hierarchical redundancy allocation for multi-level reliability systems

Information Sciences 288 (2014) 174–193

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Hierarchical redundancy allocation for multi-level reliabilitysystems employing a bacterial-inspired evolutionary algorithm

http://dx.doi.org/10.1016/j.ins.2014.07.0550020-0255/� 2014 Elsevier Inc. All rights reserved.

E-mail address: [email protected]

Tsung-Jung HsiehComputational Intelligence Technology Center, Industrial Technology Research Institute (ITRI), No. 195, Section 4, Chung Hsing Road, Chutung, Hsinchu31040, Taiwan, ROC

a r t i c l e i n f o a b s t r a c t

Article history:Received 14 August 2013Received in revised form 29 June 2014Accepted 27 July 2014Available online 11 August 2014

Keywords:ReliabilityOptimizationRedundancy allocationBacterial-inspired evolutionary algorithmMulti-level system

A high level of reliability is a crucial factor in many real-world systems. Specifically, redun-dancy allocation problems (RAPs) have attracted much attention in the last three decadesfor their comprehensive applications in various engineering systems. RAPs have been pro-ven to be NP-hard and numerous meta-heuristic methods have been proposed to addressthem. To date, a significant number of successful research endeavors regarding RAPs in sin-gle-level systems have been conducted; however, most real complex systems involve mul-tiple levels. Consequently, RAPs on multi-level systems (labeled here as MLRAPs) aredeemed to be more realistic and challenging. As such, this paper proposes a bacterial-inspired evolutionary algorithm (BiEA) for addressing MLRAPs. Apart from designing themutation operation of the canonical bacteria evolutionary algorithm (BEA) to be adaptiveto the solution encoding of MLRAPs, two new search operators, dynamic gene transfer anda niching scheme, are introduced in the BEA. As a result, the BiEA is dedicated to MLRAPdesigns with near-optimal solutions. Case studies on MLRAP designs show the BiEA outper-forms current state-of-the-art approaches in regard to two representative experiments.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

In many real-world systems, the reliability of components is fixed and the only way to enhance system reliability is toincrease the redundancy of the components. However, the extension of redundancy requires additional resources to be madeavailable. Consequently, it is important to search for optimal allocation methods regarding the redundancy of componentsunder given resource constraints, such as cost and weight. Such problems are generally referred to as redundancy allocationproblems (RAPs) [11].

Since most modern systems consist of several levels, research on RAPs is gradually extending to other domains. Therefore,RAP research originates from the studies of combinations of pure parallel or series connections to a larger scale of componentcombinations, under several limitations such as costs, weights, and volumes [14]. Recently, the research topic was extendedto a multi-level view of what is essentially practical (i.e., most multi-level configurations in existing systems); accordingly,more challenging RAPs with multi-level systems were proposed. RAPs with multi-level systems will be more complicatedbecause the changes of component numbers in the sublevels may directly influence its upper and lower levels. These prob-lems led us to investigate the interior configuration and to design an encoding mechanism for multi-level configurations. Weexpect the proposed algorithm will enhance efforts in decreasing computational time and stabilizing solutions.

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Previous studies in RAPs focused on system configurations such as series–parallel, parallel-series, general networks, andk-out-of-n, detailed in [19,43], where the most studied configuration is the parallel-serial system. For parallel-serial systems,although an RAP is sometimes formulated as a nonlinear integer programming problem with the objective to maximize thesystem’s reliability under certain constraints, it is actually more realistically formulated as a min cost subject to a reliabilityconstraint [4,21,38]. In addition, parallel-serial systems of RAPs are simply treated with the same type of components. Fol-lowing rapid advancements in technology, RAPs are being designed by considering different types of components in a sub-system or system. Moreover, when a parallel-serial system is studied, the redundancy could always be simply allocated tothe components, meaning system reliability was only considered on a single level [47].

In real-world systems, multi-level redundancy designs are commonly observed, such as in current computing systems,communication systems, control systems, and critical power systems [45]. Here, the entire system is represented at the top-most level, i.e., the system level, and the components are at the lowest level, while subsystems are at the levels between theentire system and the components. In a multi-level system, the entire system, subsystems and components are all calledunits, and redundancy can be allocated to any level [45].

Fig. 1 depicts the schematic diagram of a four-level system. In the literature, studies on reliability issues pertaining tomulti-level systems are often labeled as multi-level redundancy allocation problems (MLRAPs). In MLRAPs, several assump-tions are made as in [17]: (1) for a unit that is not at the lowest level, its subunits are in serial formation and the numberof these subunits is fixed; (2) the redundancy can be allocated to the units on any level; (3) the quality (reliability and cost)of each component is given in advance.

According to [2], RAPs can be posed as a general nonlinear integer programming problem which is NP-hard. As a result,they have garnered increasing interest from the computer science and engineering communities working on related prob-lems. To date, approaches proposed for solving RAPs include exact methods [12,35], max–min methods [38] and other heu-ristic or meta-heuristic algorithms [3,8,20,21,23,25,28,34,37,41,46,48]. Heuristic methods have been prolifically utilized dueto their robust quality and ability to rapidly converge on high-quality solutions, especially large-scale problems that aresometimes considered difficult to solve when employing conventional methods [1,7,24,39,44,50,51].

In regard to the contributions of heuristic methods proposed in the literature, Levitin [22] was among the first to studythe use of the genetic algorithm (GA) for MLRAPs. Subsequently, Yun et al. [52,53] extended the scale of the MLRAPs for con-sideration, and later introduced several methods to resolve them [16,18,49]. More recently, Kumar et al. [17,18] furtherinvestigated the MLRAPs by using the proposed hierarchical structure, and enhanced the number of system levels so theproblems were closer to real-world systems. For the proposed problems in [17], Wang et al. [47] presented a novel memeticalgorithm (MA) to solve these problems; the results showed the proposed MA outperformed the approach proposed in [17].

With regard to the characteristics of MLRAPs, the quality of the solutions places a heavy emphasis on the ability of solu-tion exploitation because all units on any level have an opportunity to be allocated. In addition, the redundancy of a certainlevel greatly influences the redundancy of its sublevel, if it exists. Therefore, a powerful local search scheme is essential tofind a near-optimal solution from known solutions. Nevertheless, when the known solutions have poor quality, the problemof local trapping may cause a disappointing performance. In this case, a system is required to reduce local trapping and facil-itate the local search process.

For this reason, the bacteria evolutionary algorithm (BEA) [29,30] is modified (herein called the BiEA) for the adaptation ofsolution encoding in MLRAPs. Thus, we propose a novel bacterial-inspired evolutionary algorithm (BiEA) for solving MLRAPs.The backbone of the BiEA comprises two search operators, namely the dynamic gene transfer (DGT) and niching scheme. Here,the proposed DGT and niching scheme inspired by [40] are employed to preserve the population diversity and enhance thesolution quality during the search process, respectively. The effectiveness of the niching scheme has been shown in numer-ical problems [13]; this paper is its first practical application. The operations of the BiEA will be detailed in later sections.

System U2

U1

U3

U12

U11

U13

System level

Subsystem levels

Component level

Fig. 1. The structure of a four-level system [17].

Table 1The related works of MLRAPs.

