research article a novel resource-leveling approach for...

Research ArticleA Novel Resource-Leveling Approach for Construction ProjectBased on Differential Evolution

Hong-Hai Tran and Nhat-Duc Hoang

Department of Technology and Construction Management Faculty of Building and Industrial ConstructionNational University of Civil Engineering Room 307 A1 Building No 55 Giai Phong Road Hanoi 10000 Vietnam

Correspondence should be addressed to Nhat-Duc Hoang ducxd85yahoocom

Received 30 November 2013 Revised 12 March 2014 Accepted 5 April 2014 Published 21 May 2014

Academic Editor Manoj Jha

Copyright copy 2014 H-H Tran and N-D HoangThis is an open access article distributed under the Creative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

In construction engineering project schedules are commonly established by the critical pathmethod Nevertheless these schedulesoften lead to substantial fluctuations in the resource profile that are not only impractical but also costly for the contractors to executeTherefore in order to smooth out the resource profile construction managers need to perform resource-leveling proceduresThis paper proposes a novel approach for resource leveling named as resource leveling based on differential evolution (RLDE)The performance of the RLDE is compared to that of Microsoft Project software the genetic algorithm and the particle swarmoptimization algorithm Experiments have proved that the newly developedmethod can deliver themost desirable resource-levelingresult Thus the RLDE is an effective method and it can be a useful tool for assisting managersplanners in the field of projectmanagement

1 Introduction

In todayrsquos market condition the survivability of any con-struction contractor essentially depends on its capability ofmanaging resources Ineffective resource management esca-lates the operational expense or even gives rise to financialand scheduling problems [1 2] Without doubt the excessrequirement of resources in the construction site leads to theextension of project duration As the construction contractorfails to deliver the project by the prespecified date the ownermay suffer from financial losses due to the nonavailabilityof the facility [3] In addition construction delays oftenbring about legal disputes among parties higher overheadcosts and degraded reputation and they occasionally resultin project failures [4 5] As a consequence resource man-agement is an essential task that needs to be implementedthoroughly in the planning phase

Basically resources in construction projects consist ofmanpower equipment materials money and expertiseNeedless to say the proper management of these resourcesholds the key to the successful accomplishment of any project[3] However construction schedules generated by networkscheduling techniques are often characterized by undesirableresource fluctuations that are impractical and costly for

the contractors to implement [6] The reason is that it isexpensive to hire and to lay off workers on a short-termbasis according to the fluctuations in the resource profileAdditionally if the resources cannot be managed efficientlythey may exceed the supply capability of the contractor andlead to schedule delay Finally the contractor must maintaina number of idle resources during low-demand periodsThese facts undoubtedly cause profit damage for constructioncompanies Thus construction managers mandatorily needto perform schedule-adjusting processes to alleviate fluctua-tions in the resource profile during the project execution

The process of smoothing out resources is well known asthe resource-leveling problem [6ndash8] In resource leveling theobjective is to minimize the peak demand and fluctuationsin the pattern of the resource usage [9] This process aimsto minimize variations in the resource profile by shiftingnoncritical activities within their available floats and keep theproject duration unchanged

Initially the mathematical approaches were used to solvethe resource-leveling problems because they can provideoptimal solutions to the problem at hand Nonethelessthese methods become impractical when the size of projectnetwork reaches a considerably large numberThis is because

Hindawi Publishing CorporationJournal of Construction EngineeringVolume 2014 Article ID 648938 7 pageshttpdxdoiorg1011552014648938

2 Journal of Construction Engineering

resource leveling belongs to the class of combinatorial prob-lem Hence the increasing number of decision variablescauses the problem solving to become infeasible As aconsequencemathematicalmethods are not computationallytractable for real-life projects [10]

Other researchers attempted to utilize heuristic methodsin solving the resource-leveling problem [9 11] Despitethe simplicity of resource-leveling heuristics and their wideimplementation on commercial project management soft-ware (eg Microsoft Project) the result oftentimes can-not satisfy project managers It is because the heuristicapproaches operate on the basis of prespecified rules Thustheir performances are dependent on specific types of prob-lem and on the set of implemented rules Hence they canonly deliver feasible solutions and by no means guarantee anoptimum solution [12]

