Mean–range based distribution-free procedures to minimize ‘overage’ and ‘underage’ costs

We discuss several mean-range based distribution-free decision procedures to minimize several ‘overage’ and ‘underage’ cost functions. For a general cost function, we identify the most favorable distribution and the least favorable distribution associated with the random variable of interest and determine the upper and lower bounds for the cost function. For the quadratic cost function, we recommend the min–max distribution-free decision. For the linear cost function, we identify the range of potential optimal decisions and recommend a hybrid distribution-free decision that has several desirable properties. We provide several numerical examples to demonstrate the robustness of the proposed distribution-free decisions.