An efficient two phase approach for solving reliability–redundancy allocation problem using artificial bee colony technique

Computers & Operations Research 40 (2013) 2961–2969

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An efficient two phase approach for solving reliability–redundancy allocation problem using artificial bee colony technique Harish Garg n, Monica Rani, S.P. Sharma Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, Uttarakhand, India

art ic l e i nf o

a b s t r a c t

Available online 20 July 2013

The main goal of the present paper is to present a two phase approach for solving the reliability– redundancy allocation problems (RRAP) with nonlinear resource constraints. In the first phase of the proposed approach, an algorithm based on artificial bee colony (ABC) is developed to solve the allocation problem while in the second phase an improvement of the solution as obtained by this algorithm is made. Four benchmark problems in the reliability–redundancy allocation and two reliability optimization problems have been taken to demonstrate the approach and it is shown by comparison that the solutions by the new proposed approach are better than the solutions available in the literature. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Series system Redundancy allocation Reliability optimization Bee colony

1. Introduction The system reliability optimization is very important in the real world applications and the various kinds of systems have been studied in the literature for decades. To design a highly reliable system there are mainly two ways of improving the system reliability. One is—adding redundant components, and the other is— increasing the component reliability. Both ways usually increase the resources (cost, volume, weight, etc.). Therefore, at the stage of designing a highly reliable system, an important problem is to get the balance between reliability and other resources [1]. Besides the above two ways, the combination of the two approaches and reassignment of interchangeable elements is other feasible ways for increasing the system reliability [1,2]. Such problem of maximizing system reliability through redundancy and component reliability choices is called “reliability–redundancy allocation problem (RRAP)”. The redundancy optimization problem is usually formulated as a non-linear integer problem, which is in general difficult to solve due to the considerable amount of computational effort required to find the exact solution. The general mathematical formulation of the reliability–redundancy allocation problem is Maximize

Rs ðr 1 ; r 2 ; …; r m ; n1 ; n2 ; …; nm Þ

subject to 0≤r i ≤1;

gðr 1 ; r 2 ; …; r m ; n1 ; n2 ; …; nm Þ≤b

i ¼ 1; 2; …; m

1≤ni ≤ni;max

ni ∈Zþ ; r i ∈½0; 1⊂Rþ

n

Corresponding author. Tel.: +91 9897599923. E-mail addresses: [email protected], [email protected] (H. Garg), [email protected] (M. Rani). URLS: http://sites.google.com/site/harishg58iitr/ (H. Garg), http://sites.google. com/site/monicaiitr/ (M. Rani). 0305-0548/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cor.2013.07.014

where gðÞ is the set of constraint functions usually associated with the system's weight, volume and cost; Rs ðÞ is the objective function for the overall system reliability; ri and ni are the reliability and the number of redundant components in the ith subsystem respectively; m is the number of subsystems in the system and b is the vector of resource limitation. This problem is an NP problem and belongs to the category of constrained nonlinear mixed-integer optimization problems because the number of redundancy ni is the positive integer values and the component reliability ri are the real values between 0 and 1. The goal of the problem is to determine the number of components ni and the components' reliability ri in each subsystem so as to maximize the overall system reliability. During the last two decades, numerous reliability design techniques have been introduced to solve these problems. These techniques can be classified as implicit enumeration, dynamic programming, branch and bound technique, linear programming, Lagrangian multiplier method, heuristic methods and so on. To solve this type of problem, Kuo et al. [3], Tillman et al. [4] have extensively reviewed the several optimization techniques for system reliability design. Nakagawa [5] compared three heuristic methods (Nakagawa–Nakashima, Gopal–Aggarwal–Gupta, Sharma–Venkateswaran) for solving reliability optimization problems with nonlinear constraints. Their effectiveness, measured in terms of computation time, optimality rate, and relative error, is evaluated on several sets of randomly generated test problems with nonlinear constraints for series systems. After combining Lagrange multiplier and branch and bound algorithms, Kohda and Inoue [6] gave a heuristic approach in which new criterion of local optimality was presented. They showed that their method generates solutions which are optimal in 2-neighborhood for the redundancy optimization problem. Kuo et al. [7] proposed a heuristic algorithm for a series system

