harmony search algorithm: application to the redundancy optimization problem

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Harmony search algorithm: applicationto the redundancy optimizationproblemNabil Nahas a & Dao Thien-My aa Mechanical Engineering Department , École de TechnologieSupérieure (ÉTS); 1100 , Notre-Dame Street West, Montreal,Québec, H3C 1K3, CanadaPublished online: 14 Jun 2010.

To cite this article: Nabil Nahas & Dao Thien-My (2010) Harmony search algorithm: application tothe redundancy optimization problem, Engineering Optimization, 42:9, 845-861

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Engineering OptimizationVol. 42, No. 9, September 2010, 845–861

Harmony search algorithm: application to the redundancyoptimization problem

Nabil Nahas* and Dao Thien-My

Mechanical Engineering Department, École de Technologie Supérieure (ÉTS); 1100, Notre-Dame StreetWest, Montreal (Québec) H3C 1K3, Canada

(Received 26 April 2009; final version received 3 November 2009 )

The redundancy optimization problem is a well known NP-hard problem which involves the selection ofelements and redundancy levels to maximize system performance, given different system-level constraints.This article presents an efficient algorithm based on the harmony search algorithm (HSA) to solve thisoptimization problem. The HSA is a new nature-inspired algorithm which mimics the improvization processof music players. Two kinds of problems are considered in testing the proposed algorithm, with the firstlimited to the binary series–parallel system, where the problem consists of a selection of elements andredundancy levels used to maximize the system reliability given various system-level constraints; the secondproblem for its part concerns the multi-state series–parallel systems with performance levels ranging fromperfect operation to complete failure, and in which identical redundant elements are included in order toachieve a desirable level of availability. Numerical results for test problems from previous research arereported and compared. The results of HSA showed that this algorithm could provide very good solutionswhen compared to those obtained through other approaches.

Keywords: redundancy allocation; series–parallel systems; harmony search algorithm; universal gener-ating function

1. Introduction

The redundancy optimization problem (ROP) is very important in various types of electricaland mechanical systems. For example, in the telecommunications and aerospace industries mostsystems require very high reliability, and redundancy is indeed a necessity to reach required levelsof reliability. In many practical systems, the overall system is partitioned into a specific numberof subsystems where different component types are available. The system reliability depends thenon the reliability of each subsystem. Several options are available to improve the system reliabilitybut two important criteria are often considered: (i) increasing the component reliabilities and (ii)providing redundancy in various subsystems.

The ROP has been shown to be an NP-hard problem (Chern 1992) and many different opti-mization approaches and formulations have been used to solve it (Kuo and Prasad 2000). Such

*Corresponding author. Email: [email protected]

ISSN 0305-215X print/ISSN 1029-0273 online© 2010 Taylor & FrancisDOI: 10.1080/03052150903468746http://www.informaworld.com

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846 N. Nahas and D. Thien-My

approaches used to determine very good solutions for the ROP include dynamic programming(Bellman and Dreyfus 1958, Fyffe et al. 1968, Nakagawa and Miyazaki 1981, Yalaoui et al.2005), mixed-integer and nonlinear programming (Tillman et al. 1977) and integer programming(Ghare and Taylor 1969, Bulfin and Liu 1985, Misra and Sharma 1991, Gen et al. 1993) but thesetraditional approaches restrict the search space to include just solutions in which only one elementtype can be chosen for each subsystem. In the case of different elements, some metaheuristicshave been proposed to solve this problem; these include the genetic algorithm (GA) (Painton andCampbell 1995, Yokota et al. 1995, Coit and Smith 1996, Yokota et al. 1996), the tabu searchalgorithm (TS) and the ant colony optimization metaheuristic (Glover 1986, Liang and Smith2004, Nahas et al. 2007, Zhao et al. 2007). These works examined the traditional binary-statereliability problem, which assumes that a system and its elements may only be in two states (i.e.good or failed).

