A GA optimization model for workgroup-based repetitive scheduling (WoRSM)

Advances in Engineering Software 40 (2009) 212–228

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Advances in Engineering Software journal homepage: www.elsevier.com/locate/advengsoft

A GA optimization model for workgroup-based repetitive scheduling (WoRSM) Rong-yau Huang a,1, Kuo-Shun Sun b,* a b

Institute of Construction Engineering and Management, National Central University, No. 300, Jhongda Road, Jhongli City, Taoyuan County 32001, Taiwan, ROC Department of Air Transportation, Kainan University, No. 1, Kainan Road, Luzhu, Taoyuan County 33857, Taiwan, ROC

a r t i c l e

i n f o

Article history: Received 23 March 2007 Received in revised form 19 November 2007 Accepted 19 January 2008 Available online 11 June 2008 Keywords: Workgroup-based repetitive scheduling Genetic algorithm Optimization

a b s t r a c t Most construction repetitive scheduling methods developed so far have been based on the premise that a repetitive project is comprised of many identical production units. Recently Huang and Sun [Huang RY, Sun KS. Non-unit based planning and scheduling of repetitive construction project. J Constr Eng Manage ASCE 2006;132(6):585–97] developed a workgroup-based repetitive scheduling method that takes the view that a repetitive construction project consists of repetitive activities of workgroups. Instead of repetitive production units, workgroups with repetitive or similar activities in a repetitive project are identified and employed in the planning and scheduling. The workgroup-based approach adds more flexibility to the planning and scheduling of repetitive construction projects and enhances the effectiveness of repetitive scheduling. This work builds on previous research and develops an optimization model for workgroup-based repetitive scheduling. A genetic algorithm (GA) is employed in model formation for finding the optimal solution. A chromosome representation, as well as specification of other parameters for GA analysis, is described in the paper. A sample case study is used for model validation and demonstration. Results and findings are reported. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction The idea of repetitive scheduling originated in the manufacturing industry, in connection with the use of mass production line units. These production line units are identical. Most of the construction repetitive scheduling methods developed so far also maintain the premise that a repetitive project consists of many identical production units. A unit network is employed to represent production activities and is repeated for production units. In the traditionally defined repetitive project, as the example shown in Fig. 1, there are two steps to define a repetitive project. At first, a unit-network is defined that is assembled by six activities and their relationships. Secondly, the traditionally defined repetitive project is assembled as the sequence of several unit-networks. Normally a crew is assigned to each of the repetitive activities in the unit network. The crew performs the same production unit activity consecutively and continuously. Each repeated activity is performed with the same sequential order of production units. In practice, however, the production units in many repetitive construction projects may not be identical. For instance, the depth and the soil conditions encountered when placing each pile will

* Corresponding author. Tel.: +886 3 3412500x6091; mobile: +886 9 28276675; fax: +886 9 45567617. E-mail addresses: [email protected] (R.-y. Huang), [email protected] (K.-S. Sun). 1 Tel.: +886 3 4227151x34108; fax: +886 3 4276408. 0965-9978/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.advengsoft.2008.01.010

not be exactly the same in a piling project; the number of manholes and the number of pipe sections will not normally be the same in a pipeline-laying project, so they cannot be classified as traditionally defined repetitive production units; the interior design of each house may differ in a multi-housing project, and therefore the required workload, as well as the time duration and cost, will differ. Furthermore, even a typical repetitive construction project with many identical production units is very likely to contain portions of work of a non-repetitive nature. Huang and Sun [1] developed a workgroup-based repetitive scheduling method in 2006. This approach takes the view that a repetitive construction project consists of workgroups of repeating activities. Instead of repetitive production units, workgroups with repetitive or similar activities are identified (as shown in Fig. 2) in a repetitive project and are employed in planning and scheduling. Activities in a repetitive project are grouped into workgroups according to their functionality by the rule that they used for the same resource group. Each workgroup contains activities that can be performed by using the resources from the same resource group, but possibly different resource usage, construction conditions, time, costs, and so on. The logical relations between the workgroups are defined. However, the sequential work order among activities in a workgroup is not fixed. In other words, there is no ‘‘hard logic” constraint for the sequential work order; activities in a workgroup can be performed in an arbitrary order as specified by the planner. Let us look at some special situations. For example, activity A1-9 and A3-5 do not have any relations, and

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B

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(a)Unit network or single repetitive network 1st unit

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F1 Relationship between units

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(b)Traditional-defined repetitive Project that combined by several unit networks Fig. 1. Example of repetitive project with the unit network sequence format (data source: Harris and Ioannou [20] and coordination of this study).

A2-A4

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(b) Physical logical relationships between each activity Fig. 2. An example of a workgroup-based repetitive project with six workgroups.

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activity A3-4, A3-7, A3-8, and A4-3 do not have any predecessors. These are very often seen in a project that combines several different kinds of work that use the same resource group. For instance, the workgroup A might include all excavation activities, the activities A-1 to A-8 are the excavation of the housing foundation but the activities A-9 are an excavation of a scenery pond outdoors. In addition, activity A3-4, A3-7, A3-8, and A4-3 are perhaps some work that was interrupted by an early project and added in this project. In a repetitive project in practice, to make a long story short, the actual situation is usually more complex than what is represented in the traditional scheme. Using the model proposed in this study, the planner can consider the relationships between activities and resource used more close to the real phenomenon. The relationships shown in Fig. 2b are not arbitrary. They represent the practical situation by actual physical constrains or logical relationships like CPM. An example of sewer project including 14 manholes and 13 pipeline segments is used to demonstrate that the more realistic representation of repetitive projects is possible. The system layout of the sewer project is shown in Fig. 3 and the relationship between activities in the sewer system project is shown in Fig. 4. In addition, multiple resource crews employing different construction methods, and/or combinations of equipment and labour may perform repetitive activities in the same workgroup, depending on the needs of the project and the market availability of work crews. For instance, in a 100-unit-housing project, two separate crews (subcontractors) employing different excavation methods can be called upon to perform the activity of excavation for the 100 housing units. Different activity time durations and costs are obtained as a result. In fact, there are many kinds of resources that can execute the same activities. For example, excavation of a housing foundation can be excavated by the power shovels that have different capacity. In general, each resource (or crew, subcontractor) in a resource group has its unique approach to complete its work, which depends on its capability, proficiency, or experience, and it will have an influence on the cost and duration of the exe-

