Resource Leveling Based on Backward Controlling Activity in ...

Research ArticleResource Leveling Based on Backward Controlling Activity inLine of Balance

Lihui Zhang Yaping Tang and Jianxun Qi

School of Economics and Management North China Electric Power University Beijing 102206 China

Correspondence should be addressed to Yaping Tang 1105205461qqcom

Received 28 October 2016 Accepted 18 January 2017 Published 21 February 2017

Academic Editor Alessandro Tasora

Copyright copy 2017 Lihui Zhang 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

The line of balance method that provides continuous and uninterrupted use of resources is one of the best methods for repetitiveproject resource management This paper develops a resource leveling algorithm based on the backward controlling activity in lineof balance The backward controlling activity is a kind of special activity and if its duration is prolonged the project duration couldbe reduced It brings two advantages to the resource leveling both the resource allocated on the backward activity and the projectduration are reduced A resource leveling algorithm is presented which permits the number of crews of the backward controllingactivity to be reduced until the terminal situation is reached where the backward controlling activity does not exist or the numberof crews cannot be reducedThat adjustment enables the productivity of all activities to be consistent An illustrative pipeline projectdemonstrates the improvement in resource leveling And this study designed aMATLAB program to execute the design algorithmThe proposed model could help practitioners to achieve the goals of both resource leveling and project duration reduction withoutincreasing any resource

1 Introduction

Repetitive projects consist of group of activities that involverepetitive ldquounitsrdquo of construction elements in constructionprojects such as highways high-rise buildings tunnels andpipelines [1ndash3] Resources are essential for any constructionproject and available resources must be matched with therequirements of all activities within a project for balancein resource allocation The objective of resource leveling isto achieve the most efficient resource consumption withoutincreasing the prescribed make span of the project whichmeans minimizing variations and the peaks and valleys inresource usage that occur during project execution [4 5]

Methods that calculate resource leveling fall broadly intothree categories analytical methods heuristic methods andmetaheuristic methods [6ndash8] Geng et al [6] introduceda directional ant colony optimization algorithm to solvenonlinear resource leveling problems The algorithm canefficiently improve the convergence rate and the quality ofsolutions for real project scheduling Rieck et al [7] presentednew mixed-integer linear model formulations and domain-reducing preprocessing techniques The authors proposeda cutting plan to strengthen the models and to reduce

peaks and valleys for resource requirements at particularpoints Kyriklidis et al [8] proposed an effective resolutionof resource leveling optimization problems by using natureinspired intelligent methodologies

Since the 1950s many lectures on network-based meth-ods have proposed resource leveling For example Younis andSaad [9] proposed a model to deal with multiple resourceleveling and the model formulation is interfaced by theCPM scheduling results Neumann and Zimmermann [10]presented polynomial heuristic procedures for different typesof resource leveling problems for projects withminimum andmaximum time lags between project activities Ponz-Tiendaet al [11] presented an adaptive genetic algorithm to levelresources in networks The authors provided a flexible andpowerful decision support system that enables practitionersto choose between different feasible solutions to a problemin realistic environments However CPM was found to beineffective for projects with repetitive characteristics wherethe same basic unit is repeated several times [12]

Many studies have attempted to find an optimum solutionfor resource leveling in repetitive project scheduling Georgy[13] presented a genetic algorithm for resource leveling usinga linear scheduling method (LSM) The authorrsquos algorithm

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 7545980 9 pageshttpsdoiorg10115520177545980

2 Mathematical Problems in Engineering

1

2

3

4

Time

Uni

ts

Crew 2

Crew 3

Crew 1

Crew 1

Ri

Figure 1 The relation of crews and rate

adopted function optimizers to overcome the primary short-coming of mathematical solutions for the resource levelingproblem but it required more resource usage and addedvariables Elwany et al [14] represented a linear programmingmodel for single resource allocation and smoothening inrepetitive construction projects Lucko [15] built on an analy-sis of the criticality of linear schedules with unique singularityfunctions The author proposed equation models such as thefirst moment area to minimize the objective function towarda level profile but the model is complex for exchangingobjective function

Line of balance (LOB) iswidely used in repetitive projectsLOB performs resource allocation but ignores resourceleveling Damci et al [16] developed a genetic algorithm-based resource leveling model in LOB schedulesThe authorsobserved that the proposed resource levelingmodel provideda smoother resource utilization histogram while maintainingoptimum productivity and meeting project duration Theproposed model in this current study is of benefit to contrac-tors because it provides a resource leveling procedure for arepetitive project without any productivity loss

This paper develops a resource leveling algorithm in LOBFor this purpose it introduces a kind of special activitydefined as the backward controlling activity by Zhang and Qi[17]The variation in the duration of the backward controllingactivity changes the project duration in an opposite directionThe proposed algorithm in this paper is designed to levelresources by reducing the number of crews participating onthe backward controlling activitiesThe algorithm is tested byan illustrative pipeline project

