A Quasi-extreme Reduction for Interval Transportation Problems

Transportation problems provide a classic linear programming model used in many areas of operations research, such as inventory control, logistics or supply chain management. The goal of a transportation problem is to find a minimum-cost transportation plan for shipping a given commodity from a set of sources to a set of destinations. Since the input data of such models are not always known exactly in practice, we adopt the approach of interval programming, which handles uncertainty in the supply, demand and cost parameters by assuming that only lower and upper bounds on these quantities are given. One of the main tasks in interval programming is to compute bounds on the values that are optimal for some realization of the interval coefficients. While the best optimal value of an interval transportation problem can be computed by a single linear program, finding the worst (finite) optimal value is a much more challenging task. For interval transportation problems that are immune against the “more-for-less” paradox, it was recently proved that the worst optimal value can be found by considering only quasi-extreme scenarios, in which all coefficients in the model but one are set to the lower or upper bounds. We strengthen the former result and show that an analogous property also holds true for general interval transportation problems. Then, we utilize the obtained characterization to derive an exact method for computing the worst optimal value.