Memetic Search for the Generalized Quadratic Multiple Knapsack Problem

Abstract

The generalized quadratic multiple knapsack problem (GQMKP) extends the classical quadratic multiple knapsack problem (QMKP) with setups and knapsack preference of the items. The GQMKP can accommodate a number of real-life applications and is computationally difficult. In this paper, we demonstrate the interest of the memetic search approach for approximating the GQMKP by presenting a highly effective memetic algorithm (denoted by MAGQMK). The algorithm combines a backbone-based crossover operator (to generate offspring solutions) and a multi-neighborhood simulated annealing procedure (to find high quality local optima). To prevent premature convergence of the search, MAGQMK employs a quality-anddistance pool updating strategy. Extensive experiments on two sets of 96 benchmarks show a remarkable performance of the proposed approach. In particular, it discovers improved best solutions in 53 and matches the best known solutions for 39 other cases. A case study on a pseudo real-life problem demonstrates the efficacy of the proposed approach in practical situations. Additional analyses show the important contribution of the novel general-exchange neighborhood, the backbone-based crossover operator as well as the quality-and-distance pool updating rule to the performance of the proposed algorithm.

Opposition-Based Memetic Search for the Maximum Diversity Problem

“…The memetic algorithm framework (MA) [36], [27] is a well-known hybrid search approach combining populationbased search and local optimization. MA has been successfully applied to tackle numerous classical NP-hard problems [9], [24], such as graph coloring [31], graph partitioning [7], [16] and generalized quadratic multiple knapsack [10] as well as the maximum diversity problem [11], [53].…”

“…This pool updating strategy simultaneously considers the solution quality and the distance between individuals in the population to guarantee the population diversity. Similar quality-and-distance pool updating strategies have been used in memetic algorithms in [10], [31], [48], [53].…”

“…. , S p } and one solution S i in P , the average distance between S i and the remaining individuals in P is computed by [10].…”

“…Our tabu search procedure relies on the parametric constrained neighborhood whose size is controlled by the parameter ρ. To highlight the effect of this parameter and determine a proper value, we ran the tabu search procedure to solve the first 10 instances of MDG-a (i.e., MDG-a 21 ∼ MDG-a 30) with ρ ∈ [1,10]. Each instance was independently solved until the number of iterations reached M axIter.…”

“…The minimum distance is defined as the minimum distance between any two individuals in the population, i.e., M D = min i =j∈{1,2,...,p} D(S i , S j ). Correspondingly, the AD is the average distance between all individuals in the population, as defined by Equation (10).…”