Due to the limitations of the mathematic and heuristicmethods the application of evolutionary algorithms (EAs)for resource leveling has attracted more attention in recentyears [13 14] EAs inspired by the natural evolution ofspecies have been successfully utilized to tackle optimizationproblems in diverse fields [15] Evolutionary computationis characterized by iterative progresses used to guide therandomly initiated population to the final optimal solution

Recently differential evolution (DE) [16ndash18] has drawnincreasing interest of researchers who have explored itscapability in a wide range of problems DE is a population-based stochastic search engine which is efficient and effectivefor global optimization in the continuous domain [19] It usesmutation crossover and selection operators at each gener-ation to move its population toward the global optimumSuperior performance of DE over other algorithms has beenverified in many reported research works [15 17 20 21]Nevertheless none of the previous works has investigated thepotentiality of DE in solving resource-leveling problem

Thus this paper proposes a novel approach for resourceleveling named as resource leveling based on differentialevolution (RLDE) RLDE is developed based on the DEalgorithm as the search engine The rest of the paper isorganized as follows In Section 2 the research methodologyis presented The structure of the proposed RLDE is shownin Section 3 The performance of the newly developed modelis demonstrated in Section 4 Finally some conclusions arementioned in the last section of this paper

2 Research Methodology

21 The Resource-Leveling Problem In the resource-levelingproblem the objective is to reduce the peak resource demandand smooth out day-to-day resource consumptions withinthe required project duration and with the assumption ofunlimited resource availability Hence the resource levelingcan be expressed as an optimization problem within whichthe following cost function is minimized [9 12]

119891 =119879

sum119894=1

(119910119894minus 119910119906)2 (1)

where 119879 denotes the project duration 119910119894represents the total

resource requirements of the activities performed at time unit119894 and 119910

119906represents a uniform resource level given as follows

119910119906=sum119879

119894=1119910119894

119879 (2)

Equation (1) means that if the value of the objective func-tion 119891 is small the actual resource profile of the project willbe close to the form of the ideal resource profile In the idealresource profile the resource demand is constant throughoutthe project duration Moreover (1) can be rewritten in thefollowing form

119891 =119879

sum119894=1

1199102119894minus 2119910119906

119879

sum119894=1

119910119894+ 1199102119906 (3)

Because activity duration and rate of resource for eachactivity are fixed 119910

119906and sum

119879

119894=1119910119894are constant Thus the cost

function can be expressed as follows

119891 =119879

sum119894=1

1199102119894 (4)

22 Differential Evolution Differential evolution (DE) is anevolutionary algorithm proposed by Storn and Price [16 17]which is designed for real parameter optimization problemsThe algorithm of DE is depicted in Algorithm 1 In this figureit is noted that NP represents the size of the population119883119895119894

is the 119895th decision variable of the 119894th individual inthe population 119892 is the current generation and 119863 denotesthe number of decision variables rand

119895(0 1) is a uniform

random number lying between 0 and 1 and rnb(119894)

is arandomly chosen index of (1 2 NP)

DE is based on the utilization of a novel crossover-mutation operator based on the linear combination of threedifferent individuals and one subject-to-replacement parent(or target vector) [20] The crossover-mutation operatoryields a trial vector (or child vector) which will competewith its parent in the selection operator These two termstrial vector and child vector can be used interchangeablyThe selection process is performed via selection between theparent and the corresponding offspring [22]

In the selection process the trial vector is compared to thetarget vector (or the parent) using the greedy criterion [17]These two terms target vector and parent vector can be usedsynonymously Under the greedy criterion the fitness value isthe only information that determines which vector is allowedto survive in the next generation (see Figure 2) Thus if thetrial vector can yield a lower objective function value than itsparent then the trial vector supersedes the target vector Theselection operator is expressed as follows

119883119894119892+1

= 119880119894119892

if 119891 (119880119894119892) le 119891 (119883

119894119892)