With the multi-state reliability problem, the system as well as its components are characterizedby various performance levels, and system reliability is defined as the ability of the system to meeta given demand. A good review of multi-state systems (MSSs) literature can be found in Levitinand Lisnianski (2001) and Levitin (2005). The ROP for MSSs was first introduced by Ushakov(1987), who used the universal generating function method for the reliability calculation. Levitinet al. (1997) and Levitin et al. (1998) applied genetic algorithms for finding the minimal costof a series–parallel MSS configuration under availability constraints while Ouzineb et al. (2008)proposed the tabu search algorithm to solve this problem.

This article aims to develop an efficient algorithm based on the harmony search algorithm(Geem et al. 2001, Lee and Geem 2005) to solve the ROP.

The harmony search algorithm conceptualizes a group of musicians seeking together to obtaina better state of harmony. The idea of mimicking the natural musical performance processes inorder to solve combinatorial optimization problems was only recently proposed. The HSA hasbeen successfully applied to various problems such as the travelling salesman problem (Geemet al. 2001), artificial neural networks (Geem et al. 2002), the vehicle routing problem (Geem et al.2005a), and the water network design problem (Geem 2006). Also, the harmony search algorithmhas been extensively applied to various real-world problems such as the optimum design of trussstructures (Saka 2009), aquifer parameter and zone structure identification (Ayyaz 2007), transportenergy demand problems (Ceylan et al. 2008), the structural design problem (Geem et al. 2005b),parameter calibration of the flood routing model (Kim et al. 2001) and parameter calibration ofthe rainfall-runoff model (Paik et al. 2005).

The remainder of this article is organized as follows. In Section 2, the proposed heuristicapproach is described by presenting the harmony search algorithm. In Section 3, the first prob-lem considered is presented, which involves the selection of elements and redundancy levelsto maximize system reliability, given weight and cost constraints and how the HSA has beenadapted to this problem. Section 4 presents a description of the ROP for series–parallel multi-state systems, in which elements of the same type are used to provide redundancy. The universalmoment generating function (UMGF) method used to estimate the system availability, as well asthe HSA proposed to solve this problem, are also presented in Section 4. The conclusion is givenin Section 5.

2. The harmony search algorithm

The harmony search algorithm is a new metaheuristic developed by Geem et al. (2001), and isbased on natural musical performance processes that occur when music players improvize thepitches of their instruments to obtain better harmony.A global optimum can be seen as a ‘musically

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Engineering Optimization 847

Figure 1. General structure of harmony memory.

pleasing harmony’, and the pitch of each musical instrument determines the aesthetic quality, anobjective function value which is determined by the set of values of the decision variables (Geemet al. 2001, Lee and Geem 2004, Geem 2008).

The following are the main steps of the HSA (Geem et al. 2001, Lee and Geem 2004).

Step 1. Parameter initialization. In this step, HSA parameters are specified: harmony memorysize (HMS) (number of solutions in memory), given harmony memory considering rate(HMCR), pitch adjusting rate (PAR), and stopping criterion.

Step 2. Initialization of a harmony memory (HM). In this step, a set of HMS solutions is randomlygenerated and for each structure i (i = 1, . . . , HMS), the objective function fi is evalu-ated. Figure 1 presents the general structure of the HM. This memory can be consideredas a matrix containing a set of solutions or harmonies.

Step 3. Improvization of a new harmony from HM. A new harmony (solution) x = (x1, . . . , xn) isgenerated from the HM with a probability HMCR. By using the HMCR parameter, eachvariable xi (i = 1, . . . , n) is randomly chosen from the vector (y1i , . . . , yHMSi) of HM.

Step 4. If the new harmony is feasible and better than the minimum harmony in HM (for maxi-mization problems), include the new harmony in HM and exclude the minimum harmonyfrom HM.

Step 5. If the stopping criterion is not satisfied, go to Step 2.

