Elsevier

Computers & Chemical Engineering

Volume 71, 4 December 2014, Pages 347-361
Computers & Chemical Engineering

Game-theoretic modeling and optimization of multi-echelon supply chain design and operation under Stackelberg game and market equilibrium

https://doi.org/10.1016/j.compchemeng.2014.08.010Get rights and content

Highlights

  • Bilevel MINLP model for design and planning of non-cooperative supply chains.

  • Stackelberg game and generalized Nash equilibrium in multi-echelon supply chains.

  • Global optimization by KKT transformation and improved branch-and-refine algorithm.

  • County-level case study on a potential biofuel supply chain in Illinois.

Abstract

We propose a bilevel mixed-integer nonlinear programming (MINLP) model for the optimal design and planning of non-cooperative supply chains from the manufacturer's perspective. Interactions among the supply chain participants are captured through a single-leader–multiple-follower Stackelberg game under the generalized Nash equilibrium assumption. Given a three-echelon superstructure, the lead manufacturer in the middle echelon first optimizes its design and operational decisions, including facility location, sizing, and technology selection, material input/output and price setting. The following suppliers and customers in the upstream and downstream then optimize their transactions with the manufacturer to maximize their individual profits. By replacing the lower level linear programs with their KKT conditions, we transform the bilevel MINLP into a single-level nonconvex MINLP, which is further globally optimized using an improved branch-and-refine algorithm. We also present two case studies, including a county-level biofuel supply chain in Illinois, to illustrate the application of the proposed modeling and solution methods.

Introduction

When entering a business, a manufacturer is often encountered with questions such as where to locate the plants, what sizes the plants should be, which conversion technology to choose, how much to produce, and how to set the transfer prices (Grossmann, 2005). Although there is a large body of literature on the modeling and optimization of supply chain design and operations, most of these works view the supply chain from a centralized perspective and integrate the various components of the supply chain into a monolithic model (Muñoz et al., 2013, Papageorgiou, 2009, Shah, 2005). Under this approach, it was implicitly assumed that the decision maker has full control over the entire supply chain so that all the strategic and operational decisions can be implemented successfully. However, the management over a supply chain is often decentralized in practice (Cachon and Netessine, 2004). In other words, different stakeholders may be in charge of different entities in the supply chain, and these stakeholders may even have conflicting interests against each other. These supply chain participants would strive to maximize their own benefits and compete with their peers, thus leading to a non-cooperative supply chain (Facchinei and Kanzow, 2007).

The goal of this work is to develop a novel game-theoretic modeling and optimization framework that addresses non-cooperative supply chain design and planning from a manufacturer's perspective. In a non-cooperative supply chain, all of the participants act selfishly and are solely driven by their own objective. The players make decisions independently without collaboration or communication (Nash, 1951). This is different from a cooperative supply chain, where the players are willing to negotiate with each other and arrive at a unanimous agreement (Nagarajan and Sošić, 2008). Recent works on cooperative supply chains include the works by Gjerdrum et al., 2001, Gjerdrum et al., 2002, Zhang et al. (2013), Fernandes et al. (2013), Banaszewski et al. (2013), and Yue and You (2014). On the other hand, representative works on non-cooperative supply chains include the works by Bard et al. (2000), Ryu et al. (2004), and Zamarripa et al., 2012, Zamarripa et al., 2013. To the best of our knowledge, most existing works on the optimal design and planning of non-cooperative supply chains are restricted to a rather simple supply chain structure, instead of the multi-echelon network considered in this work. Furthermore, in previous works, linearization assumptions and simplifications have been applied in order to keep the model tractable, which might lose the generality of the mathematical model. Therefore, this work aims to fill these research gaps by both proposing a novel single-leader–multiple-follower game-theoretic framework for general three-echelon supply chain networks and developing an effective global optimization strategy to address the resulting mathematical model.

Specifically, we consider a three-echelon supply chain superstructure, which includes a set of candidate sites for building manufacturing facilities in the middle echelon, as well as a set of given upstream suppliers and downstream customers. A single player – a manufacturer – is in charge of all the manufacturing facilities, while each supplier and customer is considered to be an independent player. The manufacturer is assumed to be the supply chain leader, who makes the decisions on supply chain design and strategic planning first. The suppliers and customers are assumed to be the followers who then maximize their own profits given the leader's decisions. We model the leader–follower relationship as a single-leader–multiple-follower Stackelberg game. We model the competition among suppliers and customers under the assumption of generalized Nash equilibrium.

Without loss of generality, we formulate the problem above into a bilevel mixed-integer nonlinear program (MINLP). In the upper level problem, the manufacturer optimally determines the location, capacity, and technology of the manufacturing facilities, as well as the operational plans and transfer prices in order to maximize the total profit generated from all the manufacturing facilities. Discrete variables are employed to select among the candidate sites, conversion technologies, etc. Capital cost economies of scale are captured by using a nonlinear power function. To calculate the transfer payments of materials, bilinear terms are also included. Therefore, the upper level problem is nonlinear, nonconvex, and has combinatorial features. Given the manufacturer's decisions in the upper level problem, each follower in the lower level problem optimizes its transactions with the installed manufacturing facilities to maximize its own profit. Specifically, the suppliers optimize the amount of raw materials to be sold to the manufacturing facilities and the customers optimize the amount of products to purchase from the manufacturing facilities. All of the lower level problems are formulated as linear programs (LPs).