119883119894119892

if 119891 (119880119894119892) gt 119891 (119883

119894119892)

(5)

where 119883119894119892

represents the parent vector at generation 119892 119880119894119892

denotes the trial vector at generation 119892 119883119894119892+1

is the chosenindividual which survives to the next generation (119892 + 1)

Journal of Construction Engineering 3

Initialize population of NP individualsDo

For each individual 119895 in the populationGenerate three random integers 119903

1 1199032 and 119903

3isin (1NP)

with 1199031

= 1199032

= 1199033

= 119895Generate random integer 119894rand isin (1 119863)For each parameter 119894

119880119895119894119892

= 1198831198951199033 119892

+ 119891 times (1198831198951199031 119892

minus 1198831198951199032 119892

) if rand119895(0 1) lt Cr or 119895 = 119903119899119887(119894)

119883119895119894119892

otherwiseEnd ForReplace119883

119894with the offspring 119880

119894if 119880119894is better

End ForUntil the stopping condition is met

Algorithm 1 The differential evolution algorithm

Population of NP solutions

Yes

Searching termination

Optimal activity start time

No

Optimal project schedule

Mutation

Crossover

DE optimizer

Project informationoptimizerrsquos parameter setting

Scheduling module

Selection

[X11 X12 X1D]

[X1 X2 XD]

[XNP1 XNP2 XNPD]

Figure 1 The resource leveling based on differential evolution (RLDE)

The optimization process can terminate when the stop-ping criterion ismetThe user can set the type of this stoppingcondition Commonly the maximum generation (119866max) orthe maximum number of function evaluations (NFE) canbe used as the stopping condition When the optimizationprocess terminates the final optimal solution will be readilypresented to the user

3 Resource Leveling Based onDifferential Evolution (RLDE)

This section of the paper is dedicated to describing the RLDEresource-leveling approach (see Figure 1) It is noticed thatthe RLDE is developed based on DE as the optimization

techniqueThe objective of this optimizationmodel is tomin-imize daily fluctuations in the project resource utilizationThis goal is achieved by altering the schedules of noncriticalactivities in the project within their available floats Thus theschedules of critical activities are unchanged and the totalproject duration remains the same as the critical pathmethod(CPM) based schedule

Herein the resource leveling is accomplished by min-imizing the fluctuations of resource requirements and adesirable uniform resource level The model requires inputsof project information including the activity relationshipthe activity duration and the resource demand In additionthe user also needs to provide the parameter setting for thesearch engine such as the maximum number of searching


Table 1 Metrics for performance measurement

Performance measurement Notation Calculation

Fitness function 119891 119891 =119879

sum119896=1

(119910119896)2

Maximum resource demand RDmax RDmax = 119910max

Cumulative variation of the resource demand betweenconsecutive periods CRV CRV =

119879minus1

sum119896=1

1003816100381610038161003816119910119896+1 minus 119910119896

1003816100381610038161003816

Maximum variation of the resource demand betweentwo consecutive periods RVmax RVmax = max 119879minus1

119896=11003816100381610038161003816119910119896+1 minus 119910

119896

1003816100381610038161003816

Note 119879 is the total project duration 119910119896 represents the resource requirements at day kth 119910max denotes the peak of the resource demand throughout the projectexecution

generation (119866max) the population size (NP) the mutationscale (119865) and the crossover probability (Cr) With theseinputs the scheduling module can carry out the calculationprocess to obtain the CPM schedule which specifies the earlystart and the late start of each activity If all the necessaryinformation is provided the model is capable of operatingautomatically without usersrsquo interventions

Before the searching process can commence an initialpopulation of feasible solutions is created by a uniform ran-dom generator A solution for the resource-leveling problemis represented as a vector with119863 elements as follows

119883 = [1198831198941 1198831198942 119883

119894119863] (6)

where 119863 is the number of decision variables of the problemat hand It is obvious that119863 is also the number of noncriticalactivities in the project network The index 119894 denotes the 119894thindividual in the populationThe vector119883 represents the starttime of 119863 activities in the network Since the original DEoperates with real-value variables a function is employed toconvert those activitiesrsquo start times from real values to integervalues within the feasible domain Consider

