Generalized quadratic multiple knapsack problem and two solution approaches Tugba Saraç n , Aydin Sipahioglu Industrial Engineering Department of Eskisehir Osmangazi University, Meselik, 26480 Eskisehir Turkey article info Available online 31 August 2013 Keywords: Generalized Quadratic Multiple Knapsack Problem (G-QMKP) F-MSG Genetic Algorithm (GA) Production with plastic injection Combinatorial optimization abstract The Quadratic Knapsack Problem (QKP) is one of the well-known combinatorial optimization problems. If more than one knapsack exists, then the problem is called a Quadratic Multiple Knapsack Problem (QMKP). Recently, knapsack problems with setups have been considered in the literature. In these studies, when an item is assigned to a knapsack, its setup cost for the class also has to be accounted for in the knapsack. In this study, the QMKP with setups is generalized taking into account the setup constraint, assignment conditions and the knapsack preferences of the items. The developed model is called Generalized Quadratic Multiple Knapsack Problem (G-QMKP). Since the G-QMKP is an NP-hard problem, two different meta-heuristic solution approaches are offered for solving the G-QMKP. The first is a genetic algorithm (GA), and the second is a hybrid solution approach which combines a feasible value based modified subgradient (F-MSG) algorithm and GA. The performances of the proposed solution approaches are shown by using randomly generated test instances. In addition, a case study is realized in a plastic injection molding manufacturing company. It is shown that the proposed hybrid solution approach can be successfully used for assigning jobs to machines in production with plastic injection, and good solutions can be obtained in a reasonable time for a large scale real-life problem. & 2013 Elsevier Ltd. All rights reserved. 1. Introduction The knapsack problem (KP) is a well-known combinatorial opti- mization problem. The classical KP seeks to select, from a finite set of items, the subset, which maximizes a linear function of the items chosen, subject to a single capacity constraint. In many real life applications, it is important that the profit of packing should also reflect how well the given items fit together. One formulation of such interdependence is the quadratic knapsack problem (QKP). The QKP asks to maximize a quadratic objective function subject to a single capacity constraint. Portfolio management problems, the determina- tion of the optimal sites for communication satellite earth stations or railway stations are good examples of the QKP [8]. The QKP was introduced and solved using a branch-and-bound algorithm by Gallo, Hammer and Simeone in 1980. Later, different branch and bound based solution techniques were offered by Chaillou, Hansen and Mahieu [7], by Michelon and Veuilleux [22] and by Billionnet and Calmels [3]. An exact algorithm was developed by [5] and by Billionnet and Soutif in 2003. In 2005, a greedy Genetic Algorithm was proposed by Julstrom. A survey of upper bounds presented in the literature has been given, and the relative tightness of several of the bounds has been shown by Pisinger [23]. In the same year, Xie and Liu [30] presented a mini-swarm approach for the QKP. In 2009, Sipahioglu and Saraç examined the performance of the modified subgradient algorithm (MSG) to solve the 0-1 QKP and they showed that the MSG algorithm can be successfully used for solving the QKP. The Quadratic multiple knapsack problem (QMKP) extends the QKP with k knapsacks, each with its own capacity c k . Hiley and Julstrom [13] proposed the first study regarding QMKP in the literature. The paper introduced three heuristic approaches, namely the greedy heuristic, the stochastic hill-climber and the Genetic Algorithm (GA). The greedy heuristic fills the knapsacks one item at a time, always choosing the unassigned item with the highest profit/ weight ratio of values to other items with a weight smaller than the remaining capacity of the knapsack. The hill-climber's neighbor operator removes objects from each knapsack, and then refills the knapsack greedily as in the greedy heuristic. The hill-climber's neighbor operator also serves as the GA's mutation. Saraç and Sipahioglu [25] proposed a hybrid genetic algorithm to solve the QMKP. They developed a specialized crossover operator to maintain the feasibility of the chromosomes and presented two distinct mutation operators with different improvement techniques from the non-evolutionary heuristic. They also showed that their GA was more successful than the GA presented by Hiley and Julstrom [13], especially in the case where the number of knapsacks (k) increases. In 2007, Singh and Baghel proposed a steady-state grouping genetic algorithm for the QMKP. Like Hiley and Julstrom [13], they also Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/caor Computers & Operations Research 0305-0548/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cor.2013.08.018 n Corresponding author. Tel.: þ90 2222393750. E-mail addresses: tsarac@ogu.edu.tr, tugba.sarac@gmail.com (T. Saraç), asipahi@ogu.edu.tr (A. Sipahioglu). Computers & Operations Research 43 (2014) 78–89