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and obtained solutions close to the optimal one via Lagrange multiplier. Misra and Sharma [8] proposed an algorithm and solved problems by integer programming, which serves as an algorithm searching for nearby boundary of the domain of feasible solutions. Prasad and Kuo [9] pointed out that the algorithm gave by Misra and Sharma sometimes cannot yield an optimal solution, and suggested a method for searching of the upper limit of a reliability objective function. Kim and Yum [10] solved the reliability optimization problem of a series-parallel system by using heuristic algorithms. The method proposed by them allows excursions over a bounded infeasible region, and hence gives the global optimal solution. On the basis of computational results they proved that the approach was faster and straightforward than any other heuristic method. Shi [11] developed a heuristic method with separable, monotonic nondecreasing constraints function following the approach of adjusting unitincrement with time. The iterative heuristic method and surrogate dual approach are used to solve this mixed-integer reliability design problems by Hikita et al. [2], Xu et al. [12] respectively. However, the heuristic techniques require derivatives for all non-linear constraint functions, that are not derived easily because of the high computational complexity. To overcome this difficulty metaheuristics have been selected and successfully applied to handle a number of reliability optimization problems. These heuristics include genetic algorithms (GA) [13,14], simulated annealing (SA) [15], particle swarm optimization (PSO) [16,17], immune algorithm (IA) [18], artificial bee colony (ABC) [19,20], etc. Yokota et al. [13], Painton and Campbell [14] and Hsieh et al. [21] applied genetic algorithms to solve these mixed-integer reliability optimization problems. Coit and Smith [22] combined GA and neural network (NN) to tackle the series-parallel redundancy problem. Chen [18] applied the IA for solving the reliability– redundancy allocation problem. Coelho [16] proposed an efficient PSO algorithm based on Gaussian distribution and chaotic sequence to solve the reliability–redundancy optimization problems. Yeh and Hsieh [19] developed a penalty guided artificial bee colony algorithm (ABC) for solving the reliability optimization problems. Gupta and Agarwal [23], Levitin et al. [24] presented penalty guided genetic algorithm making use of a heterogeneous collection of components to provide redundancy in a subsystem, with application to multi-state power systems. The aim of their problem is to minimize the cost subject to system-levelperformance constraints and has been solved for various levels of reliability requirements. Gupta and Agarwal [25] presented a heuristic algorithm to solve the two power system problems by allowing subsystem structures to be composed of maximum two types of components from the list of available products and suggested a binomial probability based formula to evaluate system reliability. Further, for these systems Agarwal and Sharma [26] adapted ACO and demonstrated its efficiency over prevalent methods. Wu et al. [27] proposed an improved particle swarm optimization algorithm for solving the reliability problems. Hsieh and You [20] proposed an immune based two-phase approach to solve the reliability–redundancy allocation problem. In the first phase, an immune based algorithm (IA) is developed to solve the allocation problem, and in the second phase a new procedure is presented to improve the solutions by the IA. Garg and Sharma [17] proposed a multiobjective reliability–redundancy allocation problem of the system using particle swarm optimization. The conflictness between the objectives are handled with the help of defining their linear as well as non-linear membership functions. In the light of the advantages of the meta-heuristics techniques, the presented paper discusses the two phase approach for the reliability–redundancy allocation problem. In the first phase, the optimal solution of the reliability–redundancy allocation problem has been obtained with one of the meta-heuristic technique namely artificial bee colony (ABC) while in the second phase the

component reliability allocation is improved after fixing the number of component redundancy as obtained during Phase I. Four benchmark problems of reliability–redundancy allocation and two reliability optimization problems are solved with the proposed technique and it is observed that our results are all better than the existing results in the literature.

2. Problem formulation: reliability–redundancy allocation problem Before introducing the reliability–redundancy allocation problem, we define the following assumptions and notations that have been used in the entire paper. 2.1. Assumptions

 If a component of any subsystem fails to function, the entire system will not be damaged or fail.

 All redundancies are active redundancy with out repair.  The components and system have only two states – operating state or failure state.

2.2. Notations m M ni n

number of subsystems in the system. number of constraints. the number of components in subsystem i; 1≤i≤m . ðn1 ; n2 ; …; nm Þ, the vector of redundancy allocation for the system. ri reliability of each components in subsystem i; 1≤i≤m: r ðr 1 ; r 2 ; …; r m Þ, the vector of component reliabilities for the system. gj the jth constraint function, j ¼ 1; 2; …; M. wi the weight of each component in subsystem i; 1≤i≤m: ci the cost of the each component in subsystem i; 1≤i≤m: vi the volume of each component in subsystem i; 1≤i≤m: Ri ¼ 1ð1r i Þni is the reliability of the ith subsystem 1≤i≤m: Qi 1Ri is the unreliability of the ith subsystem. ni;max maximum number of components in subsystem i; 1≤i≤m. Rs the system reliability. C; W the upper limit of the system's cost, weight respectively. S feasible search space.

In the present paper, four benchmark problems of the reliability–redundancy allocation problem have been studied. The first three problems with nonlinear constraints used by Hikita et al. [2], Xu et al. [12], Chen [21], Yeh and Hsieh [18], Hsieh and You [20], Hsieh et al. [19] are a series system, series-parallel system and complex (bridge) system respectively. The fourth problem, investigated by Yokota et al. [13], Coelho [18], Chen [16], Yeh and Hsieh [19], Hsieh and You [20], Dhingra [28] is of overspeed protection system. The main aim of these problems is to maximize the systems' reliability subject to multiple nonlinear constraints and can be stated as the mixed-integer nonlinear programming problems. For each problem both, the component reliabilities and redundancy allocations are to be decided simultaneously. The four reliability–redundancy allocation problems are formulated below. Problem 1. Series system (Fig. 1a) [2,15,18–21] 5

Maximize

Rs ðr; nÞ ¼ ∏ ½1ð1r i Þni  i¼1

H. Garg et al. / Computers & Operations Research 40 (2013) 2961–2969

1

1

2

3

4

2

2963

5

2

1

5

3 5 3

4

4

Gas Turbine

Mechanical and electrical overspeed detection

V1

V2

V3

V4

Air Fuel Mixture Fig. 1. (a) Series, (b) series-parallel, (c) bridge and (d) overspeed gas turbine systems.