2 Line of Balance Scheduling

LOB is a method for managing the information concerninghow many crews should be employed in each activity andhow to arrange these crews in units and assumes that a unitshould be completed by one crew To set up an LOB schedulevisually Figure 1 shows an LOB representation in which ahorizontal line represents time a vertical line represents aunit and each sloping bar represents one activity The slopeof each bar represents the planned rate of each activity

119877119894 =119899 minus 1119879119871 minus 119879119897 (1)

The productivity of the activity can be related to the numberof crews and expressed as follows

119877119894 =119862119894119863119894 (2)

119877119894 represents the productivity of the activity 119894 119879119871 is the targetdeadline of activity 119894 119879119897 is the duration of the last activityin first unit and 119899 is the number of units 119862119894 is the numberof crews arranged in activity 119894 and 119863119894 is the duration of theactivity 119894 in a unit

According to (2) the number of crews can be obtainedfrom 119862119894 = 119877119894 times 119863119894

This paper considers the finish-to-start relationshipbetween activities which means an activity in each unit canonly start after the completion of its predecessor activitiesin the same unit Accordingly the start and finish times ofactivity 119894 in each unit should be delayed by an additionalperiod Δ 119894 Start time 119878119894119895 and finish time 119865119894119895 of activity 119894 in unit119895 can be shown as follows

119878119894119895 =1119877119894(119895 minus 1) + Δ 119894

119865119894119895 = 119878119894119895 + 119863119894

Δ 119894 = max119903isin119875119894119895=12119869

(119865119903119895 minus 119878119894119895)

(3)

where 119875119894 is the set of its predecessor activitiesThis paper assumes that the principles of ldquooptimum crew

sizerdquo and ldquonature rhythmrdquo are also applied [18] ldquoOptimumcrew sizerdquo is a principle that implies that productivity willsuffer if the crew size is different than the optimum crew sizeThe principle of ldquonatural rhythmrdquo allows shifting of the starttimes of an activity forward or backward for different unitsof production by changing the number of crews employedAn increase in the number of working teams would increaseproductivity In Figure 2(a) the productivity of the activityis one unit per day when only one crew is utilized If twocrews are employed the productivity becomes two units perday Figure 2(c) shows that if the production rate of theactivity fails to achieve two units per day this leads to anincrease in worker hours per unit because of the idle time ofcrews

To calculate the daily resource usage for an activity thescheduler must know the duration of the activity for a unitthe productivity and the number of crews active in a day Inpractical engineering productivity is difficult to determinehowever required worker hours for an activity in a unit thenumber of workers per crew and the number of workinghours per day are easily obtained Therefore the duration ofan activity 119894 in a unit is calculated by the followingmagnitude

119863119894 =Rh119894119908119894 times ℎ

Re119894119905 = 119862119894119905 times 119908119894(4)

where Rh119894 represents required worker hours of activity 119894 in aunit by the optimum crew size 119908119894 is the number of workers

Mathematical Problems in Engineering 3

1 2 3 4 5

2

3

4

5

1

Time (days)

Crew 1

Crew 1

Crew 1

Crew 1

Uni

t

(a)

1 2 3 4 5

2

3

4

5

1

Crew 2

Crew 1

Crew 1

Crew 2

Time (days)

Uni

t

(b)

Time (days)1 2 3 4 5

2

3

4

5

1

Uni

t

Crew 1

Crew 1

Crew 2

Crew 2

Idle time

(c)

Figure 2 LOB scheduling with different productivity

per crew for activity 119894 ℎ is the number of working hours perday Re119894119905 is the resource usage required for activity 119894 in day 119905and 119862119894119905 is the number of crews active in day 119905

3 Backward Controlling Activity in LOB

Researchers have already found an unexpected phenomenonthat the extension of the duration of some activities wouldshorten the project duration [19 20] In network models thistype of activity is defined as the backward critical activityby Elmaghraby and Kamburowski [21] Zhang and Qi [17]proposed a method for identifying three different types ofcontrolling activities in LSM that is the forward controllingactivity the backward controlling activity and the pointcontrolling activity The forward controlling activity is thecontrolling activity we usually refer to If it is delayed theproject will be delayed The point controlling activity isthe activity that only its start time or finish time changewill affect the project duration The backward controllingactivity is special If its duration is prolonged the projectduration could be reduced The essential feature of thebackward controlling activity is that it has a higher pro-duction rate than its preceding and succeeding controllingactivity

These three types of controlling activities exist in LOBtoo On the controlling path if the production rate of anactivity is speedier than its preceding and succeeding activityit is identified as the backward controlling activity If theduration of the backward controlling activity is prolonged itmeans that the production rate is reduced and then its firstunit can be started earlier without violating the constraintfrom the preceding controlling activity Thus the succeedingactivity could start earlier and the project could be finishedearlier