cuted activity. Fig. 2 only sketches the map that displays the relationships between activities. The attributes of the non-identical activities in a workgroup will be shown in the data sheets of the activities’ duration and cost, which will show the differences due to each resource. Finally, to consider the use of multiple resources, the workgroup-based repetitive scheduling method takes into account the incurred cost and time of routing the resource crews among various activities. The workgroup-based method thus adds more flexibility to the planning and scheduling of repetitive construction projects and enhances the effectiveness of repetitive scheduling. This work builds on the previous research and develops an optimization model for workgroup-based repetitive scheduling. Our objective is to maximize total project NPV while conforming to the proper work sequence between workgroups in a repetitive project and maintaining the work continuity of resources. A genetic algorithm (GA) is employed in model formation for finding the optimal solution. A chromosome representation as well as specification of other parameters for GA analysis is described in the paper. A sample case study is used for model validation and demonstration. Results and findings are reported. This paper is a revised and updated version of Ref. [2]. 2. Literature review 2.1. General repetitive scheduling The idea of repetitive scheduling was designed for the planning and scheduling of mass production lines of identical units in manufacturing industries. The production process usually consists of a series of workstations equipped with resources (e.g. equipment, labourer, etc.). Resources are normally stationed on the plant floor with no movement. A typical repetitive scheduling method is the Line-Of-Balance method (LOB) [3–11], which provides a simple method for scheduling and controlling the production progress. However, its modelling and application to construction projects re-

Fig. 3. System layout of sewer project.

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Activity group:A1 Manhole Execavation

Activity group:A2 Pipeline Drilling

Activity group:A3 Manhole Construction

Resource group:R1 Resource: R1-1, R1-2

Resource group:R2 Resource: R2-1, R2-2

Resource group :R3 Resource: R3-1, R3-2

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(a) Sequence of activity groups and corresponding resource groups Activity group:A1

Activity group:A2

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A1-14 (b) Logical relationship between activities Fig. 4. Relationship between activities in sewer system project.

quires the simplification of repetitive activities so that they have a single duration, the same resource usage, and a fixed production order. This somewhat limits the applicability of the LOB method in construction, especially for more diversified and complex repetitive projects. Subsequent construction repetitive scheduling methods [12– 21] have tried to ease the single duration constraint for repetitive activities. Production units need not be identical, but merely similar. El-Rayes and Moselhi [18] for example, designated operations by a work group with the same duration as a ‘‘Typical Repetitive

Activity” and one with a different duration as a ‘‘Non-typical Repetitive Activity”. They also pointed out that, since ‘‘Non-typical Repetitive Activity” is common in repetitive projects, it is thus inadequate to treat repetitive operations within one work group as the same, and the difference between each work item should be considered during pre-construction planning. Nonetheless, all these methods are still based on the premise that a repetitive project involves repetitive processing units. Huang and Sun [1] developed a workgroup-based repetitive scheduling method. The method takes the view of repetition in activities in

workgroups, and provides more flexibility in the planning and scheduling of repetitive construction projects. A comparison of the repetitive scheduling methods has been developed, with each of them featuring unique functions and/or applications, as shown in Table 1. A brief description of the workgroup-based repetitive scheduling method is included in the next section. Yes No Yes No No Yes

Yes Yes Yes Suggested Suggested Yes Yes

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No No No No No Yes

Carr and Meyer [4] O’Brien [5] Selinger [12] Johnston [13] Stradal and Cacha [32] Arditi and Albulak [7] Chrzanowski and Johnston [17] Reda [9] El-Rays and Moselhi [18] Harmelink and Rowings [34] Harris and Ioannou [20] Hegazy and Wassef [11] Huang and Sun [2]

Yes No Yes Yes Yes No Yes Yes Yes Yes Yes No RPM Resource-driven scheduling Linear scheduling model RSM Repetitive non-serial activity scheduling WoRSM (origin: non-unit-based repetitive project scheduling)

No Yes No Yes No Yes

No No No No No No No Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes LOB VPM Const. planning LSM Time space scheduling LOB LSM

No No Yes Yes Yes No Yes

Fixed work sequence (4) Unit-based (3) Method (2) Author(s) (1)

Table 1 Comparison of the repetitive project scheduling methods

Non-typical activity (5)

Multiple resource assignment (6)

2.2. Optimal methods for scheduling Many scheduling techniques have been created to solve linear type repetitive scheduling problems. These techniques reveal their limitations, however, in solving the optimal scheduling of a project [22]; optimal solutions generally are not guaranteed. Analytical methods, such as linear and dynamic programming, have been adapted to find optimal solutions to linear scheduling problems. Nevertheless, they cannot solve large and complex problems effectively, and are incapable of finding optimal scheduling of workgroup-based repetitive projects. GAs are directed randomized search procedures. They derive their power from the mechanics of natural selection and the survival-of-the-fittest principle [23]. GAs have been widely used in many areas such as constrained or unconstrained optimization, scheduling and sequencing, transportation, reliability optimization, artificial intelligence, and many others [23]. Some research has been done in the optimization of construction and manufacturing scheduling using GAs [24–28]. Hegazy and Wassef [11] used a GA to solve a repetitive non-serial project with cost optimization. Leu and Hwang [29] proposed an optimal repetitive scheduling model for precast production using a GA-based search technique. In addition, the goal of most repetitive project scheduling optimization techniques is to complete the project at the earliest possible time or at minimum cost, subject to the constraints of the logical order of activities and the availability of required resources [11]. Several researchers [30,31] have argued that the proper goal of project scheduling should be to maximize the net present value (NPV) of the project using the sum of positive and negative discounted cash flows (DCF) throughout the life cycle of the project [32]. The purpose of the paper is to present information about how to plan and control resource allocation and activity alignment with regard to a repetitive construction project optimally. A new optimal workgroup-based repetitive scheduling model based on the concept of GA is proposed in this work. The goal of optimization is to maximize the net present value of the overall project while considering resource constraints, activity alignment, and resource continuity.

3. Workgroup-based repetitive project scheduling Basic scheduling procedures for workgroup-based repetitive projects are described below. Readers are referred to Huang and Sun [1] for more detail. Step 1. Identification of workgroups and their sequential relationships. Activities in a repetitive project are grouped into workgroups according to the rule that they used the same resource group. It is also possible to designate separate workgroups for each non-repetitive activity. In addition, a network describing the sequential relationship of all the workgroups is created. Fig. 2 shows an example of a workgroup-based repetitive project consisting of six workgroups. Fig. 2a depicts the project network including the workgroups; Fig. 2b shows the detail logical relationships among activities in workgroups.

