A distributionally robust production planning model for maximizing customer satisfaction with budget and carbon emissions constraints

This paper investigates a new stochastic green production planning problem without full knowledge about the probability distribution on customer demands. The only information regarding the demand is the means and covariance matrix. The goal of the study is to provide an eco-conscious manufacturer with solution methods capable of yielding, within a reasonable amount of computation time, a robust solution that maximizes customer satisfaction while taking into account the constraints of budgets and periodic carbon caps spontaneously set out of environmental awareness. We formulate the problem into a novel distributionally robust chance-constrained optimization model under demand uncertainty set. To effectively solve the model, two heuristics are proposed. The widely used sample average approximation (SAA) scheme is first deployed as a benchmark method. However, the computation time increases rapidly with the problem size. In some instances, even feasible solutions cannot be found within a reasonable amount of computation time. We then develop an approximated mixed-integer reformulation on the basis of second order cone program (SOCP). A real-world case is solved and extensive numerical experiments show that the SOCP heuristic is superior over the SAA heuristic in terms of computation time while the solution quality is only slightly lower in most of instances. Finally, sensitivity analysis on budgets, periodic carbon caps, the maximum budget overrun risk level, initial inventory levels, and the magnitude of demand uncertainty are conducted and useful managerial insights are derived.