Years Related works

2003 Levitin [22] utilized the genetic algorithm (GA) for solving multi-level protection cost minimization problems subject to survivabilityconstraints

2004 Yun et al. [52] extended the problem scale of MLRAPs; however, the proposed method was under a strong restriction that only one levelamong the component, module, and system could be a candidate for redundancy

2007 Yun et al. [53] modified their former work in [52]; however, the relaxation of the assumptions and application to more complex examples canbe made so as to be closer to real-world systems

2008 Kumar et al. [17] re-formulated MLRAPs so the redundancy allocation can be performed at any level of a multi-level system2010 Wang et al. [47] presented a novel memetic algorithm (MA) to solve these problems in [17]; the results showed the proposed MA

outperformed the approach proposed in [17]

Table 2The related works of BiEA.

Years Related works

1995 The predecessor of the bacteria evolutionary algorithm (BEA), the pseudo bacterial genetic algorithm (PBGA) was proposed by Furuhashiet al. [10]

1997 Nawa et al. [31–33] adopted PBGA to settle the problems of model identification and fuzzy system parameter discovery1998 Nawa and Furuhashi [29] proposed BEA by improving PBGA. That is, the BEA actually has the same features (bacterial mutation

operation) of the PBGA, but has incorporated a new operation, called the gene transfer operation2000–2011 Only a few applications of the BEA in data processing and fuzzy system parameter discovery [5,6,15,26]

176 T.-J. Hsieh / Information Sciences 288 (2014) 174–193

The bacteria evolutionary algorithm (BEA) is inspired by the evolutionary process of bacterial gene recombination mech-anism configurations, which uses bacterial mutation as a local improvement mechanism regarding the acquisition of traitsby recipient cells. This operation substitutes the crossover in GA, allowing the recombination of information between differ-ent individuals, with the expectation this will lead to the creation of superior individuals [29,30]. Previously there have beenonly a few applications of the BEA in data processing and fuzzy system parameter discovery [5,6,9,15,26].

To clearly realize the previous related works of MLRAPs and BiEA as mentioned above, they are organized with Table 1and 2, respectively.

In this work, based on the hierarchical genotype representation of the variables employed in [17], a modified BEA (calledthe BiEA) is proposed for MLRAPs. The BiEA is compared with the MA [47] on two representative multi-level systems. Note,the setting of the first experiment (Problem A) is adopted in [47]; this is more difficult because one component in [47] ismore costly than that in [17] (detailed later).

The main contributions of this paper are summarized as follows:

(1) This is the first successful application of the proposed niching scheme used in engineering; in addition, from the exper-iment discussed later, the proposed dynamic gene transfer (DGT) also makes progress in providing better solutions.

(2) A control mechanism is put forth to search the near-optimal configuration (solution) of multi-level systems. Fuzzycoding [29,30] in the original BEA inspired the conception of the design; this paper modifies the coding in a way thatextends it to the system encoding of component positions and numbers. This study plays an important role in systemdesign. Most modern systems are a multi-level system; to the best of our knowledge, no study exists that attemptsconfiguration design of a multi-level system in terms of automatic changes. In addition, critical issues, such as powersystem [45] and smart grid [36,54], are also multi-level systems. Therefore, the idea of the proposed method also hasgreat potential for energy conservation.

The rest of this paper is organized as follows. Section 2 describes the background information of multi-level redundancyallocation problems. In Section 3, the proposed BiEA for MLRAPs is presented and two representative MLRAPs are utilized toshow the performance comparison in Section 4. Section 5 offers conclusions.

2. Problem description

A multi-level serial system comprises the entire system at the topmost level (the system level), the components at thelowest level, and the subsystems are between the above two levels. For the problems investigated here, the multi-level serialsystems are constricted in a hierarchical form and the entire system, subsystems and components are all referred to as units[17]. Fig. 2, illustrates a general multi-level serial system where U1 is a system unit and contains n1 serial subunits. Similarly,U1i has n1i serial subunits, i = 1, 2, . . ., n1.

Fig. 3 provides an illustration of redundancy allocation on a bi-level serial system [17] which consists of a system unit U1

and two subunits, U11 and U12. After redundancy is allocated to this system, the redundancy of U1 is two, U11 and U2

1. UnderU1

1, the redundancy of U11 and U12 are three and one, respectively. Under U21, the redundancy of U11 and U12 are both two. The

corresponding final system is drawn at the bottom of Fig. 3.

Fig. 2. A general multi-level serial system [17].

Fig. 3. An example of redundancy allocation in bi-level series configuration.


In multi-level serial systems, the reliability of components, i.e., the units at the lowest level, is predefined and used tocalculate the entire reliability. Specifically, assume there is a system Ui with ni subunits; the reliability of Ui can becalculated:

Ri ¼Yni

m¼1

1�Yxi;m

j¼1

1� Rji;m

� �" #; ð1Þ


where Rji;m denotes the reliability of each redundancy of Ui,m, and xi,m is the redundancy of Ui,m. For example, the reliability of

the entire system given in Fig. 3 is:

Rsys ¼ 1� 1� 1�Y3

j¼1

ð1� Rj1;1Þ

!� R1

1;2

� � !� 1� 1�

Y2

j¼1

1� Rj1;1

� � !� 1�

Y2

j¼1

1� Rj1;2

� � ! !: ð2Þ

Notably, a system with higher reliability is subject to cost. Realistically, the total cost of a system should not exceed a pre-defined value. In traditional RAPs, the system cost is simply the sum of the cost of each component. However, for multi-levelsystems, additional costs need to be considered to reflect the hierarchical structure, such as the addition or duplication ofredundant units or the increase of components [17]. Practical systems, taking power systems as an example [6,27], needadditional cost to ensure reliable system matches. In general, the cost of a system Ui in a multi-level serial system can becalculated as:

Ci ¼Xni

m¼1

Xxi;m

j¼1

Cji;m

!þ ki;m� �xi;m : ð3Þ

where xi,m is the redundancy of Ui,m, Cji;m is the cost of Uj

i;m, and ki,m the additional costs of the mth subunit of Ui. For example,the cost of the entire system in Fig. 3 is:

Csys ¼X3

j¼1

Cj1;1 þ ðk1;1Þ3 þ C1

1;2 þ ðk1;2Þ1 !

þX2

j¼1

Cj1;1 þ ðk1;1Þ2 þ

X2

j¼1

Cj1;2 þ ðk1;2Þ2

!: ð4Þ

In sum, suppose x is a variable of an MLRAP which consists of all redundancy states in the system units. The MLRAPs stud-ied is formulated as [17]:

Max Rsys ¼ RðxÞs:t: Csys ¼ CðxÞ 6 C0

1 6 xi;m 6 di

8><>: ; ð5Þ

where C0 is the predefined maximum cost allowed, and di is the given maximum redundancy of subunits Ui,m of the entiresystem Ui.

3. BiEA for MLRAPs

In early works related to BEA, the pseudo bacterial genetic algorithm (PBGA) [10] was first proposed to settle the problemsof model identification and fuzzy system parameter discovery [31–33]. The BEA actually has the same features (bacterialmutation operation) of the PBGA, but has incorporated a new operation, called the gene transfer operation, inspired by amicrobial evolution phenomenon. Specifically, the bacterial mutation (solution exploration) performs a scheme of local opti-mization, and the gene transfer operation (solution exploitation) allows the chromosomes (or solutions) to directly transferinformation to other counterparts in the population.

