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Expert Systems With Applications

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Project resources scheduling and leveling using Multi-Attribute Decision Models: Models implementation and case study

Ch. Markou, G.K. Koulinas, A.P. Vavatsikos

^∗

Production and Management Engineering Dept., Democritus University of Thrace, Vas. Sofias 12, Xanthi 67100, Greece

a rt i c l e i nf o

Article history:

Received 19 September 2016 Revised 16 December 2016 Accepted 25 January 2017 Available online 2 February 2017 Keywords:

Project management Project scheduling Resource leveling

Multi-Attribute Decision Models

a b s t ra c t

ProjectschedulingisoneofthemostvitalprocessesinProjectManagement.Itisawidelydiscussedtopic inacademicandpracticalcirclesduetoitsimportanceand complexity.Manpower,machines,materials andequipmentareusedfortheexecutionofprojectactivities,butthesemostlyhavelimitedavailability, whichcanconstrainprojectschedulingprocedures.Projectresources mightexceedorfallshortofthe resource demandinaproject’stimehorizon.Theseconsiderations present issuestoprojectmanagers whomusttrytoproperlyallocateamongthesedemandsinordertoachieveanearoptimalutilization duringaproject’slifetime.Resourcelevelingisamongthegreatestchallengesfacedbyprojectmanagers asthesuccessofaprojectlargelydependsonit.Thisisbecausepeaksandvalleysintheresourceus- agehistogramareresponsibleforcostoverrunsduetothenecessaryrecruitment,dismissalandtraining ofthepersonnelMoreover,issuesmayariseregardingtheefficient managementofavailableresources giventhatlarge peakscorrespondtofluctuationsinresourceallocationduringaproject’slifecycleor constructionperiod.Toaddresstheseissues,resourcelevelingprovidesproceduresandframeworksthat ensuretheefficientmanagementofresourcestoobtainsmoothresourceusageprofiles.Theseprocedures attempttoidentifyactivitiesthatshouldbedelayedtoresolveresourceover-allocationsundertimeand cost constraints. Given theexistence ofa varietyof availablerulesthat could befollowed byproject managerstoprioritizeactivities,thepaperathandexaminestheimplementationoffiveMulti-Attribute DecisionMaking modelsandhowtheyperformintheschedulingofasolarpark constructionproject.

NamelythesemodelsaretheWeightedSumMethod,AnalyticHierarchyProcess,PROMETHEE,TOPSIS, OrderedWeightedAverage(OWA)andHybridWeightedAverage(HWA).Finally,thederivedresultsare discussedincomparisonwiththoseobtainedbythestandardresourcelevelingproceduresofMS-Project.

© 2017 Elsevier Ltd. All rights reserved.

1. Introduction

The efficient management of available resources is one ofthe greatestandmostcomplexproblemsthatprojectmanagers(PMs) haveto overcome.TheResource LevelingProblem(RLP)isa clas- sicresourcemanagementproblemfacedbypractitioners,managers andresearchers.Resourcelevelingaimstominimizepeaksandval- leysintheresourcehistogramwithoutincreasingtheprojectdura- tionbeyondtheoriginalcriticalpathduration(Harris,1990;Shtub, Bard, & Globerson, 2005). However, the problem can also come tobear incases oflimited resources, which oftenlead to exten- sions of the initial project duration (Hiyassat, 2001; Neumann &

Zimmermann,2000).Duringthepastsixdecades,severaldifferent approacheshave beendeveloped forsolving theRLP. Exact algo- rithmshavebeenproposedintheliterature,includingintegerand

∗ Corresponding author.

E-mail addresses: charalamposmarkou@outlook.com (Ch. Markou), gkoulina@pme.duth.gr (G.K. Koulinas),avavatsi@pme.duth.gr (A.P. Vavatsikos).

dynamic programming techniques (Bandelloni, Tucci, & Rinaldi, 1994;Neumann&Zimmermann,2000).Theseapproachesaresuit- able for small-sized networks due to the so-called “combinato- rialexplosion” phenomenon.Severalheuristicprocedureshavealso beendevelopedto overcometheRLP,mostofthemare basedon shifting heuristicsor priority rule methods (Burgess & Killebrew, 1962;Neumann&Zimmermann,2000).Inaddition,metaheuristic approachessuchasgeneticalgorithms(Kyriklidis&Dounias,2016;

Kyriklidis,Vassiliadis,Kirytopoulos,& Dounias,2014;Leu,Yang, &

Huang,2000;Li&Demeulemeester,2016;Ponz-Tienda,Yepes,Pel- licer, & Moreno-Flores,2013) andsimulated annealingalgorithms (Anagnostopoulos&Koulinas,2010;Son&Skibniewski,1999)have attemptedto findanoptimum solutiontothisproblem. Recently, hyperheuristicalgorithms proposed totreat theRLP andresource allocation problems (Anagnostopoulos & Koulinas, 2010; Koulinas

& Anagnostopoulos, 2011; Koulinas, Kotsikas, & Anagnostopoulos, 2014)haveofferedsomepromisingresults.Thebasicideaofthese approachesistocreatearesource profilebasedontheearlystart

http://dx.doi.org/10.1016/j.eswa.2017.01.035 0957-4174/© 2017 Elsevier Ltd. All rights reserved.

Fig. 1. RLP solution options.

schedulecalculatedfromtheCriticalPathMethod(CPM),andthen shiftnoncriticalactivitiesaccordingtofixedheuristicrules.

ThispaperdescribestheapplicationofMulti-AttributeDecision Making (MADM)methodsto definethe activityprioritiestotreat resourcelevelingunderconstraintswhenpriorityrulemethodsare implemented.Theproposed frameworkaimstooptimizeresource usagewithoutexceedingapre-determinedresourcelimit.Thisgoal isaccomplishedbyallowingPMs’participationtotheprioritiesde- terminationphaseof.Avarietyofwell-establishedMADMmodels isimplemented enablingtheperformance ofavariety ofdecision attitudes.

2. ResourceLevelingProblem

Achieving smooth resource profiles is a major goal during project implementation. RLP is a technique that adjust activi- ties’ start andfinish datesaccording toresource constraintsaim- ing to balance their demand with the available supply (PMBOK 2013). Fluctuations in resource usage histograms represent addi- tional costs in a resource usage profile. An ideal resource usage profile has a continuousandplateau-shaped resource histograms whereinstancesofbothoverandunder-allocationareminimized.

