An interactive branch-and-bound algorithm for bicriterion nonconvex/mixed integer programming

Although there has been extensive research on interactive multiple objective decision making in the last two decades, there is still a need for specialized interactive algorithms that exploit the relatively simple structure of bicriterion programming problems. This article develops an interactive branch-and-bound algorithm for bicriterion nonconvex programming problems. The algorithm searches among only the set of nondominated solutions since one of them is a most preferred solution that maximizes the overall value function of the decision maker over the set of achievable solutions. The interactive branch-and-bound algorithm requires only pairwise preference comparisons from the decision maker. Based on the decision maker’s responses, the algorithm reduces the set of nondominated solutions and terminates with his most preferred nondominated solution. Branching corresponds to dividing the subset of nondominated solutions considered at a node into two subsets. The incumbent solution is updated based on the preference of the decision maker between two nondominated solutions. Fathoming decisions are based on the decision maker’s preference between the incumbent solution and the ideal solution of the node in consideration.