A novel artificial fish swarm algorithm for solving large-scale reliability–redundancy application problem

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Research article

A novel artificial fish swarm algorithm for solving large-scale reliability–redundancy application problem Qiang He a,1, Xiangtao Hu b,n, Hong Ren a,nn, Hongqi Zhang b,2 a b

Urban Construction and Environmental Engineering, Chongqing University, Chongqing 400044, China No. 38 Research Institute of CETC, Hefei 230088, China

art ic l e i nf o

a b s t r a c t

Article history: Received 18 October 2013 Received in revised form 5 May 2015 Accepted 14 September 2015 This paper was recommended for publication by Dr. Q.-G. Wang.

A novel artificial fish swarm algorithm (NAFSA) is proposed for solving large-scale reliability-redundancy allocation problem (RAP). In NAFSA, the social behaviors of fish swarm are classified in three ways: foraging behavior, reproductive behavior, and random behavior. The foraging behavior designs two position-updating strategies. And, the selection and crossover operators are applied to define the reproductive ability of an artificial fish. For the random behavior, which is essentially a mutation strategy, the basic cloud generator is used as the mutation operator. Finally, numerical results of four benchmark problems and a large-scale RAP are reported and compared. NAFSA shows good performance in terms of computational accuracy and computational efficiency for large scale RAP. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Reliability–redundancy allocation problem Artificial fish swarm algorithm GA Cloud model

1. Introduction As systems have grown more complex, the consequences of their unreliable behavior have become severe in terms of cost, effort, lives, etc. Thus, the interests in assessing system reliability and the need for improving the reliability of products and systems have become very important [1,2]. Generally, the system reliability can be enhanced by two major approaches. The first approach is increasing the reliability of system components, and the second approach is using redundant components in various subsystems. However, using the first approach often come to a deadlock that the required reliability enhancement may be never attainable even though the most currently reliable elements are used, and in the second approach, the system reliability enhancement often lead to the increase of the cost, weight, volume, etc. simultaneously. Besides the above two routes, an effective tradeoff approach is the conjunction of the two approaches and reassignment of interchangeable elements. This is called the reliability– redundancy allocation problem (RAP) which was first introduced by Misra and Ljubojevic [3–5]. It involves setting reliability objectives for n

Corresponding author. Mobile: þ 86 18156036871. Corresponding author. Tel.: þ 86 55165391776. E-mail addresses: [email protected] (Q. He), [email protected] (X. Hu), [email protected] (H. Ren), [email protected] (H. Zhang). 1 Mobile: þ 86 18225855208. 2 Tel: þ 86 55165391771. nn

components or subsystems in order to meet the resource consumption constraint, e.g. the total cost. The classical RAP can be stated as following nonlinearly mixedinteger programming model. Max Rs ¼ f ðr; nÞ; Subject to g ðr; nÞ r l

ð1Þ

0 r r i r 1; ni A positive integer; 1 r ir m; where ni and ri are the number of redundancy components and the corresponding component reliability in the ith subsystem, respectively; f ð U Þ is the objective function for the overall system reliability; gð UÞ is the constraint function such as cost, volume, and (or) weight; l is the resource limitation; m is the number of subsystems. RAP has been an active area of research over the past four decades. Many efforts have been devoted to the optimization techniques for RAP [4–9]. Recently, with the advent of artificial intelligence technologies, several meta-heuristics have been proposed and successfully applied to handle a number of reliability optimization problems. These heuristics include PSO (particle swarm optimization) based optimization techniques [10–14], ABC (artificial bee colony) based optimization techniques [15–17], IA (immune algorithm) based optimization techniques [18,19], GA based optimization techniques [20–23], and other conventional search techniques[24–27], etc. These meta-heuristics are based more on artificial reasoning than classical mathematics based optimization. As demonstrated in the literature, the optimization techniques above show good performance in getting an optimal or

http://dx.doi.org/10.1016/j.isatra.2015.09.015 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.

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near optimal solution for various RAPs. From the comparison of four benchmark problems, IPSO proposed by Wu, et al. [12], ABC2P (for short in this paper) by Harish et al. [17], IB2P (for short in this paper) proposed by Hsieh et al. [19], and AR-ICA proposed by Leonardo et al. [26] are confirmed the superiority in terms of both best solution and robustness. IPSO designs two position updating strategies and introduces a mutation operator after position updating. And it is considered as a simple but powerful tool for solving various practical optimization problems. IB2P is a new immune based two-phase approach to solve the reliability–redundancy allocation problem. In the first phase, an immune based algorithm is developed to solve the allocation problem as same as other meta-heuristics algorithms, while in the second phase an improvement of the solution as obtained by this algorithm is made. Harish et al. used the similar idea of IB2P and proposed a new two phase approach based on ABC. AR-ICA is a new approach based on imperialist competitive algorithm and an attraction–repulsion mechanism. Although these previously-developed algorithms have shown good performance for solving RAPs, they also have some weakness such as the lower robustness, premature convergence of solution, not using a prior knowledge, not exploiting local search information, et al. Thus, a few works are directed toward exact solutions since the RAP is assumed to be a NP-hard (non-deterministic polynomial-time hard) problem [8,28]. Furthermore, these heuristic techniques are difficult to deal with large scale RAPs, and cannot effectively balance the contradiction problem of accuracy and efficiency. In order to improve this kind of barrier above mentioned, this paper introduces a novel approach named NAFSA, which is a new variation of artificial fish swarm algorithm (AFSA) [29–31]. Inspired by the social behaviors of fish swarm, NAFSA designs three classes of artificial fish behavior: foraging behavior, reproductive behavior, and random behavior. Moreover, five nonlinearly mixed-integer RAPs, where both the number of redundancy components and the corresponding component reliability in each subsystem are to be decided simultaneously, are considered and solved by proposed approach. This paper is organized as follows. In Section 2, we will briefly introduce the basic AFSA. The new approach is presented in Section 3. Next, some benchmark problems of RAP are illustrated. In Section 5, numerical results of the proposed approaches and the other typical approaches are reported and compared. Finally, short conclusions are summarized in Section 6.

