Multi-objective Fuzzy Reliability Optimization Model : A Parametric Geometric Programming Approach

This paper presents a multi-objective reliability optimization model taking system reliability and cost of a series system as objective functions. Due to the vagueness of judgements of the decision maker, the objective as well as constraint goal can involve many uncertain factor and other imprecise parameters with vague in nature in a reliability optimization model. Thus the model is formulated in fuzzy environment by considering cost coefficients and the exponential factor as triangular fuzzy number. Here, the nearest interval approximation method is applied to make the fuzzy model in crisp in nature. There are two types of parametric geometric programming technique is used to solve the proposed model. The performance of these two types of solution approach is evaluated by numerical example at the end of this paper.


Introduction
Since 1960, due to the importance of reliability in various kinds of system, like communication and transportation system, reliability optimization has attracted many researchers.In the past half century, numerous well written books on reliability optimization have been made available.The basic goal of reliability optimization problem is always to maximize the reliability of the system and to minimize the system cost.W.kuo, V.R.Prsad in [2], [4] presented some method for solving reliability optimization model.Geometric programming (GP) has been used in various fields for solving a special type of non-linear programming and becomes a very useful tool for solving optimization problems.Duffin, Preterson & Zener [11] discussed their basic theories on GP in their books and also S.Islam and T.K.Roy [1] presented some theories and applications on modified GP.When GP is applied in reliability optimization models, due to some uncertainty in judgements, there are some coefficients and parameter, which are always imprecise with vague in nature.Zimmerman [10] first applied the fuzzy set theory concept with some suitable linear membership functions to solve linear http://www.ispacs.com/journals/jfsva/2016/jfsva-00308/International Scientific Publications and Consulting Services programming problem with several objective functions.Later Cao [9] first introduced fuzzy geometric programming (FGP) problem and solve fuzzy reliability optimization models using FGP problem and solution techniques on series system.S.Islam [8] discussed the fuzzy geometric programming approach to a fuzzy EPQ model and later, many research work have been done using fuzzy parametric geometric programming (FPGP) method on single objective function.S.Dey and T.K.Roy [5] applied fuzzy parametric geometric programming in structural model and also FPGP is provided in [6], [7].But this is very rare when FPGP is applied to a multi-objective problem.In this paper, a multi-objective reliability optimization model is considered, where to maximize the system reliability and to minimize the cost of the system.We have introduced the cost coefficients and exponential factor as triangular fuzzy number.The weighted sum method is firstly applied to convert the multi-objective fuzzy reliability optimization model to a single objective model.To make the fuzzy model into crisp, the nearest interval approximation method is used.Here, we applied two types of parametric geometric approach to solve the proposed multi-objective model.

Mathematical model
Let us consider the series system of reliability with N-components with each component has reliability   for the i-th subsystem / component (for i=1,2,…,N).Now the reliability of the system is given by

𝛼 𝑖 𝑁 𝑖=1
Where   is represents the total cost of the system.Here   is the proportionality constant and   is the exponential factor relates the cost and reliability of each component.

Multi-objective Reliability optimization model
According to Tillman [2], in a single-objective optimization problems, either the system reliability is maximized subject to limits on resource consumption, or the consumption of the resource is minimized subject to a lower limit system reliability and other resource constraints.Now for designing a reliability system, when the limits of the resource consumption cannot be determined properly, it is always better to consider multiple objective approach to design a system.Thus we consider the following reliability optimization model and our object is to maximize the system reliability and to minimize the cost of the system, which is very much stable.Subject to, 0.5 ≤  , ≤   ≤ 1; i=1,2,…,N; (2.1) Where  , is the lower bound of the reliability of each component (for i=1,2,…,N).To solve the above multi-objective problem using geometric programming approach, the problem should be in minimization form.Thus, the suitable form of optimization model is taken as Min   ( 1 ,

Weighted Sum Method
The most common approach to multi-objective optimization is the weighted sum (WS) method.The basic idea of weighted sum method is to convert the multi-objective problem (MOP) into a single objective optimization problem by using convex combination of objectives.
Given a subset X of R P and p functions   : X → R for k = 1,2,…,p.We define MOP as follows: where F: X→ R P is the objective function vector and X = { = ( 1 ,  2 , … ,   ) ∈ R n }.
Thus the WS method solves the following scalar optimization problem where w k ≥ 0, for k =1,2, …., p.
It must be noted that using the WS method, it is impossible to obtain points on non-convex portions of the pareto optimal set in the criterion space.Thus using WS method, (2.1.2) reduces to a single objective function as follows - (2.

