Retail Space Analytics 3031270576, 9783031270574

Table of contents :
Preface
References
Contents
Effect of Customer Travel Behavior on Grid Layout and Shelf Space Allocation in Retail Facilities
1 Introduction
2 Prior Literature on Retail Facility Layout Design
3 Modeling Framework for Design and Evaluation of Grid Layouts
3.1 Layout Generation and Evaluation
3.2 Solution Methodology
4 Customer's Route Selection Strategy
4.1 Computational Results and Discussion
5 Conclusion
References
A Solver-Free Heuristic for Store-Wide Shelf Space Allocation
1 Introduction
2 Literature Review
3 Problem Statement and Optimization Model
3.1 Notation
3.2 Optimization Model for Shelf Space Allocation
4 Solver-Free Heuristic
5 Application to a Supermarket in Beirut
6 Conclusions and Directions for Future Research
References
In-Store Traffic Density Estimation
1 Introduction and Motivation
2 Literature Review
3 Shopping Basket Data and Store Description
4 Methodology
4.1 Traffic Density: Predictor Variables
4.2 Shelf-to-Shelf Distance Matrix
4.3 Regression Model
4.3.1 Support Vector Regression
4.3.2 Regression Tree
4.3.3 Kernel Regression
4.3.4 Gaussian Processes
4.3.5 Beta Regression
5 Results
6 Conclusions
References
A Simulation Based Tool to Guide Periodic Changes in a Supermarket Layout
1 Introduction
2 Relevant Literature
2.1 The Facility Design Problem
2.2 Store Layout
2.3 Impulse Purchases
2.4 Shelf Space Allocation
2.5 Travel Patterns
3 Optimization Framework
4 Block Layout Optimization
4.1 K-Medoids
4.2 Customer Profiles
4.3 Simulation
4.3.1 Generating Customer Shopping Lists
4.3.2 Customer Paths Generation
5 Detailed Layout
6 Evaluating Different Block Layouts
7 Case Study of a Grocery Store in Western New York
7.1 Data Collection and Analysis
7.1.1 Supermarket Layout
7.1.2 Customer Path Data
7.1.3 Customer Transaction Data: Creation of Customer Clusters and Must-Have and Impulse Item Designation
7.2 Computational Study
7.3 Managerial Insights
7.3.1 Changes in Customer Lifestyles
7.3.2 Impact of Item Prices
7.3.3 Changes in List of Products
8 Conclusions and Future Research
References
Data-Driven Analytical Grocery Store Design
1 Introduction
2 Literature Review
3 The Case Study
4 Methodology
4.1 The Apriori Algorithm
4.2 The High Utility Data Mining Algorithm
4.3 Characterizing Adjacencies
4.4 Shelf Space Allocation and the Revenue Function
5 Discrete-Event Simulation Modeling of the Migros Store
6 Tabu Search Optimization for Store Layout Design
7 Computational Experience
8 Conclusions and Discussion
References
Optimizing Stock-Keeping Unit Selection for Promotional Display Space at Grocery Retailers
1 Introduction
2 Literature Review
3 Methodology
3.1 Direct-Static Methodology
3.1.1 Sales Response Function
3.1.2 Incremental Display Profit
3.1.3 Static Optimization of SKU Choice
3.2 Hierarchical-Static Methodology
3.2.1 Sales Response Function
3.2.2 Incremental Display Profit
3.2.3 Static Optimization of the SKU Choice
4 Data Description
4.1 Estimation Data
4.2 Optimization Data (Single Store)
5 Application and Assessment of Methodology
5.1 Direct Estimation/Optimization
5.2 Hierarchical Estimation/Optimization
5.3 Static Benchmark Comparison
6 Dynamic Optimization
6.1 Application of the Dynamic Optimization
6.2 Benchmark Comparison for Dynamic Case
7 Conclusion
References
Merchandise Placement Optimization
1 The Value of Placement
1.1 Plan Modification Optimization (``PMO'')
1.2 Empirical Results
2 Forecasting the Value of Placement
2.1 Biclustering of Locational Impact on Demand
Demand Normalization and the Fractional Response Matrix
The Standard Response Matrix
The Biclustering Algorithm
2.2 Empirical Results
Example 1: 5 Items/3 Fixtures/Max 3 Clusters
Example 2: Disguised Store Data with 45 Items/7 Fixtures/Max 20 Clusters
2.3 A Note on Data Scarcity
The Fixture-Oriented Regression Model
Tackling the Sparsity of Fixture-IC Coefficient Matrices with Biclustering
3 Conclusions
Appendix
References
Problems and Opportunities of Applied Optimization Models in Retail Space Planning
1 Introduction
2 Shelf Space Allocation in Practice
3 State-of-the-Art Shelf Space Optimization Approaches
4 Blending Research and Practice
5 Improvements of Shelf Space Optimization in Practice
5.1 Additional Model Features
5.1.1 Integrating Shelf Space, Shelf Types and Store Layout Planning
5.1.2 Accounting for Product Grouping
5.1.3 Determine the Efficient Product Allocation Type
5.1.4 Extending Approaches to Multi-Store Concepts
5.1.5 Consider Omnichannel Opportunities
5.2 Enhanced Demand Effects
5.2.1 Explore the Effectiveness of Various Demand Effects
5.2.2 Consider Complementary Effects and Cross-Selling
5.2.3 Integrating Assortment Decisions if Decision-Relevant
5.3 Planning Approaches and Scope
5.3.1 Replanning Requires Rebuilding
5.3.2 Consider the Objectives of Different Stakeholders
6 Conclusion
References

Citation preview

International Series in Operations Research & Management Science

Ahmed Ghoniem Bacel Maddah   Editors

Retail Space Analytics

International Series in Operations Research & Management Science Founding Editor Frederick S. Hillier, Stanford University, Stanford, CA, USA

Volume 339

Series Editor Camille C. Price, Department of Computer Science, Stephen F. Austin State University, Nacogdoches, TX, USA Editorial Board Members Emanuele Borgonovo, Department of Decision Sciences, Bocconi University, Milan, Italy Barry L. Nelson, Department of Industrial Engineering & Management Sciences, Northwestern University, Evanston, IL, USA Bruce W. Patty, Veritec Solutions, Mill Valley, CA, USA Michael Pinedo, Stern School of Business, New York University, New York, NY, USA Robert J. Vanderbei, Princeton University, Princeton, NJ, USA Associate Editor Joe Zhu, Foisie Business School, Worcester Polytechnic Institute, Worcester, MA, USA

The book series International Series in Operations Research and Management Science encompasses the various areas of operations research and management science. Both theoretical and applied books are included. It describes current advances anywhere in the world that are at the cutting edge of the field. The series is aimed especially at researchers, advanced graduate students, and sophisticated practitioners. The series features three types of books: • Advanced expository books that extend and unify our understanding of particular areas. • Research monographs that make substantial contributions to knowledge. • Handbooks that define the new state of the art in particular areas. Each handbook will be edited by a leading authority in the area who will organize a team of experts on various aspects of the topic to write individual chapters. A handbook may emphasize expository surveys or completely new advances (either research or applications) or a combination of both. The series emphasizes the following four areas: Mathematical Programming: Including linear programming, integer programming, nonlinear programming, interior point methods, game theory, network optimization models, combinatorics, equilibrium programming, complementarity theory, multiobjective optimization, dynamic programming, stochastic programming, complexity theory, etc. Applied Probability: Including queuing theory, simulation, renewal theory, Brownian motion and diffusion processes, decision analysis, Markov decision processes, reliability theory, forecasting, other stochastic processes motivated by applications, etc. Production and Operations Management: Including inventory theory, production scheduling, capacity planning, facility location, supply chain management, distribution systems, materials requirements planning, just-in-time systems, flexible manufacturing systems, design of production lines, logistical planning, strategic issues, etc. Applications of Operations Research and Management Science: Including telecommunications, health care, capital budgeting and finance, economics, marketing, public policy, military operations research, humanitarian relief and disaster mitigation, service operations, transportation systems, etc. This book series is indexed in Scopus.

Ahmed Ghoniem • Bacel Maddah Editors

Retail Space Analytics

Editors Ahmed Ghoniem Operations & Information Management Department, Isenberg School of Management University of Massachusetts Amherst Amherst, MA, USA

Bacel Maddah Department of Industrial Engineering and Management American University of Beirut Beirut, Lebanon

ISSN 0884-8289 ISSN 2214-7934 (electronic) International Series in Operations Research & Management Science ISBN 978-3-031-27057-4 ISBN 978-3-031-27058-1 (eBook) https://doi.org/10.1007/978-3-031-27058-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed 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

Preface

Global retail sales amount to 27.34 trillion dollars, with nearly 20 trillion dollars in brick-and-mortar stores and about 6 trillion dollars in e-commerce. Whereas the growth of online sales has threatened sales in physical stores, causing alarm and the closure of numerous stores in the USA and elsewhere, traditional retailing continues to show a solid growth worldwide. For example, grocery sales amounted in 2021 (Statista, 2022) to $1375 million in supermarkets and neighborhood stores (with a projection of $1615 million in 2026), $719 million in hyper-stores (projected to be $816 million in 2026), and $468 million in e-commerce (with a remarkable projection of $809 million for 2026). It is vital for retail stores to continue to offer an attractive assortment of products and an appealing in-store experience for customers. This edited volume presents, through its eight chapters, exciting developments in retail analytics with a focus on new trends in shelf space allocation, targeting store-wide shelf space and endcap planning. Driven by modern-day sensory technologies, customer trackers, and transactional tools, this field has witnessed the development of novel data-driven models, predictive analytics, and optimization approaches that can leverage largescale data collected in retail stores in general and grocery stores in particular. The emerging retail space planning literature is scattered among different body of works that examine different complementary aspects of this multidisciplinary problem. The notion of impulse buying, which, for example, can amount to 50% of sales in supermarkets, has come to the front scene, along with the use of promotional displays (which can result in lifting sales by over 25% for certain products). This book captures state-of-the-art contributions to store shelf space planning, a major departure in scope and complexity from earlier works that have addressed planogram planning with a few shelves optimized in isolation. We hope that researchers and industry practitioners alike will find in this volume an insightful discussion of the interplay between store-wide product allocation, in-store traffic modeling, promotional and end-cap allocations, shelf orientation, and layout planning. In Chap. 1, A. Reha Botsalı, Georgia-Ann Klutke, and Brett A. Peters investigate customer in-store travel behavior as a function of the store layout, with a focus on v

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grid layouts, and shelf space allocation. In particular, the authors demonstrate that the expected impulse buying depends on the customer route selection strategy and the extent of the customer’s planned shopping list. Larger planned shopping lists result in greater customer travel and opportunities for impulse buying. The authors also offer the important insight that the presence of alternative shopping routes due to the store layout has a detrimental effect on the expected impulse buying. As a consequence, the layout planning in stores that use a grid layout should be carefully optimized in a manner that controls customer travel, limits alternative instore shopping routes, and maximizes the expected impulse buying. This naturally paves the way for future research on shelf space planning under alternative layout designs, such as serpentine layouts. In Chap. 2, Tulay Flamand, Ahmed Ghoniem, and Bacel Maddah propose a solver-free heuristic for an elaborate store-wide space planning model proposed in Flamand et al. (2023). The key distinctive feature in Flamand et al. (2023) is the traffic density along a store shelf which is evaluated as a function of both the location of the shelf in the store and the allocation of products to different shelves. Capturing the endogenous product allocation component of the traffic leads to a complex nonlinear integer program that Flamand et al. (2023) solve by proposing an involved linearization and bounding approximation scheme. The scheme produces high-quality solutions, with a measurable optimality gap, within an hour CPU time limit. In this chapter, the authors propose a variable neighborhood search heuristic that is shown to produce solutions of comparable quality to those in Flamand et al. (2023) within a couple of minutes. This chapter seeks to help practitioners adopt a “high-resolution” version of store-wide planning approach with a modest computational overhead. In Chap. 3, Jimmy Azar and Hoda Daou analyze predictive models for in-store traffic. Specifically, they target estimating the traffic density along a store shelf as a function of both the location of the shelf and the allocation of products throughout the store with the objective of improving over the work of Farley and Ring (1966) and the recent work by Flamand et al. (2023). Azar and Daou present an interesting simplification for estimating the dependent variable of the traffic regression based on a demand filtering approach. They apply their methodology to a large set of basket data obtained from a supermarket in Beirut for which they compare different regression approaches. They find that non-parametric approaches (such as regression trees) are promising. As Chap. 2 proposes simplification to the solution approach of Flamand et al. (2023) that could benefit practitioners, this chapter offers further practical ideas on implementing the framework of Flamand et al. (2023), especially on the predictive front. Combining ideas from both chapters is an exciting prospect. In Chap. 4, Jessica Dorismond, Jose Walteros, and Rajan Batta present an elaborate simulation and optimization framework for sequentially determining the location of “blocks” (groups of similar shelves), and subsequently the allocation of products in each block. Customer profiles and shopping paths are generated using Monte Carlo simulation in a mechanism similar to that in Chap. 1. However, Dorismond et al. avail of real-world basket data and propose an interesting

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clustering approach to segment the customers. They also employ customer surveys and in-store experiments to validate their model, while applying their approach to a supermarket in Western New York, in a commendable effort. The main managerial insight from this chapter is the interaction of the decisions on block location and product allocation to blocks, which makes developing a joint optimization model a worthwhile future research direction. On the methodology front, this chapter successfully utilizes a variable neighborhood search (VNS) in the first stage problem of block layout similar in spirit to that in Chap. 2. This suggests that VNS offers a promising heuristic approach for store-wide planning. In Chap. 5, Elif Ozgurmus and Alice Smith propose a practical data-driven approach for the grocery store layout optimization problem, with a successful application to and implementation at Migros, a major store in Turkey. The analysis of demand sales informs the placement of product categories (or departments) throughout the store, in a manner that optimizes the expected impulse buying revenue and customer convenience (based on the proximity of product categories that have high affinity), using an effective Tabu search metaheuristic. The proposed solutions yield substantial improvements over the current block layout configuration and higher revenues at the Migros store in Antalya, despite strong competition by other local stores. As a result, management is considering its adoption at additional store branches in Turkey. In Chap. 6, Olga Pak, Mark Ferguson, Olga Perdikaki, and Su-Ming Wu examine the problem of product allocation to end-caps, an important promotional space in grocery stores that typically enjoys double the traffic that is witnessed in the inner shelves of the store. Surprisingly, however, the problem has received little attention thus far in the academic literature. They propose a data-driven tool for decision-makers by which the relative sales lift of a candidate SKU (stock-keeping unit) is estimated and then fed into an optimization model that selects the most promising SKUs to display in end-caps. The proposed methodology can be applied in a direct manner, when the number of SKUs is manageable, or approximately in a hierarchical manner, when the number of SKUs is too large. The authors use retail CPG sales scanner data on beer sales from hundreds of grocery store chains in the New England region of the USA. The proposed methodology yields a relative improvement of about 17.5% over a benchmark where the store managers select the best-selling SKUs per week to be placed on promotional display. In Chap. 7, Wei Ke presents effective tools for evaluating the financial implications of a series of product moves inside a store with the objective of identifying an “optimized” modification of the space allocation plan. This work has several interesting features. First, unlike the works in earlier chapters of this book, it does not require a specific store format (e.g., a grid layout) and is developed with fashion retailing in mind, while the earlier chapters target supermarkets. Second, it utilizes a novel type of data that retailers have recently started to collect, i.e., the history of product (SKU) locations in the store, in order to predict (via linear regression) the lift (or lack of) in demand from placing a product at a specific store location (fixture). Third, it presents an innovative biclustering (over location and SKU) approach to compensate for the lack of historical data on potential SKU and fixture pairs. Finally,

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it utilizes data from multiple stores of the same chain to further compensate for the scarcity of product location data and allow a statistically valid calibration of the regression model. A similar multi-store data and regression concept is effectively utilized in Chap. 6 in the context of promotional spaces. This suggests that multistore data can be promising in developing predictive traffic models. In Chap. 8, Tobias Düsterhöft and Alexander Hübner present a thorough account of the steps of the space planning process based on a close collaboration with a major retailer in Germany, with a focus on assigning SKUs in a category to a shelf segment (the planogram planning problem). The authors present an insightful review of the classic and modern approaches in the literature that can be used in planogram management and classify them into seminal, advanced, or applied works. They also discuss the limitation of each class of models in the literature in light of the practice at their industry partner, and then delineate an excellent road map for future research that can bridge the gap between theory and practice. This chapter also offers an in-depth understanding of the visual effects, logistical considerations, and merchandising rules involved in the practice of retail space planning. Amherst, MA, USA Beirut, Lebanon

Ahmed Ghoniem Bacel Maddah

References Farley, J. U., & Ring, L. W. (1966). A stochastic model of supermarket traffic flow. Operations Research, 14(4), 555–567. Flamand, T., Ghoniem, A., & Maddah, B. (2023). Store-wide shelf space allocation with ripple effects driving traffic. Operations Research. Statista (2022). Grocery sales worldwide in 2021 with a forecast for 2026, by channel (in billion U.S. dollars). https://www.statista.com/statistics/1288909/ grocery-sales-worldwide-by-channel/

Contents

Effect of Customer Travel Behavior on Grid Layout and Shelf Space Allocation in Retail Facilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Reha Botsalı, Georgia-Ann Klutke, and Brett A. Peters 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Prior Literature on Retail Facility Layout Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Modeling Framework for Design and Evaluation of Grid Layouts . . . . . . . . 3.1 Layout Generation and Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Solution Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Customer’s Route Selection Strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Computational Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 4 7 9 10 12 18 19

A Solver-Free Heuristic for Store-Wide Shelf Space Allocation . . . . . . . . . . . . Tulay Flamand, Ahmed Ghoniem, and Bacel Maddah 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Problem Statement and Optimization Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Optimization Model for Shelf Space Allocation . . . . . . . . . . . . . . . . . . . . . . 4 Solver-Free Heuristic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Application to a Supermarket in Beirut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions and Directions for Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

In-Store Traffic Density Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jimmy Azar and Hoda Daou 1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Shopping Basket Data and Store Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Traffic Density: Predictor Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.2 Shelf-to-Shelf Distance Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Regression Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Support Vector Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Regression Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Kernel Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Gaussian Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 Beta Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Simulation Based Tool to Guide Periodic Changes in a Supermarket Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jessica Dorismond, Jose L. Walteros, and Rajan Batta 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Relevant Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Facility Design Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Store Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Impulse Purchases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Shelf Space Allocation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Travel Patterns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Optimization Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Block Layout Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 K-Medoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Customer Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Generating Customer Shopping Lists . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Customer Paths Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Detailed Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Evaluating Different Block Layouts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Case Study of a Grocery Store in Western New York . . . . . . . . . . . . . . . . . . . . . . . 7.1 Data Collection and Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Supermarket Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Customer Path Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Customer Transaction Data: Creation of Customer Clusters and Must-Have and Impulse Item Designation . . . . . . 7.2 Computational Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Managerial Insights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Changes in Customer Lifestyles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Impact of Item Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Changes in List of Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Conclusions and Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Data-Driven Analytical Grocery Store Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elif Danisman and Alice E. Smith 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Apriori Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The High Utility Data Mining Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Characterizing Adjacencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Shelf Space Allocation and the Revenue Function . . . . . . . . . . . . . . . . . . . . 5 Discrete-Event Simulation Modeling of the Migros Store . . . . . . . . . . . . . . . . . . 6 Tabu Search Optimization for Store Layout Design . . . . . . . . . . . . . . . . . . . . . . . . . 7 Computational Experience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Conclusions and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimizing Stock-Keeping Unit Selection for Promotional Display Space at Grocery Retailers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Olga Pak, Mark Ferguson, Olga Perdikaki, and Su-Ming Wu 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Direct-Static Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Sales Response Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Incremental Display Profit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Static Optimization of SKU Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Hierarchical-Static Methodology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Sales Response Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Incremental Display Profit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Static Optimization of the SKU Choice . . . . . . . . . . . . . . . . . . . . . . . 4 Data Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Estimation Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Optimization Data (Single Store) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Application and Assessment of Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Direct Estimation/Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Hierarchical Estimation/Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Static Benchmark Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Dynamic Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Application of the Dynamic Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Benchmark Comparison for Dynamic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Merchandise Placement Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wei Ke 1 The Value of Placement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Plan Modification Optimization (“PMO”) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Forecasting the Value of Placement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Biclustering of Locational Impact on Demand . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 A Note on Data Scarcity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems and Opportunities of Applied Optimization Models in Retail Space Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tobias Düsterhöft and Alexander Hübner 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Shelf Space Allocation in Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 State-of-the-Art Shelf Space Optimization Approaches. . . . . . . . . . . . . . . . . . . . . 4 Blending Research and Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Improvements of Shelf Space Optimization in Practice . . . . . . . . . . . . . . . . . . . . . 5.1 Additional Model Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Integrating Shelf Space, Shelf Types and Store Layout Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Accounting for Product Grouping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Determine the Efficient Product Allocation Type . . . . . . . . . . . . . 5.1.4 Extending Approaches to Multi-Store Concepts . . . . . . . . . . . . . . 5.1.5 Consider Omnichannel Opportunities . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Enhanced Demand Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Explore the Effectiveness of Various Demand Effects . . . . . . . . 5.2.2 Consider Complementary Effects and Cross-Selling . . . . . . . . . 5.2.3 Integrating Assortment Decisions if Decision-Relevant . . . . . . 5.3 Planning Approaches and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Replanning Requires Rebuilding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Consider the Objectives of Different Stakeholders. . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Effect of Customer Travel Behavior on Grid Layout and Shelf Space Allocation in Retail Facilities A. Reha Botsalı, Georgia-Ann Klutke, and Brett A. Peters

1 Introduction Determining the physical layout and design of manufacturing facilities has occupied industrial engineers for many decades. Beginning in the 1960s, plant layout problems have received attention from the academic research community, and there are now hundreds of papers and scores of textbooks addressing this subject. More recently, attention has been given to the efficient design of facilities that support the logistical aspects of the manufacturing enterprise, notably warehouses and crossdocking facilities. Somewhat surprisingly, there has been little research within the industrial engineering community that specifically optimizes the layout of retail facilities. According to U.S. Bureau of Labor Statistics data as of June 2020, the retail industry in the U.S. has 14,398,300 employees. The National Retail Federation states that retail supports 42 million jobs and represents $2.6 trillion of annual GDP in the United States. Considerable evidence from marketing studies has demonstrated the importance of product exposure and placement to sales (Dreze et al., 1994; Chen et al., 1999); thus it could be argued that the efficient design of retail facilities may be even more important to their profitable operation than for manufacturing facilities. Retail facility design differs significantly from the conventional manufacturing and warehouse design problems addressed in the literature. A key difference lies in

A. Reha Botsalı () Department of Industrial Engineering, Necmettin Erbakan University, Konya, Turkey G.-A. Klutke National Science Foundation, Alexandria, VA, USA B. A. Peters University of Wisconsin at Milwaukee, Milwaukee, WI, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Ghoniem, B. Maddah (eds.), Retail Space Analytics, International Series in Operations Research & Management Science 339, https://doi.org/10.1007/978-3-031-27058-1_1

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the objective function. Traditional layout design studies mainly focus on minimizing the material handling cost in the facility. Assuming there is an interaction cost (e.g., a travel distance) between the layout components (departments, machines, cells, etc.), the objective of the design is to minimize cost. In contrast, retail facility design is driven by a revenue-based objective, such as annual store sales or profit per square foot. In addition to the performance metrics employed, understanding customer behavior in retail facilities complicates the layout design problem further. Retail facilities are primarily self-service facilities, as customers largely attend to their own individual shopping needs in mostly an independent manner (with the possible exception of checkout). Thus, the analog of material flow in a manufacturing environment is customer traffic in a retail environment. Since traffic patterns are also determined by individual customers, they tend to be significantly more variable than material flow patterns in manufacturing facilities. Customers often enter a retail store with a list of potential products in mind, but as they traverse the store, they may decide to purchase other items beyond their initial list. These unplanned purchases, which are also called impulse purchases, depend heavily on product exposure during the time spent in the store. As such, the walkways inside the retail store direct the customer traffic and provide a mechanism for the retailer to control the manner in which the customers are exposed to products. On the other hand, customers place value on their shopping time and, if they perceive that it takes too long to find their intended purchases in a store, they may decide to shop elsewhere. Thus a retail store layout design model should balance the trade-off between the time spent in the store by the customer and the sales revenue obtained through impulse purchases. Retail facility layout design is a relatively unexplored research area, although store-wide shelf space allocation problems have received increasing attention in recent years. Our goal in this chapter is to develop new models and solution techniques to assist designers of retail stores, with a focus on grid layouts. Our primary interest is in studying the impact of customers’ route selection strategies on revenue. The paper is organized as follows. In the next section, we summarize previous works in the retailing and marketing literature that has addressed facility design. Section 3 presents our modeling framework for grid-based retail facilities. We describe an algorithm to generate and evaluate different designs. In Sect. 4, we describe four different shopping route selection scenarios and study their impact on overall revenue. In Sect. 5, we present our conclusions and discuss their implications in retail store layout design.

2 Prior Literature on Retail Facility Layout Design There are many different study fields related to retail facility layout design. Among these, product assortment and shelf space allocation problems are two important issues in which retailers are interested. In reality, these two are closely related to each other. Since each product in the assortment requires a minimum

Effect of Customer Travel Behavior on Grid Layout and Shelf Space Allocation. . .

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amount of shelf space, the assortment size is directly proportional to the shelf space requirements of the products. On the other hand, retailers generally have fixed amount of shelf space to allocate for different products. These facts make the literature focus on the integration of the product assortment and shelf space allocation models into one single model. Ideally, this single model should find the optimal product assortment and the shelf space allocated for each product in the assortment. The objective is generally maximizing the profit. Including the pioneering study of Corstjens and Doyle (1981) that uses geometric programming, there are many studies proposing several models and applying different methods such as simulated annealing (e.g., Borin et al., 1994), genetic algorithms (e.g., Urban, 1998), and empirical data analysis (e.g., Chiang and Wilcox, 1997; Desmet and Renaudin, 1998; Ozgormus and Smith, 2020) to solve these problems. Nierop et al. (2008) give information on prior shelf space allocation models along with their model. Ghoniem et al. (2016b) solve the shelf space allocation problem on a store level using mixed integer programming and later in another study Ghoniem et al. (2016a) show that it is possible to solve the same problem more efficiently using branch and price algorithm. Flamand et al. (2016) analyze the shelf space allocation problem for the whole store and propose a 0–1 integer programming model to maximize impulse purchase revenue based on customer traffic inside the store. In this study, we assume that for each product category, the product assortment and minimum allocated shelf space requirements are known in advance, which can be estimated, if necessary, by these shelf space allocation and product assortment models. Impulse purchases are another issue related to retail facility design. These purchases are generally considered to be purchases that are made by the customers without prior intention, although there are several specific definitions that have been used. Kollat and Willet (1967) define impulse purchase as the difference between the products purchased and the products planned to be purchased before entering the store, which seems to be the most general definition of impulse purchase. Studies (e.g., Bellenger et al., 1978; McGoldrick, 1982) show that different product categories have different impulse purchase rates. As an example, according to POPAI/duPont Studies (1978), 51% of pharmaceuticals and 61% of healthcare and beauty aid products are purchased without prior intention. Considering the potential influence of impulse purchases on sales revenue, retailers should locate products with high impulse purchase rates on high customer traffic areas in the store in order to increase impulse purchase revenue. In other words, impulse purchase revenue depends on the layout configuration of the store. In retailing literature, there are several studies analyzing the influence of store layout design on customer behavior through consumer surveys and empirical data analysis. These studies show a high variety with respect to the context in which they analyze the relationship. As an example, Iyer (1989) and Park et al. (1989) analyze the joint effect of time pressure and store layout knowledge on customer purchase behavior. Floch (1988) studies the layout design problem from a consumer psychology perspective, and she suggests that the store layout should be a combination of different layout patterns that fit the preferences of different customer

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groups. In this study, we analyze the effect of customer’s route choice behavior on impulse purchase revenue and customer travel distance within the layout design. Depending on the traffic flow pattern inside the store, the three common layout types given in retailing literature are grid, free-flow and boutique/racetrack layouts. Grid layout routes the traffic in straight, rectangular fashion. This layout type is often used by food retailers, discount stores, hardware stores and other convenience oriented stores whereas free-flow layout is often used in department stores, clothing stores and other browsing oriented stores Berman and Evans (1989). In contrast to the aisle structure of grid layout, free-flow layout allows customers to be able to move in several directions. Finally, boutique/racetrack layout consists of main aisles/customer paths and designated store areas, each specialized in a particular product category, along these often loop-based paths (Vrechopoulos et al., 2004). In an early study, Farley and Ring (1966) analyze the traffic flow in a supermarket by dividing the supermarket into regions and estimating the transition probabilities between these regions. Recently, Radio Frequency Identification (RFID) makes it more feasible to follow the actual traffic patterns in the retail facilities. As an example, Hui et al. (2009) and Larson et al. (2005) use retail store customer traffic data obtained by RFID and show that customers are likely to make their travel in the stores in an efficient manner. In order to control traffic flow, there are some standard guidelines that are used by retailers in store layout design. For example, it is preferred to locate high demand products (also referred to as staples or power items, such as milk and bread in a grocery store) in the back corners of the store to pull the customers through the store and encourage them to browse the store more. In product location decisions, complementary relationships (e.g., coffee and sugar, cereal and milk) is another point retailers take into account in terms of closeness of products. Despite such guidelines for store layout design, there is not an analytical layout design model for retail stores in retailing literature. In the rest of this chapter, we focus on the customer travel behavior and its impact on the store layout aspect of retail facility design. We assume that product category assortment decisions are already made and the products are represented on category levels. Under this assumption, we present a model for designing grid layouts, which is one of the most common layout types used in retail stores.

3 Modeling Framework for Design and Evaluation of Grid Layouts Since impulse purchases depend on the products exposed to the customers, it seems reasonable to posit that control of the customer flow through the store is an important mechanism to increase revenue. As suggested by the literature, we assume that a customer enters the store with a list of products he intends to purchase (or, at the least, to peruse). We refer to these products as shopping list items. In order to

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complete shopping, the customer has to visit the shelves on which these shopping list items are located. Consequently, the location of shopping list items guides the customer flow through the store and provides opportunities to increase revenue by displaying items not on the customer’s shopping list. Other than the location of shopping list items, the layout design model should clearly identify the walkways that the customer can use in the store. These walkways define the set of possible routes the customer may follow. By locating products with high impulse purchase rates along the walkways, it is possible to increase the visibility of these products, and in turn the potential impulse purchase revenue. In practice, end-of-aisle locations in grid layouts are high customer traffic areas. To take advantages of these areas, end caps are often placed at the end of shelf rows to display samples of items that have high impulse purchase rates or are in promotion. Thus, these high traffic areas also impact revenue and have a prominent role in the layout design. More information on end-of-aisle display problem is given by (Pak et al., 2019). In the grid layout, the walkways are defined by the aisles between the shelf rows. Different grid layouts can be formed by manipulating the length and orientation of aisles, as shown in Fig. 1. In order to capture these aspects of the layout design, we define a basic layout component, which we refer to as a grid unit. A grid unit contains a number of parallel shelf rows and aisle clearances between these shelf rows. There are end caps at the end of shelf rows. The grid unit has a square shape and its dimensions are proportional to the number of shelf rows it contains. If there are n shelf rows in a grid unit, then this grid unit is composed of .n2 unit squares. In Fig. 2, grid unit examples having two and three shelf rows are presented. The final layout consists of a set of connected grid units and perimeter shelves along the boundaries of the store. Because the grid units are square, they can be rotated, which results in different layout configurations with respect to aisle orientations. In addition, given that two neighboring grid units have the same

Fig. 1 Different grid layout realizations

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orientation, one or more of the corresponding shelf rows in these grid units can be connected, resulting in longer aisles and shelf rows. Once orientations of the grid units are fixed and connected shelf row decisions are made, the walkways the customer can use are clearly defined in the final layout due to the aisle structure inherited in each grid unit. We consider this walkway skeleton as a network on which the customer can use different routes to complete his shopping. On this network, the node locations are determined by the corners of the unit squares, which the grid units are composed of. Depending on the shopping route, the customer may see the products located on the shelves along the aisles he passes through. In Fig. 3, examples for shelf row connection and walkway network are presented for a sample layout. It is possible to find the minimum number of grid units required for a given net shelf space requirement in a retail store. However, when the minimum number of grid units is used, generally the resulting floor plan takes the shape of a long rectangle with a narrow width due to the effect of shelf space gain from perimeter

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shelves. To prevent unrealistic floor plans, a shape constraint can be introduced. After fixing the number of grid units and the shape of the floor plan, it is possible to analyze alternative layout configurations for the retail store. The details of how to generate different layout configurations using grid units are presented in the next section.

3.1 Layout Generation and Evaluation We assume that assorted product categories in the retail store are known in advance and they have fixed shelf space requirements, which are represented by shelf row lengths. In the final layout, each product category should get at least as much shelf space as defined by its fixed shelf row length. As an example, in Fig. 4, each letter Walkway Network

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on the shelf rows represents a product category and has certain shelf row length allocated to it. During shopping, the customer can purchase a product impulsively only if he notices the product, which depends on two conditions. First, the customer should pass near the shelf row on which the product is located. Second, he should see the product on the shelf. It is possible that the customer can pass near a product’s location without noticing it. Phillips and Bradshaw (1993) give information on customers’ visual perception such as angle of vision, visual clutter and visual cues, and state that customers looking for a specific product tend to exclude other products in their visual processing even if they see it. In order to estimate the probability of a product being noticed, we extend the visibility factor definition of Botsalı and Peters (2005), which is an estimate for the probability that the customer notices a product from a particular product category. The visibility factor (.vp ) for product category p is calculated by: vp = min{1,

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In Eq. (2), the average impulse purchase revenue contribution of product category p is assumed to be equal to the multiplication of its visibility probability (.vp ), the average rate a customer makes a purchase from product category p impulsively after noticing it (.ip ) and the average revenue per purchase from product category p (.rp ). .ip and .rp parameter values can be estimated by analyzing studies related to impulse purchases (e.g., Bellenger et al., 1978; McGoldrick, 1982) and marketing sales data, respectively. Equation (2) can be used to evaluate a layout configuration in terms of average impulse purchase revenue for a given shopping route scenario. If the customer

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makes purchases based on a shopping list, it is possible to generate alternative shopping routes for the customer. In the next section, we provide our solution methodology to find a layout that maximizes average impulse purchase revenue when the customer’s shopping list information is available.

3.2 Solution Methodology Due to the characteristics of the problem, it is difficult to formulate a mathematical programming model that considers all aspects of grid layout design. On the other hand, grid unit based layout design allows us to use meta-heuristic search algorithms to evaluate different layout configurations. Assuming we have information on how the customer selects his shopping route in the store for a given shopping list, we can compare different layout designs based on their average impulse purchase revenue. Flow chart for a possible meta-heuristic search algorithm is given in Fig. 5. We use a simulated annealing algorithm to generate layouts for a given shopping list. The details of the algorithm are given in Sect. 4.1. At each iteration of the algorithm, there are four types of layout modifications to generate alternative

Fig. 5 Search algorithm for layout design

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layout configurations. A layout can be modified by swapping the locations of two product categories on the shelf rows, changing the displayed samples of product categories located on the end caps, changing the orientation of grid units or connecting/disconnecting shelf rows on neighboring grid units.

4 Customer’s Route Selection Strategy In general, it is difficult to model and analyze the path of a customer as he makes his way through the store. Different customers almost certainly employ different strategies in selecting their routes. Assuming that customers have the same familiarity with the store, some customers may choose to move from one item on the list to the next nearest item, for example, while others may internally “compute” an efficient overall path for all items on the list. What is clear, however, is that the route chosen by the customer through the store strongly influences the potential impulse purchase revenue obtained from a particular layout. Consequently, it is preferable to have a layout design that is robust in terms of impulse purchase revenue with respect to different route selection strategies. To study the effect of individual behavior on shopping revenue, we consider four different route selection scenarios. In reality, customers likely do not use any one given strategy exclusively, and often customers may miss items on the shopping list and have to retrace their steps. So while these strategies may be viewed as an approximation of reality, nevertheless, we believe that they do characterize important aspects of how customer routes through the store are chosen. In the first scenario, the customer always makes purchases by moving to the closest remaining shopping list item’s location in the store. We refer to this route selection strategy as the nearest neighbor strategy and denote it by the acronym NNH. In the second scenario, the customer minimizes his overall shopping tour length (for items on the shopping list). This strategy is denoted by MIN. In the third scenario, the customer is equally likely to use NNH and MIN strategies. We refer to this strategy as mixed strategy and denote it by MIX. Finally, in the last scenario, we try to capture uncertainty in the customer’s route selection process. We assume that after making a purchase, the customer can go to the location of any of the remaining shopping list items with some probability that is inversely proportional to the distance to that item from the customer’s current location. If an item is closer, the probability that the customer will purchase this item next is higher than if it is farther away. This distance-based route selection strategy is denoted by DIST. When the customer moves from one location to another one, we assume that he always uses the shortest path. To illustrate, Fig. 6 portrays two different route selection strategies on the sample layout of Fig. 4. In general, retail stores have more than one entrance/exit point with designated checkout areas. This is also reflected on the sample layout of Fig. 4 with two different entrance/exit points. The test data set for this study is chosen so as to classify the products in terms of low and high impulse purchase rates, so that we may investigate the relationship

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between impulse purchase rates and product locations in the final layout. There are 100 product categories with (pre-determined) shelf length requirements of 2, 4 or 6 units. The impulse purchase rate and revenue parameter pairs .(ip , rp ) for product categories (p) are distributed as (0.8, 20) for 18 categories, (0.1, 20) for 25 categories, (0.1, 10) for 25 categories and (0.8, 10) for 25 categories. The remaining 7 product categories are reserved for shopping list items. Assuming that a shopping list item cannot be purchased impulsively, the revenue and impulse purchase rates for these product categories are set to zero. Although in reality shopping list contents are quite variable from customer to customer, here we use the deterministic shopping list approach by defining a set of product categories with high demands that tend to be common to many customer shopping lists. As mentioned earlier, in retailing literature such high demand product categories are referred to as power products (e.g., bread, milk, etc.), and they are used to partially control the customer traffic in the store. The store layout consists of 21 grid units (.3 × 7). Each grid unit is a .10 × 10 units square and contains two shelf rows. The width and length of a shelf row is 3 and 8 units, respectively. The depth of an end cap is 1 unit. In visibility factor calculations,

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we take the reference aisle length (L) as 26 units, which is the maximum possible vertical shelf row length. Customers can enter the store using any of the two doors located on the same side with equal probability and after completing all the purchases, leave the store using the closest exit. Our computational results are presented in the next section.

4.1 Computational Results and Discussion In the simulated annealing algorithm initial and final temperatures are set to 1 × 10 and .1 × 10−3 , respectively, based on our preliminary experimentation. The cooling rate is taken as 0.7 and at each temperature, 20 iterations are made. The initial layout is created randomly, so in the initial layout, grid unit orientations, connected/disconnected shelf rows, and the product category allocations to shelves are determined randomly. One iteration consists of 100 pair wise product category exchanges on the shelf rows, 20 product category changes on the end caps, one grid unit orientation change and one connected/disconnected shelf row modification. During pairwise product category shelf space location changes, the pair of product categories is chosen randomly. If there is enough shelf space for the pairwise product category exchange, then the layout change is allowed. During search, a new layout is accepted as the best layout if its objective value (.OVnew ) calculated by the Eq. (2) is greater than the objective value (.OVold ) of the current best layout. Otherwise, the new layout is taken to be the best layout with probability .e(OVnew −OVold )/CT where CT is the current temperature. We consider seven different shopping list scenarios based on the number of items in the shopping list. For each shopping list scenario, we consider four different shopping route selection strategies, which are discussed in the previous section. For a fixed shopping list length and route selection strategy, the average result at the end of five computer runs is taken for the analysis. For NNH, MIN and MIX shopping route selection strategies, on average a computer run takes between 3–5 hours on a Pentium IV 2.8 Ghz computer with 512 MB RAM. When DIST shopping route selection strategy is used, the average impulse purchase revenue of a layout configuration is calculated by simulation. During simulation, 25 shopping routes are generated by using the distance based probability values. The final average impulse purchase revenue is taken as the overall average of the average impulse purchase revenues obtained from each of these 25 shopping routes. Due to the simulation process, a typical computer run for DIST strategy takes more than eight hours. Our computational results are summarized as graphical plots in Fig. 7 where each point shows the average result for the five layouts that are generated for a fixed shopping list and shopping route selection strategy scenario. The plots demonstrate that impulse purchase revenue and travel distance are positively correlated with shopping list length, as expected. As the size of the shopping list increases, the customer visits more of the store, increasing his chance of making impulse purchases. On the other hand, the magnitude of the increase

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in impulse purchase revenue and travel distance depends on the shopping route selection strategy of the customer. If the customer minimizes his shopping tour length, as in the case of MIN strategy, he sees fewer products and makes less impulse purchases. Yet, if the customer uses NNH or MIX strategies to determine his shopping tour, then he travels more compared to MIN strategy, sees more products, and thus spends more on impulse purchases. For the DIST strategy, it appears that the customer travels more on average, but makes fewer impulse purchases compared to other route selection strategies. This is most likely due to the large number of possible routes available to the customer. In the NNH, MIN, and MIX strategies, there are only a few route alternatives that are used for a given layout configuration and shopping list. By locating the product categories with high impulse purchase rates on the shelves along these possible shopping routes, the retailer can maximize the impulse purchase revenue by ensuring that most customers notice these products. For the DIST strategy, however, the customer chooses one of many alternate routes. Since the number of product categories with high impulse purchase rate is limited, the retailer has to allocate these product categories on the shelves along only some of the possible shopping routes. Consequently, this results in less impulse purchase revenue since more customers are likely to miss these products than compared to the case where there are only a few shopping routes alternatives. From this observation, we can conclude that to maximize impulse purchase revenue, the store design should keep the customer flow as dense as possible to ensure maximum customer exposure to the high impulse products. Another observation is related to the orientation of grid units relative to the shopping routes. In the final layouts, generally the end caps contain samples from the product categories with high impulse purchase rates and they are located along the shopping route of the customer. This is similar to the real life practice in grocery stores, where end caps are located along the main aisles the customers use most often. Although it is not considered in our model, retail stores also put extra information signs near end caps for the product categories located on that shelf row. This also has positive effect on impulse purchase revenue.