2. Basic artificial fish swarm algorithm AFSA is a new swarm intelligent optimization algorithm based on the nature fish-swarm behavior, which was first proposed by Li [29,30]. The algorithm takes full advantage of the concentrated emerging mechanism of the individual intelligence to find the global optimal solution, and does not require the gradient information of the objective function. In AFSA, each artificial fish searches for food based on its own way, such as foraging behavior, swarming behavior, following behavior and random behavior. Moreover, each artificial fish allows mutual information communications until to achieve a global optimum. The basic idea of AFSA can be described as follows: in an ndimensional space, we assume that there is a fish swarm with N artificial fish. Let X¼ (x1, x2, …, xn) is the position of an artificial fish, and Y¼ f (X) is the fitness or objective function at Position X. Let dij ¼ || Xi–Xj|| denotes the distance between the position Xi and Xj, and Visual and Step stand for the perception range and the moving step of the artificial fish, respectively. The fish-swarm behaviors include foraging behavior, swarming behavior, following behavior and random behavior.

(1) Foraging behavior Let Xi be the current state of an artificial fish, and select a state Xj randomly in its Visual range. If YjoYi (for the minimization problem), the artificial fish moves a Step in the direction of (Xj–Xi). Otherwise, select a state Xj randomly again and judge whether it satisfies the forward condition. If it cannot be satisfied after a pre-set try-number times, the random behavior is performed. The foraging behavior follows the following rule: 

Xi ¼

8 < X i þ Step U X j  X i U rand;

if ðY j o X i Þ

: random behavior;

otherwise

dij

ð2Þ

where X~ i is the next state of the artificial fish, rand is uniformly generated in the range of [0, 1]. (2) Swarming behavior In the fish swarm, each artificial fish Xi should explore the central position Xc of NF artificial fish in its current neighborhood (dij o Visual). If (Yc/NF 4 δYi), the artificial fish Xi will go forward a step to the Xc. Mathematical expression of the swarming behavior is as follows: 

(

Xi ¼

X i þ Step U X cdic X i Urand;

  if Y c =N F o δ U Y i

foragying behavior;

otherwise

ð3Þ

where δ A ð0; 1Þ indicates the food concentration. (3) Following behavior Suppose Xlbest is the local best companion in the current neighborhood of Xi. If (Ylbest/NF 4 δYi), the artificial fish Xi will try to move a step in the direction (Xlbest–Xi). Mathematic description of following behavior can be stated by: 

Xi ¼

(

 Xi X i þ Step U X lbest U rand; d

  if Y lbest =N F o δ UY i

foraging behavior;

otherwise

i;lbest

ð4Þ

(4) Random behavior The artificial fish chooses a position randomly in its Visual range, and then it moves towards the position. It is a default behavior. (5) Behavior selection For each artificial fish, the above four behavior are carried out and compared, respectively. However, only the best behavior is selected to update the current state of artificial fish. (6) Bulletin Bulletin is used to record the optimal state Xbest in the fish swarm. Each artificial fish compares its own state with the bulletin after making a step. If its state is better, the bulletin will be updated. In summary, AFSA adopts the social behavior of fish swarm for solving optimization problems, and takes full advantage of the fish self-information and environment information to adjust the searching orientation for achieving the balance of diversity and convergence. Thus, the artificial fish finally get to the place (global extremum) where the food is the most abundant. Although AFSA has received a lot of attention regarding its potential as global optimization techniques for optimization problems, AFSA inevitably risks converging on a sub-optimum like other metaheuristics. This feature is known as the premature convergence in the case of complex optimization problems, and its efficiency becomes unbearable.

