An optimal stocking problem to minimize the expected time to sellout

We consider an assortment planning problem where the objective is to minimize the expected time to sell all items in the assortment. We provide several structural results for the optimal assortment. We present a heuristic policy, which we prove is asymptotically optimal. We also show that there are alternate objective criteria under which the problem simplifies considerably.     

Assortment planning for vertically differentiated products under a consider-then-choose model

Consider-then-choose models, borne out by empirical literature in marketing and psychology, note that customers follow a two-stage procedure to choose among alternatives. In this paper, we consider the assortment optimization problem of a retailer who manages a category of vertically differentiated products under customers’ consider-then-choose behavior. We characterize some structural results of the optimal assortment and find that the problem can be solved as the shortest path problem. Also, we develop an efficient algorithm to identify an optimal assortment.     

Assortment optimization under the multinomial logit model with product synergies

In synergistic assortment optimization, a product’s attractiveness changes as a function of which other products are offered. We represent synergy structure graphically. Vertices denote products. An edge denotes synergy between two products, which increases their attractiveness when both are offered. Finding an assortment to maximize retailer’s expected profit is NP-hard in general. We present efficient algorithms when the graph is a path, a tree, or has low treewidth. We give a linear program to recover the optimal assortment for paths.     

Assortment optimization under a multinomial logit model with position bias and social influence

Motivated by applications in retail, online advertising, and cultural markets, this paper studies the problem of finding an optimal assortment and positioning of products subject to a capacity constraint in a setting where consumers preferences can be modeled as a discrete choice under a multinomial logit model that captures the intrinsic product appeal, position biases, and social influence. For the static problem, we prove that the optimal assortment and positioning can be found in polynomial time. This is despite the fact that adding a product to the assortment may increase the probability of selecting the no-choice option, a phenomenon not observed in almost all models studied in the literature. We then consider the dynamics of such a market, where consumers are influenced by the aggregate past purchases. In this dynamic setting, we provide a small example to show that the natural and often used policy known as popularity ranking, that ranks products in decreasing order of the number of purchases, can reduce the expected profit as times goes by. We then prove that a greedy policy that applies the static optimal assortment and positioning at each period, always benefits from the popularity signal and outperforms any policy where consumers cannot observe the number of past purchases (in expectation).     

Assortment and inventory decisions with multiple quality levels

We consider the assortment and inventory decisions of a retailer under a locational consumer choice model where products can be differentiated both horizontally (e.g., color of a product) and vertically (e.g., quality of a product). The assortment and quantity decisions affect customer choice and, hence, the demand and sales for each product. In this paper, we investigate two different environments where product availability and assortment affect consumer choice and demand in different ways: make-to-order (MTO) and make-to-stock (MTS). In the MTO environment, customers order and purchase their most preferred product; that is, stockouts do not occur. In the MTS model, customers buy their most preferred product if it is in stock or do not buy if it is out of stock. In both environments we find conditions under which it is optimal to carry assortments of only a single quality level. In the MTS case, we show that an assortment of mixed quality levels can be optimal only within a narrow range of parameters.     

An approximately global optimization method for assortment problems

Assortment problems occur when we want to cut a number of small rectangular pieces from a large rectangle to get the minimum area within the rectangle. Recently, Chen et al. proposed a useful model for assortment problems. Although Chen et al.'s model is quite promising to find solutions, there are two inadequacies in their model: firstly, the objective function in their model is a polynomial term, which may not lead to a globally optimal solution; secondly, too many 0–1 variables are used to formulate the non-overlapping constraints. We propose a new method to reformulate an assortment model. Our model is not only able to find the approximately global optimal solution, but involves less 0–1 variables for formulating non-overlapping constraints.     

Private labels and retail assortment planning: a differential evolution approach

Despite the longstanding recognition of the importance of product assortment planning (PAP), existing literature has failed to provide satisfactory solutions to a great deal of problems that reside in this area of research. The issue of optimal assortment planning in the retail sector becomes even more important in periods of economic crisis, as retailers must adapt their product portfolios to new evolving patterns of consumer buying behaviour and reduced levels of consumer’s purchasing power. Private labels (PLs) typically experience significant growth in times of recession, due to their low prices, and the reduced disposable income of households. In this direction, the present paper introduces differential evolution to assist retailers in adapting their product portfolios in periods of economic recession and facilitate strategic PAP decisions, related to (a) optimal variety of PL product categories, (b) optimal service level of PL merchandise within a product category, and hence, (c) optimal balance between PLs and National Brands in a retailer’s product portfolio. The interrelated issue of assortment adaptation across different store formats is also considered. Economic recessions contribute to the prolonged upward evolution in PL share, and hence, our mechanism facilitates decisions that are nowadays more important than ever before. The proposed mechanism is illustrated through an implementation to an empirical dataset derived from a random sample of 1928 consumers who participated in a large-scale computer assisted telephone survey during the current economic crisis period.     