Resourcelevelingisrequiredtomeetlimitationsofresourceavail- ability by reducing the peak of daily resource requirements and determinetheminimumamountofresourcesnecessarytoaccom- plish a project by a specified finish date. To model the RLP, an acyclicactivity-on-node(AoN)representationofaprojectnetwork is used. The network consists of n activities (nodes) and prece- dence relations (arcs) betweenactivities. We denote d_i andf_i the duration and the finish time of activity i, respectively, and as- sumethatnode1(n)isthedummyactivityoftheprojectbeginning (completion) with no ingoing (outgoing) arcs. The RLP can then bedefinedasanoptimizationmathematicalprogrammingproblem accordingtoEqs.(1)–(5).

minMx=

fn

t=1

u²_t (1)

subjectto

fi≤fj−dj forallprecedencerelations

(

ⁱ,j

)

⁽²⁾

f1=0, d1=0, dn=0 (3)

fn≤f_n^P (4)

i∈Pt

uit≤Ufort=1,...,fn (5)

Thesolutionprocedureaimstofindafeasibleschedulesothat the moment Mxaround thehorizontal axis ofthe resource usage histogramis minimized(Eq.(1)). Eq.(2) assures that precedence relations arenot violated and Eq.(3)establishes that the project startsattimemomentzero(0).AsingleresourceRisassignedto eachactivity,andtheusageofthatresourceisassumedtoremain constant throughout the progress of the activity. Resource avail- abilityUisalsoconstant duringexecutionoftheproject.Thesum ofresourceusage u_itofallongoing activitiesP_tinanytimeperiod tcannotexceedU(Eq.(5)).

3. Methodologicalframework 3.1. Decisionflowchart

Theproposedframeworkaimstoelicitpriorities,whenpriority- rules heuristic procedures are implemented, in order to improve the shape ofthe resource usage histogram(Fig.1). Forexample, MS-Projectallowsuserstosetprioritiesforspecifictaskstocontrol howtheyareleveledinrelationtooneanother.Prioritiesarespec- ifiedeitherasnumbers(0–1000)oraslinguisticvalues(lowestto highest)withthe“highest” (or1000)prioritycorrespondingto"Do notlevel.” Becausetaskprioritieshaveanimpactontheschedule, itispossible toaffectlevelingby alteringassignedpriorities.The finalscheduleofactivitiesandCPMcalculationisaccomplishedby MS-Project.However,thelackofcoherentproceduresforestablish- ing prioritiesandtheintuitional considerationofa variety ofac- tivitiesdelayselection rules,canresultintheappearanceofblack boxes duringtheprioritization phase.Giventhat solutions to the RLPaimtorankadiscretesetofalternatives (activities)underthe consideration of a range of decision criteria (priority rules), this paperexamines howa variety ofMADMmodels performin elic- itingprojectactivityprioritiesandproviding flexibilityduringthe resourceallocationresolutionprocedure.Onthecontrarystandard heuristic approaches have no prior knowledge about the search spaceandthespecificcharacteristicsofeachproblem,andsothey need to be run severaltimes in orderto achieve a nearoptimal solution.

3.2.Activitiesprioritizationcriteria-rules

Tosupport PMs during the prioritization process, a variety of priorityrulesexistthatassist activityselection duringtheheuris- ticprocedure(Table1).Priorityrulesdeterminetheorderinwhich activity will be selected during the heuristic procedure, ranking them in a priority list with respect to their dependencies. As such, an activity cannot be selected before its predecessor activ- ity.Selectingpriorityrulesduringtheprojectschedulingphase is

Table 1

Major priority rules.

Activity based rules Network based rules Critical path based rules Resource based rules

Shortest Processing Time (SPT) Most Immediate Successors (MIS) Earliest Start Time (EST) Greatest Resource Demand (GRD)

Longest Processing Time (LPT) Most Total Successors (MTS) Earliest Finish Time (EFT) Greatest Cumulative Resource Demand (GCRD) Random (RND) Least Non-Related Jobs (LNRJ) Latest Start Time (LST) Resource Equivalent Duration (RED).

Greatest Rank Positional Weight (GRPW) Latest Finish Time (LFT) Weighted Resource Utilization and Precedence (WRUP).

Greatest Rank Positional Weight ^∗(GRPW ^∗) Minimum Slack (MSLK)

Fig. 2. Educational example illustration.

based on the project’s characteristics; rules are chosen to facil- itate the scheduling process depending on the assumptions and the experience of the project scheduling team. According to the literature,priority rules areclassified intofourgeneral categories basedon the type of informationneeded to constructthose pri- oritylists(Demeulemeester&Herroelen,2002).Namely,theseare (a)Activity-basedrules,(b)Network-basedrules,(c)CriticalPath- basedrulesand(d)Resource-basedrules.

Activity-basedrulesintroducethreedifferentpriorityrules:SPT, orShortestProcessing Time;LPT,orLongestProcessingTime;and RND, or Random. The first two are based on activity durations whilethe last priorityrule is usedwhen there islimited knowl- edge and information about the project. In these cases, RND is usedas a benchmarkagainst which the other priority rules may becompared(Hartmann,1999).

Network-based rules include priority rules based on a project networkdiagram.Therearefivedifferentrulesbasedontheprece- dence relationships of project activities: MIS, or Most Immedi- ate Successors; MTS, or Most Total Successors, LNRJ, or Least Non-Related Jobs; GRPW, or Greatest Rank Positional Weight;

and GRPW^∗, or Greatest Rank Positional Weight^∗. The first two network-basedpriorityrulesusedirectprecedencerelationsofthe activitiestocreateaprioritylist.Thethirdruledeterminestheac- tivitiesthat canbe scheduled inparallel withthe considered ac- tivity.The lasttwo priorityrules weight theprecedencerelations withthe duration of the corresponding activities. The difference betweenthem is that GRPW considers the durationssum ofthe activitiesimmediatesuccessorswhileGRPW^∗examinesallitssuc- cessorsdurationssum(Demeulemeester&Herroelen,2002)

Critical path-based rules consists of priority rules based on a project’scriticalpath.TheseincludeEST,orEarliestStartTime;EFT, orEarliestFinishTime;LST,orLatestStartTime;LFT,orLatestFin- ishTime;andMSLK,orMinimumSlack.Theserulescanbecalcu- latedatthestartoftheschedulingprocess.

