Multi-criteria reliability optiMization for a coMplex systeM with a bridge structure in a fuzzy environMent : a fuzzy Multi-criteria genetic algorithM approach

Reliability is central to productivity and effectiveness of real world industrial systems [22, 35]. To maximize productivity, the systems should always be available. However, it is difficult for an industrial system, comprising several complex components to survive over time since its reliability directly depends on the characteristics of its components. Failure is inevitable, such that system reliability optimization has become a key subject area in industry. Developing effective system reliability optimization is imperative. Two approaches for system reliability improvement are: (i) using redundant elements in subsystems, and (ii) increasing the reliability of system components. Reliability-redundancy allocation maximizes system reliability via redundancy and component reliability choices [23], with restrictions on cost, weight, and volume of the resources. The aim is to find a trade-off between reliability and other resource constraints [22]. Thus, for a highly reliability system, the main problem is to balance reliability enhancement and resource consumption. A number of approaches in the literature focus on the application of metaheuristic methods for solving system reliability optimization problems [9, 7, 27, 15, 33, 34, 10, 13]. However, real-life reliability optimization problems are complex:


Introduction
Reliability is central to productivity and effectiveness of real world industrial systems [22,35].To maximize productivity, the systems should always be available.However, it is difficult for an industrial system, comprising several complex components to survive over time since its reliability directly depends on the characteristics of its components.Failure is inevitable, such that system reliability optimization has become a key subject area in industry.Developing effective system reliability optimization is imperative.Two approaches for system reliability improvement are: (i) using redundant elements in subsystems, and (ii) increasing the reliability of system components.
Reliability-redundancy allocation maximizes system reliability via redundancy and component reliability choices [23], with restrictions on cost, weight, and volume of the resources.The aim is to find a trade-off between reliability and other resource constraints [22].Thus, for a highly reliability system, the main problem is to balance reliabil-ity enhancement and resource consumption.A number of approaches in the literature focus on the application of metaheuristic methods for solving system reliability optimization problems [9,7,27,15,33,34,10,13].However, real-life reliability optimization problems are complex: management goals and the constraints are often imprecise; (i) problem parameters as understood by the decision maker may (ii) be vague; and, historical data is often imprecise and vague.(iii) Uncertainties in component reliability may arise due to variability and changes in the manufacturing processes that produce the system component.Such uncertainties in data cannot be addressed by probabilistic methods which deal with randomness.Therefore, the concept of fuzzy reliability is more promising [2,4,5,6,30,31].Contrary to sciENcE aNd tEchNology probabilistic models, fuzzy theoretic approaches address uncertainties that arise from vagueness of human judgment and imprecision [26,3,28,1,13,14].
A number of methods and applications have been proposed to solve fuzzy optimization problems by treating parameters (coefficients) as fuzzy numerical data.[31,11,20,21,24].In a fuzzy multicriteria environment, simultaneous reliability maximization and cost minimization requires a trade-off approach.Metaheuristics are a potential application method for such complex problems [9].Population based metaheuristics are appropriate for finding a set of solutions that satisfy the decision maker's expectations.This calls for interactive fuzzy multi-criteria optimization which incorporates preferences and expectations of the decision maker, allowing for expert judgment.Iteratively, it becomes possible to obtain the most satisfactory solution.
In light of the above issues, the aim of this research is to address the system reliability optimization problem for a complex bridge system in a fuzzy multi-criteria environment.Specific objectives of the research are (1) to develop a fuzzy multi-criteria decision model for the problem; (2) to use an aggregation method to transform the model to a single-criteria optimization problem; and, (3) to develop a multicriteria optimization method for the problem.
The rest of the paper is organized as follows: The next section describes the problem formulation for the complex bridge system.Section 3 provides a general fuzzy multi-criteria optimization modelling approach.In Section 4, a fuzzy multi-criteria genetic algorithm approach is proposed.Computational experiments, results and discussions are presented in Section 5. Section 6 concludes the paper.

Problem formulation
This section presents the mathematical formulation for the reliability optimization for a complex bridge system.In the real world, a typical complex bridge system [23] comprises five components or subsystems.The general structure of the complex bridge system is illustrated in Fig. 1.
The aim is to maximize system reliability, subject to multiple linear constraints.In this respect, we present the following notations and assumptions;

Notations:
m the number of subsystems in the system n i the number of components in subsystem i, 1≤ i ≤ m n ≡(n 1 , n 2 , …, n m ), the vector of the redundancy allocation for the system r i the reliability of each component in subsystem i, 1≤ i ≤ m r ≡ (r 1 , r 2 , …, r m ), the vector of the component reliabilities for the system q i =1 -r i , the failure probability of each component in subsystem the reliability of subsystem i, 1 i m ≤ ≤ R s the system reliability g i the i th constraint function the upper limit on the sum of the subsystems' products of volume and weight C the upper limit on the cost of the system W the upper limit on the weight of the system b the upper limit on the resource α i , β i , parameters in constraint functions of subsystem i