In the beginning, BEA was applied to discover the combination of input variables of fuzzy rules [29,30]; this paper mod-ified the BEA and applied it to the encoding of MLRAP units. Moreover, to enhance the search ability both in solution explo-ration and exploitation, the dynamic gene transfer (DGT) operation and a niching scheme are presented to improve the qualityof the solutions. The proposed modified version is referred to as the bacterial-inspired evolutionary algorithm (BiEA).

3.1. General solution expression

Fig. 4 illustrates an example of the hierarchical representation for a tri-level redundancy configuration. The basic struc-ture is given in Fig. 4(a). Fig. 4(b) draws the final system form after redundancy allocation and its corresponding solutionrepresentation in the hierarchical structure is shown in Fig. 4(c). There, n denotes the number of connecting subunits, forexample, U1 has three subunits and thus n = 3; s stands for the redundancy allocated to it, i.e., the number of componentswith parallel connection in a unit. For example, the s of U11 is 2, meaning there will be two parallel-connecting components

to form U12. A variable in the form of xji;m is a positive integer and represents the redundancy allocated to the mth subunit of

the jth redundant unit of Ui. In other words, the variables xji;m are to be optimized in an MLRAP because the units at the lowest

level, i.e., the component level, have no subunit and only parameter s is given.

3.2. Solution encoding in BiEA

As stated above, the variables xji;m are to be optimized in an MLRAP and the upper bound of xj

i;m should be provided. For the

purpose of convenient description, the upper bound of xji;m is assumed to be Max�xj

i;m ¼ 3. The solution encoding in the BiEA

(a)

(b)

(c)

Fig. 4. An example of the hierarchical representation scheme on a tri-level serial system: (a) basic configuration; (b) system after redundancy allocations on(a); (c) hierarchical representation [47].


is similar to fuzzy rules; the difference is such that we segment a specific range of dimensions (here labeled as grids) to rep-resent a unit or subunit.

Fig. 4(a)’s corresponding full design is shown in Fig. 5(a), which considers each state of the redundancy in a system.Fig. 5(b) is the corresponding solution encoding in BiEA for Fig. 5(a), where grid 1 corresponds to system unit U1, and alwayshas value 1; grids 2 to 4 correspond to subunits U11, U12 and U13, respectively; grids 5–6, grids 7–8, and grids 9–10 individ-ually correspond to the subunits constructed by components U111 and U112. Similarly, grids 11–12, grids 13–14, and grids15–16 individually correspond to the subunits constructed by components U121 and U122; grids 17–18, grids 19–20, and grids21–22 individually correspond to the subunits constructed by components U131 and U132.

In addition, Fig. 5(b) contains two controllers concerning the solution combination. First, the value in each grid controlsthe number of redundancies; Second, the binary-value parameters A1 to A9 control the existence of subunits constructed bycomponents, i.e., the nine respective inner dotted frames in Fig. 5(a). For example, Figs. 6(a) and (b) show a certain solutionand its corresponding redundancy allocation, respectively. Since grid 2 is two (grid 1 is always 1), the first two (A1 and A2)remain and A3 is set to zero; similarly, only A7 remains because grid 4 is one. As a result, grids 9 to 10 and grids 19 to 22 willnot exist in the final solution. In this case, this solution can be written as: [(1)(231)(2132)(231113)(22)] in the form of[(x1)(x11x12x13)(x111x112)(x121x122)(x131x132)]. The original solution encoding is: [(1)(231)(213200)(231113)(220000)], wherethe zero will be omitted in the final solution representation.

(a)

(b)

Fig. 5. BiEA solution encoding based on Fig. 4: (a) system after full redundancy allocations; (b) hierarchical BiEA solution encoding.


3.3. Fitness function

To guide the search towards unexplored regions more efficiently, the penalty strategy is applied. For solutions with exces-sive cost, a penalized reliability is calculated for the system. For an experiment, with the total system cost Csys = C(x) thatexceeds a given value C0, the original objective (5) is rewritten as the following penalty form: [14]

Max Rp ¼Rsys if CðxÞ 6 C0

Rsys � ðCðxÞ � C0Þ � g otherwise

�ð6Þ

The penalty parameter g is a shrinking factor, i.e., a very small positive value depending on the applied problem is given. Thispenalty function is expected to make promising feasible and infeasible regions of the search space be explored efficiently andeffectively to identify an optimal or near-optimal solution.

3.4. Bacterial mutation operation

Before going through the evolutionary operations of the BiEA, the initial population needs to be generated first. Assume Mis the size of the population, and the M initial solutions are generated by randomly giving a positive integer within

½1;Max�xji;m� to each grid (except grid 1). Next, the bacterial mutation operation is applied to each chromosome (solution)

one by one. Suppose a chromosome was cut into n parts (n refers to the number of system level) and each part is furtherdivided into grid groups according to the number of the subunits it forms. For example, continue to the tri-level case(n = 3) in Fig. 4(a) and the corresponding full design in Fig. 5(a). The size of each chromosome can be computed as:

1þPn�2

i¼1 NiðMax�xji;mÞ

i�1þ Nn�1 ðMax�xj

i;mÞðn�1Þ�1

, where N1 to Nn�2 represent the number of subunits in each subunit level(between the topmost level and the lowest level) and Nn-1 is the number of components. Therefore, the solution size ofthe tri-level case in Fig. 4(a) will be 1 + 3 � 30 + 6 � 31 = 22, and each chromosome is cut into three parts, i.e., grid 1, grids2–4 and grids 5–22. Then, it is clear grid 1 is no longer further divided into any grid group; grids 2–4 form the only U1 suchthat they cannot have further divisions. Grids 5–22 form three subunits U11, U12 and U13; thus grids 5–22 are further divided

(a)

(b)

Fig. 6. An example of the BiEA solution encoding based on Fig. 4(a): (a) hierarchical BiEA solution encoding; (b) the corresponding system after redundancyallocations.


into three parts. In this case, we have five grid groups: grid 1 (the first grid group), grids 2–4, grids 5–10, grids 11–16 andgrids 17–22.

As can be seen from Fig. 7, parameter d denotes the solution dimension (e.g., d = 22 in Fig. 5); when the mutation operationbegins, the first chromosome is chosen and it is reproduced in m clones. The ith grid group (i – 1) is randomly chosen for eachclone and one grid in that grid group is selected at random to be mutated by a given random positive integer less or equal toMaxxj

i;m. Then, each of the m clones are substituted into (6) for a fitness evaluation procedure. Subsequently, the most elite indi-vidual among the m clones is selected, and the ith grid group of the selected chromosome is transferred to the rest of the m � 1ones, i.e., the ith grid groups of all the clones is replaced by the ith grid groups of the selected chromosome. This process: muta-tion ? evaluation ? selection ? replacement is repeated. The mutation is applied to another non-overlapped randomly chosengrid group. When all grid groups have been processed, the best chromosome from the m individuals is selected to remain in thepopulation and the other m � 1 individuals are eliminated. This operation is applied to all the M chromosomes.

3.5. Dynamic gene transfer operation (DGT)

The original gene transfer operation is described below.After all solutions go through the mutation operation, sort the population by fitness value and separate it into two halves;

the half with the individuals with a better fitness value is called the superior half, and the other is called the inferior half. Next,randomly choose a chromosome from the superior half, named the source chromosome, and another chromosome from theinferior half, named the destination chromosome; finally, randomly choose a part (called a grid group in BiEA encoding) fromthe source chromosome that will be transferred to the destination chromosome. Repeat the above procedure Ninf times,where Ninf is the number of solutions in the inferior half per generation.