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mating right for reproduction. The offspring of artificial fish can be obtained by:

3. Novel artificial fish swarm algorithm In this section, a novel artificial fish swarm algorithm by imitating the social behavior of fish swarm will be presented for solving RAP. Generally, each fish swarm in nature face two basic tasks: survival and reproduction. To survive, each individual in the population will gather to the place with abundant food. On the other hand, the primary meaning of survival is to continue population through reproduction. Driven by above two purposes, the fish swarm has to continue to explore the unknown world for food, and fight for the right to mate. In other words, the purposes lead to two social behaviors of fish swarm: foraging behavior and reproductive behavior, respectively. Furthermore, the fish swarm should have other characteristics except above behaviors, such as wandering, playing, escaping, etc. To help make things easier, we collectively call them as random behavior. Given above consideration, this article classifies the social behavior of fish into three categories: foraging behavior, reproductive behavior and random behavior. Inspired by these behaviors, we develop a NAFSA for solving RAPs. (1) Foraging behavior Let Xi is the current position of an artificial fish, and we assume that each artificial fish can apperceive the global best position Xbest recorded in bulletin and the local best position Xlbest. Then, the next position of artificial fish can be obtained by: (  Xi UStep  rand ; if ðran o ηÞ X i þ X lbest di;lbest X~ i ¼ Xi þ

X best  X i UStep  rand ; di;best

otherwise

3

ð5Þ

where ran is a random number in the range of [0, 1], the parameter η is defined as decision probability: rffiffiffiffiffiffiffiffiffiffiffi t ð6Þ η ¼ 1 T where t indicates the current iteration number, and the T stands for the maximum iteration number. In this paper, the local best position Xlbest is defined by the following rules: for each Xk (k ai), calculate and compare the dki, and Xk corresponding to the smaller dki is chose as the neighborhood of Xi. The neighborhood size is represented by the symbol Num_nb. According to Eq. (5), if probability is satisfied, the next position will locate at a random position between Xi and Xlbest. Otherwise, the current position will locate at a random position between Xi and Xbest. In other words, the artificial fish tend to explore the neighborhood in early iteration, and adjust its current position according to the successful neighbors; in late iterations, it adjusts its current position according to the global best position with a large probability, which indicates that it prefers to imitate global successful companions in this stage. (2) Reproductive behavior At the case of sexual reproduction, the offspring often have common father or mother. For example, queen bee is the common mother of the swarm, and drones as the offspring father get the mating right by competition. In this paper, this mechanism is introduced into the artificial fish swarm, and the selection and crossover operator in GA are applied to define the reproductive ability of an artificial fish. Firstly, the artificial fish at global best position Xbest is chosen as the female parent, and then some (Num_sp) artificial fish are chosen as the male parent based on the roulette wheel selection operator. Finally, the individual at better position enjoy priority

X inext ðsÞ ¼ X mi ðsÞ  ð1  randÞ þ X best ðsÞ  rand

ð7Þ

where X mi is the male parent, X inext is the offspring, s indicates the sth component of X. (3) Random behavior The random behavior is essentially a mutation strategy to prevent the fish swarm from trapping into a sub-optimum. We introduce the cloud theory into the random behavior. Cloud model proposed by Li et al. [32] can synthetically describe the randomness and fuzziness of concepts and implement the uncertain transformation between a qualitative concept and its quantitative instantiations. Due to the properties of randomness and stable tendency, the cloud mode has found broad application [33,34]. Let Xi be the current state of an artificial fish. We apply the normal cloud model to generate the cloud drop near Xi. The procedure of normal cloud model can be summed as follows:

 Let Ex ¼Xi, En ¼ Rx /c1, He ¼ En /c2, where Rx is the range of X, c1and c2 are the coefficients;

 Let En is the mathematical expectation, He is the standard deviation, and generate a normal random variable EX;

 Let Ex is the mathematical expectation, EX is the standard deviation, and generate a normal random variable Xnew;

 Let:

Ψ ¼ exp 

  ! 1 X  Ex 2 2 EX

ð8Þ

if Ψ 4 rand, Xnew will be a new position of artificial fish, otherwise, choose a random position;

 Repeat above step until num_c cloud drops are generated. For each artificial fish, the random behavior is performed based on the normal cloud model. The objective function of new generated cloud will be calculated, and the new best result Ynew is compared with the Yi. If Ynew oYi, Xi will be updated by Xnew. Otherwise, the current position of artificial fish will remain unchanged. (4) Procedure of NAFSA Given above consideration, the foraging behavior, reproductive behavior, and random behavior of artificial fish is defined. By combining the artificial fish behavior, a novel artificial fish swarm algorithm is presented. The basic procedure of NAFSA is given as shown in Fig. 1, where parameter num is used to record the iteration number. In general, NAFSA works as follows. Step 1. Initialize the problem The problem is defined as minð  Rs Þ subject to 0 r ri r1 and ni A positive integer (1 rirm). The basic algorithm parameters are also specified in this step. They are the population size N, the maximal number of iteration (T), the parameter Step, num1, num2, num_c, Num_nb, Num_sp, c1 and c2. Step 2. Initialize the population Each initial artificial fish in the population is randomly generated from a uniform distribution in the ranges of variables.

