Optimization of Time Dependent Fuzzy Multi-Objective Reliability Redundancy Allocation Problem for n-Stage Series Parallel System

This study introduces a time dependent fuzzy multi-objective reliability redundancy allocation problem (TF-MORRAP) for the n -stage (level) series parallel system. System reliability maximization and system cost minimization according to time by optimizing the redundant components counting at every stage of the system is the main objective of this study. This optimization is done by satisfying the entropy constraints with limited redundant components at every stage and in the whole system. The reliability and cost of every component are represented as triangular fuzzy numbers (TFN) to handle the uncertainty of input information of the system. According to time the component reliability and cost decrease by some factor of their previous existing value. This factor follows the change in the length of radius of the inverse logarithmic spiral with respect to angle which is regarded as time here. The proposed problem is analysed by using an over-speed protection system of a gas turbine. We compare the membership values of optimal solutions obtained by using two well-known techniques namely Non-dominated sorting genetic algorithm-II (NSGA-II) and a multi-objective particle swarm optimization algorithm called NF-MOPSO. Various performances of the algorithms are compared to solve the aforementioned problem by using some performance metrics. NF-MOPSO shows the high satisfaction level of objective functions and better performance than NSGA-II.


Introduction
In this study a n stage series parallel reliability redundancy allocation model (RRAM) is considered [1].Reliability can be improved in this model by allocating extra identical redundant components in each stage.A TF-MORRAP is proposed for the aforementioned model.Objectives of the problem are system reliability maximization and system cost minimization.The constraints maintained in this problem are entropy constraint, limited number of components in each stage and limited number of components in the full system.The main goal of this study is to optimize the number of redundant components at each stage of the system such that we can achieve the objectives of the said problem by satisfying the aforementioned constraints with respect to time.Entropy constraint is chosen to measure the scattering of allocation in different levels of the system [1].Here, the uncertainty of components reliability and cost are handled by using TFN [2] to make the TF-MORRAP more realistic and flexible.In general the component reliability and component cost decrease by the same factor of their previous existing value according to time.So the decreasing nature follows the radius of the inverse logarithmic spiral with respect to angle which is regarded as time here.The radius of the inverse logarithmic spiral from origin meets it at distances decreases in geometric progression according to the angle [3].
Multi-objective particle swarm optimization (MOPSO) [4] algorithm is a popular meta-heuristic technique used to solve multi-objective optimization problems (MOOP).NSGA-II [5] is a fast and elitist multi-objective evolutionary algorithm (MOEA) used to solve MOOP.A benchmark model "over-speed protection system of a gas turbine" [6] is considered to illustrate the proposed problem.Pareto-optimal solutions are obtained from the TF-MORRAP of the protection system on some values of time parameter using NF-MOPSO [7] and NSGA-II [5] algorithms.A comparative study is performed among them depending on the best Pareto-optimal solutions and their membership values.It is observed that the NF-MOPSO algorithm shows the high satisfaction level of system reliability and system cost in most of the time values.The average value of the best Pareto-optimal solutions and their membership values for all time values are higher in NF-MOPSO algorithm than NSGA-II algorithm.Performance metrics such as ER (Error ratio) [8], MID (Mean ideal distance) [9] and IGD (Inverted generational distance) [10] are used to compare the convergence rate, diversity, accuracy and other features of the algorithms to solve the proposed TF-MORRAP of series parallel system.The NF-MOPSO algorithm shows better performance than NSGA-II.
The rest of the paper is organized as follows.The related works exist in Section 2. Section 3 described the model formulation with inverse logarithmic spiral and the construction of TF-MORRAP.Section 4 described the solving methodology of the proposed problem using NF-MOPSO and NSGA-II algorithms.The n stage series-parallel RRAM is illustrated in Section 5 from the case study of the over-speed protection system of a gas turbine.Section 6 describes the numerical and graphical presentation of the solutions obtained from the TF-MORRAP of protection systems using NF-MOPSO and NSGA-II algorithms.Finally, conclusions are drawn in Section 7.