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Our results also demonstrate that a layout that is generated for a specific route selection strategy may not perform well when customers use a different route selection strategy. In Fig. 8, the average impulse purchase revenue amounts obtained from different route selection strategies on the layouts generated for NNH and MIN strategies are shown. The plots show that assuming the customer uses either NNH or MIN strategy in shopping route selection process may result in poor performance if the customers actually use a different route. In order to overcome this problem, more robust layouts can be generated using the MIX route strategy. In Fig. 9, average impulse purchase revenue and travel distance plots on the layouts generated for MIX strategy are given. The results show that the layouts generated for MIX route selection strategy maximize the average impulse purchase revenue over all of the customer route selection strategies. This result is due to the MIX strategy considering the possibility of other strategies (NNH and MIN) during the layout generation process. This has positive influence on the average impulse purchase revenue of all the strategies including DIST. We are also interested in the relationship between the size of the layout and different route selection strategies. In order to analyze this, we used two other data sets representing a small and a large layout, respectively. The small layout consists of 4 grid units (.2 × 2) and 50 products whereas the large layout consists of 45 grid units (.5 × 9) and 250 products. Similar to previous data set, products have shelf length requirements of random 2, 4, and 6 units. In small data set 4 products and in large data set 12 products are reserved for shopping list. For the rest of the products, the impulse purchase rates are set to 0.2 or 0.8, and similarly revenue parameters are set to 10 or 20 units. respectively. In large data set, one computer run could take up to 15 hours. For this reason, based on our previous experience, we only generate layouts using NNH and MIX strategies that seem to be the most favorable ones to generate layouts. Figures 10, 11, 12, 13, and 14 display our results for the small data set, and Figs. 15, 16, and 17 display our results for the large data set. Five computer runs are taken for each scenario and their averages are used in the plots. Average Impulse Purchase Revenue (MIN) 240 220 200 180 160 140 120 100 80 60

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As seen in Figs. 10, 11, 12, 13, and 14, when the store area is small, the deviation in impulse purchase revenue and travel distance with respect to different route selection strategies is low. This is expected, because in a small store there are not many route alternatives for the customer and different route selection strategies give

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the same route most of the time for a fixed shopping list. We can conclude that when the store area is not very large, given that the location of high demand items are chosen carefully, customer’s travel behavior does not greatly affect the impulse purchase revenue and travel distance. As the layout size gets larger, impulse purchase revenue and travel distance become more sensitive to the customer travel behavior. This is observed in Figs. 15, 16, and 17. Thus, as the store size gets larger, it is critical for the retailer to understand customer travel behavior in order to maximize impulse purchase revenue.

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5 Conclusion In this study, we analyze a layout design problem for retail stores. The unique characteristics of this problem, such as profit maximizing objective and customer decisions in shopping route selection, make it different than traditional layout design problems previously studied in IE/OR literature. Considering these differences, we first analyze the characteristics of retail facility layout design. Later, we focus on retail store layout design and present a layout design algorithm that is specific to retail stores based on one of the most common layout strategies. Our computational results show that there is a correlation between planned purchase decisions and impulse purchases. If the customer enters the store with a longer shopping list, then he browses the store more and makes more impulse purchases. At the same time, impulse purchases depend on how the customer selects his shopping route in the store. If the shopping route is longer, there is more chance the customer can make an impulse purchase. The uncertainty in customer behavior is not good for the retailer. If the retailer does not have full information about how customers select shopping routes in the store, this has negative effect on store’s impulse purchase revenue. Consequently, the retailer may wish to use a hybrid route representation, such as the MIX strategy, to generate layouts, since it provides a more robust solution in the face of variable customer routes. Additionally, we can conclude that it is to the retailer’s benefit to minimize the number of alternative shopping routes for the customers. At this point, different layout designs that have more control on customer traffic, such as serpentine layouts, provide an attractive research direction. As is common in practice, end-of-aisle locations provide a good opportunity to increase impulse purchase revenue of the store. The end caps at the end of shelf rows are used to display samples for product categories with high impulse purchase rates. This increases the visibility of these product categories and, consequently, the impulse purchase revenue.

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We are interested in a number of possible extensions that are applicable to this study. Alternative retail layout strategies other than grid layout design should be considered. In this study we did not consider the relationship between product categories. In reality, correlated product categories are located close to each other in the store. We believe our solution algorithm can be extended to consider these relationships by adding extra constraints during layout generation process. Related to uncertainty, although we assume that we have the shopping list information of the customers, this is just an aggregate estimate of high demand items. Probabilistic shopping list characterizations may represent good future research areas as well. Acknowledgments This research is partially funded by National Science Foundation Grant No. 0223488

References Bellenger, D., Robertson, D., & Hirschman, E. (1978). Impulse buying varies by product. Journal of Advertising Research, 18(6), 15–18. Berman, B., & Evans, J. (1989). Retail management– A strategic approach. New York: Macmillan. Borin, N., Farris, P., & Freeland, J. (1994). A model for determining retail product category assortment and shelf space allocation. Decision Sciences, 25(3), 359–384. Botsalı, A., & Peters, B. (2005). A network based layout design model for retail stores. In IERC Proceedings. Chen, Y., Hess, J., Wilcox, R., & Zhang, Z. (1999). Accounting profits versus marketing profits: A relevant metric for category management, Marketing Science, 18(3), 208–229. Chiang, J., & Wilcox, R. (1997). A cross-category analysis of shelf space allocation, product variety, and retail margins. Marketing Letters, 8(2), 183–191. Corstjens, M., & Doyle, P. (1981). A model for optimizing retail space allocations. Management Science, 27(7), 822–833. Desmet, P., & Renaudin, V. (1998). Estimation of product category sales responsiveness to allocated shelf space. International Journal of Research in Marketing, 15(5), 443–457. Dreze, X., Hoch, S., & Purk, M. (1994). Shelf management and space elasticity. Journal of Retailing, 70(4), 301–326. Farley, J. U., & Ring, L. W. (1966). A stochastic model of supermarket traffic flow. Operations Research, 14(4), 555–567. Flamand, T., Ghoniem, A., & Maddah, B. (2016). Promoting impulse buying by allocating retail shelf space to grouped products. Journal of the Operational Research Society, 67(7), 953–969. Floch, J. (1988). The contribution of a structural semiotics to the design of a hypermarket. International Journal of Research in Marketing, 4, 233–252. Ghoniem, A., Flamand, T., & Haouari, M. (2016a). Exact solution methods for a generalized assignment problem with location/allocation considerations, INFORMS Journal on Computing, 28(3), 589–602. Ghoniem, A., Flamand, T., & Haouari, M. (2016b). Optimization-based very large-scale neighborhood search for generalized assignment problems with location/allocation considerations. INFORMS Journal on Computing, 28(3), 575–588. Hui, S. K., Fader, P. S., & Bradlow, E. T. (2009). The traveling salesman goes shopping: The systematic deviations of grocery paths from TSP optimality. Marketing Science, 28(3), 566– 572. Iyer, E. (1989). Unplanned purchasing: Knowledge of shopping environment and time pressure. Journal of Retailing, 65(1), 41–57.

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Kollat, D., & Willet, R. (1967). Customer impulse purchasing behavior. Journal of Marketing Research, 4, 21–31. Larson, J. S., Bradlow, E. T., & Fader, P. S. (2005). An exploratory look at supermarket shopping paths, International Journal of Research in Marketing, 22, 395–414. McGoldrick, P. (1982). How unplanned are impulse purchases? Retail & Distribution Management, 10, 27–31. Nierop, E. V., Fok, D., & Franses, P. (2008). Interaction between shelf layout and marketing effectiveness and its impact on optimizing shelf arrangements. Marketing Science, 27(6), 1065– 1082. Ozgormus, E., & Smith, A. E. (2020). A data-driven approach to grocery store block layout. Computers & Industrial Engineering, 139, 1–12. Pak, O., Ferguson, M., Perdikaki, O., & Wu, S.-M. (2019). Optimizing sku selection for promotional display space at grocery retailers. Journal of Operations Management, 66(5), 501– 533. Park, C., Iyer, E., & Smith, D. (1989). The effects of situational factors on in store grocery shopping behavior: The role of store environment and time availability for shopping. Journal of Consumer Research, 15, 422–433. Phillips, H., & Bradshaw, R. (1993). How customers actually shop: Customer interaction with the point of sale. Journal of the Market Research Society, 35(1), 51–62. POPAI/duPont Studies. (1978). Marketing emphasis. In Product marketing, pp. 61–64. Urban, T. (1998). An inventory-theoretic approach to product assortment and shelf-space allocation. Journal of Retailing, 74(1), 15–35. Vrechopoulos, A. P., OKeefe, R. M., Doukidis, G. I., & Siomkos, G. J. (2004). Virtual store layout: An experimental comparison in the context of grocery retail. Journal of Retailing, 80, 13–22.

A Solver-Free Heuristic for Store-Wide Shelf Space Allocation Tulay Flamand, Ahmed Ghoniem, and Bacel Maddah

1 Introduction We investigate the problem of re-configuring shelf space in a grocery store, with the objective of maximizing the expected profit resulting from impulse buying in a customer’s shopping basket (Bellenger et al., 1978; Kollat and Willett, 1967; Phillips and Bradshaw, 1993; Piron, 1991). This is achieved by allocating space to grouped product categories that reflect an existing configuration or to newly formed groups of products that ought to be on the same shelf based on managerial directives, product affinity, and the possibility of cross-selling (Flamand et al., 2016). The store-wide shelf space optimization is conducted with the purpose of guiding instore traffic, improving overall product visibility, and maximizing impulse buying. Our work builds upon a recent in-store traffic model with ripple effects by Flamand et al. (2023), whereby traffic along a shelf depends endogenously on the allocation of product categories throughout the entire store and exogenously on the proximity of the shelf to store entrances or exits. Flamand et al. (2023) propose a mixed-integer nonlinear optimization model that is partially linearized and approxi-

T. Flamand Department of Economics and Business, Colorado School of Mines, Golden, CO, USA e-mail: [email protected] A. Ghoniem () Operations & Information Management Department, Isenberg School of Management, University of Massachusetts Amherst, Amherst, MA, USA e-mail: [email protected] B. Maddah Department of Industrial Engineering and Management, American University of Beirut, Beirut, Lebanon e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Ghoniem, B. Maddah (eds.), Retail Space Analytics, International Series in Operations Research & Management Science 339, https://doi.org/10.1007/978-3-031-27058-1_2

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mated in order to prescribe an improved store-wide shelf space configuration, based on pre-grouped product categories that serve as an input. In this chapter, we extend this work and develop a heuristic that is based on a variable neighborhood search and does not require the use of an optimization solver. Instead, through its neighborhood exploration, it employs a total enumeration of possible assignments for a group of product categories to a subset of shelves that can display them. We demonstrate in our computational study that this heuristic provides high-quality primal bounds from its generated feasible solutions. The classical shelf space allocation literature conventionally addresses the allocation of a set of products, typically a specific set of stock keeping units (SKUs), to a few shelves taken in isolation (see Bianchi-Aguiar et al., 2021 and Mou et al., 2017 for recent reviews, and Bianchi-Aguiar et al., 2016 for a practical perspective). This, however, ignores the effect of the physical store layout on traffic, as well as the indirect traffic effects stemming from other products allocated elsewhere in the store. Our work falls, in contrast, under the umbrella of the “store-wide” shelf space allocation problem addressed more recently in the literature. This trend acknowledges the interplay between the store layout, shelf space allocation, and the visibility of products to the shopper. Due to the computational challenge posed by such store-wide approaches, a substream of this literature has defined the popularity of a shelf solely as a function of its location in the store, e.g. its proximity to store entrances or exits (e.g. Botsali, 2007; Flamand et al., 2016, 2018). Although this simplification yielded encouraging preliminary results on the effectiveness of the store-wide allocation approach, it is only in the recent work by Flamand et al. (2023) that the traffic along a shelf was comprehensively modeled as a function of (1) its physical location in the layout and (2) a traffic component resulting from store-wide ripple effects due to shoppers passing by a shelf in order to reach products allocated elsewhere in the store. To avail of this novel predictive in-store traffic model with ripple effects, Flamand et al. (2023) developed a mixed-integer nonlinear program that seeks to prescribe better store configurations via a linearization and approximation methodology. The latter yields both dual and primal bounds on an objective function that maximizes the mean impulse buying profit in a shopping basket. Our work in this chapter complements these recent efforts by proposing a new heuristic approach that is based on a neighborhood search and yields high quality solutions, and thus good primal bounds, that can be empirically validated using the dual bounds from Flamand et al. (2023). The remainder of this chapter is organized as follows. Section 2 offers a brief review of the relevant literature. A formal problem statement is provided in Sect. 3, along with a cursory presentation of the mixed-integer nonlinear program for storewide space planning in Flamand et al. (2023). A neighborhood search heuristic approach is devised in Sect. 4. Section 5 presents our computational results based on a supermarket in Beirut (Lebanon), along with sensitivity analysis with regard to key parameters in the model and managerial insights. Section 6 concludes the chapter with a summary of our findings and directions for future research.

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2 Literature Review In the following, we present a brief review of the recent studies on the store-wide shelf space allocation problem. Ke and van Ryzin (2011) develop an algorithm that estimates the effect of a change in product locations (moves) on profitability, within a fashion retailing setting. The algorithm is used to identify the combination of moves that maximizes revenue. Botsali (2007) investigates decisions pertaining both to store-wide allocation of product categories and store layout. The traffic density of a shelf segment depends only on its location in the store. A simulated annealing heuristic, based on simulated shopping lists and customer behaviour, generates a layout and assigns products to shelves. Flamand et al. (2016) develop mathematical programming approach for store-wide allocation of product categories to shelves under a layout-based traffic density similar to Botsali (2007), and assume that products are divided into groups, where each group may be assigned to one shelf. The authors apply their approach to a supermarket in New England and report a significant improvement in impulse profit of 32% on average over the current practice. Flamand et al. (2018) extend the work of Flamand et al. (2016) to include assortment planning, in terms of which products categories to offer in the store. Moreover, Ben Abdelaziz et al. (2022) extend the same work to account for shopping convenience by developing an optimization model with two objectives, maximizing impulse profit and minimizing customer walking. Applying the model to a large supermarket in the Normandy area, improvements on both objectives are reported, with one Pareto solution yielding 82% increase in impulse profit and 11% decrease in walking over the current store practice. The most impactful extension of Flamand et al. (2016) is probably that by Flamand et al. (2023) where the model is enhanced to include the effect of product allocations across the store on traffic, which is referred to as “ripple effects.” For example, this traffic model can capture the increase in traffic in shelves in the middle of the store as a result of relocating a fast moving product from the front to the back of store. Such effects are captured with a predictive regression model calibrated on real consumer basket data collected from a supermarket in Beirut. This nonlinear regression traffic component adds to the model complexity. Flamand et al. (2023) adopt an involved bounding and linearization scheme to obtain near-optimal solutions in a reasonable time, and report an improvement of 65% in impulse profit over the current practice for the store in Beirut. This chapter offers a remedy to the computational overhead in Flamand et al. (2023) via a solver-free variable neighborhood search. Unlike the more elaborate approach in Flamand et al. (2023), the VNS approach does not allow the estimation of an optimality gap. However, the numerical results reported in this chapter are highly encouraging and indicate a near-optimal performance of the VNS heuristic. This chapter is motivated by successful implementation of VNS approaches in Ghoniem et al. (2016a) for solving multiple knapsack problems with the knapsacks having different attractiveness akin to multiple store shelves in this chapter (see also Ghoniem et al., 2016b).

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Other related recent works include Dorismond (2019) who develop two models for shelf space allocation and store layout, with the store traffic generated based on an elaborate Monte Carlo simulation. In the first model, departments are allocated to the floor plan and thereafter products are assigned to shelves in these departments. The second model is concerned with the allocation of impulse products to end caps and other promotion areas. Another relevant work on end cap allocations that avail of multiple store data is that of Pak et al. (2020). Of interest are the papers by Ostermeier et al. (2021) and Irion et al. (2011) which present models for jointly deciding on the amount of space to allocate to categories in a store and the SKUs in each category, assuming that the location of the categories in the store are fixed. Further, Ozgormus and Smith (2020) and Yapicioglu and Smith (2012) address store layout planning with two objectives related to profitability and proximity of commonly purchased products. These works specifically decide where to assign departments in a retail store floor plan using a Tabu Search. Finally, it is worth mentioning the promising work of Mowrey et al. (2018, 2019) and Guthrie and Parikh (2020) on estimating the visibility of products in a retail store based on the manner in which people shop and the configuration of the store shelves.

3 Problem Statement and Optimization Model In this section, we formally introduce our notation and the store-wide shelf space management problem with impulse buying maximization. Without loss of generality, and for illustrative purposes, our discussion is placed in the context of a grocery store and can be specialized for other brick-and-mortar retail businesses. We address a setting where a retailer seeks to optimize shelf space allocation in order to guide in-store traffic in a manner that maximizes the expected impulse buying under a given grocery store layout. We consider grid layouts, which are popular in grocery stores, that comprise internal and peripheral shelves, end caps, and a network of walkways defined by aisles, as depicted in Fig. 1. The store includes refrigerated, frozen, and ordinary shelves. Refrigerators are used for perishable product categories such as dairy products, juice, or egg. Freezers offer such product categories as frozen vegetables, ice cream, and pre-cooked meals, among others. Ordinary shelves exhibit all other product categories that can stay at ambient temperature, e.g., grains, pasta, flour, oil, etc.

3.1 Notation We define the following notation in Table 1, which is adopted from Flamand et al. (2023). Demand product is a resultant of both planned and impulse purchases. Products having high demand, also known as fast-movers (e.g., milk, bread, or eggs), usually

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Fig. 1 The grid store layout of B Supermarket, Beirut (Source: Flamand et al., 2023)

have low impulse purchase rates and are rather purchased on a regular basis as part of a customer’s shopping list. Such categories constitute a large portion of the total store demand volume: They exhibit stable demand patterns and drive traffic to their respective shelves. The allocation of such categories is therefore critical in directing traffic throughout the store. Products having lower demand, on the other hand, are referred to as slow-movers (e.g, cakes or cosmetics), a subset of which may have higher impulse rates (Kollat and Willett, 1967). Sales of slow-movers are more likely to benefit from changes in shelf space allocation, in comparison with fast-movers (Cox, 1970 and Curhan 1972) and should be made more visible to consumers in order to trigger impulse purchases. In contrast, the revenue resulting from planned or fast-moving products may be viewed as a “constant,” regardless of shelf-space allocation decisions. We, therefore, focus in this work on the profit due to impulse buying.

3.2 Optimization Model for Shelf Space Allocation In this section, we present an optimization model that involves the predictive traffic density construct and preprocessed values of .spb , i.e., an optimal space allocation of product p to shelf b by examining every feasible group-shelf combination, as in Flamand et al. (2023). The impulse buying maximization problem (IBMP) under a

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Table 1 Model notation Sets .B

Set of all store shelves. Set of all product categories. Set of all pre-formed groups of product categories. Set of product categories in group g, .∀g ∈ J . Subset of product groups in an input layout that can be feasibly allocated to shelf b, .∀b ∈ B. Set of shelves which are adjacent to shelf b, .∀b ∈ B.

.P .J .Zg .Gb .Rb

Parameters .Cb .λp .λw

and .up

.p .ρp .ip .kb

∈ [0, 1] ∈ [0, 1]

.dqr .mg .qbr

∈ {0, 1}

.α .β1 .β2

Decision variables ∈ {0, 1} .spb .b norm .∈ [0,1] .b .fqr .xbg

.Fqb .θb

Length (capacity) of shelf b, .∀b ∈ B. Basket sales volume of product p, .∀p ∈ P . Basket sales volume of a subset of products .w ⊆ P . Minimum and maximum shelf space requirement for product p, .∀p ∈ P . Unit profit margin for product p, .∀p ∈ P , estimated as the mean profit margins of SKUs in product category p. Impulse purchase rate of product p, .∀p ∈ P . The layout-based customer traffic density along shelf b, .∀b ∈ B. The shortest walking distance between shelf q and shelf r. The total number of people shopping for the product categories in group g, .∀g ∈ J . .qbr =1 if the shortest path between shelves b and r includes shelf q, .∀b ∈ B , q ∈ Rb , r ∈ B : r = b, q. Intercept of the beta regression model for the traffic density. Coefficient of the layout-based component in the beta regression model for the traffic density. Coefficient of the allocation based component in the beta regression model for the traffic density. if and only if group g is assigned to shelf b, .∀b ∈ B, g ∈ Gb . The shelf space allocated to product p along shelf b, .∀p ∈ P , b ∈ B. The allocation-based customer traffic density along shelf b, .∀b ∈ B. Normalized value of .b . The force (amount of attraction) exerted on shelf q by shelf r, .∀q, r ∈ B |q = r. The force on shelf b exerted through any adjacent shelf q, .∀b ∈ B, q ∈ Rb . The traffic density along shelf b, that is the likelihood of shelf b to be visited by a customer, .∀b ∈ B. .xbg =1

A Solver-Free Heuristic for Store-Wide Allocation

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given store layout is modeled as the following 0–1 mixed-integer nonlinear program: IBMP: Maximize

 

.

spb xbg θb. Cb

ρp ip

b∈B g∈Gb p∈Zg

subject to



xbg = 1,

(1a)

∀g ∈ Gb.

(1b)

∀b ∈ B.

(1c)

b∈B



xbg = 1,

g∈Gb

 g∈Gq

fbq =

mg xqg

Fbq =

∀b, q ∈ B|b = q.

,

dbq 

∀b ∈ B, q ∈ Rb.

bqr fbr ,

(1d) (1e)

r∈B|r=b&q



b =

Fqb ,

∀b ∈ B.

(1f)

q∈Rb

norm = b

b

q∈B

θb =

q

∀b ∈ B.

,

e(α+β1 kb +β2 b

norm )

1 + e(α+β1 kb +β2 b

norm )

,

(1g) ∀b ∈ B.

x binary, f, F, , norm , θb ≥ 0.

(1h) (1i)

The objective function (1a) represents the average impulse buying profit per customer. Constraint (1b) guarantees that each group is assigned to a single shelf, while Constraint (1c) ensures that each shelf can accommodate only one group. Constraints (1d)–(1f) compute a force exerted on a shelf, reflective of its expected traffic, as a function of direct traffic that is due to products allocated to it, and indirect traffic that passes through it to get to other shelves in the store. Constraint (1g) computes normalized values for these forces, which are used as an estimate for the allocation-based traffic component at each shelf. Constraint (1h) combines the location- and allocation-based traffic components using a regression model in Flamand et al. (2023). Constraint (1i) enforces logical binary and non-negativity restrictions on the decision variables.

4 Solver-Free Heuristic The proposed model (IBMP) introduced in Sect. 3.2 poses computational challenges due to its nonlinearity that precludes the use of solvers for mixed-integer (linear) problems, such as CPLEX and GUROBI. Further, KNITRO, a solver that can handle

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Fig. 2 Overall flow of the solver-free constructive heuristic algorithm

mixed-integer nonlinear models, could not solve even toy-sized instances of the proposed model. In this section, we develop a solver-free constructive heuristic methodology in order to tackle the computational challenge that the original nonlinear model poses. Specifically, we propose a variable neighborhood search (VNS) algorithm (Hansen et al., 2010) that decomposes the problem into smaller, manageable subsets of shelves. The key steps of the heuristic are summarized in Fig. 2 and involve the two following procedures: • Step 1 – Initialization: To initialize the heuristic, in the context of a real-life grocery store, the current store allocation may be used as an initial feasible solution. • Step 2 – Evaluation: In the given initial solution, products that are allocated to the same shelf form a “group.” This step seeks to improve the current best feasible solution by iteratively selecting pair of shelves and swap their allocated groups

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to evaluate whether there occurs an improvement in the overall solution or not. The details of Step 2 are as follows:

Step 2a: Shelves with their allocated content (i.e., group) are sorted in a nonincreasing order of their contribution to the overall objective value calculated by the nonlinear objective function (1a). Then, they are equally partitioned into two bins sorted in descending contribution order. If the total number of shelves is not divisible by two, one bin may have one more shelf than the other.

Step 2b: From each bin, one shelf is randomly extracted, forming a subset of two shelves and their associated allocated product groups. While selecting each pair, it should be possible to swap the contents of the two shelves based on their types (i.e., frozen, refrigerated, or ordinary) and capacities.

Step 2c: The product groups on the shelves which are selected in Step 2b, are swapped and the resulting contribution to the overall objective value is calculated. While swapping product groups, the optimal shelf space for each product category is determined using the space allocation algorithm in Flamand et al. (2023).

Step 2d: If the objective value improves, this change is adopted and the best feasible solution (i.e., incumbent) is updated; otherwise no change is enforced. If, at any iteration, the number of shelves not selected yet becomes smaller than two or if no further pairs can be formed due to size or capacity incompatibility, the algorithm checks if at least one of the stopping criteria is met. If a stopping criterion is met, the algorithm terminates; otherwise, Step 2 is repeated. The stopping criterion is met when all possible pairs are traversed a given number of times.

5 Application to a Supermarket in Beirut Our study is based on the same data in Flamand et al. (2023) which was obtained from “B Supermarket” in Beirut. B Supermarket has a grid layout and 34 shelves, as illustrated in Fig. 1, and offers 64 product categories that are classified into 21 groups, as summarized in Table 2. The traffic model was obtained using a beta regression model, where the dependent variable (i.e., overall traffic density of each shelf) was estimated using 40,000 customer receipts, with the assumption that customers minimize their walking while shopping, and two independent variables that relate to the two traffic components of location and product allocation of each shelf (Flamand et al., 2023). In order to determine the product groups to be used in the model, Flamand et al. (2023) first considered two rule-of-thumb tactics, namely, Fast Movers Spread and Fast Movers Back that could potentially increase the traffic density in the store and compared their resulting objective values with the current allocation. Specifically, Fast Movers Back, where the fast movers are allocated to the back of the store,

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Table 2 Groups and their associated categories in B Supermarket, Beirut (Source: Flamand et al., 2023) # 1

Group Alcohol

2 3 4 5 6 7 8 9 10

Bread Candy Canned&ready made food Cereals Chocolate Cigarettes Cold beverages Cookies Dairy

11 12

Deli Desserts

13 14 15 16 17

Meat Hot beverages Nuts&potato chips Pasta Powders

18 19 20 21

Sauces Sea food Vegetables Water

Product Arak (1), Beer (2), Champagne (3), Energy drinks (4), Vodka (5), Whiskey (6), Wine (7) Croissant (8), Bread (9), Sandwich (10), Bagel (11), Toast (12) Chewing Gum (13), Lollipop (14), Marshmallow (15), Candy (16) Canned vegetables (17), Ready made food (18) Hot cereals (19), Cold cereals (20) Chocolate (21), Chocolate chips (22) Cigarettes (23), Hooka products (24) Iced tea (25), Juice (26), Soda (27) Cookies (28), Biscuits (29), Kaak (30) Butter (31), Cheese (32), Eggs (33), Milk (34), Packed cheese (35), Specialty cheese (36) Canned meat (37), Sliced deli (38) Boxed desserts (39), Cakes (40), Ready made desserts (41), Dessert spreads (42) Packed meat (43), Unpacked meat (44) Coffee (45), Tea (46) Chips (47), Nuts (48), Popcorn (49) Pasta (50) Grain (51), Rice (52), Soup (53), Spice (54), Sugar-salt (55), Flour (56) Creams (57), Dips (58), Oil (59), Sweet sauce (60) Canned sea food (61), Frozen sea food (62) Vegetables (63) Water (64)

Table 3 Comparison of CPLEX vs. solver-free heuristic on the instances considering static location vs. variable location Configuration

Location

Allocation

Input allocation Configuration 1 Configuration 2 Optimized allocation

Static Variable Static Variable

(.p (.p (.p (.p

= up ) = up ) < up ), 10% < up ), 10%

Objective value($) CPLEX Solver-free heuristic 1.38 1.82 1.81 1.43 1.89 1.88

yielded the highest objective value (Flamand et al., 2023). Therefore, Flamand et al. (2023) considered the product groups in the Fast Movers Back tactic. Tables 3, 4, 5, and 6 compare the performance of CPLEX, solving the approximated and linearized IBMP (Flamand et al., 2023), with the performance of the

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Table 4 Comparison of CPLEX vs. solver-free heuristic on the instances considering the effect of the flexibility of min/max space requirements on the solution Configuration

Location

Allocation

Configuration 3 Configuration 4

Variable Variable

(.p < up ), 15% (.p < up ), 20%

Objective value($) CPLEX Solver-free heuristic 1.93 1.91 1.96 1.95

Table 5 Comparison of CPLEX vs. solver-free heuristic on the instances considering the effect of impulse purchase rates on the solution Configuration Configuration 5 Configuration 6 Configuration 7 Configuration 8

Objective value($) High-impulse products Low-impulse products CPLEX Solver-free heuristic .+5% .−5% 1.94 1.92 .+10% .−10% 1.99 1.98 .−5% .+5% 1.84 1.83 .−10% .+10% 1.78 1.77

Table 6 Comparison of CPLEX vs. solver-free heuristic on the instances considering the effect of profit margins on the solution Configuration Configuration 9 Configuration 10 Configuration 11 Configuration 12

Objective value($) High-impulse products Low-impulse products CPLEX Solver-free heuristic .+5% .−5% 1.95 1.95 .+10% .−10% 2.00 2.00 .−5% .+5% 1.84 1.83 .−10% .+10% 1.78 1.77

proposed solver-free heuristic. The instances provided in Tables 3, 4, 5, and 6 are those generated for the sensitivity analysis in Flamand et al. (2023). Table 3 compares the results for four instances. The Input Allocation represents the “Fast Movers Back” rule-of-thumb. The other three instances use the same product groups as given in Fast Movers Back. Configuration 1 investigates the problem, where (.p = up ) for every product p, and the decision is to determine the assignment of product groups to shelves (i.e., variable allocation) only. Configuration 2 examines the problem where the decision is made on the shelf space amount between .p and .up for each product p only, assuming that p stays on its currently assigned shelf (i.e., static location). The Optimized Allocation optimizes both decisions jointly. In addition, Table 4 compares the results on the instances generated to test the effect of employing different ranges on .p and .up values. Specifically, Configurations 3–4 in Table 4 consider the problem, where the .p and .up values are calculated by .±15%, .±20% of their shelf space amounts given in the Input Allocation, respectively. Tables 5 and 6 compare the results based on instances generated to test the effect of impulse purchase rates and profit margins on the solution quality. To

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generate Configurations 5–8 in Table 5, products are sorted according to the nondecreasing order of their impulse rates (.ip ). For Configurations 5–6, impulse rates of the first half are increased by 5 and 10% respectively, while those of the second half are decreased by 5 and 10% respectively. For Configurations 7–8, this process is reversed; that is, impulse rates of the first half are decreased by 5 and 10% respectively, while those of the second half are increased by 5 and 10% respectively. While generating Configurations 9–12 in Table 6, the same process is implemented to divide products into two halves based on their impulse purchase rates and to change their associated profit margins (.ρp ). The comparison of objective values with respect to the approximated model and solver-free heuristic in Tables 3, 4, 5, and 6 shows that the solver-free heuristic provides comparable (for some instances, the same) solutions. While CPLEX provides all the solutions in one CPU-hour time limit, the solver-free heuristic provides all reported solutions only in a few minutes. Therefore, the solver-free heuristic is proved to be useful in obtaining near-optimal solutions in manageable CPU times. In addition, results in Table 3 shows that Configuration 1 provides a comparable solution to the Optimized Allocation by significantly improving the solution of Input Allocation. In contrast, Configuration 2 does not yield much improvement over the Input Allocation. This implies that optimizing the allocation of product groups to shelves provides the significant portion of the overall improvement in the optimized solution as opposed to optimizing the shelf space amount of product categories on their assigned shelves. Results in Table 4 shows that the difference in solutions of Configurations 3 and 4 is modest that implies the model is not too sensitive to the changes in bounds of the shelf space amounts. Therefore, the approximated model as well as the solver-free heuristic is robust. Results in Tables 5-6 further testify to the robustness of the model, as changes in the impulse purchase rates and profit margins of products do not lead to significant changes in impulse profit. Overall, the sensitivity analysis shows that both the approximated model and the solver-free heuristic are robust. However, while the approximated model provides solutions in one CPU-hour time limit, the solver-free heuristic provides comparable, and in some cases, the same solutions only in a few minutes.

6 Conclusions and Directions for Future Research This chapter proposed a practical solution methodology for the problem of allocating grouped products to shelves throughout a store. This work builds upon recent developments by Flamand et al. (2023) where in-store traffic was estimated with a regression model and a prescriptive solution was obtained using a mixed-integer nonlinear formulation, which helped bracket the optimal objective value via a lower (primal) and an upper (dual) bound. Armed with the latter, we propose in this work

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an attractive solution methodology for practitioners based on a neighborhood search heuristic that does not necessitate the use of a commercial solver. The heuristic is both efficient and effective: It produces within a few seconds near-optimal solutions (which are validated using the dual bound in Flamand et al., 2023). This methodology can be applied to an entire store or partially to a few select shelves. It can be also judiciously adapted in the future to large department stores and home improvement stores in Western countries. This methodology could be also adjusted to investigate how store-wide shelf space allocation can be optimized to balance the expected impulse buying and the shopper’s convenience (Ben Abdelaziz et al., 2022). Due to the computational efficacy of the proposed methodology, it lends itself to the inclusion of additional features in the problem scope, such as deciding about the physical shelf orientation (Mowrey et al., 2018) and store layout (Botsali, 2007).

References Bellenger, D. N., Robertson, D. H., & Hirschman, E. C. (1978). Impulse buying varies by product. Journal of Advertising Research, 18(6), 15–18. Ben Abdelaziz, F., Maddah, B., Flamand, T., & Azar, J. (2022). Store-wide space planning balancing impulse and convenience. Working paper, American University of Beirut. Bianchi-Aguiar, T., Hübner, A., Carravilla, M. A., & Oliveira, J. F. (2021). Retail shelf space planning problems: A comprehensive review and classification framework. European Journal of Operational Research, 289, 1–16. Bianchi-Aguiar, T., Silva, E., Guimarães, L., Carravilla, M. A., Oliveira, J. F., Amaral, J. G., Liz, J., & Lapela, S. (2016). Using analytics to enhance a food retailer’s shelf-space management. Interfaces, 46, 424–444. Botsali, A. R. (2007). Retail facility layout design. PhD Dissertation, Texas A&M University. Cox, K. K. (1970). The effect of shelf space upon sales of branded products. Journal of Marketing Research, 7, 55–58. Dorismond, J. (2019). Data-driven models for promoting impulse items in supermarkets. PhD dissertation, State University of New York at Buffalo. Flamand, T., Ghoniem, A., Haouari, M., & Maddah, B. (2018). Integrated assortment planning and store-wide shelf space allocation: An optimization-based approach. Omega, 81, 134–149. Flamand, T., Ghoniem, A., & Maddah, B. (2016). Promoting impulse buying by allocating retail shelf space to grouped products. Journal of the Operational Research Society, 67(7), 953–969. Flamand, T., Ghoniem, A., & Maddah, B. (2023). Store-wide shelf space allocation with ripple effects driving traffic. Operations Research. https://doi.org/10.1287/opre.2023.2437 Ghoniem, A., Flamand, T., & Haouari, M. (2016a). Optimization-based very large-scale neighborhood search for generalized assignment problems with location/allocation considerations. INFORMS Journal on Computing, 28(3), 575–588. Ghoniem, A., Flamand, T., & Haouari, M. (2016b). Exact solution methods for a generalized assignment problem with location/allocation considerations. INFORMS Journal on Computing, 28(3), 589–602. Guthrie, B., & Parikh, P. J. (2020). The Rack orientation and curvature problem for retailers. IISE Transactions, 52(10), 1081–1097. Hansen, P., Mladenovic, N., & Perez, J. A. M. (2010). Variable neighbourhood search: Methods and applications. Annals of Operations Research, 175, 367–407.

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Irion, J., Lu, J.-C., Al-Khayyal, F., & Tsao, Y. C. (2011). A hierarchical decomposition approach to retail shelf space management and assortment decisions. Journal of the Operational Research Society, 62, 1861–1870. Ke, W., & Van Ryzin, G. (2011). Optimization of product placement in a retail environment. Manuscript, Graduate School of Business, Columbia University. Kollat, D. T., & Willett, R. P. (1967). Customer impulse purchasing behaviour. Journal of Marketing Research, 4, 21–31. Mou, S., Robb, D. J., & DeHoratius, N. (2017). Retail store operations: Literature review and research directions. European Journal of Operational Research, 265, 399–422. https://doi.org/ 10.1016/j.ejor.2017.07.003 Mowrey, C. H., Parikh, P. J., & Gue, K. R. (2019). The impact of rack layout on visual experience in a retail store. INFOR, 57(1), 75–98. Mowrey, C. H., Parikh, P. J., & Gue, K. R. (2018b). A model to optimize rack layout in a retail store. European Journal of Operational Research, 271(3), 1100–1112. Ostermeier, M., Düsterhöft, T., & Hübner, A. (2021). A model and solution approach for store-wide shelf space allocation. Omega, 102, 102425. Ozgormus, E., & Smith, A. E. (2020). A data-driven approach to grocery store block layout. Computers & Industrial Engineering, 139, 105562. Pak, O., Ferguson, M., Perdikaki, O., & Wu, S. M. (2020). Optimizing stock-keeping unit selection for promotional display space at grocery retailers. Journal of Operations Management, 66, 501–533. Phillips, H., & Bradshaw, R. (1993). How customers actually shop: Customer interaction with the point of sale. Journal of the Market Research Society, 35(1), 51–62. Piron, F. (1991). Defining impulse purchasing. In R. H. Holman & M. R. Solomon (Eds.), Advances in consumer research (vol. 18, pp. 509–514). Provo, UT: Association for Consumer Research. Yapicioglu, H., & Smith, A. E. (2012). Retail space design considering revenue and adjacencies using a racetrack aisle network. IIE Transactions, 44, 446–458.

In-Store Traffic Density Estimation Jimmy Azar and Hoda Daou

1 Introduction and Motivation Understanding and analyzing customers’ shopping behavior is an important topic for both industry practitioners and academic researchers. Identifying the drivers and understanding the dynamics behind consumers’ behavior inside stores can help draw important managerial insights and therefore help in critical decisions regarding store layout and product placement. Several previous studies have explored shopping path behavior; most involve path records of shoppers’ movements in a spatial configuration. Such data captures and describes interactions of consumers with their environment and thus is very valuable in marketing research and can give deeper insights into consumers’ motivations and what influences their purchasing decisions. Being able to estimate in-store traffic density is key to understanding and implementing optimal product allocation that can increase profit or promote impulse buying while enhancing the shopping experience. The importance of estimating traffic or the transition flows from one store area to another was realized as early as in the work of Farley and Ring (1966). To obtain such estimates in Farley and Ring (1966), research assistants were hired to secretly follow customers around a store to mark their movements and transitions from one area to another. In today’s terms, this would be the equivalent of radio frequency identification (RFID) tagging of shopping carts. This active approach of identifying individual shopping paths is one way traffic density may be estimated in a store, however it requires prior setup

J. Azar Department of Industrial Engineering and Management, American University of Beirut, Beirut, Lebanon H. Daou () Suliman S. Olayan School of Business, American University of Beirut, Beirut, Lebanon e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Ghoniem, B. Maddah (eds.), Retail Space Analytics, International Series in Operations Research & Management Science 339, https://doi.org/10.1007/978-3-031-27058-1_3

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and planning and is therefore not available in retrospective studies. On the other hand, e-receipts are always available and provide vital information for a store. Can traffic density be estimated directly from e-receipt information? In this chapter, we address the problem of predicting traffic density in a store layout which is an important component in models that aim to optimize profit or impulse purchase rates such as in Flamand et al. (2016, 2023). There are several methodologies in the literature that attempt to study or model traffic flow. Some methods use radio frequency identification for obtaining shopping paths (Larson et al., 2005); others solve a traveling salesman problem to map the shopping paths (Flamand et al., 2023; Hui et al., 2009), or model the transition probabilities from one area of the store to another (Farley and Ring, 1966). These methods do not come without assumptions or difficulties in estimation. Predicting store traffic based on RFID data requires actively tagging shopping carts and careful processing of the data to avoid anomalous signals and the inclusion of abandoned carts for instance. Using e-receipts of basket data and regression analysis as in Flamand et al. (2023) simplifies the process. Following the work of Flamand et al. (2023), in this chapter we propose a simpler way to estimate traffic densities based on real data acquired from a supermarket in Lebanon and without the need to reconstruct shopping paths. We also propose alternate regression models, such as support vector regression, regression trees, kernel regression, and Gaussian processes to predict traffic density. Hyperparameter tuning was carried out for each method, and test results show that regression tree and kernel regression methods outperform SVM and Gaussian processes. Note that some models may take into account deliberate reallocation of products from time to time to force customers into exploring other areas of the store which they may not have been accustomed to visiting. This chapter has a focus on estimating traffic density based on a given shelf allocation so that the estimation may be applied wherever needed in models that aim for example to maximize profit or modify shopping paths. Future work (as discussed in Sect. 6) could embed the traffic density estimation approach, which is the topic of this chapter, in an optimization framework that induces longer walking paths for customers. This chapter is structured as follows. Section 2 presents relevant work in the literature. The dataset (store layout and client basket data) and methodology used are described in Sects. 3 and 4 respectively. Section 5 presents a summary of the results and analysis. We conclude this chapter in Sect. 6 with a summary of our findings and suggestions for future work.

