research article a new biobjective model to...

Research ArticleA New Biobjective Model to Optimize IntegratedRedundancy Allocation and Reliability-Centered MaintenanceProblems in a System Using Metaheuristics

Shima MohammadZadeh Dogahe1 and Seyed Jafar Sadjadi2

1Department of Industrial Engineering Islamic Azad University Science and Research Branch Hesarak Tehran 1477893855 Iran2Center of Excellence in Advanced Manufacturing and Optimization School of Industrial EngineeringIran University of Science and Technology Tehran Iran

Correspondence should be addressed to Shima MohammadZadeh Dogahe shima488gmailcom

Received 5 February 2015 Revised 27 May 2015 Accepted 17 June 2015

Academic Editor Babak Shotorban

Copyright copy 2015 S MohammadZadeh Dogahe and S J Sadjadi This is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited

A novel integrated model is proposed to optimize the redundancy allocation problem (RAP) and the reliability-centeredmaintenance (RCM) simultaneously A system of both repairable and nonrepairable components has been considered Inthis system electronic components are nonrepairable while mechanical components are mostly repairable For nonrepairablecomponents a redundancy allocation problem is dealt with to determine optimal redundancy strategy and number of redundantcomponents to be implemented in each subsystem In addition a maintenance scheduling problem is considered for repairablecomponents in order to identify the best maintenance policy and optimize system reliability Both active and cold standbyredundancy strategies have been taken into account for electronic components Also net present value of the secondary costincluding operational and maintenance costs has been calculated The problem is formulated as a biobjective mathematicalprogramming model aiming to reach a tradeoff between system reliability and cost Three metaheuristic algorithms are employedto solve the proposed model Nondominated Sorting Genetic Algorithm (NSGA-II) Multiobjective Particle Swarm Optimization(MOPSO) and Multiobjective Firefly Algorithm (MOFA) Several test problems are solved using the mentioned algorithms to testefficiency and effectiveness of the solution approaches and obtained results are analyzed

1 Introduction and Literature Review

In general reliability is defined as ability of a system to meetrequired performance standards under specified conditionsduring a determined time horizon It has a significanteffect on manufacturing cost companyrsquos fame productionefficiency and environment and so forthThere are twomainapproaches to enhance system reliability implementing aproper maintenance policy and using effective redundancystrategies Applying these approaches leads to increases insystem reliability along with increasing costs of other resour-cesThus reaching a tradeoff between system reliability costvolume weight and so forth is significantly important [1]

There are two types of maintenance policies correc-tive maintenance (CM) and preventive maintenance (PM)

In corrective maintenance the system is repaired or replacedafter failure However prescheduled periodic maintenanceactions are taken in preventivemaintenance It is obvious thatpreventive maintenance prevents major failures that imposehigh costs on the system In recent years many authorsconducted variety of research on preventive maintenancescheduling problems A mathematical model has been pro-posed by Goel and Gupta [2] for a multistate system withrepair and replacement policy Goel et al [3] presented a for-mulation for design production and maintenance planningto incorporate the reliability allocation problem at the designstageThen a simultaneous optimization framework has beenemployed to solve the proposed model Tsai et al [4] studiedthe preventive maintenance scheduling problem for a mul-ticomponent system by assuming three maintenance actions

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 396864 16 pageshttpdxdoiorg1011552015396864

2 Mathematical Problems in Engineering

mechanical service repair and replacement Periodic preven-tive maintenance actions have been considered in order tomaximize availability of the system Mohanta et al [5] haveemployed bothGAandhybridGASA techniques to optimizemaintenance scheduling for a power plant and compared theobtained results by the algorithms

Martorell et al [6] proposed a multiobjective model tooptimize the maintenance scheduling problem by integrat-ing human and material resources Reliability availabilitymaintainability and cost were considered as objectives ofthe model and the Genetic Algorithm was used to solvethe problem Another multiobjective maintenance modelfor a series-parallel system was investigated by Certa et al[7] They implemented periodic PM policy and consideredmaintenance cost andmakespan as objectives of the problemAn effective Pareto optimal frontier approach was appliedto solve the multiobjective problem Moghaddass et al [8]focused on finding an optimal tradeoff between design ofa repairable multistate system with binary components andits maintenance strategy Also they considered both activeand standby redundancy strategies Doostparast et al [9]developed a reliability-based approach to optimize preventivemaintenance scheduling in coherent systems A system iscalled coherent when its performance is related to all of thecomponents In other words for coherent systems each com-ponent is relevant and system structure function ismonotoneand nondecreasing [10] They studied periodic PM perfor-mance in three types of coherent systems and used a Simu-lated Annealing (SA) algorithm to solve the problem tryingto minimize total costs along with meeting the minimumpredetermined reliability level Several other studies dealingwith preventive maintenance scheduling problem have beendone by [11ndash13] during the last two decades

However selecting a proper maintenance policy is notall we can do to maximize system reliability Identifying andimplementing the best redundancy strategy is another wayto optimize system reliability One of the famous problemsin field of reliability optimization is redundancy allocationproblem (RAP) Redundant components are incorporatedinto the system to back up different parts of the systemand prevent system breakdown under different redundancystrategies There are two main redundancy strategies (1)active redundancy in which all redundant components areimplemented in a parallel structure together from time zeroand only one component is required to work at any giventime and (2) standby redundancy inwhich a sequential orderis determined for using the redundant components at compo-nent failure time Three variants of the standby redundancystrategy are called cold warm and hot Each strategy canbe implemented in a different part of a system RAPs areproved to beNP-hard byChern [14]Thereforemetaheuristicalgorithms have been widely used in the literature to solvesuch problems Literature of the redundancy allocation prob-lem (RAP) could be reviewed from several points of viewIn this paper literature of the problem has been accuratelyreviewed by assuming three main characteristics objectivesof the problem applied solution algorithms and consideredredundancy strategies

Coit [15] studied the redundancy optimization problemusing integer programming approach and logarithm functionto develop an equivalent formulation of the problem andobtained high quality solutions He assumed nonconstantcomponent hazard functions Erlang distribution for compo-nent time-to-failure imperfect switching and multiple com-ponent choices for each subsystem Cold standby redundancystrategy is considered for a nonrepairable series-parallelsystem Zhao and Liu [16] proposed a stochasticmodel for theredundancy optimization problem aiming atmaximizing sys-tem lifetime or system reliability by considering both activeand standby redundancy strategiesThey used stochastic sim-ulation Genetic Algorithm (GA) and Neural Network (NN)to develop a hybrid intelligent algorithm to solve the problemLiang and Smith [17] proposed an Ant Colony Optimiza-tion (ACO) Algorithm to solve the redundancy allocationproblem (RAP) for a series-parallel systemThe objective is tomaximize system reliability when active redundancy strategyhas been implemented in the system Restrictions are set onsystem cost and system weight in addition They found theACO algorithm to be very effective and efficient for solvingNP-hard reliability design problems because it brings GAflexibility robustness and ease of implementation along withimproving its random behavior Tavakkoli-Moghaddam et al[18] studied RAP for a series-parallel system by consideringboth active and standby redundancy strategies They formu-lated the problem as a nonlinear integer programmingmodeland used a Genetic Algorithm to solve the NP-hard problemand maximize system reliability Sadjadi and Soltani [19]proposed a heuristic and a hybrid GA for the RAP in a series-parallel system to maximize its reliability Parameters of theproposed hybrid GA are calibrated using Taguchirsquos robustdesign method to enhance efficiency and effectiveness of thealgorithm Solving numerical examples indicated that theproposed heuristic method is time-efficient and producescomparable solutions to the hybrid GA in terms of qualityKumar et al [20] studied amultiobjectivemultilevel RAP andproposed multiobjective hierarchical Genetic Algorithms tosolve two numerical examples They integrated the hierar-chical genotype encoding scheme with two multiobjectiveGenetic Algorithms Beji et al [21] proposed a hybrid meta-heuristic algorithm based on Particle Swarm Optimization(PSO) and local search for RAP in a series-parallel system andtried to maximize system reliability

A large number of studies on redundancy allocationproblem have been conducted after 2010 Among those whostudied multiobjective RAP (MORAP) Zio and Bazzo [22]used a level diagram analysis of Pareto solutions to assist thedecision maker in selecting hisher preferred system designin terms of reliability and availability Soylu and Ulusoy [23]applied UTADIS sorting procedure to categorize the Paretosolutions obtained by augmented epsilon constraint methodinto preference ordered classes They considered maximiza-tion of the minimum system reliability along with minimiza-tion of the overall system cost and weight as objectives Safari[24] studied a MORAP by considering system reliability andoverall system cost as objectives and both active and standbystrategies as candidate redundancy strategies He used aNon-dominated Sorting Genetic Algorithm (NSGA-II) to solve

Mathematical Problems in Engineering 3

the multiobjective RAP Khalili-Damghani and Amiri [25]applied three solutionmethods epsilon constraint multistartpartial bound enumeration algorithm and Data Envelop-ment Analysis (DEA) to optimize system reliability cost andweight in a RAP for a series-parallel system Chambari et al[26] studied a biobjective RAP trying to maximize systemreliability and minimize overall cost along with makingdecision about using active andor standby redundancystrategies for a system with nonrepairable components Theyproposed twometaheuristics NSGA-II andMOPSO to solvethe problem However Zoulfaghari et al [27] consideredsystem reliability and availability as objectives of a RAP withboth repairable and nonrepairable components and proposeda mixed integer nonlinearprogramming (MINLP) model forthe problem Cao et al [28] used a decomposition-basedexact approach to solve a multiobjective RAP of a mixedsystem trying to optimize system reliability cost and weightGarg and Sharma [29] studied a multiobjective RAP withnonstochastic uncertain parameters by considering systemreliability and cost as objectivesThey formulated the problemas a fuzzy multiobjective optimization problem

Firefly Algorithm as a new metaheuristic optimizationmethod was introduced in 2008 by Yang [30] dos SantosCoelho et al introduced a modified FA approach combinedwith chaotic sequences to optimize reliability-redundancyproblem [31] Many other authors proposed metaheuristicalgorithms for RAP Sadjadi and Soltani [32] developed aheuristic method and a honey bee mating algorithm to solvethe large-scale RAP Hsieh and Yeh [33] applied a penaltyguided bee colony algorithm for RAP in a series-parallelsystem Several other metaheuristic algorithms have beenproposed by other researchers in [34ndash38]

In this paper a novel mathematical model of a systemof repairable and nonrepairable components is formulatedThe model contains two objectives firstly it aims to select aproper redundancy strategy for nonrepairable part of the sys-tem and secondly it offers amaintenance policy for repairablepart of the system Minimizing net present value of totalcost and maximizing system reliability are objectives of theproblem In addition different types of redundancy strate-gies repair and replacement actions are considered in orderto model the problem as realistic as possible Other practicalconstraints such as available budget for purchasing redundantcomponents volume weight and maximum allowed failurerate in each inspection period are taken into account Dueto NP-hardness of the problem the authors tried to employmetaheuristic methods to solve proposed modelThree com-mon solution approaches called NSGA-II MOPSO andMOFA were selected based on the Vanoye and Parra clas-sification Ruiz-Vanoye and Dıaz-Parra [39] classified meta-heuristics into three groups metaheuristics based on genetransfer (like Genetic Algorithms) metaheuristics based oninteractions among individual insects (eg Ant ColonyHoney Bees and Firefly Algorithms) and metaheuristicsbased on biological aspects of alive beings (such as SimulatedAnnealing Tabu Search and Particle Swarm OptimizationAlgorithms)

Remainder of the paper is organized as follows In Sec-tion 2 mathematical formulation of the problem is proposed

followed by detailed explanation of objective functions andconstraints Three metaheuristic algorithms are presented inSection 3 to solve the proposed model A set of numericalexamples have been solved using the metaheuristics inSection 4 Then obtained results are indicated and computa-tional analysis is carried out Finally a summary of the paperand conclusions have been presented in Section 5

2 Problem Formulation

In this section a new integrated mathematical model isproposed for redundancy allocation and reliability-centeredmaintenance problems Objective of the reliability problemscould be one or a set of the following objectives maximizingsystem reliability andminimizing cost weight and volume ofthe system In this paper system reliability and costs includ-ing maintenance and operational costs are considered asobjectives

In most articles the system under study includes eitherrepairable or nonrepairable components However systemsusually consist of repairable and nonrepairable componentssimultaneously in real world [27] Generally components ofelectronic devices are not repairable and should be replacedby new ones after failure However components of mechan-ical systems are usually repairable and repairing or replacingthe broken component after failure brings the system back tothe normal condition

In this paper a system of electronic andmechanical com-ponents has been considered Figure 1 represents configura-tion of the system Two approaches are applied to achieve thehighest possible system reliability (1) maximizing reliabilityof each component by using a diverse set of high quality andreliable redundant parallelized components (heterogeneousredundancy) (2) choosing optimalmaintenance policies It isobvious that the first approach can be used for nonrepairableelectronic components and the second one is applied onrepairable mechanical components

Assumptions The following assumptions have been takeninto account in proposing mathematical formulation of theproblem under study

(i) The system is comprised of two subsystems in seriesmechanical components and electronic components(Figure 1) Mechanical components are repairablewhile electronic components are nonrepairable

(ii) Selecting optimal maintenance policy for mechanicalcomponents is considered in order to maximizesystem reliability

(iii) Selecting a proper redundancy strategy active orcold standby and determining the number of redun-dant components in the electronic section is takeninto account aimed at improving system reliabilitySelecting active redundancy strategy adds operationalcosts to the system cost while selecting cold standbyredundancy strategy threatens system performanceby imperfect switching


Electronic section Mechanical section

1 2

Ver 1

Ver 2

Ver 3

Ver 1

Ver 2

Ver 3

Ver 1

Ver 2

Ver 3

Ver S1 Ver S2 Ver Si

M

Figure 1 A system of mechanical and electronic subsystems

(iv) It is possible to use different type of components withdifferent initial and operational costs and failure ratesfor the electronic subsystem

(v) Since unstable market and economic conditions mayhave serious effects on results inflation rate and timevalue of money are considered in computations

(vi) Required resources such as financial resourceshuman resources volume and weight are knowndeterministically

(vii) Time-to-failure distributions of components are inde-pendent

(viii) A fixed amount of budget is available at time zero topurchase electronic components (initial cost)

(ix) Secondary cost of the system is calculated by takingoperational costs of the electronic subsystem andmaintenance costs of the mechanical subsystem intoaccount during the system running period (missiontime)

(x) The system mission time is finite(xi) Repair and replacement times and restoration times

are calculated as the system downtime cost(xii) In cold standby strategy redundant components do

not fail before their activation In addition failurerates in active redundancy strategy are larger thancold standby strategy because active redundant com-ponents are exposed to the operational stresses (ie1205821198941198951 ge 1205821198941198952)

(xiii) Failure detection mechanism and switching areimperfect

21 Mathematical Model The constraints of the problem canbe formulated as follows

2sum

119896=1

119878119894

sum

119895=1

119864

sum

119894=1119908119894119895sdot 119909119894119895119896le 119882 (1)

2sum

119896=1

119878119894

sum

119895=1

119864

sum

119894=1V119894119895sdot 119909119894119895119896le 119881 (2)

2sum

119896=1

119878119894

sum

119895=1119909119894119895119896le 119873max119894 forall119894 = 1 119864 (3)

119909119894119895119896le 119872119910

119894119896forall119894 = 1 119864 119895 = 1 119878

119894 119896 = 1 2 (4)

2sum

119896=1119910119894119896= 1 forall119894 = 1 119864 (5)

119871

sum

119897=1119909119898119903119897119905+

119878119903

sum

119906=1119909119903119903119906119905

le 1

forall119903 = 1 119872 119905 = 1 119898119879

(6)

119872

sum

119903=1(

119871

sum

119897=1ℎ119903119897sdot 119909119898119903119897119905+

119878119903

sum

119906=1ℎ119903119906sdot 119909119903119903119906119905) le 119867

119900

forall119905 = 1 119898119879

(7)

120582119903119905

= (120582119903119905minus1 + 120578119903) (1minus(

119871

sum

119897=1119909119898119903119897119905+

119878119903

sum

119906=1119909119903119903119906119905))

+

119871

sum

119897=1119909119898119903119897119905120582119903119897+

119878119903

sum

119906=1119909119903119903119906119905120582119903119906

forall119903 = 1 119872 119905 = 1 119898119879

(8)

120582119903119905le 120582max119903 forall119903 = 1 119872 119905 = 1 119898119879 (9)

119864

sum

119894=1

119878119894

sum

119895=1

2sum

119896=1119909119894119895119896119888119894119895le 1198610 (10)

119909119898119903119897119905isin 0 1

119909119903119903119906119905

isin 0 1

119910119894119896isin 0 1

119909119894119895119896isin 119885+

(11)


Constraints (1) and (2) ensure that total weight and volumeof the electronic components are lower than the maximumallowed amounts Constraint (3) represents the maximumallowed number of redundant components in each subsys-tem Constraints (4) and (5) ensure that each subsystemuses only one redundancy strategy Constraint (6) states thatcomponents of the mechanical section can be repaired orreplaced only at inspection points Constraint (7) ensures thatthe number of required operators to perform maintenanceactions does not exceed the number of available operatorsConstraint (8) calculates failure rates of mechanical compo-nents in each period Failure rate is increased by 120578

119903if no repair

or replacement action is taken on the component at period 119905and is changed into 120582

119903119897or 120582119903119906if the component is repaired or

replaced at period 119905 respectively Constraint (9) ensures thatfailure rates of mechanical components in each period do notexceed the maximum allowed amount Constraint (10) rep-resents the maximum available budget for purchasing elec-tronic components at the beginning of the mission Finallyconstraint (11) represents domain restrictions of the decisionvariables

22 Objective Functions The proposed model contains twoobjectives

(1) Maximizing system reliability in each period byselecting optimal redundancy strategy and mainte-nance policy

(2) Minimizing secondary cost of the system includingoperational costs of the electronic subsystem andmaintenance costs of the mechanical subsystem dur-ing the system mission

In the following of this section detailed explanations on theintroduced objectives have been proposed

221 System Reliability The system under study consistsof two electronic and mechanical subsystems connected inseries according to Figure 1Thus system reliability is equal to119877119878(119905) = 119877

119864(119905)times119877

119872(119905) where119877

119864(119905) and119877

119872(119905) are reliability of

electronic and mechanical subsystem respectively

Electronic Subsystem The electronic part consists of 119864 sub-systems connected in series Each subsystem can use eitheractive or cold standby redundancy strategy to maximize reli-ability of the subsystem As discussed previously electroniccomponents are nonrepairable and failed components canonly be replaced by redundant components until all redun-dant components are used Generally cold standby redun-dancy leads to higher system reliability in comparison withactive redundancy [15]

