions is a fuzzy number or when the technological coefficients are fuzzy numbers or when both previous cases are present. This work only focuses on analyzing the first case, due to the fact that the parameter to be determined is the demand and corresponds to the right side of a restriction. Such types of restrictions are called flexible restrictions. A linear programming problem with flexible restrictions is defined as follows. Maximize n

z = ∑c j x j j =1



(8.6)

Subject to n

∑a

ij



x j ≤ Bi , i = 1, 2, 3,…, m

j =1

x j ≥ 0,

(8.7)

j = 1, 2, 3,…, n



(8.8)

Considering that Bi is a fuzzy number of trapezoidal form (TrFN), with the following membership function:

8  Determining the Demand in Inventory Policies for Mexican Companies Using Fuzzy… 149

 1, x ≤ di   − x + di + pi Bi =  , di ≤ x ≤ di + pi pi   x ≥ di + pi 0, 



(8.9)

where x ∈ ℜ . As can be seen in Fig. 8.8, the graphic form of the fuzzy number is linear and descendent of di to (di + pi)di. Once the type of fuzzy number that will be used to represent the parameter under uncertainty has been defined, to solve FLP, it is necessary to calculate the fuzzy set of optimal values; thus, it is necessary to calculate the superior limit (zu) and inferior (zl) for the objective function. The way to calculate these limits is found when solving a problem LP for each z, as follows: LP problem to obtain zl Maximize n

z = ∑c j x j j =1



(8.10)

subject to n

∑a

ij

x j ≤ di , i = 1, 2, 3,…, m

j =1



x j ≥ 0,



(8.11)

j = 1, 2, 3,…, n

(8.12)



LP problem to obtain zu Maximize

1 0.8 0.6 0.4 0.2

pi

0 0

di

Fig. 8.8  Fuzzy number for the FLP problem. (Source: Mula et al. (2007))

di+pi

150

R. Aceves-García and Z. E. Alarcón-Bernal n

z = ∑c j x j j =1



(8.13)

subject to n

∑a

ij



x j ≤ di + pi , i = 1, 2, 3,…, m

j =1

(8.14)

x j ≥ 0,



j = 1, 2, 3,…, n

(8.15)



It can be seen that each new LP considers the limits of the fuzzy number. The fuzzy set of optimal values is represented by G, which is defined as follows:



 1, zu ≤ CX  CX − z  l , zl < CX < zu G ( x) =  z z −  u l  0, CX ≤ zl 

(8.16)

Introducing a new variable, λ where λ ∈ [0,1], we have the following LP problem (see Zimmermann 1993). Maximize

λ



(8.17)

subject to

λ≤

CX − zl zu − zl

(8.18)

n



λ pi + ∑aij x j ≤ di + pi , i = 1, 2, 3,…, m j =1

λ x j ≥ 0,

j = 1, 2, 3,…, n



(8.19) (8.20)

where λ represents the maximum grade of membership, inside the fuzzy set of optimal values (G(x)), value which varies between zl and zu (see Martínez Fonseca 2001). Reordering the objective function, we obtain the following: Maximize

λ

(8.21)

subject to

λ ( zu − zl ) − CX ≤ zl



(8.22)

8  Determining the Demand in Inventory Policies for Mexican Companies Using Fuzzy… 151 n

λ pi + ∑aij x j ≤ di + pi , i = 1, 2, 3,…, m j =1



(8.23)

An MRP model is presented next, considering the flexible restrictions, that is, as a fuzzy linear programming model. Thereafter, a numeric example is presented, which allows for an analysis of the results, considering the demand of a number under uncertainty. To solve it, the material studied in this section will be applied. The following model was proposed in (Kaufmann and Gil 1987). However, for the purpose of this work, some modifications have been done. As observed, the model handles capacity and demand restrictions; in demand restrictions, the restriction is balance between what we have, what is owed, and what is required. For capacity restriction, the model includes overtime costs and idle time; however, such restriction may not contemplate such terms, if the company that utilizes it does not cover such costs or if it is not required to analyze them in the system. Therefore, capacity restriction will be defined only according to the time required to produce the item or items and by the produced quantity in a period, which cannot go over the available capacity in such period. The model’s formulation is as follows: Minimize I

