Reliability optimization of series–parallel systems with mixed redundancy strategy in subsystems

Reliability Engineering and System Safety 130 (2014) 132–139

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Reliability optimization of series–parallel systems with mixed redundancy strategy in subsystems Mostafa Abouei Ardakan n, Ali Zeinal Hamadani Department of Industrial and Systems Engineering, Isfahan University of Technology, 84156-83111 Isfahan, Iran

art ic l e i nf o

a b s t r a c t

Article history: Received 15 March 2013 Received in revised form 29 May 2014 Accepted 2 June 2014 Available online 11 June 2014

Traditionally in redundancy allocation problem (RAP), it is assumed that the redundant components are used based on a predefined active or standby strategies. Recently, some studies consider the situation that both active and standby strategies can be used in a specific system. However, these researches assume that the redundancy strategy for each subsystem can be either active or standby and determine the best strategy for these subsystems by using a proper mathematical model. As an extension to this assumption, a novel strategy, that is a combination of traditional active and standby strategies, is introduced. The new strategy is called mixed strategy which uses both active and cold-standby strategies in one subsystem simultaneously. Therefore, the problem is to determine the component type, redundancy level, number of active and cold-standby units for each subsystem in order to maximize the system reliability. To have a more practical model, the problem is formulated with imperfect switching of cold-standby redundant components and k-Erlang time-to-failure (TTF) distribution. As the optimization of RAP belongs to NP-hard class of problems, a genetic algorithm (GA) is developed. The new strategy and proposed GA are implemented on a well-known test problem in the literature which leads to interesting results. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Reliability optimization Redundancy allocation problem Series–parallel system Mixed redundancy strategy Genetic algorithm

1. Introduction Reliability optimization, which is one of the most important types of optimization problems, has attracted many researchers attention. Real-world applications of this problem can be found in most of industrial systems such as telecommunication systems, transformation systems, electrical systems, Space exploration and satellite systems [1,2]. In order to improve the reliability of a specific system, one can choose one of the following options: (a) increasing the component reliability; (b) using the redundant components in parallel; (c) a combination of component reliability enhancement and using the redundant components in parallel; and (d) reassignment of interchangeable components [3]. The second option which is called the redundancy allocation problem (RAP) is a more challengeable topic for researchers because of its wide scope [4]. Generally there are two different classes for RAP. For the first class, there are discrete component choices with known characteristics such as reliability, cost, weight, etc. In this class the objective is to determine which components must be used and what should be the corresponding redundancy levels. For the second class, component reliability is not known in advance and treated as a design variable and component cost, weight, volume, etc., is defined as an

n

Corresponding author. E-mail address: [email protected] (M. Abouei Ardakan).

http://dx.doi.org/10.1016/j.ress.2014.06.001 0951-8320/& 2014 Elsevier Ltd. All rights reserved.

increasing function of component reliability [5]. This paper pertains to the first type of problem. It has been proved that RAP belongs to the NP-hard class of optimization problems [4]. Therefore, it is too difficult to solve such problems with traditional optimization methods and meta-heuristic algorithms have been widely used in the literature to tackle these sorts of problems. In RAPs, different kinds of redundancy strategies are used to improve the reliability of a system. There are generally two types of redundancy strategies, namely, active and standby. In active redundancy strategy, all the redundant components operate together from time zero, and only one of these components is necessary at any special time. There are three variants of the standby redundancy called cold, warm, and hot standby. In cold-standby redundancy the redundant components are shielded from the operational stresses associated with system operation and therefore the component does not fail before its operation. In the warm-standby redundancy, the components are more affected by operational stresses than the cold-standby components. In the hot-standby redundancy, the occurrence of failure in component does not depend on whether the component is idle or in operation. The mathematical formulation for hot-standby strategy is the same with active redundancy case. In the standby redundancy strategy, the redundant components are sequentially used in the system at failure times of operating component by switching to one of the redundant components to continue the system operation [6,7]. In this paper a new redundancy strategy which is a combination of active and cold-standby strategies, called mixed strategy, is introduced.