Let119867 subsystems use active redundancy and 119862 (119862 = 119864 minus

119867) subsystems use cold standby redundancyThus reliabilityof the electronic subsystem is calculated according to thefollowing equation

119877119864 (119905) = 119877Hot (119905) 119877Cold (119905) (12)

119877Hot(119905) is reliability of the subsystem under active redun-dancy strategy that is calculated by considering it as

a subsystem with a number of redundant components andcalculated as follows

119877Hot (119905) = prod119894isin119867

(1minus (1minus 119903119894119895 (119905))119909119894119895119896) (13)

where 119903119894119895(119905) is the reliability of component 119895 in the subsystem

119894 at period 119905 Also reliability of a system under cold standbyredundancy is calculated as (14)

119877Cold (119905)

= prod

119894isin119862

(119903119894119895 (119905) +

119909119894119895119896minus1

sum

119899=1int

119879

119900

120588119894 (119906) 119891

119899

119894119895(119906) 119903119894119895 (119879 minus 119906) 119889119906)

(14)

where 119891119899119894119895(119905) is the density function of 119899th failure for type

119895 of the subsystem 119894 Also 120588119894(119906) refers to switch reliability

Coit [15] calculated a lower bound for 119877Cold(119905) and proposedan approximation of the system reliability by consideringimperfect switching using the following equation

Cold (119905)

= prod

119894isin119862

(119903119894119895 (119905) + 120588119894 (119905)

119909119894119895119896minus1

sum

119899=1int

119879

119900

119891119899

119894119895(119906) 119903119894119895 (119879 minus 119906) 119889119906)

(15)

According to Coitrsquos approximation [26] and since mostelectronic devices have exponential failure distributions andsystem working time is equal to the sum of working timesof components time-to-failure distribution function of eachcomponent is Erlang (120582 119896) thus reliability of the electronicsubsystem is calculated as follows

119877119864 (119905) = prod

119894isin119867

(1minus119878119894

prod

119895=1(1minus 119890minus120582119894119895119896 sdot119905

119896119894119895minus1

sum

119897=0

(120582119894119895119896119905)119897

119897)

119909119894119895119896

)prod

119894isin119862

(1

minus

119878119894

prod

119895=1(1minus 119890minus120582119894119895119905(

119896119894119895minus1

sum

119897=0

(120582119894119895119896119905)119897

119897+ 120588119894 (119905)

119896119894119895119909119894119895119896minus1

sum

119897=119896119894119895

(120582119894119895119896119905)119897

119897)))

(16)

Mechanical Subsystem The mechanical subsystem includes119872 devices connected in series Therefore all devices shouldbe in the working condition to have an active subsystem dur-ing the mission time In order to prevent system shutdownscheduled maintenance actions should be taken Reliabilityof the mechanical subsystem is calculated by multiplyingreliability of all components as follows

119877119872 (119905) =

119872

prod

119903=1119901119903 (119905) (17)

where 119901119903(119905) indicates reliability of the component 119903 at period

119905 Each time period is divided into 119898 equal intervals forinspection Thus we have119898 times 119879 inspections during the timehorizon All components are inspected in each interval andone of the following maintenance actions is performed oneach component (1) inspection (2) preventive repair and (3)preventive replacement [9] Each action has a special effect on


reliability of the component with a different cost dependenton the amount of required resources Lifetime of themechan-ical devices has Weibull (120572 1120582) distribution function Thusreliability of the mechanical subsystem is calculated as fol-lows

119877119872 (119905) =

119872

prod

119903=1119890minus(120582119903119905119905)

120572

(18)

In order to calculate reliability of the system in each perioda few points should be considered (1) each componentdeteriorates after activation with an increase in its failurerate (2) failure rate of a component is reduced or fixed byperforming maintenance actions [3]

222 System Cost Total cost of the system (119862119879) could be

divided into two parts initial costs and secondary costsInitial costs (119862

119868) include purchasing costs of the electronic

devices while secondary costs (119862119878) refer to operational costs

of the electronic subsystem (119862119874119878) [40] plus inspection and

maintenance costs of the mechanical subsystem (119862119872119878) The

objective is to minimize secondary costs (119862119878= 119862119874

119878+ 119862119872

119878)

according to restrictions on the initial budget for purchasingcosts and other constraints

Initial Cost Initial cost is calculated based on the purchasecost and number of the components implemented in elec-tronic subsystem at the beginning of running the system

119862119868=

119864

sum

119894=1

119878119894

sum

119895=1

2sum

119896=1119909119894119895119896119888119894119895 (19)

Secondary Cost As mentioned before secondary cost issum of the operational costs of the electronic componentsand maintenance and inspection costs of the mechanicalcomponents Operational cost is calculated by multiplyingthe number of operating components by unit operationalcost ( 119888

119894119895) Number of operating components in each period is

determined according to the redundancy strategy used in thesystem If active redundancy is used operational cost is calcu-lated for all working components because all redundant com-ponents are operating from time zero in this strategy How-ever under cold standby redundancy operational cost is onlycalculated for the operating redundant component becauseone component is required to be operating in this strategy

It should be noted that in active redundancy expectedvalue for failure of each component in each period (120582

119894119895119896) is

deducted from the total amount Using compound interestrate operational costs of different periods are converted tothe present time and net present value of the total operationalcost is calculated using the following equation

119862119874

119878

=

119898119879

sum

119905=1

119864

sum

119894=1

119878119894

sum

119895=1

2sum

119896=11198881015840

119894119895((119909119894119895119896minus [(119905 minus 1) 120582119894119895119896]) 119910119894119896 + (1 minus 119910119894119896))

sdot (1+ 119868)119905

(20)

Maintenance costs of the mechanical components consist ofrepairreplacement costs plus system downtime cost due toperforming maintenance actions Repairing the componentadds a repair cost while replacing it adds a purchasing cost tothe secondary cost System downtime cost is calculated basedon the selected maintenance Present value of maintenancecosts is calculated using (21) and added to the secondary costs

119862119872

119878=

119898119879

sum

119905=1

119872

sum

119903=1(

119871

sum

119897=1(119888

1119903119897+ 119888

3119903119897) 119909119898119903119897119905

+

119878119903

sum

119906=1(119888

2119903119906+ 119888

4119903119906) 119909119903119903119906119905) (1+ 119868)119905

(21)

3 Solution Approaches

As mentioned in Introduction and Literature Review thesolution methods were selected based on the Ruiz-Vanoyeand Dıaz-Parra [39] metaheuristics classification In this sec-tion themechanisms of three algorithms NSGA-IIMOPSOandMOFA are explained then in Section 4 some numericalexamples based on the proposed model are generated andare solved using these methods Finally obtained results byalgorithms have been compared and analyzed

The proposed model for the problem contains two con-flicting objectives We try to make a tradeoff between theseobjectives to achieve a desired level of optimality for eachobjective One of the common approaches to solve multiob-jective problems is the weighted-sum method that convertsthe problem into a single objective problem by making aweighted linear combination (WLC) of objectives Althoughit is a very popular method due to its simplicity and easeof implementation it has some major disadvantages such asdetermining weight of each objective and lack of informationabout it Thus another method called Pareto set has beendeveloped Pareto setmethodproduces set of solutionswithinthe feasible region of the problem that dominate other feasiblesolutions The nondominated solution sets are called Paretooptimal solutions and other inferior solutions are calleddominated solutions The decision maker selects the final setof Pareto optimal solutions according to hisher preferencesand considered criteria In this paper three metaheuristicalgorithms have been employed to produce Pareto optimalsolutions

31 Nondominated Sorting Genetic Algorithm (NSGA-II) TheNSGA-II is developed for solving multiobjective problemsby adding two operators to the classic Genetic Algorithm(GA) to find Pareto optimal sets instead of finding theunique optimal solution [41]The additional operators are (1)ranking operator which assigns a rank to each member of thegeneration based on nondominated sorting and (2) diversityoperator which increases diversity of the produced solutionswith equal ranks In the following of this section procedureof the proposed NSGA-II has been presented

In our study the proposed chromosome has two partsThe first part represents the electronic section and the secondpart represents themechanical sectionThe electronic section


21212222111Cutting point1111122221221

Child 2Child 1Parent 2Parent 1

DCDADCBA


Figure 2 Crossover in Genetic Algorithm

Table 1 Electronic section matrix

119894119895

1 2 sdot sdot sdot 119878119894

Redundancy strategy1 2 0 sdot sdot sdot 1 12 1 1 sdot sdot sdot 2 2

sdot sdot sdot

119864 0 2 sdot sdot sdot 1 1

contains 119864 rows which refer to the number of electronicsubsystems and 119878

119894+1 columnswhere 119878

119894columns are allocated

to different types of components and the last column repre-sents the selected redundancy strategy If active redundancyis selected the related element in the last column is equal to 1while selecting cold standby redundancy makes this elementequal to 2 Element (119894 119895) indicates the number of imple-mented redundant components for component type 119895 of sub-system 119894 In other words if element (2 3) is equal to 1 it meansthat one redundant component is implemented for compo-nent type 3 in subsystem 2 (Table 1)

The proposed chromosome for the mechanical section isindicated by twomatrixes with119872 rows representingmechan-ical subsystems and 119898119879 columns representing inspectionintervals Elements of the firstmatrix take three values 0 1 or2 Number 0 indicates that no repair or replacement action isrequired for that subsystem Number 1 refers to replacementand number 2 refers to repair action The second matrixdetermines type of the replacement and repair actions

Crossover Operator The proposed matrix for the electronicsection is divided into two parts in order to performcrossover The first part is related to selecting a redundancystrategy for each subsystem that is the last column while thesecond part is related to determining the number of redun-dant components for components of each subsystem thatis (119894 119895) elements The chromosome is cut from a randomlyselected point to diversify the selected redundancy strategiesThen the resulting two slices from parent chromosomes areinterchanged This process changes the selected redundancystrategy for some subsystems For the second part of the chro-mosome a vertical cut is applied from a randomly selected

pointThen the resulting two slices are interchanged Figure 2indicates the applied crossover operation

Mutation Operator Mutation process for the first chromo-some that is electronic section is performed as explained inthe following First a subsystem is selected randomly Nextone of the allowed components is selected for the selectedsubsystem Then a random number between 1 and 119873max119894 isassigned to the selected subsystem and component Finallyredundancy strategy is selected randomly For themechanicalsection mutation process is started by selecting a subsystemand a period randomly Then selected maintenance policyand type of repairreplacement actions are changed

32 Particle Swarm Algorithm The Multiobjective ParticleSwarm Optimization (MOPSO) is a metaheuristic algorithmcapable of producing high quality nondominated Paretooptimal solutions with high diversities for multiobjectiveproblems The MOPSO is widely used by researchers due toits simplicity and successful performance in continuous opti-mization problemsThe idea of this algorithm is inspired by aswarm of birds looking for food [42] In this algorithm eachfeasible solution is indicated as a particle with known velocityand fitness value Particles move in the search space andobtained results are classified based on a fitness criterion atthe end of each time interval Particles are changed intoparticles with higher fitness values gradually based on thefollowing model

119894 (119905 + 1) = 119882times

119894 (119905) +11986211199031 (119894 minus 119894 (119905))

+11986221199032 (119892 minus 119894 (119905)) 119894 = 1 2 119898

119894 (119905 + 1) = 119894 (119905) + 119894 (119905 + 1) 119894 = 1 2 119898

(22)

where11988211986211198622 and 119905 represent inertia weight cognitive fac-

tor social factor and iteration number respectively Also 1199031

and 1199032are randomnumbers in [0 1] and

119894and

119894are velocity

and position vector of 119894th particles respectively Multiob-jective approach of this algorithm (MOPSO) was proposedby Coello Coello and Lechuga in 2002 [43] The improvedversion of this algorithm in which constraint-handlingmechanism and a mutation operator have been consideredwas presented in 2004 [44]


Define objective functions 1198911(119909) 119891

2(119909) 119891

119896(119909) where 119909 = (119909

1 1199092 119909

119889)119879

Generate initial population of fireflies 119909119894= (119894 = 1 119899)

Formulate light intensity 119868Define absorption coefficient 120574 randomization parameter 120572

0 120598119894 vector of random numbers and maximum iteration

While 119905 ltMaximum Iterationfor 119894 119895 = 1 119899 (all 119899 fireflies)

Evaluate approximations PF119894and PF

119895to the Pareto Front

if PF119895dominates PF

119894

Move firefly 119894 towards 119895 using (24)if pervious position doesnrsquot dominate new one

New position replaced with old oneend if

end ifend for

Update and pass non dominated solution to next generationUpdate 119905 = 119905 + 1End while

Algorithm 1

In MOPSO all objective functions are calculated andevaluated for each particle and the nondominated solutions(based onPareto set concept) found by the particles are storedin a repository The size of repository is limited and is set bydecision maker In addition the search space is divided intohypercubes with a fitness value Fitness value is inversely pro-portional to the number of particles it contains [45]Then theselection method is used to choose a hypercube and the bestnondominated position (called leader) and finally the veloc-ities and positions of the particles are updated This processis repeated for a certain number of iterations

33 Firefly Algorithm Firefly Algorithm (FA) is the lastmethod applied in this study It was introduced by Yang in2010 [30] as a new approach for solving complex and con-tinuous problems The algorithm is inspired by the flashingbehavior of fireflies to attract each other Considering the fol-lowing three rules Firefly Algorithm introduced by Yang

(1) All fireflies are unisex and attracting a firefly byanother one is regardless of its sex

(2) Attractiveness is proportional to the brightness andboth of these features will decrease with increasingdistance Less bright fireflies are always attracted tothe brighter one and if there is no brighter one themove will be randomly

(3) The brightness of fireflies is defined according to theobjective function (like fitness function in GeneticAlgorithm)

Variation of light intensity and formulation of the attractive-ness of the FA are two important issues Brightness of eachfirefly at place 119909 is defined as 119868(119909) prop 119891(119909) and attractiveness120573(119903) is defined with respect to distance of the firefly 119894 fromfirefly 119895 that is calculated as following equation

120573 (119903) = 1205730119890minus120574119903119898

(119898 ge 1) (23)

where 1205730is attractiveness at 119903 = 0 and 120574 is fixed light absorp-

tion coefficient Based on the Cartesian distance the distancebetween the firefly 119894 and firefly 119895 at 119909

119894and 119909

119895is obtained as

119903119894119895= 119909119894minus 119909119895 = radicsum

119889

119896=1(119909119894119896minus 119909119895119896)2 where 119909

119894119896is the 119896th

component of 119909119894in spatial coordinate Calculating distances

is not limited only to the Euclidean coordinate but given thetype of the problem it can be defined differently for exampleas time interval Moving of firefly 119894 towards more attractivefirefly 119895 is calculated according to

119909119894+1 = 119909119894 +1205730119890

minus1205741199032119894119895 (119909119895minus119909119894) + 1205720120598119894 (24)

The second term of (24) is movement because of attractionand the third term is the random movement where 120572

0is a

randomization parameter and 120598119894is vector of randomnumbers

with a Gaussian or uniform distribution Although the basicdesign of this method was for continuous problems recentstudies have shown that this approach is also very efficient indiscrete problems [46]

Single objective form of Firefly Algorithm (FA) wasdeveloped to Multiobjective Firefly Algorithm (MOFA) in2013 by Yang [47] In this approach nondominated solutionsare detected based on the objective function value The mainsteps of the MOFA can be summarized as shown in Algo-rithm 1

Since that variables of our problem are binary and integerand MOPSO algorithm and MOFA find solutions in contin-uous space round function is used to convert real numberto the integer and to change detected solutions to the binarystyle sigmoid function is applied

4 Numerical Examples

In this section assigning proper values to the parametersof algorithms and using comparison metrics for evaluatingsolutionmethods are discussed firstThen three different setsof test problems (small medium and large size) are tackled


Table 2 Range of the main parameters

NSGA-II MOPSO MOFAParameter Range Parameter Range Parameter RangePop size (119873pop) 20ndash100 Pop size (119873pop) 20ndash100 Pop size (119873pop) 10ndash100Max iteration number (Maxit) 50ndash200 Max iteration number (Maxit) 50ndash200 Max iteration number (Maxit) 50ndash200Cross rate (Cr) 05ndash09 Inertia weight (119882) 04ndash09 Randomization parameter (prop0) 01ndash09Mutation rate (Mr) 001ndash03 Cognitive factor (119862

1) 1-2 Fixed light absorption coefficient (120574) 1ndash3

Social factor (1198622) 1-2

Table 3 Optimum values of the algorithms parameters

NSGA-II MOPSO MOFAPop size 100 Pop size 100 Pop size 25Max iteration number 50 Max iteration number 50 Max iteration number 200Cross rate 09 119882 09 prop0 01Mutation rate 03 119862

11 120574 1

1198622

2

and solved using the chosen solution methods Finallyobtained results by each algorithm have been compared andthe obtained results for an example are explained

41 Setting Parameters of the Algorithms Setting propervalues for the control parameters of metaheuristic algorithmshas a significant effect on their desirable performance Well-tuned parameters empower the algorithms in producing bet-ter solutions within shorter computation times Thus settingproper values for control parameters is a critical task [48]A few additional techniques are applied to tune parametersOne of the common methods is response surface methodol-ogy (RSM) RSM is a mathematical and statistical techniquethat examines the relationship between one ormore responsevariables and the set of parameters (input variables) influenc-ing them Using this method the levels of parameters thatoptimize the response variables are identified In the first stepmain parameters and range of them are determined In thisstudy we used related literature to identify these parametersand their range Table 2 shows range of the main parametersof each algorithm

In the next step the response variables should be deter-mined Three performance metrics are chosen as responsevariables which are CPU time number of nondominatedsolutions (NNS) and Diversification Metric (DM) Thesemetrics were selected based on the two features the con-vergence speed and diversity of the detected solutionsMore details on the performance metrics can be found inSection 42

Central composite design (CCD) with 6 center points isapplied for the experiments Experiments are run by DesignExpert 9 According to the number of input variables andtype of the design different number of experiments should berun For instance in case of the four parameters and 6 centerpointsrsquo design 46 experiments are required After performingthe experiments analysis of variance (ANOVA) is appliedto fit an adequate model to the experimental data Last stepis setting goals for responses to generate optimal condition

