Decentralized decision-making and protocol design for recycled material flows

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Abstract

Reverse logistics networks often consist of several tiers with independent members competing at each tier. This paper develops a methodology to examine the individual entity behavior in reverse production systems. We consider two tiers in the network, collectors and processors. The collectors determine individual flow functions that relate the flow they provide each processor to the overall vector of prices that the processors determine. Because the exact final prices are unknown, each collector solves a robust optimization formulation where the prices paid by the processors are assumed to be within given ranges. The processors compete for the flow from the collectors until the Nash equilibrium is reached in this competitive tier, which sets the vector of prices to be offered to the collectors. To demonstrate the approach, a numerical example is given for a prototypical recycling network.

Introduction

Maximizing the efficiency of recycled material flows is growing in urgency due to high demands in many raw material markets and the increasing concern for environmental impact of disposal. Supply chains are evolving from “open loop” unidirectional flows of materials, parts, and products from suppliers to end customers into more complex “closed loop”-linked forward and reverse arcs (Fleischmann et al., 2000; Guide and Harrison, 2003; Realff et al., 2004; de la Fuente et al., 2008). Forward production systems are being expanded to incorporate reverse production systems (RPS) that include sorting, demanufacturing and/or refurbished processes in reverse logistics systems.

Most of the research on RPS design views the system in a centralized way; the key assumption is that one planner has the requisite information about all the participating entities and seeks the optimal solution for the entire system (see Ammons et al., 2001; Shih, 2001; Barros et al., 1998; Assavapokee et al., 2008). Wang et al. (2004) remark upon the three major drawbacks of centralized supply chain optimization models: (1) By ignoring the independence of the supply chain members, the competitive behavior between entities may lower the system efficiency and hence a centralized model may not capture the appropriate bargaining mechanisms that can mitigate the competitive behavior; (2) The cost of information processing may be expensive and the central decision maker must gather all the information from every entity; and (3) The computation of solutions to centralized optimization models can be very challenging.

Many emerging RPS structures consist of several independent entities where individual entrepreneurs have their own profit functions and often are unwilling to reveal their own information to each other or the public. This type of system behavior is decentralized. Often the decision variables for each entity in a decentralized system are also influenced by other entities’ decisions, coupling prices between members of the same tier, and flows between supply chain tiers. In this paper, we focus on decentralized decision-making and protocol design for the RPS with two tiers. The two tiers represent the collectors, who interact directly with the source of recycled items, and the processors who purchase the items from the collectors and convert them into more fungible commodities that are sold to customers.

The concept of equilibrium has been widely applied in many fields: traffic network equilibrium (Sheffi, 1985) and economic models (Cournot, 1838; Bertrand, 1883; Stackelberg, 1934). The Cournot- (see Hobbs, 2001) and the Stackelberg-type (Savaskan et al., 2004) models are two commonly used equilibrium models in decentralized systems. However, in practice the Cournot-type model may incur information divulgence problems because it requires the collection of optimality conditions from different entities in order to establish the equilibrium solution. Conceptually, the solution procedure implies that entities need to pass the information of their optimality conditions to some invisible hand in the system, which requires the willingness to share information among participants with a centralized body in order to obtain the equilibrium solution. Furthermore, the Stackelberg-type model (a leader–follower problem) may have implicit solution problems in a multiple-entity case since the leader considers the follower's optimal response to its decision under the Stackelberg model framework. Technically, this means the leader substitutes the follower's optimal response function into its problem, and hence must have knowledge of it. This type of models is solved by the backward induction (Fudenberg and Tirole, 1991). Nevertheless, an implicit solution may be reached in a multiple-entity case due to the property of substitution for optimal responses. In addition, we doubt whether the leader will have knowledge of the follower's optimal response in real-world decentralized systems. Instead, to avoid the problems of information divulgence and implicit solutions, we develop an explicit decision-making mechanism for calculating the optimal (self-interest) acquisition prices and the independent optimal flow determination for recycled materials in a decentralized RPS.