2 Literature Review Early work by Farley and Ring (1966) partitions the store layout into zones or areas, and proceeds to model traffic using a matrix of area-to-area transition probabilities. Actual traffic flows were measured through observation by noting the transition of customers from one area to another as they shopped. This provided the ‘ground-

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truth’ values for the dependent variable, namely the transition probabilities. Inspired by Newtonian mechanics of gravitation and force, the independent (predictor) variables in Farley and Ring (1966) consisted of (1) .φij : the normalized force exerted on area i through area j , (2) .ξij : a measure of perimeter effect of a consumer’s early shopping path and (3) .γij : a measure of the effect of angular rotation on traffic. A linear model was then fitted using the least squares method and was used to predict the transition probabilities .Pij given .φij , .ξij , and .γij . A model was constructed separately on each of five different stores, and the coefficient of determination, .R 2 , ranged between 0.61 and 0.74. In our aim to estimate traffic density associated with a given shelf location and corresponding product allocation, there are a number of obstacles when following a similar approach based on transition probabilities as in Farley and Ring (1966). The resulting stochastic matrix would assume that customers ‘cycle’ indefinitely from one area to another in the store if there are no absorbing states added. The addition of an absorbing state would make the stochastic matrix not regular and therefore stationary probabilities (representing traffic density) would not exist. Using RFID technology and by tracking the exact movements of shoppers, Sorenson (2003) captured and analyzed more than 200,000 shopping paths (Sorenson, 2003). A sample of this dataset was later used in a clustering algorithm for the purpose of grouping similar behavior and identifying relevant patterns of consumers’ shopping behavior (Larson et al., 2005). Path Tracker dataset was also used by Hui et al. (2009) to study three different situational factors that could potentially impact consumers’ in-store shopping decisions: (1) time pressure, (2) shopping basket and licensing , and (3) social pressure from other shoppers. Hui et al. (2009) integrate these factors into one model and use empirical analysis to support and analyze their impact on shoppers’ behavior using an individuallevel probability model. A grocery trip is segmented into a series of visit, shop, and buy decisions, each of which is driven by latent attractions of categories and zones. Through hypothesis testing, they examine how the three situational factors (perceived time pressure, licensing, and social influence of other shoppers) influence each of these decisions. Such integrated probability models account for all aspects (visit, shop, and buy) of a grocery shopping path and allows testing different behavioral theories that lead to important managerial implications. Another set of shopping paths was also captured by equipping shopping carts with RFID devices; done as part of the ‘Future Japanese Store Project’ (Yada, 2011). Product section visiting patterns were expressed by character strings, and these were then used to build a classification model that would capture key insights of the behavioral patterns of shoppers (Yada et al., 2007). By focusing on product sections Yada et al. (2007), have to a priori choose the size and layout of the different aerial units used in the analysis. However, ideally one would want to explore different resolutions and let the model automatically determine the layout of these units from the data itself. Hui and Bradlow (2012) use a bayesian multi-resolution approach where different configurations of aerial units represent different “models” under consideration. The purpose of the approach is to select the configuration (“model”) that has the highest posterior model probability across all possible configurations,

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given the sales data and the shopping paths captured in store and derive different empirical insights obtained under different resolutions. Yet another approach to obtain traffic densities and flow paths relying on RFID tags attached to shopping carts was presented in Larson et al. (2005). The paths were clustered using k-medoids clustering to identify ‘canonical path types’ typical of shopping travel paths. With RFID tracking, the true traffic densities can be obtained, however in the absence of such tracking, other estimation methods need to be adopted, some of which may rely upon the e-receipts of consumers. In estimating traffic densities in Flamand et al. (2023), for each shopping basket, a traveling salesman problem (TSP) was solved to determine the shopping path taken such that it passes through the locations of all bought items. This assumes that customers shop in a manner that minimizes their walking distance, although this is generally not the case in reality. Some paths will deviate significantly from the TSP path especially those corresponding to larger shopping baskets (Hui et al., 2009), whereas smaller deviations can be associated with purchasing from among frequently bought products (Hui et al., 2009). Thus with data consisting of 40,000 e-receipts, just as many TSPs were solved in Flamand et al. (2023), and then the traffic density, .θb , at any given shelf b was obtained by dividing the number of times Shelf b was visited by the total number of paths. The independent variables used in Flamand et al. (2023) were .kb and .φb , where .kb = 1/ min(di , do ), i.e. the reciprocal of the minimum distance from Shelf b to either the entrance or exit, and .φb was based on the same analysis of forces and shelf interactions implemented in Farley and Ring (1966). Finally a logistic model was fit to the data. The resulting sigmoidal shape which the model assumes is restrictive and may be improved by considering more flexible non-parametric approaches to regression. Going beyond estimation of traffic density, previous work aims at increasing purchase rates and boosting profits by focusing on store layout designs (Ozgormus and Smith, 2020), increasing the visibility of impulse items (Dorismond, 2019), and promotions and their display space (Pak et al., 2019). Recent work proposes an effective analytical method to design grocery store layouts considering impulse purchase rates, departmental adjacency, and space constraints (Ozgormus and Smith, 2020). The aim is to solve grocery store block layout problems taking into account revenue generation and adjacency of departments. Ozgormus and Smith (2020) use data from a major grocery store retailer in Turkey to evaluate the market basket data using association rule mining for the purpose of establishing relationships among departments, and identify opportunities for increasing impulse purchases. Dorismond (2019) focuses on impulse items and proposes a model to increase visibility of impulse items and their placement in promotional areas. Using optimization tools and supermarket transaction data, the optimal layout of the entire store is modeled by taking into account customers purchase patterns in terms of ‘must have’ and ‘impulse products’. Pak et al. (2019) suggest a decision support tool to optimally select products for promotional display space. Testing on real transactions from grocery store sales, the proposed optimization tool can result in a significant lift in sales (with an average of 27% across all SKUs) .

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3 Shopping Basket Data and Store Description Our data was collected from a supermarket in Lebanon. The store consists of three floors (ground floor, first floor, and second floor) shown in Fig. 1 with 55 shelves in total spread as follows: 10 shelves on the ground floor, 19 shelves on the first floor, and 26 shelves on the second floor. The checkout area is located on the ground floor to the center left in the layout. The floors are connected by main stairs and an elevator; the distance between two consecutive floors is 4.59 m if stairs are used, while the corresponding vertical distance is 3.25 m in the case of using the elevator. Our data in particular consists of 51,379 e-receipts collected from the supermarket over a period of 2 months. Purchased items were grouped into 66 product categories, denoted as products hereafter. Also extracted from the data is the total sales volume .SVt over the mentioned period, in addition to the sales volume .SVp for each product. Note that volume here is measured in number of shopping baskets, not number of product units. The product visibility factor .vp ∈ [0, 1] was also measured for each of the 66 products as the fraction of shelf space allocated to a product out of the total shelf capacity where the product is located. Moreover, a survey involving 108 consumers was conducted to estimate the impulse purchase rate, .ip ∈ [0, 1], for the different product categories, .p ∈ P. The consumers were

Fig. 1 Store layout: ground floor, first floor, and second floor

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Fig. 1 (continued)

asked at checkout which items they had bought in an unplanned way. The data was then used to estimate .ip as the fraction of unplanned purchases for p over the total number of purchases for Product p.

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4 Methodology In this section, we present our research methodology. In particular, Sect. 4.1 explains how predictor variables are derived for fitting a regression model. Section 4.2 presents the method for measuring all shelf-to-shelf distances which are required for computing the influence (in the sense of traffic density) of each visited shelf on all other shelves in the store. The source of this influence is based on historical sales volumes accounted for on a given shelf, which is then propagated to all other shelves from most neighboring to the farthest ones with decreasing effect, based on distance. Section 4.3 describes the regression models built and how their hyperparameters are tuned.

4.1 Traffic Density: Predictor Variables We assume the traffic density at a given shelf location is a function of (1) location with respect to the entrance/exit reference of the store and (2) the allocated products designated to that shelf. Note that the high volume of a category captain would naturally induce high traffic in the neigborhood of the shelf where it is allocated. Areas close to the entrance or exit of the store often have relatively high traffic densities (Hui et al., 2009). Our first independent variable is therefore modeled using the relation: .kq = 1/ min(di , do ) as in Flamand et al. (2023), where .di and .do are the shortest feasible (Manhattan) distances from location q to the entrance and exit. The dependence of traffic densities on product allocation can be viewed in terms of sales volume at a given shelf as well as the sales volume at neighboring shelves. This latter can lead to high traffic densities at a given shelf due to its neighbors having large transaction or sales volume. Thus our second independent variable will be the estimated transaction volume, .λq , around the location of interest, q. The transaction volume, .tb , at any given shelf, b, containing a list  of products, .cb , is the sum of all transaction volumes of the products, i.e. .tb = p∈cb SVp . To account for the impact of neighboring shelves, the variable .λq is estimated by summing transaction volumes occurring  at all shelves .b ∈ B, each normalized by the distance to location q, hence .λq = b∈B tb /(dbq + 1), that is, the traffic influence around any shelf decays by the function .f (x) = 1/(x + 1), .x ∈ [0, ∞]. Therefore, adjacent shelves that are close to q will have a larger weight as opposed to shelves that are relatively distant from q. Shelf-to-shelf distances are computed as described in the next section. Estimating in-store traffic density using RFID tags can be regarded as an active approach to collecting data. Without RFID tracking, one can use a more passive approach to estimating density, for instance by reconstructing shopping paths from consumer receipt information such as in Flamand et al. (2023). In this work, we propose another approach which is also based on receipt information, but which does not require reconstructing shopping paths around the store. To obtain the traffic

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densities at any given shelf q, we make use of  a model proposed in Flamand et al. (2016) for defining impulse-buying profit as: . p ρp ip vp , where .ρp is the profit margin for product p, .ip is the impulse purchase rate for product p, and .vp is the visibility factor for product p within the shelf it is located on. In our case, we wish to find the traffic densities and we have the impulse purchase rates .ip from survey data, in addition to the sales volume of product p. Therefore, we use the following model to find the density, .Dpq , at the shelf q where product p is located: .SVp = SVt × Dpq × vp × ip . The variable .SVt is the total sales volume, .SVp is the sales volume for Product p, .vp is the visibility factor computed as the shelf space allocated to p divided by the total shelf space of q. Intuitively, the equation describes that .SVp is obtained by ‘filtering’ .SVt through the density, .Dpq , the visibility, .vp , and the impulse rate, .ip . In other words, a consumer will buy product p impulsively if the customer passes by the shelf q, is able to view the product, and has the tendency to buy that product impulsively. For multiple products the  on Shelf q, we compute  traffic density at q as a weighted average: .Dq = ( p∈cb vp × Dpq )/( p∈cb vp ). Obtaining .Dq in this manner gives us an estimate of the ground-truth traffic density values, i.e. the dependent variable.

4.2 Shelf-to-Shelf Distance Matrix The store layout consists of three different floors containing 55 shelves in total (see Fig. 1). The corresponding maps including shelf boundaries were set up in digital format to allow for measuring distances between shelves as well as distances between any shelf and the main store entrance or exit.  The distance .dbb between shelves .b ∈ B and .b ∈ B is obtained by a sequential region-growing algorithm that avoids intermediate obstacles between source and destination locations (i.e. shelves). To the best of our knowledge, the proposed method is a novel approach to solving this type of problem. The distance is measured between the centers of the two shelves by starting at a pixel representing one shelf center and then applying sequential dilations using a cross (‘+’) structuring element while subtracting the binary image of the store layout after each dilation step in order to avoid growing regions through obstacles along the path. This way the grown region is forced to pass around obstacles rather than pass through them. Once the grown region reaches the pixel representing the other shelf center, the procedure halts and returns the number of steps (dilations) it required to reach the destination. This number (in pixels) is exactly the Manhattan (city block) or .L1 norm distance between the two shelf   centers. With 55 shelves in the store, the total number of distances computed is . 55 2 = 1485. Note that for the purpose of speeding up computations without affecting end results, the image used for the store layout was reduced in size to 1/5 of the original size before measuring distances between shelves. The resulting distances (in pixels) were finally restored to scale through multiplication by a factor of 5, in addition to being multiplied by the corresponding map-scale factor to transform them into a measure of meters.

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Each shelf was numbered from 1 to 55, and proceeding sequentially, distances between each shelf and those shelves with higher number were computed. For shelves .b1 and .b2 that lie on different floors I and F (respectively), first the distance from the shelf .b1 to the main stairs or elevator on I (whichever is smaller) is computed. The smallest distance is chosen based on the assumption that clients would choose the closest path to their destination, which is on a different floor. Then, this is added to the distance required to traverse the stairs or elevator to arrive at F , in addition to the distance from the stairs at F to .b2 or from the elevator at F to .b2 , whichever of stairs or elevator was selected in the first step on floor I . The distance corresponding to taking the stairs between two consecutive floors is 4.59 m while the vertical distance corresponding to taking the elevator is 3.25 m. A subset of the final distance matrix which includes the first 20 shelves is shown as a heatmap in Fig. 2.

0

5.67 4.25 11.22 14.49 16.56 13.4 15.58 13.51 11.22 23.01 24.97 23.99 23.01 22.03 22.46 23.34 24.1 23.77 24.75

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10.9 13.29 13.4 10.24 12.42 10.35 6.97 19.85 21.81 20.83 19.85 18.87 19.3 20.18 20.94 20.61 21.59 0

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24.97 25.41 21.81 29.44 28.31 26.89 23.73 23.95 21.88 23.08 3.49 23.99 24.43 20.83 28.46 27.33 25.91 22.75 22.97 20.9 22.1

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Fig. 2 Heatmap of the shelf-to-shelf distance matrix (in meters) for shelves 1–20

shelf 13 shelf 14 shelf 15 shelf 16 shelf 17 shelf 18

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In measuring predictor variable, .kq , the same procedure is applied except that distances to shelf q from both the main entrance and exit on the ground floor are computed, and whichever distance is smaller is selected. .kq is then the reciprocal of this minimal distance.

4.3 Regression Model We normalize our data and build a regression model to predict .Dq given predictor variables .kq and .λq using several methods which we compare to each other.

4.3.1

Support Vector Regression

Support vector regression (SVR) Smola and Schölkopf (2004) is characterized by the use of kernels to map the data into an often higher-dimensional space before solving the machine learning task as a convex optimization problem. The regression problem is a generalization of the classification problem (SVM), in which the model returns a continuous-valued output, as opposed to an output from a finite set. To build the optimal SVR model, the hyperparameters that define the model depending on the training dataset need to be optimized. Hyperparameters include: 1. The kernel function commonly used kernel functions are: (a) Linear, (b) Polynomial, (c) Sigmoid and (d) Radial Basis (RBF). SVR maps the data, using predefined kernel functions, into a new often higher-dimensional space, where a linear model would be able to fit well to the data. Thus, the selection and settings of the kernel function are crucial for optimality. 2. The maximum allowed error .: a parameter that ensures the error (.i ) is less than a certain value. This is known as the principle of maximal margin. 3. The cost parameter c: a tuneable parameter used to avoid over-fitting. This parameter gives more weight to either minimizing the flatness, or the error, for the optimization problem. For example, a larger c gives more weight to minimizing the error, forcing the algorithm to fit the input data more accurately and therefore potentially overfit. 4. The gamma parameter .γ : the parameter of a Gaussian kernel that controls the shape of the ‘peaks’ of the Gaussian function. Setting .γ is required only for the Gaussian radial basis function kernel. An R implementation of SVM is used which is based on Chang and Lin (2011). We divide our regression dataset randomly into a training set and test set with this latter comprising 20% of the total dataset size. In order to tune the hyperparameters and choose optimal values, we use the training set to perform five-fold crossvalidation using two different kernel functions: linear and radial. The following selected ranges are used: .γ ∈ [0.1, 5], . ∈ [0, 1], and .c ∈ [1, 1000]. The best resulting model is selected and used to predict the values of the test set. The

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Table 1 Error (.± standard deviation) SVM Regression tree Kernel regression Gaussian process Beta regression

RMSE 0.2233 (.±0.1026) 0.1936 (.±0.0859) 0.2136 (.±0.0683) 0.2366 (.±0.0826) 0.2483 (.±0.0787)

MAE 0.1050 (.±0.0295) 0.0613 (.±0.0289) 0.0840 (.±0.0382) 0.0948 (.±0.0429) 0.1713 (.±0.0366)

SMAPE 0.5241 (.±0.0760) 0.4475 (.±0.0844) 0.4829 (.±0.0867) 0.5784 (.±0.0679) 0.5768 (.±0.0819)

Bold is for the optimal value

entire process is repeated 20 times, and the averages of the following test errors are reported (along with the standard deviation) in Table 1: root mean square error (RMSE), median absolute error (MAE), and symmetric mean absolute percentage error (SMAPE; as fraction). Figure 3a shows the predicted response surface along with the original data points.

4.3.2

Regression Tree

A regression tree (RT) Breiman et al. (1984) is a non-parametric decision tree where the target variable can take continuous values. RT uses a binary recursive partitioning scheme by splitting among all the possible splits at each node repeatedly, starting with the root node. In regression modeling problems, split points are chosen based on minimizing an error metric (such as sum of squared error) over the training samples. To build the optimal RT using the training dataset, certain hyperparameters need to be tuned. Choosing proper values help set the complexity of the tree while controlling overfitting and thus allowing the trained model to generalize well on test set. The hyperparameters used in our training are the following: 1. The maximum depth of the tree ‘maxdepth’: parameter to set the maximum allowed depth of the tree. Depth is the length of the longest path from a root node to a leaf node. In general, the deeper the tree is allowed to grow, the more complex the model. 2. The cost complexity parameter ‘cp’: The complexity parameter is the minimum improvement in the model needed at each node. If any split at the node does not increase the overall .R 2 of the model by at least cp, then that split is considered not worth pursuing. 3. The minimum split value ‘minsplit’: the minimum number of samples required to attempt a split, otherwise a terminal node is created. The same procedure is adopted for testing and hyperparameter tuning as in SVR. The regression tree used in this case is an R implementation of CART (Breiman et al., 1984), and the range of hyperparameters were set as follows: .maxdepth ∈ [3, 11], .cp ∈ [0.01, 0.09], and .minsplit ∈ [2, 10]. Average test errors can be found in Table 1 and the response surface in Fig. 3b.

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Fig. 3 Response surface: (a) SVR (.kernel = radial, .c = 1, .γ = 0.357, . = 0) (b) Regression tree (.cp = 0.03, .maxdepth = 11, .minsplit = 2) (c) kernel regression (.h = 0.0214) (d) Gaussian process (.l = 0.0725, .σn = 0.407) (e) Beta regression (logit link function)

4.3.3

Kernel Regression

We also attempt kernel smoothing for regression which was first introduced by Nadaraya (1964) and Watson (1964). The kernel smoothed response at query point q is given by:  xi − q yi iK h   . y (q) =  xi − q iK h 



The weights are given by the kernel function .K(r) = exp(−r 2 ); .xi are training inputs and .yi are the training targets. The method has a single hyperparameter h called the bandwidth which defines the standard deviation of the Gaussian kernel we choose to use. Large values of h can result in underfitting (high bias; low variance), whereas small values of h can result in overfitting (low bias; high variance). For selection of the hyperparameter h and testing we use repeated random subsampling validation, i.e. Monte Carlo cross-validation. We randomly sample 20% of the dataset to create a test set .T at the outset (denote the remaining part

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as .D). We select 300 values for h chosen in equal steps from 0.01 to 0.5. For each value of h, we repeat the following 20 times: we randomly split .D into 20% validation set and 80% training set, and we train kernel regression and measure the performance over the validation set in terms of median absolute error (MAE). An optimal value for h is then selected. We then test the performance over the test set using this optimal value for h. To ensure the results are insensitive to the initial random split between .T and .D, the entire process is repeated 20 times (i.e. random data splitting, hyperparameter tuning over the specified ranges, and testing), and the averages of various test errors are reported (along with the standard deviation) in Table 1. Figure 3c shows the predicted response surface.

4.3.4

Gaussian Processes

A Gaussian process is a distribution over functions and may be used in regression. The posterior distribution for the output (response) .f∗ corresponding to test data .X∗ , given training data .X with known target values .y is given by a normal distribution: f∗ |X∗ , X, y ∼ N (K(X∗ , X)[K(X, X) + σn2 I]−1 y, K(X∗ , X∗ )

.

− K(X∗ , X)[K(X, X) + σn2 I]−1 K(X, X∗ )) The mean function is thus given by: .K(X∗ , X)[K(X, X) + σn2 I]−1 y, whereas the covariance matrix is: .K(X∗ , X∗ ) − K(X∗ , X)[K(X, X) + σn2 I]−1 K(X, X∗ ), where 2 .K(x, y) represents the covariance between x and y cases, .σn is the noise variance, and .I is the identity matrix. For more details on the application of Gaussian processes in machine learning, the reader is referred to Rasmussen and Williams (2006). We use a Gaussian process for regression with a simplified RBF (radial basis function) kernel. The hyperparameters are listed below: 1. Characteristic length-scale used in the radial basis function, l. Large values of l indicate a wide kernel and would result in high bias but low variance, whereas small value of l indicate a narrow kernel and would result in low bias but high variance. 2. Standard deviation of Gaussian-distributed noise, .σn , which accounts for the presence of noise when fitting to the training set. Small values would cause a tight fit over the training samples, whereas larger values allow for a margin of error around each sample point and can result in a smoother response. For selection of hyperparameters and testing, the procedure used is similar to that applied with kernel regression. In this case, we select 20 values for the length-scale hyperparameter l of the RBF kernel chosen in equal steps from 0.05 to 0.5, and likewise for the standard deviation of Gaussian-distributed noise, .σn . The average test errors are reported in Table 1 and the predicted response surface is shown in Fig. 3d.

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Beta Regression

The previous models are compared with beta regression (Ferrari and Cribari-Neto, 2004) as utilized in Flamand et al. (2023). Beta regression assumes the dependent variable is beta-distributed and a logit link function was used in this case as in Flamand et al. (2023). This means that .ln(Dq /(1 − Dq )) = β0 + β1 kq + β2 λq . Therefore, Dq =

.

exp(β0 + β1 kq + β2 λq ) . 1 + exp(β0 + β1 kq + β2 λq )

This type of parametric regression assumes a rather restrictive (sigmoid-shaped) model compared to the other methods experimented with in this work. There are no hyperparameters to tune in this case. For testing beta regression, we use Monte Carlo cross-validation. The average test errors are shown in Table 1 along with the predicted response surface in Fig. 3e.

5 Results The test errors for the various methods are summarized in Table 1. The results across the different optimized methods are close to each other. However, we note that root mean square error (RMSE) is sensitive to outliers. Therefore based on the median absolute error (MAE), the regression tree and kernel regression method outperform SVM and Gaussian processes, which also holds true for the symmetric mean absolute percentage error (SMAPE).

6 Conclusions Estimating in-store traffic density is an important component when optimizing product allocation in a retail setting. One approach for obtaining ground-truth estimates is through using RFID tags affixed to shopping carts to track consumers as they shop around the store. This would require careful data cleaning and processing given for example that shopping carts can be abandoned leading to long idle times. Alternatively, in the absence of RFID tags, traffic density approximations may be obtained by reconstructing shopping paths from consumer receipts. This more passive approach can be carried out by solving a traveling salesman problem (TSP) for each receipt, although the assumptions behind treating shopping paths as a TSP can significantly deviate from reality, particularly in the case of larger shopping baskets. In this work, we have explored another way for approximating traffic density also from receipt information but based on product sales volumes pertaining to any given shelf as an indicator of visiting frequency. We have explored

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several regression methods for modeling traffic density data such as SVM, Gaussian processes, beta regression, kernel regression, and regression trees. Among these, the latter two resulted in the lowest out-of-sample error measures. In general, our work has shown that in this context non-parametric approaches can be more suitable than parametric ones due to their less restrictive nature, especially when we do not know apriori whether the data can fit under a particular model. Ultimately, the aim behind predicting traffic density is to utilize the regression model when optimizing product allocation so as to maximize impulse-buying profit for instance. We end this chapter with a proposition that differs from the standard approach on how to use the traffic density regression model for this purpose. Instead of treating the model endogenously, i.e. as part of the optimization model, we present an idea for an alternative approach which simplifies the optimization by decoupling regression from the optimization model. Traffic density clearly depends on location (to entrance/exit) and on the allocated products at that location. At the same time, the (optimal) allocation of products depends on traffic density. This is akin to estimating two inter-dependent quantities. Such types of recursive problems can be solved using the Expectation-Maximization (EM) algorithm, which is utilized in many areas including clustering in machine learning. Future work will involve developing an EM-style method for solving the product allocation and traffic density estimation problems in an iterative manner.

References Breiman, L., Friedman, J. H., Olshen, R. A., & Stone, C. J. (1984). Classification and regression trees. Belmont, CA: Wadsworth. Chang, C.-C., & Lin, C.-J. (2011). LIBSVM: A library for support vector machines. ACM Transactions on Intelligent Systems and Technology, 2, 27:1–27:27. http://www.csie.ntu.edu. tw/~cjlin/libsvm Dorismond, J. P. (2019). Data-driven models for promoting impulse items in supermarkets. PhD thesis, The State University of New York at Buffalo. Farley, J. U., & Ring, L. W. (1966). A stochastic model of supermarket traffic flow. Operations Research, 14(4), 555–567. Ferrari, S. L., & Cribari-Neto, F. (2004). Beta regression for modelling rates and proportions. Journal of Applied Statistics, 31(7), 799–815. Flamand, T., Ghoniem, A., & Maddah, B. (2016). Promoting impulse buying by allocating retail shelf space to grouped product categories. Journal of Operational Research Society, 67, 953– 969. Flamand, T., Ghoniem, A., & Maddah, B. (2023). Store-wide shelf space allocation with ripple effects driving traffic. Operations Research. https://doi.org/10.1287/opre.2023.2437 Hui, S. K., & Bradlow, E. T. (2012). Bayesian multi-resolution spatial analysis with applications to marketing. Quantitative Marketing and Economics, 10, 419–452. Hui, S. K., Bradlow, E. T., & Fader, P. S. (2009). Testing behavioral hypotheses using an integrated model of grocery store shopping path and purchase behavior. Journal of Consumer Research, 36(3), 478–493. Hui, S. K., Fader, P. S., & Bradlow, E. T. (2009). The traveling salesman goes shopping: The systematic deviations of grocery paths from tsp optimality. Marketing Science, 28(3), 556–572.

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Larson, J. S., Bradlow, E. T., & Fader, P. S. (2005). An exploratory look at supermarket shopping paths. International Journal of Research in Marketing, 22(4), 395–414. Nadaraya, E. A. (1964). On estimating regression. Theory of Probability and Its Applications, 9, 141–142. Ozgormus, E., & Smith, A. (2020). A data-driven approach to grocery store block layout. Computers & Industrial Engineering, 139, 105562. Pak, O., Ferguson, M., Perdikaki, O., Su-Ming, W. (2019). Optimizing stock-keeping unit selection for promotional display space at grocery retailers. Journal of Operations Management, 66, 501–533. Rasmussen, C. E., & Williams, C. K. I. (2006). Gaussian processes for machine learning. Cambridge: MIT Press. Smola, A. J., & Schölkopf, B. (2004). A tutorial on support vector regression. Statistics and Computing, 14, 199–222. Sorenson, H. (2003). The science of shopping. Marketing Research, 15(3), 30–35. Watson, G. S. (1964). Smooth regression analysis. Sankhy¯a: The Indian Journal of Statistics, Series A, 26, 359–372. Yada, K. (2011). String analysis technique for shopping path in a supermarket. Journal of Intelligent Information Systems, 36(3), 385–402. Yada, K., Ip, E., & Katoh, N. (2007). Is this brand ephemeral? A multivariate tree-based decision analysis of new product sustainability. Journal of Decision Support Systems, 44(1), 223–234.

A Simulation Based Tool to Guide Periodic Changes in a Supermarket Layout Jessica Dorismond, Jose L. Walteros, and Rajan Batta

1 Introduction The significance of this research is to provide a simulation-based tool to optimize the block and detailed facility design of a supermarket. The supermarket industry is a vital sector for economies worldwide that is plagued with high competition and tight profit margins, while serving demanding customers. Recent advances in marketing research reveal that encouraging customers to walk longer paths when they visit the store often increases spending, as longer paths expose them to more products. Supermarkets frequently use this paradigm to increase the likelihood that customers purchase items that are not initially on their shopping list (Hui et al., 2009b), i.e. impulse sales. This is a challenging task as the layout must ensure that customers encounter products that may trigger impulse sales, without exceeding the point at which the extra length becomes noticeable and burdensome for customers. Our chapter systematically studies the implementation of this “longer path” paradigm using customer and supermarket data. The first goal of our chapter is to use customer and store level data to drive a simulation model that can be used to predict revenue generated due to impulse item sales, for a given supermarket layout. The second goal of our chapter is to improve the supermarket’s block layout by using a variable neighborhood search algorithm that employs the simulation tool to evaluate each layout that is searched by the algorithm. The third goal of our chapter is to optimize the detailed (shelf) layouts for each block of the supermarket, using the simulation tool to predict the customer types that visit each block. The fourth goal of our chapter is to provide managerial insights that aim at determining when a block

J. Dorismond · J. L. Walteros · R. Batta () Department of Industrial and Systems Engineering, University at Buffalo, The State University of New York, Buffalo, NY, USA e-mail: [email protected]; [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Ghoniem, B. Maddah (eds.), Retail Space Analytics, International Series in Operations Research & Management Science 339, https://doi.org/10.1007/978-3-031-27058-1_4

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layout should be changed, when a detailed layout should be changed, and when customer data should be reanalyzed to improve layout design for the supermarket. Our work on supermarket layout design has close relationship to facility design methods, which have been traditionally used to find layouts that minimize the material handling cost in a factory. In such models the data that is needed is the interdepartmental material flow matrix, and the distance matrix. The interdepartmental material flow matrix is usually measured in terms of pallets, trips, or pounds per day for every period, and the data may change from period to period due to the following factors (Palekar et al., 1992): – Product related: Changes that may include the design of a product, the quantity to be produced, or may involve the introduction of a new product. – Process related: Changes that may include the process plan for some products or may relate to the introduction of new special purpose machinery. – Production related: Machine breakdowns requiring long periods of downtime, changes in schedules, unavailability of materials and/or tools may be included. – Management related: Changes that may include the organization structure that in turn may be caused by changes in management philosophy. – Demand related: May include seasonal fluctuations in demand and growth or decline in the demand for a product. They also may involve special one-time orders for particular items. We believe that the factors that affect the interdepartmental material flow matrix of a manufacturing plant, as introduced in Palekar et al. (1992), are similar to the traffic flow of a supermarket. A product related issue would be the introduction of new items or category of items that causes customers to alter their normal paths (e.g., a new health food section). A production related issue is similar to shortages in an item or discontinuation of an item that can affect the customer flow of a supermarket. A management related issue could be the switch of managers and a change of philosophy on how the supermarket should be run. A demand related issue could be due to change in demand of a product due to different seasons and change in customer buying patterns. We therefore use the well-established path followed in plant layout analysis as a driving force for our research on supermarket layout analysis. From a marketing perspective, the two primary performance measures to evaluate layout design effectiveness of a supermarket are impulse item visibility and customer travel distance. To study these two performance measures, we develop a simulation model that mimics the behavior of customers inside of a supermarket layout. Customer travel behavior inside of a supermarket is influenced by many different factors, such as the placement of departments within the physical store (i.e. supermarket block layout) and the allocations of various items on the shelves of the supermarket (i.e. supermarket detailed layout). We use simulated customer paths to develop estimates for visibility of different areas during their shopping trip. In traditional facility layout design models, the block layout and the detailed layout of a plant do not have a direct correlation. The relative placement of each department (block layout) is first determined, followed by a detailed layout for each

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department which spatially arranges the machines in each department. Changes in the block layout do not impact the detailed layouts for each department in a traditional facility layout design. However, in a supermarket, the block and detailed layouts impact each other drastically. A change in the block layout alters customer flow patterns. Since customer flow patterns are used as an input to develop detailed layouts in a supermarket, there is an interaction between block layouts and detailed layouts in a supermarket. This research can be used to study the impact of various actions on the layout of a supermarket, such as: (1) introducing a new set of items, (2) deleting a set of items, (3) increase or decrease in the percentage of a customers who are interested in purchasing an item, (4) consideration of a flexible layout that works well across several likely customer shopping patterns. The rest of this book chapter is organized as follows: Sect. 2 reviews the relevant literature in this area. Section 3 presents the optimization framework that is used. Section 4 discusses the block layout optimization problem, and Sect. 5 discusses the process of developing a detailed layout. Section 6 details our method of evaluating different block layouts. Our model and results are demonstrated through a case study that is based on a grocery store setting in Sect. 7. Finally, conclusions and future research are presented in Sect. 8.

2 Relevant Literature To develop our model and approach we utilize concepts from traditional facilities design, previous studies in store layout, impulse purchases, shelf space allocation, and customer travel patterns. Relevant contributions from these areas are reviewed in this section.

2.1 The Facility Design Problem A facility layout is a strategic decision-making tool that influences the profits and operations of an organization. Traditional facility layout methodologies often involve the arrangement of different facilities so that the total cost to move materials between facilities is minimized (Heragu and Kusiak, 1991). The intent is to find the most efficient arrangement of m indivisible departments with unequal area requirements within a facility (Mawdesley et al., 2002). Strategic placement of facilities contributes to the overall efficiency of operations and can reduce up to 50% the total operating expenses (Tompkins et al., 1996). In contrast to a manufacturing facility, we apply facilities design principles to find the best block and detailed layouts of a supermarket. In lieu of minimizing the material handling cost, our objective is to maximize supermarket sales, which translates to maximization of impulse item purchasing.

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2.2 Store Layout Facility design problems frequently involve the determination of site locations, a block layout, and detailed layouts. The model in this chapter focuses on finding the optimal block layout for the supermarket and detailed layouts for various blocks of the supermarket. The most common block layouts for stores are grid layout, freeform layout, racetrack layout, and serpentine layout (Li, 2011). The block layout of a retail store specifies the location of every department and is determined by using various facilities design methods. There are two methods that are commonly used to determine block layouts, and Relationship (REL) diagram which works from qualitative data assessments and the quadratic assignment problem (QAP) which works from quantitative data assessments. The REL diagram is an activity relationship diagram that describes the spatial relationship between departments. The quadratic assignment problem (QAP) allocates a set of facilities to a set of locations, where the distance and flow between the facilities define the cost function. A grid layout is the typical layout of a grocery store and is preferred in a grocery store atmosphere because the grid layout is excellent for customers who have preplanned purchases (Li, 2011). Figure 1a is a visual representation of a grid layout. The serpentine layout is a special case of the grid form. This layout is rarely used in retail stores and has few entrances and exits. This layout forces customers to walk only one route throughout the grocery store that traverses all the floor space. Two stores that have adopted a serpentine layout are Aldi and Ikea (Li, 2011). The racetrack layout leads customers along specific paths to visit as many store sections or departments as possible. The main aisle facilitates customers around the store and is mainly used in department stores (Li, 2011; Yapicioglu and Smith, 2012a,b). The freeform layout has a free-flowing pattern, where customers are able to move around freely throughout the store, and have long aisles that are parallel to each other. This type of layout is appropriate for fashion stores, because usually customers who enter this store do not have a pre-planned purchase, and the free flow layout give customers the ability to move around the retail store freely. The most recent works done in store layout design are the works by Li (2011), Botsali (2007), Ozgormus (2015), and Ozgormus and Smith (2020). Li (2011) optimizes the store layout by having a floor plan that promotes sales and is accomplished by maximizing the area exposure and optimizing the adjacency preference of all departments. Botsali (2007) analyze the grid, serpentine and huband-spoke layouts, and evaluate the performance of mentioned layouts concerning impulse purchase revenue and customer travel distance. Ozgormus (2015) and Ozgormus and Smith (2020) research the integration of stochastic simulation and optimization to solve grocery store layout problems. In addition to these works, Hui et al. (2009a) model shopper travel behavior within a supermarket through a simulation model and conclude that relocating three destination categories increases impulse buying by 7.2% in that specific supermarket.

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Fig. 1 Different types of store layouts. (a) Grid. (b) Serpentine. (c) Racetrack (Levy et al., 2012). (d) Freeform

2.3 Impulse Purchases The phrase “unseen is unsold” is one theory that describes the cause of “impulse buying,” because shoppers often use physical products in the store as external memory cues that simulate new needs or trigger forgotten needs (Hui et al., 2013). The memory cues in retail stores can also be referred to as the exposure to in-store stimuli hypothesis (Kollat & Willett, 1967). This theory assumes that impulse items are caused by memory cues in the store such as product displays, shelf layouts, atmospheres and promotional advertisements. Hui et al. (2013) conducted a field experiment to evaluate the effectiveness of mobile promotions in a retail store. The mobile promotion used coupons

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to deviate customers path, by making the customer path longer inside of the grocery store. They found that this resulted in a substantial increase in unplanned spending ($21.29) as they often used a coupon for an unplanned category near their planned path ($13.83). A simulation created by the authors suggest that strategically promoting three product categories through mobile promotion could increase unplanned spending by 16.1%, and found that relocating three destination categories increases impulse buying by 7.2%. Flamand et al. (2016) create a groupbased model that allocates product category groups to shelf and subsets of the groups to aisles. The objective of their model is to maximize the impulse purchase reward, and they report a 32% increase in impulse buying for a store in New England.

2.4 Shelf Space Allocation Strategic placement of items on a shelf influences the sales of items. Therefore, shelf space allocation is regularly manipulated by retailers to increase sales and profits (Curhan, 1972). For instance, products with high gross margins frequently are displayed in supermarkets at highly visible areas, such as locations at eye-level and high traffic areas. Overall, shelf allocation is an important tool in attracting customer’s attention and impacts the retailer’s profit, since the visibility of an item relies heavily on its shelf placement and space allocation. Retailers tend to allocate more space to brands that have a higher contribution to their profit. The impact of visual experience of a customer on retail sales has been studied by Mowrey et al. (2017) and more recently by Guthrie (2018). The three different shelves in a grocery store are often categorized as either bull’s-eye black, kids’ eye-level shelf, top shelf, and bottom line. The bull’s-eye block is the prime shelf level, and items placed on this level are within the customers eye-level; these items tend to have more visibility than the other shelves. Manufacturers often give retailers supplementary money to place items on this shelf. Moreover, the kids level are usually places that are visible to children, where they can reach out and grab products that are appealing to them. Shoppers with children usually spend 10–40% more than other customers. Lastly, bulk and store brands items are located on the bottom shelf, since customers who are hunting for a good deal will always find them. Coskun (2012) created a shelf allocation maximization model that improves a retailer’s profitability by using adjustable shelf heights. The proposed model increased profitability by about 7% compared to the conventional models that exist in the academic literature. The shelf allocation model captures the visibility of the impulse items in a grocery store. The importance of incorporating a shelf allocation model is because an impulse item may be in a prime location, however, if the item is not at the eye level then the customer will be less likely to see the item. There has been a significant amount of work in recent years on optimization of shelf layouts. Readers are referred to papers by Murray et al. (2010), Flamand et al. (2018), Pak et al. (2019), and Ostermeier et al. (2021).

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2.5 Travel Patterns There is relatively scant literature available on how shoppers traverse a grocery store. A theory postulated by Hui et al. (2009b) is that shoppers establish a visit sequence that is close to that of the traveling salesman path to pick up their “must have” items from the store. However, they suggest that when traversing between two consecutive “must have” items in this sequence shoppers tend not to travel on the shortest point-to-point path, i.e. they have significant travel deviations. A separate theory postulated by Larson et al. (2005) assumes that shoppers always travel to the closest location among the locations that they have left to visit. By this theory, a shopper goes to their closest “must have” item when they enter the store, and from there move to the “must have” item (from among those remaining) that is closest to the first “must have” item visited. This is equivalent to using the closest neighbor heuristic for the traveling salesman problem.

3 Optimization Framework The elements of the optimization framework for this study is shown in Fig. 2. It has two main components. The first component is described in Sect. 4, where the objective is to optimize the block layout of the supermarket. Customer shopping paths are a function of the block layout, since a customer’s travel path is determined by the blocks to which items (at the SKU level) on their shopping list belong. Thus, the store block layout determines customer traffic flow and hence impacts the different visibility areas inside the store. The second component is described in Sect. 5, where we use a shelf allocation model that optimizes the visibility of impulse items on a supermarket shelf, thus optimizing each “block” of the block layout. The goal of the shelf allocation process is to maximize the visibility of impulse items—we place them on shelves where customers can easily see. As an example, placing impulse items on the bottom shelves make them not easily visible to customers even in high traffic areas. Customer flows from the block layout optimization provide the input data for the shelf optimization phase. Thus, the two components, block layout optimization and shelf layout optimization, are related.

4 Block Layout Optimization The key to our block layout optimization process is our ability to score a block layout. The scoring of a block layout is based on the amount of impulse purchases that customers are expected to make when that block layout is used by the supermarket. An accurate prediction of customer flows that are generated due to the block layout is necessary to determine the expected value of the impulse purchases supported by

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Customer's Profiles

Block Layout

Detailed Layout

Simulation

Shelf Allocation Model

Develop a simulation that incorporates the various stochastic variables of a supermarket to model a customer's walking path in a supermarket ~ 1. Generating customers (entities) from using different customer profiles. Each customer will have a randomly generated shopping list. ~ 2. Predict the traffic flow of customers within the supermarket, and number of times customers pass their impulse items

Customer's paths

Developed a mathematical model for the shelf space allocation problem where the objective is to maximize the visibility of impulse items. ~ 1. The shelf allocation model will use the customer's path information to know how many people passed by the different shelves in the supermarket. ~ 2. The score of the layout will be determined mainly by how many times a customer passed by their impulse items throughout their trip.

Score

Fig. 2 Optimization framework

the block layout. We design a simulation model that approximated the stochastic behavior of customers within a supermarket, given a specific block layout. A simulation allow us to create several variations of customers by generating shopping lists and predicting their paths using probabilistic modeling. Customer flows of the discrete event simulation are simulated by using a path generation algorithm. Obviously, we cannot individually generate a separate path for each customer. Rather, we cluster customers into groups based on historical shopping patterns. Each customer in the simulation is an entity for the simulation model. To create an entity, the simulation uses data that is pre-clustered. In Sect. 4.1, we describe the K-medoids method we use to cluster the customer data. Then, in Sect. 4.2 we will describe how we build customer profiles. Lastly, in Sect. 4.3, we describe how the simulation models customer travel behavior inside the supermarket.

4.1 K-Medoids A clustering method circumvents the problem of determining the correlation between products because items that correlate are clustered together. K-medoids is the method of choice instead of K-means. We use a similarity metric from Erkut and Verter (1998) to compare each transaction instead of using a distance metric, and the update step for K-medoids ensures that transactions are clustered within an appropriate profile. The initial step of the algorithm is to find the “best” three hundred shopping lists (we used 300 because is roughly 10% of the transactions that we had available). The “best” shopping lists are defined as the lists that have a high average similarity score when you compare them to all the other shopping list. Then, K shopping lists from the best 300 shopping list become the initial medoids. Each shopping list is assigned to most similar medoid. Next, the algorithm iterates through all the shopping lists and assigns it to a cluster. When all the clusters are generated, the shopping list

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that has the highest similarity average score within their cluster becomes the new medoid. The algorithm keeps clustering the shopping lists with the newly updated medoids until the medoids do not change anymore.