M. Abouei Ardakan, A. Zeinal Hamadani / Reliability Engineering and System Safety 130 (2014) 132–139

There are two scenarios for failure detection and switching in cold-standby strategy. These are classified as Cases 1 and 2. For Case 1, the failure detection and switching hardware (or software) is continually monitoring system performance to detect a failure and to activate the redundant component. It is assumed that, switch failure can occur at any time and switch reliability, is a non-increasing function of time ðρi ðtÞÞ and does not depend on the number of required switches. In Case 2, failure of the switching will happen when the switch is required with a constant probability ðρi Þ. Reliability optimization problems have been generally formulated by considering active redundancy using different kinds of models and solution methods. For this purpose, exact optimization methods like dynamic programming [8,9], integer programming [10], Lagrangean multipliers [11], column generation [12] and various types of the meta-heuristic algorithms such as GA [13], ant colony optimization [14], Immune algorithm [15], the improved surrogate constraint method [16], variable neighborhood search (VNS) algorithms [17], Tabu search (TS) algorithm [18], and particle swarm optimization [19] are used to maximize the system reliability. For the cold-standby redundancy strategy, fewer studies have been conducted. Robinson and Neuts [20] studied cold-standby redundancy system with non-repairable components. Also imperfect switching problem was studied by Shankar and Gururajan [21] and Gurov and Utkin [22]. For the case of series–parallel system, Coit [2] presents an integer programming solution to the RAP when the system only uses the cold-standby redundancy. He considered imperfect switching and the k-Erlang distribution for TTF of components. Recently few studies have been conducted by considering active and cold-standby redundancies in a specific system simultaneously in order to design more realistic models for RAP. Coit and Liu [23] presented a new mathematical model for RAP to determine the optimal system design configuration by considering predetermined redundancy strategy (active or cold-standby) for each subsystem. In 2003, Coit [5] presented an integer programming method for solving the problem in which selection of active or cold-standby redundancy strategy for each subsystem is a decision variable. TavakkoliMoghaddam et al. [7] proposed a GA to solve this problem. Also, the presented mathematical model by coit [5] is extended in multiobjective assumption by Safari [24] and Chambari et al. [1]. In the latest articles, the redundancy strategy for each subsystem is either active or standby and their aim is to determine the best strategy for subsystems. As an extension to this assumption, in this paper a novel strategy that is a combination of traditional active and cold-standby strategies is introduced. The rest of the paper is organized as follows. In Section 2 the mixed redundancy strategy is described in details. Section 3 presents the modeling of the problem, and in Section 4, the developed GA is introduced for solving the proposed model when both active and cold-standby redundancy can be selected for individual subsystems. Section 5 considers a well-known numerical example to demonstrate the interesting result of new redundancy strategy and the efficiency of the proposed methodology through computational experiment. The final conclusion is given in Section 6.

2. Mixed redundancy strategy As it is mentioned in the previous section, the mixed redundancy strategy for a system is a combination of active and standby strategies. In such a system, each subsystem i can have different numbers of active and cold-standby redundancies represented as nAi and nSi , respectively. The aim is to determine the proper values of these numbers in order to maximize the reliability of the system. The redundancy level in each subsystem i is then a decision variable (ni ) which is not known in advance and will be calculated as ni ¼ nAi þ nSi .

133

Fig. 1. Series–parallel system.

Table 1 Redundancy strategies. Scenarios

nAi

nSi

Redundancy strategy

S1 S2 S3 S4

¼1 41 ¼1 41

¼0 ¼0 Z1 Z1

No redundancy Active redundancy strategy Cold-standby redundancy strategy Mixed redundancy strategy

Coit [5] investigated the relationship between reliability of cold-standby strategy and active redundancy strategy in a specific subsystem. He demonstrated that for each component j, there is a maximum redundancy level where cold-standby reliability is greater than or equal to active reliability. If this maximum level is denoted as n0ij , for ni r n0ij cold-standby is preferable and for ni 4n0ij active redundancy will be preferable. This result reveals that both active and standby strategies have their own strengths and weaknesses and a combined strategy could have a better performance. The aim of this paper is to find a scientific answer for the above claim. Fig. 1 demonstrates the structure of a series–parallel system with the proposed redundancy strategy. As it is shown in this figure, each subsystem has different levels of active and coldstandby redundancies and initially nAi primary components are online and operating in subsystem i. It is essential to notice that, only one component is necessary to work at any special time for operation of each sub systems. At the starting time of the process, all the active components are online and working; as the last online active component fails, it is replaced by one of the available cold-standby units in the prescribed order. The entire system fails when ni components fail in any of the subsystem. It is assumed that each subsystem has at least one operating unit and the number of active units should be greater than or equal to 1 (i.e., nAi Z1) and can have any number of cold-standby units (i.e., nSi Z0). For a specific subsystem in the series–parallel system, there are four different redundancy scenarios as described in Table 1. As it is shown in this table the mixed redundancy is a more general form of active and standby strategies and applies them simultaneously.

3. Problem formulation It is assumed that for each subsystem, there are mi functionally equivalent component choices (types) that can be selected. Also, in each subsystem only one component type is allowed to be used.

