An exact parallel objective space decomposition algorithm for solving multi-objective integer programming problems

The set of all nondominated solutions for a multi-objective integer programming (MOIP) problem is finite if the feasible region is bounded, and it may contain unsupported solutions. Finding these sets is NP-hard for most MOIP problems and current methods are unable to scale with the number of objectives. We propose a deterministic exact parallel algorithm for solving MOIP problems with any number of objectives. The proposed algorithm generates the full set of nondominated solutions based on intelligent iterative decomposition of the objective space utilizing a particular scalarization scheme. The algorithm relies on a set of rules that exploits regional dominance relations among the decomposed partitions for pruning. These expediting rules are both used as part of a pre-solve step as well as judiciously employed throughout the parallel running threads. Using an extensive test-bed of MOIP instances with three, four, five, and six objectives, the performance of the proposed algorithm is evaluated and compared with leading benchmark algorithms for MOIPs. Results of the experimental study demonstrate the effectiveness of the proposed algorithm and the computational advantage of its parallelism.