Research Article A Discrete-Binary Transformation of the ...downloads.hindawi.com/journals/mpe/2015/276234.pdf · A Discrete-Binary Transformation of the Reliability Redundancy Allocation

Research ArticleA Discrete-Binary Transformation of the ReliabilityRedundancy Allocation Problem

Marco Caserta1 and Stefan Voszlig23

1 IE Business School IE University Maria de Molina 31B 28006 Madrid Spain2Institute of Information Systems University of Hamburg Von-Melle-Park 5 20146 Hamburg Germany3Escuela de Ingeniera Industrial Pontificia Universidad Catolica de Valparaıso Avenida Brasil 2241 2362807 Valparaıso Chile

Correspondence should be addressed to Stefan Voszlig stefanvossuni-hamburgde

Received 10 December 2014 Accepted 4 April 2015

Academic Editor Wei-Chiang Hong

Copyright copy 2015 M Caserta and S Voszlig This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Given a reliability redundancy optimization problem in its discrete version it is possible to transform such integer problem intoa corresponding binary problem in log-time A simple discrete-binary transformation is presented in this paper The proposedtransformation is illustrated using an example taken from the reliability literature An immediate implication is that a standardexact dynamic programming approach may easily solve instances to optimality that were usually only solved heuristically

1 Introduction

Dealing with system reliability is a central issue in a varietyof fields Hardware and software reliability are of paramountimportance since hardware and software components arepervasive in modern society The current competitive busi-ness environment is placing further emphasis on effectiveproduct and system design In particular in engineeringdesign with reliability in mind it is most important toimprove the competitive position and to save in engineeringdesign and warranty costs Academics as well as practitionershave devoted and still put special attention to the advance-ment of reliability design and analysis methods for complexsystems both hardware and software Reliability is a strategicissue in a number of industries such as for example theaerospace automotive civil defense telecommunicationsand power industries where advanced systems like spaceshuttle aerospace propulsion nanocomposite structure andbioengineering systems are designed and developed withreliability inmind Electronics mechanics computer scienceand industrial engineering are just other fields interested incurrent studies on reliability

Design with reliability in mind provides a number ofadvantages spanning from the ability to produce safer and

obviously more reliable products to the improvement in thecompetitive position via significant reduction of costs

One of the recent trends observed in the field of reliabilityis related to the growth in size and complexity of the systemsstudied Due to the fact that hardware and software systemskeep growing in size and complexity there is a need todesign and develop efficient methods that is algorithmsfor reliability problems More precisely software solutionsthat are aimed at reducing the time required to designcomplex reliability systems that are able to deal with largescale models and that are robust with respect to the systemconfiguration are deemed vital

An immediate idea or technique commonly used toincrease reliability of a complex system is via redundancyallocation The underlying hypothesis is that the reliabilityvalue is directly correlated to the number of redundantcomponents placed in each stage of the system Howeverowing to a set of constraints describing limitations in termsof cost physical space availability and similar limitations theoptimal allocation of redundant components is a hard task

Usually algorithms developed for the reliability redun-dancy allocation problem (RAP) work on the discrete versionof the problem (see eg [1ndash7]) In this paper we present asimple transformation into the binary version of the problem

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 276234 6 pageshttpdxdoiorg1011552015276234

2 Mathematical Problems in Engineering

i = 1 i = 2

y11 y21

y22y12

i = n

yn1

yn2

y1s1y2s2

yns119899

Figure 1 RAP series-parallel system

Based on that without loss of generality one can work withthe binary version of the RAP and use this for the futuredevelopment of algorithms As a result we can show thatproblem instances that are still solved heuristically in manyrecent papers (see eg [8ndash10]) can be solved to optimalitywith a straightforward dynamic programming algorithm inreasonable computational time The proposed discretizationis supporting the dynamic programming problem as it allowstreating incorporated problems of knapsack type directly asbinary knapsack problems In that sense it allows binarizinga set of subproblems and then solving the general RAP viadynamic programming by linking or combining a set ofbinary knapsack problems

In the next section we first provide a mathematicalformulation for the RAP In the sequel we transform theinteger version of the problem into its analogous binary ver-sion (log-time transformation) in the spirit of for examplethe authors of [11] who developed similar ideas regardingthe knapsack problem After that we show that a standarddynamic programming approach is able to quickly solveinstances to optimality that were usually out of reach for exactapproaches We close with some conclusions

