Multi-objective Knapsack Problem Research Articles

MEMOTS: a memetic algorithm integrating tabu search for combinatorial multiobjective optimization

We present in this paper a new multiobjective memetic algorithm scheme called MEMOX. In current multiobjective memetic algorithms, the parents used for recombination are randomly selected. We improve this approach by using a dynamic hypergrid which allows to select a parent located in a region of minimal density. The second parent selected is a solution close, in the objective space, to the first parent. A local search is then applied to the offspring. We experiment this scheme with a new multiobjective tabu search called PRTS, which leads to the memetic algorithm MEMOTS. We show on the multidimensional multiobjective knapsack problem that if the number of objectives increase, it is preferable to have a diversified research rather using an advanced local search. We compare the memetic algorithm MEMOTS to other multiobjective memetic algorithms by using different quality indicators and show that the performances of the method are very interesting.

Solving efficiently the 0–1 multi-objective knapsack problem

In this paper, we present an approach, based on dynamic programming, for solving the 0–1 multi-objective knapsack problem. The main idea of the approach relies on the use of several complementary dominance relations to discard partial solutions that cannot lead to new non-dominated criterion vectors. This way, we obtain an efficient method that outperforms the existing methods both in terms of CPU time and size of solved instances. Extensive numerical experiments on various types of instances are reported. A comparison with other exact methods is also performed. In addition, for the first time to our knowledge, we present experiments in the three-objective case.

Local dominance and local recombination in MOEAs on 0/1 multiobjective knapsack problems

This work studies and compares the effects on performance of local dominance and local recombination applied with different locality in multiobjective evolutionary algorithms on combinatorial 0/1 multiobjective knapsack problems. For this purpose, we introduce a method that creates a neighborhood around each individual and assigns a local dominance rank after alignment of the principle search direction of the neighborhood by using polar coordinates in objective space. For recombination a different neighborhood determined around a random principle search direction is created. The neighborhood sizes for dominance and recombination are separately controlled by two different parameters. Experimental results show that the optimum locality of dominance is different from the optimum locality of recombination. Additionally, it is shown that the performance of the algorithm that applies local dominance and local recombination with different locality is significantly better than the performance of algorithms applying local dominance alone, local recombination alone, or dominance and recombination globally as conventional approaches do.

Multi-start and path relinking methods to deal with multiobjective knapsack problems

This paper deals with a multiobjective combinatorial optimization problem called Extended Knapsack Problem. By applying multi-start search and path relinking we rapidly guide the search toward the most balanced zone of the Pareto-optimal front. The Pareto relation is applied in order to designate a subset of the best generated solutions to be the current efficient set of solutions. The max-min criterion with the Hamming distance is used as a measure of dissimilarity in order to find diverse solutions to be combined. The performance of our approach is compared with several state-of-the-art MOEAs for a suite test problems taken from the literature.

An Evolutionary Metaheuristic for Approximating Preference-Nondominated Solutions

We propose an evolutionary metaheuristic for approximating the preference-nondominated solutions of a decision maker in multiobjective combinatorial problems. The method starts out with some partial preference information provided by the decision maker, and utilizes an individualized fitness function to converge toward a representative set of solutions favored by the information at hand. The breadth of the set depends on the precision of the partial information available on the decision maker’s preferences. The algorithm simultaneously evolves the population of solutions out toward the efficient frontier, focuses the population on those segments of the efficient frontier that will appeal to the decision maker, and disperses it over these segments to have an adequate representation. Simulation runs carried out on randomly generated instances of the multiobjective knapsack problem and the multiobjective spanning-tree problem have found the algorithm to yield highly satisfactory results.

Controlling Dominance Area of Solutions in Multiobjective Evolutionary Algorithms and Performance Analysis on Multiobjective 0/1 Knapsack Problems

This work proposes a method to control the dominance area of solutions in order to induce appropriate ranking of solutions for the problem at hand, enhance selection, and improve the performance of MOEAs on combinatorial optimization problems. The proposed method can control the degree of expansion or contraction of the dominance area of solutions using a user-defined parameter S. Modifying the dominance area of solutions changes their dominance relation inducing a ranking of solutions that is different to conventional dominance. In this work we use 0/1 multiobjective knapsack problems to analyze the effects on solutions ranking caused by contracting and expanding the dominance area of solutions and its impact on the search performance of a multi-objective optimizer when the number of objectives, the size of the search space, and the feasibility of the problems vary. We show that either convergence or diversity can be emphasized by contracting or expanding the dominance area. Also, we show that the optimal value of the area of dominance depends strongly on all factors analyzed here: number of objectives, size of the search space, and feasibility of the problems.