However, two issues can be improved regarding the original gene transfer operation. First, the significant elites are notnecessary in the half portion, e.g., they may exist in the first 40% of the elites or other proportions. Second, the inferior halfdoes not necessarily receive the better chromosomes from the superior half because the selection of the grid groups is ran-dom. As a result, by employing a probability scheme, a dynamic genetic transfer operation is presented here to select the supe-rior and inferior parts flexibly and transfer their genes. The following details provide the specific steps of the dynamic genetictransfer operation:

Fig. 7. Bacterial mutation operation.


Step 1. When all chromosomes complete the bacterial mutation, they are evolved through a probability-based selection.Each chromosome is selected according to a probability proportional to the quality of fitness via (7), and the probability Ph

for each solution (h = 1, 2, . . ., M) is then determined:

Ph ¼fitnesshPMi¼1fitnessi

ð7Þ

c ¼ M � c ð8Þ

Step 2. In (8), a parameter cut would be set to decide the proportion of individuals with better fitness to be chosen. Thechosen individuals are called superiors; conversely, the rest are called inferiors. Parameter � is a randomly selecteddynamic factor within [0.4,0.6] in each run, to decide the value of cut for each run. The range [04,0.6] is set to avoidthe number of superiors being extremely large or small in the dynamic module.Step 3. Probabilities P1, P2, . . ., PM are rearranged by directly proportioning to fitness, and thus obtain pr1, pr2, . . ., prM,where pr1 P pr2 P . . . P prM . In this case, when solutions with the corresponding probability are larger or equal to prcut,they will be selected to be superiors, while the rest of the chromosomes are inferiors.Step 4. One grid is randomly chosen for transference from a chromosome in the superiors to another in the inferiors. Themethod of selecting chromosomes for superiors and inferiors follows the quality of fitness. After the transference, continueto the next chromosome in the inferiors for the same process. Repeat the above process until all of the chromosomes inthe inferiors are completed.

For a more detailed elucidation, the following is a simple example of the dynamic genetic transfer operation.Suppose M = 6 and d is 3; the solutions obtained from the bacterial mutation operation are:S1 = (5.35,0.12,7.91), S2 = (1.75,0.02,1.97), S3 = (0.41,2.27,0.12), S4 = (5.5,8.98,0.21), S5 = (4.75,0.18,2.33), S6 = (5.67,

1.94,8.60), and the corresponding probabilities via (7) are assumed P1 = 0.43, P2 = 0.171, P3 = 0.057, P4 = 0.280, P5 = 0.062,and P6 = 0.503. Thus, (pr1,pr2,pr3,pr4,pr5,pr6) = (P6,P1,P4,P2,P5,P3) = (0.503,0.43,0.280,0.171,0.062,0.057). Assume � in thisrun is 0.402, so cut is 0.402 � 6 = 2 (round-off) and prcut = pr2 = 0.43. In this case, the superiors are (S6, S1) and the inferiorsare (S4,S2,S5,S3). Assume the randomly selected dimensions for inferiors are (3,1,1,2), then S4, S2, S5 and S3 becomeS4 = (5.5,8.98,8.60), S2 = (5.35,0.02,1.97), S5 = (5.67,0.18,2.33) and S3 = (0.41,0.12,0.12).

3.6. Niching-inspired searching

One of the key elements in avoiding the less promising local optimums of a difficult optimization problem is to preservethe population diversity during the search [40]. Inspired by [42], this work presents a modified simple niching scheme orig-inally used in addressing the cluster problem [42]. The introduced niching scheme is designed to not only preserve the solu-tion diversity, but also to avoid the problem of trapping into the local optimum [13].

Search

A

B

Radius: ratio× AB

The trapping solution

Selection group

Fig. 8. Illustration of niching method [13].


In Fig. 8, suppose the dotted outline is the search space, and the longest distance between two boundaries is line AB,which is determined through Euclidean distance, where A = (lb1, lb2, . . ., lbd), B = (ub1, ub2, . . ., ubd), and lbi and ubi are thelower and upper bounds for dimension i, respectively. In the search process, the cycle times numi of a chromosome Si whosefitness cannot be improved is recorded. When numi exceeds a predefined value num, Si will be sent to the niching scheme forfurther improvement. The steps are as follows:

For a certain run, suppose a chromosome Si has retained the same fitness value for num cycles, i.e., numi = num; then Si isimpelled to implement the niching process.

Step 1. An appropriate proportion of line AB is chosen to be the radius of center point Xi. The appropriate proportion of lineAB is set to ratio� AB; where ratio is a parameter within (0, 1).Step 2. The chromosomes in the area (sphere) swept through by radius ratio� AB is called the selection group.Step 3. The following (9) is applied to all the chromosomes in the selection group, including Si itself, and rand() is within[�1, 1].

Vsj ¼ Xsj þ randðÞXsj ð9Þ

Step 4. The execution via (9): For a chromosome Xs in the selection group, two positive integers p and q ðp 6 qÞ are ran-domly selected to form a region for variation. With the exception of the elements of Xs that do not belong to the variationregion, all other elements of Xs implement (9). Namely, the Xs becomes a new chromosome: Vs = (Xs1,Xs2, . . ., Vsp, Vs(p+1), . . . Vsq, Xs(q+1), . . ., Xid).Step 5. As all chromosomes in the selection group go through (9), they may have a new form through the niching scheme;then, if Vi has a greater fitness value than Xi, Vi replaces Xi.

Therefore, by implementing the niching scheme, the solution diversity can be preserved and the local trapping solutions canalso be derived.

3.7. Summary of BiEA

The summarization of the above techniques forming the proposed BiEA is presented here for the purpose of descriptionclarity. The BiEA begins with a random initialization and then, new individuals are generated through each evolved operationiteratively. At each generation, the mutation operation and DGT operation are applied to the population of new candidates.Subsequently, a niching scheme is implemented (if required) to specific solutions for further improvement. After that, the bestaspects of the solutions are selected to survive into the next generation. This process is repeated for a predefined number ofgenerations and the best individual in the final population will be the final solution to the MLRAPs.

4. Case studies

In this section, the performance of BiEA is evaluated in regard to three cases of MLRAPs taken from [47,17]. The first case(labeled as Case 1) involves comparing BiEA’s results with those of MA [47], the most current algorithm used for MLRAPs; theexperimental designs follow [47]. The second case (labeled as Case 2) involves evaluating the performance of BiEA and HGAby employing the three experimental designs in [17]. The third case (labeled as Case 3) involves the performance validationsof a DGT operation and a niching scheme.


The MLRAPs used in this section are two multi-level serial systems, as illustrated in Figs. 9 and 10. The first system(labeled Problem A) comprises three levels and the second (labeled Problem B) consists of four levels. Regarding Max�xj

i;m;

a large value may entail excessive cost; conversely, a small value would not produce a sufficiently high reliability. Thus,for both systems, the redundancy allocated to a unit is set at 1, 2 or 3, i.e., Max�xj

i;m ¼ 3. In addition, a sign in dark shadeis used for the expression of a better value; for further validation, the statistic, Wilcoxon rank sum test (with a significancelevel 0.05), was utilized to compare the overall performance of the two methods. For each cost constraint, the significantlybetter result is highlighted in boldface.

