Multi-objective Knapsack Problem Research Articles

Role of combinatorial optimization problems for the efficient decision making of a company- Case study-

All company seeks to improve their performance through achieving several objectives to stay in the labor market and increase competitiveness. The problem lies in the fact that it often relies on traditional methods for efficient decision-making, causing a waste of time and effort. The paper aims to find the best car service stations among several proposed stations by the Naftal Company, which seeks to achieve the two most important objectives; maximizing lubricants and tires revenues, and maximizing fuel and LPG revenues for the year 2019. The new applied case study was done with the help of the binary multi-objective knapsack problem, which is considered one of the most important problems in multi-objective combinatorial optimization problems. The problem was solved by the epsilon constraint method using MATLAB and CPLEX software. The study results confirmed that the proposed efficient solution set (Pareto optimal solutions) provides the decision maker with a comprehensive view of how to make their decision in a scientific, thoughtful, logical, and convincing manner that suits his situation and puts him at ease, away from trial and error or relying on self-experience.

Supported nondominated points as a representation of the nondominated set: An empirical analysis

AbstractThe nondominated set of a multiple objective discrete optimization problem is known to contain unsupported nondominated points, which outnumber the supported ones and are more difficult to obtain. We treat supported nondominated points as a representation and analyse their quality using different metrics beyond their sheer numbers. Under different data generation schemes on multiobjective knapsack and assignment problems, we observe that supported nondominated points almost always provide a good representation of the entire nondominated set.

A Review of the Multitasking Allocation Model for Vertical Farming System

Vertical Farming System (VFS) can produce food efficiently in times of dramatic climate change, regional conflicts that affect food transportation, or space exploration. The high degree of automation in the system, a large number of sensors and robots work together, so an efficient task allocation model can significantly enhance operational efficiency, reduce energy consumption, and elevate the overall economic performance of VFS.With this background, the purpose of this review is to describe the current available multi-task allocation models applicable to VFS or related contexts. Through a literature analysis over the past 5 years, researchers have developed and analyzed numerous multi-task allocation models in the relevant fields, such as Multi-Objective Knapsack Problem (MOKP) models and Vehicle Routing Problem (VRP) models. Depending on the nature of the problem and the constraints of the environment, diverse types of multi-task allocation models have been proposed. Tasks generated within the VFS must be completed within specified time limits, and certain tasks require collaborative efforts among different robots. Taking into consideration these task requirements and drawing upon existing multi-task allocation models proposed by other researchers, this review propose the VFSRTP (VFS Robot Task Planning) model. It is expected that this work will have a positive impact on the research of VFS multi-task allocation model, and provide some model descriptions and guidelines.

Hybridization of Evolutionary Algorithm and Deep Reinforcement Learning for Multiobjective Orienteering Optimization

Multi-objective orienteering problems (MO-OPs) are classical multi-objective routing problems and have received much attention in recent decades. This study seeks to solve MO-OPs through a problem-decomposition framework, that is, an MO-OP is decomposed into a multi-objective knapsack problem (MOKP) and a traveling salesman problem (TSP). The MOKP and TSP are then solved by a multi-objective evolutionary algorithm (MOEA) and a deep reinforcement learning (DRL) method, respectively. While the MOEA module is for selecting cities, the DRL module is for planning a Hamiltonian path for these cities. An iterative use of these two modules drives the population towards the Pareto front of MO-OPs. The effectiveness of the proposed method is compared against NSGA-II and NSGA-III on various types of MO-OP instances. Experimental results show that our method performs best on almost all the test instances, and has shown strong generalization ability.

A universal large-scale many-objective optimization framework based on cultural learning

When solving large-scale many-objective optimization problems (LMaOPs), due to the large number of variables and objectives involved, the algorithm is faced with a very high-dimensional and complex search space, which is difficult to be explored with limited resources. To address these issues, this paper proposes a universal large-scale many-objective optimization framework based on cultural learning (UCLMO). First, a universal framework is proposed, and multi-objective optimizers can be embedded into the framework to accelerate the convergence. Moreover, inspired by cultural learning, an individual selection strategy based on historical knowledge is proposed to promote the diversity of the population, and an assisted evolution strategy based on normative knowledge is presented to accelerate the convergence of the algorithm. Experiments have been conducted on multi-objective knapsack problems and LMaOPs with decision variables ranging from 500 to 1500, and the number of objectives ranging from 5 to 15. The experimental results verify the superiority and competitiveness of the proposed UCLMO framework in solving LMaOPs compared with state-of-the-art algorithms.