119883119894119895= Round (LB (119895) + rand [0 1] times (UB (119895) minus LB (119895)))

(7)

where 119883119894119895is the start time of activity 119895 at the individual 119894th

rand[0 1] denotes a uniformly distributed random numberbetween 0 and 1 LB(119895) and UB(119895) are early start and late startof the activity 119895

The search engine DE takes into account the resultobtained from the schedulingmodule and it shifts noncriticalactivities within their float values to seek for an optimalproject schedule In our research the following objectivefunction and constraints are employed

119891 =119879

sum119896=1

(119910119896)2 (8)

subject to

ST119894minus ES119894le TF119894 ST

119894ge 0 119894 = 1 2 119863 (9)


resource requirements of the activities performed at time unit119896 ST119894is the start time of activity 119894 ES

119894and TF

119894represent early

start and total float of activity 119894 respectively119863 is the numberof activities in the network

After the searching process terminates an optimal solu-tion is identified The project schedule and its correspondingresource histogram are then constructed based on the opti-mal start times of all activitiesThe user can assess the qualityof a project schedule using a set of metrics (see Table 1) Itis worth noticing that the fitness function stated in (8) isthe objective function for the search engine DE Meanwhilethe set of metrics shown in Table 1 includes the fitnessfunction (see (8)) and other criteria (the maximum resourcedemand the cumulative variation of the resource demandbetween consecutive periods and the maximum variation ofthe resource demand between two consecutive periods)

It is noted that the fitness function utilized in our researchaims at obtaining an optimized resource profile that is asclose to the ideal resource profile as possibleThus this fitnessfunction is the most crucial criterion and it is very similarto that of the minimum moment method used in [9 12]However the three additional criteria illustrated in Table 1can provide more insights about the optimized resourceprofile for users

4 Experimental Result

In this section a construction project adapted from Sears etal [23] is used to demonstrate the capability of the newlydeveloped RLDE The project consists of 44 activities (seeTable 2) In this experiment 119877 denotes a resource used inthe project The resource profile of project before resource-leveling process is shown in Figure 2

RLDE is applied to reduce the resource fluctuation in thisproject Parameter setting for the DE optimizer is shown inTable 3 The projectrsquos resource profile after being optimizedby the RLDE is depicted in Figure 3 The optimal solutionswhich specify the optimal start times of activities are listedin Table 4 The optimal results obtained from the new modelare listed as follows 119891 = 13839 RDmax = 19 RVmax = 7 andCRV = 57

The results comparison between the RLDE andMicrosoftProject 2007 (MP 07) is shown in Table 5The fitness functionof the proposed method (13839) is better than that of MP 07(15245)This means that the resource profile produced by theRLDE is more stable than that obtained from the benchmarkapproach Noticeably the proposed RLDE can reduce the


Table 2 Project information

Activity Duration Predecessor Resource demand (119877)1 1 02 10 1 53 5 1 24 15 1 35 3 1 26 10 1 27 15 2 68 7 3 109 3 5 610 3 5 211 2 5 212 3 4 10 11 613 2 10 114 2 8 12 515 3 12 13 216 1 14 617 1 15 718 1 16 719 4 7 9 17 18 1320 2 15 18 921 2 19 422 1 20 623 3 21 824 1 22 325 4 23 24 826 2 25 727 25 6 528 3 23 629 3 23 230 3 26 931 3 30 1032 3 30 333 2 27 29 30 434 1 32 035 4 33 136 3 34 35 1237 3 36 1238 3 28 31 37 339 5 28 31 37 840 1 36 241 3 38 39 40 1042 1 41 343 6 42 344 1 43 0

peak of resource demand much better than the commercialsoftware The maximum resource variation between twoconsecutive days of the RLDE is 7 which is lower than that ofMP 07 (RVmax = 12) Moreover in terms of the cumulativevariation of the resource demand between two consecutive

