Quadratic Multiple Knapsack Problem Research Articles

Lagrangian matheuristics for the Quadratic Multiple Knapsack Problem

The Quadratic Multiple Knapsack Problem (QMKP) is a challenging combinatorial optimization problem combining the well-known Quadratic Knapsack Problem with the Multiple Knapsack Problem. After a long stream of research devoted to metaheuristic approaches for large-scale instances, only recently some authors started to study the mathematical properties of the QMKP and proposed exact solution methods for benchmark instances of smaller size. In this paper, we propose the first matheuristic approach for solving the QMKP approximately. The proposed method exploits the strength of a Lagrangian relaxation for the natural quadratic formulation of the QMKP to derive heuristic solutions. Postoptimization local search procedures are embedded in the final framework. Experimental studies show that the resulting deterministic matheuristic approach consistently delivers solutions of very good quality in short computing times. • First matheuristic for the Quadratic Multiple Knapsack Problem. • Competitive with currently best performing exact method. • Tighter Lagrangian bounds on medium-size instances.

A Genetic algorithm combined with mathematical programming to solve Generalised Quadratic Multiple Knapsack Problem

A Genetic algorithm combined with mathematical programming to solve Generalised Quadratic Multiple Knapsack Problem

QMKPy: A Python Testbed for the Quadratic Multiple Knapsack Problem

QMKPy provides a Python framework for modeling and solving the quadratic multiple knapsack problem (QMKP). It is primarily aimed at researchers who develop new solution algorithms for the QMKP. QMKPy therefore mostly functions as a testbed to quickly implement novel algorithms and compare their results with existing ones. However, the package also already includes implementations of established algorithms for those who only need to solve a QMKP as part of their research.

Effect of learning strategies in an evolutionary method: the case of the bi-objective quadratic multiple knapsack problem

Effect of learning strategies in an evolutionary method: the case of the bi-objective quadratic multiple knapsack problem

A hybrid population-based algorithm for the bi-objective quadratic multiple knapsack problem

In this paper, the bi-objective quadratic multiple knapsack problem is tackled with a hybrid population-based method. The proposed method starts by computing two reference solutions, where a specialized powerful mono-objective algorithm is used. From both reference solutions, a starting population is built by using a series of perturbations around the solutions. Next, the so-called non-sorting genetic process is combined with a new drop/rebuild operator for generating a series of populations till converging toward an approximate Pareto front with high density. The performance of the hybrid population based algorithm (namely HBPA) is evaluated on a set of benchmark instances of the literature containing both medium and large-scale instances. Its provided results are compared to those achieved by the best methods available in the literature. Encouraging results have been obtained.

A branch-and-bound algorithm for the quadratic multiple knapsack problem

We present a new branch-and-bound algorithm for the Quadratic Multiple Knapsack Problem. A key component of our algorithm is a new upper bound that divides the pairwise item values among individual items, estimates the maximum potential value contributed by each individual item, and calculates the upper bound via a transportation model. A local search is used to adjust the division of pairwise item values in order to improve the upper bound. Reduced costs from the solution of the transportation problem are used to forbid some item-to-knapsack assignments. Information from the upper bound is also used in selecting the item to branch on next. Computational experiments are carried out on three sets of benchmark instances from the literature, two sets of smaller instances with 20–35 items and 3–10 knapsacks per instance, and one set of larger instances with 40–60 items and 3–10 knapsacks per instance. Our algorithm finds and verifies optimal solutions for all small benchmark instances in less than one hour of CPU time apiece and outperforms the best previously proposed algorithms for the problem in terms of both the average computation time and the number of instances for which optimality is verified. For the larger benchmark instances, our algorithm obtains smaller average gaps than the best previously proposed methods.

A hybrid evolutionary search for the generalized quadratic multiple knapsack problem

Knapsack problems are useful models that can formulate many real-life applications. The generalized quadratic multiple knapsack problem (GQMKP) extends the well-known quadratic multiple knapsack problem by taking into account setups and knapsack preference of the items. In this study, an efficient hybrid evolutionary search algorithm (HESA) is proposed to tackle GQMKP, which relies on a knapsack-based crossover operator to generate new offspring solutions, and an adaptive feasible and infeasible tabu search to improve new offspring solutions. Other new features of HESA include a dedicated strategy to ensure a diversified and high-quality initial population, and a streamlining technique to speed up the evaluations of candidate solutions. The experiments on two sets of 96 benchmark instances as well as one large-scale real-life instance show that the proposed algorithm outperforms the state-of-the-art algorithms from the literature. In particular, HESA finds 44 improved best-known solutions (new lower bounds) (for more than 45% cases). The key components of the algorithm are studied to assess their effects on the algorithm’s performance.