For example there is a project including activities A Band C and all of them are in the controlling path as shownin Figure 3 The productivity of activity B is 43 units perday which is greater than the productivity of activities Aand C at 12 units per day Therefore activity A and C areforward controlling activities and activity B is a backwardcontrolling activity The project duration could be reducedif we accelerate activity A and C by allocating more crewson them which is the most common way to shorten theproject duration But there is another way to achieve thegoal by reducing crews on the backward controlling activitythat is activity B After two crews on activity B are firedthe succeeding controlling activity C can be started 119905 daysearlier and the project can be finished 119905 days earlier as shown


Days1

2

3

4

5

6

7U

nits

A B C

(a)

t

Days

1

2

3

4

5

6

7

Uni

ts

A B C

(b)

Figure 3 The backward controlling activity in LOB

in Figure 3(b) In a certain range reducing the productivityof a backward controlling activity can shorten the projectduration

However a backward controlling activity does not alwaysshorten the project duration after reducing crews If activity119894 is a backward controlling activity a minimum slope existsAfter adjustment the productivity of activity 119894 cannot slowmore than the minimum slope which can be expressed asfollows

MR = 119899 minus 1119878119906119899 minus 119863119894 minus 1198651199011

(5)

where MR is the minimum rate 119899 is the number of units119901 and 119906 are the preceding controlling activity and thesucceeding controlling activity of activity 119894119863119894 is the durationof the activity 119894 in a unit 119878119906119899 is the start time of the activity 119906in the last unit and 1198651199011 is the finish time of the activity 119901 inthe first unit

In Figure 3(b) the slope of the thick black line representsthe minimum production rate The productivity of activity Bcannot be slower than the minimum production rate or theproject duration will be prolonged

4 Algorithm for Resource Leveling

The backward controlling activity brings two advantages toresource leveling On the one hand the crews are reducedon the backward controlling activity which means possibleleveler resource usage on the other hand the project durationcould be shortened which means the project will not bedelayed The proposed algorithm in this paper aims toachieve resource leveling by reducing crews on the backwardcontrolling activities and shifting its start times Figure 4shows the flowchart of the algorithm

The productivity of all crews that consist of an optimumnumber of workers will be up to the maximum because moreor fewer workers than the optimum crew size would resultin lower productivity To ensure productivity change linearlyall units in any activity are always executed by the optimum

crew size The following principles are followed to level theresource of a whole project

(1) Only the backward controlling activities are can-didates for adjustment for instance activity B inFigure 3The productivity of these activities should begreater than the preceding controlling activities andsucceeding controlling activities

(2) If the number of crews in the backward activity isgreater than one and the production of the activity isnot slower than the minimum rate after reducing onecrew then one crew of the backward controlling activ-ity could be fired and the number of crews should beupdated If any of these two conditions is not metthe identified backward controlling activity cannotbe adjusted When all of the backward controllingactivities are adjusted the sum of absolute values ofthe deviations between resource usage on any day andthe average resource usage should be calculated

(3) When a cycle is finished the deviation values ofpostresource leveling and preresource leveling shouldbe compared and the smaller value should be notedStep 2 should not be repeated until the terminalsituation is met

(4) There are two terminal situations First the backwardcontrolling activities no longer exist Second thebackward controlling activities do exist but only onecrew is employed Either situation would disrupt thecomputing stop

Various objective functions are presented in previousstudies for resource leveling Resource leveling is tominimizeresource fluctuations within the deadline Demeulemeesterand Herroelen [22] defined ldquoresource leveling aims at com-pleting the project within its deadline with a resource usagewhich is as level as possible over the entire project horizonrdquoThe resource leveling problem can be formulated conceptu-ally as follows


Input information such as number of available crews duration of single unit

Calculate rate getthe initial schedule

Next activityNo

Reduce one crew

Yes

Meeting the duration andsmoother schedule

Back to the last scheduleNo

Update the number of crews and note the better resource

result

Yes

End of activities

No

Terminal situations

No

End

Yes

Yes

Ri gt Riminus1 amp Ri gt Ri+1

Figure 4 Flowchart of the algorithm

Minimize119898

sum119898=1

120575119899

sum119905=1

119888119896 (119906119896119905)

subject to 119891119894 le 119891119895 minus 119889119895

119891119899 le 120575119899

(6)

where 120575119899 is the project deadline119891119894 is the finish time of activity119894 (0 lt 119894 le 119899) and 119906119896119905 is the availability of the resource 119896 in theperiod 119905

Damci and Polat [23] investigated the effects of differ-ent objective functions on resource utilization histogramsin CPMs They studied nine different resource levelingobjective functions generating different resource utilizationhistograms To prove the proposedmodelrsquos effect this currentstudy uses the same objective function as Damci et al [16]

The objective function minimizes the sum of the absolutedeviations between daily resource requirements and theaverage resource requirement This is one of the most com-monly used objective functions for resource leveling in linearscheduling methods [14] The formulation can be expressedas

