Production, Manufacturing, Transportation and Logistics
Capacitated strategic assortment planning under explicit demand substitution

https://doi.org/10.1016/j.ejor.2021.02.026Get rights and content

Highlights

  • Examines optimal assortment under exogenous demand and explicit demand substitution.

  • Resulting capacitated assortment problem is proven to be computationally complex.

  • Optimal assortment is in popular set, if a higher margin product has a lower demand.

  • Assortment capacity is fully utilized if substitution ratio is below a threshold level.

  • Approximate policies offered with numerically and theoretically shown performances.

Abstract

Buyers have easier access to a variety of products with the rise of multi-channel distribution strategies and the increase in new product introductions. On the other hand, firms experience greater pressure in offering the correct product variety given that the manufacturing infrastructure often imposes physical and financial constraints in attaining variety. This study examines a firm’s optimal assortment planning problem under an exogenous demand model, where each customer has a predetermined preference for each product from a potential set. Proportional demand substitutions are allowed from out-of-assortment products to those available. We show that the problem is NP-complete. We also show that an optimal assortment is composed of some number of the highest margin products, if one product having a higher margin than another implies that the former product has a lower demand rate than the latter. The firm’s assortment capacity is fully utilized at the optimum if the customers’ substitution ratio does not exceed a particular threshold. We also introduce several approximate assortment policies that can be easily implemented, and test these policies through extensive numerical analyses. The results reveal that some of the policies can provide less than a 1% profit gap with an optimal solution for a 20-product set. The policy’s performance highly depends on the firm’s assortment capacity-to-product set size ratio. Moreover, we provide performance bounds for two of these well-performing approximate policies.

Introduction

When the Ford Motor Company revealed a newly revised Ford Fusion sedan in March 2018, they also announced that they standardized many features of the vehicle—for example, Ford’s Co-Pilot360 driver-assist technology—while leaving only a few additional options (Ford, 2018). The company states that this strategy substantially decreased the number of orderable configurations for the Ford Fusion, from approximately 2000 to 36. By decreasing these configurations, they decreased their manufacturing complexity with the aim to reduce costs.

In parallel to the reduced number of configurations per model, the number of car models produced in each plant also significantly decreased with a similar incentive of obtaining a leaner production plant operating at lower costs. Choudhary, Hasija, and Netessine (2018) report that between 1998 and 2006, the number of models produced in assembly plants by three major U.S. automobile manufacturers—Chrysler, Ford, and General Motors—is decreasing. For example, while Ford manufactured up to nine models per plant in 1998, the maximum number of models per plant decreased to three in 2006. The authors note that each plant should reach a particular level of variety to balance the demand satisfaction benefits and excessive set-up costs.

One critical decision for the automotive industry involves assigning models and their subsequent configurations to production plants. These assignments are determined by tooling and capacity investment decisions that must be made between one to three years before production begins. Further, this may require substantial investment, such as funds for a new assembly line, additional tooling, or employee training (Jordan & Graves, 1995). The problem of assigning products to automobile production plants is a strategic assortment-planning problem that does not often consider inventory.

Generally, the firm’s limited assortment, or set of products offered at any time, should be carefully set consistent with the firm’s strategy. Assortment planning is the process of deciding (i) the number of categories, called the breadth; (ii) the number of products in each category, called the depth; and (iii) the corresponding inventory levels for each product to be offered at any time. An assortment of a certain size involves its relevant operational complexities and costs as well as customer sales potential. Further, assortment planning aims to offer an optimal variety of products to customers to maximize the total profit from sales relative to the given costs and limitations of this variety.

Briesch, Chintagunta, and Fox (2009) report that customers’ brand choice decisions can be more sensitive to assortments than to prices. Further, firms must attempt to implement periodical assortment planning to consider customers’ changing preferences over time, seasons, and the launch of new products on the market (Kök, Fisher, & Vaidyanathan, 2015). The assortment-planning decision is complex due to several trade-offs between having a rich versus limited assortment. From the customer’s perspective, on the one hand, a rich and variable assortment draws higher customer traffic (Timonina-Farkas, Katsifou, & Seifert, 2020). For example, expanding an assortment in a retail setting can decrease consumers’ search even with unprofitable products, consequently increasing profit (Cachon, Terwiesch, & Xu, 2005). On the other hand, a narrow assortment can make customers’ decision easier thus increasing the probability of purchase (de Vries-van Ketel, 2006, Mantrala, Levy, Barbara, Fox, Gaidarev, Dankworth, Shah, 2009). Moreover, Boatwright and Nunes (2001) reveal that reducing the assortment by up to 54% increases average sales by 11%.