2 A Nonlinear Formulation

Let us consider the RAP for the series-parallel configurationsystem where 119899 different subsystems are placed in seriesand within each system 119904

119894parallel components are available

as illustrated in Figure 1 The objective of the problem is todetermine which components and how many replications ofeach available component should be selected tomaximize theoverall system reliability The problem is complicated by theexistence of knapsack-type constraints typically describinglimitations in terms of volume weight and cost

A nonlinear integer formulation for the RAP is

RAP

max 119877 =

119899

prod

119894=1(1 minus

119904119894

prod

119896=1(1 minus 119877

119894119896)119910119894119896

)

st119899

sum

119894=1

119904119894

sum

119896=1119892119902

119894119896119910119894119896

le 119887119902 119902 = 1 119876

119910119894119896

isin N 119894 = 1 119899 119896 = 1 119904119894

(1)

where 119904119894denotes the number of parallel components within

subsystem 119894 119877119894119896

isin [0 1] is the reliability of component 119896

within subsystem 119894 with 119894 = 1 119899 and 119896 = 1 119904119894 119892119902119894119896

gt 0accounts for the usage of resource 119902 of component 119896 withinsubsystem 119894 (eg volume and cost) and 119887

119902isin R+provides

the maximum availability of resource 119902 with 119902 = 1 119876Within the set of decision variables each 119910

119894119896indicates how

many times component 119896 of subsystem 119894 is present in anoptimal configuration It is easy to see that in order to achievea nonnull system reliability the constraintsum

119896119910119894119896

ge 1must beimplicitly satisfied for each subsystem 119894

3 Transformation between the Discrete RAPand the Binary RAP

Let us indicate for any 119894 = 1 119899 and 119902 = 1 119876 with

119892119902

119894= min 119892

119902

119894119896 119896 = 1 119904

119894 (2)

the minimum amount of resource 119902 required to select at leastone component within each subsystem 119894 In addition let 119887

119902

indicate the total amount of resource 119902 available and

119887

119902

119894= 119887119902minus sum

119908 =119894

119892119902

119908 (3)

the maximum amount of resource 119902 that could be devoted tosubsystem 119894 while ensuring that all the other subsystems willhave enough spare resources to select at least one componentAn upper bound on the number of replications of eachcomponent 119896 within subsystem 119894 is given by

119906119894119896

= lfloormin119902

119887

119902

119894

119892119902

119894119896

rfloor (4)

Consequently possible encoding for the binary RAP isobtained by creating lceillog2(119906119894119896 + 1)rceil binary variables for eachcomponent within the system For the sake of readability letus focus on a single component 119896 belonging to subsystem 119894

and on a single resource 119902 Consequently let us omit indexes119894 and 119902 and let 119910

119896indicate the 119896th component of subsystem

119894 Finally let us indicate with 119899 the total number of binaryvariables needed to encode the discrete variable 119910

119896 that is

119899 = lceillog2 (119906119896 + 1)rceil (5)

The discrete variable 119910119896 with 0 le 119910

119896le 119906119896 is

substituted by a set of 119899 binary variables that is 119910119896

=

(1199091198961 119909

119896ℎ 119909

119896119899) Each of these 119899 binary ldquocomponentsrdquo

corresponds to 119899119896ℎreplications of component 119896 where

119899119896ℎ

=

2ℎminus1 if ℎ lt 119899

119906119894119896minus

119899minus1sum

ℎ=12ℎminus1 if ℎ = 119899

(6)

with ℎ = 1 119899 Consequently we have that

119909119896ℎ

=

1 if 119899119896ℎ

replications of 119896 are taken

0 otherwise(7)

Mathematical Problems in Engineering 3

Therefore coefficients 119899119896ℎare defined such that they sum

up to 119906119896 that is the maximum number of replications of

component 119896 of subsystem 119894 Thus

119910119896=

119899

sum

ℎ=1119899119896ℎ119909119896ℎ

(8)

can take any integer value between 0 and 119906119896

Finally each binary variable 119909119896ℎ

is characterized by thefollowing coefficients

119886119896ℎ

= 119899119896ℎ119892119896

119903119896ℎ

= 1 minus (1 minus 119877119896)119899119896ℎ

(9)

where 119877119896and 119892

119896indicate the reliability value and the amount

of resource 119902 used by component 119896 within subsystem 119894respectively

Each subsystem 119894 can thus be expressed through the useof a set of binary variables Let us now reintroduce subindex119894 Let us indicate with 119899

