Product selection and space allocation in supermarkets

A model and an algorithm are proposed for the simultaneous optimal selection among a given set of products of the assortment of products to be sold in a supermarket and the allocation of shelf space to these products. The model takes into account the space elasticity of the sales as well as the constraints that any chosen product must receive a minimum shelf space and that the shelf space allocated to each product must be equal to an integer number of facings.Computational results are presented, including a comparison with the results given by several rules for space allocation currently used. Some marketing implications are discussed.     

Shelf-space allocation of national and private brands

We propose a game-theoretic model in which one national-brand manufacturer, acting as a leader, maximizes her own profit and one retailer, selling the national brand and her private label and acting as a follower, maximizes her category profit. We characterize the resulting Stackelberg equilibrium in terms of the amount of shelf space allocated to these brands as well as their prices. The results suggest that the allocation of the shelf space depends on the quality of the private label. In our framework, quality is measured by the baseline sales (or brand equity), the degree of brand substitution and the price positioning.     

An efficient algorithm to allocate shelf space

Shelf space is one of the most important resources of a retail firm. This paper formulates a model and proposes an approach which is similar to the algorithm used for solving a knapsack problem. Subject to given constraints, the proposed heuristic allocates shelf space item by item according to a descending order of sales profit for each item per display area or length. Through the use of simulations, the performances of objective value and the computational efficiency of this method are evaluated. Three options are also proposed for improving the heuristics. Compared to an optimal method, the improved heuristic is shown to be a very efficient algorithm which allocates shelf space at near-optimal levels.     

Retail replenishment models with display-space elastic demand

We study a single store multi-product inventory problem in which product sales are a composite function of shelf space. Since sales tend to deplete the amount of product on display, the effective shelf space assigned to the product diminishes with time unless replenishment occurs. We consider the problem of optimal replenishment times under these conditions. We assume a linear dependence of sales rate to effective shelf space in all our analysis. We present exact and approximate solutions for the single product and multi-product cases. For the single product case, we study the effect of space elasticity, cross elasticity and empty space elasticity on the optimal replenishment period. For the multi-product case we present a computationally attractive method using matrix exponentials and develop error bounds for this method.     

A Dynamic Model for Strategically Allocating Retail Space

The allocation of shelf space is a major determinent of a retailer's sales and operating costs. All the existing models of this problem focus on static optimization. But to the retailer, anticipating and adapting to new tastes and changing product life cycles is the central strategic problem. This paper shows how the static model can be extented to incorporate such market dynamics. The new model is more plausible than earlier static models in encouraging the retailer to allocate more space to new products and divest earlier from declining ones.     

基于“进场费+收益共享”契约的供应链协调策略研究

在超市等自助零售环境下,影响产品需求的往往是产品的感知质量,而产品感知质量不仅取决于产品质量水平,还受货架空间展示量的影响。本文基于价格和感知质量依赖的需求函数,分别建立了一体化供应链集中决策模型以及考虑是否存在进场费的寄售契约下零售商主导的Stackelberg博弈模型,研究了制造商和零售商的产品定价和货架空间展示策略,并给出了进场费+收益共享的供应链协调策略。研究结果表明分散供应链环境下产品的最优零售价格并不总高于集中决策下产品的最优定价,也就是说提升感知质量在一定程度上可以缓解双重边际效应。另外,收取进场费的收益共享模式可以实现供应链的完美协调,这在理论上说明了进场费模式在一定的市场环境下是合理的。     

Goal programming model for grocery shelf space allocation

The highly competitive grocery retail industry has annual sales of roughly half a trillion dollars in the US. While gross margins average about 28% of sales, net profits after taxes are only 1% industry-wide, causing retailers to continually search for operational improvements that increase profitability and improve customer service. One important decision that affects both of these goals is how to allocate shelf space to different products.     

基于时序矩阵分解的缺失销售数据估计

企业的历史销售记录是供应链优化研究的基础数据来源,然而,在日常的研究中,几乎所有可以通过公开途径获得的销售记录都是高度不完整的,这为研究者开展工作带来了极大的不便。为解决此问题,本文提出,以销售数据集中已有的数据为基础,使用面向时序数据的矩阵分解模型MAFTIS对其缺失的部分进行估算,从而把残缺的数据集补全完整。进一步地,为提高MAFTIS的计算效率,本文还为该模型设计了一种基于交替最小二乘法的求解策略MAFTIS^ALS。在评估实验中,MAFTIS^ALS被用于三个真实销售数据集的缺失记录估计,结果显示,与其它估计模型相比,MAFTIS^ALS能获得更准确的估计结果,并且具有更高的收敛速度。     