Strength Pareto Evolutionary Algorithm Based on New Fitness Strategy for Multi Objective Knapsack Problem

problems. Many different experiments have shown that the evolutionary algorithm Strength Pareto Evolutionary Algorithm 2 (SPEA2) outperforms other approaches, and this makes it an excellent candidate for inclusion in the final design. According to Strength, one of SPEA2's core concepts, the population is sorted into niches based on the Pareto Front idea. With regard to outcomes this technique has a flaw that is mitigated by adding a fitness density estimator. Weakness in Strength is addressed with an approach called Strength by objective, which aims to include solution who do not dominate or are dominated by others inside the process. In this paper, the results will show a clear superiority of the proposed method comparing with the original method in solving the multi-objective Knapsack problem using three sizes of the Knapsack and the problem size of 750 items which is generated randomly. The comparison results between the proposed method and the other algorithms show outperformance of the Proposed algorithm using the dominance inductor showed the superiority of the proposed algorithm by a percentage of more than 1 %, Thanks to a highly diverse population and the inclusion of solutions that can be improved, the (SPEA2) algorithm's performance has been vastly enhanced by the Objective sorting mechanism

Objective space division-based hybrid evolutionary algorithm for handing overlapping solutions in combinatorial problems.

Overlapping solutions occur when more than one solution in the space of decisions maps to the same solution in the space of objectives. This situation threatens the exploration capacity of Multi-Objective Evolutionary Algorithms (MOEAs), preventing them from having a good diversity in their population. The influence of overlapping solutions is intensified on multi-objective combinatorial problems with a low number of objectives. This paper presents a hybrid MOEA for handling overlapping solutions that combines the classic NSGA-II with a strategy based on Objective Space Division (OSD). Basically, in each generation of the algorithm, the objective space is divided into several regions using the nadir solution calculated from the current generation solutions. Furthermore, the solutions in each region are classified into non-dominated fronts using different optimization strategies in each of them. This significantly enhances the achieved diversity of the approximate front of non-dominated solutions. The proposed algorithm (called NSGA-II/OSD) is tested on a classic Operations Research problem: the Multi-Objective Knapsack Problem (0-1 MOKP) with two objectives. Classic NSGA-II, MOEA/D and Global WASF-GA are used to compare the performance of NSGA-II/OSD. In the case of MOEA/D two different versions are implemented, each of them with a different strategy for specifying the reference point. These MOEA/D reference point strategies are thoroughly studied and new insights are provided. This paper analyses in depth the impact of overlapping solutions on MOEAs, studying the number of overlapping solutions, the number of solution repairs, the hypervolume metric, the attainment surfaces and the approximation to the real Pareto front, for different sizes of 0-1 MOKPs with two objectives. The proposed method offers very good performance when compared to the classic NSGA-II, MOEA/D and Global WASF-GA algorithms, all of them well-known in the literature.

Fitness Landscape Analysis on Binary Dynamic Optimization Problems

Fitness Landscape Analysis (FLA) denotes the task of analyzing black-box optimization problems and capturing their characteristic features with the goal of providing additional information, that may help in algorithm selection, parametrization or guidance. Many real-world optimization tasks require dynamic on-going optimization and a plethora of meta-heuristic algorithms has been introduced for this task. However, most analysis focuses on static problems or dynamic optimization tasks without time-linkage, where the dynamic changes of the problem are independent of the decisions taken by the optimizer, but many real-world optimization problems display very heavy dependence on previous states and decisions. In this paper, the techniques of the static FLA are combined with dynamic and domain specific measures and applied to two dynamic problems. A time-linked dynamic OneMax problem and a dynamic multi-objective knapsack problem are presented and the impact of time-linkage on their FLA features is analyzed.