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M. Abouei Ardakan, A. Zeinal Hamadani / Reliability Engineering and System Safety 130 (2014) 132–139

Each choice has different levels of reliability, weight, cost and other characteristics and there is an unlimited supply of each choice of components. Furthermore, there are system-level constraints and the problem is to select the component choices, the redundancy type (active, cold-standby or mixed) and the levels of redundancy to maximize the system reliability. Finally, it is assumed that the series–parallel system has imperfect switching system and failure detection and switching hardware is continually monitoring system performance. The new mathematical model and its notations for a series– parallel system with s subsystems and two separable linear constraints on cost and weight by considering the mixed redundancy strategy is presented as the following integer non-linear programming model. 3.1. Notations

s

∑ cizi ni þ

i¼1

∑

i A fS [ Mg

s

∑ wizi ni þ

i¼1

cswitch;i r C;

∑

ni A f1; 2; :::; nmax ;i g;

ð2Þ

zi A f1; 2; :::; mi g

ð3Þ

wswitch;i r W;

i A fS [ Mg

ni ¼ nA;i þ nS;i

8 i ¼ 1; 2 ; :::; s:

ð4Þ

In Eq. (1), the objective function contains the component type, redundancy strategy, the redundancy level and the quantity of active and standby components in each subsystem to achieve maximum system reliability. Constraints (2) and (3) consider the limitations on cost and weight, respectively. For optimization of the objective function, Rðt; z; nA ; ns Þ, Eqs. (5) and (7) are presented for the system reliability in two cases as the following. The detailed procedure of obtaining the following equations is presented in Appendix A. Case 1. continuous detection and switching

s ni

Number of subsystems Number of components used in subsystemi ðiA f1; 2; …; sgÞ nAi Number of active components used in subsystem i nSi Number of standby components used in subsystem i A Set of all subsystems using active redundancy S Set of all subsystems using standby redundancy N Set of all subsystems with no redundancy M Set of all subsystems using mixed redundancy nmax ; i Upper bound for ni mi Number of available component choices for the subsystem i zi Index of component choice used for the subsystem iðzi A f1; 2; :::; mi gÞ z Set of zi ðz1 ; z2 ; :::; zS Þ nA Set of nA;i ðnA;1 ; nA;2 ; :::; nA;S Þ nS Set of nA;i ðnA;1 ; nA;2 ; :::; nA;S Þ T Mission time Rðt; z; nA ; nS Þ System reliability at time t for designing vectors z, nA and nS r ij ðtÞ Reliability at time t for the jth available component for subsystem i Scale and shape parameters for the gamma λij ; kij distribution C, W System-level constraint limits for cost and weight cij ; wij Cost and weight for the jth available component for the subsystem i ρi ðtÞ Failure-detection/switching reliability at time t for Case 1 ρi Failure-detection/switching reliability at time t for Case 2 cswitch;i Cost of switch used in subsystem i wswitch;i Weight of switch used in subsystem i ðjÞ pdf for the jth failure arrival for subsystemi when f ik component type k is used Max; nAi pdf for the maximum active component failure f ik time in subsystem i when component type k is used

Rðt; z; nA ; ns Þ ¼ ∏ ð1  ð1  r izi ðtÞÞni Þ iAA

ni  1

Z

 ∏ r izi ðtÞ þ ∑ iAS

j¼1

t

0

! ðjÞ ρi ðuÞr izi ðt  uÞf izi ðuÞdu

 ∏ r izi ðtÞ  ∏ Rizi ;Mix ðtÞ iAN

ð5Þ

iAM

where, Z Rizi ;Mix ðtÞ ¼ ð1  ð1  r izi ðtÞÞnAi Þ þ nszi  1 Z t

þ ∑

j¼1

Z

0

t t1

t

0

Max; nAi

ρi ðuÞr izi ðt  uÞ  f izi ðjÞ

ðuÞdu

Max; nAi

ρi ðuÞr izi ðt  uÞ  f izi ðu  t 1 Þ  f izi

ðt 1 Þdudt 1

In Eq. (5), ðjÞ

f izi ¼pdf for the jth failure arrival for subsystemi, i.e., sum of j iid failure times of the zi th component for subsystem i and, Max; nAi f izi ¼pdf for the maximum failure times of nAi number of iid failures of the zi th component for subsystem i which is calculated as follow: Max; n

f izi

ðtÞ ¼ n½F izi ðtÞn  1 f izi ðtÞ

ð6Þ

in Eq. (6), f izi ðtÞ is the density function of time-to-failure for the zi th component of subsystem i and F izi ðtÞ is its cumulative distribution function. In this paper it is assumed that the components time-to-failure follow a gamma distribution and therefore, ðjÞ f izi is a gamma distribution. For a gamma distribution with parameters λij and kij , the probability density and reliability function are given by, f ij ðtÞ ¼

kij  1 λij kij t kij  1 e  λij t ðλij tÞl ; r i ðt; jÞ ¼ e  λij t ∑ Γðkij Þ l! l¼0

Case 2. switch activation only in response to a failure occurrence Rðt; z; nA ; ns Þ ¼ ∏ ð1  ð1  r izi ðtÞÞni Þ iAA

