Automation in Construction 18 (2009) 942–949

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Automation in Construction

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Resource-constrained scheduling for continuous repetitive projects withtime-based production units

Machine Hsie a, Ching-Jung Chang a, I-Tung Yang b,⁎, Chun-Yen Huang a

a Department of Civil Engineering, National Chung Hsing University, No. 250 Kuo Kuang Road, Taichung 402, Taiwanb Department of Construction Engineering, National Taiwan University of Science and Technology, No. 43 Keelung Road, Taipei 106, Taiwan

⁎ Corresponding author. Tel.: +886 2 27376569; fax:E-mail address: ityang@mail.ntust.edu.tw (I.-T. Yang

0926-5805/$ – see front matter © 2009 Elsevier B.V. Adoi:10.1016/j.autcon.2009.04.006

a b s t r a c t

a r t i c l e i n f o

Article history:Accepted 23 April 2009

Keywords:Repetitive projectResource-constrained schedulingEvolutionary strategyOptimization

In construction projects of highways, pipelines, and tunnels, labor and equipment continuously move in alinear geographic layout. This class of continuous repetitive projects encounters the resource-constrainedproblem when there are limits on the availabilities of resources (labor and equipment). Conventionalscheduling models divide continuous repetitive projects into space segments. The premise is that crewswould maintain the same production rate in each space segment. However, when the length of segment isindivisible by the production rate, this assumption leads to an inefficient schedule which asks the crews tochange their production rate, a reflection of their size, composition, and associated equipment, in the middleof a time period. Unproductive time would then be spent in extra preparation and unnecessary warming up.Another drawback is that production units divided in space cannot be directly linked to the time-basedpayment schedule. In light of these shortcomings, this paper presents a scheduling model to find the optimalset of production rates in different time periods for each crew, considering limited availability of resources. Tobe practical, the proposed model addresses work continuity while maintaining lead-time and lead-distancebetween operations. The optimization problem is solved by an evolutionary strategy algorithm, which is easyto program and takes less execution time, with no need for selection and crossover process. A real-life projectis used to validate the performance of the proposed model in terms of effectiveness, efficiency, and stability.

© 2009 Elsevier B.V. All rights reserved.

Table 1Example of crew composition.

Activity Crew composition Resource Productionrate(m/day)Excavator Truck Common laborer

Available resource 2 4 8

A Va1 1 2 2 100Va2 2 4 4 120

1. Introduction

Many construction projects involve continuous movement ofworkers andmachines in a linear geographic layout, such as highways,tunnels, and pipelines. As crews “parade” the site one after the otherduring progress, work accomplishment in this class of repetitiveprojects is not measured in discrete work units, but rather continu-ouslywith certain amounts of distance and time interval kept betweenoperations [1,2]. This class of projects is named as “continuous repeti-tive projects”, as opposed to “discrete repetitive projects”, whereworkunits are separable, such as floors in multistory buildings or houses inhousing communities [3].

Since there is no uniform repetition of a module network in con-tinuous repetitive projects, traditional scheduling techniques, such asCPM (critical path method) and Gantt chart, have long been criticizedfor their inadequacy in accurately modeling production of continuousrepetitive projects [4,5]. Alternatives were proposed to provide thevisual benefit of showing production rates and locations of crews on atwo-dimensional chart: line-of-balance [6], linear scheduling method[7–9], and repetitive scheduling method [3]. According to the conver-gence or divergence of production lines, [10] defined control points

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and the controlling sequence, which should be carefully distinguishedfrom the set of critical activities [11]. Subsequently, [12] developed analgorithm to automatically identify the controlling sequences andcritical activities.

For optimization purpose, numerous mathematical programmingand meta-heuristics techniques were applied to schedule repetitiveprojects: linear programming [13,14], dynamic programming [15–18],integer programming [19], neural networks [20], object-orientedscheduling [21], and genetic algorithms [22,23].

Resource usage plays an important role in successfully manage-ment of continuous repetitive projects. Thus, a vast amount of efforthas gone into handling limited resource availability with sharing

B Vb1 0 0 4 60Vb2 0 0 5 80Vb3 0 0 8 100

Fig. 1. Selection of production rates and consideration of resource availabilities.

943M. Hsie et al. / Automation in Construction 18 (2009) 942–949

strategies, such as reusable steel forms [24] and maintaining workcontinuity of crews [25].

