The multiobjective multidimensional knapsack problem: a survey and a new approach

Abstract

The knapsack problem (KP) and its multidimensional version (MKP) are basic problems in combinatorial optimization. In this paper, we consider their multiobjective extension (MOKP and MOMKP), for which the aim is to obtain or approximate the set of efficient solutions. In the first step, we classify and briefly describe the existing works that are essentially based on the use of metaheuristics. In the second step, we propose the adaptation of the two‐phase Pareto local search (2PPLS) to the resolution of the MOMKP. With this aim, we use a very large scale neighborhood in the second phase of the method, that is the PLS. We compare our results with state‐of‐the‐art results and show that the results we obtained were never reached before by heuristics for biobjective instances. Finally, we consider the extension to three‐objective instances.

Good Laboratory Practice for optimization research

“…• Assignment Problems - (Cattrysse and Van Wassenhove, 1992, Loiola et al, 2007, Pentico, 2007 • Bioinformatics - (B lazewicz et al, 2005(B lazewicz et al, , Lukasiak et al, 2010 • Forecasting - Fildes (1979Fildes ( , 1985, Fildes et al (2008), Ghobbar and Friend (2003), Trapero et al (2015) • Job Shop Scheduling - (B lazewicz et al, 1996, Brucker et al, 2007, Cheng et al, 1999a,b, Kiran and Smith, 1984 • Knapsack Problem - (Lin, 1998, Lust and Teghem, 2012, Pisinger, 2007, Salkin and Kluyver, 1975, Wilbaut et al, 2008 • Personnel Scheduling - (Ernst et al, 2004, Miller, 1976 • Sports Scheduling - (Kendall et al, 2010, Rasmussen andTrick, 2008) • Traveling Salesman Problem - (Bellmore and Nemhauser, 1968, Burkard et al, 1998, Cook, 2011, Lust and Teghem, 2010 • Vehicle Routing - (Archetti and Speranza, 2008, Braysy et al, 2004, Braysy and Gendreau, 2005a,b, Cordeau et al, 2002, Doerner and Schmid, 2010, Gendreau et al, 2008, Jaillet and Wagner, 2008, Laporte, 2009, Vidal et al, 2013 Of course, there are many other areas which also come under the optimization research radar. Indeed, it is difficult to think of a discipline/sector that either does not currently use optimization research, or could benefit from some form of optimization.…”

An Analysis of Local Search for the Bi-objective Bidimensional Knapsack Problem

“…, λ z } is generated, where z is the number of input solutions. For each λ ∈ Λ, a greedy solution is generated using one of the following heuristic functions [12]: …”

“…When analyzing runtimes (see also Fig. 2, right), a candidate list of size 50 combined with removal-full and insertion-first is a setting that takes computation times similar to those that are used as time limits in the analysis of state-of-the-art algorithms [2,12].…”

“…The bi-objective bidimensional knapsack problem is a widely-used bi-objective benchmark problem [14,12], and is a special case of the general multi-objective multidimensional knapsack problem (moMKP), which is formalized as follows: where each item i has k profits and m costs, f c is the c-th component of the objective vector f , n is the number of items, p c i is the c-th profit of item i, w j i is Table 1. Methods used for computing the weights used by SolutionOrdering.…”

Combining Two Search Paradigms for Multi-objective Optimization: Two-Phase and Pareto Local Search

“…Finally, Lust and Teghem [49] tackled multi-objective multi-dimensional knapsack problems. In these problems, two or more types of profits are associated to each item, each type of profit representing a different objective, and the goal is to determine a subset of items to place in a knapsack that maximizes, in the Pareto sense, the sum of profits for each different objective.…”