Z

ni  1

 ∏ r izi ðtÞ þ ∑ ρi j iAS

j¼1

t 0

! ðjÞ

f izi ðuÞr izi ðt  uÞdu

 ∏ r izi ðtÞ  ∏ Rizi ;Mix ðtÞ iAN

ð7Þ

iAM

where, Z Rizi ;Mix ðtÞ ¼ ð1  ð1  r izi ðtÞÞnAi Þ þ ρi

3.2. Mathematical model

nszi  1

The proposed mathematical model for this paper is as follows: max Rðt; z; nA ; ns Þ s:t:

ð1Þ

þ ∑ ρji j¼1

Z 0

t

Z

t t1

t 0

Max; nAi

r izi ðt  uÞ  f izi ðjÞ

ðuÞdu

Max; nAi

r izi ðt  uÞ  f izi ðu  t 1 Þ  f izi

ðt 1 Þdudt 1

in this paper, continuous detection and switching is considered (Case 1). As it is mentioned by Coit [2], it is difficult to determine a

M. Abouei Ardakan, A. Zeinal Hamadani / Reliability Engineering and System Safety 130 (2014) 132–139

135

closed form for equations similar to Eq. (5). Therefore, a conve~ nient lower bound on system reliability,RðtÞ, can be determined as, ~ z; nA ; ns Þ ¼ ∏ ð1  ð1  r iz ðtÞÞni Þ Rðt; i iAA

Z

ni  1

 ∏ r izi ðtÞ þ ρi ðtÞ ∑ iAS

j¼1

t

0

! ðjÞ

r izi ðt  uÞf izi ðuÞdu

 ∏ r izi ðtÞ  ∏ Rizi ;Mix ðtÞ iAN

ð8Þ

Fig. 3. Representation of a solution.

iAM

where,

4.2. Fitness function Z

Rizi ;Mix ðtÞ ¼ ð1  ð1  r izi ðtÞÞnAi Þ þ ρi ðtÞ nszi  1 Z t

þ ρi ðtÞ ∑

j¼1

0

Z

t

t1

t 0

Max; nAi

r izi ðt  uÞ  f izi ðjÞ

ðuÞdu

Max; nAi

r izi ðt  uÞ  f izi ðu  t 1 Þ  f izi

ðt 1 Þdudt 1

this equation is a an estimation for Eq. (5), because ρi ðtÞ r ρi ðuÞfor all u rt. As it is mentioned in the study of Tavakkoli-Moghaddam et al. [7], it is very hard to obtain the exact solution for reliability optimization problems. Moreover, for the proposed mathematical model, the complex non-linear form of objective function makes the problem more difficult to solve and exact techniques are not necessarily desirable. Accordingly, in this paper a GA, as an efficient metaheuristic algorithm for reliability optimization, is developed.

Fitness function is equal to the sum of objective function (reliability) and the penalty of constraints violation. In other words, the problem constraints are added to the objective function in such a way that if one solution goes beyond the constraints, a relatively large amount of penalty is added to the objective function. This penalty provides the feasibility of the final solution while keeping the search in the infeasible space of the problem. The search in the infeasible space leads to an appropriate diversity for GA. 4.3. Initial population In order to produce initial population, Pop chromosomes are generated randomly and legally. In this paper, population size (Pop) is equal to 20. This value is selected after some preliminary experiments. 4.4. Selection

4. Genetic algorithm Genetic algorithm (GA) is one of the most efficient meta-heuristic methods for solving combinatorial optimization problems introduced by Holland [25]. In this section, the proper design of the GA for implementing on the proposed model is explained. To use GA in its best form; first of all, features of this algorithm should be designed based on the problem trait. These features are chromosomes (solution encoding), fitness function, initial population, selection of parents for generation of new population, crossover, mutation, and stopping criteria, which are explained in the following subsections.

In order to select the required chromosomes for crossover and mutation operators, the tournament selection is used based on the following steps. The fitness function is calculated for all the existing chromosomes (Pop) in the present population, then from the Pop chromosomes, k ¼ 4 chromosomes are selected randomly and compared with each other based on fitness function. A chromosome with the largest fitness function is selected as the parent for generating a new population. This procedure will be repeated Pop times until Pop parents are selected for the next generation. After the selection process, crossover and mutation operators will be used. 4.5. Crossover

4.1. Chromosome definition In the proposed GA, the solution encoding (chromosome) represent as a 3  s matrix where s is the number of subsystems and the first, second, and third rows of this matrix are the types of selected components, the number of selected components for active strategy, and the number of selected components for standby strategy, respectively. Fig. 2 represents a chromosomes structure considered for this problem with s¼14. This figure demonstrates a solution in which the first subsystem (s¼1), is using three of the forth type of components in parallel with one active and two standby (cold-standby redundancy strategy); the second subsystem, uses three of the first type of components in parallel with all active (active redundancy strategy); for the forth subsystem, three of the second type of components are suggested in parallel with two active and one standby component (mixed redundancy strategy), etc. Fig. 3 represents the structure of this solution. 1 2 3

4

5 6 7

8 9 10 11 12 13 14

Type of selected components 1 4 1 2 2 3 2 1 4 6 1 4 3 3 2 Number of active component 2 1 3 2 2 3 1 1 3 2 2 1 2 2 3 Number of standby component 3 2 0 0 1 2 1 3 1 2 1 3 2 2 1 Fig. 2. Chromosome representation (solution encoding).