From an operational perspective, firms may also experience space and budget constraints regarding the variety offered. Each product requires substantial investment, such as funds for new assembly lines, additional tooling, or employee training. For example, Toyota Europe announced that they will invest 300 million euros in its plant in France to build a platform to enable the production of new Toyota models (Toyota-Europe, 2018). After a product is included in the assortment, operational costs per product are incurred because of material handling and warehousing, as well as merchandize presentation (Smith & Agrawal, 2000), record-keeping, and reordering. While it might be more challenging to compute the fixed assortment costs per product, a constrained assortment size often inevitably arises in practice. Subsequently, manufacturers are limited in the number of assembly lines they can place into service, and each line can only produce a few models. Each offered product involves handling, replenishment, and inventory costs, as a certain quantity of inventory should be maintained. All products in an assortment are intertwined through the total available budget. If all products have symmetrical or similar space and/or financial needs, then assortment constraints can be reduced to a cardinality constraint.

Given that the assortment size will be limited because of both demand and cost perspectives, firms should also consider the behaviors of consumers faced with a limited variety. When customers visit a firm, they typically demand a specific product, and its unavailability may lead them to consider either leaving the firm without purchasing, or switching to another product. The act of switching to an alternative product when the favored product is unavailable is known as substitution (Shin, Park, Lee, & Benton, 2015). This can occur when either a shortage exists in the product’s inventory, called stockout-based substitution, or the product is not offered within the assortment, called assortment-based substitution. Corsten and Gruen (2004) report that almost half of customers may tend to switch to different products when their favorite is unavailable. Customers’ substitution-behavior effects can also further complicate firms’ assortment choices. A survey of U.S. vehicle dealers indicated that 15–30% of customers switched from the car they originally sought to one available on the lot (Stalk, Stephenson, & King, 1997). Further, Mahajan and van Ryzin (2001b) indicate that firms can stock relatively more quantities of popular products and relatively fewer unpopular ones under substitution than in the event in which substitutions are not allowed; thus, inventory will be more evenly spread across variants.

This paper examines the optimal assortment of a manufacturing firm whose assortment size constrained by manufacturing infrastructure requirements (Hart & Rafiq, 2006) and that explicitly considers customers’ substitution behavior. It is studied as a strategic problem, thus tactical level inventory and/or production capacity decisions are not incorporated at assortment planning stage. The existence of product substitution complicates the assortment planning problem. We actually show that the problem is NP-complete, which invalidates the use of simple “greedy” algorithms for optimal solution. So, we aim to illustrate the properties of optimal assortments to understand the effects of substitution and capacity constraints on the choice of assortment.

We demonstrate that the optimal assortment contains most preferred products, and that its capacity is always fully utilized if products have different customer preference rates but equal profit margins. When products also vary in their profit margins, the optimal assortment cannot be obtained with a “greedy” algorithm. If all products can be sorted monotonically in increasing order of their profit margins and decreasing order of their demand probabilities, the optimal assortment is composed of some number of most dominant (profitable) products, which can be lower than the assortment capacity. It is shown that by keeping a low profit margin, but high demand product out of the assortment, its demand can be directed to higher margin substitutes.

If all products do not posit monotonic ordering of profit margins and demand probabilities, some number of highly dominant products can omitted from the assortment under a high substitution ratio, which increases the probability of retaining high-margin substitutes. It is proven that when the substitution ratio is smaller, it is more likely that the assortment capacity is fully utilized and an optimal assortment will be composed of the most profitable products. We introduce an upper limit on the substitution ratio below which the assortment capacity of the firm is always fully utilized at optimality. Alternatively, when the assortment capacity is high, capacity utilization may decrease and the optimal assortment may include some less profitable products while excluding some that are more profitable to direct customers to high-margin products in the assortment. We prove that the firm may take the risk of a shallower assortment and expect customers to substitute their demands with those that are high-margin under a high substitution ratio.

Next, our work benefits from obtained optimality properties to introduce seven heuristic assortment-planning algorithms, the complexities of which vary according to their optimality properties and differ in a range from simple sorting to complete profit computation. Our numerical analyses demonstrate that a firm should consider the product set’s size, the assortment capacity, and computational capability when deciding not only whether to use a heuristic policy, but also which to utilize.

We contribute to strategic assortment-planning literature by analyzing a generalized exogenous demand model with product-specific demand rates and profit margins under assortment capacity. We analytically show how substitution ratio affects the product choice as well as the optimal assortment capacity utilization. Moreover, the approximate algorithms we introduce are efficient in computation and effective in obtaining nearly optimal assortments proved by their performance bounds. The assortment optimization problem can also be formulated as a mixed-integer model, but the resulting problem can still be computationally very challenging to solve (Chung, Ahn, & Jasin, 2019) and would not provide the insights on optimal assortments that we obtain in our current study.