We note that the resulting bilevel program cannot be handled directly using off-the-shelf optimization solvers. However, for the cases that all the lower level problems are LPs, we can reformulate the bilevel MINLP problem into an equivalent single-level MINLP by replacing each follower's optimization problem with the corresponding Karush–Kuhn–Tucker (KKT) conditions (Bard, 1998). Though solvable, the resulting single-level MINLP problem can still be computationally intractable due to the presence of concave and bilinear terms as well as integer variables. To facilitate the solution process, we further propose an improved branch-and-refine algorithm which is based on a class of SOS1 (specially ordered set of type 1) piecewise linear formulations. The algorithm takes advantage of powerful mixed-integer linear programming (MILP) solvers and globally optimizes the nonconvex MINLP problem efficiently in finite iterations. To illustrate the application of the proposed modeling and optimization framework, we also present two case studies, including a county-level case study on a potential biofuel supply chain in the state of Illinois.

Major novelties of this work are summarized as follows:

  • A novel bilevel MINLP model is proposed for the design and strategic planning of non-cooperative supply chains;

  • An extension of Stackelberg game and generalized Nash equilibrium is made to multi-echelon supply chains;

  • An efficient global optimization strategy is developed for the bilevel MINLP using the KKT transformation and the improved branch-and-refine algorithm;

  • A county-level case study on a potential cellulosic bioethanol supply chain in Illinois is presented.

The rest of this paper is organized as follows. We first provide a brief introduction to the concepts of Stackelberg game and generalized Nash equilibrium in Section 2. We then present a general problem statement in Section 3 and a generic model formulation in Section 4. The application on cellulosic biofuel supply chains is covered by Sections 5 Specific problem statement, 6 Specific model formulation, followed by the solution strategies presented in Section 7. Two case studies are given in Section 8, with discussions on the results and implications.

Section snippets

Stackelberg game

The first Stackelberg game was described by the German economist Heinrich Freiherr von Stackelberg in 1934, who studied the competition between two firms selling a homogeneous good (Von Stackelberg et al., 2010). The concept of Stackelberg game was then extended to a variety of research fields and applications to study the situation where a leader–follower relationship is observed (Chu and You, 2014, Chu et al., 2014). A standard Stackelberg game involves two players: a leader and a follower.

General problem statement

We address the optimal design and planning of non-cooperative supply chains from a manufacturer's perspective. Fig. 1 shows a three-echelon supply chain superstructure. The first echelon consists of a set of individual suppliers. The second echelon includes a set of candidate sites for building the manufacturing facilities. The third echelon consists of a set of individual customers. One decision maker, the manufacturer, is in charge of all of the manufacturing facilities belonging to one

General model formulation

According to the problem statement above, this is a Stackelberg game involving a single leader and multiple followers. Assuming that the competition among the followers leads to the generalized Nash equilibria, we can integrate the mathematical model for the standard Stackelberg game (1), (2), (3), (4), (5) and the generalized Nash equilibrium model (7), (8), thus leading to the following game-theoretic model formulation.maxxXF(x,y)s.t.Gi(x,y)0,i=1,,mHj(x,y)=0,j=1,,rwhereysolves

Specific problem statement

We now apply the proposed game-theoretic model to study the optimal design and planning of biomass-to-ethanol supply chains. Cellulosic-biomass-derived fuel ethanol provides a promising solution to the increasing concerns on climate change, energy security, and the diminishing supply of fossil fuels. As a number of pilot-scale biorefineries have been built and have been in operation for years all over the world, there are mature technologies available in the fuel ethanol industry. Large-scale

Specific model formulation

The problem stated above can be regarded as a Stackelberg game with the biorefinery investor as the single leader and the multiple biomass suppliers and biofuel customers as the followers, under the assumption of normalized Nash equilibrium. Therefore, a bilevel MINLP problem (P0) is proposed, with the upper level problem corresponding to the biorefinery investor's MINLP problem, and the lower level problems corresponding to the suppliers’ and customers’ LP problems. Based on the KKT conditions

Solution strategy

Although global optimizers for nonconvex MINLP problems are available off-the-shelf, their computational performance on large-scale applications is somewhat lacking. To facilitate the solution of the nonconvex MINLP problem (PS), we present an improved branch-and-refine algorithm in this section. The algorithm takes advantage of the powerful MILP solvers (e.g., CPLEX) and returns the global optimal solution to the nonconvex MINLP problem by iteratively solving a sequence of MILP subproblems.

Case studies

To illustrate the application of the proposed modeling framework and global optimization algorithm, we present two case studies on the optimal design and planning of biofuel supply chains. The small-scale illustrative example compares the differences between the centralized and decentralized design, and shows the computational efficiency of the proposed algorithm. The county-level case study on a potential biofuel supply chain in Illinois demonstrates the capability of this game-theoretic

Conclusion

A novel game-theoretic modeling and optimization framework was proposed for the design and strategic planning of non-cooperative three-echelon supply chains. As the leader, the manufacturer determined the location, capacity, and technology of manufacturing facilities, strategic plans on material input/output, as well as the transfer prices. As the follower, each supplier or customer maximized its own profit by optimizing the material transactions with the installed manufacturing facilities. We

Acknowledgement

We gratefully acknowledge the financial support from the Institute for Sustainability and Energy at Northwestern University (ISEN).

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