A joint optimisation model for inventory replenishment,product assortment,shelf space and display area allocation decisions

In this paper, we propose an optimisation model to determine the product assortment, inventory replenishment, display area and shelf space allocation decisions that jointly maximize the retailer’s profit under shelf space and backroom storage constraints. The variety of products to be displayed in the retail store, their display locations within the store, their ordering quantities, and the allocated shelf space in each display area are considered as decision variables to be determined by the proposed integrated model. In the model formulation, we include the inventory investment costs, which are proportional to the average inventory, and storage and display costs as components of the inventory costs and make a clear distinction between showroom and backroom inventories. We also consider the effect of the display area location on the item demand. The developed model is a mixed integer non-linear program that we solved using LINGO software. Numerical examples are used to illustrate the developed model.     

A piecewise linearization for retail shelf space allocation problem and a local search heuristic

Retail shelf space allocation problem is well known in literature. In this paper, we make three contributions to retail shelf space allocation problem considering space elasticity (SSAPSE). First, we reformulate an existing nonlinear model for SSAPSE to an integer programming (IP) model using piecewise linearization. Second, we show that the linear programming relaxation of the proposed IP model produces tight upper bound. Third, we develop a heuristic that consistently produces near optimal solutions for randomly generated instances of problems with size (products, shelves) varying from (25, 5) to (200, 50) within a minute of CPU time.     

Heuristic and meta-heuristic approaches for multi-period shelf-space optimization: the case of motion picture retailing

The retail industry is in a highly competitive situation currently. The success of the industry depends upon the efficient allocation of products in the shelf space. Several previous authors have developed mathematical models for optimal shelf-space allocation. We extend the prior research in the direction of the multi-period problem and introduce more realistic characteristics, such as product demand perishability, pricing contract and cross-elasticity. The new characteristics help us address the case of the real-life movie allocation problem in multiplexes. We formulate a linear integer programming model to represent the problem. The proposed model shows a potential benefit of at least 11% increase in revenue for a multiplex theatre situation as compared to the existing methods. We also propose two greedy heuristics and a genetic algorithm to solve the same problem. A computational study shows that the genetic algorithm performs better than the existing method.     

The shelf space and pricing strategies for a retailer-dominated supply chain with consignment based revenue sharing contracts

Consider a two-echelon supply chain consisting of two manufacturers and a dominant retailer, such as big supermarkets like Walmart. Under a consignment contract with revenue sharing, the two manufacturers sell through the retailer two substitutable products whose demands are dependent on their shelf space and sales prices. The two manufacturers may compete horizontally for shelf space and pricing by three scenarios: Nash game, Stackelberg game, and collusion, and play vertically the retailer-Stackelberg game with the retailer. For each of these horizontal scenarios, we present all participators’ equilibrium strategies and their corresponding profits, based on which the impacts of manufacturers’ cost difference and moving sequence are investigated. Additionally, we discuss whether a horizontal collusion among manufacturers occurs when they choose their scenarios and whether centralization is always beneficial for the entire chain under the considered consignment contract. The study reveals the following results: (i) When the manufacturers compete horizontally, the high-cost manufacturer always sets a high-price and less shelf space strategy, while the low-cost manufacturer always adopts a low-price and more shelf space strategy, which is not affected by their moving sequence. If they collude horizontally, it is just reverse. (ii) When the two manufacturers compete horizontally, all participators’ equilibrium strategies and their corresponding profits are significantly influenced by manufacturers’ moving sequence. (iii) A horizontal collusion between the manufacturers can occur only when their cost difference is relatively small; this finding supplements existing literature. (iv) When the cost difference between manufacturers is relatively big, then centralization may be detrimental to the entire chain, which can explain why several supply chains adopt vertical competition strategies in practice. In addition, we find that these results still hold for the limited shelf space scenario and shelf-space limitation enhances the horizontal and vertical competition intensity by increasing shelf space fee.     