Time

Reso

urce

dem

and

0

5

10

15

20

25

30

1 7 13 19 25 31 37 43 49 55 61 67

Figure 2 The original resource profile

02468

101214161820

1 7 13 19 25 31 37 43 49 55 61 67Time

Reso

urce

dem

and

Figure 3 The optimal resource profile produced by the RLDE

Table 3 The RLDErsquos parameter setting

Parameters Notation SettingPopulation size NP 8 times DMaximum generation Gmax 1000Mutation scale F 05Crossover probability Cr 08Note119863 is the number of decision variables

days the proposed approach is also significantly better thanMP 07

To better verify the performance of the proposed RLDEits performance is compared to that of the genetic algorithm(GA) [24] and the particle swarm optimization algorithm(PSO) [25] The GA and PSO algorithms are both exten-sively utilized in the field of optimization and they havebeen applied to solve the resource-leveling problem It isnoted that in this experiment the population size and themaximum number of generations of the two benchmarkalgorithms are set similar to that of the RLDE Thus com-paring the performance of the RLDE with that of the GA andPSO algorithms can point out the advantage of the proposedtechnique for tackling the problem at hand

As can be seen from Table 5 the proposed resource-leveling approach has outperformed the GA and PSO algo-rithms The fitness functions of the RLDE GA and PSO are


Table 4 The optimal solutions of the RLDE method

Activity Start time1 12 23 24 25 26 27 128 129 910 611 712 1913 1814 2215 2116 2417 2618 2519 2720 3121 3122 3323 3324 3425 3626 4027 2028 4529 3930 4231 4832 4533 4534 5035 4736 5137 5438 5839 5740 5741 6242 6543 6644 72

13830 13945 and 13965 respectively This outcome indicatesthat the proposed method can produce a resource profilethat is less fluctuating than those yielded by the GA andPSO algorithms In addition the newly proposed approachalso yields the best performances in terms of the peak of

Table 5 Result comparison

Approach Performance119891 119910max RVmax CRV

Early start schedule 15339 28 13 151Microsoft Project 2007 15245 24 12 139PSO 13965 20 7 72GA 13945 21 7 95RLDE 13839 19 7 57

the resource demand and the cumulative variation of theresource demand between two consecutive days

5 Conclusion

This study investigates the capability of the DE algorithmin solving resource-leveling problem and proposes a newresource-leveling approach named as RLDE to tackle thecomplexity of the problem at hand The experiments andthe result comparisons have demonstrated that the newlyproposed RLDE can overcome the commonly usedMicrosoftProject software as well as the GA and PSO algorithms Thisfact has strongly proved that the RLDE is a promising tool toassist project managers in dealing with the resource-levelingproblemThe future works of this research include evaluatingthe performance of the new approach with different forms offitness function reported in the literature and utilizing noveltechniques to enhance the searching capability of the DEalgorithm

Conflict of Interests

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

References

[1] F A Karaa and A Y Nasr ldquoResource management in construc-tionrdquo Journal of Construction Engineering andManagement vol112 no 3 pp 346ndash357 1986

[2] J Wu and Q An ldquoNew approaches for resource allocation viadea modelsrdquo International Journal of Information Technologyand Decision Making vol 11 no 1 pp 103ndash117 2012

[3] GMaged ldquoEvolutionary resource scheduler for linear projectsrdquoAutomation in Construction vol 17 no 5 pp 573ndash583 2008

[4] S AAssaf and SAl-Hejji ldquoCauses of delay in large constructionprojectsrdquo International Journal of Project Management vol 24no 4 pp 349ndash357 2006

[5] D Arditi and T Pattanakitchamroon ldquoSelecting a delay analysismethod in resolving construction claimsrdquo International Journalof Project Management vol 24 no 2 pp 145ndash155 2006

[6] K El-Rayes and D H Jun ldquoOptimizing resource leveling inconstruction projectsrdquo Journal of Construction Engineering andManagement vol 135 no 11 pp 1172ndash1180 2009

[7] S E Christodoulou G Ellinas and A Michaelidou-KamenouldquoMinimummomentmethod for resource leveling using entropymaximizationrdquo Journal of Construction Engineering and Man-agement vol 136 no 5 pp 518ndash527 2010


