Interval Multi-objective Optimization Problems Research Articles

Decomposition-based interval multi-objective evolutionary algorithm with adaptive adjustment of weight vectors and neighborhoods

Interval multi-objective optimization problems (IMOPs) are one of the most critical optimization problems in practical applications. However, compared to deterministic multi-objective optimization problems (MOPs), there are few researchs addressing IMOP. In addition, the uncertainty contained in the problem makes the distribution of the population more challenging. Therefore, this paper proposed a decomposition-based interval multi-objective evolutionary algorithm with adaptive adjustment of weight vectors and neighborhoods (IMOEA/D-AWN). Firstly, an interval sparsity level function (ISL) is constructed to measure the density of individuals, and a comprehensive ranking of interval sparsity ranking and interval uncertainty ranking is proposed. For the purpose of improving the distribution of the population while reducing its uncertainty, based on the above comprehensive ranking, a new adaptive adjustment weight vector strategy guided by interval elite population is designed. Besides, an adaptive adjustment neighborhoods strategy is designed. This strategy adjusts individuals' neighborhood size according to the number of iterations to improve the efficiency of evolution. Finally, the IMOEA/D-AWN is evaluated on 17 interval benchmark test problems and a collaborative computation offloading optimization problem, and compared with four advanced multi-objective evolutionary algorithms with interval parameters (IMOEAs). Experimental results show that this algorithm performs well in convergence, diversity, and uncertainty.

Interval multi-objective grey wolf optimization algorithm based on fuzzy system

PurposeThe interval multi-objective optimization problems (IMOPs) are universal and vital uncertain optimization problems. In this study, an interval multi-objective grey wolf optimization algorithm (GWO) based on fuzzy system is proposed to solve IMOPs effectively.Design/methodology/approachFirst, the classical genetic operators are embedded into the interval multi-objective GWO as local search strategies, which effectively balanced the global search ability and local development ability. Second, by constructing a fuzzy system, an effective local search activation mechanism is proposed to save computing resources as much as possible while ensuring the performance of the algorithm. The fuzzy system takes hypervolume, imprecision and number of iterations as inputs and outputs the activation index, local population size and maximum number of iterations. Then, the fuzzy inference rules are defined. It uses the activation index to determine whether to activate the local search process and sets the population size and the maximum number of iterations in the process.FindingsThe experimental results show that the proposed algorithm achieves optimal hypervolume results on 9 of the 10 benchmark test problems. The imprecision achieved on 8 test problems is significantly better than other algorithms. This means that the proposed algorithm has better performance than the commonly used interval multi-objective evolutionary algorithms. Moreover, through experiments show that the local search activation mechanism based on fuzzy system proposed in this study can effectively ensure that the local search is activated reasonably in the whole algorithm process, and reasonably allocate computing resources by adaptively setting the population size and maximum number of iterations in the local search process.Originality/valueThis study proposes an Interval multi-objective GWO, which could effectively balance the global search ability and local development ability. Then an effective local search activation mechanism is developed by using fuzzy inference system. It closely combines global optimization with local search, which improves the performance of the algorithm and saves computing resources.

A Similarity-Based Cooperative Co-Evolutionary Algorithm for Dynamic Interval Multiobjective Optimization Problems

Dynamic interval multiobjective optimization problems (DI-MOPs) are very common in real-world applications. However, there are few evolutionary algorithms (EAs) that are suitable for tackling DI-MOPs up to date. A framework of dynamic interval multiobjective cooperative co-evolutionary optimization based on the interval similarity is presented in this paper to handle DI-MOPs. In the framework, a strategy for decomposing decision variables is first proposed, through which all the decision variables are divided into two groups according to the interval similarity between each decision variable and interval parameters. Following that, two subpopulations are utilized to cooperatively optimize decision variables in the two groups. Furthermore, two response strategies, i.e., a strategy based on the change intensity and a random mutation strategy, are employed to rapidly track the changing Pareto front of the optimization problem. The proposed algorithm is applied to eight benchmark optimization instances as well as a multiperiod portfolio selection problem and compared with five state-of-the-art EAs. The experimental results reveal that the proposed algorithm is very competitive on most optimization instances.

Cooperative Artificial Bee Colony Algorithm With Multiple Populations for Interval Multiobjective Optimization Problems

In practical engineering optimization problems (such as risk assessments), the parameters of the objective functions can be intervals because of noise and uncertainty; however, such problems cannot be solved by traditional multiobjective optimization methods. Yet, very little study has addressed interval multiobjective optimization methods compared to traditional multiobjective optimization methods. Therefore, a novel interval multiobjective optimization method called the Interval Cooperative Multiobjective Artificial Bee Colony Algorithm (ICMOABC) based on multiple populations for multiple objectives and interval credibility is proposed. Interval credibility is selected as the interval dominant method. Interval credibility is easy to combine with multiobjective optimization methods because it can describe the mean and width of intervals without increasing the dimension of the objective functions. The proposed algorithm has M single-objective optimization subpopulations updated by artificial bee colony algorithm, meaning it uses evolutionary resources more efficiently. In order to bring in diversity, the elitist learning strategy is used in the archive. The results of ICMOABC on various benchmark problems sets with different characteristics demonstrate its superior performance compared to some state-of-the-art algorithms.

Interval Multiobjective Optimization With Memetic Algorithms.

