Simulated Annealing Genetic Algorithm Based Schedule Risk Management of IT Outsourcing Project

18 Pages·2017·2.94 MB·English

Hindawi
Mathematical Problems in Engineering
Volume 2017, Article ID 6916575, 17 pages
https://doi.org/10.1155/2017/6916575
Research Article
Simulated Annealing Genetic Algorithm Based Schedule Risk
Management of IT Outsourcing Project
FuqiangLu,HualingBi,MinHuang,andShupengDuan
CollegeofInformationScienceandEngineering,NortheasternUniversity,Shenyang110819,China
CorrespondenceshouldbeaddressedtoHualingBi;[email protected]
Received 10 April 2017; Revised 20 July 2017; Accepted 3 August 2017; Published 28 September 2017
AcademicEditor:M.L.R.Varela
Copyright©2017FuqiangLuetal.ThisisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,
whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.
IToutsourcingisaneffectivewaytoenhancethecorecompetitivenessformanyenterprises.ButthescheduleriskofIToutsourcing
projectmaycauseenormouseconomiclosstoenterprise.Inthispaper,theDistributedDecisionMaking(DDM)theoryandthe
principal-agenttheoryareusedtobuildamodelforscheduleriskmanagementofIToutsourcingproject.Inaddition,ahybrid
algorithm combining simulated annealing (SA) and genetic algorithm (GA) is designed, namely, simulated annealing genetic
algorithm(SAGA).Theeffectoftheproposedmodelonthescheduleriskmanagementproblemisanalyzedinthesimulation
experiment.Meanwhile,thesimulationresultsofthethreealgorithmsGA,SA,andSAGAshowthatSAGAisthemostsuperior
onetotheothertwoalgorithmsintermsofstabilityandconvergence.Consequently,thispaperprovidesthescientificquantitative
proposalforthedecisionmakerwhoneedstomanagethescheduleriskofIToutsourcingproject.
1.Introduction and the distribution characteristics of the IT outsourcing
activities. In recent years, principal-agent theory has been
Withtheincreasingdevelopmentofinformationtechnology,
widelyemployedtosolvetheproblemofriskmanagementof
IT outsourcing has been developing rapidly. It is currently
IToutsourcingandgoodresultshavebeenachievedthrough
being used as an important strategy by many companies
thesestudies.
to focus on the core competency, reduce cost, and increase
Earl et al. concluded some adverse consequences of IT
profit.InEuropeandotherdevelopedcountries,eithersmall
outsourcing based on existing literatures [13–16]. Then, he
businesses or large multinational companies always give
argued that the risk of IT outsourcing came from enter-
the noncore business to external professional company [1–
prises, agents, and the process of IT activities and pro-
4]. According to Gartner, one of the leading information
technology research firms, global spending for IT services posed corresponding risk management measures based on
was approximately $932 billion in 2013 and is expected to principal-agent theory [17–19]. Bahli and Rivard proposed
growto$967billionin2014,agrowthof3.8%from2013[5]. a scenario-based conceptualization of the IT outsourcing
AlthoughIToutsourcinghasmanyadvantagesincluding risk and applied it to the specific context of IT outsourcing
reducing cost and enhancing the core competence, there usingtransactioncostandagencytheory[20].Osei-Bryson
also exist some problems that need to be solved urgently, and Ngwenyama offered a method and some mathemati-
especially the problem of managing the schedule risk of IT cal models for analyzing risks and constructing incentive
outsourcing, which may bring about huge loss to company. contracts for information system outsourcing [21]. Sanfa et
Consequently,itisveryvitaltoresearchhowtomanagethe al. analyzed risk factors of producer services outsourcing
scheduleriskofIToutsourcing. fromtheperspectiveofengineeringandaffordedmanagersa
Researchers have done a lot of related research [6–12]. theoreticmethodtomanageoutsourcingrisksbydesigning
But most methods and models proposed in the literature the incentive and monitoring mechanism of the producer
only discuss the risk management issues on project itself services outsourcing contract [22]. Xianli et al. present the
and ignore the cooperation between principal and agent ideaofapplyingDDM(distributeddecisionmaking)tothe
2 MathematicalProblemsinEngineering
riskmanagementofvirtualenterprisesanddesignincentive agent𝑖onlyallocatesriskmanagementcapitaltoactivities1,
and punishment mechanism in the principal-agent model 2,3,6,8,and9.
[23].
