Research ArticleCritical Chain Design Structure Matrix Method for ConstructionProject Scheduling under Rework Scenarios

Guofeng Ma Keke Hao Yu Xiao and Tiancheng Zhu

Department of Construction Management and Real Estate Tongji University Shanghai 200092 China

Correspondence should be addressed to Keke Hao 1631029tongjieducn

Received 5 October 2018 Accepted 1 April 2019 Published 21 April 2019

Academic Editor Emilio Jimenez Macıas

Copyright copy 2019 Guofeng Ma et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Rework risks have been a major challenge in the construction industry that constantly affects project schedules and threatenson-time project completion Traditional project scheduling methods are not capable of modeling rework relationships betweenactivities and mitigating the impact of resulting uncertainties during the development of project schedules To address thischallenge a critical chain design structure matrix (CCDSM) method is proposed in this paper The CCDSM method aims todevelop construction project schedules that are adaptive to rework scenarios and robust against rework risksThe CCDSMmethodmodels and displays large-scale rework relationships among activities and introduces a new rework buffer to quantitatively representthe impact of rework instances in project schedules A max-plus algorithm is adopted in CCDSM to transform complex logicrelationships into simple matrix operations reducing computational load of schedule generation A case study was conducted todemonstrate the implementation of the CCDSMmethod and assess its effectiveness in addressing rework risksThe results showedthat the CCDSM is a promising tool to generate schedules which could improve on-time project completion rate and reduceimpacts of varying rework scenarios on project execution

1 Introduction

Rework has been regarded as one of themajor challenges thatcan adversely affect project performance in the constructionindustry [1] Hwang et al [2] surveyed about Client-RelatedRework (CRR) in 381 projects conducted by 51 constructioncompanies and concluded that more than 80 companiesand 59 projects experienced CRR which increased projectcosts by 71 and caused 33 weeksrsquo delay on average Simpehet al [3] investigated 78 construction professionals and foundthat if the mean rework cost of a project reached 512 thelikelihood of the project exceeding its budget would be ashigh as 76

Extensive efforts have been made to manage rework-caused project delay and minimize the rework impact inconstruction projects When used to predict the projectduration as well as its on-time completion probability tradi-tional scheduling techniques such as CPM PERT and GanttCharts cannot quantitatively measure activity rework timeand its impact on project completion time [4] and henceare not able to directly take into account the rework risks

during project planning Although buffers in the CCPMcan aggregate uncertainties in project execution which arereflected as blocks of resource or time redundancy in projectschedules it is still difficult to model large-scale reworkrelationships between activities using the CCPM [5] Tofurther address this challenge a design structure matrix(DSM) method has been introduced which is designedto represent the dependency or information flow betweenactivities providing an effective representation of reworkinstances [5] However one limitation of the DSM methodexists that the matrix format is too concise to fully displaycertain attributes of activities which leads the DSM methodusually to be adapted as a project duration calculator ratherthan a process control tool [6] Overall most of the currenttools for managing rework are reactive can barely computeand diagram rework instances with satisfactory accuracyand focus on measuring the rework risk of activity durationrather than that of overall project completion time Due tothese limitations although in many cases project managersexpect rework to occur they lack reliable tools to fully assessand address these risks at the planning stage of the projects

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 1595628 14 pageshttpsdoiorg10115520191595628

2 Mathematical Problems in Engineering

helliphellip

helliphellip

1d

1d 2d nd

2d nd nST1ST 2ST

PB or FB

Figure 1 Concentration principle of buffer determination

and to mitigate potential impacts of rework with proactivemeasures

Accordingly a new method termed critical chain designstructure matrix (CCDSM) is proposed in this paper Builton the strength of the CCPM andDSM in project schedulingthe CCDSM aims to integrate these two methods to over-come their respective limitations in developing constructionproject schedules that are adaptive to rework instances TheCCDSM is designed in such a way that it not only modelslarge-scale rework relationships among activities improv-ing the accuracy and effectiveness of project managementbut also displays various attributes of activities enablinganalysis and visualization of the project schedule underrework scenarios Moreover the CCDSM adopts max-plusalgorithm a straightforward linear algebra algorithm totransform complex logic relationships into simple matrixoperations which effectively reduces the computational loadof generating schedules The detailed design of the CCDSMand a case study that demonstrates the effectiveness of theCCDSM in addressing rework risks in construction projectsare presented in the remainder of this paper

2 Review of Relevant Works

21 Buffer Management and Critical Chain Scheduling in theCCPM The CCPM which has proved to be a powerfultool for project scheduling in many fields [7 8] has beenincreasingly used in project scheduling and subject to anumber of refinements and extensions [9] There are twomain components in the implementation of the CCPMincluding buffer management and critical chain scheduling

There are three major types of buffers including resourcebuffer (RB) feeding buffer (FB) and project buffer (PB)in CCPM which are distinguished by their positions andfunctions in the schedules [10] RBs are set to protect thecritical chain from the tightness of critical resources Theyare used as warnings consuming no time FBs are set toprotect the critical chain from the variation of tasks noton the critical chain and are placed where noncritical andcritical activities converge PBs are placed at the end of thecritical chain to protect against exceeding project deliverydates The concentration principle of buffer determinationin CCPM is shown as Figure 1 where 119889119894 represents theestimated duration of activity i and 119878119879119894 denotes the safetytime of 119894 In fact another two major time buffering methodsexisted for developing reliable schedules and protect themfrom effects of uncertainties One is optimization-based timebuffering method such as starting time criticality and Tabusearch which insert buffers in front of activities dispersedly

to ensure that each activity can be conducted on timein accordance with the project schedules However it isunknown whether such scattered buffers can ensure on-time completion of the entire project [11] The other issimulation-based time buffering method However a majordrawback of simulation-basedmethods exists that simulationis highly computationally demanding which could becomea significant problem in practice when large projects areinvolved or when project managers want to perform what-ifanalysis for a wide variety of scenarios [12]TheCCPM-basedtime buffering used in this study has several advantages Itcalculates time buffers efficiently provides explicit protectionagainst stochastic variation and presents a set of reasonablecontrol guidelines to ensure on-time completion of the entireproject [13]

Determining the size of buffers depends on variousfactors such as managerial experience and preferences of theproject team project circumstances personnel and equip-ment capabilities and so on [14] A number of buffer sizingapproaches have been developed in CCPM among which thecut and paste method (CampPM) and the root square errormethod (RSEM) are widely used The CampPM reduces thesafety time of each activity by 50 and estimates half ofthe sum of safety time to be a buffer However the CampPMadopts a linear procedure and the size of the calculatedbuffer increases linearly with the length of the critical chainThe RSEM calculates the square root of the total squares ofthe difference between two estimates a safe estimate andan average estimate for each activity in the critical chainas the buffer size [15] Tukel et al [16] argued that theRSEM could be less affected by the length of the criticalchain than the CampPM In addition Icmeli and Erenguc[17] proposed a resource utilization factor (RF) to calculatebuffers Roghanian et al [18] took into account variation offuzzy numbers for required resources when buffer sizingGhaari and Emsley [9] employed Monte Carlo simulationmethod to validate the efficiency of CampPM in buffer siz-ing and put forward buffer sizing schemes in multiprojectenvironments considering the level of resource capacity Amajor limitation of these existing buffer sizing approacheshowever is that they operate based on an assumption thatproject activity durations are independent of each other (Liet al 2012) These approaches therefore are not appropriateto cope with rework which involves extensively the mutualrelationships between activities In a recent study Zhang etal [19] proposed a CCPM-based scheduling method that byintegrating the DSM took into account rework risks in thecalculation of resource tightness and information constraintsHowever this method failed to recognize that the rework

Mathematical Problems in Engineering 3

A1 A1

A1

A2 X

A1 A1

A1

A2 X

A1 A2

(a) Sequential

A1 A2

A1

A2

A1 A2

A1

A2

A1

A2

(b) Parallel

A1 A2

A1 X

A2 X

A1 A2

A1 X

A2 X

A1

A2

(c) Coupled

Figure 2 Different types of relationship between activities represented in the DSM

risks could also impact the critical chain and it identified thecritical chain in a traditional manner

With respect to critical chain scheduling Francisco andAlfonso [20] extended traditional critical chain schedulingmethods by using critical set and critical cloud to avoidambiguity in the identification of critical tasks Ma et al [21]proposed a critical chain scheduling method with a scenario-based proactive robustness optimization The method wasable to yield higher probabilities of on-time project com-pletion than traditional CCPM-based scheduling methodsGoto [22] developed a scheduling framework for projectsunder limited resources by representing the CCPM methodin max-plus algebra to achieve a short lead time Salama etal [23] took into account the impacts of resource continuityand variability on activity durations and integrated linearscheduling and CCPM methods to identify multiple criticalsequences Despite active research in CCPM-based linearprocess scheduling in recent years the challenge of reworkin CCPM-based schedules has barely been addressed [24]where further research is needed

22 DSM as a Process Management Technique Since Steward[25] first introduced the DSM for the square-matrix-basedmodels of processes process architecture DSM models (pro-cess DSMs) have received the most attention among differentapplication areas of DSM [6]TheDSMuses matrix format touniformly represent three different logical relationships anddependencies including sequential parallel and coupledbetween activities as shown in Figure 2 An off-diagonalelement 119860 119894119895 of the DSM matrix represents with a mark ora number the information flow from activity 119895 to activity 119894or the dependency relationship between these two activitiesIf no dependency exists between activities 119895 and 119894 it willbe null or zero Diagonal element 119860 119894119894 is occasionally usedto represents certain characteristics such as duration of theactivity 119894 or simply nothing in most cases

In this matrix-based display elements above the diagonalare usually regarded as information feedback or iterativeactivities relationship [5] Such display provides a useful tool

for analyzing rework and has given rise to a number of rele-vant studiesThefirstDSM-based discrete eventMonteCarlosimulation model was proposed by Browning [26] and laterextended by Browning and Eppinger [27] to estimate projectduration and cost and their variances They argued thatprocesseswith the fewest feedbackmarks in theDSMmaynotbe necessarily optimal and could be sped up with appropriateincrease in overlapping and iterations [6] They introducedrework probability (RP) matrix rework impact (RI) matrixand learning curve (LC) to represent and calculate reworkduration [27] For the element in row 119894 and column 119895 in[119877119875]119894119895 if 119895 gt 119894 which means the element belongs to uppertriangular it describes the probability that the completionof activity 119894 causing rework of activity j if 119895 lt 119894 whichmeans the element belongs to lower triangular it describesthe probability that after activity 119895 completes rework theactivity 119894 will be influenced [119877119868]119894119895 represents the possibilityof activity 119894 to be reworked when rework is caused by activity119895 for 119894 119895 = 1 119899 119871119862119894 represents the ratio of activity 119894 to bereworked because the participants may benefit from learningand adaption when rerunning the activity Based on theirdefinition the values of the elements in these twomatrices arebetween 0 and 1 The values of elements in the two matricescan be collected from historical data or estimated basedon risk preferences of the project team Then the expectedrework time of activity 119894 caused by activity 119895 can be calculatedby 119877119875119894119895 times 119877119868119894119895 times 119889119894 times 119871119862119894 where 119889119894 represents the durationof activity i

A number of other DSM simulations have adopted theabove framework with certain extensions to account foradditional constraints in process scheduling For instanceCho and Eppinger [28] proposed a heuristic for solvingstochastic resource-constrained project scheduling problemsin an iterative project network Levardy and Browning [29]accounted for technical performance characteristics besidesduration and cost by setting up a superset of general classesof activities each with modes that vary in terms of inputsduration cost and expected benefits Meanwhile a numberof studies have looked into transferring process DSM to a

4 Mathematical Problems in Engineering

A

B

A

B

tAtB

dA

dB

Figure 3 Information transfer process with overlapping

Table 1 Nomenclature for all parameters in the paper

Parameter Notation Parameter Notation

119860Initial state matrix[119860]119894119894 shows duration of activity119894 for the non-diagonal elements if there isinformation delivered from 119895 to 119894 then the value

of [119860] 119894119895 is non empty else it is empty

119878119863119864119878119894 The activity duration variance matrixEarliest start time of activity i

119875119894119895 The predecessor time factor from j to i 119864119865119894 Earliest finish time of activity i119878119894119895 The successor time factor from j to i 119879119879 The length of critical path

119863 Activity time matrix the diagonal elements areactivity duration and the non-diagonal elements

are minus infinity119865 Activity deviation matrix [F]119894119895 shows the

deviation between activity j and activity i

119877 Activity relationship matrix if there isinformation delivered from j to i then the value of[R]119894119895 is 0 else it is 120576 1199090 0 vector the number of dimension equals

the number of activities

119909+119864 The early finish time for activity 119909minus119864 The early start time for activity

[ ]119899times1 N-dimension vector of certain value 119877119879119894The first and second rework time as wellas rework time the last critical activitycause other activities to generate

respectively for i=123

119890 The activity element in matrix R but the value is 0in the algorithm 119879119865 The total float of each activity

wFor the diagonal elements if the correspondingactivity is on the critical path then the value is 0

or the value is 120576 v

For the diagonal elements if thecorresponding activity is on the

non-critical path then the value is 0 orthe value is 120576

120576 Negative infinity in max-plus [119877119875]119894119895 Rework probability matrix[119877119868]119894119895 Rework impact matrix 119877119908119861119894 Rework buffer of activity iPT Estimated project completion time

more effective process management tool that is closer totraditional process management tools and expressions Atime factor was therefore introduced [30] The time factoris defined based on the overlapping relationship between theactivities As shown in Figure 3 119889119894 and 119889j are defined as theduration of activities 119894 and j respectively 119905119894 as the durationbetween the start of activity 119894 and the time when informationtransition from activity 119894 to activity 119895 is completed and 119905119895as the duration between the start of activity j and the timethe information transition is completed A predecessor timefactor is defined as 119875119895119894 = 119905119894119889119894 and a successor time factor as119878119895119894 = 119905119895119889119895 where 0 lt 119875 le 1 0 le 119878 lt 1 Two matrices Pand S are defined by these two factors The project duration

is then calculated according to (1) and a nomenclature isprovided in Table 1

119864119878lowast119894 = max 119864119878119895 + 119875119894119895 times 119875119895119895 minus 119878119894119895 times 119878119894119894119864119878119894 = max (119864119878lowast119894 0)119864119865119894 = 119864119878119894 + 119860 119894119894119879119879 = max [119864119865119894]

(1)

In short prior research has explored the possibil-ity of using the DSM to predict and control rework inproject scheduling and planning and has achieved noticeableprogressHoweverDSM lacks the capability of functioning as

Mathematical Problems in Engineering 5

a stand-alone project management technique for that DSMrsquosmatrix-based expression is difficult to be effectively convertedto network diagram-based expression sometimes leadingto ambiguity and confusion [5] Therefore it is challengingto apply DSM to the process management of constructionprojects and DSM needs to be extended and improved withfurther research efforts

3 The Critical Chain Design StructureMatrix Method

In this section we are going to demonstrate how to usethe CCDSM method to generate the project schedule Weimprove the max-plus algorithm to reduce computationalload in generating project schedules and introduce reworkbuffers to evaluate and address rework risks

31 Construction Project Scheduling with the Max-Plus Algo-rithm A discrete event system is state-discrete and event-driven Its state evolution depends on asynchronous discreteevents occurring at discrete points over time [31] Simplediscrete event systems are usually linear systems in whichstate and output variables for all possible input variables andthe initial state satisfy superposition principle of the systemAconstruction project can generally be seen as a linear systemthat consists of a number of discrete events

The max-plus algorithm provides an alternative way todescribe the discrete event system Based on the max-plusalgorithm the status of all discrete events denoted as vector119909(119896) in a system can be used to describe the status ofthe system and adjacent statuses are interchangeable with alimited number of linear changes as follows

119909 (1) = 1198601119909 (0)119909 (2) = 1198602119909 (1)

119909 (119896) = 119860119896119909 (119896 minus 1)

(2)

Based on the above equation if the linear changes at everystage 1198601 1198602 119860119896 are known then the final status of thesystem 119909(119896) can be derived from the initial status of thesystem 119909(0) Using this as a starting point the max-plusalgorithm introduces four types of operations to describe thediscrete event systems For two 119898 times 119898 matrices 119883 and 119884denoted as 119883119884 isin 119874119898times119898 whose elements are nonnegativenumbers or negative infinity these operations are defined asfollows

[X oplus Y]119894119895 = max ([X]119894119895 [Y]119894119895)[X and Y]119894119895 = min ([X]119894119895 [Y]119894119895)[X otimes Y]119894119895 = 119898⨁

119896=1

([X]119894119896 + [Y]119896119895)

[X ⊙ Y]119894119895 = 119898⋀119896=1

(minus [X]119894119896 + [Y]119896119895)

(3)

where 119883 otimes 119884 could be further simplified as 119883119884 and 119883119899 =119883119883 sdot sdot sdot 119883⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119899

To apply the max-plus algorithm to schedule manage-ment three new matrices need to be developed to describerelevant project information These matrices include (1)activity durationmatrix denoted asD which is developed byfilling the diagonal elements with each activityrsquos duration andthe nondiagonal elements with negative infinity (2) activitydeviation matrix denoted as F which is developed based onthe rule that [119865]119894119895 represents the deviation of the start time ofactivity 119895 and activity i and (3) activity relationship matrixdenoted as R which is developed based on the rule that ifthere is information delivered from activity 119895 to activity ithen the value of [119877]119894119895 is 0 otherwise the value is negativeinfinity

According to the max-plus algorithm [32] we have thefollowing

The early finish time for each activity is calculated as

119909+119864 = 119863 otimes (119865)lowast otimes 1199090 (4)

where (119865)lowast = 119890 oplus 119865 oplus oplus (119865)119897minus1 The (119894 119895)-th element ofmatrix (119865)lowast means the largest deviation of the start time ofactivity 119895 and activity i if activities 119894 and 119895 are on one ormore paths at the same time otherwise it is 120576 For simplicitythe elements of vector 1199090 are set as e which means thebeginning of activities in the project will not be affected byother construction projects

The length of critical path denoted as the maximum ofearly finish time for all activities is calculated as

119879119879 = max [119909+119864] (5)

The early start time is calculated as the difference betweenearly finish time for each activity and its duration accordingto

119909minus119864 = 119863 ⊙ 119909+119864 (6)

The late start time of activity 119894 is calculated as thedifference between length of critical path and the sum ofactivity 119894rsquos duration as well as subsequent critical activitiesrsquodurations according to

119909minus119871 = [119863 otimes (119865)lowast]119879 ⊙ [1199090 ⊙ 119879119879] (7)

The total float is calculated as

[119879119865]119894 = [119909minus119864]119894 ⊙ [119909minus119871]119894 (8)

The critical chain is then determined by the set ofactivities 120572 that satisfy 120572 | [119879119865]120572 = 0

It needs to be noted that when determining floats themax-plus algorithm is run in simple matrix form whichsignificantly improves the computational efficiency and ismore applicable to large-scale projects [33] The definitionsof parameters in the above equations are summarized in thenomenclature in Table 1

6 Mathematical Problems in Engineering

rework time

i

j

(a) First rework

j

k

i

rework time

(b) Second rework caused by information tran-sition from its precedent activity

rework time

i

j

k

(c) Second rework caused by information feed-back from its successive activity

Figure 4 First rework and second rework

32 Determining Project Rework Buffer It is widely believedthat information uncertainty is themain cause of rework [34]Such uncertain information may transmit from upstreamactivities to downstream activities or feedback in the oppo-site direction creating information flow Information flowinteractions lead to rework risks in construction projects andconsequently bring about more rework time There are twomain situations that rework exists(1)First rework as depicted in Figure 4(a) i is a precedentactivity of j and they perform sequentially according to rela-tionships After it is completed the performance informationof 119895 will be generated and transmitted to i which may resultin rework of 119894 The first rework time shown as the shadowarea in Figure 4(a) can be calculated by [27]

[1198771198791]119894119895 = 119877119875119894119895 sdot 119877119868119894119895 sdot 119871119862119894 sdot 119863119894119894 (9)

Then the total first rework time of 119894 caused by all itsdownstream activities can be obtained by

[1198771198791]119894 =119899sum119895=119894+1

119877119875119894119895 sdot 119877119868119894119895 sdot 119871119862119894 sdot 119863119894119894 (10)

where 119899 represents the number of activities in the schedule(2) Second rework as depicted in Figures 4(b) and 4(c)after completing the rework of 119895 caused by 119896 119895 transmits orfeeds back some revised information to 119894 which may causerework of 119894 Such two forms are defined as the second reworkin the paper The second rework times shown as the shadowareas in Figures 4(b) and 4(c) can be calculated by

[1198771198792]119894119895119896 = 119877119875119894119895 sdot 119877119875119895119896 sdot 119877119868119894119895 sdot 119871119862119894 sdot 119863119894119894 (11)

Then the total second rework time of 119894 caused by all itsdownstream activities can be obtained by

[1198771198792]119894 =119896minus1sum119895=1

119899sum119896=119894+1

119877119875119895119896 sdot 119877119875119894119895 sdot 119877119868119894119895 sdot 119871119862119894 sdot 119863119894119894 (12)

To proactively account for the uncertainty caused byinteractivity relationships and the associated rework risks arework buffer (119877119908119861) is introduced in the proposed CCDSMConceptually 119877119908119861 assesses and compresses the durationuncertainty caused by rework and is placed after the comple-tion of rework activities in project schedules119877119908119861 is designedto warn project participants from how much workload may

increase for each rework activity so as to prepare the projectparticipants for sufficient time and resources

The rework buffer of 119894 can be obtained by

119877119908119861119894 = [1198771198791]119894 + [1198771198792]119894 (13)

For the last critical activity 120579 in the schedule consideringall reworks it causes other activities to generate will take placeafter its completion andhave an impact onproject completiontime therefore its rework buffer is defined as the sum of thetotal rework time it causes other activities to generate and itstotal rework time caused by other activities

The total rework time it causes all other activities togenerate can be calculated by

[1198771198793]120579 ==119899sum119894=1

119877119875119894120579 sdot 119877119868119894120579 sdot 119871119862119894 sdot 119863119894119894 + 119899sum119894=1

119899sum119895=1

119877119875119894119895sdot 119877119875119895120579 sdot 119877119868119894119895 sdot 119871119862119894 sdot 119863119894119894

(14)

The rework buffer of 120579 then can be obtained by

119877119908119861120579 = [1198771198791]120579 + [1198771198792]120579 + [1198771198793]120579 (15)

33 Determining Project Buffers and Feeding Buffers inMax-Plus Representation

331 Determining Project Buffers Previous studies on theCCPM-max-plus representation adopted the CampPMmethodfor calculating project buffers shown as (16) and feedingbuffers [33]

119875119861 = max [(119863 otimes 119908 otimes 119877)lowast (119863 otimes 119908) 1199090]3 (16)

where (119894 119895)-th element of matrix (119863 otimes 119908 otimes 119877)lowast means thecumulative time of duration of activity 120572 which is on thecritical chain and range from 119894 + 1 to j (119894 119895)-th element ofmatrix119863 otimes119908 satisfies that if activity 119894 is on the critical chainthen (119894 119895)-th element is duration of activity I otherwise it is120576

However the CampPM method lacks sound mathematicalfoundation and overestimates project durations resulting ina waste of time and resources Alternatively the CampPM isreplaced by the RSEM which is based on the large numberlaw and central limit theorem in the CCDSM to calculate thePB

Mathematical Problems in Engineering 7

119875119861 = radic sum119894isin119862119875

119878119863119894 (17)

where 119878119863119894 is the variance of duration of activity 119894 on thecritical path

To calculate the PB in max-plus representation a matrixdenoted as 119879119901 is introduced For the element in row i andcolumn 119894 minus 1 in [119879119901]119894119895 if activity 119894 is critical activity it willbe filled with the variance of duration of activity i else it willbe zero For other elements in [119879119901]119894119895 it is filled with zeroThePB then can be represented in max-plus algebra as

119875119861 = radic1198621198790 otimes 119879lowast119901 otimes 1198620 (18)

where 1198621198790 = (119890 119890 119890)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119899+1

332 Determining Feeding Buffers In order to insert feedingbuffers in place to protect critical path from delays innoncritical paths two operators denoted as diag() and arefirst introduced For a vector 119909 a newmatrix can be obtainedby the operator diag() as [diag()]119894119895 = 119909119894 119894119891 119894 = 119895 120576 119890119897119904119890 and a new vector can be obtained by the operator as[119909]119894 = 120576 119894119891 [119909]119894 = 120576 119890 119894119891 [119909]119894 = 120576

In preparation of the location and size of the feedingbuffer another two vectors are introduced denoted as 119908 andV Matrix119908 is developed based on the rule that for a diagonalelement if the corresponding activity is on the critical paththen the element value is 0 otherwise the element value is 120576Matrix V is developed based on the rule that for a diagonalelement if the corresponding activity is on the noncriticalpath then the value is 0 otherwise the value is 120576 Moreoveran adjacency matrix 119877 is transformed from matrix 119865 by[119877]119894119895 = 119890 119894119891 [119865]119894119895 = 120576 120576 119890119897119904119890 Vector V1015840 is then introducedto locate the feeding buffers as follows

[V1015840]120574= [119877119879120573120572 otimes 119908]120574 (19)

where 119877120573120572 is the adjacency matrix which represents transi-tions from noncritical activities to critical ones and can beobtained by 119877120573120572 = diag(119908) otimes 119877 otimes diag(V) 120582 represents thenoncritical activity one ofwhose successors is critical activityTherefore a feeding buffer should be inserted behind theactivity 120574

The feeding buffer can then be determined by squarerooting of the sum of [119878119863]120578 where 120578 is the set of activitieson a certain noncritical chain The formula is expressed inmax-plus algebra as

1198651198610 = radicdiag [119877120573120573 otimes diag (V1015840) otimes diag (119878119863)]lowast otimes 1198621 otimes diag [diag (119878119863) otimes diag (V1015840) otimes 1198621] otimes 1198621 (20)

Additionally the calculation of feeding buffer should beadjusted using the following equation to meet the constraintthat the size of feeding buffer cannot exceed total float

[119865119861]120574 = min ([1198651198610]120574 [119879119865]120574) (21)

34 Implementation of the CCDSM with Max-Plus Lin-ear Expression The max-plus-based implementation of theCCDSM is summarized in the following seven steps It needsto be noted that as the max-plus algorithm cannot solvenonlinear conversions such as multiplication the input datafor the project rework problem including matricesA P S RPand RI needs to be preprocessed

Step 1 (preprocess119860) Extract all diagonal elements of matrix119860 and form a diagonal matrix D

Step 2 (preprocess 119875 and 119878) Combine matrices 119875 and 119878 togenerate matrix 119865 according to the following equation

[119865]119894119895 = [119875]119894119895 times [119875]119895119895 minus [119878]119894119895 times [119878]119894119894 (22)

Extract all nonnull elements ofmatrix119865 and use them to formmatrix 119877 Then update R by replacing all nonnull elementswith 119890 and all null elements with 120576Step 3 (preprocess RP and RI) Based on (9)-(15) mergematrices RP and RI to generate the vector 119877119908119861 whose 119894-thelement is the corresponding rework buffer of activity i

Step 4 (calculate the length of critical path) Based on (4)-(8)the length of critical pathTT and other parameters includingthe total float time TF and critical path are determined

Step 5 (calculate matrices 119908 and V) According to the criticalpath determined in Step 4matrices119908 and V can be generatedMeanwhile the activity duration variance matrix SD canbe generated based on data collected from the project tocalculate the buffers size

Step 6 (calculate buffer sizes) Based on (17)-(21) projectbuffer PB and feeding buffers FBs are calculated with RSEMand represented in max-plus algebra

Step 7 (generate the schedule) As the principle of CCPMthe feeding buffers should be placed on the noncritical chainsprior to the joints of the critical chain and noncritical chainsand the project buffer should be placed at the end of theschedule to protect the whole project process Then theestimated project time PT can be calculated based on

119875119879 = 119879119879 + 119875119861 + 119877119908119861120579 (23)

4 Case Study

In this section the proposed CCDSM method is imple-mented in a case project to demonstrate its feasibility andeffectiveness in addressing rework risks in project schedule

8 Mathematical Problems in Engineering

A B C D E F G H

A 8 X X X X

B X 8 X X X X

C 4 X X

D X 13 X

E X X X X 24 X

F X X X X 35 X

G X X X X 8

H X X X X

I J K L M N O

I X X X X X

J X X X X X X

K X X X X X

L X X X X X X X

M X

N X

O X X X X X X X X

X X X 4 X

X X X X X X 2

X X 4 X X

13 X

X X 8 X X

13

X X 5 X X X

X

X

X

X

X

13

X X X

X

X X

X X

X

X

X

P Q R S

X X

X X

X X

X X

X X

X X

X

X

P X X X X X X X X

Q X X X X X X X X

R X X X X X X X

S

X X X X X X

X

X X X X X X X

X X X X X X X

X X 4

X X X 4

X 4 X

3 X

(a) Matrix A

A B C D E F G H

A 10 09 07 08

B 09 06 07 06

C 07 09 10

D 07 06

E 08 06 06 08 06

F 09 09 08 08 06

G 06 07 06 06

H 06 06 06 06

I J K L M N O

I 06 09 09 06 10

J 05 09 09 07 07 08

K 06 07 06 06 08

L 06 10 08 07 07 06

M 05

N 06 05

O 07 07 08 06 06 09 09 05

08 08 08 08

06 10 06 08 06 06

09 10 09 09

05

08 06 06 10

07 06 09 06 06

09

08

08

10

07

07 08 06

09 07

09 09

09

08

08

07

P Q R S

10

09

09

10

09

05

06

08

P 09 06 07 08 09 06 07 05

Q 08 09 10 09 10 05 10

R 10 07 07 07 06 06 08

S

10 06 08 06 08 09

09

07 08 09 06 10 05 09

06 08 06 10 09 06 07

07 07

09 08 08

08 07

09

(b) Matrix P

A B C D E F G H

A 05 03 02 02

B 01 04 02 05 05

C 01 02

D 01 03

E 04 02 04

F 04 04 04 04 04

G 03 05 05 04

H 04 03 01 05

I J K L M N O

I 01 01 05 02 04

J 03 04 02

K 05 02 03 01 04

L 02 04 03 05 04 03 02

M 01

N 04

O 04 04 02 03 01 02 04

02 02 04 02

04 05 04 03 01 05

02 04 01 03

03

02 04 01 01

03 01 03 01 01

04

03

03

04

01

04 02 03

04

03 04

03 01

02

05

03

P Q R S

01 02

05

04

01 02

04 02

05

01

P 02 05 05 02 03 02 04

Q 03 04 01 02 02

R 02 03 01 04 02 04 01

S

04 03 02 01 01 02

02

01 04 03 01 01 03 04

01 01 04 02 04 02 02

02 01

02 04 03

01 04

01

(c) Matrix S

A B C D E F G HA 01 02 03

B 04 02 01 01

C 02 01

D 03 04

E 06 03 05 05 04

F 03 02 05 05 03

G 02 05 05 02

H 04 02 06 03

I J K L M N O

I 02 03 03 04 02

J 03 03 04 05 03 05

K 04 05 02 06 03

L 03 05 05 03 04 04 03

M 02

N 03

O 05 03 02 05 04 05 06 0306 03 05 01

03 02 05 06 03 05

03 04 01

0203 05 02 0403 04 04 03 03

03

0103 02 02

02

01 04

02

03

0503

P Q R S

02 0102 02

01 03

0301 0202 03

01

02

P 04 04 06 02 04 04 05 06

Q 03 06 05 03 06 05 04 05

R 03 05 03 03 03 05 05

S

02 06 04 03 04 0404

03 06 04 05 03 02 0404 05 06 05 04 03 03

03 0605 02 04

03 0201

(d) Matrix RP

A B C D E F G H

A 008 001 001 009

B 020 014 016 020 007

C 007 008

D 008 023

E 010 001 009 020 013

F 003 025 013 008 023

G 025 002 003 015

H 018 005 022 001

I J K L M N O

I 013 023 020 011 021

J 002 017 007 025 003 024

K 001 008 012 015 002

L 006 013 016 014 003 023 009

M 001

N 006

O 005 016 011 004 014 019 010 015

021 014 005 003

007 004 019 002 005 016

018 009 013 022

012

004 020 015 022

003 022 013 004 023

013

006

023

017

019

005 023 021

018

023 006

023 017

004

004

011

P Q R S

010 009

010 009

022 009

016 010

014 013

004 018

001

018

P 007 014 014 021 023 012 011 014

Q 03 008 025 001 022 014 002 008

R 012 014 006 021 015 021 018

S

022 001 006 020 008 003

014

022 001 006 010 008 019 001

016 011 021 001 021 001 015

009 025

021 018 019

009 019

014

(e) Matrix RI

Figure 5 Project information in the case project

management The settings of the case project are first pre-sented followed by descriptions of implementation of theCCDSM in the case project The schedule generated with theCCDSMmethod is assessed in detail and compared with theschedules generated with traditional CCPM- and DES-basedmethods

41 Case Project Settings The case used in this study wasderived from a modular real estate development projectfirstly introduced in [35] and further described in [36]The matrix 119860 of the case is shown in Figure 5(a) Thiscase consisted of 19 major activities and 183 interactivityrelationships including 65 rework relationships representedby elements above the diagonal of matrix 119860 It is assumedthat rework can propagate up to twice to avoid infinite loopin the computation The parameters for each activity and theinteractivity relationships were derived based on literatureand empirical evidence and are shown in Figures 5(b) 5(c)5(d) and 5(e) and summarized in Table 2 Large-scale reworkrelationships and complex connections between activitieswere observed in the case project which was representativeof typical construction projects in reality

42 Implementation of the CCDSM Method The proposedCCDSM method was implemented in the case project fol-lowing the steps explained in the last section and the results

were shown below Noticeably for better understandingof scheduling process the implementation process of theCCDSMmethod is reorganized as below

Step 1 Matrices 119863 and 119877 were derived from matrix Aas shown in Figure 6 The duration of each activity waspresented in the diagonal of the matrix D

Step 2 Matrix F (see Figure 7) was derived from matrices119875 and 119878 according to (22) This step calculated the activitydeviation matrix based on the predecessor time matrix andthe successor time matrix

Step 3 Optimize activity sequence and update matrices D RRP RI and 119865 according to new activity sequence To reduceproject rework and obtain near-optimal project completiontime the genetic algorithm (GA) was applied to optimizeactivity sequence The GA is a metaheuristic method thatsearches for optimal solutions using processes similar tothose in natural selections and genetics [37] In the paperminimization of total length of rework path first proposed byGebala and Eppinger [38] was used as the objective functionto calculate the optimal sequence of activities

119891 = 119899sum119894=1

119899sum119895=119894+1

(119895 minus 119894) sdot 119908 (119894 119895) (24)

Mathematical Problems in Engineering 9

Table 2 Activity parameters in the case project

ID Activity Learning curve Duration (day)Min Likely Max

A Perform prelim mkt analysis 05 5 8 15B Evaluate marketability options 06 5 8 15C Engage feasibility consultants 03 3 4 7D Evaluate planning amp zoning process 05 10 13 20E Perform massing study 06 20 24 35F Develop conceptual design 08 30 35 50G Identify external stakeholders 04 5 8 15H Identify permits amp approvals 07 10 13 20I Complete phase 1 ESA 03 10 13 20J Evaluate consultants amp contractors 06 10 13 20K Obtain rough construction costs 05 5 8 15L Determine highest amp best use 05 3 5 9M Identify debt options 04 2 4 8N Identify equity options 05 3 4 5O Update financial underwriting 05 1 2 3P Reevaluate organization strategy 06 2 3 5Q Estimate schedule 06 3 4 7R Gain control of site andor client 06 3 4 7S Review and approve 07 3 4 7

F C A I J R K PF 30C 3A 5I 10J 10R 3K 5P

D M O B H E N

DMOBHEN

203

10

21

10

5

2

L G Q S

LGQS

33

53

(a) Matrix D

F C A I J R K PFCA e eI eJ e eR e e eK e e e eP e e e e e e

D M O B H E N

D e eM eO e e e e eB e e eH e e eE e eN e e

e e e ee e

e

e eee

L G Q S

L e e e e eG e e eQ e e e e e e e eS e e

e e e e e e ee

e e ee e e e e e e

e ee

e

e

e

e

e

(b) Matrix R (element 120576 is omitted)

Figure 6 Matrix D amp R

where 119908(119894 119895) represents rework probability of activity 119894caused by activity 119895

The parameter settings were selected as follows pop-ulation size set as 50 number of generations set as 150crossover probability set as 095 and mutation probabilityset as 008 The GA process reported an optimal schedule as[119875 119876119867 119864 119877 119861 119862119870119873 119871119872 119865119863 119878 119860 119866 119869 119868 119874] The GAconvergence process is shown in Figure 8

Step 4 119877119908119861 was generated from matrices RP and RI basedon (9)-(15) The goal of this step was to factor in andcalculate the rework time of the case project 119877119908119861 was[3 4 6 10 5 1 1 3 1 2 1 15 1 1 1 1 1 1 9]Step 5 To generate reliable project schedules the most likelyduration of each activity which has been widely used andaccepted by project teams in prior research [39] was selected

10 Mathematical Problems in Engineering

A B C D E F G HA 1 10 15 28

B 7 1 13 23 2

C 13 8

D 2 1

E -3 1 3 10 15

F -6 -11 -4 5 -

12G 3 2 6

H -3 4 14

I J K L M N O

I 2 11 16 3 8

J 5 -1 8 18 28 4

K 2 5 2 15 28

L 4 6 1 10 16 27 7

M 13

N 12

O 5 6 2 8 16 35 7 76 4 2 1

8 12 4 4 3 2

7 4 4 1

-19 5 3 29 5 3 2 1

1174-19

-8 -1 -99

-1 -8

8 9

-1

57

P Q R S

3 33 4

1 4

-1 33 2

2

-1

2

P 8 5 2 11 23 22 6 6

Q 7 7 2 14 23 37 4 13

R 7 5 11 17 23 4 10

S

11 4 4 2 3 11

9 10 6 3 4 19 10 4 5 3 2 1

2 32 2 3

2 22

Figure 7 Matrix F (element 120576 is omitted)

20

30

40

50

60

70

80

Estim

ated

Fee

dbac

k Le

ngth

50 100 1500Generation

Figure 8 Convergence curve in the GA-based optimization

to calculate the estimated project duration The critical chainwas reported as [119867 119864 119861 119870119873 119871 119865 119860 119866 119869 119874]Step 6 The activity duration variance matrix SD was gen-erated with 1000 Monte Carlo simulations of the activitiesrsquoduration that follows a beta distribution

Step 7 Buffers were calculated based on (17)-(21) Theproject buffer was 36 days and the feeding buffer was[3 4 0 0 38 0 4 0 0 0 6 0 10 38 0 0 0 0 2 0]Step 8 The estimated project duration was calculated as 102days based on (23) The project buffer was placed at the endof the schedule and the feeding buffer was placed on thenoncritical chains prior to the joints of the critical chain andnoncritical chains following the principle of CCPM

It was worth mentioning that overlapping was observedamong various activities including critical activities in the

above schedule which illustrated real-world high-frequencyinteractions among activities in the project

To assess the performance of the CCDSM method andthe quality of the project schedule it generated the executionof case project was simulated 1000 times using Monte Carloapproach where the sampling of actual activity durationsfollowed beta distribution The simulated actual projectdurations were depicted in Figure 9 It can be seen in thefigure that under rework assumption the process scheduleof 102 days had a completion probability of 981 indicatingthat the proposed CCDSM method was able to providesatisfactory assurance of on-time project completion undervarying scheduling scenarios

43 Comparison of the CCDSM Method with TraditionalCCPM and DES Methods This subsection further comparesthe performance of the CCDSM method with traditional

Mathematical Problems in Engineering 11

75 80 85 90 95 100 105 1100

20

40

60

80

100

120

140

Duration

CCDSM method=102

0

01

02

03

04

05

06

07

08

09

1

981

CCDSM method=102

Cum

ulat

ive P

roba

bilit

y

Freq

uenc

y

80 85 90 95 100 105 11075Duration

Figure 9 Simulated actual project durations and on-time project completion probability of the CCDSM-based schedule

75 80 85 90 95 100 105 1100

01

02

03

04

05

06

07

08

09

1

Duration

981

CCDSM method=102

493

CCPM method=93

Cum

ulat

ive P

roba

bilit

y

Figure 10 Comparison of on-time project completion probabilities between the CCDSM- and CCPM-based schedules

CCPM and DES methods to assess its effectiveness inaddressing rework risks

A CCPM-based schedule was generated with tradi-tional buffers including project buffer and feeding bufferand without rework buffer The results indicated that theestimated project durations of the CCPM-based schedulewere 93 days including project buffers of 36 days Whenthe estimated project duration of CCPM-based schedulewas superimposed in the Monte Carlo-simulated projectdurations as depicted in Figure 10 it can be seen thatthe on-time completion probability was 493 The resultssuggested that the CCDSM-based schedule which took intoconsideration rework relationships and had a duration of 102days significantly outperformed the CCPM-based schedulein ensuring on-time completion of the case project that wasfaced with typical rework risks

Furthermore a DES-based schedule was generated byimplementing DES algorithm in which activity durationsfollowing a beta distribution were generated at random tocompute the project duration When the DES algorithm was

used to generate project schedules under rework scenarios aprobabilistic judgment is performed firstly on whether or noteach activity causes rework and then an iterative simulationprocess began until all activities were finished Specificallythe DES algorithm proposed by Browning and Eppinger[27] was adopted in this study with the following minormodifications to make it more applicable to the case projectand comparable to the CCDSM method (1) the optimizedactivity sequence in Section 42 was employed in the DESprocess (2) activity 119894 did not started until STS logic tie withits each predecessor activity 119895 that is [119865]119894119895 was satisfied (3)rework probability of activity 119894 would decrease by 50 eachtime it reworked [28]