Branch and solve strategies-based algorithm for the quadratic multiple knapsack problem

Suppose a manager has to assign agents for multiple projects, where each agent has its own budget. The manager knows the salary and productivity of each agent, both individually and in pairs. The goal of the manager is to assign a subset of agents to each project at the same time in such a way that the total productivity, without exceeding the budget of any project, is maximised. This problem can be formulated as a quadratic multiple knapsack, an NP-hard problem. This paper investigates the use of branch and solve strategies in order to solve large-scale quadratic multiple knapsack problems. An enhanced fix and solve solution procedure is developed, which is embedded in the local branching-based method, where the branches reflect intensification and diversification search around a solution. The proposed method is analyzed on a set of benchmark instances taken from the literature and new generated large-scale instances. Its provided solution values are compared to those achieved by more recent algorithms available in the literature and the state-of-the-art exact Cplex solver. The experimental study shows that the method is able to reach new solutions, match several best available solutions and, outperforms the Cplex on the new hardness instances.

Polynomial-size formulations and relaxations for the quadratic multiple knapsack problem

The Quadratic Multiple Knapsack Problem generalizes, simultaneously, two well-known combinatorial optimization problems that have been intensively studied in the literature: the (single) Quadratic Knapsack Problem and the Multiple Knapsack Problem. The only exact algorithm for its solution uses a formulation based on an exponential-size number of variables, that is solved via a Branch-and-Price algorithm. This work studies polynomial-size formulations and upper bounds. We derive linear models from classical reformulations of 0-1 quadratic programs and analyze theoretical properties and dominances among them. We define surrogate and Lagrangian relaxations, and we compare the effectiveness of the Lagrangian relaxation when applied to a quadratic formulation and to a Level 1 reformulation linearization that leads to a decomposable structure. The proposed methods are evaluated through extensive computational experiments.

A matheuristic for the 0–1 generalized quadratic multiple knapsack problem

In this study, we address the 0–1 generalized quadratic multiple Knapsack problem. We use a linearization technique of the existing mathematical model and we propose a new matheuristic that we called Matheuristic Variable Neighborhood Search combining variable neighborhood search with integer programing to solve the large sized instances. The matheuristic considers a local search technique with an adaptive perturbation mechanism to assign the classes to different knapsacks, and then once the assignment is identified, applies the IP to select the items to allocate to each knapsack. Experimental results obtained on a wide set of benchmark instances clearly show the competitiveness of the proposed approach compared to the best state-of-the-art solving techniques.

A Repair-based approach for stochastic quadratic multiple knapsack problem

In this paper, a new problem called stochastic quadratic multiple knapsack problem (SQMKP) is studied. SQMKP is the extension of the quadratic multiple knapsack problem (QMKP) with stochastic factors. Different from QMKP, the objective of SQMKP is no longer to find a single global optimal solution, but to find a solution with the best expected quality under all possible cases. We use the recoverable robustness technique to tackle these stochastic factors. The idea of the recoverable robustness is to first find a solution that is feasible for the static QMKP, and then adjust the solution through a simple recovery algorithm according to the emerging stochastic scenarios. Following the idea of the recoverable robustness technique, we propose a repair-based approach (RBOA) which involves three key components – an instance generator that is able to create a set of representative scenarios, a static QMKP algorithm that is able to provide a set of potentially robust solutions and a simple fast repair algorithm that can quickly restore the feasibility of a solution with respect to a given stochastic scenario. Experimental results on a set of stochastic instances extended from 90 well-known QMKP benchmarks demonstrate the efficacy of the proposed RBOA.

A multi-start iterated local search algorithm for the generalized quadratic multiple knapsack problem

The quadratic multiple knapsack problem (QMKP) is a variant of the classical knapsack problem where multiple knapsacks are considered and the objective is to maximize a quadratic objective function subject to capacity constraints. The generalized quadratic multiple knapsack problem (G-QMKP) extends the QMKP by considering the setups, assignment conditions and the knapsack preferences of the items. In this study, a multi-start iterated local search algorithm (MS-ILS) in w the variable neighborhood descent (VND) algorithm is integrated with an adaptive perturbation mechanism is proposed to solve the G-QMKP. The multi-start implementation together with the adaptive perturbation mechanism enables the search process to explore different search regions in the search space while VND is applied to obtain high-quality solutions from the examined regions. Another remarkable feature of MS-ILS is its simplicity and adaptiveness that ease its implementation. The proposed MS-ILS is tested on G-QMKP benchmark instances derived from the literature. The results indicate that the developed MS-ILS can produce high-quality solutions for the G-QMKP. Particularly, it has been observed that the developed MS-ILS has improved the best known solutions for 35 out of 48 large-size G-QMKP instances.

Memetic Search for the Generalized Quadratic Multiple Knapsack Problem

The generalized quadratic multiple knapsack problem (GQMKP) extends the classical quadratic multiple knapsack problem 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 multineighborhood simulated annealing procedure (to find high quality local optima). To prevent premature convergence of the search, MAGQMK employs a quality-and-distance (QD) 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 QD pool updating rule to the performance of the proposed algorithm.