Inthispaper,webuildatwo-levelprincipal-agentmodel
2.2.Assumptions
combined with reward and punishment mechanism for
schedule risk management of IT outsourcing project based (1)Subprojects are serial relation in the IT developing
on the Distributed Decision Making (DDM) theory and process.
principal-agent theory [24–26]. According to the feature of
(2)Thechangeofcompletionprobabilityordurationof
problem,theSAGAisdesignedtosolvetheproposedmodel
subproject will not have an effect on other subpro-
andtheoptimalplanofmanagingscheduleriskisgivenbased
jects; that is, subprojects are independent of each
on the simulation analysis. The purpose of this paper is to
other.
providecrucialdecisionsupportforthepeoplewhoneedto
managethescheduleriskofIToutsourcingproject. (3)Thecriticalpathofsubprojectwillnotbecomenon-
The remainder of this paper is structured as follows. criticalpathundertheinfluenceofriskmanagement
Section1presentsthescheduleriskmanagementmodelofIT capital.
outsourcingproject.InSection2,thedesignofalgorithmis
(4)Thedurationandrisklossofprojectonlyareaffected
given.Inaddition,numericalexamplesandresultsanalyzed
byriskmanagementcapital.
are depicted in Section 3. Finally, conclusion is given in
Section4. (5)Duetotheinformationasymmetry,therisklossper
unit of subproject is clear to the agents, but the
principalonlymastersitsdistributionfunction.
2.ScheduleRiskManagementModelof
ITOutsourcingProject
2.3. The Two-Level Principal-Agent Model. Based on the
2.1. Problem Description. For IT outsourcing, principal DDMtheoryandprincipal-agenttheory,atwo-levelschedule
divides a whole project into some serial subprojects in the risk management model of IT outsourcing project is built
IT developing process, as shown in Figure 1. The definition [27,28].Inthetop-level,thedecisionmakeristheprincipal
of serial subprojects is that subproject 𝑖 (𝑖 = 2,...,𝐼) is whodetermineshowtoallocatetheriskmanagementcapital
performedaftercompletionofsubproject𝑖−1. among agents. The objective of top-level is to maximize
Thescheduleriskisreflectedintwoaspectsofduration the profit of principal, and the reward and punishment
and risk loss. Each subproject has an initial duration and mechanism is introduced into the model. In the base-level,
initial risk loss. In order to effectively manage the schedule
the decision maker is the agents who determine the best
risk, the reward and punishment mechanism is added in
combinationofriskmanagementmeasureofsubproject.The
outsourcing contract; that is, if the project is completed
objectiveofbase-levelistomaximizetheagent’sbenefit.
in advance, the agent is rewarded; otherwise the agent is
punished. Each subproject will be contracted with different
agents, and a typical principal-agent relationship between 2.3.1.Top-Level
principal and agent will be generated. For the relationship
VariableDefinition
betweenprincipalandagents,seeFigure2.
The optimal solution of top-level model is the optimal 𝑥𝑖:riskmanagementcapitalofsubproject𝑖
combination of risk management capital, and the optimal
solution of base-level model is the optimal combination of 𝑦̂𝑖: the predicted combination of risk management
riskmanagementmeasureofsubproject.Inthedecisionmak- measureofsubproject𝑖
ingprocess,theprincipaltransfersriskmanagementcapital 𝑏𝑖:theagent’sprofitsharingcoefficientofsubproject𝑖
to the predicted base-level model. The optimal solution of
top-levelmodelisobtainedbasedonthegoalofmaximizing 𝐼:thenumberofagentsorsubprojects
the profit of the principal and the information returned 𝑡𝑖(𝑦̂𝑖):predicteddurationofsubproject𝑖
from the predicted base-level model. Then, the optimal
solutionoftop-levelmodelistransferredtotherealbase-level 𝑇0:planneddurationofsubproject𝑖
𝑖
model.Undertheconstraintofriskmanagementcapital,the Δ𝐿𝑖(𝑦̂𝑖):predictedsavedrisklossofsubproject𝑖
agents obtain the best control measure combination of the
subprojectaccordingtothegoalofmaximizingtheprofit.The 𝐵𝑖(𝑥𝑖,𝑦̂𝑖):predictedprofitofagent𝑖
informationexchangeprocessbetweentheprincipalandthe 𝑋max:riskmanagementcapitalbudget
agentsisshowninFigure3.
For the IT development, the duration of the subproject ℎ𝑖(𝑦̂𝑖):predictedrewardandpunishmentfunctionof
isdeterminedbythedurationoftheactivitiesonthecritical subproject𝑖
path. Hence we only consider the schedule of the activities
onthecriticalpath.Figure4showsthenetworkdiagramof AL𝑖:theaspirationlevelofagent𝑖
subproject 𝐼, in which the critical path is 1–3-4–6-7-8. So 𝑒𝑖:additionalprofitpertimeunitofsubproject𝑖.
MathematicalProblemsinEngineering 3
Project
Subproject 1 Subproject 2 Subprojecti SubprojectI
Figure1:Thediagramofprojectconstruction.