119894119896the number of binary variables used

to encode the discrete variable 119910119894119896 that is

119899119894119896

= lceillog2 (119906119894119896 + 1)rceil (10)

Each subsystem 119894 is defined by 119898119894

= sum119904119894

119896=1 119899119894119896 binaryvariables Let us call these variables in a progressive fashion1199091198941 119909119894119895 119909119894119898119894 In addition let us indicate the progres-

sive position of vector x where the block of binary variablescorresponding to a discrete decision variable 119910

119894119896begins with

119901119894119896

=

119896

sum

119908=1119899119894119908

(11)

with 119896 = 1 119904119894 Therefore the binary vector x

119894 that is the

set of binary variables used to describe subsystem 119894 is definedin the following way

x119894= (119909

1198941 1199091198941199011198941⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1199101198941

| 1199091198941199011198941+1 1199091198941199011198942⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1199101198942

| ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119910119894119896

|

119909119894119901119894119904119894minus1+1

119909119894119901119894119904119894⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119910119894119904119894

)

(12)

while vector x accounting for the whole system is defined asx = (x1 x119899)

Finally note that the RAP can be transformed into itscorresponding binary version in O(119898) where

119898 =

119899

sum

119894=1

119904119894

sum

119896=1lceillog2 (119906119894119896 + 1)rceil =

119899

sum

119894=1119898119894 (13)

It is easy to see that since 119899119894119896is the minimum amount

of binary variables required to encode the correspondingdiscrete variable any other encoding will introduce the sameamount of binary variables [11]

With this encoding we finally define theRAP in its binaryversion

RAP-B

max 119877 =

119899

prod

119894=1(1 minus

119898119894

prod

119895=1(1 minus 119903

119894119895)

119909119894119895)

st119899

sum

119894=1

119898119894

sum

119895=1119886119902

119894119895119909119894119895le 119887119902 119902 = 1 119876

119909119894119895isin B 119894 = 1 119899 119895 = 1 119898

119894

(14)

where each 119909119894119895 119886119902119894119895 and 119903

119894119895are defined as in (7) and (9)

In line with what is mentioned in [3] for the RAP modelRAP-B enables consideration of (i) multiple componentchoice for each subsystemmdashthat is each component canbe selected more than oncemdashand (ii) component mixingwithin a subsystemmdashthat is more than one component persubsystem can be selected

An Example Consider the first subsystem of the benchmarkinstances of [12] We present the encoding of variables11991011 11991014 accounting for the number of replications ofcomponents 1 4 within subsystem 1 In Table 1 we sum-marize the relevant data where1198771119896 119892

11119896 and 119892

2119894119896represent the

reliability cost and weight of component 119896within subsystem1 with 119896 = 1 4

Let us first consider component 119896 = 1 From [12]the maximum number of replications for each componentis equal to 4 that is 119906

119894119896= 4 Therefore we have 119899 =

lceillog2(4 + 1)rceil = 3 Consequently we encode the discretevariable 11991011 through the use of three binary variables that is11991011 = (11990911 11990912 11990913) Using (6) we compute the coefficientscorresponding to the number of replications associated witheach binary variable that is 11989911 = 1 11989912 = 2 and 11989913 = 1Therefore as in (7) each binary variable has the followingmeaning

11990911 =

1 if 1 replication of component 1 is taken

0 otherwise

11990912 =

1 if 2 replications of component 1 are taken

0 otherwise

11990913 =

1 if 1 replication of component 1 is taken

0 otherwise(15)

It is easy to see that every possible value of 11991011 can beexpressed through an appropriate choice of values of 11990911 11990912

4 Mathematical Problems in Engineering

Table 1 Input data of subsystem 1 of the instance from [12]

119894 = 1 119896 = 1 411991011 11991012 11991013 11991014

1198771119896 090 093 091 09511989211119896 1 1 2 2

11989221119896 3 4 2 5

and 11990913The coefficients of the binary variables are computedusing (9)