The fourth category includes priority rules based on the re- source utilization of project activities. These are GRD, or Great- est Resource Demand; GCRD, or Greatest Cumulative Resource Demand; RED, or Resource Equivalent Duration; and WRUP, or WeightedResourceUtilizationandPrecedence.WRUP,proposedby

UlusoyandÖzdamar (1995)andÖzdamar(1999),istheweighted sum ofthe number of successors (weight=0.7) and the average resourceutilizationforallresourcetypes(weight=0.3).

Theselectedheuristic rulesare consideredtobe themostim- portant and most frequently used among seventy three priority rulesevaluatedinKlein(2000).Anypriorityruleapplicationleads to a schedule construction by selecting and programmingactivi- tiesprioritization accordingtothe criterion definedbyeach rule.

Forexample,applying theMSLK rule meansthat withrespect to the precedencerelations, the first activityselected to be delayed istheone havingthesmallestslack. Inthat manneralist ofpri- oritized activities is the output. Similarly, applying the rule GRD results to a different list, asit selects activity in each loop with respect to the precedence relations and the largest demand for resources. Fig. 2 illustrates an educational example ofthe above rules implementation to a project with a resource constraint at seven units. Each priority rule implementation provides resolu- tion tothe RLP,howeverthederived schedules are different.The first outcome resolves RLPwithout provokingoverrun to the ini- tialtimewindow,whilethesecondexpandstheprojects’duration by two time units. Thus the indirect-costs will remain constant while possible penalty clausing risk is avoided with the derived MSLK-basedschedule.Nevertheless,withrespecttoresourcesallo- cationGRD-based schedulepresentssmootherresource allocation profileanditdemandsoneunitlessthantheresourceavailability.

Asa resulttheprojectminimizesunder-allocationresource usage issues andmoreoverit can adapteasierto address unpredictable resources shortageduringprojectexecution. Moreover theMSLK- based schedule can considered risk-averse since technicallynone oftheactivities canbedelayedwithoutproducing over-allocation issues.

3.3. Multi-AttributeDecisionModels

These models provide sophisticated tools for handling semi- constructeddecisionproblems whereavariety ofalternatives are to be evaluated under the consideration of a variety of criteria and they are suitable for selecting and ranking candidate solu- tions.The performedframework aimstoestablishactivitiesprior-

ities using MADM models. During the years a variety of MADM approaches have been developed (e.g. Figuera, Greco, & Ehrgott, 2005; Hwang&Yoon 1981). Ingeneral, thesemodels canbe cat- egorized as compensatory or non-compensatory decision models (Yoon & Hwang1995). The above distinction ismade on theba- sis ofdecision models’ allowance trade-offs among attributes. In a compensatory decision model inferior performances of an al- ternative to costcriteria (descendcriterion type)can be counter- balanced by additionalsuperiority to benefitcriteria (ascend cri- terion type). Onthe contrary non-compensatorydecision models provide proceduresthat allows DecisionMakers’ (DMs)tohandle the levelofthe permitted trade-offs.Withrespect tothe current framework a varietyofapproachesshould beprovided to PMsto enablethemwiththeabilitytodefinetheirdecisionattitudewith respecttotheanalysisenvironment.Inparticular,thispaperexam- ines a varietyofboth compensatory andnon-compensatorydeci- sionmodelsimplementationtoresolveRLPthrougharealcaseex- ample. WeightedSum Method(WSM), AnalyticHierarchy Process (AHP) (Saaty, 1980), Preference Ranking Organization Method for Enrichment Evaluation(PROMETHEE)(Brans,Vincke, &Mareschal, 1986) areperformedascompensatorymodels representatives.On the other hand Technique for Order Preference by Similarity to IdealSolutions(TOPSIS) (Huang&Yoon, 1981),OrderedWeighted Average(OWA)withlinguisticquantifiers (Malczewski, 2006) and Hybrid WeightedAverage(HWA) (Xu &Da,2003) standsasnon- compensatoryapproaches.

3.3.1. WeightedSumMethod

The WSMisthemostcommonlyused MADMmodelandmay be expressed as follows: givena decision problemwith m alter- natives(activities)andncriteria(priorityrules), thescoreofeach alternative is derived from theweighted sumof its standardized performance to the analysiscriteria(Eq.(6)). Giventhat the per- formances ofalternativestothe analysiscriteriaareestimatedon differentscales ofmeasurement,standardization – alsoknown as utilizationornormalization– allowstheusertoobtaincomparable andconsistentscalesofpreferenceintensityforcomparison.Thisis achievedthrough maximumscore,vector, orscore-rangenormal- izationapproaches(Hwang,Lai,&Liu,1993;Yoon&Hwang,1995) presentedinEqs.(7)–(9),wherex_ijistheperformanceofithalter- nativetothejthcriterionandmax/minf_jthemaximum/minimum observed performance the same criterion (i=1,…,m; j=1,…,n).

ThenWSMscoreisestimatedaccordingtoEq.(6). U_i^wsm=

n

j=1

u_{i j}= n

j=1

w_j×r_{i j} (6)

ri j= x_{i j}

x^max_j (7)

r_{i j}= xi j

m i=1x²_{i j}

(8)

r_{i j}= xi j−x^min_j

x^max_j −x^min_j (9)

Whereu_ij istheweightedstandardized performanceoftheith alternativetothejthcriterionforeachi=1,2,...,m,r_ij thestan- dardized performanceandw_jisthecriterion weight.Thebest al- ternativehasthehighestoverallvalue(Chang&Yeh,2001).

