Enhancing Benders decomposition algorithm to solve a combat logistics problem

Abstract

This paper proposes a multi-time period, two-stage stochastic programming model for the design and management of a typical combat logistics problem. The design shall minimize the total path setup cost, commodity preposition and processing costs, and expected transportation, storage, and shortage costs across all possible path failure scenarios. Due to the complexity associated with solving the model, we propose an accelerated Benders decomposition algorithm to solve the model in a realistic-size network problem within a reasonable amount of time. The Benders decomposition algorithm incorporates several algorithmic improvements such as pareto-optimal cuts, multi-cuts, knapsack inequalities, integer cuts, input ordering, mean-value cuts, and the rolling horizon heuristic. Computational experiments are performed to assess the efficiency of different enhancement techniques within the Benders decomposition algorithm.

Appendices

Appendix A

1.1 Input ordering

Jans and Desrosiers (2013) observed that the order in which the input data are loaded into the model can have a major impact on the linear programming (LP) relaxation, node exploration, and ultimately the solution time. We use this concept to order some input parameters while solving [RMP]. To the best of the authors’ knowledge, we are the first to apply an input ordering concept in solving a Benders master problem. The idea lies in the fact that assigning critical combat nodes first may help solvers (e.g., CPLEX, Gurobi) to obtain the lower bound quickly. Based on the structure of our problem, we propose two different ways to sort the input data, besides random ordering obtained from the initial data instances.

• Customer demand$d_{mjt}$sorting: This technique ranks the combat nodes according to their commodity demand throughout the entire planning horizon. The nodes with higher demands will be ranked first in the input file in an attempt to quickly obtain a lower bound for [RMP].

• Risk parameter, $\beta _{ijpt}$, sorting: This technique sorts the paths $p \in {\mathcal {P}}$ between supply sites $i \in {\mathcal {I}}$ and combat nodes $j \in {\mathcal {J}}$ based on their average risk values ${\overline{\beta }}_{ijpt}$ throughout the entire planning horizon. The paths with lower risk values will be sorted first in the input file in an attempt to quickly obtain a lower bound for [RMP].

The last set of experiments examine the impact of input order technique on solving Benders master problem [RMP] and the overall algorithmic performance. Table 9 presents the impact of customer demand $d_{mjt}$ and risk parameter $\beta _{ijpt}$ sorting on overall Benders decomposition algorithm. We use Type D+PO+Integer variant of the Benders decomposition algorithm (see Table 5) and set sample size $N = 50$ to run these experiments. Note although the results produced by the Type D+PO+Integer+RH algorithm in Table 5 are encouraging (with respect to both running time and optimality gap), we still decided to carry forward with Type D+PO+Integer algorithm in Table 9 primarily due to its ability to produce a valid optimality gap for the overall problem CLM. Results indicate that both $d_{mjt}$ and $\beta _{ijpt}$ sorting provide slightly better computational performance in solving Benders decomposition algorithm over random sorting. Results in Table 9 indicate that the average running time of the Type D+PO+Integer variant of the Benders decomposition algorithm can be dropped down to 5804 and 5893 CPU seconds, respectively, from 5956 CPU seconds if $d_{mjt}$ and $\beta _{ijpt}$ input ordering are applied over random ordering. Note that between $d_{mjt}$ and $\beta _{ijpt}$ ordering, $d_{mjt}$ demonstrates slightly higher computational performance over $\beta _{ijpt}$. This may be due to the fact that the nodes with higher demands, once explored by the solver, may generate a quick lower bound in solving [RMP] of the Benders decomposition algorithm. Overall, we observe that instead of using random ordering, input ordering on specific parameters (e.g., $d_{mjt}$, $\beta _{ijpt}$) slightly improves the computational performance of the basic Benders decomposition algorithm.

Table 9 Experimental results for the three different input ordering techniques: random, $d_{mjt}$, and $\beta _{ijpt}$

Appendix B

See Table 10.

Table 10 Classes of combat supplies (NATO 2012)