Optimal Planning and Scheduling for Repetitive Construction Projects Khalied Hyari1 and Khaled El-Rayes, M.ASCE2 Abstract: This paper presents a multiobjective optimization model for the planning and scheduling of repetitive construction projects. The model enables construction planners to generate and evaluate optimal construction plans that minimize project duration and maximize crew work continuity, simultaneously. The computations in the present model are organized in three major modules: scheduling, optimization, and ranking modules. First, the scheduling module uses a resource-driven scheduling algorithm to develop practical schedules for repetitive construction projects. Second, the optimization module utilizes multiobjective genetic algorithms to search for and identify feasible construction plans that establish optimal tradeoffs between project duration and crew work continuity. Third, the ranking module uses multiattribute utility theory to rank the generated plans in order to facilitate the selection and execution of the best overall plan for the project being considered. An application example is analyzed to illustrate the use of the model demonstrate its new capabilities in optimizing the planning and scheduling of repetitive construction projects. DOI: 10.1061/共ASCE兲0742-597X共2006兲22:1共11兲 CE Database subject headings: Road construction; Highway construction; Scheduling; Resource management; Optimization; Evolutionary computation; Computer models; Construction management.
Introduction Examples of repetitive construction projects include highways, tunnels, bridges, railways, pipeline networks, sewer mains, highrise buildings, and housing development projects. In this class of projects, construction crews are often required to repeat the same work in various locations of the project, moving from one location to another. Due to this frequent crew movement, available scheduling methods for repetitive construction projects focus on maximizing crew work continuity by enabling each crew to finish work in one location of the project and move promptly to the next in order to minimize work interruptions. The application of work continuity improves the overall productivity of construction crews due to: 共1兲 minimizing their idle time during their frequent movements on site; and 共2兲 maximizing their benefits from learning curve effects 共Ashley 1980; El-Rayes 2001兲. Despite the advantages of crew work continuity, its strict application can lead to a longer overall project duration 共Selinger 1980; Russell and Caselton 1988; El-Rayes and Moselhi 2001兲. This led to a number of research studies that investigated the impact of 共Russell and Caselton 1988; Senouci and Eldin 1996; crew work continuity on the planning and scheduling of repetitive construction projects; Harris and Ioannou 1998; El-Rayes 2001; El-Rayes and Moselhi 2001; Hegazi and Wassef兲. 1
Assistant Professor, Dept. of Civil Engineering, The Hashemite Univ., P.O. Box 150459, Zarqa 13115, Jordan. E-mail:
[email protected] 2 Assistant Professor, Dept. of Civil and Environmental Engineering, Univ. of Illinois at Urbana-Champaign, Urbana, IL 61801 共corresponding author兲. E-mail:
[email protected] Note. Discussion open until June 1, 2006. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on February 2, 2005; approved on May 9, 2005. This paper is part of the Journal of Management in Engineering, Vol. 22, No. 1, January 1, 2006. ©ASCE, ISSN 0742-597X/2006/1-11–19/$25.00.
Available planning and scheduling models that focused on minimizing the duration of repetitive construction projects can be grouped into two main categories: 共1兲 models that provide strict compliance with crew work continuity 共Selinger 1980兲; and 共2兲 models that allow interruptions to crew work continuity 共Russell and Caselton 1988; El-Rayes 2001兲. All these models are capable of generating a single optimal solution that minimizes the duration of the project being considered. For example, Fig. 1 shows the single optimal solution produced by one model from each of the above two categories. Although both models analyzed the same application example, the first category model 共Selinger 1980兲 that did not allow interruptions produced a project duration of 117.9, while the second category model 共El-Rayes and Moselhi 2001兲 that allowed 15 days of interruptions generated a project duration of 106.8 as shown in solutions A and B in Fig. 1, respectively. The two schedules shown in Fig. 1 illustrate two possible tradeoffs between the two important and conflicting objectives of minimizing the project duration and maximizing crew work continuity. As stated earlier, maximizing crew work continuity can produce significant improvements in construction productivity and reductions in crew utilization costs, however it can also lead to a longer overall project duration and an increase in project overhead costs. As such, construction planners need to evaluate all these feasible tradeoffs in order to select a construction plan that strikes an optimal balance between minimizing the project duration and maximizing its crew work continuity. Available planning and scheduling models for repetitive construction are incapable of generating these feasible tradeoffs as they can only produce a single optimal solution as shown in Fig. 2. There is a pressing need for advanced models that can help construction planners in generating and evaluating all the feasible tradeoff solutions shown in Fig. 2 in order to select an optimal plan that satisfies the specific requirements of the project being considered.