10 20 30 40 5050

875

125

1625

200 Desirability

05

06

0607

07

2 2

2 2Prediction 0768474

Max

it

Npop

Figure 3 Counter plot for MOFA desirability versus Maxit and119873pop

(optimal level of the parameters) Here we aim to minimizeCPU time and maximize NNS and DM

Figure 3 shows counter plot of RSM results for MOFAapproach Desirability displays the amount of goals that havebeen met It can be concluded from Figure 3 that the highestdesirability is obtained when119873pop level is medium andMaxitis high RSM is applied for all threemethods and the optimumvalues for the algorithms parameters are presented in Table 3

Other parameters are set according to the literatures asfollows

MOPSO 1199031 1199032= rand[0 1] repository size = 500 and

MOFA 120598119894= [rand minus 12] 1205730 = 1

42 Performance Measures According to Deb et al [49]there are two main features that must be taken into accountin order to evaluate performance of metaheuristic algo-rithms in solving multiobjective optimization problems (1)convergence to the Pareto optimal set and (2) diversity of


Table 4 Test problems dimensions

Case Number of electroniccomponents 119894

Number of mechanicalcomponents 119903

Number of componenttypes in the electronic

subsystem 119895

Number of componenttypes in the mechanical

subsystem 119906Small size lowastrand[3 8] rand[3 7] rand[2 4] rand[2 4]Medium size rand[8 13] rand[7 11] rand[2 4] rand[2 4]Large size rand[13 18] rand[11 15] rand[2 4] rand[2 4]lowastThis function returns an integer number between lower and upper bounds randomly

Table 5 Mean and standard deviation of the metrics for the different sizes of the test problems

Problem size Time (seconds) NNS DM MSNSGA-II MOPSO MOFA NSGA-II MOPSO MOFA NSGA-II MOPSO MOFA NSGA-II MOPSO MOFA

Small 191(026)

078(023)

350(116)

229(1354)

123(345)

33(126)

5535(3022)

6069(2054)

2260(1172)

001493(00257)

007999(00542)

004230(00376)

Medium 259(021)

145(046)

699(205)

1285(560)

1190(380)

385(114)

3902(1732)

6579(2350)

2865(1271)

00238(00367)

01562(02004)

00482(00303)

Large 3149(691)

2235(936)

9125(2548)

1560(622)

1284(419)

411(149)

5135(2577)

9744(3093)

3435(1381)

000419(00048)

014625(00285)

007408(00418)

lowastNumbers in the parentheses are standard deviation

the produced Pareto optimal set Numerous metrics havebeen proposed for quantifying these features in the literatureYu and Gen [50] and Zitzler and Thiele [51] proposed somecriteria such as number of nondominated solutions (NNS)Error Ratio (ER) and Generational Distance (GD) for mea-suring accuracy of algorithms Higher values for NNS andlower values for GD and ER are preferred and show thatthe algorithm under analysis performed better in providinga set of Pareto optimal solutions In addition DiversificationMetric (DM) Spacing Metric (SM) and Maximum SpreadMetrics (MS) have been proposed to check ability of the algo-rithms in providing a diverse set of Pareto optimal solutionsIn this study we utilized DM and MS to compare algorithmsperformance which are calculated using

DM = [

119873

sum

119894=1max (1003817100381710038171003817119909119894 minus119910119894

1003817100381710038171003817)]

12

MS = [119872

sum

119898=1(|119873|

max119894=1

119891119894

119898minus|119873|

min119894=1

119891119894

119898)

2

]

12

(25)

where 119909119894minus 119910119894 is the Euclidean distance between of the

nondominated solution 119909119894and the nondominated solution 119910

119894

and 119873 denote number of the Pareto solutions [52] The MSpresents the distance between the boundary solutions in theobtained nondominated solutions (NDS)where119891119894

119898is the119898th

objective function value for 119894th Pareto solution

43 Numerical Examples In order to examine performanceof the algorithms three sets of test problems with differentsizes (small medium and large) are simulated The dimen-sions of these sets are shown in Table 4 The properties ofthe test problemswere generated using random functions Allthree algorithms are programmed in MATLAB and the testproblems have been solved on a PCwith 4GBRAM180GHz

CPU Using the algorithmrsquos parameters in Table 3 twenty dif-ferent problems in each set are solved by NSGA-II MOPSOandMOFA All problems are run for five times and the meanof the obtained values for the metrics is considered for algo-rithmrsquos performance comparison In each run 500 solutionsare generated (according to the number of population andmaximum iterations) that included infeasible dominatedand similar solutions Detected solutions by the algorithmsare checked for feasibility firstly and if all the constraintsare satisfied the generated solution is accepted otherwiseanother solution would be found This process has beencoded in MATLAB Mean and standard deviation values ofCPU time NNS DM and MS for each case are presented inTable 5

Four metrics are considered to evaluate algorithms per-formance (Table 5) These are CPU time NNS DM and MSThe lower values for time and the higher values for the restof the metrics are desirable The first rank of the CPU timemetric in all cases (small medium and large sized instances)belongs to MOPSO approach It means that MOPSO is moretime-efficient Second rank belongs to NSGA-II and MOFArepresents a very longCPU time in comparisonwith the othercompeting methods

It also can be concluded from Table 5 that the average ofNNS in the NSGA-II method is higher than other methodsThis means that the NSGA-II can generate more nondomi-nated solutions on average However MOPSO has providedclose values of this metric to NSGA-II in medium and largesized cases

Obtained values for the DM and MS indicate higherdiversity solutions of the produced nondominated solutionsby MOPSO in all three cases This means that the MOPSOmethod has a wider spread

Our model has two conflicting objectives reliability andcost As the reliability of the system goes higher its costincreases too We try to find an optimal tradeoff between


Table 6 Boundary values for reliability

Method Small size problems Medium size problems Large size problemsNSGA-II MOPSO MOFA NSGA-II MOPSO MOFA NSGA-II MOPSO MOFA

Mean 0991548 0945431 096181 0985369 0914826 0938127 099101 086809 090353Lower bound 0981583 0908162 0939269 0970327 0860268 0909683 098771 080074 086255Upper bound 0996518 0988152 0980975 0994157 0974125 0957834 099190 094698 093663

Table 7 Boundary values for cost



Table 8 Mean and standard deviation values for performance metrics

Algorithm NSGA-II MOPSO MOFAMean Std Dev Mean Std Dev Mean Std Dev

Time 274 012 149 006 684 049NNS 2040 519 1320 365 370 106DM 4496 1079 9146 1511 3017 1221MS 00068 00165 01436 00132 00508 00218

Table 9 Objectives boundary values for the sample

Method Reliability CostNSGA-II MOPSO MOFA NSGA-II MOPSO MOFA

Mean 0999421 087153 090719 384862 390048 428562Lower bound 0999825 0802065 0841807 3760333 3485712 4044209Upper bound 0999833 0977786 0951062 4027349 4488861 458562

these objectives To compare quality of solutions generatedby three approaches mean and lower and upper bounds ofobjectives are reported in Tables 6 and 7 It should be notedthat the average values of the parameters had been calculatedin each case

As it can be concluded from Tables 6 and 7 MOPSOmethod provides wider ranges of the objective functions butthe quality of the solutions which are detected by NSGA-II isbetter in other words solutions with higher system reliabilityand lower values of cost were generated by NSGA-II

In remainder of this section an example is consideredbased on the proposed model and the results are explainedTables 11 and 12 provide data of failure rate Weibull distri-bution parameters repair replacement and downtime costsand required human resources (number of operators) foreach maintenance action in the mechanical section Theelectronic section includes 14 subsystems connected in seriesinwhich each subsystem can have 3 or 4 types of componentsAlso the mechanical section includes 11 repairable andreplaceable subsystems Table 13 presents upper bounds forsome problem parameters

This example is solved using three approaches for tentimes and the mean and standard deviation values for

the performance metrics and objective functions boundaryare reported in Tables 8 and 9

As concluded before MOPSO method provides bettervalues for the metrics (time NNS DM and MS) and widerboundary values for the objectives but as Table 9 showsNSGA-II approach detects Pareto solutions lead to highersystem reliability and lower cost

The proposed model in this paper aims at finding propermaintenance policies and effective redundancy strategiesTable 10 represents one of the nondominated solutions thatare found by NSGA-II for the mentioned example Theselected maintenance policies and redundancy strategiesincluding number and type of the redundant components andthe implemented redundancy strategy in the electronic sec-tion along with the selected repair and replacement actionsin the mechanical section are shown in Table 10 The firstfour columns in the electronic section represent the numberof redundant components and the last column indicates theimplemented redundancy strategy in each subsystem whereletter A means active strategy and letter C indicates coldstrategy In the mechanical subsystem 119883119903 and 119883119898 indicatereplacement and repair action in each period respectively119883119903and 119883119898 are followed by a number which determines


Table 10 Selected maintenance actions and redundancy strategies by the NSGA-II


119894119895

119903119905

1 2 3 4 Strategy 1 2 3 4 5 6 7 8 9 10 11 12 13 14 151 0 0 1 0 A 1 00 00 00 1198831199031 00 1198831199032 00 1198831199032 00 00 1198831199032 1198831198981 00 1198831199031 00

2 1 1 0 0 C 2 00 00 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 1198831199032 1198831199031 1198831198982 00

3 1 0 0 0 A 3 00 1198831199031 00 1198831199031 00 00 1198831199031 1198831199032 00 00 1198831199032 1198831199032 1198831198982 1198831199032 00

4 0 1 0 0 C 4 00 00 00 1198831199032 00 00 1198831198982 00 00 1198831198982 00 1198831199032 1198831199032 1198831198982 00

5 0 1 0 0 A 5 00 1198831199031 00 00 1198831199032 00 00 1198831199031 00 00 1198831199031 00 1198831199032 1198831199031 00

6 0 1 0 1 A 6 00 1198831199032 00 1198831199032 1198831199032 00 00 1198831199032 00 00 1198831199031 00 1198831198982 1198831198982 00

7 0 0 1 0 A 7 00 1198831199032 00 00 1198831198982 00 00 1198831199032 00 00 1198831199031 00 1198831199032 1198831198982 1198831199032

8 0 1 0 0 A 8 00 1198831199032 00 00 1198831199032 1198831199032 00 00 1198831198982 00 00 1198831199032 00 1198831199032 00

9 1 0 1 0 A 9 00 1198831199032 1198831198982 1198831199032 00 00 1198831199032 00 00 1198831199031 00 1198831199031 00 1198831199032 00

10 1 0 0 0 C 10 00 1198831199032 00 1198831199032 00 00 1198831199031 00 00 1198831199031 00 1198831199032 1198831199031 1198831199032 00

11 0 0 1 0 C 11 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 1198831199031 1198831198982 00

12 2 0 0 1 A13 0 1 0 0 C14 1 0 0 0 A

3800 3850 3900 3950 4000 4050 410009997

09997

09997

09997

09998

09998

09998

09998

09998

09999

a

(a)

3600 3700 3800 3900 4000 4100 4200 4300 4400 450008

082

084

086

088

09

092

094

096

098

b

(b)

Figure 4 Reliability-cost Pareto solutions obtained by (a) NSGA-II and (b) MOPSO

the type of replacementrepair action For instance 1198831199032means replacement type two is required and element 00indicates that no repair or replacement action has been takenin that period Figure 4 shows Pareto solutions obtained byNSGA-II and MOPSO methods

5 Conclusion

In this paper a biobjective reliability model by integrat-ing redundancy allocation problem (RAP) and reliability-centered maintenance (RCM) problem for a system of non-repairable electronic components and repairable mechanicalcomponents has been proposed Objectives of the problemaremaximizing the system reliability andminimizing the sys-tem operational and maintenance costs In order to improvesystem reliability active and cold standby redundancy strate-gies and periodic maintenance actions are considered for

electronic section and mechanical section respectively Totalsystem cost includes initial costs for purchasing the requiredequipment and secondary costs such as operational costs ofthe electronic section and maintenance costs of the mechan-ical section Initial costs are taken into account by settinga budget constraint and operational costs are considered asthe second objective of the problem Three metaheuristicalgorithms NSGA-II MOPSO and MOFA are used to solvethe proposed model Different sets of test problems (smallmedium and large size) have been generated for evaluatingsolution methods Obtained results indicate that MOPSOalgorithm requires less time to produce Pareto optimal setswith high diversities according to the DM and MS NSGA-II outperforms MOPSO and MOFA in terms of generatingmore nondominated solutions (NNS) with better values forsystem reliability and cost Finally an example was solved andthe obtained results were explained


Table 11 Component data for electronic subsystem

(a)

119894Choice 1 (119895 = 1) Choice 2 (119895 = 2) Choice 3 (119895 = 3) Choice 4 (119895 = 4)

1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895

1 000532 0004 2 0000726 00012 1 000499 0004 2 000818 0007 32 000818 0007 3 0000619 00001 1 000431 0003 2 mdash mdash mdash3 00133 0012 3 00110 0011 3 00124 0011 3 000466 0004 24 000741 0006 2 00124 0012 3 000683 0006 2 mdash mdash mdash5 000619 0005 1 000413 0004 2 000818 0007 3 mdash mdash mdash6 000436 0003 3 000567 0005 3 000268 0002 2 0000408 00004 17 00105 001 3 000466 0004 2 000394 0003 2 mdash mdash mdash8 00105 001 3 000105 00006 1 00105 001 3 mdash mdash mdash9 000268 0002 2 0000101 000005 1 000408 0003 1 0000943 000005 110 00141 0013 3 000683 0006 2 000105 00005 1 mdash mdash mdash11 000394 0003 2 000355 0003 2 000314 0002 2 mdash mdash mdash12 000236 0001 1 000769 0007 2 00133 0012 3 00110 001 313 000215 0001 2 000536 0005 3 000665 0006 3 mdash mdash mdash14 00110 0001 3 000834 0003 1 000355 0003 2 000436 0004 3

(b)


119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

1 10 1 3 3 10 1 4 4 20 2 2 2 20 2 5 52 20 2 8 8 10 1 10 10 10 1 9 9 mdash mdash mdash mdash3 20 2 7 7 30 3 5 5 10 1 6 6 40 4 4 44 30 3 5 5 40 4 6 6 50 5 4 4 mdash mdash mdash mdash5 20 2 4 4 20 2 3 3 30 3 5 5 mdash mdash mdash mdash6 30 3 5 5 30 3 4 4 20 2 5 5 20 2 4 47 40 4 7 7 40 4 8 8 50 5 9 9 mdash mdash mdash mdash8 30 3 4 4 50 5 7 7 60 6 6 6 mdash mdash mdash mdash9 20 2 8 8 30 3 9 9 40 4 7 7 30 3 8 810 40 4 6 6 40 4 5 5 50 5 6 6 mdash mdash mdash mdash11 30 3 5 5 40 4 6 6 50 5 6 6 mdash mdash mdash mdash12 20 2 4 4 30 3 5 5 40 4 6 6 50 5 7 713 20 2 5 5 30 3 5 5 20 2 6 6 mdash mdash mdash mdash14 40 4 4 6 40 4 6 7 50 5 6 6 60 6 5 9

Nomenclature

Indices

119894 Electronic components 119894 isin 1 2 119864119903 Mechanical components 119903 isin 1 2 119872

119896 Redundancy strategy 1 for active and 2 for coldstandby 119896 isin 1 2

119895 Component type in the electronic subsystem119895 isin 1 2 119878

119894

119906 Component type in the mechanical subsystem119906 isin 1 2 119878

119903

119878119894 Number of available types for component 119894 inelectronic subsystem

119878119903 Number of available types for component 119903 inthe mechanical subsystem

119897 Type of maintenance activities performed oncomponent 119903 119897 isin 1 2 119871

119905 Time 119905 isin 1 2 119898119879

Parameters

119881119882 Maximum allowed weight and volume forsystem

119873max119894 Maximum allowed number of components insubsystem 119894

1198610 Initial available budget to purchase electroniccomponents

1198670 Available operators to perform maintenanceactivities in each period


Table 12 Component data for mechanical subsystem

Parameters 119903

1 2 3 4 5 6 7 8 9 10 11120582119903119905

mdash 00004 00004 00004 00004 00004 00004 00004 00004 00004167 00004167 00004167

120582119903119897

119897 = 1 000056 000056 000056 000056 000056 000056 000056 000056 00005834 00005834 00005834119897 = 2 000085 000085 000085 000085 000085 000085 000085 000085 00008784 00008784 00008784

120582119903119906

119906 = 1 00004 00004 00004 00004 00004 00004 00004 00004 00004167 00004167 00004167119906 = 2 00005 00005 00005 00005 00005 00005 00005 00005 00005167 00005167 00005167

120572119903

mdash 25 25 25 25 25 25 25 26 26 26 24

1198881

119903119897

119897 = 1 2 15 25 3 18 35 3 2 25 15 2119897 = 2 1 1 15 15 1 2 15 1 15 05 1

1198882

119903119906

119906 = 1 4 3 5 6 4 7 6 4 35 3 45119906 = 2 3 15 3 2 2 4 3 15 2 15 25

ℎ119903119897

119897 = 1 1 1 2 1 2 1 1 2 1 1 2119897 = 2 2 1 1 1 1 2 1 1 2 1 1

ℎ119903119906

119906 = 1 1 2 1 2 2 1 2 2 1 2 1119906 = 2 1 2 2 1 2 2 1 2 2 1 2

1198883

119903119897

119897 = 1 2 2 2 2 2 2 2 2 2 2 2119897 = 2 15 15 15 1 1 15 15 1 1 15 15

1198884

119903119906

119906 = 1 8 7 6 8 9 7 6 5 7 6 9119906 = 2 3 2 5 4 5 5 2 3 3 2 3

Table 13 Upper bound of parameters

Parameter 119879 119872 119873max 119894 120582max119903 1198610

119867119900

119882 119881 120578119903

119868 120588119894(119905)

Value 5 3 4 00009 1000 20 200 200 0001 003 099

119908119894119895 Weight of type 119895 of component 119894 in the electronic

subsystemV119894119895 Volume of type 119895 of component 119894 in the elec-

tronic subsystem119888119894119895 Purchasing cost for type 119895 of component 119894

1198881015840

119894119895 Operational cost for type 119895 of component 119894 in

active redundancy1198881119903119897 Cost of repair type 119897 for component 119903

1198882119903119906 Replacement cost for type 119906 of component 119903

1198883119903119897 Cost of system downtime due to performing

repair type 119897 on component 1199031198884119903119906 Cost of system downtime due to replacing for

type 119906 of component 119903ℎ119903119897 Number of operators required to perform repair

type 119897 on component 119903ℎ119903119906 Number of operators required to replace type 119906

of component 119903120582119894119895119896 Failure rate of type 119895 of component 119894 when

redundancy strategy 119896 is implemented120582119903119897 Failure rate of component 119903 after repair type 119897

