332247 Robust Scenario Formulations for Strategic Supply Chain Optimization Under Uncertainty
Strategic supply chain optimization (SCO) problems are often modelled as a two-stage optimization problem, in which the first-stage variables represent decisions on the development of the supply chain and the second-stage variables represent decisions on the operations of the supply chain. When uncertainty is explicitly considered, the problem becomes an intractable infinite-dimensional optimization problem, which is usually solved approximately via a scenario [1] or a robust [2] approach. The scenario approach only addresses part of the uncertainty realizations, so it cannot guarantee the feasibility and optimality of the solution; the robust approach addresses the worst-case uncertainty realization, so its solution is feasible but often overly conservative.
This paper proposes a novel synergy of the scenario and robust approaches for stochastic mixed-integer linear programming (MILP) problems arising from strategic SCO under uncertainty. Two formulations are developed, namely, naïve robust scenario formulation and affinely adjustable robust scenario formulation. In these two formulations, a scenario represents a group of uncertainty realizations instead of a single uncertainty realization. It is shown that both formulations can be reformulated into tractable deterministic optimization problems if the uncertainty is bounded with the infinity-norm. In addition, the uncertain equality constraints can be reformulated into tractable deterministic constraints without assumption of the uncertainty region. Note that the two proposed formulations hold a decomposable structure similar to that of classical scenario formulation, so the resulting problems can be solved efficiently by classical Benders decomposition/L-shaped method [3] [4].
Case studies of a classical farm planning problem [1] and an energy and bioproduct SCO problem [5] demonstrate the advantages of the proposed formulations over the classical scenario formulation. The proposed formulations not only can generate solutions with guaranteed feasibility or indicate infeasibility of a problem, but also can achieve optimal expected economic performance with smaller numbers of scenarios. In addition, affinely adjustable robust scenario formulation outperforms naive robust scenario formulation by consistently providing good prediction of the expected profit to be achieved, although this formulation generally leads to larger optimization problems and requires longer solution times.
References
[1] Birge, J. R.; Louveaux, F. Introduction to stochastic programming, Second Edition; Springer: New York, 2011.
[2] Ben-Tal, A.; Ghaoui, L. E. Robust Optimization; Princeton University Press: New Jersey, 2009.
[3] Benders, J. F. Partitioning procedures for solving mixed-variables programming problems. Numer. Math. 1962, 4, 238-252.
[4] Van Slyke R. M.; Wets R. L-shaped linear programs with applications to optimal control and stochastic programming. SIAM J. Appl. Math. 1969, 17 (4), 638-633.
[5] Čuček, L.; Lam, H.; Klemes, J.; Varbanov, P.; Kravanja, Z. Synthesis of regional networks for the supply of energy and bioproducts. Clean Tech. and Environ. Pol. 2010, 12, 635-645.
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