119911 = min119879

sum119905=1

1003816100381610038161003816PR119905 minus Ave1003816100381610038161003816 (7)

where 119905 is the day under consideration 119879 is the projectduration PR119905 is the resource usage required on day 119905 and Aveis the average resource usage requirement for the duration ofthe project

Tomaintain job continuity for the same repetitive activitythe typical scheduling precedence relationship between the


Table 1 The Information for the pipeline project

Activity Required workerhours to finish unit

Number ofworkers

Daily workinghours Duration (days) Number of crews Productivity

rate(A) Locating andclearing 96 6 8 2 2 1

(B) Excavating 64 8 8 1 2 2(C) Laying aggregate 80 10 8 1 3 3(D) Laying pipes 84 7 8 15 2 133(E) Testing 80 10 8 1 4 4(F) Backfilling 96 6 8 2 5 25(G) Compacting 144 9 8 2 2 1

controlling activities can be finish-to-start This implies thata successor activity can start only once its predecessor hasfinished The formulation can be shown as the followingequation

119878119894119895 gt 119865119901119895

119865119872119873 lt 119879(8)

where 119901 represents all predecessors of the activity 119894 and 119865119872119873represents finish time of the last activity in the last unit

5 Example

A pipeline project presented by Damci et al [16] is used totest the proposed resource leveling model The pipeline is26 km in length and expected to be completed in 65 daysand consists of seven consecutive activities (1) locating andclearing (activity A) (2) excavating (activity B) (3) layingaggregate (activity C) (4) laying pipes (activity D) (5) testing(activity E) (6) backfilling (activity F) and (7) compacting(activity G) The initial schedule was accelerated using theprinciples of Tokdemir et al [24] that is by increasing thenumber of crews in selected activities to complete the projectin 65 days without resource limitations Table 1 shows theproject information Although other resources are necessaryto complete this pipeline projectrsquos activities only workerswere considered in this study for demonstration purposes

Figure 5(a) shows the initial scheduling and Figure 5(b)shows the scheduling given by Damci et al [16] Theyobtained a better result than the initial schedule by adjustingthe number of crews in some units of activity B and activityF Figure 5(c) shows the scheduling given by this study TheLOB chart obtained by the proposed algorithm in this studyis totally different from the former schedule The project canbe completed in 48 days after resource leveling The numberof employed crews is 2 1 1 1 1 2 2 arranged on activities fromlocating and clearing (activity A) to compacting (activity G)All activities have an equal productivity of one unit per dayexcept activity D which is executed in 067 units per day

The total workforce used in the project is composed of thesum of the workers used in each activity through the entireproject (2093 workers)The total absolute value of deviationsfrom the average is 1390 and the maximum number ofworkers is 102 in the initial schedules as shown in Figure 6(a)

The average resource usage of 33 is calculated by dividingthe total resources (2093 workers) used by the whole projectduration of 65 days

Damci et al [16] achieved a better result although theyonly adjusted two activities The authors have the same totalresource usage and project duration as the initial scheduleThe total absolute value of the deviations between resourceusage on any day and the average resource usage is 1037 [16]which represents an improvement in resource leveling of 25compared to the initial schedule as shown in Figure 6(b)And the maximum resource usage is reduced from 102 to89

This study designed a MATLAB program to execute thedesign algorithm The major benefit of this tool is high-speed computing and user-friendly language The use of aMATLAB program to solve this problem requires the inputof the initial number of crews and productivity In this caseafter performing a set of adjustments based on backwardcontrolling activities the proposed model achieves betterleveling and reduced project duration The duration of theproject is 48 days after resource leveling and the averageresource usage is 436 workers as shown in Figure 6(c)Because the worker value cannot be noninteger the result isrounded up to 44The sumof the absolute deviations betweendaily resource usage and average resource usage is 591 whichshows a leveling of 575 compared with the initial scheduleand which shows a leveling of 43 compared with Damciet alrsquos [16] The maximum resource usage is reduced fromDamci et alrsquos result 89 to 77

The proposed resource leveling algorithm follows theprincipal of adjusting the backward controlling activity Oncethe number of crews is reduced in a backward controllingactivity its preceding controlling activity or succeedingcontrolling activity may become the backward controllingactivity Because the backward activity is represented by theactivities that have a higher productivity than the precedingand succeeding controlling activity the productivity of allactivities would tend to achieve resource leveling after a set ofadjustments because of a reduction in crews The advantagesof the proposedmodel in this study are as follows First effec-tive resource leveling is obtained while the project durationcan meet or even beat the deadline Second the algorithmprocess limits the complex principles required to adjust theLOB schedule which simplifies practical operations


1

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45 50 55 60 65

ABC DE F GUnit

Days

Kilo

met

ers o

f pip

elin

e

(a) The initial LOB schedule

Kilo

met

ers o

f pip

elin

e

1

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45 50 55 60 65

ABC DE F G

Days

(b) LOB scheduling in Damci et alrsquos study

1

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45 50

A B C D E F G

Days

Kilo

met

ers o

f pip

elin

e

(c) LOB scheduling in this study

Figure 5 The LOB chart with different scheduling

102030405060708090

100

5 10 15 20 25 30 35 40 45 50 60 6555Time (days)

