Genetic algorithm based multi-objective reliability optimization in interval environment

Abstract

In most of the real world design or decision making problems involving reliability optimization, there are simultaneous optimization of multiple objectives such as the maximization of system reliability and the minimization of system cost, weight and volume. In this paper, our goal is to solve the constrained multi-objective reliability optimization problem of a system with interval valued reliability of each component by maximizing the system reliability and minimizing the system cost under several constraints. For this purpose, four different multi-objective optimization problems have been formulated with the help of interval mathematics and our newly proposed order relations of interval valued numbers. Then these optimization problems have been solved by advanced genetic algorithm and the concept of Pareto optimality. Finally, to illustrate and also to compare the results, a numerical example has been solved.

Introduction

Most of the real-world design or decision making problems involving reliability optimization require the simultaneous optimization of more than one objective function. Mostly, the reliability optimization problems have been formulated by researchers by single objective optimization problems. An early study in this field was reported by Sakawa (2002). For simultaneous maximization of system reliability and minimization of system cost of reliability allocation he formulated and solved a multi-objective problem using a surrogate worth trade-off method (Sakawa, 1978). Around the same time, Inagaki, Inoue, and Akashi (1978) solved a different problem by maximizing the system reliability and minimizing the system cost and weight by implementing an interactive optimization method. To develop an overview on the trend of research in this area, one may refer to the works of Park, 1987, Dhingra, 1992, Rao and Dhingra, 1992, Srinivas and Deb, 1994, Ravi et al., 2000, Huang et al., 2005, Coit and Konak, 2006 and others. In the recent years, Taboada and Coit (2007) proposed a new method which is based on the sequential combination of multi-objective evolutionary algorithms and data clustering on the prospective solutions. In addition Taboada, Baheranwala, Coit, and Wattanapongsakorn (2007) proposed two different approaches to reduce the size of the Pareto optimal set for multi-objective reliability optimization design problems. Out of those two approaches, in the first approach, a pseudo-ranking scheme to select the solutions by the decision maker according to their objective function priorities was developed. On the other hand in their second approach, demonstrates the use of data mining clustering techniques to group the data with the implementation of k-means algorithm to find the clusters of similar solutions. In the same year, Ramirez-Marquez and Coit (2007) proposed a multi-state component critical analysis for the improvement of reliability in multi-state systems. In the same area, Taboada, Espiritu, and Coit (2008a) presented an extension and applied a previously developed multi-objective evolutionary algorithm for solving the design allocation problems of multi-state series–parallel system for power system. Taboada, Espiritu, and Coit (2008b) solved multiple objective multi-state reliability optimization design problems by maximizing system reliability and minimizing both the system cost and weight. In the year 2009, Li, Liao, and Cioit (2009) proposed a two-stage approach for multi-objective decision making with applications to system reliability optimization. Ramirez-Marquez and Rocco (2010) developed a new evolutionary optimization technique for multi-state two-terminal reliability allocation in multi-objective problems. With the view to identify the combination of component failures that provide maximum reduction of network performance, Rocco, Ramirez-Marquez, Salazar, and Hernandez (2010) studied the vulnerability analysis of a complex network. Several researchers have solved reliability optimization problems with single objective (Aggarwal and Gupta, 2005, Coit and Smith, 1996, Gopal et al., 1980, Ha and Kuo, 2006, Hikita et al., 1992, Kim and Yum, 1993, Kuo and Prasad, 2000, Kuo et al., 2001, Misra and Sharma, 1991, Nakagawa and Nakashima, 1977). Most of the reliability optimization problems with single objectives or multi-objective are based on the assumption of fixed/constant reliabilities of components which lie between zero and one. However, in real-life situations, the reliability of an individual component may not be fixed. It may vary due to several reasons, such as improper storage facilities, the human factor and other factors relating to environment. There is no technology by which different components can be produced with exactly identical reliabilities. So, the reliability of each component is sensible and it may be treated as a positive imprecise number instead of a fixed real number. To tackle the problem with such imprecise numbers, generally stochastic, fuzzy and fuzzy- stochastic approaches are applied and the corresponding problems are converted to deterministic problems for the purpose of solving. In the stochastic approach, the parameters are assumed to be random variables with known probability distributions. In the fuzzy approach, the parameters, constraints and goals are considered as fuzzy sets with known membership functions or fuzzy numbers. On the other hand, in the fuzzy-stochastic approach, some parameters are viewed as fuzzy sets/fuzzy numbers and others as random variables. However, it is a formidable task for a decision maker to specify the appropriate membership function for a fuzzy approach and probability distribution for a stochastic approach and both for the fuzzy stochastic approach. So, to avoid these difficulties for handling the imprecise numbers by different approaches, one may use an interval number to represent the imprecise number, as this representation is the most significant representation among others. Studies of the system reliability by considering the component reliabilities as imprecise have already been initiated by some researchers like Coolen and Newby, 1994, Utkin and Gurov, 1999, Utkin and Gurov, 2001, Gupta et al., 2009, Bhunia et al., 2010 and Sahoo, Bhunia, and Roy (2010). In the single objective optimization, one attempts to obtain the best design or decision, which is usually a global minimum or the global maximum depending on whether the optimization problem is of minimization or maximization type. On the other hand for the multiple objectives, there may not exist one solution which is best (global minimum or maximum) with respect to all the objectives. In multi-objective optimization, there exists a set of solutions which are superior to the rest of the solutions in the search space when all the objectives are considered, but are inferior to other solutions in the space in one or more objectives (not all). These solutions are known as Pareto optimal solutions or nondominated solutions (Srinivas and Deb (1994)) and the rest of the solutions are known as dominated solutions. Since none of the solutions in the nondominated set can be considered as absolutely better than one another, any one of them is an acceptable solution. As reliability of each component is interval valued, therefore, the system reliability would be interval valued. In this paper, GA-based approach has been presented for solving the multi-objective reliability optimization with interval objectives. The objectives considered here, are the maximization of the system reliability and minimization of the system cost. Also, we have considered the cost coefficient as interval valued. For this purpose several problems having multi-objective reliability optimization problems with interval valued objectives have been formulated and solved. In this connection, we have also developed the definition of Pareto optimality in interval environment. To obtain the optimal solution of multi-objective optimization problem we have converted the same into a single objective constrained optimization problem. Further, the reduced optimization problem has been converted into unconstrained optimization problem by using penalty function technique. For solving such typical problems, we have developed a real coded elitist GA with tournament selection, uniform crossover and one-neighborhood mutation. Finally, to illustrate the different approaches based on different multi-objective optimization techniques, a numerical example has been solved and to investigate the overall performance of the proposed GA based penalty technique for solving multi-objective optimization problems, sensitivity analyses have been carried out graphically.

The organization of the paper is given as follows. In Section 2, the assumptions and notations are given. The details of finite interval mathematics and interval order relations are given in Section 3. Section 4 presents the details of multi-objective optimization and problem formulation in interval environment. In Section 5, genetic algorithm based constraints handling approach is discussed. Section 6 presents numerical example and sensitivity analysis to illustrate the proposed GA based penalty technique. In Section 7, concluding remarks is presented to draw the conclusion from this research work.