To compare the CCDSM-based schedule with the DES-based schedule the execution of the case project usingthe DES-based schedule was simulated 1000 times usingMonte Carlo approach and the simulated actual projectdurations are depicted in Figure 11 The results showed thatthe estimated project duration of the DES-based schedulewas 115 days which was 13 days longer than that of the

12 Mathematical Problems in Engineering

70 80 90 100 110 120 1300

010203040506070809

1

Duration

981

DES method=115Cu

mul

ativ

e Pro

babi

lity

Figure 11 On-time project completion probability of the DES-based schedule

CCDSM-based schedule with the same on-time completionprobability of 981 In other words the CCDSM methodoutperformed the DES method by ensuring the same on-time completion probability with a shorter project scheduleMoreover the CCDSM method was also advantageous overthe DES-based method in that it enabled effective controlof project schedules during project implementation whichcould be done by monitoring consumption of time buffersleading to reduction in project overrun risks

5 Conclusions

Construction projects are constantly challenged by reworkrisks which have largely remained unaddressed by existingproject schedulingThis paper proposed the CCDSMmethodfor developing project schedules that are adaptive to reworkinstances during project execution This method modelslarge-scale rework relationships among activities with theintroduction of a new rework buffer in traditional CCPMmethod The method allows analysis and visualization ofthe schedules and utilizes max-plus algorithm to transformcomplex logic relationships into simple matrix operationsreducing the computational load of generating process sched-ules A case study was conducted to demonstrate the imple-mentation of the CCDSMmethod and assess its effectivenessin addressing rework risksThe results showed that CCDSM-based schedule outperformed the CCPM-based schedule inensuring on-time completion of the case project that wasfaced with typical rework risks In addition in comparison tothe DESmethod the CCDSM performed better by providinga smaller project duration with the same probability of on-time project completion

The CCDSM method contributes to construction man-agement in the following ways First the CCDSM methodabsorbs risks of rework and various other uncertainties togenerate the reliable project schedule enabling managersto predict project durations more accurately before projectsstart Based on accurate prediction of project durationsthe negotiation of contracts and supply of resources canbe conducted in a more reasonable manner avoiding costsoverruns schedule delays and even project failures Second

based on generated project schedule with time bufferseffectivemonitoring and control of construction progress canbe realized Overall speaking the proposed CCDSMmethodprovides a promising solution to mitigate rework risks inconstruction projects and protect projects from undesirabledelays caused by possible rework instances

Meanwhile the CCDSM bears several limitations thatwould benefit from future improvement First only the STSlogic tie between activities is tested in this study Techniquesfor integrating multiple logic ties including STS STF FTSand FTF deserve further attention Second resource conflictsand levering are considered beyond the scope of this studybut it is a critical challenge that should be addressed inthe context of CCDSM-based scheduling in future researchThird the parameters required to implement the CCDSMmethod are hard to obtain for a specific project To employCCPM or max-plus algorithm for project scheduling it isnecessary to obtain the estimated activity durations undervarious completion rates and the logic tie between activitiesSimilarly the rework relative parameters including RP RIand LC are required for implementation of DSM-basedmethod As an integration method of CCPM max-plus algo-rithm and DSM the implementation of CCDSM requiresall the above parameters to be obtained Lastly methods fordynamic updating of buffer sizes during project execution areneeded in order to better prepare project teams for changingproject environment and dynamic project progress

Data Availability

All data used to support the findings of this study are includedwithin the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (NSFC) [Grant no 71671128]

Mathematical Problems in Engineering 13

References

[1] Y Li and T R Taylor ldquoModeling the impact of design reworkon transportation infrastructure construction project perfor-mancerdquo Journal of Construction Engineering and Managementvol 140 no 9 p 04014044 2014

[2] B Hwang X Zhao and K J Goh ldquoInvestigating the client-related rework in building projects The case of SingaporerdquoInternational Journal of Project Management vol 32 no 4 pp698ndash708 2014

[3] E K Simpeh R Ndihokubwayo P E Love and W DThwala ldquoA rework probability model a quantitative assessmentof rework occurrence in construction projectsrdquo InternationalJournal of Construction Management vol 15 no 2 pp 109ndash1162015

[4] P W Morris J K Pinto and S Jonas 13e Oxford Handbookof Project Management Oxford University Press Oxford UK2012

[5] S D Eppinger and T R Browning Design Structure MatrixMethods and Applications MIT Press Books 2012

[6] T R Browning ldquoDesign structure matrix extensions and inno-vations a survey and new opportunitiesrdquo IEEE Transactions onEngineering Management vol 63 no 1 pp 27ndash52 2016

[7] A Kulkarni D K Yadav and H Nikraz ldquoAircraft maintenancechecks using critical chain project pathrdquo Aircra EngineeringAerospace Technology vol 89 no 6 pp 879ndash892 2017

[8] J Trojanowska and E Dostatni ldquoApplication of the theory ofconstraints for project managementrdquoManagement and Produc-tion Engineering Review vol 8 no 3 pp 87ndash95 2017

[9] M Ghaari and M Emsley ldquoBufer sizing in CCPM portfolioswith dierent resource capacitiesrdquo International Journal of Infor-mation Technology Project Management vol 8 no 3 pp 40ndash512017

[10] E M Goldratt Critical Chain A Business Novel MA NorthRiver Press Great Barrington United States 1997

[11] S Van de Vonder E Demeulemeester and W HerroelenldquoProactive heuristic procedures for robust project schedulingAn experimental analysisrdquo European Journal of OperationalResearch vol 189 no 3 pp 723ndash733 2008

[12] O Lambrechts E Demeulemeester and W Herroelen ldquoTimeslack-based techniques for robust project scheduling subject toresource uncertaintyrdquo Annals of Operations Research vol 186no 1 pp 443ndash464 2011

[13] D Trietsch and K R Baker ldquoPERT 21 Fitting PERTCPMfor use in the 21st centuryrdquo International Journal of ProjectManagement vol 30 no 4 pp 490ndash502 2012

[14] GMaAWangN Li LGu andQAi ldquoImproved critical chainproject management framework for scheduling constructionprojectsrdquo Journal of Construction Engineering andManagementvol 140 no 12 p 04014055 2014

[15] R C Newbold Project Management in 13e Fast Lane Applying13e 13eory of Constraints CRC Press Boca Raton FloridaLondon 1998

[16] O I Tukel W O Rom and S D Eksioglu ldquoAn investigation ofbuffer sizing techniques in critical chain schedulingrdquo EuropeanJournal of Operational Research vol 172 no 2 pp 401ndash4162006

[17] O Icmeli and S S Erenguc ldquoA branch and bound procedurefor the resource constrained project scheduling problem withdiscounted cash flowsrdquoManagement Science vol 42 no 10 pp1395ndash1408 1996

[18] E Roghanian M Alipour and M Rezaei ldquoAn improved fuzzycritical chain approach in order to face uncertainty in projectschedulingrdquo International Journal of ConstructionManagementvol 18 no 1 pp 1ndash13 2017

[19] J Zhang X Song and E Dıaz ldquoProject buffer sizing of a criticalchain based on comprehensive resource tightnessrdquo EuropeanJournal of Operational Research vol 248 no 1 pp 174ndash182 2016

[20] F A Rivera and A Duran ldquoCritical clouds and critical sets inresource-constrained projectsrdquo International Journal of ProjectManagement vol 22 no 6 pp 489ndash497 2004

[21] G Ma L Gu and N Li ldquoScenario-based proactive robustoptimization for critical-chain project schedulingrdquo Journal ofConstruction Engineering and Management vol 141 no 10 p04015030 2015

[22] H Goto ldquoForward-compatible framework with critical-chainproject management using a max-plus linear representationrdquoOPSEARCH vol 54 no 1 pp 201ndash216 2017

[23] T Salama A Salah and O Moselhi ldquoIntegration of linearscheduling method and the critical chain project managementrdquoCanadian Journal of Civil Engineering vol 45 no 1 pp 30ndash402018

[24] C W Hu X D Chen and L H Wu ldquoThe setting method ofproject buffer in critical chain management of mould manufac-turing project based reworkingrdquo Advanced Materials Researchvol 317-319 pp 418ndash422 2011

[25] D V Steward Systems Analysis and Management StructureStrategy and Design IrwinMcGraw-Hill Boston 1981

[26] T R Browning Modeling and analyzing cost schedule andperformance in complex system product development [PhDthesis]Massachusetts Institute of Technology CambridgeMas-sachusetts USA 1998

[27] T R Browning and S D Eppinger ldquoModeling impacts ofprocess architecture on cost and schedule risk in productdevelopmentrdquo IEEE Transactions on Engineering Managementvol 49 no 4 pp 428ndash442 2002

[28] S H Cho and S D Eppinger ldquoA simulation-based processmodel for managing complex design projectsrdquo IEEE Transac-tions on Engineering Management vol 52 no 3 pp 316ndash3282005

[29] V Levardy and T R Browning ldquoAn adaptive process modelto support product development project managementrdquo IEEETransactions on Engineering Management vol 56 no 4 pp600ndash620 2009

[30] J U Maheswari and K Varghese ldquoProject scheduling usingdependency structure matrixrdquo International Journal of ProjectManagement vol 23 no 3 pp 223ndash230 2005

[31] C G Cassandras and S Lafortune Introduction to DiscreteEvent Systems Springer New York NY USA 2nd edition 2008

[32] S Yoshida H Takahashi and H Goto ldquoModified max-pluslinear representation for inserting time buffersrdquo in Proceedingsof the IEEE International Conference on Industrial Engineeringamp Engineering Management IEEE 2010

[33] S Yoshida H Takahashi and H Goto ldquoResolution of timeand worker conflicts for a single project in a max-plus linearrepresentationrdquo Industrial Engineering amp Management Systemsvol 10 no 4 pp 279ndash287 2011

[34] S Muralidharan ldquoFactors affecting rework in construc-tion projectrdquo International Journal of Engineering Sciences ampResearch Technology vol 1 no 5 pp 578ndash584 2016

[35] B B E Bulloch and J Sullivan Application of the DesignStructure Matrix (DSM) to the real estate development process

14 Mathematical Problems in Engineering

[Master thesis] Massachusetts Institute of Technology Cam-bridge Massachusetts USA 2009

[36] S D Eppinger S Bonelli andAM Gonzalez ldquoManaging itera-tions in the modular real estate development processrdquo ReducingRisk in Innovation Proceedings of the 15th International DSMConference Melbourne Australia 29-30 August 2013 pp 37ndash442013

[37] D T Pham and D Karaboga Intelligent Optimisation Tech-niques Genetic Algorithms Tabu Search Simulated AnnealingandNeural Networks Springer Science amp BusinessMedia 2000

[38] D A Gebala and S D Eppinger ldquoMethods for analyzing designproceduresrdquo in Proceedings of the AMSE 13ird InternationalConference On Design13eory andMethodology vol 31 pp 227ndash233 Miami Florida 1991

[39] O Hazır M Haouari and E Erel ldquoRobust scheduling androbustness measures for the discrete timecost trade-off prob-lemrdquo European Journal of Operational Research vol 207 no 2pp 633ndash643 2010

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Mathematical Problems in Engineering 3

A1 A1

A1

A2 X

A1 A1

A1

A2 X

A1 A2

(a) Sequential

A1 A2

A1

A2

A1 A2

A1

A2

A1

A2

(b) Parallel

A1 A2

A1 X

A2 X

A1 A2

A1 X

A2 X

A1

A2

(c) Coupled

Figure 2 Different types of relationship between activities represented in the DSM

risks could also impact the critical chain and it identified thecritical chain in a traditional manner

With respect to critical chain scheduling Francisco andAlfonso [20] extended traditional critical chain schedulingmethods by using critical set and critical cloud to avoidambiguity in the identification of critical tasks Ma et al [21]proposed a critical chain scheduling method with a scenario-based proactive robustness optimization The method wasable to yield higher probabilities of on-time project com-pletion than traditional CCPM-based scheduling methodsGoto [22] developed a scheduling framework for projectsunder limited resources by representing the CCPM methodin max-plus algebra to achieve a short lead time Salama etal [23] took into account the impacts of resource continuityand variability on activity durations and integrated linearscheduling and CCPM methods to identify multiple criticalsequences Despite active research in CCPM-based linearprocess scheduling in recent years the challenge of reworkin CCPM-based schedules has barely been addressed [24]where further research is needed

22 DSM as a Process Management Technique Since Steward[25] first introduced the DSM for the square-matrix-basedmodels of processes process architecture DSM models (pro-cess DSMs) have received the most attention among differentapplication areas of DSM [6]TheDSMuses matrix format touniformly represent three different logical relationships anddependencies including sequential parallel and coupledbetween activities as shown in Figure 2 An off-diagonalelement 119860 119894119895 of the DSM matrix represents with a mark ora number the information flow from activity 119895 to activity 119894or the dependency relationship between these two activitiesIf no dependency exists between activities 119895 and 119894 it willbe null or zero Diagonal element 119860 119894119894 is occasionally usedto represents certain characteristics such as duration of theactivity 119894 or simply nothing in most cases

In this matrix-based display elements above the diagonalare usually regarded as information feedback or iterativeactivities relationship [5] Such display provides a useful tool

for analyzing rework and has given rise to a number of rele-vant studiesThefirstDSM-based discrete eventMonteCarlosimulation model was proposed by Browning [26] and laterextended by Browning and Eppinger [27] to estimate projectduration and cost and their variances They argued thatprocesseswith the fewest feedbackmarks in theDSMmaynotbe necessarily optimal and could be sped up with appropriateincrease in overlapping and iterations [6] They introducedrework probability (RP) matrix rework impact (RI) matrixand learning curve (LC) to represent and calculate reworkduration [27] For the element in row 119894 and column 119895 in[119877119875]119894119895 if 119895 gt 119894 which means the element belongs to uppertriangular it describes the probability that the completionof activity 119894 causing rework of activity j if 119895 lt 119894 whichmeans the element belongs to lower triangular it describesthe probability that after activity 119895 completes rework theactivity 119894 will be influenced [119877119868]119894119895 represents the possibilityof activity 119894 to be reworked when rework is caused by activity119895 for 119894 119895 = 1 119899 119871119862119894 represents the ratio of activity 119894 to bereworked because the participants may benefit from learningand adaption when rerunning the activity Based on theirdefinition the values of the elements in these twomatrices arebetween 0 and 1 The values of elements in the two matricescan be collected from historical data or estimated basedon risk preferences of the project team Then the expectedrework time of activity 119894 caused by activity 119895 can be calculatedby 119877119875119894119895 times 119877119868119894119895 times 119889119894 times 119871119862119894 where 119889119894 represents the durationof activity i

A number of other DSM simulations have adopted theabove framework with certain extensions to account foradditional constraints in process scheduling For instanceCho and Eppinger [28] proposed a heuristic for solvingstochastic resource-constrained project scheduling problemsin an iterative project network Levardy and Browning [29]accounted for technical performance characteristics besidesduration and cost by setting up a superset of general classesof activities each with modes that vary in terms of inputsduration cost and expected benefits Meanwhile a numberof studies have looked into transferring process DSM to a

4 Mathematical Problems in Engineering

A

B

A

B

tAtB

dA

dB

Figure 3 Information transfer process with overlapping

Table 1 Nomenclature for all parameters in the paper

Parameter Notation Parameter Notation

119860Initial state matrix[119860]119894119894 shows duration of activity119894 for the non-diagonal elements if there isinformation delivered from 119895 to 119894 then the value

of [119860] 119894119895 is non empty else it is empty

119878119863119864119878119894 The activity duration variance matrixEarliest start time of activity i

119875119894119895 The predecessor time factor from j to i 119864119865119894 Earliest finish time of activity i119878119894119895 The successor time factor from j to i 119879119879 The length of critical path

119863 Activity time matrix the diagonal elements areactivity duration and the non-diagonal elements

are minus infinity119865 Activity deviation matrix [F]119894119895 shows the

deviation between activity j and activity i

119877 Activity relationship matrix if there isinformation delivered from j to i then the value of[R]119894119895 is 0 else it is 120576 1199090 0 vector the number of dimension equals

the number of activities

119909+119864 The early finish time for activity 119909minus119864 The early start time for activity

[ ]119899times1 N-dimension vector of certain value 119877119879119894The first and second rework time as wellas rework time the last critical activitycause other activities to generate

respectively for i=123

119890 The activity element in matrix R but the value is 0in the algorithm 119879119865 The total float of each activity

wFor the diagonal elements if the correspondingactivity is on the critical path then the value is 0

or the value is 120576 v

For the diagonal elements if thecorresponding activity is on the

non-critical path then the value is 0 orthe value is 120576

120576 Negative infinity in max-plus [119877119875]119894119895 Rework probability matrix[119877119868]119894119895 Rework impact matrix 119877119908119861119894 Rework buffer of activity iPT Estimated project completion time

more effective process management tool that is closer totraditional process management tools and expressions Atime factor was therefore introduced [30] The time factoris defined based on the overlapping relationship between theactivities As shown in Figure 3 119889119894 and 119889j are defined as theduration of activities 119894 and j respectively 119905119894 as the durationbetween the start of activity 119894 and the time when informationtransition from activity 119894 to activity 119895 is completed and 119905119895as the duration between the start of activity j and the timethe information transition is completed A predecessor timefactor is defined as 119875119895119894 = 119905119894119889119894 and a successor time factor as119878119895119894 = 119905119895119889119895 where 0 lt 119875 le 1 0 le 119878 lt 1 Two matrices Pand S are defined by these two factors The project duration

is then calculated according to (1) and a nomenclature isprovided in Table 1

119864119878lowast119894 = max 119864119878119895 + 119875119894119895 times 119875119895119895 minus 119878119894119895 times 119878119894119894119864119878119894 = max (119864119878lowast119894 0)119864119865119894 = 119864119878119894 + 119860 119894119894119879119879 = max [119864119865119894]

(1)

In short prior research has explored the possibil-ity of using the DSM to predict and control rework inproject scheduling and planning and has achieved noticeableprogressHoweverDSM lacks the capability of functioning as

Mathematical Problems in Engineering 5

a stand-alone project management technique for that DSMrsquosmatrix-based expression is difficult to be effectively convertedto network diagram-based expression sometimes leadingto ambiguity and confusion [5] Therefore it is challengingto apply DSM to the process management of constructionprojects and DSM needs to be extended and improved withfurther research efforts

3 The Critical Chain Design StructureMatrix Method

In this section we are going to demonstrate how to usethe CCDSM method to generate the project schedule Weimprove the max-plus algorithm to reduce computationalload in generating project schedules and introduce reworkbuffers to evaluate and address rework risks

31 Construction Project Scheduling with the Max-Plus Algo-rithm A discrete event system is state-discrete and event-driven Its state evolution depends on asynchronous discreteevents occurring at discrete points over time [31] Simplediscrete event systems are usually linear systems in whichstate and output variables for all possible input variables andthe initial state satisfy superposition principle of the systemAconstruction project can generally be seen as a linear systemthat consists of a number of discrete events

The max-plus algorithm provides an alternative way todescribe the discrete event system Based on the max-plusalgorithm the status of all discrete events denoted as vector119909(119896) in a system can be used to describe the status ofthe system and adjacent statuses are interchangeable with alimited number of linear changes as follows

119909 (1) = 1198601119909 (0)119909 (2) = 1198602119909 (1)

119909 (119896) = 119860119896119909 (119896 minus 1)

(2)

Based on the above equation if the linear changes at everystage 1198601 1198602 119860119896 are known then the final status of thesystem 119909(119896) can be derived from the initial status of thesystem 119909(0) Using this as a starting point the max-plusalgorithm introduces four types of operations to describe thediscrete event systems For two 119898 times 119898 matrices 119883 and 119884denoted as 119883119884 isin 119874119898times119898 whose elements are nonnegativenumbers or negative infinity these operations are defined asfollows

[X oplus Y]119894119895 = max ([X]119894119895 [Y]119894119895)[X and Y]119894119895 = min ([X]119894119895 [Y]119894119895)[X otimes Y]119894119895 = 119898⨁

119896=1

([X]119894119896 + [Y]119896119895)

[X ⊙ Y]119894119895 = 119898⋀119896=1

(minus [X]119894119896 + [Y]119896119895)

(3)

where 119883 otimes 119884 could be further simplified as 119883119884 and 119883119899 =119883119883 sdot sdot sdot 119883⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119899

To apply the max-plus algorithm to schedule manage-ment three new matrices need to be developed to describerelevant project information These matrices include (1)activity durationmatrix denoted asD which is developed byfilling the diagonal elements with each activityrsquos duration andthe nondiagonal elements with negative infinity (2) activitydeviation matrix denoted as F which is developed based onthe rule that [119865]119894119895 represents the deviation of the start time ofactivity 119895 and activity i and (3) activity relationship matrixdenoted as R which is developed based on the rule that ifthere is information delivered from activity 119895 to activity ithen the value of [119877]119894119895 is 0 otherwise the value is negativeinfinity

According to the max-plus algorithm [32] we have thefollowing

The early finish time for each activity is calculated as

119909+119864 = 119863 otimes (119865)lowast otimes 1199090 (4)

where (119865)lowast = 119890 oplus 119865 oplus oplus (119865)119897minus1 The (119894 119895)-th element ofmatrix (119865)lowast means the largest deviation of the start time ofactivity 119895 and activity i if activities 119894 and 119895 are on one ormore paths at the same time otherwise it is 120576 For simplicitythe elements of vector 1199090 are set as e which means thebeginning of activities in the project will not be affected byother construction projects

The length of critical path denoted as the maximum ofearly finish time for all activities is calculated as

119879119879 = max [119909+119864] (5)

The early start time is calculated as the difference betweenearly finish time for each activity and its duration accordingto

119909minus119864 = 119863 ⊙ 119909+119864 (6)

The late start time of activity 119894 is calculated as thedifference between length of critical path and the sum ofactivity 119894rsquos duration as well as subsequent critical activitiesrsquodurations according to

119909minus119871 = [119863 otimes (119865)lowast]119879 ⊙ [1199090 ⊙ 119879119879] (7)

The total float is calculated as

[119879119865]119894 = [119909minus119864]119894 ⊙ [119909minus119871]119894 (8)

The critical chain is then determined by the set ofactivities 120572 that satisfy 120572 | [119879119865]120572 = 0

It needs to be noted that when determining floats themax-plus algorithm is run in simple matrix form whichsignificantly improves the computational efficiency and ismore applicable to large-scale projects [33] The definitionsof parameters in the above equations are summarized in thenomenclature in Table 1

6 Mathematical Problems in Engineering

rework time

i

j

(a) First rework

j

k

i

rework time

(b) Second rework caused by information tran-sition from its precedent activity

rework time

i

j

k

(c) Second rework caused by information feed-back from its successive activity

Figure 4 First rework and second rework

32 Determining Project Rework Buffer It is widely believedthat information uncertainty is themain cause of rework [34]Such uncertain information may transmit from upstreamactivities to downstream activities or feedback in the oppo-site direction creating information flow Information flowinteractions lead to rework risks in construction projects andconsequently bring about more rework time There are twomain situations that rework exists(1)First rework as depicted in Figure 4(a) i is a precedentactivity of j and they perform sequentially according to rela-tionships After it is completed the performance informationof 119895 will be generated and transmitted to i which may resultin rework of 119894 The first rework time shown as the shadowarea in Figure 4(a) can be calculated by [27]

[1198771198791]119894119895 = 119877119875119894119895 sdot 119877119868119894119895 sdot 119871119862119894 sdot 119863119894119894 (9)

Then the total first rework time of 119894 caused by all itsdownstream activities can be obtained by

[1198771198791]119894 =119899sum119895=119894+1

119877119875119894119895 sdot 119877119868119894119895 sdot 119871119862119894 sdot 119863119894119894 (10)

where 119899 represents the number of activities in the schedule(2) Second rework as depicted in Figures 4(b) and 4(c)after completing the rework of 119895 caused by 119896 119895 transmits orfeeds back some revised information to 119894 which may causerework of 119894 Such two forms are defined as the second reworkin the paper The second rework times shown as the shadowareas in Figures 4(b) and 4(c) can be calculated by

[1198771198792]119894119895119896 = 119877119875119894119895 sdot 119877119875119895119896 sdot 119877119868119894119895 sdot 119871119862119894 sdot 119863119894119894 (11)

Then the total second rework time of 119894 caused by all itsdownstream activities can be obtained by

[1198771198792]119894 =119896minus1sum119895=1

119899sum119896=119894+1

119877119875119895119896 sdot 119877119875119894119895 sdot 119877119868119894119895 sdot 119871119862119894 sdot 119863119894119894 (12)

To proactively account for the uncertainty caused byinteractivity relationships and the associated rework risks arework buffer (119877119908119861) is introduced in the proposed CCDSMConceptually 119877119908119861 assesses and compresses the durationuncertainty caused by rework and is placed after the comple-tion of rework activities in project schedules119877119908119861 is designedto warn project participants from how much workload may

increase for each rework activity so as to prepare the projectparticipants for sufficient time and resources

The rework buffer of 119894 can be obtained by

119877119908119861119894 = [1198771198791]119894 + [1198771198792]119894 (13)

For the last critical activity 120579 in the schedule consideringall reworks it causes other activities to generate will take placeafter its completion andhave an impact onproject completiontime therefore its rework buffer is defined as the sum of thetotal rework time it causes other activities to generate and itstotal rework time caused by other activities

The total rework time it causes all other activities togenerate can be calculated by

[1198771198793]120579 ==119899sum119894=1

119877119875119894120579 sdot 119877119868119894120579 sdot 119871119862119894 sdot 119863119894119894 + 119899sum119894=1

119899sum119895=1

119877119875119894119895sdot 119877119875119895120579 sdot 119877119868119894119895 sdot 119871119862119894 sdot 119863119894119894

(14)

The rework buffer of 120579 then can be obtained by

119877119908119861120579 = [1198771198791]120579 + [1198771198792]120579 + [1198771198793]120579 (15)

33 Determining Project Buffers and Feeding Buffers inMax-Plus Representation

331 Determining Project Buffers Previous studies on theCCPM-max-plus representation adopted the CampPMmethodfor calculating project buffers shown as (16) and feedingbuffers [33]

119875119861 = max [(119863 otimes 119908 otimes 119877)lowast (119863 otimes 119908) 1199090]3 (16)

where (119894 119895)-th element of matrix (119863 otimes 119908 otimes 119877)lowast means thecumulative time of duration of activity 120572 which is on thecritical chain and range from 119894 + 1 to j (119894 119895)-th element ofmatrix119863 otimes119908 satisfies that if activity 119894 is on the critical chainthen (119894 119895)-th element is duration of activity I otherwise it is120576

However the CampPM method lacks sound mathematicalfoundation and overestimates project durations resulting ina waste of time and resources Alternatively the CampPM isreplaced by the RSEM which is based on the large numberlaw and central limit theorem in the CCDSM to calculate thePB

Mathematical Problems in Engineering 7

119875119861 = radic sum119894isin119862119875

119878119863119894 (17)

where 119878119863119894 is the variance of duration of activity 119894 on thecritical path

To calculate the PB in max-plus representation a matrixdenoted as 119879119901 is introduced For the element in row i andcolumn 119894 minus 1 in [119879119901]119894119895 if activity 119894 is critical activity it willbe filled with the variance of duration of activity i else it willbe zero For other elements in [119879119901]119894119895 it is filled with zeroThePB then can be represented in max-plus algebra as

119875119861 = radic1198621198790 otimes 119879lowast119901 otimes 1198620 (18)

where 1198621198790 = (119890 119890 119890)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119899+1

332 Determining Feeding Buffers In order to insert feedingbuffers in place to protect critical path from delays innoncritical paths two operators denoted as diag() and arefirst introduced For a vector 119909 a newmatrix can be obtainedby the operator diag() as [diag()]119894119895 = 119909119894 119894119891 119894 = 119895 120576 119890119897119904119890 and a new vector can be obtained by the operator as[119909]119894 = 120576 119894119891 [119909]119894 = 120576 119890 119894119891 [119909]119894 = 120576

In preparation of the location and size of the feedingbuffer another two vectors are introduced denoted as 119908 andV Matrix119908 is developed based on the rule that for a diagonalelement if the corresponding activity is on the critical paththen the element value is 0 otherwise the element value is 120576Matrix V is developed based on the rule that for a diagonalelement if the corresponding activity is on the noncriticalpath then the value is 0 otherwise the value is 120576 Moreoveran adjacency matrix 119877 is transformed from matrix 119865 by[119877]119894119895 = 119890 119894119891 [119865]119894119895 = 120576 120576 119890119897119904119890 Vector V1015840 is then introducedto locate the feeding buffers as follows

[V1015840]120574= [119877119879120573120572 otimes 119908]120574 (19)

where 119877120573120572 is the adjacency matrix which represents transi-tions from noncritical activities to critical ones and can beobtained by 119877120573120572 = diag(119908) otimes 119877 otimes diag(V) 120582 represents thenoncritical activity one ofwhose successors is critical activityTherefore a feeding buffer should be inserted behind theactivity 120574

The feeding buffer can then be determined by squarerooting of the sum of [119878119863]120578 where 120578 is the set of activitieson a certain noncritical chain The formula is expressed inmax-plus algebra as

1198651198610 = radicdiag [119877120573120573 otimes diag (V1015840) otimes diag (119878119863)]lowast otimes 1198621 otimes diag [diag (119878119863) otimes diag (V1015840) otimes 1198621] otimes 1198621 (20)

Additionally the calculation of feeding buffer should beadjusted using the following equation to meet the constraintthat the size of feeding buffer cannot exceed total float

[119865119861]120574 = min ([1198651198610]120574 [119879119865]120574) (21)

34 Implementation of the CCDSM with Max-Plus Lin-ear Expression The max-plus-based implementation of theCCDSM is summarized in the following seven steps It needsto be noted that as the max-plus algorithm cannot solvenonlinear conversions such as multiplication the input datafor the project rework problem including matricesA P S RPand RI needs to be preprocessed

Step 1 (preprocess119860) Extract all diagonal elements of matrix119860 and form a diagonal matrix D

Step 2 (preprocess 119875 and 119878) Combine matrices 119875 and 119878 togenerate matrix 119865 according to the following equation

[119865]119894119895 = [119875]119894119895 times [119875]119895119895 minus [119878]119894119895 times [119878]119894119894 (22)

Extract all nonnull elements ofmatrix119865 and use them to formmatrix 119877 Then update R by replacing all nonnull elementswith 119890 and all null elements with 120576Step 3 (preprocess RP and RI) Based on (9)-(15) mergematrices RP and RI to generate the vector 119877119908119861 whose 119894-thelement is the corresponding rework buffer of activity i

Step 4 (calculate the length of critical path) Based on (4)-(8)the length of critical pathTT and other parameters includingthe total float time TF and critical path are determined

Step 5 (calculate matrices 119908 and V) According to the criticalpath determined in Step 4matrices119908 and V can be generatedMeanwhile the activity duration variance matrix SD canbe generated based on data collected from the project tocalculate the buffers size

Step 6 (calculate buffer sizes) Based on (17)-(21) projectbuffer PB and feeding buffers FBs are calculated with RSEMand represented in max-plus algebra

Step 7 (generate the schedule) As the principle of CCPMthe feeding buffers should be placed on the noncritical chainsprior to the joints of the critical chain and noncritical chainsand the project buffer should be placed at the end of theschedule to protect the whole project process Then theestimated project time PT can be calculated based on

119875119879 = 119879119879 + 119875119861 + 119877119908119861120579 (23)

4 Case Study

In this section the proposed CCDSM method is imple-mented in a case project to demonstrate its feasibility andeffectiveness in addressing rework risks in project schedule

8 Mathematical Problems in Engineering

A B C D E F G H

A 8 X X X X

B X 8 X X X X

C 4 X X

D X 13 X

E X X X X 24 X

F X X X X 35 X

G X X X X 8

H X X X X

I J K L M N O

I X X X X X

J X X X X X X

K X X X X X

L X X X X X X X

M X

N X

O X X X X X X X X

X X X 4 X

X X X X X X 2

X X 4 X X

13 X

X X 8 X X

13

X X 5 X X X

X

X

X

X

X

13

X X X

X

X X

X X

X

X

X

P Q R S

X X

X X

X X

X X

X X

X X

X

X

P X X X X X X X X

Q X X X X X X X X

R X X X X X X X

S

X X X X X X

X

X X X X X X X

X X X X X X X

X X 4

X X X 4

X 4 X

3 X

(a) Matrix A

A B C D E F G H

A 10 09 07 08

B 09 06 07 06

C 07 09 10

D 07 06

E 08 06 06 08 06

F 09 09 08 08 06

G 06 07 06 06

H 06 06 06 06

I J K L M N O

I 06 09 09 06 10

J 05 09 09 07 07 08

K 06 07 06 06 08

L 06 10 08 07 07 06

M 05

N 06 05

O 07 07 08 06 06 09 09 05

08 08 08 08

06 10 06 08 06 06

09 10 09 09

05

08 06 06 10

07 06 09 06 06

09

08

08

10

07

07 08 06

09 07

09 09

09

08

08

07

P Q R S

10

09

09

10

09

05

06

08

P 09 06 07 08 09 06 07 05

Q 08 09 10 09 10 05 10

R 10 07 07 07 06 06 08

S

10 06 08 06 08 09

09

07 08 09 06 10 05 09

06 08 06 10 09 06 07

07 07

09 08 08

08 07

09

(b) Matrix P

A B C D E F G H

A 05 03 02 02

B 01 04 02 05 05

C 01 02

D 01 03

E 04 02 04

F 04 04 04 04 04

G 03 05 05 04

H 04 03 01 05

I J K L M N O

I 01 01 05 02 04

J 03 04 02

K 05 02 03 01 04

L 02 04 03 05 04 03 02

M 01

N 04

O 04 04 02 03 01 02 04

02 02 04 02

04 05 04 03 01 05

02 04 01 03

03

02 04 01 01

03 01 03 01 01

04

03

03

04

01

04 02 03

04

03 04

03 01

02

05

03

P Q R S

01 02

05

04

01 02

04 02

05

01

P 02 05 05 02 03 02 04

Q 03 04 01 02 02

R 02 03 01 04 02 04 01

S

04 03 02 01 01 02

02

01 04 03 01 01 03 04

01 01 04 02 04 02 02

02 01

02 04 03

01 04

01

(c) Matrix S

A B C D E F G HA 01 02 03

B 04 02 01 01

C 02 01

D 03 04

E 06 03 05 05 04

F 03 02 05 05 03

G 02 05 05 02

H 04 02 06 03

I J K L M N O

I 02 03 03 04 02

J 03 03 04 05 03 05

K 04 05 02 06 03

L 03 05 05 03 04 04 03

M 02

N 03

O 05 03 02 05 04 05 06 0306 03 05 01

03 02 05 06 03 05

03 04 01

0203 05 02 0403 04 04 03 03

03

0103 02 02

02

01 04

02

03

0503

P Q R S

02 0102 02

01 03

0301 0202 03

01

02

P 04 04 06 02 04 04 05 06

Q 03 06 05 03 06 05 04 05

R 03 05 03 03 03 05 05

S

02 06 04 03 04 0404

03 06 04 05 03 02 0404 05 06 05 04 03 03

03 0605 02 04

03 0201

(d) Matrix RP

A B C D E F G H

A 008 001 001 009

B 020 014 016 020 007

C 007 008

D 008 023

E 010 001 009 020 013

F 003 025 013 008 023

G 025 002 003 015

H 018 005 022 001

I J K L M N O

I 013 023 020 011 021

J 002 017 007 025 003 024

K 001 008 012 015 002

L 006 013 016 014 003 023 009

M 001

N 006

O 005 016 011 004 014 019 010 015

021 014 005 003

007 004 019 002 005 016

018 009 013 022

012

004 020 015 022

003 022 013 004 023

013

006

023

017

019

005 023 021

018

023 006

023 017

004

004

011

P Q R S

010 009

010 009

022 009

016 010

014 013

004 018

001

018

P 007 014 014 021 023 012 011 014

Q 03 008 025 001 022 014 002 008

R 012 014 006 021 015 021 018

S

022 001 006 020 008 003

014

022 001 006 010 008 019 001

016 011 021 001 021 001 015

009 025

021 018 019

009 019

014

(e) Matrix RI

Figure 5 Project information in the case project

management The settings of the case project are first pre-sented followed by descriptions of implementation of theCCDSM in the case project The schedule generated with theCCDSMmethod is assessed in detail and compared with theschedules generated with traditional CCPM- and DES-basedmethods

41 Case Project Settings The case used in this study wasderived from a modular real estate development projectfirstly introduced in [35] and further described in [36]The matrix 119860 of the case is shown in Figure 5(a) Thiscase consisted of 19 major activities and 183 interactivityrelationships including 65 rework relationships representedby elements above the diagonal of matrix 119860 It is assumedthat rework can propagate up to twice to avoid infinite loopin the computation The parameters for each activity and theinteractivity relationships were derived based on literatureand empirical evidence and are shown in Figures 5(b) 5(c)5(d) and 5(e) and summarized in Table 2 Large-scale reworkrelationships and complex connections between activitieswere observed in the case project which was representativeof typical construction projects in reality

42 Implementation of the CCDSM Method The proposedCCDSM method was implemented in the case project fol-lowing the steps explained in the last section and the results

were shown below Noticeably for better understandingof scheduling process the implementation process of theCCDSMmethod is reorganized as below

Step 1 Matrices 119863 and 119877 were derived from matrix Aas shown in Figure 6 The duration of each activity waspresented in the diagonal of the matrix D

Step 2 Matrix F (see Figure 7) was derived from matrices119875 and 119878 according to (22) This step calculated the activitydeviation matrix based on the predecessor time matrix andthe successor time matrix

Step 3 Optimize activity sequence and update matrices D RRP RI and 119865 according to new activity sequence To reduceproject rework and obtain near-optimal project completiontime the genetic algorithm (GA) was applied to optimizeactivity sequence The GA is a metaheuristic method thatsearches for optimal solutions using processes similar tothose in natural selections and genetics [37] In the paperminimization of total length of rework path first proposed byGebala and Eppinger [38] was used as the objective functionto calculate the optimal sequence of activities

119891 = 119899sum119894=1

119899sum119895=119894+1

(119895 minus 119894) sdot 119908 (119894 119895) (24)

Mathematical Problems in Engineering 9

Table 2 Activity parameters in the case project

ID Activity Learning curve Duration (day)Min Likely Max

A Perform prelim mkt analysis 05 5 8 15B Evaluate marketability options 06 5 8 15C Engage feasibility consultants 03 3 4 7D Evaluate planning amp zoning process 05 10 13 20E Perform massing study 06 20 24 35F Develop conceptual design 08 30 35 50G Identify external stakeholders 04 5 8 15H Identify permits amp approvals 07 10 13 20I Complete phase 1 ESA 03 10 13 20J Evaluate consultants amp contractors 06 10 13 20K Obtain rough construction costs 05 5 8 15L Determine highest amp best use 05 3 5 9M Identify debt options 04 2 4 8N Identify equity options 05 3 4 5O Update financial underwriting 05 1 2 3P Reevaluate organization strategy 06 2 3 5Q Estimate schedule 06 3 4 7R Gain control of site andor client 06 3 4 7S Review and approve 07 3 4 7

F C A I J R K PF 30C 3A 5I 10J 10R 3K 5P

D M O B H E N

DMOBHEN

203

10

21

10

5

2

L G Q S

LGQS

33

53

(a) Matrix D

F C A I J R K PFCA e eI eJ e eR e e eK e e e eP e e e e e e

D M O B H E N

D e eM eO e e e e eB e e eH e e eE e eN e e

e e e ee e

e

e eee

L G Q S

L e e e e eG e e eQ e e e e e e e eS e e

e e e e e e ee

e e ee e e e e e e

e ee

e

e

e

e

e

(b) Matrix R (element 120576 is omitted)

Figure 6 Matrix D amp R

where 119908(119894 119895) represents rework probability of activity 119894caused by activity 119895

The parameter settings were selected as follows pop-ulation size set as 50 number of generations set as 150crossover probability set as 095 and mutation probabilityset as 008 The GA process reported an optimal schedule as[119875 119876119867 119864 119877 119861 119862119870119873 119871119872 119865119863 119878 119860 119866 119869 119868 119874] The GAconvergence process is shown in Figure 8

Step 4 119877119908119861 was generated from matrices RP and RI basedon (9)-(15) The goal of this step was to factor in andcalculate the rework time of the case project 119877119908119861 was[3 4 6 10 5 1 1 3 1 2 1 15 1 1 1 1 1 1 9]Step 5 To generate reliable project schedules the most likelyduration of each activity which has been widely used andaccepted by project teams in prior research [39] was selected

10 Mathematical Problems in Engineering

A B C D E F G HA 1 10 15 28

B 7 1 13 23 2

C 13 8

D 2 1

E -3 1 3 10 15

F -6 -11 -4 5 -

12G 3 2 6

H -3 4 14

I J K L M N O

I 2 11 16 3 8

J 5 -1 8 18 28 4

K 2 5 2 15 28

L 4 6 1 10 16 27 7

M 13

N 12

O 5 6 2 8 16 35 7 76 4 2 1

8 12 4 4 3 2

7 4 4 1

-19 5 3 29 5 3 2 1

1174-19

-8 -1 -99

-1 -8

8 9

-1

57

P Q R S

3 33 4

1 4

-1 33 2

2

-1

2

P 8 5 2 11 23 22 6 6

Q 7 7 2 14 23 37 4 13

R 7 5 11 17 23 4 10

S

11 4 4 2 3 11

9 10 6 3 4 19 10 4 5 3 2 1

2 32 2 3

2 22

Figure 7 Matrix F (element 120576 is omitted)