1

Num

ber o

f wor

kers

Average numberof workers

Units

(a) The resource histogram by the initial schedule

5 10 15 20 25 30 35 40 45 50 60 6555Time (days)

1

102030405060708090

100


Num

ber o

f wor

kers

Units

(b) The resource histogram of schedules in Damci et alrsquos study

5 10 15 20 25 30 35 40 45 50 60 6555Time (days)

1


102030405060708090

100 Units

Num

ber o

f wor

kers

(c) The resource histogram of schedules in this paper

Figure 6 The resource histogram of different schedules


Of course the proposedmethod has limitations the effectof resource leveling depends on whether there are backwardcontrolling activities in LOB and whether it can be adjusted

6 Conclusion

An appropriate scheduling method is crucial for successfulconstruction project completion LOB is one of themost suit-able methods for resourcemanagement in repetitive projectsbut resource leveling has not been adequately considered inthe LOB schedule This paper developed a resource levelingalgorithmbased on backward controlling activity in LOBThepresented algorithm for resource leveling achieved a betterresult by reducing the number of crews in the backwardcontrolling activity enabling a shift in start and finish timesMoreover the project can be finished ahead of the deadline

It should be noted that the proposed method works onlywhen the backward controlling activities exist and when theycan be adjusted because themethod only considers adjustingthe backward controlling activities And the overall qualityof resource leveling depends on the backward controllingactivities in the initial scheduling But in repetitive projectsthe backward controlling activity exists commonly becausethe productivities of all activities are usually different It isvery rare that the productivities of all activities are equal orbecome bigger and bigger or smaller and smaller Meanwhileonly a single resource and an application project with 7activities are discussed in this paper Further study will con-sider developing the method for multiple resources levelingproblems and large-scale repetitive projects

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors would like to acknowledge the Natural Sci-ence Foundation of China (Contract no 71271081) and theFundamental Research Funds for the Central Universities(13ZD08)

References

[1] D B Ashley ldquoSimulation of repetitive-unit constructionrdquoAmerican Society of Civil Engineers Journal of the ConstructionDivision vol 106 no 2 pp 185ndash194 1980

[2] S M Amini and G Heravi ldquoSchedule compression for con-struction projects by interruption in LOB schedulingrdquo AACEInternational Transactions vol 4 pp 1ndash18 2009

[3] D Arditi P Sikangwan and O B Tokdemir ldquoSchedulingsystem for high rise building constructionrdquo Construction Man-agement and Economics vol 20 no 4 pp 353ndash364 2002

[4] K Neumann C Schwindt and J Zimmermann Projectscheduling with time windows and scarce resources SpringerBerlin Germany Second edition 2003

[5] R B Harris Precedence and Arrow Networking Techniques forConstruction John Wiley amp Sons New York NY USA 1978

[6] J-Q Geng L-P Weng and S-H Liu ldquoAn improved antcolony optimization algorithm for nonlinear resource-levelingproblemsrdquo Computers and Mathematics with Applications vol61 no 8 pp 2300ndash2305 2011

[7] J Rieck J Zimmermann and T Gather ldquoMixed-integer linearprogramming for resource leveling problemsrdquoEuropean Journalof Operational Research vol 221 no 1 pp 27ndash37 2012

[8] C Kyriklidis V Vassiliadis K Kirytopoulos and G DouniasldquoHybrid nature-inspired intelligence for the resource levelingproblemrdquoOperational Research vol 14 no 3 pp 387ndash407 2014

[9] M A Younis and B Saad ldquoOptimal resource leveling of multi-resource projectsrdquo Computers and Industrial Engineering vol31 no 1-2 pp 1ndash4 1996

[10] K Neumann and J Zimmermann ldquoResource levelling forprojects with schedule-dependent time windowsrdquo EuropeanJournal of Operational Research vol 117 no 3 pp 591ndash605 1999

[11] J L Ponz-Tienda V Yepes E Pellicer and J Moreno-FloresldquoThe Resource Leveling Problem with multiple resources usingan adaptive genetic algorithmrdquoAutomation inConstruction vol29 no 1 pp 161ndash172 2013

[12] I-T Yang and C-Y Chang ldquoStochastic resource-constrainedscheduling for repetitive construction projects with uncertainsupply of resources and fundingrdquo International Journal ofProject Management vol 23 no 7 pp 546ndash553 2005

[13] M E Georgy ldquoEvolutionary resource scheduler for linearprojectsrdquoAutomation in Construction vol 17 no 5 pp 573ndash5832008

[14] M H Elwany I E Korish M A Barakat and S M HafezldquoResource smoothening in repetitive projectsrdquo Computers andIndustrial Engineering vol 35 no 3-4 pp 415ndash418 1998