电子商务环境下考虑退货的选址库存问题模型与算法

选址库存问题(location inventory problem, LIP)是物流系统集成的经典问题之一,也是企业需要面对的管理决策难题。本文考虑在电子商务环境下无质量缺陷的退货商品可简单再包装后重新进入销售市场这一现实情况,对设施选址和库存控制进行集成优化,构建随机需求下有退货的LIP模型。针对此问题求解的复杂性,设计了改进的自适应混合差分进化算法对模型进行整体求解。最后,通过多组算例验证了模型和算法的实用性和优越性,可为设施选址、库存控制和商品配送回收决策提供重要参考依据。     

An Artificial Allocations Based Solution to the Label Switching Problem in Bayesian Analysis of Mixtures of Distributions

Label switching is a well-known problem occurring in MCMC outputs in Bayesian mixture modeling. In this article we propose a formal solution to this problem by considering the space of the artificial allocation variables. We show that there exist certain subsets of the allocation space leading to a class of nonsymmetric distributions that have the same support with the symmetric posterior distribution and can reproduce it by simply permuting the labels. Moreover, we select one of these distributions as a solution to the label switching problem using the simple matching distance between the artificial allocation variables. The proposed algorithm can be used in any mixture model and its computational cost depends on the length of the simulated chain but not on the parameter space dimension. Real and simulated data examples are provided in both univariate and multivariate settings. Supplemental material for this article is available online.     

Mixture Modeling for Longitudinal Data

In this article, we propose an unbiased estimating equation approach for a two-component mixture model with correlated response data. We adapt the mixture-of-experts model and a generalized linear model for component distribution and mixing proportion, respectively. The new approach only requires marginal distributions of both component densities and latent variables. We use serial correlations from subjects’ subgroup memberships, which improves estimation efficiency and classification accuracy, and show that estimation consistency does not depend on the choice of the working correlation matrix. The proposed estimating equation is solved by an expectation-estimating-equation (EEE) algorithm. In the E-step of the EEE algorithm, we propose a joint imputation based on the conditional linear property for the multivariate Bernoulli distribution. In addition, we establish asymptotic properties for the proposed estimators and the convergence property using the EEE algorithm. Our method is compared to an existing competitive mixture model approach in both simulation studies and an election data application. Supplementary materials for this article are available online.     

An objective space cut and bound algorithm for convex multiplicative programmes

Multiplicative programming problems are global optimisation problems known to be NP-hard. In this paper we propose an objective space cut and bound algorithm for approximately solving convex multiplicative programming problems. This method is based on an objective space approximation algorithm for convex multi-objective programming problems. We show that this multi-objective optimisation algorithm can be changed into a cut and bound algorithm to solve convex multiplicative programming problems. We use an illustrative example to demonstrate the working of the algorithm. Computational experiments illustrate the superior performance of our algorithm compared to other methods from the literature.     

A new benchmarking approach in Cold Chain

The dark area of Supply Chain Management (SCM) is inadequate attentions to those products that have limitations such as shelf life, need to special equipments and facilities for sales, storage and distribution and so on. For this reason the concept of Cold Chain Management (CCM) was emerged. The main objective of this paper is to develop a linear pair model for selecting the best sales agents as a “Benchmark” in the presence of non-discretionary factors and imprecise data under Free Disposability assumption.     

The location and allocation of products and product families on retail shelves

In this paper, we model a shelf-management problem in which individual products are categorized as part of a product family. It is well known that a product’s shelf location has a significant impact on sales for many retail items. We develop a continuous as well as a discrete model with postprocessing to optimize product placement with consideration given to maintaining the grouping of product families. Computational results are reported on test problems as well as real-world beverage placement problems.     

推广成本信息不对称下推广商与酒店间的合作契约设计

为应对日趋激烈的酒店市场竞争、加强对销售渠道的控制,酒店急需与推广商合作以提高自有销售网站的需求水平。文章基于推广成本信息不对称,构建了一个占主导的酒店和一家推广商关于需求推广合作的博弈模型,分析了信息不对称情形下推广商的谎报动机,指出合作契约设计的必要性。并提出一套契约菜单,在保证推广商依据自身实际成本类型进行契约选择的前提下,实现了酒店利润的最大化。最后通过算例分析,验证了所设计的合作契约的有效性,并给出了参数灵敏性分析。     

Finding market structure by sales count dynamics—Multivariate structural time series models with hierarchical structure for count data—

In this paper, we propose a multivariate time series model for sales count data. Based on the fact that setting an independent Poisson distribution to each brand’s sales produces the Poisson distribution for their total number, characterized as market sales, and then, conditional on market sales, the brand sales follow a multinomial distribution, we first extend this Poisson–multinomial modeling to a dynamic model in terms of a generalized linear model. We further extend the model to contain nesting hierarchical structures in order to apply it to find the market structure in the field of marketing. As an application using point of sales time series in a store, we compare several possible hypotheses on market structure and choose the most plausible structure by using several model selection criteria, including in-sample fit, out-of-sample forecasting errors, and information criterion.