20

30

40

50

60

70

80

Estim

ated

Fee

dbac

k Le

ngth

50 100 1500Generation

Figure 8 Convergence curve in the GA-based optimization

to calculate the estimated project duration The critical chainwas reported as [119867 119864 119861 119870119873 119871 119865 119860 119866 119869 119874]Step 6 The activity duration variance matrix SD was gen-erated with 1000 Monte Carlo simulations of the activitiesrsquoduration that follows a beta distribution

Step 7 Buffers were calculated based on (17)-(21) Theproject buffer was 36 days and the feeding buffer was[3 4 0 0 38 0 4 0 0 0 6 0 10 38 0 0 0 0 2 0]Step 8 The estimated project duration was calculated as 102days based on (23) The project buffer was placed at the endof the schedule and the feeding buffer was placed on thenoncritical chains prior to the joints of the critical chain andnoncritical chains following the principle of CCPM

It was worth mentioning that overlapping was observedamong various activities including critical activities in the

above schedule which illustrated real-world high-frequencyinteractions among activities in the project

To assess the performance of the CCDSM method andthe quality of the project schedule it generated the executionof case project was simulated 1000 times using Monte Carloapproach where the sampling of actual activity durationsfollowed beta distribution The simulated actual projectdurations were depicted in Figure 9 It can be seen in thefigure that under rework assumption the process scheduleof 102 days had a completion probability of 981 indicatingthat the proposed CCDSM method was able to providesatisfactory assurance of on-time project completion undervarying scheduling scenarios

43 Comparison of the CCDSM Method with TraditionalCCPM and DES Methods This subsection further comparesthe performance of the CCDSM method with traditional

Mathematical Problems in Engineering 11

75 80 85 90 95 100 105 1100

20

40

60

80

100

120

140

Duration

CCDSM method=102

0

01

02

03

04

05

06

07

08

09

1

981

CCDSM method=102

Cum

ulat

ive P

roba

bilit

y

Freq

uenc

y

80 85 90 95 100 105 11075Duration

Figure 9 Simulated actual project durations and on-time project completion probability of the CCDSM-based schedule

75 80 85 90 95 100 105 1100

01

02

03

04

05

06

07

08

09

1

Duration

981

CCDSM method=102

493

CCPM method=93

Cum

ulat

ive P

roba

bilit

y

Figure 10 Comparison of on-time project completion probabilities between the CCDSM- and CCPM-based schedules

CCPM and DES methods to assess its effectiveness inaddressing rework risks

A CCPM-based schedule was generated with tradi-tional buffers including project buffer and feeding bufferand without rework buffer The results indicated that theestimated project durations of the CCPM-based schedulewere 93 days including project buffers of 36 days Whenthe estimated project duration of CCPM-based schedulewas superimposed in the Monte Carlo-simulated projectdurations as depicted in Figure 10 it can be seen thatthe on-time completion probability was 493 The resultssuggested that the CCDSM-based schedule which took intoconsideration rework relationships and had a duration of 102days significantly outperformed the CCPM-based schedulein ensuring on-time completion of the case project that wasfaced with typical rework risks

Furthermore a DES-based schedule was generated byimplementing DES algorithm in which activity durationsfollowing a beta distribution were generated at random tocompute the project duration When the DES algorithm was

used to generate project schedules under rework scenarios aprobabilistic judgment is performed firstly on whether or noteach activity causes rework and then an iterative simulationprocess began until all activities were finished Specificallythe DES algorithm proposed by Browning and Eppinger[27] was adopted in this study with the following minormodifications to make it more applicable to the case projectand comparable to the CCDSM method (1) the optimizedactivity sequence in Section 42 was employed in the DESprocess (2) activity 119894 did not started until STS logic tie withits each predecessor activity 119895 that is [119865]119894119895 was satisfied (3)rework probability of activity 119894 would decrease by 50 eachtime it reworked [28]

To compare the CCDSM-based schedule with the DES-based schedule the execution of the case project usingthe DES-based schedule was simulated 1000 times usingMonte Carlo approach and the simulated actual projectdurations are depicted in Figure 11 The results showed thatthe estimated project duration of the DES-based schedulewas 115 days which was 13 days longer than that of the

12 Mathematical Problems in Engineering

70 80 90 100 110 120 1300

010203040506070809

1

Duration

981

DES method=115Cu

mul

ativ

e Pro

babi

lity

Figure 11 On-time project completion probability of the DES-based schedule

CCDSM-based schedule with the same on-time completionprobability of 981 In other words the CCDSM methodoutperformed the DES method by ensuring the same on-time completion probability with a shorter project scheduleMoreover the CCDSM method was also advantageous overthe DES-based method in that it enabled effective controlof project schedules during project implementation whichcould be done by monitoring consumption of time buffersleading to reduction in project overrun risks

5 Conclusions

Construction projects are constantly challenged by reworkrisks which have largely remained unaddressed by existingproject schedulingThis paper proposed the CCDSMmethodfor developing project schedules that are adaptive to reworkinstances during project execution This method modelslarge-scale rework relationships among activities with theintroduction of a new rework buffer in traditional CCPMmethod The method allows analysis and visualization ofthe schedules and utilizes max-plus algorithm to transformcomplex logic relationships into simple matrix operationsreducing the computational load of generating process sched-ules A case study was conducted to demonstrate the imple-mentation of the CCDSMmethod and assess its effectivenessin addressing rework risksThe results showed that CCDSM-based schedule outperformed the CCPM-based schedule inensuring on-time completion of the case project that wasfaced with typical rework risks In addition in comparison tothe DESmethod the CCDSM performed better by providinga smaller project duration with the same probability of on-time project completion

The CCDSM method contributes to construction man-agement in the following ways First the CCDSM methodabsorbs risks of rework and various other uncertainties togenerate the reliable project schedule enabling managersto predict project durations more accurately before projectsstart Based on accurate prediction of project durationsthe negotiation of contracts and supply of resources canbe conducted in a more reasonable manner avoiding costsoverruns schedule delays and even project failures Second

based on generated project schedule with time bufferseffectivemonitoring and control of construction progress canbe realized Overall speaking the proposed CCDSMmethodprovides a promising solution to mitigate rework risks inconstruction projects and protect projects from undesirabledelays caused by possible rework instances

Meanwhile the CCDSM bears several limitations thatwould benefit from future improvement First only the STSlogic tie between activities is tested in this study Techniquesfor integrating multiple logic ties including STS STF FTSand FTF deserve further attention Second resource conflictsand levering are considered beyond the scope of this studybut it is a critical challenge that should be addressed inthe context of CCDSM-based scheduling in future researchThird the parameters required to implement the CCDSMmethod are hard to obtain for a specific project To employCCPM or max-plus algorithm for project scheduling it isnecessary to obtain the estimated activity durations undervarious completion rates and the logic tie between activitiesSimilarly the rework relative parameters including RP RIand LC are required for implementation of DSM-basedmethod As an integration method of CCPM max-plus algo-rithm and DSM the implementation of CCDSM requiresall the above parameters to be obtained Lastly methods fordynamic updating of buffer sizes during project execution areneeded in order to better prepare project teams for changingproject environment and dynamic project progress

Data Availability

All data used to support the findings of this study are includedwithin the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (NSFC) [Grant no 71671128]

Mathematical Problems in Engineering 13

References

[1] Y Li and T R Taylor ldquoModeling the impact of design reworkon transportation infrastructure construction project perfor-mancerdquo Journal of Construction Engineering and Managementvol 140 no 9 p 04014044 2014

[2] B Hwang X Zhao and K J Goh ldquoInvestigating the client-related rework in building projects The case of SingaporerdquoInternational Journal of Project Management vol 32 no 4 pp698ndash708 2014

[3] E K Simpeh R Ndihokubwayo P E Love and W DThwala ldquoA rework probability model a quantitative assessmentof rework occurrence in construction projectsrdquo InternationalJournal of Construction Management vol 15 no 2 pp 109ndash1162015

[4] P W Morris J K Pinto and S Jonas 13e Oxford Handbookof Project Management Oxford University Press Oxford UK2012

[5] S D Eppinger and T R Browning Design Structure MatrixMethods and Applications MIT Press Books 2012

[6] T R Browning ldquoDesign structure matrix extensions and inno-vations a survey and new opportunitiesrdquo IEEE Transactions onEngineering Management vol 63 no 1 pp 27ndash52 2016

[7] A Kulkarni D K Yadav and H Nikraz ldquoAircraft maintenancechecks using critical chain project pathrdquo Aircra EngineeringAerospace Technology vol 89 no 6 pp 879ndash892 2017

[8] J Trojanowska and E Dostatni ldquoApplication of the theory ofconstraints for project managementrdquoManagement and Produc-tion Engineering Review vol 8 no 3 pp 87ndash95 2017

[9] M Ghaari and M Emsley ldquoBufer sizing in CCPM portfolioswith dierent resource capacitiesrdquo International Journal of Infor-mation Technology Project Management vol 8 no 3 pp 40ndash512017

[10] E M Goldratt Critical Chain A Business Novel MA NorthRiver Press Great Barrington United States 1997

[11] S Van de Vonder E Demeulemeester and W HerroelenldquoProactive heuristic procedures for robust project schedulingAn experimental analysisrdquo European Journal of OperationalResearch vol 189 no 3 pp 723ndash733 2008

[12] O Lambrechts E Demeulemeester and W Herroelen ldquoTimeslack-based techniques for robust project scheduling subject toresource uncertaintyrdquo Annals of Operations Research vol 186no 1 pp 443ndash464 2011

[13] D Trietsch and K R Baker ldquoPERT 21 Fitting PERTCPMfor use in the 21st centuryrdquo International Journal of ProjectManagement vol 30 no 4 pp 490ndash502 2012

[14] GMaAWangN Li LGu andQAi ldquoImproved critical chainproject management framework for scheduling constructionprojectsrdquo Journal of Construction Engineering andManagementvol 140 no 12 p 04014055 2014

[15] R C Newbold Project Management in 13e Fast Lane Applying13e 13eory of Constraints CRC Press Boca Raton FloridaLondon 1998

[16] O I Tukel W O Rom and S D Eksioglu ldquoAn investigation ofbuffer sizing techniques in critical chain schedulingrdquo EuropeanJournal of Operational Research vol 172 no 2 pp 401ndash4162006

[17] O Icmeli and S S Erenguc ldquoA branch and bound procedurefor the resource constrained project scheduling problem withdiscounted cash flowsrdquoManagement Science vol 42 no 10 pp1395ndash1408 1996

[18] E Roghanian M Alipour and M Rezaei ldquoAn improved fuzzycritical chain approach in order to face uncertainty in projectschedulingrdquo International Journal of ConstructionManagementvol 18 no 1 pp 1ndash13 2017

[19] J Zhang X Song and E Dıaz ldquoProject buffer sizing of a criticalchain based on comprehensive resource tightnessrdquo EuropeanJournal of Operational Research vol 248 no 1 pp 174ndash182 2016

[20] F A Rivera and A Duran ldquoCritical clouds and critical sets inresource-constrained projectsrdquo International Journal of ProjectManagement vol 22 no 6 pp 489ndash497 2004

[21] G Ma L Gu and N Li ldquoScenario-based proactive robustoptimization for critical-chain project schedulingrdquo Journal ofConstruction Engineering and Management vol 141 no 10 p04015030 2015

[22] H Goto ldquoForward-compatible framework with critical-chainproject management using a max-plus linear representationrdquoOPSEARCH vol 54 no 1 pp 201ndash216 2017

[23] T Salama A Salah and O Moselhi ldquoIntegration of linearscheduling method and the critical chain project managementrdquoCanadian Journal of Civil Engineering vol 45 no 1 pp 30ndash402018

[24] C W Hu X D Chen and L H Wu ldquoThe setting method ofproject buffer in critical chain management of mould manufac-turing project based reworkingrdquo Advanced Materials Researchvol 317-319 pp 418ndash422 2011

[25] D V Steward Systems Analysis and Management StructureStrategy and Design IrwinMcGraw-Hill Boston 1981

[26] T R Browning Modeling and analyzing cost schedule andperformance in complex system product development [PhDthesis]Massachusetts Institute of Technology CambridgeMas-sachusetts USA 1998

[27] T R Browning and S D Eppinger ldquoModeling impacts ofprocess architecture on cost and schedule risk in productdevelopmentrdquo IEEE Transactions on Engineering Managementvol 49 no 4 pp 428ndash442 2002

[28] S H Cho and S D Eppinger ldquoA simulation-based processmodel for managing complex design projectsrdquo IEEE Transac-tions on Engineering Management vol 52 no 3 pp 316ndash3282005

[29] V Levardy and T R Browning ldquoAn adaptive process modelto support product development project managementrdquo IEEETransactions on Engineering Management vol 56 no 4 pp600ndash620 2009

[30] J U Maheswari and K Varghese ldquoProject scheduling usingdependency structure matrixrdquo International Journal of ProjectManagement vol 23 no 3 pp 223ndash230 2005

[31] C G Cassandras and S Lafortune Introduction to DiscreteEvent Systems Springer New York NY USA 2nd edition 2008

[32] S Yoshida H Takahashi and H Goto ldquoModified max-pluslinear representation for inserting time buffersrdquo in Proceedingsof the IEEE International Conference on Industrial Engineeringamp Engineering Management IEEE 2010

[33] S Yoshida H Takahashi and H Goto ldquoResolution of timeand worker conflicts for a single project in a max-plus linearrepresentationrdquo Industrial Engineering amp Management Systemsvol 10 no 4 pp 279ndash287 2011

[34] S Muralidharan ldquoFactors affecting rework in construc-tion projectrdquo International Journal of Engineering Sciences ampResearch Technology vol 1 no 5 pp 578ndash584 2016

[35] B B E Bulloch and J Sullivan Application of the DesignStructure Matrix (DSM) to the real estate development process

14 Mathematical Problems in Engineering

[Master thesis] Massachusetts Institute of Technology Cam-bridge Massachusetts USA 2009

[36] S D Eppinger S Bonelli andAM Gonzalez ldquoManaging itera-tions in the modular real estate development processrdquo ReducingRisk in Innovation Proceedings of the 15th International DSMConference Melbourne Australia 29-30 August 2013 pp 37ndash442013

[37] D T Pham and D Karaboga Intelligent Optimisation Tech-niques Genetic Algorithms Tabu Search Simulated AnnealingandNeural Networks Springer Science amp BusinessMedia 2000

[38] D A Gebala and S D Eppinger ldquoMethods for analyzing designproceduresrdquo in Proceedings of the AMSE 13ird InternationalConference On Design13eory andMethodology vol 31 pp 227ndash233 Miami Florida 1991

[39] O Hazır M Haouari and E Erel ldquoRobust scheduling androbustness measures for the discrete timecost trade-off prob-lemrdquo European Journal of Operational Research vol 207 no 2pp 633ndash643 2010

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4 Mathematical Problems in Engineering

A

B

A

B

tAtB

dA

dB

Figure 3 Information transfer process with overlapping

Table 1 Nomenclature for all parameters in the paper

Parameter Notation Parameter Notation

119860Initial state matrix[119860]119894119894 shows duration of activity119894 for the non-diagonal elements if there isinformation delivered from 119895 to 119894 then the value

of [119860] 119894119895 is non empty else it is empty

119878119863119864119878119894 The activity duration variance matrixEarliest start time of activity i

119875119894119895 The predecessor time factor from j to i 119864119865119894 Earliest finish time of activity i119878119894119895 The successor time factor from j to i 119879119879 The length of critical path

119863 Activity time matrix the diagonal elements areactivity duration and the non-diagonal elements

are minus infinity119865 Activity deviation matrix [F]119894119895 shows the

deviation between activity j and activity i

119877 Activity relationship matrix if there isinformation delivered from j to i then the value of[R]119894119895 is 0 else it is 120576 1199090 0 vector the number of dimension equals

the number of activities

119909+119864 The early finish time for activity 119909minus119864 The early start time for activity

[ ]119899times1 N-dimension vector of certain value 119877119879119894The first and second rework time as wellas rework time the last critical activitycause other activities to generate

respectively for i=123

119890 The activity element in matrix R but the value is 0in the algorithm 119879119865 The total float of each activity

wFor the diagonal elements if the correspondingactivity is on the critical path then the value is 0

or the value is 120576 v

For the diagonal elements if thecorresponding activity is on the

non-critical path then the value is 0 orthe value is 120576

120576 Negative infinity in max-plus [119877119875]119894119895 Rework probability matrix[119877119868]119894119895 Rework impact matrix 119877119908119861119894 Rework buffer of activity iPT Estimated project completion time

more effective process management tool that is closer totraditional process management tools and expressions Atime factor was therefore introduced [30] The time factoris defined based on the overlapping relationship between theactivities As shown in Figure 3 119889119894 and 119889j are defined as theduration of activities 119894 and j respectively 119905119894 as the durationbetween the start of activity 119894 and the time when informationtransition from activity 119894 to activity 119895 is completed and 119905119895as the duration between the start of activity j and the timethe information transition is completed A predecessor timefactor is defined as 119875119895119894 = 119905119894119889119894 and a successor time factor as119878119895119894 = 119905119895119889119895 where 0 lt 119875 le 1 0 le 119878 lt 1 Two matrices Pand S are defined by these two factors The project duration

is then calculated according to (1) and a nomenclature isprovided in Table 1

119864119878lowast119894 = max 119864119878119895 + 119875119894119895 times 119875119895119895 minus 119878119894119895 times 119878119894119894119864119878119894 = max (119864119878lowast119894 0)119864119865119894 = 119864119878119894 + 119860 119894119894119879119879 = max [119864119865119894]

(1)

In short prior research has explored the possibil-ity of using the DSM to predict and control rework inproject scheduling and planning and has achieved noticeableprogressHoweverDSM lacks the capability of functioning as

Mathematical Problems in Engineering 5

a stand-alone project management technique for that DSMrsquosmatrix-based expression is difficult to be effectively convertedto network diagram-based expression sometimes leadingto ambiguity and confusion [5] Therefore it is challengingto apply DSM to the process management of constructionprojects and DSM needs to be extended and improved withfurther research efforts

3 The Critical Chain Design StructureMatrix Method

In this section we are going to demonstrate how to usethe CCDSM method to generate the project schedule Weimprove the max-plus algorithm to reduce computationalload in generating project schedules and introduce reworkbuffers to evaluate and address rework risks

31 Construction Project Scheduling with the Max-Plus Algo-rithm A discrete event system is state-discrete and event-driven Its state evolution depends on asynchronous discreteevents occurring at discrete points over time [31] Simplediscrete event systems are usually linear systems in whichstate and output variables for all possible input variables andthe initial state satisfy superposition principle of the systemAconstruction project can generally be seen as a linear systemthat consists of a number of discrete events

The max-plus algorithm provides an alternative way todescribe the discrete event system Based on the max-plusalgorithm the status of all discrete events denoted as vector119909(119896) in a system can be used to describe the status ofthe system and adjacent statuses are interchangeable with alimited number of linear changes as follows

119909 (1) = 1198601119909 (0)119909 (2) = 1198602119909 (1)

119909 (119896) = 119860119896119909 (119896 minus 1)

(2)

Based on the above equation if the linear changes at everystage 1198601 1198602 119860119896 are known then the final status of thesystem 119909(119896) can be derived from the initial status of thesystem 119909(0) Using this as a starting point the max-plusalgorithm introduces four types of operations to describe thediscrete event systems For two 119898 times 119898 matrices 119883 and 119884denoted as 119883119884 isin 119874119898times119898 whose elements are nonnegativenumbers or negative infinity these operations are defined asfollows

[X oplus Y]119894119895 = max ([X]119894119895 [Y]119894119895)[X and Y]119894119895 = min ([X]119894119895 [Y]119894119895)[X otimes Y]119894119895 = 119898⨁

119896=1

([X]119894119896 + [Y]119896119895)

[X ⊙ Y]119894119895 = 119898⋀119896=1

(minus [X]119894119896 + [Y]119896119895)

(3)

where 119883 otimes 119884 could be further simplified as 119883119884 and 119883119899 =119883119883 sdot sdot sdot 119883⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119899

To apply the max-plus algorithm to schedule manage-ment three new matrices need to be developed to describerelevant project information These matrices include (1)activity durationmatrix denoted asD which is developed byfilling the diagonal elements with each activityrsquos duration andthe nondiagonal elements with negative infinity (2) activitydeviation matrix denoted as F which is developed based onthe rule that [119865]119894119895 represents the deviation of the start time ofactivity 119895 and activity i and (3) activity relationship matrixdenoted as R which is developed based on the rule that ifthere is information delivered from activity 119895 to activity ithen the value of [119877]119894119895 is 0 otherwise the value is negativeinfinity

According to the max-plus algorithm [32] we have thefollowing

The early finish time for each activity is calculated as

119909+119864 = 119863 otimes (119865)lowast otimes 1199090 (4)

where (119865)lowast = 119890 oplus 119865 oplus oplus (119865)119897minus1 The (119894 119895)-th element ofmatrix (119865)lowast means the largest deviation of the start time ofactivity 119895 and activity i if activities 119894 and 119895 are on one ormore paths at the same time otherwise it is 120576 For simplicitythe elements of vector 1199090 are set as e which means thebeginning of activities in the project will not be affected byother construction projects

The length of critical path denoted as the maximum ofearly finish time for all activities is calculated as

119879119879 = max [119909+119864] (5)

The early start time is calculated as the difference betweenearly finish time for each activity and its duration accordingto

119909minus119864 = 119863 ⊙ 119909+119864 (6)

The late start time of activity 119894 is calculated as thedifference between length of critical path and the sum ofactivity 119894rsquos duration as well as subsequent critical activitiesrsquodurations according to

119909minus119871 = [119863 otimes (119865)lowast]119879 ⊙ [1199090 ⊙ 119879119879] (7)

The total float is calculated as

[119879119865]119894 = [119909minus119864]119894 ⊙ [119909minus119871]119894 (8)

The critical chain is then determined by the set ofactivities 120572 that satisfy 120572 | [119879119865]120572 = 0

It needs to be noted that when determining floats themax-plus algorithm is run in simple matrix form whichsignificantly improves the computational efficiency and ismore applicable to large-scale projects [33] The definitionsof parameters in the above equations are summarized in thenomenclature in Table 1

6 Mathematical Problems in Engineering

rework time

i

j

(a) First rework

j

k

i

rework time

(b) Second rework caused by information tran-sition from its precedent activity

rework time

i

j

k

(c) Second rework caused by information feed-back from its successive activity

Figure 4 First rework and second rework

32 Determining Project Rework Buffer It is widely believedthat information uncertainty is themain cause of rework [34]Such uncertain information may transmit from upstreamactivities to downstream activities or feedback in the oppo-site direction creating information flow Information flowinteractions lead to rework risks in construction projects andconsequently bring about more rework time There are twomain situations that rework exists(1)First rework as depicted in Figure 4(a) i is a precedentactivity of j and they perform sequentially according to rela-tionships After it is completed the performance informationof 119895 will be generated and transmitted to i which may resultin rework of 119894 The first rework time shown as the shadowarea in Figure 4(a) can be calculated by [27]

[1198771198791]119894119895 = 119877119875119894119895 sdot 119877119868119894119895 sdot 119871119862119894 sdot 119863119894119894 (9)

Then the total first rework time of 119894 caused by all itsdownstream activities can be obtained by

[1198771198791]119894 =119899sum119895=119894+1

119877119875119894119895 sdot 119877119868119894119895 sdot 119871119862119894 sdot 119863119894119894 (10)

where 119899 represents the number of activities in the schedule(2) Second rework as depicted in Figures 4(b) and 4(c)after completing the rework of 119895 caused by 119896 119895 transmits orfeeds back some revised information to 119894 which may causerework of 119894 Such two forms are defined as the second reworkin the paper The second rework times shown as the shadowareas in Figures 4(b) and 4(c) can be calculated by

[1198771198792]119894119895119896 = 119877119875119894119895 sdot 119877119875119895119896 sdot 119877119868119894119895 sdot 119871119862119894 sdot 119863119894119894 (11)

Then the total second rework time of 119894 caused by all itsdownstream activities can be obtained by

[1198771198792]119894 =119896minus1sum119895=1

119899sum119896=119894+1

119877119875119895119896 sdot 119877119875119894119895 sdot 119877119868119894119895 sdot 119871119862119894 sdot 119863119894119894 (12)

To proactively account for the uncertainty caused byinteractivity relationships and the associated rework risks arework buffer (119877119908119861) is introduced in the proposed CCDSMConceptually 119877119908119861 assesses and compresses the durationuncertainty caused by rework and is placed after the comple-tion of rework activities in project schedules119877119908119861 is designedto warn project participants from how much workload may

increase for each rework activity so as to prepare the projectparticipants for sufficient time and resources

The rework buffer of 119894 can be obtained by

119877119908119861119894 = [1198771198791]119894 + [1198771198792]119894 (13)

For the last critical activity 120579 in the schedule consideringall reworks it causes other activities to generate will take placeafter its completion andhave an impact onproject completiontime therefore its rework buffer is defined as the sum of thetotal rework time it causes other activities to generate and itstotal rework time caused by other activities

The total rework time it causes all other activities togenerate can be calculated by

[1198771198793]120579 ==119899sum119894=1

119877119875119894120579 sdot 119877119868119894120579 sdot 119871119862119894 sdot 119863119894119894 + 119899sum119894=1

119899sum119895=1

119877119875119894119895sdot 119877119875119895120579 sdot 119877119868119894119895 sdot 119871119862119894 sdot 119863119894119894

(14)

The rework buffer of 120579 then can be obtained by

119877119908119861120579 = [1198771198791]120579 + [1198771198792]120579 + [1198771198793]120579 (15)

33 Determining Project Buffers and Feeding Buffers inMax-Plus Representation

331 Determining Project Buffers Previous studies on theCCPM-max-plus representation adopted the CampPMmethodfor calculating project buffers shown as (16) and feedingbuffers [33]

119875119861 = max [(119863 otimes 119908 otimes 119877)lowast (119863 otimes 119908) 1199090]3 (16)

where (119894 119895)-th element of matrix (119863 otimes 119908 otimes 119877)lowast means thecumulative time of duration of activity 120572 which is on thecritical chain and range from 119894 + 1 to j (119894 119895)-th element ofmatrix119863 otimes119908 satisfies that if activity 119894 is on the critical chainthen (119894 119895)-th element is duration of activity I otherwise it is120576

However the CampPM method lacks sound mathematicalfoundation and overestimates project durations resulting ina waste of time and resources Alternatively the CampPM isreplaced by the RSEM which is based on the large numberlaw and central limit theorem in the CCDSM to calculate thePB

Mathematical Problems in Engineering 7

119875119861 = radic sum119894isin119862119875

119878119863119894 (17)

where 119878119863119894 is the variance of duration of activity 119894 on thecritical path

To calculate the PB in max-plus representation a matrixdenoted as 119879119901 is introduced For the element in row i andcolumn 119894 minus 1 in [119879119901]119894119895 if activity 119894 is critical activity it willbe filled with the variance of duration of activity i else it willbe zero For other elements in [119879119901]119894119895 it is filled with zeroThePB then can be represented in max-plus algebra as

119875119861 = radic1198621198790 otimes 119879lowast119901 otimes 1198620 (18)

where 1198621198790 = (119890 119890 119890)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119899+1

332 Determining Feeding Buffers In order to insert feedingbuffers in place to protect critical path from delays innoncritical paths two operators denoted as diag() and arefirst introduced For a vector 119909 a newmatrix can be obtainedby the operator diag() as [diag()]119894119895 = 119909119894 119894119891 119894 = 119895 120576 119890119897119904119890 and a new vector can be obtained by the operator as[119909]119894 = 120576 119894119891 [119909]119894 = 120576 119890 119894119891 [119909]119894 = 120576

In preparation of the location and size of the feedingbuffer another two vectors are introduced denoted as 119908 andV Matrix119908 is developed based on the rule that for a diagonalelement if the corresponding activity is on the critical paththen the element value is 0 otherwise the element value is 120576Matrix V is developed based on the rule that for a diagonalelement if the corresponding activity is on the noncriticalpath then the value is 0 otherwise the value is 120576 Moreoveran adjacency matrix 119877 is transformed from matrix 119865 by[119877]119894119895 = 119890 119894119891 [119865]119894119895 = 120576 120576 119890119897119904119890 Vector V1015840 is then introducedto locate the feeding buffers as follows

[V1015840]120574= [119877119879120573120572 otimes 119908]120574 (19)

where 119877120573120572 is the adjacency matrix which represents transi-tions from noncritical activities to critical ones and can beobtained by 119877120573120572 = diag(119908) otimes 119877 otimes diag(V) 120582 represents thenoncritical activity one ofwhose successors is critical activityTherefore a feeding buffer should be inserted behind theactivity 120574

The feeding buffer can then be determined by squarerooting of the sum of [119878119863]120578 where 120578 is the set of activitieson a certain noncritical chain The formula is expressed inmax-plus algebra as

1198651198610 = radicdiag [119877120573120573 otimes diag (V1015840) otimes diag (119878119863)]lowast otimes 1198621 otimes diag [diag (119878119863) otimes diag (V1015840) otimes 1198621] otimes 1198621 (20)

Additionally the calculation of feeding buffer should beadjusted using the following equation to meet the constraintthat the size of feeding buffer cannot exceed total float

[119865119861]120574 = min ([1198651198610]120574 [119879119865]120574) (21)

34 Implementation of the CCDSM with Max-Plus Lin-ear Expression The max-plus-based implementation of theCCDSM is summarized in the following seven steps It needsto be noted that as the max-plus algorithm cannot solvenonlinear conversions such as multiplication the input datafor the project rework problem including matricesA P S RPand RI needs to be preprocessed

Step 1 (preprocess119860) Extract all diagonal elements of matrix119860 and form a diagonal matrix D

Step 2 (preprocess 119875 and 119878) Combine matrices 119875 and 119878 togenerate matrix 119865 according to the following equation

[119865]119894119895 = [119875]119894119895 times [119875]119895119895 minus [119878]119894119895 times [119878]119894119894 (22)

Extract all nonnull elements ofmatrix119865 and use them to formmatrix 119877 Then update R by replacing all nonnull elementswith 119890 and all null elements with 120576Step 3 (preprocess RP and RI) Based on (9)-(15) mergematrices RP and RI to generate the vector 119877119908119861 whose 119894-thelement is the corresponding rework buffer of activity i

Step 4 (calculate the length of critical path) Based on (4)-(8)the length of critical pathTT and other parameters includingthe total float time TF and critical path are determined

Step 5 (calculate matrices 119908 and V) According to the criticalpath determined in Step 4matrices119908 and V can be generatedMeanwhile the activity duration variance matrix SD canbe generated based on data collected from the project tocalculate the buffers size

Step 6 (calculate buffer sizes) Based on (17)-(21) projectbuffer PB and feeding buffers FBs are calculated with RSEMand represented in max-plus algebra

Step 7 (generate the schedule) As the principle of CCPMthe feeding buffers should be placed on the noncritical chainsprior to the joints of the critical chain and noncritical chainsand the project buffer should be placed at the end of theschedule to protect the whole project process Then theestimated project time PT can be calculated based on

119875119879 = 119879119879 + 119875119861 + 119877119908119861120579 (23)

4 Case Study

In this section the proposed CCDSM method is imple-mented in a case project to demonstrate its feasibility andeffectiveness in addressing rework risks in project schedule

8 Mathematical Problems in Engineering

A B C D E F G H

A 8 X X X X

B X 8 X X X X

C 4 X X

D X 13 X

E X X X X 24 X

F X X X X 35 X

G X X X X 8

H X X X X

I J K L M N O

I X X X X X

J X X X X X X

K X X X X X

L X X X X X X X

M X

N X

O X X X X X X X X

X X X 4 X

X X X X X X 2

X X 4 X X

13 X

X X 8 X X

13

X X 5 X X X

X

X

X

X

X

13

X X X

X

X X

X X

X

X

X

P Q R S

X X

X X

X X

X X

X X

X X

X

X

P X X X X X X X X

Q X X X X X X X X

R X X X X X X X

S

X X X X X X

X

X X X X X X X

X X X X X X X

X X 4

X X X 4

X 4 X

3 X

(a) Matrix A

A B C D E F G H

A 10 09 07 08

B 09 06 07 06

C 07 09 10

D 07 06

E 08 06 06 08 06

F 09 09 08 08 06

G 06 07 06 06

H 06 06 06 06

I J K L M N O

I 06 09 09 06 10

J 05 09 09 07 07 08

K 06 07 06 06 08

L 06 10 08 07 07 06

M 05

N 06 05

O 07 07 08 06 06 09 09 05

08 08 08 08

06 10 06 08 06 06

09 10 09 09

05

08 06 06 10

07 06 09 06 06

09

08

08

10

07

07 08 06

09 07

09 09

09

08

08

07

P Q R S

10

09

09

10

09

05

06

08

P 09 06 07 08 09 06 07 05

Q 08 09 10 09 10 05 10

R 10 07 07 07 06 06 08

S

10 06 08 06 08 09

09

07 08 09 06 10 05 09

06 08 06 10 09 06 07

07 07

09 08 08

08 07

09

(b) Matrix P

A B C D E F G H

A 05 03 02 02

B 01 04 02 05 05

C 01 02

D 01 03

E 04 02 04

F 04 04 04 04 04

G 03 05 05 04

H 04 03 01 05

I J K L M N O

I 01 01 05 02 04

J 03 04 02

K 05 02 03 01 04

L 02 04 03 05 04 03 02

M 01

N 04

O 04 04 02 03 01 02 04

02 02 04 02

04 05 04 03 01 05

02 04 01 03

03

02 04 01 01

03 01 03 01 01

04

03

03

04

01

04 02 03

04

03 04

03 01

02

05

03

P Q R S

01 02

05

04

01 02

04 02

05

01

P 02 05 05 02 03 02 04

Q 03 04 01 02 02

R 02 03 01 04 02 04 01

S

04 03 02 01 01 02

02

01 04 03 01 01 03 04

01 01 04 02 04 02 02

02 01

02 04 03

01 04

01

(c) Matrix S

A B C D E F G HA 01 02 03

B 04 02 01 01

C 02 01

D 03 04

E 06 03 05 05 04

F 03 02 05 05 03

G 02 05 05 02

H 04 02 06 03

I J K L M N O

I 02 03 03 04 02

J 03 03 04 05 03 05

K 04 05 02 06 03

L 03 05 05 03 04 04 03

M 02

N 03

O 05 03 02 05 04 05 06 0306 03 05 01

03 02 05 06 03 05

03 04 01

0203 05 02 0403 04 04 03 03

03

0103 02 02

02

01 04

02

03

0503

P Q R S

02 0102 02

01 03

0301 0202 03

01

02

P 04 04 06 02 04 04 05 06

Q 03 06 05 03 06 05 04 05

R 03 05 03 03 03 05 05

S

02 06 04 03 04 0404

03 06 04 05 03 02 0404 05 06 05 04 03 03

03 0605 02 04

03 0201

(d) Matrix RP

A B C D E F G H

A 008 001 001 009

B 020 014 016 020 007

C 007 008

D 008 023

E 010 001 009 020 013

F 003 025 013 008 023

G 025 002 003 015

H 018 005 022 001

I J K L M N O

I 013 023 020 011 021

J 002 017 007 025 003 024

K 001 008 012 015 002

L 006 013 016 014 003 023 009

M 001

N 006

O 005 016 011 004 014 019 010 015

021 014 005 003

007 004 019 002 005 016

018 009 013 022

012

004 020 015 022

003 022 013 004 023

013

006

023

017

019

005 023 021

018

023 006

023 017

004

004

011

P Q R S

010 009

010 009

022 009

016 010

014 013

004 018

001

018

P 007 014 014 021 023 012 011 014

Q 03 008 025 001 022 014 002 008

R 012 014 006 021 015 021 018

S

022 001 006 020 008 003

014

022 001 006 010 008 019 001

016 011 021 001 021 001 015

009 025

021 018 019

009 019

014

(e) Matrix RI

Figure 5 Project information in the case project

management The settings of the case project are first pre-sented followed by descriptions of implementation of theCCDSM in the case project The schedule generated with theCCDSMmethod is assessed in detail and compared with theschedules generated with traditional CCPM- and DES-basedmethods

41 Case Project Settings The case used in this study wasderived from a modular real estate development projectfirstly introduced in [35] and further described in [36]The matrix 119860 of the case is shown in Figure 5(a) Thiscase consisted of 19 major activities and 183 interactivityrelationships including 65 rework relationships representedby elements above the diagonal of matrix 119860 It is assumedthat rework can propagate up to twice to avoid infinite loopin the computation The parameters for each activity and theinteractivity relationships were derived based on literatureand empirical evidence and are shown in Figures 5(b) 5(c)5(d) and 5(e) and summarized in Table 2 Large-scale reworkrelationships and complex connections between activitieswere observed in the case project which was representativeof typical construction projects in reality

42 Implementation of the CCDSM Method The proposedCCDSM method was implemented in the case project fol-lowing the steps explained in the last section and the results

were shown below Noticeably for better understandingof scheduling process the implementation process of theCCDSMmethod is reorganized as below

Step 1 Matrices 119863 and 119877 were derived from matrix Aas shown in Figure 6 The duration of each activity waspresented in the diagonal of the matrix D

Step 2 Matrix F (see Figure 7) was derived from matrices119875 and 119878 according to (22) This step calculated the activitydeviation matrix based on the predecessor time matrix andthe successor time matrix

Step 3 Optimize activity sequence and update matrices D RRP RI and 119865 according to new activity sequence To reduceproject rework and obtain near-optimal project completiontime the genetic algorithm (GA) was applied to optimizeactivity sequence The GA is a metaheuristic method thatsearches for optimal solutions using processes similar tothose in natural selections and genetics [37] In the paperminimization of total length of rework path first proposed byGebala and Eppinger [38] was used as the objective functionto calculate the optimal sequence of activities

119891 = 119899sum119894=1

119899sum119895=119894+1

(119895 minus 119894) sdot 119908 (119894 119895) (24)

Mathematical Problems in Engineering 9

Table 2 Activity parameters in the case project

ID Activity Learning curve Duration (day)Min Likely Max

A Perform prelim mkt analysis 05 5 8 15B Evaluate marketability options 06 5 8 15C Engage feasibility consultants 03 3 4 7D Evaluate planning amp zoning process 05 10 13 20E Perform massing study 06 20 24 35F Develop conceptual design 08 30 35 50G Identify external stakeholders 04 5 8 15H Identify permits amp approvals 07 10 13 20I Complete phase 1 ESA 03 10 13 20J Evaluate consultants amp contractors 06 10 13 20K Obtain rough construction costs 05 5 8 15L Determine highest amp best use 05 3 5 9M Identify debt options 04 2 4 8N Identify equity options 05 3 4 5O Update financial underwriting 05 1 2 3P Reevaluate organization strategy 06 2 3 5Q Estimate schedule 06 3 4 7R Gain control of site andor client 06 3 4 7S Review and approve 07 3 4 7

F C A I J R K PF 30C 3A 5I 10J 10R 3K 5P

D M O B H E N

DMOBHEN

203

10

21

10

5

2

L G Q S

LGQS

33

53

(a) Matrix D

F C A I J R K PFCA e eI eJ e eR e e eK e e e eP e e e e e e

D M O B H E N

D e eM eO e e e e eB e e eH e e eE e eN e e

e e e ee e

e

e eee

L G Q S

L e e e e eG e e eQ e e e e e e e eS e e

e e e e e e ee

e e ee e e e e e e

e ee

e

e

e

e

e

(b) Matrix R (element 120576 is omitted)

Figure 6 Matrix D amp R

where 119908(119894 119895) represents rework probability of activity 119894caused by activity 119895

The parameter settings were selected as follows pop-ulation size set as 50 number of generations set as 150crossover probability set as 095 and mutation probabilityset as 008 The GA process reported an optimal schedule as[119875 119876119867 119864 119877 119861 119862119870119873 119871119872 119865119863 119878 119860 119866 119869 119868 119874] The GAconvergence process is shown in Figure 8

Step 4 119877119908119861 was generated from matrices RP and RI basedon (9)-(15) The goal of this step was to factor in andcalculate the rework time of the case project 119877119908119861 was[3 4 6 10 5 1 1 3 1 2 1 15 1 1 1 1 1 1 9]Step 5 To generate reliable project schedules the most likelyduration of each activity which has been widely used andaccepted by project teams in prior research [39] was selected

10 Mathematical Problems in Engineering

A B C D E F G HA 1 10 15 28

B 7 1 13 23 2

C 13 8

D 2 1

E -3 1 3 10 15

F -6 -11 -4 5 -

12G 3 2 6

H -3 4 14

I J K L M N O

I 2 11 16 3 8

J 5 -1 8 18 28 4

K 2 5 2 15 28

L 4 6 1 10 16 27 7

M 13

N 12

O 5 6 2 8 16 35 7 76 4 2 1

8 12 4 4 3 2

7 4 4 1

-19 5 3 29 5 3 2 1

1174-19

-8 -1 -99

-1 -8

8 9

-1

57

P Q R S

3 33 4

1 4

-1 33 2

2

-1

2

P 8 5 2 11 23 22 6 6

Q 7 7 2 14 23 37 4 13

R 7 5 11 17 23 4 10

S

11 4 4 2 3 11

9 10 6 3 4 19 10 4 5 3 2 1

2 32 2 3

2 22

Figure 7 Matrix F (element 120576 is omitted)