Improved Memetic NSGA-II Using a Deep Neighborhood Search

Elitist non-sorted genetic algorithms as part of Pareto-based multi-objective evolutionary algorithms seems to be one of the most efficient algorithms for multi-objective optimization. However, it has some shortcomings, such as low convergence accuracy, uneven Pareto front distribution, and slow convergence. A number of review papers using memetic technique to improve NSGA-II have been published. Hence, it is imperative to improve memetic NSGA-II by increasing its solving accuracy. In this paper, an improved memetic NSGA-II, called deep memetic non-sorted genetic algorithm (DM-NSGA-II), is proposed, aiming to obtain more non-dominated solutions uniformly distributed and better converged near the true Pareto-optimal front. The proposed algorithm combines the advantages of both exact and heuristic approaches. The effectiveness of DM-NSGA-II is validated using well-known instances taken from the standard literature on multi-objective knapsack problem. As will be shown, the performance of the proposed algorithm is demonstrated by comparing it with M-NSGA-II using hypervolume metric.

Solving binary multi-objective knapsack problems with novel greedy strategy

Solving binary multi-objective knapsack problems with novel greedy strategy

Min cost improvement and max gain stability in multicriteria sorting methods on combinatorial domains

AbstractVarious multicriteria sorting methods have been proposed in the literature to assign the feasible alternatives into predefined categories. We consider here problems involving a set of totally ordered categories representing different achievement levels in the satisfaction of criteria. As in many existing methods, the assignment rule of an alternative to a category is based on the comparison of its performance vector to reference profiles defining lower bounds of the categories. Within this standard setting we address a new problem that consists in finding how to modify a given solution, within a combinatorial set of alternatives, to upgrade it in the upper category (or higher) at minimum cost. We also consider the problem of identifying the sequence of solutions that minimize the total cost while satisfying some budget constraint at every step, and the problem of determining how to modify the current solution to save money while staying in the same category. We first propose a general approach based on mixed integer (linear or quadratic) programming to solve these problems. Then, we implement this approach on various multiobjective combinatorial problems, such as multi‐agent assignment problems and multiobjective knapsack problems. Numerical tests are provided to establish the feasibility of the approach on instances of different sizes.

Explainable Interactive Evolutionary Multiobjective Optimization

We present a new interactive procedure for evolutionary multiobjective optimization involving the decision rule preference model in the search for the best compromise solution. As usual in the interactive multiobjective optimization, preference elicitation phases alternate with optimization phases, however, in this evolutionary procedure, one preference elicitation phase falls into many iterations of the evolutionary multiobjective optimization. In the preference elicitation phase, the Decision Maker (DM) is asked to select those solutions she considers relatively good in the current population. Using the Dominance-based Rough Set Approach (DRSA) this information is represented in terms of ``if ..., then ... decision rules which represent DM’s preferences. They are used in the next phases of the evolutionary multiobjective optimization to iteratively converge towards the part of the Pareto front containing the best compromise solution. Besides guiding the search process, the decision rules can be read as arguments explaining the DM's choices. In this way, the proposed method implements the postulate of transparency and explainability expected from interactive procedures, because the DM has a chance to understand how her answers given in the preference elicitation phase are translated into guidelines for the algorithm in the optimization phase, i.e., what is the impact of the answers on newly proposed compromise solutions. This can be considered the distinctive aspect of what we define explainable multiobjective optimization evolutionary approach (XIMEA). In particular, the method proposed in this paper is called XIMEA-DRSA. From the decision psychology point of view, in line with the main results of behavioral sciences, the decision rules support the DM to construct and learn her preferences during the decision process. For its explanatory feature resulting from the conjunction of evolutionary multiobjective optimization with DRSA, the method has been called XIMEA-DRSA. The computational effectiveness of the method has been tested on both continuous and discrete multiobjective optimization problems. In the latter case, the XIMEA-DRSA method has been applied to the multiobjective knapsack problem.