120582119903119906 Failure rate of type119906 of component 119903 at time zero

and after each replacement120582max119903 Maximum allowed failure rate for eachmechan-

ical component in each period120578119903 Rate of increase in failure rate for mechanical

component 119903

119868 Compound interest rate based on time periods119879 System mission time119898 Number of inspections during each time unit

Decision Variables

119909119894119895119896 Number of components 119894 with type 119895 used in

redundancy strategy 119896119910119894119896 1 if redundancy strategy 119896 is selected for

subsystem 119894 otherwise 0119909119898119903119897119905 If repair type 119897 is performed on component 119903 inperiod 119905 equals 1 otherwise 0

119909119903119903119906119905 If type 119906 of component 119903 is replaced in period 119905equals 1 otherwise 0

120582119903119905 Failure rate of component 119903 at period 119905

Conflict of Interests

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

Acknowledgment

The authors are grateful to the anonymous reviewer for theprecious and constructive comments which led to significantimprovement in the paper


References

[1] M Gen and Y Yun ldquoSoft computing approach for reliabilityoptimization state-of-the-art surveyrdquo Reliability Engineering ampSystem Safety vol 91 no 9 pp 1008ndash1026 2006

[2] L R Goel and R Gupta ldquoMulti-standby systemwith repair andreplacement policyrdquo Microelectronics Reliability vol 23 no 5pp 805ndash808 1983

[3] H D Goel J Grievink and M P C Weijnen ldquoIntegrated opti-mal reliable design production and maintenance planning formultipurpose process plantsrdquo Computers and Chemical Engi-neering vol 27 no 11 pp 1543ndash1555 2003

[4] Y-T Tsai K-S Wang and L-C Tsai ldquoA study of availability-centered preventive maintenance for multi-component sys-temsrdquo Reliability Engineering and System Safety vol 84 no 3pp 261ndash270 2004

[5] D K Mohanta P K Sadhu and R Chakrabarti ldquoDeterministicand stochastic approach for safety and reliability optimizationof captive power plant maintenance scheduling using GASA-based hybrid techniques a comparison of resultsrdquo ReliabilityEngineering amp System Safety vol 92 no 2 pp 187ndash199 2007

[6] S Martorell M Villamizar S Carlos and A Sanchez ldquoMain-tenance modeling and optimization integrating human andmaterial resourcesrdquo Reliability Engineeringamp System Safety vol95 no 12 pp 1293ndash1299 2010

[7] A Certa G Galante T Lupo and G Passannanti ldquoDetermi-nation of Pareto frontier in multi-objective maintenance opti-mizationrdquo Reliability Engineering amp System Safety vol 96 no7 pp 861ndash867 2011

[8] RMoghaddass M J Zuo andM Pandey ldquoOptimal design andmaintenance of a repairable multi-state system with standbycomponentsrdquo Journal of Statistical Planning and Inference vol142 no 8 pp 2409ndash2420 2012

[9] M Doostparast F Kolahan and M Doostparast ldquoA reliability-based approach to optimize preventivemaintenance schedulingfor coherent systemsrdquo Reliability Engineering amp System Safetyvol 126 pp 98ndash106 2014

[10] F J Samaniego System Signatures and their Applications in Engi-neering Reliability vol 110 of International Series In OperationsResearchampManagement Science Springer New York NY USA2007

[11] M H Moradi and A Khandani ldquoEvaluation economic andreliability issues for an autonomous independent network ofdistributed energy resourcesrdquo International Journal of ElectricalPower and Energy Systems vol 56 pp 75ndash82 2014

[12] A Oyarbide-Zubillaga A Goti and A Sanchez ldquoPreventivemaintenance optimisation of multi-equipment manufacturingsystems by combining discrete event simulation and multi-objective evolutionary algorithmsrdquo Production PlanningampCon-trol vol 19 no 4 pp 342ndash355 2008

[13] P Hilber V Miranda M A Matos and L Bertling ldquoMultiob-jective optimization applied tomaintenance policy for electricalnetworksrdquo IEEE Transactions on Power Systems vol 22 no 4pp 1675ndash1682 2007

[14] M S Chern ldquoOn the computational complexity of reliabilityredundancy allocation in a series systemrdquo Operations ResearchLetters vol 11 no 5 pp 309ndash315 1992

[15] D W Coit ldquoCold-standby redundancy optimization for nonre-pairable systemsrdquo IIE Transactions vol 33 no 6 pp 471ndash4782001

[16] R Zhao and B Liu ldquoStochastic programming models for gen-eral redundancy-optimization problemsrdquo IEEE Transactions onReliability vol 52 no 2 pp 181ndash191 2003

[17] Y-C Liang and A E Smith ldquoAn ant colony optimizationalgorithm for the redundancy allocation problem (RAP)rdquo IEEETransactions on Reliability vol 53 no 3 pp 417ndash423 2004

[18] R Tavakkoli-Moghaddam J Safari and F Sassani ldquoReliabilityoptimization of series-parallel systems with a choice of redun-dancy strategies using a genetic algorithmrdquo Reliability Engineer-ing amp System Safety vol 93 no 4 pp 550ndash556 2008

[19] S J Sadjadi andR Soltani ldquoAn efficient heuristic versus a robusthybrid meta-heuristic for general framework of serialmdashparallelredundancy problemrdquoReliability Engineering and System Safetyvol 94 no 11 pp 1703ndash1710 2009

[20] R Kumar K Izui M Yoshimura and S Nishiwaki ldquoMulti-objective hierarchical genetic algorithms for multilevel redun-dancy allocation optimizationrdquoReliability Engineeringamp SystemSafety vol 94 no 4 pp 891ndash904 2009

[21] N Beji B Jarboui M Eddaly and H Chabchoub ldquoA hybridparticle swarm optimization algorithm for the redundancyallocation problemrdquo Journal of Computational Science vol 1 no3 pp 159ndash167 2010

[22] E Zio and R Bazzo ldquoLevel diagrams analysis of pareto frontfor multiobjective system redundancy allocationrdquo ReliabilityEngineering amp System Safety vol 96 no 5 pp 569ndash580 2011

[23] B Soylu and S K Ulusoy ldquoA preference ordered classificationfor a multi-objective maxndashmin redundancy allocation prob-lemrdquoComputersampOperations Research vol 38 no 12 pp 1855ndash1866 2011

[24] J Safari ldquoMulti-objective reliability optimization of series-parallel systems with a choice of redundancy strategiesrdquo Reli-ability Engineering and System Safety vol 108 pp 10ndash20 2012

[25] K Khalili-Damghani and M Amiri ldquoSolving binary-statemulti-objective reliability redundancy allocation series-parallelproblem using efficient epsilon-constraint multi-start partialbound enumeration algorithm and DEArdquo Reliability Engineer-ing and System Safety vol 103 pp 35ndash44 2012

[26] A Chambari S H A Rahmati A A Najafi and A KarimildquoA bi-objective model to optimize reliability and cost of systemwith a choice of redundancy strategiesrdquo Computersamp IndustrialEngineering vol 63 no 1 pp 109ndash119 2012

[27] H Zoulfaghari A Z Hamadani and M A Ardakan ldquoBi-objective redundancy allocation problem for a system withmixed repairable and non-repairable componentsrdquo ISA Trans-actions vol 53 no 1 pp 17ndash24 2014

[28] D Cao A Murat and R B Chinnam ldquoEfficient exact opti-mization of multi-objective redundancy allocation problems inseries-parallel systemsrdquo Reliability Engineering amp System Safetyvol 111 pp 154ndash163 2013

[29] H Garg and S P Sharma ldquoMulti-objective reliability-redun-dancy allocation problem using particle swarm optimizationrdquoComputers amp Industrial Engineering vol 64 no 1 pp 247ndash2552013

[30] X S Yang Nature-Inspired Metaheuristic Algorithm LuniverPress 2010

[31] L dos Santos Coelho D L de Andrade Bernert and V CMariani ldquoA chaotic firefly algorithm applied to reliability-redundancy optimizationrdquo in Proceedings of the IEEE Congressof Evolutionary Computation (CEC rsquo11) pp 517ndash521 NewOrleans La USA June 2011


[32] S J Sadjadi and R Soltani ldquoAlternative design redundancyallocation using an efficient heuristic and a honey bee matingalgorithmrdquo Expert Systems with Applications vol 39 no 1 pp990ndash999 2012

[33] T-J Hsieh and W-C Yeh ldquoPenalty guided bees search forredundancy allocation problems with a mix of components inseriesmdashparallel systemsrdquo Computers and Operations Researchvol 39 no 11 pp 2688ndash2704 2012

[34] L D Afonso V C Mariani and L D S Coelho ldquoModifiedimperialist competitive algorithm based on attraction andrepulsion concepts for reliability-redundancy optimizationrdquoExpert Systems with Applications vol 40 no 9 pp 3794ndash38022013

[35] L Wang and L-P Li ldquoA coevolutionary differential evolutionwith harmony search for reliability-redundancy optimizationrdquoExpert Systems with Applications vol 39 no 5 pp 5271ndash52782012

[36] D E Fyffe W W Hines and N K Lee ldquoSystem reliabilityallocation and a computational algorithmrdquo IEEE Transactionson Reliability vol 17 no 2 pp 64ndash69 1968

[37] 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

[38] M A Ardakan and A Z Hamadani ldquoReliability-redundancyallocation problem with cold-standby redundancy strategyrdquoSimulation Modelling Practice and Theory vol 42 pp 107ndash1182014

[39] J A Ruiz-Vanoye and O Dıaz-Parra ldquoSimilarities betweenmeta-heuristics algorithms and the science of liferdquo CentralEuropean Journal of Operations Research vol 19 no 4 pp 445ndash466 2011

[40] G Levitin L Xing and Y Dai ldquoCold vs hot standby missionoperation cost minimization for 1-out-of-N systemsrdquo EuropeanJournal of Operational Research vol 234 no 1 pp 155ndash162 2014

[41] N Srinivas and K Deb ldquoMuiltiobjective optimization usingnondominated sorting in genetic algorithmsrdquo EvolutionaryComputation vol 2 no 3 pp 221ndash248 1994

[42] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE Conference on Neural NetworksPiscataway NJ USA 1998

[43] C A Coello Coello andM S Lechuga ldquoMOPSO a proposal formultiple objective particle swarm optimizationrdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo02) pp1051ndash1056 Piscataway NJ USA May 2002

[44] C A Coello Coello G T Pulido andM S Lechuga ldquoHandlingmultiple objectives with particle swarm optimizationrdquo IEEETransactions on Evolutionary Computation vol 8 no 3 pp256ndash279 2004

[45] K E Parsopoulos and M N Vrahatis ldquoMultiobjective particleswarm optimization approachesrdquo inMulti-Objective Optimiza-tion in Computational Intelligence Theory and Practice chapter2 pp 20ndash42 University of New SouthWales Sydney Australia2008

[46] M K Sayadi A Hafezalkotob and S G Naini ldquoFirefly-inspiredalgorithm for discrete optimization problems an applicationto manufacturing cell formationrdquo Journal of ManufacturingSystems vol 32 no 1 pp 78ndash84 2013

[47] X-S Yang ldquoMultiobjective firefly algorithm for continuousoptimizationrdquo Engineering with Computers vol 29 no 2 pp175ndash184 2013

[48] A A Najafi S T A Niaki and M Shahsavar ldquoA parameter-tuned genetic algorithm for the resource investment problemwith discounted cash flows and generalized precedence rela-tionsrdquo Computers amp Operations Research vol 36 no 11 pp2994ndash3001 2009

[49] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multiobjective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002

[50] X Yu and M Gen in Introduction to Evolutionary Algorithmschapter 6 Springer London UK 2010

[51] E Zitzler and L Thiele ldquoMulti-objective optimization usingevolutionary algorithmsmdasha comparative case studyrdquo in ParallelProblem Solving from NaturemdashPPSN V vol 1498 of LectureNotes in Computer Science pp 292ndash301 Springer Berlin Ger-many 1998

[52] K Khalili-Damghani A-R Abtahi and M Tavana ldquoA newmulti-objective particle swarmoptimizationmethod for solvingreliability redundancy allocation problemsrdquo Reliability Engi-neering amp System Safety vol 111 pp 58ndash75 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


mechanical service repair and replacement Periodic preven-tive maintenance actions have been considered in order tomaximize availability of the system Mohanta et al [5] haveemployed bothGAandhybridGASA techniques to optimizemaintenance scheduling for a power plant and compared theobtained results by the algorithms

Martorell et al [6] proposed a multiobjective model tooptimize the maintenance scheduling problem by integrat-ing human and material resources Reliability availabilitymaintainability and cost were considered as objectives ofthe model and the Genetic Algorithm was used to solvethe problem Another multiobjective maintenance modelfor a series-parallel system was investigated by Certa et al[7] They implemented periodic PM policy and consideredmaintenance cost andmakespan as objectives of the problemAn effective Pareto optimal frontier approach was appliedto solve the multiobjective problem Moghaddass et al [8]focused on finding an optimal tradeoff between design ofa repairable multistate system with binary components andits maintenance strategy Also they considered both activeand standby redundancy strategies Doostparast et al [9]developed a reliability-based approach to optimize preventivemaintenance scheduling in coherent systems A system iscalled coherent when its performance is related to all of thecomponents In other words for coherent systems each com-ponent is relevant and system structure function ismonotoneand nondecreasing [10] They studied periodic PM perfor-mance in three types of coherent systems and used a Simu-lated Annealing (SA) algorithm to solve the problem tryingto minimize total costs along with meeting the minimumpredetermined reliability level Several other studies dealingwith preventive maintenance scheduling problem have beendone by [11ndash13] during the last two decades

However selecting a proper maintenance policy is notall we can do to maximize system reliability Identifying andimplementing the best redundancy strategy is another wayto optimize system reliability One of the famous problemsin field of reliability optimization is redundancy allocationproblem (RAP) Redundant components are incorporatedinto the system to back up different parts of the systemand prevent system breakdown under different redundancystrategies There are two main redundancy strategies (1)active redundancy in which all redundant components areimplemented in a parallel structure together from time zeroand only one component is required to work at any giventime and (2) standby redundancy inwhich a sequential orderis determined for using the redundant components at compo-nent failure time Three variants of the standby redundancystrategy are called cold warm and hot Each strategy canbe implemented in a different part of a system RAPs areproved to beNP-hard byChern [14]Thereforemetaheuristicalgorithms have been widely used in the literature to solvesuch problems Literature of the redundancy allocation prob-lem (RAP) could be reviewed from several points of viewIn this paper literature of the problem has been accuratelyreviewed by assuming three main characteristics objectivesof the problem applied solution algorithms and consideredredundancy strategies

Coit [15] studied the redundancy optimization problemusing integer programming approach and logarithm functionto develop an equivalent formulation of the problem andobtained high quality solutions He assumed nonconstantcomponent hazard functions Erlang distribution for compo-nent time-to-failure imperfect switching and multiple com-ponent choices for each subsystem Cold standby redundancystrategy is considered for a nonrepairable series-parallelsystem Zhao and Liu [16] proposed a stochasticmodel for theredundancy optimization problem aiming atmaximizing sys-tem lifetime or system reliability by considering both activeand standby redundancy strategiesThey used stochastic sim-ulation Genetic Algorithm (GA) and Neural Network (NN)to develop a hybrid intelligent algorithm to solve the problemLiang and Smith [17] proposed an Ant Colony Optimiza-tion (ACO) Algorithm to solve the redundancy allocationproblem (RAP) for a series-parallel systemThe objective is tomaximize system reliability when active redundancy strategyhas been implemented in the system Restrictions are set onsystem cost and system weight in addition They found theACO algorithm to be very effective and efficient for solvingNP-hard reliability design problems because it brings GAflexibility robustness and ease of implementation along withimproving its random behavior Tavakkoli-Moghaddam et al[18] studied RAP for a series-parallel system by consideringboth active and standby redundancy strategies They formu-lated the problem as a nonlinear integer programmingmodeland used a Genetic Algorithm to solve the NP-hard problemand maximize system reliability Sadjadi and Soltani [19]proposed a heuristic and a hybrid GA for the RAP in a series-parallel system to maximize its reliability Parameters of theproposed hybrid GA are calibrated using Taguchirsquos robustdesign method to enhance efficiency and effectiveness of thealgorithm Solving numerical examples indicated that theproposed heuristic method is time-efficient and producescomparable solutions to the hybrid GA in terms of qualityKumar et al [20] studied amultiobjectivemultilevel RAP andproposed multiobjective hierarchical Genetic Algorithms tosolve two numerical examples They integrated the hierar-chical genotype encoding scheme with two multiobjectiveGenetic Algorithms Beji et al [21] proposed a hybrid meta-heuristic algorithm based on Particle Swarm Optimization(PSO) and local search for RAP in a series-parallel system andtried to maximize system reliability

A large number of studies on redundancy allocationproblem have been conducted after 2010 Among those whostudied multiobjective RAP (MORAP) Zio and Bazzo [22]used a level diagram analysis of Pareto solutions to assist thedecision maker in selecting hisher preferred system designin terms of reliability and availability Soylu and Ulusoy [23]applied UTADIS sorting procedure to categorize the Paretosolutions obtained by augmented epsilon constraint methodinto preference ordered classes They considered maximiza-tion of the minimum system reliability along with minimiza-tion of the overall system cost and weight as objectives Safari[24] studied a MORAP by considering system reliability andoverall system cost as objectives and both active and standbystrategies as candidate redundancy strategies He used aNon-dominated Sorting Genetic Algorithm (NSGA-II) to solve

















1 2

Ver 1

Ver 2

Ver 3

Ver 1

Ver 2

Ver 3

Ver 1

Ver 2

Ver 3


M













2sum

119896=1

119878119894

sum

119895=1

119864

sum

119894=1119908119894119895sdot 119909119894119895119896le 119882 (1)

2sum

119896=1

119878119894

sum

119895=1

119864

sum

119894=1V119894119895sdot 119909119894119895119896le 119881 (2)

2sum

119896=1

119878119894

sum

119895=1119909119894119895119896le 119873max119894 forall119894 = 1 119864 (3)

119909119894119895119896le 119872119910

119894119896forall119894 = 1 119864 119895 = 1 119878

119894 119896 = 1 2 (4)

2sum

119896=1119910119894119896= 1 forall119894 = 1 119864 (5)