The crossover operator is taken place with a predefined rate of 0.8. By using the crossover operator, 4 offspring will be generated from each 2 selected parents. The 2 parents and 4 offspring create 6 chromosomes on which the 2 premier chromosomes based on their fitness function will be selected for transferring to the next generation. As a result, there would be Pop populations at the end of crossover operation. In order to produce these 4 offspring from 2 selected parents, the double-point crossover and a modified version of max–min crossover [7] are used which are shown in Figs. 4 and 5, respectively. In max–min crossover the subsystems with the lowest and highest reliability amongst the candidate solutions are determined and all relative genes for each parent are exchanged with the same genes in other parent. 4.6. Mutation Mutation operator is used with a predefined rate of 0.3. The main purpose of applying mutation operator is to increase diversity and avoid trapping into local optimization. For implementing mutation, 30% of parents are selected randomly and the genes values of these chromosomes are changed by a probability of 0.2. Then fitness function is calculated for the muted chromosome and is compared with the fitness function of the chromosome

136

M. Abouei Ardakan, A. Zeinal Hamadani / Reliability Engineering and System Safety 130 (2014) 132–139

1 2 3 4 5

6 7 8 9 10 11 12 13 14

1 2

3 4 5

6 7 8 9 10 11 12 13 14

1 4 1 2 2 4 2 3 1 5 6 4 3 3 2 2 1 3 2 2 1 2 3 2 2 2 1 2 1 3

2 4 1 4 3 2 1 4 6 1 2 5 1 4 1 3 2 2 3 1 1 3 2 2 1 3 2 1

3 2 0 0 1 1 1 1 0 3 3 3 2 3 1

2 0 2 3 2 1 3 1 2 1 0 3 1 3

Fig. 4. Double-point crossover operator.

4 1 2 2 3 2 1 4 6 1 4 3 3 2 1 3 2 2 3 1 1 3 2 2 1 2 1 3

4 1 1 2 4 2 1 4 6 1 4 3 3 2 1 3 2 2 1 1 1 3 2 2 1 2 1 3

2 0 0 1 2 1 3 1 2 1 3 2 3 1

2 0 2 1 1 1 3 1 2 1 3 2 3 1

min

max

2 4 1 4 4 2 3 1 5 6 2 5 1 4 1 3 2 2 1 2 3 2 2 2 1 3 2 1

2 4 1 4 4 2 3 1 5 6 4 3 1 4 1 3 2 2 1 2 3 2 2 2 1 2 2 1

2 0 2 3 1 1 1 0 3 3 0 3 1 3

2 0 2 3 1 1 1 0 3 3 3 2 1 3

min max Fig. 5. Max–min crossover operator.

4 1 2 2 3 2 1 4 6 1 4 3 3 2

4 1 2 2 4 2 1 4 6 1 4 3 3 2

1 3 2 2 3 1 1 3 2 2 1 2 1 3

1 3 3 2 1 1 1 3 2 2 1 2 1 3

2 0 0 1 2 1 3 1 2 1 3 2 3 1

2 0 1 1 2 1 3 1 2 1 3 2 3 1

min

max Fig. 6. Max–min mutation operator.

of pre-mutation. If the fitness function of new chromosome is more than the previous one, the previous chromosome will be replaced by the new generated offspring. Otherwise, the previous chromosome remains as the superior one. In this paper, the max– min mutation operator [7], also is used. In this operator, for each candidate solution, the subsystems with the highest and lowest reliability are mutated randomly. The values of gens for these subsystems are randomly changed at a mutation rate of 0.2. As an example, the max–min mutation is depicted in Fig. 6. Crossover and mutation operators transfer one generation to the next one. For this purpose, the Pop best solutions amongst the previous generation and the new offspring are retained to form the next generation. 4.7. Stopping criteria The GA process is terminated after a predefined number of iteration (MaxGen). After a few preliminary tests, it is found that 40 iterations are sufficient.