Principal Top-level
Base-level
Agent 1 Agent 2 ··· AgentI
Figure2:Therelationshipbetweenprincipalandagents.
Principal
Top-level
Predict agenti
Real agenti Base-level
Figure3:Theinformationexchangeprocessbetweentheprincipalandtheagents.
𝐼
max ∑𝑏𝑖Δ𝐿𝑖(𝑦̂𝑖)−𝑥𝑖+ℎ𝑖(𝑦̂𝑖)−𝑒𝑖(𝑡𝑖(𝑦̂𝑖)−𝑇𝑖0)− (1) (3) iFnodrimcautelas(in2)ceinntdiviceacteosmppaarttiibciilpitayticoonncstornasintrta;ifnotr;mfourlmau(4la)
𝑖=1
indicates risk management capital constraint; formula (5)
s.t. 𝐵𝑖(𝑥𝑖,𝑦̂𝑖)≥AL𝑖 (2) indicates that the risk management capital 𝑥𝑖 is a natural
integer,whichisthedecisionvariableinthetop-levelmodel.
𝐵𝑖(𝑥𝑖,𝑦̂𝑖∗)≥𝐵𝑖(𝑥𝑖,𝑦̂𝑖) (3)
2.3.2. Predicted Base-Level. In the predicted base-level, the
𝐼
∑𝑥𝑖 ≤𝑋max (4) adgeecnistio𝑖,nfomraekxearmspalree.theagents,andthereare𝐼agents.Take
𝑖=1
𝑥𝑖 ∈𝑁+. (5) VariableDefinition
𝑌𝑖𝑗:thenumberofmanagementmeasuresofactivityj
The objective of top-level shown in formula (1) is to 𝐽𝑖:thenumberofactivitiesofsubprojecti
maximizetheprofitofprincipal;therewardandpunishment 𝐿𝑖:initialrisklossofsubprojecti
mechanism is fulfilled by item ℎ𝑖(𝑦̂𝑖). The operation (𝑥)− is 𝜉𝑖:predictedrisklossperunitofsubprojecti
definedas
𝑞𝑖:rewardandpunishmentstandardofsubprojecti
{𝑥, 𝑥<0, 𝑐𝑖(𝑡𝑖(𝑦̂𝑖)):predictedcostofsubprojecti
(𝑥)− ={{0, else. (6) 𝑑su𝑖(b𝑦̂p𝑖r)o:pjercetdiictedinvestedriskmanagementcapitalof
4 MathematicalProblemsinEngineering
Activity 3 Activity 6
4
Activity 1 Activity 2 Activity 4 Activity 8 Activity 9
1 2 3 6 7 8
Activity 5 Activity 7
5
Figure4:Thenetworkdiagramofsubproject𝑖.
𝛼𝑖:confidencelevel
𝐶max:theupperboundofcostofsubprojecti.
𝑖
𝐵𝑖(𝑥𝑖,𝑦̂𝑖)=max 𝐵𝑖 (7)
s.t. pr{𝑏𝑖Δ𝐿𝑖(𝑦̂𝑖)−𝑐𝑖(𝑡𝑖(𝑦̂𝑖))+𝑥𝑖−𝑑𝑖(𝑦̂𝑖)−ℎ𝑖(𝑦̂𝑖)≥𝐵𝑖}≥𝛼𝑖 (8)
ℎ𝑖(𝑦̂𝑖)=𝑞𝑖(𝑡𝑖(𝑦̂𝑖)−𝑇𝑖0) (9)
Δ𝐿 (𝑦̂)=𝐿 −𝑎 (𝑡 (𝑦̂)−𝑇0)+ (10)
𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖
𝑑𝑖(𝑦̂𝑖)≤𝑥𝑖 (11)
𝑐𝑖(𝑡𝑖(𝑦̂𝑖))≤𝐶𝑖max (12)
𝑦̂𝑖 =(𝑦̂𝑖1,𝑦̂𝑖2,...,𝑦̂𝑖𝐽) (13)
𝑖
𝑦̂𝑖𝑗 ∈{1,2,...,𝑌𝑖𝑗}. (14)
Formula (7) indicates that the objective of predicted base- 2.3.3. Real Base-Level. In the real base-level, the decision
level is to maximize agent’s benefit. Formula (8) indicates makers are the agents, and there are 𝐼 agents. Take agent 𝑖,
chance constraint. Formula (9) indicates the reward and forexample.