A hybrid system for multiobjective problems – A case study in NP-hard problems

In attempt to solve multiobjective problems, various mathematical and stochastic methods have been developed. The methods operate based on mathematical models while in most cases these models are drastically simplified imagine of real world problems. In this study, a hybrid intelligent system is used instead of mathematical models. The main core of the system is fuzzy rule base which maps decision space ( Z) to solution space ( X). The system is designed on noninferior region and gives a big picture of this region in the pattern of fuzzy rules. Since some solutions may be infeasible; then specified feedforward neural network is used to obtain noninferior solutions in an exterior movement. In addition, numerical examples of well-known NP-hard problems (i.e. multiobjective traveling salesman problem and multiobjective knapsack problem) are provided to clarify the accuracy of developed system.

Analysis of a Multiobjective Evolutionary Algorithm on the 0–1 knapsack problem

Multiobjective Evolutionary Algorithms (MOEAs) are increasingly being used for effectively solving many real-world problems, and many empirical results are available. However, theoretical analysis is limited to a few simple toy functions. In this work, we select the well-known knapsack problem for the analysis. The multiobjective knapsack problem in its general form is NP-complete. Moreover, the size of the set of Pareto-optimal solutions can grow exponentially with the number of items in the knapsack. Thus, we formalize a ( 1 + ε ) -approximate set of the knapsack problem and attempt to present a rigorous running time analysis of a MOEA to obtain the formalized set. The algorithm used in the paper is based on a restricted mating pool with a separate archive to store the remaining population; we call the algorithm a Restricted Evolutionary Multiobjective Optimizer (REMO). We also analyze the running time of REMO on a special bi-objective linear function, known as LOTZ (Leading Ones : Trailing Zeros), whose Pareto set is shown to be a subset of the knapsack. An extension of the analysis to the Simple Evolutionary Multiobjective Optimizer (SEMO) is also presented. A strategy based on partitioning of the decision space into fitness layers is used for the analysis.

A Hybrid Fine-Tuned Multi-Objective Memetic Algorithm

In this paper, we propose a hybrid fine-tuned multi-objective memetic algorithm hybridizing different solution fitness evaluation methods for global exploitation and exploration. To search across all regions in objective space, the algorithm uses a widely diversified set of weights at each generation, and employs a simulated annealing to optimize each utility function. For broader exploration, a grid-based technique is adopted to discover the missing nondominated regions on existing tradeoff surface, and a Pareto-based local perturbation is performed to reproduce incrementing solutions trying to fill up the discontinuous areas. Additional advanced feature is that the procedure is made dynamic and adaptive to the online optimization conditions based on a function of improvement ratio to obtain better stability and convergence of the algorithm. Effectiveness of our approach is shown by applying it to multi-objective 0/1 knapsack problem (MOKP).

An efficient, adaptive parameter variation scheme for metaheuristics based on the epsilon-constraint method

This paper discusses methods for generating or approximating the Pareto set of multiobjective optimization problems by solving a sequence of constrained single-objective problems. The necessity of determining the constraint value a priori is shown to be a serious drawback of the original epsilon-constraint method. We therefore propose a new, adaptive scheme to generate appropriate constraint values during the run. A simple example problem is presented, where the running time (measured by the number of constrained single-objective sub-problems to be solved) of the original epsilon-constraint method is exponential in the problem size (number of decision variables), although the size of the Pareto set grows only linearly. We prove that––independent of the problem or the problem size––the time complexity of the new scheme is O( k m−1 ), where k is the number of Pareto-optimal solutions to be found and m the number of objectives. Simulation results for the example problem as well as for different instances of the multiobjective knapsack problem demonstrate the behavior of the method, and links to reference implementations are provided.

Genetic Algorithm for Optimization of Infrastructure Investment Under Time‐Resource Constraints

Abstract: This work shows the infrastructure investment decision is essentially a 0‐1, nonlinear, multiobjective knapsack problem. It argues that, without making substantial simplification on some of the practical considerations, a conventional mathematical optimization approach may not be suitable for solving the problem. Further, the use of explicit heuristics may sometimes be unwieldy dependent on the system parameters. An innovative approach based on the concept of genetic algorithms is proposed in the work to tackle the nonlinearity in the decision problem. Based on the experimental results, the proposed approach is capable of producing good quality results. The key success feature of the proposed approach resides in the unique and innovative modeling of system parameters by genetic coding in the algorithms.

Screening Alternatives In Multiple Criteria Subset Selection

New techniques are presenled to reduce the number of feasible alternatives in certain multiple criteria subset selection problems, thereby making it less difficult to find a good subset. The class of m-best alternatives problems is defined and the relation between dominance and potential optimalily explored in the context of this class. A program is proposed to identify whether an individually dominated alternative can belong to an optimal subset satisfying certain pre-specified constraints. The extension of the proposed method to multi-objective knapsack problems is considered. Two examples illustrate the screening procedure for m-best alternatives problems and multi-objective knapsack problems.