Fuzzy Reliability Optimization Model
In practical life, the objective as well as constraint goal in reliability optimization problem may be involve uncertain parameters and also due to some uncertainty in judgements, it is not always possible to get consequential data for the reliability, thus using fuzzy number, it can be possible for the decision maker to specify the membership functions of the constraints and the goal of the optimization model.Thus considering the cost coefficients   and the exponential factor   as triangular fuzzy number, the above reliability optimization problem (3.3)  ) Subject to,   > 0, (1 ≤ p ≤ n, 1 ≤ j ≤ m); (5.6) Here  ̃ and  ̃ are the triangular fuzzy numbers.Now using β-cut of fuzzy number, the above problem reduces to - (5. (5.9) Case I: When  ≥ ( + ), the dual problem presents system of linear equations for the dual variables.A solution vector for the dual variable exists.
Case II: When  ≤ ( + ), in this case generally no solution vector for the dual variable exists.However, using either the least square method or Max-Min method one can get n approximate solution vector for the system.It is a system of linear equations in log (  ) (1 ≤  ≤ ; 1 ≤  ≤ ).Since there are more primal variables   than the number of terms T, many solutions   may exists.So, to find the optimal primal variables   , it remains to minimum the primal objective functions with respect to reduce  − (≠ 0) variaables.When  −  = 0 i.e. number of primal variables = number of log-linear equations, primal variables can be determined uniquely from log-liner equations.

Type (I) (A). Parametric interval-valued function:
Let [p,q] be an interval, where (p > 0,  > 0).The parametric interval-valued function for any interval [p,q] can be taken as g(t) = p 1−s q s for s ∈ [0,1], which is strictly monotone, continuous function and its inverse exists.

(B). Parametric approach using parametric interval-valued function:
The reliability optimization model in (4.5) reduces to a parametric programming by replacing  ̂ = (   )

Type (I):
Table 1: The fuzzy input data of the coefficients  ̃ and the factor  ̃ for the type (I) parametric geometric programming approach is given in the following table Table 2: The optimal solutions of the fuzzy reliability optimization model for the same weightages of system reliability ( 1 ) and the system cost ( 2 ) by type (I) interval-valued parametric programming approach is presented in the following table (for  1 = 0.5,  2 = 0.5) Table 2 shows that system cost increases with the parameter , but system reliability decreases as the parameter  increases.

Type (II):
Now for the type (II) parametric geometric programming approach, the fuzzy input data of the coefficients  ̃ and the factor  ̃ is given by  ̃1 = 10 + (1 − )0.5 ,  ̃2 = 10 + (1 − )0.6 ,  ̃1 =  ̃2 = 0.The optimal solutions of the fuzzy reliability optimization model for the same weightages of system reliability (w 1 ) and the system cost (w 2 ) by type (II) parametric programming approach is presented in the following table (for w 1 = 0.5, w 2 = 0.5): Table 3 shows that system reliability increases with the parameterβ, but system cost decreases as the parameter β increases.

Conclusions
Here we introduced two types of parametric-geometric programming technique to find the optimal solutions of the multi-objective reliability optimization model.The parametric geometric programming method provides an alternative approach to the proposed model.The weighted sum method is used to combine multi-objective functions into a single objective function.Also here we consider the cost coefficients and the exponential factor as triangular fuzzy number.The aim of this paper is to find the optimal solutions of the model which maximize the system reliability and to minimize the cost of the system.The method presented is quite general and practical which can be applied to the model in other areas of operation research and engineering sciences.