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Fig. 1. Minimum duration solution provided by existing models
Objective The objective of this paper is to present the development of a multiobjective optimization model for planning and scheduling of repetitive construction projects. The model provides the capability of: 共1兲 generating a set of feasible construction plans that establish optimal tradeoffs between project duration 共D*兲 and crew work continuity 共R*兲, as shown in Fig. 2; 共2兲 identifying an optimal level of resource utilization for each construction activity 共i兲 in the project; and 共3兲 ranking the generated optimal plans to facilitate the selection of the best overall plan for executing the project 共see Fig. 3兲. In order to provide this output, the present model requires construction planners to input readily available data including: 共1兲 project data that specifies the number of repetitive activities in the project 共I兲 and the job logic and interrelationships among them; and 共2兲 activity data that specifies the number of repetitive sections 共J兲, the quantity of work 共Qi,j兲 of activity 共i兲 in each unit 共j兲, available crew formations 共ni = 1 – Ni兲 that can perform each activity 共i兲 and their daily productivity rates 共Pi,n兲, as shown in Fig. 3. The planning and scheduling computations in the present model are organized in three major modules: 共1兲 a scheduling
module that develops practical schedules for repetitive construction projects; 共2兲 a multiobjective optimization module that searches for and identifies near optimal tradeoffs between project duration and crew work continuity; and 共3兲 a ranking and selection module that ranks the generated solutions according to the specific requirements of the project being considered 共see Fig. 3兲. The following sections present a detailed description of these three major modules.
Scheduling Module The main objective of this module is to develop practical schedules for repetitive construction projects. The module is designed to provide 共1兲 flexibility in considering typical and nontypical repetitive activities that have identical and varying durations in different repetitive units 共Selinger 1980; Russell and Caselton 1988; Moselhi and El-Rayes 1993兲; and 共2兲 practicality in complying with all scheduling constraints, including crew work continuity, crew availability, and job logic constraints 共El-Rayes and Moselhi 1998; El-Rayes 2001兲. The module is also designed to compute: 共1兲 the scheduled start 共Si,j兲 and finish 共Fi,j兲 dates of
Fig. 2. Tradeoffs between project duration and crew work continuity 12 / JOURNAL OF MANAGEMENT IN ENGINEERING © ASCE / JANUARY 2006
Fig. 3. Optimization model for scheduling repetitive construction
construction for each activity 共i兲 in each repetitive unit 共j兲; 共2兲 the total project duration 共D兲; and 共3兲 the total number of crew interruption days 共R兲 at the project level. To this end, the scheduling computations in this module are performed using the following eight major steps 共see Fig. 4兲: 1. Calculate the construction duration 共di,j兲 of activity 共i兲 in each repetitive unit 共j兲, using its quantity of work 共Qi,j兲 and the daily productivity rate 共Pi,n兲 of the selected crew option as shown in Eq. 共1兲 and the example foundation activity in Fig. 4 di,j =
Qi,j Pi,n
共1兲
2.
