Computational experiment of critical event tabu search for the general integer multidimensional knapsack problem

Abstract

In this paper, we propose a critical event tabu search meta-heuristic for the general integer multidimensional knapsack problem (GMDKP). Variations of GMDKP have enormous applications, and often occur as a sub-problem of more general combinatorial problems. For the special case of binary multidimensional knapsack problems (BMDKP) there are variety of heuristics, mostly sophisticated meta-heuristics, which provides good solutions to the problem. However, to date there is no method that can provide reasonable solutions to realistic size GMDKP. To the best of our knowledge there are only three heuristics published in the literature for GMDKP, and all three are simple greedy heuristics. There is no meta-heuristic available that effectively provides good solutions for large-scale GMDKP. One successful meta-heuristic that has proven to be highly effective in solving combinatorial optimization is a variation of tabu search known as the critical event tabu search (CETS). CETS was originally proposed for the BMDKP with considerable success afterwards. In CETS, clever use of surrogate programming is embedded as choice rules to obtain high quality solutions. The main purpose of this paper is to design the meta-heuristic CETS for the GMDKP using variety of different surrogate choice rules. Extensive computational experiment for large-scale problems are presented. Our procedures open the door for further applications of meta-heuristics to general integer programs.

Appendix

An explanation of use of non-parametric Friedman test with its companion utility ranking process, [adapted from, Golden and Stewart (1985)].

k, :

number of heuristics,

n, :

number of problem instances solved by heuristics,

$R_{ij},$ :

rank of heuristic $j (j=1,{\ldots },k$) compared to all heuristics for problem instance $i (i=1,{\ldots }n$), (lowest value gets 1 in the case of ties average ranks are used),

$$\begin{aligned} R_j= & {} \mathop \sum \nolimits _{i=1}^n {R_{ij} } ,\hbox { }j=1,\ldots ,k, \\ A_F= & {} \mathop \sum \nolimits _{i=1}^n {\mathop \sum \nolimits _{j=1}^k {(R_{ij} )^{2}} } , \\ B_F= & {} (1/n)\mathop \sum \nolimits _{j=1}^k {(R_j )^{2}} , \end{aligned}$$

$T_F =\frac{(n-1)\{B_F -nk(k+1)^{2}/4\}}{A_F -B_F },$ the test statistic,

The null hypothesis is rejected at level $\alpha$ if the test statistic $T_F$ exceeds the $1-\alpha$ quantity of F distribution with $k-1$ and $(n-1)(k-1)$ degrees of freedom, $F(\alpha ,(n-1)(k-1))$.

If the null hypothesis is rejected that all heuristics are equally accurate at $\alpha$ level then we have reason to perform further analysis to identify the ‘best’ of the heuristics. This is done based on expected utility approach defined as follows (Table 3).

A and B, two positive constant numbers, say 600 and 100, respectively,

$\bar{{x}}_j =\mathop \sum \nolimits _{i=1}^n {x_{ij} } /n,$ average performance of heuristic j, where $x_{ij}$ is value of result of heuristic j applied to problem instance i,

$$\begin{aligned} s_j^2= & {} \frac{\mathop \sum \nolimits _{i=1}^n {(x_{ij} -\bar{{x}}_j )^{2}} }{n}, \\ \hat{{b}}_j= & {} s_j^2 /\bar{{x}}_j , \\ \hat{{c}}_j= & {} (\bar{{x}}_j /s_j )^{2}, \end{aligned}$$

$UTILITY_j =A-B(1-\alpha \hat{{b}}_j )^{-\hat{{c}}},$ utility of heuristic j, (largest is better performing heuristic)

Table 3 Analysis of Friedman test, and its companion utility ranking process for performance of 4 heuristics for 10 problems with $s_i \in [0.6,0.8]$