In this article using Cuckoo Optimization Algorithm and simple additive weighting method the hybrid COAW algorithm is presented to solve multi-objective problems. Cuckoo algorithm is an efficient and structured method for solving nonlinear continuous problems. The created Pareto frontiers of the COAW proposed algorithm are exact and have good dispersion. This method has a high speed in finding the Pareto frontiers and identifies the beginning and end points of Pareto frontiers properly. In order to validation the proposed algorithm, several experimental problems were analyzed. The results of which indicate the proper effectiveness of COAW algorithm for solving multi-objective problems.
There are many methods for solving nonlinear constrained programming problems such as Newton, Genetic algorithm, the algorithm of birds and so on. In this paper using the emerging Cuckoo Optimization Algorithm and simple additive weighting a method to solve multi-objective problems is presented.
In single-objective optimization, it is assumed that the decision makers communicate only with one goal like: profit maximization, cost minimization, waste minimization, share minimization and so on. But in the real world it is not possible to consider single goals and usually more than one goal are examined. For example, in the control of the projects if only the time factor is considered, other objectives such as cost and quality are ignored and the results are not reliable. So it is necessary to use multi-objective optimization problems.
Ehrgott and Gandibleux presented a detailed approximation method regarding the problems related to combinatorial multi-objective optimization [1]. Klein and Hannan for multiple objective integer linear programming problems (MOILP) presented and algorithm in which some additional restrictions is used to remove the known dominant solutions [2]. Sylva and Crema offered a method to find the set of dominant vectors in multiple objective integer linear programming problems [3]. Arakawa et al. used combined general data envelopment analysis and Genetic Algorithm to produce efficient frontier in multi-objective optimization problems [4].
Deb analyzed the solution of multi-objective problems by evolutionary algorithms [5]. Reyesseerra and Coello Coello analyzed the solution of multi-objective problems by particle swarm [6]. Cooper et al. have worked on the solution of multi-objective problems by the DEA and presenting an application [7]. Pham and Ghanbarzadeh solved multi-objective problems by bee algorithm [8]. Nebro et al. analyzed a new method based on particle swarm algorithm for solving multiobjective optimization problems [9]. Gorjestani et al. proposed a COA multi objective algorithm using DEA method [10].
For multi-objective optimization problems usually it is not possible to obtain the optimal solution that simultaneously optimizes all the targets in question. Therefore we should try to find good solutions rather than the optimal ones known as Pareto frontier. Given that so far the Simple Additive Weighting method is not used in meta-heuristic, especially cuckoo algorithms, this paper presents a combined method.
The first section introduces Cuckoo optimization algorithm, then in the second section Simple Additive Weighting (SAW) method is discussed as a combined method for solving multiobjective described. Finally, the fourth section provides the proposed implemented approach, numerical results and a comparison which is made with other methods.
Cuckoo optimization algorithm was developed by Xin-She Yang and Suash Deb in 2009. Thence Cuckoo optimization algorithm was presented by Ramin Rajabioun in 2011 [11]. Cuckoo algorithm flowchart is as figure 1. This algorithm applied in several researches such as production planning problem [12], portfolio selection problem [13], evaluation of organization efficiency [14], evaluation of COA [15] and so on. For more information about the algorithm refer to [11].
SAW is one of the most practical methods designed for decision-making with multiple criteria presented by Hong and Eun in 1981. In this method which is also known as weighted linear combination after scaling the decision matrix by weighted coefficients of criteria, the free scale weighted decision matrix id obtained and according to this scale the score of each option is selected. The most important feature of this method is the simple application because of its mathematical logic.
Assuming the multiple target model (1) and defining the parameters w 1 and w 2 which are the weight of the objective functions and defined based on the importance of the functions by the decision maker, the model can be converted to single-objective models (2):
In these models x …
x are objective functions. is the weight defined by the importance of the decision maker.
In this section we present the method algorithm are as follows. Also the flowchart of COAW algorithm is as figure 2.
Step1 Different random w
In this section we present the method COAW which is proposed in this paper. The steps of this algorithm are as follows. Also the flowchart of COAW algorithm is as figure 2.
In this section in order to validat problems are presented in Table 1.
In this section in order to validation the COAW algorithm some test problems are analyzed.
esented in Table 1. Given that determining input parameters is one of the effective problems in meta-heuristic algorithms, so the parameters of the algorithm are presented as follows: the number of initial population=5, minimum number of eggs for each cuckoo= 2, maximum number of eggs for each cuckoo =4, maximum iterations of the Cuckoo Algorithm=50, number
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