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4.1. The first four RAPs P1. Complex (bridge) system (Fig. 2) P1 is a nonlinear mixed integer-programming problem for a complex (bridge) system with five subsystems. The complex (bridge) system optimization problem is given as follows: max : Rs ¼ R1 R2 þ R3 R4 þR1 R4 R5 þ R2 R3 R5  R1 R2 R3 R4 R1 R2 R3 R5  R1 R2 R4 R5  R1 R3 R4 R5 R2 R3 R4 R5 þ 2R1 R2 R3 R4 R5 m X s:t:g 1 ¼ wi v2i n2i V r 0 i¼1

g2 ¼ g3 ¼

m X i¼1 m X



αi 

 1000 βi ½ni þ expð0:25ni Þ  C r0 lnðr i Þ

wi ni expð0:25ni Þ  W r 0

i¼1

0 rr i r 1; ni A Z þ ; 1 r i rm Fig. 1. Flowchart of NAFSA.

Step 3. Calculate the fitness In RAP, for each artificial fish, the fitness is calculated as follows: ( P if g ¼0  Rs  Rs ¼ P  ð9Þ otherwise g where Rs is calculated as formula (1), g~ is defined as follows: g~ ¼ maxð0; g  lÞ

ð10Þ

Apparently, the fitness has following features: (1) When the artificial fish belongs to the feasible space, the range of R~ s is [  1,0]; (2) Otherwise, the punishment is increased with the growth of constraint violates degrees, and the range of punishment is (0,1). Step 4. Foraging behavior and reproductive behavior When num1 onum rnum2, the reproductive behavior is carried out. Otherwise, perform foraging behavior. It is should be noted that the num assign zero again when num4num2. Thus, the foraging behavior and reproductive behavior alternately run during iteration in accordance with the natural rule. Step 5. Random behavior Random behavior is a default behavior. Step 6. Check the stopping criterion If the stopping criterion is satisfied, the computation is terminated. Otherwise, the Step 3–Step 5 is repeated.

where, m is the number of subsystems in the system; ni is the number of components in subsystem i; qi ¼1  ri is the failure probability of each component in subsystem i; Ri(ni)¼1  qni i is the reliability of subsystem i; wi and ci is the weight and cost of each component in subsystem i, respectively. Furthermore, V, C and W are the upper limit on the volume, cost and weight of system. The parameter βi and αi are the physical features of system components. The input parameters of the complex (bridge) system are shown in Table 1. P2. Series system (Fig. 3) P2 is a nonlinear mixed integer-programming problem for a series system with five subsystems. And the problem formulation is given as follows: m

max : Rs ¼ ∏ Ri ðni Þ s:t: g 1 ¼

i¼1 m X

wi v2i n2i  V r 0

i¼1

g2 ¼ g3 ¼

m X i¼1 m X



αi 

 1000 βi ½ni þ expð0:25ni Þ  C r0 lnðr i Þ

wi ni expð0:25ni Þ  W r 0

i¼1

0 rr i r 1; ni A Z þ ; 1 r i rm The input parameters of P2 are the same as those of P1, as shown in Table 1. P3. Series–parallel system (Fig. 4) P3 is a nonlinear mixed integer-programming problem for a series–parallel system with five subsystems. And the problem formulation is given as follows: max : Rs ¼ 1  ð1  R1 R2 Þ½1  ðR3 þ R4  R3 R4 ÞR5  m X s:t: g 1 ¼ wi v2i n2i  V r 0 i¼1

g2 ¼

m X i¼1



αi 

 1000 βi ½ni þ expð0:25ni Þ  C r0 lnðr i Þ

4. Numerical examples In order to validate the performance of proposed NAFSA, five RAPs of different levels of complexity from the literature are selected as the benchmark. The first four well-known problems are a series system, series–parallel system, complex (bridge) system, and overspeed protection system, respectively. And the fifth problem is a large-scale system. The complete formulations of the test problems are given below.

1

2 5

3

4

Fig. 2. The schematic diagram of P1.

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Table 1 Data used in P1 and P2.

Gas Turbine

i

105 αi

βi

wi v2i

wi

V

C

W

1 2 3 4 5

2.330 1.450 0.541 8.050 1.950

1.5 1.5 1.5 1.5 1.5

1 2 3 4 2

7 8 8 6 9

110 110 110 110 110

175 175 175 175 175

200 200 200 200 200

Mechanical and electrical overspeed detection

V1

V2

V3

V4

Air Fuel Mixture 1

2

3

4

5

Fig. 5. The schematic diagram of P4.

Fig. 3. The schematic diagram of P2.

1

Table 3 Data used in P4.

2

i

105αi

βi

vi

wi

V

C

W

5

1 2 3 4

1.0 2.3 0.3 2.3

1.5 1.5 1.5 1.5

1 2 3 2

6 6 8 7

250 250 150 150

400 400 400 400

500 500 500 500

3 4 Fig. 4. The schematic diagram of P3.

Table 4 Data used in P5.