One of the most important and widely faced optimization problems in real applications is the interval multiobjective optimization problems (IMOPs). The state-of-the-art evolutionary algorithms (EAs) for IMOPs (IMOEAs) need a great deal of objective function evaluations to find a final Pareto front with good convergence and even distribution. Further, the final Pareto front is of great uncertainty. In this paper, we incorporate several local searches into an existing IMOEA, and propose a memetic algorithm (MA) to tackle IMOPs. At the start, the existing IMOEA is utilized to explore the entire decision space; then, the increment of the hypervolume is employed to develop an activation strategy for every local search procedure; finally, the local search procedure is conducted by constituting its initial population, whose center is an individual with a small uncertainty and a big contribution to the hypervolume, taking the contribution of an individual to the hypervolume as its fitness function, and performing the conventional genetic operators. The proposed MA is empirically evaluated on ten benchmark IMOPs as well as an uncertain solar desalination optimization problem and compared with three state-of-the-art algorithms with no local search procedure. The experimental results demonstrate the applicability and effectiveness of the proposed MA.

Multi-objective reliability redundancy allocation in an interval environment using particle swarm optimization

Most of the existing works addressing reliability redundancy allocation problems are based on the assumption of fixed reliabilities of components. In real-life situations, however, the reliabilities of individual components may be imprecise, most often given as intervals, under different operating or environmental conditions. This paper deals with reliability redundancy allocation problems modeled in an interval environment. An interval multi-objective optimization problem is formulated from the original crisp one, where system reliability and cost are simultaneously considered. To render the multi-objective particle swarm optimization (MOPSO) algorithm capable of dealing with interval multi-objective optimization problems, a dominance relation for interval-valued functions is defined with the help of our newly proposed order relations of interval-valued numbers. Then, the crowding distance is extended to the multi-objective interval-valued case. Finally, the effectiveness of the proposed approach has been demonstrated through two numerical examples and a case study of supervisory control and data acquisition (SCADA) system in water resource management.

Evolutionary algorithms with user’s preferences for solving hybrid interval multi-objective optimization problems

Hybrid interval multi-objective optimization problems are common in real-world applications. These problems involve both explicit and implicit objectives, and the values of these objectives are intervals. Few previous methods are suitable for them. An evolutionary algorithm with a large population and a user's interval preferences was presented to effectively solve the problems in this paper. In the proposed algorithm, a similarity-based strategy was employed to estimate the interval values of implicit objectives of evolutionary individuals that the user had not evaluated in order to alleviate user fatigue; the user's preferences to different objectives were expressed precisely as intervals by solving an auxiliary optimization problem; a sorting scheme based on the user's preferences was proposed to guide the population evolving toward the user's preferred regions. We applied the proposed method to an interior layout problem, which is a typical optimization problem with both interval parameters in the explicit objective and interval value of the implicit objective. The proposed algorithm was compared with four other optimization algorithms on the interior layout problem. Experimental results validated its effectiveness and superiority over the compared algorithms in terms of solution quality and the number of user's evaluations.

A study of interval metric and its application in multi-objective optimization with interval objectives

The aim of this paper is to discuss the optimality of interval multi-objective optimization problems with the help of different interval metric. For this purpose, we have proposed the new definitions of interval order relations by modifying the existing definitions and also modified different definitions of interval mathematics. Using the definitions of interval order relations and interval metric, the multi-objective optimization problem is converted into single objective optimization problem by different techniques. Then the corresponding problems have been solved by hybrid Tournament Genetic Algorithm with whole arithmetic crossover and double mutation (combination of non-uniform and boundary mutations). To illustrate the methodology, five numerical examples have been solved and the computational results have been compared. Finally, to test the efficiency of the proposed hybrid Tournament Genetic Algorithm, sensitivity analyses have been carried out graphically with respect to genetic algorithm parameters.

Interactive evolutionary algorithms with decision-maker׳s preferences for solving interval multi-objective optimization problems

Interval multi-objective optimization problems (IMOPs), whose parameters are intervals, are considerably ubiquitous in real-world applications. Previous evolutionary algorithms (EAs) aim at finding the well-converged and evenly-distributed Pareto front. An EA incorporating with a decision-maker (DM)׳s preferences was presented in this study to obtain a Pareto-optimal subset that meets the DM׳s preferences. In this algorithm, the DM׳s preferences in terms of the relative importance of objectives were interactively input, and the corresponding preferred regions were then obtained. Based on these regions, solutions with the same rank were further distinguished to guide the search towards the DM׳s preferred region. The proposed method was empirically evaluated on four IMOPs and compared with other state-of-the-art methods. The experimental results demonstrated the simplicity and the effectiveness of the proposed method.

Evolutionary algorithms with preference polyhedron for interval multi-objective optimization problems

Multi-objective optimization problems (MOPs) with interval parameters are ubiquitous in real-world applications. Existing evolutionary optimization methods, however, aim to obtain a set of well-converged and evenly-distributed Pareto-optimal solutions. In this paper, we presented a novel evolutionary algorithm (EA) that interacts with a decision maker (DM) during the optimization process to obtain the most preferred solution. The theory of preference polyhedron for the optimization problem with interval parameters was developed first. An interactive evolutionary algorithm (IEA) for MOPs with interval parameters was then proposed based on the theory of preference polyhedron. The algorithm periodically provides a set of non-dominated solutions to the DM; a preference polyhedron is constructed in the objective space by taking the worst solution chosen by the DM as the vertex. The solutions with the same rank are sorted based on the polyhedron constructed. Finally, our method was empirically evaluated on several MOPs with interval parameters where the value functions were used to simulate the DM’s responses. The numerical results indicated that our method is simpler and more efficient than the a posteriori method.