4.1. Case 1

The experimental design follows the scheme in [47]: (1) Single case study, (2) Performance over different cost constraintsand (3) Comparison with different system parameters. The data descriptions of the components for two problems are listedin Table 3. Note, the additional cost k of component U113 in Problem A used in this paper is 4 [47]; that is, it is more expensivethan [17] and so the problem becomes more difficult and challenging.

4.1.1. Problem ASince the previous research [47] was undertaken on Problems A and B, the parameter settings used previously are used

directly here. Besides, the proposed BiEA contains a few control parameters to be set in advance. For a fair comparison, thesame population size and maximum generation numbers in [47] were adopted in this case. Specifically, the population sizewas 100 and the maximum generation was 500; in addition, the number of clone m, num and ratio are set at 20, 4 and 0.2,respectively (this setting is also for cases 2 and 3).

(1) Single Case Study: The aim of this experiment is to observe the convergence behavior of BiEA in a single case. The samesettings follow [47], i.e., the cost constraint is 300, the maximum generation number was 500 and other essentialparameter information is listed in Table 3. The best solution obtained in each generation was recorded, and the cor-responding system reliability was calculated. The convergence curves of MA and BiEA are drawn in Fig. 11, where bothmethods exhibit a similar trend and rapid convergence within 30 generations. Nevertheless, BiEA exhibits a morerapid convergence speed than MA within 30 generations and the generations 100–200; finally, they achieve the sameoptimal reliability with fewer generations. The result revealed BiEA can converge to an optimal reliability with lesscomputational time in terms of the same anticipated reliability.

U1

U11 U12 U13

U111 U113 U121 U122 U131 U132U112

Fig. 9. The multi-level configuration of Problem A.

U1

U11 U12

U111 U112 U121 U122

U1111 U1112 U1121 U1122 U1211 U1212 U1221 U1222

Fig. 10. The multi-level configuration of Problem B.

Table 3The input data of the multi-level systems on Problem A and Problem B [47].

Problem A Problem B

Unit Reliability Cost k Unit Reliability Cost k

U111 0.9000 5 3 U1111 0.9000 7 4U112 0.9500 6 4 U1112 0.8000 6 4U113 0.8500 5 4 U1121 0.7500 8 4U121 0.9000 6 4 U1122 0.9500 5 4U122 0.8500 7 4 U1211 0.7000 9 4U131 0.9000 8 3 U1212 0.9000 6 4U132 0.8000 7 4 U1221 0.8500 5 4

U1222 0.8000 8 4


(2) Performance over Different Cost Constraints: This experiment shows a further comparison under various cost con-straints. Specifically, twenty cost constraints between 150 and 340 are used to test the performances of MA and BiEA.All of the settings of the system parameters are also shown in Table 3.

For each cost constraint, both MA and BiEA are run ten times with different initial populations. The results are summa-rized in Table 4, in which the columns headed Best solutions present the reliability, the used cost and the corresponding bestsolution in the ten independent runs. The results are provided in Fig. 12. In addition, the columns headed Overall Performancepresent the average reliability of the solutions obtained over the ten runs. The corresponding variances are also given toobserve the robustness of the methods.

From the dark areas for each cost constraint, the results obtained from BiEA are better or equal to those from MA. Addi-tionally, BiEA presents more robust results by smaller variances. Further, the average result of the significantly bettermethod is highlighted in boldface. The data clearly shows the proposed BiEA outperformed MA in nearly two-thirds ofthe scenarios.

(3) Comparison with Different System Parameters: The third experiment is designed to test whether the practice of the BiEAadapts to different system parameters. We adopt the same experimental design in [47]: ten test instances were gen-erated at random by resetting the reliability of the components. Specifically, the reliability of each component wasrandomly chosen from the set {0.80,0.85,0.90,0.95}. If an algorithm is robust, the results can reflect the quality of ran-domly generated initial populations. Both MA and BiEA undergo each instance for ten runs. The best result and averageresult for each run are presented in Table 5. The dark areas show the reliability and robustness obtained from BiEAboth outperform MA in the majority of test cases. In addition, it is clear the BiEA outperformed or equaled MA forall 20 cases, in terms of the quality of overall performance (highlighted in boldface). The result shows BiEA can bemore adaptable to system parameters under components with different reliabilities.

4.1.2. Problem BFor the first experiment, the cost constraint was set to 500 following the previous works [17]; the system parameters are

given in Table 3. The maximum generation number was set to 500 for BiEA and MA. The results of this experiment are pre-sented in Fig. 13. The results also show both algorithms exhibit rapid convergence while BiEA clearly outperforms in thissingle case as compared to Problem A.

For the second experiment, 15 different cost constraints were chosen from 200 to 900. The same information listed inTable 3 is used. The results are presented in Table 6 and Fig. 14, in which it can be observed BiEA has a clear and obvious

0.88

0.9

0.92

0.94

0.96

0.98

1

1 101 201 301 401 501

Generation Number

Syst

em R

elia

bilit

y

BiEA

MA

Fig. 11. Convergence plot of BiEA and MA on Problem A.

0.75

0.8

0.85

0.9

0.95

1

150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340

Cost Constraint

Syst

em R

elia

bilit

y

MA

BiEA

Fig. 12. Comparison between the MA and BiEA under different constraints on Problem A.

Table 4Results of BiEA and MA on Problem A with different constraints, where ‘‘best solutions’’ and ‘‘overall performance’’ indicate the best solution and the averagesover the 10 runs, respectively.

The average computational cost of MA and BiEA are 69.26 and 86.09 (s).a The average variance of MA in Overall Performance.b The average variance of BiEA in Overall Performance.


advantage in this more complicated system as compared to Problem A. For the third experiment, ten test instances were gen-erated by randomly choosing the reliability of each component from the set {0.70,0.75,0.80,0.85,0.90}. Both MA and BiEAwere implemented on these instances for ten runs. The results shown in Table 7 reveal similar conclusions to those in Prob-lem A; these can be discussed in the following aspects.

As seen in Fig. 13, the performance difference in this Single Case Study is easily observed. BiEA shows a more rapid con-vergence trend than MA in most of the generations, and the difference increases with each successive generation. The resultreveals BiEA has rapid convergence speed and can achieve higher anticipated reliability than MA. Further, in the design ofPerformance over Different Cost Constraints, for each cost constraint, the results obtained from BiEA are better than those fromMA in most of the constraints by viewing the number of dark areas. Moreover, for robustness, BiEA presents more robustresults because of the smaller variances (5.85E�7 is much smaller than 7.01E�5, as can be seen from Table 6). Finally,the Wilcoxon rank sum test explains the average result of the significantly better method is highlighted in boldface. Thus,the proposed BiEA outperformed MA in nearly all scenarios. Although the average computation cost of BiEA needs a littlemore execution time than MA, BiEA can achieve the same reliability as MA in a shorter computational time because ofthe trait of rapid convergence. Further, this shows the proposed method possesses the characteristics of robustness,

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1 50 99 148 197 246 295 344 393 442 491

Generation Number

Syst

em R

elia

bilit

y

BiEA

MA

Fig. 13. Convergence plot of BiEA and MA on Problem B.

Table 5Results of BiEA and MA on Problem A with different system parameters, where ‘‘best solutions’’ and ‘‘overall performance’’ indicate the best solution and theaverages over the 10 runs, respectively.