119871

sum

119897=1119909119898119903119897119905+

119878119903

sum

119906=1119909119903119903119906119905

le 1

forall119903 = 1 119872 119905 = 1 119898119879

(6)

119872

sum

119903=1(

119871

sum

119897=1ℎ119903119897sdot 119909119898119903119897119905+

119878119903

sum

119906=1ℎ119903119906sdot 119909119903119903119906119905) le 119867

119900

forall119905 = 1 119898119879

(7)

120582119903119905

= (120582119903119905minus1 + 120578119903) (1minus(

119871

sum

119897=1119909119898119903119897119905+

119878119903

sum

119906=1119909119903119903119906119905))

+

119871

sum

119897=1119909119898119903119897119905120582119903119897+

119878119903

sum

119906=1119909119903119903119906119905120582119903119906

forall119903 = 1 119872 119905 = 1 119898119879

(8)

120582119903119905le 120582max119903 forall119903 = 1 119872 119905 = 1 119898119879 (9)

119864

sum

119894=1

119878119894

sum

119895=1

2sum

119896=1119909119894119895119896119888119894119895le 1198610 (10)

119909119898119903119897119905isin 0 1

119909119903119903119906119905

isin 0 1

119910119894119896isin 0 1

119909119894119895119896isin 119885+

(11)



119903if no repair









119864(119905)times119877

119872(119905) where119877

119864(119905) and119877






119877119864 (119905) = 119877Hot (119905) 119877Cold (119905) (12)



119877Hot (119905) = prod119894isin119867

(1minus (1minus 119903119894119895 (119905))119909119894119895119896) (13)



119877Cold (119905)

= prod

119894isin119862

(119903119894119895 (119905) +

119909119894119895119896minus1

sum

119899=1int

119879

119900

120588119894 (119906) 119891

119899

119894119895(119906) 119903119894119895 (119879 minus 119906) 119889119906)

(14)




Cold (119905)

= prod

119894isin119862

(119903119894119895 (119905) + 120588119894 (119905)

119909119894119895119896minus1

sum

119899=1int

119879

119900

119891119899

119894119895(119906) 119903119894119895 (119879 minus 119906) 119889119906)

(15)


119877119864 (119905) = prod

119894isin119867

(1minus119878119894

prod

119895=1(1minus 119890minus120582119894119895119896 sdot119905

119896119894119895minus1

sum

119897=0

(120582119894119895119896119905)119897

119897)

119909119894119895119896

)prod

119894isin119862

(1

minus

119878119894

prod

119895=1(1minus 119890minus120582119894119895119905(

119896119894119895minus1

sum

119897=0

(120582119894119895119896119905)119897

119897+ 120588119894 (119905)

119896119894119895119909119894119895119896minus1

sum

119897=119896119894119895

(120582119894119895119896119905)119897

119897)))

(16)


119877119872 (119905) =

119872

prod

119903=1119901119903 (119905) (17)





119877119872 (119905) =

119872

prod

119903=1119890minus(120582119903119905119905)

120572

(18)









119878+ 119862119872

119878)



119862119868=

119864

sum

119894=1

119878119894

sum

119895=1

2sum

119896=1119909119894119895119896119888119894119895 (19)





119894119895119896) is


119862119874

119878

=

119898119879

sum

119905=1

119864

sum

119894=1

119878119894

sum

119895=1

2sum

119896=11198881015840


sdot (1+ 119868)119905

(20)


119862119872

119878=

119898119879

sum

119905=1

119872

sum

119903=1(

119871

sum

119897=1(119888

1119903119897+ 119888

3119903119897) 119909119898119903119897119905

+

119878119903

sum

119906=1(119888

2119903119906+ 119888

4119903119906) 119909119903119903119906119905) (1+ 119868)119905

(21)







21212222111Cutting point1111122221221


DCDADCBA




119894119895

1 2 sdot sdot sdot 119878119894


sdot sdot sdot











119894 (119905 + 1) = 119882times

119894 (119905) +11986211199031 (119894 minus 119894 (119905))

+11986221199032 (119892 minus 119894 (119905)) 119894 = 1 2 119898

119894 (119905 + 1) = 119894 (119905) + 119894 (119905 + 1) 119894 = 1 2 119898

(22)




119894and

119894are velocity




2(119909) 119891

119896(119909) where 119909 = (119909

1 1199092 119909

119889)119879








119894



end ifend for


Algorithm 1







120573 (119903) = 1205730119890minus120574119903119898

(119898 ge 1) (23)



119894and 119909


119903119894119895= 119909119894minus 119909119895 = radicsum

119889

119896=1(119909119894119896minus 119909119895119896)2 where 119909

119894119896is the 119896th



119909119894+1 = 119909119894 +1205730119890

minus1205741199032119894119895 (119909119895minus119909119894) + 1205720120598119894 (24)


0is a














11 120574 1

1198622

2





10 20 30 40 5050

875

125

1625

200 Desirability

05

06

0607

07

2 2


Max

it

Npop






MOFA 120598119894= [rand minus 12] 1205730 = 1







subsystem 119895





Small 191(026)

078(023)

350(116)

229(1354)

123(345)

33(126)

5535(3022)

6069(2054)

2260(1172)

001493(00257)

007999(00542)

004230(00376)

Medium 259(021)

145(046)

699(205)

1285(560)

1190(380)

385(114)

3902(1732)

6579(2350)

2865(1271)

00238(00367)

01562(02004)

00482(00303)

Large 3149(691)

2235(936)

9125(2548)

1560(622)

1284(419)

411(149)

5135(2577)

9744(3093)

3435(1381)

000419(00048)

014625(00285)

007408(00418)



DM = [

119873

sum

119894=1max (1003817100381710038171003817119909119894 minus119910119894

1003817100381710038171003817)]

12

MS = [119872

sum

119898=1(|119873|

max119894=1

119891119894

119898minus|119873|

min119894=1

119891119894

119898)

2

]

12

(25)



119894


119898is the119898th

















Time 274 012 149 006 684 049NNS 2040 519 1320 365 370 106DM 4496 1079 9146 1511 3017 1221MS 00068 00165 01436 00132 00508 00218














119894119895

119903119905

1 2 3 4 Strategy 1 2 3 4 5 6 7 8 9 10 11 12 13 14 151 0 0 1 0 A 1 00 00 00 1198831199031 00 1198831199032 00 1198831199032 00 00 1198831199032 1198831198981 00 1198831199031 00

2 1 1 0 0 C 2 00 00 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 1198831199032 1198831199031 1198831198982 00

3 1 0 0 0 A 3 00 1198831199031 00 1198831199031 00 00 1198831199031 1198831199032 00 00 1198831199032 1198831199032 1198831198982 1198831199032 00

4 0 1 0 0 C 4 00 00 00 1198831199032 00 00 1198831198982 00 00 1198831198982 00 1198831199032 1198831199032 1198831198982 00

5 0 1 0 0 A 5 00 1198831199031 00 00 1198831199032 00 00 1198831199031 00 00 1198831199031 00 1198831199032 1198831199031 00

6 0 1 0 1 A 6 00 1198831199032 00 1198831199032 1198831199032 00 00 1198831199032 00 00 1198831199031 00 1198831198982 1198831198982 00

7 0 0 1 0 A 7 00 1198831199032 00 00 1198831198982 00 00 1198831199032 00 00 1198831199031 00 1198831199032 1198831198982 1198831199032

8 0 1 0 0 A 8 00 1198831199032 00 00 1198831199032 1198831199032 00 00 1198831198982 00 00 1198831199032 00 1198831199032 00

9 1 0 1 0 A 9 00 1198831199032 1198831198982 1198831199032 00 00 1198831199032 00 00 1198831199031 00 1198831199031 00 1198831199032 00

10 1 0 0 0 C 10 00 1198831199032 00 1198831199032 00 00 1198831199031 00 00 1198831199031 00 1198831199032 1198831199031 1198831199032 00

11 0 0 1 0 C 11 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 1198831199031 1198831198982 00

12 2 0 0 1 A13 0 1 0 0 C14 1 0 0 0 A

3800 3850 3900 3950 4000 4050 410009997

09997

09997

09997

09998

09998

09998

09998

09998

09999

a

(a)

3600 3700 3800 3900 4000 4100 4200 4300 4400 450008

082

084

086

088

09

092

094

096

098

b

(b)



5 Conclusion





(a)


1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895


(b)


119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895


Nomenclature

Indices




119894


119903




119905 Time 119905 isin 1 2 119898119879

Parameters







Parameters 119903

1 2 3 4 5 6 7 8 9 10 11120582119903119905

mdash 00004 00004 00004 00004 00004 00004 00004 00004 00004167 00004167 00004167

120582119903119897

119897 = 1 000056 000056 000056 000056 000056 000056 000056 000056 00005834 00005834 00005834119897 = 2 000085 000085 000085 000085 000085 000085 000085 000085 00008784 00008784 00008784

120582119903119906

119906 = 1 00004 00004 00004 00004 00004 00004 00004 00004 00004167 00004167 00004167119906 = 2 00005 00005 00005 00005 00005 00005 00005 00005 00005167 00005167 00005167

120572119903

mdash 25 25 25 25 25 25 25 26 26 26 24

1198881

119903119897

119897 = 1 2 15 25 3 18 35 3 2 25 15 2119897 = 2 1 1 15 15 1 2 15 1 15 05 1

1198882

119903119906

119906 = 1 4 3 5 6 4 7 6 4 35 3 45119906 = 2 3 15 3 2 2 4 3 15 2 15 25

ℎ119903119897

119897 = 1 1 1 2 1 2 1 1 2 1 1 2119897 = 2 2 1 1 1 1 2 1 1 2 1 1

ℎ119903119906

119906 = 1 1 2 1 2 2 1 2 2 1 2 1119906 = 2 1 2 2 1 2 2 1 2 2 1 2

1198883

119903119897

119897 = 1 2 2 2 2 2 2 2 2 2 2 2119897 = 2 15 15 15 1 1 15 15 1 1 15 15

1198884

119903119906

119906 = 1 8 7 6 8 9 7 6 5 7 6 9119906 = 2 3 2 5 4 5 5 2 3 3 2 3



119867119900

119882 119881 120578119903

119868 120588119894(119905)

Value 5 3 4 00009 1000 20 200 200 0001 003 099




1198881015840













component 119903


Decision Variables








Acknowledgment



References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























1 2

Ver 1

Ver 2

Ver 3

Ver 1

Ver 2

Ver 3

Ver 1

Ver 2

Ver 3


M













2sum

119896=1

119878119894

sum

119895=1

119864

sum

119894=1119908119894119895sdot 119909119894119895119896le 119882 (1)

2sum

119896=1

119878119894

sum

119895=1

119864

sum

119894=1V119894119895sdot 119909119894119895119896le 119881 (2)

2sum

119896=1

119878119894

sum

119895=1119909119894119895119896le 119873max119894 forall119894 = 1 119864 (3)

119909119894119895119896le 119872119910

119894119896forall119894 = 1 119864 119895 = 1 119878

119894 119896 = 1 2 (4)

2sum

119896=1119910119894119896= 1 forall119894 = 1 119864 (5)

119871

sum

119897=1119909119898119903119897119905+

119878119903

sum

119906=1119909119903119903119906119905

le 1

forall119903 = 1 119872 119905 = 1 119898119879

(6)

119872

sum

119903=1(

119871

sum

119897=1ℎ119903119897sdot 119909119898119903119897119905+

119878119903

sum

119906=1ℎ119903119906sdot 119909119903119903119906119905) le 119867

119900

forall119905 = 1 119898119879

(7)

120582119903119905

= (120582119903119905minus1 + 120578119903) (1minus(

119871

sum

119897=1119909119898119903119897119905+

119878119903

sum

119906=1119909119903119903119906119905))

+

119871

sum

119897=1119909119898119903119897119905120582119903119897+

119878119903

sum

119906=1119909119903119903119906119905120582119903119906

forall119903 = 1 119872 119905 = 1 119898119879

(8)

120582119903119905le 120582max119903 forall119903 = 1 119872 119905 = 1 119898119879 (9)

119864

sum

119894=1

119878119894

sum

119895=1

2sum

119896=1119909119894119895119896119888119894119895le 1198610 (10)

119909119898119903119897119905isin 0 1

119909119903119903119906119905

isin 0 1

119910119894119896isin 0 1

119909119894119895119896isin 119885+

(11)



119903if no repair









119864(119905)times119877

119872(119905) where119877

119864(119905) and119877






119877119864 (119905) = 119877Hot (119905) 119877Cold (119905) (12)



119877Hot (119905) = prod119894isin119867

(1minus (1minus 119903119894119895 (119905))119909119894119895119896) (13)



119877Cold (119905)

= prod

119894isin119862

(119903119894119895 (119905) +

119909119894119895119896minus1

sum

119899=1int

119879

119900

120588119894 (119906) 119891

119899

119894119895(119906) 119903119894119895 (119879 minus 119906) 119889119906)

(14)




Cold (119905)

= prod

119894isin119862

(119903119894119895 (119905) + 120588119894 (119905)

119909119894119895119896minus1

sum

119899=1int

119879

119900

119891119899

119894119895(119906) 119903119894119895 (119879 minus 119906) 119889119906)

(15)


119877119864 (119905) = prod

119894isin119867

(1minus119878119894

prod

119895=1(1minus 119890minus120582119894119895119896 sdot119905

119896119894119895minus1

sum

119897=0

(120582119894119895119896119905)119897

119897)

119909119894119895119896

)prod

119894isin119862

(1

minus

119878119894

prod

119895=1(1minus 119890minus120582119894119895119905(

119896119894119895minus1

sum

119897=0

(120582119894119895119896119905)119897

119897+ 120588119894 (119905)

119896119894119895119909119894119895119896minus1

sum

119897=119896119894119895

(120582119894119895119896119905)119897

119897)))

(16)


119877119872 (119905) =

119872

prod

119903=1119901119903 (119905) (17)





119877119872 (119905) =

119872

prod

119903=1119890minus(120582119903119905119905)

120572

(18)









119878+ 119862119872

119878)



119862119868=

119864

sum

119894=1

119878119894

sum

119895=1

2sum

119896=1119909119894119895119896119888119894119895 (19)





119894119895119896) is


119862119874

119878

=

119898119879

sum

119905=1

119864

sum

119894=1

119878119894

sum

119895=1

2sum

119896=11198881015840


sdot (1+ 119868)119905

(20)


119862119872

119878=

119898119879

sum

119905=1

119872

sum

119903=1(

119871

sum

119897=1(119888

1119903119897+ 119888

3119903119897) 119909119898119903119897119905

+

119878119903

sum

119906=1(119888

2119903119906+ 119888

4119903119906) 119909119903119903119906119905) (1+ 119868)119905

(21)







21212222111Cutting point1111122221221


DCDADCBA




119894119895

1 2 sdot sdot sdot 119878119894


sdot sdot sdot











119894 (119905 + 1) = 119882times

119894 (119905) +11986211199031 (119894 minus 119894 (119905))

+11986221199032 (119892 minus 119894 (119905)) 119894 = 1 2 119898

119894 (119905 + 1) = 119894 (119905) + 119894 (119905 + 1) 119894 = 1 2 119898

(22)




119894and

119894are velocity




2(119909) 119891

119896(119909) where 119909 = (119909

1 1199092 119909

119889)119879








119894



end ifend for


Algorithm 1







120573 (119903) = 1205730119890minus120574119903119898

(119898 ge 1) (23)



119894and 119909


119903119894119895= 119909119894minus 119909119895 = radicsum

119889

119896=1(119909119894119896minus 119909119895119896)2 where 119909

119894119896is the 119896th



119909119894+1 = 119909119894 +1205730119890

minus1205741199032119894119895 (119909119895minus119909119894) + 1205720120598119894 (24)


0is a














11 120574 1

1198622

2





10 20 30 40 5050

875

125

1625

200 Desirability

05

06

0607

07

2 2


Max

it

Npop






MOFA 120598119894= [rand minus 12] 1205730 = 1







subsystem 119895





Small 191(026)

078(023)

350(116)

229(1354)

123(345)

33(126)

5535(3022)

6069(2054)

2260(1172)

001493(00257)

007999(00542)

004230(00376)

Medium 259(021)

145(046)

699(205)

1285(560)

1190(380)

385(114)

3902(1732)

6579(2350)

2865(1271)

00238(00367)

01562(02004)

00482(00303)

Large 3149(691)

2235(936)

9125(2548)

1560(622)

1284(419)

411(149)

5135(2577)

9744(3093)

3435(1381)

000419(00048)

014625(00285)

007408(00418)



DM = [

119873

sum

119894=1max (1003817100381710038171003817119909119894 minus119910119894

1003817100381710038171003817)]

12

MS = [119872

sum

119898=1(|119873|

max119894=1

119891119894

119898minus|119873|

min119894=1

119891119894

119898)

2

]

12

(25)



119894


119898is the119898th

















Time 274 012 149 006 684 049NNS 2040 519 1320 365 370 106DM 4496 1079 9146 1511 3017 1221MS 00068 00165 01436 00132 00508 00218














119894119895

119903119905

1 2 3 4 Strategy 1 2 3 4 5 6 7 8 9 10 11 12 13 14 151 0 0 1 0 A 1 00 00 00 1198831199031 00 1198831199032 00 1198831199032 00 00 1198831199032 1198831198981 00 1198831199031 00

2 1 1 0 0 C 2 00 00 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 1198831199032 1198831199031 1198831198982 00

3 1 0 0 0 A 3 00 1198831199031 00 1198831199031 00 00 1198831199031 1198831199032 00 00 1198831199032 1198831199032 1198831198982 1198831199032 00

4 0 1 0 0 C 4 00 00 00 1198831199032 00 00 1198831198982 00 00 1198831198982 00 1198831199032 1198831199032 1198831198982 00

5 0 1 0 0 A 5 00 1198831199031 00 00 1198831199032 00 00 1198831199031 00 00 1198831199031 00 1198831199032 1198831199031 00

6 0 1 0 1 A 6 00 1198831199032 00 1198831199032 1198831199032 00 00 1198831199032 00 00 1198831199031 00 1198831198982 1198831198982 00

7 0 0 1 0 A 7 00 1198831199032 00 00 1198831198982 00 00 1198831199032 00 00 1198831199031 00 1198831199032 1198831198982 1198831199032

8 0 1 0 0 A 8 00 1198831199032 00 00 1198831199032 1198831199032 00 00 1198831198982 00 00 1198831199032 00 1198831199032 00