5. A numerical example In order to prove the advantage of new redundancy strategy (mixed strategy) compared with the traditional active or standby strategies, a well-known example from the literature is considered. This example has been adapted from the example provided by Fyffe

et al. [8], and its adjusted version given by Coit [2]. The example has been used by other researchers such as Coit [5], TavakkoliMoghaddam et al. [7], Safari [24], and Chambari et al. [1]. In this example a series–parallel system with 14 subsystems is considered. For each subsystem, there are three or four component choices with predetermined cost, weight and k-Erlang distribution parameters which are presented in Table 2. The objective is to maximize system reliability at a given 100 units of time by considering the constraints for system cost (C¼130) and system weight (W¼170). For each subsystem, active, cold-standby or mixed redundancy can be used. Also, it is assumed that the failure detection and switching hardware is continually monitoring system performance to detect a failure and to activate the redundant component (Case 1). Therefore, switch reliability is represented by a continuous decreasing function. The reliability of a switch (at 100 units of time) is 0.99 for all subsystems. The maximum number of components within any subsystem is six (nmax,i ¼6). The component k-Erlang distribution parameters added by Coit [2] leads to the same component reliability for t¼ 100 as Fyffe et al. [8] have used in their paper. Since, GA is a stochastic search algorithm, to find a good solution for the problem, five trials are performed and the best solution amongst the five is considered as the final solution. The maximum reliability obtained by the proposed GA is used to compare the performance of proposed mixed redundancy strategy to the optimal solutions for the series–parallel system in different strategies such as active, cold-standby and either active or coldstandby strategies.

M. Abouei Ardakan, A. Zeinal Hamadani / Reliability Engineering and System Safety 130 (2014) 132–139

137

Table 2 Component data for example. Choice 1 (j¼ 1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Choice 2 (j¼2)

Choice 3 (j¼ 3)

Choice 4 (j¼ 4)

λij

kij

cij

wij

λij

kij

cij

wij

λij

kij

cij

wij

λij

kij

cij

wij

0.00532 0.00818 0.0133 0.00741 0.00619 0.00436 0.0105 0.015 0.00268 0.0141 0.00394 0.00236 0.00215 0.011

2 3 3 2 1 3 3 3 2 3 2 1 2 3

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

3 8 7 5 4 5 7 4 8 6 5 4 5 6

0.000726 0.000619 0.011 0.0124 0.00431 0.00567 0.00466 0.00105 0.000101 0.00683 0.00355 0.00769 0.00436 0.00834

1 1 3 3 2 3 2 1 1 2 2 2 3 1

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

4 10 5 6 3 4 8 7 9 5 6 5 5 7

0.00499 0.00431 0.0124 0.00683 0.00818 0.00268 0.00394 0.0105 0.000408 0.00105 0.00314 0.0133 0.00665 0.00355

2 2 3 2 3 2 2 3 1 1 2 3 3 2

2 1 1 5 3 2 5 6 4 5 5 4 2 5

2 9 6 4 5 5 9 6 7 6 6 6 6 6

0.00818 – 0.00466 – – 0.000408 – – 0.000943 – – 0.011 – 0.00436

3 – 2 – – 1 – – 1 – – 3 – 3

2 – 4 – – 2 – – 3 – – 5 – 6

5 – 4 – – 4 – – 8 – – 7 – 9

Table 3 Comparison between proposed mixed redundancy and other redundancy strategies. Strategy: i

Active [8] zi

ni

1 3 3 2 1 2 3 4 3 4 3 3 5 2 3 6 2 2 7 1 2 8 1 4 9 3 2 10 2 3 11 1 2 12 1 4 13 2 2 14 3 2 System reliability 0.9700 System weight 170 System cost 119

Cold-standby [2]

Active and standby [5]

Active and standby [7]

Active and standby [24]

Proposed mixed

zi

ni

zi

Redundancy

zi

Redundancy

zi

Redundancy

zi

3 2 3 3 3 2 2 2 2 3 2 2 2 2

3 4 1 2 4 3 3 3 2 3 2 2 1 2 3 2 1 2 2 3 3 2 4 2 2 2 3 2 0.9875 170 123

Active Standby Active Standby Active Standby Standby Standby Standby Standby Standby Standby Active Standby

1 2 1 2 4 3 3 3 2 2 2 2 1 2 1 3 1 2 1 2 1 4 1 3 3 2 3 2 0.9705 170 104

Standby Active Active Standby Active Active Standby Standby Active Standby Standby Standby Standby Active

3 4 1 2 3 2 1 3 2 2 3 2 2 2 3 2 1 2 2 3 1 2 3 2 2 2 1 2 0.9722 170 95

Standby Active Standby Standby Standby Standby Standby Standby Active Active Active Standby Active Standby