Related work
Recently several types of RRAM and reliability redundancy allocation problems (RRAP) are proposed on series parallel systems or other systems.Maneckshaw et al. [11] used a multi-objective RRAM with constraints in their study and optimized the objectives using evolutionary algorithms.Zarei et al. [12] defined a MOOP representing objectives as pressure deficit minimization and cost minimization in the whole network and solved it using MOPSO and NSGA-II algorithms.Kumar et al. [6] constructed a fuzzy based MORRAP for an over speed protection system and it is solved by using their proposed hybrid NSGA-II algorithm.Roy et al. [1] calculated maximum system reliability and minimum system cost of a MORRAP for a series parallel system maintaining entropy constraint.Cao et al. [13] introduced a MORRAP with interval-valued parameters to optimize the system reliability and cost maintaining entropy as a constraint.Du et al. [14] presented a non-homogeneous RRAP for multi-state series-parallel systems.Huang et al. [15] formulated a model of fuzzy constraints for a series-parallel RRAP.Li et al. [16] designed a global reliability with RRAP and it is solved by using an updated swarm optimization technique.Guilani et al. [17] introduced a mathematical model for the RRAPs.By maintaining the limits on weight Kundu [18] simultaneously optimized system reliability with system cost of a MORRAP for a series-parallel system.
In recent years various MOPSO algorithms and NSGA-II algorithms are used to solve various MOOP.De et al. [19] introduced a fuzzy rank-based MOPSO algorithm to solve a fuzzy MORRAP for x j −out−of −m j series-parallel system.Li et al. [20] improved the search effectiveness and efficiency of the MOPSO algorithm in their study.Mahapatra et al. [21] used MOPSO algorithm to solve a MORRAP under a hesitant fuzzy environment.Nshimirimana et al. [7] introduced the NF-MOPSO algorithm to solve MORRAP.Muaddi et al. [22] used MOPSO in their study to optimize the reliability and cost of MOOP of grid connected and stand-alone systems.Davoudi et al. [23] introduced a MOPSO algorithm to solve their proposed optimization problem.Zand et al. [24] used a mixed MOPSO with NSGA-II algorithm to solve their proposed bi-objective nonlinear redundancy allocation problem.He et al. [25] proposed a MOPSO-LS (Local search strategy) to identify the key quality characteristic from the production process.NSGA-II with crowding distance is introduced by Deb et al. [5] to optimize the reliability and other important measures of MOOP.Liu et al. [26] introduced a multi-objective optimization model of the greenhouse light environment and solved it by using an improved NSGA-II algorithm.Nath et al. [27] formulated the multi-objective RRAP with different structures and solved it using NSGA-II and NSGA-III algorithms.To perform a specific task by optimizing the collective expertise and team cost, Yadav et al. [28] selected the best team of workers using their proposed NSGA-II based algorithm.Brentan et al. [29] found out the optimal solutions of the MOOP of water quality sensor placement using NSGA-II.Pal et al. [30] used variable length NSGA-II to find out the optimal portfolio of stocks.Wang et al. [31] constructed the MOOP of a multi-type production system with optimal redundancy strategy and solved this problem using NSGA-II.
3 Model with inverse logarithmic spiral and its TF-MORRAP

Model with inverse logarithmic spiral
Fig. 1 shows a n stage series parallel system [1].In this system n stages exist in series and at each i th (i = 1, 2..n) stage x i components are connected in parallel.Among x i components (x i − 1) components are redundant.The aim is to optimize the redundant components counting x i at i th (i = 1, 2, ..n) stage by maintaining the aforementioned objectives and constraints.
Fig. 1: n stage series parallel system.
The polar equation of the inverse logarithmic spiral is r = pe −qθ [3], where p and q are arbitrary constants.When the angle θ increases, the radius vector r decreases in geometric progression.In general component reliability and cost will decrease by the same factor of their previous existing value with respect to time t.These factors are changed by following the radius of the inverse logarithmic spiral r = pe −qt , where angel θ is regarded as time t.The decreasing rate can be controlled by the spiral parameter q.Without loss of generality it is assumed that p = 1.

Reliability and cost functions on time
System reliability of the n stage series parallel model is defined from [1] as Since the reliability of each component decreases by following the radius of the inverse logarithmic spiral with respect to time, therefore the reliability of each component at i th stage becomes R i e −qt .The system reliability function at time t is defined in equation 1.
Similarly the cost of each component at i th stage on time t becomes C i e −qt .The total cost of x i components at i th stage becomes C i e −qt x i .Since the cost of every component is directly proportional to the reliability, therefore the total cost of the x i components at i th stage on time t becomes C i e −qt x i (R i e −qt ) ai .
Here a i is the shape parameter.Shape parameters maintain various properties of the components in different stages.Hence the system cost function at time t is defined in equation 2.
For time t = 0 the initial reliability R i , cost C i occurred and their corresponding rate of decreasing is controlled by time t and arbitrary constant q.