Calculate the earliest start 共SCrew关i,j兴兲 and finish 共FCrew关i,j兴兲 times for activity 共i兲 in unit 共j兲 that satisfy crew availability and crew work continuity constraints, assuming that the activity has no predecessors and accordingly its first repetitive unit 共j = 1兲 can start at time zero as shown in Eq. 共2兲. SCrew关i,j兴 and FCrew关i,j兴 are calculated using Eqs. 共3兲 and 共4兲 that consider 共1兲 the availability the crew after the completion of work in its previously assigned repetitive unit 共FCrew关i,j−1兴兲; and 共2兲 the specified crew interruption time applied after the completion of the previous unit j − 1 共Interi,j−1兲, as shown in Fig. 4. Although the computations of SCrew关i,j兴 and FCrew关i,j兴 are formulated to satisfy crew availability and crew work
Fig. 4. Scheduling computations for foundation activity JOURNAL OF MANAGEMENT IN ENGINEERING © ASCE / JANUARY 2006 / 13
continuity constraints, they do not comply with the job logic constraint. This constraint requires that activity i 共e.g., foundation兲 can start only after the completion of its predecessor activity i − 1 共e.g., excavation兲 in each repetitive unit 共j兲. As such, the following three steps are designed to shift SCrew关i,j兴 and FCrew关i,j兴 to a later date in order to further comply with the job logic constraint 共see Fig. 4兲:
3.
4.
SCrew关i,1兴 = 0
共2兲
SCrew关i,j兴 = FCrew关i,j−1兴 + Interi,j−1
共3兲
FCrew关i,j兴 = SCrew关i,j兴 + di,j
共4兲
Identify the earliest start time 共SLogic关i,j兴兲 that satisfies the job logic and precedence relationships between activity 共i兲 and all its predecessors. These relationships can be either finish to start, start to start, finish to finish or start to finish with or without lag time. For example, if the precedence relationship between two activities is finish to start, then the successor activity 共i兲 in each unit 共j兲 can start only after the completion of its predecessor activity 共i − 1兲 and the lag time 共lagi,i−1兲 between the two activities 共i.e., SLogic关i,j兴 艌 Fi−1,j + lagi,i−1兲. Calculate the time difference 共⌬i,j兲 between the earliest start time that satisfies job logic 共SLogic关i,j兴兲 and that which com-
plies with crew availability and work continuity 共SCrew关i,j兴兲 as shown in Eq. 共5兲. This time difference represents the required delay in SCrew关i,j兴 and FCrew关i,j兴 to further satisfy the job logic constraint, as shown in Fig. 4 ⌬i,j = SLogic关i,j兴 − SCrew关i,j兴 5.
6.
Identify the maximum time difference from those calculated in the previous step 共Shifti兲 using Eq. 共6兲. As shown in Fig. 4, this time shift is then used to delay SCrew关i,j兴 and FCrew关i,j兴 in order to identify the scheduled start 共Si,j兲 and finish 共Fi,j兲 times that satisfy all scheduling constraints for activity 共i兲 in each repetitive unit 共j兲 using Eqs. 共7兲 and 共8兲 Shifti = MaxJj=1关⌬i,j兴
共6兲
Si,j = SCrew关i,j兴 + Shifti
共7兲
Fi,j = FCrew关i,j兴 + Shifti
共8兲
Identify the total crew interruption days in activity i 共Interi兲 using the following equation:
Fig. 5. Multiobjective optimization module 14 / JOURNAL OF MANAGEMENT IN ENGINEERING © ASCE / JANUARY 2006
共5兲
J
Interi =
Interi,j 兺 j=1
共9兲
7.
Compute the total project duration 共D兲 by identifying the time difference between the scheduled finish time of the last activity in its last repetitive unit 共FI,J兲 and the start of the first activity in the first unit 共S1,1兲, as shown in the following equation: 共10兲
D = FI,J − S1,1 8.