In their experiments, the authors introduced another control parameter called pitch adjustingrate, used to escape local optima and improve the solutions obtained (Geem et al. 2001, Lee andGeem 2004). The PAR is used after a new harmony is constructed, and can be applied to eachcomponent of this harmony. In this case, the value of each component is modified to match itsneighbouring value, with a probability of PAR. Figure 2 shows the optimization procedure of theHSA.

3. The redundancy optimization problem: series–parallel binary-state system case

3.1. Problem formulation

The system considered consists of n components in series. For each component i (i = 1, . . . , n)there are various versions of elements, which are proposed by the suppliers in the market. Eachelement is characterized by its cost and weight according to its version. Each component i containsa different number of elements connected in parallel, and different versions of elements can be

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Figure 2. Harmony search algorithm optimization procedure.

placed in parallel. A component i is functioning properly if at least ki of its pi elements areoperational (k-out-of-n:G).

Assumptions

• Elements and the system may experience only two possible states: good and failed.• The system weight and cost are linear combinations of the elements’ weight and cost.• The element attributes (reliability, cost and weight) are known and deterministic.• Failed elements do not damage the system, and are not repaired.• All redundancy is active: failure rates of elements when not in use are the same as when in use.• The supply of elements is unlimited.• Failures of individual elements are s-independent.

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Notation

Rsys overall reliability of the series–parallel systemR∗ optimal solutionC cost constraintW weight constraintn number of componentsi index for componentsai number of available element choices (i.e. versions) for component i

rij reliability of element j available for component i

wij weight of element j available for component i

cij cost of element j available for component i

xij number of element j used in component i

xi (xi1, . . . , xiai)

pi total number of elements used in component i

pmax maximum number of elements in parallelki minimum number of elements in parallel required for component i

k (k1, . . . , kn)

Ri(xi |ki) reliability of component i, given ki

Ci(xi) total cost of component i

Wi(xi) total weight of component i

The redundancy optimization problem is formulated such as to maximize system reliabilitygiven restrictions on system cost and weight. That is,

Maximize Rsys =n∏

i=1

Ri(xi |ki) (1)

subject ton∑

i=1

Ci(xi ) ≤ C (2)

n∑i=1

Wi(xi ) ≤ W (3)

ki ≤ pi ≤ pmax ∀i = 1, . . . , n. (4)

Constraints (2) and (3) represent the budget and the weight constraints, respectively. Constraint(4) limits the maximum number of elements allowed to be in parallel.

3.2. Solution methodology

3.2.1. The harmony search algorithm for the ROP

Harmony memory matrix

To apply the harmony search algorithm to the ROP, it is necessary adequately to represent theharmony matrix. In the proposed algorithm, each row of the matrix is composed of n sets ofvariables (a total of n.pmax variables), with each set representing a configuration of one component.The value of a variable corresponds to a version choice. Figure 3 shows the adopted structure of theharmony memory matrix for the ROP, where the variables are represented by yk

ij (i = 1, . . . , HMS;

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Figure 3. Harmony memory matrix for the ROP: mixed parallel elements case.

k = 1, . . . , n; j = 1, . . . , pmax). Each variable of the row i can take any value between 0 and ak ,and the value 0 means that no version has been chosen.

Generation of feasible initial solutions

To obtain an initial feasible solution, the number of elements to be connected in parallel in eachcomponent is limited to a maximum of two. For each row i (i = 1, . . . , HMS) of the HM, aninitial feasible solution is generated as follows.

Step 1. For each component k (k = 1, . . . , n), choose randomly one or two variables to bedifferent from 0.

Step 2. For each chosen variable, randomly assign a version between 1 and ak .

New solution construction

A new solution X = (x11 , . . . , x

1pmax

, . . . , xn1 , . . . , xn

pmax) is generated from the HM by using the

HMCR and PAR parameters. These parameters help the algorithm to obtain globally and locallyimproved solutions, respectively. The solution is constructed as follows:

xkl ∈

{{yk

1l , yk2l , . . . , y

kHMSl

}, with probability HMCR

{1, . . . , ak} , with probability (1-HMCR).