The average computational cost of MA and BiEA are 65.44 and 85.92 (s).a The average variance of MA in overall performance.b The average variance of BiEA in overall performance.


rapidness and precision in solution search based on the above statement. Finally, in the design of Comparison with DifferentSystem Parameters, the dark areas illustrate the reliability and robustness obtained from BiEA both outperform MA in most ofinstances, as can be seen in Table 7. Similarly, the results revealed BiEA is more adaptable to system parameters under com-ponents with different reliabilities.

4.2. Case 2

In case 2, the BiEA is used to solve MLRAPs and compares the obtained solutions in Problem A with those obtained by theHGA [17]. Since [47,17] have the same setting of Problem B as shown in Table 3, case 2 is a comparison of BiEA and HGA inProblem A, based on the setting in [17]. Specifically, the input information is also listed in Table 3, except the additional costk of component U113 in Problem A is 3. The experimental designs in [17] are similar to Case 1 with slight differences, asdescribed below.

First, the Single Case Study involves observing the convergence of the optimal solutions. Fig. 15 shows the results whenusing BiEA and HGA. The cost constraint for this case is 240. Fig. 15 indicates the BiEA has faster convergence speed anda better convergence trend from start to end under the same computational environment.

In the second experiment, Performance over different cost constraints, similarly, there are 20 cost constraints and ten 500-generation trials performed by using BiEA and HGA. For each constraint, the best solution of the ten trials was chosen as theoptimal solution, and recorded in Table 8. For an integrated observation, Fig. 16 draws the curves of the optimal solutionsobtained using BiEA, average BiEA and HGA. Clearly, BiEA and average BiEA are very close; this signifies a small variation,as the same as the reflection in Table 8. More importantly, the HGA curve is clearly below the other two curves in mostof the cost constraints.

Finally, ten cases with different unit reliability values, as described in Case 1, were performed; the cost constraint washeld to a value of 300. In the same manner as before, ten 500-generation trials for each of these ten cases were carried

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

200 250 300 350 400 450 500 550 600 650 700 750 800 850 900

Cost Constraint

Syst

em R

elia

bilit

y

MA

BiEA

Fig. 14. Comparison between the MA and BiEA under different constraints on Problem B.

Table 6Results of BiEA and MA on Problem B with different constraints, where ‘‘best solutions’’ and ‘‘overall performance’’ indicate the best solution and the averagesover the 10 runs, respectively.

The average computational cost of MA and BiEA are 160.48 and 183.06 (s).a The average variance of MA in overall performance.b The average variance of BiEA in overall performance.

Table 7Results of BiEA and MA on Problem B with different system parameters, where ‘‘best solutions’’ and ‘‘overall performance’’ indicate the best solution and theaverages over the 10 runs, respectively.

The average computational cost of MA and BiEA are 164.94 and 178.78 (s).a The average variance of MA in Overall Performance.b The average variance of BiEA in Overall Performance.


0.75

0.8

0.85

0.9

0.95

1

1 51 101 151 201 251 301 351 401 451 501

Generation

Syst

em r

elia

bilit

y

HGA

BiEA

Fig. 15. Convergence plot of BiEA and HGA for Problem A.

Table 8Results of BiEA for Problem A with different constraints, where ‘‘best solutions’’ and ‘‘overall performance’’ indicate the best solution and the averages over the10 runs, respectively.

Cost Best solutions Overall performanceCons. BiEA BiEA

Reliability Cost Detailed Solutions Mean Variance[(x1)(x11 x12 x13)(x111 x112 x113)(x121 x122)(x131 x132)]

150 0.831629 149 [(1)(222)(112111)(1111)(1111)] 0.830863 3.30E�6160 0.857618 160 [(1)(221)(111112)(1111)(22)] 0.857618 0170 0.868025 170 [(1)(231)(111111)(111111)(22)] 0.867889 1.41E�7180 0.888767 179 [(1)(222)(111212)(1111)(1112)] 0.888672 8.13E�8190 0.903187 190 [(1)(221)(111212)(1211)(22)] 0.903187 0200 0.917436 193 [(1)(123)(222)(1112)(111111)] 0.914465 9.8E�7210 0.933252 209 [(1)(212)(212111)(22)(1122)] 0.933252 0220 0.949237 219 [(1)(223)(212111)(2211)(111111)] 0.943162 9.22E�6230 0.960420 225 [(1)(232)(112212)(111111)(2211)] 0.957449 1.00E�6240 0.960194 238 [(1)(223)(111212)(1122)(111112)] 0.957649 8.85E�7250 0.970484 250 [(1)(232)(212212)(111111)(2221)] 0.969468 1.14E�7260 0.975446 260 [(1)(223)(212212)(2211)(111112)] 0.973773 1.19E�6270 0.979184 270 [(1)(223)(222111)(2211)(221111)] 0.977592 1.76E�6280 0.982124 274 [(1)(223)(212212)(1122)(111122)] 0.980799 8.00E�7290 0.984468 288 [(1)(232)(212212)(111211)(2222)] 0.983626 7.87E�8300 0.986727 295 [(1)(233)(212222)(111211)(221111)] 0.986489 5.67E�8310 0.989283 306 [(1)(232)(212212)(221111)(2222)] 0.989011 7.57E�8320 0.991553 313 [(1)(233)(222212)(221111)(112211)] 0.990164 5.82E�7330 0.992975 324 [(1)(232)(121222)(112211)(2222)] 0.992907 4.24E�8340 0.994561 340 [(1)(333)(212212111)(221111)(112212)] 0.993842 7.56E�8

The average computational cost is 87.51 (s).

0.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340

Cost Constrains

Syst

em r

elia

bilit

y

HGA

BiEAAvg_BiEA

Fig. 16. Comparison between the BiEA, average BiEA, and HGA under different constraints for Problem A.


Table 9Results of BiEA and HGA for Problem A with different system parameters, where the presented values are the best solutions over the 10 runs.

The average computational cost of HGA and BiEA are 95.60 and 88.93 (s).

Table 10Results of mutation operation combined with the dynamic and original gene transfer operation for Problem A with different constraints, where ‘‘best solutions’’indicate the best solution over the 10 runs.

The average computational cost of the original and dynamic are 79.52 and 81.24 (s).

0.84

0.86

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0.92

0.94

0.96

0.98

1 51 101 151 201 251 301 351 401 451

BiEA

Dynamic

Original

Fig. 17. Convergence plot of BiEA, mutation operation combined with dynamic and original gene transfer operation for Problem A.


0.7

0.75

0.8

0.85

0.9

0.95

1

150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340

Cost constrainsSy

stem

rel

iabi

lity

Original

Dynamic

BiEA

Fig. 18. Comparison between the BiEA, mutation operation combined with dynamic and original gene transfer operation under different constraints forProblem A.

Table 11Results of mutation operation combined with dynamic and original gene transfer operation on Problem A with different system parameters, where ‘‘bestsolutions’’ indicate the best solution over the 10 runs.

The average computational cost of the original and dynamic are 81.45 and 82.19 (s).


out, and the best solution was chosen as the optimal solution for each case. Table 9 summarizes the optimal solutionsobtained from BiEA and HGA for Problem A. The BiEA has an overall outperformance with a large difference in results.