20

30

40

50

60

70

80

Estim

ated

Fee

dbac

k Le

ngth

50 100 1500Generation

Figure 8 Convergence curve in the GA-based optimization

to calculate the estimated project duration The critical chainwas reported as [119867 119864 119861 119870119873 119871 119865 119860 119866 119869 119874]Step 6 The activity duration variance matrix SD was gen-erated with 1000 Monte Carlo simulations of the activitiesrsquoduration that follows a beta distribution

Step 7 Buffers were calculated based on (17)-(21) Theproject buffer was 36 days and the feeding buffer was[3 4 0 0 38 0 4 0 0 0 6 0 10 38 0 0 0 0 2 0]Step 8 The estimated project duration was calculated as 102days based on (23) The project buffer was placed at the endof the schedule and the feeding buffer was placed on thenoncritical chains prior to the joints of the critical chain andnoncritical chains following the principle of CCPM

It was worth mentioning that overlapping was observedamong various activities including critical activities in the

above schedule which illustrated real-world high-frequencyinteractions among activities in the project

To assess the performance of the CCDSM method andthe quality of the project schedule it generated the executionof case project was simulated 1000 times using Monte Carloapproach where the sampling of actual activity durationsfollowed beta distribution The simulated actual projectdurations were depicted in Figure 9 It can be seen in thefigure that under rework assumption the process scheduleof 102 days had a completion probability of 981 indicatingthat the proposed CCDSM method was able to providesatisfactory assurance of on-time project completion undervarying scheduling scenarios

43 Comparison of the CCDSM Method with TraditionalCCPM and DES Methods This subsection further comparesthe performance of the CCDSM method with traditional

Mathematical Problems in Engineering 11

75 80 85 90 95 100 105 1100

20

40

60

80

100

120

140

Duration

CCDSM method=102

0

01

02

03

04

05

06

07

08

09

1

981

CCDSM method=102

Cum

ulat

ive P

roba

bilit

y

Freq

uenc

y

80 85 90 95 100 105 11075Duration

Figure 9 Simulated actual project durations and on-time project completion probability of the CCDSM-based schedule

75 80 85 90 95 100 105 1100

01

02

03

04

05

06

07

08

09

1

Duration

981

CCDSM method=102

493

CCPM method=93

Cum

ulat

ive P

roba

bilit

y

Figure 10 Comparison of on-time project completion probabilities between the CCDSM- and CCPM-based schedules

CCPM and DES methods to assess its effectiveness inaddressing rework risks

A CCPM-based schedule was generated with tradi-tional buffers including project buffer and feeding bufferand without rework buffer The results indicated that theestimated project durations of the CCPM-based schedulewere 93 days including project buffers of 36 days Whenthe estimated project duration of CCPM-based schedulewas superimposed in the Monte Carlo-simulated projectdurations as depicted in Figure 10 it can be seen thatthe on-time completion probability was 493 The resultssuggested that the CCDSM-based schedule which took intoconsideration rework relationships and had a duration of 102days significantly outperformed the CCPM-based schedulein ensuring on-time completion of the case project that wasfaced with typical rework risks

Furthermore a DES-based schedule was generated byimplementing DES algorithm in which activity durationsfollowing a beta distribution were generated at random tocompute the project duration When the DES algorithm was

used to generate project schedules under rework scenarios aprobabilistic judgment is performed firstly on whether or noteach activity causes rework and then an iterative simulationprocess began until all activities were finished Specificallythe DES algorithm proposed by Browning and Eppinger[27] was adopted in this study with the following minormodifications to make it more applicable to the case projectand comparable to the CCDSM method (1) the optimizedactivity sequence in Section 42 was employed in the DESprocess (2) activity 119894 did not started until STS logic tie withits each predecessor activity 119895 that is [119865]119894119895 was satisfied (3)rework probability of activity 119894 would decrease by 50 eachtime it reworked [28]

To compare the CCDSM-based schedule with the DES-based schedule the execution of the case project usingthe DES-based schedule was simulated 1000 times usingMonte Carlo approach and the simulated actual projectdurations are depicted in Figure 11 The results showed thatthe estimated project duration of the DES-based schedulewas 115 days which was 13 days longer than that of the

12 Mathematical Problems in Engineering

70 80 90 100 110 120 1300

010203040506070809

1

Duration

981

DES method=115Cu

mul

ativ

e Pro

babi

lity

Figure 11 On-time project completion probability of the DES-based schedule

CCDSM-based schedule with the same on-time completionprobability of 981 In other words the CCDSM methodoutperformed the DES method by ensuring the same on-time completion probability with a shorter project scheduleMoreover the CCDSM method was also advantageous overthe DES-based method in that it enabled effective controlof project schedules during project implementation whichcould be done by monitoring consumption of time buffersleading to reduction in project overrun risks

5 Conclusions

Construction projects are constantly challenged by reworkrisks which have largely remained unaddressed by existingproject schedulingThis paper proposed the CCDSMmethodfor developing project schedules that are adaptive to reworkinstances during project execution This method modelslarge-scale rework relationships among activities with theintroduction of a new rework buffer in traditional CCPMmethod The method allows analysis and visualization ofthe schedules and utilizes max-plus algorithm to transformcomplex logic relationships into simple matrix operationsreducing the computational load of generating process sched-ules A case study was conducted to demonstrate the imple-mentation of the CCDSMmethod and assess its effectivenessin addressing rework risksThe results showed that CCDSM-based schedule outperformed the CCPM-based schedule inensuring on-time completion of the case project that wasfaced with typical rework risks In addition in comparison tothe DESmethod the CCDSM performed better by providinga smaller project duration with the same probability of on-time project completion

The CCDSM method contributes to construction man-agement in the following ways First the CCDSM methodabsorbs risks of rework and various other uncertainties togenerate the reliable project schedule enabling managersto predict project durations more accurately before projectsstart Based on accurate prediction of project durationsthe negotiation of contracts and supply of resources canbe conducted in a more reasonable manner avoiding costsoverruns schedule delays and even project failures Second

based on generated project schedule with time bufferseffectivemonitoring and control of construction progress canbe realized Overall speaking the proposed CCDSMmethodprovides a promising solution to mitigate rework risks inconstruction projects and protect projects from undesirabledelays caused by possible rework instances

Meanwhile the CCDSM bears several limitations thatwould benefit from future improvement First only the STSlogic tie between activities is tested in this study Techniquesfor integrating multiple logic ties including STS STF FTSand FTF deserve further attention Second resource conflictsand levering are considered beyond the scope of this studybut it is a critical challenge that should be addressed inthe context of CCDSM-based scheduling in future researchThird the parameters required to implement the CCDSMmethod are hard to obtain for a specific project To employCCPM or max-plus algorithm for project scheduling it isnecessary to obtain the estimated activity durations undervarious completion rates and the logic tie between activitiesSimilarly the rework relative parameters including RP RIand LC are required for implementation of DSM-basedmethod As an integration method of CCPM max-plus algo-rithm and DSM the implementation of CCDSM requiresall the above parameters to be obtained Lastly methods fordynamic updating of buffer sizes during project execution areneeded in order to better prepare project teams for changingproject environment and dynamic project progress

Data Availability

All data used to support the findings of this study are includedwithin the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (NSFC) [Grant no 71671128]

Mathematical Problems in Engineering 13

References

[1] Y Li and T R Taylor ldquoModeling the impact of design reworkon transportation infrastructure construction project perfor-mancerdquo Journal of Construction Engineering and Managementvol 140 no 9 p 04014044 2014

[2] B Hwang X Zhao and K J Goh ldquoInvestigating the client-related rework in building projects The case of SingaporerdquoInternational Journal of Project Management vol 32 no 4 pp698ndash708 2014

[3] E K Simpeh R Ndihokubwayo P E Love and W DThwala ldquoA rework probability model a quantitative assessmentof rework occurrence in construction projectsrdquo InternationalJournal of Construction Management vol 15 no 2 pp 109ndash1162015

[4] P W Morris J K Pinto and S Jonas 13e Oxford Handbookof Project Management Oxford University Press Oxford UK2012

[5] S D Eppinger and T R Browning Design Structure MatrixMethods and Applications MIT Press Books 2012

[6] T R Browning ldquoDesign structure matrix extensions and inno-vations a survey and new opportunitiesrdquo IEEE Transactions onEngineering Management vol 63 no 1 pp 27ndash52 2016

[7] A Kulkarni D K Yadav and H Nikraz ldquoAircraft maintenancechecks using critical chain project pathrdquo Aircra EngineeringAerospace Technology vol 89 no 6 pp 879ndash892 2017

[8] J Trojanowska and E Dostatni ldquoApplication of the theory ofconstraints for project managementrdquoManagement and Produc-tion Engineering Review vol 8 no 3 pp 87ndash95 2017

[9] M Ghaari and M Emsley ldquoBufer sizing in CCPM portfolioswith dierent resource capacitiesrdquo International Journal of Infor-mation Technology Project Management vol 8 no 3 pp 40ndash512017

[10] E M Goldratt Critical Chain A Business Novel MA NorthRiver Press Great Barrington United States 1997

[11] S Van de Vonder E Demeulemeester and W HerroelenldquoProactive heuristic procedures for robust project schedulingAn experimental analysisrdquo European Journal of OperationalResearch vol 189 no 3 pp 723ndash733 2008

[12] O Lambrechts E Demeulemeester and W Herroelen ldquoTimeslack-based techniques for robust project scheduling subject toresource uncertaintyrdquo Annals of Operations Research vol 186no 1 pp 443ndash464 2011

[13] D Trietsch and K R Baker ldquoPERT 21 Fitting PERTCPMfor use in the 21st centuryrdquo International Journal of ProjectManagement vol 30 no 4 pp 490ndash502 2012

[14] GMaAWangN Li LGu andQAi ldquoImproved critical chainproject management framework for scheduling constructionprojectsrdquo Journal of Construction Engineering andManagementvol 140 no 12 p 04014055 2014

[15] R C Newbold Project Management in 13e Fast Lane Applying13e 13eory of Constraints CRC Press Boca Raton FloridaLondon 1998

[16] O I Tukel W O Rom and S D Eksioglu ldquoAn investigation ofbuffer sizing techniques in critical chain schedulingrdquo EuropeanJournal of Operational Research vol 172 no 2 pp 401ndash4162006

[17] O Icmeli and S S Erenguc ldquoA branch and bound procedurefor the resource constrained project scheduling problem withdiscounted cash flowsrdquoManagement Science vol 42 no 10 pp1395ndash1408 1996

[18] E Roghanian M Alipour and M Rezaei ldquoAn improved fuzzycritical chain approach in order to face uncertainty in projectschedulingrdquo International Journal of ConstructionManagementvol 18 no 1 pp 1ndash13 2017

[19] J Zhang X Song and E Dıaz ldquoProject buffer sizing of a criticalchain based on comprehensive resource tightnessrdquo EuropeanJournal of Operational Research vol 248 no 1 pp 174ndash182 2016

[20] F A Rivera and A Duran ldquoCritical clouds and critical sets inresource-constrained projectsrdquo International Journal of ProjectManagement vol 22 no 6 pp 489ndash497 2004

[21] G Ma L Gu and N Li ldquoScenario-based proactive robustoptimization for critical-chain project schedulingrdquo Journal ofConstruction Engineering and Management vol 141 no 10 p04015030 2015

[22] H Goto ldquoForward-compatible framework with critical-chainproject management using a max-plus linear representationrdquoOPSEARCH vol 54 no 1 pp 201ndash216 2017

[23] T Salama A Salah and O Moselhi ldquoIntegration of linearscheduling method and the critical chain project managementrdquoCanadian Journal of Civil Engineering vol 45 no 1 pp 30ndash402018

[24] C W Hu X D Chen and L H Wu ldquoThe setting method ofproject buffer in critical chain management of mould manufac-turing project based reworkingrdquo Advanced Materials Researchvol 317-319 pp 418ndash422 2011

[25] D V Steward Systems Analysis and Management StructureStrategy and Design IrwinMcGraw-Hill Boston 1981

[26] T R Browning Modeling and analyzing cost schedule andperformance in complex system product development [PhDthesis]Massachusetts Institute of Technology CambridgeMas-sachusetts USA 1998

[27] T R Browning and S D Eppinger ldquoModeling impacts ofprocess architecture on cost and schedule risk in productdevelopmentrdquo IEEE Transactions on Engineering Managementvol 49 no 4 pp 428ndash442 2002

[28] S H Cho and S D Eppinger ldquoA simulation-based processmodel for managing complex design projectsrdquo IEEE Transac-tions on Engineering Management vol 52 no 3 pp 316ndash3282005

[29] V Levardy and T R Browning ldquoAn adaptive process modelto support product development project managementrdquo IEEETransactions on Engineering Management vol 56 no 4 pp600ndash620 2009

[30] J U Maheswari and K Varghese ldquoProject scheduling usingdependency structure matrixrdquo International Journal of ProjectManagement vol 23 no 3 pp 223ndash230 2005

[31] C G Cassandras and S Lafortune Introduction to DiscreteEvent Systems Springer New York NY USA 2nd edition 2008

[32] S Yoshida H Takahashi and H Goto ldquoModified max-pluslinear representation for inserting time buffersrdquo in Proceedingsof the IEEE International Conference on Industrial Engineeringamp Engineering Management IEEE 2010

[33] S Yoshida H Takahashi and H Goto ldquoResolution of timeand worker conflicts for a single project in a max-plus linearrepresentationrdquo Industrial Engineering amp Management Systemsvol 10 no 4 pp 279ndash287 2011

[34] S Muralidharan ldquoFactors affecting rework in construc-tion projectrdquo International Journal of Engineering Sciences ampResearch Technology vol 1 no 5 pp 578ndash584 2016

[35] B B E Bulloch and J Sullivan Application of the DesignStructure Matrix (DSM) to the real estate development process

14 Mathematical Problems in Engineering

[Master thesis] Massachusetts Institute of Technology Cam-bridge Massachusetts USA 2009

[36] S D Eppinger S Bonelli andAM Gonzalez ldquoManaging itera-tions in the modular real estate development processrdquo ReducingRisk in Innovation Proceedings of the 15th International DSMConference Melbourne Australia 29-30 August 2013 pp 37ndash442013

[37] D T Pham and D Karaboga Intelligent Optimisation Tech-niques Genetic Algorithms Tabu Search Simulated AnnealingandNeural Networks Springer Science amp BusinessMedia 2000

[38] D A Gebala and S D Eppinger ldquoMethods for analyzing designproceduresrdquo in Proceedings of the AMSE 13ird InternationalConference On Design13eory andMethodology vol 31 pp 227ndash233 Miami Florida 1991

[39] O Hazır M Haouari and E Erel ldquoRobust scheduling androbustness measures for the discrete timecost trade-off prob-lemrdquo European Journal of Operational Research vol 207 no 2pp 633ndash643 2010

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Mathematical Problems in Engineering 5

a stand-alone project management technique for that DSMrsquosmatrix-based expression is difficult to be effectively convertedto network diagram-based expression sometimes leadingto ambiguity and confusion [5] Therefore it is challengingto apply DSM to the process management of constructionprojects and DSM needs to be extended and improved withfurther research efforts

3 The Critical Chain Design StructureMatrix Method

In this section we are going to demonstrate how to usethe CCDSM method to generate the project schedule Weimprove the max-plus algorithm to reduce computationalload in generating project schedules and introduce reworkbuffers to evaluate and address rework risks

31 Construction Project Scheduling with the Max-Plus Algo-rithm A discrete event system is state-discrete and event-driven Its state evolution depends on asynchronous discreteevents occurring at discrete points over time [31] Simplediscrete event systems are usually linear systems in whichstate and output variables for all possible input variables andthe initial state satisfy superposition principle of the systemAconstruction project can generally be seen as a linear systemthat consists of a number of discrete events

The max-plus algorithm provides an alternative way todescribe the discrete event system Based on the max-plusalgorithm the status of all discrete events denoted as vector119909(119896) in a system can be used to describe the status ofthe system and adjacent statuses are interchangeable with alimited number of linear changes as follows

119909 (1) = 1198601119909 (0)119909 (2) = 1198602119909 (1)

119909 (119896) = 119860119896119909 (119896 minus 1)

(2)

Based on the above equation if the linear changes at everystage 1198601 1198602 119860119896 are known then the final status of thesystem 119909(119896) can be derived from the initial status of thesystem 119909(0) Using this as a starting point the max-plusalgorithm introduces four types of operations to describe thediscrete event systems For two 119898 times 119898 matrices 119883 and 119884denoted as 119883119884 isin 119874119898times119898 whose elements are nonnegativenumbers or negative infinity these operations are defined asfollows

[X oplus Y]119894119895 = max ([X]119894119895 [Y]119894119895)[X and Y]119894119895 = min ([X]119894119895 [Y]119894119895)[X otimes Y]119894119895 = 119898⨁

119896=1

([X]119894119896 + [Y]119896119895)

[X ⊙ Y]119894119895 = 119898⋀119896=1

(minus [X]119894119896 + [Y]119896119895)

(3)

where 119883 otimes 119884 could be further simplified as 119883119884 and 119883119899 =119883119883 sdot sdot sdot 119883⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119899

To apply the max-plus algorithm to schedule manage-ment three new matrices need to be developed to describerelevant project information These matrices include (1)activity durationmatrix denoted asD which is developed byfilling the diagonal elements with each activityrsquos duration andthe nondiagonal elements with negative infinity (2) activitydeviation matrix denoted as F which is developed based onthe rule that [119865]119894119895 represents the deviation of the start time ofactivity 119895 and activity i and (3) activity relationship matrixdenoted as R which is developed based on the rule that ifthere is information delivered from activity 119895 to activity ithen the value of [119877]119894119895 is 0 otherwise the value is negativeinfinity

According to the max-plus algorithm [32] we have thefollowing

The early finish time for each activity is calculated as

119909+119864 = 119863 otimes (119865)lowast otimes 1199090 (4)

where (119865)lowast = 119890 oplus 119865 oplus oplus (119865)119897minus1 The (119894 119895)-th element ofmatrix (119865)lowast means the largest deviation of the start time ofactivity 119895 and activity i if activities 119894 and 119895 are on one ormore paths at the same time otherwise it is 120576 For simplicitythe elements of vector 1199090 are set as e which means thebeginning of activities in the project will not be affected byother construction projects

The length of critical path denoted as the maximum ofearly finish time for all activities is calculated as

119879119879 = max [119909+119864] (5)

The early start time is calculated as the difference betweenearly finish time for each activity and its duration accordingto

119909minus119864 = 119863 ⊙ 119909+119864 (6)

The late start time of activity 119894 is calculated as thedifference between length of critical path and the sum ofactivity 119894rsquos duration as well as subsequent critical activitiesrsquodurations according to

119909minus119871 = [119863 otimes (119865)lowast]119879 ⊙ [1199090 ⊙ 119879119879] (7)

The total float is calculated as

[119879119865]119894 = [119909minus119864]119894 ⊙ [119909minus119871]119894 (8)

The critical chain is then determined by the set ofactivities 120572 that satisfy 120572 | [119879119865]120572 = 0

It needs to be noted that when determining floats themax-plus algorithm is run in simple matrix form whichsignificantly improves the computational efficiency and ismore applicable to large-scale projects [33] The definitionsof parameters in the above equations are summarized in thenomenclature in Table 1

6 Mathematical Problems in Engineering

rework time

i

j

(a) First rework

j

k

i

rework time

(b) Second rework caused by information tran-sition from its precedent activity

rework time

i

j

k

(c) Second rework caused by information feed-back from its successive activity

Figure 4 First rework and second rework

32 Determining Project Rework Buffer It is widely believedthat information uncertainty is themain cause of rework [34]Such uncertain information may transmit from upstreamactivities to downstream activities or feedback in the oppo-site direction creating information flow Information flowinteractions lead to rework risks in construction projects andconsequently bring about more rework time There are twomain situations that rework exists(1)First rework as depicted in Figure 4(a) i is a precedentactivity of j and they perform sequentially according to rela-tionships After it is completed the performance informationof 119895 will be generated and transmitted to i which may resultin rework of 119894 The first rework time shown as the shadowarea in Figure 4(a) can be calculated by [27]

[1198771198791]119894119895 = 119877119875119894119895 sdot 119877119868119894119895 sdot 119871119862119894 sdot 119863119894119894 (9)

Then the total first rework time of 119894 caused by all itsdownstream activities can be obtained by

[1198771198791]119894 =119899sum119895=119894+1

119877119875119894119895 sdot 119877119868119894119895 sdot 119871119862119894 sdot 119863119894119894 (10)

where 119899 represents the number of activities in the schedule(2) Second rework as depicted in Figures 4(b) and 4(c)after completing the rework of 119895 caused by 119896 119895 transmits orfeeds back some revised information to 119894 which may causerework of 119894 Such two forms are defined as the second reworkin the paper The second rework times shown as the shadowareas in Figures 4(b) and 4(c) can be calculated by

[1198771198792]119894119895119896 = 119877119875119894119895 sdot 119877119875119895119896 sdot 119877119868119894119895 sdot 119871119862119894 sdot 119863119894119894 (11)

Then the total second rework time of 119894 caused by all itsdownstream activities can be obtained by

[1198771198792]119894 =119896minus1sum119895=1

119899sum119896=119894+1

119877119875119895119896 sdot 119877119875119894119895 sdot 119877119868119894119895 sdot 119871119862119894 sdot 119863119894119894 (12)

To proactively account for the uncertainty caused byinteractivity relationships and the associated rework risks arework buffer (119877119908119861) is introduced in the proposed CCDSMConceptually 119877119908119861 assesses and compresses the durationuncertainty caused by rework and is placed after the comple-tion of rework activities in project schedules119877119908119861 is designedto warn project participants from how much workload may

increase for each rework activity so as to prepare the projectparticipants for sufficient time and resources

The rework buffer of 119894 can be obtained by

119877119908119861119894 = [1198771198791]119894 + [1198771198792]119894 (13)

For the last critical activity 120579 in the schedule consideringall reworks it causes other activities to generate will take placeafter its completion andhave an impact onproject completiontime therefore its rework buffer is defined as the sum of thetotal rework time it causes other activities to generate and itstotal rework time caused by other activities

The total rework time it causes all other activities togenerate can be calculated by

[1198771198793]120579 ==119899sum119894=1

119877119875119894120579 sdot 119877119868119894120579 sdot 119871119862119894 sdot 119863119894119894 + 119899sum119894=1

119899sum119895=1

119877119875119894119895sdot 119877119875119895120579 sdot 119877119868119894119895 sdot 119871119862119894 sdot 119863119894119894

(14)

The rework buffer of 120579 then can be obtained by

119877119908119861120579 = [1198771198791]120579 + [1198771198792]120579 + [1198771198793]120579 (15)

33 Determining Project Buffers and Feeding Buffers inMax-Plus Representation

331 Determining Project Buffers Previous studies on theCCPM-max-plus representation adopted the CampPMmethodfor calculating project buffers shown as (16) and feedingbuffers [33]

119875119861 = max [(119863 otimes 119908 otimes 119877)lowast (119863 otimes 119908) 1199090]3 (16)

where (119894 119895)-th element of matrix (119863 otimes 119908 otimes 119877)lowast means thecumulative time of duration of activity 120572 which is on thecritical chain and range from 119894 + 1 to j (119894 119895)-th element ofmatrix119863 otimes119908 satisfies that if activity 119894 is on the critical chainthen (119894 119895)-th element is duration of activity I otherwise it is120576

However the CampPM method lacks sound mathematicalfoundation and overestimates project durations resulting ina waste of time and resources Alternatively the CampPM isreplaced by the RSEM which is based on the large numberlaw and central limit theorem in the CCDSM to calculate thePB

Mathematical Problems in Engineering 7

119875119861 = radic sum119894isin119862119875

119878119863119894 (17)

where 119878119863119894 is the variance of duration of activity 119894 on thecritical path

To calculate the PB in max-plus representation a matrixdenoted as 119879119901 is introduced For the element in row i andcolumn 119894 minus 1 in [119879119901]119894119895 if activity 119894 is critical activity it willbe filled with the variance of duration of activity i else it willbe zero For other elements in [119879119901]119894119895 it is filled with zeroThePB then can be represented in max-plus algebra as

119875119861 = radic1198621198790 otimes 119879lowast119901 otimes 1198620 (18)

where 1198621198790 = (119890 119890 119890)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119899+1

332 Determining Feeding Buffers In order to insert feedingbuffers in place to protect critical path from delays innoncritical paths two operators denoted as diag() and arefirst introduced For a vector 119909 a newmatrix can be obtainedby the operator diag() as [diag()]119894119895 = 119909119894 119894119891 119894 = 119895 120576 119890119897119904119890 and a new vector can be obtained by the operator as[119909]119894 = 120576 119894119891 [119909]119894 = 120576 119890 119894119891 [119909]119894 = 120576

In preparation of the location and size of the feedingbuffer another two vectors are introduced denoted as 119908 andV Matrix119908 is developed based on the rule that for a diagonalelement if the corresponding activity is on the critical paththen the element value is 0 otherwise the element value is 120576Matrix V is developed based on the rule that for a diagonalelement if the corresponding activity is on the noncriticalpath then the value is 0 otherwise the value is 120576 Moreoveran adjacency matrix 119877 is transformed from matrix 119865 by[119877]119894119895 = 119890 119894119891 [119865]119894119895 = 120576 120576 119890119897119904119890 Vector V1015840 is then introducedto locate the feeding buffers as follows

[V1015840]120574= [119877119879120573120572 otimes 119908]120574 (19)

where 119877120573120572 is the adjacency matrix which represents transi-tions from noncritical activities to critical ones and can beobtained by 119877120573120572 = diag(119908) otimes 119877 otimes diag(V) 120582 represents thenoncritical activity one ofwhose successors is critical activityTherefore a feeding buffer should be inserted behind theactivity 120574

The feeding buffer can then be determined by squarerooting of the sum of [119878119863]120578 where 120578 is the set of activitieson a certain noncritical chain The formula is expressed inmax-plus algebra as

1198651198610 = radicdiag [119877120573120573 otimes diag (V1015840) otimes diag (119878119863)]lowast otimes 1198621 otimes diag [diag (119878119863) otimes diag (V1015840) otimes 1198621] otimes 1198621 (20)

Additionally the calculation of feeding buffer should beadjusted using the following equation to meet the constraintthat the size of feeding buffer cannot exceed total float

[119865119861]120574 = min ([1198651198610]120574 [119879119865]120574) (21)

34 Implementation of the CCDSM with Max-Plus Lin-ear Expression The max-plus-based implementation of theCCDSM is summarized in the following seven steps It needsto be noted that as the max-plus algorithm cannot solvenonlinear conversions such as multiplication the input datafor the project rework problem including matricesA P S RPand RI needs to be preprocessed

Step 1 (preprocess119860) Extract all diagonal elements of matrix119860 and form a diagonal matrix D

Step 2 (preprocess 119875 and 119878) Combine matrices 119875 and 119878 togenerate matrix 119865 according to the following equation

[119865]119894119895 = [119875]119894119895 times [119875]119895119895 minus [119878]119894119895 times [119878]119894119894 (22)

Extract all nonnull elements ofmatrix119865 and use them to formmatrix 119877 Then update R by replacing all nonnull elementswith 119890 and all null elements with 120576Step 3 (preprocess RP and RI) Based on (9)-(15) mergematrices RP and RI to generate the vector 119877119908119861 whose 119894-thelement is the corresponding rework buffer of activity i

Step 4 (calculate the length of critical path) Based on (4)-(8)the length of critical pathTT and other parameters includingthe total float time TF and critical path are determined

Step 5 (calculate matrices 119908 and V) According to the criticalpath determined in Step 4matrices119908 and V can be generatedMeanwhile the activity duration variance matrix SD canbe generated based on data collected from the project tocalculate the buffers size

Step 6 (calculate buffer sizes) Based on (17)-(21) projectbuffer PB and feeding buffers FBs are calculated with RSEMand represented in max-plus algebra

Step 7 (generate the schedule) As the principle of CCPMthe feeding buffers should be placed on the noncritical chainsprior to the joints of the critical chain and noncritical chainsand the project buffer should be placed at the end of theschedule to protect the whole project process Then theestimated project time PT can be calculated based on

119875119879 = 119879119879 + 119875119861 + 119877119908119861120579 (23)

4 Case Study

In this section the proposed CCDSM method is imple-mented in a case project to demonstrate its feasibility andeffectiveness in addressing rework risks in project schedule

8 Mathematical Problems in Engineering

A B C D E F G H

A 8 X X X X

B X 8 X X X X

C 4 X X

D X 13 X

E X X X X 24 X

F X X X X 35 X

G X X X X 8

H X X X X

I J K L M N O

I X X X X X

J X X X X X X

K X X X X X

L X X X X X X X

M X

N X

O X X X X X X X X

X X X 4 X

X X X X X X 2

X X 4 X X

13 X

X X 8 X X

13

X X 5 X X X

X

X

X

X

X

13

X X X

X

X X

X X

X

X

X

P Q R S

X X

X X

X X

X X

X X

X X

X

X

P X X X X X X X X

Q X X X X X X X X

R X X X X X X X

S

X X X X X X

X

X X X X X X X

X X X X X X X

X X 4

X X X 4

X 4 X

3 X

(a) Matrix A

A B C D E F G H

A 10 09 07 08

B 09 06 07 06

C 07 09 10

D 07 06

E 08 06 06 08 06

F 09 09 08 08 06

G 06 07 06 06

H 06 06 06 06

I J K L M N O

I 06 09 09 06 10

J 05 09 09 07 07 08

K 06 07 06 06 08

L 06 10 08 07 07 06

M 05

N 06 05

O 07 07 08 06 06 09 09 05

08 08 08 08

06 10 06 08 06 06

09 10 09 09

05

08 06 06 10

07 06 09 06 06

09

08

08

10

07

07 08 06

09 07

09 09

09

08

08

07

P Q R S

10

09

09

10

09

05

06

08

P 09 06 07 08 09 06 07 05

Q 08 09 10 09 10 05 10

R 10 07 07 07 06 06 08

S

10 06 08 06 08 09

09

07 08 09 06 10 05 09

06 08 06 10 09 06 07

07 07

09 08 08

08 07

09

(b) Matrix P

A B C D E F G H

A 05 03 02 02

B 01 04 02 05 05

C 01 02

D 01 03

E 04 02 04

F 04 04 04 04 04

G 03 05 05 04

H 04 03 01 05

I J K L M N O

I 01 01 05 02 04

J 03 04 02

K 05 02 03 01 04

L 02 04 03 05 04 03 02

M 01

N 04

O 04 04 02 03 01 02 04

02 02 04 02

04 05 04 03 01 05

02 04 01 03

03

02 04 01 01

03 01 03 01 01

04

03

03

04

01

04 02 03

04

03 04

03 01

02

05

03

P Q R S

01 02

05

04

01 02

04 02

05

01

P 02 05 05 02 03 02 04

Q 03 04 01 02 02

R 02 03 01 04 02 04 01

S

04 03 02 01 01 02

02

01 04 03 01 01 03 04

01 01 04 02 04 02 02

02 01

02 04 03

01 04

01

(c) Matrix S

A B C D E F G HA 01 02 03

B 04 02 01 01

C 02 01

D 03 04

E 06 03 05 05 04

F 03 02 05 05 03

G 02 05 05 02

H 04 02 06 03

I J K L M N O

I 02 03 03 04 02

J 03 03 04 05 03 05

K 04 05 02 06 03

L 03 05 05 03 04 04 03

M 02

N 03

O 05 03 02 05 04 05 06 0306 03 05 01

03 02 05 06 03 05

03 04 01

0203 05 02 0403 04 04 03 03

03

0103 02 02

02

01 04

02

03

0503

P Q R S

02 0102 02

01 03

0301 0202 03

01

02

P 04 04 06 02 04 04 05 06

Q 03 06 05 03 06 05 04 05

R 03 05 03 03 03 05 05

S

02 06 04 03 04 0404

03 06 04 05 03 02 0404 05 06 05 04 03 03

03 0605 02 04

03 0201

(d) Matrix RP

A B C D E F G H

A 008 001 001 009

B 020 014 016 020 007

C 007 008

D 008 023

E 010 001 009 020 013

F 003 025 013 008 023

G 025 002 003 015

H 018 005 022 001

I J K L M N O

I 013 023 020 011 021

J 002 017 007 025 003 024

K 001 008 012 015 002

L 006 013 016 014 003 023 009

M 001

N 006

O 005 016 011 004 014 019 010 015

021 014 005 003

007 004 019 002 005 016

018 009 013 022

012

004 020 015 022

003 022 013 004 023

013

006

023

017

019

005 023 021

018

023 006

023 017

004

004

011

P Q R S

010 009

010 009

022 009

016 010

014 013

004 018

001

018

P 007 014 014 021 023 012 011 014

Q 03 008 025 001 022 014 002 008

R 012 014 006 021 015 021 018

S

022 001 006 020 008 003

014

022 001 006 010 008 019 001

016 011 021 001 021 001 015

009 025

021 018 019

009 019

014

(e) Matrix RI

Figure 5 Project information in the case project

management The settings of the case project are first pre-sented followed by descriptions of implementation of theCCDSM in the case project The schedule generated with theCCDSMmethod is assessed in detail and compared with theschedules generated with traditional CCPM- and DES-basedmethods

41 Case Project Settings The case used in this study wasderived from a modular real estate development projectfirstly introduced in [35] and further described in [36]The matrix 119860 of the case is shown in Figure 5(a) Thiscase consisted of 19 major activities and 183 interactivityrelationships including 65 rework relationships representedby elements above the diagonal of matrix 119860 It is assumedthat rework can propagate up to twice to avoid infinite loopin the computation The parameters for each activity and theinteractivity relationships were derived based on literatureand empirical evidence and are shown in Figures 5(b) 5(c)5(d) and 5(e) and summarized in Table 2 Large-scale reworkrelationships and complex connections between activitieswere observed in the case project which was representativeof typical construction projects in reality

42 Implementation of the CCDSM Method The proposedCCDSM method was implemented in the case project fol-lowing the steps explained in the last section and the results

were shown below Noticeably for better understandingof scheduling process the implementation process of theCCDSMmethod is reorganized as below

Step 1 Matrices 119863 and 119877 were derived from matrix Aas shown in Figure 6 The duration of each activity waspresented in the diagonal of the matrix D

Step 2 Matrix F (see Figure 7) was derived from matrices119875 and 119878 according to (22) This step calculated the activitydeviation matrix based on the predecessor time matrix andthe successor time matrix

Step 3 Optimize activity sequence and update matrices D RRP RI and 119865 according to new activity sequence To reduceproject rework and obtain near-optimal project completiontime the genetic algorithm (GA) was applied to optimizeactivity sequence The GA is a metaheuristic method thatsearches for optimal solutions using processes similar tothose in natural selections and genetics [37] In the paperminimization of total length of rework path first proposed byGebala and Eppinger [38] was used as the objective functionto calculate the optimal sequence of activities

119891 = 119899sum119894=1

119899sum119895=119894+1

(119895 minus 119894) sdot 119908 (119894 119895) (24)

Mathematical Problems in Engineering 9

Table 2 Activity parameters in the case project

ID Activity Learning curve Duration (day)Min Likely Max

A Perform prelim mkt analysis 05 5 8 15B Evaluate marketability options 06 5 8 15C Engage feasibility consultants 03 3 4 7D Evaluate planning amp zoning process 05 10 13 20E Perform massing study 06 20 24 35F Develop conceptual design 08 30 35 50G Identify external stakeholders 04 5 8 15H Identify permits amp approvals 07 10 13 20I Complete phase 1 ESA 03 10 13 20J Evaluate consultants amp contractors 06 10 13 20K Obtain rough construction costs 05 5 8 15L Determine highest amp best use 05 3 5 9M Identify debt options 04 2 4 8N Identify equity options 05 3 4 5O Update financial underwriting 05 1 2 3P Reevaluate organization strategy 06 2 3 5Q Estimate schedule 06 3 4 7R Gain control of site andor client 06 3 4 7S Review and approve 07 3 4 7

F C A I J R K PF 30C 3A 5I 10J 10R 3K 5P

D M O B H E N

DMOBHEN

203

10

21

10

5

2

L G Q S

LGQS

33

53

(a) Matrix D

F C A I J R K PFCA e eI eJ e eR e e eK e e e eP e e e e e e

D M O B H E N

D e eM eO e e e e eB e e eH e e eE e eN e e

e e e ee e

e

e eee

L G Q S

L e e e e eG e e eQ e e e e e e e eS e e

e e e e e e ee

e e ee e e e e e e

e ee

e

e

e

e

e

(b) Matrix R (element 120576 is omitted)

Figure 6 Matrix D amp R

where 119908(119894 119895) represents rework probability of activity 119894caused by activity 119895

The parameter settings were selected as follows pop-ulation size set as 50 number of generations set as 150crossover probability set as 095 and mutation probabilityset as 008 The GA process reported an optimal schedule as[119875 119876119867 119864 119877 119861 119862119870119873 119871119872 119865119863 119878 119860 119866 119869 119868 119874] The GAconvergence process is shown in Figure 8

Step 4 119877119908119861 was generated from matrices RP and RI basedon (9)-(15) The goal of this step was to factor in andcalculate the rework time of the case project 119877119908119861 was[3 4 6 10 5 1 1 3 1 2 1 15 1 1 1 1 1 1 9]Step 5 To generate reliable project schedules the most likelyduration of each activity which has been widely used andaccepted by project teams in prior research [39] was selected

10 Mathematical Problems in Engineering

A B C D E F G HA 1 10 15 28

B 7 1 13 23 2

C 13 8

D 2 1

E -3 1 3 10 15

F -6 -11 -4 5 -

12G 3 2 6

H -3 4 14

I J K L M N O

I 2 11 16 3 8

J 5 -1 8 18 28 4

K 2 5 2 15 28

L 4 6 1 10 16 27 7

M 13

N 12

O 5 6 2 8 16 35 7 76 4 2 1

8 12 4 4 3 2

7 4 4 1

-19 5 3 29 5 3 2 1

1174-19

-8 -1 -99

-1 -8

8 9

-1

57

P Q R S

3 33 4

1 4

-1 33 2

2

-1

2

P 8 5 2 11 23 22 6 6

Q 7 7 2 14 23 37 4 13

R 7 5 11 17 23 4 10

S

11 4 4 2 3 11

9 10 6 3 4 19 10 4 5 3 2 1

2 32 2 3

2 22

Figure 7 Matrix F (element 120576 is omitted)

20

30

40

50

60

70

80

Estim

ated

Fee

dbac

k Le

ngth

50 100 1500Generation

Figure 8 Convergence curve in the GA-based optimization

to calculate the estimated project duration The critical chainwas reported as [119867 119864 119861 119870119873 119871 119865 119860 119866 119869 119874]Step 6 The activity duration variance matrix SD was gen-erated with 1000 Monte Carlo simulations of the activitiesrsquoduration that follows a beta distribution

Step 7 Buffers were calculated based on (17)-(21) Theproject buffer was 36 days and the feeding buffer was[3 4 0 0 38 0 4 0 0 0 6 0 10 38 0 0 0 0 2 0]Step 8 The estimated project duration was calculated as 102days based on (23) The project buffer was placed at the endof the schedule and the feeding buffer was placed on thenoncritical chains prior to the joints of the critical chain andnoncritical chains following the principle of CCPM

It was worth mentioning that overlapping was observedamong various activities including critical activities in the

above schedule which illustrated real-world high-frequencyinteractions among activities in the project

To assess the performance of the CCDSM method andthe quality of the project schedule it generated the executionof case project was simulated 1000 times using Monte Carloapproach where the sampling of actual activity durationsfollowed beta distribution The simulated actual projectdurations were depicted in Figure 9 It can be seen in thefigure that under rework assumption the process scheduleof 102 days had a completion probability of 981 indicatingthat the proposed CCDSM method was able to providesatisfactory assurance of on-time project completion undervarying scheduling scenarios

43 Comparison of the CCDSM Method with TraditionalCCPM and DES Methods This subsection further comparesthe performance of the CCDSM method with traditional

Mathematical Problems in Engineering 11

75 80 85 90 95 100 105 1100

20

40

60

80

100

120

140

Duration

CCDSM method=102

0

01

02

03

04

05

06

07

08

09

1

981

CCDSM method=102

Cum

ulat

ive P

roba

bilit

y

Freq

uenc

y

80 85 90 95 100 105 11075Duration

Figure 9 Simulated actual project durations and on-time project completion probability of the CCDSM-based schedule

75 80 85 90 95 100 105 1100

01

02

03

04

05

06

07

08

09

1

Duration

981

CCDSM method=102

493

CCPM method=93

Cum

ulat

ive P

roba

bilit

y

Figure 10 Comparison of on-time project completion probabilities between the CCDSM- and CCPM-based schedules

CCPM and DES methods to assess its effectiveness inaddressing rework risks

A CCPM-based schedule was generated with tradi-tional buffers including project buffer and feeding bufferand without rework buffer The results indicated that theestimated project durations of the CCPM-based schedulewere 93 days including project buffers of 36 days Whenthe estimated project duration of CCPM-based schedulewas superimposed in the Monte Carlo-simulated projectdurations as depicted in Figure 10 it can be seen thatthe on-time completion probability was 493 The resultssuggested that the CCDSM-based schedule which took intoconsideration rework relationships and had a duration of 102days significantly outperformed the CCPM-based schedulein ensuring on-time completion of the case project that wasfaced with typical rework risks