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Step 2. Development of routings of resources. When scheduling the workgroup-based repetitive project, two major decision variables for each activity, resource assignment and priority for activities within a workgroup, must be preset by the planner. Since multiple resource crews can be employed for work, each workgroup is assigned a resource group, which includes the available resource crews for that particular workgroup. As shown in Fig. 5c, resource group 2, including R2-1 and R2-2, is assigned to work on activities in workgroup 2. Once the scheduler has decided on the resource assignment and the priority alignment of activities in a workgroup, routings of resource for each resource type can be determined and used for scheduling. A resource crew is mobilized from outside to work continuously and sequentially on the assigned activities in accordance with activity priority, and eventually demobilized after the work has been completed. In Fig. 5d, for instance, resource type R2-1 is assigned to work

on activities A2-1, A2-3, and A2-4. The pre-determined priority for activities in workgroup 2 is A2-1, A2-2, A2-3, and then A2-4. Therefore, resource R2-1 will first perform A2-1, then A2-3 and finally A2-4. A routing of resource for resource R2-1 including the mobilization, demobilization, and all movement between activities is developed and shown in Fig. 5d. Step 3. Scheduling of each routing of resource and the project. After the formulation of routings of resource, the following subsequent steps are executed to schedule each routing of each resource, following the work order of workgroups: 1. Calculation of a base schedule: By setting the time for a resource entering the project site to day 0, the start and the finish time of each action in the routing of resource can be calculated. For instance, Fig. 5e shows the work sheet for positioning the routing of resource R2-1. Row (1) and (2) list all the actions in the routing and their respective durations. By setting day 0 as the

Unit 23

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(a) Project Schedule forn routing of Resource Preceding R2-1

(c) Resource and priority assignment

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5 10 Time (d) Routing of Resource R2-1 on Base Schedule

In-> A2-1 2 0 2

A2-1-> A2-3

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(e) Calculation Table for positioning Routing of Resource R2-1 from Base Schedule to total project schedule

Out

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22-5=17 (Max) 22 26

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Legend Move activity

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(f) Positioning Current Routing of Resource into total Project Schedule Fig. 5. Illustration of routing of resource scheduling (R2-1).

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starting day for mobilization of resource R2-1 to the jobsite, the base start and finish days of actions in the routing of resource R2-1 can be calculated, as shown in row (3). 2. Calculation of the earliest possible starting time of each activity: The earliest possible starting time of an activity can be determined based on the latest finish time of the preceding activities. Following the example of routing of resource R2-1, we see in Fig. 5b the logical relationship between activities in workgroups 1 and 2. The preceding activities for A2-1, A2-3, and A2-4 are identified and listed in row (4) of Fig. 5e, respectively. In addition, Fig. 5a shows the scheduling of the routings of resource in the preceding workgroup. In this case, since only one resource performs activities in workgroup 1, there is only one routing of each resource developed for workgroup 1. The final finishing times of the preceding activities for A2-1, A2-3, and A2-4 are calculated from Fig. 5a, and shown in row (5) of Fig. 5e. Furthermore, the latest finish time of the preceding activities is also the earliest start time of the activity. 3. Determination of the earliest possible start time for the routing of resource: The earliest possible start time for the routing of resource R2-1 can be calculated and determined by subtracting the start time of an activity in the base schedule from its earliest start time from sub-step 2. The earlier possible time for the routing of resource to start can be determined by checking the schedule of all the preceding activities for each activity in the routing of resource. For example, for routing of resource R2-1, the earliest possible starting time for activity A2-3 is on the 22nd day (row (5) of Fig. 5e). Given the base schedule, the starting time of activity A2-3 will be on the 5th day (row (3)). Therefore, the earliest possible start time for routing of resource R2-1 is on the 17th (22-5) day, based on the constraints for activity A2-3. In a similar fashion, the earliest possible start times for routing of resource R2-1 are calculated based on activities A2-1 and A2-4, and are the 16th and the 8th day, respectively. Thus, the 17th day (the highest of the 17th, 16th, and 8th days) becomes the earliest possible starting time for routing of resource R2-1. Meanwhile, since the 17th day was originally governed by activity A1-5, a controlling logical relationship is

Resource group 1

1

formed between activity A1-5 and A2-3, which in turn establishes a critical logical constraint between the two routings of resource (Fig. 5f). 4. Calculation of the project schedule for the routing of resource: With the determination of the earliest possible starting time of a routing of resource from sub-step 3, the start and finish time of each work action in the routing of resource can now be calculated, in accordance with their durations. Calculation results are shown as row (7) of Fig. 5e for the example of routing of resource R2-1, and the scheduling flow is shown as Fig. 5f. 5. Repetition of sub-steps 1-4 for each routing of resource in the project, following the sequential order of the workgroups until they are all scheduled. 4. Development of a GA-based optimization model The genetic algorithm (GA) method is employed in this study to develop an optimization model for workgroup-based repetitive scheduling. In GA, potential solutions to a problem are represented as a population of chromosomes in the GA method, and each chromosome stands for a possible solution. The chromosomes evolve through successive generations. The offspring chromosomes are created by merging two parent chromosomes using a crossover operator, or modifying a chromosome using a mutation operator. During each generation, the chromosomes are evaluated on their performances with respect to fitness functions. Fitter chromosomes have higher survival probabilities. After several generations, chromosomes in the new generation may be identical, or certain termination conditions may be met. The final chromosomes hopefully represent optimal or near-optimal solutions to a problem. The genetic algorithms approach was distinctive because it was interested in formally modelling the process of evolution rather than just solving engineering problems. The mapping of the problem representation to reality could be seriously lacking, such as when solved some engineering problem by Laplace transform or Fourier transform.

Resource group 2

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Chromosome-segment Chromosome-segment for activity group 2 for activity group 3 Chromosome for total project

Fig. 6. Illustration of a chromosome for a repetitive project with three workgroups.

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The GA method consists of four major elements: chromosome representation, application of genetic operators, fitness calculation, and selection/replacement strategy.

Fig. 6 shows an example of the chromosome representation for a repetitive project involving three workgroups. There are three segments in the entire chromosome, with segments 1, 2, and 3 representing the workgroups 1, 2, and 3, respectively. For workgroup 1 there are five activities and two available resources. Five gene couples represent the assigned resource number and work sequence number for activities 1–5. In this case, resource 1 is assigned to work on activities 1, 4 and 5, and resource 2 on activity 2 and 3. Activity 1–5 is performed, respectively, in the sequence of second (2), first (1), fourth (4), third (3), and fifth (5). Segments for the workgroup 2 and 3 are encoded in a similar fashion. No infeasible solution can be generated during the process of evolution with this design of chromosome representation, and substantial computation time spent searching for the optimal solution of the problem is saved.

