Local Branching Strategy-Based Method for the Knapsack Problem with Setup

Abstract

In this paper, we propose to solve the knapsack problem with setups by combining mixed linear relaxation and local branching. The problem with setups can be seen as a generalization of 0–1 knapsack problem, where items belong to disjoint classes (or families) and can be selected only if the corresponding class is activated. The selection of a class involves setup costs and resource consumptions thus affecting both the objective function and the capacity constraint. The mixed linear relaxation can be viewed as driving problem, where it is solved by using a special blackbox solver while the local branching tries to enhance the solutions provided by adding a series of invalid / valid constraints. The performance of the proposed method is evaluated on benchmark instances of the literature and new large-scale instances. Its provided results are compared to those reached by the Cplex solver and the best methods available in the literature. New results have been reached.

LP relaxation and dynamic programming enhancing VNS for the multiple knapsack problem with setup

“…To assess the performance of the LP&DP-VNS, we carried out extensive experiments on 200 KPS and 360 MKPS benchmark instances, in addition to newly generated instances. The KPS benchmark instances are proposed by Chebil and Khemakham (2015) and considered by Khemakhem and Chebil (2016), Lahyani et al (2019), Della et al (2017), Pferschy and Scatamacchia (2017), Amri (2019), and Boukhari et al (2020). The MKPS benchmark instances are proposed by Lahyani et al (2019) and considered by Adouani et al (2020), Amiri and Barkhi (2020) Amiri and Barkhi (2020) but not published nor provided by th, and Boukhari et al (2022a).…”

“…The KPS benchmark instances are provided in Chebil and Khemakhem (2015) and considered by Khemakhem and Chebil (2016), Della et al (2017), Pfershy andScatamacchia (2017), Furini et al (2018), Amiri (2019), Lahyani et al (2019), andBoukhari et al (2020) (available at https://padlet.com/adouaniyassine/s4l3hkwhdtk9i4r6). It contains 200 instances randomly generated with N in {5, 10, 20, 30}, and n i in {500, 1000, 2500, 5000, 10000}.…”

“…The LP&DP-VNS results of is developed to deal with MKPS and are reduced to LP&DP to be tested on the well-known KPS. Table 3 summarizes LP&DP in comparison to the most recent and effective algorithms in literature dealing with KPS: DP of Chebil and Khemakhem (2015), TSC of Khemakhem and Chebil (2016), EA of Della et al (2017), DP of Pfershy and Scatamacchia (2017), IDP of Furini et al (2018), Lag of Amiri ( 2019), M red of Lahyani et al (2019), and LBBM of Boukhari et al (2020). Each row presents a set of 10 instances with the same values of N and n i .…”

“…The KPS has been the subject of intensive studies in the past decades (see e.g., Chajakis and Guinard, 1994;Yang and Bulfin, 2009;Michel et al, 2009;Chebil and Khemakhem, 2015;Della et al, 2017;Al-Maliky et al, 2018;Furini et al, 2018;Amri, 2019;Boukhari et al, 2020). Chajakis et al (1994) proposed a dynamic programming (DP) algorithm and two versions of a two-phase enumerative scheme for handling KPS.…”

“…They tested the M red on KPS benchmark instances and show its efficiency and effectiveness in comparison to existing state-of-the-art algorithms. Boukhari et al (2020) proposed a local branching strategy-based method (LBBM) that combines mixed linear relaxation and local branching. The LBBM is a competitive method for KPS: It solves to optimality the majority of the KPS benchmark instances, but its average runtime remains too much higher when compared to existing state-of-the-art algorithms.…”