penalty function based on the duration. Formula (10) indi-
catesthesavedrisklossofsubproject;theoperation(𝑥)+ is VariableDefinition
definedas
𝑦𝑖:theactualcombinationofriskmanagementmea-
sureofsubprojecti
{𝑥, 𝑥>0,
(𝑥)+ ={ (15) 𝑎𝑖:risklossperunitofsubprojecti
{0, else. Δ𝐿𝑖(𝑦𝑖):actualsavedrisklossofsubprojecti
𝑡𝑖(𝑦𝑖):actualdurationofsubprojecti
Formula(11)indicatesthatthesumofusedriskmanage- ℎ𝑖(𝑦𝑖):actualrewardandpenaltyfunctionofsubpro-
mentcapitalisnotgreaterthantheriskmanagementcapital
jecti
whichisallocatedtosubproject;formula(13)representsaset
ofthepredictedbase-levelvariables;formula(14)represents 𝑑𝑖(𝑦𝑖): actual invested risk management capital of
the value range of 𝑦̂𝑖𝑗 that is the decision variable in the subprojecti
predictedbase-levelmodel. 𝑐𝑖(𝑡𝑖(𝑦𝑖)):actualcostofsubprojecti.
MathematicalProblemsinEngineering 5
𝐵𝑖(𝑥𝑖∗,𝑦𝑖)=max 𝑏𝑖Δ𝐿𝑖(𝑦𝑖)−𝑐𝑖(𝑡𝑖(𝑦𝑖))𝑥𝑖∗−𝑑𝑖(𝑦𝑖)−ℎ𝑖(𝑦𝑖) (16)
s.t. ℎ𝑖(𝑦𝑖)=𝑞𝑖(𝑡𝑖(𝑦𝑖)−𝑇𝑖0) (17)
Δ𝐿 (𝑦)=𝐿 −𝑎 (𝑡 (𝑦)−𝑇0)+ (18)
𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖
𝑑𝑖(𝑦𝑖)≤𝑥𝑖∗ (19)
𝑐𝑖(𝑡𝑖(𝑦𝑖))≤𝐶𝑖max (20)
𝑦𝑖 =(𝑦𝑖1,𝑦𝑖2,...,𝑦𝑖𝐽) (21)
𝑖
𝑦𝑖𝑗 ∈{1,2,...,𝑌𝑖𝑗}. (22)
Formula (16) indicates that the objective of real base- 3.1.EncodingScheme. Intop-level,eachchromosomerepre-
levelistomaximizeagent’sbenefit;formula(17)indicatesthe sentedbyrealnumberisacombinationofriskmanagement
rewardandpenaltyfunctionbasedontheduration;formula capitalandthelengthofchromosomerepresentsthenumber
(18)indicatesthesavedrisklossofsubproject;formula(19) ofagents.Thetop-levelencodingschemeofSAGAisshown
indicatesthatthesumofusedriskmanagementcapitalisnot inFigure5.Wecanseethatthereare4agents,and$580.2is
greaterthantheriskmanagementcapitalwhichisallocatedto allocatedtoagent1;$600.9isallocatedtoagent2;therestcan
subproject;formula(20)representsthecostthatshouldnot bedoneinthesamemanner.
bebeyondtheupperboundofcostforsubproject𝑖;formula In base-level, each chromosome represented by real
(21)representsasetoftherealbase-levelvariables;formula numberisacombinationofriskmanagementcapitalandthe
(22) represents the value range of 𝑦𝑖𝑗 that is the decision lengthofchromosomerepresentsthenumberoftheactivities
variableintherealbase-levelmodel.
on the critical path. Take the base-level encoding scheme
showninFigure6asexample;itcanbeseenthatthereare5
3.AlgorithmDesign activitiesonthecriticalpath,andmeasure2isusedtomanage
theriskofactivity1,measure4isusedtomanagetheriskof
The top-level model is an integer programming problem,
activity2,andsoon.
andthebase-levelmodel(includingthepredictedbase-level
model)isacombinatorialoptimizationproblem.Thewhole
3.2.PopulationInitialization. Initialpopulationisgenerated
problem is a NP hard problem, because the base-level is
randomly. Punishment strategy is adopted to deal with the
embeddedinthetop-level.So,weusegeneticalgorithm(GA)
constraints, so we do not have to judge whether the initial
tosolvetheprobleminthispaper.ItiswellknownthatGA
solutionmeetstheconstraintconditions.