4.1. Chromosome representation The chromosome is designed to represent the two decision variables in workgroup-based repetitive project scheduling, which are assignment of resources and determination of work priority for activities in each workgroup. The chromosome consists of the same number of segments as the number of workgroups identified in the repetitive project. In each chromosome segment, a pair of genes (gene couple) represents the assigned resource number and work sequence number of an activity. The numbers of gene couples in a chromosome segment is therefore equal to the number of activities in the corresponding workgroup.

Parent chromosome 1 Resource Assignment Genes Sequential Priority Genes

Parent chromosome 2

Workgroup 1

Workgroup 2

Workgroup 3

Workgroup 1

Workgroup 2

Workgroup 3

1 2 2 1 1 2 1 4 3 5

1 2 1 1 2 3 4 1

2 3 2 1 1 3 5 2 1 4

2 2 1 1 2 3 2 5 1 4

1 2 2 1 4 3 1 2

1 3 3 2 1 5 1 3 4 2

1 2 1 1 2 3 4 1

2 3 2 1 1 3 5 2 1 4

external crossover point

New chromosome 1

New chromosome 2

1 2 2 1 1 2 1 4 3 5

1 2 2 1 4 3 1 2

1-1

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2 2 1 1 2 3 2 5 1 4

2-2

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Fig. 7. Illustration of an external crossover operation.

Parent sub-chromosome 1

Step 0: initial sub-chromosome

Resource Assignment Sequential Priority

Internal crossover for Resource Assignment Step 1a: Select crossover point

Step 2b: Crossover operation 1.1-point crossover 2.n-point crossover 3.uniform crossover

Step 3c: combination the new strings

Step 4: combination the new sub-chromosome s

Activity group 1

1 2 2 1 1 2 1 4 3 5

2 2 1 1 2 3 2 5 1 4

Resource Assignment Part

1 2 2 1 1

Parent sub-chromosome 2

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1-point crossover as the example

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Order

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Step 2b: Order Crossover operation

1 1 2

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New Resource Assignment Part 1

New Resource Assignment Part 2

New sub-chromosome 1

1 2 1 1 2 5 1 4 3 2

5 1 4 3 2 New Sequential Priority Part 1

New sub-chromosome 2

2 2 2 1 1 5 1 4 3 2

Fig. 8. Illustration of an internal crossover operation.

5 1 4 3 2 New Sequential Priority Part 2

Step 3c: combination the new strings

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4.2. Genetic operators The crossover of genes is performed in two parts: external crossover with a probability rate of Pec and internal crossover with a probability rate of Pic. External crossover applies a single point scheme on segments (workgroups). A connection between segments is randomly selected and all the segments after the connection point are exchanged between two chromosomes. An example of the external crossover with three workgroups is depicted in Fig. 7. The crossover point in this example is the connection between workgroups 1 and 2. Segments 2 and 3 in the respective chromosome are exchanged. On the other hand, internal crossover is applied for each of the workgroup segments with a crossover rate of Pic. It includes both crossovers of genes of resource assignment and of genes of activity work priority. A single point scheme is again applied for the crossover of genes of resource assignment. However, a conventional order-based scheme is applied for the crossover of genes of activity work priority. This is done by randomly selecting an arbitrary part from the first parent and copying this to the first child. The remaining genes that are not in the copied part are then sequenced to the first child according to the order of the exact same genes on the second parent. Finally, the sequenced genes are placed in the first child starting immediately from the end of the copied part and wrapping around the copied part. The same procedure is repeated with the roles of the first and second parents exchanged to produce the other child. Fig. 8 demonstrates an internal crossover for workgroup 1. Similarly, mutation is also performed in two parts: external mutation with a probability rate of Pem and internal mutation with a probability rate of Pim. Internal mutation is applied to each segment of a chromosome. For mutation of the resource assignment genes, a gene is randomly selected and its resource assignment is changed to a randomly selected one from the rest of the available resources. On the other hand, for mutation of activity work priority genes, two genes are randomly selected and their assigned work orders are swapped. Fig. 9 shows an illustration of an internal mutation for a segment (workgroup).

As in the case of external mutation, one segment is randomly selected and the same procedure as for internal mutation is applied. 4.3. Fitness calculation The objective of the optimization model is to maximize the revenue of a repetitive project. A net present value of the project revenue is employed as the fitness value for GA computation. The project’s NPV is calculated from its discounted cash flows under a minimum attractive rate of return (MARR) preset by the planner. The cash flow of the project includes cash inflow and outflow incurred in each payment period throughout the entire project. The cash inflow of each payment period is usually described by the project contract; it typically includes periodic payments for completed work and other conditional payments such as the advance payment, return of retaining fee, and return of warranty fee. For its part, the cash outflow of each payment period includes the direct cost and indirect cost of the project, including the expense of routing the resources between activities. In addition, the penalty for delay of the project is also included in the calculation of the project NPV. The project NPV must be calculated based on the characteristics of the specific project contract. Fig. 10 shows the calculation worksheet for the project NPV. The calculations are based on scheduling results and consist of three steps, which are described in detail below. Step 1: Cash outflow calculation: Based on the given schedule, we can find the cash outflow by summing the resource implementation/movement/entry cost Ai, the activity direct cost Bi, the resource-related indirect cost Ci, the workgroup-related indirect cost Di, the project-related indirect cost Ei, and the penalty for project delay Fi over each payment period. In addition, the contractors usually deposit a warranty fee with the owner following the final approval of the project.

Parent sub-chromosome 1 Activity group 1 Step 0: initial sub-chromosome

Resource Assignment Sequential Priority

Internal mutation for Resource Assignment Resource Assignment Part

1 2 2 1 1 2 1 4 3 5

Sequential Priority Part

Step 1a: Select mutation point

1 2 2 1 1

2 1 4 3 5

Step 2a: Mutation operation

1 2 1 1 1

2 5 4 3 1

Step 3: combination the new sub-chromosome

New sub-chromosome

1 2 1 1 1 2 5 4 3 1 Fig. 9. Illustration of an internal mutation operation.

Internal mutation for Sequential Priority Step 1b: Select mutation pairs

Step 2b: Order mutation operation

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221

Fig. 10. Calculation worksheet for project NPV.