3.3.2. AnalyticHierarchyProcess

The Analytic Hierarchy Process (AHP) is a decision making modeldevelopedby Saaty(1980).Itisusedin decisionproblems where thereare multiple conflicting attributes. Its objectiveis to

identifythe best alternative by rankingall ofthe proposed solu- tions when all of the problemcriteria are considered simultane- ously.TheAHPinvolvestheimplementationoffoursteps.Thefirst stepistoformhierarchicalstructures(hierarchies)inordertoan- alyze the decision making problem by its constituent parts. The problem is defined in a hierarchical breakdown structure where theoverallanalysisobjectivecreatesthetoplevelofthehierarchy, followedbylevelsofsub-objectives anddecisioncriteria.Alterna- tive scenarios arefound atthe bottomlevel. Problemcriteriaare identifiedbyexpertjudgmentsbasedonthecharacteristicsofthe decisionathand.Thesecondstepistoestablishtheframeworkfor criterionweightelicitation.Thisestimationisachievedbyforming pairwise comparisonmatrices (denotedasA) that allow decision analysiscriteriaevaluationby comparing theimportance ofchild nodeswithrespecttotheirparentnodes.Pairwisecomparisonsare madeusing the linguisticterms of a DMs judgments,which are quantifiedusinga numericalscaleof1to9.Thevalue ofa_ij ofA representstheimportanceofcriterioniovercriterionj,withtheir respectiveimportanceelicitedfromtheprincipaleigenvectorofthe pairwisecomparisonsmatrix.Thethirdstepistoassessthestan- dardizedperformancesofalternatives.Thisisachievedbyforming analternativespairwise comparisonmatrixforeverydecisioncri- terion.Finally, thealternatives are ranked accordingtotheir nor- malizedadditive value (Eq.(6)), wherew_j isthe overallcriterion weightwithrespecttotheanalysis’sprimary objective.Eachpair- wisecomparisonmatrixintheAHPmethodundergoesconsistency testing,leadingtostableresults.TheAHPmethod,therefore,mod- elsthedecisionproblemasahierarchicalstructurethatillustrates theprobleminasimpleway.

3.3.3. PROMETHEE

ThePROMETHEEis amethodusedtoconstructoutrankingre- lationsbetweenasetofalternatives tofindtheoptimumsolution (Brans et al., 1986). This framework is able to combine both in- formationbetween-criteria, using criterion weights, andinforma- tion within-criterion, using the amplitude of the deviations be- tween each pair (i,k) of alternatives over N analysis criteria (Eq.

(10)) (Brans & Mareschal, 2005). To address the scaling effect, PROMETHEEprovidessixdifferenttypes(Usual,U-Shape,V-Shape, Level,V-Shapewithindifference andGaussian)ofgeneralizedcri- teria(preferencefunctions)F_jtoestimate theindifferencearea of onealternativeovertheotherundertheconsiderationofcriterion j(Eq.(11)).

d_j

(

^i,k

)

^=xi j−x_{k j} (10)

P_j=F_j

d_j

(

^i,k

)

(11) Thedegree ofpreferenceofalternativeiover alternativekfor alltheanalysiscriteriaisthenderivedaccordingtoEq.(12).When

π

^{(i,k) tends to zero,}

π

^{indicates a weak preference of i over k}

while values closer to 1 indicate a strong preference (Brans &

Mareschal,2005).Toconstructtheoutrankinggraph,PROMETHEE calculatestheoutgoingandincomingflowforeachnodeaccording toEqs.13and14:

π (

^i,k

)

⁼ n

j=1

w_jP_j

(

^i,k

)

⁽¹²⁾

⁺

(

ⁱ

)

= 1 n−1

m

k=1

π (

i,k

)

⁽¹³⁾

⁻

(

ⁱ

)

= 1 n−1

m

k=1

π (

^k,i

)

⁽¹⁴⁾

The PROMETHEE method is capable of providing both partial (PROMETHEEI) andfull (PROMETHEEII) rankings ofalternatives.

Partialrankingisobtainedusingpairwisecomparisonsoftheout- goingandincomingalternativeflows.Throughpartialranking,the methodcanconcludethatonealternativeispreferred(P)overthe other (Eq.(15)), that the two alternatives are indifferent (I) (Eq.

(16))orthatthetwoalternativesareincomparable(R)(Eq.(17)).

iPk=

⎧ ⎨

⎩

⁺

(

ⁱ

)

^>

⁺

(

^k

)

^and

⁻

(

ⁱ

)

^<

⁻

(

^k

)

⁺

(

ⁱ

)

⁼

⁺

(

^k

)

^and

⁻

(

ⁱ

)

^<

⁻

(

^k

)

⁺

(

ⁱ

)

^>

⁺

(

^k

)

^and

⁻

(

ⁱ

)

⁼

⁻

(

^k

)

(15)

iIk=

⁺

(

ⁱ

)

⁼

⁺

(

^k

)

^and

⁻

(

ⁱ

)

⁼

⁻

(

^k

)

⁽¹⁶⁾ iRk=

₊

(

ⁱ

)

^>

⁺

(

^k

)

^and

⁻

(

ⁱ

)

^>

⁻

(

^k

)

⁺

(

ⁱ

)

^<

⁺

(

^k

)

^and

⁻

(

ⁱ

)

^<

⁻

(

^k

)

⁽¹⁷⁾

PROMETHEE II is able torank alternatives by estimatingeach alternative’snetflowasthedifference betweenpositive andneg- ativeflows (Eq.(18)) and then ordering those net flows. In con- trasttopartial ranking,PROMETHEE II doesnotprovide informa- tionregardingpossibleincomparabilities.The alternativewiththe highestnetflowisconsideredtobethemostpreferredalternative (Eq.(19)), while alternatives withequal netflows are considered indifferent(Eq.(20)).