thatwasfirstintroducedbyHollandisveryeffectiveforsolv-
ingcombinatorialoptimizationproblems.Forexample,GA
has been successfully applied in solving traveling salesman 3.3. Fitness Function. Considering the proposed optimiza-
problem, knapsack problem, bin packing problem, and so tionproblemswithconstraints,wesetupafitnessfunction
on.However,thedisadvantageofGAisthatthelocalsearch withpunishmenttermtoevaluateindividuals.Thetop-level
fitnessfunctionisgivenas
capabilityisnotstrong[29–33].Simulatedannealing(SA)is
ageneralrandomsearchalgorithm,whichisanextensionof
𝐼
the local search algorithm [34–37]. Considering the strong 𝐹𝑇 =𝑓𝑇−∑(𝛼𝑖(𝐵𝑖(𝑥𝑖,𝑦̂𝑖)−AL𝑖))
localsearchcapabilityofSA,wedesignedahybridalgorithm 𝑖=1
named simulated annealing genetic algorithm (SAGA) by (23)
𝐼
combiningsimulatedSAwithGA. −𝛽(∑𝑥 −𝑋max),
𝑖
The overall thought of SAGA is simple. Firstly, some
𝑖=1
initialsolutionsofGAaregeneratedrandomly.Afteraperiod
ofiteration,somesuperiorsolutionsareproduced.Then,sort where
thecorrespondingfitnessvalueofthesesuperiorsolutionsin
𝐼
descendingorder.Andthenselectthesolutionsintop10%as 𝑓𝑇 =∑𝑏𝑖Δ𝐿𝑖(𝑦̂𝑖)−𝑥𝑖+ℎ𝑖(𝑦̂𝑖)−𝑒𝑖(𝑡𝑖(𝑦̂𝑖)−𝑇𝑖0)−. (24)
theinitialsolutionsofSA.Wetrytofindthebestsolutionof
𝑖=1
the proposed problem around these superior solutions. For
the random variables of the predicted base-level model, we Function (24) is top-level objective function; 𝛼𝑖 and 𝛽 are
embedMonteCarloSimulationinSAGA. punishmentcoefficients,respectively.
6 MathematicalProblemsinEngineering
Agent 1 2 3 4
Capital 580.2 600.9 643.7 576
Figure5:Thetop-levelencodingschemeofSAGA.
Activity 1 2 3 4 5
Measure 2 4 2 3 1
Figure6:Thebase-levelencodingschemeofSAGA.
Thebase-levelfitnessfunctionisgivenas 3.5. Crossover. Double-point crossover is adopted in this
paper, which is beneficial for keeping excellent individual.
𝐹 =𝑓 −𝛾 (𝑑 (𝑦)−𝑥∗)+
𝐵𝑖 𝐵𝑖 𝑖 𝑖 𝑖 𝑖 Figure 7 shows the example of the double-point crossover
−𝛿 (𝑐 (𝑡 (𝑦))−𝐶max)+, (25) operator, where 𝑃1 = (1,3,3,2,4) and 𝑃2 = (2,2,5,1,3)
𝑖 𝑖 𝑖 𝑖 𝑖 areparentchromosomes.So,thegeneratedchildrenchromo-
where somesare𝐶1 =(1,2,5,1,4)and𝐶2 =(2,3,3,2,3).
𝑓𝐵𝑖 =𝑏𝑖Δ𝐿𝑖(𝑦𝑖)−𝑐𝑖(𝑡𝑖(𝑦𝑖))+𝑥𝑖∗−𝑑𝑖(𝑦𝑖)−ℎ𝑖(𝑦𝑖). (26) 3.6. Mutation. Reversal mutation is adopted in this paper
Function (25) is base-level objective function; 𝛾𝑖 and 𝛿𝑖 are [35]. Under the condition of satisfying the mutation rate,
randomlyselecttwopointsintheparentandsortthegenes
punishmentcoefficients,respectively.
between these two points in reverse order. The reversal
TheMonteCarloSimulationmethodisusedtodealwith
mutation operator of SAGA is shown in Figure 8. Parent
therandomvariable[37–40].Theprocessofcalculatingthe
chromosome 𝑃 = (1,4,2,3,1) is selected for mutation
fitness of the predicted base-level model by Monte Carlo
operation.Andthegeneratedchildrenchromosomeis𝑃󸀠 =
Simulationisshownasfollows.
(1,2,4,3,1).
Step1. Set𝑄󸀠isequalto[𝛼𝑖𝑄];𝑄issamplingnumber.
3.7.NeighborhoodDefinition. Thebit𝑥𝑖 isselectedfromthe
Step2. Samples𝜉𝑖1,𝜉𝑖2,...,𝜉𝑖𝑄aregeneratedbynormaldistri- current state 𝑥 = [𝑥1,𝑥2,...,𝑥𝑚], and the value of 𝑥𝑖 is
butionfunction𝑁(𝑎𝑖,0.01). changedintherangeofitsvalue.So,aneighborhoodsolution
isgenerated.
Step3. Calculate𝐹𝑡 thatisthefitnessvalueofthepredicted
𝐵𝑖
base-levelmodelbyformula(26),𝑡=1,2,...,𝑄.