In–Out Algorithm for assortment planning under a ranking-based consumer choice model

We consider a retailer’s assortment planning problem under a ranking-based consumer choice model. The retailer incurs a fixed carrying cost per product offered, a substitution penalty cost for each customer who does not purchase his first choice, and a penalty cost on lost sales. We develop an effective In–Out Algorithm to identify the optimal solution. The extensive numerical study shows that the algorithm performs well, and is more than 10,000 times faster than enumeration on problems with 20 products.     

Robust assortment optimization using worst-case CVaR under the multinomial logit model

We consider robust assortment optimization problems with partial distributional information of parameters in the multinomial logit choice model. The objective is to find an assortment that maximizes a revenue target using a distributionally robust chance constraint, which can be approximated by the worst-case Conditional Value-at-Risk. We show that our problems are equivalent to robust assortment optimization problems over special uncertainty sets of parameters, implying the optimality of revenue-ordered assortments under certain conditions.     

Assortment planning in fashion retailing: methodology,application and analysis

Assortment planning is the process conducted by the retailer to determine the number and types of products in a line. Key questions that arise in this process include choosing the inventory depth and variety breadth, and the mix between basic and fashion merchandise of the assortment to maximize expected profits. We describe a method for resolving these questions. Using demand forecasts derived from historical sales patterns, we use a nonlinear integer programming model to make the assortment choice. Efficient heuristics are developed to solve this problem. We applied our method at a large catalog retailer specializing in women’s apparel. We compared our method to the existing rules used by this retailer and found that it could choose the assortment in a manner that reduces markdowns due to excessive inventory and lost margins due to stockouts by enough to increase profits by at least 40%. Insights are developed to better understand why products are included in an assortment and the implications of this choice on the realized profit. We extend our model to include shelf space constraints and the effect of assortment choice on product demand.     

Product sequencing and pricing under cascade browse model

We study the assortment optimization problem under the newly proposed cascade browse model. We propose a constant approximate solution to this problem. As a byproduct, we propose the first fully polynomial-time approximation scheme (FPTAS) for the classic assortment optimization problem subject to one capacity constraint and one cardinality constraint. We also studied a joint pricing and sequencing problem under the above model and develop a constant approximate solution to this problem.     

Pricing to accelerate demand learning in dynamic assortment planning for perishable products

Retailers, from fashion stores to grocery stores, have to decide what range of products to offer, i.e., their product assortment. Frequent introduction of new products, a recent business trend, makes predicting demand more difficult, which in turn complicates assortment planning. We propose and study a stochastic dynamic programming model for simultaneously making assortment and pricing decisions which incorporates demand learning using Bayesian updates. We show analytically that it is profitable for the retailer to use price reductions early in the sales season to accelerate demand learning. A computational study demonstrates the benefits of such a policy and provides managerial insights that may help improve a retailer’s profitability.     

Optimizing product assortment under customer-driven demand substitution

The problem of product assortment and inventory planning under customer-driven demand substitution is analyzed and a mathematical model for this problem is provided in this paper. Realistic issues in a retail context such as supplier selection, shelf space constraints, and poor quality procurement are also taken into account. The performance of three modified models, one that neglects customers’ substitution behavior, another that excludes supplier selection decision, and one that ignores shelf space limitations, are analyzed separately with computational experiments. The results of the analysis demonstrate that neglecting customer-driven substitution or excluding supplier selection or ignoring shelf space limitations may lead to significantly inefficient assortments. The effects of demand variability and substitution cost on optimal assortment and supplier selection decisions as well as on the optimal revenue are also investigated. The main contribution of this paper is the development of a practical and flexible model to aid retailers in finding optimal assortments to maximize the expected profit.     