9 1 0 1 0 A 9 00 1198831199032 1198831198982 1198831199032 00 00 1198831199032 00 00 1198831199031 00 1198831199031 00 1198831199032 00

10 1 0 0 0 C 10 00 1198831199032 00 1198831199032 00 00 1198831199031 00 00 1198831199031 00 1198831199032 1198831199031 1198831199032 00

11 0 0 1 0 C 11 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 1198831199031 1198831198982 00

12 2 0 0 1 A13 0 1 0 0 C14 1 0 0 0 A

3800 3850 3900 3950 4000 4050 410009997

09997

09997

09997

09998

09998

09998

09998

09998

09999

a

(a)

3600 3700 3800 3900 4000 4100 4200 4300 4400 450008

082

084

086

088

09

092

094

096

098

b

(b)



5 Conclusion





(a)


1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895


(b)


119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895


Nomenclature

Indices




119894


119903




119905 Time 119905 isin 1 2 119898119879

Parameters







Parameters 119903

1 2 3 4 5 6 7 8 9 10 11120582119903119905

mdash 00004 00004 00004 00004 00004 00004 00004 00004 00004167 00004167 00004167

120582119903119897

119897 = 1 000056 000056 000056 000056 000056 000056 000056 000056 00005834 00005834 00005834119897 = 2 000085 000085 000085 000085 000085 000085 000085 000085 00008784 00008784 00008784

120582119903119906

119906 = 1 00004 00004 00004 00004 00004 00004 00004 00004 00004167 00004167 00004167119906 = 2 00005 00005 00005 00005 00005 00005 00005 00005 00005167 00005167 00005167

120572119903

mdash 25 25 25 25 25 25 25 26 26 26 24

1198881

119903119897

119897 = 1 2 15 25 3 18 35 3 2 25 15 2119897 = 2 1 1 15 15 1 2 15 1 15 05 1

1198882

119903119906

119906 = 1 4 3 5 6 4 7 6 4 35 3 45119906 = 2 3 15 3 2 2 4 3 15 2 15 25

ℎ119903119897

119897 = 1 1 1 2 1 2 1 1 2 1 1 2119897 = 2 2 1 1 1 1 2 1 1 2 1 1

ℎ119903119906

119906 = 1 1 2 1 2 2 1 2 2 1 2 1119906 = 2 1 2 2 1 2 2 1 2 2 1 2

1198883

119903119897

119897 = 1 2 2 2 2 2 2 2 2 2 2 2119897 = 2 15 15 15 1 1 15 15 1 1 15 15

1198884

119903119906

119906 = 1 8 7 6 8 9 7 6 5 7 6 9119906 = 2 3 2 5 4 5 5 2 3 3 2 3



119867119900

119882 119881 120578119903

119868 120588119894(119905)

Value 5 3 4 00009 1000 20 200 200 0001 003 099




1198881015840













component 119903


Decision Variables








Acknowledgment



References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119903if no repair









119864(119905)times119877

119872(119905) where119877

119864(119905) and119877






119877119864 (119905) = 119877Hot (119905) 119877Cold (119905) (12)



119877Hot (119905) = prod119894isin119867

(1minus (1minus 119903119894119895 (119905))119909119894119895119896) (13)



119877Cold (119905)

= prod

119894isin119862

(119903119894119895 (119905) +

119909119894119895119896minus1

sum

119899=1int

119879

119900

120588119894 (119906) 119891

119899

119894119895(119906) 119903119894119895 (119879 minus 119906) 119889119906)

(14)




Cold (119905)

= prod

119894isin119862

(119903119894119895 (119905) + 120588119894 (119905)

119909119894119895119896minus1

sum

119899=1int

119879

119900

119891119899

119894119895(119906) 119903119894119895 (119879 minus 119906) 119889119906)

(15)


119877119864 (119905) = prod

119894isin119867

(1minus119878119894

prod

119895=1(1minus 119890minus120582119894119895119896 sdot119905

119896119894119895minus1

sum

119897=0

(120582119894119895119896119905)119897

119897)

119909119894119895119896

)prod

119894isin119862

(1

minus

119878119894

prod

119895=1(1minus 119890minus120582119894119895119905(

119896119894119895minus1

sum

119897=0

(120582119894119895119896119905)119897

119897+ 120588119894 (119905)

119896119894119895119909119894119895119896minus1

sum

119897=119896119894119895

(120582119894119895119896119905)119897

119897)))

(16)


119877119872 (119905) =

119872

prod

119903=1119901119903 (119905) (17)





119877119872 (119905) =

119872

prod

119903=1119890minus(120582119903119905119905)

120572

(18)









119878+ 119862119872

119878)



119862119868=

119864

sum

119894=1

119878119894

sum

119895=1

2sum

119896=1119909119894119895119896119888119894119895 (19)





119894119895119896) is


119862119874

119878

=

119898119879

sum

119905=1

119864

sum

119894=1

119878119894

sum

119895=1

2sum

119896=11198881015840


sdot (1+ 119868)119905

(20)


119862119872

119878=

119898119879

sum

119905=1

119872

sum

119903=1(

119871

sum

119897=1(119888

1119903119897+ 119888

3119903119897) 119909119898119903119897119905

+

119878119903

sum

119906=1(119888

2119903119906+ 119888

4119903119906) 119909119903119903119906119905) (1+ 119868)119905

(21)







21212222111Cutting point1111122221221


DCDADCBA




119894119895

1 2 sdot sdot sdot 119878119894


sdot sdot sdot











119894 (119905 + 1) = 119882times

119894 (119905) +11986211199031 (119894 minus 119894 (119905))

+11986221199032 (119892 minus 119894 (119905)) 119894 = 1 2 119898

119894 (119905 + 1) = 119894 (119905) + 119894 (119905 + 1) 119894 = 1 2 119898

(22)




119894and

119894are velocity




2(119909) 119891

119896(119909) where 119909 = (119909

1 1199092 119909

119889)119879








119894



end ifend for


Algorithm 1







120573 (119903) = 1205730119890minus120574119903119898

(119898 ge 1) (23)



119894and 119909


119903119894119895= 119909119894minus 119909119895 = radicsum

119889

119896=1(119909119894119896minus 119909119895119896)2 where 119909

119894119896is the 119896th



119909119894+1 = 119909119894 +1205730119890

minus1205741199032119894119895 (119909119895minus119909119894) + 1205720120598119894 (24)


0is a














11 120574 1

1198622

2





10 20 30 40 5050

875

125

1625

200 Desirability

05

06

0607

07

2 2


Max

it

Npop






MOFA 120598119894= [rand minus 12] 1205730 = 1







subsystem 119895





Small 191(026)

078(023)

350(116)

229(1354)

123(345)

33(126)

5535(3022)

6069(2054)

2260(1172)

001493(00257)

007999(00542)

004230(00376)

Medium 259(021)

145(046)

699(205)

1285(560)

1190(380)

385(114)

3902(1732)

6579(2350)

2865(1271)

00238(00367)

01562(02004)

00482(00303)

Large 3149(691)

2235(936)

9125(2548)

1560(622)

1284(419)

411(149)

5135(2577)

9744(3093)

3435(1381)

000419(00048)

014625(00285)

007408(00418)



DM = [

119873

sum

119894=1max (1003817100381710038171003817119909119894 minus119910119894

1003817100381710038171003817)]

12

MS = [119872

sum

119898=1(|119873|

max119894=1

119891119894

119898minus|119873|

min119894=1

119891119894

119898)

2

]

12

(25)



119894


119898is the119898th

















Time 274 012 149 006 684 049NNS 2040 519 1320 365 370 106DM 4496 1079 9146 1511 3017 1221MS 00068 00165 01436 00132 00508 00218














119894119895

119903119905

1 2 3 4 Strategy 1 2 3 4 5 6 7 8 9 10 11 12 13 14 151 0 0 1 0 A 1 00 00 00 1198831199031 00 1198831199032 00 1198831199032 00 00 1198831199032 1198831198981 00 1198831199031 00

2 1 1 0 0 C 2 00 00 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 1198831199032 1198831199031 1198831198982 00

3 1 0 0 0 A 3 00 1198831199031 00 1198831199031 00 00 1198831199031 1198831199032 00 00 1198831199032 1198831199032 1198831198982 1198831199032 00

4 0 1 0 0 C 4 00 00 00 1198831199032 00 00 1198831198982 00 00 1198831198982 00 1198831199032 1198831199032 1198831198982 00

5 0 1 0 0 A 5 00 1198831199031 00 00 1198831199032 00 00 1198831199031 00 00 1198831199031 00 1198831199032 1198831199031 00

6 0 1 0 1 A 6 00 1198831199032 00 1198831199032 1198831199032 00 00 1198831199032 00 00 1198831199031 00 1198831198982 1198831198982 00

7 0 0 1 0 A 7 00 1198831199032 00 00 1198831198982 00 00 1198831199032 00 00 1198831199031 00 1198831199032 1198831198982 1198831199032

8 0 1 0 0 A 8 00 1198831199032 00 00 1198831199032 1198831199032 00 00 1198831198982 00 00 1198831199032 00 1198831199032 00

9 1 0 1 0 A 9 00 1198831199032 1198831198982 1198831199032 00 00 1198831199032 00 00 1198831199031 00 1198831199031 00 1198831199032 00

10 1 0 0 0 C 10 00 1198831199032 00 1198831199032 00 00 1198831199031 00 00 1198831199031 00 1198831199032 1198831199031 1198831199032 00

11 0 0 1 0 C 11 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 1198831199031 1198831198982 00

12 2 0 0 1 A13 0 1 0 0 C14 1 0 0 0 A

3800 3850 3900 3950 4000 4050 410009997

09997

09997

09997

09998

09998

09998

09998

09998

09999

a

(a)

3600 3700 3800 3900 4000 4100 4200 4300 4400 450008

082

084

086

088

09

092

094

096

098

b

(b)



5 Conclusion





(a)


1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895


(b)


119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895


Nomenclature

Indices




119894


119903




119905 Time 119905 isin 1 2 119898119879

Parameters







Parameters 119903

1 2 3 4 5 6 7 8 9 10 11120582119903119905

mdash 00004 00004 00004 00004 00004 00004 00004 00004 00004167 00004167 00004167

120582119903119897

119897 = 1 000056 000056 000056 000056 000056 000056 000056 000056 00005834 00005834 00005834119897 = 2 000085 000085 000085 000085 000085 000085 000085 000085 00008784 00008784 00008784

120582119903119906

119906 = 1 00004 00004 00004 00004 00004 00004 00004 00004 00004167 00004167 00004167119906 = 2 00005 00005 00005 00005 00005 00005 00005 00005 00005167 00005167 00005167

120572119903

mdash 25 25 25 25 25 25 25 26 26 26 24

1198881

119903119897

119897 = 1 2 15 25 3 18 35 3 2 25 15 2119897 = 2 1 1 15 15 1 2 15 1 15 05 1

1198882

119903119906

119906 = 1 4 3 5 6 4 7 6 4 35 3 45119906 = 2 3 15 3 2 2 4 3 15 2 15 25

ℎ119903119897

119897 = 1 1 1 2 1 2 1 1 2 1 1 2119897 = 2 2 1 1 1 1 2 1 1 2 1 1

ℎ119903119906

119906 = 1 1 2 1 2 2 1 2 2 1 2 1119906 = 2 1 2 2 1 2 2 1 2 2 1 2

1198883

119903119897

119897 = 1 2 2 2 2 2 2 2 2 2 2 2119897 = 2 15 15 15 1 1 15 15 1 1 15 15

1198884

119903119906

119906 = 1 8 7 6 8 9 7 6 5 7 6 9119906 = 2 3 2 5 4 5 5 2 3 3 2 3



119867119900

119882 119881 120578119903

119868 120588119894(119905)

Value 5 3 4 00009 1000 20 200 200 0001 003 099




1198881015840













component 119903


Decision Variables








Acknowledgment



References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119877119872 (119905) =

119872

prod

119903=1119890minus(120582119903119905119905)

120572

(18)









119878+ 119862119872

119878)



119862119868=

119864

sum

119894=1

119878119894

sum

119895=1

2sum

119896=1119909119894119895119896119888119894119895 (19)





119894119895119896) is


119862119874

119878

=

119898119879

sum

119905=1

119864

sum

119894=1

119878119894

sum

119895=1

2sum

119896=11198881015840


sdot (1+ 119868)119905

(20)


119862119872

119878=

119898119879

sum

119905=1

119872

sum

119903=1(

119871

sum

119897=1(119888

1119903119897+ 119888

3119903119897) 119909119898119903119897119905

+

119878119903

sum

119906=1(119888

2119903119906+ 119888

4119903119906) 119909119903119903119906119905) (1+ 119868)119905

(21)







21212222111Cutting point1111122221221


DCDADCBA




119894119895

1 2 sdot sdot sdot 119878119894


sdot sdot sdot











119894 (119905 + 1) = 119882times

119894 (119905) +11986211199031 (119894 minus 119894 (119905))

+11986221199032 (119892 minus 119894 (119905)) 119894 = 1 2 119898

119894 (119905 + 1) = 119894 (119905) + 119894 (119905 + 1) 119894 = 1 2 119898

(22)




119894and

119894are velocity




2(119909) 119891

119896(119909) where 119909 = (119909

1 1199092 119909

119889)119879








119894



end ifend for


Algorithm 1







120573 (119903) = 1205730119890minus120574119903119898

(119898 ge 1) (23)



119894and 119909


119903119894119895= 119909119894minus 119909119895 = radicsum

119889

119896=1(119909119894119896minus 119909119895119896)2 where 119909

119894119896is the 119896th



119909119894+1 = 119909119894 +1205730119890

minus1205741199032119894119895 (119909119895minus119909119894) + 1205720120598119894 (24)


0is a














11 120574 1

1198622

2





10 20 30 40 5050

875

125

1625

200 Desirability

05

06

0607

07

2 2


Max

it

Npop






MOFA 120598119894= [rand minus 12] 1205730 = 1







subsystem 119895





Small 191(026)

078(023)

350(116)

229(1354)

123(345)

33(126)

5535(3022)

6069(2054)

2260(1172)

001493(00257)

007999(00542)

004230(00376)

Medium 259(021)

145(046)

699(205)

1285(560)

1190(380)

385(114)

3902(1732)

6579(2350)

2865(1271)

00238(00367)

01562(02004)

00482(00303)

Large 3149(691)

2235(936)

9125(2548)

1560(622)

1284(419)

411(149)

5135(2577)

9744(3093)

3435(1381)

000419(00048)

014625(00285)

007408(00418)



DM = [

119873

sum

119894=1max (1003817100381710038171003817119909119894 minus119910119894

1003817100381710038171003817)]

12

MS = [119872

sum

119898=1(|119873|

max119894=1

119891119894

119898minus|119873|

min119894=1

119891119894

119898)

2

]

12

(25)



119894


119898is the119898th

















Time 274 012 149 006 684 049NNS 2040 519 1320 365 370 106DM 4496 1079 9146 1511 3017 1221MS 00068 00165 01436 00132 00508 00218














119894119895

119903119905

1 2 3 4 Strategy 1 2 3 4 5 6 7 8 9 10 11 12 13 14 151 0 0 1 0 A 1 00 00 00 1198831199031 00 1198831199032 00 1198831199032 00 00 1198831199032 1198831198981 00 1198831199031 00

2 1 1 0 0 C 2 00 00 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 1198831199032 1198831199031 1198831198982 00

3 1 0 0 0 A 3 00 1198831199031 00 1198831199031 00 00 1198831199031 1198831199032 00 00 1198831199032 1198831199032 1198831198982 1198831199032 00

4 0 1 0 0 C 4 00 00 00 1198831199032 00 00 1198831198982 00 00 1198831198982 00 1198831199032 1198831199032 1198831198982 00

5 0 1 0 0 A 5 00 1198831199031 00 00 1198831199032 00 00 1198831199031 00 00 1198831199031 00 1198831199032 1198831199031 00

6 0 1 0 1 A 6 00 1198831199032 00 1198831199032 1198831199032 00 00 1198831199032 00 00 1198831199031 00 1198831198982 1198831198982 00

7 0 0 1 0 A 7 00 1198831199032 00 00 1198831198982 00 00 1198831199032 00 00 1198831199031 00 1198831199032 1198831198982 1198831199032

8 0 1 0 0 A 8 00 1198831199032 00 00 1198831199032 1198831199032 00 00 1198831198982 00 00 1198831199032 00 1198831199032 00

9 1 0 1 0 A 9 00 1198831199032 1198831198982 1198831199032 00 00 1198831199032 00 00 1198831199031 00 1198831199031 00 1198831199032 00

10 1 0 0 0 C 10 00 1198831199032 00 1198831199032 00 00 1198831199031 00 00 1198831199031 00 1198831199032 1198831199031 1198831199032 00

11 0 0 1 0 C 11 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 1198831199031 1198831198982 00

12 2 0 0 1 A13 0 1 0 0 C14 1 0 0 0 A

3800 3850 3900 3950 4000 4050 410009997

09997

09997

09997

09998

09998

09998

09998

09998

09999

a

(a)

3600 3700 3800 3900 4000 4100 4200 4300 4400 450008

082

084

086

088

09

092

094

096

098

b

(b)



5 Conclusion





(a)


1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895


(b)


119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895


Nomenclature

Indices




119894


119903




119905 Time 119905 isin 1 2 119898119879

Parameters







Parameters 119903

1 2 3 4 5 6 7 8 9 10 11120582119903119905

mdash 00004 00004 00004 00004 00004 00004 00004 00004 00004167 00004167 00004167

120582119903119897

119897 = 1 000056 000056 000056 000056 000056 000056 000056 000056 00005834 00005834 00005834119897 = 2 000085 000085 000085 000085 000085 000085 000085 000085 00008784 00008784 00008784

120582119903119906

119906 = 1 00004 00004 00004 00004 00004 00004 00004 00004 00004167 00004167 00004167119906 = 2 00005 00005 00005 00005 00005 00005 00005 00005 00005167 00005167 00005167

120572119903

mdash 25 25 25 25 25 25 25 26 26 26 24

1198881

119903119897

119897 = 1 2 15 25 3 18 35 3 2 25 15 2119897 = 2 1 1 15 15 1 2 15 1 15 05 1

1198882

119903119906

119906 = 1 4 3 5 6 4 7 6 4 35 3 45119906 = 2 3 15 3 2 2 4 3 15 2 15 25

ℎ119903119897

119897 = 1 1 1 2 1 2 1 1 2 1 1 2119897 = 2 2 1 1 1 1 2 1 1 2 1 1

ℎ119903119906

119906 = 1 1 2 1 2 2 1 2 2 1 2 1119906 = 2 1 2 2 1 2 2 1 2 2 1 2

1198883

119903119897

119897 = 1 2 2 2 2 2 2 2 2 2 2 2119897 = 2 15 15 15 1 1 15 15 1 1 15 15

1198884

119903119906

119906 = 1 8 7 6 8 9 7 6 5 7 6 9119906 = 2 3 2 5 4 5 5 2 3 3 2 3



119867119900

119882 119881 120578119903

119868 120588119894(119905)