Time based MORRAP formulation in crisp and fuzzy environment
The time based MORRAP for the n stage series parallel system in crisp environment is defined in equation 3 by using equation 1 and 2.
The TF-MORRAP for the n stage series parallel system is defined in equation 4 by using equation 3.
Subject to same constraints as in equation 3

Methodology
In our methodology to solve the proposed TF-MORRAP defined in equation 4 using NSGA-II and NF-MOPSO algorithms initially the sections 4.1, 4.2, 4.3 and 4.4 must be followed sequentially.

Extreme values evaluation of each objective function at time t
For time parameter t, every objective function is solved separately with the same constraints defined in equation 4 taken as a single objective optimization problem.Let X 1 be the optimal solution of the reliability objective function and Y 1 be the optimal solution of the cost objective function.The interval of extreme values of the system reliability is In this study the lower and upper value of each objective function is calculated by using particle swarm optimization (PSO) [32] algorithm for time values t = 0, 1, 2, ...10.The value of PSO parameters are considered as follows: the value of cognitive acceleration is 0.5, the value of global acceleration is 0.5, the upper limit of inertia weight is 0.9 and the lower limit of inertia weight is 0.2.

Construction of Membership functions
In this study at the time of applying the membership functions, the TFN form of reliability R = (r 1 , r 2 , r 3 ) and cost C = (c 1 , c 2 , c 3 ) are converted to the corresponding crisp (defuzzified) value using center of gravity (COG) [33] technique defined in equation 5.
Here, the TFN form of system reliability and system cost are represented by ordered triplets R S = (R min , R mid , R max ) and C S = (C min , C mid , C max ).The advantage of choosing TFN is that the decisionmakers may set the modal values R mid and C mid in the suitable position of the uncertain intervals [R min , R max ] and [C min , C max ] respectively to achieve the appropriate level of satisfaction.Decision-makers may set R mid towards R max and C mid towards C min to achieve into the higher level of satisfaction for the proposed problem.Here we set R mid = R max − (Rmax−Rmin) 8 * 3, C mid = C min + (Cmax−Cmin) 8 * 3. The membership functions for the crisp value of system reliability and system cost are defined in equation 6.
Here x represents the crisp (defuzzified) value of a TFN evaluated by using equation 5.

Formulation of the proposed problem in the form of fuzzy objectives
Maximization of membership functions for the objective functions gives the best satisfaction level [6].A time based fuzzy MOOP is constructed from equation 4 shown in equation 7.In this problem the best satisfaction level is achieved by the maximization of the membership functions of all objective functions.
Subject to same constraints as in equation 4 Here R * and C * represents the defuzzified values of the reliability and cost fuzzy objective functions respectively.

Non-dominated sorting technique
Here, in NSGA-II and NF-MOPSO algorithms the non-dominated solution (best alternative) [5] are found out among different feasible solutions depending on the best membership values of the objective functions.
It is performed by using the membership vector defined in equation 7. Let (y 1 , y 2 ) and (z 1 , z 2 ) be the membership vector of two feasible solutions Y and Z. Then Y dominates Z (Y ≻ Z) if y i ≥ z i for i = 1, 2 and at least one i exists such that y i > z i .

Fuzzy ranking method
Based on the satisfaction level of each solution of a solution set the rank will be provided in fuzzy ranking method (FRM) [6].The maximum ranked solution indicates the most compromise solution.The formula to calculate the best satisfaction level (membership value) using FRM is shown in equation 8 Here M represents the Pareto optimal solutions set.

Solving methodology using NSGA-II algorithm
Algorithm 1 : Solving of TF-MORRAP using NSGA-II algorithm Require: N number of random solutions (chromosomes) of dimension n with time t Ensure: Pareto optimal solutions with best alternative with time t 1: Initialization: Initialize the number of stages or dimension of solution vector n, lower limit L and upper limit U of total allowable components, lower limit S l and upper limit S u of sample space, shape array [a 1 , a 2 , a 3 ..a n ], triangular fuzzy shaped components reliability and cost as ( R 1 , R 2 , R 3 , ..., R n ) and ( C 1 , C 2 , C 3 , ..., C n ), population size N , total generation (maximum number of iteration) max gen, maximum number of independent trials max trial, maximum time limit max time.

