Multi-period supply chain network equilibrium with capacity constraints and purchasing strategies

Abstract

In this paper, we propose a capacitated supply chain network equilibrium model in which three tiers of decision makers (manufacturers, retailers and consumers at demand markets) seek to determine their optimal plans over a multi-period planning horizon. Unlike other studies in the extent literature, we use a new concept of purchasing strategy to model the strategic behavior of retailers and consumers at demand markets in a capacitated supply chain network. A purchasing strategy denotes an ordered set of manufacturers (or retailers) from which each retailer (or each consumer at a demand market) prefers to purchase a product. We show that the equilibrium conditions governing the multi-period capacitated supply chain network equilibrium problem can be formulated as a variational inequality in terms of strategies and strategic flows. To find a solution to this variational inequality, we propose an iterative algorithm that generates strategies for retailers and consumers as required by solving a dynamic program. We prove that the solution set is nonempty and provide a numerical example to illustrate the validity of our model.

Introduction

A supply chain is a network of manufacturers, transporters and retailers that perform the functions of production, storage, transportation, and sale of a particular product. Many researchers describe the various networks that underly supply chain management as optimization problems (e.g. Lee and Billington (1993), Stadler and Kigler (2000), Hensher et al., 2001, Geunes and Pardalos, 2003). To study decentralized decision making and competition in the supply chain, Lederer and Li (1997) applied game theory to model competition between firms that produce goods or services for customers who are sensitive to delay time. Cachon and Zipkin (1999) studied inventorying decision-making in a two-stage serial supply chain. Bernstein and Federgruen (2003) modeled retail market competition in the case of a single supplier and multiple retailers in a multi-period environment. Perakis and Sood (2006) studied multi-period pricing at retail markets using a robust optimization approach.

Nagurney et al. (2002) proposed the first supply chain network equilibrium model, which was multitiered and involved competition among decision-makers in a given tier, but cooperation between tiers of decision makers, consisting of manufacturers, retailers and consumers at the demand markets. Using a properly constructed supernetwork, Nagurney (2006) demonstrated that this problem can be reformulated and solved as a traffic network equilibrium problem in terms of paths and path flows, which has opened up the study of supply chain networks as transportation networks, a subject with a much longer history and literature. Chen and Chou, 2004, Liu and Nagurney, 2006 extended the model to a dynamic setting where decision makers seek to determine their optimal plans over a multi-period (time-dependent) planning horizon. Chen and Chou (2005) proposed a capacitated static supply chain equilibrium model where shadow prices are added to capacitated path costs with which the traffic equilibrium conditions are satisfied.

In this paper, we develop a new capacitated multi-period supply chain network equilibrium model in which manufacturers, retailers and consumers at the demand markets are non atomic and are located at distinct tiers of the network, and decisions are made in distinct time periods over a given planning horizon. In the first tier, there are m manufacturers competing for orders from retailers. These manufacturers have sufficient information about the future and seek to maximize their own profits over the planning horizon by making optimal production, transaction, and inventory decisions. Each manufacturer’s production and inventory decisions must be compatible with its production and storage capacities, respectively.

The second tier consists of n retailers. Retailers purchase the product from the manufacturers during each period and the total amount unsold at the end of each period is carried over to the next period as inventory. In addition to the cost of the product itself, each retailer also incurs costs associated with holding inventory and handling the product. Similar to the manufacturers, each retailer competes with other retailers and wishes to maximize its profit over the planning horizon by making optimal decisions that must be compatible with its storage capacity.

The third, and last, tier consists of k demand markets, each of which consists of a number of homogenous consumers. Consumers in each demand market determine their consumption levels and consider both the prices charged by the retailers and the transaction costs of obtaining the product. At equilibrium, consumers in each market have no financial incentive to change the quantities with which they will purchase the product.

Unlike other studies using the path-based approach, we model the optimizing behavior of retailers and consumers at demand markets using the concept of purchasing “strategy”. In the presence of capacities, retailers adopt strategies in purchasing the product from their preferred manufacturers. A purchasing strategy is shown as an ordered set of manufacturers, and the order in which the manufacturers are listed indicates the retailer’s preference. For example, one retailer may prefer to order from manufacturer A rather than B because A might have a better reputation, a more convenient location, or a better price. In this case, the retailer would order from B only when A did not have sufficient supply to satisfy the order. If, for instance, the total capacity of manufacturer A is equal to 10 and the total amount of product needed for the retailer is 25, then the retailer will purchase 10 units from manufacturer A and buys the remaining 15 units from manufacturer B. As a consequence, the behavior of the retailer is stochastic because it depends on the probability to buy the product from manufacturer A. Similar to retailers, consumers in each market use purchasing strategies, i.e., each consumer has a preference list of retailers, and the decision to purchase the product depends on the product availability at the retailers. If a consumer, who prefers retailer C the most tries to purchase the product and retailer C does not have it, then the individual would go to the retailer next on his or her preference list.

By combining purchasing strategies for retailers and consumers, we define an end-to-end supply chain strategy and show, through a supernetwork construction similar to the one used in the model of Liu and Nagurney (2006), that the conditions governing the multi-period supply chain network equilibrium problem can be formulated as a transportation equilibrium model, which leads to a new variational inequality involving a vector-value function of expected strategy costs. To calculate the expected cost of strategies, we develop a loading process that computes link probabilities to capture the strategic behavior of retailers and consumers in a capacitated supply chain network.

This paper is organized as follows: In Section 2, we present the multi-period supply chain network model with capacity constraints. In Section 3, we introduce the notion of purchasing strategies for retailers and consumers and establish that the proposed multi-period supply chain network equilibrium model can be formulated as a traffic network equilibrium model, which leads to a variational inequality in terms of strategies and strategic flows. Section 3 also discusses the existence and uniqueness of an equilibrium solution. Section 4 proposes a solution algorithm that generates strategies for retailers and consumers as required by solving a dynamic program. To illustrate the effectiveness of the algorithm, Section 5 presents numerical results from a small supply chain network. Finally, Section 6 concludes the paper.