For each variable xkl (k = 1, . . . , n and l = 1, . . . , pmax), a value is chosen with the rate HMCR

from the historic values stored in the matrix HM or from the possible versions from 0 and ak withthe rate (1 − HMCR). The new solution obtained is then examined to determine whether or notit should be pitch-adjusted. This action is performed at the PAR rate by proceeding as follows

(1) Randomly choose a component k (k ∈ {1, . . . , n}).(2) Randomly choose two elements xk

i and xkj (i ∈ {1, . . . , pmax}; j ∈ {1, . . . , pmax}and i �= j )

from the component k.(3) Exchange the two values of xk

i and xkj .

In the next section, the proposed harmony search algorithm is demonstrated on 33 test problemsused in previous research, and is compared to the best solutions found in the literature. Beforepresenting the results obtained, the influence of the HMCR, PAR and HMS parameters on theperformance of the proposed algorithm will be analysed.

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3.3. Test problems and results

To evaluate the performance of the HSA, the test problems originally proposed by Fyffe et al.(1968) and modified by Nakagawa and Miyazaki (1981) are used. Initially, Fyffe et al. (1968)specified constraint limits of 130 units of system cost, 170 units of system weight with ki = 1.Thirty-three variations of the original problem were proposed by Nakagawa and Miyazaki (1981),in which the weight constraint W varied from 159 to 191 units. Table 1 presents the element cost,weight and reliability values (Fyffe et al. 1968). The system is designed with 14 components, andall the algorithms were implemented using MATLAB on a 2 GHz Dual Core PC.

For the proposed approach, various types are allowed to reside in parallel (Coit and Smith1996, Kulturel-Konak et al. 2003, Liang and Smith 2004). (Coit and Smith 1996) first solved theROP with a genetic algorithm without restricting the search space. Kulturel-Konak et al. (2003)solved this problem by proposing an efficient tabu search algorithm, while Liang and Smith (2004)developed an ant colony optimization approach improved by a local search. Considering elementmixing, the search space size is larger than 7.6 × 1033. Previous approaches are chosen for thesake of comparison as element mixing is allowed.

In meta-heuristics such as HSA, ant colony optimization and genetic algorithms, a number ofparameters must be tuned in order to achieve good performance. The user-specified parametersof the HSA were varied to establish the most beneficial values for the optimization process. TheHSA was then implemented and simulations were run to set the parameters. The parameters con-sidered here were HMCR, PAR and HMS. Several values were tested for each parameter whileall the others were held constant (over twenty trials for each setting in order to obtain statis-tical information about the average evolution). Various tests were performed for this problem,with C = 130 and W = 191, in order to select appropriate values for the parameters used inthe search. The stopping criterion was satisfied whenever the maximum number of iterations,set at 2 × 106, was reached, or the rows of the matrix representing the solutions converged tothe same solution. Figure 4 shows the results produced by the HSA with the HMCR rangingfrom 0.6 to 0.99, and without considering the pitch adjusting rate (i.e. PAR = 0). The size ofthe matrix HMS was set to 50. From Figure 4, it can be seen that the algorithm considerablyimproves the results if parameter HMCR is adjusted adequately. It appears that HMCR playsan essential role in terms of the quality of the results obtained. The algorithm yields the bestsolutions with HMCR = 0.96, with an average number of evaluated solutions not exceeding

Table 1. Component parameters for testing problem (Fyffe et al. 1968).