Step 2: Cash inflow calculation: Based on the given schedule, the project is paid in each payment period for the completed activities Ii. Nevertheless, the actual payment Ji must take into account the overheads, the advance payment, if any, the retaining fee, and the time lag of the payment. Therefore, the actual payment Ji is equal to Ii * (1 + overhead cost ratio) * (1 advanced payment ratio retaining fee ratio), and the payment will be received in the next period or whatever period is specified in the contract. In addition, the advance payment (K), if there is one, is usually paid by the owner at the beginning of the project. The deposited retaining fee (L) is returned to the contractor at time of the last payment. The deposited warranty fee (M) is returned to the contractor after the warranty period has expired. Step 3: Net discounted cash flow calculation: The net cash flow per period Pi can be obtained by subtracting the cash outflow Ni from the cash inflow Ii. The net cash flow Pi is then discounted by the given MARR (minimum attractive rate of return) to the present value Qi. The net present value R is then obtained by summing up Qi in each period. The cash flow items and payment conditions used in NPV calculations may vary depending on different project contracts. For the purpose of GA computation, all the NVP values in each generation are shifted to the non-negative range in order to perform roulette wheel selection.

The needs to meet a given deadline for a project is not preset as a constraint but implied in the optimization process by the given delay penalty. If the planner places the most importance on the given deadline, he can set a very high value in the delay penalty to avoid selecting an ill-fitting schedule.

4.4. GA-based solver A workgroup-based optimum repetitive project scheduling system (WoRSM) is developed by combining the above-mentioned GA processes. This system consists of three modules: new chromosome generation, fitness calculation, and selection. Each module has its own function. The new chromosome generation module is responsible for randomly generating the initial population and performing crossover and mutation of each generation. The fitness calculation module is responsible for computing the NPV of each solution (chromosome). The calculated fitness values are then used for the selection of mating chromosomes for generating the next generation. The roulette wheel selection mechanism and replacement elitism are employed. Fig. 11 shows an operation flowchart for the WoRSM model. The WoRSM model was implemented using Borland C++ Builder 6.0 Enterprise Edition with the BDE (Borland Database Engine). The computing platform was a PC with a Pentium IV 2.0 GHz microprocessor, 256;MB RAM, and Microsoft Windows XP operating system.

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New chromosome Generation Sub-System

Start Generate Initial Population

N

External Crossover ?

Internal Crossover ?

Y

Y Internal Crossover

Fitness Calculation Sub-System Fitness Calculation

External Crossover

N

Optimum Result Output

End

Fitness Evaluation

N

External Mutation ?

N

N

Internal Mutation ?

Elitism ? Y

Y Y

External Mutation

Elitism Selection

Internal Mutation

Select Sub-System Selection/ Replacement

Terminated ?

Y

N Fig. 11. GA WoRSM operation flowchart.

5. The sample cases study

Table 2 Basic project data for sample case

5.1. A simple sample case A simple sample case, as shown in Fig. 12, has three workgroups, 14 activities, and 16 logical relationships. At first, the simple resource assignment is specified for each workgroup to examine the efficacy of the optimization scheme. Only one resource crew is made available to work on each of the workgroups 1 (R1-1) and 3 (R3-1), and two resource crews are made available on the workgroup 2 (R2-1 and R2-2). Subsequently the sample case will be tested in three scenarios to verify the developed WoRSM. The basic project and the workgroup data for the sample case are shown in Tables 2 and 3, respectively. The durations and the cost for each activity, as well as the action of the resource movement, are shown in Tables 4 and 5, respectively. It should be noted

Workgroup 1

Workgroup 2

A 3-1 A 2-1

A 1-2

Data

Unit

Project duration

35

Contract price

1800

Time units Cost units

Delay penalty ratio (per time unit) Maximum delay penalty Contractor’s project overhead cost

0.30% 9.00% 10

MARR for the project Ratio for paying the contractor’s overhead and markup Duration per payment period

10% 20%

Lag of periods for receiving payment Ratio of advance payment Number of periods needed for the project’s final approval Ratio of retained fee

2 20% 2

Ratio for warranty fee Number of warranty periods

5% 10

A 3-2

A 1-4 A 1-5

10%

Time units Periods Contract price Periods Payment a period Contract price Periods

Number of activity

Resource code (indirect cost, $/ time unit)

Overhead cost of contractor ($/ time unit)

Contract unit price for each activity finished ($)

A1 A2

5 4

1 2

80 150

A3

5

R1-1(1) R2-1(0.5) R2-1(0.5) R3-1(2)

1.5

100

A 3-4 A 2-4

Direct cost

Workgroup

A 3-3 A 2-3

$/time unit

Table 3 Workgroup data for sample case

A 2-2 A 1-3

Contract price Contract price

10

Workgroup 3

(a) Sequence of workgroups A 1-1

Item

A 3-5 (b ) Logical relationship between activities

Fig. 12. Basic logical relations for activities in sample case: (a) sequence of workgroups; (b) logical relationship between activities.

that in both tables the time or cost from activity x to the same activity x means the duration or cost for activity x. For instance, the time ‘‘from activity A1-2” ‘‘to activity A1-2” in Table 4 is 2 time units, which represents the duration of activity A1-2.

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R.-y. Huang, K.-S. Sun / Advances in Engineering Software 40 (2009) 212–228 Table 4 Duration data for activity and resource movement in sample case

Table 5 Cost data for activity and rsource movement in sample case

To activity ? (a) Resource R1-1 From activity

In 1-1 1-2 1-3 1-4 1-5

To activity

Out – 1 1 1 1 1

1-1

1-2

1 2 1 1 1 1

1-3

1 1 2 1 1 1

1-4

1 1 1 2 1 1

1 1 1 1 2 1

1-5 1 1 1 1 1 2

(a) Resource R1-1 From activity

To activity

(b) Resource R2-1 From activity

Out

2-1

2-2

2-3

2-4

In 2-1 2-2 2-3 2-4

– 1 1 1 1

1 6 1 1 1

1 1 6 1 1

1 1 1 6 1

1 1 1 1 6

In 2-1 2-2 2-3 2-4 To activity ?

(d) Resource R3-1 From activity

In 3-1 3-2 3-3 3-4 3-5

2-1

– 1 1 1 1

Out – 1 1 1 1 1

1-1

1-2

1-3

1-4

1-5

In 1-1 1-2 1-3 1-4 1-5

– 15 15 15 15 15

15 30 15 15 15 15

15 15 30 15 15 15

15 15 15 30 15 15

15 15 15 15 30 15

15 15 15 15 15 30

(b) Resource R2-1 From activity

?

Out

2-1

2-2

2-3

2-4

In 2-1 2-2 2-3 2-4

– 15 15 15 15

15 75 15 15 15

15 15 75 15 15

15 15 15 75 15

15 15 15 15 75

Out

2-1

2-2

2-3

2-4

– 15 15 15 15

15 75 15 15 15

15 15 75 15 15

15 15 15 75 15

15 15 15 15 75

To activity Out

?

Out

To activity

?

To activity

(c) Resource R2-2 From activity

?