Improvement of the branch and bound algorithm for solving the knapsack linear integer problem

The paper presents a new reformulation approach to reduce the complexity of a branch and bound algorithm for solving the knapsack linear integer problem. The branch and bound algorithm in general relies on the usual strategy of first relaxing the integer problem into a linear programing (LP) model. If the linear programming optimal solution is integer then, the optimal solution to the integer problem is available. If the linear programming optimal solution is not integer, then a variable with a fractional value is selected to create two sub-problems such that part of the feasible region is discarded without eliminating any of the feasible integer solutions. The process is repeated on all variables with fractional values until an integer solution is found. In this approach variable sum and additional constraints are generated and added to the original problem before solving. In order to do this the objective bound of knapsack problem is quickly determined. The bound is then used to generate a set of variable sum limits and four additional constraints. From the variable sum limits, initial sub-problems are constructed and solved. The optimal solution is then obtained as the best solution from all the sub-problems in terms of the objective value. The proposed procedure results in sub-problems that have reduced complexity and easier to solve than the original problem in terms of numbers of branch and bound iterations or sub-problems. The knapsack problem is a special form of the general linear integer problem. There are so many types of knapsack problems. These include the zero-one, multiple, multiple-choice, bounded, unbounded, quadratic, multi-objective, multi-dimensional, collapsing zero-one and set union knapsack problems. The zero-one knapsack problem is one in which the variables assume 0 s and 1 s only. The reason is that an item can be chosen or not chosen. In other words there is no way it is possible to have fractional amounts or items. This is the easiest class of the knapsack problems and is the only one that can be solved in polynomial by interior point algorithms and in pseudo-polynomial time by dynamic programming approaches. The multiple-choice knapsack problem is a generalization of the ordinary knapsack problem, where the set of items is partitioned into classes. The zero-one choice of taking an item is replaced by the selection of exactly one item out of each class of items

Solution approaches for equitable multiobjective integer programming problems

We consider multi-objective optimization problems where the decision maker (DM) has equity concerns. We assume that the preference model of the DM satisfies properties related to inequity-aversion, hence we focus on finding nondominated solutions in line with the properties of inequity-averse preferences, namely the equitably nondominated solutions. We discuss two algorithms for finding good subsets of equitably nondominated solutions. The first approach is an extension of an interactive approach developed for finding the most preferred nondominated solution when the utility function is assumed to be quasiconcave. We find the most preferred equitably nondominated solution when the utility function is assumed to be symmetric quasiconcave. In the second approach we generate an evenly distributed subset of the set of equitably nondominated solutions to be considered further by the DM. We show the computational feasibility of the two algorithms on equitable multi-objective knapsack problem, in which projects in different categories are to be funded subject to a limited budget. We perform experiments to show and discuss the performances of the algorithms.

Solving single and multi-objective 01 Knapsack Problem using Harmony Search Algorithm

Solving single and multi-objective 01 Knapsack Problem using Harmony Search Algorithm

Multiobjective cooperative swarm intelligence algorithm for uplink resource allocation in LTE‐A networks

AbstractNowadays, Long‐Term Evolution Advanced (LTE‐A) network is the primary innovation in 4G networks. The LTE‐A networks convey exceptional data rates and low latency for few sorts of application. Sometimes, in LTE‐A, the multiobjective uplink resource allocation is the perplexing optimization problem, and this problem is considered as 0‐1 multiobjective knapsack trouble. In order to overcome this trouble, a multiobjective cooperative swarm intelligence algorithm for resource allocation is proposed in this paper. In our proposed work, hybrid firefly algorithm (FFA) and particle swarm optimization (PSO) algorithm are utilized to solve 0‐1 multiobjective knapsack problem. Initially, a priority and urgency factor (urgency of packets)–based user ranking and quantifying scheme is designed for scheduling process. After the scheduling process, the resource allocations employ three objective functions, such as maximization of resource utilization, maximization of quality of service (QoS), and interference minimization. The optimization problem is overcome by using the hybrid FFA and PSO algorithm. The experimental outcomes demonstrate that it has the best QoS and less interference in the resource allocation in LTE‐A network than state‐of‐art methods in the proposed strategy.