T

R

T

z = ∑∑ ( cpi pit + cii INVTit + crd i Rd it ) + ∑∑ ( ctocrt Tocc rt + ctex rt Tex rt ) j =1 t =1

r =1 t =1

(8.24)

subject to

INVTi ,t −1 − Rd i ,t −1 + pi ,t −TSi + RPi ,t − INVTi ,t + Rd i ,t = di ,t ; ∀i, ∀t



Rd i ,T = 0; ∀i



pit , INVTi ,t , Rd i ,t , Toc rt , Tex rt ≥ 0; ∀i, ∀r , ∀t



(8.25) (8.26)



(8.27)

The objective function of this model includes, as first part, the minimization of the sum of the costs, such as costs for producing the produced quantity, inventory costs for material quantity found in inventory, and the costs for missing material for each item. In the second part of the function, the sum of the costs originated by overtime and idle time is minimized by the resources. The first restriction is to cover the demand, which is a balance between what comes in, what goes out, and what is withheld of the product (i), during a period of time (t). The second restriction is necessary for when the company has a limited capacity, as it includes the capacity to produce the resources, in a given case that the company includes overtime policies, are considered into this restriction.

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The third restriction for each product (i) allows for the delays to be covered in the last period and, lastly, the non-negative restrictions. Considering a fuzzy number does exist which represents the uncertainty and as a consequence, the demand’s behavior is as follows.  1,   − x + di + pi , Bi ( x ) =  pi   0, 



if x ≤ di if di ≤ x ≤ di + pi if x ≥ di + pi

(8.28)

The model is defined as shown below, where we can see that the objective function and the demand restriction are modified; the other restrictions are defined in the same way. Maximize (8.29)

λ

Subject to I

T

λ ( zu − zi ) − ∑∑ ( cpi pit + cii INVTit + crd i Rd it ) j =1 t =1 R

(8.30)

T

+ ∑∑ ( cttoc rt Toc rt + ctex rt Tex rt ) ≤ − zi



r =1 t =1

(



)

λ d(it )u − d(it )l + INVTi ( t −1) − Rd i ( t −1) + pit − TSi + RPit − INVTit + Rd it

≤ dit ; ∀i, ∀t I



∑AR

ir

(8.31)

Pit + Toc rt − Tex rt = CAPrt , ∀r , ∀t

i =1

(8.32)



Rd iT = 0; ∀i

(8.33)



pit , INVTit , Rd it , Toc rt , Tex rt ≥ 0; ∀i, ∀r , ∀t

(8.34)

Following is an example of the MRP model, considering the unknown demand and without capacity restrictions; the problem is solved as a fuzzy MRP model. The modifications that were done in the objective function and the restrictions that cover the demand are explained, and finally the obtained result is presented, considering the demand as a fuzzy number. Due to the extension of the modified code, such code is not presented. The software utilized to solve it was LINGO 6.0 (LINGO/PC s.f.). The problem considers 11 items, of which only one is the final product; to build such item, it is necessary to mix and assemble with the scrap items. In the following figure, the list of materials is introduced (Fig. 8.9).

