Stochastic Programming Approaches to Multi-product Inventory Management Problems with Substitution
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The presence of substitution among multiple similar products plays an important role in inventory management. It has been observed in the literature that incorporating the impact of substitution among products can substantially improve the profit and reduce the understock or overstock risk. This thesis focuses on exploring and exploiting the impact of substitution on inventory management problems by theoretically analyzing mathematical models and developing efficient solution approaches. To that end, we address four problems. In the first problem, we study different pricing strategies and the role of substitution for new and remanufactured products. Our work presents a two-stage model for an original equipment manufacturer (OEM) in this regard. A closed-form one-to-one mapping of product designs onto the optimal product strategies is developed, which provides useful information for the retailer. Our second problem is a multi-product newsvendor problem with customer-driven demand substitution. We completely characterize the optimal order policy when the demand is known and reformulate this nonconvex problem as a binary quadratic program. When the demand is stochastic, we formulate the problem as a two-stage stochastic program with mixed integer recourse, derive several necessary optimality conditions, prove the submodularity of the profit function, develop polynomial-time approximation algorithms, and show their performance guarantees. Our numerical investigation demonstrates the effectiveness of the proposed algorithms and, furthermore, reveals several useful findings and managerial insights. In the third problem, we study a robust multi-product newsvendor model with substitution (R-MNMS), where both demand and substitution rates are uncertain and are subject to cardinality-constrained uncertainty set. We show that for given order quantities, computing the worst-case total profit, in general, is NP-hard, and therefore, address three special cases for which we provide closed-form solutions. In practice, placing an order might incur a fixed cost. Motivated by this fact, our fourth problem extends the R-MNMS by incorporating fixed cost (denoted as R-MNMSF) and develop efficient approaches for its solution. In particular, we propose an exact branch-and-cut algorithm to solve small- or medium-sized problem instances of the R-MNMSF, and for large-scale problem instances, we develop an approximation algorithm. We further study the effects of the fixed cost and show how to tune the parameters of the uncertainty set.
General Audience Abstract
In a multi-product supply chain, the substitution of products arises if a customer's first-choice product is out-of-stock, and she/he have to turn to buy another similar product. It has been shown in the literature that the presence of product substitution reduces the assortment size, and thus, brings in more profit. %and reduce the inventory level. However, how to quantitatively study and analyze substitution effects has not been addressed in the literature. This thesis fills this gap by developing and analyzing the profit model, and therefore, providing judicious decisions for the retailer to make in order to maximize their profit. In our first problem, we consider substitution between new products and remanufactured products. We provide closed-form solutions, and a mapping that can help the retailer in choosing optimal prices and end-of-life options given a certain product design. In our second problem, we study multi-product newsvendor model with substitution. We first show that, when the probability distribution of customers' demand is known, we can tightly approximate the proposed model as a stochastic integer program under discrete support. Next, we provide effective solution approaches to solve the multi-product newsvendor model with substitution. In practice, typically, there is a limited information available on the customers' demand or substitution rates, and therefore, for our third problem, we study a robust model with a cardinality uncertainty set to account for these stochastic demand and substitution rates. We give closed-form solutions for the following three special cases: (1) there are only two products, (2) there is no substitution among different products, and (3) the budget of uncertainty is equal to the number of products. Finally, similar to many inventory management problems, we include a fixed cost in the robust model and develop efficient approaches for its solution. The numerical study demonstrates the effectiveness of the proposed methods and the robustness of our model. We further illustrate the effects of the fixed cost and how to tune the parameters of the uncertainty set.
- Doctoral Dissertations