4.2 Customer Profiles Transactional customer data can be used to define different customer profiles. The transactional data is a record of purchases made by the customer—i.e. how much did the customer spend, what time did the customer check out from the store, and what items did the customer buy (Lingras et al., 2003). Each customer profile consists of must-have items and impulse items. Suppose that there are K profiles of customers, indexed by .k = 1, . . . , K and let .θk = 1, . . . , K define the set of must-have items for customer profile k. Also, let .βk = 1, . . . , K define the set of impulse items for customer profile k. The set of must-have and impulse items for a specific customer profile are determined by examining their purchase history (specifically, items that are purchased with significant frequency are considered to be must-have and the other are considered to be impulse). We use transactional data from a supermarket (a data set with 3423 transactions) along with a K-medoids method will find different customer profiles. An example of customers profiles is as follows: Customer Profile I: Must-have items .θ1 = {I-2, I-11, I-12, I-13, I-23, I-25} and Impulse items .β1 = {I-7, I-28} Customer Profile II: Must-have items .θ2 = {I-2, I-11, I-12, I-20, I-22, I-30} and Impulse items .β2 = {I-15, I-16} Customer Profile III: Must-have items .θ3 = {I-2, I-3, I-11, I-12, I-20, I-22} and Impulse items .β3 = {I-24, I-29}

4.3 Simulation The simulation evaluates the score of the block layout. The customers in the simulation are independent entities that move around the store to pick up their musthave items and impulse items, in accordance with the probabilistic choice rule in selecting the sequence of must-have items. Details are provided in Sect. 4.3.2. The method that creates entities in the simulation will be described in Sect. 4.3.1. A flowchart of the simulation is shown in Fig. 3.

Start, Read Data, Create Customer

Assign Customer Shopping List

Fig. 3 Simulation flow chart

Enter Supermarket

Which zone to visit?

Go to Supermaket zone

Got all items in shopping list?

Exit Supermarket

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Generating Customer Shopping Lists

Once customer transactions are clustered, we have all the information needed to generate customer shopping lists. Each cluster is a collection of shopping lists, where the algorithm first creates a list of items that appear in the cluster of customers and then finds the number of times the items appear in the shopping list of the customer. The probability that a customer will have a specific item on their shopping list will depend on the number of times it appears in each shopping list. Note that we ignore the number of times an item appears on one transaction, because we are concerned with the visibility of items within the store, and not in predicting the total revenue of the supermarket. Ultimately, increasing the visibility of various products throughout the store, especially customer impulse items will encourage impulse purchases and increase the sales of the supermarket.

4.3.2

Customer Paths Generation

Previous models in the academic literature use the traveling salesman path generated from the must-have items in the customer’s shopping list as the path that the customer will take when traversing the store. However, the simulation generates paths by looking forward just one step at a time. This is achieved by using a negative −αxi exponential function. The probability function that is used is: .P (xi ) = αeαe−αxi . The variable .xi represents the probability of choosing block i next out all the N block the customer plans to visit. The value .α is the penalization coefficient; it penalizes farther blocks by forcing them to have a lower probability. This is a more realistic way of capturing customer movement in a store, since customers are likely to think sequentially about must-have items that they need to purchase when planning a path as opposed to optimizing the path using all must-have items in their list. The decision-making process of a customer viz-a-viz their path is as follows: The entering customer already has a pre-planned shopping list (must-have items) and determines the next block of the store to visit by using a probability function. Thus customers are assumed to be more likely to travel to a nearer block for their next must-have item purchase.

5 Detailed Layout In a plant layout situation, a detailed layout corresponds to arrangement of machines in a specific block. In the supermarket context, a block has shelves. We have to decide which block each item is placed on and the quantity of each item (total space for all assigned items to a block cannot exceed its space). This optimization of items on shelves is referred to in the literature as shelf layout optimization. The goal of our detailed layout module is to maximize the visibility of impulse items in each

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block of the supermarket. The layout of each block is optimized separately. We now present the optimization model for a single block. Consider a block with N items and M storage locations (shelves) like the one in Fig. 4. We let .βi denote the rate at which customers (for whom item i is an impulse item) pass by. It is important to note that the block layout determines customer paths and hence the .βi values. These values are determined by running the simulation for a given block layout. Our first set of decision variables are the binary variables .xij , which are equal to 1 if item i is placed on shelf j , and 0 otherwise. Our second set of decision variables are integer variables .fij , which denote the number of units of item i that are placed to shelf j . To make our model realistic we place lower and upper limits on the .fij values if .xij is equal to 1. The visibility level of an item depends on which shelf it is placed on and how much space is allocated to it. We let .vj denote the visibility of shelf j . We are now ready to present our detailed layout formulation, where a block consists of a group of vertical shelves, one above the other, forming a planogram. (P )

.

maximize



(1)

βi vj fij .

i∈N j ∈M

subject to



wi fij ≤ Wj ,

∀j ∈ M.

(2)

xij = 1,

∀i ∈ N .

(3)

i∈N M  j

Li xij ≤ fij ≤ Ui xij , ∀i ∈ N, j ∈ M.

(4)

fij ∈ {0, 1, 2, ..},

∀i ∈ N, j ∈ M.

(5)

xij ∈ {0, 1},

∀i ∈ N, j ∈ M

(6)

The formulation .(P ) is explained as follows: The objective function (1) maximizes the visibility of impulse items. Constraint (2) makes sure that items placed on shelf j will not exceed that shelf capacity, .Wj . Constraint (3) ensures that each item is assigned to exactly one shelf. Constraint (4) provides an upper and lower bound on the units of an item that can be assigned to a shelf.

6 Evaluating Different Block Layouts A naive way is to obtain a score for each possible block layout (there are an exponential number) by running the simulation and picking the one with the maximum score. This method may be possible for small supermarkets that only have a few blocks. However, if the number of blocks in a supermarket is large we will need a heuristic to find a near-optimal block layout. Furthermore, since

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Fig. 4 Detailed layout

optimizing the block layout requires the input of the simulation, using other type of optimization methods like integer programming is computationally challenging. The meta-heuristic that we used is a variable neighborhood search algorithm (Mladenovi`c and Hansen, 1997). Neighborhood search represents an intuitive way to improve a block layout and are very popular in the plant layout domain. Unlike a standard neighborhood search algorithm, a variable neighborhood search algorithm randomly selects between the solutions in the neighborhood and is less likely to get stuck in a locally optimal solution. The variable neighborhood search must consider the following restrictions before swapping block locations: size, location, and adjacency requirements. As an example of an adjacency requirement a block that contains a food item should not be kept next to a block that contains a rodent spray item. Consider a block layout z. For this block layout z we define a neighborhood structure that contains layouts .Nzk (k = 1, ..., kmax ). A local search algorithm is one in which .kmax = 1, i.e. the neighborhood contains only one solution (usually the one that gives us the maximum improvement through a simple interchange of blocks). In a variable neighborhood search we randomly select between solutions contained in the neighborhood. Our procedure is as follows: – Initialization: The starting point of the algorithm is the current block layout of the supermarket. A local search is run to obtain an incumbent solution, a. This incumbent a is further improved in the main steps described below. – Main Steps: (1) Randomly select a block layout b in the neighborhood of the incumbent a (using a specified value of k). (2) Perform a local search starting with block layout b to obtain a block layout c. If c is an improvement over b, we repeat steps 1 and 2 till the time limit is exceeded.

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Since the optimization framework is a constructive solution methodology, the starting solution will be the current layout of the supermarket. The simulation will first run for the current layout of the supermarket and give it a score based on the predicted supermarket flows and visibility level of each of the customer’s impulse items. Deterministic methods can be used to get a starting solution for the simulation. However, the supermarket must find a way to build the different profiles for the customers. In the VNS algorithm, we define a Neighborhood change to be a layout change that has more than two swaps. Two departments should be the same size and have the same requirements to be eligible to swap. For instance, some items are required to be refrigerated, frozen to be kept at room temperature.

7 Case Study of a Grocery Store in Western New York To investigate the practicality of our approach we tested this methodology for a supermarket in Western New York.

7.1 Data Collection and Analysis For our case study we assembled and analyzed data from three different sources: – The supermarket layout was obtained from the supermarket management in Western New York, and was used to construct a network that represents the supermarket. – Actual customer path data was obtained by observation, and was used to validate the path generation model used in Sect. 4.3.2. – Customer transaction data was obtained from another public source and was used for creation of customer cluster profiles and specification of which items are must-have and which are impulse, for each customer profile.

7.1.1

Supermarket Layout

The supermarket layout was provided in the form of a CAD drawing, through which dimensions could be inferred. The layout of the supermarket is discretized by separating the areas of the supermarket into blocks. An undirected network of the supermarket was created by using the Networkx package in python to capture customer movement inside of the store. Figure 5 shows the network of the supermarket. Each shelf is represent by a node. In the network model, edges represent the walkways that customers can use during their shopping trip, and nodes represents a shelf. A block of a layout is represented by a collection of adjacent

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Fig. 5 Supermarket network

shelves. For example, the frozen dinner block can be represented by the collection of shelves 216, 217 and 218.

7.1.2

Customer Path Data

We conducted our empirical study over the three month period from June to August 2016. The store is a medium sized supermarket separated into 228 blocks. Our key objective for conducting this empirical study was to verify that the model we used in Sect. 4.3.2 for customer path generation matches closely with actual customer paths. To empirically verify this, we identified 100 customer volunteers to participate in our study at this supermarket. Each of these customers was given a survey, whose questions were based on a study conducted by Hui et al. (2009b). The survey, which was conducted when they entered the store, asked the following questions: (1) Do you have a shopping list today? (2) How familiar are you with the store? (3) What items are on your shopping list? After they had answered these questions, a list of product categories in the store was provided to them and they selected all the products they intended to purchase. The customers were then free to shop, and once they were finished, we checked their receipts to see what they acquired during their shopping trip. This gave us the list of impulse items that they purchased. We captured the actual path of the 100 customers by discretely following each of these customers as they walked through the store. We found that out of the 100 customers surveyed at the supermarket, 59 bought impulse items that were not originally on their shopping list. On average, customers who did “impulse buy” bought three more items. Figure 6 illustrates the difference

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Fig. 6 Planned vs purchased sales

between the number of items customers planned to get versus the number of items the customers bought.

7.1.3

Customer Transaction Data: Creation of Customer Clusters and Must-Have and Impulse Item Designation

Customer transaction data was not provided to us by the store where the empirical study on customer paths was conducted, due to confidentiality reasons. We obtained customer transaction data from the company dunnhumby, which has data from a group of 2500 households who are frequent shoppers at a supermarket. The source file had information on the supermarket, all the purchases that the customer made inside of the store, and the demographics and direct marketing contact history for select households. We used this data set to cluster the different customers for the simulation and matched up the product category with the supermarket layout we have and used the items they sold at their supermarket for the shelf allocation model. It is not evident how many clusters should be used. One method to decide on the appropriate number of clusters is to use the elbow method. Towards this end, we ran the k-medoids clustering algorithm on the data set for ranges of k from 5 to 95 in increments of 5. For each value of k, we calculate the weighted average similarity score for each point in each cluster. As shown in Fig. 7, the elbow occurs at 50 clusters, which is the number that we used. Once customers were partitioned into 50 clusters, we need to determine the musthave and impulse items associated with each of these clusters. To do this we used the following rule: If an item is purchased by customers in the cluster less that 20% of the time, it is labeled as an impulse item for customers in this cluster. Otherwise, it is labeled as a must-have item for customers in this cluster. We note that an item could be labeled as a must-have item for one or more clusters and simultaneously be labeled as an impulse item for other clusters. A sample cluster is shown in Table 1, in which only 4 of the 25 items in the cluster are must-have items.

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Fig. 7 k-medoids clusters Table 1 Cluster 33 information Cluster profile no 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33

Item Vitamins Vegetables/Potatoes Automotive Cakes Baby Seasonal CocoaMixes DeliMeat Magazines BottledWater Cream Candles/Accessories Apparel ElectricalSupplies PreparedFoods Candy Fruits Cards Cookies Coffee/Tea Cereal SaltySnacks CannedMilk Beer Floral

Probability 3% 3% 3% 3% 3% 3% 3% 3% 3% 3% 3% 5% 5% 5% 5% 8% 8% 8% 10% 10% 15% 20% 23% 35% 53%

Impulse or must have Impulse Impulse Impulse Impulse Impulse Impulse Impulse Impulse Impulse Impulse Impulse Impulse Impulse Impulse Impulse Impulse Impulse Impulse Impulse Impulse Impulse Must have Must have Must have Must have

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7.2 Computational Study The store layout has 228 blocks (departments), and within these blocks there are 1466 different items that are allocated. Our first task is to establish a score for the current layout. To do this we ran 15 simulations, each with between 10,000 and 14,000 customers. Each created customer is assigned to one of the 50 clusters based on the cluster probabilities. Once we know which cluster a customer belongs to, a shopping list is generated using the item probabilities related to the cluster. This shopping list includes both must-have and impulse items. After a customer’s shopping list is generated, the path that they take through the store is simulated and these paths are used to generate the score, which reflects the impulse item purchases made. Taking the average across all of the simulations we obtain the score for the initial layout. The next step is to use the VNS method to improve the block layout. The detailed layouts associated with each block are held constant at this point. After we have completed improvements in the block layout, the detailed layout is optimized for the final block layout obtained. When scoring the new layouts, the same seeds to generate customers are used. This provides a consistent scoring method across all layouts (Table 2). Our optimization scheme found a layout that had a 5% impulse score increase. The swaps that made the biggest impact on the layout were the ones that changed the location of common impulse items. For instance, the locations for Cocoa Mix, Pickles, Sugar, and Sports Drinks were swapped, and each of the locations for these items was placed closer to the end-cap aisles. These items are impulse items for many of the 50 different customer profiles.

7.3 Managerial Insights This section develops managerial insights to identify when to change a supermarket layout, both at a detailed level (shelf layout) and at the block level (Zhou & Wong, 2004). The results obtained from our analysis answers the question of how much difference in the customer buying pattern would need to happen before we should consider changing either type of layouts. For small changes in the customer buying pattern, one could merely change the detailed layouts to capture the shift in impulse buying behavior. For substantial changes in customer buying pattern, one can do a modification of the block layout followed, of course, by a change in the detailed layouts. To establish different rules we perform data analysis to identify scenarios when the layout of the supermarket should be changed. The form of the data that is needed for our analysis is shown in Table 3. The first column numbers the k clusters. The second column contains the list of products that belong to that cluster, with the third column providing the purchase probabilities for each of these items. The final

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Table 2 Products used in simulation Product category groups in a supermarket store in Western New York, the United States Air fresheners Coffee/Tea Nuts Apparel Cookies Oral care Baby Cooking oil Orange juice Baked goods Crackers Pancake mix Bakeware Cream Paper towels Baking products Deli meat Pasta Bath soaps Dish soap/detergent Pasta sauce Dry desserts Peanut butter and jelly Bath tissue Eggs Pet supplies Battery Beer Electrical supplies Pharmacy Bird seeds Ethnic meals Pickle/Relish/Pkld Veg Fish Plastic cups Bleach Bottled water Floral Precut fresh salad mix Flour Prepared foods Bread Bread and Bagels Frozen bread /pies Powder/crystal drink mix Frozen breakfast Refrigerated dough products Breading coatings crumbs Frozen dinners Rice Breakfast bar Breakfast sausage/sandwiches Frozen fish Rice cakes Bug Frozen fruits/toppings Salad dressing Cake mixes Frozen meat/seafood Salty snacks Cakes Frozen pizza Sauces Candles/accessories Frozen potatoes School supplies Frozen snacks Seasonal Candy Frozen vegetables Shampoo/conditioner Canned beans Fruit juices/drinks Snacks Canned fruit Canned meat/fish Fruits Soft drinks Canned milk Gravy Spices/seasoning Health/beauty Sports drink Canned pickles/olive Canned soup Household cleaners Sugar Canned vegetables Ice Sweet snacks Ice cream Trash Cards Cat food Laundry Tupperware Cat litter Magazines Vegetables/potatoes Margarine/butter Vinegar Cereal Cheese Meat Vitamins Napkins Yogurt Cocoa mixes

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Table 3 Simulation input information Clusters 1 2 3 ... k

List of products [1,3,8] [3,6,10] [5,7,11] ... [2,3]

Probability of purchasing item [.q1 ,.q3 ,.q8 ] [.q3 ,.q6 ,.q10 ] [.q5 ,.q7 ,.q11 ] ... [.q2 ,.q3 ]

Probability of cluster .p1 .p2 .p3

... .pk

Fig. 8 Percentage of survey respondents that said they sought these characteristics in their grocery purchases

column contains the probability that a randomly selected customer belongs to a specific cluster.

7.3.1

Changes in Customer Lifestyles

The percentage of customers in each cluster can change is due to lifestyle changes. Many American are changing to healthier diets. Figure 8 shows that in a relatively short time span, shoppers have become more focused on the healthiness of the goods they are buying. Another reason for a change in lifestyle is due to additional members being added to a household. A very common case is the addition of a child in a household, which undoubtedly changes the types of items that the household purchases. To test the impact of lifestyle changes we tested various scenarios. The first scenario that was tested was increasing the percentage of customers in the most popular cluster, and seeing how it affected the overall traffic of the layout. Our

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results showed that increasing the percentage of customers who belong to the most popular cluster decreased the score of the layout to 76% of its original value, because the major item bought by people who were in the most popular cluster is water, which has low revenue associated with it. The net effect of this change was that the overall traffic went down for the supermarket, since majority of the customers were only coming to buy one item. The second scenario tested was increasing the percentage of customers in the least popular cluster. This has a positive affect on the overall layout because the number of items customers purchased in this cluster is large. This increased the score of the layout by 5%. The net effect of this change was that the overall traffic went up for the supermarket, as customers were passing by their impulse items more frequently. The managerial insight gained from the analysis of these two scenarios is that changes in the percentage of customers who belong to a particular cluster can have both a positive or a negative impact on the layout. If there is a negative effect, changes in both the block and detailed layouts may be necessary to maintain the projected profitability associated with the layout.

7.3.2

Impact of Item Prices

Item prices vary, and customer demand for an item is a function of its price, as shown in the typical demand curve displayed in Fig. 9. A change in demand for an item implies a change in the probability of its selection when it is part of a shopping list for a customer. We ran the simulation for seven different runs, and for each run we decreased the probability of purchasing an item by 0.01%. The results are shown in Fig. 10, whose plot indicates a negative slope. The overall score for the layout decreased to up 50% in the last scenario. The managerial insight gained here is that a decrease in a item’s demand would likely require either a block or detailed layout change. Fig. 9 Demand curve

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Average Layout Score for Decreasing Probabilities 800 700

Average Layout Score

600 500 400 300 200 100

1

2

3

4 Scenarios

5

6

7

Fig. 10 Average layout score of the scenarios Table 4 Simulation input information

# of items 1 items 2 items 3 items

% Change of score 12% 13% 12%

For a small decrease in item purchasing probability (less than 0.03%) a supermarket manager could get by with changing just the shelf layout. However, for situations with a large decrease in item purchasing probability (0.03% and up) a block layout change should be considered.

7.3.3

Changes in List of Products

The list of products associated with any specific cluster may change because of lifestyle changes, seasonality considerations, shortage of products, change of price of a product, or the change of demand for the product (Table 4). We tested three different scenarios for this case. For the first scenario, we change only one item in each shopping list and run the simulation 14 times. Averaging these 14 runs, we found that the score of the layout decreased by 12%. For the second scenario, we changed two items in each shopping list; if the shopping list had one item we changed that one item. We found that the average score of the layout for

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this scenario decreased by 13%. For the third scenario, we changed three item in each shopping list; if the shopping list had only 2 items, then we changed those two items, and if it had only one item, then we changed that one item. For this scenario, the average layout score went down by 12%. We believe that a 12% to 13% change in score is significant. The managerial insight that one can derive is that if the items in customer shopping lists change significantly it is beneficial to re-cluster customer transactions and build new profiles for the layout analysis. The new clusters should be used to develop a new block layout and the subsequent detailed layouts.

8 Conclusions and Future Research This chapter provides a data-driven algorithmic framework for a supermarket to generate higher impulse revenue. Transactional customer data helps create clusters for a simulation model that predicts the flows of customers in a supermarket. This is used to guide in the optimization of the block layout. A shelf allocation model uses information from the simulation to optimize the shelf space and provide the layout a score. A case study based on a supermarket in Western New York demonstrated data collection schemes and the usefulness of the model and yielded a layout that had a 5% score increase. Our first managerial insight is that unlike traditional layout problem, where block and detailed layouts do not affect each other, in a supermarket setting the block and detailed layouts are not independent. This implies that a change in the block layout can change the detailed layout. Our second managerial insight is that changes in the customer buying patterns can imply a benefit in changing either the block, detailed, or both layouts. The framework in this chapter has its limitations. A VNS algorithm is used to create an improved block layout. Other heuristics could be attempted. There can also be a more in-depth empirical study to further evaluate the way customers walks in the supermarket. Also, the case study could be improved by using data of customers and customer loyalty information.

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Data-Driven Analytical Grocery Store Design Elif Danisman and Alice E. Smith

1 Introduction The retail industry is an important sector driving economies all over the world. It has been a major employer for decades and is notable for intense competition, stringent profit margins, and demanding customers. For the marking situation of today’s buyers, retailers are seeking new strategies to increase both revenues and customer satisfaction. The decisive goal of managers is to boost the gross profit margin through sales in order to retain or increase their market share. According to recent studies in retailing literature (e.g., Danisman & Smith, 2020; Flamand et al., 2018; Yapicioglu & Smith, 2012a, 2012b; Li, 2010) there is a strong relationship between the store layout and the shopper’s purchasing behavior. Much of the sales and marketing efforts for any retailer is related to its layout and the design of where to locate merchandise displays (Liao & Tasi, 2019). If retailers can find the locations within the stores where much of the sales activities occur and where consumers usually visit, they can understand where to locate products and promotions more effectively. Up to now, however, store managers generally make decisions about how to change in store layout ad hoc based on their judgement and expertise. In the retail industry, a well-known and crucial segment is supermarkets and grocery stores with a market size, measured by revenue, of $678.4 billion in 2020 (https://www.ibisworld.com). Grocery store managers can use some operational and marketing tools to improve the profitability of their facilities. Store layout

E. Danisman ˙ Industrial Engineering Department, Izmir Demokrasi University, Izmir, Turkey e-mail: [email protected] A. E. Smith () Industrial and Systems Engineering, Auburn University, Auburn, AL, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Ghoniem, B. Maddah (eds.), Retail Space Analytics, International Series in Operations Research & Management Science 339, https://doi.org/10.1007/978-3-031-27058-1_5

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is a key planning consideration that enables a store to improve sales volume per square foot of the selling space of the store (Pak, 2020). Therefore, the floor plan is important not only to maximize use of the available space but also to present goods advantageously (Liao & Tasi, 2019). Big data analysis, grounded in data mining, provides an important analytical tool for the discovery of hidden knowledge obtained by analyzing, understanding, or even visualizing data gathered from business and retail firms (Liao et al., 2008). Market Basket Analysis can aid the retailer to identify commonly purchased products in order to discover cross-selling opportunities, improve store layout, and manage the inventory (Tanuja & Govindarajulu, 2017). In the retailing literature, there are many methods including association rules, classification analysis, and probability heuristic analysis (Maita et al., 2015). Knowledge of the customer extracted by using data mining can be a powerful tool to design the business processes of retailing companies. A methodological approach analyzing in-store customer behavior with the purpose of optimizing space and store performance can greatly benefit retail managers. This chapter develops and implements a model to design grocery store block layouts by specifying the size and the location of the product category areas (“departments”). The aim is to improve revenue by especially concentrating on impulse purchases, where the purchase decision is spontaneous and occurs in the store (Abratt & Goodey, 1990; Kollat & Willet, 1967). We use customer market basket data to help determine the best product placement using data-driven techniques. By mining the market basket data (items purchased per customer visit), we can discern relationships among departments and identify chances for improving the magnitude of impulse purchases. Our partner is a major grocery store retailer, Migros which operates 2230 stores in 81 provinces of Turkey. Migros also has 41 Ramstores outside of Turkey. Together, these stores contain a total area of 2,508,743 square meters (Migros, www.migroskurumsal.com). Migros is the largest retailer in Turkey, and their data and insights from knowledgeable managers there were incorporated throughout. The rest of the chapter is organized as follows: a literature review of grocery store layout problem is given in Sect. 2. Section 3 introduces the demonstration case—a middle sized Migros grocery store in Turkey. The data-driven approach is covered in Sect. 4 along with a description of the discrete-event simulation modeling of the store and a tabu search metaheuristic for block layout. The computational results are presented in Sect. 5. Finally, Sect. 6 gives conclusions and potential future work.

2 Literature Review Companies are continually faced with dynamic market and technology requirements (Klausnitzer & Lasch, 2019). Facility design problems (FLP) deal with these challenges by re-determination of site locations, block layout, and detailed layouts

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(Tompkins et al., 2010). Despite the large amount of research and approaches for manufacturing companies FLP, there is little related literature for retail industries. The retail literature can be classified into two main streams: shelf space allocation problems (Ghoniem et al., 2016a, 2016b; Flamand et al., 2016, 2018; Ostermeier et al., 2021; Karki et al., 2021; Czerniachowska & Hernes, 2021) and store layout problems. The model in this chapter focuses on finding the optimal store layout for a supermarket. Despite the influential proportion of shelf space allocation problems and marketing strategies in the literature, the number of papers addressing block layout of retail stores is quite limited. However, designing a grocery store layout that increases sales is a challenging task. There are essential requirements to consider for finding a successful design such as size, space, and adjacencies. The past two decades witnessed a proliferation of research on retail operations (see Caro et al., 2020). The number of publications has appeared at a fairly steady pace, with a recent peak in 2018. The topics covered by these articles are pricing, assortment optimization, marketing, inventory management, and online retailing. There are a few studies addressing the block layout problem including those by Botsali and Peters (2005); Surjandari and Seruni (2010); Li (2010); Yapicioglu and Smith (2012a, 2012b); Ozcan and Esnaf (2013); Danisman and Smith (2020); Dorismond (2019), Guthrie and Parikh (2020), and Hirpara and Parikh (2021). According to (O’Brien, 2018) customers spent on average $5400 annually on impulse purchases of food, shoes, clothing, and household items. Impulse purchases are chosen unexpectedly in the store (Abratt & Goodey, 1990; Kollat & Willet, 1967). Thus, there is considerable scholarly research that explores the triggers of consumer impulse buying (see Iyer et al., 2020). A store’s layout influences the customer’s exposure to goods and thus affects the impulse purchases (Inglay & Dhalla, 2010). This motivates researchers to maximize revenue especially that of impulse purchases, as an objective function. An initial paper addressing the retail store layout problem is by Botsali and Peters (2005) which aims to maximize the expected impulse purchases of customers according to the locations of product categories. Later, Peng (2011) in his master thesis proposed an algorithm and a metaheuristic approach to maximize the impulse purchases of a grocery store which uses a grid layout. Another study from Aloysius and Binu (2013) presents an approach to maximize impulse purchases by using market basket data analysis of consumers. The latest study by Hirpara and Parikh (2021), proposes a model for the retail facility layout problem considering shopper paths that places departments to maximize expected per shopper impulse revenue. They present the impact of changes in shopper paths on the impulse revenue of departments by applying the approach to a U.S. retail store. Unlike the above-mentioned researchers, impulse purchase has been used to calculate total sales indirectly in some retail papers. For instance, a paper from Pinto et al. (2015) integrates regression models and a particle swarm metaheuristic to design space distributions of product categories for a retail store with the aim of maximizing sales. An empirical study is conducted and results show that its designs are both useful and “interesting” by business specialists. Bhadury et al. (2018) shows a p-dispersion model to optimize the placement of products in a retail store with the

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goal of maximizing total profit from the sale of impulse items using a simulated annealing metaheuristic. Actual data from a grocery store in the western region of New York is used to showcase the approach. Their limitations are that layout designs are all grids and dimensions of the departments are assumed to be equal sized or known a priori. Traffic density and the exposure of the layout has direct influence on the sales and revenue. Yapicioglu and Smith (2012a, 2012b) developed a bi-objective model that maximizes both the revenue and adjacency by considering traffic zones and departmental exposure. Guthrie and Parikh (2020), estimate exposure of the retail lack layout in 3D and validate their approach through human participants in a virtual layout. Their model analyzes the impact of three primary parameters of a layout, i.e., rack orientation, rack curvature, and rack height. A follow up study by Karki et al. (2021) combines rack configuration and shelf space allocation for retailers and solves the model by using a particle swarm optimization algorithm. Another study by Mowrey et al. (2019) shows the influence of varying angles of rack layout in the exposure of retail stores and illustrates the use of this approach to analyze a real store layout. Optimization models have been developed in retail stores layout analysis by some researchers. In Flamand et al. (2016), the problem of allocating grouped product categories to shelves and aisles is studied. They adapted the MIP model of Ghoniem et al. (2014) to a more tractable algorithmic approach by pre-computing the value of assigning certain product categories to aisles and shelves. The usefulness of the proposed optimization model is demonstrated by examining a case study motivated by a grocery store in New England. Another related example is from Ozcan and Esnaf (2013). These authors consider a bookstore layout and use a mixed integer mathematical model along with tabu search and a genetic algorithm for optimization of its design. Their approach includes the specific requirements of bookstore shelves. Further, they use association rules for the determination of the position of the books within a grid layout. Considering the overall literature, not so many papers have addressed grocery store layout and among those the data-driven aspects are not prominent. The papers impose a variety of constraints and limitations that make them less usable in practice. These include considering solely a grid layout and only allowing equal area departments. Grocery stores (and most other stores) usually have a combinations layout with both grid and racetrack elements along with varying sizes of departments. We do not depend on the grid only and equal area layout assumptions of these prior papers. Another critical aspect of this chapter is explicitly using the relationship between store layout and consumer behavior (as evidenced by the data). Some retail management papers have considered this point however they use a largely qualitative approach. In summary, four important distinctions between this chapter and the previous literature are: (i) Allowing a reasonable and common form for the department layout (we use a racetrack surrounding a grid), (ii) Letting departments vary in size and shape according to the objective function and arrange them with respect to preferred adjacencies and locational constraints (an example is oven equipment should be on an outer wall), (iii) Developing data driven approaches

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to relate sales to inferring a customer’s path within the store, and (iv) Verifying and validating the complete model using both discrete-event simulation along with real (that is, live) sales data after the implementation of the proposed layout at an actual store.

3 The Case Study We applied our methodology in a case study—a medium sized Migros store in the southern part of Turkey, namely, in the Antalya area. This store was selected by the Migros marketing and planning staff as it has not been renewed for 10 years. It is in city center and one of the most popular stores, especially in summer (Antalya is a noted resort area). The store is 1300 square meters and is categorized as a 3M Migros (Migros stores range from 1M, the smallest, to 5M, the largest). This Antalya store has ten straight aisles, a racetrack aisle, and one entrance and one exit as shown in Fig. 1. Since this is a touristic area, summer beach products are sold in a “beach products” department in this store. This is included in the “seasonal” category. We model the store as a combined grid/racetrack layout with departments of different sizes. However, the height and depth of all departments are the same while the length changes depending on the department. Each department has a pre-selected minimum area. This is to display essential and best-selling products. Departments face each other in pairs and are separated by an aisle within the grid section (inner section) of the layout. This type of layout design is the most commonly found in retail, especially for grocery stores. The company’s financial department collects revenue data for 29 product categories as shown in Table 1. Note that, the “oil and spices” category includes beans, lentils, and dried food and the “soft drinks” category includes juices. These comprise the 29 departments which optimize in terms of size and location within the store.

Fig. 1 Current layout of the Migros 3M grocery store in Antalya, Turkey with both grid and racetrack aisles

80 Table 1 Departments of a typical Migros store

E. Danisman and A. E. Smith

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

Product category Alcoholic Drinks/Tobacco Baby Products Bakery Barbecue Books/Magazines Cheese/Olives/Breakfast Chocolate/Cookies Cosmetics Dairy/Milk/Yogurt Deli/Side Dishes Detergents Electronics Fish Frozen Foods/Eggs Fruits/Vegetables Glassware Home Appliances Household Meat/Poultry Section Milk (on shelf, not refrigerated) Oil/Spices/Beans/Lentils/Dried Food Paper Products Pet/Hobby Seasonal/Beach Snacks/Nuts Soft Drinks/Juices Tea/Coffee/Canned Food Textile/Shoes Toys

4 Methodology The market basket data collected using association rule mining was used in determining consumer purchasing patterns and the relationships among departments and revenue. Using this analysis, we specified an adjacency matrix for the departments. This matrix defines the preferences for department proximity in a pairwise fashion. This adjacency forms one of the two objectives. Identifying buying characteristics of customers by analyzing the transactional data is crucial for every successful retail business. The well-known combination of beer and diapers is just an example of an association rule found by data scientists. The most commonly used algorithm to generate association rules is the Apriori algorithm however it has some limitations when it comes to analyzing customer transactions. It does not consider the purchase quantities and all items are viewed as having the same importance. For this reason,

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another methodology, the high utility mining algorithm, was also used. We use both of these algorithms to identify product categories that should be placed near each other in the layout.

4.1 The Apriori Algorithm Association rule mining is a powerful tool used to identify correlations or patterns between objects. Market basket analysis is a prominent way to derive associations by using the customers’ baskets to infer their buying habits (Karthiyayini & Balasubramanian, 2016). Association rules work primarily with the measures of support, confidence, and lift (Berry & Linoff, 2004; Zhang & Zhang, 2002). Lift provides information on whether an association exists and if the lift value concludes an association rule exists, the support value is relevant. Support is the probability that a set of items co-occurs with another set of items. Finally, confidence is calculated and this is the probability that a set of items is selected given that another set of items has already been selected (Aguinis et al., 2013). Let us assume that A and B are two different products in a retail store. Lift is defined as P(A∩B)/(P(A)*P(B)). A value greater than 1 indicates that the presence of A has increased the probability that product B will occur also in this transaction. Support is defined as P(A∩B), the probability that A and B co-occur. It measures the frequency of the rule within the transactions. Confidence, P(A∩B)/P(A), is the probability that a customer will select a set of items, given that this customer has already selected another set of items. It calculates the percentage of transactions containing A which also contain B. Various minimum threshold values for support and confidence can be found in the literature (Aguinis et al., 2013; Goh & Ang, 2007; Yang et al., 2007). Considering growth in the size of the data set (e.g., millions of transactions), the support value’s usefulness decreases. In these situations, support values can be rather low because there are so many other transactions and these act as noise in the data set. In this chapter, we have selected a minimum lift value of 1.1, a minimum confidence level of 30%, and a minimum support value of 5%, which are consistent with the literature concerning market basket data sets of our magnitude. Recall that support is only considered if both the minimum lift value and confidence value are satisfied. A calendar years’ worth of data was obtained from the company, and this dataset has almost 700,000 customers with 5.5 million products. Calculations of support, confidence, and lift for the Migros 3M store reveals that the largest lift value is oil/spices with deli. The second largest is household with deli, and the third largest is textile with beach products. The largest support value is between chocolate and soft drinks. Note that lift and support are symmetric, but confidence is not. For example, the oil/spices with coffee/tea confidence value is not the same as the coffee/tea with oil/spices confidence value. Table 2 lists the results which meet or exceed the thresholds. These results were shared with the headquarters staff at Migros, and

82 Table 2 The top ten rules for cut off values of lift 1.1, confidence 0.3, and support 0.05 from the Migros data mining effort

E. Danisman and A. E. Smith 1 2 3 4 5 6 7 8 9 10

Bakery-Fruit and Vegetables Meat and Milk Soft Drinks and Chocolate and Cookies Cosmetics and Chocolate and Cookies Detergent and Milk Paper Products and Soft Drinks Egg and Fruits Tea/Coffee/Canned Food-and Bakery Chocolate Cookies and Tea/Coffee/Canned Food Paper Products and Chocolate and Cookies

they concurred that the results are aligned with those obtained by their marketing consulting company.

4.2 The High Utility Data Mining Algorithm The Apriori algorithm is a prominent technique in data mining and knowledge discovery and has numerous applications in business, science, and other domains. The main objective of algorithm is to identify frequently occurring patterns of item sets; however, Apriori treats all the items equally by only considering if an item is present in a transaction or not. It does not reflect any other factor such as portion of revenue or profit (Liu et al., 2005). Sometimes, a retail business will be interested in knowing its most impactful customers (customers who contribute a major fraction of the profits to the company). These are the customers who might buy full priced, high margin, and/or gourmet products. These may be absent from most transactions because most customers do not buy these items. Since association rule mining algorithms do not consider some relevant factors such as the price, utility, importance, weight, or cost of items, they could miss patterns involving these impactful buyers (Gan et al., 2018). High utility item set mining can be used to facilitate business decisions and thereby adapt business strategies to increase sales and profit. A simple example is given in Table 3 to clarify the difference between the Apriori algorithm and high utility mining. If we have five transactions composed of three different items, according to the Apriori algorithm, item B would be the most frequent product since it is sold 5 units out of the five transactions. The revenue from B is only 1 unit. However, in high utility mining the revenue, or profit, of the item is also considered so the transactions with item C are weighted more since C has 5 units of revenue. Our retail partner, Migros, desired to identify the mostly valued/profitable item sets so we also mined the customer data by using the fast high utility mining algorithm developed by Liu et al. (2005). In this study, a two-phase algorithm which discovers all high utility items in a database using a transaction-weighted downward closure property is proposed. In the first phase, from the transaction database, the

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Table 3 A transaction database example (a) Transaction table where each row is a transaction. Columns represent the number of items of a particular transaction Transactions Item A Item B Item C 1 1 Transaction 1 – Transaction 2 1 1 – Transaction 3 – 2 – 1 1 Transaction 4 1 Transaction 5 – – 2 (b)The utility table where the right column shows the profit of each item per unit Item Revenue Item A 3 1 Item B 5 Item C (c) Transaction utility (TU) of the transaction database Transaction Utility 6 Transaction 1 Transaction 2 4 Transaction 3 2 9 Transaction 4 10 Transaction 5

quantity of the item and the revenue obtained from that item is selected. The utility of that item is defined as quantity of the item i times its unit revenue. Then by considering a threshold value the algorithm finds the item sets whose utility values are above a user specified threshold. Phase I may overestimate some low utility item sets, but it never underestimates any item sets. In phase II, one more database scan is performed to filter the overestimated item sets by using the transaction weighted downward closure property. This removes numerous candidates early on and finds the complete set of high utility items. Therefore, we used this algorithm for the analysis in this chapter. According to the calculations of our data set from 1 year, the top 10 valued item sets obtained with a 1.5% minimum utility threshold are listed below in order of utility (Table 4).

4.3 Characterizing Adjacencies After we obtained the association rules by the Apriori algorithm and High utility mining, we merged the two lists and those pairs appearing on either list were given priority to be close to each other. If a pair appeared on both lists, then the higher rank of either list was used. The ranks on the two lists were used to establish the translation to a REL chart. REL charts (see, for example, Heragu, 1997 or Tompkins

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Table 4 The top ten rules for a cut off value of 1.5, utility (revenue-based quantity) from the Migros data mining effort

1 2 3 4 5 6 7 8 9 10

Cheese and Tea/Coffee/Canned Food Alcoholic Drinks and Soft Drinks Cheese and Oil Spices Tea/Coffee/Canned Food and Chocolate Cookies Detergent and Oil Spices Cheese and Soft Drinks Cosmetics and Chocolate Cookies Soft Drinks and Snacks/Nuts Paper Products and Tea/Coffee/Canned Food Meat and Soft Drinks

et al., 2010) are widely used in facility layout problems and they usually define the closeness ratings given in Table 3. The top two department pairs obtained from both Apriori and High utility mining, are considered ‘absolutely necessary’. The rest of them are rated as ‘especially important’ since they are on the lists. Furthermore, the store manager’s preferences are incorporated into the adjacency matrix in Table 5. For example, the manager states that baby products should not be next to detergent and pet food, so we added this insight to the chart as “undesirable”. He also mentioned the relationship between the fruit and vegetable department with the fish department. Locating these two departments next to each other allows customers to shop the fresh foods while waiting for the fish to be prepared. Departments are considered adjacent if they share a common edge or if they are across the aisle from each other. If the racetrack aisle separates departments, they are not considered adjacent. The REL chart for the 29 product category departments is given in Table 3. We use the layout efficiency adjacency metric devised by Yapicioglu and Smith (2012a, 2012b) which measures how the proposed layout performs by calculating the relative difference between this candidate design and an ideal design, that is, one that perfectly fulfills all adjacency preferences. The layout adjacency efficiency is denoted by ε. The cij notation is the REL value and whether it is positive (cij > 0) or cij negative (cij < 0). The xij indicator variables are whether departments are adjacent (1) or not (0). The number of departments is denoted by n. n−1 

ε=

 n−1 n     + − 1 − xij xij − cij cij

n  

i=1 j =i+1

n−1 

i=1 j =i+1

n 

i=1 j =i+1

+ cij

−

n−1 

n 

i=1 j =i+1

(1) − cij

The maximum value for ε is 1. The denominator is the largest possible value of the adjacency score for a given matrix of adjacency ratings (i.e., a layout where all department pairs with c+ ij are adjacent and all department pairs with c− ij are not). The numerator is the calculated adjacency score of the candidate layout.