We consider manufacturing firms’ strategic assortment decisions, thus excluding inventory decisions in assortment planning. The proposed methodologies are also applicable to other assortment problems without significant inventory concerns during the assortment planning stage, such as when either the inventory management is relatively easier as with slow-moving goods or all inventory is not carried on shelves, but in a depot with limited shelf facings (Kök et al., 2015). For example, Fisher and Vaidyanathan (2014) study the assortment-planning problem for slow-moving products with no inventory concerns motivated by retailers who carry a fixed, often small inventory for each SKU. Feldman and Topaloglu (2017) provide a detailed list of similar assortment problems independent of inventory decisions.

While we primarily express our motivation using automobile manufacturers, the problem setting and results are largely generalizable to other manufacturing industries where the inclusion of each product in a production assortment requires substantial investment. For example, Akçay and Tan (2008) state that small to medium-sized enterprises (SMEs), and textile manufacturers in particular, collect orders from large buyers and procure accordingly (i.e., make-to-order firms). However, each SME specializes in the production of certain types of fabric and subsequently constructs their assortments accordingly. Thus, each SME commits to producing a limited number of varieties due to the significant assortment costs per product. Tan and Akçay (2014) provide the furniture industry as an example, in which a manufacturer has a production catalog with a limited number of models, and each model may require a specific expertise and tooling set-up. Thus, our model and insights are also valid for manufacturers operating in these industries.

The remainder of this paper is organized as follows. In Section 2, we review related studies in literature. In Section 3, we present the problem and show that it is NP-Complete. We reveal the analytical properties of an optimal assortment in Section 4. We provide numerical analysis results to further explain and illustrate the optimal assortment’s properties in Section 5. Next, we introduce several approximate assortment policies that are easy to use and compare their performances with system parameters in Section 6. Finally, Section 7 presents our final remarks by summarizing our findings and obtained insights from the perspective of subsequent research.

Section snippets

Literature review

Assortment planning has two main inputs. One of them is customer-related, as assortment affects customer traffic and sales. The other is operations-related, as the assortment size determines many cost terms, such as handling, shelving, and replenishment, as discussed in Section 1. Thus, both operations management and marketing researchers work on assortment planning. This section presents a brief survey of related literature. Extensive reviews of assortment planning literature are provided by

Assortment planning model

We model the assortment-planning problem of a manufacturing firm that needs to determine its limited product portfolio due to significant investment limitations, which is ultimately treated as a cardinality constraint. Each product in the possible set has a predetermined customer preference and a profit margin. The goal is to select the right products to be offered in the assortment to maximize the firm’s total profit from sales relative to the capacity limitation.

The set of all potential

Properties of optimal assortments

In Section 3, we proved that assortment problem we study is NP-complete. So, this section aims to obtain structural properties optimal assortment. We first analyze the assortment-planning problem under symmetric product profit margins (ri=r), in which products only differ in their customer demand rates αi as noted by Cachon et al. (2005) and Alptekinoglu and Grasas (2014). Symmetrical profit margins result that the firm’s optimal assortment includes a set of its most popular products. Thus, the

Computational insights on optimal assortments

This section investigates the optimal assortments’ sensitivity to the changes in three parameters: the substitution ratio θ, the variance of product demand rates VAR(α), and the variance of product profit margins VAR(r). The product set’s size is |N|=10 for the numerical tests in this section. For each sensitivity analysis, we use 8 different levels of the parameter under test and observe the changes in the optimal solutions’ properties.

We generated the problem instances in three steps.

Approximate assortment policies

Firms should make essentially two decisions during assortment planning under capacity constraints: how much of the capacity to use and which products to include in the assortment set. The decision as to whether to use the complete capacity depends on the product substitution expectations. Section 5 demonstrates that as the substitution rate θ decreases, the number of products kept in the optimal solution also decreases. The decision regarding the products to include in the assortment set

Concluding remarks

This paper examines the strategic assortment optimization problem of a firm. It is a strategic level decision, because manufacturing infrastructure investment is based on the assortment selected. Our proposed methodology is also applicable to other problem settings without significant inventory concerns during the assortment optimization stage. We consider the cardinality constraint on the assortment and customer demand is defined with an exogenous demand model, where each customer has a

Acknowledgment

This research has been partially supported by the Scientific and Technological Research Council of Turkey (TUBITAK) Grant 110M488.

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