^net=

⁺

(

ⁱ

)

⁻

⁻

(

^k

)

⁽¹⁸⁾

iPk=

(

ⁱ

)

^>

(

^k

)

⁽¹⁹⁾

iIk=

(

ⁱ

)

=

(

^k

)

⁽²⁰⁾

3.3.4. TOPSIS

TOPSIS is amethod forrankingalternatives accordingto their distancefromthepositive andnegativeidealsolutions. TOPSIS is basedonthe concept that themostpreferable alternativeshould beonethatisascloseaspossibletotheidealsolutionandasfar aspossiblefromtheanti-idealsolution(Yoon&Hwang,1995).The separationmeasure isobtainedusingtheEuclideandistancemet- ric.ToimplementTOPSIS,sixstepsarerequired.Theperformance ofalternativesshouldfirstbestandardizedtoobtainacomparable scaleformeasurement.Then, theweightednormalizedmatrix u_ij canbe obtainedaccordingto Eq.(21),wherew_jis theweight of criterionj.

ui j=wj∗ri j (21)

As long as the weighted normalized matrix is formed, both ideal and anti-ideal solutions can be reached. In particular, the idealsolutionisobtainedbythealternatives’highestscoresinben- efitattributessetI’andthelowestscoresinthecostattributesset I” (Eq.(22)).Correspondingly,theanti-idealsolutionisobtainedin ananalogouswayaccordingtoEq.(23)(Yoon&Hwang,1995).

A⁺=

u⁺₁,....,u⁺_m

=

max_iu_{i j}

|

^j∈I

,

min_iu_{i j}

|

^j∈I

(22)

A⁻=

u⁻₁,....,u⁻_m

=

miniui j

|

^j∈I

,

maxiui j

|

^i∈I

(23) Oncebothidealandanti-idealsolutionsareestimated,thedis- tancesofalternative solutionsfromthemare calculatedusingthe Euclideandistancemetric(Eqs.24and25).Finally,arankingofal- ternativesisobtainedthroughtheestimationofrelative closeness c_iusingEq.(26).

D⁺_i =

_n

j=1

u_{i j}−u⁺_i

2

(24)

D⁻_i =

_n

j=1

u_{i j}−u⁻_i

2

(25)

c_i= D⁻_i

D⁺_i +D⁻_i

⁽²⁶⁾

3.3.5. OrderedWeightedAverage(OWA)

TheOWAcanillustratetheDM’sriskattitudebyintroducingan aggregation operator based on linguistic quantifiers. The method usestwo differenttypesofweights:criterionweights aredefined by the DMduring problem quantification andorder weights are obtained by an aggregationoperator when considering the DM’s degree of optimism. This operator can be a Regular Increasing Monotone (RIM) quantifier, as shown in Eq. (27) (Yager, 1996).

Here,Q(r) is a fuzzy set in interval [0,1]. By adjusting parameter

α

^{,wecangeneratedifferentsetsofweights regardingaDM’srisk}

attitude. For

α

=1,the risk attitude ofthe DM is considered in- different.As

α

^{tends to0, risk acceptance increases,indicating a}

veryoptimisticDM.As

α

^{tendstoinfinity,theDM’sriskacceptance}

decreases and the DM has a pessimistic attitude (risk aversion) (Malczewski, 2006).Giventhoseconsiderations,orderweights are obtainedaccordingtoEq.(28).

Q

(

^r

)

=r^a,a>0 (27)

uj=

_j

k=1

u_k

a

−

_j

−1

k=1

u_k

a

(28)

As soon asboth order u_j and criterion weights w_j are calcu- lated,theOWAindexofeach alternativeiscalculatedbyordering thecriteriavaluesinadecreasingorder.Criterionweightsarethen analogouslyreorderedand,finally,thealternativerankingsareob- tained according to Eq. (29)(Yager, 1988). The OWA can be im- plemented when considering differentrisk scenarios by applying differentparametersof

α

^.

OWA_i= n

j=1

⎛

⎜ ⎜

⎝

^uj

υ

j

n j=1

u_j

υ

j

⎞

⎟ ⎟

⎠

^{zi j (29)}

3.3.6. HybridWeightedAverage(HWA)

The HWA is a method similar to OWA, considering both cri- terion values and criterion importance (Xu & Da, 2003). HWA is an aggregation operator and can be computed using Eq. (30). Here, v_j is the order weights calculated in the same wayas the OWA,b_j isthebiggestvalue ofsetnw_ia_i inascending order,W= (^w1,w₂,...,w_n)^{T denote the criterion weights, a}i(ⁱ=1,2,...,n)^, and n is a balance parameter. If W=(^w1,w₂,...,wn)^{T tends} to (¹n,¹n,...,¹n)^{T, then nw}na_i=a_i and the HWA is equal to the OWA(Mirabi,Mianabadi,Zarghami,Sharifi,&Mostert,2014;Xu&

Da,2003).

HWA_v,w

(

^a1,a2,...,an

)

⁼ n

j=1

v

jb_j (30)

4. Frameworkimplementation

ToillustratehowMADMmethodsaddressRLP,we applythese approaches to a casestudy regardingthe construction of a solar park(100kWcapacity) in theregion of Thessaloniki inNorthern Greece.Thearea ofthe proposed constructionlocation 19,856m² and the actual construction size of the installation is 1600m².

Table 2

Solar park project WBS and scheduling information.

Level 1 project Level 2 work packages Level 3 activities Relationship with predecessor activity

Relationship with successor activity

Duration

1. Landscaping 1.1 Site preparation and leveling – 1.2SS 2

1.2 Fence installation 1.1SS 2.1SS 1

2. Earthworks 2.1 Base holes excavation 1.2SS 2.2SS + 1d;3.1FS 1

2.2 Ditches excavation 2.1SS + 1d

2.3SS + 1d;2.4SS + 1d;5.1SS 5

2.3 Concrete implementation 2.2SS + 1d 5.8FS 2

2.4 Pillar excavations 2.2SS + 1d 5.8FS 1

3. Bases construction 3.1 Bases construction 2.1FS 3.2FS 2

3.2 Bases audit 3.1FS 5.6FS 1

Solar park project 4. Panel delivery 4.1 Panel delivery 5FS 0

5. Electrical infrastructure 5.1 Grounding 2.2SS 5.2SS + 1d;5.3FF 6

5.2 DC cables installation 5.1SS + 1d 5.3SS + 1d;5.4SS;5.9FS 7

5.3 AC cables installation 5.2SS + 1d;5.1FF 5.7FS 6

5.4 Security System cables 5.2SS 6.1SF 1

5.5 Panel & Inverter delivery 5.6FS;5.7FS 1

5.6 Panel installation 3.2FS;5.5FS 5.9FS 2

5.7 Inverter installation 5.3FS;5.5FS 5.9SS 1

5.8 Pillars installation 2.3FS;2.4FS 5.9FS 1

5.9 Connection of modules 5.2FS;5.7SS;5.6FS;5.8FS 6.1FS;7.1FS 2 6. Security system 6.1 Surveillance system installation 5.4SF;5.9FS 7.1SS 1