[15] G Lucko ldquoIntegrating efficient resource optimization andlinear schedule analysis with singularity functionsrdquo Journal ofConstruction Engineering and Management vol 137 no 1 pp45ndash55 2011

[16] A Damci D Arditi and G Polat ldquoResource leveling in line-of-balance schedulingrdquo Computer-Aided Civil and InfrastructureEngineering vol 28 no 9 pp 679ndash692 2013

[17] L H Zhang and J X Qi ldquoControlling path and controllingsegment analysis in repetitive scheduling methodrdquo Journal ofConstruction Engineering and Management vol 138 no 11 pp1341ndash1345 2012

[18] D Arditi O B Tokdemir and K Suh ldquoChallenges in line-of-balance schedulingrdquo Journal of Construction Engineering andManagement vol 128 no 6 pp 545ndash556 2002

[19] K G Mattila and D M Abraham ldquoLinear scheduling pastresearch efforts and future directionsrdquoEngineering Constructionand Architectural Management vol 5 no 3 pp 294ndash303 1998

[20] R B Harris and P G Ioannou ldquoScheduling projects withrepeating activitiesrdquo Journal of Construction Engineering andManagement vol 124 no 4 pp 269ndash278 1998

[21] S E Elmaghraby and J Kamburowski ldquoThe analysis of activ-ity networks under generalized precedence relations (GPRs)rdquoManagement Science vol 38 no 9 pp 1245ndash1263 1992

[22] E L Demeulemeester andW SHerroelenProject SchedulingmdashA Research Handbook Department of Applied EconomicsKatholieke Universiteit Leuven Belgium 2002

[23] A Damci andG Polat ldquoImpacts of different objective functionson resource leveling in construction projects a case studyrdquoJournal of Civil Engineering and Management vol 20 no 4 pp537ndash547 2014


[24] O B Tokdemir D Arditi and C Balcik ldquoALISS advancedlinear scheduling systemrdquo Construction Management and Eco-nomics vol 24 no 12 pp 1253ndash1267 2006

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1

2

3

4

Time

Uni

ts

Crew 2

Crew 3

Crew 1

Crew 1

Ri

Figure 1 The relation of crews and rate

adopted function optimizers to overcome the primary short-coming of mathematical solutions for the resource levelingproblem but it required more resource usage and addedvariables Elwany et al [14] represented a linear programmingmodel for single resource allocation and smoothening inrepetitive construction projects Lucko [15] built on an analy-sis of the criticality of linear schedules with unique singularityfunctions The author proposed equation models such as thefirst moment area to minimize the objective function towarda level profile but the model is complex for exchangingobjective function

Line of balance (LOB) iswidely used in repetitive projectsLOB performs resource allocation but ignores resourceleveling Damci et al [16] developed a genetic algorithm-based resource leveling model in LOB schedulesThe authorsobserved that the proposed resource levelingmodel provideda smoother resource utilization histogram while maintainingoptimum productivity and meeting project duration Theproposed model in this current study is of benefit to contrac-tors because it provides a resource leveling procedure for arepetitive project without any productivity loss

This paper develops a resource leveling algorithm in LOBFor this purpose it introduces a kind of special activitydefined as the backward controlling activity by Zhang and Qi[17]The variation in the duration of the backward controllingactivity changes the project duration in an opposite directionThe proposed algorithm in this paper is designed to levelresources by reducing the number of crews participating onthe backward controlling activitiesThe algorithm is tested byan illustrative pipeline project

2 Line of Balance Scheduling

LOB is a method for managing the information concerninghow many crews should be employed in each activity andhow to arrange these crews in units and assumes that a unitshould be completed by one crew To set up an LOB schedulevisually Figure 1 shows an LOB representation in which ahorizontal line represents time a vertical line represents aunit and each sloping bar represents one activity The slopeof each bar represents the planned rate of each activity

119877119894 =119899 minus 1119879119871 minus 119879119897 (1)

The productivity of the activity can be related to the numberof crews and expressed as follows

119877119894 =119862119894119863119894 (2)

119877119894 represents the productivity of the activity 119894 119879119871 is the targetdeadline of activity 119894 119879119897 is the duration of the last activityin first unit and 119899 is the number of units 119862119894 is the numberof crews arranged in activity 119894 and 119863119894 is the duration of theactivity 119894 in a unit

According to (2) the number of crews can be obtainedfrom 119862119894 = 119877119894 times 119863119894

This paper considers the finish-to-start relationshipbetween activities which means an activity in each unit canonly start after the completion of its predecessor activitiesin the same unit Accordingly the start and finish times ofactivity 119894 in each unit should be delayed by an additionalperiod Δ 119894 Start time 119878119894119895 and finish time 119865119894119895 of activity 119894 in unit119895 can be shown as follows

119878119894119895 =1119877119894(119895 minus 1) + Δ 119894

119865119894119895 = 119878119894119895 + 119863119894

Δ 119894 = max119903isin119875119894119895=12119869

(119865119903119895 minus 119878119894119895)