In order to search for nondominated construction plans that specify the optimal values for the aforementioned two resource utilization variables, the present multiobjective optimization module is implemented using a multiobjective genetic algorithm 共Zitzler and Thiele 1999; Deb et al. 2000; Coello and Lechuga 2001; El-Rayes and Kandil 2005; El-Rayes and Khalafallah 2005; Kandil and El-Rayes 2005兲. This algorithm adopts the concept of Pareto optimality to support multiobjective optimization, and the survival of the fittest criteria to evolve solutions over generations in order to yield near optimal solutions. The present module starts the optimization process by randomly generating a number of initial resource utilization plans 共k = 1 – K兲, where each represents a random selection of a crew formation option 共ni兲 and a crew interruption vector 共vi兲 for each repetitive activity 共i兲 in the project. These randomly generated plans form the parent population 共Pt兲 of the first generation 共t = 1兲 that evolves into a near optimal solution after a number of predetermined generations 共T兲. As shown in Fig. 4, this optimization process is performed using the following four major cyclical steps: 1. Calculate the project duration 共Dk兲 and total interruption days 共Rk兲 for each resource utilization plan 共k = 1 – K兲 in the first generation 共t = 1兲, using the earlier described scheduling module for repetitive construction projects. As shown in Fig. 4, the scheduling module returns the computed values of the project duration and total interruption days for each resource utilization plan after considering all the scheduling constraints of crew availability, crew work continuity and job logic. 2. Perform genetic operators 共i.e., selection, crossover, and mutation兲 to create a new child population. These three operators adopt the survival of the fittest principle to provide an effective evolutionary mechanism for generating improved resource utilization plans over a number of generations. First, the selection operator favors solutions with better fitness values to move to the reproduction phase. Second, the crossover operator swaps a chunk of the genetic material between the mated pairs of solution in the reproduction phase. Third, the mutation operator is used to change the genetic materials in the produced child population randomly to avoid convergence to local optimal solutions. 3. Evaluate the fitness functions 共i.e., project duration and total crew interruption days兲 for each solution in the newly created child population Ct in a similar process to that described in step 共1兲. 4. Combine child and parent populations 共Ct and Pt兲 to form newly combined populations, and then select the best 50% of the members of the combined population to form a new parent population for the next generation. This process represents a strong form of elitism as it enables preserving the
Calculate the total number of interruption days 共R兲 for all crews in the project using the following equation: I
R=
Interi 兺 i=1
共11兲
Multiobjective Optimization Module The primary purpose of this module is to search for and identify a set of optimal construction plans that simultaneously minimize project duration and maximize crew work continuity for repetitive construction projects. Each of these identified optimal plans provides a unique and nondominated tradeoff between project duration and crew work continuity. An optimal plan is identified to be nondominated if no other plan can provide a better solution in both objectives simultaneously 共i.e., a combination of shorter project duration and higher level of crew work continuity兲. For example, all the generated plans in Fig. 2 are nondominated because no other plans can be found to provide a better performance in both objectives simultaneously. Conversely, plan C is dominated by all the generated plans in group D because each of these plans provides a combination of a shorter project duration 共less than 110.4 days兲 and a higher level of crew work continuity 共less than 16 days of interruption兲, as shown in Fig. 2. The present optimization module is designed to preclude all dominated solutions in order to enable the generation of only the non-dominated optimal construction plans 共see Fig. 2兲. In this optimization module, each of the identified optimal construction plans specifies an optimal solution for two important resource utilization variables: 共1兲 crew formation 共ni兲 that represents the availability of alternative crew utilization options for activity 共i兲 with varying daily productivity rates 共Pi,n兲; and 共2兲 crew interruption vector 共vi兲 that represents the interruption days to the work continuity of crew 共ni兲 when it moves from one repetitive section 共j兲 to the next 共El-Rayes and Moselhi 2001兲. This module requires construction planners to specify all feasible crew formations 共ni兲 for each activity 共i兲 in the project, however it does not require them to provide data for the feasible crew interruption vectors 共vi兲 which are automatically generated and evaluated by the module.
