Multiple Knapsack Problem Research Articles

Merging valid inequalities over the multiple knapsack polyhedron

This paper provides the theoretical foundations for generating a new class of valid inequalities for integer programming problems through inequality merging. The inequality merging technique combines two low dimensional inequalities of a multiple knapsack problem, potentially yielding a valid inequality of higher dimension. The paper describes theoretical conditions for validity of the merged inequality and shows that the validity of a merged cover inequality may be verified in quadratic time. Conditions under which a valid merged inequality is facet defining are also presented. The technique is demonstrated through a multiple knapsack example. The example also demonstrates that inequality merging yields a new class of valid inequalities that are fundamentally different from other known techniques.

On Merging Cover Inequalities for Multiple Knapsack Problems

This paper describes methods to merge two cover inequalities and also simultaneously merge multiple cover inequalities in a multiple knapsack instance. Theoretical results provide conditions under which merged cover inequalities are valid. Polynomial time algorithms are created to find merged cover inequalities. A computational study demonstrates that merged inequalities improve the solution times for benchmark multiple knapsack instances by about 9% on average over CPLEX with default settings.

Implicit cover inequalities

In this paper we consider combinatorial optimization problems whose feasible sets are simultaneously restricted by a binary knapsack constraint and a cardinality constraint imposing the exact number of selected variables. In particular, such sets arise when the feasible set corresponds to the bases of a matroid with a side knapsack constraint, for instance the weighted spanning tree problem and the multiple choice knapsack problem. We introduce the family of implicit cover inequalities which generalize the well-known cover inequalities for such feasible sets and discuss the lifting of the implicit cover inequalities. A computational study for the weighted spanning tree problem is reported.

Balanced Quantum-Inspired Evolutionary Algorithm for Multiple Knapsack Problem

Multiple Knapsack Problem, a generalization of more popular 0/1 Knapsack Problem, is NP-hard and considered harder than simple Knapsack Problem. 0/1 Multiple Knapsack Problem has many applications in disciplines related to computer science and operations research. Quantum Inspired Evolutionary Algorithms (QIEAs), a subclass of Evolutionary algorithms, are considered effective to solve difficult problems particularly NP-hard combinatorial optimization problems. A hybrid QIEA is presented for multiple knapsack problem which incorporates several features for better balance between exploration and exploitation. The proposed QIEA, dubbed QIEA-MKP, provides significantly improved performance over simple QIEA from both the perspectives viz., the quality of solutions and computational effort required to reach the best solution. QIEA-MKP is also able to provide the solutions that are better than those obtained using a well known heuristic alone.

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.

A combined approach for fuzzy multi-objective multiple knapsack problems for defence project selection

In this study, a model representing military requirements as scenarios and capabilities is offered. Pair-wise comparisons of scenarios are made according to occurrence probabilities by using the Analytical Hierarchy Process (AHP). The weights calculated from AHP are used as the starting weights in a Quality Function Deployment (QFD) matrix. QFD is used to transfer war fighter requirements into the benefit values of projects. Two levels of QFD matrices are used to evaluate new capability areas versus capabilities and capabilities versus projects. The benefit values of the projects are used in a multi-objective problem (multi-objective multiple knapsack problem) that considers the project benefit, implementation risks and environmental impact as multiple objectives. Implementation risk and environmental impact values are also calculated using the same combined AHP and QFD methodology. Finally, the results of the fuzzy multi-objective goal programming suggest a list of projects that offers optimal benefit when carried out within multiple budgets.

Upper and lower bounding procedures for the multiple knapsack assignment problem

We formulate the multiple knapsack assignment problem (MKAP) as an extension of the multiple knapsack problem (MKP), as well as of the assignment problem. Except for small instances, MKAP is hard to solve to optimality. We present a heuristic algorithm to solve this problem approximately but very quickly. We first discuss three approaches to evaluate its upper bound, and prove that these methods compute an identical upper bound. In this process, reference capacities are derived, which enables us to decompose the problem into mutually independent MKPs. These MKPs are solved euristically, and in total give an approximate solution to MKAP. Through numerical experiments, we evaluate the performance of our algorithm. Although the algorithm is weak for small instances, we find it prospective for large instances. Indeed, for instances with more than a few thousand items we usually obtain solutions with relative errors less than 0.1% within one CPU second.

Parallel Machine Selection and Job Scheduling to Minimize Sum of Machine Holding Cost, Total Machine Time Costs, and Total Tardiness Costs

This paper is concerned with scheduling of a set of single-operation tasks on a set of parallel machines where subcontracting is allowed. The objective is to choose a subset of machines/subcontractors from a set of available machines/subcontractors to perform all jobs to minimize sum of several costs. Processing time of jobs is assumed to be equal. Lower and upper bound for number of jobs assigned to a machine/subcontractor is considered. We first present a comprehensive survey of applications and models. We show special case of the problem when lower bound for number of jobs assigned to each machine/subcontractor is equal to zero is equivalent to single-sink fixed-charge transportation problem (SSFCT). This proves NP-hardness of the problem. Efficient dynamic programming algorithm for this special case is presented. Complicating issues regarding the general case with nonzero lower bounds for number of jobs assigned to machines/subcontractors is discussed. We transfer the general problem to multiple choice knapsack problem (MCKP) that can be solved efficiently using available algorithms. Several new problems are introduced. Complexity of each problem is resolved. Transformation to MCKP is provided that allows available algorithms to solve the problems. The main contribution of this paper is to establish theoretical results regarding the solution of these difficult problems.

A New Artificial Fish Swarm Algorithm for the Multiple Knapsack Problem

A New Artificial Fish Swarm Algorithm for the Multiple Knapsack Problem

Multiple criteria dynamic programming and multiple knapsack problem

The aim of the paper is to show relations between two different types of optimization problems: multiple criteria dynamic programming (MCDP) and integer linear programming (ILP) in a form of multiple knapsack (MK) problem. Moreover, the paper presents how to use MCDP methods in order to solve (MK) problems.

A quantum-inspired Tabu search algorithm for solving combinatorial optimization problems

In this study, we propose a novel quantum-inspired evolutionary algorithm (QEA), called quantum inspired Tabu search (QTS). QTS is based on the classical Tabu search and characteristics of quantum computation, such as superposition. The process of qubit measurement is a probability operation that increases diversification; a quantum rotation gate used to searching toward attractive regions will increase intensification. This paper will show how to implement QTS into NP-complete problems such as 0/1 knapsack problems, multiple knapsack problems and the traveling salesman problem. These problems are important to computer science, cryptography and network security. Furthermore, our experimental results on 0/1 knapsack problems are compared with those of other heuristic algorithms, such as a conventional genetic algorithm, a Tabu search algorithm and the original QEA. The final outcomes show that QTS performs much better than other heuristic algorithms without premature convergence and with more efficiency. Also on multiple knapsack problems and the traveling salesman problem QTS verify its effectiveness.

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.

Algorithms for scheduling of chemotherapy plans

Chemotherapy is used to control and cure cancer by using drugs to destroy cancer cells. Treatment schedules for chemotherapy may vary depending on the type of cancer, the goals of treatment, the type of chemotherapy and the patient's state of health. Chemotherapy is usually given in cycles of a treatment-period followed by a rest-period. An oncologist decides the choice of a particular regimen; however, modifications to drug dose and schedule are often necessary because of variabilities in the health of an individual patient. Therefore an orderly execution of chemotherapy regimens requires management, scheduling and allocation of the resources available. Chemotherapy scheduling is an optimization problem. In this paper, a two-phase approach has been adopted to deal with the problem. An adaptive negative-feedback scheduling algorithm is proposed for the first phase to control the load on the system. Two heuristics based on the ‘Multiple Knapsack Problem’ have been evaluated for the second phase to assign patients to specific infusion seats. The overall design has been put to test at a local chemotherapy center and has yielded good results for patient waiting times, orderly execution of chemotherapy regimen and utilization of infusion chairs.

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.

Tabu-enhanced iterated greedy algorithm: A case study in the quadratic multiple knapsack problem

Iterated greedy search is a simple and effective metaheuristic for combinatorial problems. Its flexibility enables the incorporation of components from other metaheuristics with the aim of obtaining effective and powerful hybrid approaches. We propose a tabu-enhanced destruction mechanism for iterated greedy search that records the last removed objects and avoids removing them again in subsequent iterations. The aim is to provide a more diversified and successful search process with regards to the standard destruction mechanism, which selects the solution components for removal completely at random.We have considered the quadratic multiple knapsack problem as the application domain, for which we also propose a novel local search procedure, and have developed experiments in order to assess the benefits of the proposal. The results show that the tabu-enhanced iterated greedy approach, in conjunction with the new local search operator, effectively exploits the problem-knowledge associated with the requirements of the problem considered, attaining competitive results with regard to the corresponding state-of-the-art algorithms.

Modelling and a tabu search solution for the slab reallocation problem in the steel industry

This paper considers a slab reallocation problem arising from operations planning in the steel industry. The problem involves reallocating steel slabs to customer orders to improve the utilisation of slabs and the level of customer satisfaction. It can be viewed as an extension of a multiple knapsack problem. We firstly formulate the problem as an integer nonlinear programming (INLP) model. With variable replacement, the INLP model is then transformed into a mixed integer linear programming (MILP) model, which can be solved to optimality by MILP optimisers for very small instances. To obtain satisfactory solutions efficiently for practical-sized instances, a heuristic algorithm based on tabu search (TS) is proposed. The algorithm employs multiple neighbourhoods including swap, insertion and ejection chain in local search, and adopts solution space decomposition to speed up computation. In the ejection chain neighbourhood, a new and more effective search method is also proposed to take advantage of the structural properties of the problem. Computational experiments on real data from an advanced iron and steel company in China show that the algorithm generates very good results within a short time. Based on the model and solution approach, a decision support system has been developed and implemented in the company.

Introduction to special xQx issue

The Unconstrained Binary Quadratic Programming (UBQP) problem is notable for encompassing a remarkable range of applications in combinatorial optimization. As observed in Kochenberger et al. (2013), classes of problems that can be formally expressed using UBQP formulations include: Quadratic Assignment Problems, Capital Budgeting Problems, Multiple Knapsack Problems, Task Allocation Problems (distributed computer systems), Maximum Diversity Problems, P-Median Problems, Asymmetric Assignment Problems, Symmetric Assignment Problems, Side Constrained Assignment Problems, Quadratic Knapsack Problems, Constraint Satisfaction Problems (CSPs), Set Partitioning Problems, Fixed Charge Warehouse Location Problems, Maximum Clique Problems, Maximum Independent Set Problems, Maximum Cut Problems, Graph Coloring Problems, Graph Partitioning Problems, Number Partitioning Problems, and Linear Ordering Problems. Even more remarkable is the fact that, once given a UBQP formulation, these problems can be solved by a UBQP method which is not specialized to exploit the problem domain of any individual class of problems, to yield solutions whose quality in many cases rivals or even surpasses the quality of the solutions produced by the best specialized methods, while achieving this outcome with an efficiency that likewise rivals or surpasses the efficiency of leading specialized methods. Moreover, these outcomes typically dominate those produced by current state-of-the-art commercial solvers for mixed integer linear and quadratic optimization, sometimes consuming two orders of magnitude less solution time to yield solutions superior to those of competing approaches. In other cases, where specialized methods edge out general UBQP methods for certain classes of problems, the UBQP formulation often still

On the multicriteria allocation problem

We consider multicriteria allocation problems with linear sum objectives. Despite the fact that the single objective allocation problem is easily solvable, we show that already in the bicriteria case the problem becomes intractable, is NP-hard and has a non-connected efficient set in general. Using the equivalence to appropriately defined multiple criteria multiple-choice knapsack problems, an algorithm is suggested that uses partial dominance conditions to save computational time. Different types of enumeration schemes are discussed, for example, with respect to the number of necessary filtering operations and with regard to possible parallelizations of the procedure.

A discrete artificial bee colony for multiple Knapsack problem

Multiple Knapsack problem (MKP) is a most popular multiple subset selection problem that belongs to the class of NP–Complete problems. The aim is to assign optimal subsets among all original items to some knapsacks, such that the overall profit of all selected items be maximised, while the total weight of all assigned items to any knapsack does not exceed the allowable capacity of it. Artificial bee colony (ABC) algorithm is a new meta–heuristic with a stochastic search strategy. In ABC, the neighbourhood area of any best–found solution is searched by the employed bees to achieve better solutions. This paper presents a discrete ABC algorithm for the MKP. In this approach, a hybrid probabilistic mutation scheme is performed for searching the neighbourhood of food sources. The proposed algorithm can guide the search space quickly and improve the local search ability. Experimental results demonstrate that the presented approach has improved the quality and convergence speed than other evolutionary algorithms.

User QoS enhanced web service composition framework in cloud platforms

Recently, deploying web services in cloud platforms has become a popular issue that attracts plenty of attentions. Unlike typical service platforms, cloud system provides an elastic infrastructure for service provision and security isolation. As a result, the service composition in a cloud platform becomes more flexible and negotiable, especially when users have multiple QoS constraints with various weights. To address the problem of service composition with multiple QoS constraints in cloud systems, we propose a QoS-enhanced web service composition framework, in which evolution strategy is applied to solve the optimal service composition which is formulated as multiple choice multiple dimension knapsack problem. Based on the proposed service composition, a QoS negation mechanism is also designed with aiming to adapt the dynamical and elastic cloud environments. Extensive experiments are conducted to investigate the effectiveness of the proposed framework, and the results show that it can significantly reduce the costs of large-scale service-based application in terms of various QoS metrics.