Assumptions
The availability of the components is unlimited; 1.
The weight and product of weight and square of the volume of 2.
the components are deterministic; The redundant components of individual subsystems are iden-3.
tical; Failures of individual components are independent; 4.
All failed components will not damage the system and are not 5. repaired.
The problem can be formulated as a mixed integer nonlinear programming model as follows [8,34,35]: where, η(•) denotes the system reliability, and expressions g 1 (•), g 2 (•), and g 3 (•) represent the total volume, cost, and weight of the system, respectively.
In the next section, we propose a general approach to fuzzy multicriteria optimization, in the context of system reliability optimization.

Fuzzy multi-criteria optimization modelling
In a fuzzy environment, the aim is to find a trade-off between reliability, cost, weight and volume.A common approach is to simultaneously maximize reliability and minimize cost.Constraints are transformed into objective functions, so that reliability and other cost functions can be optimized jointly.This is achieved through the use of membership functions, which are easily applicable and adaptable to the real life decision process.
In general, the fuzzy multi-criteria optimization problem can be represented by the following [13,29]; where, x = (x 1 , x 2 ,…,x Q ) T , is a vector of decision variables that optimize a vector of objective functions, ῆ(x) = {ῆ 1 (x), ῆ 2 (x),…,ῆ h (x)} are sciENcE aNd tEchNology h individual objective functions over the decision space X; υ q and ῡ q are lower and upper bounds on the decision variable x q , respectively.Note that expressions g 1 (•), g 2 (•) and g 3 (•) in ( 1) are converted into objective functions.
Fuzzy set theory permits gradual assessment of membership, in terms of a suitable function that maps to the unit interval [0,1].Membership functions such as Generalized Bell, Gaussian, Triangular and Trapezoidal can represent the fuzzy membership [31].Linear membership functions can provide good quality solutions with much ease, including the widely recommended triangular and trapezoidal membership functions [6,8,11,30,31].Thus, we use linear functions to define fuzzy memberships of objective functions (or decision criteria).
Let a t and b t denote the minimum and maximum feasible values of each objective function ῆ t (x), t = 1,2,…,h, where h is the number of objective functions.Let µ η t denote the membership function corresponding to the objective function f t .Then, the membership function corresponding to minimization and maximization is defined based on satisfaction degree.Fig. 2 illustrates the linear membership functions defined for minimization and maximization.
As shown in Fig. 2(a), the linear membership function is suitable for representing cost functions that should be minimized.The membership function is represented as follows; The linear membership function shown in Fig. 2(b) is suitable for representing profit functions that should be maximized.The membership function is represented as follows; Having defined the fuzzy model using membership functions, the corresponding crisp model is formulated.Fuzzy evaluation enables FMGA to cope with infeasibilities which is otherwise impossible with crisp formulation.This gives the algorithm speed and flexibility, which ultimately improves the search power of the algorithm.

A fuzzy multi-criteria genetic algorithm approach
FMGA is an improvement from the classical genetic algorithm (GA).GA is a stochastic global optimization technique that evolves a population of candidate solutions by giving preference of survival to quality solutions, while allowing some low quality solutions to survive, to maintain diversity in the population [18].Each candidate solution is coded into a string of digits, called chromosomes.New offspring are obtained from probabilistic genetic operators, such as selection, crossover (at probability p c ), mutation (at a probability p m ), and inversion [16].A comparison of new and old (parent) candidates is done based on a given fitness function, retaining the best performing candidates into the next population.Characteristics of candidate solutions are passed through generations using genetic operators.The overall flow of the FMGA is presented in Fig. 3.

Chromosome coding
Traditionally, candidate solutions were encoded as binary strings.In the FMGA, each candidate solution is encoded into a chromosome using the variable vectors n and r.An integer variable n i is coded as a real variable and transformed to the nearest integer value upon objective function evaluating.