4.3. Case 3

The aim of this case is to determine the effectiveness of the DGT operation and niching scheme. To achieve this, the exper-iment was designed to be a comparison between the original BEA (i.e. mutation operation combined with the original genetransfer operation, and simply called Original) and the mutation operation combined with DGT (simply labeled as Dynamic).For the purpose of consistency, the Dynamic and Original performed Problem A as described in Case 2, so the differenceamong BiEA, Dynamic and Original can reveal the progress of the DGT and niching scheme. All settings and experimentaldesigns here were the same as Case 2. Fig. 17 shows the result of a Single Case Study. It can be seen the Dynamic and Originalhave similar convergence speed; however, Dynamic reaches a better convergence result. More importantly, the BiEA curveconverges to a much better result compared to the other two algorithms.

In the second experiment, the best solution of the ten trials was chosen as the optimal solution and the results arerecorded in Table 10, in which the Dynamic outperformed the Original in nearly all cost constraints. Specifically, Fig. 18clearly shows the difference of performance among the three algorithms and reveals two findings: First, the Dynamic per-formed better or equal to the Original in all 20 constraints; the second is the niching scheme made obvious progress based onthe solutions obtained by Dynamic, which shows the niching scheme efficiently perform the solution exploration and avoidthe trapping trouble.

Table 11 presented the results in the third experiment: ten cases with different unit reliability values. As stated in Sec-tion 3.5, the significant elites are not necessary in the half portion, and we knew the Dynamic outperformed the Original innearly all cost constraints in Case 2, the results in Table 11 again support this argument from the significant superiority in alltest constraints.

5. Conclusions

MLRAPs are frequently encountered in real-world systems. However, few studies have been conducted to examine themin this environment. This paper aimed to investigate the RAPs in multi-level systems and a BiEA was proposed to solve MLR-


APs based on the hierarchical solution representation. Case studies on two system configurations reveal our BiEA has thebetter performance than two state-of-the-art algorithms, MA and HGA in most of the case designs. In addition, to validatethe effectiveness of the two main members of BiEA: DGT operation and niching scheme, extra cases studies were designed toprove they are the significant superiors. However, in terms of CPU communing, although BiEA spent less time than HGA, theCPU time of BiEA is a little longer than MA. Even though the average computation cost of BiEA requires further research, BiEAcan achieve the same reliability as MA in less computational time due to its rapid convergence. Therefore, this shows theproposed BiEA is rapid. Further, experimental results reveal BiEA is robust because BiEA presents a smaller average variancewhen compared to MA. Additionally, the Wilcoxon rank sum test shows the average result of the significantly better method ishighlighted in boldface, which shows the proposed BiEA outperformed MA in nearly all experimental scenarios. For futurework, to retain the effectiveness of BiEA, the evolutionary operations can be simpler; besides, different forms of MLRAPs canbe extended to more complicated configurations or the difficulty of case designs increased so the system designs could moreclosely match real-world systems.

Acknowledgments

This research was supported by the Computational Intelligence Technology Center (CITC), Industrial Technology ResearchInstitute (ITRI), Taiwan, ROC, under project code D301AR2N10.

References

[1] O. Castillo, G. Huesca, F. Valdez, Evolutionary computing for topology optimization of type-2 fuzzy controllers, Stud. Fuzz. Soft Comput 208 (2007)163–178.

[2] M.S. Chern, On the computational complexity of reliability redundancy allocation in a series system, Operat. Res. Lett. 11 (1992) 309–315.[3] D.W. Coit, A.E. Smith, Reliability optimization of series-parallel systems using a genetic algorithm, IEEE Trans. Reliab. 45 (2) (June 1996) 254–260.[4] D.W. Coit, A.E. Smith, Penalty guided genetic search for reliability design optimization, Comput. Ind. Eng. 30 (4) (1996) 895–904.[5] S. Das, A. Chowdhury, A. Abraham, A bacterial evolutionary algorithm for automatic data clustering, IEEE Congress on Evolutionary Computation,

Trondheim, Norway, 2009, pp. 2403–2410. May18–21.[6] M.A. Delucchi, M.Z. Jacobson, Providing all global energy with wind, water, and solar power, Part II: Reliability, system and transmission costs, and

policies, Energy Pol. 39 (2011) 1170–1190.[7] M. Dowlatshahi, H. Nezamabadi-Pour, M. Mashinchi, A discrete gravitational search algorithm for solving combinatorial optimization problems, Inf.

Sci. 258 (2014) 94–107.[8] C. Elegbede, C. Chu, K. Adajallah, F. Yalaoui, Reliability allocation through cost minimization, IEEE Trans. Reliab. 52 (1) (Mar. 2003) 106–111.[9] P. Foldesi, L.T. Koczy, J. Botzheim, Fuzzy extension for Kano’s model using bacterial evolutionary algorithm, in: 3rd International Symposium on

Computational Intelligence and Intelligent Informatics, Agadir, Morocco, Mar 28–30, 2007, pp. 147–151.[10] T. Furuhashi, Y. Miyata, K. Nakaoka, Y. Uchikawa, A new approach to genetic based machine learning for efficient finding of fuzzy rules, Lecture Notes

in Artificial Intelligence, vol. 1011, Springer-Verlag, Berlin, Germany, 1995, pp. 173–189.[11] D.E. Fyffe, W.W. Hines, N.K. Lee, System reliability allocation and a computational algorithm, IEEE Trans. Reliab. 17 (2) (Jun. 1968) 64–69.[12] C. Ha, W. Kuo, Reliability redundancy allocation: an improved realization for nonconvex nonlinear programming problems, Eur. J. Oper. Res. 171 (1)

(2006) 24–38.[13] T.J. Hsieh, C.L. Cheng, W.C. Yeh, A hybrid Niching-based evolutionary PSO for numerical optimization problems, in: 2012 IEEE International Conference

on Computational Intelligence and Cybernetics (CyberneticsCom), 2012, pp. 133-137.[14] T.-J. Hsieh, W.-C. Yeh, Penalty guided bees search for redundancy allocation problem with a mix of components in series-parallel systems, Comput.

Oper. Res. 39 (11) (2012) 2688–2704.[15] T. Inoue, T. Furuhashi, H. Maeda, M. Takaba, A study on Bacterial Evolutionary Algorithm engine for interactive nurse scheduling support system, in:

26th Annual Conference of the IEEE Industrial Electronics Society, Nagoya, Japan, Oct. 22–28, 2000, pp. 651–565.[16] K.-W. Jang, J.-H. Kim, A tabu search for multiple multi-level redundancy allocation problems in series-parallel systems, Int. J. Ind. Eng. 18 (2011) 2386–

2393.[17] R. Kumar, K. Izui, M. Yoshimura, S. Nishiwaki, Multilevel redundancy allocation optimization using hierarchical genetic algorithm, IEEE Trans. Reliab.