11990311 = 1 minus (1 minus 11987711)11989911

= 09

11990312 = 1 minus (1 minus 11987711)11989912

= 099

11990313 = 1 minus (1 minus 11987711)11989913

= 09

119886111 = 11989911119892

111 = 1 times 1 = 1

119886112 = 11989912119892

111 = 2 times 1 = 2

119886113 = 11989913119892

111 = 1 times 1 = 1

119886211 = 11989911119892

211 = 1 times 3 = 3

119886212 = 11989912119892

211 = 2 times 3 = 6

119886213 = 11989913119892

211 = 1 times 3 = 3

(16)

In a similar way we encode variables 11991012 11991013 and 11991014Consequently the whole set of discrete variables referring tosubsystem 1 is encoded using 12 binary variables as follows

x1 = (11990911 11990913⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

11991011

| 11990914 11990916⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

11991012

| 11990917 11990919⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

11991013

|

119909110 119909112⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

11991014

)

(17)

4 Utilizing the Transformation

Dynamic programming (DP) has been used for the RAPby for example the authors in [12ndash14] who present DPschemes for the single-constraint version of the RAP Theinherent difficulty of using DP in the multiconstraint case isrelated to the growth of the dimensions of the table for theDP return function More precisely both space complexityand computational complexity of the DP scheme grow withO(prod119876

119902=1119887119902) where 119876 indicates the number of knapsack-typeconstraints Consequently even with only a few constraintsthe use of a DP approach can be cumbersome though usingthe above transformation enables its use in settings thatseemed out of reach without it

In the next subsection we describe a DP algorithm forthe RAP followed by a subsection presenting and discussingnumerical results

41 A Dynamic Programming Approach We present astraightforward DP recursion for the RAP (note that this

recursion can be used in various settings eg as subroutinefor metaheuristics see [15])We first separate each subsystemand transform the original problem into a series of ldquoinde-pendentrdquo knapsack problems (similar to [16]) The objectivehere is to maximize the reliability of subsystem 119894 indicatedwith 119877

119894 The objective function of each subsystem can easily

be linearized and the reliability of single subsystem 119894 can berewritten as

1198771015840

119894=

119898119894

sum

119895=11199031015840

119894119895119909119894119895 (18)

where 1198771015840119894= minus ln(1minus119877

119894) and 119903

1015840

119894119895= minus ln(1minus 119903

119894119895) The problem of

finding the optimal allocation of redundant components forsubsystem 119894 with 119894 = 1 119899 can be written as follows

RAP (119894)

max119898119894

sum

119895=11199031015840

119894119895119909119894119895

st119898119894

sum

119895=1119886119902

119894119895119909119894119895le 119887119902

119902 = 1 119876

119909119894119895isin 0 1 119895 = 1 119898

119894

(19)

Problem RAP(119894) is equivalent to a standard multidimen-sional knapsack problem with 119876 constraints which can besolved in pseudopolynomial time using DP (O(119898

119894119887) where

119887 = prod119902119887119902) Let us indicate with 119891

119894(119896 d) the optimal

objective function value of the RAP(119894) when only the first 119896components are considered with 119896 = 1 119898

119894 and d119879 =

(1198891 119889119876) such that 119889119902

= 0 119887119902 A forward recursion

formula for this problem is

119891119894 (119896 d)

=

0 if d lt a119894119896

1199031015840

119894119896otherwise

for 119896 = 1

max 1199031015840119894119896+ 119891119894(119896 minus 1 d minus a

119894119896)

119891119894 (119896 minus 1 d) for 119896 = 2 119898

119894

(20)

where a119879119894119896

= (1198861119894119896 119886

119876

119894119896) indicates the resource consumption

of component 119896 with 119896 = 1 119898119894 Thus 119891

119894(d) = 119891

119894(119898119894 d)

indicates the optimal solution with respect to subsystem 119894

when d units of resources are used In order to solve the RAPlet us now define 119877(119904 d) as the optimal objective functionvalue of the RAP when the first 119904 subsystems are consideredwith 119904 = 1 119899 and d119879 = (1198891 119889119876) such that 119889