Element choices

Component, i ri1 ci1 wi1 ri2 ci2 wi2 ri3 ci3 wi3 ri4 ci4 wi4

1 0.95 2 5 0.93 1 4 0.91 2 2 0.90 1 32 0.95 2 8 0.94 1 10 0.93 1 93 0.92 4 4 0.90 3 5 0.87 1 6 0.85 2 74 0.87 4 6 0.85 5 4 0.83 3 55 0.95 3 5 0.94 2 4 0.93 2 36 0.99 3 5 0.98 3 4 0.97 2 5 0.96 2 47 0.94 5 9 0.92 4 8 0.91 4 78 0.91 6 6 0.90 5 7 0.81 3 49 0.99 3 9 0.97 2 8 0.96 4 7 0.91 3 8

10 0.90 5 6 0.85 4 5 0.83 4 611 0.96 5 6 0.95 4 6 0.94 3 512 0.90 5 7 0.85 4 6 0.82 3 5 0.79 2 413 0.99 3 5 0.98 2 5 0.97 2 614 0.99 6 9 0.95 5 6 0.92 4 7 0.90 4 6

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Figure 4. Influence of the HMCR parameter.

7.5 × 105. The effect of the pitch adjusting rate is represented graphically in Figure 5. The testsshow that PAR should be set between 0.1 and 0.3, as the algorithm performs very well withPAR = 0.2. Several values were tested for the HMS parameter, while all the others were heldconstant (HMCR = 0.96 and PAR = 0.2). The tests show that the best results are obtained forHMS ≥ 30. Figure 6 shows that the HMS parameter clearly influences the number of solu-tions evaluated, and that HMS = 50 yields a good result. Figure 7 shows an example of theevolution of the best solution using the proposed algorithm when HMS = 50. The proposedalgorithm converges quickly to the best solution. These parameter values are used for all the testproblems.

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Figure 5. Influence of the PAR parameter.

As mentioned earlier, the proposed algorithm will be compared to the best solutions foundby genetic algorithms (Coit and Smith 1996), tabu search (Kulturel- Konak et al. 2003) and antcolony optimization (ACO) (Liang and Smith 2004). Table 2 shows the system reliability of thebest solution for each of the 33 instances. The table also presents the average and the maximumfor the best solutions found in 20 trials as well as the standard deviations of the 20 solutions.The standard deviation is an important measure of algorithm robustness, and from Table 2 it canbe seen that the standard deviation is low, with the average computation time not exceeding 200seconds for any instance.

Comparing the results obtained by this approach with those of previous works (Coitand Smith 1996, Kulturel-Konak et al. 2003, Liang and Smith 2004). Table 2 shows thefollowing.

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Figure 6. Influence of the HMS parameter on the number of iterations.

• In 22 of 33 cases examined, the HSA found the best known results (BKR) while the rest are alllower, but much closer to the BKR.

• Cases 24 to 29 and cases 22, 31 and 33 are as good as those found by the genetic algorithmof (Coit and Smith 1996), while the rest (i.e. 21 instances) are all better, except for cases 9, 10and 32.

• In 5 of the 33 test cases, the HSA outperforms the tabu search algorithm of Kulturel-Konaket al. (2003), although it is very close, but at a lower reliability in 9 instances.

• In 6 of the 33 test cases, the solutions found by the HSA algorithm are better than those obtainedby the ant colony optimization of Liang and Smith (2004), while in 11 instances the resultsobtained by HSA are much closer than those they found.

In the next section, the redundancy optimization problem when the series–parallel systemsconsidered are multi-state systems and the elements connected in parallel are assumed to beidentical will be studied.

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Figure 7. Convergence results.

Table 2. Comparison of the HSA and the solutions found in the literature (20 trials).