While forward and reverse supply chains share many similarities, there are significant differences. For forward supply chain systems, the material flow volumes are usually assumed to be functions of all prices in the final market (Nicholson, 2002; Corbett and Karmarkar, 2001). Once the historical data of demand and prices are available, the quantity and price relationship can be predicted since retailers face a considerable number of customers and perfect market assumptions are not unreasonable. However, for the RPS, the number of entities in the network is relatively small compared with a forward supply chain network. The relationship of the quantity and price in certain parts of the supply chain cannot been derived due to the lack of data. Instead, we present a robust approach to determine the relationship between the material flow volume and price between the collection and processing tier of the supply chain.

The remainder of the paper is organized as follows. In Section 2, we give a brief literature review. In Section 3, we provide the formal definition of our two-tier problem: the upstream and downstream entities and their connection. In 4 The upstream model: price-flow contract, 5 The downstream model: the equilibrium price, we develop mathematical models for upstream and downstream entities to determine the price and flow decisions in a decentralized RPS. In Section 6, we apply the algorithm to a numerical example to determine the equilibrium product prices and resulting flows, and also provide a discussion of the model and results. Section 7 presents conclusions and also suggests directions for future research.

Section snippets

Literature review

The past decade has seen an enormous increase in research on reverse logistics management issues. Flapper, 1995, Flapper, 1996, Fleischmann et al. (2000), and Guide and Harrison (2003) give systematic overviews and challenges of the logistic aspects of reuse and recycling in closed loop supply chains. Much of the research in RPS tends to be product, or system, specific due to the various features and complexities needed to handle the different recycling and reuse scenarios. Research on

A two-tier RPS problem: upstream and downstream

A RPS to reuse or recycle end-of-life products is a network of transportation logistics and processing functions that collect, refurbish, and demanufacture. In general, several entities in different tiers compose a network of collection and processing steps, connected by a transportation logistics system. In this paper, for simplicity, we assume a basic RPS consisting of two tiers of multiple facilities, one collection and one processing, facing sources and demand markets. Material flow

The upstream model: price-flow contract

In this section, we present a robust optimization model for the independent upstream entity to determine the robust price-flow contract between upstream and downstream tiers. For simplicity, we refer to the price-flow contract as the flow function. We depict the upstream and downstream sites as nodes and the material flows as links in Fig. 1. Specifically, we consider m upstream sites who are involved in the collection of end-of-life products, which can then be acquired by n downstream sites. A

The downstream model: the equilibrium price

Downstream sites are involved in transactions with upstream sites and customers in final demand markets since they wish to obtain recycled items from upstream tier and sell the materials or sub-components after refurbished/demanufacturing processes. Downstream sites make decisions on their own acquisition prices subject to their constraints of processing capacities, transportation capacities, and technology restrictions. We develop an equilibrium model of competitive downstream sites to

A numerical example

This example, depicted in Fig. 3, illustrates the application of the above upstream and downstream models. There are three collection sites, i=1, 2, and 3 in the upstream tier and three processing sites, j=1, 2, and 3 in the downstream tier. The collection sites collect end-of-life products from sources and ship them to processing sites. The transportation costs per unit flow between collection and processing sites are given in Table 1.

The final market prices for processing sites, j=1, 2, and 3

Conclusions and extensions

This paper presents a decentralized perspective for reverse production systems where each independent entity considers its own objective function and is subject to its own constraints. Meanwhile, the objective function of each entity not only depends on its own decision variables but also depends on decision variables of other entities. In this paper, we focus on a two-tier reverse production system involving the price and material flow decisions where the price-flow contract is determined by

Acknowledgments

We appreciate the efforts and expertise of the previous anonymous reviewers and are thankful for his/her careful comments and thoughtful reading of the paper. We have found them greatly valuable to improving the current manuscript. This research has been partially supported by the National Science Foundation under Grants DMI#0200162 and SBE-0123532. The authors are grateful for the generous interaction and guidance provided from many industry experts, including Julian Powell of Zentech, Carolyn

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