Global optimization in multiproduct and multipurpose batch design under uncertainty

This paper addresses the design of multiproduct and multipurpose batch plants with uncertainty in both product demands and processing parameters. The uncertain demands may be described by any continuous/discrete probability distribution. Uncertain processing parameters are handled in a scenario-based approach. Through the relaxation of the feasibility requirement, the design problem with a fixed number of pieces of equipment per stage is formulated as a single large-scale nonconvex optimization problem. This problem is solved using a branch and bound technique in which a convex relaxation of the original nonconvex problem is solved to provide a lower bound on the global solution. Several different expressions for the tight convex lower bounding functions are proposed. Using these expressions, a tight lower bound on the global optimum solution can be obtained at each iteration. The alpha BB algorithm is subsequently employed to refine the upper and lower bounds and converge to the global solution. The tight lower bounds and the efficiency of the proposed approach is demonstrated in several example problems. These case studies correspond to large-scale global optimization problems with nonconvex constraints ranging in number from 25 to 3750, variables ranging from 30 to 15636 and nonconvex terms ranging from 50 to 15000. It is shown that such large-scale multiproduct and multipurpose batch design problems can be solved to global optimality with reasonable computational effort.