A continuous repetitive project can be broken down into pieces ofproduction units either in terms of space segments, such as stations orkilometers, or of time periods, such as days or weeks. Conventionally,production units of continuous repetitive projects are defined indiscrete space segments, similar to discrete repetitive projects. Thisimplies that a crew would maintain the same production rate thatreflects its size, composition, and associated equipment within eachspace segment. Yet, the assumption often renders the schedule ineffi-cient when the length of space segment is invisible by the productionrate. In this case, a crewwould need to change its size, composition orassociated equipment in the middle of a time period, causing un-productive preparation and warming up. Another disadvantage is thatproduction units in space segments cannot be directly linked to the

Fig. 2. Lead-distance and lead-time between production lines.

daily or weekly payment schedule, thereby involving extra adminis-trative work.

The present study aims to develop a new optimization model tominimize the project duration by automatically searching for theoptimal set of production rates of crews in different time periods. Thefundamental principle is for crews to maintain the same productionrate in each time period to attain maximum efficiency, i.e. no need forchanging tools, extra preparation, or warming up in the middle of atime period. The proposed model encompasses previous efforts byaddressing the following practical constraints: (1) limited availabilityofmultiple resources, (2)work continuity if required, and (3) specifiedlead-distance and lead-time between operations. To tackle the intrac-table combinatorial problem, whose problem spacewould grow expo-nentially as the number of activities increases, evolutionary strategieshave been employed to perform the optimization.

To be practical, the proposedmodel can address the following real-life situations: (1) crews may have variable production rates, changecomposition, and perform variable work quantities in different loca-tions; (2) crews may start and finish at different locations and skipcertain portions of the project; and (3) an activity may have multiplepredecessors and successors.

This paper comprises five sections. Section 2 describes theproposed model and its mathematical representation. Section 3 intro-duces Evolutionary Strategy and its implementation in the presentstudy. In Section 4, the performance of the proposed model is

Table 2Illustration of production plans.

No. of plan Production rate

Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7

1 80 80 80 80 80 80 20a

2 80 80 80 80 80 100b –

3 80 80 80 80 160 20a –

4 80 80 80 160 80 20a –

5 80 80 160 80 80 20a –

6 80 160 80 80 80 20a –

7 160 80 80 80 80 20a –

8 80 80 80 160 100b – –

9 80 80 160 80 100b – –

10 80 160 80 80 100b – –

11 160 80 80 80 100b – –

12 80 80 160 160 20a – –

13 80 160 80 160 20a – –

14 80 160 160 80 20a – –

15 160 80 80 160 20a – –

16 160 80 160 80 20a – –

17 160 160 80 80 20a – –

18 80 160 160 100b – – –

19 160 80 160 100b – – –

20 160 160 80 100b – – –

21 160 160 160 20a – – –

a Crew finish the remaining 20 m at 80 m/day.b Crew finish the remaining 100 m at 160 m/day.

Table 3Ditch upgrade project.

ID Activity Resource Prod. rate(M/day)

No. ofpossiblecombinations

Availability

Excavator Truck Rebarcrew

Formworkcrew

2 4 7 9

1 Removingexisting ditch

1 0 0 0 150 3442 0 0 0 300

2 Excavation 1 2 0 0 100 1442 4 0 0 200

3 Precast concrete 0 0 0 2 400 14 Rebar frame

(ditch wall)0 0 4 0 100 496710 0 5 0 1270 0 6 0 1530 0 7 0 180

5 Formwork andconcrete(ditch wall)

0 0 0 5 140 8080 0 0 7 1700 0 0 9 200

6 Form stripping(ditch wall)

0 0 0 2 50 286570 0 0 4 100

7 Formwork(ditch cover)

0 0 0 4 250 2410 0 0 5 2800 0 0 6 3100 0 0 7 3400 0 0 8 3700 0 0 9 400

8 Rebar frame(ditch cover)

0 0 3 0 150 10090 0 4 0 2000 0 5 0 2500 0 6 0 3000 0 7 0 350

9 Pour concrete 0 0 0 3 200 820 0 0 4 2500 0 0 5 300

10 Form stripping 0 0 0 5 150 30570 0 0 6 182.50 0 0 7 2150 0 0 8 247.50 0 0 9 280

11 Backfill 1 2 0 0 250 82 4 0 0 450

944 M. Hsie et al. / Automation in Construction 18 (2009) 942–949

demonstrated through a ditch upgrade project. Section 5 concludesthe present study.

2. Model formulation

Activities in continuous repetitive projects are performed by crews,i.e., workers and machines. A crew composition may have variousoptions, each of which possesses a specific production rate measuredin distance/time, e.g., km/week. To attain the greatest efficiency, theproposed model adopts the resource pooling principle. Combinationsof resources are automatically selected from a common pool to mini-mize the project duration. This is considered superior to traditional

Table 4Lead-time and lead-distance between activities.

