Discrete location theory

We present a linear programming based branch-and-bound algorithm for a class of mixed integer optimization problems with a bi-linear objective function and linear constraints. This class of optimization problems can be viewed as a special case of the problem of optimization over the set of efficient solutions in bi-objective optimization. It is known that when there exists no integer decision variable, such a problem can be solved in polynomial time. In fact, in such a case, the problem can be transformed into a Second-Order Cone Program (SOCP) and so it can be solved efficiently by a commercial solver such as CPLEX SOCP solver. However, in a recent study, it is shown that such a problem can be solved even faster in practice by using a bi-objective linear programming based algorithm. So, in this study, we embed that algorithm in an effective branch-and-bound framework to solve mixed integer instances. We also develop several enhancement techniques including preprocessing and cuts. A computational study demonstrates that the proposed branch-and-bound algorithm outperforms a commercial mixed integer SOCP solver. Moreover, the effect of different branching and node selecting strategies is explored.