Initialization and evaluation
An initial population of the desired size, pop, is randomly generated.FMGA then computes the objective function for each string (chromosome).The string is then evaluated according to the overall objective function in the model.
To improve flexibility and to incorporate the decision maker's preferences into the model, we introduce user-defined weightings, w = {w 1 , w 2 ,…,w h }.We use the max-min operator to aggregate the membership functions of the objective functions, incorporating expert opinion.Thus, from models (1) and (2), constraints g 1 (•), g 2 (•), and g 3 (•) which represent volume, cost, and weight, respectively, are transformed into objective functions using the fuzzy membership functions.This leads to a multiple criteria system reliability optimization model, consisting of five criteria namely, reliability, volume, cost, and weight.In addition, the model is converted into a single objective optimization model as follows: Here, signifies a set of fuzzy regions that satisfy the objective functions λ t which denote the degree of satisfaction of the t th objective; x is a vector of decision variables; w t is the weighting of the t th objective function; and symbol "˄" is the aggregate min operator.Thus, expression (1˄λ 1 (x)/w 1 ) gives the minimum between 1 and λ 1 (x)/w 1 .Though λ 1 (x) are in the range [0,1], the value of λ 1 (x)/w 1 may exceed 1, howbeit, the final value of (1˄λ 1 (x)/ w 1 ) will always lie in [0,1].A FMGA approach is used to solve the model.

Selection and crossover
Several selection strategies have been suggested in [16].The remainder stochastic sampling without replacement is preferred; each chromosome j is selected and stored in the mating pool according to the expected count e j ; where, f j is the objective function value of the j th chromosome.Each chromosome receives copies equal to the integer part of e i , while the fractional part is treated as success probability of obtaining additional copies of the same chromosome into the mating pool.
Genes of selected parent chromosomes are partially exchanged to produce new offspring.We use an arithmetic crossover operator which defines a linear combination of two chromosomes [25] [29].Assume a crossover probability of 0.41.Let p 1 and p 2 be two parents randomly selected for crossover.Then, the resulting offspring, q 1 and q 2 , are given by the following expression; where, ε represents a random number in the range [0,1].

Mutation
Mutation is applied to every new chromosome so as to maintain diversity of the population, howbeit, at a very low probability.A uniform mutation probability rate of 0.032 is applied.

. Replacement
At each generation or iteration, new offspring may be better or worse.As a result, nonperforming chromosomes should be replaced.A number of replacement strategies exist in the literature, e.g., probabilistic replacement, crowding strategy, and elitist strategy [26].The proposed FMGA uses a hybrid of these strategies.

Termination
The FMGA algorithm uses two termination criteria to stop the iterations: when the number of generations exceeds the user-defined maximum iterations, and when the average change in the fitness of the best solution over specific generations is less than a small number, which is 10 -5 .

Computational illustrations
This section presents the computational experiments, results and discussions based on benchmark problems in [17,19].

Computational experiments
We use the parameter values in [23] and define the specific instances of the problems as shown in Table 1.
The FMGA was implemented in JAVA on a 3.06 GHz speed processor with 4GB RAM.The FMGA crossover and mutation parameters were set at 0.45 and 0.035, respectively.A two-point crossover was used in this application.The population size was set to 20, and the maximum number of generations was set at 500.The termination criteria was controlled by either the maximum number of iterations, or the order of the relative error set at 10 -5 , whichever is earlier.Whenever the best fitness f * at iteration t is such that |f t -f * | < ε is satisfied, then five best solutions are selected; where ε is a small number, which was set at value ε = 10 -5 for the computational experiments.
Expression ( 5) is used to solve benchmark problems.A fuzzy region of satisfaction is constructed for each criterion, that is, system reliability, cost, volume, and weight, denoted by λ 1 , λ 2 , λ 3 , and λ 4 , respectively.By using the constructed membership functions together with their corresponding weighting vectors, an equivalent crisp optimization formulation is obtained [29]; The set ω = {ω 1 , ω 2 , ω 3 , ω 4 } are user-defined weightings in the range [0.2,1] that indicate the bias towards specific decision criteria.To illustrate, given the weighting set ω = [1,1,1,1], the expert user expects no bias towards any criterion.On the contrary, set ω = [1,0.4,0.4,0.4],indicates preferential bias towards the region with higher reliability values as compared to the rest of the criteria equally weighted at 0.4.Consequently, the decision process considers the expert opinion and preferences of the decision maker.
Rather than prescribing a single solution to the user or decision maker, the FMGA interactively provides a population of near-optimal sciENcE aNd tEchNology

Experiment 1 results
Figure shows a plot of the variation of the best fitness in each generation over a run time of 250 generations.After 250 generations, the following solution is obtained as the best solution: the maximum system reliability is R s = 0.999958830.The reliability for the 5 constituent components are r 1 = 0.81059326:3, r 2 = 0.85436730, r 3 = 0.88721528, r 4 = 0.72126594 and r 5 = 0.71732358.The resulting system cost C s = 175.000.
It can be seen that the algorithm converged to a desirable solution within about 200 iterations (generations).This indicates the potential of the algorithm in terms of computational efficiency.

