On the integrated production and distribution problem with bidirectional flows

The integrated production and distribution problem with bidirectional flows is a complicated optimization problem, usually with large problem sizes when encountered in practice. In this study, we propose a partial linear programming relaxation-based heuristic approach to solve a variation of this problem. The approach is called a partial relaxation in the sense that it relaxes the integer requirements only on selected variables. We also report on the gaps between the optimal solution and the heuristic solution provided by this partial relaxation, including analytical gaps for a special case and empirical gaps for randomly generated test cases. Our study of this problem was motivated by the operational planning problem of a medical equipment leasing network that involves a forward flow for new and refurbished devices and a reverse flow for used devices to be returned to suppliers over a multiple time-period planning horizon.