5:
Set trial count variable tr count ← 1 and Pareto optimal set P areto opt sol ← ∅ 6: while tr count ≤ max trial do 7: Generate N chromosomes randomly of dimension n satisfying the constraints of equation 7 and store into a set P op set.

8:
Set the generation counting variable gen count ← 0 9: while gen count < max gen do 10: Applying NSGA-II algorithm on the time based fuzzy MOOP defined in equation 7 based on whole arithmetic crossover [34], polynomial mutation [35], fast non-dominated sorting technique (section 4.4) and crowding distance technique to get the Pareto fronts from P op set.

11:
Replace all N chromosomes of P op set by the new N chromosomes collecting from the Pareto fronts. 12: gen count ← gen count + 1 Display the Pareto optimal solutions with time t from P areto opt sol set whose membership vector (calculated from equation 7) components are non zero.

18:
Choose the best suitable solution among the non zero Pareto optimal solutions using fuzzy ranking method (equation 8 in section 4.5) and display it with time t.

Solving methodology using NF-MOPSO algorithm
Algorithm 2 : Solving of TF-MORRAP using NF-MOPSO algorithm Require: N number of random particles (solutions) of dimension n with time t Ensure: Pareto optimal solutions with best alternative with time t 1: Initialization: Initialize the number of stages or dimension of solution vector n, lower limit L and upper limit U of total allowable components, lower limit S l and upper limit S u of sample space, shape array [a 1 , a 2 , a 3 ..a n ], triangular fuzzy shaped components reliability and cost as ( R 1 , R 2 , R 3 , ..., R n ) and ( C 1 , C 2 , C 3 , ..., C n ), population size N , total generation (maximum number of iteration) max gen, maximum number of independent trials max trial, maximum time limit max time.Set trial count variable tr c ount ← 1 and Pareto optimal set P areto opt sol ← ∅ 6: while tr count ≤ max trial do 7: Global best solution set Glbest ← ∅ 8: Generate N particles (or solutions) randomly of dimension n satisfying the constraints of equation (7).Construct the personal best repository individually for each particle and initialize it by the particle itself.Update the global best solution set Glbest by applying the non-dominated solutions technique defined in section 4.4 on the particles of personal best set and the existing particles of the Glbest set.

9:
Set the generation counting variable gen count ← 0 10: while gen count < max gen do 11: Applying NF-MOPSO algorithm on the time based fuzzy MOOP defined in equation 7 based on Gaussian mutation operator, death penalty as a constraint handling technique, a hypercube which is used to store the non-dominated solutions, roulette-wheel as a selection procedure and non-dominated sorting technique (section 4.4) to get the upgraded positions of N particles of the swarm and upgraded global best set Glbest (Pareto optimal solutions set).

12:
gen count ← gen count + 1 Display the Pareto optimal solutions with time t from P areto opt sol set whose membership vector (calculated from equation 7) components are non zero

18:
Choose the best suitable solution among the non zero Pareto optimal solutions using fuzzy ranking method (equation 8 in section 4.5) and display it with time t 19: t ← t + 1 20: end while 5 Illustrative example for a series parallel RRAM The efficiency and applicability of the proposed TF-MORRAP is illustrated through a benchmark model shown in Figure 2.

Over-speed protection system of a gas turbine
Figure 2 shows the model of an over-speed protection system of a gas turbine [6] containing five control valves from V 1 to V 5 .It has a mechanical and electrical mechanism which is used to detect the speed continuously.If an over-speed is detected then the supply of fuel to the turbine becomes cut-off by the 5 control valves.This protection system behaves like a series-parallel system as in Figure1.In this system (x i − 1) redundant valves are connected in parallel at each i th stage to increase its reliability.To increase the flexibility of the protection system the input information of reliability and cost of each valve are designed as TFN form.
The TF-MORRAP of the protection system is defined in equation 9.The aim is to optimize the number of active valves x i (i = 1, 2, ..5) within the search space such that the objectives and constraints of equation 9 are maintained.Fig. 2: Over-speed protection system of gas turbine.

