Abstract
The effectiveness and efficiency of optimization algorithms might deteriorate when solving large-scale binary integer programs (BIPs). Consequently, researchers have tried to fix the values of certain variables called adjunct variables, and only optimize a small problem version formed from the remaining variables called core variables, by relying on information obtained from the BIP’s LP-relaxation solution. The resulting reduced problem is called a core problem (CP), and finding an optimal solution to a CP does not mean finding an optimal solution to the BIP unless the adjunct variables are fixed to their optimal values. Thus, in this work, we borrow several concepts from local search (LS) heuristics to move from a CP to a neighbouring CP to find a CP whose optimal solution is also optimal for the BIP. Thus, we call our framework CORE-LP-LS. We also propose a new mechanism to choose core variables based on reduced costs. To demonstrate and test the CORE-LP-LS framework, we solve a set of 126 multidemand multidimensional knapsack problem (MDMKP) instances. We solve the resulting CPs using two algorithms, namely, commercial branch and bound solver and the state-of-the-art meta-heuristic algorithm to solve MDMKP. As a by-product to our experiments, the CORE-LP-LS framework variants found 28 new best-known solutions and better average solutions for most of the solved instances.
Highlights
The purpose of the core concept to solve a binary integer program (BIP), as stated in [41], is ǣto reduce the original problem by only considering a core of items for which it is hard to decide if they belong to the optimal solution or not, whereas the variables for all items outside the core are fixed to certain values, i.e., 0 and 1 for binary integer programs (BIPs).ǥ based on the core concept, variables can belong to one of two sets: the core var iables’ set Ncore and the adjunct variables’ set Nadj, while the BIP vari ables’ set is N = Ncore ∪ Nadj
The CORE-LP-local search (LS) framework is a solution methodology by which we can avoid solving a large BIP and instead solve several core problem (CP) derived from the BIP
Classifying variables as core or adjunct ones depends on the reduced costs that we obtain from the LP-relaxation solutions
Summary
We use an exact solver in this experiment because we want to block the effects of the algorithm used to solve the CPs. On the other hand, the efficiency of the CORE-LP-LS framework, its ability to find high-quality solutions in short times, is checked by replacing the exact solver with the state-of-the-art algorithm to solve MDMKP. The efficiency of the CORE-LP-LS framework, its ability to find high-quality solutions in short times, is checked by replacing the exact solver with the state-of-the-art algorithm to solve MDMKP For this purpose, we use the two-stage tabu search (TSTS) algorithm of [31] to solve the CPs. we denote this algorithm by CORE-LP-LS-TSTS.
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