ID Activity Precedence

LT (days) LD (m)

1 Removing existing ditch 0 502 Excavation 0 2003 Precast concrete 1 04 Rebar frame (ditch wall) 0 1005 Formwork and concrete (ditch wall) 1 06 Form stripping (ditch wall) 0 2007 Formwork (ditch cover) 0 08 Rebar frame (ditch cover) 0 109 Pour concrete 1 010 Form stripping 0 011 Backfill – –

approaches thatwould spend a significant amount of time in schedulingtens or hundreds of resources sequentially, i.e., one after the other.

The selection of crew compositions can be illustrated in the fol-lowing example. A 500-meter continuous repetitive project involvestwo successive activities A and B and requires usages of excavators,trucks, and common laborers. For technical reasons, the required lead-time between these two activities is 1 day. Table 1 lists the optionsof crew compositions and the corresponding production rates. Ifcrew composition Va1 is selected to performed activity A, it will take

Fig. 3. Illustration of activities: excavation, formwork, ditch cover.

Fig. 4. Convergence history varying mutation rates.

945M. Hsie et al. / Automation in Construction 18 (2009) 942–949

500/100=5 days. On the other hand, crew composition Vb2 will finishactivity B in 500/80=6.25 days. Since activity B proceeds slower thanactivity A, the production lines diverge and the control point is at thebeginning. The project duration is therefore 7.25 days, as shown inFig. 1(a). In practice, the duration of activity B may be rounded up to7 days if crews were paid in full days; the project durationwould thenbe 8 days.

Table 1 also lists the availability of individual resources. Due to thelimited availability, expediting the progress of an activity may actuallyprolong the project duration. Using the same example above,accelerating activity B from 80 m/day to 100 m/day would lead tothe activity duration of 5 days. Nevertheless, both activities demand10 common laborers in total, which exceeds the availability of 8

Fig. 5. Project sche

laborers. Hence, activity B has to be postponed from the dashed line tothe full line in Fig. 1(b). Consequently, the project duration islengthened to 10 days. A wiser alternative is to let activity B beperformed at 80 m/day during Day 2 through Day 5, withoutexceeding the resource availability, and increase the pace to 100 m/day afterwards. In this way, the project duration can be aptly crashedto 5+(500−4⁎80)/100=6.8 days in Fig. 1(c).

The example above illustrates the inherent complexity in schedulingcontinuous repetitive projects with limited resource availability. Here,the traditional principle “working at the same pace” advocated bythe line-of-balance approach would not lead to the optimal schedule,if limited resource availability has to be taken into consideration.

The proposedmodel is to geometrically arrange production lines ofcrews to minimize the project duration. The production lines aredrawn in a two-dimensional chart with the X-axis plotting space andthe Y-axis plotting time. The production line may be crooked as thecrew changes its production rate during progress. The location ofchange occurring is called the “change point”. The specified lead-distance will be checked at every change point to make sure it is noshorter than the horizontal distance between the production lines, asillustrated in Fig. 2(a). In contrast, the lead-time represents theshortest vertical distance between the production lines as shown inFig. 2(b).

The optimization objective is

Minimize Max Yi;mn o

8 i = 1;2:: :::n ð1Þ

where Yi,m is the Y coordinate of activity i at the last change pointm; this represents the finish time of activity i. The objective is tominimize the latest finish times among all the activities.

dule (56 days).

946 M. Hsie et al. / Automation in Construction 18 (2009) 942–949

The Y coordinate of production line i at the (j+1)th change pointis determined by adding the duration between the (j+1)th and jthchange points to the Y coordinate at the jth change point. Theduration is then determined by the set of production rates associatedwith selected crew compositions.

Yi;j + 1 = Yi;j + Ti;j 8j = 1;2; N m − 1 ð2Þ

Ti;j = Li;j = Pi;j 8i = 1;2 N ::n; j = 1;2; N m ð3Þ

where Ti,j denotes the duration of activity i between the jth and (j+1)th change points (the jth segment); Li,j is the length of the jth segmentfor activity i; and Pi,j represents the production rate of activity i in thejth segment. The production rate is determined by the selected crewsize and composition Oij based on a correspondence table.

Oi;jYPi;j 8 i = 1;2:: N n; j = 1;2; N m ð4Þ

The selected crew size and composition also determines thenumbers of required resources for each activity:

Oi;jYVi;j;q 8i = 1;2:: N n; j = 1;2; N m ð5Þ

where subscript q represents resource q used in the jth segment ofactivity i.