Furthermore a DES-based schedule was generated byimplementing DES algorithm in which activity durationsfollowing a beta distribution were generated at random tocompute the project duration When the DES algorithm was

used to generate project schedules under rework scenarios aprobabilistic judgment is performed firstly on whether or noteach activity causes rework and then an iterative simulationprocess began until all activities were finished Specificallythe DES algorithm proposed by Browning and Eppinger[27] was adopted in this study with the following minormodifications to make it more applicable to the case projectand comparable to the CCDSM method (1) the optimizedactivity sequence in Section 42 was employed in the DESprocess (2) activity 119894 did not started until STS logic tie withits each predecessor activity 119895 that is [119865]119894119895 was satisfied (3)rework probability of activity 119894 would decrease by 50 eachtime it reworked [28]

To compare the CCDSM-based schedule with the DES-based schedule the execution of the case project usingthe DES-based schedule was simulated 1000 times usingMonte Carlo approach and the simulated actual projectdurations are depicted in Figure 11 The results showed thatthe estimated project duration of the DES-based schedulewas 115 days which was 13 days longer than that of the

12 Mathematical Problems in Engineering

70 80 90 100 110 120 1300

010203040506070809

1

Duration

981

DES method=115Cu

mul

ativ

e Pro

babi

lity

Figure 11 On-time project completion probability of the DES-based schedule

CCDSM-based schedule with the same on-time completionprobability of 981 In other words the CCDSM methodoutperformed the DES method by ensuring the same on-time completion probability with a shorter project scheduleMoreover the CCDSM method was also advantageous overthe DES-based method in that it enabled effective controlof project schedules during project implementation whichcould be done by monitoring consumption of time buffersleading to reduction in project overrun risks

5 Conclusions

Construction projects are constantly challenged by reworkrisks which have largely remained unaddressed by existingproject schedulingThis paper proposed the CCDSMmethodfor developing project schedules that are adaptive to reworkinstances during project execution This method modelslarge-scale rework relationships among activities with theintroduction of a new rework buffer in traditional CCPMmethod The method allows analysis and visualization ofthe schedules and utilizes max-plus algorithm to transformcomplex logic relationships into simple matrix operationsreducing the computational load of generating process sched-ules A case study was conducted to demonstrate the imple-mentation of the CCDSMmethod and assess its effectivenessin addressing rework risksThe results showed that CCDSM-based schedule outperformed the CCPM-based schedule inensuring on-time completion of the case project that wasfaced with typical rework risks In addition in comparison tothe DESmethod the CCDSM performed better by providinga smaller project duration with the same probability of on-time project completion

The CCDSM method contributes to construction man-agement in the following ways First the CCDSM methodabsorbs risks of rework and various other uncertainties togenerate the reliable project schedule enabling managersto predict project durations more accurately before projectsstart Based on accurate prediction of project durationsthe negotiation of contracts and supply of resources canbe conducted in a more reasonable manner avoiding costsoverruns schedule delays and even project failures Second

based on generated project schedule with time bufferseffectivemonitoring and control of construction progress canbe realized Overall speaking the proposed CCDSMmethodprovides a promising solution to mitigate rework risks inconstruction projects and protect projects from undesirabledelays caused by possible rework instances

Meanwhile the CCDSM bears several limitations thatwould benefit from future improvement First only the STSlogic tie between activities is tested in this study Techniquesfor integrating multiple logic ties including STS STF FTSand FTF deserve further attention Second resource conflictsand levering are considered beyond the scope of this studybut it is a critical challenge that should be addressed inthe context of CCDSM-based scheduling in future researchThird the parameters required to implement the CCDSMmethod are hard to obtain for a specific project To employCCPM or max-plus algorithm for project scheduling it isnecessary to obtain the estimated activity durations undervarious completion rates and the logic tie between activitiesSimilarly the rework relative parameters including RP RIand LC are required for implementation of DSM-basedmethod As an integration method of CCPM max-plus algo-rithm and DSM the implementation of CCDSM requiresall the above parameters to be obtained Lastly methods fordynamic updating of buffer sizes during project execution areneeded in order to better prepare project teams for changingproject environment and dynamic project progress

Data Availability

All data used to support the findings of this study are includedwithin the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (NSFC) [Grant no 71671128]

Mathematical Problems in Engineering 13

References

[1] Y Li and T R Taylor ldquoModeling the impact of design reworkon transportation infrastructure construction project perfor-mancerdquo Journal of Construction Engineering and Managementvol 140 no 9 p 04014044 2014

[2] B Hwang X Zhao and K J Goh ldquoInvestigating the client-related rework in building projects The case of SingaporerdquoInternational Journal of Project Management vol 32 no 4 pp698ndash708 2014

[3] E K Simpeh R Ndihokubwayo P E Love and W DThwala ldquoA rework probability model a quantitative assessmentof rework occurrence in construction projectsrdquo InternationalJournal of Construction Management vol 15 no 2 pp 109ndash1162015

[4] P W Morris J K Pinto and S Jonas 13e Oxford Handbookof Project Management Oxford University Press Oxford UK2012

[5] S D Eppinger and T R Browning Design Structure MatrixMethods and Applications MIT Press Books 2012

[6] T R Browning ldquoDesign structure matrix extensions and inno-vations a survey and new opportunitiesrdquo IEEE Transactions onEngineering Management vol 63 no 1 pp 27ndash52 2016

[7] A Kulkarni D K Yadav and H Nikraz ldquoAircraft maintenancechecks using critical chain project pathrdquo Aircra EngineeringAerospace Technology vol 89 no 6 pp 879ndash892 2017

[8] J Trojanowska and E Dostatni ldquoApplication of the theory ofconstraints for project managementrdquoManagement and Produc-tion Engineering Review vol 8 no 3 pp 87ndash95 2017

[9] M Ghaari and M Emsley ldquoBufer sizing in CCPM portfolioswith dierent resource capacitiesrdquo International Journal of Infor-mation Technology Project Management vol 8 no 3 pp 40ndash512017

[10] E M Goldratt Critical Chain A Business Novel MA NorthRiver Press Great Barrington United States 1997

[11] S Van de Vonder E Demeulemeester and W HerroelenldquoProactive heuristic procedures for robust project schedulingAn experimental analysisrdquo European Journal of OperationalResearch vol 189 no 3 pp 723ndash733 2008

[12] O Lambrechts E Demeulemeester and W Herroelen ldquoTimeslack-based techniques for robust project scheduling subject toresource uncertaintyrdquo Annals of Operations Research vol 186no 1 pp 443ndash464 2011

[13] D Trietsch and K R Baker ldquoPERT 21 Fitting PERTCPMfor use in the 21st centuryrdquo International Journal of ProjectManagement vol 30 no 4 pp 490ndash502 2012

[14] GMaAWangN Li LGu andQAi ldquoImproved critical chainproject management framework for scheduling constructionprojectsrdquo Journal of Construction Engineering andManagementvol 140 no 12 p 04014055 2014

[15] R C Newbold Project Management in 13e Fast Lane Applying13e 13eory of Constraints CRC Press Boca Raton FloridaLondon 1998

[16] O I Tukel W O Rom and S D Eksioglu ldquoAn investigation ofbuffer sizing techniques in critical chain schedulingrdquo EuropeanJournal of Operational Research vol 172 no 2 pp 401ndash4162006

[17] O Icmeli and S S Erenguc ldquoA branch and bound procedurefor the resource constrained project scheduling problem withdiscounted cash flowsrdquoManagement Science vol 42 no 10 pp1395ndash1408 1996

[18] E Roghanian M Alipour and M Rezaei ldquoAn improved fuzzycritical chain approach in order to face uncertainty in projectschedulingrdquo International Journal of ConstructionManagementvol 18 no 1 pp 1ndash13 2017

[19] J Zhang X Song and E Dıaz ldquoProject buffer sizing of a criticalchain based on comprehensive resource tightnessrdquo EuropeanJournal of Operational Research vol 248 no 1 pp 174ndash182 2016

[20] F A Rivera and A Duran ldquoCritical clouds and critical sets inresource-constrained projectsrdquo International Journal of ProjectManagement vol 22 no 6 pp 489ndash497 2004

[21] G Ma L Gu and N Li ldquoScenario-based proactive robustoptimization for critical-chain project schedulingrdquo Journal ofConstruction Engineering and Management vol 141 no 10 p04015030 2015

[22] H Goto ldquoForward-compatible framework with critical-chainproject management using a max-plus linear representationrdquoOPSEARCH vol 54 no 1 pp 201ndash216 2017

[23] T Salama A Salah and O Moselhi ldquoIntegration of linearscheduling method and the critical chain project managementrdquoCanadian Journal of Civil Engineering vol 45 no 1 pp 30ndash402018

[24] C W Hu X D Chen and L H Wu ldquoThe setting method ofproject buffer in critical chain management of mould manufac-turing project based reworkingrdquo Advanced Materials Researchvol 317-319 pp 418ndash422 2011

[25] D V Steward Systems Analysis and Management StructureStrategy and Design IrwinMcGraw-Hill Boston 1981

[26] T R Browning Modeling and analyzing cost schedule andperformance in complex system product development [PhDthesis]Massachusetts Institute of Technology CambridgeMas-sachusetts USA 1998

[27] T R Browning and S D Eppinger ldquoModeling impacts ofprocess architecture on cost and schedule risk in productdevelopmentrdquo IEEE Transactions on Engineering Managementvol 49 no 4 pp 428ndash442 2002

[28] S H Cho and S D Eppinger ldquoA simulation-based processmodel for managing complex design projectsrdquo IEEE Transac-tions on Engineering Management vol 52 no 3 pp 316ndash3282005

[29] V Levardy and T R Browning ldquoAn adaptive process modelto support product development project managementrdquo IEEETransactions on Engineering Management vol 56 no 4 pp600ndash620 2009

[30] J U Maheswari and K Varghese ldquoProject scheduling usingdependency structure matrixrdquo International Journal of ProjectManagement vol 23 no 3 pp 223ndash230 2005

[31] C G Cassandras and S Lafortune Introduction to DiscreteEvent Systems Springer New York NY USA 2nd edition 2008

[32] S Yoshida H Takahashi and H Goto ldquoModified max-pluslinear representation for inserting time buffersrdquo in Proceedingsof the IEEE International Conference on Industrial Engineeringamp Engineering Management IEEE 2010

[33] S Yoshida H Takahashi and H Goto ldquoResolution of timeand worker conflicts for a single project in a max-plus linearrepresentationrdquo Industrial Engineering amp Management Systemsvol 10 no 4 pp 279ndash287 2011

[34] S Muralidharan ldquoFactors affecting rework in construc-tion projectrdquo International Journal of Engineering Sciences ampResearch Technology vol 1 no 5 pp 578ndash584 2016

[35] B B E Bulloch and J Sullivan Application of the DesignStructure Matrix (DSM) to the real estate development process

14 Mathematical Problems in Engineering

[Master thesis] Massachusetts Institute of Technology Cam-bridge Massachusetts USA 2009

[36] S D Eppinger S Bonelli andAM Gonzalez ldquoManaging itera-tions in the modular real estate development processrdquo ReducingRisk in Innovation Proceedings of the 15th International DSMConference Melbourne Australia 29-30 August 2013 pp 37ndash442013

[37] D T Pham and D Karaboga Intelligent Optimisation Tech-niques Genetic Algorithms Tabu Search Simulated AnnealingandNeural Networks Springer Science amp BusinessMedia 2000

[38] D A Gebala and S D Eppinger ldquoMethods for analyzing designproceduresrdquo in Proceedings of the AMSE 13ird InternationalConference On Design13eory andMethodology vol 31 pp 227ndash233 Miami Florida 1991

[39] O Hazır M Haouari and E Erel ldquoRobust scheduling androbustness measures for the discrete timecost trade-off prob-lemrdquo European Journal of Operational Research vol 207 no 2pp 633ndash643 2010

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6 Mathematical Problems in Engineering

rework time

i

j

(a) First rework

j

k

i

rework time

(b) Second rework caused by information tran-sition from its precedent activity

rework time

i

j

k

(c) Second rework caused by information feed-back from its successive activity

Figure 4 First rework and second rework

32 Determining Project Rework Buffer It is widely believedthat information uncertainty is themain cause of rework [34]Such uncertain information may transmit from upstreamactivities to downstream activities or feedback in the oppo-site direction creating information flow Information flowinteractions lead to rework risks in construction projects andconsequently bring about more rework time There are twomain situations that rework exists(1)First rework as depicted in Figure 4(a) i is a precedentactivity of j and they perform sequentially according to rela-tionships After it is completed the performance informationof 119895 will be generated and transmitted to i which may resultin rework of 119894 The first rework time shown as the shadowarea in Figure 4(a) can be calculated by [27]

[1198771198791]119894119895 = 119877119875119894119895 sdot 119877119868119894119895 sdot 119871119862119894 sdot 119863119894119894 (9)

Then the total first rework time of 119894 caused by all itsdownstream activities can be obtained by

[1198771198791]119894 =119899sum119895=119894+1

119877119875119894119895 sdot 119877119868119894119895 sdot 119871119862119894 sdot 119863119894119894 (10)

where 119899 represents the number of activities in the schedule(2) Second rework as depicted in Figures 4(b) and 4(c)after completing the rework of 119895 caused by 119896 119895 transmits orfeeds back some revised information to 119894 which may causerework of 119894 Such two forms are defined as the second reworkin the paper The second rework times shown as the shadowareas in Figures 4(b) and 4(c) can be calculated by

[1198771198792]119894119895119896 = 119877119875119894119895 sdot 119877119875119895119896 sdot 119877119868119894119895 sdot 119871119862119894 sdot 119863119894119894 (11)

Then the total second rework time of 119894 caused by all itsdownstream activities can be obtained by

[1198771198792]119894 =119896minus1sum119895=1

119899sum119896=119894+1

119877119875119895119896 sdot 119877119875119894119895 sdot 119877119868119894119895 sdot 119871119862119894 sdot 119863119894119894 (12)

To proactively account for the uncertainty caused byinteractivity relationships and the associated rework risks arework buffer (119877119908119861) is introduced in the proposed CCDSMConceptually 119877119908119861 assesses and compresses the durationuncertainty caused by rework and is placed after the comple-tion of rework activities in project schedules119877119908119861 is designedto warn project participants from how much workload may

increase for each rework activity so as to prepare the projectparticipants for sufficient time and resources

The rework buffer of 119894 can be obtained by

119877119908119861119894 = [1198771198791]119894 + [1198771198792]119894 (13)

For the last critical activity 120579 in the schedule consideringall reworks it causes other activities to generate will take placeafter its completion andhave an impact onproject completiontime therefore its rework buffer is defined as the sum of thetotal rework time it causes other activities to generate and itstotal rework time caused by other activities

The total rework time it causes all other activities togenerate can be calculated by

[1198771198793]120579 ==119899sum119894=1

119877119875119894120579 sdot 119877119868119894120579 sdot 119871119862119894 sdot 119863119894119894 + 119899sum119894=1

119899sum119895=1

119877119875119894119895sdot 119877119875119895120579 sdot 119877119868119894119895 sdot 119871119862119894 sdot 119863119894119894

(14)

The rework buffer of 120579 then can be obtained by

119877119908119861120579 = [1198771198791]120579 + [1198771198792]120579 + [1198771198793]120579 (15)

33 Determining Project Buffers and Feeding Buffers inMax-Plus Representation

331 Determining Project Buffers Previous studies on theCCPM-max-plus representation adopted the CampPMmethodfor calculating project buffers shown as (16) and feedingbuffers [33]

119875119861 = max [(119863 otimes 119908 otimes 119877)lowast (119863 otimes 119908) 1199090]3 (16)

where (119894 119895)-th element of matrix (119863 otimes 119908 otimes 119877)lowast means thecumulative time of duration of activity 120572 which is on thecritical chain and range from 119894 + 1 to j (119894 119895)-th element ofmatrix119863 otimes119908 satisfies that if activity 119894 is on the critical chainthen (119894 119895)-th element is duration of activity I otherwise it is120576

However the CampPM method lacks sound mathematicalfoundation and overestimates project durations resulting ina waste of time and resources Alternatively the CampPM isreplaced by the RSEM which is based on the large numberlaw and central limit theorem in the CCDSM to calculate thePB

Mathematical Problems in Engineering 7

119875119861 = radic sum119894isin119862119875

119878119863119894 (17)

where 119878119863119894 is the variance of duration of activity 119894 on thecritical path

To calculate the PB in max-plus representation a matrixdenoted as 119879119901 is introduced For the element in row i andcolumn 119894 minus 1 in [119879119901]119894119895 if activity 119894 is critical activity it willbe filled with the variance of duration of activity i else it willbe zero For other elements in [119879119901]119894119895 it is filled with zeroThePB then can be represented in max-plus algebra as

119875119861 = radic1198621198790 otimes 119879lowast119901 otimes 1198620 (18)

where 1198621198790 = (119890 119890 119890)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119899+1

332 Determining Feeding Buffers In order to insert feedingbuffers in place to protect critical path from delays innoncritical paths two operators denoted as diag() and arefirst introduced For a vector 119909 a newmatrix can be obtainedby the operator diag() as [diag()]119894119895 = 119909119894 119894119891 119894 = 119895 120576 119890119897119904119890 and a new vector can be obtained by the operator as[119909]119894 = 120576 119894119891 [119909]119894 = 120576 119890 119894119891 [119909]119894 = 120576

In preparation of the location and size of the feedingbuffer another two vectors are introduced denoted as 119908 andV Matrix119908 is developed based on the rule that for a diagonalelement if the corresponding activity is on the critical paththen the element value is 0 otherwise the element value is 120576Matrix V is developed based on the rule that for a diagonalelement if the corresponding activity is on the noncriticalpath then the value is 0 otherwise the value is 120576 Moreoveran adjacency matrix 119877 is transformed from matrix 119865 by[119877]119894119895 = 119890 119894119891 [119865]119894119895 = 120576 120576 119890119897119904119890 Vector V1015840 is then introducedto locate the feeding buffers as follows

[V1015840]120574= [119877119879120573120572 otimes 119908]120574 (19)

where 119877120573120572 is the adjacency matrix which represents transi-tions from noncritical activities to critical ones and can beobtained by 119877120573120572 = diag(119908) otimes 119877 otimes diag(V) 120582 represents thenoncritical activity one ofwhose successors is critical activityTherefore a feeding buffer should be inserted behind theactivity 120574

The feeding buffer can then be determined by squarerooting of the sum of [119878119863]120578 where 120578 is the set of activitieson a certain noncritical chain The formula is expressed inmax-plus algebra as

1198651198610 = radicdiag [119877120573120573 otimes diag (V1015840) otimes diag (119878119863)]lowast otimes 1198621 otimes diag [diag (119878119863) otimes diag (V1015840) otimes 1198621] otimes 1198621 (20)

Additionally the calculation of feeding buffer should beadjusted using the following equation to meet the constraintthat the size of feeding buffer cannot exceed total float

[119865119861]120574 = min ([1198651198610]120574 [119879119865]120574) (21)

34 Implementation of the CCDSM with Max-Plus Lin-ear Expression The max-plus-based implementation of theCCDSM is summarized in the following seven steps It needsto be noted that as the max-plus algorithm cannot solvenonlinear conversions such as multiplication the input datafor the project rework problem including matricesA P S RPand RI needs to be preprocessed

Step 1 (preprocess119860) Extract all diagonal elements of matrix119860 and form a diagonal matrix D

Step 2 (preprocess 119875 and 119878) Combine matrices 119875 and 119878 togenerate matrix 119865 according to the following equation

[119865]119894119895 = [119875]119894119895 times [119875]119895119895 minus [119878]119894119895 times [119878]119894119894 (22)

Extract all nonnull elements ofmatrix119865 and use them to formmatrix 119877 Then update R by replacing all nonnull elementswith 119890 and all null elements with 120576Step 3 (preprocess RP and RI) Based on (9)-(15) mergematrices RP and RI to generate the vector 119877119908119861 whose 119894-thelement is the corresponding rework buffer of activity i

Step 4 (calculate the length of critical path) Based on (4)-(8)the length of critical pathTT and other parameters includingthe total float time TF and critical path are determined

Step 5 (calculate matrices 119908 and V) According to the criticalpath determined in Step 4matrices119908 and V can be generatedMeanwhile the activity duration variance matrix SD canbe generated based on data collected from the project tocalculate the buffers size

Step 6 (calculate buffer sizes) Based on (17)-(21) projectbuffer PB and feeding buffers FBs are calculated with RSEMand represented in max-plus algebra

Step 7 (generate the schedule) As the principle of CCPMthe feeding buffers should be placed on the noncritical chainsprior to the joints of the critical chain and noncritical chainsand the project buffer should be placed at the end of theschedule to protect the whole project process Then theestimated project time PT can be calculated based on

119875119879 = 119879119879 + 119875119861 + 119877119908119861120579 (23)

4 Case Study

In this section the proposed CCDSM method is imple-mented in a case project to demonstrate its feasibility andeffectiveness in addressing rework risks in project schedule

8 Mathematical Problems in Engineering

A B C D E F G H

A 8 X X X X

B X 8 X X X X

C 4 X X

D X 13 X

E X X X X 24 X

F X X X X 35 X

G X X X X 8

H X X X X

I J K L M N O

I X X X X X

J X X X X X X

K X X X X X

L X X X X X X X

M X

N X

O X X X X X X X X

X X X 4 X

X X X X X X 2

X X 4 X X

13 X

X X 8 X X

13

X X 5 X X X

X

X

X

X

X

13

X X X

X

X X

X X

X

X

X

P Q R S

X X

X X

X X

X X

X X

X X

X

X

P X X X X X X X X

Q X X X X X X X X

R X X X X X X X

S

X X X X X X

X

X X X X X X X

X X X X X X X

X X 4

X X X 4

X 4 X

3 X

(a) Matrix A

A B C D E F G H

A 10 09 07 08

B 09 06 07 06

C 07 09 10

D 07 06

E 08 06 06 08 06

F 09 09 08 08 06

G 06 07 06 06

H 06 06 06 06

I J K L M N O

I 06 09 09 06 10

J 05 09 09 07 07 08

K 06 07 06 06 08

L 06 10 08 07 07 06

M 05

N 06 05

O 07 07 08 06 06 09 09 05

08 08 08 08

06 10 06 08 06 06

09 10 09 09

05

08 06 06 10

07 06 09 06 06

09

08

08

10

07

07 08 06

09 07

09 09

09

08

08

07

P Q R S

10

09

09

10

09

05

06

08

P 09 06 07 08 09 06 07 05

Q 08 09 10 09 10 05 10

R 10 07 07 07 06 06 08

S

10 06 08 06 08 09

09

07 08 09 06 10 05 09

06 08 06 10 09 06 07

07 07

09 08 08

08 07

09

(b) Matrix P

A B C D E F G H

A 05 03 02 02

B 01 04 02 05 05

C 01 02

D 01 03

E 04 02 04

F 04 04 04 04 04

G 03 05 05 04

H 04 03 01 05

I J K L M N O

I 01 01 05 02 04

J 03 04 02

K 05 02 03 01 04

L 02 04 03 05 04 03 02

M 01

N 04

O 04 04 02 03 01 02 04

02 02 04 02

04 05 04 03 01 05

02 04 01 03

03

02 04 01 01

03 01 03 01 01

04

03

03

04

01

04 02 03

04

03 04

03 01

02

05

03

P Q R S

01 02

05

04

01 02

04 02

05

01

P 02 05 05 02 03 02 04

Q 03 04 01 02 02

R 02 03 01 04 02 04 01

S

04 03 02 01 01 02

02

01 04 03 01 01 03 04

01 01 04 02 04 02 02

02 01

02 04 03

01 04

01

(c) Matrix S

A B C D E F G HA 01 02 03

B 04 02 01 01

C 02 01

D 03 04

E 06 03 05 05 04

F 03 02 05 05 03

G 02 05 05 02

H 04 02 06 03

I J K L M N O

I 02 03 03 04 02

J 03 03 04 05 03 05

K 04 05 02 06 03

L 03 05 05 03 04 04 03

M 02

N 03

O 05 03 02 05 04 05 06 0306 03 05 01

03 02 05 06 03 05

03 04 01

0203 05 02 0403 04 04 03 03

03

0103 02 02

02

01 04

02

03

0503

P Q R S

02 0102 02

01 03

0301 0202 03

01

02

P 04 04 06 02 04 04 05 06

Q 03 06 05 03 06 05 04 05

R 03 05 03 03 03 05 05

S

02 06 04 03 04 0404

03 06 04 05 03 02 0404 05 06 05 04 03 03

03 0605 02 04

03 0201

(d) Matrix RP

A B C D E F G H

A 008 001 001 009

B 020 014 016 020 007

C 007 008

D 008 023

E 010 001 009 020 013

F 003 025 013 008 023

G 025 002 003 015

H 018 005 022 001

I J K L M N O

I 013 023 020 011 021

J 002 017 007 025 003 024

K 001 008 012 015 002

L 006 013 016 014 003 023 009

M 001

N 006

O 005 016 011 004 014 019 010 015

021 014 005 003

007 004 019 002 005 016

018 009 013 022

012

004 020 015 022

003 022 013 004 023

013

006

023

017

019

005 023 021

018

023 006

023 017

004

004

011

P Q R S

010 009

010 009

022 009

016 010

014 013

004 018

001

018

P 007 014 014 021 023 012 011 014

Q 03 008 025 001 022 014 002 008

R 012 014 006 021 015 021 018

S

022 001 006 020 008 003

014

022 001 006 010 008 019 001

016 011 021 001 021 001 015

009 025

021 018 019

009 019

014

(e) Matrix RI

Figure 5 Project information in the case project

management The settings of the case project are first pre-sented followed by descriptions of implementation of theCCDSM in the case project The schedule generated with theCCDSMmethod is assessed in detail and compared with theschedules generated with traditional CCPM- and DES-basedmethods

41 Case Project Settings The case used in this study wasderived from a modular real estate development projectfirstly introduced in [35] and further described in [36]The matrix 119860 of the case is shown in Figure 5(a) Thiscase consisted of 19 major activities and 183 interactivityrelationships including 65 rework relationships representedby elements above the diagonal of matrix 119860 It is assumedthat rework can propagate up to twice to avoid infinite loopin the computation The parameters for each activity and theinteractivity relationships were derived based on literatureand empirical evidence and are shown in Figures 5(b) 5(c)5(d) and 5(e) and summarized in Table 2 Large-scale reworkrelationships and complex connections between activitieswere observed in the case project which was representativeof typical construction projects in reality

42 Implementation of the CCDSM Method The proposedCCDSM method was implemented in the case project fol-lowing the steps explained in the last section and the results

were shown below Noticeably for better understandingof scheduling process the implementation process of theCCDSMmethod is reorganized as below

Step 1 Matrices 119863 and 119877 were derived from matrix Aas shown in Figure 6 The duration of each activity waspresented in the diagonal of the matrix D

Step 2 Matrix F (see Figure 7) was derived from matrices119875 and 119878 according to (22) This step calculated the activitydeviation matrix based on the predecessor time matrix andthe successor time matrix

Step 3 Optimize activity sequence and update matrices D RRP RI and 119865 according to new activity sequence To reduceproject rework and obtain near-optimal project completiontime the genetic algorithm (GA) was applied to optimizeactivity sequence The GA is a metaheuristic method thatsearches for optimal solutions using processes similar tothose in natural selections and genetics [37] In the paperminimization of total length of rework path first proposed byGebala and Eppinger [38] was used as the objective functionto calculate the optimal sequence of activities

119891 = 119899sum119894=1

119899sum119895=119894+1

(119895 minus 119894) sdot 119908 (119894 119895) (24)

Mathematical Problems in Engineering 9

Table 2 Activity parameters in the case project

ID Activity Learning curve Duration (day)Min Likely Max

A Perform prelim mkt analysis 05 5 8 15B Evaluate marketability options 06 5 8 15C Engage feasibility consultants 03 3 4 7D Evaluate planning amp zoning process 05 10 13 20E Perform massing study 06 20 24 35F Develop conceptual design 08 30 35 50G Identify external stakeholders 04 5 8 15H Identify permits amp approvals 07 10 13 20I Complete phase 1 ESA 03 10 13 20J Evaluate consultants amp contractors 06 10 13 20K Obtain rough construction costs 05 5 8 15L Determine highest amp best use 05 3 5 9M Identify debt options 04 2 4 8N Identify equity options 05 3 4 5O Update financial underwriting 05 1 2 3P Reevaluate organization strategy 06 2 3 5Q Estimate schedule 06 3 4 7R Gain control of site andor client 06 3 4 7S Review and approve 07 3 4 7

F C A I J R K PF 30C 3A 5I 10J 10R 3K 5P

D M O B H E N

DMOBHEN

203

10

21

10

5

2

L G Q S

LGQS

33

53

(a) Matrix D

F C A I J R K PFCA e eI eJ e eR e e eK e e e eP e e e e e e

D M O B H E N

D e eM eO e e e e eB e e eH e e eE e eN e e

e e e ee e

e

e eee

L G Q S

L e e e e eG e e eQ e e e e e e e eS e e

e e e e e e ee

e e ee e e e e e e

e ee

e

e

e

e

e

(b) Matrix R (element 120576 is omitted)

Figure 6 Matrix D amp R

where 119908(119894 119895) represents rework probability of activity 119894caused by activity 119895

The parameter settings were selected as follows pop-ulation size set as 50 number of generations set as 150crossover probability set as 095 and mutation probabilityset as 008 The GA process reported an optimal schedule as[119875 119876119867 119864 119877 119861 119862119870119873 119871119872 119865119863 119878 119860 119866 119869 119868 119874] The GAconvergence process is shown in Figure 8

Step 4 119877119908119861 was generated from matrices RP and RI basedon (9)-(15) The goal of this step was to factor in andcalculate the rework time of the case project 119877119908119861 was[3 4 6 10 5 1 1 3 1 2 1 15 1 1 1 1 1 1 9]Step 5 To generate reliable project schedules the most likelyduration of each activity which has been widely used andaccepted by project teams in prior research [39] was selected

10 Mathematical Problems in Engineering

A B C D E F G HA 1 10 15 28

B 7 1 13 23 2

C 13 8

D 2 1

E -3 1 3 10 15

F -6 -11 -4 5 -

12G 3 2 6

H -3 4 14

I J K L M N O

I 2 11 16 3 8

J 5 -1 8 18 28 4

K 2 5 2 15 28

L 4 6 1 10 16 27 7

M 13

N 12

O 5 6 2 8 16 35 7 76 4 2 1

8 12 4 4 3 2

7 4 4 1

-19 5 3 29 5 3 2 1

1174-19

-8 -1 -99

-1 -8

8 9

-1

57

P Q R S

3 33 4

1 4

-1 33 2

2

-1

2

P 8 5 2 11 23 22 6 6

Q 7 7 2 14 23 37 4 13

R 7 5 11 17 23 4 10

S

11 4 4 2 3 11

9 10 6 3 4 19 10 4 5 3 2 1

2 32 2 3

2 22

Figure 7 Matrix F (element 120576 is omitted)

20

30

40

50

60

70

80

Estim

ated

Fee

dbac

k Le

ngth

50 100 1500Generation

Figure 8 Convergence curve in the GA-based optimization

to calculate the estimated project duration The critical chainwas reported as [119867 119864 119861 119870119873 119871 119865 119860 119866 119869 119874]Step 6 The activity duration variance matrix SD was gen-erated with 1000 Monte Carlo simulations of the activitiesrsquoduration that follows a beta distribution

Step 7 Buffers were calculated based on (17)-(21) Theproject buffer was 36 days and the feeding buffer was[3 4 0 0 38 0 4 0 0 0 6 0 10 38 0 0 0 0 2 0]Step 8 The estimated project duration was calculated as 102days based on (23) The project buffer was placed at the endof the schedule and the feeding buffer was placed on thenoncritical chains prior to the joints of the critical chain andnoncritical chains following the principle of CCPM

It was worth mentioning that overlapping was observedamong various activities including critical activities in the

above schedule which illustrated real-world high-frequencyinteractions among activities in the project

To assess the performance of the CCDSM method andthe quality of the project schedule it generated the executionof case project was simulated 1000 times using Monte Carloapproach where the sampling of actual activity durationsfollowed beta distribution The simulated actual projectdurations were depicted in Figure 9 It can be seen in thefigure that under rework assumption the process scheduleof 102 days had a completion probability of 981 indicatingthat the proposed CCDSM method was able to providesatisfactory assurance of on-time project completion undervarying scheduling scenarios

43 Comparison of the CCDSM Method with TraditionalCCPM and DES Methods This subsection further comparesthe performance of the CCDSM method with traditional

Mathematical Problems in Engineering 11

75 80 85 90 95 100 105 1100

20

40

60

80

100

120

140

Duration

CCDSM method=102

0

01

02

03

04

05

06

07

08

09

1

981

CCDSM method=102

Cum

ulat

ive P

roba

bilit

y

Freq

uenc

y

80 85 90 95 100 105 11075Duration

Figure 9 Simulated actual project durations and on-time project completion probability of the CCDSM-based schedule

75 80 85 90 95 100 105 1100

01

02

03

04

05

06

07

08

09

1

Duration

981

CCDSM method=102

493

CCPM method=93

Cum

ulat

ive P

roba

bilit

y

Figure 10 Comparison of on-time project completion probabilities between the CCDSM- and CCPM-based schedules

CCPM and DES methods to assess its effectiveness inaddressing rework risks

A CCPM-based schedule was generated with tradi-tional buffers including project buffer and feeding bufferand without rework buffer The results indicated that theestimated project durations of the CCPM-based schedulewere 93 days including project buffers of 36 days Whenthe estimated project duration of CCPM-based schedulewas superimposed in the Monte Carlo-simulated projectdurations as depicted in Figure 10 it can be seen thatthe on-time completion probability was 493 The resultssuggested that the CCDSM-based schedule which took intoconsideration rework relationships and had a duration of 102days significantly outperformed the CCPM-based schedulein ensuring on-time completion of the case project that wasfaced with typical rework risks

Furthermore a DES-based schedule was generated byimplementing DES algorithm in which activity durationsfollowing a beta distribution were generated at random tocompute the project duration When the DES algorithm was

used to generate project schedules under rework scenarios aprobabilistic judgment is performed firstly on whether or noteach activity causes rework and then an iterative simulationprocess began until all activities were finished Specificallythe DES algorithm proposed by Browning and Eppinger[27] was adopted in this study with the following minormodifications to make it more applicable to the case projectand comparable to the CCDSM method (1) the optimizedactivity sequence in Section 42 was employed in the DESprocess (2) activity 119894 did not started until STS logic tie withits each predecessor activity 119895 that is [119865]119894119895 was satisfied (3)rework probability of activity 119894 would decrease by 50 eachtime it reworked [28]

To compare the CCDSM-based schedule with the DES-based schedule the execution of the case project usingthe DES-based schedule was simulated 1000 times usingMonte Carlo approach and the simulated actual projectdurations are depicted in Figure 11 The results showed thatthe estimated project duration of the DES-based schedulewas 115 days which was 13 days longer than that of the

12 Mathematical Problems in Engineering

70 80 90 100 110 120 1300

010203040506070809

1

Duration

981

DES method=115Cu

mul

ativ

e Pro

babi

lity

Figure 11 On-time project completion probability of the DES-based schedule

CCDSM-based schedule with the same on-time completionprobability of 981 In other words the CCDSM methodoutperformed the DES method by ensuring the same on-time completion probability with a shorter project scheduleMoreover the CCDSM method was also advantageous overthe DES-based method in that it enabled effective controlof project schedules during project implementation whichcould be done by monitoring consumption of time buffersleading to reduction in project overrun risks

5 Conclusions

Construction projects are constantly challenged by reworkrisks which have largely remained unaddressed by existingproject schedulingThis paper proposed the CCDSMmethodfor developing project schedules that are adaptive to reworkinstances during project execution This method modelslarge-scale rework relationships among activities with theintroduction of a new rework buffer in traditional CCPMmethod The method allows analysis and visualization ofthe schedules and utilizes max-plus algorithm to transformcomplex logic relationships into simple matrix operationsreducing the computational load of generating process sched-ules A case study was conducted to demonstrate the imple-mentation of the CCDSMmethod and assess its effectivenessin addressing rework risksThe results showed that CCDSM-based schedule outperformed the CCPM-based schedule inensuring on-time completion of the case project that wasfaced with typical rework risks In addition in comparison tothe DESmethod the CCDSM performed better by providinga smaller project duration with the same probability of on-time project completion

The CCDSM method contributes to construction man-agement in the following ways First the CCDSM methodabsorbs risks of rework and various other uncertainties togenerate the reliable project schedule enabling managersto predict project durations more accurately before projectsstart Based on accurate prediction of project durationsthe negotiation of contracts and supply of resources canbe conducted in a more reasonable manner avoiding costsoverruns schedule delays and even project failures Second

based on generated project schedule with time bufferseffectivemonitoring and control of construction progress canbe realized Overall speaking the proposed CCDSMmethodprovides a promising solution to mitigate rework risks inconstruction projects and protect projects from undesirabledelays caused by possible rework instances

Meanwhile the CCDSM bears several limitations thatwould benefit from future improvement First only the STSlogic tie between activities is tested in this study Techniquesfor integrating multiple logic ties including STS STF FTSand FTF deserve further attention Second resource conflictsand levering are considered beyond the scope of this studybut it is a critical challenge that should be addressed inthe context of CCDSM-based scheduling in future researchThird the parameters required to implement the CCDSMmethod are hard to obtain for a specific project To employCCPM or max-plus algorithm for project scheduling it isnecessary to obtain the estimated activity durations undervarious completion rates and the logic tie between activitiesSimilarly the rework relative parameters including RP RIand LC are required for implementation of DSM-basedmethod As an integration method of CCPM max-plus algo-rithm and DSM the implementation of CCDSM requiresall the above parameters to be obtained Lastly methods fordynamic updating of buffer sizes during project execution areneeded in order to better prepare project teams for changingproject environment and dynamic project progress

Data Availability

All data used to support the findings of this study are includedwithin the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (NSFC) [Grant no 71671128]

Mathematical Problems in Engineering 13

References

[1] Y Li and T R Taylor ldquoModeling the impact of design reworkon transportation infrastructure construction project perfor-mancerdquo Journal of Construction Engineering and Managementvol 140 no 9 p 04014044 2014

[2] B Hwang X Zhao and K J Goh ldquoInvestigating the client-related rework in building projects The case of SingaporerdquoInternational Journal of Project Management vol 32 no 4 pp698ndash708 2014

[3] E K Simpeh R Ndihokubwayo P E Love and W DThwala ldquoA rework probability model a quantitative assessmentof rework occurrence in construction projectsrdquo InternationalJournal of Construction Management vol 15 no 2 pp 109ndash1162015

[4] P W Morris J K Pinto and S Jonas 13e Oxford Handbookof Project Management Oxford University Press Oxford UK2012

[5] S D Eppinger and T R Browning Design Structure MatrixMethods and Applications MIT Press Books 2012

[6] T R Browning ldquoDesign structure matrix extensions and inno-vations a survey and new opportunitiesrdquo IEEE Transactions onEngineering Management vol 63 no 1 pp 27ndash52 2016

[7] A Kulkarni D K Yadav and H Nikraz ldquoAircraft maintenancechecks using critical chain project pathrdquo Aircra EngineeringAerospace Technology vol 89 no 6 pp 879ndash892 2017

[8] J Trojanowska and E Dostatni ldquoApplication of the theory ofconstraints for project managementrdquoManagement and Produc-tion Engineering Review vol 8 no 3 pp 87ndash95 2017

[9] M Ghaari and M Emsley ldquoBufer sizing in CCPM portfolioswith dierent resource capacitiesrdquo International Journal of Infor-mation Technology Project Management vol 8 no 3 pp 40ndash512017

[10] E M Goldratt Critical Chain A Business Novel MA NorthRiver Press Great Barrington United States 1997

[11] S Van de Vonder E Demeulemeester and W HerroelenldquoProactive heuristic procedures for robust project schedulingAn experimental analysisrdquo European Journal of OperationalResearch vol 189 no 3 pp 723ndash733 2008

[12] O Lambrechts E Demeulemeester and W Herroelen ldquoTimeslack-based techniques for robust project scheduling subject toresource uncertaintyrdquo Annals of Operations Research vol 186no 1 pp 443ndash464 2011

[13] D Trietsch and K R Baker ldquoPERT 21 Fitting PERTCPMfor use in the 21st centuryrdquo International Journal of ProjectManagement vol 30 no 4 pp 490ndash502 2012

[14] GMaAWangN Li LGu andQAi ldquoImproved critical chainproject management framework for scheduling constructionprojectsrdquo Journal of Construction Engineering andManagementvol 140 no 12 p 04014055 2014

[15] R C Newbold Project Management in 13e Fast Lane Applying13e 13eory of Constraints CRC Press Boca Raton FloridaLondon 1998

[16] O I Tukel W O Rom and S D Eksioglu ldquoAn investigation ofbuffer sizing techniques in critical chain schedulingrdquo EuropeanJournal of Operational Research vol 172 no 2 pp 401ndash4162006

[17] O Icmeli and S S Erenguc ldquoA branch and bound procedurefor the resource constrained project scheduling problem withdiscounted cash flowsrdquoManagement Science vol 42 no 10 pp1395ndash1408 1996