3.8. Neighborhood Movement. The Metropolis criterion is
Step4. The𝑄󸀠thbiggestelementof{𝐹𝑡 ,𝐹𝑡 ,...,𝐹𝑡 }canbe adoptedinthispaper.Iftheobjectivevalueofneighborhood
𝐵𝑖 𝐵𝑖 𝐵𝑖 solution is smaller than the current solution’s objective
usedasthefitnessvalueofthe𝑖thpredictedbase-levelmodel,
value,thecurrentsolutionisreplacedbytheneighborhood
whichcanbeseenbythelawoflargenumbers.
solution.Otherwise,thecurrentsolutionmovesaccordingto
acertainprobability.
3.4.Selection. Thispapertakesproportionalselectionstrat-
egy [41, 42]. For each individual, the probability of being
3.9. Thermal Equilibrium. Thermal equilibrium is achieved
selectedistheproportionofitsfitnesstothesumofallindi-
whenthepresetnumberofinternalloopisreached.
viduals’fitness.Then,theprobabilityofselectedindividual𝐼
isgivenby
3.10. Cooling Rule. Reduce 𝑇𝑘 by multiplying a number r,
𝑃 = 𝐹𝑖 , which is in the range [0,1] and close to 1. Cooling rule is
𝑖 ∑𝑁𝑃𝐹 (27) shownas
𝑖=1 𝑖
where𝐹𝑖isthefitnessofindividual𝑖andNPisthepopulation 𝑇𝑘+1 =𝑇𝑘, (28)
size. Here, we adopt well-known Roulette Wheel scheme. where𝑇𝑘isthecurrenttemperature,𝑇0isinitialtemperature,
In order to prevent the best individual in each generation 𝑘istheiterativeindex,and𝑟∈(0.95,0.99).
frombeingdestroyed,elite-preservationstrategyisalsoused.
Thatistosay,thebestindividualofeachgenerationdirectly 3.11.ProcedureofSAGA. TheflowchartofSAGAisshownin
becomes one of individuals in the next generation without Figure9.TheprocessoftheSAGAismadeupoftwoparts,
crossoverandmutationoperation. GAandSA.InFigure9,theleftpartistheprocessofGA;GA
MathematicalProblemsinEngineering 7
P1 1 3 3 2 4 C1 1 2 5 1 4
P2 2 2 5 1 3 C2 2 3 3 2 3
Figure7:Thedouble-pointcrossoveroperatorofSAGA.
P 1 4 2 3 1 P 1 4 2 3 1
Figure8:ThereversalvariationoperatorofSAGA.
isusedfirstlytofindagoodsolutionoftheproblem.Tocover shown in formula (29), and risk loss function is shown in
thedefectofGAonthelocalareasearch,SAisusedtofindan formula(30),respectively,
optimalsolutionaroundthe“goodsolution”findbyGA.The
𝐽
pthreocloescsalosfeSaArchisosfhSoAw,nthoenptrhoeporisgehdtaplgarotriothfmtheenfidgeudr.e.After 𝑡1(𝑦1)= ∑1𝑡10𝑗exp(−𝑦1𝑗𝜎1𝑗), (29)
𝑗=1
4.NumericalExamples 𝐿 (𝑦 )=𝑎 (𝑡 (𝑦̂ )−𝑇0)+, (30)
1 1 1 1 1 1
4p.r1i.nEcixpaamlpanled1.1aTgaeknett(h𝐼e=IT1o)ufotsroaunrceinxagmpprolej.ecTht iencplruindicnipga1l where 𝜎1𝑗 is control parameter, which is used to represent
thatamanagementmeasurehasdifferenteffectsondifferent
outsources the whole project to the agent. Schedule risk is activities.Thesavedrisklossofprojectisshownasfollows:
reflected by duration and risk loss of project. It is assumed
thatiftheprojectiscompletedbeforeplannedduration,there Δ𝐿1(𝑦1)=𝑎1(𝑇1−𝑇10)−𝐿1(𝑦1). (31)
willbenoriskloss.Onthecontrary,theprojectwilllose$80
perday;namely,𝑎1 =80.Duetotheinformationasymmetry, If the project can be completed before the planned
the agent is clear for parameter 𝑎1, but the principal only duration, the principal will obtain additional profit. It is
knows its distribution function, which is approximated as assumedthattheadditionalprofit𝑒1is$25pertimeunit.So,
𝑁(80,0.01). The planned duration of project 𝑇0 is 400, the therewardandpunishmentfunctionaboutdurationisshown
1
initialdurationofproject𝑇1 is500,andtheinitialduration informula(32),whichisdesignedbytheprincipal
ofactivities𝑡0 is{55,60,65,70,75,80,95}.Theprofitsharing
coefficientof1p𝑗rincipal𝑏1is0.51,theexpectedprofitofagent ℎ1(𝑦1)=𝑞1(𝑡1(𝑦1)−𝑇10). (32)
AL1Thise2r0e0i0s,aanndincitoinalfisdcehnecdeulleevreils𝛼k,iws0h.i9c.hcanbeseenfrom Itisassumedthattherewardandpunishmentparameter𝑞1is
$21.Ifthesubprojectiscompletedinadvance,theagentcan
the parameters of duration. The principal prepares $900 to
manage schedule risk, 𝑋max = 900. The project consists of get$21perday;otherwise,hewilllose$21perday.