For any time period, the usage of each resource should not exceedits availability:

X8i

Vi;j;qVRq 8 j = 1;2 N ::m; q = 1; N Q ð6Þ

where Rq represents the availability of resource q.The X coordinate of production line i at the (j+1)th change point

equals the X coordinate at the jth change point plus the production

Fig. 6. Project sche

ratemultiplying by a unit of time. For example, if the production rate is10 km/week and the current location is 100 km, the X-coordinate atthe next week will be 100+10×1=110 km.

Xi;j + 1 = Xi;j + Pi;j × UT� �

8j = 1;2; N m − 1 ð7Þ

where UT stands for one time unit.The lead-time between production lines i and k is maintained by a

vertical distance as follows:

Yk;j − Yi;;jzLTi;k 8 i = 1;2 N ::n − 1; j = 1;2; N m; kaSucc if g ð8Þ

where the time difference between the Y coordinate of productionline i and its successor k at the jth change point must be no less thanthe lead-time LTi,k.

Similarly, the lead-distance between production lines i and k ismaintained by keeping a proper horizontal distance between twoproduction lines. Note that the succeeding production line is on theleft-hand side of the preceding production line.

Xi;;j − Xk;jzLDi;k 8i = 1;2 N ::n − 1; j = 1;2; N m; kaSucc if g ð9Þ

3. Solution procedures

Since crews are allowed to maintain the same composition andproduction rate within a time period, each production line consists oflinear segments, whose slopes are determined by the designatedproduction rates. In practice, the choices of production rates constitutea discrete problem space, so the optimization problem has a combi-natorial nature. The combinatorial nature makes the optimizationproblem difficult as the search space would grow rapidly when there

dule (34 days).

947M. Hsie et al. / Automation in Construction 18 (2009) 942–949

are hundreds of activities, each of which involves several options ofproduction rates and various types of resources.

The present study employs Evolutionary Strategies (ES) to solvethe combinatorial optimization problem. ES was preliminarily devel-oped in the 1960s in Germany [26]. In a modern version of ES, knownas the (μ+λ) strategy, μ parents produce λ offspring. In succeedinggenerations, the best of the λ offspring and μ parents survive until thenext generations [27]. As opposed to genetic algorithms (GA), the ESalgorithm is asexual since it requires only the mutation operatorwithout the process of selection and crossover. With fewer tuningparameters to adjust, the ES algorithm is considered simple, and yetable to find competitive solutions of combinatorial problems in areasonable amount of computing time [28,29]. Moreover, with noselection and crossover, the ES algorithm is easier to program andtakes less execution time for the same generations.

The proposed algorithm takes the following input: (1) projectlength, (2) work contents of activities and inter-precedence relation-ships, (3) feasible crew compositions and production rates for everyactivity, and (4) availabilities of resources.

The algorithm includes six steps as follows:

1. Initialize a population with np chromosomes.2. Compute the individual fitness values, i.e., project duration, of

chromosomes.3. Select the best ne individuals as the elite parents.4. Perform mutation on each elitist to produce no offspring.5. With ne×no offspring and ne parents, return to Step 2 and continue.6. Stop when the termination criterion is met.

The coding of chromosomes is to reflect the selection of productionrates. A chromosome string contains n genes, which represents achoice of production plan for n individual activities. For example, thechromosome string of {9, 48, 36} denotes that the 9th production plan

Fig. 7. Project sched

of activity A is selected, alongwith the 48th production plan of activityB and the 36th production plan of activity C. The production plan, onthe other hand, is a permutation set of feasible production rates tocomplete one activity. Suppose activity A can be performed at twodifferent production rates: 80 or 160 m/day in a 500-meter longproject. There would be 21 possible production plans, as shown inTable 2. Taking the second plan as an example, the crew works at80 m/day for 5 days and finishes the remaining 100 m (500−80⁎5=100) at 160 m/day in the 6th day.

Themutationmechanism in Step 4 is controlled by amutation rate.A uniformly random number between 0 and 1 is generated for eachgene. If the random number is greater than the mutation rate, thecorresponding gene would mutate to one of the other productionplans. For instance, a set of random numbers {0.16, 0.88} is generatedwhile the mutation rate is 0.2, then only the first gene would bemutated. It is evident that broader exploration is encouraged by ahigher mutation rate.

The termination criterion can be of various forms, such as maxi-mum generations or allowable running time. The optimization canalso be stopped earlier when improvement between generations isonly marginal, i.e., reaching convergence.

4. Practical examples

The proposed algorithm is tested on a real-life ditch upgradeproject. The project is 1093 m in length and includes 11 activities.Table 3 lists the activities, related resources, and feasible options ofproduction rates. Table 4 indicates the required lead-times or lead-distances between activities, which are performed in a sequentialmanner, one after another. Note that the proposed algorithm is notrestricted to sequential activities; it can be used to model projectswith multiple activity chains. Fig. 3 shows the pictures of three

ule (29.6 days).

