A Multiparametric Mixed-integer Bi-level Optimization Strategy for Supply Chain Planning Under Demand Uncertainty
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2017 Supply chain planning problems with seasonal demand variability can be often expressed within a hierarchical structure, where optimal decisions at an aggregate upper level provide constraints for the decision making at a lower level. In this work, we are addressing the case of a hierarchical distribution-production planning problem, where each decision level is controlled by a different company, each trying to optimize its own objective. This type of problems can be posed as bi-level programming problems, and since discrete decisions are involved the resulting formulations typically correspond to bi-level mixed-integer linear programming problems (B-MILP). The solution of these problems is very challenging, and typically requires the use of global optimization techniques, even for the derivation of approximate solutions. To overcome this, we propose the use of a novel algorithm capable of providing the exact, global and parametric solution of bi-level programming problems for the solution of distribution-production planning problems under demand uncertainty. The main idea of our approach is to treat the lower production planning level as a multi-parametric programming problem in which the distribution center demand (optimization variable of the upper level distribution planning problem) is considered as a parameter. The resulting exact parametric solutions are then substituted into the upper level distribution planning problem, which can be solved as a set of single-level deterministic programming problems. Through the use of this algorithm, we are able to derive the exact solution of a distribution-production planning problem with or without demand uncertainty.