7 activities,𝐽1 = 7. There are 10 available risk management
measures for each activity, 𝑌1𝑗 = 10, 𝑗 ∈ {1,2,...,7}. It is 4.1.1. The Parameters of SAGA. The parameters of SAGA
assumedthattheagentmustchooseanumberedmeasurefor mainlyincludepopulationsize(𝑁𝑃),maximumgenerations
eachactivityandthattheselectednumberedmeasurecanbe (𝑁𝐺),crossoverrate(𝑃𝑐),mutationrate (𝑃𝑚),initialtemper-
selectedmultipletimes. ature (𝑇𝑖), termination temperature (𝑇𝑠), and internal loop
Riskmanagementmeasureistheonlyvariablethataffects number (𝑁𝑖). The parameters have a significant impact on
the duration and risk loss of project, which can be seen the performance of algorithm. For example, big population
from the proposed model. It is assumed that the impact size may lead to slower convergence speed but can avoid
of risk management measure on the project is enhanced suboptimal solution. Small population size may lead to
with the increase of the number of measures. For example, prematurebutcanensurethequickspeedofconvergence.
for an activity, the implementation of measure 7 will result Inthispaper,NGisdeterminedby
in a shorter duration and a smaller risk loss than the
1500
implementationofmeasure2.So,themonotonicdecreasing 𝑁𝐺= . (33)
function is used to represent the duration function that is 𝑁𝑃
8 MathematicalProblemsinEngineering
Start
Population initialization
t=0
SA
Yes
Iterations?
No
Calculating fitness Select the initial solution of SA
Selection For each initial solutionqi
Double-point crossover
Generate neighborhood solutionqi.
Calculate fitness differenceΔf
Reversal variation
Yes
Update population Δf<0?
No
Save contemporary
Yes
optimal individual exp(−Δf/T)>?
No qi=qi
t=t+1
No
Thermal equilibrium?
Yes
Update current temperatureT
No
T<T?
0
Yes
End
Figure9:TheflowchartofSAGA.
Generally,Tsisasmallerpositivenumber;let𝑇𝑠=1here.So, (2)Foreachcombinationofparameter,runalgorithm20
weonlyneedtotesttheother5parameterswiththefollowing times.Thereare25combinations.
method.
(1)Provide two values for each parameter. One is rela- (3)Thefinalparametercombinationistheonewithbest
tivelysmall,andtheotherisrelativelybig. averagefitnessvalue.
MathematicalProblemsinEngineering 9
400 2450
395
390 nt 2400
e
g
a
on of
ati 385 fit
ur ne
D e
b
380 The 2350
375
370 2300
14 16 18 20 22 24 26 14 16 18 20 22 24 26
q1 q1
Figure10:Theinfluenceof𝑞1onduration. Figure11:Theinfluenceof𝑞1ontheprofitofagent.
3620
ThetestingprocessisshowninTable1.Thefinalcombi-
nationis{30,50,0.6,0.1,100,10}.
3600
4.1.2. Simulation Results. The simulation results of example
1 are shown in Table 2. For real results, agent is allocated pal 3580
ci
$597andheselectsmeasures2 1 10 1 10 10 10tomanage prin
risk.Afterthat,thedurationofprojectisreducedfrom500 of 3560
daysto377.41days;thatistosay,theprojectiscompletedin fit
e
n
advance.Forindexdurationandriskloss,theschedulerisk e
b
ofIToutsourcingprojecthasbeenwellmanaged. The 3540
It can be seen that the predicted results and real results
are different, such as profit of principal, because there is 3520
informationasymmetrybetweenprincipalandagent,andthe
principal’s prediction is not completely correct in the real
3500
situation,whichalwaysplaysanimportantroleintheprocess 14 16 18 20 22 24 26
of schedule risk management. If there is no prediction, the q
1
wholeprojectwouldbecompletelyoutofcontrol;theeffect
ofscheduleriskmanagementwillbeworsethanitisnow. Figure12:Theinfluenceof𝑞1ontheprofitofprincipal.