Estimating the market shares of stores based on the shopper's search and purchase behavior

Suppose that a consumer has decided to shop around several retail stores in an attempt to find a desired product or service. From his or her past shopping experience, the shopper may know: (1) the assortment size of each store, (2) the search cost per visit, and (3) the price variation among the stores. For such a situation, we first consider the optimal sequence of stores and the optimal search strategy from the shopper's search-theoretic point of view. Based on the assumption that shoppers follow the optimal sequencing and search strategy that maximizes his or her expected net gain, we then estimate the market share of each store in the market area. With a numerical example, we finally analyze the effects of the assortment size, the search cost, and the price variation on the market shares of existing retail stores. Based on the shopper's search and purchase model, we attempt to explain: (1) why shoppers visit bigger stores first, (2) why they visit fewer stores if the search cost is relatively higher than the product price, and (3) why they shop around more stores if the price variation among the stores is large.     

基于Logit离散选择模型的品类优化问题综述

品类优化问题(Assortment Optimization Problem)是收益管理的经典问题.它研究零售商在满足运营约束的前提下,应如何从给定产品集合中选择一个子集提供给消费者,以最大化预期收益.该问题的核心在于如何准确地刻画消费者在面对细分产品时的选择行为、建立相应的优化模型并设计高效率的求解算法.基于Logit离散选择模型的品类优化问题:首先,介绍了基于Multinomial Logit模型的品类优化问题.然后介绍了两个更复杂的变种:第一个是基于两层以及多层Nested Logit模型的品类优化问题,这类问题可合理刻画细分产品之间的"替代效应";第二个是基于Mixtures of Multinomial Logits模型的品类优化问题,这类问题可充分考虑消费者群体的异质性.随后,介绍了数据驱动的品类优化问题的相关进展.最后,指出该问题未来可能的若干研究方向.     

A multi-objective programming approach to 1.5-dimensional assortment problem

In this paper we study a 1.5-dimensional cutting stock and assortment problem which includes determination of the number of different widths of roll stocks to be maintained as inventory and determination of how these roll stocks should be cut by choosing the optimal cutting pattern combinations. We propose a new multi-objective mixed integer linear programming (MILP) model in the form of simultaneously minimization two contradicting objectives related to the trim loss cost and the combined inventory cost in order to fulfill a given set of cutting orders. An equivalent nonlinear version and a particular case related to the situation when a producer is interested in choosing only a few number of types among all possible roll sizes, have also been considered. A new method called the conic scalarization is proposed for scalarizing non-convex multi-objective problems and several experimental tests are reported in order to demonstrate the validity of the developed modeling and solving approaches.     

Assortment optimization with repeated exposures and product-dependent patience cost

In this paper, we study the assortment optimization problem faced by many online retailers such as Amazon. We develop a cascade multinomial logit model, based on the classic multinomial logit model, to capture the consumers' purchasing behavior across multiple stages. Unlike most of existing studies, our model allows for repeated exposures of a product. In addition, each consumer has a patience budget that is sampled from a known distribution and each product is associated with a patience cost, which is the required amount of the cognitive efforts on browsing that product. We propose an approximation solution to the assortment optimization problem under cascade multinomial logit model.     

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.     

模糊最优归类模型在服务过程质量评价中的应用

介绍模糊最优归类模型,针对服务质量差距环式模型,应用模糊最优归类模型计算服务过程各环节质量指标值及各环节之间差距值,并且引入模糊最优归类模型提高差距分辨的方法,以提高质量指标值计算的水平.     

A neural network system for solving an assortment problem in the steel industry

A neural network model for solving an assortment problem found in the iron and steel industry is discussed in this paper. The problem arises in the yard where steel plate is cut into rectangular pieces. The neural network model can be categorized as a Hopfield model, but the model is expanded to handle inequality constraints. The idea of a penalty function is used. A large penalty is applied to the network if a constraint is not satisfied. The weights are updated based on the penalty values. A special term is added to the energy function of the network to guarantee the convergence of the neural network which has this feature. The performance of the neural network was evaluated by comparison with an existing expert system. The results showed that the neural network has the potential to identify in a short time near-optimal solutions to the assortment problem. The neural network is used as the core of a system for dealing with the assortment problem. In building the neural networks system for practical use, there were many implementation issues. Some of them are presented here, and the fundamental ideas are explained. The performance of the neural network system is compared to that of the expert system and evaluated from the practical viewpoint. The results show that the neural network system is useful in handling the assortment problem.