1 2

1

4

4

3

3 1

2

2

5

4 Fig. 2. Systematic diagram of (a) space capsule and (b) bridge network systems.

5

g 1 ðr; nÞ ¼ ∑ vi n2i V≤0

subject to

ð1Þ

where

Q i ¼ 1Ri ¼ ð1r i Þni

i¼1

5

βi

g 2 ðr; nÞ ¼ ∑ αi ð1000=ln r i Þ ½ni þ expðni =4ÞC≤0

ð2Þ

i¼1 5

g 3 ðr; nÞ ¼ ∑ wi ni expðni =4ÞW≤0 i¼1

r i ∈½0; 1⊂Rþ ; 1≤ni ≤5; ni ∈Zþ ; i ¼ 1; 2; …; 5

0:5≤r i ≤1;

Problem 2. Series-parallel system (Fig. 2b) [2,21,18,19,15,20] Rs ðr; nÞ ¼ 1ð1R1 R2 Þ½1ðR3 þ R4 R3 R4 ÞR5 

Maximize

subject to

g 1 ðr; nÞ; g 2 ðr; nÞ; g 3 ðr; nÞ

ðas specified by ð1Þ–ð3Þ respectivelyÞ r i ∈½0; 1⊂Rþ ; 1≤ni ≤5; ni ∈Zþ ; i ¼ 1; 2; …; 5

0:5≤r i ≤1; where

Ri ¼ 1ð1r i Þni

ð3Þ

Problem 4. Overspeed protection system (Fig. 1d) [13,15,16,18– 20,28] The fourth problem is considered for the reliability–redundancy allocation problem of the overspeed protection system for a gas turbine. Overspeed detection is continuously provided by the electrical and mechanical systems. When an overspeed occurs, it is necessary to cut off the fuel supply. For this purpose, four control valves (V1–V4) must close. The control system is modeled as a four-stage series system. The objective is to determine an optimal level of ri and ni at each stage i such that the system reliability is maximized. This reliability problem is formulated as follows: 4

Maximize

Rs ðr; nÞ ¼ ∏ f1ð1r i Þni g i¼1

4

g 1 ðr; nÞ ¼ ∑ vi n2i V ≤0

subject to

i¼1

4

Problem 3. Complex (bridge) system (Fig. 1c) [2,15,16,18–21] Maximize

Rs ðr; nÞ ¼ R5 ð1Q 1 Q 3 Þð1Q 2 Q 4 Þ þQ 5 ½1ð1R1 R2 Þð1R3 R4 Þ

subject to

g 1 ðr; nÞ; g 2 ðr; nÞ; g 3 ðr; nÞ

ðas specified by ð1Þ–ð3Þ respectivelyÞ 0:5≤r i ≤1;

r i ∈½0; 1⊂Rþ ; 1≤ni ≤5; ni ∈Zþ ; i ¼ 1; 2; …; 5

g 2 ðr; nÞ ¼ ∑ αi ð1000=ln r i Þβi ½ni þ expðni =4ÞC≤0 i¼1 4

g 3 ðr; nÞ ¼ ∑ wi ni expðni =4ÞW≤0 i¼1

0:5≤r i ≤1; r i ∈½0; 1⊂Rþ ; 1≤ni ≤10; ni ∈Zþ ; i ¼ 1; 2; …; 4 where vi is the volume of each component at stage i, wi is the weight of each component at the stage i, Q i ¼ 1Ri is the failure

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probability of each component in subsystem i. The factor expðni =4Þ accounts for the interconnecting hardware. The parameters βi and αi are the physical feature (shaping and scaling factor) of the cost– reliability curve of each component in stage i. V is the upper limit on the volume; C is the upper limit on the cost of the system, and W is the upper limit on the weight of the system. Constraints g 1 ðr; nÞ is the volume constraints, g 2 ðr; nÞ is a cost constraints while g 3 ðr; nÞ is a weight constraints. The value of the input parameters defining the specific instances of these first four problems has been the same as taken in [2,12,13,15,16,18,19,21,27–30], and are shown in Tables 1–3. Problem 5. Life support system in a space capsule (Fig. 2a) [4,31–37]. This problem is a continuous nonlinear optimization problem. The source of this problem is Ravi et al. [34]. The schematic of the complex system is usually found in (i) the communication system of two-man space capsule and (ii) the high-pressure oxygen support system in a space capsule. The problem is to determine the minimum cost of a life support system in a space capsule subject to the constraints on the reliability of the system. The objective is to find the minimum cost of the system as well as maximum reliability. The schematic diagram is shown in Fig. 2a. The system has four components, each having component reliability r i ; i ¼ 1; 2; 3; 4 such that the system reliability is given by Rs ¼ 1r 3 ½ð1r 1 Þð1r 4 Þ2 ð1r 3 Þ½1r 2 ð1ð1r 1 Þð1r 4 ÞÞ2