4.1.3. Model Comparison. Model I refers to the proposed
duration.The reason forthe above phenomenonis that the
modelinSection2.3;ModelIIistheoneinwhichthereisno
rewardandpunishmentitem.ModelIisModelII,whenthe agentwillnotmakeeffortstoshortenduration,ifthereward
parameter𝑞1 =0.ForthesimulationresultsinTable3,from is not greater than the cost because of duration reduction.
index3to8,ModelIisbetterthanModelII.Therefore,itis When 𝑞1 ≥ 𝑒1 = 25, the reward of agent is greater than
significanttojointherewardandpunishmentmechanismin the additional profit of principal due to shorting duration,
themodel. resulting in the decline of principal’s profit. Obviously, this
is not acceptable to the principal. So, we must determine
4.1.4.TheParameterofRewardandPunishmentMechanism. thespecificvalueof𝑞1 intheinterval(𝑐1,𝑒1)basedonthe4
Ifparameter𝑞1isnotsetreasonably,therewardandpunish- indexes:duration,savedriskloss,profitofagent,andprofit
mentwillnotreallyplayitsrole.Therefore,theeffectofthe of principal. Considering managing the schedule risk of IT
parameter𝑞1istestedontheproposedmodel.Thetestresults outsourcing project, we set 𝑞1 to be 21. Finally, the way for
areshowninTable4.Theinfluenceof𝑞1ontheduration,the setting the reward and punishment mechanism’s parameter
profitofagent,andtheprofitofprincipalareshowninFigures issummarizedasfollows.
10–12,respectively.
(1)Setthetwovalues𝑐and𝑒,𝑐<𝑒.
Wecanget someinformationfromTable4andFigures
10–12. When 𝑞1 ≤ 𝑐1 = 16 where 𝑐1 is additional cost per (2)Determinethespecificvalueof𝑞1intheinterval[𝑐,𝑒]
day due to duration reduction, total risk loss is saved, but basedonthe4indexes:duration,recoveredriskloss,
durationisnotsignificantlyreducedcomparedtotheplanned profitofagent,andprofitofprincipal.
10 MathematicalProblemsinEngineering
Table1:TheparametersofSAGA.
NP NG Pc Pm Ti Ni Fitness
30 50 0.9 0.1 130 20 2417.8
30 50 0.9 0.1 130 10 2352.3
30 50 0.9 0.1 100 20 2344.9
30 50 0.9 0.1 100 10 2327.5
30 50 0.9 0.01 130 20 2417.3
30 50 0.9 0.01 130 10 2407.6
30 50 0.9 0.01 100 20 2327.9
30 50 0.9 0.01 100 10 2423.4
30 50 0.6 0.1 130 20 2334.8
30 50 0.6 0.1 130 10 2338.3
30 50 0.6 0.1 100 20 2348.7
30 50 0.6 0.1 100 10 2441.1
30 50 0.6 0.01 130 20 2417.4
30 50 0.6 0.01 130 10 2339.2
30 50 0.6 0.01 100 20 2326.3
30 50 0.6 0.01 100 10 2404.1
50 30 0.9 0.1 130 20 2319.5
50 30 0.9 0.1 130 10 2408.2
50 30 0.9 0.1 100 20 2328.6
50 30 0.9 0.1 100 10 2328.2
50 30 0.9 0.01 130 20 2402.0
50 30 0.9 0.01 130 10 2316.8
50 30 0.9 0.01 100 20 2322.8
50 30 0.9 0.01 100 10 2317.0
50 30 0.6 0.1 130 20 2394.7
50 30 0.6 0.1 130 10 2396.3
50 30 0.6 0.1 100 20 2324.5
50 30 0.6 0.1 100 10 2326.3
50 30 0.6 0.01 130 20 2361.7
50 30 0.6 0.01 130 10 2319.3
50 30 0.6 0.01 100 20 2367.9
50 30 0.6 0.01 100 10 2321.6
Table2:Thesimulationresultsofexample1.
Index Predictedresults Realresults
Allocatedcapital 597 597
Measurecombination 1 10 1 1 9 10 10 2 1 10 1 10 10 10
Duration 380.52 377.41
Finishtheprojectbeforeplannedduration Y Y
Savedriskloss 7985.6 8000
Savetotalriskloss? Y Y
Profitofagent 2430.69 2440.50
Profitofprincipal 3565.15 3564.98
4.1.5.PerformanceComparisonofAlgorithms. Inthispaper, risk loss, total risk loss saved, profit of agent, and profit of
weusethreealgorithms,SA,GA,andSAGAtocarryoutthe principal,theconclusioncanbedrawn thatSAGAisbetter
simulationcalculationfortheproposedmodel.Thesimula- thanGAandSA
tion results are compared in Table 5. Focusing on indexes, Theconvergenceprocessof3algorithmsinexample1is
duration,finishingtheprojectbeforeplannedduration,saved shown in Figure 13. For convergent speed, SA achieves its