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Table 5 Typical adjacency scores for a REL chart

1 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

3

4 5 6

0 0 0 25 25 0

0 0 0 0

0 0

7 8 0 0

9 0

0 0 25 0

0 0 0 0 0 25 0 0

Rang

Definion

Value

A

Absolutely Necessary

125

E

Especially Important

25

I

Important

5

O, U

Ordinary Closeness

0

X

Undesirable

-25

10 11 12 13 14 15 16 17 18 19 20 21 0

0

0

0

0

0

0

0

0 0 0 25

0

0

25 25 25 0

25 25 25 0

0

0

0

0

0

0

0

0

0

0

0

25 0

0

0

0

0

0

25 0

0

0

0

0

0

0

0

0

0

0

0

25 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

25 25 125 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

25 0 0 0 0 25 0 0

0

0 0

0

0

0 25 0

0 25 0

0 0 0 25 0 0

0

0

0

0 25 0

0 0 -25 25 0 0

0

0

0

0

0

0

0

25 0

0

0

0

0

0

0

0

0

0 0 0 0 0 25 25 25

0

0

0

25 0

0 0

0

25 25

0

0

0

0

0

28 29

0 25 0 25 0 0

0

0

22 23 24 25 26 27

25

0 25

0

25 0

25 25 25

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 25 0 -25 25 25

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

125 0

0

0

0

25 0

25

25 0 0 0 25 0 0

0

0

0

0 0

0 0 0 25 25 0

0 25 25 25 0

0

0

0 0 25 0 25 25 25

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

25

0

0

0

0

0

0

0

0

0 0

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4.4 Shelf Space Allocation and the Revenue Function Since demand is non-linear relative to shelf space allocation, the revenue function must consider the space elasticities of product categories. Space elasticity measures the change in unit sales for a unit change in shelf space allocated (Curhan, 1972). As with previous studies in the literature, the concept of diminishing returns in revenue with respect to length of the shelf is used in this chapter. Inspired by the model proposed by Danisman and Smith (2020) and the view of the marketing managers of Migros, the estimated space elasticity, β, of each department is given in Table 6. These are used in the revenue objective function in Eq. (2) which calculates the total revenue of department i using a base amount by product category (the first term in the equation) and an increasing, but with decreasing rate, of revenue based on the area for that product category above its lower bound. The variable s is the shelf space with upper (U) and lower (L) bounds and r is the revenue per unit space for that department.  βi Ri ri siL + ri si − siL

(2)

siL ≤ si ≤ siU

(3)

5 Discrete-Event Simulation Modeling of the Migros Store To provide strong validation of our approach, a discrete event simulation model is developed to estimate the total revenue of the store for a given layout. Initially, the simulation model creates a shopping list dynamically for each buyer by using the probabilities calculated by the analysis of the market basket data from the Migros store. With the data of the monetary value of the purchases from each department by each consumer, the unit revenue for each department was calculated—that is, the average amount spent per customer visit. We used triangular distributions for the amount spent and also for the probabilities to visit by department. Triangular distributions are straightforward and need few estimation parameters. Annual data was analyzed and both optimistic and pessimistic revenue values per department were estimated using to the amount bought by the customers, the price of the items, and insights of store managers. The simulated customer moves through the store using the shortest path and selects items on her/his shopping list. A product’s impulse rate determines the likelihood that the shopper will make additional (impulse) purchases along the route by passing by other items and displays. Because of the lack of knowledge concerning which purchases in the market basket data are planned or are impulse, we assumed impulse rates of the departments estimated by the store management using a 5-point Likert-type scale ranging from 1 (very low) to 5 (very high). This scale from Verplanken and Herabadi (2001) shows the impulse purchase tendency

Data-Driven Analytical Grocery Store Design Table 6 Space elasticity βi for each of the 29 departments

Department Alcoholic Drinks/Tobacco Baby Products Bakery Barbecue Books/Magazines Cheese/Olives/Breakfast Chocolate/Cookies Cosmetics Dairy/Milk/Yogurt Deli/Side Dishes Detergent Electronics Fish Frozen Food Fruits/Vegetables Glassware Home Appliances Household Meat/Poultry Section Milk(on shelf) Oil/Spices (Beans/Lentils/Dried Food) Paper Products Pet/Hobby Seasonal/Beach Snacks/Nuts Soft Drinks/Juices Tea/Coffee/Canned Food Textile/Shoes Toys

87 Space elasticity (β) 0.95 0.85 0.95 0.75 0.85 0.95 0.95 0.95 0.95 0.95 0.85 0.75 0.65 0.75 0.95 0.85 0.75 0.85 0.75 0.95 0.95 0.95 0.65 0.85 0.95 0.85 0.95 0.75 0.65

of customers per department. It is assumed that the customer will spend on average 10% of average sales times the impulse rate of that department that is along the shopping route. The impulse purchase rates used are in Table 7. Whenever a customer moves through a department, a random number is generated between 0 and 1. This number is compared to an upper threshold value for that department, which is based on its likelihood of impulse purchases. For instance, there is a threshold value of 0.1 for an impulse rate of 1 and a value of 0.8 for impulse rate 5. If the random number is below the threshold, the shopper makes some impulse purchase in that department. For instance, the average purchased amount for alcoholic drinks/tobacco is 5.80 TL (Turkish Lira) and this department is assigned an impulse rate of 3 from the Likert scale. The average impulse amount totals 3 × 5.80 × 0.10 = 1.74 TL. Using a triangular distribution, the impulse amount would be (1.50, 1.74, 2.5) TL in the simulation. The minimum and maximum values

88 Table 7 Likert scale assigned impulse purchase rates by department where 1 is lowest and 5 is highest

E. Danisman and A. E. Smith No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

Product category Alcoholic Drinks/Tobacco Baby Products Bakery Barbecue Books/Magazines Cheese/Olives/Breakfast Chocolate/Cookies Cosmetics Dairy/Milk/Yogurt Deli/Side Dishes Detergents Electronics Fish Frozen Foods/Eggs Fruits/Vegetables Glassware Home Appliances Household Meat/Poultry Section Milk(on shelf) Oil/Spices (Beans/Lentils/Dried Food) Paper Products Pet/Hobby Seasonal/Beach Snacks/Nuts Soft Drinks/Juices Tea/Coffee/Canned Food Textile/Shoes Toys

Impulse rate 3 1 5 2 3 3 5 5 4 4 1 2 3 2 4 4 3 3 2 1 1 1 3 4 5 3 1 5 3

are assigned by considering the average spent amount from the market basket data. Thus, impulse are assigned to each simulated customer. A comparison of actual revenue by department and the expected revenue from the simulation is shown in Table 8. They are quite congruent. The simulation agrees statistically with the real sales data. The marketing staff at Migros concurred that this revenue projection is in line with their expectations. From a one-way analysis of variance (ANOVA) there is no statistically significant difference between the actual store data and the simulation models at 99% confidence (Table 9). The simulation model can be used for further analysis of the recommended store designs and also validates the algorithmic approach. Objective function (Eq. 2) does not replicate the stochastic simulation exactly, but the two approaches output very similar results. It is not computationally practical to run a stochastic simulation during the optimization of the design of the store’s block layout because the simulation would need to

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Table 8 Comparison of actual revenues per day with simulated revenues per day by department in Turkish Lira Product category Alcoholic Drinks/Tobacco Baby Products Bakery Barbecue Books/Magazines Cheese/Olives/Breakfast Chocolate/Cookies Cosmetics Dairy/Milk/Yogurt Deli/Side Dishes Detergents Electronics Fish Frozen Foods/Eggs Fruits/Vegetables Glassware Home Appliances Household Meat/Poultry Section Milk(on shelf) Oil/Spices (Beans/Lentils/Dried Food) Paper Products Pet/Hobby Seasonal/Beach Snacks/Nuts Soft Drinks/Juices Tea/Coffee/Canned Food Textile/Shoes Toys Total

Total revenue Actual value 12, 846 2275 2601 87 302 2412 4524 3947 3219 12, 411 3100 1373 2888 1153 7803 1396 375 1867 11, 178 1739 4220 2063 479 3774 7549 4006 3130 1675 592 104, 984

Simulated value 13, 056 2276 2305 57 404 2237 4557 3779 3256 12, 689 3063 1044 2859 1353 7672 1262 337 1973 11, 341 1566 4430 2071 614 4041 7547 3921 3019 1687 528 104, 944

be run thousands of times. Therefore, it is vital to know that this straightforward algorithmic model of revenue is congruent with the revenues actually realized. Equation (2) is used in the tabu search optimization described in the next section. The discrete-event simulations are used for the final candidate store designs to more completely understand the expected performance of the layout design.

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Table 9 ANOVA results for store data versus simulation of Table 8 Between groups Within groups Total

Sum of squares 27.586 695, 596, 956.759 695, 596, 984.345

df 1 56 57

Mean square 27.586 12, 421, 374.228

F .000

Sig. .999

6 Tabu Search Optimization for Store Layout Design To optimize the block layout of the Migros store, we chose to use a tabu search (TS) metaheuristic. TS was selected since it has been used efficiently and effectively for many combinatorial optimization problems. As in our earlier study of grocery store layout problems (Danisman & Smith, 2020), the layout design uses two objectives— revenue generation and adjacency satisfaction—and TS can be easily adapted for this. The primary motivation of bi-objective optimization is that revenue and adjacency often conflict. The multinomial tabu search (MTS) algorithm developed by Kulturel-Konak et al. (2006) was used. This identifies the Pareto set of nondominated solutions and the details are below: (a) Solution representation: The layout is encoded as a 3 by 29 matrix since there are 29 departments in the store. The first row of the matrix is the bay number assigned to a department. Departments are either placed in a single racetrack bay along the perimeter of the store, or in one of the grid bays where there are reciprocal aisles. Because of the preferences of store management. A department can only be in a single bay. All bays are arbitrarily numbered. The second row of the matrix is the ordering of the departments beginning from next to the entrance and continuing counterclockwise around the racetrack bay. With the grid aisles, the ordering starts from top to bottom. The third row is the length of the department in the bay (remember that widths and heights of the racks are identical across all departments so only length is variable). Table 10 shows an example. To start the algorithm, the user must input the total bay lengths, the department minimum and maximum lengths (i.e., sizes), and, from the store actual data, the unit revenues per length of department, the β of each department (space elasticity of each department), and the adjacency matrix (the ε values). (b) Constraints: Length and adjacency. Departments must be within the range of their minimum and maximum lengths—and the total of department lengths must equal the total length available in the bay. Departments which have a REL score of −25 must not be placed next to each other. (c) Initial solution: A random department sequence string (that is the second row of the encoding) is generated. Using that department string, all possible bay strings (that is, the first row of the encoding) are tried. The total minimum lengths of departments in the bay must be ≤ bay length ≤ total maximum lengths of departments in the bay. The algorithm searches until no feasible bay

12 5 7

12 12 6

12 17 2

12 23 9

12 4 4

12 16 27

12 24 6

12 25 9

12 1 14

12 13 6

12 3 3

12 19 14

12 15 10

12 9 12

12 10 23

12 14 12

11 29 9

11 2 13

10 8 33

9 7 27

8 27 22

7 21 22

6 28 11

5 18 11

4 11 22

3 22 22

2 26 33

1 20 5

1 25 6

1 6 11

Table 10 Solution encoding of Design 3 (adjacency = 0.74) where the top row is the bay number, the second row is the sequence within the bay, and the third row is the length of the department

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(d)

(e)

(f)

(g)

(h)

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strings and unassigned departments exist. Then, the length string (that is, the third row of the encoding) is developed. From all possible bay strings and the given department string, specify the lengths of departments by considering the minimum and maximum length constraints. Finally, calculate the total revenue for each candidate layout. Choose the layout giving the maximum total revenue. Move operator: A swap operator is used which exchanges the location of two departments. The number of solutions reachable using the swap operator equals n × (n − 1)/2 = 29 × 28/2 = 300. After the swap, the best bay arrangement and department lengths for that sequence are identified using the procedures of step c above. Tabu list entries: The most recently swapped department pairs are stored in the tabu list, prohibiting the pair from being swapped again during its duration on the tabu list. The length of the tabu list varies uniform randomly between 15 and 30 entries. This is a typical method in TS, which engenders robustness of the search. Objective functions: Randomly alternates between maximizing total revenue (TR) and maximizing the adjacency efficiency, ε, according to the method of Kulturel-Konak et al. (2006). This generates the non-dominated set of designs which have the highest total revenue and the highest adjacency score. Aspiration criterion: If a candidate solution dominates at least one of the nondominated solutions then that move to that solution is allowed even if it is on the tabu list. Termination criteria: A maximum number of iterations (e.g., 1000 iterations) is used as a first termination criterion. As a second criterion, if no new nondominated solutions have been found for a certain number of consecutive moves (such as 50), the search terminates. These termination criteria are typical mechanisms in TS. The flow chart of tabu search algorithm developped for this problem is given in Fig. 2.

7 Computational Experience We used the tabu search to optimize the layout of the Migros 3M store in Antalya. As the results in Table 10 show, we found three non-dominated solutions in terms of the adjacency and the total revenue by using the deterministic objective function (Eq. 2), as reported in Table 11. Before, explaining these designs, we should clarify the change in store length, which was 378 m in the current layout. There was an additional mobile rack in front of the cosmetics department which was used only for that department. The managers decided to change this mobile rack to an additional grid rack that could be used for any product category and the proposed layouts assumed this change. The first layout has an expected revenue per day of 109,908 TL with 0.68 adjacency (maximum revenue), the second has 0.82 adjacency and expected revenue of 108,522 TL (maximum adjacency), and the third has

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MARKET BASKET DATA ANALYSIS bay capacity department lengths units revenues space elasticity adjacency matrix solution representation

START OPTIMIZATION generate random solution (department string)

generate neighborhood of permutations (swap departments is department string) generate all possible bay structures

generate best length string

choose the best non tabu solution if it is tabu, check aspiration criterian update tabu list (department pairs swapped)

update the best solution

termination criterion N Y report the best solution

END OPTIMIZATION

SIMULATION OF PROPOSED LAYOUT

Fig. 2 Flowchart of the tabu search optimization algorithm

expected revenue of 109,836 TL and 0.74 adjacency. Depending on the relative preferences for either revenue or adjacency, the store management can choose the final layout from the set of three non-dominated solutions found by the TS. In this case of Antalya, the managers preferred the third layout which is shown in Fig. 3 because of the desire of locating pet products or detergent far from baby products. Note that all three optimized designs dominate the current store layout in terms of both adjacency and expected revenue.

Alcoholic Drinks/Tobacco Baby Products Bakery Barbecue Books/Magazines Cheese/Olives/Breakfast Chocolate/Cookies Cosmetics Dairy/Milk/Yogurt Deli/Side Dishes Detergents Electronics Fish Frozen Foods/Eggs Fruits/Vegetables Glassware Home Appliances Household Meat/Poultry Section Milk (on shelf, not refrigerated) Oil/Spices/Beans/Lentils/Dried Food Paper Products Pet/Hobby

Product category

Current design Revenue Length 12,846 14 2275 11 2601 3 87 4 302 7 2412 6 4524 17 3947 32 3219 12 12,411 23 3100 22 1373 6 2888 6 1153 12 7803 10 1396 29 375 3 1867 9 11,178 14 1739 5 4220 22 2063 11 479 9

Design 1 Revenue 12,846 2689 2601 65 302 3216 5855 4440 3219 12,411 2677 1144 2888 1153 7803 1300 375 2282 11,178 1391 4604 2063 426 Length 14 13 3 3 7 8 22 36 12 23 19 5 6 12 10 27 3 11 14 4 24 11 8

Design 2 Revenue 12,846 2689 2601 65 302 2814 5322 4687 3219 12,411 3100 1831 2888 1153 7803 1252 375 2074 11,178 2087 4220 2251 426 Length 14 13 3 3 7 7 20 38 12 23 22 8 6 12 10 26 3 10 14 6 22 12 8

Table 11 Expected revenue of the three non-dominated designs found by the tabu search versus that of the current store layout Design 3 Revenue 12,846 2689 2601 87 302 4422 7185 4070 3219 12,411 3100 1373 2888 1153 7803 1300 250 2282 11,178 1739 4220 2063 479 Length 14 13 3 4 7 11 27 33 12 23 22 6 6 12 10 27 2 11 14 5 22 11 9

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Seasonal/Beach Snacks/Nuts Soft Drinks/Juices Tea/Coffee/Canned Food Textile/Shoes Toys Total revenue Adjacency

3774 7549 4006 3130 1675 592 104,984 0.48

11 5 22 22 22 9 378

3774 9059 4552 3557 1447 592 109,908 0.68

11 6 25 25 19 9 390

3774 9059 4006 3130 1675 526 108,522 0.82

11 6 22 22 22 8 390

2059 7549 6009 3130 838 592 109,836 0.74

6 5 33 22 11 9 390

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Fig. 3 Chosen layout (design 3) from Table 11 above

To verify and detail the expected revenues, the layout selected by the managers was simulated in SIMIO for 10 replications and 10,000 h run using the simulation model described in Sect. 5. (It takes about 2 h for this simulation to complete.) Table 12 gives the expected revenues of each department for the proposed layout from the detailed simulation. It agrees well with that found by the deterministic objective function. According to the results of the ANOVA in Table 13 for the proposed layout, here is no statistically significant difference between the store revenue used in the tabu search from the deterministic algorithm and the discrete-event simulation estimates of these. The project with the store began in January of 2018 and the new layout, designed by the procedure described in this chapter, was implemented in March of 2019. The sales data from this new design was compared with the previous sales data as shown in Table 10. The following are the results of analysis of two designs. The basket size of sales per customer is now 61.9 TL which means an increase of 3.8% in real value (accounting for the inflation present in Turkey over that year) since the new layout was applied (considering 5 months of sales ending in July 2019). This inflationadjusted 3.8% rate is an impressive increase considering the low profit margins in the grocery store retail segment. The average daily number of customers that visited the store stayed about the same when compared with the same months in 2018. The reason behind this is explained by the managers who said that two new stores from a competing grocery store chain opened at the beginning of 2019 in the same region. So, even though the shopping options of the customers increased, they loyally kept shopping at this 3M store after the new layout arrangement. Also, the managers mentioned that a new Migros store (small 1M size) opened 1 km away from this 3M store which could have influenced the number of customers that visit per day (Table 14).

Data-Driven Analytical Grocery Store Design Table 12 The simulated expected revenues of the proposed layout (Tabu 3)

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Product category Alcoholic Drinks/Tobacco Baby Products Bakery Barbecue Books/Magazines Cheese/Olives/Breakfast Chocolate/Cookies Cosmetics Dairy/Milk/Yogurt Deli/Side Dishes Detergent Electronics Fish Fruits/Vegetables Glassware Home Appliances Household Meat/Poultry Section Milk (on shelf) Oil/Spices (Beans/Lentils/Dried Food) Paper Products Pet Hobby Seasonal/Beach Snacks/Nuts Soft Drinks/Juices Tea/Coffee/Canned Food Textile/Shoes Toys Total revenue

Simulation results 13, 288 2445 1688 341 61 4522 2154 7344 3211 13, 112 3654 1245 2997 8088 1466 127 2133 11, 276 1698 5043 2234 407 2113 7908 6344 3448 897 613 109, 857

Table 13 ANOVA results for store data versus simulation of Table 11 Between groups Within groups Total

Sum of squares 502, 944.845 745, 223, 176.276 745, 726, 121.121

df 1 56 57

Mean square 502, 944.845 13, 307, 556.719

F .038

Sig. .847

98 Table 14 Monthly revenue comparisons between the existing layout in 2018 and the newly implemented layout in 2019 (blue shaded rows)

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Month-Year

Daily Average Average Daily Average Number of Basket Size Revenue Customers (TL)

March 2018

104,455

2064

50.6

April 2018

102,796

2116

48.5

May.2018

108,026

2110

51.1

June 2018

111,070

2153

51.5

March 2019

124,947

2038

61.3

April 2019

118,811

2095

56.7

May.2019

123,525

2136

57.8

June 2019

130,647

2111

61.9

8 Conclusions and Discussion This chapter explains a data-driven approach for the grocery store block layout problem and the proposed approach has been applied to an actual grocery store located in Turkey. The approach identifies a Pareto optimal set of block layout designs considering both overall revenues and adjacency objectives. One of the resulting designs was implemented in the store with striking results. Sales significantly rose each month despite increased competition in the local area. The method, although applied to a specific store, is general and allows for flexibility by incorporating both a racetrack aisle and grid aisles. Impulse purchase rates and unit sales per display areas can easily be customized to a given store, as can the set of departments and the REL values between department pairs. The approach is pragmatic and effective, key attributes when dealing with the retail industry. These results impressed the Migros managers, and they decided to use the same approach to change the layout of a store in Isparta, another city in southern Turkey. As a follow up study, the comparison between the customer segments of these two stores and the total revenue change could be analyzed after implementation of the new layout in Isparta. Another enhancement would be to track customers as they move through the store so that more accurate paths could be established. We used the shortest path among the departments from which the shoppers bought goods, but we know that this is a lower bound and many paths will include additional movements and aisles.

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Optimizing Stock-Keeping Unit Selection for Promotional Display Space at Grocery Retailers Olga Pak, Mark Ferguson, Olga Perdikaki, and Su-Ming Wu

1 Introduction The grocery store sector is one of the major players in the US retail and foodservice industry. Although the retail brick-and-mortar industry continues its downfall with more than 9300 physical stores shut down in 2019 alone (Peterson, 2019), the grocery store industry continues to strive. According to the US Census, monthly food sales at grocery stores are at $62 billion (3 billion higher from last year) (US Census Bureau, 2020). The footage of retail food space per person is nearly 30 times higher than in 1950 (Haddon and Jargon, 2017). Most grocery stores have a very common layout. The store perimeter typically carries fresh, high-margin foods like produce, bakery items, meats, dairy products, and deli, whereas the center aisles (twelve or more) are located inside the perimeter and carry shelf-stable fast-moving consumer goods ranging from beverages to canned soups to toiletries. A combination of factors over the years (e.g., nationwide anti-obesity campaigns to encourage shopping on the outside store perimeter (Mayo Clinic, 2018)) have changed the way consumers physically move around the store. Studies show that an overwhelming majority of grocery shoppers shop the perimeter

O. Pak () Pennsylvania State University, State College, PA, USA e-mail: [email protected] M. Ferguson · O. Perdikaki University of South Carolina, Columbia, SC, USA e-mail: [email protected]; [email protected] S.-M. Wu Oracle Retail, Waltham, MA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Ghoniem, B. Maddah (eds.), Retail Space Analytics, International Series in Operations Research & Management Science 339, https://doi.org/10.1007/978-3-031-27058-1_6

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of a grocery store and avoid entering the center aisles (Sorensen, 2008; Shop Association, 2016). As a result of this shopping behavior, products located on promotional endcap displays, which are located on the outside edges of a store’s shelf space perimeter, enjoy substantially more face time (Larson et al., 2005). A recent industry study of two million grocery store shoppers shows that products placed on these displays are seen by nearly twice as many store visitors as products that are located in the inner aisles of the store (Wade, 2014). Grocery store managers can take advantage of this behavior. By placing a product in these areas, they can drive impulse purchases for new or existing products by exposing the products to potential customers who may not have originally planned to purchase an item from that particular product category. An example of an unplanned impulse buy is when a father stops by a store to pick up some baby formulae but, upon passing by a promotional display space of beer, decides to include a six-pack of beer in his purchase. Given the low-cost but high-return nature of promotional display space, it is surprising that, while the academic literature and retail software solution providers offer a variety of optimization solutions for the inner shelf-space management and assortment optimization (Chong et al., 2001; Kok and Fisher, 2007; Rooderkerk et al., 2013; Geismar et al., 2015), there is very little guidance for retailers on how to optimally determine when and what products to place on their promotional display space. In this chapter, we offer two distinct methodologies (Direct and Hierarchical) that allow a local store manager to estimate the relative lift of an SKU and develop an optimization framework to identify the most display-profitable products to place on a promotional end-cap display. To build our models, we use retail CPG sales scanner data on beer sales from hundreds of grocery stores from various grocery store chains in the New England region of the USA (Bronnenberg et al., 2008). We find that the estimated average display effect (i.e., sales lift) for the beer category across all SKUs and all weeks is 27%, which makes the promotional display a very effective tool for stimulating incremental product sales.

2 Literature Review Our methodology provides a decision support tool and joins a list of other papers that develop various decision support tools in the assortment optimization literature. However, our work is different from others in that we focus on the selection of SKUs from a given assortment to be placed on promotional display space, whereas others are concerned with regular shelf-space. Kok and Fisher (2007) propose an SKUspecific product assortment optimization by developing an iterative heuristic in the form of a knapsack problem. Rooderkerk et al. (2013) use store-level scanner data to develop an attribute-based demand estimation model and a profit-maximizing product assortment heuristic. Chong et al. (2001) develop a nested multinomial

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logit (NMNL) model to identify the optimal brand-level product assortment and use household shopping data to estimate the model. Boada-Collado and Martínezde-Albéniz (2014) offer a variation of Kok and Fisher (2007) assortment planning problem by optimizing SKU-level assortment using choice modeling in a multiperiod setting. Fisher and Vaidyanathan (2014) present a product assortment decision support tool for SKUs that have never been carried before. Other studies have added a shelf space dimension to the assortment problem. For example, Hubner and Schaal (2017) maximize a retailer’s profit by simultaneously choosing the assortment and shelf-space under stochastic and space-elastic demand. For a more comprehensive review of the product assortment literature, we refer the reader to Kok et al. (2008), Maddah et al. (2011), Kok et al. (2015), and Mou et al. (2018). Another relevant stream of articles looks into price-promotion planning, where various methodologies are proposed to make price-promotion scheduling more costeffective and profitable. Cohen et al. (2017) develop a profit-maximizing price promotion optimization problem and show that their optimized promotion schedule can improve a retailer’s profit by 3%. Baardman et al. (2018) model flyer and TV commercial promotions decisions as a non-linear bipartite matching-type problem. Their optimized promotion assignments result in up to 9% profit improvement. Natter et al. (2007) provide a decision support tool for price promotion activities. An actual business implementation of their solution leads to their partner’s profit and sales increase of 8.1% and 2.1%, respectively. Other relevant work includes Allenby (1989), Ailawadi et al. (2006), Ailawadi et al. (2007), Dawes (2012), and Gong et al. (2015) in marketing and Flamand et al. (2016), Flamand et al. (2018), Mowrey et al. (2018), Ozgormus and Smith (2020), Botsali (2007), and Dorismond (2019) in retail layout optimization. Only two prior studies, Ma and Fildes (2017) and Cetin et al. (2020), focus on promotional display allocation decisions. Ma and Fildes (2017) simultaneously optimize three different types of promotional activities such as price, display, and feature advertising for SKUs that have been placed on promotional display at a given store location. Since most SKUs might have never been put on display at a single store location, this can significantly limit the selection of SKUs available for display consideration. Besides, they only estimate the display effect at the aggregate level, as opposed to our week-SKU level. Cetin et al. (2020) propose a stylized optimization model without an estimation procedure. They show that low popularity/high margin products are better candidates for promotional displays to promote impulse buys, whereas high popularity/low margin products should be kept in the store’s inner-aisles. Cetin et al. (2020), however, do not demonstrate how to estimate the parameters needed for their stylized model.

3 Methodology Let’s assume that once deciding on a product category, a store manager seeks to identify a particular profit-maximizing SKU to put on a promotional display. An effective way to do this is to calculate an additional sales lift the store manager

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will get from placing each SKU on display, and once known, choose the SKUs that provide the highest lift. Thus, we first, estimate the relative lift of each SKU candidate for promotional display, which is later fed into an optimization framework to identify the most profitable one. The computational challenge arises when the number of SKUs that can be selected across the different subcategories and evaluated simultaneously becomes too large to efficiently estimate the needed model coefficients. To accommodate for this, we distinguish between two distinct scenarios that each require a unique handling approach. In the first scenario, we limit the number of working SKUs to evaluate. In this case, the manager can simultaneously evaluate all the selected SKUs in all the subcategories, thus, leveraging the acrosssubcategory variation in sales, in addition to the within-subcategory variation, which results in higher efficiency of the estimates and a better precision (see Sect. 3.1). The Direct approach may be more suitable in situations where grocery retailers can restrict the potential SKU candidates to put on a promotional display to a reasonable size in exchange for better precision of the estimated parameters. In the second scenario, if the number of SKUs becomes too large, the store manager can use the existing subcategory information and evaluate each subcategory separately. This approach comes at a loss of capturing across-subcategory variation in sales, during the estimation process (see Sect. 3.2). The Hierarchical approach may be better suited in situations where grocery retailers would like to evaluate a much wider variety of SKUs as potential candidates at the expense of lower estimation precision. Figure 1 summarizes the two approaches.

Fig. 1 Difference between the methods

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3.1 Direct-Static Methodology 3.1.1

Sales Response Function

Measurement of the incremental sales lift requires the estimation of an econometric model. Our econometric model (1) is built upon the original Scan*Pro model, which incorporates various promotional instruments, including display, as predictors, and allows the use of syndicated sales scanner data (Wittink et al., 1988; Foekens et al., 1994). Weekly unit sales of an SKU on promotional display are captured by controlling for seasonality and marketing-related activities (i.e., discounts, temporary price reductions, advertisements, coupons) at the store/week level. Since the majority of grocery store purchases are done in smaller quantities, the dependent variable is log-transformed to mitigate the positively skewed distribution of sales in our dataset. The total display effect is captured through the main display effect and two related interaction terms, Display-Week and Display-SKU, since marketing mix instruments can be time-dependent (Mela et al., 1997) and vary by product (Blattberg et al., 1995). Weekly indicators not only account for seasonality in the consumption of the product (Fok et al., 2007) but also for potentially unobserved weekly effects, such as manufacturer advertising. The model also accounts for subcategory seasonality by including the interaction term Subcategory-Week.1 The SKU’s price is captured indirectly through its percentage discount, also referred by Nijs et al. (2001) and Raju (1992) as “promotional depth”.2 Inclusion of store dummies controls for store-specific fixed effects. These store-specific fixed effects are critical, as they eliminate bias that might otherwise occur from factors such as store size and unobservable manager skills that could affect both store sales and promotional display decisions. For a detailed description of the notations we use in this chapter please refer to Table 1. Intercept

Log Unit Sales

   ln Sj ti



=

δ0

Display

SKU

% Discount

          + δ1z Zj z + δ2 Dj ti + δ3 Hj ti z∈U

Week

Marketing Mix       T   + δ4t  Wt  t + δ5m Mj mti t  =1

m∈M

1 As opposed to SKU seasonality, subcategory seasonality reduces the number of interactions that have to be estimated and prevents overfitting. 2 An interaction term between display and price reduction turned out to be insignificant and thus is not included.

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Table 1 Description of variables and symbols Description Variable .ln Sj ti .Aj a .Dj ti .Wt  t .Mj mti .Hj ti .Zj z .Bi  i

Log unit sales of SKU j for .j = 1, 2, . . . , J in store i for .i = 1, 2, . . . , I in week t for .t = 1, 2..., T Indicator variable, .= 1 when SKU j is part of subcategory .Va Indicator variable, .= 1 when SKU j at store i in week t is put on display; 0 otherwise Indicator variable, .= 1 if .t = t  ; 0 otherwise Indicator variable, .= 1 when marketing mix instrument .m ∈ M is applied to SKU j at store i in week t Size of price reduction in cents for SKU j at store i in week t Indicator variable for SKUs, = 1 if j equals SKU z; 0 otherwise (one “dummy” represents all “other” SKUs not included in the consideration set) Indicator variable, .= 1 if .i = i  ; 0 otherwise

Symbol .xj t .j ti .bj .kt .cj t .qj ti .lj ti .πj ti .oj ti .U .Ud .V

C .Va .Dti .M

T .T˜ .Qj

R .

Binary decision variable which equals to 1 if SKU j is placed on display in week t, 0 otherwise The incremental profit obtained from placing SKU j on display at store i in week t The maximum number of times an SKU j can be promotionally displayed across the time horizon. The maximum number of displays available in week t A binary indicator, equals to 1 if SKU j goes from being on display to being off display in week t The base demand for SKU j in week t at store i The display lift for SKU j in week t at store i The profit margin of SKU j sold in week t at store i A manufacturer-sponsored trade promotion for SKU j sold in week t at store i A set of SKU candidates evaluated to be placed on a promotional display with the proposed approach SKU j is a legitimate candidate for display d A set of SKU candidates evaluated to be placed on a promotional display with the Hierarchical approach The number of subcategories A disjoint union partitioning .V into C subcategories; .V = V1 ∪ V2 ∪ · · · ∪ VC The set of displays available at week t at store i The set of all marketing mix instruments including temporary price reduction, coupon, feature and advertisement excluding promotional display The number of weeks of history used in the estimation The number of weeks to optimize for in the (dynamic) optimization A consecutive set of weeks The cost the retailer incurs every time a product on display is replaced Smearing correction factor

Optimizing SKU Selection for Promotional Display Space at Grocery Retailers Display-Week

     T      δ6z Dj ti Zj z + δ7t  Dj ti Wt  t

Display-SKU

+

 

109

t  =1

z∈U Subcategory-Week

Store

     Error Term   T C  I     δ8at  Aj a Wt  t + δ9i  Bi  i + ej ti + a=1 t  =1

(1)

i  =1

In addition to an SKU’s own-marketing mix effects, our model could also account for cross-marketing mix effects. For a detailed discussion on cross-marketing mix effects, readers are encouraged to refer to Pak et al. (2019). After estimating our model on a large dataset that includes many stores and grocery chains, we next apply these estimated coefficients to the data from each individual store to calculate the incremental display profit (see below). After choosing the SKUs for promotional display for each week in the planning horizon, we repeat this procedure for all the stores that we desire a solution for. 3.1.2

Incremental Display Profit

Incremental display profit (2) is calculated using the estimated parameters of the SKU-level sales response function in (1). Specifically, the self-display profit is a combination of a product’s own base demand, its own display lift, and its own profit margin.3 j ti = qj ti (lj ti − 1)πj ti 

(2)

.

The base demand .qj ti (3) represents the unit sales of the SKU of interest in the absence of any external influences of any marketing-mix-related activities, and is calculated by subtracting from the log-transformed unit sales the estimated effects of related activities such as display, feature, and price reduction. The .δi are the corresponding estimated parameters obtained from regression (1). Base Demand .

   ln qj ti =

Log Unit Sales

   ln Sj ti

Display

Marketing Mix

% Discount

         − δˆ2 Dj ti − δˆ3 Hj ti − δˆ5m Mj mti m∈M Display-Week

      T     δˆ7t  Dj ti Wt  t δˆ6z Dj ti Zj z −

Display-SKU

−

  z∈U

(3)

t  =1

3 Errors associated with the exponential re-transformation of predicted estimates are offset with the smearing correction factor . (Duan, 1983).

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In accordance with the definition of .lj ti in (2), the SKU-level display lift .lj ti (4) is defined as the exponential of the sum of main display and partial display effects (by week and by product), which is given in terms of “total return”. Display Lift .

Display

Display-SKU

            ln lj ti = δˆ2 Dj ti + δˆ6z Dj ti Zj z +

 T 

Display-Week

    δˆ7t  Dj ti Wt  t

(4)

t  =1

z∈U

The profit margin .πj ti is determined by the store manager, and typically represents the dollar percentage the manager is expected to earn off the full charging price of a product.

3.1.3

Static Optimization of SKU Choice

We first provide a static optimization model that a retailer can use to identify the SKUs that will be placed on promotional display at a weekly level. The optimization model can be run independently for every store i in every week t. For this reason, we drop the subscripts i and t. The objective function of the optimization model takes .j ti (or .j ) as an input (see (2)). It can also account for any trade promotion allowance .oj ti (or .oj ) offered by a manufacturer to the retailer for placing an SKU of interest on promotional display at week t in store i: .maxxj d

subject to

(j + oj )xj d j ∈Ud ,d∈D . j ∈U xj d ≤ 1, ∀d ∈ D, d . d∈D xj d ≤ 1, ∀j ∈ Ud , .

.xj d

∈ {0, 1}.

Maximize incremental display profit Only one product per display SKU j can be placed on only one display each week Binary solution if SKU j is on display d

Here, the binary decision variable .xj d is 1 if SKU j can be placed on display d (at store i in week t), and we only have such decision variables for those combinations of SKU j and display d where SKU j is a legitimate candidate for display d. For example, some displays may be freezers, and thus only SKUs requiring refrigeration will be candidates. Some displays may have restrictions on the size of items that may be displayed on them. The retailer may also desire certain SKUs to always be displayed next to a particular merchandise, or certain SKUs not to be displayed too close to the entrance for security reasons.

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We now discuss the constraints that we impose. The first constraint states that each display d can have at most one SKU on it,4 while the second constraint states that each SKU can be on at most one display. Thus, this problem becomes a variation of the maximum weighted bipartite matching problem, where the left-hand nodes are SKUs and the right-hand nodes are displays and the weight of an edge between SKU j and display d is the incremental profit .j + oj . The bipartite graph only contains an edge between SKU j and display d if SKU j is allowed to be on display d (i.e., if the decision variable .xj d is present in the above objective function).

3.2 Hierarchical-Static Methodology One problem with the previously described Direct-Static estimation approach is that a product subcategory/category may contain so many different individual SKUs that the model (1) cannot be estimated in a reasonable amount of time. One solution to this problem is to limit the number of potential SKUs under consideration for promotional display. Of course, this solution may inadvertently leave out the profitmaximizing SKU since such an SKU it is not known a priori. Another option for easing the estimation challenge is to use a Hierarchical estimation approach. In contrast to the Direct approach, the Hierarchical approach allows the retailer to consider a broader selection of SKUs per subcategory. However, in this twostep approach, each subcategory has to be evaluated and optimized to find a subcategory-optimal SKU first, and only then, a profit-maximizing SKU among all subcategory-optimal SKUs per week is chosen (see Fig. 1). Neither estimation methodology necessarily dominates the other. The assortment strategy of a retailer should determine which of the two methodologies fits one’s business goals best. For small scale stores that only offer a narrow selection of products within a narrow selection of product categories, the Direct approach might be more suitable to optimize all the potential display candidates at once. For large scale retailers that offer a wide selection of products within a large set of product categories, the Hierarchical approach might be more preferable as it allows consideration of the widest selection of SKUs within each subcategory for the profit-maximizing display decision.

3.2.1

Sales Response Function

The sales response function for the Hierarchical approach is very similar to the Direct approach model. The main difference is that there is no longer a SubcategoryWeek effect since this method estimates each subcategory separately. Hence,

4 Our methodology can be easily extended to cases where more than one SKU can be located on the same promotional display space.

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the sales response function is estimated for SKUs within each subcategory, and the number of regressions being estimated equals the number of subcategories considered.

3.2.2

Incremental Display Profit

Using the estimates obtained from (3.2.1), the incremental profit is determined for each subcategory. Profits are calculated as in (2).

3.2.3

Static Optimization of the SKU Choice

For each store i, week t, and subcategory .Va , we find the best SKU with the highest incremental profit in that subcategory .ja = arg maxj ∈Va j ti . We perform the same optimization as in Sect. 3.1.3 but now we only use SKUs .ja . .maxxj d a

subject to

(ja + oja )xja d ja ∈Ud ,d∈D . j ∈U xja d ≤ 1, ∀d ∈ D, a d . d∈D xja d ≤ 1, ∀ja ∈ Ud , .

.xja d

∈ {0, 1}.

Maximize incremental display profit Only one product on display A specific SKU can be placed on only one display each week Binary solution if SKU .ja is on display d

4 Data Description 4.1 Estimation Data For parameter estimation, we use a sample of syndicated retail sales scanner data collected by IRI, known as the IRI Marketing Data Set (Bronnenberg et al., 2008). Unlike a retailer’s proprietary data that is limited to the sales of one particular retailer, syndicated data comes from a variety of retail sources and is gathered by a third party (e.g. IRI or Nielsen). Syndicated data represents a richer source of information because it covers more than one retailer, thus, containing information on more products across more markets and numerous grocery store chains. We illustrate our methodology using a sample from the New England region, which covers a variety of retail outlets with more than 102 stores in total for all fifty-two weeks of the latest available year, 2011, and is recorded at an SKU/store/week level, with a total of 387,228 sales observations. We focus on the sales of beer/malt beverages since this product category is typically one of the most popular impulse buy purchases, and is a popular choice for promotional display spaces (Bell et al., 2009). The beers across subcategories

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Table 2 New England data set summary Subcategory Subpremium Premium Superpremium Craft Import Total

Observations 35,653 80,699 70,830 116,008 84,038 387,228

Unit sales 149,554 720,379 332,791 632,439 432,586 2,267,749

SKU count 59 75 85 352 159 730

SKU count on display 10 45 44 105 60 264

significantly vary in the data set. Some beers are mass-produced like Subpremium, Premium, Super Premium, while others are more niche like Craft and Import. Due to a wide variety of package sizes sold (from single bottles to 36-can packs), we only consider the most popular package sizes: 6-, 12-, 18-, and 24-unit products.5 We exclude unusually expensive transactions with a unit cost greater than $1 per ounce (e.g., Samuel Adams’ Utopia at $150 per 24 oz. bottle). Additionally, we exclude oddly shaped and rarely purchased product packages like party balls and kegs, since our focus is on endcap displays and these package types are often put on floor displays. A summary of our dataset is provided in Table 2. Although the top-selling subcategory by volume is Premium (720,379 units sold), Craft has the most diverse set of beer SKUs at 352. Each observation captures detailed information about a purchase at the SKU/store/week level, including brand name, units sold, dollars paid, any marketing mix activity associated with the transaction (i.e., whether the SKU was on promotional display, whether there was any price reduction, whether it was featured in a store flyer or advertised in other ways). All beer SKUs are also uniquely identified using four nominal attributes: calorie content (light, regular), container type (bottle, can), package size (6-, 12-, 18-, 24-pack), and brand (thirty-three brands in total). An example of a typical transaction is 21 units of SKU 00-01-18200-00016 (Budweiser, can, 6-pack) sold for $125.79 in week 16 at store .#250872, with no promotional activity associated with this transaction (i.e., no price promotion, no promotional display, and no coverage in any store flyers). Table 3 provides descriptive statistics of the dataset for select variables that we include in our estimation model. Table 4 provides overall information on transactions with promotional display activity.

4.2 Optimization Data (Single Store) We assume that only one SKU is placed on a given promotional end-cap every week and illustrate the selection of the optimal SKU to be placed on one promotional end-cap every week for a randomly chosen store i located in New England. Overall, 5 Our

main results continue to hold when we include the full set of package types.

