Multi-Criteria Sorting with Category Size Restrictions

We consider the multi-criteria sorting problem where alternatives that are evaluated on multiple criteria are assigned into ordered categories. We focus on the sorting problem with category size restrictions, where the decision maker (DM) may have some concerns or constraints on the number of alternatives that should be assigned to some of the categories. We develop an approach based on the UTADIS method that fits an additive utility function to represent the decision maker's preferences. We introduce additional variables and constraints to enforce the restrictions on the sizes of categories. The new formulation reduces the number of binary variables and hence decreases the computational effort compared to the existing approaches in the literature. We further improve the computational efficiency by developing lower and upper bounds on the rank of each alternative in order to narrow down the set of categories that each alternative can be assigned to. We demonstrate our approach on two applications from practice.