8  Determining the Demand in Inventory Policies for Mexican Companies Using Fuzzy… 153

MESA 1

MEZCLATINTILLA 2

ALCOHOL 9

TINTILLA 10

Producto Final

PINTURAACABADO 11

ROBLE 7

ROBLE 8

MESASINACABAR 3

Cobón 5

CLAVOS 4

Productos intermedios

UA 6

Productos básicos

Fig. 8.9  Materials to produce a table. (Source: Mula et al. (2007))

The demand for the final product for the following four periods is shown in Table 8.7. The data presented in this problem is estimated data. For example, production capacity restrictions are not considered; what is considered is to cover the demand and to fulfill product production due to other subproducts. In Table 8.8, costs of production, storage, and scraps for each product are presented, as well as the initial inventory. The example does not consider material delivery for scheduled receiving. With this data, the model can be formulated to solve it with LINGO. The number of periods modeled are T = 1, 2, 3, 4; the number of products is 11, for which I = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. The objective function seeks to minimize the sum of the costs, for which this is defined as follows: (cost to produce the product i times the production of i) + (cost for inventory of i times the inventory of i) + (cost for scraps of i times scraps of i) for everything i and for everything t. The variables are defined for all the products and for all the periods, for example, p11 means the production of the product i = 1 in the period t = 1, variable inv34 means the inventory of the product i = 3 in the period t = 4, and so on. As observed in the previous model, restrictions were added which allow for the determination of the demand of the products, which depend of the final product. For each product i, four restrictions were added to cover the demand, of t = 1...4. The characteristics of the model are that there are a total of 131 variables and 57 restrictions. The model was solved in 50 iterations, resulting in a value in the objective function of $ 951,824.00 (Mexican pesos).

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Table 8.7  Demand for the final product for the four periods

Period 1 2 3 4

Final product 100 160 160 240

Source: Own elaboration

Table 8.8  Costs and initial inventory of the products Product 1 2 3 4 5 6 7 8 9 10 11

Cost to produce $ 100 40 30 2 10 10 120 130 50 70 100

Cost to store $ 4 5 5 0.3 5 3 6.48 6.48 2.22 2.22 2.22

Cost for scraps $ 2 8 2 5 5 7 58 58 2 0 25

Initial inventory (units) 0 0 0 250 10 10 15 15 5 10 5

8.3.1  Demand as a Fuzzy Number The use of the demand as a fuzzy number means that modifications need to be made, so it is necessary to add a new variable and to modify the restrictions of the demand. For this, the model for each one of the limits needs to be solved. Obtain the differences and solve the model again. First, it is necessary to define the demand as a fuzzy number; in such case, it is defined as a trapezoidal fuzzy number. Then the objective function’s limits are calculated, to alter define a model which includes both limits in the restrictions. For the period t  =  1, the demand behaves as shown in the following figure (Fig. 8.10). With membership function, where x ∈ ℜ



1,   − x + 250  Bi ( x ) =  ,  150  0,

if x ≤ 100 if 100 ≤ x ≤ 250 if x ≥ 250

For the following periods, the data are shown in Table 8.9.

(8.35)

8  Determining the Demand in Inventory Policies for Mexican Companies Using Fuzzy… 155 1 0.8 0.6 0.4 0.2

150

0 0

100

250

Fig. 8.10  Behavior of the demand for period t = 1. (Source: Elaborated by the authors) Table 8.9  Data per period

Period 2 3 4

di 160 160 240

di + pi 280 280 360

The following modification is made on one of the restrictions.

i = 1

(8.36)



= inv10 0= , rd10 0, inv14 = 0

(8.37)



X f ∗ (150 ) + ( inv10 − rd10 + p11 − inv11 ) ≤ d11





X f ∗ (120 ) + ( inv11 − rd11 + p12 − inv12 ) ≤ d12





X f ∗ (120 ) + ( inv12 − rd12 + p13 − inv13 ) ≤ d13





X f ∗ (120 ) + ( inv13 − rd13 + p14 − inv14 ) ≤ d14



(8.38) (8.39) (8.40) (8.41)

As observed in the previous restrictions, the value found in parenthesis is the result of the difference between the limits of the fuzzy number. Letter x is equal to the Greek letter λ (new variable). The value of the right is the superior limit. For the following restrictions, the same procedure is carried out. It is necessary to consider that the demand of the other products depends on the demand of product 1; thus, in the model, the value in parenthesis is defined by the inferior demand and the demand with the growth. The objective function changes to a function of maximizing the new variable, and the previous changes, using the zu and zl.