(3)

where 119875119894 is the set of its predecessor activitiesThis paper assumes that the principles of ldquooptimum crew

sizerdquo and ldquonature rhythmrdquo are also applied [18] ldquoOptimumcrew sizerdquo is a principle that implies that productivity willsuffer if the crew size is different than the optimum crew sizeThe principle of ldquonatural rhythmrdquo allows shifting of the starttimes of an activity forward or backward for different unitsof production by changing the number of crews employedAn increase in the number of working teams would increaseproductivity In Figure 2(a) the productivity of the activityis one unit per day when only one crew is utilized If twocrews are employed the productivity becomes two units perday Figure 2(c) shows that if the production rate of theactivity fails to achieve two units per day this leads to anincrease in worker hours per unit because of the idle time ofcrews

To calculate the daily resource usage for an activity thescheduler must know the duration of the activity for a unitthe productivity and the number of crews active in a day Inpractical engineering productivity is difficult to determinehowever required worker hours for an activity in a unit thenumber of workers per crew and the number of workinghours per day are easily obtained Therefore the duration ofan activity 119894 in a unit is calculated by the followingmagnitude

119863119894 =Rh119894119908119894 times ℎ

Re119894119905 = 119862119894119905 times 119908119894(4)

where Rh119894 represents required worker hours of activity 119894 in aunit by the optimum crew size 119908119894 is the number of workers


1 2 3 4 5

2

3

4

5

1

Time (days)

Crew 1

Crew 1

Crew 1

Crew 1

Uni

t

(a)

1 2 3 4 5

2

3

4

5

1

Crew 2

Crew 1

Crew 1

Crew 2

Time (days)

Uni

t

(b)


2

3

4

5

1

Uni

t

Crew 1

Crew 1

Crew 2

Crew 2

Idle time

(c)








Days1

2

3

4

5

6

7U

nits

A B C

(a)

t

Days

1

2

3

4

5

6

7

Uni

ts

A B C

(b)





(5)















Next activityNo

Reduce one crew

Yes




result

Yes

End of activities

No

Terminal situations

No

End

Yes

Yes



Minimize119898

sum119898=1

120575119899

sum119905=1

119888119896 (119906119896119905)


119891119899 le 120575119899

(6)




119911 = min119879

sum119905=1

1003816100381610038161003816PR119905 minus Ave1003816100381610038161003816 (7)






Number ofworkers





119878119894119895 gt 119865119901119895

119865119872119873 lt 119879(8)


5 Example









1

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45 50 55 60 65

ABC DE F GUnit

Days

Kilo

met

ers o

f pip

elin

e


Kilo

met

ers o

f pip

elin

e

1

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45 50 55 60 65

ABC DE F G

Days


1

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45 50

A B C D E F G

Days

Kilo

met

ers o

f pip

elin

e



102030405060708090

100

5 10 15 20 25 30 35 40 45 50 60 6555Time (days)

1

Num

ber o

f wor

kers


Units


5 10 15 20 25 30 35 40 45 50 60 6555Time (days)

1

102030405060708090

100


Num

ber o

f wor

kers

Units


5 10 15 20 25 30 35 40 45 50 60 6555Time (days)

1


102030405060708090

100 Units

Num

ber o

f wor

kers





6 Conclusion



Competing Interests


Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 4 5

2

3

4

5

1

Time (days)

Crew 1

Crew 1

Crew 1

Crew 1

Uni

t

(a)

1 2 3 4 5

2

3

4

5

1

Crew 2

Crew 1

Crew 1

Crew 2

Time (days)

Uni

t

(b)


2

3

4

5

1

Uni

t

Crew 1

Crew 1

Crew 2

Crew 2

Idle time

(c)








Days1

2

3

4

5

6

7U

nits

A B C

(a)

t

Days

1

2

3

4

5

6

7

Uni

ts

A B C

(b)





(5)















Next activityNo

Reduce one crew

Yes




result

Yes

End of activities

No

Terminal situations

No

End

Yes

Yes



Minimize119898

sum119898=1

120575119899

sum119905=1

119888119896 (119906119896119905)


119891119899 le 120575119899

(6)




119911 = min119879

sum119905=1

1003816100381610038161003816PR119905 minus Ave1003816100381610038161003816 (7)






Number ofworkers





119878119894119895 gt 119865119901119895

119865119872119873 lt 119879(8)


5 Example









1

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45 50 55 60 65

ABC DE F GUnit

Days

Kilo

met

ers o

f pip

elin

e


Kilo

met

ers o

f pip

elin

e

1

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45 50 55 60 65

ABC DE F G

Days


1

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45 50

A B C D E F G

Days

Kilo

met

ers o

f pip

elin

e



102030405060708090

100

5 10 15 20 25 30 35 40 45 50 60 6555Time (days)