7. Testing 7.1 Test runs 5.9FS;6.1SS 8.1FS + 1d 1

8. Project completion 8 .1 Connection with national grid 7.1FS + 1d 8.2FS 1

8.2 Closing proceedings 8.1FS 9.1FS 2

9. Project delivery 9.1 Project takeover instruction 8.2FS – 0

While a feasibilitystudyshould be conductedas well – covering theproject’stechnicalcharacteristicsandimplementationmethod- ology – this paper focuses on the project management frame- worksothoseelementsareoutofscopeandconsequentlynotin- cluded. Inthe earliest stagesof theproject life cycle, theproject scopeandsuccesscriteriahavetobeestablishedandagreedupon by all stakeholders. Duringthe planningphase of the project,all necessary workrequiredto completetheproject,accordingto its scope, should be identified anddocumented in the Work Break- down Structure (WBS) and WBS Dictionary. WBS is a hierarchi- cal decompositionofaproject’sscope,includingalltheworkthat needstobeexecutedtocompletetheproject,achievetheproject’s objectivesandmeetitssuccesscriteria.Table2presentstheWBS, activities,schedulinglimitationsandduration.

The EarlyStartSchedule(ESS) constructedby CPMhas atotal durationof16timeunits,butthisisnotconsideredtobeafeasi- bleschedulebecauseresourceavailabilitywasnotconsidered.The ESS is shownin Fig. 3.To solve thisissueand constructa feasi- ble project schedule withregards to resource availability (R=5), activities mustbe shifted (delayed)in time. Tosolve these over- allocations,MS-Projectoffersresourcelevelingoptionstoitsusers.

Selectingthe“Standard” optionthroughtheleveling window,MS- Project provides aproject schedulewithrespectto resource con- straints.Thisfeasibleprojectschedule,afterthelevelingprocedure, hasaminimumdurationof22timeunits.Duringthisprocess,the priority rules that select the order of activities to be scheduled areunknown.Fig.4showstheprojectscheduleafterMS-Project’s standardleveling.

To remedy this issue, MS-Project provides the “Priority Stan- dard” leveling option, which allows users to insert their own weights and thus determine the priority list of project activities forscheduling.Theproposedmethodologyconsiderspriorityrules fromtheliteratureasproblemcriteriaandprojectactivitiesasthe alternative solutions.The derived resultsrepresentthepriorityof eachactivitytobeusedbyMS-ProjecttorunthePriorityStandard Levelingprocess that generates theprojectschedule.The priority rules forsolving RLPwithresourceconstraintsselectthe orderof activities to be scheduled; this is done duringthe heuristic pro- cedure froma prioritylistconsistingof theprojectactivities. We

considerninepriorityrulesasproblemsub-criteriawithinthefour generalcriteria(ProblemCriteria).

These priority rules are treated as decision criteria and their weights are defined according to the DM’s experience andjudg- ment.CriterionweightsareobtainedaccordingtoAHPbycompar- ingdecisioncriteriainpairstoestimatetheirrelative importance.

Table3presentstheanalysiscriteriaandtheirweights.Afterdefin- ing all criteria, sub-criteriaand their weights aswell asalterna- tive solutions, the decision matrix is obtained (Table 4) and the problembecomesquantified.Activitiesperformancetothepriority rules revealsthat there is nosuperiority ofan activity to all the decision rules thus neither a dominant solution nor a dominant activities ranking exists. Ifit had an obvious solution wouldhad arisen. Additionally, there is no unique activities performance to anyoftheexaminedpriorityruleswhichcouldresulttoitsexclu- sionfromtheanalysis.Tosolvetheproblem,eachMADMmethod is implemented to generate a priority number for each activity.

Thus, sixdifferentpriority listswill be estimatedto usethem in theprioritystandardleveling operationinMS-Project.The results ofeachprioritylistwillbecomparedanddiscussedtoderivecon- clusions.

5. Resultsanddiscussion

Todifferentiate the results obtained by WSM withrespect to AHP, criterionweights have amore generalizedform. All thead- ditive utility approaches are implemented by estimation of the weightednormalizedsumof theattributesrowdata. TheTOPSIS isperformedastheEuclideandistanceseparationsfromboth the idealandtheanti-idealsolutionssummation.ForthePROMETHEE II, a preferencefunction isdefined foreach criterion, which rep- resents the DM’s preference of an activity when compared to all other activities.DMs usetheir judgment andpriorexperience to define the type of each problem criterion and quantify each criterion’s function thresholds in accordance with project data.

Table5providesinformationaboutthetype ofeach criterionand thethresholdsusedforthesolarparkproject.

Thepreferenceindexforeachalternativecanbecalculatedfrom the pairwise comparison of all alternatives over all problem cri- teria. To derive the priority of each activity, the PROMETHEE II

Fig. 3. Early start schedule before leveling.

completeranking method isused by calculating thenet flow for each criterion. The OWA addressesthe risk anduncertainty atti- tudesofDMsinthe decisionmaking process.Alinguisticquanti- fier(Q(^p)=p^a,a>0)isusedtoillustratethreedifferentscenarios basedon different DMrisk attitudes: Extremely Optimistic, Neu- tralandExtremelyPessimistic.TocomputetheOWAscoreforeach

scenario, adescending orderof activityscores is usedto reorder thecriterionweights.SimilartotheOWA,theHWAconsidersboth thepositions ofalternatives intheorder preferenceandcriterion weights.OrderweightsarecalculatedinthesamewayastheOWA.

The balance coefficient n equals 10. Table 6 provides the results derived fromthe examinedmethodologicalframeworksand,with

Fig. 4. Early start schedule after leveling using MS-project.

Table 3

Decision criteria weights.