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The modified model has the following characteristics, 132 variables and 59 restrictions, and was solved in 54 iterations. The value of the objective function is $ 940,077.54 (Mexican pesos). By comparing the results, it can be observed that a better solution was obtained, by making the restrictions more flexible.

8.4  Conclusions From the analysis of this work, it can be concluded that in the Mexican companies, the use of models for inventory administration is a problem, because they generally don’t have databases to administer such models and they see this as an additional expense for the company. They use the experience and empirical knowledge of the ones in charge of inventory administration to determine any necessary parameter for decision-making, specifically to estimate the demand, and generally the security inventories and the amount of requested materials are excessive. As established in this study, a simple tool, of easy application and not so costly, which allows for the solution of this problem of Mexican companies, is the fuzzy sets. With such tool it is possible to determine the demand for inventory systems, even when there is no statistical information, but incorporating the experience and knowledge of the administration as empirical information. Since it is very important to determine the demand and its behavior in the inventory control, using fuzzy sets to estimate it proves to be a very promising alternative for the Mexican industry.

References H.  Behret, C.  Kahraman, A fuzzy optimization model for single-period inventory problem. Proceeding of the World Congress on Engineering, Vol. 11, 6–8 July, London, U.K. (2011) N. Brahimi, S. Dauzere-Peres, N. Najid, A. Nordli, Single item lot sizing problems. Eur. J. Oper. Res. 168(1), 1–16 (2006) J.  Campos, Arte en la administración de inventarios de materias primas. (2010). Online http:// www.logistica.enfasis.com/notas/7284-arte-la-administracion-inventarios-materiasprimas, consulting date: May 26 B.B.  Flores Brito, MRP y EOQ bajo un entorno de incertidumbre aplicando conjuntos borrosos. Tesis para obtener el grado de maestría. Programa de Posgrado en Ingeniería. UNAM. Noviembre (2010) Guiffrida, Alfred L.  Kent State University Kent, Ohio, Chapter 8: Fuzzy inventory models, in Inventory Management: Non-Classical Views, ed. by M. Y. Jaber, (CRC Press, Boca Raton, 2010), pp. 173–190 F. Harris, How many parts to make at once. Factory, The Magazine of Management, 1913, 10(2): 135–136, 152; reprinted in Operations Research. 38(6), 947–950 (1990) A.  Kaufmann, J.  Gil, Técnicas operativas de gestión para el tratamiento de la incertidumbre (Hispano Europea, S. A, España, 1987)

8  Determining the Demand in Inventory Policies for Mexican Companies Using Fuzzy… 157 H. Lee, J. Wu, A study on inventory replenishment policies in a two-echelon supply chain system. Comput. Ind. Eng. 51(2), 257–263 (2006) P. Martínez Fonseca, Conjuntos Borrosos y Algunas Aplicaciones (Tesis UNAM, México, 2001) J.  Mula, R.  Poler, J.P.  Garcia, Material requirement planning with fuzzy constraints and fuzzy coefficients. Fuzzy Set. Syst. 158, 783–793 (2007). no. 150 J. Mula, R. Poler, J.P. Garcia-Sabater, Capacity and material requirement planning modelling by comparing deterministic and fuzzy models. Int. J. Prod. Res. 46(20) (2008) J. Orlicky, Material Requirements Planning (McGraw Hill, London, 1975) P.C.  Pendharkar, A fuzzy linear programming model for production planning in coal mines. Comput. Oper. Res. 24, 1141–1149 (1997) C. Soodong, J. Noble, Determination of economic order quantities (EOQ) in an integrated material flow system. Int. J. Prod. Res. 38(14), 3203–3226 (2000). Taylor and Francis Ltd P. Vasant, Optimization in Product Mix Problem Using Fuzzy Linear Programming. Department of Mathematics, American Degree Program Nilai International College Malaysia, 1–25 (2004) L.A.  Zadeh, The concept of a lingüistic variable and its application to approximate reasoning. Inform. Sci. 8, 199–249 (1975) H.J. Zimmermann, Fuzzy Set Theory and Its Applications, 2nd edn. (Kluwer Academic Publishers, Boston, 1993)