GA TS ACO Harmony search algorithm

W BKR Rmax Rmax Rmax Rmax Rav Std. dev. Time (s)

191 0.986811 0.98675 0.986811 0.986745 0.986811 0.986079 0.00059 110190 0.986416 0.98603 0.986416 0.985905 0.986316 0.985406 0.00070 85189 0.985922 0.98556 0.985922 0.985773 0.985922 0.984851 0.00068 130188 0.985378 0.98503 0.985378 0.985329 0.985378 0.984367 0.00051 110187 0.984688 0.98429 0.984688 0.984688 0.984688 0.983895 0.00061 130186 0.984176 0.98362 0.984176 0.983801 0.984176 0.983252 0.00059 190185 0.983505 0.98311 0.983505 0.983505 0.983436 0.982385 0.00046 110184 0.982994 0.98239 0.982994 0.982994 0.982417 0.981970 0.00030 160183 0.982256 0.98190 0.982256 0.982206 0.981876 0.981319 0.00040 120182 0.981518 0.98102 0.981518 0.981468 0.980729 0.980439 0.00021 110181 0.981027 0.98006 0.981027 0.980681 0.980681 0.979830 0.00099 130180 0.980290 0.97942 0.980290 0.980290 0.980009 0.977661 0.00291 130179 0.979505 0.97906 0.979505 0.979505 0.979115 0.978197 0.00091 100178 0.978400 0.97810 0.978400 0.978400 0.978208 0.977704 0.00061 75177 0.977596 0.97715 0.977474 0.977596 0.977498 0.977068 0.00034 100176 0.976690 0.97642 0.976690 0.976494 0.976690 0.976302 0.00038 90175 0.975708 0.97552 0.975708 0.975708 0.975708 0.975431 0.00043 120174 0.974926 0.97435 0.974788 0.974926 0.974926 0.974587 0.00041 120173 0.973827 0.97362 0.973827 0.973827 0.973827 0.973681 0.00038 100172 0.973027 0.97266 0.973027 0.973027 0.973027 0.972775 0.00061 85171 0.971929 0.97186 0.971929 0.971929 0.971929 0.971498 0.00063 95170 0.970760 0.97076 0.970760 0.970760 0.970760 0.969468 0.00153 120169 0.969291 0.96922 0.969291 0.969291 0.969291 0.968138 0.00124 100168 0.968125 0.96813 0.968125 0.968125 0.968125 0.967115 0.00084 115167 0.966335 0.96634 0.966335 0.966335 0.966335 0.965383 0.00082 100166 0.965042 0.96504 0.965042 0.965042 0.965042 0.963835 0.00072 140165 0.963712 0.96371 0.963712 0.963712 0.963712 0.962491 0.00066 130164 0.962422 0.96242 0.962422 0.962422 0.962422 0.961086 0.00042 125163 0.960642 0.96064 0.959980 0.960642 0.960642 0.959463 0.00076 140162 0.959188 0.95912 0.958205 0.959188 0.959188 0.957764 0.00083 130161 0.958034 0.95803 0.958034 0.958034 0.958034 0.956621 0.00058 100160 0.955714 0.95567 0.955714 0.963712 0.955604 0.955393 0.00041 120159 0.954564 0.95432 0.954325 0.954564 0.954325 0.953784 0.00063 150

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4. The redundancy optimization problem: series–parallel multi-state system case

4.1. MSS reliability estimation by using universal moment generating function

Consider a series–parallel system of n components Ci(i = 1, . . . , n) in series. The entire systemis assumed to have K different states corresponding to different output performance levels, and iscalled a multi-state system (MSS). Within MSS, reliability is related to the ability of the systemto satisfy the required performance level (demand) at a given time t . MSS reliability is given asfollows (Xue and Yang 1995):

E(t, W) = Pr{G(t) ≥ W }, (5)

where G(t) represents the output performance of the system at time t , and W is the requiredperformance level. Let Gk be the output performance level of the system’s kth state and qk(t)

the probability Pr{G(t) = Gk} (for k = 1, . . . , K). The output performance distribution (OPD)of the system can be then determined based on the two sets G and q such that

G = {Gk : 1 ≤ k ≤ K} (6)

qk = {qk(t) : 1 ≤ k ≤ K}. (7)

The MSS reliability can be expressed as the probability of the output performance level G(t)

being greater than or equal to the demand W , that is

E(t, W) = Pr{G(t) ≥ W } =∑

Gk≥W

qk(t). (8)