[18] E Roghanian M Alipour and M Rezaei ldquoAn improved fuzzycritical chain approach in order to face uncertainty in projectschedulingrdquo International Journal of ConstructionManagementvol 18 no 1 pp 1ndash13 2017

[19] J Zhang X Song and E Dıaz ldquoProject buffer sizing of a criticalchain based on comprehensive resource tightnessrdquo EuropeanJournal of Operational Research vol 248 no 1 pp 174ndash182 2016

[20] F A Rivera and A Duran ldquoCritical clouds and critical sets inresource-constrained projectsrdquo International Journal of ProjectManagement vol 22 no 6 pp 489ndash497 2004

[21] G Ma L Gu and N Li ldquoScenario-based proactive robustoptimization for critical-chain project schedulingrdquo Journal ofConstruction Engineering and Management vol 141 no 10 p04015030 2015

[22] H Goto ldquoForward-compatible framework with critical-chainproject management using a max-plus linear representationrdquoOPSEARCH vol 54 no 1 pp 201ndash216 2017

[23] T Salama A Salah and O Moselhi ldquoIntegration of linearscheduling method and the critical chain project managementrdquoCanadian Journal of Civil Engineering vol 45 no 1 pp 30ndash402018

[24] C W Hu X D Chen and L H Wu ldquoThe setting method ofproject buffer in critical chain management of mould manufac-turing project based reworkingrdquo Advanced Materials Researchvol 317-319 pp 418ndash422 2011

[25] D V Steward Systems Analysis and Management StructureStrategy and Design IrwinMcGraw-Hill Boston 1981

[26] T R Browning Modeling and analyzing cost schedule andperformance in complex system product development [PhDthesis]Massachusetts Institute of Technology CambridgeMas-sachusetts USA 1998

[27] T R Browning and S D Eppinger ldquoModeling impacts ofprocess architecture on cost and schedule risk in productdevelopmentrdquo IEEE Transactions on Engineering Managementvol 49 no 4 pp 428ndash442 2002

[28] S H Cho and S D Eppinger ldquoA simulation-based processmodel for managing complex design projectsrdquo IEEE Transac-tions on Engineering Management vol 52 no 3 pp 316ndash3282005

[29] V Levardy and T R Browning ldquoAn adaptive process modelto support product development project managementrdquo IEEETransactions on Engineering Management vol 56 no 4 pp600ndash620 2009

[30] J U Maheswari and K Varghese ldquoProject scheduling usingdependency structure matrixrdquo International Journal of ProjectManagement vol 23 no 3 pp 223ndash230 2005

[31] C G Cassandras and S Lafortune Introduction to DiscreteEvent Systems Springer New York NY USA 2nd edition 2008

[32] S Yoshida H Takahashi and H Goto ldquoModified max-pluslinear representation for inserting time buffersrdquo in Proceedingsof the IEEE International Conference on Industrial Engineeringamp Engineering Management IEEE 2010

[33] S Yoshida H Takahashi and H Goto ldquoResolution of timeand worker conflicts for a single project in a max-plus linearrepresentationrdquo Industrial Engineering amp Management Systemsvol 10 no 4 pp 279ndash287 2011

[34] S Muralidharan ldquoFactors affecting rework in construc-tion projectrdquo International Journal of Engineering Sciences ampResearch Technology vol 1 no 5 pp 578ndash584 2016

[35] B B E Bulloch and J Sullivan Application of the DesignStructure Matrix (DSM) to the real estate development process

14 Mathematical Problems in Engineering

[Master thesis] Massachusetts Institute of Technology Cam-bridge Massachusetts USA 2009

[36] S D Eppinger S Bonelli andAM Gonzalez ldquoManaging itera-tions in the modular real estate development processrdquo ReducingRisk in Innovation Proceedings of the 15th International DSMConference Melbourne Australia 29-30 August 2013 pp 37ndash442013

[37] D T Pham and D Karaboga Intelligent Optimisation Tech-niques Genetic Algorithms Tabu Search Simulated AnnealingandNeural Networks Springer Science amp BusinessMedia 2000

[38] D A Gebala and S D Eppinger ldquoMethods for analyzing designproceduresrdquo in Proceedings of the AMSE 13ird InternationalConference On Design13eory andMethodology vol 31 pp 227ndash233 Miami Florida 1991

[39] O Hazır M Haouari and E Erel ldquoRobust scheduling androbustness measures for the discrete timecost trade-off prob-lemrdquo European Journal of Operational Research vol 207 no 2pp 633ndash643 2010

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Mathematical Problems in Engineering 7

119875119861 = radic sum119894isin119862119875

119878119863119894 (17)

where 119878119863119894 is the variance of duration of activity 119894 on thecritical path

To calculate the PB in max-plus representation a matrixdenoted as 119879119901 is introduced For the element in row i andcolumn 119894 minus 1 in [119879119901]119894119895 if activity 119894 is critical activity it willbe filled with the variance of duration of activity i else it willbe zero For other elements in [119879119901]119894119895 it is filled with zeroThePB then can be represented in max-plus algebra as

119875119861 = radic1198621198790 otimes 119879lowast119901 otimes 1198620 (18)

where 1198621198790 = (119890 119890 119890)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟119899+1

332 Determining Feeding Buffers In order to insert feedingbuffers in place to protect critical path from delays innoncritical paths two operators denoted as diag() and arefirst introduced For a vector 119909 a newmatrix can be obtainedby the operator diag() as [diag()]119894119895 = 119909119894 119894119891 119894 = 119895 120576 119890119897119904119890 and a new vector can be obtained by the operator as[119909]119894 = 120576 119894119891 [119909]119894 = 120576 119890 119894119891 [119909]119894 = 120576

In preparation of the location and size of the feedingbuffer another two vectors are introduced denoted as 119908 andV Matrix119908 is developed based on the rule that for a diagonalelement if the corresponding activity is on the critical paththen the element value is 0 otherwise the element value is 120576Matrix V is developed based on the rule that for a diagonalelement if the corresponding activity is on the noncriticalpath then the value is 0 otherwise the value is 120576 Moreoveran adjacency matrix 119877 is transformed from matrix 119865 by[119877]119894119895 = 119890 119894119891 [119865]119894119895 = 120576 120576 119890119897119904119890 Vector V1015840 is then introducedto locate the feeding buffers as follows

[V1015840]120574= [119877119879120573120572 otimes 119908]120574 (19)

where 119877120573120572 is the adjacency matrix which represents transi-tions from noncritical activities to critical ones and can beobtained by 119877120573120572 = diag(119908) otimes 119877 otimes diag(V) 120582 represents thenoncritical activity one ofwhose successors is critical activityTherefore a feeding buffer should be inserted behind theactivity 120574

The feeding buffer can then be determined by squarerooting of the sum of [119878119863]120578 where 120578 is the set of activitieson a certain noncritical chain The formula is expressed inmax-plus algebra as

1198651198610 = radicdiag [119877120573120573 otimes diag (V1015840) otimes diag (119878119863)]lowast otimes 1198621 otimes diag [diag (119878119863) otimes diag (V1015840) otimes 1198621] otimes 1198621 (20)

Additionally the calculation of feeding buffer should beadjusted using the following equation to meet the constraintthat the size of feeding buffer cannot exceed total float

[119865119861]120574 = min ([1198651198610]120574 [119879119865]120574) (21)

34 Implementation of the CCDSM with Max-Plus Lin-ear Expression The max-plus-based implementation of theCCDSM is summarized in the following seven steps It needsto be noted that as the max-plus algorithm cannot solvenonlinear conversions such as multiplication the input datafor the project rework problem including matricesA P S RPand RI needs to be preprocessed

Step 1 (preprocess119860) Extract all diagonal elements of matrix119860 and form a diagonal matrix D

Step 2 (preprocess 119875 and 119878) Combine matrices 119875 and 119878 togenerate matrix 119865 according to the following equation

[119865]119894119895 = [119875]119894119895 times [119875]119895119895 minus [119878]119894119895 times [119878]119894119894 (22)

Extract all nonnull elements ofmatrix119865 and use them to formmatrix 119877 Then update R by replacing all nonnull elementswith 119890 and all null elements with 120576Step 3 (preprocess RP and RI) Based on (9)-(15) mergematrices RP and RI to generate the vector 119877119908119861 whose 119894-thelement is the corresponding rework buffer of activity i

Step 4 (calculate the length of critical path) Based on (4)-(8)the length of critical pathTT and other parameters includingthe total float time TF and critical path are determined

Step 5 (calculate matrices 119908 and V) According to the criticalpath determined in Step 4matrices119908 and V can be generatedMeanwhile the activity duration variance matrix SD canbe generated based on data collected from the project tocalculate the buffers size

Step 6 (calculate buffer sizes) Based on (17)-(21) projectbuffer PB and feeding buffers FBs are calculated with RSEMand represented in max-plus algebra

Step 7 (generate the schedule) As the principle of CCPMthe feeding buffers should be placed on the noncritical chainsprior to the joints of the critical chain and noncritical chainsand the project buffer should be placed at the end of theschedule to protect the whole project process Then theestimated project time PT can be calculated based on

119875119879 = 119879119879 + 119875119861 + 119877119908119861120579 (23)

4 Case Study

In this section the proposed CCDSM method is imple-mented in a case project to demonstrate its feasibility andeffectiveness in addressing rework risks in project schedule

8 Mathematical Problems in Engineering

A B C D E F G H

A 8 X X X X

B X 8 X X X X

C 4 X X

D X 13 X

E X X X X 24 X

F X X X X 35 X

G X X X X 8

H X X X X

I J K L M N O

I X X X X X

J X X X X X X

K X X X X X

L X X X X X X X

M X

N X

O X X X X X X X X

X X X 4 X

X X X X X X 2

X X 4 X X

13 X

X X 8 X X

13

X X 5 X X X

X

X

X

X

X

13

X X X

X

X X

X X

X

X

X

P Q R S

X X

X X

X X

X X

X X

X X

X

X

P X X X X X X X X

Q X X X X X X X X

R X X X X X X X

S

X X X X X X

X

X X X X X X X

X X X X X X X

X X 4

X X X 4

X 4 X

3 X

(a) Matrix A

A B C D E F G H

A 10 09 07 08

B 09 06 07 06

C 07 09 10

D 07 06

E 08 06 06 08 06

F 09 09 08 08 06

G 06 07 06 06

H 06 06 06 06

I J K L M N O

I 06 09 09 06 10

J 05 09 09 07 07 08

K 06 07 06 06 08

L 06 10 08 07 07 06

M 05

N 06 05

O 07 07 08 06 06 09 09 05

08 08 08 08

06 10 06 08 06 06

09 10 09 09

05

08 06 06 10

07 06 09 06 06

09

08

08

10

07

07 08 06

09 07

09 09

09

08

08

07

P Q R S

10

09

09

10

09

05

06

08

P 09 06 07 08 09 06 07 05

Q 08 09 10 09 10 05 10

R 10 07 07 07 06 06 08

S

10 06 08 06 08 09

09

07 08 09 06 10 05 09

06 08 06 10 09 06 07

07 07

09 08 08

08 07

09

(b) Matrix P

A B C D E F G H

A 05 03 02 02

B 01 04 02 05 05

C 01 02

D 01 03

E 04 02 04

F 04 04 04 04 04

G 03 05 05 04

H 04 03 01 05

I J K L M N O

I 01 01 05 02 04

J 03 04 02

K 05 02 03 01 04

L 02 04 03 05 04 03 02

M 01

N 04

O 04 04 02 03 01 02 04

02 02 04 02

04 05 04 03 01 05

02 04 01 03

03

02 04 01 01

03 01 03 01 01

04

03

03

04

01

04 02 03

04

03 04

03 01

02

05

03

P Q R S

01 02

05

04

01 02

04 02

05

01

P 02 05 05 02 03 02 04

Q 03 04 01 02 02

R 02 03 01 04 02 04 01

S

04 03 02 01 01 02

02

01 04 03 01 01 03 04

01 01 04 02 04 02 02

02 01

02 04 03

01 04

01

(c) Matrix S

A B C D E F G HA 01 02 03

B 04 02 01 01

C 02 01

D 03 04

E 06 03 05 05 04

F 03 02 05 05 03

G 02 05 05 02

H 04 02 06 03

I J K L M N O

I 02 03 03 04 02

J 03 03 04 05 03 05

K 04 05 02 06 03

L 03 05 05 03 04 04 03

M 02

N 03

O 05 03 02 05 04 05 06 0306 03 05 01

03 02 05 06 03 05

03 04 01

0203 05 02 0403 04 04 03 03

03

0103 02 02

02

01 04

02

03

0503

P Q R S

02 0102 02

01 03

0301 0202 03

01

02

P 04 04 06 02 04 04 05 06

Q 03 06 05 03 06 05 04 05

R 03 05 03 03 03 05 05

S

02 06 04 03 04 0404

03 06 04 05 03 02 0404 05 06 05 04 03 03

03 0605 02 04

03 0201

(d) Matrix RP

A B C D E F G H

A 008 001 001 009

B 020 014 016 020 007

C 007 008

D 008 023

E 010 001 009 020 013

F 003 025 013 008 023

G 025 002 003 015

H 018 005 022 001

I J K L M N O

I 013 023 020 011 021

J 002 017 007 025 003 024

K 001 008 012 015 002

L 006 013 016 014 003 023 009

M 001

N 006

O 005 016 011 004 014 019 010 015

021 014 005 003

007 004 019 002 005 016

018 009 013 022

012

004 020 015 022

003 022 013 004 023

013

006

023

017

019

005 023 021

018

023 006

023 017

004

004

011

P Q R S

010 009

010 009

022 009

016 010

014 013

004 018

001

018

P 007 014 014 021 023 012 011 014

Q 03 008 025 001 022 014 002 008

R 012 014 006 021 015 021 018

S

022 001 006 020 008 003

014

022 001 006 010 008 019 001

016 011 021 001 021 001 015

009 025

021 018 019

009 019

014

(e) Matrix RI

Figure 5 Project information in the case project

management The settings of the case project are first pre-sented followed by descriptions of implementation of theCCDSM in the case project The schedule generated with theCCDSMmethod is assessed in detail and compared with theschedules generated with traditional CCPM- and DES-basedmethods

41 Case Project Settings The case used in this study wasderived from a modular real estate development projectfirstly introduced in [35] and further described in [36]The matrix 119860 of the case is shown in Figure 5(a) Thiscase consisted of 19 major activities and 183 interactivityrelationships including 65 rework relationships representedby elements above the diagonal of matrix 119860 It is assumedthat rework can propagate up to twice to avoid infinite loopin the computation The parameters for each activity and theinteractivity relationships were derived based on literatureand empirical evidence and are shown in Figures 5(b) 5(c)5(d) and 5(e) and summarized in Table 2 Large-scale reworkrelationships and complex connections between activitieswere observed in the case project which was representativeof typical construction projects in reality

42 Implementation of the CCDSM Method The proposedCCDSM method was implemented in the case project fol-lowing the steps explained in the last section and the results

were shown below Noticeably for better understandingof scheduling process the implementation process of theCCDSMmethod is reorganized as below

Step 1 Matrices 119863 and 119877 were derived from matrix Aas shown in Figure 6 The duration of each activity waspresented in the diagonal of the matrix D

Step 2 Matrix F (see Figure 7) was derived from matrices119875 and 119878 according to (22) This step calculated the activitydeviation matrix based on the predecessor time matrix andthe successor time matrix

Step 3 Optimize activity sequence and update matrices D RRP RI and 119865 according to new activity sequence To reduceproject rework and obtain near-optimal project completiontime the genetic algorithm (GA) was applied to optimizeactivity sequence The GA is a metaheuristic method thatsearches for optimal solutions using processes similar tothose in natural selections and genetics [37] In the paperminimization of total length of rework path first proposed byGebala and Eppinger [38] was used as the objective functionto calculate the optimal sequence of activities

119891 = 119899sum119894=1

119899sum119895=119894+1

(119895 minus 119894) sdot 119908 (119894 119895) (24)

Mathematical Problems in Engineering 9

Table 2 Activity parameters in the case project

ID Activity Learning curve Duration (day)Min Likely Max

A Perform prelim mkt analysis 05 5 8 15B Evaluate marketability options 06 5 8 15C Engage feasibility consultants 03 3 4 7D Evaluate planning amp zoning process 05 10 13 20E Perform massing study 06 20 24 35F Develop conceptual design 08 30 35 50G Identify external stakeholders 04 5 8 15H Identify permits amp approvals 07 10 13 20I Complete phase 1 ESA 03 10 13 20J Evaluate consultants amp contractors 06 10 13 20K Obtain rough construction costs 05 5 8 15L Determine highest amp best use 05 3 5 9M Identify debt options 04 2 4 8N Identify equity options 05 3 4 5O Update financial underwriting 05 1 2 3P Reevaluate organization strategy 06 2 3 5Q Estimate schedule 06 3 4 7R Gain control of site andor client 06 3 4 7S Review and approve 07 3 4 7

F C A I J R K PF 30C 3A 5I 10J 10R 3K 5P

D M O B H E N

DMOBHEN

203

10

21

10

5

2

L G Q S

LGQS

33

53

(a) Matrix D

F C A I J R K PFCA e eI eJ e eR e e eK e e e eP e e e e e e

D M O B H E N

D e eM eO e e e e eB e e eH e e eE e eN e e

e e e ee e

e

e eee

L G Q S

L e e e e eG e e eQ e e e e e e e eS e e

e e e e e e ee

e e ee e e e e e e

e ee

e

e

e

e

e

(b) Matrix R (element 120576 is omitted)

Figure 6 Matrix D amp R

where 119908(119894 119895) represents rework probability of activity 119894caused by activity 119895

The parameter settings were selected as follows pop-ulation size set as 50 number of generations set as 150crossover probability set as 095 and mutation probabilityset as 008 The GA process reported an optimal schedule as[119875 119876119867 119864 119877 119861 119862119870119873 119871119872 119865119863 119878 119860 119866 119869 119868 119874] The GAconvergence process is shown in Figure 8

Step 4 119877119908119861 was generated from matrices RP and RI basedon (9)-(15) The goal of this step was to factor in andcalculate the rework time of the case project 119877119908119861 was[3 4 6 10 5 1 1 3 1 2 1 15 1 1 1 1 1 1 9]Step 5 To generate reliable project schedules the most likelyduration of each activity which has been widely used andaccepted by project teams in prior research [39] was selected

10 Mathematical Problems in Engineering

A B C D E F G HA 1 10 15 28

B 7 1 13 23 2

C 13 8

D 2 1

E -3 1 3 10 15

F -6 -11 -4 5 -

12G 3 2 6

H -3 4 14

I J K L M N O

I 2 11 16 3 8

J 5 -1 8 18 28 4

K 2 5 2 15 28

L 4 6 1 10 16 27 7

M 13

N 12

O 5 6 2 8 16 35 7 76 4 2 1

8 12 4 4 3 2

7 4 4 1

-19 5 3 29 5 3 2 1

1174-19

-8 -1 -99

-1 -8

8 9

-1

57

P Q R S

3 33 4

1 4

-1 33 2

2

-1

2

P 8 5 2 11 23 22 6 6

Q 7 7 2 14 23 37 4 13

R 7 5 11 17 23 4 10

S

11 4 4 2 3 11

9 10 6 3 4 19 10 4 5 3 2 1

2 32 2 3

2 22

Figure 7 Matrix F (element 120576 is omitted)

20

30

40

50

60

70

80

Estim

ated

Fee

dbac

k Le

ngth

50 100 1500Generation

Figure 8 Convergence curve in the GA-based optimization

to calculate the estimated project duration The critical chainwas reported as [119867 119864 119861 119870119873 119871 119865 119860 119866 119869 119874]Step 6 The activity duration variance matrix SD was gen-erated with 1000 Monte Carlo simulations of the activitiesrsquoduration that follows a beta distribution

Step 7 Buffers were calculated based on (17)-(21) Theproject buffer was 36 days and the feeding buffer was[3 4 0 0 38 0 4 0 0 0 6 0 10 38 0 0 0 0 2 0]Step 8 The estimated project duration was calculated as 102days based on (23) The project buffer was placed at the endof the schedule and the feeding buffer was placed on thenoncritical chains prior to the joints of the critical chain andnoncritical chains following the principle of CCPM

It was worth mentioning that overlapping was observedamong various activities including critical activities in the

above schedule which illustrated real-world high-frequencyinteractions among activities in the project

To assess the performance of the CCDSM method andthe quality of the project schedule it generated the executionof case project was simulated 1000 times using Monte Carloapproach where the sampling of actual activity durationsfollowed beta distribution The simulated actual projectdurations were depicted in Figure 9 It can be seen in thefigure that under rework assumption the process scheduleof 102 days had a completion probability of 981 indicatingthat the proposed CCDSM method was able to providesatisfactory assurance of on-time project completion undervarying scheduling scenarios

43 Comparison of the CCDSM Method with TraditionalCCPM and DES Methods This subsection further comparesthe performance of the CCDSM method with traditional

Mathematical Problems in Engineering 11

75 80 85 90 95 100 105 1100

20

40

60

80

100

120

140

Duration

CCDSM method=102

0

01

02

03

04

05

06

07

08

09

1

981

CCDSM method=102

Cum

ulat

ive P

roba

bilit

y

Freq

uenc

y

80 85 90 95 100 105 11075Duration

Figure 9 Simulated actual project durations and on-time project completion probability of the CCDSM-based schedule

75 80 85 90 95 100 105 1100

01

02

03

04

05

06

07

08

09

1

Duration

981

CCDSM method=102

493

CCPM method=93

Cum

ulat

ive P

roba

bilit

y

Figure 10 Comparison of on-time project completion probabilities between the CCDSM- and CCPM-based schedules

CCPM and DES methods to assess its effectiveness inaddressing rework risks

A CCPM-based schedule was generated with tradi-tional buffers including project buffer and feeding bufferand without rework buffer The results indicated that theestimated project durations of the CCPM-based schedulewere 93 days including project buffers of 36 days Whenthe estimated project duration of CCPM-based schedulewas superimposed in the Monte Carlo-simulated projectdurations as depicted in Figure 10 it can be seen thatthe on-time completion probability was 493 The resultssuggested that the CCDSM-based schedule which took intoconsideration rework relationships and had a duration of 102days significantly outperformed the CCPM-based schedulein ensuring on-time completion of the case project that wasfaced with typical rework risks

Furthermore a DES-based schedule was generated byimplementing DES algorithm in which activity durationsfollowing a beta distribution were generated at random tocompute the project duration When the DES algorithm was

used to generate project schedules under rework scenarios aprobabilistic judgment is performed firstly on whether or noteach activity causes rework and then an iterative simulationprocess began until all activities were finished Specificallythe DES algorithm proposed by Browning and Eppinger[27] was adopted in this study with the following minormodifications to make it more applicable to the case projectand comparable to the CCDSM method (1) the optimizedactivity sequence in Section 42 was employed in the DESprocess (2) activity 119894 did not started until STS logic tie withits each predecessor activity 119895 that is [119865]119894119895 was satisfied (3)rework probability of activity 119894 would decrease by 50 eachtime it reworked [28]

To compare the CCDSM-based schedule with the DES-based schedule the execution of the case project usingthe DES-based schedule was simulated 1000 times usingMonte Carlo approach and the simulated actual projectdurations are depicted in Figure 11 The results showed thatthe estimated project duration of the DES-based schedulewas 115 days which was 13 days longer than that of the

12 Mathematical Problems in Engineering

70 80 90 100 110 120 1300

010203040506070809

1

Duration

981

DES method=115Cu

mul

ativ

e Pro

babi

lity

Figure 11 On-time project completion probability of the DES-based schedule

CCDSM-based schedule with the same on-time completionprobability of 981 In other words the CCDSM methodoutperformed the DES method by ensuring the same on-time completion probability with a shorter project scheduleMoreover the CCDSM method was also advantageous overthe DES-based method in that it enabled effective controlof project schedules during project implementation whichcould be done by monitoring consumption of time buffersleading to reduction in project overrun risks

5 Conclusions

Construction projects are constantly challenged by reworkrisks which have largely remained unaddressed by existingproject schedulingThis paper proposed the CCDSMmethodfor developing project schedules that are adaptive to reworkinstances during project execution This method modelslarge-scale rework relationships among activities with theintroduction of a new rework buffer in traditional CCPMmethod The method allows analysis and visualization ofthe schedules and utilizes max-plus algorithm to transformcomplex logic relationships into simple matrix operationsreducing the computational load of generating process sched-ules A case study was conducted to demonstrate the imple-mentation of the CCDSMmethod and assess its effectivenessin addressing rework risksThe results showed that CCDSM-based schedule outperformed the CCPM-based schedule inensuring on-time completion of the case project that wasfaced with typical rework risks In addition in comparison tothe DESmethod the CCDSM performed better by providinga smaller project duration with the same probability of on-time project completion

The CCDSM method contributes to construction man-agement in the following ways First the CCDSM methodabsorbs risks of rework and various other uncertainties togenerate the reliable project schedule enabling managersto predict project durations more accurately before projectsstart Based on accurate prediction of project durationsthe negotiation of contracts and supply of resources canbe conducted in a more reasonable manner avoiding costsoverruns schedule delays and even project failures Second

based on generated project schedule with time bufferseffectivemonitoring and control of construction progress canbe realized Overall speaking the proposed CCDSMmethodprovides a promising solution to mitigate rework risks inconstruction projects and protect projects from undesirabledelays caused by possible rework instances

Meanwhile the CCDSM bears several limitations thatwould benefit from future improvement First only the STSlogic tie between activities is tested in this study Techniquesfor integrating multiple logic ties including STS STF FTSand FTF deserve further attention Second resource conflictsand levering are considered beyond the scope of this studybut it is a critical challenge that should be addressed inthe context of CCDSM-based scheduling in future researchThird the parameters required to implement the CCDSMmethod are hard to obtain for a specific project To employCCPM or max-plus algorithm for project scheduling it isnecessary to obtain the estimated activity durations undervarious completion rates and the logic tie between activitiesSimilarly the rework relative parameters including RP RIand LC are required for implementation of DSM-basedmethod As an integration method of CCPM max-plus algo-rithm and DSM the implementation of CCDSM requiresall the above parameters to be obtained Lastly methods fordynamic updating of buffer sizes during project execution areneeded in order to better prepare project teams for changingproject environment and dynamic project progress

Data Availability

All data used to support the findings of this study are includedwithin the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (NSFC) [Grant no 71671128]

Mathematical Problems in Engineering 13

References

[1] Y Li and T R Taylor ldquoModeling the impact of design reworkon transportation infrastructure construction project perfor-mancerdquo Journal of Construction Engineering and Managementvol 140 no 9 p 04014044 2014

[2] B Hwang X Zhao and K J Goh ldquoInvestigating the client-related rework in building projects The case of SingaporerdquoInternational Journal of Project Management vol 32 no 4 pp698ndash708 2014

[3] E K Simpeh R Ndihokubwayo P E Love and W DThwala ldquoA rework probability model a quantitative assessmentof rework occurrence in construction projectsrdquo InternationalJournal of Construction Management vol 15 no 2 pp 109ndash1162015

[4] P W Morris J K Pinto and S Jonas 13e Oxford Handbookof Project Management Oxford University Press Oxford UK2012

[5] S D Eppinger and T R Browning Design Structure MatrixMethods and Applications MIT Press Books 2012

[6] T R Browning ldquoDesign structure matrix extensions and inno-vations a survey and new opportunitiesrdquo IEEE Transactions onEngineering Management vol 63 no 1 pp 27ndash52 2016

[7] A Kulkarni D K Yadav and H Nikraz ldquoAircraft maintenancechecks using critical chain project pathrdquo Aircra EngineeringAerospace Technology vol 89 no 6 pp 879ndash892 2017

[8] J Trojanowska and E Dostatni ldquoApplication of the theory ofconstraints for project managementrdquoManagement and Produc-tion Engineering Review vol 8 no 3 pp 87ndash95 2017

[9] M Ghaari and M Emsley ldquoBufer sizing in CCPM portfolioswith dierent resource capacitiesrdquo International Journal of Infor-mation Technology Project Management vol 8 no 3 pp 40ndash512017

[10] E M Goldratt Critical Chain A Business Novel MA NorthRiver Press Great Barrington United States 1997

[11] S Van de Vonder E Demeulemeester and W HerroelenldquoProactive heuristic procedures for robust project schedulingAn experimental analysisrdquo European Journal of OperationalResearch vol 189 no 3 pp 723ndash733 2008

[12] O Lambrechts E Demeulemeester and W Herroelen ldquoTimeslack-based techniques for robust project scheduling subject toresource uncertaintyrdquo Annals of Operations Research vol 186no 1 pp 443ndash464 2011

[13] D Trietsch and K R Baker ldquoPERT 21 Fitting PERTCPMfor use in the 21st centuryrdquo International Journal of ProjectManagement vol 30 no 4 pp 490ndash502 2012

[14] GMaAWangN Li LGu andQAi ldquoImproved critical chainproject management framework for scheduling constructionprojectsrdquo Journal of Construction Engineering andManagementvol 140 no 12 p 04014055 2014

[15] R C Newbold Project Management in 13e Fast Lane Applying13e 13eory of Constraints CRC Press Boca Raton FloridaLondon 1998

[16] O I Tukel W O Rom and S D Eksioglu ldquoAn investigation ofbuffer sizing techniques in critical chain schedulingrdquo EuropeanJournal of Operational Research vol 172 no 2 pp 401ndash4162006

[17] O Icmeli and S S Erenguc ldquoA branch and bound procedurefor the resource constrained project scheduling problem withdiscounted cash flowsrdquoManagement Science vol 42 no 10 pp1395ndash1408 1996

[18] E Roghanian M Alipour and M Rezaei ldquoAn improved fuzzycritical chain approach in order to face uncertainty in projectschedulingrdquo International Journal of ConstructionManagementvol 18 no 1 pp 1ndash13 2017

[19] J Zhang X Song and E Dıaz ldquoProject buffer sizing of a criticalchain based on comprehensive resource tightnessrdquo EuropeanJournal of Operational Research vol 248 no 1 pp 174ndash182 2016

[20] F A Rivera and A Duran ldquoCritical clouds and critical sets inresource-constrained projectsrdquo International Journal of ProjectManagement vol 22 no 6 pp 489ndash497 2004

[21] G Ma L Gu and N Li ldquoScenario-based proactive robustoptimization for critical-chain project schedulingrdquo Journal ofConstruction Engineering and Management vol 141 no 10 p04015030 2015

[22] H Goto ldquoForward-compatible framework with critical-chainproject management using a max-plus linear representationrdquoOPSEARCH vol 54 no 1 pp 201ndash216 2017

[23] T Salama A Salah and O Moselhi ldquoIntegration of linearscheduling method and the critical chain project managementrdquoCanadian Journal of Civil Engineering vol 45 no 1 pp 30ndash402018

[24] C W Hu X D Chen and L H Wu ldquoThe setting method ofproject buffer in critical chain management of mould manufac-turing project based reworkingrdquo Advanced Materials Researchvol 317-319 pp 418ndash422 2011

[25] D V Steward Systems Analysis and Management StructureStrategy and Design IrwinMcGraw-Hill Boston 1981

[26] T R Browning Modeling and analyzing cost schedule andperformance in complex system product development [PhDthesis]Massachusetts Institute of Technology CambridgeMas-sachusetts USA 1998

[27] T R Browning and S D Eppinger ldquoModeling impacts ofprocess architecture on cost and schedule risk in productdevelopmentrdquo IEEE Transactions on Engineering Managementvol 49 no 4 pp 428ndash442 2002

[28] S H Cho and S D Eppinger ldquoA simulation-based processmodel for managing complex design projectsrdquo IEEE Transac-tions on Engineering Management vol 52 no 3 pp 316ndash3282005

[29] V Levardy and T R Browning ldquoAn adaptive process modelto support product development project managementrdquo IEEETransactions on Engineering Management vol 56 no 4 pp600ndash620 2009

[30] J U Maheswari and K Varghese ldquoProject scheduling usingdependency structure matrixrdquo International Journal of ProjectManagement vol 23 no 3 pp 223ndash230 2005

[31] C G Cassandras and S Lafortune Introduction to DiscreteEvent Systems Springer New York NY USA 2nd edition 2008

[32] S Yoshida H Takahashi and H Goto ldquoModified max-pluslinear representation for inserting time buffersrdquo in Proceedingsof the IEEE International Conference on Industrial Engineeringamp Engineering Management IEEE 2010

[33] S Yoshida H Takahashi and H Goto ldquoResolution of timeand worker conflicts for a single project in a max-plus linearrepresentationrdquo Industrial Engineering amp Management Systemsvol 10 no 4 pp 279ndash287 2011

[34] S Muralidharan ldquoFactors affecting rework in construc-tion projectrdquo International Journal of Engineering Sciences ampResearch Technology vol 1 no 5 pp 578ndash584 2016

[35] B B E Bulloch and J Sullivan Application of the DesignStructure Matrix (DSM) to the real estate development process

14 Mathematical Problems in Engineering

[Master thesis] Massachusetts Institute of Technology Cam-bridge Massachusetts USA 2009

[36] S D Eppinger S Bonelli andAM Gonzalez ldquoManaging itera-tions in the modular real estate development processrdquo ReducingRisk in Innovation Proceedings of the 15th International DSMConference Melbourne Australia 29-30 August 2013 pp 37ndash442013

[37] D T Pham and D Karaboga Intelligent Optimisation Tech-niques Genetic Algorithms Tabu Search Simulated AnnealingandNeural Networks Springer Science amp BusinessMedia 2000

[38] D A Gebala and S D Eppinger ldquoMethods for analyzing designproceduresrdquo in Proceedings of the AMSE 13ird InternationalConference On Design13eory andMethodology vol 31 pp 227ndash233 Miami Florida 1991

[39] O Hazır M Haouari and E Erel ldquoRobust scheduling androbustness measures for the discrete timecost trade-off prob-lemrdquo European Journal of Operational Research vol 207 no 2pp 633ndash643 2010

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

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OptimizationJournal of

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Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

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Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Mathematical Problems in Engineering

A B C D E F G H

A 8 X X X X

B X 8 X X X X

C 4 X X

D X 13 X

E X X X X 24 X

F X X X X 35 X

G X X X X 8

H X X X X

I J K L M N O

I X X X X X

J X X X X X X

K X X X X X

L X X X X X X X

M X

N X

O X X X X X X X X

X X X 4 X

X X X X X X 2

X X 4 X X

13 X

X X 8 X X

13

X X 5 X X X

X

X

X

X

X

13

X X X

X

X X

X X

X

X

X

P Q R S

X X

X X

X X

X X

X X

X X

X

X

P X X X X X X X X

Q X X X X X X X X

R X X X X X X X

S

X X X X X X

X

X X X X X X X

X X X X X X X

X X 4

X X X 4

X 4 X

3 X

(a) Matrix A

A B C D E F G H

A 10 09 07 08

B 09 06 07 06

C 07 09 10

D 07 06

E 08 06 06 08 06

F 09 09 08 08 06

G 06 07 06 06

H 06 06 06 06

I J K L M N O

I 06 09 09 06 10

J 05 09 09 07 07 08

K 06 07 06 06 08

L 06 10 08 07 07 06

M 05

N 06 05

O 07 07 08 06 06 09 09 05

08 08 08 08

06 10 06 08 06 06

09 10 09 09

05

08 06 06 10

07 06 09 06 06

09

08

08

10

07

07 08 06

09 07

09 09

09

08

08

07

P Q R S

10

09

09

10

09

05

06

08

P 09 06 07 08 09 06 07 05

Q 08 09 10 09 10 05 10

R 10 07 07 07 06 06 08

S

10 06 08 06 08 09

09

07 08 09 06 10 05 09

06 08 06 10 09 06 07

07 07

09 08 08

08 07

09

(b) Matrix P

A B C D E F G H

A 05 03 02 02

B 01 04 02 05 05

C 01 02

D 01 03

E 04 02 04

F 04 04 04 04 04

G 03 05 05 04

H 04 03 01 05

I J K L M N O

I 01 01 05 02 04

J 03 04 02

K 05 02 03 01 04

L 02 04 03 05 04 03 02

M 01

N 04

O 04 04 02 03 01 02 04

02 02 04 02

04 05 04 03 01 05

02 04 01 03

03

02 04 01 01

03 01 03 01 01

04

03

03

04

01

04 02 03

04

03 04

03 01

02

05

03

P Q R S

01 02

05

04

01 02

04 02

05

01

P 02 05 05 02 03 02 04

Q 03 04 01 02 02

R 02 03 01 04 02 04 01

S

04 03 02 01 01 02

02

01 04 03 01 01 03 04

01 01 04 02 04 02 02

02 01

02 04 03

01 04

01

(c) Matrix S

A B C D E F G HA 01 02 03

B 04 02 01 01

C 02 01

D 03 04

E 06 03 05 05 04

F 03 02 05 05 03

G 02 05 05 02

H 04 02 06 03

I J K L M N O

I 02 03 03 04 02

J 03 03 04 05 03 05

K 04 05 02 06 03

L 03 05 05 03 04 04 03

M 02

N 03

O 05 03 02 05 04 05 06 0306 03 05 01

03 02 05 06 03 05

03 04 01

0203 05 02 0403 04 04 03 03

03

0103 02 02

02

01 04

02

03

0503

P Q R S

02 0102 02

01 03

0301 0202 03

01

02

P 04 04 06 02 04 04 05 06

Q 03 06 05 03 06 05 04 05

R 03 05 03 03 03 05 05

S

02 06 04 03 04 0404

03 06 04 05 03 02 0404 05 06 05 04 03 03

03 0605 02 04

03 0201

(d) Matrix RP

A B C D E F G H

A 008 001 001 009

B 020 014 016 020 007

C 007 008

D 008 023

E 010 001 009 020 013

F 003 025 013 008 023

G 025 002 003 015

H 018 005 022 001

I J K L M N O

I 013 023 020 011 021

J 002 017 007 025 003 024

K 001 008 012 015 002

L 006 013 016 014 003 023 009

M 001

N 006

O 005 016 011 004 014 019 010 015

021 014 005 003

007 004 019 002 005 016

018 009 013 022

012

004 020 015 022

003 022 013 004 023

013

006

023

017

019

005 023 021

018

023 006

023 017

004

004

011

P Q R S

010 009

010 009

022 009

016 010

014 013

004 018

001

018

P 007 014 014 021 023 012 011 014

Q 03 008 025 001 022 014 002 008

R 012 014 006 021 015 021 018

S

022 001 006 020 008 003

014

022 001 006 010 008 019 001

016 011 021 001 021 001 015

009 025

021 018 019

009 019

014

(e) Matrix RI

Figure 5 Project information in the case project

management The settings of the case project are first pre-sented followed by descriptions of implementation of theCCDSM in the case project The schedule generated with theCCDSMmethod is assessed in detail and compared with theschedules generated with traditional CCPM- and DES-basedmethods

41 Case Project Settings The case used in this study wasderived from a modular real estate development projectfirstly introduced in [35] and further described in [36]The matrix 119860 of the case is shown in Figure 5(a) Thiscase consisted of 19 major activities and 183 interactivityrelationships including 65 rework relationships representedby elements above the diagonal of matrix 119860 It is assumedthat rework can propagate up to twice to avoid infinite loopin the computation The parameters for each activity and theinteractivity relationships were derived based on literatureand empirical evidence and are shown in Figures 5(b) 5(c)5(d) and 5(e) and summarized in Table 2 Large-scale reworkrelationships and complex connections between activitieswere observed in the case project which was representativeof typical construction projects in reality

42 Implementation of the CCDSM Method The proposedCCDSM method was implemented in the case project fol-lowing the steps explained in the last section and the results

were shown below Noticeably for better understandingof scheduling process the implementation process of theCCDSMmethod is reorganized as below

Step 1 Matrices 119863 and 119877 were derived from matrix Aas shown in Figure 6 The duration of each activity waspresented in the diagonal of the matrix D

Step 2 Matrix F (see Figure 7) was derived from matrices119875 and 119878 according to (22) This step calculated the activitydeviation matrix based on the predecessor time matrix andthe successor time matrix

Step 3 Optimize activity sequence and update matrices D RRP RI and 119865 according to new activity sequence To reduceproject rework and obtain near-optimal project completiontime the genetic algorithm (GA) was applied to optimizeactivity sequence The GA is a metaheuristic method thatsearches for optimal solutions using processes similar tothose in natural selections and genetics [37] In the paperminimization of total length of rework path first proposed byGebala and Eppinger [38] was used as the objective functionto calculate the optimal sequence of activities

119891 = 119899sum119894=1

119899sum119895=119894+1

(119895 minus 119894) sdot 119908 (119894 119895) (24)

Mathematical Problems in Engineering 9

Table 2 Activity parameters in the case project

ID Activity Learning curve Duration (day)Min Likely Max

A Perform prelim mkt analysis 05 5 8 15B Evaluate marketability options 06 5 8 15C Engage feasibility consultants 03 3 4 7D Evaluate planning amp zoning process 05 10 13 20E Perform massing study 06 20 24 35F Develop conceptual design 08 30 35 50G Identify external stakeholders 04 5 8 15H Identify permits amp approvals 07 10 13 20I Complete phase 1 ESA 03 10 13 20J Evaluate consultants amp contractors 06 10 13 20K Obtain rough construction costs 05 5 8 15L Determine highest amp best use 05 3 5 9M Identify debt options 04 2 4 8N Identify equity options 05 3 4 5O Update financial underwriting 05 1 2 3P Reevaluate organization strategy 06 2 3 5Q Estimate schedule 06 3 4 7R Gain control of site andor client 06 3 4 7S Review and approve 07 3 4 7

