Efficient optimization in stochastic production planning problems with product substitution

We consider the stochastic production planning problem with product substitution, which can be decomposed into several optimization subproblems with sequential decisions. The decision variables in each time period include (1) the product substitution decision and (2) the recipe input quantity decision. The goal is to minimize the total of production cost, holding cost, and shortage cost, while achieving a service level for demand satisfaction. Since this optimization model involves analytically intractable probabilistic formulation, traditional mathematical programming techniques cannot be readily applied. We develop the deterministic SPLINE and the stochastic R-SPLINE algorithms for different scenarios. The probability generating function is embedded into the deterministic algorithm to exactly calculate the desired performance measures, which is reasonable when dealing with independent data (with a small number of product classes as well). The stochastic R-SPLINE algorithm uses simulation to estimate the desired probabilistic measures, allowing correlations between different production recipes as well as between different demand classes. We also present a convergence analysis for the stochastic R-SPLINE algorithm. Experimental results demonstrate the efficiency of the developed algorithms compared to other existing approaches.