Experiment 2 results
Computational results from experiment 2 showed the performance of FMGA as compared to other best known algorithms.The best five FMGA solutions were compared with the best results obtained from the literature [8] [35].
Tables 3 presents the best five FMGA solutions, and the best known solutions obtained from [8] (with system reliability R s = 0.999958830, cost C s = 175.00,weight W s = 195.7352300,and V s = 92.00).It can be seen that, based on system reliability, cost, weight and volume, the five FMGA solutions are better than the best known results, except for a single weight value from solution S 1 (that is, 196.988273245) which is slightly higher than the best known (that is, 195.7352300).Further, all the five best FMGA solutions outperformed the solutions in [35], based on all performance criteria.This indicates that, overall, the FMGA performs better than the previous algorithms.
Table 4 presents the percentage improvement of the FMGA solutions, using the best known results as benchmarks.The improvements in reliability, cost, weight and volume are denoted by I R , I C , I W and solutions.The algorithm enables the decision maker to specify the minimum and maximum values of objective functions in terms of reliability η 1 , cost η 2 , volume η 3 , and weight η 4 .Table 2 provides a list of selected minimum and maximum values of the objective functions for the complex bridge Two experiments were conducted: Experiment 1 and Experiment 2.

Experiment 1
The aim of experiment 1 was to demonstrate the performance of the FMGA algorithm over time.As such, the algorithm was executed for 500 iterations, to show the results of intermediate solutions over time.A graphical analysis of the results was presented to show the performance behaviour of the algorithm.

Experiment 2
This purpose of experiment 2 was to make a comparative analysis of the performance of the FMGA algorithm against best known algorithms in the literature.Thus, the algorithm was executed 25 times, and the best five solutions were selected.The experimental results were compared with best known algorithms in [17] and [19], based on four performance criteria namely, reliability R s , cost C s , weight W s , and volume V s .
For further comparative analysis, an improvement measure is defined R s , C s , W s and V s values obtained.Thus, for each value, the percentage improvement I is defined according to the following expression: where, v s and v best represent the FMGA solution value and the best known solution from literature.Computational results and the ensuing discussions are presented in the next section.

Computational results and discussions
This section presents the results of the computational experiments outlined in the previous section.Overall, the proposed algorithm is more reliable and effective than existing algorithms in the literature.The algorithm offers a number of practical advantages to the decision maker, including the following: The FMGA method addresses the conflicting multiple objec-• tives of the problem, giving a trade-off between the objectives; The approach accommodates the decision maker's fuzzy pref-• erences; The method gives a population of alternative solutions, rather • than prescribe a single solution; The method is practical, flexible and easily adaptable to prob-• lem situations.
In view of the above advantages, FMGA is a useful decision support tool for the practicing decision maker in system reliability optimization, especially in a fuzzy environment.

Conclusions
Decision makers in system reliability optimization seek to satisfice reliability enhancement and cost minimization.In a fuzzy environment, management goals and constraints are often imprecise and conflicting.One most viable and useful option is to us a fuzzy satisfic-ing approach that includes the preferences and expert judgments of the decision maker.This study provided a multi-criteria non-linear mixed integer program for reliability optimization of a complex bridge system.Using fuzzy multi-criteria evaluation, the model is converted into a singleobjective model.Thus, FMGA uses fuzzy evaluation to find the fitness of candidates in each population.Illustrative computation experiments showed that the FMGA approach is highly capable of providing near optimal solutions.Contrary to single-objective approaches which optimize system reliability only, FMGA provides satisficing solutions in the presence of fuzzy multiple criteria.Furthermore, the algorithm provides a population of good alternative solutions, which offers the decision maker a wide choice of practical solutions and an opportunity to consider other practical factors not included in the formulation.Therefore, the approach gives a robust method for system reliability optimization.
A fuzzy based approach is especially essential, given that, at design stage, the desired design information is not precisely known, which makes the problem rather ill-structured.As such, reliance on human experience and expert information is unavoidable.FMGA uses fuzzy theory concepts to effectively model the vagueness and imprecision of the expert knowledge, taking into account the conflicting multiple criteria.Computational results and comparative analysis showed that the proposed algorithm is more effective than best known algorithms in the literature.

Acknowledgement
This research work was financed by the University of Johannesburg, Johannesburg, South Africa, in the year 2015.

Fig. 1 .
Fig. 1.The schematic diagram of the complex bridge system

Table 1 .
Basic data used for the bridge complex system

Table 2 .
Minimum and maximum values of objective functions

Table 3 .
FMGA performance against other algorithms

Table 4 .
Percentage improvement of FMGA solutions over best known results