Table 2 Data used in P3. i

105 αi

βi

wi v2i

wi

V

C

W

1 2 3 4 5

2.500 1.450 0.541 0.541 2.100

1.5 1.5 1.5 1.5 1.5

2 4 5 8 4

3.5 4.0 4.0 3.5 4.5

180 180 180 180 180

175 175 175 175 175

100 100 100 100 100

g3 ¼

m X

wi ni expð0:25ni Þ W r0

i¼1

0 r r i r1; ni A Z þ ; 1 ri r m The input parameters of P3 are shown in Table 2. P4. Overspeed protection system for a gas turbine (Fig. 5) Overspeed detection is continuously provided by the electrical and mechanical systems, as shown in Fig. 5. When an overspeed occurs, it is necessary to cut off the fuel supply. For more details and references about the P4, readers should refer to [12]. And the problem formulation is as follow: m   max : Rs ¼ ∏ 1  ð1  r i Þni i¼1

m X

s:t: g 1 ¼

vi n2i  V r 0

i¼1

g2 ¼

m X i¼1

g3 ¼

m X



αi 

β i T ½ni þ expð0:25ni Þ C r0 lnðr i Þ

wi ni expð0:25ni Þ W r0

i

105 αi

βi

vi

wi

V

C

W

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0.6 0.1 1.2 0.3 2.9 1.7 2.6 2.5 1.3 1.8 2.4 1.3 1.2 2.1 0.9 1.3 1.9 2.7 2.8 1.5

1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5

2 5 5 4 4 1 1 4 4 3 3 1 1 3 4 5 1 4 2 1

8 9 6 10 8 9 9 7 9 8 9 8 7 10 6 7 7 8 9 9

600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600 600

700 700 700 700 700 700 700 700 700 700 700 700 700 700 700 700 700 700 700 700

900 900 900 900 900 900 900 900 900 900 900 900 900 900 900 900 900 900 900 900

5. Results and discussion To demonstrate the superiority of the proposed approach in solving RAPs, we select the other five algorithms for comparison. They are the IPSO proposed by Wu et al. [12], ABC2P by Harish et al. [17], IAs proposed by Chen [18], IB2P proposed by Hsieh et al. [19], and AR-ICA proposed by Leonardo et al. [26], respectively. All above five RAPs are used as the benchmark to validate the performance of the algorithms. 5.1. Results of the first four RAPs

i¼1

0:5 r r i r 1  10  6 ; ni A Z þ ; 1 r ni r 10 The input parameters of P4 are shown in Table 3. 4.2. Large-scale RAP At the base of P4, we define a large-scale RAP as P5, where the number of subsystems is 20. The input parameters of P5 are uniformly generated in the given range, as shown in Table 4.

The parameters of the algorithms are set as shown in Table 5. All results are computed by Intel-Pentium T4300 PC with 2 GB RAM under WINXP platform and programs are coded in MATLAB 2007. The determination of proposed algorithm's parameters is a significant problem for the NAFSA implementation. However, there is no formal methodology to solve the problem because different value-combinations of the parameters result to different characteristics as well as different performance of NAFSA. Therefore, one should note that the best values for NAFSA parameters

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are case-dependent and based upon the experience from preliminary runs. 100 times experiments are carried out for each RAPs, and the results are shown in Tables 6–9, in which the best solutions of each problem are reported. Furthermore, MPI (maximum possible improvement) can be used to measure the amount improvement of the solutions found by the proposed approaches to the previous best known solutions. MPI is defined as: MPI ð%Þ ¼ ðRnew  Rchen Þ=ð1  Rchen Þ s s s

ð11Þ

represents the best system reliability obtained by the where Rnew s new proposed algorithm or any other method in the literature, and Rchen indicates that obtained by Chen [18]. SD indicates the stans dard deviation, and it is given as: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 100 u1 X ðRs  R~ s Þ2 ; ð12Þ SD ¼ t 99 1 where R~ s is the average optimum value. From Tables 6–9, we observe that:

calculation accuracy is shown as follows: ABC2P 4 IB2P 4 NAFSA 4IPSO 4 AR  ICA 4 IAs (3) Similar to P2, the new approach acquired more exact solutions than IAs, IPSO and ICA-AR for P3 in Table 8. Furthermore, the best system reliability obtained by the new approach is 0.999976648, and it is very close to the best solution by IB2P (0.999976649) and ABC2P (0.999976649054). The order of calculation accuracy is shown as follows: ABC2P 4 IB2P 4 NAFSA 4IPSO 4 ICA  AR 4 IAs (4) Compared with the solutions of P4 in Table 9, NAFSA made the most relevant improvement to the previous best-known solutions in terms of MPI. And IB2P can find quite close results to that obtained by NAFSA, but the mean CPU time was more than 3 min. The order of calculation accuracy is shown as follows: NAFSA 4 ABC2P 4 IB2P 4ICA  AR 4 IPSO 4 IAs

(1) For P1, It can be clearly seen from Table 6 that the solution found by the new approach is significantly better than solutions by IAs and IB2P, although this difference is not so significant when compared with AR-ICA, IPSO and ABC2P. The order of calculation accuracy is shown as follows: NAFSA 4 ABC2P 4AR  ICA ¼ IPSO 4IB2P 4 IAs (2) Table 7 depicts that the solution of P2 obtained by the new approach is better than the previously known solutions by IAs, IPSO and ICA-AR. Although it is just slightly worse than that by IB2P and ABC2P, the difference is very tiny. The order of Table 5 Parameters of NAFSA.

