The quadratic multiple knapsack problem and three heuristic approaches to it

Abstract

The quadratic multiple knapsack problem extends the quadratic knapsack problem with K knapsacks, each with its own capacity C k . A greedy heuristic fills the knapsacks one at a time with objects whose contributions are likely to be large relative to their weights. A hill-climber and a genetic algorithm encode candidate solutions as strings over {0, 1, . . . , K} with length equal to the number of objects. The hill-climber's neighbor operator is also the GA's mutation. In tests on 60 problem instances, the GA performed better than the greedy heuristic on the smaller instances, but it fell behind as the numbers of objects and knapsacks grew. The hill-climber always outperformed the greedy heuristic, and on the larger instances, also the GA.

An evolutionary path relinking approach for the quadratic multiple knapsack problem

“…• Set I: This set consists of 60 well-known benchmarks which are commonly used for the QMKP algorithm assessment in the literature (Chen & Hao, 2015;García-Martínez et al, 2014a,b;Sundar & Singh, 2010;Saraç & Sipahioglu, 2007;Singh & Baghel, 2007;Hiley & Julstrom, 2006). Built from the quadratic knapsack problem (QKP) instances introduced in (Billionnet & Soutif, 2015) which can be download from: http://cedric.…”

“…The quadratic multiple knapsack problem (QMKP) is a well-known combinatorial optimization problem (Hiley & Julstrom, 2006). Given a set of knapsacks of limited capacity and a set of objects (or items), each object is associated with a weight, an individual profit and a pairwise profit with any other object.…”

“…The QMKP aims to determine a max-profit assignment (packing) of objects to the knapsacks subject to the capacity constraint of each knapsack. The QMKP has a number of relevant applications where resources with different levels of interaction have to be distributed among different tasks (Hiley & Julstrom, 2006).…”

“…Due to their quadratic nature, these problems are known to be extremely difficult. For this reason, approximate algorithms based on metaheuristics like evolutionary algorithms, constitute a very popular approach for tackling these problems (Misevicius, 2004;Hiley & Julstrom, 2006;Saraç & Sipahioglu, 2007;Singh & Baghel, 2007;Soak & Lee, 2012).…”

“…On the other hand, some effort has been devoted to develop heuristics which aim to provide satisfactory suboptimal solutions in acceptable computing time, but without provable optimal guarantee of the attained solutions. Among these heuristics, neighborhood search and constructive/destructive search approaches are very popular, including hill-climbing (Hiley & Julstrom, 2006), tabu-enhanced iterated greedy search (García-Martínez et al, 2014a), strategic oscillation (García-Martínez et al, 2014b) and responsive iterated threshold search (Chen & Hao, 2015). Population-based algorithms constitute another class of popular tools for addressing the QMKP, such as genetic algorithms (Hiley & Julstrom, 2006;Saraç & Sipahioglu, 2007), memetic algorithms (Singh & Baghel, 2007;Soak & Lee, 2012) and artificial bee colony algorithms (Sundar & Singh, 2010).…”