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Top1. Introduction
Maximization of system reliability over redundancy at every stage of operation can be formulated as non-linear integer programming problem involving nonlinear constraints. The redundancy allocation problem is a quite effective and important topic in the areas of reliability optimizations. The problem in reliability design in which redundant components are added is often termed as the redundancy allocation problem (RAP). The basic objective of this problem is to raise the system reliability by increasing the reliability of each subsystem so as to attain a prefixed reliability plan for system as a whole, subject to various available resource constraints on the system/subsystem. Across the preceding, most of the reliability engineers/researchers/planners have given more attention in solving this problem. On the whole, RAP has been formulated as a non-linear integer/mixed-integer programming problem. Chern (1992) has investigated that even a simple redundancy allocation problem in series systems with linear constraints is NP- hard. And it is well studied and summarized by Tillman et al. (1977), Kuo et al. (2001) and Kuo and Prasad (2001). For solving such problems, several deterministic methods, like heuristic methods (Kim &Yum, 1993; Nakagawa & Nakashima, 1977) mixed-integer nonlinear programming (Tillman et al., 1977), reduced gradient method (Hwang et al., 1979), integer programming (Misra & Sharma, 1991), linear programming approach (Kim & Yum, 1993), dynamic programming method (Kuo et al., 2001), branch and bound method (Sun & Li, 2002) were used in the initial stage of development. However, these methods have some advantages and disadvantages also. From the computational point of view NP-hard problems are classified as highly complex and it is very unlikely that simple and efficient algorithm can be employed to find out exact optimal solutions. Therefore, it is very rational to depend on heuristic methods which give a reasonably good solution but not the exact one with less computational complexities. Several heuristic methods have been developed in the last few decades for solving the RAP and after the development of evolutionary algorithms; researchers gave their prior attention to use these algorithms in solving RAP. These algorithms are of heuristic in nature and have the characteristic of wider flexibility. Moreover, these algorithms require lesser assumptions for solving complicated optimization problems. In addition, these algorithms are applicable in a search space where variables are either discrete, continuous or both i.e. mixed. For these features, the researchers have been attracted to employ such heuristic algorithms for solving highly complex types of optimization problems. Lots of works are available in the existing literature in the areas of reliability optimization and also in redundancy allocation problems. Most of the studies on reliability optimization, the crisp environment has been considered in which the design parameters of the problem have been considered as precise valued. This implies that every probability involved for determining the values of the parameters is perfectly known. However, it is quite contradictory to the real life situation as nothing is absolutely determinable and measurable perfectly. So, there should be some ambiguities in estimating the parameters. That is why the impreciseness came into consideration in the study of reliability optimization. Generally, different approaches have been used to represent impreciseness. These approaches are mainly fuzzy, interval, stochastic/probabilistic. The reliability of a component of a system and also other design parameters may be imprecise numbers those can be represented by different environment, like fuzzy (Mahapatra & Roy, 2012; Mahato et al., 2013) stochastic and interval approaches. In the area of reliability optimization of interval approach one may refer to the works of Feizollahi and Modarres (2012), Sahoo et al. (2010, 2012 & 2014), Mahato et al. (2012) and Bhunia et al. (2010). Also, some works are available in intuitionistic fuzzy environment.