An ejection chain approach for the quadratic multiple knapsack problem

In an algorithm for a problem whose candidate solutions are selections of objects, an ejection chain is a sequence of moves from one solution to another that begins by removing an object from the current solution. The quadratic multiple knapsack problem extends the familiar 0–1 knapsack problem both with several knapsacks and with values associated with pairs of objects. A hybrid algorithm for this problem extends a local search algorithm through an ejection chain mechanism to create more powerful moves. In addition, adaptive perturbations enhance the diversity of the search process. The resulting algorithm produces results that are competitive with the best heuristics currently published for this problem. In particular, it improves the best known results on 34 out of 60 test problem instances and matches the best known results on all but 6 of the remaining instances.

The bi-objective quadratic multiple knapsack problem: Model and heuristics

The single objective quadratic multiple knapsack problem (QMKP) is a useful model to formulate a number of practical problems. However, it is not suitable for situations where more than one objective needs to be considered. In this paper, we extend the single objective QMKP to the bi-objective case such that we simultaneously maximize the total profit of the items packed into the knapsacks and the ’makespan’ (the gain of the least profit knapsack). Given the imposing computational challenge, we propose a hybrid two-stage (HTS) algorithm to approximate the Pareto front of the bi-objective QMKP. HTS combines two different and complementary search methods — scalarizing memetic search (first stage) and Pareto local search (second stage). Experimental assessments on a set of 60 problem instances show that HTS dominates a standard multi-objective evolutionary algorithm (NSGA II), and two simplified variants of HTS. We also present a comparison with two state-of-the-art algorithms for the single objective QMKP to assess the quality of the extreme solutions of the approximated Pareto front.

An evolutionary path relinking approach for the quadratic multiple knapsack problem

The quadratic multiple knapsack problem (QMKP) is a challenging combinatorial optimization problem with numerous applications. In this paper, we propose the first evolutionary path relinking approach (EPR) for solving the QMKP approximately. This approach combines advanced features both from the path relinking (PR) method and the responsive threshold search algorithm. Thanks to the tunneling property which allows a controlled exploration of infeasible regions, the proposed EPR algorithm is able to identify very high quality solutions. Experimental studies on the set of 60 well-known benchmarks and a new set of 30 large-sized instances show that EPR outperforms several state-of-the-art algorithms. In particular, for the 60 conventional benchmarks, it discovers 10 improved results (new lower bounds) and matches the best known result for the remaining 50 cases. More significantly, EPR demonstrates remarkable efficacy on the 30 new larger instances by easily dominating the current best performing algorithms across the whole instance set. Key components of the algorithm are analyzed to shed lights on their impact on the proposed approach.

Hybridization of tabu search with feasible and infeasible local searches for the quadratic multiple knapsack problem

The quadratic multiple knapsack problem (QMKP) concerns assigning a set of objects, which interact among themselves through paired profit values, to a set of capacity-constrained knapsacks such that the overall profit of the objects included in the knapsacks is maximized and the total weight of the objects in each knapsack does not exceed the capacity of the knapsack. In this paper we present a highly effective tabu search (TS) approach for QMKP that incorporates a hybridization scheme combining both feasible and infeasible local searches. The feasible local search focuses its search on the most relevant feasible solutions, while the infeasible local search explores a large search space composed of both feasible and infeasible solutions, and employs several tailored move selection rules to keep the infeasible solutions close to the feasible regions located in promising areas. Extensive computational results on a set of 60 benchmark instances in the literature illustrate that the new TS approach compares very favorably with the current state-of-the-art solution methods for QMKP. In particular, the TS approach finds improved best solutions for ten instances. We also analyze the hybridization scheme in the TS approach to ascertain its effect on the performance of the solution method.

Iterated responsive threshold search for the quadratic multiple knapsack problem

The quadratic multiple knapsack problem (QMKP) consists in assigning objects with both individual and pairwise profits to a set of limited knapsacks in order to maximize the total profit. QMKP is a NP-hard combinatorial optimization problem with a number of applications. In this paper, we present an iterated responsive threshold search (IRTS) approach for solving the QMKP. Based on a combined use of three neighborhoods, the algorithm alternates between a threshold-based exploration phase where solution transitions are allowed among those satisfying a responsive threshold and a descent-based improvement phase where only improving solutions are accepted. A dedicated perturbation strategy is utilized to ensure a global diversification of the search procedure. Extensive experiments performed on a set of 60 benchmark instances in the literature show that the proposed approach competes very favorably with the current state-of-the-art methods for the QMKP. In particular, it discovers 41 improved lower bounds and attains all the best known results for the remaining instances. The key components of IRTS are analyzed to shed light on their impact on the performance of the algorithm.

Strategic oscillation for the quadratic multiple knapsack problem

The quadratic multiple knapsack problem (QMKP) consists in assigning a set of objects, which interact through paired profit values, exclusively to different capacity-constrained knapsacks with the aim of maximising total profit. Its many applications include the assignment of workmen to different tasks when their ability to cooperate may affect the results.

Generalized quadratic multiple knapsack problem and two solution approaches

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.