Index

A Advanced calculators, 81, 83 Aquifer entrance, 33 exits, 33 functions, 33 future events, 34 hypotheses, 34 subject matters, 34 themes, 34 AVT problematic, 35 B Benchmarking, 15 Bi-level programming, 117–119 application, 127 average demand, 127 complexity, 118 micro-warehouses, 128 properties, 118 solution, 128 types, 119 Bi-level programming problems (BLPP) definitions, 121 discrete, 122, 123 Bi-level search services model algorithm, 125, 126 problem, 124 solution, 125, 126 Block consulting process, 52 Boundary definition, 55 Business to business (BtoB), 68 Business to consumer (BtoC), 68 Buyer usefulness matrix, 76

C Checkland’s methodology, 24, 25 Cognitive integration (IC), 90 Cognitive perspective, 84 Cognitive processes, 81 Conceptualizations, 82 Consultancy, 48 in academic research, 49 activity, 48 consulting process, 49 critical thinking, 55 customers, 48, 49 implementation of tools, 50 ineffectiveness, 50 judgment, 58 mental structure, 55 in Mexican context, 48 monopolize, 49 PyME Fund, 48 strategic aspects, 55 theoretical-methodological limitations, 49 Consulting practice, 62 Consulting process, 59 definition, 62 diagnosis phase, 51 dimensions, 53 intervention procedures, 52 Kurb’s consulting process, 51 literature, 50 placid context, 53 problem’s solution, 51 stakeholders share interests, 53 Corporate Resources Management (CRM), 136 Critical thinking, 55

© Springer Nature Switzerland AG 2021 P. E. Balderas-Cañas, G. de las N. Sánchez-Guerrero (eds.), Problem Solving In Operation Management, https://doi.org/10.1007/978-3-030-50089-4

159

Index

160 Cross impact analysis, 39 Customer–consultant information exchange, 52 Customer–consultant relationship, 51 D Daily average demand, 129 Data organization, 91 Decision variables, 130 Decision-making, 105, 109 park location, 106 public spaces, 105 urban areas, 106 Decision variables, 130 Delegación Cuauhtémoc, 112 Demand of representations (DR), 96 Differential calculus, 81 Distribution, 120, 123, 127, 128, 130–132 Dynamic diagnosis, 4 Dynamic environment, 106 E Economic Order Quantity (EOQ) equation, 137–139 analysis of results, 146 average cost, 140 deficit, 140 fuzzy, 138, 139, 142, 143, 145 parameters, 137 solution, 141 Economic rupture (ER), 72 F Fitness landscape, 10 Flexible linear programming (FLP), 148 Forecast, 22 Forecast integration, 34, 36 Formulation, 120 Future image, 37 Fuzzy number, 154–156 G Geographic Information System (GIS), 108 Goal or multi-criteria programming (GP), 108 Guiding prediction, 22 H Heavy trends, 31 Hegelian system, 57

I Innovation, 65 achieve, 69 advantages, 68 avoid, 69 challenges, 68 classification, 66, 67 consumer’s behavior, 68 definitions, 66 dimensions, 67 dynamics, 73 problems, 69 stages, 69 types, 68 Innovation process, 69–71, 73, 74, 78 elements, 72 first stage, 71 nonlinear, 70 second stage, 71 technical object, 72 tools, 71 Input variables, 6 Interactive planning process, 18 International financial system crisis, 18 Interpretative element, 83 Inventory and fuzzy demand, 139, 154 K Kantian system, 57 Kurb’s consulting process, 51 Kurtosis, 10 L Learning materials, 96 Leibniz system, 56 Linear programming (LP), 147 Location, 117, 119, 120, 128, 129 Lockean system, 56 M Material requirements planning (MRP), 138, 152 first restriction, 151 flexible restrictions, 151 inventories, 137 low use, 136 model formulation, 151 objective function, 151 second restriction, 151 solution procedure, 138 third restriction, 152 use, 138