57 (4) (2008) 650–661.[18] R. Kumar, K. Izui, M. Yoshimura, S. Nishiwaki, Multi-objective hierarchical genetic algorithms for multilevel redundancy allocation optimization,

Reliab. Eng. Syst. Saf. 94 (2009) 891–904.[19] W. Kuo, V.R. Prasad, An annotated overview of system-reliability optimization, IEEE Trans. Reliab. 49 (2) (2000) 176–187.[20] G. Levitin, A. Lisnianski, Optimal separation of elements in vulnerable multi–state systems, Reliab. Eng. Syst. Saf. 73 (1) (2001) 55–66.[21] G. Levitin, Optimal allocation of multi-state elements in a linear consecutively–connected system, IEEE Trans. Reliab. 52 (2) (Jun. 2003) 192–199.[22] G. Levitin, Optimal multilevel protection in serial-parallel systems, Reliab. Eng. Syst. Saf. 81 (1) (2003) 93–102.[23] Y.C. Liang, A.E. Smith, An ant colony optimization algorithm for the redundancy allocation problem, IEEE Trans. Reliab. 53 (3) (Sep. 2004) 417–423.[24] Y.K. Lin, C.T. Yeh, Computer network reliability optimization under double-resource assignments subject to a transmission budget, Inf. Sci. 181 (2011)

582–599.[25] M. Miwa, T. Furuhashi, M. Matsuzaki, S. Okuma, CMAC modeling using pseudo-bacterial genetic algorithm and its acceleration, in: IEEE International

Conference on Systems Man and Cybernetics (SMC), 1, 2001, pp. 250–254.[26] M. Miwa, T. Inoue, M. Matsuzaki, T. Furuhashi, S. Okuma, Fuzzy system parameters discovery by bacterial evolutionary algorithm, in: 28th Annual

Conference of the IEEE Industrial Electronics Society. Seville, Spain, Nov. 05–08, 2002, pp. 1801–1805.[27] O.E. Moya, J.A. Herrera, Long term marginal costs of reliability and inter-zone reliability transfers, in: Power Systems 9th International Conference on

Probabilistic Methods Applied to Power Systems, KTH, Stockholm, Sweden, 2006, pp. 1–8. June 11–15.[28] N. Nahas, M. Nourelfath, D. Ait-Kadi, Coupling ant colony and the degraded ceiling algorithm for the redundancy allocation problem of series–parallel

systems, Reliab. Eng. Syst. Saf. 92 (2) (2007) 211–222.[29] N.E. Nawa, T. Furuhashi, Bacterial evolutionary algorithm for fuzzy system design, in: IEEE International Conference on Systems, Man, and Cybernetics

– Intelligent Systems for Humans in a Cyber world, San Diego, October 11-14, 1998, pp. 2424-2429.[30] N.E. Nawa, T. Furuhashi, Fuzzy system parameters discovery by bacterial evolutionary algorithm, IEEE Trans. Fuzzy Syst. 7 (1999) 608–616.[31] N.E. Nawa, T. Furuhashi, A study on nonlinear model identification using pseudo-bacterial genetic algorithm, in: International Conference on Neural

Information Processing and Intelligent Information Systems, vol. 1–2, 1997, pp. 408–411.[32] N.E. Nawa, T. Hashiyama, T. Furuhashi, Y. Uchikawa, A study on fuzzy rules discovery using pseudo-bacterial genetic algorithm with adaptive operator,

in: 1997 IEEE International Conference on Evolutionary Computation, 1997, pp. 589-593.

http://refhub.elsevier.com/S0020-0255(14)00789-0/h0005









































[33] N.E. Nawa, T. Hashiyama, T. Furuhashi, Y. Uchikawa, Fuzzy logic controllers generated by pseudo-bacterial genetic algorithm with adaptive operator,in: 1997 IEEE International Conference on Neural Networks, 1997, pp. 2408–2413.

[34] M. Nourelfath, D. Ait-Kadi, Optimization of series–parallel multi–state systems under maintenance policies, Reliab. Eng. Syst. Saf. 92 (12) (2007) 1620–1626.

[35] J. Onishi, S. Kimura, R.J.W. James, Y.J. Nakagawa, Solving the redundancy allocation problem with a mix of components using the improved surrogateconstraint method, IEEE Trans. Reliab. 56 (1) (2007) 94–101.

[36] E. Ortjohann, P. Wirasanti, M. Lingemann, W. Sinsukthavorn, S. Jaloudi, D. Morton, Multi-level hierarchical control strategy for smart grid usingclustering concept, in: International Conference on Clean Electrical Power (ICCEP), 2011, pp. 648–653.

[37] J.E. Ramirez-Marques, D.W. Coit, A heuristic for solving the redundancy allocation problem for multi-state series-parallel system, Reliab. Eng. Syst. Saf.83 (3) (2004) 341–349.

[38] J.E. Ramirez-Marques, D.W. Coit, A. Konak, Redundancy allocation for series-parallel systems using a max-min approach, IIE Trans. 36 (9) (2004) 891–898.

[39] E. Rashedi, H. Nezamabadi-Pour, S. Saryazdi, GSA: a gravitational search algorithm, Inf. Sci. 179 (2009) 2232–2248.[40] B. Sareni, L. Krahenbuhl, Fitness sharing and niching methods revisited, IEEE Trans. Evol. Comput. 2 (1998) 97–106.[41] V.K. Sharma, M. Agarwal, K. Sen, Reliability evaluation and optimal design in heterogeneous multi-state series-parallel systems, Inf. Sci. 181 (2011)

362–378.[42] W. Sheng, A. Tucker, X. Liu, Clustering with niching genetic K-means algorithm, in: Proc. Genetic and Evolutionary Computation Conference, 2004, pp.

162–173.[43] F.A. Tillman, C.L. Hwang, W. Kuo, Optimization techniques for systems reliability with redundancy—a review, IEEE Trans. Reliab. 26 (1977) 148–155.[44] F. Valdez, P. Melin, O. Castillo, Evolutionary method combining particle swarm optimization and genetic algorithms using fuzzy logic for decision

making, in: IEEE International Conference on Fuzzy Systems, 2009, pp. 2114-2119.[45] W. Wang, J. Loman, P. Vassiliou, Reliability importance of components in a complex system, in: Proceedings of the Annual Reliability and

Maintainability Symposium, LA, 2004, pp. 6–11.[46] Z. Wang, T. Chen, K. Tang, X. Yao, A multi-objective approach to redundancy allocation problem in parallel-series systems, in: Proceedings of the 2009

IEEE Congress on Evolutionary Computation (CEC2009), Trondheim, Norway, May 18-21, 2009, pp. 582–589.[47] Z. Wang, K. Tang, X. Yao, A memetic algorithm for multi-level redundancy allocation, IEEE Trans. Reliab. 59 (4) (2010) 754–765.[48] W.C. Yeh, T.-J. Hsieh, Solving reliability redundancy allocation problems using an artificial bee colony algorithm, Comput. Oper. Res. 38 (11) (2011)

1465–1473.[49] W.C. Yeh, A two-stage discrete particle swarm optimization for the problem of multiple multi-level redundancy allocation in series systems, Expert

Syst. Appl. 36 (2009) 9192–9200.[50] W.C. Yeh, Simplified swarm optimization in disassembly sequencing problems with learning effects, Comput. Oper. Res. 39 (2012) 2168–2177.[51] W.C. Yeh, Novel swarm optimization for mining classification rules on thyroid gland data, Inf. Sci. 197 (2012) 65–76.[52] W.Y. Yun, J.W. kim, Multilevel redundancy optimization in series systems, Comput. Ind. Eng. 46 (2004) 337–346.[53] W.Y. Yun, Y.M. Song, H.G. Kim, Multiple multi-level redundancy allocation in series systems, Reliab. Eng. Syst. Saf. 92 (2007) 308–313.[54] Y. Zhang, S. Weiqing, L. Wang, H. Wang, R.C. Green, M. Alam, A multi-level communication architecture of smart grid based on congestion aware

wireless mesh network, in: North American Power Symposium (NAPS), 2011, pp. 1–6.























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