A naïve approach to achieve network-wide coordination is to aggregate all stakeholders’ scheduling models and solve for the global optimum. However, this is impractical because it requires sharing proprietary process details and may impose higher costs on some stakeholders compared to their status quo, thus discouraging participation. Therefore, the network-wide optimization currently faces two primary challenges, the fair distribution of total realized cost reduction and privacy concerns arising from model sharing. Researchers have addressed fairness issues in such multi-stakeholder coordinating systems by modeling them as coalitional cost games and using game-theoretic solution concepts to allocate a cost to each stakeholder (Sampat & Zavala, 2019). Widely employed solution concepts include social welfare, Rawlsian welfare, and Nash bargaining (Allman & Zhang, 2022; Marousi & Charitopoulos, 2023). Nonetheless, previous studies lack quantitative evaluation of scenarios where stakeholders may withdraw from the grand coalition (network-wide) and instead form smaller sub-coalitions driven by potentially higher individual cost savings. The formation of the grand coalition is ensured when stakeholders’ cost allocations lie in the core of the game (Maschler et al., 2013). However, currently employed solution concepts often overlook important properties such as convexity of the cost game, which is a sufficient condition for a non-empty core (Maschler et al., 2013).
The second challenge, related to model sharing, can be addressed through distributed optimization, which requires only limited data sharing and has been shown to achieve global optimality across a broad range of optimization problems (Boyd et al., 2010). Distributed optimization algorithms such as Lagrangian decomposition and the alternating direction method of multipliers (ADMM), enables stakeholders to share information limited to product shipping quantities and multipliers in the dual space. Indeed, recent studies done by Allman and Zhang (Allman & Zhang, 2020, 2022) consider ADMM for network-wide optimization; however, their work focused on small case studies, lacking conclusive insights for benchmarking industrial-scale potential and scalability. Data sharing in the network can be facilitated through an Independent Central Coordinator (ICC) or a Peer-to-Peer (P2P) scheme. Although stakeholders share the same type of information in both approaches, the latter may be preferred because participants agree only to share revenue with players with whom they already have links. Yet, even limited data sharing can inadvertently reveal sensitive process metrics, such as efficiencies and capacities. Protecting data sharing from such risks can be achieved by introducing meaningful perturbations into the process scheduling models, leading us to characterize the optimization framework under the paradigm of Differential Privacy—a concept not yet explored for chemical engineering applications (Dwork & Roth, 2014). Currently, there is no open-source platform for stakeholders to collectively test these strategies and benchmark the potential of co-optimization in their supply chain networks.
In this work, we propose a general open-source architecture to employ differential-privacy-assisted decomposition algorithms for co-optimizing industrial-scale Mixed-Integer Linear Programming (MILP) based scheduling models. Furthermore, we apply solution concepts such as the nucleolus, which ensures that the cost allocation lies within the core whenever it is non-empty, and provide a comparison with other cooperative cost-sharing solution concepts, including the Shapley value, Nash bargaining, and cost gap allocation methods (Driessen & Tijs, 1985; Maschler et al., 2013). We consider a hypothetical case study of co-optimizing three Air Separation Units and demonstrate computational convergence to the global optimum under both data sharing scenarios: ICC-enabled and P2P.
Allman, A., & Zhang, Q. (2020). Distributed cooperative industrial demand response. Journal of Process Control, 86, 81–93. https://doi.org/10.1016/J.JPROCONT.2019.12.011
Allman, A., & Zhang, Q. (2022). Distributed fairness-guided optimization for coordinated demand response in multi-stakeholder process networks. Computers & Chemical Engineering, 161, 107777. https://doi.org/10.1016/J.COMPCHEMENG.2022.107777
Boyd, S., Parikh, N., Chu, E., Eckstein, J., Boyd, S., Parikh, N., Chu, E., Peleato, B., & Eckstein, J. (2010). Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers. Foundations and Trends R in Machine Learning, 3(1), 1–122. https://doi.org/10.1561/2200000016
Driessen, T. S. H., & Tijs, S. H. (1985). The Cost Gap Method and Other Cost Allocation Methods For Multipurpose Water Projects. Water Resources Research, 21(10), 1469–1475. https://doi.org/10.1029/WR021I010P01469
Dwork, C., & Roth, A. (2014). The Algorithmic Foundations of Differential Privacy. Foundations and Trends® in Theoretical Computer Science, 9(3–4), 211–407. https://doi.org/10.1561/0400000042
Kyritsis, E., Andersson, J., & Serletis, A. (2017). Electricity prices, large-scale renewable integration, and policy implications. Energy Policy, 101, 550–560. https://doi.org/10.1016/J.ENPOL.2016.11.014
Marousi, A., & Charitopoulos, V. M. (2023). Game theoretic optimisation in process and energy systems engineering: A review. Frontiers in Chemical Engineering, 5, 1130568. https://doi.org/10.3389/FCENG.2023.1130568/REFERENCE
Maschler, M., Solan, E., & Zamir, S. (2013). Game Theory. https://doi.org/10.1017/CBO9780511794216
Sampat, A. M., & Zavala, V. M. (2019). Fairness measures for decision-making and conflict resolution. Optimization and Engineering, 20(4), 1249–1272. https://doi.org/10.1007/S11081-019-09452-3/FIGURES/3
Zhang, Q., Grossmann, I. E., Heuberger, C. F., Sundaramoorthy, A., & Pinto, J. M. (2015). Air separation with cryogenic energy storage: Optimal scheduling considering electric energy and reserve markets. AIChE Journal, 61(5), 1547–1558. https://doi.org/10.1002/AIC.14730