Value 5 3 4 00009 1000 20 200 200 0001 003 099




1198881015840













component 119903


Decision Variables








Acknowledgment



References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










21212222111Cutting point1111122221221


DCDADCBA




119894119895

1 2 sdot sdot sdot 119878119894


sdot sdot sdot











119894 (119905 + 1) = 119882times

119894 (119905) +11986211199031 (119894 minus 119894 (119905))

+11986221199032 (119892 minus 119894 (119905)) 119894 = 1 2 119898

119894 (119905 + 1) = 119894 (119905) + 119894 (119905 + 1) 119894 = 1 2 119898

(22)




119894and

119894are velocity




2(119909) 119891

119896(119909) where 119909 = (119909

1 1199092 119909

119889)119879








119894



end ifend for


Algorithm 1







120573 (119903) = 1205730119890minus120574119903119898

(119898 ge 1) (23)



119894and 119909


119903119894119895= 119909119894minus 119909119895 = radicsum

119889

119896=1(119909119894119896minus 119909119895119896)2 where 119909

119894119896is the 119896th



119909119894+1 = 119909119894 +1205730119890

minus1205741199032119894119895 (119909119895minus119909119894) + 1205720120598119894 (24)


0is a














11 120574 1

1198622

2





10 20 30 40 5050

875

125

1625

200 Desirability

05

06

0607

07

2 2


Max

it

Npop






MOFA 120598119894= [rand minus 12] 1205730 = 1







subsystem 119895





Small 191(026)

078(023)

350(116)

229(1354)

123(345)

33(126)

5535(3022)

6069(2054)

2260(1172)

001493(00257)

007999(00542)

004230(00376)

Medium 259(021)

145(046)

699(205)

1285(560)

1190(380)

385(114)

3902(1732)

6579(2350)

2865(1271)

00238(00367)

01562(02004)

00482(00303)

Large 3149(691)

2235(936)

9125(2548)

1560(622)

1284(419)

411(149)

5135(2577)

9744(3093)

3435(1381)

000419(00048)

014625(00285)

007408(00418)



DM = [

119873

sum

119894=1max (1003817100381710038171003817119909119894 minus119910119894

1003817100381710038171003817)]

12

MS = [119872

sum

119898=1(|119873|

max119894=1

119891119894

119898minus|119873|

min119894=1

119891119894

119898)

2

]

12

(25)



119894


119898is the119898th

















Time 274 012 149 006 684 049NNS 2040 519 1320 365 370 106DM 4496 1079 9146 1511 3017 1221MS 00068 00165 01436 00132 00508 00218














119894119895

119903119905

1 2 3 4 Strategy 1 2 3 4 5 6 7 8 9 10 11 12 13 14 151 0 0 1 0 A 1 00 00 00 1198831199031 00 1198831199032 00 1198831199032 00 00 1198831199032 1198831198981 00 1198831199031 00

2 1 1 0 0 C 2 00 00 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 1198831199032 1198831199031 1198831198982 00

3 1 0 0 0 A 3 00 1198831199031 00 1198831199031 00 00 1198831199031 1198831199032 00 00 1198831199032 1198831199032 1198831198982 1198831199032 00

4 0 1 0 0 C 4 00 00 00 1198831199032 00 00 1198831198982 00 00 1198831198982 00 1198831199032 1198831199032 1198831198982 00

5 0 1 0 0 A 5 00 1198831199031 00 00 1198831199032 00 00 1198831199031 00 00 1198831199031 00 1198831199032 1198831199031 00

6 0 1 0 1 A 6 00 1198831199032 00 1198831199032 1198831199032 00 00 1198831199032 00 00 1198831199031 00 1198831198982 1198831198982 00

7 0 0 1 0 A 7 00 1198831199032 00 00 1198831198982 00 00 1198831199032 00 00 1198831199031 00 1198831199032 1198831198982 1198831199032

8 0 1 0 0 A 8 00 1198831199032 00 00 1198831199032 1198831199032 00 00 1198831198982 00 00 1198831199032 00 1198831199032 00

9 1 0 1 0 A 9 00 1198831199032 1198831198982 1198831199032 00 00 1198831199032 00 00 1198831199031 00 1198831199031 00 1198831199032 00

10 1 0 0 0 C 10 00 1198831199032 00 1198831199032 00 00 1198831199031 00 00 1198831199031 00 1198831199032 1198831199031 1198831199032 00

11 0 0 1 0 C 11 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 1198831199031 1198831198982 00

12 2 0 0 1 A13 0 1 0 0 C14 1 0 0 0 A

3800 3850 3900 3950 4000 4050 410009997

09997

09997

09997

09998

09998

09998

09998

09998

09999

a

(a)

3600 3700 3800 3900 4000 4100 4200 4300 4400 450008

082

084

086

088

09

092

094

096

098

b

(b)



5 Conclusion





(a)


1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895


(b)


119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895


Nomenclature

Indices




119894


119903




119905 Time 119905 isin 1 2 119898119879

Parameters







Parameters 119903

1 2 3 4 5 6 7 8 9 10 11120582119903119905

mdash 00004 00004 00004 00004 00004 00004 00004 00004 00004167 00004167 00004167

120582119903119897

119897 = 1 000056 000056 000056 000056 000056 000056 000056 000056 00005834 00005834 00005834119897 = 2 000085 000085 000085 000085 000085 000085 000085 000085 00008784 00008784 00008784

120582119903119906

119906 = 1 00004 00004 00004 00004 00004 00004 00004 00004 00004167 00004167 00004167119906 = 2 00005 00005 00005 00005 00005 00005 00005 00005 00005167 00005167 00005167

120572119903

mdash 25 25 25 25 25 25 25 26 26 26 24

1198881

119903119897

119897 = 1 2 15 25 3 18 35 3 2 25 15 2119897 = 2 1 1 15 15 1 2 15 1 15 05 1

1198882

119903119906

119906 = 1 4 3 5 6 4 7 6 4 35 3 45119906 = 2 3 15 3 2 2 4 3 15 2 15 25

ℎ119903119897

119897 = 1 1 1 2 1 2 1 1 2 1 1 2119897 = 2 2 1 1 1 1 2 1 1 2 1 1

ℎ119903119906

119906 = 1 1 2 1 2 2 1 2 2 1 2 1119906 = 2 1 2 2 1 2 2 1 2 2 1 2

1198883

119903119897

119897 = 1 2 2 2 2 2 2 2 2 2 2 2119897 = 2 15 15 15 1 1 15 15 1 1 15 15

1198884

119903119906

119906 = 1 8 7 6 8 9 7 6 5 7 6 9119906 = 2 3 2 5 4 5 5 2 3 3 2 3



119867119900

119882 119881 120578119903

119868 120588119894(119905)

Value 5 3 4 00009 1000 20 200 200 0001 003 099




1198881015840













component 119903


Decision Variables








Acknowledgment



References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2(119909) 119891

119896(119909) where 119909 = (119909

1 1199092 119909

119889)119879








119894



end ifend for


Algorithm 1







120573 (119903) = 1205730119890minus120574119903119898

(119898 ge 1) (23)



119894and 119909


119903119894119895= 119909119894minus 119909119895 = radicsum

119889

119896=1(119909119894119896minus 119909119895119896)2 where 119909

119894119896is the 119896th



119909119894+1 = 119909119894 +1205730119890

minus1205741199032119894119895 (119909119895minus119909119894) + 1205720120598119894 (24)


0is a














11 120574 1

1198622

2





10 20 30 40 5050

875

125

1625

200 Desirability

05

06

0607

07

2 2


Max

it

Npop






MOFA 120598119894= [rand minus 12] 1205730 = 1







subsystem 119895





Small 191(026)

078(023)

350(116)

229(1354)

123(345)

33(126)

5535(3022)

6069(2054)

2260(1172)

001493(00257)

007999(00542)

004230(00376)

Medium 259(021)

145(046)

699(205)

1285(560)

1190(380)

385(114)

3902(1732)

6579(2350)

2865(1271)

00238(00367)

01562(02004)

00482(00303)

Large 3149(691)

2235(936)

9125(2548)

1560(622)

1284(419)

411(149)

5135(2577)

9744(3093)

3435(1381)

000419(00048)

014625(00285)

007408(00418)



DM = [

119873

sum

119894=1max (1003817100381710038171003817119909119894 minus119910119894

1003817100381710038171003817)]

12

MS = [119872

sum

119898=1(|119873|

max119894=1

119891119894

119898minus|119873|

min119894=1

119891119894

119898)

2

]

12

(25)



119894


119898is the119898th

















Time 274 012 149 006 684 049NNS 2040 519 1320 365 370 106DM 4496 1079 9146 1511 3017 1221MS 00068 00165 01436 00132 00508 00218














119894119895

119903119905

1 2 3 4 Strategy 1 2 3 4 5 6 7 8 9 10 11 12 13 14 151 0 0 1 0 A 1 00 00 00 1198831199031 00 1198831199032 00 1198831199032 00 00 1198831199032 1198831198981 00 1198831199031 00

2 1 1 0 0 C 2 00 00 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 1198831199032 1198831199031 1198831198982 00

3 1 0 0 0 A 3 00 1198831199031 00 1198831199031 00 00 1198831199031 1198831199032 00 00 1198831199032 1198831199032 1198831198982 1198831199032 00

4 0 1 0 0 C 4 00 00 00 1198831199032 00 00 1198831198982 00 00 1198831198982 00 1198831199032 1198831199032 1198831198982 00

5 0 1 0 0 A 5 00 1198831199031 00 00 1198831199032 00 00 1198831199031 00 00 1198831199031 00 1198831199032 1198831199031 00

6 0 1 0 1 A 6 00 1198831199032 00 1198831199032 1198831199032 00 00 1198831199032 00 00 1198831199031 00 1198831198982 1198831198982 00

7 0 0 1 0 A 7 00 1198831199032 00 00 1198831198982 00 00 1198831199032 00 00 1198831199031 00 1198831199032 1198831198982 1198831199032

8 0 1 0 0 A 8 00 1198831199032 00 00 1198831199032 1198831199032 00 00 1198831198982 00 00 1198831199032 00 1198831199032 00

9 1 0 1 0 A 9 00 1198831199032 1198831198982 1198831199032 00 00 1198831199032 00 00 1198831199031 00 1198831199031 00 1198831199032 00

10 1 0 0 0 C 10 00 1198831199032 00 1198831199032 00 00 1198831199031 00 00 1198831199031 00 1198831199032 1198831199031 1198831199032 00

11 0 0 1 0 C 11 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 1198831199031 1198831198982 00

12 2 0 0 1 A13 0 1 0 0 C14 1 0 0 0 A

3800 3850 3900 3950 4000 4050 410009997

09997

09997

09997

09998

09998

09998

09998

09998

09999

a

(a)

3600 3700 3800 3900 4000 4100 4200 4300 4400 450008

082

084

086

088

09

092

094

096

098

b

(b)



5 Conclusion





(a)


1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895


(b)


119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895


Nomenclature

Indices




119894


119903




119905 Time 119905 isin 1 2 119898119879

Parameters







Parameters 119903

1 2 3 4 5 6 7 8 9 10 11120582119903119905

mdash 00004 00004 00004 00004 00004 00004 00004 00004 00004167 00004167 00004167

120582119903119897

119897 = 1 000056 000056 000056 000056 000056 000056 000056 000056 00005834 00005834 00005834119897 = 2 000085 000085 000085 000085 000085 000085 000085 000085 00008784 00008784 00008784

120582119903119906

119906 = 1 00004 00004 00004 00004 00004 00004 00004 00004 00004167 00004167 00004167119906 = 2 00005 00005 00005 00005 00005 00005 00005 00005 00005167 00005167 00005167

120572119903

mdash 25 25 25 25 25 25 25 26 26 26 24

1198881

119903119897

119897 = 1 2 15 25 3 18 35 3 2 25 15 2119897 = 2 1 1 15 15 1 2 15 1 15 05 1

1198882

119903119906

119906 = 1 4 3 5 6 4 7 6 4 35 3 45119906 = 2 3 15 3 2 2 4 3 15 2 15 25

ℎ119903119897

119897 = 1 1 1 2 1 2 1 1 2 1 1 2119897 = 2 2 1 1 1 1 2 1 1 2 1 1

ℎ119903119906

119906 = 1 1 2 1 2 2 1 2 2 1 2 1119906 = 2 1 2 2 1 2 2 1 2 2 1 2

1198883

119903119897

119897 = 1 2 2 2 2 2 2 2 2 2 2 2119897 = 2 15 15 15 1 1 15 15 1 1 15 15

1198884

119903119906

119906 = 1 8 7 6 8 9 7 6 5 7 6 9119906 = 2 3 2 5 4 5 5 2 3 3 2 3



119867119900

119882 119881 120578119903

119868 120588119894(119905)

Value 5 3 4 00009 1000 20 200 200 0001 003 099




1198881015840













component 119903


Decision Variables








Acknowledgment



References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















11 120574 1

1198622

2





10 20 30 40 5050

875

125

1625

200 Desirability

05

06

0607

07

2 2


Max

it

Npop






MOFA 120598119894= [rand minus 12] 1205730 = 1







subsystem 119895





Small 191(026)

078(023)

350(116)

229(1354)

123(345)

33(126)

5535(3022)

6069(2054)

2260(1172)

001493(00257)

007999(00542)

004230(00376)

Medium 259(021)

145(046)

699(205)

1285(560)

1190(380)

385(114)

3902(1732)

6579(2350)

2865(1271)

00238(00367)

01562(02004)

00482(00303)

Large 3149(691)

2235(936)

9125(2548)

1560(622)

1284(419)

411(149)

5135(2577)

9744(3093)

3435(1381)

000419(00048)

014625(00285)

007408(00418)



DM = [

119873

sum

119894=1max (1003817100381710038171003817119909119894 minus119910119894

1003817100381710038171003817)]

12

MS = [119872

sum

119898=1(|119873|

max119894=1

119891119894

119898minus|119873|

min119894=1

119891119894

119898)

2

]

12

(25)



119894


119898is the119898th

















Time 274 012 149 006 684 049NNS 2040 519 1320 365 370 106DM 4496 1079 9146 1511 3017 1221MS 00068 00165 01436 00132 00508 00218














119894119895

119903119905

1 2 3 4 Strategy 1 2 3 4 5 6 7 8 9 10 11 12 13 14 151 0 0 1 0 A 1 00 00 00 1198831199031 00 1198831199032 00 1198831199032 00 00 1198831199032 1198831198981 00 1198831199031 00

2 1 1 0 0 C 2 00 00 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 1198831199032 1198831199031 1198831198982 00

3 1 0 0 0 A 3 00 1198831199031 00 1198831199031 00 00 1198831199031 1198831199032 00 00 1198831199032 1198831199032 1198831198982 1198831199032 00

4 0 1 0 0 C 4 00 00 00 1198831199032 00 00 1198831198982 00 00 1198831198982 00 1198831199032 1198831199032 1198831198982 00

5 0 1 0 0 A 5 00 1198831199031 00 00 1198831199032 00 00 1198831199031 00 00 1198831199031 00 1198831199032 1198831199031 00

6 0 1 0 1 A 6 00 1198831199032 00 1198831199032 1198831199032 00 00 1198831199032 00 00 1198831199031 00 1198831198982 1198831198982 00

7 0 0 1 0 A 7 00 1198831199032 00 00 1198831198982 00 00 1198831199032 00 00 1198831199031 00 1198831199032 1198831198982 1198831199032

8 0 1 0 0 A 8 00 1198831199032 00 00 1198831199032 1198831199032 00 00 1198831198982 00 00 1198831199032 00 1198831199032 00

9 1 0 1 0 A 9 00 1198831199032 1198831198982 1198831199032 00 00 1198831199032 00 00 1198831199031 00 1198831199031 00 1198831199032 00

10 1 0 0 0 C 10 00 1198831199032 00 1198831199032 00 00 1198831199031 00 00 1198831199031 00 1198831199032 1198831199031 1198831199032 00

11 0 0 1 0 C 11 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 1198831199031 1198831198982 00

12 2 0 0 1 A13 0 1 0 0 C14 1 0 0 0 A

3800 3850 3900 3950 4000 4050 410009997

09997

09997

09997

09998

09998

09998

09998

09998

09999

a

(a)

3600 3700 3800 3900 4000 4100 4200 4300 4400 450008

082

084

086

088

09

092

094

096

098

b

(b)



5 Conclusion





(a)


1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895


(b)


119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895


Nomenclature

Indices




119894


119903




119905 Time 119905 isin 1 2 119898119879

Parameters







Parameters 119903

1 2 3 4 5 6 7 8 9 10 11120582119903119905

mdash 00004 00004 00004 00004 00004 00004 00004 00004 00004167 00004167 00004167

120582119903119897

119897 = 1 000056 000056 000056 000056 000056 000056 000056 000056 00005834 00005834 00005834119897 = 2 000085 000085 000085 000085 000085 000085 000085 000085 00008784 00008784 00008784

120582119903119906

119906 = 1 00004 00004 00004 00004 00004 00004 00004 00004 00004167 00004167 00004167119906 = 2 00005 00005 00005 00005 00005 00005 00005 00005 00005167 00005167 00005167

120572119903

mdash 25 25 25 25 25 25 25 26 26 26 24

1198881

119903119897

119897 = 1 2 15 25 3 18 35 3 2 25 15 2119897 = 2 1 1 15 15 1 2 15 1 15 05 1

1198882

119903119906

119906 = 1 4 3 5 6 4 7 6 4 35 3 45119906 = 2 3 15 3 2 2 4 3 15 2 15 25

ℎ119903119897

119897 = 1 1 1 2 1 2 1 1 2 1 1 2119897 = 2 2 1 1 1 1 2 1 1 2 1 1

ℎ119903119906

119906 = 1 1 2 1 2 2 1 2 2 1 2 1119906 = 2 1 2 2 1 2 2 1 2 2 1 2

1198883

119903119897

119897 = 1 2 2 2 2 2 2 2 2 2 2 2119897 = 2 15 15 15 1 1 15 15 1 1 15 15

1198884

119903119906

119906 = 1 8 7 6 8 9 7 6 5 7 6 9119906 = 2 3 2 5 4 5 5 2 3 3 2 3



119867119900

119882 119881 120578119903

119868 120588119894(119905)