In sum, the solutions obtained by the new approach, NAFSA, can dominate any other methods for P1 and P4 discussed in the literature. And for P2 and P3, the best system reliabilities obtained by NAFSA are very close to the previous best-known solutions. Thus, we can concluded that NAFSA has considerable accuracy in finding optimal or near optimal for P1–P4 comparing with IB2P and ABC2P. Although IB2P seemingly has shown more outstanding performance for P2 and P3, the mean time for evaluating the problems is quite greater as compared to the NAFSA and ABC2P. For instance, the mean CPU time of IB2P for P2 during the phases I and II are 341.59 s and 53.07813 s, respectively; while that of NAFSA is only 47.84 s. It implies that IB2P will be time-consuming for the large-scale problem. 5.2. Results of P5

Parameters

Value

Population size (N) Iteration number (T) num1 and num2 Cloud model parameter c1and c2 Neighborhood size (Num_nb) Number of offspring(Num_sp) Number of cloud drops (num_c) Step

100 500 5 and 10 100 and 10 20 80 10 1

Although the new proposed approach has shown excellent performance for P1–P4, some of the improvements look extremely small. For instance, the MPI for P1–P3 are only 0.38452%, 0.00625%, and 0.2904%, respectively. Especially, the solution comparisons between NAFSA and ABC2P show that the improvement seems to be relatively tiny. Thus, we further validate the performance of the proposed approaches for a large-scale problem P5, and select IPSO, ABC2P for comparison. It should be mentioned that IB2P is also selected for comparison at first, while

Table 6 Comparison of best result for P1 with other results presented in the literature. Methods

n r

Rs MPI (%) Slack (g1) Slack (g2) Slack (g3) Mean SD Mean CPU

IAs [18]

(3,3,3,3,1) 0.812485 0.867661 0.861221 0.713852 0.756699 0.99988921 – 19 0.001494 4.264770 – – –

AR-ICA [26]

(3,3,2,4,1) 0.82764257 0.85747845 0.91419677 0.64927379 0.70409200 0.99988963 0.3791 5 4.428  10  5 1.56046629 – – –

IPSO [12]

(3,3,2,4,1) 0.82868361 0.85802567 0.91364616 0.64803407 0.70227595 0.99988963 0.3791 5 3.59  10  6 1.56046629 0.99988799 4.0163  10  5 –

IB2P [19]

ABC2P [17]

New approach NAFSA

Phase I

Phase II

Phase I

Phase II

(3,3,3,3,1) 0.814422607 0.867172241 0.859344482 0.715805054 0.741836548 0.999889112  0.088 18 0.011392 4.264770 0.999824 5.6  10  9 138.85

(3,3,3,3,1) 0.816624176 0.868767396 0.858748781 0.710279379 0.753429200 0.999889350 0.12682 18 0.000000 4.264770 0.999889350 4.0  10  20 234.22

(3,3,2,4,1) 0.827222999 0.856301308 0.914575668 0.651220477 0.701774722 0.999889596 0.3484 5 1.8746  10  6 1.560466288 0.999886526 1.186  10  5 2.0827

(3,3,2,4,1) 0.82798027626 0.85787475859 0.91418640423 0.64835538681 0.70357531105 0.99988963581 0.38434 5 3.7464  10  4 1.560466288 0.99988962347 8.667  10  9 0.3837

(3,3,2,4,1) 0.82832179189 0.85797450730 0.91422098825 0.64775717018 0.70300666185 0.99988963601 0.38452 5 1.5485  10  5 1.56046629 0.99987756441 2.1017  10  5 34.98

Please cite this article as: He Q, et al. A novel artificial fish swarm algorithm for solving large-scale reliability–redundancy application problem. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.09.015i

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7

Table 7 Comparison of best result for P2 with other results presented in the literature. Methods

n r

Rs MPI (%) Slack(g1) Slack(g2) Slack(g3) Mean SD Mean CPU

IAs [18]

(3,2,2,3,3) 0.779266 0.872513 0.902634 0.710648 0.788406 0.931678 – 27 0.001559 7.518918 – – –

AR-ICA [26]

(3,2,2,3,3) 0.779874 0.872057 0.903426 0.710960 0.786902 0.93167939 0.002 27 9.9  10  5 7.518918 – – –

IPSO [12]

(3,2,2,3,3) 0.78037307 0.87178343 0.90240890 0.71147356 0.78738760 0.93167996 0.0029 27 0.000101 7.518918 0.92847132 5.24  10  3 –

IB2P [19]

ABC2P [17]

New approach NAFSA

Phase I

Phase II

Phase I

Phase II

(3,2,2,3,3) 0.780624390 0.872299194 0.904159546 0.710647583 0.785079956 0.931662515 0.0023 27 0.00059 7.518918 0.9263678 4.35  10  5 341.59

(3,2,2,3,3) 0.779462304 0.871883456 0.902800879 0.711350168 0.787861587 0.931682340 0.0064 27 5.284  10  7 7.518918 0.93168222 1.3  10  14 53.07813

(3,2,2,3,3) 0.778383314 0.872307791 0.902426561 0.712061479 0.787402330 0.931678416 0.00061 27 2.125  10  9 7.5189182 0.929887360 2.543  10  3 2.3266