2-2

1 6 1 1 1 3-1 1 2 1 1 1 1

1 1 6 1 1 3-2 1 1 2 1 1 1

2-3 1 1 1 6 1

3-3 1 1 1 2 1 1

2-4 1 1 1 1 6

3-4 1 1 1 1 2 1

? (c) Resource R2-2 From activity

3-5 1 1 1 1 1 2

There is a total of 5,529,600 ((5!*1ˆ5)*(4!*2ˆ4)*(5!*1ˆ5)) different possible schedules in the searching space. A population size of 50, 100 generations, a crossover rate of 0.6 (both external and internal), and a mutation rate of 0.1 (both external and internal) are set for GA calculation. Fig. 13 shows the optimal schedule of the sample case. Each line represents the schedule of a routing of resource, from entering to exiting the project site. The flat segments of the line represent the elapsed times of activities, while the sloped segments depict the movement of resources between activities, or in and out of the project site. The optimum NPV to be searched is 46.0 cost units. In the optimum schedule, the project duration is 28 time units. The sequential work orders and resource assignment of activities in the project are listed as Table 6. The result is verified by exhaustive enumeration of the total possible solutions. In addition, as shown in Fig. 14, WoRSM searches through less than 0.01628% (18*50/5,529,600) of the solution space by taking 18 generations to converge on the optimal solution. The average and the minimum NPV values of all generations are improved very quickly during the initial stage, and then enter a random walk phase to search for other possible better solutions. The result demonstrates the good searching efficiency of the WoRSM system. One more resource type, R1-2, is then made available to work on activities in the workgroup 1.Three scenarios, as listed below, and are tested in the sample case. Scenario 1: The duration and cost of R1-2 are twice those of resource type R1-1. Scenario 2: The duration and cost of R1-2 are equal to those of resource type R1-1.

(d) Resource R3-1 From activity

In 2-1 2-2 2-3 2-4 To activity ?

Out

3-1

3-2

3-3

3-4

3-5

In 3-1 3-2 3-3 3-4 3-5

– 15 15 15 15 15

15 25 15 15 15 15

15 15 25 15 15 15

15 15 15 25 15 15

15 15 15 15 25 15

15 15 15 15 15 25

Scenario 3: The duration and cost of R1-2 are only half of those of resource type R1-1. The optimal schedules for the three scenarios are shown in Table 7 and Fig. 15. In Scenario 1, since resource type R1-2 is twice as expensive and requires twice the working time, project NPV decreases if R1-2 is used. As a result, R1-2 is never used in the optimal schedule, and the maximum NPV and the project duration remain the same. In the Scenario 2, since the duration and cost are the same for R1-1 and R1-2, the routings of resource R1-2 and R1-1 progress side by side and the total project duration is reduced to 25 time units (workgroup 1 is finished three time units earlier). Meanwhile, the maximum NVP rises to 61.2 because of the shortening of the project duration. In Scenario 3, since resource type R12 is less expensive and requires less working time, all activities in workgroup 1 are executed by R1-2. The project duration falls to 25 time units, the same as the one in Scenario 2. Meanwhile, the maximum NVP of the project with the optimal schedule rises to 198.2 units. The NPV of Scenario 3 is so high because the lower cost of resource, R1-2, is selected automatically by the GA optimum method that proposed by the paper. The cost of the resource R1-2 is only half of the resource R1-1. The decision support regarding the number of work crews is implied in the optimization results. In the three scenarios of the first sample, although two resources, R1-1 and R1-2, are involved in the R1 resource group, only R1-1 is adopted in Scenario 1 and R1-2 is adopted in Scenario 3. It has already provided the verified evidence that the optimization results will be a strong decision support regarding the number of work crews.

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Fig. 13. Scheduling result for sample case.

Table 6 Work orders and resource assignment in optimal schedule for sample case Workgroup activity no.

Sequence Resource

A1

A2

A3

1

2

3

4

5

1

2

3

4

1

2

3

4

5

3 1

4 1

5 1

1 1

2 1

3 1

2 2

4 1

1 2

2 1

3 1

5 1

4 1

1 1

Fig. 14. WoRSM NPV output for sample case.

5.2. A sewer project case A more practical case, a sanitary sewer network construction project that includes 14 manholes and 13 pipeline segments, is

employed to validate the developed algorithm. The construction layout is shown in Fig. 3. In the project, there are three activity groups, include manhole excavation (A1), pipeline drilling (A2), and manhole construction (A3), respectively. The contents of the

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R.-y. Huang, K.-S. Sun / Advances in Engineering Software 40 (2009) 212–228 Table 7 Optimal scheduling result for sample case with three scenarios Scenario no.

Workgroup activity no.

A1

A2

1

2

3

4

5

1

2

3

4

1

2

3

4

5

1

Sequence Resource

5 1

4 1

1 1

2 1

3 1

4 2

3 1

1 1

2 2

5 1

3 1

1 1

4 1

2

Sequence Resource

3 1

5 2

1 2

2 1

4 2

2 1

4 2

1 2

3 1

2 1

3 1

1 1

3

Sequence Resource

4 2

3 2

1 2

2 2

5 2

3 1

1 2

2 1

4 2

3 1

2 1

1 1

activity groups and their logic relationships are shown in Fig. 4; it is a summary of the construction layout for scheduling purpose. There are two resources with different capacity used for each resource workgroup in this case. The total data of project is too large to show in the article. The readers can see the detail data from the doctoral dissertation of Sun [33]. There are three sample scenarios suggested by the field engineer and experts, as listed below, that are tested to compare with the optimum result found by the WoRSM. Table 8 shows their respective data input. Scenario 1: Because the single operation items in workgroup A1 (manhole excavation) and workgroup A3 (manhole construction) generally consume less time than workgroup A2 (pipeline installation), generally we adopt the principle of production line balance, so that the working speeds of all operation items are about the same. The actual approach for the principle of production line balance is to adopt fewer resources in the operation items that consume less time for a single operation, and adopt more resources in the operation items that consume more time in single operation. Therefore, in this scenario, workgroup A1 (manhole excavation) uses only resource R1-1, workgroup A3 (manhole construction) uses only resource R3-1, and workgroup A2 (pipeline installation) uses resources R2-1 and R2-2, with alternative work depending on their codes; resource R2-1 for operation items coded in odd numbers, and resource R2-2 for operation items coded in even numbers. All workgroups are prioritized for work according to their codes of operation items. The scheduling result shows its NPV at 160048.93, and the project working duration is 70.97 days. Scenario 2: This scenario also adopts the principle of traditional production line balance, but the resource used for workgroup A1 (manhole excavation) is changed to R1-2, the resource used for workgroup A3 (manhole construction) is changed to R3-2, and workgroup A2 (pipeline installation) adopts the same resource designation as scenario 1. Prioritizing of work of operation items in various workgroups in this scenario are exactly the same as in scenario 1. The scheduling result shows this NPV is 406993.37, and the project work duration is 72.36 days. Scenario 3: Scenario 3 is a consideration of regional working divisions. The operation items are divided into two zones according to the resources in use, with the manhole MH-011 as the borderline. The pipes and manholes connected to the left side of manhole MH-011 are working zone 1 (including manhole MH-011), the resources in use in workgroups A1, A2 and A3 are,