[8] S H H Doulabi A Seifi and S Y Shariat ldquoEfficient hybridgenetic algorithm for resource leveling via activity splittingrdquoJournal of Construction Engineering and Management vol 137no 2 pp 137ndash146 2011

[9] J Son and M J Skibniewski ldquoMultiheuristic approach forresource leveling problem in construction engineering Hybridapproachrdquo Journal of Construction Engineering and Manage-ment vol 125 no 1 pp 23ndash31 1999

[10] Y Liu S Zhao X Du and S Li ldquoOptimization of resourceallocation in construction using genetic algorithmsrdquo inProceed-ings of the International Conference on Machine Learning andCybernetics (ICMLC rsquo05) vol 2 pp 3428ndash3432 August 2005

[11] R B Harris ldquoPacking method for resource leveling (pack)rdquoJournal of Construction Engineering and Management vol 116no 2 pp 331ndash350 1990

[12] T Hegazy ldquoOptimization of resource allocation and levelingusing genetic algorithmsrdquo Journal of Construction Engineeringand Management vol 125 no 3 pp 167ndash175 1999

[13] J Geng L Weng and S Liu ldquoAn improved ant colony optimi-zation algorithm for nonlinear resource-leveling problemsrdquoComputers andMathematics with Applications vol 61 no 8 pp2300ndash2305 2011

[14] S Leu C Yang and J Huang ldquoResource leveling in construc-tion by genetic algorithm-based optimization and its decisionsupport system applicationrdquo Automation in Construction vol10 no 1 pp 27ndash41 2000

[15] S Das and P N Suganthan ldquoDifferential evolution a surveyof the state-of-the-artrdquo IEEE Transactions on EvolutionaryComputation vol 15 no 1 pp 4ndash31 2011

[16] K V Price R M Storn and J A Lampinen DifferentialEvolution a Practical Approach to Global Optimization Springer2005

[17] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[18] F Neri and V Tirronen ldquoRecent advances in differential evolu-tion a survey and experimental analysisrdquo Artificial IntelligenceReview vol 33 no 1-2 pp 61ndash106 2010

[19] S Ghosh S Das A V Vasilakos and K Suresh ldquoOn con-vergence of differential evolution over a class of continuousfunctions with unique global optimumrdquo IEEE Transactions onSystems Man and Cybernetics B Cybernetics vol 42 no 1 pp107ndash124 2012

[20] R L Becerra and C A C Coello ldquoCultured differentialevolution for constrained optimizationrdquo Computer Methods inApplied Mechanics and Engineering vol 195 no 33ndash36 pp4303ndash4322 2006

[21] S Das A Abraham U K Chakraborty and A KonarldquoDifferential evolution using a neighborhood-based mutationoperatorrdquo IEEE Transactions on Evolutionary Computation vol13 no 3 pp 526ndash553 2009

[22] E Mezura-Montes C A C Coello E I Tun-Morales S DComputacion and V Tabasco ldquoSimple feasibility rules anddifferential evolution for constrained optimizationrdquo in Proceed-ings of the 3rd Mexican International Conference on ArtificialIntelligence (MICAI rsquo04) April 2004

[23] K Sears G Sears and R Clough Construction Project Man-agement A Practical Guide to Field Construction ManagementJohn Wiley and Son Hoboken NJ USA 5th edition 2008

[24] R L Haupt and S E Haupt Practical Genetic Algorithm JohnWiley and Son Hoboken NJ USA 2004

[25] M Clerc Particle Swarm Optimization ISTE 2006

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resource leveling belongs to the class of combinatorial prob-lem Hence the increasing number of decision variablescauses the problem solving to become infeasible As aconsequencemathematicalmethods are not computationallytractable for real-life projects [10]

Other researchers attempted to utilize heuristic methodsin solving the resource-leveling problem [9 11] Despitethe simplicity of resource-leveling heuristics and their wideimplementation on commercial project management soft-ware (eg Microsoft Project) the result oftentimes can-not satisfy project managers It is because the heuristicapproaches operate on the basis of prespecified rules Thustheir performances are dependent on specific types of prob-lem and on the set of implemented rules Hence they canonly deliver feasible solutions and by no means guarantee anoptimum solution [12]