Several approaches have been proposed in the literature to evaluate the MSS reliability. How-ever, it is important to have an efficient and fast procedure for estimating the MSS reliabilitywhen dealing with optimization problems. A procedure that has been proven to be very effectiveis based on the universal z-transform and has been developed by Ushakov (1987); it is widelyused in MSS performance measure evaluation. In the literature, the universal z-transform is alsocalled the universal moment generating function (UMGF) and is denoted as a u-function or az-function. The present article uses this technique to evaluate the MSS reliability, and additionaldetails concerning the use of the UMGF in MSS performance measure evaluation can be foundin the literature (Ushakov 1987, Levitin et al. 1997, 1998, Levitin and Lisnianski 2001, Levitin2005).

4.2. Problem formulation

In this article, it is assumed that every component Ci contains a number of identical elementsconnected in parallel. For each component there are a number of element versions availableon the market. Each version hi of element of type i is characterized by a nominal capacityG(hi), an availability A(hi) and a cost C(hi). The structure of the component i is defined by theversion number hi (1 ≤ hi ≤ Hi) and the number of parallel elements ri (1 ≤ ri ≤ Ri). Hi andRi represent the number of versions available for element of type i and the maximum allowednumber of parallel elements of type i, respectively. The total cost of the system is calculated as

C =n∑

i=1

(ri × C(hi)). (9)

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The problem of multi-state system reliability optimization consists in finding the minimumcost system configuration that provides the required reliability level E0 (Lisnianski et al. 1996):

(h, r) = arg{C(h, r) −→ min |E(h, r, W, t) ≥ E0}, (10)

where h = {h1, . . . , hn}, r = {r1, . . . , rn}.

4.3. The harmony search algorithm for multi-state system reliability optimization

The harmony memory matrix

In this problem, it was considered that the elements to be connected in parallel are identical. Theharmony memory matrix is then slightly different from the previous matrix. In this case, each rowof the matrix is composed of 2n variables, and each component i is represented by two variables(i.e. ri and hi). Figure 8 shows the adopted structure of the harmony memory matrix for thisproblem, where the variables are represented by yk

ij (i = 1, . . . , HMS; k = 1, . . . , n; j = 1, 2).Each variableyk

ij of row i can take any value between 0 and rk if j = 1, or between 0 andhk if j = 2.

Feasible initial solution generation

Here, the number of elements to be connected in parallel is chosen randomly as follows until afeasible solution is obtained.

(1) For each component k (k = 1, . . . , n), choose randomly a number m (1≤ m ≤ Rk) and setyk

i1 = m.(2) Assign randomly a version between 1 and Hk to yk

i2.(3) Calculate the system reliability E.(4) If E < E0 then goto step 1.

New solution construction

A new solution X = {x11 , x

12 ; . . . ; xn

1 , xn2 } is generated from the HM by using the same procedure

presented in Section 3.2.1. This solution is constructed as follows:

xki ∈

{{yk

1i , yk2i , . . . , y

kHMSi

}, with probability HMCR

{1, . . . , Fk}, with probability (1-HMCR),

with (Fk = Rk if i = 1) and (Fk = Hk if i = 2).

Figure 8. Harmony memory matrix for the ROP: identical parallel elements case.

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The newly obtained solution is examined to determine whether it should be pitch-adjusted. Thisaction is performed with the PAR rate using the following steps.

• Randomly choose a component k (k ∈ {1, . . . , n}).• Set the corresponding number of elements connected in parallel, i.e. (rk) to (rk + 1) or (rk − 1).

Test problem and results

The problem considered in this article was proposed by Levitin et al. (1997) and consists offive components connected in series, with different versions available for each component. Foreach component j (j = 1, . . . , 5) the reliability, cost and nominal performance parameters arereported in Table 3. Three different reliability constraints are used to test the algorithm: E ≥ 0.975,E ≥ 0.98 and E ≥ 0.99. Table 4 shows the system supply requirements, characterized by acumulative load-demand curve with four different levels.