Criterion Criterion weights Sub-criterion Sub-criterion weights Global weights

Activity based criteria 0 ,10 LPT 1 0 ,10

Network based criteria 0 ,36 MIS 0 ,16 0 ,06

MTS 0 ,30 0 ,11

GRPW 0 ,54 0 ,20

Critical path based criteria 0 ,15 EST 0 ,25 0 ,04

MLSK 0 ,75 0 ,11

GRD 0 ,17 0 ,06

Resource based criteria 0 ,39 GCUMRD 0 ,39 0 ,15

WRUP 0 ,44 0 ,17

respecttotheOWAandHWA,threeriskscenarios arealsoexam- ined.

The derived results provide different lists of activities scores.

The latterare adjustedaccordingly(withintherange0–1000) us- ingthemaximumscorenormalizationoption(Eq.(7))tobeconsis- tentwiththenormofinsertingprioritiesinMS-Project.Maximum scoreprocedurehasbeenchosengivenitsabilitytoretainconstant therelativeorderofmagnitudebetweennaturalandstandardized performances.TheseprioritiesareinsertedintoMS-Projectandthe resourcelevelingproceduregeneratesoneprojectscheduleforev- eryconsidered approach.Tomeasure resourceefficiency,fourdif- ferentresourcemomentshavebeencalculated(Jaejun,Kyughwan, Namyong, & Yungasang, 2005). The level of resource fluctuations throughouttheprojectlifespaniscalculatedusingthemomentof variance (Mx), which isthe squared sumof resource usage from a time period(Eq.(31)), andmomentof time(Mv),which isthe squaredsum ofthedifference ofresource usagesoftwo sequen- tialtimeperiods(Eq.(32)).Tomeasuretheuncertaintyofdemand andsupplyforfutureresources,theMyindexisused,whichiscal- culatedby the sumofthe product oftheutilized resourcetimes theirlevelofavailability(Eq.(33)).TheRRindexevaluatestherate ofresourceusageincomparisonwiththemaximumamountofre- sourcesutilized(Eq.(34)).Theresultsoftheseindexes,alongwith the total project duration for each method were compared with theMS-Projectstandardlevelingresultstoderiveconclusions.

Mx= n

i=1

(

^Di

)

^{2 (31)}

M_v= n

i=1

(

^Di−1−Di

)

^{2 (32)} My=

n

i=1

(

^Di×i

)

⁽³³⁾

RR= n

i=1

_D

i

n×maxDi

(34)

According to Table 7, the standard leveling method in MS- Project performs the bestaccordingto theMx andMv moments.

The lattermetricindicatesthat software’sdefaultoptionprovides better results in reducing resource usage fluctuations during a project’s duration, which is reasonable because these results are achieved in a schedule with a longer duration by 2 time units.

However, if the decision is to be made within an uncertain en- vironment for future resources, the pessimistic scenarios of the OWA andHWAprovide ahigher scorebecause theDM doesnot accept anyrisksinthesescenarios,insteadseekingtheminimum uncertainty forhis project.Given theshort project duration; the resourceconstraint,whichhasbeensettofiveresourceunits;and thedependenceofrelationshipsbetweenactivities,theRRindexis almostidenticalforalloftheabovemethods.Consideringthetotal projectduration,theMS-Projectstandardlevelingmethodyieldsa

Table 4

Activities performance to the priority rules under consideration (Decision matrix).

LPT MIS MTS GRPW EST MLSK GRD GCUMRD WRUP alt max max max max min min max max max

1 .1 2 1 19 3 0 0 2 3 0 ,76

1 .2 1 1 18 2 0 0 1 2 0 ,76

2 .1 1 2 17 8 0 0 1 8 1 ,46

2 .2 5 3 16 14 1 0 5 20 2 ,16

2 .3 2 1 7 3 2 4 2 3 0 ,76

2 .4 1 1 7 2 2 5 1 2 0 ,76

3 .1 2 1 8 3 1 3 2 3 0 ,76

3 .2 1 1 7 3 3 3 1 3 0 ,76

5 .1 6 2 7 19 1 0 12 38 1 ,52

5 .2 7 3 6 16 2 0 14 29 2 ,22

5 .3 6 1 6 7 3 0 12 13 0 ,82

5 .4 1 1 4 2 2 10 1 2 0 ,76

5 .5 1 2 6 4 0 6 1 4 1 ,46

5 .6 2 1 7 4 4 3 2 4 0 ,76

5 .7 1 1 5 3 5 0 1 3 0 ,76

5 .8 1 1 6 3 4 4 1 3 0 ,76

5 .9 2 2 4 4 5 0 2 4 1 ,46

6 .1 1 1 3 2 6 0 1 2 0 ,76

7 1 1 3 2 6 0 1 2 0 ,76

8 .1 1 1 2 3 7 0 1 3 0 ,76

8 .2 2 1 1 2 8 0 2 2 0 ,76

Table 5

PROMETHEE preference functions to the analysis criteria.

Criterion Criterion type Preference function LPT U-shape criterion Fj(d)= _{0 , i f d ≤2}

1 , i f d > 2 MIS V-shape criterion Fj(d)=

_{0 ,} d 1 , ,

i f d ≤0 i f 0 < d ≤1

i f d > 1 MTS V-shape with indifference

criterion

Fj(d)=

(d −⁰ 2)/ 4 i f 2 ^{i f d} < ^≤d ² ≤6 1 i f 6 < d GRPW V-shape with indifference

criterion

Fj(^d)=

(d −⁰ 3)/ 4 i f 3 ^{i f d} < ^≤d ³ ≤7 1 i f 7 < d EST U-shape criterion F_j(^d)= _{0 , i f −d ≤2}

1 , i f −d > 2 MLSK U-shape criterion Fj(d)= _{0 , i f −d ≤3}

1 , i f −d > 3 GRD U-shape criterion Fj(d)= _{0 , i f d ≤3}

1 , i f d > 3 GCUMRD V-shape criterion Fj(d)=

_{0 ,} d/ 7

1 , ,

i f d ≤0 i f 0 < d ≤7

i f d > 7 WRUP Level criterion Fj(d)=

0 i f d ≤0 . 69 1 / 2 i f 0 . 69 < d ≤1 . 51

1 i f 1 . 51 < d

projectscheduleof22time units.Onthe other hand,implemen- tationofMADMmethods reducesthe totalprojectduration by 2 timeunits,whichamountstoa10%shorterprojectduration.More- overitisnotedthatoftheexaminedmodelspresentthesameper- formancetotheanalysismetrics(durationandmoments).Theex-

Table 6

Estimated activities priorities.