3 2 1 1 4 2 3 2 2 2 2 1 1 1 1 2 1 1 2 2 3 1 1 3 2 1 3 1 0.9923 170 116

3 1 4 3 2 2 1 3 2 2 3 4 2 3 0.9863 170 123

ni

Table 3 demonstrates the comparison between the new mixed redundancy strategy and the traditional redundancy strategies. Fyffe et al. [8], solved this series–parallel system with active redundancy and find the optimal solution of the problem with system reliability of 0.9700. Coit [2] considered the problem with cold-standby redundancy and finds the optimal solution of 0.9863. In the next study, Coit [5] considered the problem when either active or cold-standby redundancy strategies can be used for each subsystem and the choice of redundancy strategy becomes a decision variable. The maximum system reliability with this assumption increases to the value of 0.9875. This result reveals that using both active and standby redundancy strategies in a system lead to higher system reliability. Some researchers such as Tavakkoli-Moghaddam et al. [7], Safari [24], and Chambari et al. [1] developed meta-heuristic algorithm for the Coit's study [5]. As it is depicted in Table 3, among the previous studies, the best solution for this system that leads to a maximum value of system reliability can be found in Coit [5], with the value of 0.9875. The best solution for this system with the new redundancy strategy is a system with system reliability of 0.99329. As expected, by using the proposed mixed redundancy strategy, higher levels of reliability can be achieved. This improvement in reliability is very important for system designers. The subsystems reliability for the proposed mixed redundancy and previous redundancy strategies is depicted

ni

ni

na

ns

Redundancy

1 1 1 1 1 1 1 1 1 1 1 1 1 1

Mixed Standby Mixed Mixed Mixed Standby Standby Mixed Standby Mixed Standby Mixed Standby Standby

in Fig. 7. This figure demonstrates that the new redundancy strategy can improve the reliability of subsystems 3, 4, 5, 8, 10 and 12. To show the advantage of mixed redundancy strategy compared to previous strategies and also the capability of proposed GA, 33 test problems which are modified by TavakkoliMoghaddam et al. [7] are considered and the comparison results are shown in Table 4. This table demonstrates that for all 33 instances of the problem, the mixed redundancy strategy leads to a considerable improvement in system reliability.

6. Conclusion In this paper, a new redundancy strategy for redundancy allocation problem (RAP) is introduced. The new redundancy strategy which is called mixed redundancy is a combination of active and cold-standby strategies which can be used in each subsystem. In order to evaluate the efficiency of the new strategy, a well-known series–parallel system is considered and a novel mathematical model is developed. The problem is formulated as a non-linear integer programming model subject to a number of given constraints. In general, RAPs are not easy to solve in real cases, especially for large scale situations. Therefore, using meta-heuristic methods for solving such a hard and complex

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1.001 1 0.999 0.998 0.997 0.996 0.995 0.994 0.993 0.992 0.991 sub 1

sub 2

sub 3

sub 4

sub 5

sub 6

sub 7

sub 8

sub 9

sub 10 sub 11 sub 12 sub 13 sub 14

Fyyfe et al., 1968

Coit, 2001

Coit, 2003

Tavakkoli-Moghaddam et al., 2008

Safari, 2012

Proposed mixed Fig. 7. The subsystems reliability.

Table 4 Performance of proposed GA for problems taken from Tavakkoli-Moghaddam et al. [7]. Problem

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

Weight constraint

159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191

Either active or standby [7]

Proposed mixed redundancy

Improvement (%)

Reliability

Weight

Cost

Reliability

Weight

Cost

0.9641 0.9629 0.9636 0.9564 0.9675 0.9619 0.9454 0.9647 0.9614 0.9669 0.9602 0.9705 0.9639 0.9608 0.9717 0.9707 0.9681 0.9703 0.9738 0.9734 0.9756 0.9834 0.9741 0.9779 0.9737 0.9804 0.9844 0.9821 0.9815 0.9771 0.9861 0.9863 0.9856

159 159 161 162 163 164 164 165 167 166 168 170 171 172 172 174 174 176 177 178 176 179 180 178 183 184 184 186 187 187 189 190 189

95 93 95 93 104 96 97 100 91 92 102 104 98 107 102 109 105 102 114 107 100 113 106 102 112 105 120 116 115 112 120 128 123

0.98364 0.98639 0.98763 0.98639 0.98564 0.98964 0.98840 0.99036 0.99126 0.99160 0.99198 0.99233 0.99230 0.99263 0.99263 0.99276 0.99284 0.99318 0.99352 0.99352 0.99417 0.99417 0.99449 0.99448 0.99448 0.99471 0.99512 0.99506 0.99513 0.99527 0.99525 0.99578 0.99592

159 160 160 160 163 164 164 166 167 168 169 170 171 172 172 174 175 176 176 176 179 179 181 182 182 184 185 185 185 188 188 188 191

114 114 115 115 118 119 118 115 121 120 117 116 119 120 120 122 123 123 121 121 124 124 126 126 126 129 129 122 125 129 129 128 128

problem is suggested. In this paper a genetic algorithm (GA), as an effective meta-heuristic algorithm for RAP, is developed. The new strategy and the proposed GA are implemented on a well-known test problem in the literature. Numerical results demonstrate that mixed redundancy strategy leads to a higher reliability compared to other used strategies. The new proposed strategy offers a greater flexibility to system designers and reliability analysts and leads to a noticeable improvement in the reliability of complex systems. For the future studies one can focuses on extending other solution methodology for the proposed mixed strategy to achieve a better result.