F C A I J R K PF 30C 3A 5I 10J 10R 3K 5P

D M O B H E N

DMOBHEN

203

10

21

10

5

2

L G Q S

LGQS

33

53

(a) Matrix D

F C A I J R K PFCA e eI eJ e eR e e eK e e e eP e e e e e e

D M O B H E N

D e eM eO e e e e eB e e eH e e eE e eN e e

e e e ee e

e

e eee

L G Q S

L e e e e eG e e eQ e e e e e e e eS e e

e e e e e e ee

e e ee e e e e e e

e ee

e

e

e

e

e

(b) Matrix R (element 120576 is omitted)

Figure 6 Matrix D amp R

where 119908(119894 119895) represents rework probability of activity 119894caused by activity 119895

The parameter settings were selected as follows pop-ulation size set as 50 number of generations set as 150crossover probability set as 095 and mutation probabilityset as 008 The GA process reported an optimal schedule as[119875 119876119867 119864 119877 119861 119862119870119873 119871119872 119865119863 119878 119860 119866 119869 119868 119874] The GAconvergence process is shown in Figure 8

Step 4 119877119908119861 was generated from matrices RP and RI basedon (9)-(15) The goal of this step was to factor in andcalculate the rework time of the case project 119877119908119861 was[3 4 6 10 5 1 1 3 1 2 1 15 1 1 1 1 1 1 9]Step 5 To generate reliable project schedules the most likelyduration of each activity which has been widely used andaccepted by project teams in prior research [39] was selected

10 Mathematical Problems in Engineering

A B C D E F G HA 1 10 15 28

B 7 1 13 23 2

C 13 8

D 2 1

E -3 1 3 10 15

F -6 -11 -4 5 -

12G 3 2 6

H -3 4 14

I J K L M N O

I 2 11 16 3 8

J 5 -1 8 18 28 4

K 2 5 2 15 28

L 4 6 1 10 16 27 7

M 13

N 12

O 5 6 2 8 16 35 7 76 4 2 1

8 12 4 4 3 2

7 4 4 1

-19 5 3 29 5 3 2 1

1174-19

-8 -1 -99

-1 -8

8 9

-1

57

P Q R S

3 33 4

1 4

-1 33 2

2

-1

2

P 8 5 2 11 23 22 6 6

Q 7 7 2 14 23 37 4 13

R 7 5 11 17 23 4 10

S

11 4 4 2 3 11

9 10 6 3 4 19 10 4 5 3 2 1

2 32 2 3

2 22

Figure 7 Matrix F (element 120576 is omitted)

20

30

40

50

60

70

80

Estim

ated

Fee

dbac

k Le

ngth

50 100 1500Generation

Figure 8 Convergence curve in the GA-based optimization

to calculate the estimated project duration The critical chainwas reported as [119867 119864 119861 119870119873 119871 119865 119860 119866 119869 119874]Step 6 The activity duration variance matrix SD was gen-erated with 1000 Monte Carlo simulations of the activitiesrsquoduration that follows a beta distribution

Step 7 Buffers were calculated based on (17)-(21) Theproject buffer was 36 days and the feeding buffer was[3 4 0 0 38 0 4 0 0 0 6 0 10 38 0 0 0 0 2 0]Step 8 The estimated project duration was calculated as 102days based on (23) The project buffer was placed at the endof the schedule and the feeding buffer was placed on thenoncritical chains prior to the joints of the critical chain andnoncritical chains following the principle of CCPM

It was worth mentioning that overlapping was observedamong various activities including critical activities in the

above schedule which illustrated real-world high-frequencyinteractions among activities in the project

To assess the performance of the CCDSM method andthe quality of the project schedule it generated the executionof case project was simulated 1000 times using Monte Carloapproach where the sampling of actual activity durationsfollowed beta distribution The simulated actual projectdurations were depicted in Figure 9 It can be seen in thefigure that under rework assumption the process scheduleof 102 days had a completion probability of 981 indicatingthat the proposed CCDSM method was able to providesatisfactory assurance of on-time project completion undervarying scheduling scenarios

43 Comparison of the CCDSM Method with TraditionalCCPM and DES Methods This subsection further comparesthe performance of the CCDSM method with traditional

Mathematical Problems in Engineering 11

75 80 85 90 95 100 105 1100

20

40

60

80

100

120

140

Duration

CCDSM method=102

0

01

02

03

04

05

06

07

08

09

1

981

CCDSM method=102

Cum

ulat

ive P

roba

bilit

y

Freq

uenc

y

80 85 90 95 100 105 11075Duration

Figure 9 Simulated actual project durations and on-time project completion probability of the CCDSM-based schedule

75 80 85 90 95 100 105 1100

01

02

03

04

05

06

07

08

09

1

Duration

981

CCDSM method=102

493

CCPM method=93

Cum

ulat

ive P

roba

bilit

y

Figure 10 Comparison of on-time project completion probabilities between the CCDSM- and CCPM-based schedules

CCPM and DES methods to assess its effectiveness inaddressing rework risks

A CCPM-based schedule was generated with tradi-tional buffers including project buffer and feeding bufferand without rework buffer The results indicated that theestimated project durations of the CCPM-based schedulewere 93 days including project buffers of 36 days Whenthe estimated project duration of CCPM-based schedulewas superimposed in the Monte Carlo-simulated projectdurations as depicted in Figure 10 it can be seen thatthe on-time completion probability was 493 The resultssuggested that the CCDSM-based schedule which took intoconsideration rework relationships and had a duration of 102days significantly outperformed the CCPM-based schedulein ensuring on-time completion of the case project that wasfaced with typical rework risks

Furthermore a DES-based schedule was generated byimplementing DES algorithm in which activity durationsfollowing a beta distribution were generated at random tocompute the project duration When the DES algorithm was

used to generate project schedules under rework scenarios aprobabilistic judgment is performed firstly on whether or noteach activity causes rework and then an iterative simulationprocess began until all activities were finished Specificallythe DES algorithm proposed by Browning and Eppinger[27] was adopted in this study with the following minormodifications to make it more applicable to the case projectand comparable to the CCDSM method (1) the optimizedactivity sequence in Section 42 was employed in the DESprocess (2) activity 119894 did not started until STS logic tie withits each predecessor activity 119895 that is [119865]119894119895 was satisfied (3)rework probability of activity 119894 would decrease by 50 eachtime it reworked [28]

To compare the CCDSM-based schedule with the DES-based schedule the execution of the case project usingthe DES-based schedule was simulated 1000 times usingMonte Carlo approach and the simulated actual projectdurations are depicted in Figure 11 The results showed thatthe estimated project duration of the DES-based schedulewas 115 days which was 13 days longer than that of the

12 Mathematical Problems in Engineering

70 80 90 100 110 120 1300

010203040506070809

1

Duration

981

DES method=115Cu

mul

ativ

e Pro

babi

lity

Figure 11 On-time project completion probability of the DES-based schedule

CCDSM-based schedule with the same on-time completionprobability of 981 In other words the CCDSM methodoutperformed the DES method by ensuring the same on-time completion probability with a shorter project scheduleMoreover the CCDSM method was also advantageous overthe DES-based method in that it enabled effective controlof project schedules during project implementation whichcould be done by monitoring consumption of time buffersleading to reduction in project overrun risks

5 Conclusions

Construction projects are constantly challenged by reworkrisks which have largely remained unaddressed by existingproject schedulingThis paper proposed the CCDSMmethodfor developing project schedules that are adaptive to reworkinstances during project execution This method modelslarge-scale rework relationships among activities with theintroduction of a new rework buffer in traditional CCPMmethod The method allows analysis and visualization ofthe schedules and utilizes max-plus algorithm to transformcomplex logic relationships into simple matrix operationsreducing the computational load of generating process sched-ules A case study was conducted to demonstrate the imple-mentation of the CCDSMmethod and assess its effectivenessin addressing rework risksThe results showed that CCDSM-based schedule outperformed the CCPM-based schedule inensuring on-time completion of the case project that wasfaced with typical rework risks In addition in comparison tothe DESmethod the CCDSM performed better by providinga smaller project duration with the same probability of on-time project completion

The CCDSM method contributes to construction man-agement in the following ways First the CCDSM methodabsorbs risks of rework and various other uncertainties togenerate the reliable project schedule enabling managersto predict project durations more accurately before projectsstart Based on accurate prediction of project durationsthe negotiation of contracts and supply of resources canbe conducted in a more reasonable manner avoiding costsoverruns schedule delays and even project failures Second

based on generated project schedule with time bufferseffectivemonitoring and control of construction progress canbe realized Overall speaking the proposed CCDSMmethodprovides a promising solution to mitigate rework risks inconstruction projects and protect projects from undesirabledelays caused by possible rework instances

Meanwhile the CCDSM bears several limitations thatwould benefit from future improvement First only the STSlogic tie between activities is tested in this study Techniquesfor integrating multiple logic ties including STS STF FTSand FTF deserve further attention Second resource conflictsand levering are considered beyond the scope of this studybut it is a critical challenge that should be addressed inthe context of CCDSM-based scheduling in future researchThird the parameters required to implement the CCDSMmethod are hard to obtain for a specific project To employCCPM or max-plus algorithm for project scheduling it isnecessary to obtain the estimated activity durations undervarious completion rates and the logic tie between activitiesSimilarly the rework relative parameters including RP RIand LC are required for implementation of DSM-basedmethod As an integration method of CCPM max-plus algo-rithm and DSM the implementation of CCDSM requiresall the above parameters to be obtained Lastly methods fordynamic updating of buffer sizes during project execution areneeded in order to better prepare project teams for changingproject environment and dynamic project progress

Data Availability

All data used to support the findings of this study are includedwithin the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (NSFC) [Grant no 71671128]

Mathematical Problems in Engineering 13

References

[1] Y Li and T R Taylor ldquoModeling the impact of design reworkon transportation infrastructure construction project perfor-mancerdquo Journal of Construction Engineering and Managementvol 140 no 9 p 04014044 2014

[2] B Hwang X Zhao and K J Goh ldquoInvestigating the client-related rework in building projects The case of SingaporerdquoInternational Journal of Project Management vol 32 no 4 pp698ndash708 2014

[3] E K Simpeh R Ndihokubwayo P E Love and W DThwala ldquoA rework probability model a quantitative assessmentof rework occurrence in construction projectsrdquo InternationalJournal of Construction Management vol 15 no 2 pp 109ndash1162015

[4] P W Morris J K Pinto and S Jonas 13e Oxford Handbookof Project Management Oxford University Press Oxford UK2012

[5] S D Eppinger and T R Browning Design Structure MatrixMethods and Applications MIT Press Books 2012

[6] T R Browning ldquoDesign structure matrix extensions and inno-vations a survey and new opportunitiesrdquo IEEE Transactions onEngineering Management vol 63 no 1 pp 27ndash52 2016

[7] A Kulkarni D K Yadav and H Nikraz ldquoAircraft maintenancechecks using critical chain project pathrdquo Aircra EngineeringAerospace Technology vol 89 no 6 pp 879ndash892 2017

[8] J Trojanowska and E Dostatni ldquoApplication of the theory ofconstraints for project managementrdquoManagement and Produc-tion Engineering Review vol 8 no 3 pp 87ndash95 2017

[9] M Ghaari and M Emsley ldquoBufer sizing in CCPM portfolioswith dierent resource capacitiesrdquo International Journal of Infor-mation Technology Project Management vol 8 no 3 pp 40ndash512017

[10] E M Goldratt Critical Chain A Business Novel MA NorthRiver Press Great Barrington United States 1997

[11] S Van de Vonder E Demeulemeester and W HerroelenldquoProactive heuristic procedures for robust project schedulingAn experimental analysisrdquo European Journal of OperationalResearch vol 189 no 3 pp 723ndash733 2008

[12] O Lambrechts E Demeulemeester and W Herroelen ldquoTimeslack-based techniques for robust project scheduling subject toresource uncertaintyrdquo Annals of Operations Research vol 186no 1 pp 443ndash464 2011

[13] D Trietsch and K R Baker ldquoPERT 21 Fitting PERTCPMfor use in the 21st centuryrdquo International Journal of ProjectManagement vol 30 no 4 pp 490ndash502 2012

[14] GMaAWangN Li LGu andQAi ldquoImproved critical chainproject management framework for scheduling constructionprojectsrdquo Journal of Construction Engineering andManagementvol 140 no 12 p 04014055 2014

[15] R C Newbold Project Management in 13e Fast Lane Applying13e 13eory of Constraints CRC Press Boca Raton FloridaLondon 1998

[16] O I Tukel W O Rom and S D Eksioglu ldquoAn investigation ofbuffer sizing techniques in critical chain schedulingrdquo EuropeanJournal of Operational Research vol 172 no 2 pp 401ndash4162006

[17] O Icmeli and S S Erenguc ldquoA branch and bound procedurefor the resource constrained project scheduling problem withdiscounted cash flowsrdquoManagement Science vol 42 no 10 pp1395ndash1408 1996

[18] E Roghanian M Alipour and M Rezaei ldquoAn improved fuzzycritical chain approach in order to face uncertainty in projectschedulingrdquo International Journal of ConstructionManagementvol 18 no 1 pp 1ndash13 2017

[19] J Zhang X Song and E Dıaz ldquoProject buffer sizing of a criticalchain based on comprehensive resource tightnessrdquo EuropeanJournal of Operational Research vol 248 no 1 pp 174ndash182 2016

[20] F A Rivera and A Duran ldquoCritical clouds and critical sets inresource-constrained projectsrdquo International Journal of ProjectManagement vol 22 no 6 pp 489ndash497 2004

[21] G Ma L Gu and N Li ldquoScenario-based proactive robustoptimization for critical-chain project schedulingrdquo Journal ofConstruction Engineering and Management vol 141 no 10 p04015030 2015

[22] H Goto ldquoForward-compatible framework with critical-chainproject management using a max-plus linear representationrdquoOPSEARCH vol 54 no 1 pp 201ndash216 2017

[23] T Salama A Salah and O Moselhi ldquoIntegration of linearscheduling method and the critical chain project managementrdquoCanadian Journal of Civil Engineering vol 45 no 1 pp 30ndash402018

[24] C W Hu X D Chen and L H Wu ldquoThe setting method ofproject buffer in critical chain management of mould manufac-turing project based reworkingrdquo Advanced Materials Researchvol 317-319 pp 418ndash422 2011

[25] D V Steward Systems Analysis and Management StructureStrategy and Design IrwinMcGraw-Hill Boston 1981

[26] T R Browning Modeling and analyzing cost schedule andperformance in complex system product development [PhDthesis]Massachusetts Institute of Technology CambridgeMas-sachusetts USA 1998

[27] T R Browning and S D Eppinger ldquoModeling impacts ofprocess architecture on cost and schedule risk in productdevelopmentrdquo IEEE Transactions on Engineering Managementvol 49 no 4 pp 428ndash442 2002

[28] S H Cho and S D Eppinger ldquoA simulation-based processmodel for managing complex design projectsrdquo IEEE Transac-tions on Engineering Management vol 52 no 3 pp 316ndash3282005

[29] V Levardy and T R Browning ldquoAn adaptive process modelto support product development project managementrdquo IEEETransactions on Engineering Management vol 56 no 4 pp600ndash620 2009

[30] J U Maheswari and K Varghese ldquoProject scheduling usingdependency structure matrixrdquo International Journal of ProjectManagement vol 23 no 3 pp 223ndash230 2005

[31] C G Cassandras and S Lafortune Introduction to DiscreteEvent Systems Springer New York NY USA 2nd edition 2008

[32] S Yoshida H Takahashi and H Goto ldquoModified max-pluslinear representation for inserting time buffersrdquo in Proceedingsof the IEEE International Conference on Industrial Engineeringamp Engineering Management IEEE 2010

[33] S Yoshida H Takahashi and H Goto ldquoResolution of timeand worker conflicts for a single project in a max-plus linearrepresentationrdquo Industrial Engineering amp Management Systemsvol 10 no 4 pp 279ndash287 2011

[34] S Muralidharan ldquoFactors affecting rework in construc-tion projectrdquo International Journal of Engineering Sciences ampResearch Technology vol 1 no 5 pp 578ndash584 2016

[35] B B E Bulloch and J Sullivan Application of the DesignStructure Matrix (DSM) to the real estate development process

14 Mathematical Problems in Engineering

[Master thesis] Massachusetts Institute of Technology Cam-bridge Massachusetts USA 2009

[36] S D Eppinger S Bonelli andAM Gonzalez ldquoManaging itera-tions in the modular real estate development processrdquo ReducingRisk in Innovation Proceedings of the 15th International DSMConference Melbourne Australia 29-30 August 2013 pp 37ndash442013

[37] D T Pham and D Karaboga Intelligent Optimisation Tech-niques Genetic Algorithms Tabu Search Simulated AnnealingandNeural Networks Springer Science amp BusinessMedia 2000

[38] D A Gebala and S D Eppinger ldquoMethods for analyzing designproceduresrdquo in Proceedings of the AMSE 13ird InternationalConference On Design13eory andMethodology vol 31 pp 227ndash233 Miami Florida 1991

[39] O Hazır M Haouari and E Erel ldquoRobust scheduling androbustness measures for the discrete timecost trade-off prob-lemrdquo European Journal of Operational Research vol 207 no 2pp 633ndash643 2010

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 9

Table 2 Activity parameters in the case project

ID Activity Learning curve Duration (day)Min Likely Max

A Perform prelim mkt analysis 05 5 8 15B Evaluate marketability options 06 5 8 15C Engage feasibility consultants 03 3 4 7D Evaluate planning amp zoning process 05 10 13 20E Perform massing study 06 20 24 35F Develop conceptual design 08 30 35 50G Identify external stakeholders 04 5 8 15H Identify permits amp approvals 07 10 13 20I Complete phase 1 ESA 03 10 13 20J Evaluate consultants amp contractors 06 10 13 20K Obtain rough construction costs 05 5 8 15L Determine highest amp best use 05 3 5 9M Identify debt options 04 2 4 8N Identify equity options 05 3 4 5O Update financial underwriting 05 1 2 3P Reevaluate organization strategy 06 2 3 5Q Estimate schedule 06 3 4 7R Gain control of site andor client 06 3 4 7S Review and approve 07 3 4 7

F C A I J R K PF 30C 3A 5I 10J 10R 3K 5P

D M O B H E N

DMOBHEN

203

10

21

10

5

2

L G Q S

LGQS

33

53

(a) Matrix D

F C A I J R K PFCA e eI eJ e eR e e eK e e e eP e e e e e e

D M O B H E N

D e eM eO e e e e eB e e eH e e eE e eN e e

e e e ee e

e

e eee

L G Q S

L e e e e eG e e eQ e e e e e e e eS e e

e e e e e e ee

e e ee e e e e e e

e ee

e

e

e

e

e

(b) Matrix R (element 120576 is omitted)

Figure 6 Matrix D amp R

where 119908(119894 119895) represents rework probability of activity 119894caused by activity 119895

The parameter settings were selected as follows pop-ulation size set as 50 number of generations set as 150crossover probability set as 095 and mutation probabilityset as 008 The GA process reported an optimal schedule as[119875 119876119867 119864 119877 119861 119862119870119873 119871119872 119865119863 119878 119860 119866 119869 119868 119874] The GAconvergence process is shown in Figure 8

Step 4 119877119908119861 was generated from matrices RP and RI basedon (9)-(15) The goal of this step was to factor in andcalculate the rework time of the case project 119877119908119861 was[3 4 6 10 5 1 1 3 1 2 1 15 1 1 1 1 1 1 9]Step 5 To generate reliable project schedules the most likelyduration of each activity which has been widely used andaccepted by project teams in prior research [39] was selected

10 Mathematical Problems in Engineering

A B C D E F G HA 1 10 15 28

B 7 1 13 23 2

C 13 8

D 2 1

E -3 1 3 10 15

F -6 -11 -4 5 -

12G 3 2 6

H -3 4 14

I J K L M N O

I 2 11 16 3 8

J 5 -1 8 18 28 4

K 2 5 2 15 28

L 4 6 1 10 16 27 7

M 13

N 12

O 5 6 2 8 16 35 7 76 4 2 1

8 12 4 4 3 2

7 4 4 1

-19 5 3 29 5 3 2 1

1174-19

-8 -1 -99

-1 -8

8 9

-1

57

P Q R S

3 33 4

1 4

-1 33 2

2

-1

2

P 8 5 2 11 23 22 6 6

Q 7 7 2 14 23 37 4 13

R 7 5 11 17 23 4 10

S

11 4 4 2 3 11

9 10 6 3 4 19 10 4 5 3 2 1

2 32 2 3

2 22

Figure 7 Matrix F (element 120576 is omitted)

20

30

40

50

60

70

80

Estim

ated

Fee

dbac

k Le

ngth

50 100 1500Generation

Figure 8 Convergence curve in the GA-based optimization

to calculate the estimated project duration The critical chainwas reported as [119867 119864 119861 119870119873 119871 119865 119860 119866 119869 119874]Step 6 The activity duration variance matrix SD was gen-erated with 1000 Monte Carlo simulations of the activitiesrsquoduration that follows a beta distribution

Step 7 Buffers were calculated based on (17)-(21) Theproject buffer was 36 days and the feeding buffer was[3 4 0 0 38 0 4 0 0 0 6 0 10 38 0 0 0 0 2 0]Step 8 The estimated project duration was calculated as 102days based on (23) The project buffer was placed at the endof the schedule and the feeding buffer was placed on thenoncritical chains prior to the joints of the critical chain andnoncritical chains following the principle of CCPM

It was worth mentioning that overlapping was observedamong various activities including critical activities in the

above schedule which illustrated real-world high-frequencyinteractions among activities in the project

To assess the performance of the CCDSM method andthe quality of the project schedule it generated the executionof case project was simulated 1000 times using Monte Carloapproach where the sampling of actual activity durationsfollowed beta distribution The simulated actual projectdurations were depicted in Figure 9 It can be seen in thefigure that under rework assumption the process scheduleof 102 days had a completion probability of 981 indicatingthat the proposed CCDSM method was able to providesatisfactory assurance of on-time project completion undervarying scheduling scenarios

43 Comparison of the CCDSM Method with TraditionalCCPM and DES Methods This subsection further comparesthe performance of the CCDSM method with traditional

Mathematical Problems in Engineering 11

75 80 85 90 95 100 105 1100

20

40

60

80

100

120

140

Duration

CCDSM method=102

0

01

02

03

04

05

06

07

08

09

1

981

CCDSM method=102

Cum

ulat

ive P

roba

bilit

y

Freq

uenc

y

80 85 90 95 100 105 11075Duration

Figure 9 Simulated actual project durations and on-time project completion probability of the CCDSM-based schedule

75 80 85 90 95 100 105 1100

01

02

03

04

05

06

07

08

09

1

Duration

981

CCDSM method=102

493

CCPM method=93

Cum

ulat

ive P

roba

bilit

y

Figure 10 Comparison of on-time project completion probabilities between the CCDSM- and CCPM-based schedules

CCPM and DES methods to assess its effectiveness inaddressing rework risks

A CCPM-based schedule was generated with tradi-tional buffers including project buffer and feeding bufferand without rework buffer The results indicated that theestimated project durations of the CCPM-based schedulewere 93 days including project buffers of 36 days Whenthe estimated project duration of CCPM-based schedulewas superimposed in the Monte Carlo-simulated projectdurations as depicted in Figure 10 it can be seen thatthe on-time completion probability was 493 The resultssuggested that the CCDSM-based schedule which took intoconsideration rework relationships and had a duration of 102days significantly outperformed the CCPM-based schedulein ensuring on-time completion of the case project that wasfaced with typical rework risks

Furthermore a DES-based schedule was generated byimplementing DES algorithm in which activity durationsfollowing a beta distribution were generated at random tocompute the project duration When the DES algorithm was

used to generate project schedules under rework scenarios aprobabilistic judgment is performed firstly on whether or noteach activity causes rework and then an iterative simulationprocess began until all activities were finished Specificallythe DES algorithm proposed by Browning and Eppinger[27] was adopted in this study with the following minormodifications to make it more applicable to the case projectand comparable to the CCDSM method (1) the optimizedactivity sequence in Section 42 was employed in the DESprocess (2) activity 119894 did not started until STS logic tie withits each predecessor activity 119895 that is [119865]119894119895 was satisfied (3)rework probability of activity 119894 would decrease by 50 eachtime it reworked [28]

To compare the CCDSM-based schedule with the DES-based schedule the execution of the case project usingthe DES-based schedule was simulated 1000 times usingMonte Carlo approach and the simulated actual projectdurations are depicted in Figure 11 The results showed thatthe estimated project duration of the DES-based schedulewas 115 days which was 13 days longer than that of the

12 Mathematical Problems in Engineering

70 80 90 100 110 120 1300

010203040506070809

1

Duration

981

DES method=115Cu

mul

ativ

e Pro

babi

lity

Figure 11 On-time project completion probability of the DES-based schedule

CCDSM-based schedule with the same on-time completionprobability of 981 In other words the CCDSM methodoutperformed the DES method by ensuring the same on-time completion probability with a shorter project scheduleMoreover the CCDSM method was also advantageous overthe DES-based method in that it enabled effective controlof project schedules during project implementation whichcould be done by monitoring consumption of time buffersleading to reduction in project overrun risks

5 Conclusions

Construction projects are constantly challenged by reworkrisks which have largely remained unaddressed by existingproject schedulingThis paper proposed the CCDSMmethodfor developing project schedules that are adaptive to reworkinstances during project execution This method modelslarge-scale rework relationships among activities with theintroduction of a new rework buffer in traditional CCPMmethod The method allows analysis and visualization ofthe schedules and utilizes max-plus algorithm to transformcomplex logic relationships into simple matrix operationsreducing the computational load of generating process sched-ules A case study was conducted to demonstrate the imple-mentation of the CCDSMmethod and assess its effectivenessin addressing rework risksThe results showed that CCDSM-based schedule outperformed the CCPM-based schedule inensuring on-time completion of the case project that wasfaced with typical rework risks In addition in comparison tothe DESmethod the CCDSM performed better by providinga smaller project duration with the same probability of on-time project completion

The CCDSM method contributes to construction man-agement in the following ways First the CCDSM methodabsorbs risks of rework and various other uncertainties togenerate the reliable project schedule enabling managersto predict project durations more accurately before projectsstart Based on accurate prediction of project durationsthe negotiation of contracts and supply of resources canbe conducted in a more reasonable manner avoiding costsoverruns schedule delays and even project failures Second

based on generated project schedule with time bufferseffectivemonitoring and control of construction progress canbe realized Overall speaking the proposed CCDSMmethodprovides a promising solution to mitigate rework risks inconstruction projects and protect projects from undesirabledelays caused by possible rework instances

Meanwhile the CCDSM bears several limitations thatwould benefit from future improvement First only the STSlogic tie between activities is tested in this study Techniquesfor integrating multiple logic ties including STS STF FTSand FTF deserve further attention Second resource conflictsand levering are considered beyond the scope of this studybut it is a critical challenge that should be addressed inthe context of CCDSM-based scheduling in future researchThird the parameters required to implement the CCDSMmethod are hard to obtain for a specific project To employCCPM or max-plus algorithm for project scheduling it isnecessary to obtain the estimated activity durations undervarious completion rates and the logic tie between activitiesSimilarly the rework relative parameters including RP RIand LC are required for implementation of DSM-basedmethod As an integration method of CCPM max-plus algo-rithm and DSM the implementation of CCDSM requiresall the above parameters to be obtained Lastly methods fordynamic updating of buffer sizes during project execution areneeded in order to better prepare project teams for changingproject environment and dynamic project progress

Data Availability

All data used to support the findings of this study are includedwithin the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (NSFC) [Grant no 71671128]

Mathematical Problems in Engineering 13

References

[1] Y Li and T R Taylor ldquoModeling the impact of design reworkon transportation infrastructure construction project perfor-mancerdquo Journal of Construction Engineering and Managementvol 140 no 9 p 04014044 2014

[2] B Hwang X Zhao and K J Goh ldquoInvestigating the client-related rework in building projects The case of SingaporerdquoInternational Journal of Project Management vol 32 no 4 pp698ndash708 2014

[3] E K Simpeh R Ndihokubwayo P E Love and W DThwala ldquoA rework probability model a quantitative assessmentof rework occurrence in construction projectsrdquo InternationalJournal of Construction Management vol 15 no 2 pp 109ndash1162015

[4] P W Morris J K Pinto and S Jonas 13e Oxford Handbookof Project Management Oxford University Press Oxford UK2012

[5] S D Eppinger and T R Browning Design Structure MatrixMethods and Applications MIT Press Books 2012

[6] T R Browning ldquoDesign structure matrix extensions and inno-vations a survey and new opportunitiesrdquo IEEE Transactions onEngineering Management vol 63 no 1 pp 27ndash52 2016

[7] A Kulkarni D K Yadav and H Nikraz ldquoAircraft maintenancechecks using critical chain project pathrdquo Aircra EngineeringAerospace Technology vol 89 no 6 pp 879ndash892 2017

[8] J Trojanowska and E Dostatni ldquoApplication of the theory ofconstraints for project managementrdquoManagement and Produc-tion Engineering Review vol 8 no 3 pp 87ndash95 2017

[9] M Ghaari and M Emsley ldquoBufer sizing in CCPM portfolioswith dierent resource capacitiesrdquo International Journal of Infor-mation Technology Project Management vol 8 no 3 pp 40ndash512017

[10] E M Goldratt Critical Chain A Business Novel MA NorthRiver Press Great Barrington United States 1997

[11] S Van de Vonder E Demeulemeester and W HerroelenldquoProactive heuristic procedures for robust project schedulingAn experimental analysisrdquo European Journal of OperationalResearch vol 189 no 3 pp 723ndash733 2008

[12] O Lambrechts E Demeulemeester and W Herroelen ldquoTimeslack-based techniques for robust project scheduling subject toresource uncertaintyrdquo Annals of Operations Research vol 186no 1 pp 443ndash464 2011

[13] D Trietsch and K R Baker ldquoPERT 21 Fitting PERTCPMfor use in the 21st centuryrdquo International Journal of ProjectManagement vol 30 no 4 pp 490ndash502 2012

[14] GMaAWangN Li LGu andQAi ldquoImproved critical chainproject management framework for scheduling constructionprojectsrdquo Journal of Construction Engineering andManagementvol 140 no 12 p 04014055 2014

[15] R C Newbold Project Management in 13e Fast Lane Applying13e 13eory of Constraints CRC Press Boca Raton FloridaLondon 1998

[16] O I Tukel W O Rom and S D Eksioglu ldquoAn investigation ofbuffer sizing techniques in critical chain schedulingrdquo EuropeanJournal of Operational Research vol 172 no 2 pp 401ndash4162006

[17] O Icmeli and S S Erenguc ldquoA branch and bound procedurefor the resource constrained project scheduling problem withdiscounted cash flowsrdquoManagement Science vol 42 no 10 pp1395ndash1408 1996

[18] E Roghanian M Alipour and M Rezaei ldquoAn improved fuzzycritical chain approach in order to face uncertainty in projectschedulingrdquo International Journal of ConstructionManagementvol 18 no 1 pp 1ndash13 2017

[19] J Zhang X Song and E Dıaz ldquoProject buffer sizing of a criticalchain based on comprehensive resource tightnessrdquo EuropeanJournal of Operational Research vol 248 no 1 pp 174ndash182 2016

[20] F A Rivera and A Duran ldquoCritical clouds and critical sets inresource-constrained projectsrdquo International Journal of ProjectManagement vol 22 no 6 pp 489ndash497 2004

[21] G Ma L Gu and N Li ldquoScenario-based proactive robustoptimization for critical-chain project schedulingrdquo Journal ofConstruction Engineering and Management vol 141 no 10 p04015030 2015

[22] H Goto ldquoForward-compatible framework with critical-chainproject management using a max-plus linear representationrdquoOPSEARCH vol 54 no 1 pp 201ndash216 2017

[23] T Salama A Salah and O Moselhi ldquoIntegration of linearscheduling method and the critical chain project managementrdquoCanadian Journal of Civil Engineering vol 45 no 1 pp 30ndash402018

[24] C W Hu X D Chen and L H Wu ldquoThe setting method ofproject buffer in critical chain management of mould manufac-turing project based reworkingrdquo Advanced Materials Researchvol 317-319 pp 418ndash422 2011

[25] D V Steward Systems Analysis and Management StructureStrategy and Design IrwinMcGraw-Hill Boston 1981

[26] T R Browning Modeling and analyzing cost schedule andperformance in complex system product development [PhDthesis]Massachusetts Institute of Technology CambridgeMas-sachusetts USA 1998

[27] T R Browning and S D Eppinger ldquoModeling impacts ofprocess architecture on cost and schedule risk in productdevelopmentrdquo IEEE Transactions on Engineering Managementvol 49 no 4 pp 428ndash442 2002

[28] S H Cho and S D Eppinger ldquoA simulation-based processmodel for managing complex design projectsrdquo IEEE Transac-tions on Engineering Management vol 52 no 3 pp 316ndash3282005

[29] V Levardy and T R Browning ldquoAn adaptive process modelto support product development project managementrdquo IEEETransactions on Engineering Management vol 56 no 4 pp600ndash620 2009

[30] J U Maheswari and K Varghese ldquoProject scheduling usingdependency structure matrixrdquo International Journal of ProjectManagement vol 23 no 3 pp 223ndash230 2005

[31] C G Cassandras and S Lafortune Introduction to DiscreteEvent Systems Springer New York NY USA 2nd edition 2008

[32] S Yoshida H Takahashi and H Goto ldquoModified max-pluslinear representation for inserting time buffersrdquo in Proceedingsof the IEEE International Conference on Industrial Engineeringamp Engineering Management IEEE 2010

[33] S Yoshida H Takahashi and H Goto ldquoResolution of timeand worker conflicts for a single project in a max-plus linearrepresentationrdquo Industrial Engineering amp Management Systemsvol 10 no 4 pp 279ndash287 2011

[34] S Muralidharan ldquoFactors affecting rework in construc-tion projectrdquo International Journal of Engineering Sciences ampResearch Technology vol 1 no 5 pp 578ndash584 2016

[35] B B E Bulloch and J Sullivan Application of the DesignStructure Matrix (DSM) to the real estate development process

14 Mathematical Problems in Engineering

[Master thesis] Massachusetts Institute of Technology Cam-bridge Massachusetts USA 2009

[36] S D Eppinger S Bonelli andAM Gonzalez ldquoManaging itera-tions in the modular real estate development processrdquo ReducingRisk in Innovation Proceedings of the 15th International DSMConference Melbourne Australia 29-30 August 2013 pp 37ndash442013

[37] D T Pham and D Karaboga Intelligent Optimisation Tech-niques Genetic Algorithms Tabu Search Simulated AnnealingandNeural Networks Springer Science amp BusinessMedia 2000

[38] D A Gebala and S D Eppinger ldquoMethods for analyzing designproceduresrdquo in Proceedings of the AMSE 13ird InternationalConference On Design13eory andMethodology vol 31 pp 227ndash233 Miami Florida 1991

[39] O Hazır M Haouari and E Erel ldquoRobust scheduling androbustness measures for the discrete timecost trade-off prob-lemrdquo European Journal of Operational Research vol 207 no 2pp 633ndash643 2010

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Mathematical Problems in Engineering

A B C D E F G HA 1 10 15 28

B 7 1 13 23 2

C 13 8

D 2 1

E -3 1 3 10 15

F -6 -11 -4 5 -

12G 3 2 6

H -3 4 14

I J K L M N O

I 2 11 16 3 8

J 5 -1 8 18 28 4

K 2 5 2 15 28

L 4 6 1 10 16 27 7

M 13

N 12

O 5 6 2 8 16 35 7 76 4 2 1

8 12 4 4 3 2

7 4 4 1

-19 5 3 29 5 3 2 1

1174-19

-8 -1 -99

-1 -8

8 9

-1

57

P Q R S

3 33 4

1 4

-1 33 2

2

-1

2

P 8 5 2 11 23 22 6 6

Q 7 7 2 14 23 37 4 13

R 7 5 11 17 23 4 10

S

11 4 4 2 3 11

9 10 6 3 4 19 10 4 5 3 2 1

2 32 2 3

2 22

Figure 7 Matrix F (element 120576 is omitted)

20

30

40

50

60

70

80

Estim

ated

Fee

dbac

k Le

ngth

50 100 1500Generation

Figure 8 Convergence curve in the GA-based optimization

to calculate the estimated project duration The critical chainwas reported as [119867 119864 119861 119870119873 119871 119865 119860 119866 119869 119874]Step 6 The activity duration variance matrix SD was gen-erated with 1000 Monte Carlo simulations of the activitiesrsquoduration that follows a beta distribution

Step 7 Buffers were calculated based on (17)-(21) Theproject buffer was 36 days and the feeding buffer was[3 4 0 0 38 0 4 0 0 0 6 0 10 38 0 0 0 0 2 0]Step 8 The estimated project duration was calculated as 102days based on (23) The project buffer was placed at the endof the schedule and the feeding buffer was placed on thenoncritical chains prior to the joints of the critical chain andnoncritical chains following the principle of CCPM

It was worth mentioning that overlapping was observedamong various activities including critical activities in the

above schedule which illustrated real-world high-frequencyinteractions among activities in the project

To assess the performance of the CCDSM method andthe quality of the project schedule it generated the executionof case project was simulated 1000 times using Monte Carloapproach where the sampling of actual activity durationsfollowed beta distribution The simulated actual projectdurations were depicted in Figure 9 It can be seen in thefigure that under rework assumption the process scheduleof 102 days had a completion probability of 981 indicatingthat the proposed CCDSM method was able to providesatisfactory assurance of on-time project completion undervarying scheduling scenarios

43 Comparison of the CCDSM Method with TraditionalCCPM and DES Methods This subsection further comparesthe performance of the CCDSM method with traditional

Mathematical Problems in Engineering 11

75 80 85 90 95 100 105 1100

20

40

60

80

100

120

140

Duration

CCDSM method=102

0

01

02

03

04

05

06

07

08

09

1

981

CCDSM method=102

Cum

ulat

ive P

roba

bilit

y

Freq

uenc

y

80 85 90 95 100 105 11075Duration

Figure 9 Simulated actual project durations and on-time project completion probability of the CCDSM-based schedule

75 80 85 90 95 100 105 1100

01

02

03

04

05

06

07

08

09

1

Duration

981

CCDSM method=102

493

CCPM method=93

Cum

ulat

ive P

roba

bilit

y

Figure 10 Comparison of on-time project completion probabilities between the CCDSM- and CCPM-based schedules

CCPM and DES methods to assess its effectiveness inaddressing rework risks

A CCPM-based schedule was generated with tradi-tional buffers including project buffer and feeding bufferand without rework buffer The results indicated that theestimated project durations of the CCPM-based schedulewere 93 days including project buffers of 36 days Whenthe estimated project duration of CCPM-based schedulewas superimposed in the Monte Carlo-simulated projectdurations as depicted in Figure 10 it can be seen thatthe on-time completion probability was 493 The resultssuggested that the CCDSM-based schedule which took intoconsideration rework relationships and had a duration of 102days significantly outperformed the CCPM-based schedulein ensuring on-time completion of the case project that wasfaced with typical rework risks

Furthermore a DES-based schedule was generated byimplementing DES algorithm in which activity durationsfollowing a beta distribution were generated at random tocompute the project duration When the DES algorithm was

used to generate project schedules under rework scenarios aprobabilistic judgment is performed firstly on whether or noteach activity causes rework and then an iterative simulationprocess began until all activities were finished Specificallythe DES algorithm proposed by Browning and Eppinger[27] was adopted in this study with the following minormodifications to make it more applicable to the case projectand comparable to the CCDSM method (1) the optimizedactivity sequence in Section 42 was employed in the DESprocess (2) activity 119894 did not started until STS logic tie withits each predecessor activity 119895 that is [119865]119894119895 was satisfied (3)rework probability of activity 119894 would decrease by 50 eachtime it reworked [28]

To compare the CCDSM-based schedule with the DES-based schedule the execution of the case project usingthe DES-based schedule was simulated 1000 times usingMonte Carlo approach and the simulated actual projectdurations are depicted in Figure 11 The results showed thatthe estimated project duration of the DES-based schedulewas 115 days which was 13 days longer than that of the

12 Mathematical Problems in Engineering

70 80 90 100 110 120 1300

010203040506070809

1

Duration

981

DES method=115Cu

mul

ativ

e Pro

babi

lity

Figure 11 On-time project completion probability of the DES-based schedule

CCDSM-based schedule with the same on-time completionprobability of 981 In other words the CCDSM methodoutperformed the DES method by ensuring the same on-time completion probability with a shorter project scheduleMoreover the CCDSM method was also advantageous overthe DES-based method in that it enabled effective controlof project schedules during project implementation whichcould be done by monitoring consumption of time buffersleading to reduction in project overrun risks

5 Conclusions

Construction projects are constantly challenged by reworkrisks which have largely remained unaddressed by existingproject schedulingThis paper proposed the CCDSMmethodfor developing project schedules that are adaptive to reworkinstances during project execution This method modelslarge-scale rework relationships among activities with theintroduction of a new rework buffer in traditional CCPMmethod The method allows analysis and visualization ofthe schedules and utilizes max-plus algorithm to transformcomplex logic relationships into simple matrix operationsreducing the computational load of generating process sched-ules A case study was conducted to demonstrate the imple-mentation of the CCDSMmethod and assess its effectivenessin addressing rework risksThe results showed that CCDSM-based schedule outperformed the CCPM-based schedule inensuring on-time completion of the case project that wasfaced with typical rework risks In addition in comparison tothe DESmethod the CCDSM performed better by providinga smaller project duration with the same probability of on-time project completion