Running time analysis of evolutionary algorithms on a simplified multiobjective knapsack problem

In this paper, the expected running time of two multiobjective evolutionary algorithms, SEMO and FEMO, is analyzed for a simple instance of the multiobjective 0/1 knapsack problem. The considered problem instance has two profit values per item and cannot be solved by one-bit mutations. In the analysis, we make use of two general upper bound techniques, the decision space partition method and the graph search method. The paper demonstrates how these methods, which have previously only been applied to algorithms with one-bit mutations, are equally applicable for mutation operators where each bit is flipped independently with a certain probability.

An Interactive Evolutionary Metaheuristic for Multiobjective Combinatorial Optimization

We propose an evolutionary metaheuristic for multiobjective combinatorial optimization problems that interacts with the decision maker (DM) to guide the search effort toward his or her preferred solutions. Solutions are presented to the DM, whose pairwise comparisons are then used to estimate the desirability or fitness of newly generated solutions. The evolutionary algorithm comprising the skeleton of the metaheuristic makes use of selection strategies specifically designed to address the multiobjective nature of the problem. Interactions with the DM are triggered by a probabilistic evaluation of estimated fitnesses, while memory structures with indifference thresholds restrict the presentation of solutions resembling those that have already been rejected. The algorithm has been tested on a number of random instances of the Multiobjective Knapsack Problem (MOKP) and the Multiobjective Spanning Tree Problem (MOST). Simulation results indicate that the algorithm requires only a small number of comparisons to be made for satisfactory solutions to be found.

Multi Agent Systems with Symbiotic Learning and Evolution -Masbiole- and Its Application.

Recently, systems are becoming more complex and larger than ever, so numerous attempts have been made to introduce biological features into artificial systems, because many biological systems in the nature exist as one of the most complex systems.Multi agent system with symbiotic learning and evolution have been recently proposed. It is named Masbiole. In this paper, Masbiole is reviewed and the method for evolving multi agent systems is proposed. From simulations on a multi objective knapsack problem, it has been clarified that Masbiole has better performance than that of conventional multi objective genetic algorithms.

Approximating Multiobjective Knapsack Problems

For multiobjective optimization problems, it is meaningful to compute a set of solutions covering all possible trade-offs between the different objectives. The multiobjective knapsack problem is a generalization of the classical knapsack problem in which each item has several profit values. For this problem, efficient algorithms for computing a provably good approximation to the set of all nondominated feasible solutions, the Pareto frontier, are studied. For the multiobjective one-dimensional knapsack problem, a practical fully polynomial-time approximation scheme (FPTAS) is derived. It is based on a new approach to the single-objective knapsack problem using a partition of the profit space into intervals of exponentially increasing length. For the multiobjective m-dimensional knapsack problem, the first known polynomial-time approximation scheme (PTAS), based on linear programming, is presented.

An interactive heuristic method for multi-objective combinatorial optimization

We have previously developed an adaptation of the simulated annealing for multi-objective combinatorial optimization (MOCO) problems to construct an approximation of the efficient set of such problem. In order to deal with large-scale problems, we transform this approach to propose an interactive procedure. The method is tested on the multi-objective knapsack problem and the multi-objective assignment problem. Scope and purpose Meta-heuristics methods are intensively used with success to solve optimization problems and especially combinatorial problems (Pirlot. EJOR 1996;92:493–511). In the case of a single objective problem, such methods compute an approximation to the unique optimal solution. Recently, some meta-heuristics have been adapted to treat multi-objective problems. These methods construct an approximation of the set of all efficient solutions. For large-scale multi-objective combinatorial problems, the number of efficient solutions may become very large. In order to help a decision maker to make a choice between these solutions, an interactive procedure is developed in this paper.

An improved interactive hybrid method for the linear multi-objective knapsack problem

In many situations, the knapsack problem in the presence of multiple, conflicting objectives frequently occurs in, for example, capital budgeting, project selection and capital investment, and budget control. Previous work was done to solve decision problems that had a weak point. An improved hybrid method is suggested to reduce the burden to the decision maker (DM) in selecting a solution and to obtain computational efficiency. In the method, bounding of the DM's utility value based on the revealed preference information is incorporated into a dynamic programming framework. In finding the most preferred solution (MPS), an implicit utility function is approximated by a linear function completely described by the scaling constants. The DM's partial preference expression is translated into a set of possible scaling constants [7]. In the stagewise solution process, an ideal objectives achievement of each partial solution is derived, and examined for whether it can give the highest utility value by comparing to the best-known objectives achievement. Elimination of partial solutions, which have proved not to lead to the MPS, is done to make the procedure more effective in finding the MPS. A suggested scheme based on the DM partial preference expression can give computational efficiency, which will be shown through computational experiments.