119902=

0 119887119902 A DP recursion for the ldquoclassicalrdquo series-parallel

RAP (ie 119896119894= 1) is the following

119877 (119904 d)

=

1198911 (d) for 119904 = 1

maxw=0d

119891119904 (w) times 119877 (119904 minus 1 d minus w) for 119904 = 2 119899

(21)

Mathematical Problems in Engineering 5

Obviously 119877(119899 b) is the optimal objective function valueof the original RAPThe overall computational complexity ofthe recursive algorithm is O(119899119887

2)

42 Numerical Results The DP algorithm described in theprevious subsection has been coded in C++ and compiledwith the GNU C++ compiler using the -O option Wepresent results for a benchmark set made up of 33 instancesoriginally proposed by [12] but also used by a number ofresearchers Best known results have been reported by [717] and consequently we will compare the results of theproposed algorithm with those reported by these authorsThe 33 variations of the RAP are series-parallel systems withtwo knapsack-type constraints representing weight and costand 14 different links That is the system characteristics are119899 = 14 119876 = 2 and 119898

119894either 3 or 4 depending on the

value of 119894 see [7 12] for details Table 2 presents numericalresults for these 33 benchmark instancesThe wall-clock timeis measured in seconds on a dual core Pentium 18GHz Linuxworkstation with 4Gb of RAM

InTable 2 the first columnprovides the instances numberand the second and third columns give the total weight119882 andcost119862 allowedWe solved all the instances to optimality usingthe DP scheme described above with the objective values andcomputational times as indicated in Table 2 Note that [7] alsofound these solutions however without being able to proveoptimality That is the benefit of our transformation lies inapplying a standard algorithm under this transformation toprove optimality of the instances within a short amount oftime

To make things more clear we should emphasize thatthere are elaborate and versatile exact algorithms availablethat solve these instances to optimality as well as variousheuristics and metaheuristics Most interesting seem twoavenues of research

The first is the fact that we have successfully applied thetransformation described in this paper in a subsequent paper[18] which not only is able to solve all benchmark instancesfrom [12] to optimality but for the first time also providesoptimal solutions to the extended benchmark instances givenin [3] that have not been solved to optimality before

The second is the observation that the benchmarkinstances from [12] are still an object of intensive investigationfor heuristics and metaheuristics For instance in a recentpaper [8] the authors provide a summary of various meta-heuristics (including variable neighborhood search tabusearch ant colony optimization a genetic algorithm andtheir own differential evolution algorithm see Tables 2 and 3of [8]) and none was able to solve all 33 benchmark instancesto optimality (and they did not try those from [3]) Whilethe computational times of our DP algorithm are higher thanthose of most of these metaheuristics it is a standard DPapproach which is easy to implement That is based on ourresults there should be no good reason to study heuristicsand metaheuristics for these instances just by themselvesunless the algorithms are applied to more general problemsor at least more difficult instances than the ones from[12]

Table 2 Results on 33 benchmark instances

Number 119882 119862 Objective Time DP1 191 130 0986811 176642 190 130 0986416 175563 189 130 0985922 177124 188 130 0985378 17225 187 130 0984688 16986 186 129 0984176 167287 185 130 0983505 16748 184 130 0982994 166929 183 129 0982256 1645210 182 130 0981518 1600811 181 129 0981027 1606812 180 128 098029 161413 179 126 0979505 1586414 178 125 09784 1533615 177 126 0977596 150616 176 124 097669 1491617 175 125 0975708 1526418 174 123 0974926 1462819 173 122 0973827 1466420 172 123 0973027 1418421 171 122 0971929 1413622 170 120 097076 1394423 169 121 0969291 138624 168 119 0968125 135625 167 118 0966335 133826 166 116 0965042 1328427 165 117 0963712 1318828 164 115 0962422 1305629 163 114 0960642 1267230 162 115 0959188 1273231 161 113 0958035 1231232 160 112 0955714 1237233 159 110 0954565 11808

5 Conclusion

In this paper we have proposed a reformulation of thereliability redundancy allocation problem based on a binarydiscretization That is we proposed a simple discrete-binarytransformation for reliability redundancy allocation Thediscretization may be characterized as known in other fieldsthough it was not applied to this particular class of problemsbefore While our aim was not to investigate any new typeof algorithm we strongly believe that future research onalgorithms for the redundancy allocation problemmay favor-ably utilize this transformation as we have shown by meansof a standard dynamic programming approach Note inpassing that we have applied this transformation in differentcontext to obtain numerical results that were out of reachto the RAP community for quite some time Moreover forfuture research it should be of interest to apply our simple

