Optimal blending under general uncertainties: A chance-constrained programming approach

Abstract

An optimization algorithm is proposed for blend planning with linear mixing law and general parameter uncertainties. The objective is to make the final product satisfy all quality specifications with high probability, and maximize the production profit by carefully determining the feedstock ratio. Conventional approaches that rely on deterministic optimization fail to account for parameter uncertainty, and thus may not generate a probabilistic feasible solution. The proposed work formulates the blend planning problem as a joint chance-constrained program (CCP). Using Boole’s inequality to decompose joint constraints and the Gaussian mixture model to characterize uncertainty distributions, a conservative deterministic approximation of CCP can be formulated. Through second-order cone relaxation, branch-and-bound, optimality-based bound tightening, and reformulate-linearization techniques, the global optimum of deterministic approximation can be found. A risk level adjustment procedure is presented to reduce the conservativeness and further improve the objective value of the solution if posterior evaluation is allowed. Two numerical cases, including steel and gasoline productions, are studied to show the solving time, probabilistic feasibility, and solution quality of the proposed optimization method.