A cut-and-solve based algorithm for the single-source capacitated facility location problem

In this paper, we present a cut-and-solve (CS) based exact algorithm for the Single Source Capacitated Facility Location Problem (SSCFLP). At each level of CS’s branching tree, it has only two nodes, corresponding to the Sparse Problem (SP) and the Dense Problem (DP), respectively. The SP, whose solution space is relatively small with the values of some variables fixed to zero, is solved to optimality by using a commercial MIP solver and its solution if it exists provides an upper bound to the SSCFLP. Meanwhile, the resolution of the LP of DP provides a lower bound for the SSCFLP. A cutting plane method which combines the lifted cover inequalities and Fenchel cutting planes to separate the 0–1 knapsack polytopes is applied to strengthen the lower bound of SSCFLP and that of DP. These lower bounds are further tightened with a partial integrality strategy. Numerical tests on benchmark instances demonstrate the effectiveness of the proposed cutting plane algorithm and the partial integrality strategy in reducing integrality gap and the effectiveness of the CS approach in searching an optimal solution in a reasonable time. Computational results on large sized instances are also presented.

► The Single Source Capacitated Facility Location Problem (SSCFLP) is studied. ► A cut-and-solve based exact method is proposed for SSCFLP. ► New Fenchel cutting plane algorithm is proposed to separate 0–1 knapsack polytopes. ► Partial integrality strategy is used to further tighten the lower bound of SSCFLP. ► Numerical experiments demonstrate the effectiveness of the proposed method.