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Table 3 Descriptive statistics for select variables (all stores) Variable Unit sales Display (Binary) Discount percentage Price reduction (Binary) Feature (Binary)

Obs 387,228 387,228 387,228 387,228 387,228

Mean 5.61 3.60% 5.60% 18.70% 8.40%

Median 3 0 4% 0 0

Min 1 0 0 0 0

Max 570 100% 77% 100% 100%

Table 4 Categorization of sales transactions Promotional display No Yes Total

Units sold 2, 120, 514 147, 235 2, 267, 749

Frequency of observations 372, 508 14, 720 387, 228

Table 5 Single store sales data summary Subcategory Subpremium Premium Superpremium Craft Import Total

Observations 1050 2007 1947 3392 2637 11,033

Unit sales 6110 53, 999 21, 233 21, 601 19, 336 122,279

SKU count 27 50 47 102 67 293

SKU count on display 0 12 0 0 0 12

293 SKUs in the five subcategories were sold at this store over the entire year (see Table 5). Note that only 12 of these 293 SKUs were selected for promotional display, reinforcing our earlier point that a single store lacks the data needed to estimate the promotional display lift for most SKUs. The top-selling subcategory by volume is Premium (53,999 units).

5 Application and Assessment of Methodology We assess the performance of our methodologies described in Sects. 3.1 and 3.2— Direct and Hierarchical—and evaluate their expected improvement in incremental store profits compared to some common benchmark heuristics used in practice. Our goal is to determine the profit-optimizing SKUs to be placed on promotional display in a store (randomly selected from our sample) and compare the performance of our methodologies against the performance of a commonly used practice in the grocery industry.

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5.1 Direct Estimation/Optimization Recall that the Direct approach selects a subset of SKUs from every subcategory for simultaneous evaluation as potential candidates to put on promotional display. We select the ten best-selling SKUs from each subcategory at our store of interest. This results in a set of fifty SKUs to ensure reasonable estimation times. Table 6 summarizes the regression results for the full model6 obtained using the sales response function (1). A set of marketing mix instruments include {Price Reduction, Discount, and Feature}. ‘Price Reduction’ is a binary indicator of a temporary price reduction that accounts for the lift in sales due to the advertisement of a price reduction. ‘Discount’—the price reduction effect—is the percent discount a given product was sold at, measured as the percentage difference between the full price (i.e., the highest retail price the product sold for at the store) and the selling price from that week. ‘Feature’ is a binary variable indicating if a product sale is associated with any of the following: a small, medium, or large advertisement, a coupon, or a rebate. The size of Discount, Price Reduction, and Feature, along with Display, have a positive and statistically significant impact on sales across all models. The main effect of Display is quite significant in magnitude i.e., providing around a 24% lift in the log of weekly sales. To calculate the incremental display profit for every SKU j in week t for store i as shown in (2), we feed the estimates of (1) into (3) and (4). Since we don’t have access to actual product profit margins, we set the profit margin equal to 25% of each SKU’s full price, which is consistent with the measurement of profit margin in prior retail operations work (Dreze et al., 1994; Campo et al., 2004; Rooderkerk et al., 2013). Our Direct optimization approach suggests that Bud Light (bottle, 18-pack) provides the highest incremental profit in almost every week (see Fig. 2) except for week 33, where the profit-maximizing SKU is Shipyard Seasonal (bottle, 12pack) assuming there is only one end-cap display available each week. The total incremental profit (across all weeks) obtained by the Direct approach is $13,980.79.

5.2 Hierarchical Estimation/Optimization The Hierarchical approach evaluates the SKUs within each subcategory separately. Specifically, for each subcategory, it selects a subset of SKUs to evaluate as potential candidates. To illustrate and evaluate the performance of the Hierarchical approach, we select the twenty most popular SKUs per subcategory. Altogether, the number of SKUs considered across all subcategories is 100. Since we estimate the sales response function for a set of candidate SKUs in each subcategory, we obtain 6 For each of the models in both the Direct and Hierarchical approaches, incomplete models were tested (not reported here for space consideration). The size and direction of coefficient estimates stay consistent across all the models.

a

a

.Dj ti Zj a

.Dj ti Wt  t

b

a

Robust stand. err. clustered at store level in parentheses Included in the model Reference category is all “other” SKUs

Log-likelihood AIC BIC 2 .R

a

a

.Wt  t

.Aj a Wt  t

a,b

.Zj a

.Dj ti

Feature

Price reduction (binary)

.Mj mti

0.0368 (0.0145) 0.2193 (0.0339) 0.2438 (0.0912) Yes Yes Yes Yes Yes .−346,316.4 692,762.7 693,469.1 0.3769

0.6948 (0.0464) 1.4058 (0.1259)

Constant

Discount (price effect)

Direct 1

ln (unit sales)

0.0400 (0.0188) 0.4394 (0.0366) 0.5407 (0.0697) Yes Yes Yes Yes Yes .−429,224.9 858,651.9 859,761.2 0.2209

Hierarchical Subcat. 1 0.9552 (0.0174) 1.4894 (0.1437) 0.0669 (0.0176) 0.3791 (0.0323) 0.5067 (0.0782) Yes Yes Yes Yes Yes .−398,047.4 796,296.9 797,406.2 0.3249

Subcat. 2 0.9591 (0.0186) 1.4331 (0.1491)

Table 6 Sales response function estimates for direct and hierarchical approaches

0.0391 (0.0190) 0.4404 (0.0369) 0.5598 (0.0692) Yes Yes Yes Yes Yes .−428,061.6 856,325.3 857,434.6 0.2251

Subcat. 3 0.9524 (0.0169) 1.5246 (0.1390) 0.0107 (0.0181) 0.3849 (0.0370) 0.5312 (0.0706) Yes Yes Yes Yes Yes .−422,051.4 844,304.8 845,414.2 0.2462

Subcat. 4 0.9492 (0.0174) 1.5860 (0.1477)

0.0405 (0.0183) 0.3999 (0.0378) 0.5204 (0.0701) Yes Yes Yes Yes Yes .−421,844.8 843,891.7 845,001 0.2469

Subcat. 5 0.9482 (0.0168) 1.5966 (0.1372)

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Fig. 2 Weekly incremental profit obtained under the direct approach

five sets of regression coefficients (Table 6, columns under “Hierarchical”). The size and direction of all coefficient estimates stay consistent across the different subcategories. The main effect of Display is quite significant in magnitude i.e., providing around a 50–55% lift in the log of weekly sales at the subcategory level. Recall that the Hierarchical optimization works as follows. First, for each week, we determine an SKU that would optimize the total incremental profit per subcategory. Once the most profitable SKU in each subcategory is determined, the most profitable among those is selected. The total incremental profit obtained using the Hierarchical approach is $15,615.87, which is about $1635.08 more than the profit obtained from the Direct approach. Note that the Hierarchical approach selects Sea Dog Wild Blueberry Wheat (i.e., the 14th highest selling SKU in the Craft subcategory) to be placed on a promotional display, an SKU that is not even in the consideration set of the Direct approach (see Fig. 3). This is a good example of one of the limitations of the Direct approach i.e., it may inadvertently leave out the profit-maximizing SKU since it considers a smaller set of SKUs as potential candidates. This highlights the main difference between the two methodologies. The preferred method will depend on the application setting since the assortment strategy of a retailer should determine which of the two methodologies fits one’s business goals best. For example, small scale stores like Aldi or Trader Joe’s that only offer a narrow selection of products within a narrow selection of product categories could

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Fig. 3 Weekly incremental profit obtained under the hierarchical approach

use the Direct approach to consider all the potential candidates for a promotional display at once. On the other hand, retailers like Walmart, which offers a wide selection of products within a wider selection of product categories, could use the Hierarchical approach to make sure that the widest selection of SKUs within each subcategory are considered.

5.3 Static Benchmark Comparison We now examine how the profits obtained using our methodology compare with the profitability of a commonly used practice by retailers. A standard and commonly used approach to choose a product for promotional display, which will serve as our benchmark, is to pick a best-selling SKU in that week. In reality, store managers do not know in advance which SKU will be the best-seller for the incoming week and need to rely on forecasts. Since we only have one year of data, we assume that store-managers have full information regarding which SKU will be best-selling for the incoming week, so this represents a conservative benchmark comparison. The set of SKUs that are assigned to a promotional display based on this commonly used approach is summarized in Table 7 and depicted in Fig. 4. The benchmark yields an annual incremental profit of $9343.50. It is interesting to note that even though we

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Table 7 SKUs chosen in benchmark Top SKU Bud Light, bottle, 18-pack

Annual unit sales 5822

Bud Light, can, 18-pack

5614

Miller Light, can, 18-pack

4166

Coors Light, can, 18-pack Michelob Ultra, bottle, 18-pack Samuel Adams Seasonal, bottle, 12-pack Shipyard Seasonal, bottle, 12-pack Miller High Life, bottle, 18-pack

3560 3054 2678

Top seller in week 0, 16, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 34, 35, 37, 40, 41, 48 1, 2, 3, 9, 11, 13, 15, 21, 24, 32, 36, 38, 46, 39 4, 5, 8, 12, 14, 17, 18, 39, 43, 44, 45, 47 19, 33, 42 6, 10, 31 50, 51

1053 959

32 7

assume the store-manager has full information while constructing our benchmark, we find that both approaches (i.e., the Hierarchical and the Direct) outperform the benchmark (see Table 8 for profit comparison)

6 Dynamic Optimization In this section we describe a dynamic version of the promotional display optimization problem to determine the SKUs to be assigned to promotional display in a single optimization run over every week in the planning horizon. Our optimization problem can be solved on a rolling horizon schedule and our model is flexible enough to incorporate as constraints some important business rules typically applied in practice. Some retailers may want to impose a limit on how frequently a particular product can appear on display. For example, knowing that frequent promotion is likely to lose some effectiveness over time, a grocer may want to impose the constraint that certain SKUs may appear on display at most twice in a month. Besides, some retailers may have some general business rules that store managers must follow, such as requiring that no single subcategory can be on display for more than two consecutive weeks. To include a greater product diversity on promotional display across time, we define a new objective function below along with the imposed constraints. The constraints that we consider in the dynamic optimization are typical constraints that are often used in the promotions/pricing literature (e.g., Elmaghraby and Keskinocak, 2003) and the shelf space7 /assortment optimization literature (e.g.,

7 This family of constraints are identified in the shelf space optimization literature as control or capacity constraints, where retailers set lower and/or upper bounds for products’ days-supply,

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Fig. 4 Weekly incremental profits obtained in benchmark

Rusmevichientong et al., 2010; Gallego and Topaloglu, 2014) to address realistic issues faced by retailers. We illustrate how these types of constraints can be adopted in a new setting to address specific issues pertinent to the promotional display optimization based on our conversations with store managers. The optimization is done independently for each store, hence, the subscript i is dropped. (Unlike the static optimization, we don’t drop the subscript t, since the decisions are no longer independent across time.)

brand share, and shelf-space exposure, used to manage operational costs associated with stockouts and replenishments.

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Table 8 SKU selection and total incremental yearly profit comparison for static optimization Direct Hierarchical Bud Light, bottle, 18-pack Bud Light, bottle, 18-pack Shipyard Seasonal, bottle, 12-pack Budweiser, bottle, 24-pack Seadog, bottle, 12-pack

$13,980.79

.maxxj t

subject to

$15,615.87

.

j ∈U

T˜



.

j ∈U xj t

T˜

.

t=1 xj t

Qj

.

t=1 (j t xj t

− R × cj t )

Benchmark Bud Light, bottle, 18-pack Bud Light, can, 18-pack Miller Light, can, 18-pack Michelob Ultra, bottle, 18-pack Miller High Life, bottle, 18-pack Coors Light, can 18-pack Shipyard Seasonal, bottle, 12-pack Samuel Adams Seasonal, bottle, 12-pack $9343.50

Maximize incremental display profit

≤ kt , ∀t < T˜

In week t, limit the maximum number of displays to .kt

≤ bj , ∀j ∈ U

SKU j is displayed at most b number of times during the planning horizon

r=0 xj,t+r

≤ 1, ∀j, ∀t

.xj t

− xj,t+1 ≤ cj t , ∀j, ∀t

.xj t

∈ {0, 1}, cj t ∈ {0, 1}

SKU j is only on display once every .Qj weeks Index of a changeover when SKU j goes off display Binary decision restrictions

Objective Function The objective function defines the incremental profitability of a display decision over the entire planning horizon .T˜ . The first term in the summation is the total incremental profit obtained from placing SKUs on promotional display in week t. The second term is the total changeover cost associated with changing a display. Updating a promotional display might be costly as removing one product and replacing it with another requires additional labor hours. Specifically, we assume that every time an SKU j has to be removed from the display, the retailer incurs some cost R. To capture the incremental profitability and the display-related changeover cost, we define .xj t as a binary decision variable which equals to 1 if SKU j is placed on display in week t, 0 otherwise; and .cj t as a binary indicator, which equals to 1 if SKU j goes from being on display to being off display in week t. Display-Space Restriction The first constraint indicates the total number of displays available per week, where .kt is the maximum number of promotional displays available in a week t (referred in operations literature as a capacity or space constraint (Rusmevichientong et al., 2010; Gallego & Topaloglu, 2014)).

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Table 9 How .cj t obtains its value from .xj t − xj,t+1 ≤ cj t .xj t

.xj,t+1

.xj,t

0 0 1 1

1 0 1 0

.−1

0 0 1

− xj,t+1

Interpretation SKU j went from not being on display to being on display SKU j was not and is not on display SKU j remains on display this week too SKU j went from being on display to not being on display

.cj t

0 0 0 1

Week Restriction The second constraint indicates that an SKU j can be promotionally displayed at most .bj number of times across the time horizon. Sparsity Restriction The third constraint requires that in every .Qj consecutive set of weeks, SKU j can only be promotionally displayed once (also referred in operations literature as a cardinality constraint (Gallego & Topaloglu, 2014)). This way, a decision-maker can impose a limitation on the successive frequency of promotional activity over time (Elmaghraby & Keskinocak, 2003). In other words, a retailer can indicate how many weeks in a row a product can be displayed.8 Changeover Cost The fourth constraint is needed to indicate a changeover, which occurs every time an SKU has to be removed from display. Here, the value of .xj t determines the value of .cj t . Table 9 illustrates how the binary “change detector” .cj t obtains its value once .xj t is known. The profit-maximizing objective function will always force .cj t equal to 0 in the first three cases. In the last case, .cj t is forced to equal 1, hence, a changeover cost is incurred. The optimization model discussed in this section is suitable for the Direct dynamic approach. We now briefly discuss how to obtain the dynamic optimization version of the Hierarchical static analog. We use the same mixed-integer program as the static approach but the incremental profit coefficients are calculated using the methodology described in the current section instead of how it has been described in Sect. 3.2.3. In the Hierarchical case, the optimization first decides which subcategory to place on display. For each week and each subcategory, the SKU (from that subcategory) is selected such that the incremental profit is maximized. Once the optimization chooses the optimal subcategory for each week, the optimal SKU for the week is the SKU selection for that subcategory.

6.1 Application of the Dynamic Optimization In this section, we analyze the incremental profitability of the dynamic optimization with a numerical example using the same profit margins .πj t as in the static

8 Alternatively, one can consider a non-sparsity constraint. In this case, a retailer can indicate how many consecutive weeks a product should not be displayed.

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Table 10 Profit comparison for dynamic optimization and dynamic benchmark

Constraint Total display/products per week (only one display in each of 52 weeks) Total weeks on display per product (a product can be displayed at most five weeks in the year) Sparsity (each product cannot be displayed more than three consecutive weeks) Changeover cost ($5) Dynamic optimization Incremental profit Changeover cost Final incremental profit (profit minus changeover cost) *Dynamic Benchmark Incremental profit Changeover cost Final incremental profit (profit minus changeover cost) Dynamic Optimization vs. Benchmark comparison

Scenario 1 

Scenario 2 

Scenario 3 











Scenario 4 (same as Static Direct) 

 7824.64

7824.64 0 7824.64

8267.08 0 8267.08

13,980 0 13,980

6339.75

6598.00 0 6598.00

7035.08 0 7035.08

10,172 0 10,172

19.4%

18.6%

17.5%

29.5%

More

More

More

More

.−255.00

7569.64

6594.75 .−255.0

optimization (see Sect. 5.1). Table 10 summarizes the results where the previously discussed sets of constraints are enabled. Since the dynamic optimization is more restrictive, it typically results in profits smaller than those shown in the static optimization. The most restrictive is Scenario 1, where a retailer can only use one product/display space per week; each beer product can only be displayed at most five weeks per year; every three weeks each product/display combination can only be used no more than once; and every time a retailer replaces one beer SKU for another SKU, they incur a changeover cost of $5. In this case, the incremental profit is .$7569.64 once an incurred .$255 total changeover cost is subtracted. The subsequent scenarios individually relax these constraints. Intuitively, the total incremental profits increase as we relax constraints. In Scenario 2, where the restrictions stay mostly the same but the retailer no longer incurs the changeover cost of $5, the realized profit is .$7824. In Scenario 3, we remove the ‘no more than three consecutive weeks’ constraint, and the realized profit is .$8267. Finally, Scenario 4 only restricts the use of one product/display space per week, which by definition is identical to the scenario examined in our static optimization, resulting in a profit of .$13,980.

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Please note the changeover cost needs to be at the level of the difference between the incremental profits of the two most profitable SKUs to affect the SKU choice. If the difference in the incremental profits between the two most profitable SKUs is small, then the changeover cost may not affect the promotional display decision. For this reason, the changeover cost also prevents the optimization model from changing the promotional display recommendation when the differences in the incremental profits are small.

6.2 Benchmark Comparison for Dynamic Case We choose a benchmark for the dynamic case that is analogous to the benchmark for the static case. Again, we assume that the store manager has no prior knowledge of the incremental profits he can obtain from placing his products on display. In this case, a reasonable approach is to have an objective that maximizes total sales units (while still obeying the constraints). The static benchmark is not applicable because it violates the dynamic constraints (as listed in Scenario 1 discussed above). For example, in the benchmark of the static case schedule (Table 7), the manager chooses bottled Bud Light (18-pack) nine weeks in a row (i.e., it is the highest seller in weeks 22–30). This violates the sparsity restriction and the total weeks on display restriction discussed earlier. Due to the need for a benchmark to satisfy the constraints, we use an integer program to choose the benchmark solution. Specifically, we choose our objective function to maximize the sales units as being the analog of the benchmark for the static case (i.e., “Value of the objective function” in Table 10). We take the solution obtained from using this objective function and calculate the actual incremental profits produced by this solution. For comparison purposes, the scenario constraints imposed for the benchmark are identical to the Dynamic Optimization. In all of the benchmark scenarios, the profits are smaller than those obtained through the Dynamic optimization (Table 10). In Scenario 1, the dynamic optimization yields a 19.4% higher incremental profit than the benchmark. In the subsequent scenarios, the profit improvements of our dynamic approach over the benchmark are 18.6%, 17.5%, and 29.5% for Scenarios 2, 3, and 4 respectively. Note that the value of the objective function (not shown here)—total units sold—is also always increasing as the constraints are relaxed. Our results corroborate the importance of estimating sales lifts and calculating incremental profits to facilitate promotional display space SKU-selection as opposed to simply selecting best-selling SKUs.

7 Conclusion Optimizing product selection for promotional display space is an important lever that grocery store managers have at their disposal to influence customers’ purchasing decisions and increase the profitability of their stores. In this study, we

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provide a decision support tool for choosing which SKUs to place on special promotional display spaces (such as end-of-aisle displays) inside a grocery store. Our methodology allows a retailer to choose an SKU for each promotional display space that results in the largest improvement in incremental profit for that particular store location. Historically, retailers have identified which SKUs to put on promotional display spaces using simple heuristics, such as picking a best-selling SKU or the same SKU that was assigned on display during that time in the previous year. For example, our methodology offers several improvements over these existing practices. Our methodology is the first to propose an estimation technique for measuring the incremental lift in sales of placing a particular SKU on promotional display space. These incremental lifts (represented by estimates of the percent increase in sales) are estimated using a sample from a national grocery store sales transaction dataset (collected by IRI), which allows us to estimate the sales lifts from a much larger set of SKUs than if the estimates were obtained using only the transaction data from a single store or store chain. This allows us to even estimate the sales lift for SKUs that have never been put on promotional display at a particular retailer. Our estimation methodology is capable of handling an extensive and complex product assortment and can be extended to capture important aspects of promotional activities such as cannibalization and halo effects (interested readers can refer to Pak et al. (2019)). Our methodology also includes the first optimization model for selecting which SKUs to put on a promotional display for each store. The optimization model includes the incremental lifts (from the estimation method) combined with the estimated base-sales rates and profit margins of each SKU so that the profit-maximizing SKU can be chosen for a promotional display space for each week of the year. Our optimization model is also flexible enough to consider several practical aspects such as common business rules that restrict the selection of the same SKU over a consecutive set of weeks, display-related changeover costs, and trade fund deals, which can provide grocers with additional profit through agreements with manufacturers concerning the placement of the manufacturers’ products on promotional displays. To demonstrate our methodology, we use retail CPG sales scanner data from multiple stores across different grocery store chains in New England. For illustration purposes, we use the beer category and focus on the optimization of major promotional displays for a given store within our dataset. We find that assigning an SKU on promotional display can result in a significant lift in sales. For instance, we estimate an average sales lift of 27.6% across all SKUs, confirming that promotional display is a very effective tool for stimulating incremental product sales. We then compare the profitability of our proposed approach with a common industry benchmark. Our benchmark selects the best-selling SKUs per week to be placed on promotional display under the assumption that a store-manager has full information regarding which SKUs will be best-selling for the incoming week. We find that our approach significantly outperforms this common practice, resulting in at least a 17.5% improvement in incremental profit.

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Our study identifies several opportunities for future research. We have focused on the selection of SKUs to be placed on promotional displays assuming that product categories have already been assigned to displays. One possible extension of our work involves developing a methodology that will determine which product categories to assign to promotional displays, which will require a category-level estimation and optimization model. This category-level model could be combined with our existing methodology to provide a comprehensive decision support tool to grocers regarding promotional display product selection. Another future research opportunity would be to develop an estimation model that would capture the decay of the display lift. As with many types of promotions, the lift from a display promotion can diminish over time if the display continues to have the same item. A model that incorporates the decay of the display lift would identify when to switch a promotional display to a new SKU so that a store manager does not have to rely on specific business rules to make such a decision.

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Merchandise Placement Optimization Wei Ke

Retailers typically use the following method for revenue management, i.e. maximize the total revenue of the merchandise they carry in their physical stores: (1) First Price Optimization, which determines what best initial price to offer for a product, (2) Markdown Price Optimization, which analyzes the timing and depth of permanent price discounts from the first price, and (3) Assortment Optimization, which considers the best mix of products to offer in a store. Extensive research already exists for these methods. In this chapter, we introduce a new angle to retail revenue maximization by first analyzing how physical placement of a store’s merchandise affects consumer demand and then proposing a pragmatic heuristic algorithm to efficiently determine the optimal physical location and space assignments (on a shelf or fixture) of a store’s displayed merchandise, while accounting for different retail floor layouts. The research work done by Abratt and Goodey (1990), Bryner (2005), and Mendelsohn (2007) summarized several well-known retail shopping patterns that offer clues on where to place certain types of merchandise in a store: • “Most shoppers turn right,” suggesting that it may be advantageous to place high margin items along the right thoroughfare in the store; • Shoppers “stick to the perimeter of the store” and “do not weave up and down the aisle,” suggesting that products placed in the center of an aisle are less frequented; • And lastly, shoppers “have a natural tendency to focus and perceive at eye level.” The impact of shelf space allocation on product sales has also been well studied in consumer behavior literature. Cox (1970) stated and verified while there is no

W. Ke () The Decision, Risk and Operations Division, Graduate School of Business, Columbia University, New York, NY, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Ghoniem, B. Maddah (eds.), Retail Space Analytics, International Series in Operations Research & Management Science 339, https://doi.org/10.1007/978-3-031-27058-1_7

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relationship between allocated shelf space and sales for a “staple product” brand, such a relationship does exist for an “impulse product” brand. Other research work, such as Abratt and Goodey (1990), Bellenger et al. (1978), Curhan (1972), Kollat and Willett (1967, 1969), Wilkinson et al. (1982), all echoed Cox (1970)’s findings. Technological advances, such as the radio frequency identification (RFID) tags, have made it possible for retailers to continuously track shelf locations of the entire merchandise assortment and measure, albeit indirectly, foot traffic levels of certain areas within a store. More direct means of tracking shopper movements have also proliferated in recent years. These include in-store video surveillance, RFID chips planted in shopping carts, or location data broadcast by smartphones. Most importantly, retailers have also begun to store shelf placement history at the stock-keeping unit (SKU) or item level—data that made this research possible—in addition to the more traditional data elements, such as planograms that do not link to detailed SKU or item numbers or sales records that do not track from which shelf the item was taken. Most research and commercial solutions on merchandise placement optimization that predate this work typically solved for the optimal planogram, instead of the assignment problem that determines the best physical location of every displayed SKU in a store. This is likely because the assignment problem formulation that optimizes merchandise placement (akin to a facilities location problem) is, by definition, .N P-hard. Botsali (2007) proposed an early attempt at formulating the computationally hard problem by characterizing store shelf locations as a stylized grid layout and optimizing it using the simulated annealing heuristic. To make it more challenging though, business rules and constraints that apply to each store’s day-to-day circumstances, are hard to fully specify in an optimization model. As a result, merchandise physical decisions today rely, for the most part, on the experiences of the retailer or store planner. The origins of this chapter are in Ke (2009). Since the pulication of Ke (2009), there has been exciting advances in defining and solving for the assignment problem formulation of the merchandise placement optimization problem. Bhadury et al. (2016) developed a p-dispersion model to optimize merchandise placement in a retail store and use the simulated annealing heuristic to address computational complexity, though their model comes with clear limitations in that they confine the store layout to a grid only while assuming the dimensions of the departments to be equal sized or fixed. Those limitations were removed by Ozgormus and Smith (2020) recently. Flamand et al. (2016), on the other hand, proposed a generalized assignment formulation to jointly optimize the assignment of grouped product categories to shelves, the specific location of a product on the shelf, and the size of the allocated shelf space. They addressed computational complexity by adopting a clever decomposition technique that iteratively solves a single-shelf, single-group subproblem. Mowrey et al. (2018) recognized the fact that store layout should precede space allocation and proposed a heuristic solution using particle swarm optimization. Flamand et al. (2018) then further expanded their previous work by integrating assortment planning decisions in addition.

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This work has two significant and practical contributions to the merchandise placement optimization field: (1) The proposed fast heuristic, called Plan Modification Optimization (“PMO”), works for store layout formats of any size or complexity without needing to simplify and stylize them, e.g. into grids, for model tractability, and (2) The proposed biclustering algorithm on both the SKU and shelf dimensions addresses the data sparsity issue, a common occurence and indeed a significant roadblock in implementing a merchandise placement optimization algorithm at any decent sized stores with a large enough assortment. Working with a commercial sponsor that manages over 300 retail outlets with different store formats at both resort destinations and shopping malls, we attempt to provide a model that characterizes the total assortment profit as a function of physical locations, amongst other variables. This, in turn, allows for experimentation and optimization of merchandise placement decisions. The model also provides a strategic retail manager with an added benefit: Placement changes, e.g. relocating a product to a high traffic area, or conversely, the back of the store where traffic is lower, may achieve comparable demand effects as pricing changes. This is particularly relevant, when the retail manager does not want a customer’s internal reference price (see Winer, 1986 for a definition) to be lowered due to a decreased sticker price, thus helping to preserve the associated brand equity. In Sect. 1, we first introduce the “plan modification optimization,” a heuristic algorithm that addresses the computational complexity and modeling difficulties of the assignment problem formulation of merchandise placement optimization. The algorithm decomposes a set of moves proposed by store planners into feasible independent components and recommends the implementation of those components that contribute positively to revenue gain. Next, in Sect. 2, we introduce a forecasting framework to complement the optimization algorithm. In reality, there is usually not enough natural experimentation that allows historically observed pairings between an SKU and any fixture location in a store—enumerations of all feasible pairings are astronomically prohibitive. Noting that (1) different SKUs respond to their placement locations in varying degrees and (2) prime real estate, such as high foot-traffic areas near the store entrance, generally lifts SKU sales, we propose a biclustering algorithm, operating on both the SKU- and the fixture-dimensions, to correlate the locational sensitivity of an SKU with the inherent financial impact of a location. This correlation provides a good estimate for forecasting the financial consequence of a move that was not observed historically. Finally, in Sect. 3, we present our conclusions and some suggestions for future research.

1 The Value of Placement The first question we must ask ourselves is whether placement is a zero-sum game from the revenue perspective. For space utilization, placement change appears to

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Stock Room

Pre-move plan $28,000

Stock Room

Original proposal $31,000

Stock Room

Modified proposal 1 $32,000 (best)

Stock Room

Modified proposal 2 $27,000

Fig. 1 Revenue zero-sum game

be a zero-sum game, because existing SKUs must be moved away from a shelf or a fixture before others can take their place. However, will aggregate sales volume also remain unchanged when sales of individual SKUs are impacted by a move? We illustrate this question in Fig. 1. In this example, we consider the inter-fixture moves of shirts, shoes and jeans. Before the move, these items have achieved sales totalling $1200; While individual sales have been impacted, the aggregate sales remain $1200 after the move—will this always be the outcome? There are well documented industry benefits from shelf space management, which determines how much space should be allocated for each product. Hansen and Heinsbroek (1979) found “about 6%” in increased profit, when taking space elasticity into consideration; Bultez and Naert (1988) reported “actual increases in profitability varying from a low of 6.9% to a high of 33.8%” due to implementation of the SH.A.R.P. method;1 Drèze et al. (1994) found “5–6% changes [in sales] due to shelf reorganization”; van Nierop et al. (2006) used a simulated annealing algorithm to generate “10–15%” profit lift. Merchandise placement optimization, which decides where each product should be located on the retail floor, brings additional financial benefits on top. Using retail merchandise data provided by a

1 SH.A.R.P., or Shelf Allocation for Retailer’s Profit, is a theoretical shelf space optimization/allocation model that accounts for demand interdependencies across and within product groups.

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multi-billion dollar entertainment company, we are able to verify revenue uplifts averaging about 5% over the existing practice, through what we call the “Plan Modification Optimization” algorithm in a series of conservative what-if analyses. We explain the idea in Sect. 1.1. And then in Sect. 1.2, we present some empirical results.

1.1 Plan Modification Optimization (“PMO”) There are two immediate hurdles to solving the full placement optimization problem: (1) The problem formulation is likely to be combinatorial and thus mathematically complex; (2) Business rules (e.g., presentation/face-out constraints, promotions/special events, marketing aesthetics) are hard to fully specify in a mathematical formulation. Therefore, we devise a suboptimal “semi-automatic” heuristic that allows us to overcome these two hurdles and assess the value of placement from historical data. It is “semi-automatic,” because this procedure requires a store planner to propose a move before it can be analyzed and modified. The key is to intelligently improve proposed placement plans through the following steps: Step 1 Decompose proposed plan into feasible independent components, or submoves; Step 2 Evaluate the revenue gain/loss of each submove; Step 3 Implement only the submoves that produce positive revenue gains. We can visualize SKU moves with the help of a network model. Let each fixture—for instance, a shelf—be represented by a capacitated node. Let stock room be an uncapacitated super-node, where we introduce new and retire old SKUs. A submove can be defined as a collection of nodes (excluding the stock room supernode) that are not connected to other nodes (i.e. a component or submove). The flow chart shown in Fig. 2 contains the details of the submove identification algorithm. As a simple example, Fig. 3 shows two submoves (marked by red and blue colors) identified in a move. A proposed placement plan can be modified by deciding to execute or not execute any combinations of all submoves. Each submove already represents a collection of feasible moves, proposed by a store planner. Hence, this circumvents the difficulties that may arise when we model business rules. Key steps of the "Plan Modification Optimization” (PMO) are exhibited in Fig. 4. Using the example shown in Fig. 3, we are able to obtain two modified plans. An improved plan can be determined by projecting the financial performance of each of the four plans (original, proposed, plus two modified plans). Figure 5 exhibits the case where an original proposal is improved by executing only the red-colored submove.

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Read in the network (adjacency matrix)

Set the submove labels for all nodes (fixtures) to unlabeled; Set the value of current submove label to 1

Initialize visit buffer as empty (visit buffer contains two fields, node and whether it has been visited)

Increment current submove label by 1

No

Are there still unlabeled nodes?

Yes

Set first unlabeled node to unvisited and add it to the visit buffer; Set first unlabeled node to current submove label

No

Are there still unvisited nodes in the visit buffer?

Yes

Output all nodes along with their submove labels

Set first unvisited node in the visit buffer to visited No

End Found?

Find in the network any node that is connected to or from this node but is not already present in the visit buffer

Yes Set all found nodes to unvisited and add them to the visit buffer; Set all found nodes to current submove label

Fig. 2 Submove identification algorithm

1.2 Empirical Results A crucial PMO step involves projecting the financial impact of all proposed store plans, original or modified. The projection should account for the impact of placement, which is obtainable through a regression model in the product space. At the moment, the regression model has two limitations: (1) it only works for historical analysis, meaning we have executed the original proposal and are now looking back to see if it could have been improved; (2) estimated impact of location represents an average value and is not item-specific. Our commercial research partner is developing a set of regression models that can trace the impact of location for each item. More sophisticated demand models built in characteristic space, e.g., Nevo (2000), and Berry et al. (1995, 2004), is also worth considering—in this case, item location can be an essential characteristic in the demand model. As part of ongoing research, however, we wish to formulate a clustering/testing methodology that will address both limitations.

Merchandise Placement Optimization

135

Fig. 3 Placement move as a network

We use the following variables to explain the sales of an individual SKU: (1) fixture placement (dummy vector), (2) display quantity or face-out, (3) unit retail price, and (4) seasonality indices. The relative impact of placement with respect to a benchmark fixture can be inferred through the regression coefficients associated with the placement dummy vector. Figure 6 shows average revenue contribution of 20 shelves at a sample store relative to the benchmark shelf, labeled 0. Average revenue contribution is defined as the difference in average revenue when an SKU is placed on a particular shelf and when it is placed on the benchmark shelf 0, all else being equal. We applied PMO to 5 stores and a total of 29 pairs of consecutive placement plans, or 29 plan changes. Here, we do not consider consecutive moves across more than 2 consecutive plans. Suppose we have consecutive plans A, B, C and D. The analyses are conducted on 3 pairs of consecutive placement plans, i.e., plans A and B, plans B and C, and finally plans C and D. Even though an improvement can be found over plan B as we analyze plans A and B, we still use the original plan B as the starting point for the analyses carried out on plans B and C. This is reasonable because PMO is used to improve a proposed plan, rather than a series of moves. The PMO procedure shows an average revenue improvement of 5% (range: 0– 24%) over the original proposal. The example in Fig. 7 shows revenue improvement for each of the 14 plan changes at a sample store. Certain information—specifically, store name and date range of each plan—is disguised due to a nondisclosure agreement with our commercial partner. PMO is a conservative, fast and scalable procedure to obtain the value of placement. It considers only a subset of all possible moves, but enumerating plans using submoves is a polynomial-time operation. The procedure can be applied to

136

W. Ke Start

Read in a pair of consecutive plans Group all items into 1) those present in both plans, 2) those retired in the next plan and 3) those introduced to the next plan Enumerate all source-sink pairs and set the corresponding entry in the adjacency matrix to 1

Initialize n-by-n adjacency matrix (n = # of unique fixtures in both plans plus 1 for stock room) with 0's

Yes

At end of list for items present in both plans?

No

Set all fixtures associated with current item in current plan as source, and those associated with current item in next plans as sink

Contents of the source different from those of the sink?

Set all fixtures associated with current item in current plan as source, and stock room as sink

Enumerate all sourcestockroom pairs and set the corresponding entry in the adjacency matrix to 1

Move to next item in the list

Set stockroom as source, and all fixtures associated with current item in next plan as sink

Enumerate all stockroomsink pairs and set the corresponding entry in the adjacency matrix to 1

Move to next item in the list

No

Move to next item in the list

Yes At end of list for items retired in the next plan?

No

Yes At end of list for items introduced to the next plans?

No

Yes Remove all adjacency matrix rows and columns that are completely 0 (i.e. fixtures that are not involved in any moves)

Temporarily remove the stock room (last row and last column) from the adjacency matrix

Input the resulting matrix into the algorithm that finds all independent submoves and labels corresponding fixtures (see separate flowchart)

Project revenue of the original proposal (all submoves executed), and that of the premove plan (no submoves executed)

Processed all submove groups?

No

Yes Best plan is found by executing all submoves that help store performance, and worst plan is found by submoves that hurt

Fig. 4 PMO algorithm

Project revenue of an intermediate plan, obtained by executing only the current submove (while keeping all other submoves unexecuted)

Subtract revenue of the premove plan from that of the current intermediate plan (positive = current submove helps store performance)

Construct and save PMO output

End

Move to next submove group

Merchandise Placement Optimization

137

Pre-move plan $28,000

Original proposal $31,000

Modified proposal 1 $32,000 (best)

Modified proposal 2 $27,000

Fig. 5 Sample output of PMO algorithm

Average Contribution w.r.t. Shelf 0 ($)

350 300 250 200 150 100 50 0 (50) (100)

01

02

03

04

05

06

07

08

09

10 11 Shelf

12

13

14

15

16

17

18

Fig. 6 Contribution of shelf placement to SKU sales

Revenue improvement over original proposal 30.00% 25.00% 20.00% 15.00% 10.00% 5.00% 0.00%

Fig. 7 Revenue improvement due to PMO

Additional revenue improvement over worst move

19

20

138

W. Ke

Stock Room

Stock Room

Fig. 8 Varying store layouts

multiple stores and inter-store moves. That said, the value of placement is likely to vary from store to store. Some stores may see higher/lower improvement due to different layout arrangements from other stores (see Fig. 8).

2 Forecasting the Value of Placement In this section, we propose a method to forecast the impact of future moves through a biclustering algorithm that operates on both the SKU- and the fixturedimensions. SKUs may get moved to a future location for the first time. In this case, we cannot rely directly upon historical observations to analyze the impact of this move, because the models that work well on historical moves, e.g., product space regressions, are not sufficient to establish a correlation between an SKU and its correspondingly unobserved future location.2 Furthermore, we note that 1. SKUs do not all respond to their placement locations in the same manner: Some SKUs may be more sensitive to locations than others, resulting in a differing change in demand due to the move; 2. Prime real estate, such as high foot-traffic areas near the store entrance, generally improves SKU sales (at varying degrees as discussed previously), but fixtures at the back usually provide no such lift, if not depressing the sales further. A biclustering algorithm, which operates on both the SKU- and the fixturedimensions and employs historically observed correlations between the locational sensitivity of an SKU and the inherent financial impact of a location, may provide a good guess of those unobserved future correlations that belong to the same cluster. 2 Demand

models in the characteristic space, e.g., Nevo (2000), and Berry et al. (1995, 2004), could provide an answer to this dilemma—we plan to investigate those models in future work.

Merchandise Placement Optimization

139

In addition, store managers are often faced with the daunting task of moving thousands of SKUs across hundreds of fixtures. It is unrealistic and almost impossible to test the financial impact of each product-location combination. A clustering methodology that groups “similar” products as well as “similar” fixtures will significantly reduce the size of the problem, and better ensure that a sufficient amount of data are collected for meaningful statistical analyses.

2.1 Biclustering of Locational Impact on Demand As its name suggests, a biclustering algorithm performs simultaneous clustering of both the rows and the columns of an input matrix. The row and column values of an input matrix suited for biclustering are usually categorical variables instead of multi-dimensional attributes. A traditional clustering algorithm, e.g. k-means, relies on creating a Euclidean metric for all attribute dimensions and does not work well with categorical variables. Biclustering as a term was first introduced by Cheng and Church (2000) in gene expression analysis, although one of the earliest biclustering algorithms was the direct clustering method introduced by Hartigan (1972)—on which our algorithm is based. A good survey of biclustering algorithms can be found in Madeira and Oliveira (2004), whereas the traditional clustering algorithms are discussed in Punj and Stewart (1983). Demand Normalization and the Fractional Response Matrix First, we formalize a set of simple regression models, developed by our commercial partner and discussed in Sect. 1.2, to tease out the locational impact on demand for each SKU. Assume there are m items and n fixtures. Let .dit be demand of SKU i at time t. Let .zij t ∈ {0, 1} represent whether SKU i was placed on fixture j at time t. Let .Hi be the set of all fixtures on which SKU i was placed historically. Let .h0i be the base location of SKU i. Let .yit be the vector of non-location explanatory variables for SKU i at time t, such as display quantity, unit retail price and seasonality indices. The multi-linear regression to normalize demand of SKU i to control for known sources of shocks can be written as:  .dit = βi0 + αij zij t + βi yit + i . (1) j ∈Hi \h0i

Let .y¯ i represent the average values of the vector of non-location dependent variables. We define the normalized demand of SKU i absent the location effect as follows: d¯i := βi0 + βi y¯ i .

.

140

W. Ke

Furthermore, we define the fractional response of SKU i when it is placed on fixture j as follows: δij :=

.

αij . d¯i

These .δij ’s can be interpreted as the percentage gain/loss in demand relative to a base location .h0i , and can be organized into the fractional response matrix, to which we apply the biclustering algorithm. This suggests that the normalized demand of SKU i on fixture j , which was historically observed, can be expressed as dij := d i + αij = d i (1 + δij ).

.

(2)

Note that the fractional response matrix contains missing cells when the historical fixture set .Hi of an SKU .i ∈ {1, . . . , m} is a subset of the full fixture set .Si := {1, . . . , n}— This occurs quite frequently, as it takes an astronomical number of trials to cover all possible item-fixture combinations.

The Standard Response Matrix Fractional responses cannot be compared from one row to another, with each row representing an SKU. This is due to the fact that the base location .h0i , which is dropped from the regression (Eq. 1) to prevent collinearity of the location vector, may be different from SKU to SKU. We wish to standardize these responses so that, for all SKUs, they represent a percentage gain/loss in demand relative to a standard location (e.g. the store-wide “average” location). Let .xij be the standard response of SKU i when it is placed on fixture j . Let .xi be the standard response of SKU i when it is placed on the base fixture .h0i . For SKU i, we then have that xij = xi + δij , ∀j ∈ Hi .

.

(3)

Let .x¯ij be the average response of the bicluster to which SKU i and fixture j belong. Let .Cij represent the ID of this bicluster. To update the standard response matrix given a clustering scheme, we solve the following quadratic program: (P)

.

min x

s.t.