(3,2,2,3,3) 0.77940356521 0.87183320141 0.90288641164 0.71139806131 0.78780854858 0.93168238767 0.00642 27 2.258957  10  10 7.518918241 0.93168235223 2.37214  10  8 0.2911

IB2P [19] Phase I

Phase II

ABC2P [17] Phase I

Phase II

(2,2,2,2,4) 0.826843262 0.851425171 0.907211304 0.874832153 0.865188599 0.999976094  2.0751 40 0.002385 1.609289 0.999951900 7.6  10  10 135.73

(2,2,2,2,4) 0.819591561 0.844951068 0.895428548 0.895522339 0.868490229 0.999976649 0.2946 40 5.9845  10  8 1.609289 0.999976649 3.0  10  21 410.375

(2,2,2,2,4) 0.822437533 0.842382359 0.897571538 0.891862761 0.868597931 0.999976609 0.1238 40 3.6006  10  7 1.609288966 0.9999647207 1.458  10  5 1.0261

(2,2,2,2,4) 0.819737753469 0.844991099776 0.895529543820 0.895433687206 0.868434824469 0.999976649054 0.29485 40 1.39152  10  10 1.609288966 0.999976649011 3.18206  10  11 0.2236

(3,2,2,3,3) 0.77938841387 0.87172098236 0.90303339184 0.71141836221 0.78778928811 0.93168226855 0.00625 27 6.7347  10  9 7.518918 0.92300041197 0.0067994 47.84

Table 8 Comparison of best result for P3 with other results presented in the literature. Methods

n r

Rs MPI (%) Slack (g1) Slack (g2) Slack (g3) Mean SD Mean CPU

IAs [18]

(2,2,2,2,4) 0.812485 0.843155 0.897385 0.894516 0.870590 0.99997658 – 40 0.002627 1.609289 – – –

AR-ICA [26]

(2,2,2,2,4) 0.82201264 0.84365640 0.89129092 0.89869886 0.86824939 0.99997661 0.1281 40 3.96  10  4 1.609289 – – –

IPSO [12]

(2,2,2,2,4) 0.81918526 0.84366421 0.89472992 0.89537628 0.86912724 0.99997664 0.2562 40 0.000561 1.609289 0.99996974 1.34  10  5 –

New approach NAFSA

(2,2,2,2,4) 0.819787575273 0.845671943728 0.894868363315 0.895908268569 0.868295830551 0.999976648004 0.2904 40 3.1248  10  8 1.609289 0.999959785424 1.285  10  5 57.61

Table 9 Comparison of best result for P4 with other results presented in the literature. Methods

n r

Rs MPI (%) Slack (g1) Slack (g2) Slack (g3) Mean SD Mean CPU a

IAs [18]

(5,5,5,5) 0.903800 0.874992 0.919898 0.890609 0.999942 50 0.002152 28.80370 – – –

AR-ICA [26]

(5,6,4,5) 0.90148988 0.85003526 0.94812952 0.88823833 0.99995467 21.85 55 0.00213782 24.8018827 – – –

IPSO [12]

(5,6,4,5) 0.90186194 0.84968407 0.94842696 0.88800590 0.99995467 21.8448 55 0.00120356 24.8018827 0.99995460 1.39  10  5 –

IB2P [19]

ABC2P [17]

New approach NAFSA

Phase I

Phase II

Phase I

Phase II

(5,5,4,6) 0.900863647 0.891220093 0.949249268 0.843612671 0.999953931 20.5707 55 0.0761580 15.3634631 0.9999062 5.7  10  9 124.7787

(5,5,4,6) 0.901588628 0.888192380 0.948166022 0.849969792 0.999954674554a 21.8527 55 0.0001250 15.3634631 0.999954673 4.14  10  18 66.54688

(5,5,4,6) 0.901840702 0.888232945 0.948285852 0.849492971 0.999954671 21.84655 55 1.747  10  9 15.36346309 0.999950515 5.543  10  9 1.1309

(5,5,4,6) 0.90162680956 0.88820835588 0.94813437788 0.84994213567 0.99995467466 21.85286 55 5.573042  10  9 15.3634630874 0.99995467461 3.38683  10  11 0.2819

(5,6,4,5) 0.90160779120 0.84993077684 0.94814603278 0.88821809379 0.99995467467 21.85288 55 4.5195  10  7 24.802 0.99995075542 4.43  10  6 17.96

In [18], it was reported 0.999954675.

experiments show that IB2P is very time-consuming for P5. The CPU time is more than 30 min. It is further confirmed that IB2P is inapplicable to the case of large-scale problem. Therefore, we have to give up IB2P for further comparison. The parameters of NAFSA are the same as shown in Table 5, except Step. We divided Step into two parameters corresponding to ri and ni (Step 1 ¼0.01 for ri and Step 2 ¼0.5 for ni ). 100 independent runs are made for the algorithms, and the results of P5

are reported in Table 10. And, five criteria (Best, Worst, Median, SD, and MPI) are used to compare the algorithm performance. Thereinto, Best, Worst, and Median stand for the best, worst and average solutions in the obtained 100-solutions, respectively. From Table 10, we observe that: 1. For NAFSA, when T¼500 and 1000, the best system reliabilities of P5 are 0.998983493020893 and 0.999999763342545, respectively.