TF-MORRAP for over-speed protection system
The TF-MORRAP of the over-speed protection system is defined in equation 9, which is constructed from equation 4 by considering n = 5.
x i ≤ U and S l ≤x i ≤ S u f or i = 1, 2, 3, .., Similarly as in equation 7 the corresponding time based fuzzy MOOP for the over-speed protection system is constructed from equation 9 and it is defined in equation 10.
Subject to same constraints as in equation 9 (10) 6 Results and Discussion

Numerical Results
For the equation 9 we considered the value of shape parameters, crisp and TFN form of component reliability and cost are shown in Table 2.Here for each component 5% uncertainty for reliability and 10% uncertainty for cost are considered.The decision maker may change these values as per their requirements.The values of another parameters of equation 9 are considered as q = 0.03, E = 1.6, L = 10, U = 30, S l = 2, S u = 10.Other parameter values used in algorithm 1, algorithm 2 and PSO [32] algorithms are maximum time value max time = 10, population size N = 60, maximum number of iteration max gen = 200, maximum independent trial max trial = 20.
PSO technique is applied by following the section 4.1 for time t=0,1,2..,10 and collect the extreme values of system reliability, system cost in Table 3.The modal values R mid and C mid are evaluated by using the formulas defined in section 4.2.Table 3: Extreme values of system reliability and system cost with their modal values.Table 4 shows the result obtained by solving the MOOP defined in equation 10 using algorithm 1 (NSGA-II) and algorithm 2 (NF-MOPSO) for time t = 0, 1, 2, ..10.From Table 4 it is observed that the high system reliability and low system cost occurred in NF-MOPSO algorithm in most of the time values.
Also on average the NF-MOPSO algorithm appears to have better results than NSGA-II algorithm to find out the optimal solutions for the problem defined in equation 10.
Figure 3 shows the graphical representation of the membership values shown in Table 4.This graph shows that high satisfaction levels occurred for NF-MOPSO technique in most of the time values.

Performance comparison
Here, the performance metrics ER [8], IGD [10] and M ID [9] are used to measure the accuracy, diversity and other features of the NF-MOPSO and NSGA-II algorithms to solve the proposed TF-MORRAP of series parallel system.Without loss of generality here we show the comparison of the performances for the arbitrary time values t = 3, 5, 7.
The ER metric indicator is defined as , where e j = 0 if the solution vector j belongs to the true Pareto-optimal set, otherwise e j = 1.Low indicator values of ER shows the better performance of an algorithm [8]. Figure 6 shows the graphical comparison of the ER indicator values of the Paretooptimal set obtained by the algorithms for time values t = 3, 5, 7. Figure 6 shows that lower ER values occurred for the NF-MOPSO algorithm and therefore it performs better than NSGA-II algorithm.

W
. Here W represent the size of the true Pareto optimal set and d j represents the Euclidean distance from the j th solution of the true Pareto optimal set to the nearest member of the Pareto-optimal set. Figure 7 shows the graphical comparison of IGD indicator values with respect to iteration numbers for m = 2. Lower IGD values are shown in Figure 7 for the NF-MOPSO algorithm.Hence NF-MOPSO shows better spread and better convergence rate of the Pareto-optimal solutions than NSGA-II algorithm.Here G be the number of objective functions, M be the Pareto-optimal front obtained by the algorithm, y z be the ideal point for z th objective function and f j z represents the fitness value of the z th objective function of j th solution in M .Figure 8 graphically compares the M ID indicator values calculated on Pareto optimal front obtained by NSGA-II and NF-MOPSO algorithms for the time values t = 3, 5, 7. Figure 8 shows the lower M ID values for the NF-MOPSO algorithm and hence its performance is better than the NSGA-II algorithm.

Conclusions
In this study, a TF-MORRAP of a n stage series-parallel system is introduced.The uncertainty nature of component reliability and cost of the system are managed by using TFN to increase its flexibility.The highest degree of satisfaction level of objective functions with respect to time are achieved at the time of

Fig. 6 :
Fig. 6: Comparison of ER values with respect to iteration number

Fig. 7 :
Fig. 7: Comparison of IGD values with respect to iteration number

Fig. 8 :
Fig. 8: Comparison of M ID metric values with respect to iteration number

Table 1
shows the notations used in the TF-MORRAP for the n stage series parallel system.

Table 1 :
Notations used in TF-MORRAP for n stage series parallel system

Table 2 :
Value of parameters used in equation 9

Table 4 :
Comparison of optimal solutions obtained by solving equation 10 (Highlighted values show better results)