(xi + δij − x¯ij )2

(i,j ) not missing



(xk + δkl − x¯ij ) = 0, ∀(i, j ) not missing

(k,l):Ckl =Cij



(i,j ) not missing

(xi + δij ) = 0

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141

The first equality condition can be interpreted as follows: The average response of a cluster is simply the simple average of all responses within that cluster. The above quadratic problem can be solved by finding the solution to the linear Lagrangian system: 

L(x, x¯ , λ, μ) =

(xi + δij − x¯ij )2

.

(i,j ) not missing





+

λij (xk + δkl − x¯ij )

(i,j ) not missing (k,l):Ckl =Cij



+μ

(xi + δij ).

(4)

(i,j ) not missing

We define the following auxiliary variables:  yij =

.

 zklij =

1, if (i, j ) is not missing; 0, otherwise;

1, if (k, l) is in the same cluster as (i, j ) and is not missing; 0, otherwise.

Then, Eq. 4 can be arranged into a linear system described in Table 1. We also note that updating the standard response assumes a given clustering scheme. In other words, the standard response matrix should be updated after each biclustering iteration. Using the final standard response matrix following the completion of the clustering steps, we can arrive at an estimate of the normalized demand for SKU i when it is placed on a new fixture j (i.e., it is not a historically observed move) by calculating: dijest := d i (1 + δijest ) = d i (1 + x ij − xi ),

.

(5)

where .x ij is the average standard response of the cluster, to which .(i, j ) belongs. Equation 5 is obtained from the relationships described in Eqs. 2 and 3.

The Biclustering Algorithm The biclustering algorithm applied to the standard response matrix is based on the direct clustering method introduced by Hartigan (1972), which uses a divideand-conquer approach. The algorithm starts by sorting the rows (SKUs) by their variances and the columns (fixtures) by their means. We do this because it makes sense for SKUs to be grouped by their location sensitivities, measured through the

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ .⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

y1j

−2 .. . −2

j

2

j

y2j

−2 .. . −2



..

..

.

.

2 j

ymj

−2 .. .

 2

−2 · · · −2

−2 .. .. .. .. . . . .    z z · · · z 1lij 2lij mlij l l l .. .. ... . .    j y1j j y2j · · · j ymj

2

−2

Table 1 Lagrangian system for updating standard response ⎛ 

−

.

k

l zklij

 

..

···

−2 .. .

.. .

2

−2 · · · −2 · · · .. .

··· ···

− k

l zklij

 

 z1lij l l z2lij .. .  l zmlij

.. .

 ··· y1j j ··· j y2j .. .  · · · j ymj

−

i

j δij yij

⎛ ⎞  −2 j δ1j ⎟⎛ ⎞  −2 δ ⎟ j 2j ⎜ ⎟ ⎟ x .. ⎟ ⎟⎜ ⎟ ⎜ . ⎟  ⎟⎜ ⎟ ⎜ ⎟ −2 j δmj ⎟⎜ ⎟ ⎜ ⎟ ⎟⎜ ⎟ ⎜ ⎜ ⎟ ⎟⎜ ⎟ ⎜ .. ⎟ ⎟⎜ ⎟ ⎜ . ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎟ ⎜ x¯ ⎟ ⎜ 2δij ⎟ ⎟⎜ ⎟ = ⎜ ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎟⎜ ⎟ ⎜ ⎟ . ⎟⎜ ⎟ ⎜ .. ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎟⎜ ⎟ ⎜ ⎟ . .. ⎟⎝ λ⎠ ⎜ ⎟ ⎟   ⎜ ⎟ δ z − ⎟ ⎝ k .l kl klij ⎠ ⎟ μ .. ⎠  

⎞

142 W. Ke

Merchandise Placement Optimization

143

variance of the standard response to different fixtures, and for fixtures to be grouped by the average level of sales lift, measured by the mean standard response across SKUs. At each iteration, the algorithm makes a row or column split over one of the existing clusters to improve total square error. The iterative process continues until we have reached a pre-specified number of clusters or we are running out of feasible clusters to split. We would like to remind the reader that the standard response updating problem (P) at each iteration minimizes total square error, given the clustering scheme arrived at in the previous step. This means solving (P) is compatible with the objective of each clustering step and as such does not affect the optimality of the clustering scheme that was input into (P) in the first place. The flow-chart of the algorithm is shown in Fig. 9. A few numerical examples are discussed in the following section.

Start Read in the item-fixture response matrix (converted to fractions)

Total # < Max # of clusters and there exists a feasible cluster ID?

Clean up the fractional response matrix by removing rows or columns that are completely missing

Initialize the standard response matrix with a single cluster Sort the rows by their variances and sort the columns by their means

Output the final clustering scheme

End

No

Set min error to the total square error of the entire response matrix Set total # of clusters to 1 and set cluster ID 1 to feasible

Yes

Set clustering changed to false

No

Enumerated all existing clusters?

No

Current cluster feasible?

Yes

Current cluster contains at least 4 nonmissing cells?

Yes

No

Set current cluster to infeasible

Yes No There are at least 2 rows in the current cluster?

Yes

Considered all row split scenarios?

No

No

Yes

Each of the two post-split parts contains at least 2 non-missing cells? Yes

Tag the first post-split part with the current cluster ID and tag the second post-split part with a new cluster ID (= current total + 1)

Compute total square error of the new clustering scheme

Update incumbent clustering scheme and min error Set clustering changed to true Track which cluster was split

There are at least 2 columns in the current cluster?

Considered all column split scenarios?

Each of the two post-split parts contains at least 2 non-missing cells?

No

No

Clustering changed?

Yes

Yes

Yes

Tag the first post-split part with the current cluster ID and tag the second post-split part with a new cluster ID (= current total + 1)

Compute total square error of the new clustering scheme

Set both post-split parts to feasible Increment total # of clusters by 1

Update the standard response matrix with the new clustering scheme

No

Fig. 9 Biclustering algorithm

No

Yes Update incumbent clustering scheme and min error Set clustering changed to true Track which cluster was split

144

W. Ke

2.2 Empirical Results Example 1: 5 Items/3 Fixtures/Max 3 Clusters We consider an example with the artificial fractional response matrix (.δij ’s) shown in Table 2, where blank entries indicate missing values. Step 0a Find the standard response matrix (.xij ’s) by solving the Lagrangian system in Table 15 (in Appendix) and applying Eq. 3. We obtain Table 3. The .xi ’s (base locations) that solved the system are x1 = − 0.13333, x2 = − 0.15000, x3 = − 0.40000, x4 = − 0.45000, x5 = − 0.45000.

.

Step 0b Sort the rows of the standard response matrix above by their variances and the columns by their means, and obtain Table 4. Step 1a Find the first row or column split that minimizes the total square error (best error = 0.03129), and obtain Table 5.

Table 2 Example 1—fractional response matrix

Table 3 Example 1 step 0a—initial standard response matrix

Table 4 Example 1 step 0b—sorted standard response matrix

.δij

Item 1 Item 2 Item 3 Item 4 Item 5

Fix 1 0.10000 0.10000 0.30000

Fix 1

Fix 2

Item 1 Item 2 Item 3 Item 4 Item 5 Mean

.−0.03333

.−0.03333

.−0.05000 .−0.10000 .−0.15000 .−0.15000 .−0.08333

.−0.09444

.xij

Fix 2

Fix 1

Item 1 Item 2 Item 3 Item 4 Item 5 Mean

.−0.03333

.−0.03333 .−0.05000

.−0.10000

.−0.10000

.−0.15000 .−0.15000 .−0.09444

0.30000 0.30000

0.30000

.xij

.−0.10000

Fix 2 0.10000

.−0.08333

Fix 3 0.20000 0.20000 0.60000 0.60000 0.60000

Fix 3 0.06667 0.05000 0.20000 0.15000 0.15000 0.12333

Variance 0.00333 0.00500 0.03000 0.04500 0.04500

Fix 3 0.06667 0.05000 0.20000 0.15000 0.15000 0.12333

Variance 0.00333 0.00500 0.03000 0.04500 0.04500

Merchandise Placement Optimization Table 5 Example 1 step 1a—standard response matrix after 1st split

Table 6 Example 1 step 1b —updated standard response matrix given 1st split

Table 7 Example 1 step 2a—standard response matrix after 2nd split

Table 8 Example 1 step 2b—updated standard response matrix given 2nd split

145 .xij

Item 1 Item 2 Item 3 Item 4 Item 5

Fix 2 .−0.03333 .−0.10000

Fix 1 .−0.03333 .−0.05000 .−0.10000

.−0.15000 .−0.15000

.xij

Fix 2

Fix 1

Item 1 Item 2 Item 3 Item 4 Item 5

.−0.05147

.−0.05147 .−0.03186

.−0.11814

.−0.11814

.−0.13186 .−0.13186

.xij

Fix 2

Fix 1

Item 1 Item 2 Item 3 Item 4 Item 5

.−0.05147

.−0.05147

.xij

Fix 2 .−0.09167

Item 1 Item 2 Item 3 Item 4 Item 5

.−0.03186 .−0.11814

.−0.11814

.−0.13186 .−0.13186

.−0.09167

Fix 1 .−0.09167 .−0.09167 .−0.09167

.−0.09167 .−0.09167

Fix 3 0.06667 0.05000 0.20000 0.15000 0.15000 Fix 3 0.04853 0.06814 0.18186 0.16814 0.16814 Fix 3 0.04853 0.06814 0.18186 0.16814 0.16814 Fix 3 0.00833 0.00833 0.20833 0.20833 0.20833

Step 1b Recompute the standard response matrix by solving the Lagrangian System in Table 16, given the updated clustering scheme. We obtain Table 6. The .xi ’s that solved the system are x1 = − 0.15147, x2 = − 0.13186, x3 = − 0.41814, x4 = − 0.43186, x5 = − 0.43186.

.

Step 2a Find the second row or column split that minimizes the total square error over the updated standard response matrix (best error = 0.01175), and obtain Table 7. Step 2b Recompute the standard response matrix by solving the Lagrangian System in Table 17, given the updated clustering scheme and stop because we have reached 3 clusters. We obtain Table 8.

146 Table 9 Example 1—forecasted fractional response matrix

W. Ke .δij

Item 1 Item 2 Item 3 Item 4 Item 5

Fix 1 0.10000 0.10000 0.30000 0.30000 0.30000

Fix 2 0.10000 0.10000 0.30000 0.30000 0.30000

Fix 3 0.20000 0.20000 0.60000 0.60000 0.60000

The .xi ’s that solved the system are x1 = − 0.19167, x2 = − 0.19167, x3 = − 0.39167, x4 = − 0.39167, x5 = − 0.39167.

.

We observe in this particular example that subtracting the final set of .xi ’s from final standard response matrix in Table 8 recovers the original fractional response matrix Table 2 for the item-fixture combinations that were historically present. Furthermore, we are also able to say, for instance, that Item 2 will receive a fractional est = −0.09167 − 0.19176 = 0.1. We lift of 0.1 when it is placed on Fixture 2, i.e. .δ22 arrive at the fractional response matrix in Table 9 with a forecast for all previously empty cells, which represents historically unobserved item-fixture combinations.

Example 2: Disguised Store Data with 45 Items/7 Fixtures/Max 20 Clusters There are a total of 161 items across 38 fixtures in the initial data that we received from our commercial partner. For simplicity, we will demonstrate the performance of the biclustering algorithm on a subset of that data containing 45 items and 7 fixtures. Item and fixture names are disguised. The exhibits are organized as follows: • • • • •

Table 10 summarizes the performance of the clustering scheme; Table 11 shows the fractional response matrix; Table 12 shows the clustered standard response matrix; Table 13 shows the clustering scheme; Table 14 shows the forecasted fractional response matrix (see Eq. 5).

We summarize the performance of the clustering scheme in Table 10, which is sorted by the standard deviation of each cluster. Most of the clusters appear to perform well with a relatively low standard deviation amongst the non-missing itemfixture combinations in the cluster. Clusters with high standard deviations could be improved if we allow more clusters (greater than the designated maximum of 20 clusters) to be made.

Merchandise Placement Optimization Table 10 Example 2—performance of standard response by cluster

147 Cluster ID 14 11 4 16 10 1 6 17 20 2 5 15 13 19 12 18 9 8 3 7

Mean −0.3497 0.5957 0.6729 −1.2101 0.0789 0.3509 0.7394 −0.2298 0.1179 0.5007 0.5907 0.0030 0.3085 −0.7636 −1.6983 −0.9623 −0.2569 1.1372 −2.4495 −1.7982

Std dev 0.0000 0.0000 0.0000 0.0000 0.0058 0.0115 0.0273 0.0361 0.0505 0.0575 0.0587 0.0710 0.0862 0.0892 0.3532 0.3532 0.3709 0.5245 0.5245 0.6396

Max −0.3497 0.5957 0.6729 −1.2101 0.0897 0.3650 0.7587 −0.2043 0.1536 0.6606 0.6908 0.0532 0.3694 −0.6390 −1.4486 −0.7126 0.1140 1.5080 −2.0786 −1.1054

Min −0.3497 0.5957 0.6729 −1.2101 0.0681 0.3369 0.7201 −0.2553 0.0822 0.4006 0.4308 −0.0472 0.2475 −0.8883 −1.9480 −1.2121 −0.6277 0.7663 −2.8203 −2.3664

Table 11 Example 2—fractional response matrix .δij

Item 1 Item 2 Item 3 Item 4 Item 5 Item 6 Item 7 Item 8 Item 9 Item 10 Item 11 Item 12 Item 13 Item 14 Item 15 Item 16 Item 17 Item 18 Item 19

Fix 1

Fix 2 0.4341

Fix 3

Fix 4 −0.6278 1.7331 −0.5763

2.5623

Fix 5

Fix 6

Fix 7 0.7190

−1.2879 0.9872

1.3962

−2.2721 0.6628

0.9544 −0.8314 −0.7683

1.1867 0.8180

0.9324

0.3524 −0.6935

0.8725 1.2337 −0.3274

−0.9275 1.6898

1.2268 −1.5355 −0.5563

−0.4472 −0.7859 −0.4966 (continued)

148

W. Ke

Table 11 (continued) .δij

Item 20 Item 21 Item 22 Item 23 Item 24 Item 25 Item 26 Item 27 Item 28 Item 29 Item 30 Item 31 Item 32 Item 33 Item 34 Item 35 Item 36 Item 37 Item 38 Item 39 Item 40 Item 41 Item 42 Item 43 Item 44 Item 45

Fix 1

Fix 2

Fix 3

−0.6499 2.5386 0.6348

Fix 4

Fix 5 −0.3522

Fix 6

Fix 7

−0.5282 0.9969 0.4220 −0.3831

1.3273 1.3966 0.9596 0.6382 0.2844

−0.5819

−0.7112 −0.8184

0.7397 0.3757

0.7895 0.7253

0.8749

1.6444 1.4157 2.1597 0.8764 −0.9289 −0.8472 −0.5649 0.8712 1.1934

0.9650 1.9132 0.8759 0.6592

0.6778 −0.6172 0.5355

−0.7529 0.3883 0.4736

−0.9359 1.1254 4.2969 1.5685 1.7574

1.5680

Table 12 Example 2—final standard response matrix Fix 2 Fix 6 Fix 5 Fix 4 Fix 3 Fix 7 Item 4 0.5724 Item 7 0.7888 Item 8 0.7888 Item 12 Item 13 0.7888 Item 15 0.7888 Item 16 0.7888 Item 17 0.3594 Item 18 0.5966 Item 19 0.5966 Item 20 0.5957 Item 24 0.5724

.xij

Fix 1

Base (.xi ) 1.0084 0.8820 0.8472 0.6729 1.3659 1.0064 1.6144 0.6352 0.7982 1.3766 1.0813 0.9460 0.8837 (continued)

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149

Table 12 (continued) .xij

Item 25 Item 26 Item 27 Item 30 Item 41 Item 42 Item 43 Item 44 Item 40 Item 36 Item 21 Item 23 Item 39 Item 1 Item 6 Item 9 Item 10 Item 38 Item 2 Item 14 Item 3 Item 11 Item 31 Item 45 Item 29 Item 37 Item 32 Item 33 Item 34 Item 28 Item 35 Item 22 Item 5

Fix 2

Fix 6

Fix 5

Fix 4

Fix 3 Fix 7 0.5966 0.5966

0.5724 0.5966 0.5724 0.7888 0.5966 0.5966 0.5966 0.5957

0.5724 0.4867 0.4550 0.6661

0.3650

0.6364 0.4377

0.3369

0.6470 0.6975 0.2475 0.3694

0.4634 0.5617 0.6889 −0.4720 0.5316 0.8967

0.4555 0.4899 0.5365 0.4954

−0.2553 −0.2425

0.4752 −0.3497 −0.3497 −1.2166 −1.2166 −0.6390 −0.8883 −0.7636 −0.7443 −0.7830

0.3594 −1.4486 −1.2126 −1.9485 −0.7126 −1.9229 −2.3664 −1.1545 −2.7862

0.8218 0.1536 0.1140

−0.2569

−0.6277 0.7256 0.7587

-2.8233

Fix 1

Base (.xi ) −0.4414 −0.8060 −0.4583 0.2150 1.4358 −1.0462 −3.7063 −0.9778 0.6729 −0.1346 1.3478 1.0149 0.0330 −0.3587 −0.0972 −0.2596 −0.6250 −0.8643 −0.4205 1.1177 −1.1445 −1.2426 0.7663 0.4161 −1.1692 −1.4065 −0.6301 −0.6451 −2.6258 −2.7987 −1.7647 −1.3674 0.1846 −1.7798 1.5826 −0.5482

Table 13 Example 2—clustering scheme Cluster ID Item 4 Item 7 Item 8 Item 12 Item 13

Fix 2 1 1 1 1 1

Fix 6 10 10 10 10 10

Fix 5 11 11 11 11 11

Fix 4 2 2 2 2 2

Fix 3 5 5 5 5 5

Fix 7 5 5 5 5 5

Fix 1 4 4 4 4 4 (continued)

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Table 13 (continued) Cluster ID Item 15 Item 16 Item 17 Item 18 Item 19 Item 20 Item 24 Item 25 Item 26 Item 27 Item 30 Item 41 Item 42 Item 43 Item 44 Item 40 Item 36 Item 21 Item 23 Item 39 Item 1 Item 6 Item 9 Item 10 Item 38 Item 2 Item 14 Item 3 Item 11 Item 31 Item 45 Item 29 Item 37 Item 32 Item 33 Item 34 Item 28 Item 35 Item 22 Item 5

Fix 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7 7 3 3 3 3

Fix 6 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 12 12 12 12 12 12 12 12 7 7 3 3 3 3

Fix 5 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 15 15 15 15 15 15 15 15 17 17 17 17 17 18 18 18 18 7 7 3 3 3 3

Fix 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 14 14 16 16 19 19 19 19 19 19

Fix 3 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 13 13 20 20 20 20 9 9 9 9 9 9 9 9 9 6 6 6

Fix 7 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 13 13 20 20 20 20 9 9 9 9 9 9 9 9 9 6 6 6

Fix 1 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

Merchandise Placement Optimization

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Table 14 Example 2—forecasted fractional response matrix est

.δij

Fix 2

Fix 6

Fix 5

Fix 4

Fix 3

Fix 7

Fix 1

Item 4 Item 7 Item 8 Item 12 Item 13 Item 15 Item 16 Item 17 Item 18 Item 19 Item 20 Item 24 Item 25 Item 26 Item 27 Item 30 Item 41 Item 42 Item 43 Item 44 Item 40 Item 36 Item 21 Item 23 Item 39 Item 1 Item 6 Item 9 Item 10 Item 38 Item 2 Item 14 Item 3 Item 11 Item 31 Item 45 Item 29 Item 37 Item 32 Item 33 Item 34

−0.6574 −0.5311 −0.4962 −1.0150 −0.6555 −1.2634 −0.2842 −0.4472 −1.0256 −0.7304 −0.5950 −0.5328 0.7924 1.1569 0.8092 0.1360 −1.0849 1.3971 4.0572 1.3288 0.4855 −0.9969 −0.6499 0.3180 0.7097 0.4341 0.6106 0.9759 1.2152 0.7714 −0.7668 1.4955 1.5935 −0.0651 1.5202 1.7574 0.9811 0.9961 2.9767 1.0005 −0.0336

−0.9295 −0.8031 −0.7683 −1.2871 −0.9275 −1.5355 −0.5563 −0.7193 −1.2977 −1.0024 −0.8671 −0.8049 0.5203 0.8848 0.5372 −0.1361 −1.3569 1.1250 3.7852 1.0567 0.2135 −1.2690 −0.9360 0.0459 0.4376 0.1761 0.3385 0.7038 0.9324 0.4993 −1.0281 −0.5538 −0.4557 −2.1143 −0.5290 −0.2918 −0.8184 −1.0531 0.6778 1.0005 −0.6017

−0.4126 −0.2863 −0.2514 −0.7702 −0.4107 −1.0186 −0.0394 −0.2024 −0.7808 −0.4856 −0.3502 −0.2880 1.0372 1.4017 1.0540 0.3808 −0.8401 1.6419 4.3020 1.5736 0.7304 −0.7521 −1.0119 −0.0300 0.3617 0.1002 0.2626 0.6279 0.8171 0.4736 −1.3475 0.9147 0.9872 −0.6459 0.9650 0.4441 −0.5819 −0.3172 1.9132 0.8759 0.6592

−0.5076 −0.3813 −0.3465 −0.8652 −0.5057 −1.1136 −0.1345 −0.2974 −0.8759 −0.5806 −0.4452 −0.3830 0.9421 1.3067 0.9590 0.2857 −0.9351 1.5469 4.2070 1.4785 0.6353 −0.8471 −0.5282 0.4220 1.0193 0.5979 0.6603 1.1867 1.3650 0.8710 −0.6278 1.6810 1.7331 0.0847 1.6444 1.0568 0.2804 −0.5649 1.4157 2.1597 0.8764

−0.4177 −0.2913 −0.2565 −0.7753 −0.4157 −1.0237 −0.0445 −0.2075 −0.7859 −0.4907 −0.3553 −0.2931 1.0321 1.3966 1.0489 0.3757 −0.8452 1.6368 4.2969 1.5685 0.7253 −0.7572 −0.4242 0.6035 0.7895 0.6879 0.8503 0.9334 1.1728 0.5384 −0.9998 1.2624 1.3605 −0.6729 0.9124 1.1496 0.3732 0.3883 2.3689 2.5419 1.5078

−0.4177 −0.2913 −0.2565 −0.7753 −0.4157 −1.0237 −0.0445 −0.2075 −0.7859 −0.4907 −0.3553 −0.2931 1.0321 1.3966 1.0489 0.3757 −0.8452 1.6368 4.2969 1.5685 0.7253 −0.7572 −0.4242 0.5577 0.9494 0.7019 0.9504 0.8725 1.2337 0.5384 −0.9998 1.2267 1.3962 −0.3021 0.9124 1.1496 0.3732 0.3883 2.3689 2.5419 1.5078

−0.3355 −0.2091 −0.1743 −0.6930 −0.3335 −0.9415 0.0377 −0.1253 −0.7037 −0.4084 −0.2731 −0.2109 1.1143 1.4789 1.1312 0.4579 −0.7629 1.7190 4.3792 1.6507 0.8075 −0.2107 0.1223 1.1042 1.4959 1.2344 1.3968 1.7621 2.0015 1.5576 0.0194 2.2817 2.3797 0.3502 2.3064 2.5436 1.7673 1.7823 3.7630 3.9359 2.9018 (continued)

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Table 14 (continued) est

.δij

Fix 2

Fix 6

Fix 5

Fix 4

Item 28 Item 35 Item 22 Item 5

−1.0820 −2.6341 −0.6696 −2.2721

−0.7112 −2.6341 −0.6696 −1.9013

−1.0820 −2.6341 −0.6696 −1.9013

0.6038 −0.9289 0.9969 −0.2154

Fix 3 1.1106 0.5548 2.5386 1.2876

Fix 7 0.7397 0.5355 2.5192 1.2876

Fix 1 2.5046 0.9526 2.9170 2.0562

2.3 A Note on Data Scarcity The SKU-oriented regression model, first described in Sect. 1.2 and later formalized in Sect. 2.1, has an important restriction: an SKU needs to be present long enough before it can be fitted by the regression model. That restriction comes from the normality assumption of the error term in a multi-linear regression. In a regression with only one independent variable, it translates to a practical minimum of around 30 weekly3 observations. Needless to say, the SKU-level regression model currently implemented by our commercial partner, has a lot more variables, plenty of which are ued to capture locational impact on a particular SKU, and thus requires a lot more observations to ensure sufficient degrees of freedom in the error term. Even with the adopted cutoff at 40 weeks, statistical implications aside, an SKU would have to be tracked for nearly 10 months before we can estimate the impact of location on its sales. A large majority of the SKUs sold by our commercial partner do have a shelf life much longer than 40 weeks, so eventually we have sufficient data to estimate locational impact. However at the inception of this project, locational data was just beginning to be properly tracked. As a result, a large number of SKUs did not have accurate location estimates. While the biclustering method in Fig. 9 is designed to accommodate a sparse SKU-fixture response matrix, arising from insufficient natural experimentation on historical moves, it still requires that some observations of the locational impact on SKU sales be present—i.e., the SKU still needs to be tracked for more than the adopted cutoff time. Until we allowed enough time for more SKUs to make the cut, implementing the biclustering algorithm was not a practical reality. Our commercial partner has therefore proposed a fixture-oriented model. Instead of forecasting the SKU demand directly, the fixture-oriented model estimates the total aggregate demand of a fixture. It also relies on the consolidation of SKUs into item clusters (“ICs”) as a first step, and therefore does not generally suffer from the same data scarcity problem as the SKU-oriented model.

3 Retailers generally track sales and inventory data on a weekly basis, possibly due to operational habit. Another reason is that one can smooth over the daily sales volatilities with weekly data.

Merchandise Placement Optimization

153

The Fixture-Oriented Regression Model SKUs are first grouped into c distinct ICs by sales volume, price, and vintage.4 At period t, let total aggregate sales volume of fixture .j ∈ {1, . . . , n} be denoted .Dj t , which is obtained by simply adding up the sales volume of each individual SKU on fixture j . .Dj t is then regressed against two sets of factors: (1) a total of q sets of ICspecific attributes .upj kt (.p ∈ {1, . . . , q} and .k ∈ {1, . . . , c}), such as total number of different SKUs in each IC, average presentation quantity and average price, both weighted by historical sales of SKUs in each IC, and (2) store-level and seasonal indices .vj t , e.g. foot traffic to the store and weather. We then multiply the predicted total sales volume of fixture j by the corresponding weighted average price to reach an estimate of the total dollar sales of the entire fixture, often referred to as the fixture productivity. The fixture-oriented regression model can be formally defined as the following: Dj t = ψ0j +

q  c 

.

φpj k upj kt + ψj vj t + j .

(6)

p=1 k=1

Tackling the Sparsity of Fixture-IC Coefficient Matrices with Biclustering For each set .p ∈ {1, . . . , q} of the IC-specific attributes from (6), we denote the fixture-IC coefficient matrix . p := [φpj k ]j =1...n,k=1...c . While this matrix does not generally suffer from data scarcity problem, i.e., every row j contains a nonzero row vector, the matrix . p is still not immune to the insufficiency of natural experimentation on historical IC moves. In other words, there may still be missing cells in . p , for example when a particular IC has never been placed on a particular fixture before. To forecast the missing coefficients, we propose applying the direct biclustering algorithm from Hartigan (1972) to . p or the corresponding standardized coefficient matrix. We refer the reader to Bring (1994) for a discussion on standardization of regression coefficients.

3 Conclusions Merchandise placement optimization is a relatively new addition to the toolkit of retail revenue management. Through a series of practical model formulations, we confirmed that merchandise placement was not a zero-sum game from a revenue perspective. The “Plan Modification Optimization” algorithm proposed in this chapter constitutes a lower bound on the revenue benefit of placement optimization, which we estimated to be roughly 5% using retail data from our commercial partner, and represented a practical (albeit partial) solution to the placement optimization problem. The PMO algorithm is designed to address the combinatorial nature of 4 That

is, how many weeks since an SKU was first introduced.

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the assignment problem formulation for merchandise placement and the difficulty in adequately specifying business rules unique to each retail environment and operation. The key is to intelligently improve a proposed placement plan by (1) decomposing it into feasible independent submoves, (2) evaluating the revenue impact of each submove, and (3) implementing only the submoves with positive revenue gains. Next, we proposed a biclustering algorithm to forecast the impact of future moves. SKUs could move to a future location for the first time. In that case, traditional models that work well on historical moves are no longer sufficient to establish a correlation between an SKU and its correspondingly unobserved location. A biclustering algorithm, which operates on both the SKU- and the fixture-dimensions, is the answer. The proposed algorithm relies on iterative error-minimizing cuts to cluster the sales responses due to location. Empirical experiments show that it works fairly well with sparse response matrices, which are characteristic of retail environments lacking sufficient natural experimentation of SKU placement changes. One interesting problem we hope to tackle in future research is the determination of an optimal number of clusters. Although this problem is likely to be .N P-hard, we hope to find a fast heuristic that can also be easily incorporated into the biclustering framework. The PMO algorithm discussed in this work is semi-automatic in nature, because it requiers a store manager to propose the next placement plan before it can start optimizing. A natural next step is to consider a fully automated placement optimization problem where SKUs are assigned to specific locations in the store, akin to a facilities location problem, at the expense of significantly increased computational complexity. The research works (Bhadury et al., 2016; Ozgormus and Smith, 2020; Flamand et al., 2016; Mowrey et al., 2018; Flamand et al., 2018) briefly discussed in the preamble of this chapter all showcased novel ideas in formulating and tackling this computationally hard problem. To make it even more interesting, let us assume that retail product demand can typically be classified into two types—intrinsic and impulse. Intrinsic demand occurs wherever the SKU is placed, whereas impulse demand depends on foot traffic and location. Along this vein, Hübner and Schaal (2017) addressed impulse demand in their shelf-space optimization model formulation by measuring so-called space elasticity and characterizing demand as stochastic. Pak et al. (2020) focused on solving for the assignment of promotional display space to trigger impulse purchases, resulting in “1.6-2 times” the profit improvement achieved by a common practice where the best-selling SKUs are placed on promotional display. These all repesent exciting directions of research for the merchandise placement optimization problem.

Appendix The Lagrangian systems used in the clustering steps of Example 1 in Sect. 2.2 are shown here in Tables 15, 16, and 17. Solutions to these systems are used to advance the clustering iterations.

6

3 3

λ53 μ

−2 −2 −2

2 2

3 3

2 2

2 2

−2 −2 ⎜ x2 ⎟ ⎜ 4 6 −2 −2 −2 ⎜ x3 ⎟ ⎜ ⎜ x4 ⎟ ⎜ 4 −2 −2 ⎜ x5 ⎟ ⎜ 4 −2 ⎜ ⎟ ⎜ 2 ⎜ x¯11 ⎟ ⎜ −2 ⎜ x¯12 ⎟ ⎜ −2 2 ⎜ x¯13 ⎟ ⎜ −2 2 ⎜ ⎟ ⎜ 2 ⎜ x¯21 ⎟ ⎜ −2 ⎜ x¯23 ⎟ ⎜ −2 2 ⎜ x¯31 ⎟ ⎜ −2 2 ⎜ ⎟ ⎜ −2 2 ⎜ x¯32 ⎟ ⎜ ⎜ x¯33 ⎟ ⎜ −2 2 ⎜ x¯42 ⎟ ⎜ −2 2 ⎜ ⎟ ⎜ −2 2 ⎜ x¯43 ⎟ = ⎜ ⎜ x¯51 ⎟ ⎜ −2 2 ⎜ x¯53 ⎟ ⎜ −2 ⎜ λ ⎟ ⎜ 3 2 3 2 2 −12 ⎜ 11 ⎟ ⎜ ⎜ λ12 ⎟ ⎜ 3 2 3 2 2 −12 ⎜ λ13 ⎟ ⎜ 3 2 3 2 2 −12 ⎜λ ⎟ ⎜ 3 2 3 2 2 −12 ⎜ 21 ⎟ ⎜ ⎜ λ23 ⎟ ⎜ 3 2 3 2 2 −12 ⎜ λ31 ⎟ ⎜ 3 2 3 2 2 −12 ⎜λ ⎟ ⎜ 3 2 3 2 2 −12 ⎜ 32 ⎟ ⎜ ⎜ λ33 ⎟ ⎜ 3 2 3 2 2 −12 ⎜ λ42 ⎟ ⎜ 3 2 3 2 2 −12 ⎜λ ⎟ ⎜ 3 2 3 2 2 −12 ⎜ 43 ⎟ ⎜ ⎝ λ51 ⎠ ⎝ 3 2 3 2 2 −12

x1

Table 15 Example 1 step 0a—Lagrangian system to initialize standard response ⎛ ⎞ ⎛

−12

2

3 2 3 2 −2 2 −12 −12

3 2 3 2 2

−12

3 2 3 2 2

−12

3 2 3 2 2

−12

3 2 3 2 2

−12

3 2 3 2 2

−12

3 2 3 2 2

−12

3 2 3 2 2

−12

3 2 3 2 2

−12

3 2 3 2 2

−12

3 2 3 2 2

−12

3 2 3 2 2

3 2 3 2 2

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

−0.8

⎞

−3.7 −3.7

⎜ −0.6 ⎟ ⎜ −2.4 ⎟ ⎜ −1.8 ⎟ ⎜ −1.8 ⎟ ⎜ ⎟ ⎜ 0.2 ⎟ ⎜ 0.2 ⎟ ⎜ 0.4 ⎟ ⎜ ⎟ ⎜ 0.2 ⎟ ⎜ 0.4 ⎟ ⎜ 0.6 ⎟ ⎜ ⎟ ⎜ 0.6 ⎟ ⎜ 1.2 ⎟ ⎜ 0.6 ⎟ ⎜ ⎟ ⎜ 1.2 ⎟ ⎜ 0.6 ⎟ ⎜ 1.2 ⎟ ⎜ −3.7 ⎟ ⎜ ⎟ ⎜ −3.7 ⎟ ⎜ −3.7 ⎟ ⎜ −3.7 ⎟ ⎜ ⎟ ⎜ −3.7 ⎟ ⎜ −3.7 ⎟ ⎜ −3.7 ⎟ ⎜ ⎟ ⎜ −3.7 ⎟ ⎜ −3.7 ⎟ ⎜ −3.7 ⎟ ⎜ ⎟ ⎝ −3.7 ⎠

⎞−1 ⎛

Merchandise Placement Optimization 155

x1 ⎜ x2 ⎟ ⎜x ⎟ ⎜ 3⎟ ⎜ x4 ⎟ ⎜ ⎟ ⎜ x5 ⎟ ⎜ x¯ ⎟ ⎜ 11 ⎟ ⎜ x¯12 ⎟ ⎜ ⎟ ⎜ x¯13 ⎟ ⎜ x¯ ⎟ ⎜ 21 ⎟ ⎜ x¯23 ⎟ ⎜ ⎟ ⎜ x¯31 ⎟ ⎜ x¯ ⎟ ⎜ 32 ⎟ ⎜ x¯33 ⎟ ⎜ ⎟ ⎜ x¯42 ⎟ ⎜ x¯ ⎟ ⎜ 43 ⎟ ⎜ x¯51 ⎟ ⎜ ⎟ ⎜ x¯53 ⎟ ⎜λ ⎟ ⎜ 11 ⎟ ⎜ λ12 ⎟ ⎜ ⎟ ⎜ λ13 ⎟ ⎜ λ21 ⎟ ⎜ ⎟ ⎜ λ23 ⎟ ⎜ ⎟ ⎜ λ31 ⎟ ⎜ λ32 ⎟ ⎜ ⎟ ⎜ λ33 ⎟ ⎜ ⎟ ⎜ λ42 ⎟ ⎜ λ43 ⎟ ⎜ ⎟ ⎜ λ51 ⎟ ⎝ ⎠ λ53 μ

−2 −2 −2

1 3

1 2

1 3

1 2

1 2

−5

2

2 −2 −2 1 −2 −2 −2 2 −2 −2 1 −2 −2 1 −7

⎜ 4 ⎜ 6 ⎜ ⎜ 4 ⎜ ⎜ 4 ⎜ −2 2 ⎜ ⎜ −2 2 ⎜ ⎜ −2 2 ⎜ −2 2 ⎜ ⎜ −2 2 ⎜ ⎜ −2 2 ⎜ −2 2 ⎜ ⎜ −2 2 ⎜ ⎜ −2 2 ⎜ −2 2 =⎜ ⎜ −2 2 ⎜ ⎜ −2 ⎜ 2 1 2 1 1 −7 ⎜ ⎜ 2 1 2 1 1 −7 ⎜ ⎜ 1 1 1 1 1 −5 ⎜ 2 1 2 1 1 −7 ⎜ ⎜ 1 1 1 1 1 −5 ⎜ ⎜ 2 1 2 1 1 −7 ⎜ 2 1 2 1 1 −7 ⎜ ⎜ 1 1 1 1 1 −5 ⎜ −7 ⎜ 2 1 2 1 1 ⎜ 1 1 1 1 1 −5 ⎜ ⎜ 2 1 2 1 1 −7 ⎝

6

2 1 2 1 1

−7

Table 16 Example 1 step 1b—Lagrangian system to update standard response given 1st split ⎛ ⎞ ⎛

−5

1 1 1 1 1

−7

2 1 2 1 1

−5

1 1 1 1 1

−7

2 1 2 1 1

−7

2 1 2 1 1

−5

1 1 1 1 1

−7

2 1 2 1 1

−5

1 1 1 1 1

−7

2 1 2 1 1

−5

1 1 1 1 1

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

⎞

−0.8 ⎜ −0.6 ⎟ ⎜ −2.4 ⎟ ⎜ ⎟ ⎜ −1.8 ⎟ ⎜ ⎟ ⎜ −1.8 ⎟ ⎜ 0.2 ⎟ ⎜ ⎟ ⎜ 0.2 ⎟ ⎜ ⎟ ⎜ 0.4 ⎟ ⎜ 0.2 ⎟ ⎜ ⎟ ⎜ 0.4 ⎟ ⎜ ⎟ ⎜ 0.6 ⎟ ⎜ 0.6 ⎟ ⎜ ⎟ ⎜ 1.2 ⎟ ⎜ ⎟ ⎜ 0.6 ⎟ ⎜ 1.2 ⎟ ⎜ ⎟ ⎜ 0.6 ⎟ ⎜ ⎟ ⎜ 1.2 ⎟ ⎜ −1.5 ⎟ ⎜ ⎟ ⎜ −1.5 ⎟ ⎜ ⎟ ⎜ −2.2 ⎟ ⎜ −1.5 ⎟ ⎜ ⎟ ⎜ −2.2 ⎟ ⎜ ⎟ ⎜ −1.5 ⎟ ⎜ −1.5 ⎟ ⎜ ⎟ ⎜ −2.2 ⎟ ⎜ ⎟ ⎜ −1.5 ⎟ ⎜ −2.2 ⎟ ⎜ ⎟ ⎜ −1.5 ⎟ ⎝ ⎠ −2.2 −3.7

⎞−1 ⎛

3 2⎟ ⎟ 3⎟ 2⎟ ⎟ 2⎟

156 W. Ke

x1 ⎜ x2 ⎟ ⎜x ⎟ ⎜ 3⎟ ⎜ x4 ⎟ ⎜ ⎟ ⎜ x5 ⎟ ⎜ x¯ ⎟ ⎜ 11 ⎟ ⎜ x¯12 ⎟ ⎜ ⎟ ⎜ x¯13 ⎟ ⎜ x¯ ⎟ ⎜ 21 ⎟ ⎜ x¯23 ⎟ ⎜ ⎟ ⎜ x¯31 ⎟ ⎜ x¯ ⎟ ⎜ 32 ⎟ ⎜ x¯33 ⎟ ⎜ ⎟ ⎜ x¯42 ⎟ ⎜ x¯ ⎟ ⎜ 43 ⎟ ⎜ x¯51 ⎟ ⎜ ⎟ ⎜ x¯53 ⎟ ⎜λ ⎟ ⎜ 11 ⎟ ⎜ λ12 ⎟ ⎜ ⎟ ⎜ λ13 ⎟ ⎜ λ21 ⎟ ⎜ ⎟ ⎜ λ23 ⎟ ⎜ ⎟ ⎜ λ31 ⎟ ⎜ λ32 ⎟ ⎜ ⎟ ⎜ λ33 ⎟ ⎜ ⎟ ⎜ λ42 ⎟ ⎜ λ43 ⎟ ⎜ ⎟ ⎜ λ51 ⎟ ⎝ ⎠ λ53 μ

−2 −2 −2

3

2

1 3

1 2

1 2

−3

2

2 −2 −2 1 −2 −2 −2 2 −2 −2 1 −2 −2 1 −7

⎜ 4 ⎜ 6 ⎜ ⎜ 4 ⎜ ⎜ 4 ⎜ −2 2 ⎜ ⎜ −2 2 ⎜ ⎜ −2 2 ⎜ −2 2 ⎜ ⎜ −2 2 ⎜ ⎜ −2 2 ⎜ −2 2 ⎜ ⎜ −2 2 ⎜ ⎜ −2 2 ⎜ −2 2 =⎜ ⎜ −2 2 ⎜ ⎜ −2 ⎜ 2 1 2 1 1 −7 ⎜ ⎜ 2 1 2 1 1 −7 ⎜ ⎜ 1 1 −2 ⎜ 2 1 2 1 1 −7 ⎜ ⎜ 1 1 −2 ⎜ ⎜ 2 1 2 1 1 −7 ⎜ 2 1 2 1 1 −7 ⎜ ⎜ 1 1 1 −3 ⎜ −7 ⎜ 2 1 2 1 1 ⎜ 1 1 1 −3 ⎜ ⎜ 2 1 2 1 1 −7 ⎝

6

−7

2 1 2 1 1

Table 17 Example 1 step 2b—Lagrangian system to update standard response given 2nd split ⎛ ⎞ ⎛

−2

1 1

−7

2 1 2 1 1

−2

1 1

−7

2 1 2 1 1

−7

2 1 2 1 1

−3

1 1 1

−7

2 1 2 1 1

−3

1 1 1

−7

2 1 2 1 1

−3

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

⎞

−0.8 ⎜ −0.6 ⎟ ⎜ −2.4 ⎟ ⎜ ⎟ ⎜ −1.8 ⎟ ⎜ ⎟ ⎜ −1.8 ⎟ ⎜ 0.2 ⎟ ⎜ ⎟ ⎜ 0.2 ⎟ ⎜ ⎟ ⎜ 0.4 ⎟ ⎜ 0.2 ⎟ ⎜ ⎟ ⎜ 0.4 ⎟ ⎜ ⎟ ⎜ 0.6 ⎟ ⎜ 0.6 ⎟ ⎜ ⎟ ⎜ 1.2 ⎟ ⎜ ⎟ ⎜ 0.6 ⎟ ⎜ 1.2 ⎟ ⎜ ⎟ ⎜ 0.6 ⎟ ⎜ ⎟ ⎜ 1.2 ⎟ ⎜ −1.5 ⎟ ⎜ ⎟ ⎜ −1.5 ⎟ ⎜ ⎟ ⎜ −0.4 ⎟ ⎜ −1.5 ⎟ ⎜ ⎟ ⎜ −0.4 ⎟ ⎜ ⎟ ⎜ −1.5 ⎟ ⎜ −1.5 ⎟ ⎜ ⎟ ⎜ −1.8 ⎟ ⎜ ⎟ ⎜ −1.5 ⎟ ⎜ −1.8 ⎟ ⎜ ⎟ ⎜ −1.5 ⎟ ⎝ ⎠ −1.8 −3.7

⎞−1 ⎛

3 2⎟ ⎟ 1 3⎟ 1 2⎟ ⎟ 1 2⎟

Merchandise Placement Optimization 157

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Problems and Opportunities of Applied Optimization Models in Retail Space Planning Tobias Düsterhöft and Alexander Hübner

1 Introduction Shelf space allocation models in the Operations Research field receive great attention among researchers and practitioners. Retailers are becoming aware of the benefits provided by innovative optimization techniques and fast computation times together with a very high availability of appropriate data (Kuhn & Sternbeck, 2013; Kök et al., 2015; Bianchi-Aguiar et al., 2021). Earlier approaches involving shelf space allocation models and studies had to struggle with insufficient computational power and scope (from today‘s perspective), and could therefore only handle limited problem sizes on restricted problem scopes (e.g., Hansen & Heinsbroek, 1979; Corstjens & Doyle, 1981 or Borin et al., 1994). With increasing processing power over time, the sizes of problem instances increased rapidly (e.g., Hansen et al., 2010; Zhao et al., 2016; Bianchi-Aguiar et al., 2016 or Hübner & Schaal, 2017a). However, there were still limitations regarding the applicability of those approaches in practice (e.g., Hübner & Kuhn, 2012). While the newer approaches and models consider a more comprehensive set of parameters (e.g., demand effects, refill frequencies or shelf stock), the actual shelf design and store layout in particular received only limited attention. The high-performing computers and extensive databases currently available imply that the fundamental requirements for advanced approaches can be achieved. Nevertheless, the practical use of shelf space optimization approaches is still difficult in most cases, and the transfer from advances in research to industry is not straightforward. In order to develop decision support and optimization models for shelf space planning, research usually starts with the analysis of a retailers’ real-world problem.