Please cite this article as: He Q, et al. A novel artificial fish swarm algorithm for solving large-scale reliability–redundancy application problem. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.09.015i

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8

Table 10 Comparison of best result for P4 with other results presented in the literature. Algorithm

T

Best

Worst

Median

SD

MPI (%)

CPU time

IPSO [12]

500 1000 1500 2000 5000 10,000 500

0.788052649018579 0.835935695089697 0.845439831327050 0.860317574372899 0.877512005817407 0.881947959876058 0.957971810968611 0.978812145313323a 0.977632359143739 0.999668695913533a 0.998983493020893 0.999999763342545

0.473723927292196 0.662076338652914 0.723131504339322 0.754982638511719 0.823173625809728 0.834943312851731 0.800210670352507

0.680627774563814 0.775480358499161 0.811092504525138 0.827000513974036 0.854544996985771 0.868463874616883 0.888738748266877

0.05677 0.03908 0.02229 0.02078 0.01137 0.01009 0.04345

0.864104061701488

0.936373726268754

0.02657

0.872996071566892 0.999651850621475

0.959155837320281 0.999842544014356

0.04245 0.00013

– 22.59 27.08 34.10 42.21 44.30 80.17 90.00 84.45 99.84 99.52 99.99

2.9 5.5 8.4 11.3 28.7 56.7 5.1 4.8 11.2 9.7 53.7 122.9

ABC2P [17]

1000 NAFSA

a

500 1000

It was obtained by Phase 2 of ABC2P.

consuming under the precondition of higher computational accuracy.

6. Conclusion

Fig. 6. The distribution of 100-times numerical results.

For ABC2P, the best system reliabilities of P5 are 0.957971810968611 and 0.977632359143739 corresponding to T¼500 and 1000 by Phase1, and they are improved by Phase 2 up to 0.978812145313323 and 0.999668695913533, respectively. For IPSO, they are 0.78805, 0.83594, 0.84543, 0.86031, 0.87751, and 0.88194 corresponding to T¼500, 1000, 1500, 2000, 5000, and 10,000, respectively. Obviously, the Best, Worst, Median, and SD obtained by NAFSA are all superior to those by IPSO and ABC2P at any given T. The standard deviations SD by NAFSA are pretty low, and it further implies that the proposed approach seems reliable to solve P5. 2. Fig. 6 shows that the distribution of 100-times numerical results when T¼ 500 tends to higher system reliability. It implies that the proposed approach seems robust with respect to the objective function. 3. It can be seen that the improvements relative to the result by IPSO at T ¼500 are very significant. For NAFSA, MPI are 99.52% and 99.99%, respectively, whereas for ABC2P they are 84.45% and 99.84%, respectively. Although the increments of all MPI decrease with the increasing of iteration number gradually, the improvement by NAFSA is not so significant between T ¼500 and 1000. It is implies that NAFSA has a faster convergence rate than IPSO and ABC2P in terms of iteration number. 4. In the case of computation efficiency, although the average CPU time by IPSO and ABC2P is much less than that by NAFSA, NAFSA performs its program within an acceptable time. For instance, the CPU time by NAFSA are 53.7 at T¼ 500 and 122.9 at T ¼1000, while those are 2.9 and 5.5 for IPSO. If jRs 1jo 0.01 is satisfied, NAFSA narrowly win IPSO in terms of CPU time. In conclusion, comparing with IPSO and ABC2P, NAFSA has shown outstanding performance in getting an optimal or near optimal solution for large scale RAPs. Furthermore, it is less time-

In this paper, we focus on the RAPs where both the redundancy and the corresponding reliability of each component in each subsystem under multiple constraints are to be decided simultaneously. And a new approach, called NAFSA, by imitating the fish swarm behaviors is proposed for solving various RAP. In NAFSA, the social behaviors of fish swarm are classified in three ways: foraging behavior, reproductive behavior, and random behavior. The basic idea of foraging behavior is borrowed from AFSA and IPSO. Moreover, GA and cloud model are applied to define the reproductive ability and random behavior of an artificial fish, respectively. Finally, numerical results of four benchmark problems and a large-scale RAP are reported and compared. As demonstrated in the previous section, the best solutions found by NAFSA are better than or tie the well-known best solutions by other heuristic methods for the test problems. Especially, NAFSA shows good performance in terms of computational accuracy and efficiency for large scale RAP. Our limiter experience suggests that the NAFSA finds solutions which are of a quality and are comparable to that of other heuristic algorithm.

Acknowledgments This work was supported by the National Defense Basic Technology Research Program of China (Grant no. Z312012B001), and the National Program on Key Basic Research Project of China (“973” Program) (Grant no. 2013CB035405).

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Please cite this article as: He Q, et al. A novel artificial fish swarm algorithm for solving large-scale reliability–redundancy application problem. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.09.015i