A3

NPV

Duration

2 1

46.0

28

4 1

5 1

61.2

25

4 1

5 1

198.2

25

respectively, R1-1, R2-1 and R3-1. The pipes and manholes on the upper side, right side and lower side of manhole MH-011 are working zone 2 (not including manhole MH-011), and the resources in use in workgroups A1, A2 and A3 are, respectively, R1-2, R2-2 and R3-2. The principle of working prioritization is based on manhole MH-011 as the starting point. The scheduling result shows the NPV is 424497.56, and the project working duration is 66.19 days. In conducting optimization algorithm in this case, the evolution generation is given at 1000, the quantity of chromosome population in each generation is 100 pieces, the internal crossover rate and external crossover rate are set at 0.7, internal mutation rate and external mutation rate are set at 0.1, and each generation reserves five elites and executes elite replacement. The maximum NPV of this case obtained after WoRSM execution of the optimization algorithm is 454583.50, the duration required for the project is 65.41 days, and its decision variables are shown in Table 8. The generation-to-generation shrinkage in this project is shown in Fig. 16. As shown, the optimization solution obtained in the final step indeed has significant improvement compared to the result obtained in the initial generation. Following analysis of the project through the scheduling planning analysis in Scenarios 1, 2 and 3 and the optimization analysis in this section, the decision variables, project duration and NPV obtained, respectively, are shown in Table 8. According to the result of execution of this case, discussions are as below: 1. Because the repetitive project scheduling of workgroups is a combination explosion issue, though it is not difficult to obtain a possible solution (to some other types of issues, it will be extremely difficult to obtain a solution), but when the scale of issue is enlarged, the space for seeking an optimal solution is far beyond the human brain and the processing capability of existing computers. Therefore, the general rule is to adopt expert judgment or a heuristic rule, or more instinctive methods to seek relatively better and feasible plans. Scenarios 1, 2 and 3 listed in this section are the scheduling result obtained by way of expert judgment or heuristic rule. 2. Though the adoption of expert judgment or heuristic rule may conveniently and quickly obtain feasible scheduling plans, the quality of the result could not be easily controlled. Take the three scenarios of this case as examples; as shown in Table 8, the NPV of scenarios 1, 2 and 3 are, respectively, 35.21%, 89.53% and 93.38% of optimization results, with significant scope of change. Therefore, it is necessary to conduct related research to establish systemized, more extensive, and more a quickly searchable optimization algorithm. 3. This case has respectively considered three scheduling scenarios established according to expert opinions, and the results of the execution of the optimization scheduling algorithm pro-

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Fig. 15. Scheduling result for sample case with three scenarios.

posed by this study. We found that the execution result of this study is indeed better than the instinctive judgment of experts or the performance set according to the heuristic rule. The main reason must be because there are more input data in use in workgroup type repetitive projects, so it is not easy to directly deduce the best solution using simple rules. The optimization algorithm proposed in this study fully employed the character-

istics of mass data processing and repetitive calculation provided by existing information technology, so that it is possible to extensively search for solution space within the limited time allowed for users, thereby providing the possibility of improvement on decision quality, and it is possible to conduct appraisal of the planning results obtained through expert judgment or the heuristic rule.

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R.-y. Huang, K.-S. Sun / Advances in Engineering Software 40 (2009) 212–228 Table 8 The decision variables and scheduling results for the tested and optimal scenarios for the sewer project WG

No.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Project duration

NPV

NPV ratio (%)

Scenarios: 1 A1 S R A2 S R A3 S R

1 1 1 1 1 1

2 1 2 2 2 1

3 1 3 1 3 1

4 1 4 2 4 1

5 1 5 1 5 1

6 1 6 2 6 1

7 1 7 1 7 1

8 1 8 2 8 1

9 1 9 1 9 1

10 1 10 2 10 1

11 1 11 1 11 1

12 1 12 2 12 1

13 1 13 1 13 1

14 1

70.97

160048.93

35.21

Scenarios: 2 A1 S R A2 S R A3 S R

1 2 1 1 1 2

2 2 2 2 2 2

3 2 3 1 3 2

4 2 4 2 4 2

5 2 5 1 5 2

6 2 6 2 6 2

7 2 7 1 7 2

8 2 8 2 8 2

9 2 9 1 9 2

10 2 10 2 10 2

11 2 11 1 11 2

12 2 12 2 12 2

13 2 13 1 13 2

14 2

72.36

406993.37

89.53

Scenarios: 3 A1 S R A2 S R A3 S R

7 1 1 1 6 1

6 1 2 1 5 1

5 1 3 1 4 1

4 1 4 1 3 1

3 1 5 1 2 1

2 1 6 1 1 1

8 2 7 2 8 2

9 2 8 2 9 2

1 1 9 2 7 1

10 2 10 2 10 2

11 2 11 2 11 2

12 2 12 2 12 2

13 2 13 2 13 2

14 2

66.19

424497.56

93.38

7 2 8 1 9 1

11 1 3 1 1 1

14 2 5 2 2 1

4 1 4 2 7 2

3 1 7 2 10 2

2 2 6 1 4 2

5 2 11 2 8 2

9 1 10 2 3 1

13 2 12 1 11 2

10 2 9 2 12 2

6 1 1 1 5 1

1 2 2 1 14 2

12 1

65.41

454583.50

100

Scenarios: optimal A1 S 8 R 2 A2 S 13 R 2 A3 S 13 R 1

14 1

14 2

14 2

6 2

Notes: WG, Workgroup; No., activity no.; S, sequence; R, resource; NPV ratio: optimization NPV/NPV * 100%.

Fig. 16. WoRSM NPV output for sewer project.

4. In the process of seeking the best solution, we can find from the generation-to-generation shrinkage in the case analysis of Fig. 16 that, the optimization algorithm proposed in this study

has found a quite satisfying solution in the evolution of about 50 generations (NPV is 424138.56, project duration is 68.27), with only slight improvement in the subsequent evolution pro-

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cess. Therefore, the user may consider giving an appropriate number of evolution generations in seeking an optimization solution, to proceed with immediate decision.