FPBH: A feasibility pump based heuristic for multi-objective mixed integer linear programming

Feasibility pump is one of the successful heuristics developed almost a decade ago for computing high-quality feasible solutions of single-objective integer linear programs, and it is implemented in exact commercial solvers such as CPLEX and Gurobi. In this study, we present the first Feasibility Pump Based Heuristic (FPBH) for approximately generating nondominated frontiers of multi-objective mixed integer linear programs with an arbitrary number of objectives. The proposed algorithm extends our recent study for bi-objective pure integer programs and employs a customized version of several existing algorithms in the literature of both single-objective and multi-objective optimization. The method has two desirable characteristics: (1) There is no parameter to be tuned by users other than the time limit; (2) It can naturally exploit parallelism. An extensive computational study shows the efficacy of the proposed method on some existing standard test instances in which the true frontier is known, and also some randomly generated instances. We also numerically show the importance of parallelization feature of FPBH and illustrate that FPBH outperforms MDLS developed by Tricoire (2012) on instances of multi-objective knapsack problem. We test the effect of using different commercial and non-commercial linear programming solvers for solving linear programs arising during the course of FPBH, and show that the performance of FPBH is almost the same in all cases. It is worth mentioning that FPBH is available as an open source Julia package, named as ‘FPBH.jl’, on GitHub. The package is compatible with the popular JuMP modeling language and supports input in LP and MPS file formats. The package can plot nondominated frontiers, can compute different quality measures (hypervolume, cardinality, coverage and uniformity), supports execution on multiple processors, and can use any linear programming solver supported by MathProgBase.jl (such as CPLEX, Clp, GLPK, etc).

Integer cat swarm optimization algorithm for multiobjective integer problems

In the literature, several variants of cat swarm optimization (CSO) algorithm are reported. However, CSO for integer multiobjective optimization problems (MOPs) has not yet been investigated. Owing to the frequent occurrence of integer MOPs and their importance in practical design problems, in this work, we investigate a new CSO approach for solving purely integer MOPs. This new approach named as multiobjective integer cat swarm optimization (MO-ICSO) algorithm incorporates the modified version of the CSO algorithm for MOPs. This approach is comprised of the concepts of rounding the floating points to the nearest integer numbers and the probabilistic updating (PU) technique. It uses the idea of Pareto dominance for finding the non-dominated solutions and an external archive for storing these solutions. We demonstrate the power of this new approach via its quantitative analysis and sensitivity test of its several parameters using different performance metrics performed over multiobjective multidimensional knapsack problem and several standard test functions. The simulation results argue that the proposed MO-ICSO approach can be a better candidate for solving the integer MOPs.

A parallel MOEA with criterion-based selection applied to the Knapsack Problem

In this paper, we propose a parallel multiobjective evolutionary algorithm called Parallel Criterion-based Partitioning MOEA (PCPMOEA), with an application to the Multiobjective Knapsack Problem (MOKP). The suggested search strategy is based on a periodic partitioning of potentially efficient solutions, which are distributed to multiple multiobjective evolutionary algorithms (MOEAs). Each MOEA is dedicated to a sole objective, in which it combines both criterion-based and dominance-based approaches. The suggested algorithm addresses two main sub-objectives: minimizing the distance between the current non-dominated solutions and the ideal point, and ensuring the spread of the potentially efficient solutions. Experimental results are included, where we assess the performance of the suggested algorithm against the above mentioned sub-objectives, compared with state-of-the-art results using well-known multi-objective metaheuristics.

A gradual weight-based ant colony approach for solving the multiobjective multidimensional knapsack problem

The multiobjective multidimensional knapsack problem (MOMKP) is an extension of the multiobjective knapsack problem that consists in selecting a subset of items in order to maximize m objective functions. The MOMKP creates an additional difficulty than the monodimensional version caused by the fact of respecting more than one constraint simultaneously. In this paper, we propose to solve the MOMKP with an ant colony optimization approach based on a gradual weight generation method, named Gw-ACO. Here, the weight vectors are gradually distributed in the objective space and change relatively to the optimization process. This enables ants to target, at each cycle, different regions in order to try to achieve almost all solutions covering the Pareto front. To evaluate the suggested Gw-ACO approach, a set of experiments is performed on MOMKP benchmark instances and compared with well-known state-of-the-art metaheuristic approaches. The obtained experimental results show that Gw-ACO is significantly better and able to achieve a well distribution all over the Pareto-optimal front.