(655g) Game-Theoretic Modeling and Optimization of Decentralized Supply Chains

Yue, D., Northwestern University
You, F., Northwestern University

When entering a business, the manufacturer is often encountered with questions such as where to locate the plants, how big the plants should be, which conversion technology to choose, how much to produce, how to set the prices, etc. [1]. Although there is a large body of literature on the modeling and optimization of supply chain design and operations in the process system engineering community and the operations research field, most of these works viewed the supply chain from a centralized perspective and integrated the various components of the supply chain into a monolithic model [2, 3]. 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 very often decentralized in practice [4, 5]. In other words, the different entities in a supply chain may be under the charge of different stakeholders, who may even have conflicting interests. These entities would strive to maximize their own benefits and compete with their peers, thus leading to a non-cooperative supply chain [6, 7].

The goal of this work is to develop a novel game-theoretic modeling and optimization framework to assist the manufacturers in optimal design and strategic planning in a three-echelon supply chain system. 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 distributors. The manufacturer is assumed as the leader in this problem, who will take actions first. The suppliers and distributors are assumed as the followers, who will then react to the leader’s decisions. We formulate a bilevel program to model the Stackelberg leader-follower relationship [8]. We model the competitions among the suppliers and distributors under the generalized Nash equilibrium assumption [9]. We also address the selection of the location, sizing, and technology of manufacturing facilities, strategic plans on material flows, as well as the determination of transfer prices. Since bilevel programs cannot be directly handled by the existing solvers, we reformulate the problem into an equivalent single-level mixed-integer nonlinear program (MINLP) by replacing the followers’ problems with the corresponding Karush-Kuhn-Tucker (KKT) conditions. However, the resulting MINLP involves bilinear and concave terms in the objective function, which makes the problem nonconvex. To facilitate the solution process, we propose an improved branch-and-refine algorithm which solves the nonconvex MINLP problem efficiently by solving a sequence of mixed-integer linear programming (MILP) problems by iteration [10, 11].

To illustrate the application of the proposed game-theoretic modeling and optimization framework, a case study on the design and strategic planning of a biofuel supply chain is presented [12-18]. The biorefinery investor tends to maximize his total profit by optimizing the location, capacity, and technology selection of the biorefineries, the strategic plan on harvesting and production, as well as the setting of transfer prices. Given the biorefinery investor’s and the competitors’ decisions, the individual supplier tends to maximize his own profit by optimizing the amount of biomass shipping to every installed biorefinery, and the individual distributor tends to maximize his own profit by optimizing the amount of biofuel to purchase from each of the installed biorefineries. Optimization results of the non-cooperative supply chain are obtained and comparison with the centralized design is presented. The proposed solution strategy is shown to be much more efficient than the existing general-purpose MINLP solvers.


[1]       I. Grossmann, "Enterprise-wide optimization: A new frontier in process systems engineering," AIChE Journal, vol. 51, pp. 1846-1857, 2005.

[2]        N. Shah, "Process industry supply chains: Advances and challenges," Computers & Chemical Engineering, vol. 29, pp. 1225-1236, 5/15/ 2005.

[3]        L. G. Papageorgiou, "Supply chain optimisation for the process industries: Advances and opportunities," Computers & Chemical Engineering, vol. 33, pp. 1931-1938, 12/10/ 2009.

[4]        J. Gjerdrum, N. Shah, and L. G. Papageorgiou, "Fair transfer price and inventory holding policies in two-enterprise supply chains," European Journal of Operational Research, vol. 143, pp. 582-599, 12/16/ 2002.

[5]        J. Gjerdrum, N. Shah, and L. G. Papageorgiou, "Transfer Prices for Multienterprise Supply Chain Optimization," Industrial & Engineering Chemistry Research, vol. 40, pp. 1650-1660, 2001/04/01 2001.

[6]        K. Yeh, J. H. Lee, C. Whittaker, and M. J. Realff, "Two Stage Bilevel Programming Approach for Representation of Biorefinery Investment Decision Making in a Pre-Established Timberlands Supply Chain," in 8th International Conference on Foundations of Computer-Aided Process Design, Cle Elum, Washington, USA, 2014.

[7]        Y. Bai, Y. Ouyang, and J.-S. Pang, "Biofuel supply chain design under competitive agricultural land use and feedstock market equilibrium," Energy Economics, vol. 34, pp. 1623-1633, 9// 2012.

[8]        J. F. Bard, Practical bilevel optimization: algorithms and applications vol. 30: Springer, 1998.

[9]        F. Facchinei and C. Kanzow, "Generalized Nash equilibrium problems," 4OR, vol. 5, pp. 173-210, 2007/09/01 2007.

[10]      D. Yue and F. You, "Planning and scheduling of flexible process networks under uncertainty with stochastic inventory: MINLP models and algorithm," AIChE Journal, vol. 59, pp. 1511-1532, 2013.

[11]      J. Gong and F. You, "Optimal Design and Synthesis of Algal Biorefinery Processes for Biological Carbon Sequestration and Utilization with Zero Direct Greenhouse Gas Emissions: MINLP Model and Global Optimization Algorithm," Industrial & Engineering Chemistry Research, vol. 53, pp. 1563-1579, 2014/01/29 2014.

[12]      D. Yue, F. You, and S. W. Snyder, "Biomass-to-bioenergy and biofuel supply chain optimization: Overview, key issues and challenges," Computers & Chemical Engineering, 2014.

[13]      F. You and B. Wang, "Life Cycle Optimization of Biomass-to-Liquid Supply Chains with Distributed–Centralized Processing Networks," Industrial & Engineering Chemistry Research, vol. 50, pp. 10102-10127, 2011/09/07 2011.

[14]      F. You, L. Tao, D. J. Graziano, and S. W. Snyder, "Optimal design of sustainable cellulosic biofuel supply chains: Multiobjective optimization coupled with life cycle assessment and input–output analysis," AIChE Journal, vol. 58, pp. 1157-1180, 2012.

[15]      K. Tong, J. Gong, D. Yue, and F. You, "Stochastic Programming Approach to Optimal Design and Operations of Integrated Hydrocarbon Biofuel and Petroleum Supply Chains," ACS Sustainable Chemistry & Engineering, vol. 2, pp. 49-61, 2014/01/06 2013.

[16]      K. Tong, M. J. Gleeson, G. Rong, and F. You, "Optimal design of advanced drop-in hydrocarbon biofuel supply chain integrating with existing petroleum refineries under uncertainty," Biomass and Bioenergy, vol. 60, pp. 108-120, 1// 2014.

[17]      D. Yue, M. A. Kim, and F. You, "Design of Sustainable Product Systems and Supply Chains with Life Cycle Optimization Based on Functional Unit: General Modeling Framework, Mixed-Integer Nonlinear Programming Algorithms and Case Study on Hydrocarbon Biofuels," ACS Sustainable Chemistry & Engineering, vol. 1, pp. 1003-1014, 2013/08/05 2013.

[18]     B. H. Gebreslassie, Y. Yao, and F. You, "Design under uncertainty of hydrocarbon biorefinery supply chains: Multiobjective stochastic programming models, decomposition algorithm, and a Comparison between CVaR and downside risk," AIChE Journal, vol. 58, pp. 2155-2179, 2012.