Branch-and-Bound for Bi-objective Integer Programming

Abstract

In bi-objective integer optimization the optimal result corresponds to a set of non-dominated solutions. We propose a generic bi-objective branch-and-bound algorithm that uses a problem-independent branching rule exploiting available integer solutions and takes advantage of integer objective coefficients. The developed algorithm is applied to bi-objective facility location problems, to the bi-objective set covering problem, as well as to the bi-objective team orienteering problem with time windows. In the latter case, lower bound sets are computed by means of column generation. Comparison to state-of-the-art exact algorithms shows the effectiveness of the proposed branch-and-bound algorithm.

A hybrid approach for biobjective optimization

“…In particular, a number of methods have been designed for multiobjective integer problems, where the integers are required to be binary, see for example [17,22,23,24]. Recently, approaches has been suggested for biobjective mixed integer programs [14] where the integers are required to be binary, for general biobjective mixed integer programs [13], and for general biobjective integer programs [9]. In particular, the method presented in [13] makes use of advanced fathoming rules, and the method presented in [9] develop a problem-independent branching rule.…”

“…Recently, approaches has been suggested for biobjective mixed integer programs [14] where the integers are required to be binary, for general biobjective mixed integer programs [13], and for general biobjective integer programs [9]. In particular, the method presented in [13] makes use of advanced fathoming rules, and the method presented in [9] develop a problem-independent branching rule. Interestingly, the method presented in [9] makes use of a combination of DSS and CSS, just like the method presented in [10].…”

“…In particular, the method presented in [13] makes use of advanced fathoming rules, and the method presented in [9] develop a problem-independent branching rule. Interestingly, the method presented in [9] makes use of a combination of DSS and CSS, just like the method presented in [10].…”

“…The above method is illustrated in Figure 9. Assume that we know a subset of the feasible set of points in criterion space consisting of 8 points: z U L = (2, 16), z A = (4, 14), z B = (5, 10), z C = (6,9), z D = (9, 8), z E = (11, 7.2), z F = (16, 5.5), z LR = (20,4). We want to divide the rectangle spanned by z U L and z LR into 3 slices.…”

“…While there has been research going on in multiobjective optimization for decades, an increased interest in exact solution of multiobjective optimization problems has flourished in the last decade (i.e. since 2006): [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. This research has however not yet lead to general practical solvers of multiobjective Mixed Integer Programming (MOMIP) models.…”