A Provably Convergent Heuristic for Bicriteria Stochastic Integer Programming

The area of combinatorial optimization has been enlarged in the recent years into two directions: First, a huge number of articles deals with multiobjective combinatorial optimization (MOCO) problems, for which techniques to determine the set of Pareto-optimal solutions have been developed (cf., e.g., [6], [3]). A major advantage of MOCO approaches is that they are able to provide the decision maker with a small set of reasonable decision alternatives, leaving to her/him the final choice on a management level, but dispensing her/him from the charge of considering, evaluating and comparing thousands or millions of possible decision options from which only a very small fraction finally turns out as promising. In this way, the MOCO decision analysis paradigm can contribute in a substantial way to the development of modern decision support systems. Especially the solution of MO problems by metaheuristic techniques has recently attracted the attention of many researchers [3]. Secondly, stochastic combinatorial optimization (SCO) problems, representing uncertainty on problem parameters by means of a stochastic model, have led to the development of a further class of techniques which apply, before or during optimization, numerical procedures or simulation for obtaining characteristic values of distributions from the underlying stochastic model (cf., e.g., [2]). Also for SCO, metaheuristic approaches find increasing interest. In the metaheuristic community, SCO is often termed noisy combinatorial optimization; a recent survey can be found in [5]. The large body of literature in both the MOCO and the SCO field could lead to the conjecture that there might also be a considerable intersection domain between both fields, i.e., articles combining multiobjective and stochastic features in combinatorial optimization. Indeed, from the perspective of various applications (e.g., in vehicle routing, project scheduling, software engineering, or health care management), it would be very desirable to be able to cope with problems that are both of a multicriteria type and incorporate uncertainty. Surprisingly, however, literature in this intersection, which we call multiobjective stochastic combinatorial optimization (MOSCO), is scarce, as has also been noted in [2]. In [4], two general-purpose metaheuristic solution algorithms SP-ACO and SP-SA determining approximations to the set of Pareto-optimal solutions for instances from a large class of MOSCO problems are presented. Some experimental results are given, but the proposed algorithms do not yet come with a convergence guarantee.