1

Num

ber o

f wor

kers


Units


5 10 15 20 25 30 35 40 45 50 60 6555Time (days)

1

102030405060708090

100


Num

ber o

f wor

kers

Units


5 10 15 20 25 30 35 40 45 50 60 6555Time (days)

1


102030405060708090

100 Units

Num

ber o

f wor

kers





6 Conclusion



Competing Interests


Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Days1

2

3

4

5

6

7U

nits

A B C

(a)

t

Days

1

2

3

4

5

6

7

Uni

ts

A B C

(b)





(5)















Next activityNo

Reduce one crew

Yes




result

Yes

End of activities

No

Terminal situations

No

End

Yes

Yes



Minimize119898

sum119898=1

120575119899

sum119905=1

119888119896 (119906119896119905)


119891119899 le 120575119899

(6)




119911 = min119879

sum119905=1

1003816100381610038161003816PR119905 minus Ave1003816100381610038161003816 (7)






Number ofworkers





119878119894119895 gt 119865119901119895

119865119872119873 lt 119879(8)


5 Example









1

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45 50 55 60 65

ABC DE F GUnit

Days

Kilo

met

ers o

f pip

elin

e


Kilo

met

ers o

f pip

elin

e

1

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45 50 55 60 65

ABC DE F G

Days


1

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45 50

A B C D E F G

Days

Kilo

met

ers o

f pip

elin

e



102030405060708090

100

5 10 15 20 25 30 35 40 45 50 60 6555Time (days)

1

Num

ber o

f wor

kers


Units


5 10 15 20 25 30 35 40 45 50 60 6555Time (days)

1

102030405060708090

100


Num

ber o

f wor

kers

Units


5 10 15 20 25 30 35 40 45 50 60 6555Time (days)

1


102030405060708090

100 Units

Num

ber o

f wor

kers





6 Conclusion



Competing Interests


Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Next activityNo

Reduce one crew

Yes




result

Yes

End of activities

No

Terminal situations

No

End

Yes

Yes



Minimize119898

sum119898=1

120575119899

sum119905=1

119888119896 (119906119896119905)


119891119899 le 120575119899

(6)




119911 = min119879

sum119905=1

1003816100381610038161003816PR119905 minus Ave1003816100381610038161003816 (7)






Number ofworkers





119878119894119895 gt 119865119901119895

119865119872119873 lt 119879(8)


5 Example









1

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45 50 55 60 65

ABC DE F GUnit

Days

Kilo

met

ers o

f pip

elin

e


Kilo

met

ers o

f pip

elin

e

1

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45 50 55 60 65

ABC DE F G

Days


1

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45 50

A B C D E F G

Days

Kilo

met

ers o

f pip

elin

e



102030405060708090

100

5 10 15 20 25 30 35 40 45 50 60 6555Time (days)

1

Num

ber o

f wor

kers


Units


5 10 15 20 25 30 35 40 45 50 60 6555Time (days)

1

102030405060708090

100


Num

ber o

f wor

kers

Units


5 10 15 20 25 30 35 40 45 50 60 6555Time (days)

1


102030405060708090

100 Units

Num

ber o

f wor

kers





6 Conclusion



Competing Interests


Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Number ofworkers





119878119894119895 gt 119865119901119895

119865119872119873 lt 119879(8)


5 Example









1

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45 50 55 60 65

ABC DE F GUnit

Days

Kilo

met

ers o

f pip

elin

e


Kilo

met

ers o

f pip

elin

e

1

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45 50 55 60 65

ABC DE F G

Days


1

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45 50

A B C D E F G

Days

Kilo

met

ers o

f pip

elin

e



102030405060708090

100

5 10 15 20 25 30 35 40 45 50 60 6555Time (days)

1

Num

ber o

f wor

kers


Units


5 10 15 20 25 30 35 40 45 50 60 6555Time (days)

1

102030405060708090

100


Num

ber o

f wor

kers

Units


5 10 15 20 25 30 35 40 45 50 60 6555Time (days)

1


102030405060708090

100 Units

Num

ber o

f wor

kers





6 Conclusion



Competing Interests


Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45 50 55 60 65

ABC DE F GUnit

Days

Kilo

met

ers o

f pip

elin

e


Kilo

met

ers o

f pip

elin

e

1

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45 50 55 60 65

ABC DE F G

Days


1

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45 50

A B C D E F G

Days

Kilo

met

ers o

f pip

elin

e



102030405060708090

100

5 10 15 20 25 30 35 40 45 50 60 6555Time (days)

1

Num

ber o

f wor

kers


Units


5 10 15 20 25 30 35 40 45 50 60 6555Time (days)

1

102030405060708090

100


Num

ber o

f wor

kers

Units


5 10 15 20 25 30 35 40 45 50 60 6555Time (days)

1


102030405060708090

100 Units

Num

ber o

f wor

kers





6 Conclusion



Competing Interests


Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











6 Conclusion



Competing Interests


Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Resource Leveling Based on Backward Controlling Activity in ...

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