In Ouzineb et al. (2008), the authors proposed a tabu search heuristic to solve this problem andcompared the obtained results with those obtained by the genetic algorithm (Levitin et al. 1997).The genetic algorithm has been re-implemented in Ouzineb et al. (2008) and the two algorithmshave been used for different running times (10, 100, 1000 and 2000 s).

Table 3. Characteristics of system elements available in the market(Levitin et al. 1997).

Component, i version rij Cij Wij (%)

1 1 0.980 0.590 1202 0.977 0.535 1003 0.982 0.470 854 0.978 0.420 855 0.983 0.400 486 0.920 0.180 317 0.984 0.220 26

2 1 0.995 0.205 1002 0.996 0.189 923 0.997 0.091 534 0.997 0.056 285 0.998 0.042 21

3 1 0.971 7.725 1002 0.973 4.720 603 0.971 3.590 404 0.976 2.420 20

4 1 0.977 0.180 1152 0.978 0.160 1003 0.978 0.150 914 0.983 0.121 725 0.981 0.102 726 0.971 0.096 727 0.983 0.071 558 0.982 0.049 259 0.977 0.044 25

5 1 0.984 0.986 1282 0.983 0.825 1003 0.987 0.490 604 0.981 0.475 51

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Table 4. Parameters of the cumulative load demand curve (Levitinet al. 1997).

Wk 100 80 50 20tk 4203 788 1228 2536

Table 5. The harmony search algorithm results.

E0 E Cmin h r

0.975 0.9774 14.45 {2,3,2,7,2} {2,2,3,3,1}0.980 0.9808 16.52 {2,5,2,7,2} {2,6,3,3,1}0.990 0.9937 17.050 {2,3,2,7,4} {2,2,3,3,3}

Table 6. Genetic, tabu search and harmony search algorithms comparison (20 trials).

HSA

GA Tabu search Av. num. ofE0 Cmin Cmin Cmin Cav Std. dev. iterations Time (s)

0.975 16.45 16.45 16.45 16.45 0 98794 650.980 16.52 16.52 16.52 16.53 0.030 97315 800.990 17.095 17.050 17.050 17.060 0.003 68016 85

The HSA, the tabu search and the genetic algorithm are compared on the basis of resultsproduced by the three instances. A value of the required reliability level E0 is assigned to eachinstance, and these values are the same as those proposed by Levitin et al. (1997). Within theharmony search algorithm, the stopping criterion adopted is satisfied either if all rows of the matrixHM converge to the same structure or if the maximum number of iterations is reached (2 × 105).In this example, the HMS, HMCR and PAR parameters are set to 30, 0.7 and 0.1, respectively.For each instance, its corresponding best structure and total cost obtained by the HSA are given inTable 5. For the sake of comparison, the best total costs obtained by the proposed algorithm andthose obtained by the tabu search algorithm and the genetic algorithm in Ouzineb et al. (2008) aregiven in Table 6. It is found that one instance has been improved by the HSA and the tabu searchalgorithm while the rest are as good as those found by the genetic algorithm. This table also givesthe average of the total cost obtained from 20 trials, as well as the standard deviation of the 20final best total costs. It can be seen that the standard deviation corresponding to each instance isvery low. Consequently, the proposed method is robust and credible. The average running timedid not exceed 90 seconds for the three instances.

5. Conclusion

In this article, a new algorithm based on the harmony search algorithm to solve the redundancyoptimization problem was explored, by considering binary and multi-state systems. In order toprove its efficiency, the HSA was compared with three other metaheuristics. The results obtainedby the harmony search algorithm to solve this problem are clearly encouraging, especially whencompared with those obtained by the genetic algorithm. In the proposed HSA approach, it hasbeen necessary to fine-tune many parameters, and so the development of a systematic procedurefor parameter fine-tuning could be investigated in future studies.

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harmony search algorithm: application to the redundancy optimization problem

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