ID WSM AHP TOPSIS PROMETHEE OWA HWA

Optimistic Neutral Pessimistic Optimistic Neutral Pessimistic

1 .1 486 494 325 320 999 494 215 550 388 155

1 .2 442 448 304 290 999 448 144 550 350 69

2 .1 631 631 404 672 999 631 193 560 524 69

2 .2 925 922 704 914 783 922 970 825 816 362

2 .3 351 348 164 89 750 348 215 330 291 155

2 .4 300 296 143 82 750 296 144 290 243 69

3 .1 376 374 185 203 875 374 215 385 301 155

3 .2 335 334 172 173 700 334 193 385 291 69

4 .1 – – – – – – – – – –

5 .1 973 978 10 0 0 998 10 0 0 978 10 0 0 10 0 0 10 0 0 603 5 .2 10 0 0 10 0 0 862 10 0 0 10 0 0 10 0 0 859 850 864 517

5 .3 595 607 412 588 999 607 859 550 544 345

5 .4 223 211 69 0 611 211 0 290 165 0

5 .5 425 409 156 280 999 409 193 560 330 69

5 .6 365 367 189 189 700 367 285 145 350 10 0 0

5 .7 342 347 191 161 999 347 193 550 291 69

5 .8 309 308 149 75 600 308 193 330 262 69

5 .9 467 466 205 324 999 466 285 560 369 155

6 .1 305 310 173 100 998 310 144 550 301 69

7 .1 305 310 173 100 998 310 144 550 301 69

8 .1 309 315 178 106 998 315 193 550 311 69

8 .2 299 307 170 59 998 307 0 550 291 0

9 .1 – – – – – – – – – –

Table 7

Comparative analysis results.

Method Mx Mv My RR Project duration

MS-Standard 260 28 551 0 .63 22

WSM 282 30 521 0 .66 20

AHP 282 30 521 0 .66 20

PROMETHEE 282 30 521 0 .66 20

TOPSIS 282 30 521 0 .66 20

OWA-Extremely optimistic 282 30 521 0 .66 20

OWA-Neutral 282 30 521 0 .66 20

OWA-Extremely pessimistic 280 32 486 0 .66 20 HWA-Extremely optimistic 282 30 521 0 .66 20

HWA-Neutral 282 30 521 0 .66 20

HWA-Extremely pessimistic 280 32 486 0 .66 20

tentoftheexaminedprojectandthelimitedoptionsforactivities scheduling restricts the area of the possible solutions. However, thisshould not leadto theconclusion thatthe derivedschedules aresame.Inparticular,withtheobviousexceptionofMS-standard result, four different schedules occur since, as it is shown in Table6,differentactivitiesareselectedfordelayby eachdecision model.Inparticular,WSM,AHP,OWA-optimisticandOWA-neutral result to the same outcome, PROMETHEE, TOPSIS, HWA-neutral producea common schedule,a third solution is obtainedby the pessimisticversionsofOWA andHWAwhile HWA-optimisticim- plementationresultstoitsown Ganttchart. Finally,theresultsof mostMADMmethodsarenoteworthybecauseinsomecasesthey yieldbetterscoresinoneormorecriteria.

6. Conclusions

Theframeworkpresenteddiscussabasicdilemmawhenexam- iningRLP solutions.Insummary,DMs havetodecide eitherona moreefficientresourceutilizationoronareductionofprojectdu- ration.To properly decide, they should also consider the project management environment to address the impacts of these two options. Therefore, DMs must consider the best option to level theirprojects basedontheir own thoughtsandpriorities. Foror- ganizations,theseimpactsare felt infinancialterms suchascost overrunsdueto poorresource utilizationora longer projectdu-

ration. To address these considerations correctly, more informa- tion should also be considered. For example, constant resource utilizationprovides an opportunity foran organization to have a smoother flow of procurement materials, thereby achieving low purchasepricesfromitssuppliers.However,areductioninproject duration provides an opportunity fororganizationsto avoidindi- rectcostsandcostsduetodelays,aswellastheabilitytoapprove moreprojectsforexecutionsoonerthanbefore.Inconclusion,the choicedilemmabetweenresourceutilizationandprojectduration fororganizationsneeds tobe resolved through theestablishment ofthedecisionattitudethatwillbefollowedunderresourcesavail- ability,timeandbudgetconstraintsandevenmorearisk-proneat- titude.

Withrespecttocriteriaandsub-criteriaimplementedtheinte- gration of MADM models to the RLP problemcan be considered sufficiently generic since it considers the most common priority rules for shifting activities. However, it can be easily adapted to meetDMs preferenceswithrespectto other priorityrulesimple- mentation.Forexamplein caseswherefewerrules areto beim- plemented theanalysis canbe utilizedby assigningzero weights tothose inexcess.Additionally theconsiderationofdifferentpri- ority rulescan beassistedsinceMS-Projecttasks formiscapable ofsupportingavarietyofindexesestimations.Theinparallelper- formance of MADM synergy and standard procedures facilitating byMS-Projectrevealedthatthefirstcanbeeffective.Thus,theso- lutionprocess gainbenefitfromthelow computationaleffortde- mandedbyMADMmodels.

RLPsolutionapproachesare dominatedbytheperformance of heuristic algorithms which have no prior knowledge about the searchspaceandthespecificcharacteristics ofeachproblem,and theyneedtorun severaltimesinordertoachieve anearoptimal solution. The proposed approach enrich the solution phase with specificknowledgeabouttheproblembyincorporatingDMspref- erential system. The framework is illustrated through a realcase studywhichhasbeenchosen withrespect toits extentandthus itcanalsoserveasaneducationalexampleeasytofollow.Finally, theproposed frameworkwouldgainadded valuebythe develop- ment ofan MS-Projectadd-in forsupportingMADM modelsper- formance. Moreoverthe implementation ofstochastic approaches willenforcetheframeworks’abilitytoassistsensitivityanalysis.

References

Anagnostopoulos, K. P. , & Koulinas, G. K. (2010). A simulated annealing hyperheuris- tic for construction resource levelling. Construction Management and Economics, 28 (2), 163