Value 5 3 4 00009 1000 20 200 200 0001 003 099




1198881015840













component 119903


Decision Variables








Acknowledgment



References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














subsystem 119895





Small 191(026)

078(023)

350(116)

229(1354)

123(345)

33(126)

5535(3022)

6069(2054)

2260(1172)

001493(00257)

007999(00542)

004230(00376)

Medium 259(021)

145(046)

699(205)

1285(560)

1190(380)

385(114)

3902(1732)

6579(2350)

2865(1271)

00238(00367)

01562(02004)

00482(00303)

Large 3149(691)

2235(936)

9125(2548)

1560(622)

1284(419)

411(149)

5135(2577)

9744(3093)

3435(1381)

000419(00048)

014625(00285)

007408(00418)



DM = [

119873

sum

119894=1max (1003817100381710038171003817119909119894 minus119910119894

1003817100381710038171003817)]

12

MS = [119872

sum

119898=1(|119873|

max119894=1

119891119894

119898minus|119873|

min119894=1

119891119894

119898)

2

]

12

(25)



119894


119898is the119898th

















Time 274 012 149 006 684 049NNS 2040 519 1320 365 370 106DM 4496 1079 9146 1511 3017 1221MS 00068 00165 01436 00132 00508 00218














119894119895

119903119905

1 2 3 4 Strategy 1 2 3 4 5 6 7 8 9 10 11 12 13 14 151 0 0 1 0 A 1 00 00 00 1198831199031 00 1198831199032 00 1198831199032 00 00 1198831199032 1198831198981 00 1198831199031 00

2 1 1 0 0 C 2 00 00 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 1198831199032 1198831199031 1198831198982 00

3 1 0 0 0 A 3 00 1198831199031 00 1198831199031 00 00 1198831199031 1198831199032 00 00 1198831199032 1198831199032 1198831198982 1198831199032 00

4 0 1 0 0 C 4 00 00 00 1198831199032 00 00 1198831198982 00 00 1198831198982 00 1198831199032 1198831199032 1198831198982 00

5 0 1 0 0 A 5 00 1198831199031 00 00 1198831199032 00 00 1198831199031 00 00 1198831199031 00 1198831199032 1198831199031 00

6 0 1 0 1 A 6 00 1198831199032 00 1198831199032 1198831199032 00 00 1198831199032 00 00 1198831199031 00 1198831198982 1198831198982 00

7 0 0 1 0 A 7 00 1198831199032 00 00 1198831198982 00 00 1198831199032 00 00 1198831199031 00 1198831199032 1198831198982 1198831199032

8 0 1 0 0 A 8 00 1198831199032 00 00 1198831199032 1198831199032 00 00 1198831198982 00 00 1198831199032 00 1198831199032 00

9 1 0 1 0 A 9 00 1198831199032 1198831198982 1198831199032 00 00 1198831199032 00 00 1198831199031 00 1198831199031 00 1198831199032 00

10 1 0 0 0 C 10 00 1198831199032 00 1198831199032 00 00 1198831199031 00 00 1198831199031 00 1198831199032 1198831199031 1198831199032 00

11 0 0 1 0 C 11 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 1198831199031 1198831198982 00

12 2 0 0 1 A13 0 1 0 0 C14 1 0 0 0 A

3800 3850 3900 3950 4000 4050 410009997

09997

09997

09997

09998

09998

09998

09998

09998

09999

a

(a)

3600 3700 3800 3900 4000 4100 4200 4300 4400 450008

082

084

086

088

09

092

094

096

098

b

(b)



5 Conclusion





(a)


1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895


(b)


119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895


Nomenclature

Indices




119894


119903




119905 Time 119905 isin 1 2 119898119879

Parameters







Parameters 119903

1 2 3 4 5 6 7 8 9 10 11120582119903119905

mdash 00004 00004 00004 00004 00004 00004 00004 00004 00004167 00004167 00004167

120582119903119897

119897 = 1 000056 000056 000056 000056 000056 000056 000056 000056 00005834 00005834 00005834119897 = 2 000085 000085 000085 000085 000085 000085 000085 000085 00008784 00008784 00008784

120582119903119906

119906 = 1 00004 00004 00004 00004 00004 00004 00004 00004 00004167 00004167 00004167119906 = 2 00005 00005 00005 00005 00005 00005 00005 00005 00005167 00005167 00005167

120572119903

mdash 25 25 25 25 25 25 25 26 26 26 24

1198881

119903119897

119897 = 1 2 15 25 3 18 35 3 2 25 15 2119897 = 2 1 1 15 15 1 2 15 1 15 05 1

1198882

119903119906

119906 = 1 4 3 5 6 4 7 6 4 35 3 45119906 = 2 3 15 3 2 2 4 3 15 2 15 25

ℎ119903119897

119897 = 1 1 1 2 1 2 1 1 2 1 1 2119897 = 2 2 1 1 1 1 2 1 1 2 1 1

ℎ119903119906

119906 = 1 1 2 1 2 2 1 2 2 1 2 1119906 = 2 1 2 2 1 2 2 1 2 2 1 2

1198883

119903119897

119897 = 1 2 2 2 2 2 2 2 2 2 2 2119897 = 2 15 15 15 1 1 15 15 1 1 15 15

1198884

119903119906

119906 = 1 8 7 6 8 9 7 6 5 7 6 9119906 = 2 3 2 5 4 5 5 2 3 3 2 3



119867119900

119882 119881 120578119903

119868 120588119894(119905)

Value 5 3 4 00009 1000 20 200 200 0001 003 099




1198881015840













component 119903


Decision Variables








Acknowledgment



References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















Time 274 012 149 006 684 049NNS 2040 519 1320 365 370 106DM 4496 1079 9146 1511 3017 1221MS 00068 00165 01436 00132 00508 00218














119894119895

119903119905

1 2 3 4 Strategy 1 2 3 4 5 6 7 8 9 10 11 12 13 14 151 0 0 1 0 A 1 00 00 00 1198831199031 00 1198831199032 00 1198831199032 00 00 1198831199032 1198831198981 00 1198831199031 00

2 1 1 0 0 C 2 00 00 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 1198831199032 1198831199031 1198831198982 00

3 1 0 0 0 A 3 00 1198831199031 00 1198831199031 00 00 1198831199031 1198831199032 00 00 1198831199032 1198831199032 1198831198982 1198831199032 00

4 0 1 0 0 C 4 00 00 00 1198831199032 00 00 1198831198982 00 00 1198831198982 00 1198831199032 1198831199032 1198831198982 00

5 0 1 0 0 A 5 00 1198831199031 00 00 1198831199032 00 00 1198831199031 00 00 1198831199031 00 1198831199032 1198831199031 00

6 0 1 0 1 A 6 00 1198831199032 00 1198831199032 1198831199032 00 00 1198831199032 00 00 1198831199031 00 1198831198982 1198831198982 00

7 0 0 1 0 A 7 00 1198831199032 00 00 1198831198982 00 00 1198831199032 00 00 1198831199031 00 1198831199032 1198831198982 1198831199032

8 0 1 0 0 A 8 00 1198831199032 00 00 1198831199032 1198831199032 00 00 1198831198982 00 00 1198831199032 00 1198831199032 00

9 1 0 1 0 A 9 00 1198831199032 1198831198982 1198831199032 00 00 1198831199032 00 00 1198831199031 00 1198831199031 00 1198831199032 00

10 1 0 0 0 C 10 00 1198831199032 00 1198831199032 00 00 1198831199031 00 00 1198831199031 00 1198831199032 1198831199031 1198831199032 00

11 0 0 1 0 C 11 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 1198831199031 1198831198982 00

12 2 0 0 1 A13 0 1 0 0 C14 1 0 0 0 A

3800 3850 3900 3950 4000 4050 410009997

09997

09997

09997

09998

09998

09998

09998

09998

09999

a

(a)

3600 3700 3800 3900 4000 4100 4200 4300 4400 450008

082

084

086

088

09

092

094

096

098

b

(b)



5 Conclusion





(a)


1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895


(b)


119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895


Nomenclature

Indices




119894


119903




119905 Time 119905 isin 1 2 119898119879

Parameters







Parameters 119903

1 2 3 4 5 6 7 8 9 10 11120582119903119905

mdash 00004 00004 00004 00004 00004 00004 00004 00004 00004167 00004167 00004167

120582119903119897

119897 = 1 000056 000056 000056 000056 000056 000056 000056 000056 00005834 00005834 00005834119897 = 2 000085 000085 000085 000085 000085 000085 000085 000085 00008784 00008784 00008784

120582119903119906

119906 = 1 00004 00004 00004 00004 00004 00004 00004 00004 00004167 00004167 00004167119906 = 2 00005 00005 00005 00005 00005 00005 00005 00005 00005167 00005167 00005167

120572119903

mdash 25 25 25 25 25 25 25 26 26 26 24

1198881

119903119897

119897 = 1 2 15 25 3 18 35 3 2 25 15 2119897 = 2 1 1 15 15 1 2 15 1 15 05 1

1198882

119903119906

119906 = 1 4 3 5 6 4 7 6 4 35 3 45119906 = 2 3 15 3 2 2 4 3 15 2 15 25

ℎ119903119897

119897 = 1 1 1 2 1 2 1 1 2 1 1 2119897 = 2 2 1 1 1 1 2 1 1 2 1 1

ℎ119903119906

119906 = 1 1 2 1 2 2 1 2 2 1 2 1119906 = 2 1 2 2 1 2 2 1 2 2 1 2

1198883

119903119897

119897 = 1 2 2 2 2 2 2 2 2 2 2 2119897 = 2 15 15 15 1 1 15 15 1 1 15 15

1198884

119903119906

119906 = 1 8 7 6 8 9 7 6 5 7 6 9119906 = 2 3 2 5 4 5 5 2 3 3 2 3



119867119900

119882 119881 120578119903

119868 120588119894(119905)

Value 5 3 4 00009 1000 20 200 200 0001 003 099




1198881015840













component 119903


Decision Variables








Acknowledgment



References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119894119895

119903119905

1 2 3 4 Strategy 1 2 3 4 5 6 7 8 9 10 11 12 13 14 151 0 0 1 0 A 1 00 00 00 1198831199031 00 1198831199032 00 1198831199032 00 00 1198831199032 1198831198981 00 1198831199031 00

2 1 1 0 0 C 2 00 00 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 1198831199032 1198831199031 1198831198982 00

3 1 0 0 0 A 3 00 1198831199031 00 1198831199031 00 00 1198831199031 1198831199032 00 00 1198831199032 1198831199032 1198831198982 1198831199032 00

4 0 1 0 0 C 4 00 00 00 1198831199032 00 00 1198831198982 00 00 1198831198982 00 1198831199032 1198831199032 1198831198982 00

5 0 1 0 0 A 5 00 1198831199031 00 00 1198831199032 00 00 1198831199031 00 00 1198831199031 00 1198831199032 1198831199031 00

6 0 1 0 1 A 6 00 1198831199032 00 1198831199032 1198831199032 00 00 1198831199032 00 00 1198831199031 00 1198831198982 1198831198982 00

7 0 0 1 0 A 7 00 1198831199032 00 00 1198831198982 00 00 1198831199032 00 00 1198831199031 00 1198831199032 1198831198982 1198831199032

8 0 1 0 0 A 8 00 1198831199032 00 00 1198831199032 1198831199032 00 00 1198831198982 00 00 1198831199032 00 1198831199032 00

9 1 0 1 0 A 9 00 1198831199032 1198831198982 1198831199032 00 00 1198831199032 00 00 1198831199031 00 1198831199031 00 1198831199032 00

10 1 0 0 0 C 10 00 1198831199032 00 1198831199032 00 00 1198831199031 00 00 1198831199031 00 1198831199032 1198831199031 1198831199032 00

11 0 0 1 0 C 11 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 00 1198831199032 00 1198831199031 1198831198982 00

12 2 0 0 1 A13 0 1 0 0 C14 1 0 0 0 A

3800 3850 3900 3950 4000 4050 410009997

09997

09997

09997

09998

09998

09998

09998

09998

09999

a

(a)

3600 3700 3800 3900 4000 4100 4200 4300 4400 450008

082

084

086

088

09

092

094

096

098

b

(b)



5 Conclusion





(a)


1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895


(b)


119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895


Nomenclature

Indices




119894


119903




119905 Time 119905 isin 1 2 119898119879

Parameters







Parameters 119903

1 2 3 4 5 6 7 8 9 10 11120582119903119905

mdash 00004 00004 00004 00004 00004 00004 00004 00004 00004167 00004167 00004167

120582119903119897

119897 = 1 000056 000056 000056 000056 000056 000056 000056 000056 00005834 00005834 00005834119897 = 2 000085 000085 000085 000085 000085 000085 000085 000085 00008784 00008784 00008784

120582119903119906

119906 = 1 00004 00004 00004 00004 00004 00004 00004 00004 00004167 00004167 00004167119906 = 2 00005 00005 00005 00005 00005 00005 00005 00005 00005167 00005167 00005167

120572119903

mdash 25 25 25 25 25 25 25 26 26 26 24

1198881

119903119897

119897 = 1 2 15 25 3 18 35 3 2 25 15 2119897 = 2 1 1 15 15 1 2 15 1 15 05 1

1198882

119903119906

119906 = 1 4 3 5 6 4 7 6 4 35 3 45119906 = 2 3 15 3 2 2 4 3 15 2 15 25

ℎ119903119897

119897 = 1 1 1 2 1 2 1 1 2 1 1 2119897 = 2 2 1 1 1 1 2 1 1 2 1 1

ℎ119903119906

119906 = 1 1 2 1 2 2 1 2 2 1 2 1119906 = 2 1 2 2 1 2 2 1 2 2 1 2

1198883

119903119897

119897 = 1 2 2 2 2 2 2 2 2 2 2 2119897 = 2 15 15 15 1 1 15 15 1 1 15 15

1198884

119903119906

119906 = 1 8 7 6 8 9 7 6 5 7 6 9119906 = 2 3 2 5 4 5 5 2 3 3 2 3



119867119900

119882 119881 120578119903

119868 120588119894(119905)

Value 5 3 4 00009 1000 20 200 200 0001 003 099




1198881015840













component 119903


Decision Variables








Acknowledgment



References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(a)


1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895

1205821198941198951 120582

1198941198952 119896119894119895


(b)


119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895

119888119894119895

1198881015840

119894119895V119894119895

119908119894119895


Nomenclature

Indices




119894


119903




119905 Time 119905 isin 1 2 119898119879

Parameters







Parameters 119903

1 2 3 4 5 6 7 8 9 10 11120582119903119905

mdash 00004 00004 00004 00004 00004 00004 00004 00004 00004167 00004167 00004167

120582119903119897

119897 = 1 000056 000056 000056 000056 000056 000056 000056 000056 00005834 00005834 00005834119897 = 2 000085 000085 000085 000085 000085 000085 000085 000085 00008784 00008784 00008784

120582119903119906

119906 = 1 00004 00004 00004 00004 00004 00004 00004 00004 00004167 00004167 00004167119906 = 2 00005 00005 00005 00005 00005 00005 00005 00005 00005167 00005167 00005167

120572119903

mdash 25 25 25 25 25 25 25 26 26 26 24

1198881

119903119897

119897 = 1 2 15 25 3 18 35 3 2 25 15 2119897 = 2 1 1 15 15 1 2 15 1 15 05 1

1198882

119903119906

119906 = 1 4 3 5 6 4 7 6 4 35 3 45119906 = 2 3 15 3 2 2 4 3 15 2 15 25

ℎ119903119897

119897 = 1 1 1 2 1 2 1 1 2 1 1 2119897 = 2 2 1 1 1 1 2 1 1 2 1 1

ℎ119903119906

119906 = 1 1 2 1 2 2 1 2 2 1 2 1119906 = 2 1 2 2 1 2 2 1 2 2 1 2

1198883

119903119897

119897 = 1 2 2 2 2 2 2 2 2 2 2 2119897 = 2 15 15 15 1 1 15 15 1 1 15 15

1198884

119903119906

119906 = 1 8 7 6 8 9 7 6 5 7 6 9119906 = 2 3 2 5 4 5 5 2 3 3 2 3



119867119900

119882 119881 120578119903

119868 120588119894(119905)

Value 5 3 4 00009 1000 20 200 200 0001 003 099




1198881015840













component 119903


Decision Variables








Acknowledgment



References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Parameters 119903

1 2 3 4 5 6 7 8 9 10 11120582119903119905

mdash 00004 00004 00004 00004 00004 00004 00004 00004 00004167 00004167 00004167

120582119903119897

119897 = 1 000056 000056 000056 000056 000056 000056 000056 000056 00005834 00005834 00005834119897 = 2 000085 000085 000085 000085 000085 000085 000085 000085 00008784 00008784 00008784

120582119903119906

119906 = 1 00004 00004 00004 00004 00004 00004 00004 00004 00004167 00004167 00004167119906 = 2 00005 00005 00005 00005 00005 00005 00005 00005 00005167 00005167 00005167

120572119903

mdash 25 25 25 25 25 25 25 26 26 26 24

1198881

119903119897

119897 = 1 2 15 25 3 18 35 3 2 25 15 2119897 = 2 1 1 15 15 1 2 15 1 15 05 1

1198882

119903119906

119906 = 1 4 3 5 6 4 7 6 4 35 3 45119906 = 2 3 15 3 2 2 4 3 15 2 15 25

ℎ119903119897

119897 = 1 1 1 2 1 2 1 1 2 1 1 2119897 = 2 2 1 1 1 1 2 1 1 2 1 1

ℎ119903119906

119906 = 1 1 2 1 2 2 1 2 2 1 2 1119906 = 2 1 2 2 1 2 2 1 2 2 1 2

1198883

119903119897

119897 = 1 2 2 2 2 2 2 2 2 2 2 2119897 = 2 15 15 15 1 1 15 15 1 1 15 15

1198884

119903119906

119906 = 1 8 7 6 8 9 7 6 5 7 6 9119906 = 2 3 2 5 4 5 5 2 3 3 2 3



119867119900

119882 119881 120578119903

119868 120588119894(119905)

Value 5 3 4 00009 1000 20 200 200 0001 003 099




1198881015840













component 119903


Decision Variables








Acknowledgment



References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a new biobjective model to...

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