C s ¼ 2K 1 r a11 þ 2K 2 r a22 þ K 3 r a33 þ 2K 4 r a44

subject to Ri;min ≤r i ≤1;

Problem 6. Complex bridge network (Fig. 2b) [31–34,38]. The bridge network is considered as a system of the five components to find out the system reliability as shown in Fig. 2b. The source of this problem is Mohan and Shanker [38]. This problem is to determine the minimum cost of a complex bridge network system with constraints on system reliability. The objective is to minimize the cost and maximize reliability at the same time. The algebraic expression for system reliability Rs of the bridge system is given as follows: Rs ðr 1 ; r 2 ; r 3 ; r 4 ; r 5 Þ ¼ r 1 r 4 þ r 2 r 5 þ r 2 r 3 r 4 þ r 1 r 3 r 5 þ2r 1 r 2 r 3 r 4 r 5 r 2 r 3 r 4 r 5 r 1 r 3 r 4 r 5 r 1 r 2 r 4 r 5 r 1 r 2 r 3 r 5 r 1 r 2 r 3 r 4 where 0≤r i ≤1 for i ¼ 1; 2; 3; 4; 5. By following the Misra [39], the cost of the ith component is taken as
 bi C i ¼ ai exp 1r i Thus the total cost of the system, which is to minimized is 5

The mathematical model for the life support system in a space capsule can be formulated using the block diagram (Fig. 2a) as follows: Minimize

where Ri;min and Rs;min are the lower bounds on the reliabilities of the ith component and the system respectively and Ri;min ¼ 0:5, Rs;min ¼ 0:9. On the other hand the different values of Ki's as 100, 100, 200, 150 respectively and all ai's are equal to 0.6.

Rs;min ≤Rs ≤1 i ¼ 1; 2; 3; 4

Cs ¼ ∑ Ci i¼1

Hence, mathematically, the reliability optimization problem for the bridge network can be formulated as follows:
 5 bi Minimize C s ¼ ∑ ai exp 1r i i¼1 subject to 0:5≤r i ≤1 0:99≤Rs ≤1 ai ¼ 1; b1 ¼ 0:0003; ∀i

Table 1 Parameter used for problems 1 and 3. i

105 αi

βi

vi

wi

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

175

110

200

C

V

W

3.1. Phase I: artificial bee colony optimization

Table 2 Parameter used for problem 2. i

105 αi

βi

vi

wi

C

V

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 3.5

175

180

100

C

V

W

400

250

500

Table 3 Parameter used for problem 4. i

105 αi

βi

vi

wi

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

3. Two phase approach

The artificial bee colony (ABC) optimization algorithm was first developed by Karaboga in 2005. Since then Karaboga and Basturk and their coauthors [40,41] have systematically studied the performance of the ABC algorithm and its extension on unconstrained optimization problems. In ABC algorithm, the bees in a colony are divided into three groups: employed bees (forager bees), onlooker bees (observer bees) and scouts. For each food source, there is only one employed bee. That is to say, the number of employed bees is equal to a number of food sources. The employed bee of a discarded food site is forced to become a scout for searching new food source randomly. Employed bees share information with the onlooker bees in a hive so that onlooker bee can choose a food source to the forager. The whole process of the algorithm may also be explained through the flowchart given in Fig. 3. In this, the first stage is the initialization stage in which food source positions are randomly selected by the bees and their nectar amounts (i.e. fitness function) are determined. Then, these bees come into the hive and share the nectar information of the sources with the bees waiting on the dance area within the hive. At the second stage, after sharing the information, every employed bee goes to the food source area visited by her at the previous cycle. Thus the probability ph of an onlooker bee choose to go the preferred food

Method

n r

Rs MPI (%) Slacks of g 1 ∼g3 Mean Std Mean CPU a

Gopal et al. [43]

Hikita et al. [44]

Xu et al. [12]

Hikita et al. [2]

Hsieh et al. [21]

Gen and Yun [45]

Chen [18] Yeh and Hsieh [19]

Wu et al. [27] Hsieh and You [20]

(3,3,2,3,2) 0.77960 0.80065 0.90227 0.71044 0.85947 0.92975 2.75072% 27

(3,2,2,3,3) 0.8 0.8625 0.90156 0.7 0.8 0.930289 1.99881% 27

(3,2,2,3,3) 0.774887 0.870065 0.898549 0.716524 0.791368 0.931451 0.33755% 27

(3,2,2,3,3) 0.77939 0.87183 0.90288 0.71139 0.78779 0.931677 0.00788% 27

(3,2,2,3,3) 0.777143 0.867514 0.896696 0.717739 0.793889 0.931363 0.46532% 27

(3,2,2,3,3) 0.779427 0.869482 0.902674 0.714038 0.786896 0.931578 0.15256% 27

(3,2,2,3,3) 0.780874 0.871292 0.902316 0.711945 0.786995 0.931676 0.00934% 27

(3,2,2,3,3) 0.779266 0.872513 0.902634 0.710648 0.788406 0.931678 0.00642% 27

(3,2,2,3,3) 0.779399 0.871837 0.902885 0.711403 0.787800 0.931682 27

(3,2,2,3,3) 0.78037307 0.87178343 0.90240890 0.71147356 0.78738760 0.931680 0.00349% 27

0.00001 10.57248 – – –

0.0265 7.518918 – – –

0.108244 7.518918 – – –

0.013773 7.518918 – – –

0 7.518918 – – –

0.121454 7.518918 – – –

0.003352 7.518918 – – –

0.001559 7.518918 – – –

 0.0002184b 7.518918 0.930580 8.14  10  4 0.696

0.121454 7.518918 – 5.2382  10  3 –

Infeasible. Violate constraint.

a

Proposed approach

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 0.02908% 27

(3,2,2,3,3) 0.779462304 0.871883456 0.902800879 0.711350168 0.787861587 0.931682340 0.00006% 27

(3,2,2,3,3) 0.778383314081 0.872307790659 0.902426561194 0.712061479297 0.787402330497 0.931678415673 0.00581% 27

(3,2,2,3,3) 0.779403565208 0.871833201410 0.902886411643 0.711398061305 0.787808548579 0.931682387672 – 27

0.00059 7.518918 0.9263678 0.0000435 341.59

0.0000005284 7.518918 0.93168222 1.3  10  14 53.07813

2.12529  10  9 7.518918241 0.92988735978 2.543  10  3 2.3266

2.258957  10  10 7.518918241 0.931682352234 2.37214  10  8 0.2911

2965

b

Kuo et al. [29]

H. Garg et al. / Computers & Operations Research 40 (2013) 2961–2969

Fig. 3. Procedure of ABC approach.

ð4Þ

source at Xh can be defined by ph ¼ f h =∑hN ¼ 1 f h , where N is the number of food sources and f h ¼ f ðX h Þ is the amount of nectar evaluated by its employed bee. If a food source is tried/foraged at a given number of explorations without improvement, then it is abandoned, and the bee at this location will move randomly to explore new locations. After a solution is generated, that solution is improved by using a local search process called greedy selection process carried out by onlooker and employed bees and is given by the following equation:

Z hj ¼ X hj þ ϕðX hj X kj Þ

where k∈f1; 2; …; Ng and j∈f1; 2; …; Dg are randomly chosen indices and D is the number of solutions parameters. Although k is determined randomly, it has to be different from h. ϕ is a random number between [  1, 1] and Zh is the solution in the neighborhood of X h ¼ ðX h1 ; X h2 ; …; X hD Þ. Except for the selected

Table 4 Optimal solutions of the problem 1.

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H. Garg et al. / Computers & Operations Research 40 (2013) 2961–2969

parameter j, all other parametric values of Zh are same as that of Xh i.e. Z h ¼ ðX h1 ; X h2 ; …; X hðj1Þ ; Z hj ; X hðjþ1Þ ; …; X hD Þ. If a particular food source solution does not improve for a predetermined iteration number then a new food source will be searched out by its associated bee and it becomes a scout. So this randomly generated food source is equally assigned to this scout and changing its status from scout to employ and hence other iteration/cycle of the algorithm begins until the termination condition, maximum cycle number (MCN) or relative error, is not satisfied.

3.1.1. Constraint handling technique The main task while solving the constrained optimization problem is to handle the constraints. In such a problem, it is not easy to find the feasible solution of the problem due to the presence of both types of constraints in the form of the equalities and inequalities. When ABC algorithms are used for constrained optimization problem then penalty function, which penalize the unfeasible function, is used to handle the constraints. Despite the popularity of penalty functions, they have several drawbacks out of which the main one is that of having too many parameters to be adjusted and finding the right combination of the same may not be easy. Also during that the search is very slow and there is no guarantee that the optima will be attained. To overcome this limitation, Deb [42] modified these algorithms using the concept of penalty functions in which an attempt to solve an unconstrained problem in a search space S using a modified objective function F such as 8 f ðxÞ if x∈S > < M FðxÞ ¼ f þ ∑ ð5Þ > : w j ¼ 1g j ðxÞ if x∉S where x are solutions obtained by ABC approaches and fw is the worst feasible solution and set it to be zero if there is no feasible solution in the population.

3.2. Phase II: improvement procedure In the second phase, we fix the number of component redundancy obtained by Phase-I and then use the following procedure

to improve the component reliability allocation. The main steps of the second phase are as follows. Step 1: Obtain the solution ðn; rÞ and Rs by Phase I. Step 2: In order to increase the efficiency of the system, the obtained r's is to be converted into closed interval ½r i ; r j  with equal spread 7 0.5% in both the directions (left and right) of r's i.e. r i ¼ 0:995r and r j ¼ 1:005r≤1 Step 3: Find max R~s ðr~ ; nÞ where r~ ∈½r i ; r j  s.t. r i ; r j ∈½0; 1 w.r.t. g1 ; g2 ; g3 . Step 4: If R~s 4 Rs and jR~s Rs j4 ∈ then Rs ⟵R~s , ðr; nÞ⟵ðr~ ; nÞ and go to next step, otherwise go to Step 3. Step 5: Report the optimal or near optimal solution. 4. Computational results The bees' particle for each problem uses the variable vectors n and r. During the evolution process, the integer variable ni are treated as real variables, and in evaluating the objective functions, the real values are transformed to the nearest integer values. In the experiment we set ∈ ¼ 107 . The presented algorithm is implemented in Matlab (MathWorks) and the program has been run on a T6400 @ 2 GHz Intel Core(TM) 2 Duo processor with 2 GB of random access memory (RAM). The values of the parameters such as population size and total evaluation number are set to be randomly as 20  D and 1000 respectively, where D is the dimension of the problem. In order to eliminate stochastic discrepancy, in each case study, 30 independent runs are made for each of the optimization methods involving 30 different initial trial solutions, and their best values are reported. The termination criterion has been set either limited to a maximum number of generations ( ¼1000) or to the order to a relative error equal to 106 , whichever is achieved first. The numerical results for the Problem 1–6 are shown in Tables 4–9, in which the best solutions of each problem are reported. For the series system (i.e. Problem 1), Table 4 shows that the best solution by the presented approach is 0.931682387672 which is better than solutions obtained by the other approaches available in the literature [2,12,18,21,27,29,43–45] with an improvement factor 2.75072%, 1.99881%, 0.33755%, 0.00788%, 0.46532%, 0.15256%, 0.00934%, 0.00642%, 0.00349%, 0.02908%, 0.00006% respectively. It should be noticed that even very small improvements in reliability

Table 5 Optimal solutions of the problem 2. Method

n r

Rs MPI (%) Slacks of g 1 ∼g3 Mean Std Mean CPU a b

Hikita et al. [2]

Hsieh et al. [21]

Chen [18]

Kim et al. [15]

Yeh and Hsieh [19]

Wu et al. [27] Hsieh and You [20]

(3,3,1,2,3) 0.83819295 0.85506525 0.87885933 0.91140223 0.85035522 0.99996875 25.27697% 53

(2,2,2,2,4) 0.785452 0.842998 0.885333 0.917958 0.870318 0.99997418 9.56256% 40

(2,2,2,2,4) 0.812485 0.843155 0.897385 0.894516 0.870590 0.99997658 0.29485% 40

(2,2,2,2,4) 0.812161 0.853346 0.897597 0.900710 0.866316 0.99997631 1.43121% 40

(2,2,2,2,4) 0.8197457 0.8450080 0.8954581 0.9009032 0.8684069 0.99997731 40

(2,2,2,2,4) 0.81918526 0.84366421 0.89472992 0.89537628 0.86912724 0.99997664 0.03875% 40

0.000011 7.110849 – – –

1.194440 1.609289 – – –

0.002627 1.609289 – – –

0.007300 1.609289 – – –

–1.469522b 1.609289 0.99997517 2.89  10  6 0.936

0.000561 1.609289 – 1.3362  10  5 –

Infeasible. Violate constraint.

a

Proposed approach

Phase I

Phase II

Phase I

Phase II

(2,2,2,2,4) 0.826843262 0.851425171 0.907211304 0.874832153 0.865188599 0.999976094 2.32181% 40

(2,2,2,2,4) 0.819591561 0.844951068 0.895428548 0.895522339 0.868490229 0.999976649 0.00023% 40

(2,2,2,2,4) 0.822437533034 0.842382359204 0.897571538285 0.891862760631 0.868597930940 0.999976609441 0.16935% 40

(2,2,2,2,4) 0.819737753469 0.844991099776 0.895529543820 0.895433687206 0.868434824469 0.999976649054 – 40

0.002385 1.609289 0.999951900 7.6  10  10 135.73

0.000000 1.609289 0.999976649 3.0  10  21 410.375

3.6006  10  7 1.609288966 0.9999647207 1.458  10  5 1.0261

1.39152  10  10 1.609288966 0.999976649011357 3.18206  10  11 0.2236

Table 6 Optimal solutions of the problem 3. Method

n r

Mean Std Mean CPU a b c

Hsieh et al. [21]

(3,3,2,3,2) 0.814483 0.821383 0.896151 0.713091 0.814091 0.99978937 47.60281% 27 0.000000 10.572475 – – –

(3,3,3,3,1) 0.814090 0.864614 0.890291 0.701190 0.734731 0.99987916 8.66914% 18 0.376347 4.264770 – – –

Chen [18]

Kim et al. [15]

Coelho [16]

Yeh and Hsieh [19]

Wu et al. [27]

(3,3,3,3,1) 0.812485 0.867661 0.861221 0.713852 0.756699 0.99988921 0.38433% 18 0.001494 4.264770 – – –

(3,3,3,3,1) 0.807263 0.868116 0.872862 0.712673 0.751034 0.99988764 1.77625% 18 0.007300 1.609289 – – –

(3,3,2,4,1) 0.826678 0.857172 0.914629 0.648918 0.715290 0.99988957 0.05958% 5 0.000339 1.560466 0.99988594 0.00000069 –

(3,3,2,4,1) 0.828087 0.857805 0.704163 0.648146 0.914240 0.99948407a 5  25.433926c 1.560466288 0.99988362 1.026  10  5 1.0956

(3,3,2,4,1) 0.82868361 0.85802567 0.91364616 0.64803407 0.70227595 0.99988963 0.00526% 5 0.00000359 1.56046629 – 4.0163  10  5 –

Chen [18]

Kim et al. [15]

Coelho [16]

Yeh and Hsieh [19]

Zou et al. [30]

b

Zou et al. [46]

(3,3,2,4,1) 0.82983999 0.85798911 0.91333926 0.64674479 0.70310972 0.99988960 0.03243% 5 0.00000594 1.56046629 0.99988263 1.6  10  5 –

Hsieh and You [20]

Proposed approach

Phase I

Phase II

Phase I

Phase II

(3,3,3,3,1) 0.814422607 0.867172241 0.859344482 0.715805054 0.741836548 0.9998891120 0.47237% 18 0.011392 4.264770 0.999824 5.6  10  9 138.8469

(3,3,3,3,1) 0.816624176 0.868767396 0.858748781 0.710279379 0.753429200 0.9998893505 0.25784% 18 0.000000 4.264770 0.9998893503 4.0  10  20 234.2188

(3,3,2,4,1) 0.827222999061 0.856301308081 0.914575667656 0.651220477286 0.701774721926 0.9998895964609 0.03564% 5 1.874635  10  6 1.560466288 0.9998865255 1.186  10  5 2.0827

(3,3,2,4,1) 0.827970276262 0.857874758586 0.914186404228 0.648355386813 0.703575311047 0.999889635809 – 5 3.74636763  10  4 1.560466288 0.99988962347 8.667  10  9 0.3837

In [19], it was reported 0.99988962. Infeasible. Violate constraint.

Table 7 Optimal solutions of the problem 4. Method

n r

Rs MPI (%) Slacks of g 1 ∼g3 Mean Std Mean CPU a b

Dhingra [28]

(6,6,3,5) 0.81604 0.80309 0.98364 0.80373 0.99961 88.37811% 65 0.064 4.348 – – –

Yokota et al. [13]

(3,6,3,5) 0.965593 0.760592 0.972646 0.804660 0.999468 a

92  70.73357b 127.583189 – – –

(5,5,5,5) 0.903800 0.874992 0.919898 0.890609 0.999942 21.85286% 50 0.002152 28.803701 – – –

(5,5,5,5) 0.895644 0.885878 0.912184 0.887785 0.999945 17.59029% 50 0.9380 28.8037 – – –

(5,6,4,5) 0.902231 0.856325 0.948145 0.883156 0.999953 3.56311% 55 0.975465 24.801882 0.999907 0.000011 –

(5,6,4,5) 0.901614 0.849920 0.948143 0.888223 0.999955 a

55  0.0003364b 24.80188272 0.9999487 9.244  10  6 0.592

(5,6,4,5) 0.90186194 0.84968407 0.94842696 0.88800590 0.99995467 0.01028% 55 0.00120356 24.8018827 0.99992624 2.8874  10  5 0.93

Wu et al. [27]

(5,6,4,5) 0.90163164 0.84997020 0.94821828 0.88812885 0.99995467 0.01028% 55 0.000009 24.081883 – 1.3895  10  5 –

Hsieh and You [20]

Proposed approach

Phase I

Phase II

Phase I

Phase II

(5,5,4,6) 0.900863647 0.891220093 0.949249268 0.843612671 0.999953931 1.61423% 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 0.999954674 0.00146% 55 0.0001250 15.3634631 0.999954673 4.14  10  18 66.54688

(5,5,4,6) 0.901840702077 0.888232944989 0.948285851863 0.849492971195 0.999954671156 0.00774% 55 1.74702  10  9 15.3634630874 0.99995051535 5.543  10  6 1.1309

(5,5,4,6) 0.901626809561 0.888208355883 0.948134377884 0.849942135673 0.999954674663 – 55 5.57304247  10  9 15.3634630874 0.9999546746091 3.38683  10  11 0.2819

H. Garg et al. / Computers & Operations Research 40 (2013) 2961–2969

Rs MPI (%) Slacks of g 1 ∼g3

Hikita et al. [2]

Infeasible solution. Violate constraint. 2967

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H. Garg et al. / Computers & Operations Research 40 (2013) 2961–2969

Table 8 Result comparison for life support system in a space capsule. Method Murty and Reddy Tillman et al. Deep and Deepti Ravi et al. [31] [4] [32] [33]

Ravi et al. [34]

Pant and Singh [35]

Ravi [36]

Rocco et al. [37] Proposed approach Phase I

0.5 0.83892 0.5 0.5 0.90000 641.823

r1 r2 r3 r4 Rs Cs

0.50001 0.84062 0.5 0.5 0.90050 642.040

0.50002256 0.83889956 0.556 0.5 0.9000010 641.824

0.50006 0.83887 0.50001 0.50002 0.90001 641.83320

0.50000 0.83892 0.5 0.5 0.90000 641.82400

0.5 0.838920 0.5 0.5 0.9 641.823562

Phase II

0.50001 0.500000009 0.5 0.5 0.838919 0.838920148 0.8389201 0.8389201 0.5 0.500000011 0.5 0.5 0.5 0.500000022 0.5 0.5 0.9 0.900000023 0.9 0.9 641.823608 641.8235769 641.82356232 641.82356232

Table 9 Result comparison for complex bridge network system. Method

r1 r2 r3 r4 r5 Rs Cs

Murty and Reddy [31]

0.93479 0.93493 0.79219 0.93487 0.93496 0.99000 5.01992

Mohan and Shanker [38]

0.93924 0.93454 0.77154 0.93938 0.92844 0.99004 5.02001

Deep and Deepti [32]

0.935359 0.934304 0.790332 0.935504 0.934575 0.99 5.01992

are critical and beneficial to system security and system efficiency. The results of the experiment for the problem 2, shown in Table 5, indicate that the best solution by the presented approach (Rs ¼0.999976649054) is much better than the solutions given by [2,15,20,21,18]. It is worth mentioning that the solution obtained by Yeh and Hsieh [19] by using ABC algorithm is not a feasible solution as it violates the cost constraint function. From Table 6 one can observe that the solution to the Problem 3 as obtained by us is relatively with more significant improvement over the solutions presented by [2,15,16,18,20,21]. It may again be pointed out that the solution by ABC algorithm, obtained by Yeh and Hsieh [19] is also infeasible, since it again violates the cost constraint function. Table 7 depicts that the solution of Problem 4 as obtained by the proposed approach is better than the previously known solutions by [13,15,18,20,28]. The optimal component redundancy by the proposed approach is (5,5,4,6) which is completely different from those from the other approaches. Here again we have observed through calculations that the solutions given by Yeh and Hsieh [19] and Yokota et al. [13] are not feasible solutions as both of these violate the cost constraint function. Moreover, the solutions found by the proposed approach for all the four problems dominate the solutions obtained by other methods discussed in literature. This confirms the superiority of the presented approach over the approaches available in the literature. On the other hand, the results computed by the proposed approach to the reliability optimization problems 5 and 6 are shown in Tables 8 and 9, in which best solutions were reported along with the results obtained for the same in the past. Critical examination of Tables 8 and 9 reveals that for Problems 5 and 6, where objective is to minimize system cost subject to the constraints on system reliability yielded by proposed approach is better than other past reported solutions [4,31–38]. To evaluate the performance of proposed approach, the following maximum possible improvement (MPI) index [18] has been used to compute the relative improvement: MPI ¼

Rs ðapproachÞRs ðotherÞ 1Rs ðotherÞ

ð6Þ

where Rs ðapproachÞ is the best-known solution obtained from proposed approach and Rs ðotherÞ is the best solution by other

Ravi et al. [33]

0.93747 0.93291 0.78485 0.93641 0.93342 0.99000 5.01993

Ravi et al. [34]

0.93635 0.93869 0.80615 0.93512 0.93476 0.9905 5.02042

Proposed approach Phase I

Phase II

0.93536408 0.93428882 0.79034429 0.93555331 0.93453440 0.99000003 5.01991891

0.93545253 0.93438965 0.79041103 0.93541044 0.93448154 0.99 5.01991875

typical approaches. Numerical results are reported in Tables 4–7 which show that proposed approach when compared with other optimization approaches leads to improvement. Clearly, greater MPI implies greater improvement. Moreover, the standard deviations of system reliabilities by proposed approach are pretty low, and it further implies that the approach seems reliable to solve the reliability–redundancy allocation problems. For example, the standard deviations of their cost functions for Problems 1–6 are 2:37214  108 , 3:18206  1011 , 8:667  109 , 3:38683  1011 , 8:4866  107 and 7:5105  106 respectively. Also, the mean time for evaluating the problems is quite less as compared to the other techniques and is shown in their respective tables. For instance, the mean CPU time for the Problem 1 during the phases I and II are 2.3266 and 0.2911 (in s) while for the problem 2, it is 1.0261 and 0.2236 respectively. Similarly for the Problem 4, the mean CPU time during phases I and II are 4.09257 and 3.59517 respectively and for Problem 5, the CPU time is 3.18895 and 1.882458 for phase I and II respectively.

5. Conclusion The goal of this paper is to present an efficient two phase approach for solving the constrained reliability–redundancy allocation problem of series, series-parallel and complex system under different resource constraints. The objective of the problem is to maximize the system reliability subject to three nonlinear resource constraints, namely cost, weight and volume. In these optimization problems, both the redundancy and the corresponding reliability of each component in each subsystem are decided simultaneously. Also the reliability optimization of space capsule and complex network systems have been considered. The resource constraints have been handled with the help of parameter-free penalty technique. The performance of the proposed algorithm is evaluated through the comparison of numerical experiments with the previous study for mixed-integer reliability problems. The best solutions found by this approach are all individually better than the well-known best solutions by other heuristic methods for mixed-integer reliability problems.

H. Garg et al. / Computers & Operations Research 40 (2013) 2961–2969

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