Due to the limitations of the mathematic and heuristicmethods the application of evolutionary algorithms (EAs)for resource leveling has attracted more attention in recentyears [13 14] EAs inspired by the natural evolution ofspecies have been successfully utilized to tackle optimizationproblems in diverse fields [15] Evolutionary computationis characterized by iterative progresses used to guide therandomly initiated population to the final optimal solution

Recently differential evolution (DE) [16ndash18] has drawnincreasing interest of researchers who have explored itscapability in a wide range of problems DE is a population-based stochastic search engine which is efficient and effectivefor global optimization in the continuous domain [19] It usesmutation crossover and selection operators at each gener-ation to move its population toward the global optimumSuperior performance of DE over other algorithms has beenverified in many reported research works [15 17 20 21]Nevertheless none of the previous works has investigated thepotentiality of DE in solving resource-leveling problem

Thus this paper proposes a novel approach for resourceleveling named as resource leveling based on differentialevolution (RLDE) RLDE is developed based on the DEalgorithm as the search engine The rest of the paper isorganized as follows In Section 2 the research methodologyis presented The structure of the proposed RLDE is shownin Section 3 The performance of the newly developed modelis demonstrated in Section 4 Finally some conclusions arementioned in the last section of this paper

2 Research Methodology

21 The Resource-Leveling Problem In the resource-levelingproblem the objective is to reduce the peak resource demandand smooth out day-to-day resource consumptions withinthe required project duration and with the assumption ofunlimited resource availability Hence the resource levelingcan be expressed as an optimization problem within whichthe following cost function is minimized [9 12]

119891 =119879

sum119894=1

(119910119894minus 119910119906)2 (1)


resource requirements of the activities performed at time unit119894 and 119910

119906represents a uniform resource level given as follows

119910119906=sum119879

119894=1119910119894

119879 (2)

Equation (1) means that if the value of the objective func-tion 119891 is small the actual resource profile of the project willbe close to the form of the ideal resource profile In the idealresource profile the resource demand is constant throughoutthe project duration Moreover (1) can be rewritten in thefollowing form

119891 =119879

sum119894=1

1199102119894minus 2119910119906

119879

sum119894=1

119910119894+ 1199102119906 (3)

Because activity duration and rate of resource for eachactivity are fixed 119910

119906and sum

119879

119894=1119910119894are constant Thus the cost

function can be expressed as follows

119891 =119879

sum119894=1

1199102119894 (4)

22 Differential Evolution Differential evolution (DE) is anevolutionary algorithm proposed by Storn and Price [16 17]which is designed for real parameter optimization problemsThe algorithm of DE is depicted in Algorithm 1 In this figureit is noted that NP represents the size of the population119883119895119894

is the 119895th decision variable of the 119894th individual inthe population 119892 is the current generation and 119863 denotesthe number of decision variables rand

119895(0 1) is a uniform

random number lying between 0 and 1 and rnb(119894)

is arandomly chosen index of (1 2 NP)

DE is based on the utilization of a novel crossover-mutation operator based on the linear combination of threedifferent individuals and one subject-to-replacement parent(or target vector) [20] The crossover-mutation operatoryields a trial vector (or child vector) which will competewith its parent in the selection operator These two termstrial vector and child vector can be used interchangeablyThe selection process is performed via selection between theparent and the corresponding offspring [22]

In the selection process the trial vector is compared to thetarget vector (or the parent) using the greedy criterion [17]These two terms target vector and parent vector can be usedsynonymously Under the greedy criterion the fitness value isthe only information that determines which vector is allowedto survive in the next generation (see Figure 2) Thus if thetrial vector can yield a lower objective function value than itsparent then the trial vector supersedes the target vector Theselection operator is expressed as follows

119883119894119892+1

= 119880119894119892

if 119891 (119880119894119892) le 119891 (119883

119894119892)