2.03 2.44 2.49 3.14 1.87 2.88 4.55 2.66 3.11 2.56 3.31 2.25 2.95 3.31 2.15 2.27 2.56 2.36 2.03 2.07 1.90 1.10 2.09 1.70 2.13 1.46 1.09 1.32 1.39 1.86 0.93 0.96 1.05

Appendix A. Determination of reliability in subsystem with mixed redundancy As it is mentioned in Section 3, the reliability of the system with continuous switching is calculated based on the following equation. ni  1

Rðt; z; nA ; ns Þ ¼ ∏ ð1  ð1  r izi ðtÞÞni Þ  ∏ r izi ðtÞ þρi ðtÞ ∑ iAA

iAS

 ∏ r izi ðtÞ  ∏ Rizi ;Mix ðtÞ iAN

iAM

j¼1

Z 0

t

! ðjÞ

r izi ðt  uÞf izi ðuÞdu

ðA:1Þ

M. Abouei Ardakan, A. Zeinal Hamadani / Reliability Engineering and System Safety 130 (2014) 132–139

in this equation, the first three segments demonstrate the reliability of subsystems with active, cold-standby and no redundancy strategies, respectively. These three segments are obtained from Coit [2]. Considering the gamma distribution as the time to failure distribution of the components, the tractable form for the second segment is given as follows (Coit [2]). ni  1

Z

∏ ðr izi ðtÞ þ ρi ðtÞ ∑

iAS

j¼1

t

0

kij ni  1

ðjÞ

r izi ðt  uÞf izi ðuÞduÞ ¼ ∏ ðr izi ðtÞþ ρi ðtÞ ∑ iAS

l ¼ kij

e  λizi t ðλizi tÞl Þ l!

ðA:2Þ Finally, the last segment of Rðt; z; nA ; ns Þ calculating the reliability of subsystem with mixed redundancy is given by the following equation. Z t Rizi ;Mix ðtÞ ¼ ð1  ð1  r izi ðtÞÞnAi Þ þ ρi ðuÞr izi ðt  uÞ 0 Z tZ t Max; nAi ð1Þ ðuÞdu þ ρi ðuÞr izi ðt  uÞ  f izi ðu  t 1 Þ f izi 0

Max; nAi f izi Max; nAi

f izi

ðnszi  1Þ

f izi

t1

Z

t

ðt 1 Þdudt 1 þ 0

Z

t1

Z

ðt 1 Þdudt 1 þ ::: þ Max; nAi

ðu  t 1 Þ  f izi

t

t 0

ð2Þ

ρi ðuÞr izi ðt  uÞ  f izi ðu  t 1 Þ Z

t

t1

ρi ðuÞr izi ðt  uÞ

ðt 1 Þdudt 1

ðA:3Þ

Therefore, we have: Z Rizi ;Mix ðtÞ ¼ ð1  ð1  r izi ðtÞÞnAi Þ þ nszi  1 Z t

þ ∑

j¼1

0

Z

t t1

0

t

Max; nAi

ρi ðuÞr izi ðt  uÞ  f izi ðjÞ

ðuÞdu

Max; nAi

ρi ðuÞr izi ðt  uÞ  f izi ðu t 1 Þ  f izi

ðt 1 Þdudt 1

In this equation, subsystem reliability is the sum of ns;zi þ1 probabilities associated with ns;zi þ1 mutually exclusive events that result in successful subsystem operation for mission time t. ð1  ð1  r izi ðtÞÞnAi Þ is the probability that no redundant components are required and at least one active component is operating until the mission time. The subsequent ns;zi additive terms (in the summation) represent the mutually exclusive probabilities that all active components are failed and the number of standby failures could be from one to ns;zi  1 such that at least one standby component is still operating at time t. Rt As an illustration, the second term (i.e., 0 ρi ðuÞr izi ðt  uÞ Max; nAi f izi ðuÞdu) is the probability that all the active redundant components fail, but the first standby component is still operating Rt Rt at time t. Also, the fourth term (i.e., 0 t 1 ρi ðuÞr izi ðt  uÞ ð2Þ Max; nAi f izi ðu  t 1 Þ  f izi ðt 1 Þdudt 1 ) shows the probability that the entire active redundant and the first and second standby redundant components fail, but the third standby component is still operating at time t.

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