6 Mathematical Problems in Engineering

transformation on a wider scale for other types of reliabilityproblems wherever deemed practical

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Azaron H Katagiri K Kato and M Sakawa ldquoA multi-objective discrete reliability optimization problem for dissim-ilar-unit standby systemsrdquo OR Spectrum vol 29 no 2 pp 235ndash257 2007

[2] A Billionnet ldquoRedundancy allocation for series-parallel sys-tems using integer linear programmingrdquo IEEE Transactions onReliability vol 57 no 3 pp 507ndash516 2008

[3] DW Coit and A Konak ldquoMultiple weighted objectives heuris-tic for the redundancy allocation problemrdquo IEEE Transactionson Reliability vol 55 no 3 pp 551ndash558 2006

[4] S Kulturel-Konak A E Smith and D W Coit ldquoEfficientlysolving the redundancy allocation problem using tabu searchrdquoIIE Transactions vol 35 no 6 pp 515ndash526 2003

[5] J E Ramirez-Marquez D W Coit and A Konak ldquoRedun-dancy allocation for series-parallel systems using a max-minapproachrdquo IIE Transactions vol 36 no 9 pp 891ndash898 2004

[6] B Soylu and S K Ulusoy ldquoA preference ordered classificationfor a multi-objective maxmin redundancy allocation problemrdquoComputers ampOperations Research vol 38 no 12 pp 1855ndash18662011

[7] P-S You and T-C Chen ldquoAn efficient heuristic for series-parallel redundant reliability problemsrdquo Computers amp Opera-tions Research vol 32 no 8 pp 2117ndash2127 2005

[8] N Beji B Jarboui P Siarry and H Chabchoub ldquoA differentialevolution algorithm to solve redundancy allocation problemsrdquoInternational Transactions in Operational Research vol 19 no6 pp 809ndash824 2012

[9] G Kanagaraj S G Ponnambalam and N Jawahar ldquoA hybridcuckoo search and genetic algorithm for reliability-redundancyallocation problemsrdquo Computers amp Industrial Engineering vol66 no 4 pp 1115ndash1124 2013

[10] E Valian and E Valian ldquoA cuckoo search algorithm by Levyflights for solving reliability redundancy allocation problemsrdquoEngineering Optimization vol 45 no 11 pp 1273ndash1286 2013

[11] S Martello and P Toth Knapsack Problems Algorithms andComputer Implementations John Wiley amp Sons 1990

[12] Y Nakagawa and S Miyazaki ldquoSurrogate constraints algorithmfor reliability optimization problems with two constraintsrdquoIEEE Transactions on Reliability vol 30 no 2 pp 175ndash180 1981

[13] K Y K Ng and N G F Sancho ldquoA hybrid lsquodynamic program-mingdepth-first searchrsquo algorithm with an application toredundancy allocationrdquo IIE Transactions vol 33 no 12 pp1047ndash1058 2001

[14] A Yalaoui E Chatelet and C Chu ldquoA new dynamic pro-gramming method for reliability amp redundancy allocation in aparallel-series systemrdquo IEEE Transactions on Reliability vol 54no 2 pp 254ndash261 2005

[15] M Caserta and S Voszlig ldquoA corridor method based hybridalgorithm for redundancy allocationrdquo Journal of Heuristics2014

[16] R L Bulfin and C Y Liu ldquoOptimal allocation of redundantcomponents for large systemsrdquo IEEE Transactions on Reliabilityvol 34 no 3 pp 241ndash247 1985

[17] J Onishi S Kimura R J W James and Y Nakagawa ldquoSolvingthe redundancy allocation problem with a mix of componentsusing the improved surrogate constraint methodrdquo IEEE Trans-actions on Reliability vol 56 no 1 pp 94ndash101 2007

[18] M Caserta and S Voszlig ldquoAn exact algorithm for the reliabilityredundancy allocation problemrdquo European Journal of Opera-tional Research vol 244 no 1 pp 110ndash116 2015

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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