119883119894119892

if 119891 (119880119894119892) gt 119891 (119883

119894119892)

(5)

where 119883119894119892

represents the parent vector at generation 119892 119880119894119892

denotes the trial vector at generation 119892 119883119894119892+1

is the chosenindividual which survives to the next generation (119892 + 1)




1 1199032 and 119903

3isin (1NP)

with 1199031

= 1199032

= 1199033


119880119895119894119892

= 1198831198951199033 119892

+ 119891 times (1198831198951199031 119892

minus 1198831198951199032 119892

) if rand119895(0 1) lt Cr or 119895 = 119903119899119887(119894)

119883119895119894119892



119894if 119880119894is better




Yes



No


Mutation

Crossover

DE optimizer


Scheduling module

Selection

[X11 X12 X1D]

[X1 X2 XD]

[XNP1 XNP2 XNPD]











sum119896=1

(119910119896)2



119879minus1

sum119896=1

1003816100381610038161003816119910119896+1 minus 119910119896

1003816100381610038161003816


119896=11003816100381610038161003816119910119896+1 minus 119910

119896

1003816100381610038161003816




119883 = [1198831198941 1198831198942 119883

119894119863] (6)



(7)




119891 =119879

sum119896=1

(119910119896)2 (8)

subject to


119894ge 0 119894 = 1 2 119863 (9)



119894and TF













Time

Reso

urce

dem

and

0

5

10

15

20

25

30

1 7 13 19 25 31 37 43 49 55 61 67


02468

101214161820

1 7 13 19 25 31 37 43 49 55 61 67Time

Reso

urce

dem

and















5 Conclusion




References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












1 1199032 and 119903

3isin (1NP)

with 1199031

= 1199032

= 1199033


119880119895119894119892

= 1198831198951199033 119892

+ 119891 times (1198831198951199031 119892

minus 1198831198951199032 119892

) if rand119895(0 1) lt Cr or 119895 = 119903119899119887(119894)

119883119895119894119892



119894if 119880119894is better




Yes



No


Mutation

Crossover

DE optimizer


Scheduling module

Selection

[X11 X12 X1D]

[X1 X2 XD]

[XNP1 XNP2 XNPD]











sum119896=1

(119910119896)2



119879minus1

sum119896=1

1003816100381610038161003816119910119896+1 minus 119910119896

1003816100381610038161003816


119896=11003816100381610038161003816119910119896+1 minus 119910

119896

1003816100381610038161003816




119883 = [1198831198941 1198831198942 119883

119894119863] (6)



(7)




119891 =119879

sum119896=1

(119910119896)2 (8)

subject to


119894ge 0 119894 = 1 2 119863 (9)



119894and TF













Time

Reso

urce

dem

and

0

5

10

15

20

25

30

1 7 13 19 25 31 37 43 49 55 61 67


02468

101214161820

1 7 13 19 25 31 37 43 49 55 61 67Time

Reso

urce

dem

and















5 Conclusion




References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













sum119896=1

(119910119896)2



119879minus1

sum119896=1

1003816100381610038161003816119910119896+1 minus 119910119896

1003816100381610038161003816


119896=11003816100381610038161003816119910119896+1 minus 119910

119896

1003816100381610038161003816




119883 = [1198831198941 1198831198942 119883

119894119863] (6)



(7)




119891 =119879

sum119896=1

(119910119896)2 (8)

subject to


119894ge 0 119894 = 1 2 119863 (9)



119894and TF













Time

Reso

urce

dem

and

0

5

10

15

20

25

30

1 7 13 19 25 31 37 43 49 55 61 67


02468

101214161820

1 7 13 19 25 31 37 43 49 55 61 67Time

Reso

urce

dem

and















5 Conclusion




References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













Time

Reso

urce

dem

and

0

5

10

15

20

25

30

1 7 13 19 25 31 37 43 49 55 61 67


02468

101214161820

1 7 13 19 25 31 37 43 49 55 61 67Time

Reso

urce

dem

and















5 Conclusion




References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















5 Conclusion




References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article a novel resource-leveling approach for...

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