Index Mathematical knowledge, 81 bidimensional connections, 92, 93 concepts, 82, 92 conceptual expectations, 90 images, 82 organization, 89 and procedural knowledge, 89 conceptualization, 85 digraphs, 81, 92 electronic media, 86 human mind, 84 internal representations, 85 mental images, 85 mental operations, 84 mental representations, 90 mental structures, 84 operations, 84 structural models, 81 tridimensional connections, 93 visualization, 82 visual processing, 86 Mathematic modeling methodology, 106, 109 Mental processes, 82 Mental structure, 58, 59 Micro-warehouses, trip time, 130 Ministry of Economy, 48 Modeling process circumstance, 106 structured problem, 106 Model of propositional analysis (MAP), 87 Morfín’s consulting process, 51 MRP fuzzy, 138, 148–150 Multi-criteria approach, 109 Multi-criteria model, 111 Multi-criteria optimization model, 112 Multi-criteria programming, 117 Multi-objective model, 110 N Networks, 117, 119 O Ochoa-Rosso method, 52 Optimization models, 107 Organizational dynamics, 6 attractors, 8 bifurcations, 8 chaos, 8 chaotic process, 7 diagnostic, 11 feasible states, 7

161 fitness landscape, 9 inertial attractor, 8 input variables, 6 methodology, 11, 12 objectives, 13 obstacles, 10 order vs. chaos, 8 output/response variables, 7 phases, 14 potential of, 9 state variables, 6 structure, 12, 13 Organizations adaptive, 9 complex system, 4 interaction process, 5 internal dynamics, 5, 6 interrelated subsystems, 4 methodological outline, 5 open system, 5 Output/response variables, 7 P Pie chart, 75 Planning, 18, 34 Predictions integration, 36, 38 Problem structuration process, 108 Programming methods, 111 R Radical innovation, 66 Ruptures, 70–74, 76–78 S Scenarios classification, 19 definition, 19, 20 environment future studies, 18 interactive planning, 18 planning, 19 states, 20 systemic construction, 21 writing, 37, 39–42 Shipping expenses, 129 Singerian system, 57 State Health Department of the State of Mexico (ISEM), 127 State variables, 6 Structuration, 107 park locations, 108 process, 107

Index

162 Systemic approach, 82 Systemic intervention, 55, 60 in actions, 54 characteristics, 54 identification, 54 methodology, 54 theoretical and methodological elements, 47 T Teaching guide, 97, 98 Teaching materials, 83 Technical elements, 74 Technical object value chart, 77 Technological analysis, 73 Technological rupture (TR), 72 Theoretical-methodological elements, 55 Toluca Valley Aquifer (AVT) scenario, 31 Trapezoidal form (TrFN), 148 Trapezoidal fuzzy number (TrFN), 141 Trend scenarios, 21 carriers, 31 aquifer area, 33 procedure to develop, 32 quantitative aspects, 31 system analysis, 33 conceptual model, 26 definition, 25 development, 24 elaboration process, 26, 27 forecast, 23 functions, 26

future study, 22 phases, 23 procedure, 28 actors/events, 30 forecast integration, 28, 29 future image construction, 29 invariants, 30 narrations, 29, 30 predictions integration, 29 system analysis, 27 variables, 30 process, 23 stages, 23 Triangular fuzzy number (TFN), 141 U Urban planning and renovation, 105 Usage rupture (UR), 72 V Visual information, 86 Visualizing a diagram, 88 Visual method, 88 Visual processing, 86 Visual reasoning, 81, 83, 87, 88 conceptual organization, 87 individual segments, 86 interpretative element, 83 qualitative studies, 82 validation, 84 visual images, 87