948 M. Hsie et al. / Automation in Construction 18 (2009) 942–949

major activities during construction: excavation, formwork of con-crete U-ditch, and ditch cover.

The proposed model has been programmed into a stand-alonesoftware package, which is used to schedule the project. The numbersof initial chromosomes, elite parents in each generation, and offspringgenerated from each elite parent are set to be 5000, 100, and 50,respectively. After 500 generations, the optimal solution is to com-plete the project in 29.6 days. By varying the mutation rate, Fig. 4compares the convergence history. The history beyond 35 days isomitted to place emphasis on the optimization result below 35 days.Before optimization, random solutions give the project duration of56 days, as shown in Fig. 5. The mutation rate of 0.19 yields the fastestconvergence and the overall best result, which improves the projectduration to 29.6 days. Figs. 6 and 7 plot the schedules duringoptimization when the project duration is improved to 34 days and to29.6 days. Also shown on the right of the schedules are the resourcehistograms that are automatically generated by the developedsoftware package. The histograms are helpful to check the variabilityof resource utilization. Observe that the resource usage has also beenleveled during optimization.

At thefirst sight, the project is quite straightforward as each activitypossesses only 1 to 5 feasible options of production rates. This level ofcomplexity is considered realistic and consistent with practice. Yet,despite the relatively small size, the problem space of this project isalready huge given that it comprises more than 1028 feasible solutions,each of which represents a combination of production plans forindividual activities in different time periods. This shows that due tothe combinatorial nature, even a small tomedium linear project wouldinvolve an intractable number of possible resource combinations.

The numbers of enumerative composition of production plans areshown in the last column of Table 3. Note that it is computationally

Fig. 8. Original project s

infeasible to find the true optimal solution, with such a huge problemspace. The proposed model, in contrast, searches only 2.5×106 solu-tions to find the optimal solution. The comparison establishes theefficiency of the proposed algorithm; it searches only an extremelysmall portion of the problem space (in the order of 10−21). Moreover,it is worthwhile to reemphasize that without selection and crossover,the ES algorithm is easier to program and takes less execution time.

Fig. 8 plots the original schedule developed by the site manager.The underlying principle is to let each activity be performed at aconstant pace. By comparing schedules in Figs. 7 and 8, it is apparentthat choosing a proper combination of production rates can effectivelyshorten the project duration. Overall, the proposed model improvesthe original schedule by reducing the project duration from 38.4 daysto 29.6 days, which denotes a momentous 30% improvement.

Whereas the improved schedule can effectively minimize theproject duration, it may be at the cost that crews would experienceacceleration and de-acceleration during the project period. To remedythis “ramp up and ramp down” scenario, the proposed model ensuresthe change of speedwould always occur at the start, not in themiddle,of a time period (could be a day or a week), thereby minimizinginterruption. In practice, the site manager can specify the intendedproduction rate at the end of previous day and stick to it during thenext whole day. According to the feedback from the site manager, ourtime-based model is much more practical than traditional location-based optimization techniques, which may ask crews to changespeed several times in the middle of a day when they pass differentlocations.

The proposedmodel relies on stochastic search, so it is necessary tovalidate whether the model is stable, i.e., can it always find goodsolutions or just by chance? In this regard, the proposed model is runfor 50 replications. The mean value of the solutions is 31.4 days while

chedule (38.4 days).

949M. Hsie et al. / Automation in Construction 18 (2009) 942–949

the standard deviation is 0.32 days. We can then draw a statisticalconclusion that the 99% confidence interval of the solution is between31.28 and 31.52 days. That is, the probability for the proposed modelto find a solution greater than 31.5 days is no more than 1%. Theevidences strongly support the stability of the proposed model.

5. Conclusion

The present study formulated a scheduling model for continuousrepetitive projects. The proposed model selects production rates, inreflection of crew composition and size, for individual activities tofind the minimum project duration under the constraints of limitedresources. Unlike conventional approaches, the proposed modeldivides production units in time periods instead of space segments.The advantage of this treatment is two-fold: (1) crews can maintainthe same size and composition in each time period to attainmaximumefficiency; and (2) production units can directly match the time-basedpayment schedule to save unnecessary administrative work. Theproposed model complies with work continuity and specified lead-distance and lead-time betweenproduction lines. The proposedmodelincorporates an ES algorithm, which is easy to program and takes lessexecution time. The proposed model has been shown effective,efficient, and stable in scheduling a real-life ditch upgrade project.

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