6. Conclusions This work builds on previous research and develops a GA-based optimization model for workgroup-based repetitive scheduling. Unlike conventional approaches, the workgroup-based repetitive scheduling method is more flexible in solving complex repetitive scheduling problems. It allows for the usage of multiple resources and the assignment of different working orders for activities in workgroups, while maintaining the work continuity of resources. The development of the GA-based optimization model (WoRSM) supplements the scheduling method by providing the capability of finding near-optimal solutions for repetitive project scheduling. Furthermore, a sample case study verifies and validates the WoRSM system. With WoRSM, a repetitive project can be scheduled in such a way that its maximum or near-maximum project net present value is obtained. It also facilitates application of the workgroup-based repetitive scheduling method. This study assumes that the arbitrary work order can be assigned in each workgroup and it can be a non-constraint optimization problem in our designed chromosome scheme. We did not discuss the situation where the physical and enforced relationships have been preset within a workgroup such as in a high-rise building. This issue will increase the additional constraints when scheduling. We therefore suggest that be discussed in a future study. For future research, a functional user interface can be developed for practical use of the GA scheduler. Moreover, clear guidelines concerning GA parameters, such as internal/external crossover rates, internal/external mutation rates, and so on, will be of great assistance to users. Finally, more capabilities, such as consideration of the project due date, the employment of stochastic activity durations, the consideration of learning curve effects, and so on, can be developed to further enhance the developed WoRSM model. Acknowledgements This work was supported in part by the National Science Council, Taiwan under Grant No. NSC 93-2211-E-008-33. The authors wish to express their gratitude for this support. References [1] Huang RY, Sun KS. Non-unit based planning and scheduling of repetitive construction project. J Constr Eng Manage ASCE 2006;132(6):585–97. [2] Huang RY, Sun KS. An optimization model for workgroup-based repetitive scheduling. In: Topping BHV, editor. Proceedings of the 10th international conference on civil, structural and environmental engineering computing. Stirling, United Kingdom: Civil-Comp Press; 2005 [paper 78].

[3] Khisty CJ. The application of the ‘‘Line-of-balance” technique to the construction industry. Indian Concr J 1970;44(7):297–300. and 319–20. [4] Carr RI, Meyer WL. Planning construction of repetitive building units. J Constr Div ASCE 1974;100(CO3):403–12. [5] O’Brien JJ. VPM scheduling for high-rise buildings. J Constr Div ASCE 1975;101(CO4):895–905. [6] Birrell GS. Construction planning – beyond the critical path. J Constr Div ASCE 1980;106(CO3):389–407. [7] Arditi D, Albulak MZ. Line-of-balance scheduling in pavement construction. J Constr Eng Manage ASCE 1986;112(3):411–24. [8] Al Sarraj ZM. Formal development of line-of-balance technique. J Constr Eng Manage ASCE 1990;116(4):689–704. [9] Reda RM. RPM: repetitive project modelling. J Constr Eng Manage ASCE 1990;116(2):316–30. [10] Ammar MA, Elbeltagi E. Algorithm for determining controlling path considering resource continuity. J Comput Civil Eng ASCE 2001;15(4):292–8. [11] Hegazy T, Wassef N. Cost optimization in projects with repetitive nonserial activities. J Constr Eng Manage ASCE 2001;127(3):183–91. [12] Selinger S. Construction planning for linear projects. J Constr Div ASCE 1980;106(CO2):195–205. [13] Johnston DW. Linear scheduling method for highway construction. J Constr Div ASCE 1981;107(CO2):247–61. [14] Moselhi O, El-Rayes K. Scheduling of repetitive projects with cost optimization. J Constr Eng Manage ASCE 1993;119(4):681–97. [15] Russell AD, Wong WCM. New generation of planning structures. J Constr Eng Manage ASCE 1993;119(2):196–214. [16] Eldin NN, Senouci AB. Scheduling and control of linear projects. Can J Civil Eng 1994;21:219–30. [17] Chrzanowski EN, Johnston DW. Application of linear scheduling. J Constr Eng Manage ASCE 1986;112(4):476–91. [18] El-Rayes K, Moselhi O. Resource-driven scheduling of repetitive activities. Constr Manage Econ 1998;16:433–46. [19] Russell AD, Caselton WF. Extensions to linear scheduling optimization. J Constr Eng Manage ASCE 1988;114(1):36–52. [20] Harris RB, Ioannou PG. Scheduling projects with repeating activities. J Constr Eng Manage ASCE 1998;124(4):269–78. [21] El-Rayes K, Moselhi O. Optimizing resource utilization for repetitive construction projects. J Constr Eng Manage ASCE 2001;127(1):18–27. [22] Lutz JD, Hijazi A. Planning repetitive construction: current practices. J Constr Manage Econ 1993;11:99–110. [23] Goldberg DE. Genetic algorithms in search, optimization, and machine learning. MA: Addison-Wesley; 1989. [24] Bean JC. Genetic algorithms and random keys for sequencing and optimization. ORSA J Comput 1994;6(2):154–60. [25] Chen CL, Vempati VS, Aljaber N. An application of genetic algorithms for flow shop problems. Eur J Oper Res 1995;80:389–96. [26] Chua DKH, Can WT, Govinan KA. time-cost tradeoff model with resource consideration using genetic algorithm. Civil Eng Syst 1997;14:291–311. [27] Reeves CR. A genetic algorithm for flowshop sequencing. Comput Oper Res 1995;22(1):5–13. [28] Feng CW, Liu L, Burns SA. Using genetic algorithms to solve construction timecost tradeoff problems. J Constr Eng Manage ASCE 1997;11(3):184–9. [29] Leu SS, Hwang ST. Optimal repetitive scheduling model with shareable resource constraint. J Constr Eng Manage ASCE 2001;127(4):270–80. [30] Abbasi GY, Arabiat YA. A heuristic to maximize the net present value for resource-constrained project-scheduling problems. Project Manage J 2001;32(2):17–24. [31] Sunde L, Lichtenberg S. Net-present-value cost-time tradeoff. Int J Project Manage 1995;13(1):45–9. [32] Etgar R, Shtub A, LeBlanc LJ. Scheduling projects to maximize net present value – the case of time-dependent, contingent cash flows. Eur J Oper Res 1996;96(1):90–6. [33] Sun KS. Scheduling and optimization of workgroup-based repetitive projects, Doctoral dissertation. Taiwan: National Central University; 2005. [34] Harmelink DJ, Rowings JE. Linear scheduling model: Development of controlling activity path. J Constr Eng Manage ASCE 1998;124(4):263–8.