The CCDSM method contributes to construction man-agement in the following ways First the CCDSM methodabsorbs risks of rework and various other uncertainties togenerate the reliable project schedule enabling managersto predict project durations more accurately before projectsstart Based on accurate prediction of project durationsthe negotiation of contracts and supply of resources canbe conducted in a more reasonable manner avoiding costsoverruns schedule delays and even project failures Second

based on generated project schedule with time bufferseffectivemonitoring and control of construction progress canbe realized Overall speaking the proposed CCDSMmethodprovides a promising solution to mitigate rework risks inconstruction projects and protect projects from undesirabledelays caused by possible rework instances

Meanwhile the CCDSM bears several limitations thatwould benefit from future improvement First only the STSlogic tie between activities is tested in this study Techniquesfor integrating multiple logic ties including STS STF FTSand FTF deserve further attention Second resource conflictsand levering are considered beyond the scope of this studybut it is a critical challenge that should be addressed inthe context of CCDSM-based scheduling in future researchThird the parameters required to implement the CCDSMmethod are hard to obtain for a specific project To employCCPM or max-plus algorithm for project scheduling it isnecessary to obtain the estimated activity durations undervarious completion rates and the logic tie between activitiesSimilarly the rework relative parameters including RP RIand LC are required for implementation of DSM-basedmethod As an integration method of CCPM max-plus algo-rithm and DSM the implementation of CCDSM requiresall the above parameters to be obtained Lastly methods fordynamic updating of buffer sizes during project execution areneeded in order to better prepare project teams for changingproject environment and dynamic project progress

Data Availability

All data used to support the findings of this study are includedwithin the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (NSFC) [Grant no 71671128]

Mathematical Problems in Engineering 13

References

[1] Y Li and T R Taylor ldquoModeling the impact of design reworkon transportation infrastructure construction project perfor-mancerdquo Journal of Construction Engineering and Managementvol 140 no 9 p 04014044 2014

[2] B Hwang X Zhao and K J Goh ldquoInvestigating the client-related rework in building projects The case of SingaporerdquoInternational Journal of Project Management vol 32 no 4 pp698ndash708 2014

[3] E K Simpeh R Ndihokubwayo P E Love and W DThwala ldquoA rework probability model a quantitative assessmentof rework occurrence in construction projectsrdquo InternationalJournal of Construction Management vol 15 no 2 pp 109ndash1162015

[4] P W Morris J K Pinto and S Jonas 13e Oxford Handbookof Project Management Oxford University Press Oxford UK2012

[5] S D Eppinger and T R Browning Design Structure MatrixMethods and Applications MIT Press Books 2012

[6] T R Browning ldquoDesign structure matrix extensions and inno-vations a survey and new opportunitiesrdquo IEEE Transactions onEngineering Management vol 63 no 1 pp 27ndash52 2016

[7] A Kulkarni D K Yadav and H Nikraz ldquoAircraft maintenancechecks using critical chain project pathrdquo Aircra EngineeringAerospace Technology vol 89 no 6 pp 879ndash892 2017

[8] J Trojanowska and E Dostatni ldquoApplication of the theory ofconstraints for project managementrdquoManagement and Produc-tion Engineering Review vol 8 no 3 pp 87ndash95 2017

[9] M Ghaari and M Emsley ldquoBufer sizing in CCPM portfolioswith dierent resource capacitiesrdquo International Journal of Infor-mation Technology Project Management vol 8 no 3 pp 40ndash512017

[10] E M Goldratt Critical Chain A Business Novel MA NorthRiver Press Great Barrington United States 1997

[11] S Van de Vonder E Demeulemeester and W HerroelenldquoProactive heuristic procedures for robust project schedulingAn experimental analysisrdquo European Journal of OperationalResearch vol 189 no 3 pp 723ndash733 2008

[12] O Lambrechts E Demeulemeester and W Herroelen ldquoTimeslack-based techniques for robust project scheduling subject toresource uncertaintyrdquo Annals of Operations Research vol 186no 1 pp 443ndash464 2011

[13] D Trietsch and K R Baker ldquoPERT 21 Fitting PERTCPMfor use in the 21st centuryrdquo International Journal of ProjectManagement vol 30 no 4 pp 490ndash502 2012

[14] GMaAWangN Li LGu andQAi ldquoImproved critical chainproject management framework for scheduling constructionprojectsrdquo Journal of Construction Engineering andManagementvol 140 no 12 p 04014055 2014

[15] R C Newbold Project Management in 13e Fast Lane Applying13e 13eory of Constraints CRC Press Boca Raton FloridaLondon 1998

[16] O I Tukel W O Rom and S D Eksioglu ldquoAn investigation ofbuffer sizing techniques in critical chain schedulingrdquo EuropeanJournal of Operational Research vol 172 no 2 pp 401ndash4162006

[17] O Icmeli and S S Erenguc ldquoA branch and bound procedurefor the resource constrained project scheduling problem withdiscounted cash flowsrdquoManagement Science vol 42 no 10 pp1395ndash1408 1996

[18] E Roghanian M Alipour and M Rezaei ldquoAn improved fuzzycritical chain approach in order to face uncertainty in projectschedulingrdquo International Journal of ConstructionManagementvol 18 no 1 pp 1ndash13 2017

[19] J Zhang X Song and E Dıaz ldquoProject buffer sizing of a criticalchain based on comprehensive resource tightnessrdquo EuropeanJournal of Operational Research vol 248 no 1 pp 174ndash182 2016

[20] F A Rivera and A Duran ldquoCritical clouds and critical sets inresource-constrained projectsrdquo International Journal of ProjectManagement vol 22 no 6 pp 489ndash497 2004

[21] G Ma L Gu and N Li ldquoScenario-based proactive robustoptimization for critical-chain project schedulingrdquo Journal ofConstruction Engineering and Management vol 141 no 10 p04015030 2015

[22] H Goto ldquoForward-compatible framework with critical-chainproject management using a max-plus linear representationrdquoOPSEARCH vol 54 no 1 pp 201ndash216 2017

[23] T Salama A Salah and O Moselhi ldquoIntegration of linearscheduling method and the critical chain project managementrdquoCanadian Journal of Civil Engineering vol 45 no 1 pp 30ndash402018

[24] C W Hu X D Chen and L H Wu ldquoThe setting method ofproject buffer in critical chain management of mould manufac-turing project based reworkingrdquo Advanced Materials Researchvol 317-319 pp 418ndash422 2011

[25] D V Steward Systems Analysis and Management StructureStrategy and Design IrwinMcGraw-Hill Boston 1981

[26] T R Browning Modeling and analyzing cost schedule andperformance in complex system product development [PhDthesis]Massachusetts Institute of Technology CambridgeMas-sachusetts USA 1998

[27] T R Browning and S D Eppinger ldquoModeling impacts ofprocess architecture on cost and schedule risk in productdevelopmentrdquo IEEE Transactions on Engineering Managementvol 49 no 4 pp 428ndash442 2002

[28] S H Cho and S D Eppinger ldquoA simulation-based processmodel for managing complex design projectsrdquo IEEE Transac-tions on Engineering Management vol 52 no 3 pp 316ndash3282005

[29] V Levardy and T R Browning ldquoAn adaptive process modelto support product development project managementrdquo IEEETransactions on Engineering Management vol 56 no 4 pp600ndash620 2009

[30] J U Maheswari and K Varghese ldquoProject scheduling usingdependency structure matrixrdquo International Journal of ProjectManagement vol 23 no 3 pp 223ndash230 2005

[31] C G Cassandras and S Lafortune Introduction to DiscreteEvent Systems Springer New York NY USA 2nd edition 2008

[32] S Yoshida H Takahashi and H Goto ldquoModified max-pluslinear representation for inserting time buffersrdquo in Proceedingsof the IEEE International Conference on Industrial Engineeringamp Engineering Management IEEE 2010

[33] S Yoshida H Takahashi and H Goto ldquoResolution of timeand worker conflicts for a single project in a max-plus linearrepresentationrdquo Industrial Engineering amp Management Systemsvol 10 no 4 pp 279ndash287 2011

[34] S Muralidharan ldquoFactors affecting rework in construc-tion projectrdquo International Journal of Engineering Sciences ampResearch Technology vol 1 no 5 pp 578ndash584 2016

[35] B B E Bulloch and J Sullivan Application of the DesignStructure Matrix (DSM) to the real estate development process

14 Mathematical Problems in Engineering

[Master thesis] Massachusetts Institute of Technology Cam-bridge Massachusetts USA 2009

[36] S D Eppinger S Bonelli andAM Gonzalez ldquoManaging itera-tions in the modular real estate development processrdquo ReducingRisk in Innovation Proceedings of the 15th International DSMConference Melbourne Australia 29-30 August 2013 pp 37ndash442013

[37] D T Pham and D Karaboga Intelligent Optimisation Tech-niques Genetic Algorithms Tabu Search Simulated AnnealingandNeural Networks Springer Science amp BusinessMedia 2000

[38] D A Gebala and S D Eppinger ldquoMethods for analyzing designproceduresrdquo in Proceedings of the AMSE 13ird InternationalConference On Design13eory andMethodology vol 31 pp 227ndash233 Miami Florida 1991

[39] O Hazır M Haouari and E Erel ldquoRobust scheduling androbustness measures for the discrete timecost trade-off prob-lemrdquo European Journal of Operational Research vol 207 no 2pp 633ndash643 2010

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 11

75 80 85 90 95 100 105 1100

20

40

60

80

100

120

140

Duration

CCDSM method=102

0

01

02

03

04

05

06

07

08

09

1

981

CCDSM method=102

Cum

ulat

ive P

roba

bilit

y

Freq

uenc

y

80 85 90 95 100 105 11075Duration

Figure 9 Simulated actual project durations and on-time project completion probability of the CCDSM-based schedule

75 80 85 90 95 100 105 1100

01

02

03

04

05

06

07

08

09

1

Duration

981

CCDSM method=102

493

CCPM method=93

Cum

ulat

ive P

roba

bilit

y

Figure 10 Comparison of on-time project completion probabilities between the CCDSM- and CCPM-based schedules

CCPM and DES methods to assess its effectiveness inaddressing rework risks

A CCPM-based schedule was generated with tradi-tional buffers including project buffer and feeding bufferand without rework buffer The results indicated that theestimated project durations of the CCPM-based schedulewere 93 days including project buffers of 36 days Whenthe estimated project duration of CCPM-based schedulewas superimposed in the Monte Carlo-simulated projectdurations as depicted in Figure 10 it can be seen thatthe on-time completion probability was 493 The resultssuggested that the CCDSM-based schedule which took intoconsideration rework relationships and had a duration of 102days significantly outperformed the CCPM-based schedulein ensuring on-time completion of the case project that wasfaced with typical rework risks

Furthermore a DES-based schedule was generated byimplementing DES algorithm in which activity durationsfollowing a beta distribution were generated at random tocompute the project duration When the DES algorithm was

used to generate project schedules under rework scenarios aprobabilistic judgment is performed firstly on whether or noteach activity causes rework and then an iterative simulationprocess began until all activities were finished Specificallythe DES algorithm proposed by Browning and Eppinger[27] was adopted in this study with the following minormodifications to make it more applicable to the case projectand comparable to the CCDSM method (1) the optimizedactivity sequence in Section 42 was employed in the DESprocess (2) activity 119894 did not started until STS logic tie withits each predecessor activity 119895 that is [119865]119894119895 was satisfied (3)rework probability of activity 119894 would decrease by 50 eachtime it reworked [28]

To compare the CCDSM-based schedule with the DES-based schedule the execution of the case project usingthe DES-based schedule was simulated 1000 times usingMonte Carlo approach and the simulated actual projectdurations are depicted in Figure 11 The results showed thatthe estimated project duration of the DES-based schedulewas 115 days which was 13 days longer than that of the

12 Mathematical Problems in Engineering

70 80 90 100 110 120 1300

010203040506070809

1

Duration

981

DES method=115Cu

mul

ativ

e Pro

babi

lity

Figure 11 On-time project completion probability of the DES-based schedule

CCDSM-based schedule with the same on-time completionprobability of 981 In other words the CCDSM methodoutperformed the DES method by ensuring the same on-time completion probability with a shorter project scheduleMoreover the CCDSM method was also advantageous overthe DES-based method in that it enabled effective controlof project schedules during project implementation whichcould be done by monitoring consumption of time buffersleading to reduction in project overrun risks

5 Conclusions

Construction projects are constantly challenged by reworkrisks which have largely remained unaddressed by existingproject schedulingThis paper proposed the CCDSMmethodfor developing project schedules that are adaptive to reworkinstances during project execution This method modelslarge-scale rework relationships among activities with theintroduction of a new rework buffer in traditional CCPMmethod The method allows analysis and visualization ofthe schedules and utilizes max-plus algorithm to transformcomplex logic relationships into simple matrix operationsreducing the computational load of generating process sched-ules A case study was conducted to demonstrate the imple-mentation of the CCDSMmethod and assess its effectivenessin addressing rework risksThe results showed that CCDSM-based schedule outperformed the CCPM-based schedule inensuring on-time completion of the case project that wasfaced with typical rework risks In addition in comparison tothe DESmethod the CCDSM performed better by providinga smaller project duration with the same probability of on-time project completion

The CCDSM method contributes to construction man-agement in the following ways First the CCDSM methodabsorbs risks of rework and various other uncertainties togenerate the reliable project schedule enabling managersto predict project durations more accurately before projectsstart Based on accurate prediction of project durationsthe negotiation of contracts and supply of resources canbe conducted in a more reasonable manner avoiding costsoverruns schedule delays and even project failures Second

based on generated project schedule with time bufferseffectivemonitoring and control of construction progress canbe realized Overall speaking the proposed CCDSMmethodprovides a promising solution to mitigate rework risks inconstruction projects and protect projects from undesirabledelays caused by possible rework instances

Meanwhile the CCDSM bears several limitations thatwould benefit from future improvement First only the STSlogic tie between activities is tested in this study Techniquesfor integrating multiple logic ties including STS STF FTSand FTF deserve further attention Second resource conflictsand levering are considered beyond the scope of this studybut it is a critical challenge that should be addressed inthe context of CCDSM-based scheduling in future researchThird the parameters required to implement the CCDSMmethod are hard to obtain for a specific project To employCCPM or max-plus algorithm for project scheduling it isnecessary to obtain the estimated activity durations undervarious completion rates and the logic tie between activitiesSimilarly the rework relative parameters including RP RIand LC are required for implementation of DSM-basedmethod As an integration method of CCPM max-plus algo-rithm and DSM the implementation of CCDSM requiresall the above parameters to be obtained Lastly methods fordynamic updating of buffer sizes during project execution areneeded in order to better prepare project teams for changingproject environment and dynamic project progress

Data Availability

All data used to support the findings of this study are includedwithin the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (NSFC) [Grant no 71671128]

Mathematical Problems in Engineering 13

References

[1] Y Li and T R Taylor ldquoModeling the impact of design reworkon transportation infrastructure construction project perfor-mancerdquo Journal of Construction Engineering and Managementvol 140 no 9 p 04014044 2014

[2] B Hwang X Zhao and K J Goh ldquoInvestigating the client-related rework in building projects The case of SingaporerdquoInternational Journal of Project Management vol 32 no 4 pp698ndash708 2014

[3] E K Simpeh R Ndihokubwayo P E Love and W DThwala ldquoA rework probability model a quantitative assessmentof rework occurrence in construction projectsrdquo InternationalJournal of Construction Management vol 15 no 2 pp 109ndash1162015

[4] P W Morris J K Pinto and S Jonas 13e Oxford Handbookof Project Management Oxford University Press Oxford UK2012

[5] S D Eppinger and T R Browning Design Structure MatrixMethods and Applications MIT Press Books 2012

[6] T R Browning ldquoDesign structure matrix extensions and inno-vations a survey and new opportunitiesrdquo IEEE Transactions onEngineering Management vol 63 no 1 pp 27ndash52 2016

[7] A Kulkarni D K Yadav and H Nikraz ldquoAircraft maintenancechecks using critical chain project pathrdquo Aircra EngineeringAerospace Technology vol 89 no 6 pp 879ndash892 2017

[8] J Trojanowska and E Dostatni ldquoApplication of the theory ofconstraints for project managementrdquoManagement and Produc-tion Engineering Review vol 8 no 3 pp 87ndash95 2017

[9] M Ghaari and M Emsley ldquoBufer sizing in CCPM portfolioswith dierent resource capacitiesrdquo International Journal of Infor-mation Technology Project Management vol 8 no 3 pp 40ndash512017

[10] E M Goldratt Critical Chain A Business Novel MA NorthRiver Press Great Barrington United States 1997

[11] S Van de Vonder E Demeulemeester and W HerroelenldquoProactive heuristic procedures for robust project schedulingAn experimental analysisrdquo European Journal of OperationalResearch vol 189 no 3 pp 723ndash733 2008

[12] O Lambrechts E Demeulemeester and W Herroelen ldquoTimeslack-based techniques for robust project scheduling subject toresource uncertaintyrdquo Annals of Operations Research vol 186no 1 pp 443ndash464 2011

[13] D Trietsch and K R Baker ldquoPERT 21 Fitting PERTCPMfor use in the 21st centuryrdquo International Journal of ProjectManagement vol 30 no 4 pp 490ndash502 2012

[14] GMaAWangN Li LGu andQAi ldquoImproved critical chainproject management framework for scheduling constructionprojectsrdquo Journal of Construction Engineering andManagementvol 140 no 12 p 04014055 2014

[15] R C Newbold Project Management in 13e Fast Lane Applying13e 13eory of Constraints CRC Press Boca Raton FloridaLondon 1998

[16] O I Tukel W O Rom and S D Eksioglu ldquoAn investigation ofbuffer sizing techniques in critical chain schedulingrdquo EuropeanJournal of Operational Research vol 172 no 2 pp 401ndash4162006

[17] O Icmeli and S S Erenguc ldquoA branch and bound procedurefor the resource constrained project scheduling problem withdiscounted cash flowsrdquoManagement Science vol 42 no 10 pp1395ndash1408 1996

[18] E Roghanian M Alipour and M Rezaei ldquoAn improved fuzzycritical chain approach in order to face uncertainty in projectschedulingrdquo International Journal of ConstructionManagementvol 18 no 1 pp 1ndash13 2017

[19] J Zhang X Song and E Dıaz ldquoProject buffer sizing of a criticalchain based on comprehensive resource tightnessrdquo EuropeanJournal of Operational Research vol 248 no 1 pp 174ndash182 2016

[20] F A Rivera and A Duran ldquoCritical clouds and critical sets inresource-constrained projectsrdquo International Journal of ProjectManagement vol 22 no 6 pp 489ndash497 2004

[21] G Ma L Gu and N Li ldquoScenario-based proactive robustoptimization for critical-chain project schedulingrdquo Journal ofConstruction Engineering and Management vol 141 no 10 p04015030 2015

[22] H Goto ldquoForward-compatible framework with critical-chainproject management using a max-plus linear representationrdquoOPSEARCH vol 54 no 1 pp 201ndash216 2017

[23] T Salama A Salah and O Moselhi ldquoIntegration of linearscheduling method and the critical chain project managementrdquoCanadian Journal of Civil Engineering vol 45 no 1 pp 30ndash402018

[24] C W Hu X D Chen and L H Wu ldquoThe setting method ofproject buffer in critical chain management of mould manufac-turing project based reworkingrdquo Advanced Materials Researchvol 317-319 pp 418ndash422 2011

[25] D V Steward Systems Analysis and Management StructureStrategy and Design IrwinMcGraw-Hill Boston 1981

[26] T R Browning Modeling and analyzing cost schedule andperformance in complex system product development [PhDthesis]Massachusetts Institute of Technology CambridgeMas-sachusetts USA 1998

[27] T R Browning and S D Eppinger ldquoModeling impacts ofprocess architecture on cost and schedule risk in productdevelopmentrdquo IEEE Transactions on Engineering Managementvol 49 no 4 pp 428ndash442 2002

[28] S H Cho and S D Eppinger ldquoA simulation-based processmodel for managing complex design projectsrdquo IEEE Transac-tions on Engineering Management vol 52 no 3 pp 316ndash3282005

[29] V Levardy and T R Browning ldquoAn adaptive process modelto support product development project managementrdquo IEEETransactions on Engineering Management vol 56 no 4 pp600ndash620 2009

[30] J U Maheswari and K Varghese ldquoProject scheduling usingdependency structure matrixrdquo International Journal of ProjectManagement vol 23 no 3 pp 223ndash230 2005

[31] C G Cassandras and S Lafortune Introduction to DiscreteEvent Systems Springer New York NY USA 2nd edition 2008

[32] S Yoshida H Takahashi and H Goto ldquoModified max-pluslinear representation for inserting time buffersrdquo in Proceedingsof the IEEE International Conference on Industrial Engineeringamp Engineering Management IEEE 2010

[33] S Yoshida H Takahashi and H Goto ldquoResolution of timeand worker conflicts for a single project in a max-plus linearrepresentationrdquo Industrial Engineering amp Management Systemsvol 10 no 4 pp 279ndash287 2011

[34] S Muralidharan ldquoFactors affecting rework in construc-tion projectrdquo International Journal of Engineering Sciences ampResearch Technology vol 1 no 5 pp 578ndash584 2016

[35] B B E Bulloch and J Sullivan Application of the DesignStructure Matrix (DSM) to the real estate development process

14 Mathematical Problems in Engineering

[Master thesis] Massachusetts Institute of Technology Cam-bridge Massachusetts USA 2009

[36] S D Eppinger S Bonelli andAM Gonzalez ldquoManaging itera-tions in the modular real estate development processrdquo ReducingRisk in Innovation Proceedings of the 15th International DSMConference Melbourne Australia 29-30 August 2013 pp 37ndash442013

[37] D T Pham and D Karaboga Intelligent Optimisation Tech-niques Genetic Algorithms Tabu Search Simulated AnnealingandNeural Networks Springer Science amp BusinessMedia 2000

[38] D A Gebala and S D Eppinger ldquoMethods for analyzing designproceduresrdquo in Proceedings of the AMSE 13ird InternationalConference On Design13eory andMethodology vol 31 pp 227ndash233 Miami Florida 1991

[39] O Hazır M Haouari and E Erel ldquoRobust scheduling androbustness measures for the discrete timecost trade-off prob-lemrdquo European Journal of Operational Research vol 207 no 2pp 633ndash643 2010

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

12 Mathematical Problems in Engineering

70 80 90 100 110 120 1300

010203040506070809

1

Duration

981

DES method=115Cu

mul

ativ

e Pro

babi

lity

Figure 11 On-time project completion probability of the DES-based schedule

CCDSM-based schedule with the same on-time completionprobability of 981 In other words the CCDSM methodoutperformed the DES method by ensuring the same on-time completion probability with a shorter project scheduleMoreover the CCDSM method was also advantageous overthe DES-based method in that it enabled effective controlof project schedules during project implementation whichcould be done by monitoring consumption of time buffersleading to reduction in project overrun risks

5 Conclusions

Construction projects are constantly challenged by reworkrisks which have largely remained unaddressed by existingproject schedulingThis paper proposed the CCDSMmethodfor developing project schedules that are adaptive to reworkinstances during project execution This method modelslarge-scale rework relationships among activities with theintroduction of a new rework buffer in traditional CCPMmethod The method allows analysis and visualization ofthe schedules and utilizes max-plus algorithm to transformcomplex logic relationships into simple matrix operationsreducing the computational load of generating process sched-ules A case study was conducted to demonstrate the imple-mentation of the CCDSMmethod and assess its effectivenessin addressing rework risksThe results showed that CCDSM-based schedule outperformed the CCPM-based schedule inensuring on-time completion of the case project that wasfaced with typical rework risks In addition in comparison tothe DESmethod the CCDSM performed better by providinga smaller project duration with the same probability of on-time project completion

The CCDSM method contributes to construction man-agement in the following ways First the CCDSM methodabsorbs risks of rework and various other uncertainties togenerate the reliable project schedule enabling managersto predict project durations more accurately before projectsstart Based on accurate prediction of project durationsthe negotiation of contracts and supply of resources canbe conducted in a more reasonable manner avoiding costsoverruns schedule delays and even project failures Second

based on generated project schedule with time bufferseffectivemonitoring and control of construction progress canbe realized Overall speaking the proposed CCDSMmethodprovides a promising solution to mitigate rework risks inconstruction projects and protect projects from undesirabledelays caused by possible rework instances

Meanwhile the CCDSM bears several limitations thatwould benefit from future improvement First only the STSlogic tie between activities is tested in this study Techniquesfor integrating multiple logic ties including STS STF FTSand FTF deserve further attention Second resource conflictsand levering are considered beyond the scope of this studybut it is a critical challenge that should be addressed inthe context of CCDSM-based scheduling in future researchThird the parameters required to implement the CCDSMmethod are hard to obtain for a specific project To employCCPM or max-plus algorithm for project scheduling it isnecessary to obtain the estimated activity durations undervarious completion rates and the logic tie between activitiesSimilarly the rework relative parameters including RP RIand LC are required for implementation of DSM-basedmethod As an integration method of CCPM max-plus algo-rithm and DSM the implementation of CCDSM requiresall the above parameters to be obtained Lastly methods fordynamic updating of buffer sizes during project execution areneeded in order to better prepare project teams for changingproject environment and dynamic project progress

Data Availability

All data used to support the findings of this study are includedwithin the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (NSFC) [Grant no 71671128]

Mathematical Problems in Engineering 13

References

[1] Y Li and T R Taylor ldquoModeling the impact of design reworkon transportation infrastructure construction project perfor-mancerdquo Journal of Construction Engineering and Managementvol 140 no 9 p 04014044 2014

[2] B Hwang X Zhao and K J Goh ldquoInvestigating the client-related rework in building projects The case of SingaporerdquoInternational Journal of Project Management vol 32 no 4 pp698ndash708 2014

[3] E K Simpeh R Ndihokubwayo P E Love and W DThwala ldquoA rework probability model a quantitative assessmentof rework occurrence in construction projectsrdquo InternationalJournal of Construction Management vol 15 no 2 pp 109ndash1162015

[4] P W Morris J K Pinto and S Jonas 13e Oxford Handbookof Project Management Oxford University Press Oxford UK2012

[5] S D Eppinger and T R Browning Design Structure MatrixMethods and Applications MIT Press Books 2012

[6] T R Browning ldquoDesign structure matrix extensions and inno-vations a survey and new opportunitiesrdquo IEEE Transactions onEngineering Management vol 63 no 1 pp 27ndash52 2016

[7] A Kulkarni D K Yadav and H Nikraz ldquoAircraft maintenancechecks using critical chain project pathrdquo Aircra EngineeringAerospace Technology vol 89 no 6 pp 879ndash892 2017

[8] J Trojanowska and E Dostatni ldquoApplication of the theory ofconstraints for project managementrdquoManagement and Produc-tion Engineering Review vol 8 no 3 pp 87ndash95 2017

[9] M Ghaari and M Emsley ldquoBufer sizing in CCPM portfolioswith dierent resource capacitiesrdquo International Journal of Infor-mation Technology Project Management vol 8 no 3 pp 40ndash512017

[10] E M Goldratt Critical Chain A Business Novel MA NorthRiver Press Great Barrington United States 1997

[11] S Van de Vonder E Demeulemeester and W HerroelenldquoProactive heuristic procedures for robust project schedulingAn experimental analysisrdquo European Journal of OperationalResearch vol 189 no 3 pp 723ndash733 2008

[12] O Lambrechts E Demeulemeester and W Herroelen ldquoTimeslack-based techniques for robust project scheduling subject toresource uncertaintyrdquo Annals of Operations Research vol 186no 1 pp 443ndash464 2011

[13] D Trietsch and K R Baker ldquoPERT 21 Fitting PERTCPMfor use in the 21st centuryrdquo International Journal of ProjectManagement vol 30 no 4 pp 490ndash502 2012

[14] GMaAWangN Li LGu andQAi ldquoImproved critical chainproject management framework for scheduling constructionprojectsrdquo Journal of Construction Engineering andManagementvol 140 no 12 p 04014055 2014

[15] R C Newbold Project Management in 13e Fast Lane Applying13e 13eory of Constraints CRC Press Boca Raton FloridaLondon 1998

[16] O I Tukel W O Rom and S D Eksioglu ldquoAn investigation ofbuffer sizing techniques in critical chain schedulingrdquo EuropeanJournal of Operational Research vol 172 no 2 pp 401ndash4162006

[17] O Icmeli and S S Erenguc ldquoA branch and bound procedurefor the resource constrained project scheduling problem withdiscounted cash flowsrdquoManagement Science vol 42 no 10 pp1395ndash1408 1996

[18] E Roghanian M Alipour and M Rezaei ldquoAn improved fuzzycritical chain approach in order to face uncertainty in projectschedulingrdquo International Journal of ConstructionManagementvol 18 no 1 pp 1ndash13 2017

[19] J Zhang X Song and E Dıaz ldquoProject buffer sizing of a criticalchain based on comprehensive resource tightnessrdquo EuropeanJournal of Operational Research vol 248 no 1 pp 174ndash182 2016

[20] F A Rivera and A Duran ldquoCritical clouds and critical sets inresource-constrained projectsrdquo International Journal of ProjectManagement vol 22 no 6 pp 489ndash497 2004

[21] G Ma L Gu and N Li ldquoScenario-based proactive robustoptimization for critical-chain project schedulingrdquo Journal ofConstruction Engineering and Management vol 141 no 10 p04015030 2015

[22] H Goto ldquoForward-compatible framework with critical-chainproject management using a max-plus linear representationrdquoOPSEARCH vol 54 no 1 pp 201ndash216 2017

[23] T Salama A Salah and O Moselhi ldquoIntegration of linearscheduling method and the critical chain project managementrdquoCanadian Journal of Civil Engineering vol 45 no 1 pp 30ndash402018

[24] C W Hu X D Chen and L H Wu ldquoThe setting method ofproject buffer in critical chain management of mould manufac-turing project based reworkingrdquo Advanced Materials Researchvol 317-319 pp 418ndash422 2011

[25] D V Steward Systems Analysis and Management StructureStrategy and Design IrwinMcGraw-Hill Boston 1981

[26] T R Browning Modeling and analyzing cost schedule andperformance in complex system product development [PhDthesis]Massachusetts Institute of Technology CambridgeMas-sachusetts USA 1998

[27] T R Browning and S D Eppinger ldquoModeling impacts ofprocess architecture on cost and schedule risk in productdevelopmentrdquo IEEE Transactions on Engineering Managementvol 49 no 4 pp 428ndash442 2002

[28] S H Cho and S D Eppinger ldquoA simulation-based processmodel for managing complex design projectsrdquo IEEE Transac-tions on Engineering Management vol 52 no 3 pp 316ndash3282005

[29] V Levardy and T R Browning ldquoAn adaptive process modelto support product development project managementrdquo IEEETransactions on Engineering Management vol 56 no 4 pp600ndash620 2009

[30] J U Maheswari and K Varghese ldquoProject scheduling usingdependency structure matrixrdquo International Journal of ProjectManagement vol 23 no 3 pp 223ndash230 2005

[31] C G Cassandras and S Lafortune Introduction to DiscreteEvent Systems Springer New York NY USA 2nd edition 2008

[32] S Yoshida H Takahashi and H Goto ldquoModified max-pluslinear representation for inserting time buffersrdquo in Proceedingsof the IEEE International Conference on Industrial Engineeringamp Engineering Management IEEE 2010

[33] S Yoshida H Takahashi and H Goto ldquoResolution of timeand worker conflicts for a single project in a max-plus linearrepresentationrdquo Industrial Engineering amp Management Systemsvol 10 no 4 pp 279ndash287 2011

[34] S Muralidharan ldquoFactors affecting rework in construc-tion projectrdquo International Journal of Engineering Sciences ampResearch Technology vol 1 no 5 pp 578ndash584 2016

[35] B B E Bulloch and J Sullivan Application of the DesignStructure Matrix (DSM) to the real estate development process

14 Mathematical Problems in Engineering

[Master thesis] Massachusetts Institute of Technology Cam-bridge Massachusetts USA 2009

[36] S D Eppinger S Bonelli andAM Gonzalez ldquoManaging itera-tions in the modular real estate development processrdquo ReducingRisk in Innovation Proceedings of the 15th International DSMConference Melbourne Australia 29-30 August 2013 pp 37ndash442013

[37] D T Pham and D Karaboga Intelligent Optimisation Tech-niques Genetic Algorithms Tabu Search Simulated AnnealingandNeural Networks Springer Science amp BusinessMedia 2000

[38] D A Gebala and S D Eppinger ldquoMethods for analyzing designproceduresrdquo in Proceedings of the AMSE 13ird InternationalConference On Design13eory andMethodology vol 31 pp 227ndash233 Miami Florida 1991

[39] O Hazır M Haouari and E Erel ldquoRobust scheduling androbustness measures for the discrete timecost trade-off prob-lemrdquo European Journal of Operational Research vol 207 no 2pp 633ndash643 2010

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 13

References

[1] Y Li and T R Taylor ldquoModeling the impact of design reworkon transportation infrastructure construction project perfor-mancerdquo Journal of Construction Engineering and Managementvol 140 no 9 p 04014044 2014

[2] B Hwang X Zhao and K J Goh ldquoInvestigating the client-related rework in building projects The case of SingaporerdquoInternational Journal of Project Management vol 32 no 4 pp698ndash708 2014

[3] E K Simpeh R Ndihokubwayo P E Love and W DThwala ldquoA rework probability model a quantitative assessmentof rework occurrence in construction projectsrdquo InternationalJournal of Construction Management vol 15 no 2 pp 109ndash1162015

[4] P W Morris J K Pinto and S Jonas 13e Oxford Handbookof Project Management Oxford University Press Oxford UK2012

[5] S D Eppinger and T R Browning Design Structure MatrixMethods and Applications MIT Press Books 2012

[6] T R Browning ldquoDesign structure matrix extensions and inno-vations a survey and new opportunitiesrdquo IEEE Transactions onEngineering Management vol 63 no 1 pp 27ndash52 2016

[7] A Kulkarni D K Yadav and H Nikraz ldquoAircraft maintenancechecks using critical chain project pathrdquo Aircra EngineeringAerospace Technology vol 89 no 6 pp 879ndash892 2017

[8] J Trojanowska and E Dostatni ldquoApplication of the theory ofconstraints for project managementrdquoManagement and Produc-tion Engineering Review vol 8 no 3 pp 87ndash95 2017

[9] M Ghaari and M Emsley ldquoBufer sizing in CCPM portfolioswith dierent resource capacitiesrdquo International Journal of Infor-mation Technology Project Management vol 8 no 3 pp 40ndash512017

[10] E M Goldratt Critical Chain A Business Novel MA NorthRiver Press Great Barrington United States 1997

[11] S Van de Vonder E Demeulemeester and W HerroelenldquoProactive heuristic procedures for robust project schedulingAn experimental analysisrdquo European Journal of OperationalResearch vol 189 no 3 pp 723ndash733 2008

[12] O Lambrechts E Demeulemeester and W Herroelen ldquoTimeslack-based techniques for robust project scheduling subject toresource uncertaintyrdquo Annals of Operations Research vol 186no 1 pp 443ndash464 2011

[13] D Trietsch and K R Baker ldquoPERT 21 Fitting PERTCPMfor use in the 21st centuryrdquo International Journal of ProjectManagement vol 30 no 4 pp 490ndash502 2012

[14] GMaAWangN Li LGu andQAi ldquoImproved critical chainproject management framework for scheduling constructionprojectsrdquo Journal of Construction Engineering andManagementvol 140 no 12 p 04014055 2014

[15] R C Newbold Project Management in 13e Fast Lane Applying13e 13eory of Constraints CRC Press Boca Raton FloridaLondon 1998

[16] O I Tukel W O Rom and S D Eksioglu ldquoAn investigation ofbuffer sizing techniques in critical chain schedulingrdquo EuropeanJournal of Operational Research vol 172 no 2 pp 401ndash4162006

[17] O Icmeli and S S Erenguc ldquoA branch and bound procedurefor the resource constrained project scheduling problem withdiscounted cash flowsrdquoManagement Science vol 42 no 10 pp1395ndash1408 1996

[18] E Roghanian M Alipour and M Rezaei ldquoAn improved fuzzycritical chain approach in order to face uncertainty in projectschedulingrdquo International Journal of ConstructionManagementvol 18 no 1 pp 1ndash13 2017

[19] J Zhang X Song and E Dıaz ldquoProject buffer sizing of a criticalchain based on comprehensive resource tightnessrdquo EuropeanJournal of Operational Research vol 248 no 1 pp 174ndash182 2016

[20] F A Rivera and A Duran ldquoCritical clouds and critical sets inresource-constrained projectsrdquo International Journal of ProjectManagement vol 22 no 6 pp 489ndash497 2004

[21] G Ma L Gu and N Li ldquoScenario-based proactive robustoptimization for critical-chain project schedulingrdquo Journal ofConstruction Engineering and Management vol 141 no 10 p04015030 2015

[22] H Goto ldquoForward-compatible framework with critical-chainproject management using a max-plus linear representationrdquoOPSEARCH vol 54 no 1 pp 201ndash216 2017

[23] T Salama A Salah and O Moselhi ldquoIntegration of linearscheduling method and the critical chain project managementrdquoCanadian Journal of Civil Engineering vol 45 no 1 pp 30ndash402018

[24] C W Hu X D Chen and L H Wu ldquoThe setting method ofproject buffer in critical chain management of mould manufac-turing project based reworkingrdquo Advanced Materials Researchvol 317-319 pp 418ndash422 2011

[25] D V Steward Systems Analysis and Management StructureStrategy and Design IrwinMcGraw-Hill Boston 1981

[26] T R Browning Modeling and analyzing cost schedule andperformance in complex system product development [PhDthesis]Massachusetts Institute of Technology CambridgeMas-sachusetts USA 1998

[27] T R Browning and S D Eppinger ldquoModeling impacts ofprocess architecture on cost and schedule risk in productdevelopmentrdquo IEEE Transactions on Engineering Managementvol 49 no 4 pp 428ndash442 2002

[28] S H Cho and S D Eppinger ldquoA simulation-based processmodel for managing complex design projectsrdquo IEEE Transac-tions on Engineering Management vol 52 no 3 pp 316ndash3282005

[29] V Levardy and T R Browning ldquoAn adaptive process modelto support product development project managementrdquo IEEETransactions on Engineering Management vol 56 no 4 pp600ndash620 2009

[30] J U Maheswari and K Varghese ldquoProject scheduling usingdependency structure matrixrdquo International Journal of ProjectManagement vol 23 no 3 pp 223ndash230 2005

[31] C G Cassandras and S Lafortune Introduction to DiscreteEvent Systems Springer New York NY USA 2nd edition 2008

[32] S Yoshida H Takahashi and H Goto ldquoModified max-pluslinear representation for inserting time buffersrdquo in Proceedingsof the IEEE International Conference on Industrial Engineeringamp Engineering Management IEEE 2010

[33] S Yoshida H Takahashi and H Goto ldquoResolution of timeand worker conflicts for a single project in a max-plus linearrepresentationrdquo Industrial Engineering amp Management Systemsvol 10 no 4 pp 279ndash287 2011

[34] S Muralidharan ldquoFactors affecting rework in construc-tion projectrdquo International Journal of Engineering Sciences ampResearch Technology vol 1 no 5 pp 578ndash584 2016

[35] B B E Bulloch and J Sullivan Application of the DesignStructure Matrix (DSM) to the real estate development process

14 Mathematical Problems in Engineering

[Master thesis] Massachusetts Institute of Technology Cam-bridge Massachusetts USA 2009

[36] S D Eppinger S Bonelli andAM Gonzalez ldquoManaging itera-tions in the modular real estate development processrdquo ReducingRisk in Innovation Proceedings of the 15th International DSMConference Melbourne Australia 29-30 August 2013 pp 37ndash442013

[37] D T Pham and D Karaboga Intelligent Optimisation Tech-niques Genetic Algorithms Tabu Search Simulated AnnealingandNeural Networks Springer Science amp BusinessMedia 2000

[38] D A Gebala and S D Eppinger ldquoMethods for analyzing designproceduresrdquo in Proceedings of the AMSE 13ird InternationalConference On Design13eory andMethodology vol 31 pp 227ndash233 Miami Florida 1991

[39] O Hazır M Haouari and E Erel ldquoRobust scheduling androbustness measures for the discrete timecost trade-off prob-lemrdquo European Journal of Operational Research vol 207 no 2pp 633ndash643 2010

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

14 Mathematical Problems in Engineering

[Master thesis] Massachusetts Institute of Technology Cam-bridge Massachusetts USA 2009

[36] S D Eppinger S Bonelli andAM Gonzalez ldquoManaging itera-tions in the modular real estate development processrdquo ReducingRisk in Innovation Proceedings of the 15th International DSMConference Melbourne Australia 29-30 August 2013 pp 37ndash442013

[37] D T Pham and D Karaboga Intelligent Optimisation Tech-niques Genetic Algorithms Tabu Search Simulated AnnealingandNeural Networks Springer Science amp BusinessMedia 2000

[38] D A Gebala and S D Eppinger ldquoMethods for analyzing designproceduresrdquo in Proceedings of the AMSE 13ird InternationalConference On Design13eory andMethodology vol 31 pp 227ndash233 Miami Florida 1991

[39] O Hazır M Haouari and E Erel ldquoRobust scheduling androbustness measures for the discrete timecost trade-off prob-lemrdquo European Journal of Operational Research vol 207 no 2pp 633ndash643 2010

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom