Instances Of Knapsack Problem Research Articles

New Enumeration Algorithm for Regular Boolean Functions

This paper introduces a new algorithm for enumerating regular Boolean functions. This algorithm exploits the equivalence between regular Boolean functions and positive threshold functions that can be used to represent instances of the knapsack problem. After proving this equivalence, this paper introduces a novel data structure that may, with further tweaking, enable faster enumeration algorithms for both regular Boolean functions and all-capacities knapsack problem instances.

Hybrid Symbiotic Organisms Search Algorithm for Solving 0-1 Knapsack Problem

We propose a new binary version of hybrid symbiotic organisms search algorithm based on harmony search with greedy strategy for solving 0-1 knapsack problems. A greedy strategy is employed to repair the infeasible solution and optimise the feasible solution. The experiments are carried out in small-scale and large-scale knapsack problem instances. We report on computational experiments which are compared with the results achieved with other state-of-the-art approaches. The results attest the performance of our approach.

Quantum-Inspired Evolutionary Algorithm for difficult knapsack problems

Quantum Inspired Evolutionary Algorithms (QIEAs) are Evolutionary Algorithms which use concepts and principles of quantum computing. The 0/1 knapsack problem (KP) is a well known combinatorial optimization problem that has been typically used to validate the performance of QIEAs. However, there are some variants of KPs called difficult knapsack problems (DKPs) that are known to be more difficult to solve. QIEAs have not yet been fully explored for solving these. In this work, an improved QIEA, called QIEA-PSA is presented. A novel method to initialize the qubit individuals based on heuristic information for the KP instance and a method for size reduction for each new generation are introduced in the presented QIEA-PSA. Experiments are carried out for several types of DKPs that are much larger in size than those attempted hitherto. QIEA-PSA provides much better solutions than QIEA with much lesser computation times. Even a serial implementation of QIEA-PSA competes favorably on the same problem instances with a parallel implementation of an exact algorithm given recently in literature. A comparison is made which shows QIEA-PSA outperforms a recently applied population based search technique to solve benchmark KP instances. The ideas used for developing QIEA-PSA are general and may be utilized with advantage on other problems.

Finding nadir points in multi-objective integer programs

We address the problem of finding the nadir point in a multi-objective integer programming problem. Finding the nadir point is not straightforward, especially when there are more than three objectives. The difficulty further increases for integer programming problems. We develop an exact algorithm to find the nadir point in multi-objective integer programs with integer-valued parameters. We also develop a variation that finds bounds for each component of the nadir point with a desired level of accuracy. We demonstrate on several instances of multi-objective assignment, knapsack, and shortest path problems that the algorithms work well.

Shuffled frog leaping algorithm and its application to 0/1 knapsack problem

This paper proposes a modified discrete shuffled frog leaping algorithm (MDSFL) to solve 01 knapsack problems. The proposed algorithm includes two important operations: the local search of the ‘particle swarm optimization’ technique; and the competitiveness mixing of information of the ‘shuffled complex evolution’ technique. Different types of knapsack problem instances are generated to test the convergence property of MDSFLA and the result shows that it is very effective in solving small to medium sized knapsack problems. Further, computational experiments with a set of large-scale instances show that MDSFL can be an efficient alternative for solving tightly constrained 01 knapsack problems.

A new algorithm for generating all nondominated solutions of multiobjective discrete optimization problems

Most real-life decision-making activities require more than one objective to be considered. Therefore, several studies have been presented in the literature that use multiple objectives in decision models. In a mathematical programming context, the majority of these studies deal with two objective functions known as bicriteria optimization, while few of them consider more than two objective functions. In this study, a new algorithm is proposed to generate all nondominated solutions for multiobjective discrete optimization problems with any number of objective functions. In this algorithm, the search is managed over (p−1)-dimensional rectangles where p represents the number of objectives in the problem and for each rectangle two-stage optimization problems are solved. The algorithm is motivated by the well-known ε-constraint scalarization and its contribution lies in the way rectangles are defined and tracked. The algorithm is compared with former studies on multiobjective knapsack and multiobjective assignment problem instances. The method is highly competitive in terms of solution time and the number of optimization models solved.

Essential Particle Swarm Optimization queen with Tabu Search for MKP resolution

The Particle Swarm Optimization (PSO) algorithm is an innovative and promising optimization technique in evolutionary computation. The Essential Particle Swarm Optimization queen (EPSOq) is one of the recent discrete PSO versions that further simplifies the PSO principles and improves its optimization ability. Hybridization is a principle of combining two (or more) approaches in a wise way such that the resulting algorithm includes the positive features of both (or all) the algorithms. This paper proposes a new heuristic approach such that various features inspired from the Tabu Search are incorporated in the EPSOq algorithm in order to obtain another improved discrete PSO version. The implementation of this idea is identified with the acronym TEPSOq (Tabu Essential Particle Swarm Optimization queen). Experimentally, this approach is able to solve optimally large-scale strongly correlated 0---1 Multidimensional Knapsack Problem (MKP) instances. Computational results show that TEPSOq has outperforms not only the EPSOq, but also other existing PSO-based approaches and some other meta-heuristics in solving the 0---1 MKP. It was discovered also that this algorithm is able to locate solutions extremely close and even equal to the best known results available in the literature.

Tailored Lagrangian Relaxation for the open pit block sequencing problem

A common strategic and tactical decision in open pit mining is to determine the sequence of extraction for notional three-dimensional production blocks so as to maximize the net present value of the extracted orebody while adhering to precedence and resource constraints. This problem is commonly formulated as an integer program with a binary variable representing if and when a block is extracted. In practical applications, the number of blocks can be large and the time horizon can be long, and therefore, instances of this NP-hard precedence-constrained knapsack problem can be difficult to solve using an exact approach. The problem is even more challenging to solve when it includes explicit minimum resource constraints. We employ three methodologies to reduce solution times: (i) we eliminate variables which must assume a value of 0 in the optimal solution; (ii) we use heuristics to generate an initial integer feasible solution for use by the branch-and-bound algorithm; and (iii) we employ Lagrangian relaxation, using information obtained while generating the initial solution to select a dualization scheme for the resource constraints. The combination of these techniques allows us to determine near-optimal solutions more quickly than solving the monolith, i.e., the original problem. We demonstrate our techniques to solve instances containing 25,000 blocks and 10 time periods, and 10,000 blocks and 15 time periods, to near-optimality.

On the calculation of stability radius for multi-objective combinatorial optimization problems by inverse optimization

This paper deals with stability analysis in multi-objective combinatorial optimization problems. The stability radius of an efficient solution is defined as the maximal adjustment of the problem parameters such that this solution remains efficient. An algorithm based on inverse optimization is proposed to compute it. The adjustment is limited to the coefficients of the objective functions and measured by the Chebyshev norm. This approach is applied to randomly generated instances of the bi-objective knapsack problem and computational results are reported. Several illustrative examples are analyzed.

A reduction approach for solving the rectangle packing area minimization problem

In the rectangle packing area minimization problem (RPAMP) we are given a set of rectangles with known dimensions. We have to determine an arrangement of all rectangles, without overlapping, inside an enveloping rectangle of minimum area. The paper presents a generic approach for solving the RPAMP that is based on two algorithms, one for the 2D Knapsack Problem (KP), and the other for the 2D Strip Packing Problem (SPP). In this way, solving an instance of the RPAMP is reduced to solving multiple SPP and KP instances. A fast constructive heuristic is used as SPP algorithm while the KP algorithm is instantiated by a tree search method and a genetic algorithm alternatively. All these SPP and KP methods have been published previously. Finally, the best variants of the resulting RPAMP heuristics are combined within one procedure. The guillotine cutting condition is always observed as an additional constraint. The approach was tested on 15 well-known RPAMP instances (above all MCNC and GSRC instances) and new best solutions were obtained for 10 instances. The computational effort remains acceptable. Moreover, 24 new benchmark instances are introduced and promising results are reported.

A Modified Binary Particle Swarm Optimization for Knapsack Problems

The Knapsack Problems (KPs) are classical NP-hard problems in Operations Research having a number of engineering applications. Several traditional as well as population based search algorithms are available in literature for the solution of these problems. In this paper, a new Modified Binary Particle Swarm Optimization (MBPSO) algorithm is proposed for solving KPs, particularly 0–1 Knapsack Problem (KP) and Multidimensional Knapsack Problem (MKP). Compared to the basic Binary Particle Swarm Optimization (BPSO), this improved algorithm introduces a new probability function which maintains the diversity in the swarm and makes it more explorative, effective and efficient in solving KPs. MBPSO is tested through computational experiments over benchmark problems and the results are compared with those of BPSO and a relatively recent modified version of BPSO namely Genotype–Phenotype Modified Binary Particle Swarm Optimization (GPMBPSO). To validate our idea and demonstrate the efficiency of the proposed algorithm for KPs, experiments are carried out with various data instances of KP and MKP and the results are compared with those of BPSO and GPMBPSO.

Computational performance of basic state reduction based dynamic programming algorithms for bi-objective 0–1 knapsack problems

This paper studies a group of basic state reduction based dynamic programming (DP) algorithms for the multi-objective 0–1 knapsack problem (MKP), which are related to the backward reduced-state DP space (BRDS) and forward reduced-state DP space (FRDS). The BRDS is widely ignored in the literature because it imposes disadvantage for the single objective knapsack problem (KP) in terms of memory requirements. The FRDS based DP algorithm in a general sense is related to state dominance checking, which can be time consuming for the MKP while it can be done efficiently for the KP. Consequently, no algorithm purely based on the FRDS with state dominance checking has ever been developed for the MKP. In this paper, we attempt to get some insights into the state reduction techniques efficient to the MKP. We first propose an FRDS based algorithm with a local state dominance checking for the MKP. Then we evaluate the relative advantage of the BRDS and FRDS based algorithms by analyzing their computational time and memory requirements for the MKP. Finally different combinations of the BRDS and FRDS based algorithms are developed on this basis. Numerical experiments based on the bi-objective KP instances are conducted to compare systematically between these algorithms and the recently developed BRDS based DP algorithm as well as the existing FRDS based DP algorithm without state dominance checking.

Solving 0-1 knapsack problems by a discrete binary version of cuckoo search algorithm

Cuckoo search (CS) is one of the most recent population-based meta-heuristics. CS algorithm is based on the cuckoo’s behaviour and the mechanism of Lévy flights. Unfortunately, the standard CS algorithm is proposed only for continuous optimisation problems. In this paper, we propose a discrete binary cuckoo search (BCS) algorithm in order to deal with binary optimisation problems. To get binary solutions, we have used a sigmoid function similar to that used in the binary particle swarm optimisation algorithm. Computational results on some knapsack problem instances and multidimensional knapsack problem instances show the effectiveness of the proposed algorithm and its ability to achieve good quality solutions.

Comparative analysis of multiobjective evolutionary algorithms for random and correlated instances of multiobjective d-dimensional knapsack problems

This study analyzes multiobjective d-dimensional knapsack problems (MOd-KP) within a comparative analysis of three multiobjective evolutionary algorithms (MOEAs): the ε-nondominated sorted genetic algorithm II ( ε-NSGAII), the strength Pareto evolutionary algorithm 2 (SPEA2) and the ε-nondominated hierarchical Bayesian optimization algorithm ( ε-hBOA). This study contributes new insights into the challenges posed by correlated instances of the MOd-KP that better capture the decision interdependencies often present in real world applications. A statistical performance analysis of the algorithms uses the unary ε-indicator, the hypervolume indicator and success rate plots to demonstrate their relative effectiveness, efficiency, and reliability for the MOd-KP instances analyzed. Our results indicate that the ε-hBOA achieves superior performance relative to ε-NSGAII and SPEA2 with increasing number of objectives, number of decisions, and correlative linkages between the two. Performance of the ε-hBOA suggests that probabilistic model building evolutionary algorithms have significant promise for expanding the size and scope of challenging multiobjective problems that can be explored.

A multi-level search strategy for the 0–1 Multidimensional Knapsack Problem

We propose an exact method based on a multi-level search strategy for solving the 0–1 Multidimensional Knapsack Problem. Our search strategy is primarily based on the reduced costs of the non-basic variables of the LP-relaxation solution. Considering that the variables are sorted in decreasing order of their absolute reduced cost value, the top level branches of the search tree are enumerated following Resolution Search strategy, the middle level branches are enumerated following Branch & Bound strategy and the lower level branches are enumerated according to a simple Depth First Search enumeration strategy. Experimentally, this cooperative scheme is able to solve optimally large-scale strongly correlated 0–1 Multidimensional Knapsack Problem instances. The optimal values of all the 10 constraint, 500 variable instances and some of the 30 constraint, 250 variable instances of the OR-Library were found. These values were previously unknown.

Synthetic Optimization Problem Generation: Show Us the Correlations!

In many computational experiments, correlation is induced between certain types of coefficients in synthetic (or simulated) instances of classical optimization problems. Typically, the correlations that are induced are only qualified—that is, described by their presumed intensity. We quantify the population correlations induced under several strategies for simulating 0–1 knapsack problem instances and also for correlation-induction approaches used to simulate instances of the generalized assignment, capital budgeting (or multidimensional knapsack), and set-covering problems. We discuss implications of these correlation-induction methods for previous and future computational experiments on simulated optimization problems.

Variable Neighbourhood Decomposition Search with Bounding for Multidimensional Knapsack Problem

Abstract In this paper we propose a new heuristic for solving multidimensional knapsack problem, based on the variable neighbourhood decomposition search principle. The set of neighbourhoods is generated by exploiting information obtained from a series of relaxations. In each iteration, we add new constraints to the problem in order to produce a sequence of lower and upper bounds around the optimal value, with the goal to reduce the gap between them. General-purpose CPLEX MIP solver is used as a black box for solving subproblems generated during the search process. With this approach, we have managed to obtain promising results on a set of large scale multidimensional knapsack problem instances. The results are comparable with the current state-of-the-art techniques for solving multidimensional knapsack problem.

DNA computing of solutions to knapsack problems

One line of DNA computing research focuses on parallel search algorithms, which can be used to solve many optimization problems. DNA in solution can provide an enormous molecular library, which can be searched by molecular biological techniques. We have implemented such a parallel search for solutions to knapsack problems, which ask for the best way to pack a knapsack of limited volume. Several instances of knapsack problems were solved using DNA. We demonstrate how the computations can be extended by in vivo translation of the DNA library into protein. This combination of DNA and protein allows for multi-criterion optimization. The knapsack computations performed can then be seen as protein optimizations, one of the most complex computations performed by natural systems.

Enumeration Methods for Repeatedly Solving Multidimensional Knapsack Sub-Problems

In order to solve large Multidimensional Knapsack problems we examine a technique which decomposes a problem instance into two parts. The first part is solved using a traditional technique, such as Dynamic Programming, to reduce the number of variables in the problem by creating a single variable with many non-dominated states. In the second part the remaining variables are determined by an algorithm that repeatedly enumerates them with different constraint and objective requirements. The constraint and objective requirements are imposed by the various non-dominated states of the variable created in the first part of this technique. The main advantage of this approach is that when memory requirements prevent traditional techniques solving a problem instance, the enumeration provides a much less memory-intensive method, enabling a solution to be found. Two approaches are proposed for repeatedly enumerating a 0/1 Multidimensional Knapsack problem. It is demonstrated how these enumeration methods, in conjunction with the Modular Approach, were used to find the optimal solutions to a number of 500-variable, 5-constraint Multidimensional Knapsack problem instances proposed in the literature. The exact solutions to these instances were previously unknown.

Generation of solved instances of Multiconstraint Knapsack problem and its applications to Private Key Cipher

Abstract This papers explores the relations between the generation of solved instances of optimization problem and the design of private key cipher. A class of ciphers derived from complementary slackness conditions of multiconstraint knapsack problem (MKP) are presented and discussed. It is shown that under plaintext attack, any algorithm that predicts the embedded key sequence generator of the ciphers also solves a sequence of instances of MKP. It is also shown that even under P=NP and additional assumption, security under plaintext attack of the proposed ciphers is equivalent to their associative key sequence generator.