T. Düsterhöft · A. Hübner () Technical University Munich, Straubing, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Ghoniem, B. Maddah (eds.), Retail Space Analytics, International Series in Operations Research & Management Science 339, https://doi.org/10.1007/978-3-031-27058-1_8

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The next step is to determine relevant decision variables and parameters, while at the same time performing abstracting the problem to a certain extent. Abstractions may have several reasons. (1) Relevant interdependencies have not been detected, resulting in unconscious abstraction. (2) While planners are aware of a dependency on the planning problem, this is considered insignificant or related data are not available. (3) The complexity of the resulting problem formulation needs to be reduced (e.g., due to computational limits). (4) The scope of the research goal requires or allows a focus on a certain sub-problem. A model is—by definition— always an abstract representation of an original, but the more pronounced this abstraction the harder it may be to apply conclusions from the results obtained to the real-world problem later on. Overall, different types of optimization models on different levels of abstraction address different problem statements. All approaches have their intentions and justifications, but they need to be differentiated in terms of their explanatory power for real-world problems. This chapter aims to bridge the gap between research and actual application of shelf space allocation models in practice. The objective is to make it viable when a retailer wants to actually use results generated with optimization models to build up the shelves of their stores. The findings are mainly based on a three-year collaboration with one of Europe’s biggest retail companies, operating hypermarkets and discount stores in several countries. Various approaches for shelf space optimization models have been discussed, discarded or implemented during this project, and some have been applied to certain stores in Europe. This chapter details the trade off between developments made via research, expectations of the retailer on strategic and managerial level, and the implementation within stores by employees. These stakeholders may have different objectives when they think of proper shelf space allocation. Research is trying to advance and solve sophisticated models in the best possible way, including a number of decisions (e.g., space allocation, assortment decisions or replenishment frequency) and demand effects (e.g., space-elastic demand, substitution effects or stochastic demand functions). On the other hand retailers want to have several allocation rules considered (e.g., product grouping within a category, minimum shelf quantity for each product, or weight restrictions) and it is possible that every model will fit the decision problem at hand (e.g., if assortment decisions should not be included). Computation time is usually of very great importance for the application in practice, and is expected to be within several seconds depending on the planning problem and frequency. Despite the fact that the planning problem is a mid-term issue, the planners want to run multiple scenarios that require a short computation time. The final step is meant to be where planograms, which are a graphical illustration on how to equip each shelf, are delivered to the stores. However, store employees may modify the planogram to their own needs, as from their point of view, other aspects define good product allocation (e.g., less replenishment effort). In combination, this leads to problems when shelf space optimization is applied in practice. Knowledge about common restrictions and requirements specified by retailers can help to develop suitable solutions and consider important aspects right from the outset.

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The contribution of this chapter is to achieve a basic understanding of the common real-world planning problem of retailers and to identify the main concerns when shelf space allocation models are to be applied in practice. The remainder provides a description of shelf space planning in practice in Sect. 2, followed by a brief discussion and classification of the literature in Sect. 3. A comparison of research approaches with requirements from retail practice is presented in Sect. 4. Afterwards, Sect. 5 presents a summary of problems and important topics for future research. Finally, Sect. 6 provides a conclusion and outlook.

2 Shelf Space Allocation in Practice This section analyzes the process of shelf space planning in practice. Professional shelf planners have to create planograms that provide a specific number of facings and a location for each product on a shelf. A facing is usually described as the first visible unit of a product on the shelf. A facing may contain several consumer units. The total number of facings then describes how much shelf space each product gets. The main task of shelf planners is the creation of individual instructions for each retail store describing how they have to build up and equip their shelves. Shelf planners have to follow several merchandising rules for these planograms. In turn, these rules are based on managerial decisions, marketing experience or sometimes rules of thumb of decision makers. An optimal shelf space allocation for a shelf planner is thus a planogram where all setup rules are fulfilled and the employees in the store are able to rebuild their shelves in accordance with the planogram. This means that the term optimality for a shelf planner does not implicitly mean the generation of maximum profits or sales, but in most cases just practicability and applicability. An example of a planogram together with its realisation within the store is provided in Fig. 1. Intensive discussions as well as close and regular communication with shelf planners at our industry partner helped to analyze the current workflow. The

Planogram Category Ketchup – Store X

Brand 2 12 Facings

Brand 1 4 Facings

Brand 3 15 Facings

Brand 4 19 Facings

Brand 5 8 Facings

Brand 6 9 Facings

Fig. 1 A planogram (left) and its physical representation within the store (right)

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purchasing and category management department decides which SKUs are sourced from their suppliers or manufacturers and is therefore responsible for the assortment decision. Once the assortment is fixed, it is handed over to the shelf-planning department; subsequent adjustments of the assortment are usually impossible at this planning step. The shelf-planning department must then decide the number of facings and the location for each product on a shelf. Further, they also need to create the layout of the corresponding shelves (i.e., the specific heights and dimensions of the shelf levels). A number of visual merchandising rules must be considered that influence the shelf layout and product allocation decisions. In total, this process includes the following hierarchical steps: 1. Category space: Initially decisions need to be made about the available shelf space for the category being considered. This is a trade off decision among all categories available. This decision is made using experience and rule of thumb due to the complexity of this decision and the lack of decision support tool. Category space is determined as being the number of shelf racks. A shelf rack is one unit of a certain shelf type which is usually between 1.25 and 2.00 m wide, depending on the type. 2. Shelf type: A specific shelf type is selected for each category. Shelf types can be regular, chilled, frozen, hanging racks, rummage tables, etc. Each shelf type requires individual treatment in terms of shelf layout and product allocation. 3. Shelf layout: Once the amount and type of shelves have been selected, the planners must decide on the layout of the shelves. For example, a regular shelf can be equipped with a certain number of shelf levels with different depths. To do this a set of rules has to be followed (e.g., certain depths are only allowed on certain levels or at minimum distances between two shelf levels). 4. Product allocation: The shelf planners start allocating products to the shelves only after all previous steps have been performed. Again, they need to consider certain optical and logistical guidelines. First, products of the same manufacturer or the same subgroup (e.g., subgroup strawberry in category marmalade) must be placed next to each other. Further, product dimensions, stacking possibilities and weights need to be considered. For example, heavy products have to be located on the bottom-level shelves while very high items are not allowed to be allocated on the top levels. Finally, a minimum shelf representation for each item must be satisfied as well as the demand between two store deliveries. On the other hand, a maximum shelf quantity must be considered, too. Usually this maximum quantity is determined on a product level using individual average daily demand multiplied by a certain number of days. The remaining shelf space is filled up by using a proportional sales rule. Once Steps 1–4 have been performed, the resulting planograms are delivered to the stores. During Step 4, the shelf layout defined in Step 3 may appear to be disadvantageous, and thus Step 3 may need to be adjusted via a further iteration. In the worst case, the total shelf space for the category may be insufficient and the process must be restarted as a whole. A range of software applications supports shelf planners in their work. These commercial tools are usually tailored to the specific needs in terms of the planning

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process. The focus of this lies on the automation of Step 4. Popular software tools are, e.g., Space Planning (from JDA Software Group), Quant Retail (from ExTech), Dot Active Category Management Software (from DotActive), Shelf Logic (from Shelf Logic) or ASO (from Nielsen). All of these applications have in common that they are designed for high user comfort. Shelf planners using these tools have several options for designing the shelf layout and drag and drop products onto the shelves including a photo-realistic presentation. The results of Steps 1–3 are usually required as input and can only be adjusted manually within these programs. Then, in most cases, there is the option of implementing the given setup rules for Step 4 and generating planograms automatically. The process described in this section may not hold true in every aspect for each retailer, but it highlights the question of how to integrate findings from research into an operational environment when certain processes need to be adhered to. Simply applying an optimization model that determines a number of facings for each product may be insufficient for the retailer due to a lack of consideration of other allocation guidelines. Consequently, optimization models need to be developed considering specific retailer requirements.

3 State-of-the-Art Shelf Space Optimization Approaches A significant number of publications have addressed the field of shelf space optimization over the last few decades (Hübner & Kuhn, 2012; Kuhn & Sternbeck, 2013; Kök et al., 2015; Bianchi-Aguiar et al., 2021). Related approaches uniformly aim to provide proposals about how many facings the items of a certain category should get. These contributions usually determine an optimal number of facings in order to maximize a target value (e.g., profit or sales) using objective functions that incorporate different demand effects. Examples of such demand effects are space elasticity (e.g., Hansen & Heinsbroek, 1979; Hübner, 2011), cross-space elasticity (e.g., Corstjens & Doyle, 1981; Irion et al., 2011) or positioning effects (e.g., Hwang et al., 2005; Düsterhöft et al., 2020). Space elasticity describes the fact that the demand for a product increases when it is assigned more shelf space and is therefore more visible for customers. The effect of increasing demand in line with an increasing number of facings is well analyzed and assumed to be one of the most important demand effects within retail stores (Chandon et al., 2009; Eisend, 2014). Cross-space elasticity implies the fact that the number of facings of one product may affect the demand for other products. The effect can express a substitution or competing demand relation between two products. Further, similar concepts are used when out-of-stock or out-of-assortment demand needs to be distributed to remaining products. However, Schaal and Hübner (2018) observe that cross-space elasticity only has limited impact on the solutions of a shelf space allocation model. The incorporation of positioning effects addresses the fact that customers may have different purchasing decision depending on whether a product is located at the beginning or the end of an aisle or at eye or knee level on a shelf. Discussions

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about the importance and impact of location-based effects reveal different results, however (e.g., Berghaus, 2005; Ghoniem et al., 2016). Besides different demand effects, the product allocation decision is assumed to be highly dependent on other planning steps. On the upstream side, assortment planning and category management are highly relevant for shelf space planning, whereas the determination of delivery and replenishment frequencies for each individual store is a downstream process. As changes in shelf space allocation can cause a great deal of work in the stores, this process is considered to be initiated only within a mid-term planning period (Hübner & Kuhn, 2012). Thus, shelf space optimization may be performed more often than assortment planning or store space planning but not as frequently as delivery and replenishment planning. Some researchers combine these dependent effects in integrated modeling approaches, whereas others focus on the core question and extend it using several important contributing factors. The following paragraph provides an exploration of different approaches published over the past few years. Early shelf space optimization models introducing space elasticity come from Hansen and Heinsbroek (1979), Corstjens and Doyle (1981) or Zufryden (1986). These models were extended over time, e.g., Borin et al. (1994) integrated assortment decision and stockout costs within their model. Urban (1998) extended existing models by considering the inventory of items in the store. An extensive model together with a linearization approach is provided by Irion et al. (2012), including a detailed cost function and cross-space elasticity. Hübner and Schaal (2017a) allow products to change their display orientation on a shelf in order to find the best possible shelf utilization. While all approaches mentioned so far use a single and one-dimensional value for shelf space, Yang (2001) created a first model that accounts for several shelf levels. Based on this idea, Hwang et al. (2005) include the demand effects of different shelf levels. Further, Hansen et al. (2010) determine an exact position for each product within the shelf and Zhao et al. (2016) also integrate spatial relationships between the products of a category. A stochastic demand function combined with replenishment decisions and different shelf levels is provided by Hübner and Schaal (2017b). While all these approaches consider several shelf levels, they model a one-dimensional value at each level and do not consider exact shelf dimensions. Bai et al. (2013) resolve this problem by integrating levels of different height within their approach, but only provide solutions for small data sets. The approach of Düsterhöft et al. (2020) takes different shelf levels into account, each with their individual three-dimensional size. Further, an optimal solution approach is provided that can be used to solve data sets with up to 300 different products in reasonable runtime. However, the dimensions of the shelf levels are assumed to be a given parameter. Coskun (2012) develops a model where the heights of two-dimensional shelf levels are flexible within certain limits. Hübner et al. (2021) introduce a shelf level dimensioning model integrated within a shelf space allocation approach that determines the number of facings on a certain shelf level and the number of shelf levels as well as their specific dimensions for several shelf racks. While the available shelf space for the category considered was a given parameter in the all approaches mentioned, the approach of Irion et al. (2011)

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touches on the question of how much shelf space to assign to each category. Flamand et al. (2016) as well as Ozgormus and Smith (2018) decide about the sequence (location) and shelf space within a store on the category level. Further, the approach of Ostermeier et al. (2021) extends store-wide shelf space planning among different categories by considering different store areas as well as individual shelf types for each category while simultaneously considering facing and assortment decisions combined with substitution effects on a product level. The literature presented so far is only an extract of many related publications in the field of shelf space optimization (readers are referred to Hübner and Kuhn (2012), Kök et al. (2015) or BianchiAguiar et al. (2021) for an extensive discussion of different shelf space models). State-of-the-art approaches follow the ideas and the concepts mentioned in this section. It should be highlighted that the key feature of all models is determining the number of facings for all items within a category, while considering several other related decisions and demand effects. The approaches and effects discussed above clearly demonstrate different levels of abstraction among shelf space optimization models. These differences can be used to classify approaches regarding a particular focus on their applicability in practical environments. A classification method for different modeling approaches is suggested in Table 1. Seminal Models Fundamental papers initially evaluate the impact of model-based decision making on a certain shelf space topic. While these approaches are usually very stylized, they focus on the key question of the research issue. They aim to emphasize the contribution of decision support systems for theoretical but practice-

Table 1 Classification of shelf space optimization approaches Model type Seminal models

Advanced models

Application models

Description Generate a basic understanding for the topic of shelf space optimization models or extend them by essential components, e.g., space elastic demand function Level of abstraction: high Build on seminal models and extend them in order to investigate the theoretical impact of certain parameters or relationships, e.g., cross-space elasticity Level of abstraction: medium Extend advanced approaches with the components required for the practical application of model results, e.g. shelf dimensions Level of abstraction: low

Examples Hansen and Heinsbroek (1979), Corstjens and Doyle (1981), Zufryden (1986)

Borin et al. (1994), Urban (1998), Hwang et al. (2005), Hansen et al. (2010), Irion et al. (2012), Hübner and Schaal (2017b)

Bai et al. (2013), Geismar et al. (2015), Bianchi-Aguiar et al. (2016), Düsterhöft et al. (2020), Hübner et al. (2020)

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oriented settings. Thus, in the area of shelf space planning, the first approaches addressed the question whether as to changes in the number of facings of products in a category enhance its overall profit. For example, Hansen and Heinsbroek (1979) started with a model where only a space-elastic demand function was included alongside a shelf space restriction. The generic model formulation is very close to the classical knapsack problem. However, these seminal models are not applicable in real scenarios, primarily due to the absence of relevant restrictions and model components due to the high level of abstraction (e.g., missing integer facings), though the path for further research is paved. Advanced Models Further developments extend the basic models by several components. They reduce the level of abstraction and include aspects from empirical consumer studies as well as from practical insights. These models usually focus on a specific extension of the available approaches. The intention here is often to investigate whether specific modifications or add-ons are significant within the planning problem. For example, Hübner and Schaal (2017b) evaluated the impact of stochastic demand functions on a category’s setup. Otherwise, these research models do not fully integrate all relevant planning issues, sticking to generic assumptions in many cases. For example, the approach of Irion et al. (2012) includes a sophisticated model with extensive demand and cost functions, crossspace elasticities and assortment decisions, but only implies a one-dimensional shelf space depiction. Application Models The final types of models match the findings of research models with requirements from practice in a more extensive manner. Their focus lies on the applicability of optimization models in real-world applications. Abstraction is further reduced to a lower level. To do so, application models extend advanced models in order to consider the most relevant aspects for practical usability, e.g., merchandising rules for shelf space planning. The focus in these cases is the transfer of research results to practical applications suitable for the real planning problem of the retailer. For this reason, application models usually focus on effects that are proven to be significant for the decision problem. These models provide essential insights into the real planning problem and their real world application, e.g., Düsterhöft et al. (2020) highlight the benefits of Operations Research driven shelf space optimization in practice by evaluating real profit improvements. In summary, all types of approaches are required in order to provide solutions for certain problems and to enhance theory and modeling techniques. Once the seminal models have evaluated the core question and decision support options are established, advanced research models can be developed. These models may appear too complex from a practical point of view. However, provide important information about relevant components. Finally, advanced models build the foundation for practical approaches that can be applied and solved at the retailer.

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4 Blending Research and Practice Knowing about the retailer’s process as well as state-of-the-art approaches in literature enables the evaluation and comparison of both perspectives. First, the workflow from retail practice (see Sect. 2) reveals certain weaknesses. Many steps are performed manually and without fundamental decision support. Adjustments early in the process cause high re-planning efforts, and significant factors (e.g., space elasticity) are often insufficiently considered. When software tools are applied, planograms can be created automatically. This does not mean however that the resulting planograms are optimized from an Operations Research point of view. Usually space is allocated in these applications following simple rules, e.g., proportion of sales of a product (Hübner, 2011). In line with observations from the industry, Irion et al. (2012) state that no commercial software incorporates the scientifically proven relationships between shelf space and sales of products such as space elasticity, cross-space elasticity or positioning effects. As a result, these applications tend to be automated support systems with a focus on graphical output rather than shelf space optimization tools. To summarize, it must be stated that neither shelf space planners nor commercial software applications incorporate scientific findings sufficiently regarding product allocation. The lack of consideration of demand and location effects seems to be a particularly major limitation of common retail practice. Further, it needs to be taken into account that in some cases the shelf planners are not provided with relevant financial figures and margins. If shelf planners do not have access to this data it is obvious that margin-based optimization approaches and profit functions are hard to implement without the collaboration of different departments within the company. Comparing the real planning process (see Sect. 2) with state-of-the-art approaches in literature (see Sect. 3) shows that there is a certain gap between the different expectations of proper shelf space planning. Figure 2 illustrates the key aspects of both sides in order to compare the diverging foci. On the x-axis shelf space allocation impacts are categorized based on whether they are considered by researchers or by retailers. The y-axis shows the current state of applicability of certain impact factors. A high practical level means that basically all information regarding the planning problem is available, while a high theoretical level indicates that effects are not verified conclusively, impacts are hard to measure, or data are hard to generate. Unfortunately, topics with a high practical impact are barely addressed in research. Another factor is that, retailers may set their focus too low if for example space elasticity is not considered for shelf space allocation. The structure of Fig. 2 could vary of course as the case arises, but it depicts the general components of shelf space optimization that are further explained and discussed in the following. The most important effect within research-based shelf space optimization models is the effect of space elasticity. The impact of extending the shelf space allocated to a product on its sales is a topic of all leading publications in the field of shelf space optimization. The effect is sufficiently documented within actual studies. Eisend

Theoretical level

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Cross-space elascity

Stochasc demand

Replenishment frequency OOS & OOA effects Inventory costs

Assortment decision

Space elascity

Logisc costs

Service level

Shelf dimensioning

Posioning effects

Shelf layout Physical setup

Focus of state-of-the-art research

Praccal level

Product grouping

Focus of current retail pracce

Fig. 2 Different foci of stakeholders on the components of shelf space planning

(2014) evaluates space elasticity within a meta-analysis and Düsterhöft et al. (2020) verify the occurrence of this effect within a one-year field test of a space-elastic shelf space optimization model in practice. Usually the intake of space elasticity results in non-linear models that quickly increase the complexity and thus need special heuristics or linearization approaches in order to generate solutions. On the other hand, it seems that shelf space planners in practice are barely aware of this effect as commercial providers of software applications do not integrate it within their tools. Many optimization approaches from the literature now incorporate the assortment decision, which originally belongs to category management. These models do not just determine how many facings a product should get but also whether a product is included in the assortment or not. In practice shelf space planners must stick to their instructions and the necessity may vary from case to case as a result. Stochastic demand functions replace deterministic demand in some publications. Stochastic demand functions and assortment decisions can lead to out-of-stock or out-of-assortment situations (OOS & OOA effects), which are also receiving growing attention in the literature. Retailers usually use separate demand planning tools that estimate values for their shelf space planning applications. As planograms are not modified in the short term in practice, but usually once or twice per year, relatively simple processes are presumed to be sufficient here. When it is assumed that shelf space given to one product affects the demand for other products due to substitution or complementary purchases, cross-space elasticity represents the correlation between all products considered within the shelf space planning process. Cross-space elasticities result in high computational complexity due to the mutual dependency of demand across products and the models are thus very hard to solve.

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This means the problem size is usually very limited when reasonable runtimes are required. While Zufryden (1986) already stated that it seems unrealistic to use crossspace elasticity on level of individual products due to the tremendous efforts of generating values, Eisend (2014) confirmed this as only a very small number of studies verify cross-space elasticity with actual values by means of meta-analysis. This may be the main reason why this effect is probably never considered in the practical field. Another way of extending shelf space models is the incorporation of replenishment frequency decisions as additional output. This corresponds to the decision on how often the shelf must be refilled during a certain period. In some cases, these approaches also differentiate between direct replenishment when a truck arrives at the store or when replenishment is made from backroom storage. Retailers have different systems for shelf replenishment. One option is that replenishment for a product is triggered whenever a certain reorder point (regarding the remaining shelf quantity) is reached. In this case, the store gets new items of this product delivered with the next regular supply. The hypermarkets surveyed in Germany do not have backroom space at all, whereas discount stores have a certain area for intermediate storage. If the replenishment decision is included, the idea arises of considering inventory costs, logistic costs and service levels. These parameters are important when shelf space optimization models aim to maximize the total profit of a store. In this case, more shelf space given to a product results in higher demand and its need to be associated with the costs for this setup as well. Retailers have a strong interest in evaluating the costs of their shelves, so Operations Research driven support is more than appreciated in this field. Positioning effects describe the fact that the specific location on the shelf also impacts an item’s demand. Following Underhill (1999), the shelf area has to be divided into zones of different attraction and Chandon et al. (2009) state that products on a top-level shelf induce higher sales than those on bottom levels. Retailers consider similar rules, e.g., that the top brand of a category has to be located at the beginning of an aisle, which in practice then also includes product grouping of all items of a category, e.g., those from the same manufacturer. Different rules for clustering products of a category can be detected in retail stores. On the one hand, the decision needs to be made whether to cluster products in terms of their manufacturer (e.g., all products of Manufacturer A) or in terms of their characteristic (e.g., all strawberry jams independently of the suppliers). However, it is possible to build horizontal clusters where all products are placed next to each other on a certain shelf level, or vertical clusters are spread over several shelf levels. The relevant literature has barely addressed product grouping, especially for problem sizes required in practice (e.g., 200 products within a category). Retailers also define rules for the shelf dimensions within their stores. These rules provide a certain degree of freedom for local adjustments, but are also mandatory for the planning process. Setting of shelf dimensions naturally impacts the product allocation of a category. Rules related to aspects such as the minimum and maximum distance between two shelf levels or the space needed for customers to retrieve products from the shelf. The approach of Hübner et al. (2021) integrates all these

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rules together within a shelf space optimization model for product allocation. Maybe one of the most important factors for shelf planners but also the most neglected aspects of scientific models is the possibility of considering a detailed shelf layout derived from shelf dimensioning within product allocation models. Shelf layout represents the planning framework for shelf planners creating planograms and is usually part of every commercial application. In the literature shelf space is otherwise very often reduced to a one-dimensional value representing the total shelf space of a category. Others may extend it to several shelf levels but then most of the time these levels are of the same dimensions. Düsterhöft et al. (2020) observe that neglecting shelf dimensions leads to inappropriate solutions when these have to be applied in a real environment. Finally, the physical setup a shelf planner must consider is barely addressed in the pertinent literature, too. This point implies facts such as that heavy items have to be located on the bottom level, total weight restrictions for shelf levels, specific options for stacking items (or case packs) one above the other, and other category-specific conditions. To summarize, we see that from a scientific viewpoint a shelf space allocation model that incorporates cross-space elasticity, an extensive cost function and stochastic demand with a one-dimensional shelf space consideration seems to be the spearhead of the current research approaches. But for shelf planners this is currently not as relevant. They expect an approach where major merchandising rules are considered and the output is transferable into a planogram for the store without any further modification. Two actions are recommended to counteract these differences: 1. A research stream within the field of shelf space optimization should focus closely on impact factors that are of high relevance for the retailer, e.g., consideration of all merchandising rules combined with a space-elastic demand function. The advantages of this scenario are that retailers can provide data and information easily and results are highly relevant in practice. Such models would be anything but trivial and would contribute to the scientific arena. 2. Further and intensive research must be spent on verifying pertinent impact factors, e.g., cross-space elasticity, substitution impacts or positioning effects. Retailers will be only be willing to open up such approaches if reliable information from research is available that the impact is significant for the decision problem and that the values for the corresponding parameters can be provided or determined easily.

5 Improvements of Shelf Space Optimization in Practice Having analysed the process of shelf space allocation in practice in Sect. 2 and stateof-the-art approaches from research in Sect. 3, an evaluation of both perspectives has been performed in Sect. 4. This section now presents several improvements and opportunities for further research in the field of shelf space optimization models.

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These are derived from the literature review and the work with the industry partner. The ten most important components of shelf space optimization, where further improvements are required in order to apply optimization models in practice, are addressed. These ten topics are structured along additional model features (Sects. 5.1.1–5.1.5), enhanced demand effects (Sects. 5.2.1–5.2.3), and planning approaches and scope (Sects. 5.3.1 and 5.3.2).

5.1 Additional Model Features 5.1.1

Integrating Shelf Space, Shelf Types and Store Layout Planning

Requirements regarding actual shelf configurations and their space should be evaluated very carefully to ensure that solutions obtained from models can actually be transferred to the stores and the shelves without any intermediate manipulation steps. A very crucial point for application models is a proper consideration of the available shelf space. Düsterhöft et al. (2020) and Hübner et al. (2021) show the necessity of integrating multi-dimensional shelf space when model solutions have to be applied in practice. Other approaches (e.g., Bai et al., 2013; Hansen et al., 2010) may also serve as a starting point depending on the practical use case. While three-dimensional shelf space models already exist, these approaches can be further extended with practical restrictions (e.g., weight restrictions, optical layout). Other approaches that also make shelf space a major subject of discussion focus on store-wide shelf space allocation among different categories (see e.g., Irion et al., 2011; Flamand et al., 2016; Flamand et al., 2018 or Ostermeier et al., 2021). In these cases the total space of the store is divided into the store’s categories. Obviously this is very closely connected to shelf space allocation of products, but current approaches still lack some important details. While Ostermeier et al. (2021) already offer the possibility of incorporating different shelf types for different categories that consume individual one-dimensional spaces of the total store stretch, a multidimensional depiction of the whole store is often required. Chilled or frozen shelves in food retail for instance, may have specific requirements regarding their location in the store. In stores selling consumer electronics usually only one facing of each product is displayed to the customers. The core problem is to determine the share of shelf space each category gets as a whole. However, shelf types of electronic goods differ very much from category to category. While mobile phones have shelf racks containing only one shelf level (and storage below), DVD’s have specific racks, small electronics are allocated in shelves with several levels, whereas products like washing machines stand on the floor and television hang on a wall. Determining the shelf space for these categories requires a very detailed consideration of the total store space, product dimensions and shelf types, but could significantly contribute to more precise store optimization. A further consideration in this regard is the allocation of shelf space to certain areas, e.g., to drive traffic within the store (see Flamand et al., 2016).

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Accounting for Product Grouping

Products are allocated to specific locations (levels) on a shelf with a certain number of facings, taking multi-dimensional shelf space into account. In this sense, solutions generated with application models are close to final planograms. But there is one reason model why solutions may not be directly applicable. When products are (more or less) randomly allocated to different shelf segments this will violate retailers’ merchandising guidelines of visually grouping products by defined attributes. Common guidelines for product grouping within one category are either by brand (all products on the same brand should stand together, e.g., Brand A, Brand B, own label, etc.) or by type (all products of the same type should stand together, e.g., strawberry jam, cherry jam, peach jam, etc.). A further differentiation can be made as to whether grouping should take place horizontally or vertically. Horizontal grouping means that all items of the group should be placed next to each other on the same shelf level. Vertical grouping implies that the group is spread over several (all) shelf levels such that a rectangular shape is created (e.g., Hübner et al., 2020). Some approaches in literature already consider neighborhood relationships between products (e.g., Hansen et al., 2010; Zhao et al., 2016), and provide exact positions of items on certain shelf levels. But these models only consider one-dimensional shelf levels. It is also important that models are be able to process large data sets, as categories may contain more than 200 different products in practice. The grouping decision needs to incorporate both, marketing and physical guidelines. Suggested restrictions would be: (1) allocate related products in groups (by brand or type), (2) within each group, locate heavy (or large) items at the bottom level and (3) consider total weight per shelf level.

5.1.3

Determine the Efficient Product Allocation Type

Retailers use different systems of placing their products on a shelf. A main difference is whether they allocate single units or full case packs. Single unit allocation provides more flexibility for shelf space planning but means more workload for setting up and refilling shelves. When a product is allocated in full case packs, each case pack contains a certain number of units, reducing the refill efforts significantly. However, a case pack consumes more shelf space than a single unit. The fact that one facing may contain several sales units is already considered in both one-dimensional (e.g., Irion et al., 2012) and multi-dimensional shelf space models (e.g., Düsterhöft et al., 2020). Alongside single or case pack allocation, retailers also can use pallets and half-pallets for fast-movers, which are then located on the bottom of a shelf rack (and require a certain shelf layout). These pallets are always used when an item’s demand is high that no reasonable number of facings would cover it for the time between two replenishments. Altogether, there are three options for allocating products: pallet, case pack or single unit. The best option for each item may be dependent on its individual demand data, product dimensions and profitability. Decision models could help to figure out which allocation type is

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the best for each product and integrate this within a shelf space planning model. A significant point is that the requirements for suitable shelf layout are considered when pallet allocation is integrated. A shelf dimensioning model (e.g., Hübner et al., 2021) would be a good starting point for this purpose. It may also be possible to integrate a decision about an alternative display orientation following Hübner and Schaal (2017a), which then depends on the allocation concept selected for each item.

5.1.4

Extending Approaches to Multi-Store Concepts

The current models and approaches are tailored to the development of (1) storeindividual planograms or (2) planograms built on average values for demand and shelf space. The first objective in (1) does not cater for achieving a common layout across stores, but is built on the situation at the store, whereas (2) may allow the creation of standard layout and customer interface. If all the stores are planned individually this results in a major planning effort and complicates the refilling processes. The logistics become more complex and costly when shelves are organized differently across stores. In this case the warehouse shelves cannot be mirrored to pick pallets for store supply in the sequence of store shelf replenishments as each store may have a different sequence. A new modeling approach would be needed to address the trade off between common and individual store layouts and shelf space.

5.1.5

Consider Omnichannel Opportunities

The growth of e-commerce goes along with changes in retail setups. Nowadays almost all retailers operate bricks-and-mortar and online channels. In such settings shelf space and assortment planners need to make a number of key decisions. Out of all potentially available products, which set of products should they list in each channel, how should they allocate space within stores and which products would they make available for buy-online-pickup-instore, ship-from-store or as digital assortment extensions? For omnichannel retailers, the assortment composition and store space assignment must be defined jointly as these decisions are interdependent when space in the store or the online warehouse is limited. A meaningful allocation of products, shelf space and inventories to the webshop and stores can be achieved to serve customers across channels by disentangling and quantifying demand transitions between channels and products (Hense & Hübner, 2022). In omnichannel retailing, not only in-channel substitutions are common, but also demand transitions between the webshop and stores and the other way around. Alongside omnichannelspecific demand, assortment and shelf space planning must consider product margins, distribution and replenishment costs as well as inventory holding costs for each channel. New comprehensive models can support retailers in product selection and shelf space assignment across channels.

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5.2 Enhanced Demand Effects 5.2.1

Explore the Effectiveness of Various Demand Effects

From a scientific viewpoint, a model that includes as many effects as possible usually seems to be sophisticated. This changes when it comes to the real application of such approaches. Obviously, a huge set of data is available nowadays as well as detailed sales figures on a product and store-individual level, but some other parameters still need to be evaluated. Once a model is set up one must also provide values for parameters that are not part of retail master data, e.g., space elasticity. When test categories are defined it is obvious that they may have individual space elasticities depending on the store or whether these are staples or luxury goods. In reality, these elasticity values are not known and can hardly be calculated without conducting further studies. Moreover, even when the parameters are found for a certain category in a certain store there is no guarantee that the same category in another store will come with the same values, e.g., wine in France might have a different elasticity than in Germany. While for many publications the proof of the existence of space elasticity (see e.g., Eisend, 2014) provides general values for it, the situation for product-dependent effects, such as cross-space elasticity is very different. Of course, for theoretical tests that aim to show the potential of an optimization model, it is possible to assume value of these parameters and then run some kind of sensitivity analysis to see how changes affect the solutions. However, retailers are very careful when they apply solutions generated by shelf space allocation models. Application models usually concentrate on verified components and parameters for this reason. There is still a need for significant research, not into integrating these effects within optimization models but rather into determining reliable values for parameters.

5.2.2

Consider Complementary Effects and Cross-Selling

The current models and practice approaches mainly focus on the effect of substitution, position or stock allocations. The complementary effects have not yet been investigated in a comprehensive manner across categories and sales channels.

5.2.3

Integrating Assortment Decisions if Decision-Relevant

A related decision problem is assortment optimization where the set of products for shelf space allocation is determined. Standalone assortment models have been developed for this reason (see Kök & Fisher, 2007 or Hübner et al., 2016). Some works already integrate these interrelated decisions within shelf space allocation models (e.g., Irion et al., 2012; Hübner & Schaal, 2017b) in order to determine both the set of products to be placed in the store and the number of facings for each of these products. Several demand effects are considered in each case that appear

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whenever a product is unavailable, such as out-of-assortment (OOA) or out-of-stock (OOS) substitution. Obviously the models aim to maximize profit for the retailer by selecting an optimal assortment from a mathematical point of view. However, apart from the fact that substitution rates are very hard to determine for a practical application, more important is the fact that delisting products for economic reasons may not be in line with retailers’ requirements to offer a large variety. Usually retailers sign contracts with suppliers specifying the assortment in the store. They also have specific merchandising guidelines regarding the products offered in their stores. Retailers will not give up a certain core assortment of low-budget products (usually own brand) within each category hoping that some customers will switch to brand products at the point of sale, especially in food retail where margins are very low and customers are price sensitive. Some customers may not visit the store any longer if low-budget products are delisted as competitors probably offer them. This leads to another important impact factor. The assortment depth has to be carefully considered when products are delisted within a category. For example, if a model is applied for the jam category and a decision is made to eliminate out all strawberry jams from all suppliers and own brand just because they appear to have a poor margin or sales values from a tactical viewpoint, retailers will reject such a decision. Models should not base assortment decisions on a pure profit basis. Overall, the decision on which products to include in the assortment when shelf space is a scarce resource is of very high relevance for retailers and they will greatly appreciate applicable approaches. Such models should at least consider the following options: (1) a core assortment that can not be delisted; (2) an inclusive restriction where Product A must be listed if Product B is listed; (3) an exclusive restriction where Product A must not be listed if Product B is listed, and (4) a subset restriction that ensures a certain size for different subgroups of products (by brand or type).

5.3 Planning Approaches and Scope 5.3.1

Replanning Requires Rebuilding

Another aspect that also accounts for specific shelf allocations is planning frequency. Usually shelf space optimization approaches are created in order to initially allocate products on an empty shelf. Once these solutions are implemented within the stores there is no further support until the model is applied again, which may result in a very different solution. In practice, assortments (independently of the specific category) underlie a certain development over time. This means that in some cases products of a category are replaced by newer versions (e.g., new packaging size, new design), while in other cases products are delisted without substitution or new products are included in the assortment. Insights from practice showed that even food retail assortments are changed by 5–10% per month. The handling of those changes implies a trade off for the retailer. On the one hand planograms need to be optimized at each point in time. This is even more important when

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Assortment Y

Assortment Z

Opmizaon

Applicaon

Evaluaon

Analysis not possible

Demand is transfered to remaining or new products

Old products delisted

Products are not available anymore

Analysis possible Demand impacts

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New products need a certain ramp-up phase

Time 1

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Fig. 3 Assortment development of a category over time

the results of shelf space models are to be evaluated. On the other hand reworking shelves within a store leads to high personnel expenditure (e.g., the full rework of the shelves of the pet food category is reported to take more than 48 employee hours in Düsterhöft et al. (2020)). Figure 3 shows an example how a category develops over time. Many changes have already occurred between the time the shelf planner creates the planogram and the time the planogram is applied in the store assortment. Further, as time passes the assortment may change more and more. Obviously, shelves have to be replanned in these cases if the changes are too extensive. Applying the same shelf space model used for initial planning may result in a solution that requires a complete rework of the existing shelves in the store. As labor is also a scarce resource in retail stores, this process is not appropriate in practice. A model that also considers the current product allocation in addition to product grouping and optimizes profit while considering the costs of reconstruction efforts would therefore be helpful for retailers. A secondary objective for such a shelf space updating model would then be to reduce the number of products affected.

5.3.2

Consider the Objectives of Different Stakeholders

Shelf space allocation models usually aim to maximize total profit for retailers. Basically all state-of-the-art approaches use a space-related demand function for this purpose. The intention is that (undecided) customers will more likely buy a product that is allocated with more facings and thus more visible on the shelf. Additionally, models provide options to ensure a certain minimum shelf quantity for each item in order to guarantee a certain service level. A more or less unrecognized factor is that the employees in the store manage the shelves as well as different suppliers of a retailer. Of course, optimization models factor in logistics costs as well. But this thought needs to go one step deeper. Planograms are created at a

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strategical planning division and then sent to the stores. Even if they consider a trade off between margin and logistic costs, this does not automatically mean that these planograms are convenient for the employees. It might appear that from a theoretical viewpoint it is more profitable to reduce facings of certain products and thus accept high replenishment efforts in order to extend the number of facings of other (more profitable) products. Further, contracts with suppliers are negotiated in great detail. Suppliers may insist on a certain minimum representation of their products on the shelf. This must then also be considered within shelf space allocation, meaning that current approaches do not cover the needs of all stakeholders within the shelf space planning process. The crucial point for shelf space optimization models is that independent of how good they are and whatever effects and costs they include, it is vital to remember that the employees in the stores are those who have to work with them in practice, and the suppliers are those who deliver the products. Further it is common practice that employees rebuild shelves following their own individual needs when planograms do not seem to be convenient from their perspective. Therefore, the best planning tool may not improve profits at all if employees do not stick to the resulting planograms. The best way to circumvent such situations is certainly to consider relevant aspects from different perspectives as further objectives.

6 Conclusion This chapter has analyzed the differences between scientific approaches and practical requirements for retail space planning. A description of processes in practice as well as a discussion and classification of state-of-the-art approaches in the literature are intended to build the foundation of a comparison of both worlds. Current approaches in the literature were classified depending on their practical applicability based on a close collaboration with a leading retail company. Different foci of shelf space allocation approaches were highlighted. Obviously there is a certain gap between the aspects researchers focus on and the requirements retailers have. The contribution of this chapter is to create awareness of practical specifications when application models are created. Finally, a list of specific topics reveal crucial points of shelf space allocation models that also serve as possible future areas of research and may help to improve benefits for both researchers and retailers. The practicability and applicability of shelf space optimization approaches in line with retailers’ requirements can still be improved. New application models can be developed following our analysis. A key finding is that emphasis is given to existing parameters (e.g., product weight) and existing resources (e.g., employees) rather than accounting for sophisticated demand effects (e.g., cross-space elasticity). Finding the right formulation for a shelf space optimization model must be based on individual applications. Nevertheless, this chapter has outlined crucial points that will hopefully help to improve the specification of future optimization approaches.

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