(52b) A Stochastic Game Theoretic Framework for Optimization of Decentralized Supply Chains Under Uncertainty

The management of supply chains is a complex decision-making process involving multiple stakeholders, each of which only controls part of the supply chain, and may pursue different objectives, thus leading to compromised solutions [1]. Nevertheless, most existing studies on optimal design and operations of supply chains rely on centralized optimization models [2-5]. Consequently, the optimal solutions of centralized models can be suboptimal or even infeasible in a decentralized, multi-stakeholder supply chain. To address this research challenge, multiple game theoretic models are developed to explicitly account for the performance of multi-stakeholder systems. Examples include optimization models for cooperative multi-enterprise supply chains based on the generalized Nash bargaining solution approach [6-9]. On the other hand, optimization models integrating Stackelberg game and Nash-equilibrium are proposed for noncooperative supply chain optimization [10-14]. However, these models assume perfect information sharing among different stakeholders. In other words, the resulting optimal decisions are based on deterministic information. In practice, various types of uncertainties, such as price and productivity fluctuation, exist concerning the decision-making processes of stakeholders. These uncertainties may significantly influence the rational behaviors of stakeholders. Therefore, it remains a research challenge to simultaneously consider decentralized features of multi-stakeholder supply chains and incorporate uncertainty in the noncooperative stakeholders’ optimal decision-making process for supply chain design and operations. To fill this knowledge gap, it is necessary to develop a holistic game theoretic model of multi-stakeholder decentralized supply chains that captures the influences of uncertainty on stakeholders’ optimal decisions in a systematical way.

In this work, we propose a novel modeling framework to investigate the influences of uncertainty in decentralized optimization of supply chains. This modeling framework integrates the Stackelberg game with stochastic programming approach into a holistic two-stage stochastic game theoretic model. Specifically, this modeling framework allows consideration of one leader and multiple followers. Following the sequence of decision making process, decision variables for both the leader and the followers are classified into design decisions that must be made “here-and-now” and operational decisions that are postponed to a “wait-and-see” mode after the realization of uncertainties. As a result, both types of stakeholders interact with each other to determine their optimal design strategies at the first stage. After uncertainties from both the leader and the followers are realized, all stakeholders then determine their operational strategies as “recourse” decisions of the uncertainty information based on their previous design decisions. Following the stochastic programming approach, uncertainties are depicted with discrete scenarios with known probabilities. The objectives of the leader and the followers are to maximize their own expected net present value (NPV). The resulting problem is formulated as a two-stage stochastic mixed-integer bilevel programming (MIBP) problem. The applicability of the proposed modeling framework is demonstrated with a large-scale application to Marcellus shale gas supply chains. By solving this optimization problem, we find that the expected NPV for the leader is $68.9 MM. The three followers corresponding to three processing plants are expected to achieve $2.57 MM, $3.39 MM, and $1.21 MM NPVs, respectively. To demonstrate the advantage of this two-stage stochastic MIBP model, we further compare the optimal results obtained in the proposed two-stage stochastic game theoretic model with those of deterministic game theoretic models with different expectation of stakeholders. Based on the optimization results, we concluded that stakeholders tended to choose more conservative strategies when considering uncertainties in the optimization of decentralized supply chains. Although the conservatism might affect the overall performance of stakeholders, it effectively hedged against the risk of extreme cases when stakeholders wrongly anticipated others’ performances.

References

[1] G. Cachon and S. Netessine, "Game Theory in Supply Chain Analysis," in Handbook of Quantitative Supply Chain Analysis. vol. 74, D. Simchi-Levi, S. D. Wu, and Z.-J. Shen, Eds., ed: Springer US, 2004, pp. 13-65.

[2] I. E. Grossmann, "Challenges in the new millennium: product discovery and design, enterprise and supply chain optimization, global life cycle assessment," Computers & Chemical Engineering, vol. 29, pp. 29-39, 2004.

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

[4] D. Yue, F. You, and S. W. Snyder, "Biomass-to-bioenergy and biofuel supply chain optimization: Overview, key issues and challenges," Computers & Chemical Engineering, vol. 66, pp. 36-56, 2014.

[5] D. Garcia and F. You, "Supply chain design and optimization: Challenges and opportunities," Computers & Chemical Engineering, vol. 81, pp. 153-170, 2015.

[6] 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.

[7] 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, 2002.

[8] D. Zhang, N. J. Samsatli, A. D. Hawkes, D. J. L. Brett, N. Shah, and L. G. Papageorgiou, "Fair electricity transfer price and unit capacity selection for microgrids," Energy Economics, vol. 36, pp. 581-593, 2013.

[9] D. Yue and F. You, "Fair profit allocation in supply chain optimization with transfer price and revenue sharing: MINLP model and algorithm for cellulosic biofuel supply chains," AIChE Journal, vol. 60, pp. 3211-3229, 2014.

[10] M. A. Zamarripa, A. M. Aguirre, C. A. Méndez, and A. Espuña, "Improving supply chain planning in a competitive environment," Computers & Chemical Engineering, vol. 42, pp. 178-188, 2012.

[11] D. Yue and F. You, "Game-theoretic modeling and optimization of multi-echelon supply chain design and operation under Stackelberg game and market equilibrium," Computers & Chemical Engineering, vol. 71, pp. 347-361, 2014.

[12] J. Gao and F. You, "Game theory approach to optimal design of shale gas supply chains with consideration of economics and life cycle greenhouse gas emissions," AIChE Journal, vol. 63, pp. 2671-2693, 2017.

[13] J. Gao and F. You, "Economic and Environmental Life Cycle Optimization of Noncooperative Supply Chains and Product Systems: Modeling Framework, Mixed-Integer Bilevel Fractional Programming Algorithm, and Shale Gas Application," ACS Sustainable Chemistry & Engineering, vol. 5, pp. 3362-3381, 2017.

[14] D. Yue and F. You, "Stackelberg-game-based modeling and optimization for supply chain design and operations: A mixed integer bilevel programming framework," Computers & Chemical Engineering, vol. 102, pp. 81-95, 2017.