Table 1. Quantities of Work and Available Crew Formation Options Repetitive activity 共i兲 Repetitive unit 共j兲 Quantity of work 共Qi,j兲 共m3兲
Excavation 共i = 1兲 1
2
3
Foundation 共i = 2兲 4
1
2
3
Columns 共i = 3兲 4
1
1,147 1,434 994 1,529 1,032 1,077 943 898 104
2 86
3
Beams 共i = 4兲 4
Slabs 共i = 5兲
1
2
3
4
1
2
129 100 85
92
101
80
0
138
2
3
4
3
4
114 145
Available crew options Crew formation 共ni兲 Productivity 共Pi,n兲 共m3 / day兲
1 91.75
1
2
3
89.77 71.81 53.86
1
2
3
5.73 6.88 8.03
1
9.9 8.49 7.07 5.66
1
2
28.73
7.76
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Table 2. Solutions Generated by Previous Optimization Models Selinger model 共1980兲 Activity 共i兲 Excavation Foundation Columns Beams Slabs Total interruptions 共R兲 Project duration 共D兲
Russell and Caselton model 共1988兲
El-Rayes and Moselhi model 共2001兲
Optimal crew formation 共ni쐓兲
Interruption days 共Interi兲
Optimal crew formation 共ni쐓兲
Interruption days 共Interi兲
Optimal crew formation 共ni쐓兲
Interruption days 共Interi兲
1 2 3 3 1
0 0 0 0 0
1 1 3 1 1
0 4 0 12 0
1 1 3 1 1
0 6 0 9 0
0 117.9
best members of the parents’ population over generations 共Deb et al. 2000兲. The above computation steps of 1–4 are repeated over a number of specified generations 共t = 1 – T兲 in order to yield a Pareto optimal set of nondominated resource utilization plans for the repetitive construction project. Each plan in this Pareto optimal set provides: 共1兲 an optimal construction plan that provides the least project duration 共D*兲 that can be achieved at a given crew work continuity level 共R*兲, as shown in Fig. 2; and 共2兲 an optimal level of resource utilization for each construction activity 共i兲 in the project that specifies the selected crew formation 共ni*兲 and crew interruption vector 共vi*兲, as shown in Fig. 3. A construction planner can select, from this set, the best overall plan that satisfies the specific project requirements, using the ranking and selection module which is described in more details in the next section.
Ranking and Selection Module The objective of this module is to rank all the optimal solutions 共g = 1 – G兲 obtained from the previous module to facilitate the selection of the best overall plan for executing the project. To accomplish this, the present module uses the multiattribute utility theory 共von Winterfeld and Edwards 1986兲 to enable construction planners to express their degree of satisfaction about the generated project durations and crew work continuity levels. For example, a linear utility function can be used to evaluate the performance 共PDg兲 of each generated project duration 共Dg兲, using a performance scale that ranges from 0 to 100% 共see Fig. 5兲. This function can be easily developed by assigning a 100% perfor-
16 110.4
15 106.8
mance level to the obtained minimum duration solution 共Min Dg兲, and a planner-specified level 共e.g., 0 or 20%兲 that reflects the degree of satisfaction with the maximum duration solution 共Max Dg兲, as shown in Fig. 5. This function can then be used to evaluate the performance of all intermediate project durations 共Dg兲 as shown in Eq. 共12兲 and Fig. 5. The performance in crew work continuity 共PRg兲 can also be evaluated in a similar way using Eq. 共13兲. For example, the utility values that represent project duration performance levels 共PDg兲 for solutions A, B, and E 共see Fig. 2兲 are 20, 100, and 70%, respectively; while those representing their crew work continuity performance levels 共PRg兲 are 0, 100, and 47%, respectively, as shown in Fig. 5. Both performance levels 共PDg and PRg兲 are then combined using an overall score 共Cg兲 that represents the combined performance of solution 共g兲 in both objectives. This overall score 共Cg兲 can be computed using planner-specified weights that reflect the relative importance of project duration 共WtD兲 and crew work continuity 共WtR兲 to the decision maker as shown in Eq. 共14兲 PDg = P Max Dg +
冋
Max Dg − Dg ⫻ 共100 % − P Max Dg兲 Max Dg − Min Dg
册
共12兲
PRg = P Max Rg +
冋
Max Rg − Rg ⫻ 共100 % − P Max Rg兲 Max Rg − Min Rg
共13兲 Cg = PDg ⫻ WtD + PRg ⫻ WtR
Fig. 6. Example utility functions for project duration and crew work continuity 16 / JOURNAL OF MANAGEMENT IN ENGINEERING © ASCE / JANUARY 2006
册
共14兲
Table 3. Optimal Solutions Generated by Present Model Selected crew formation 共ni兲
Project performance
Solution 共g兲
Project duration in days 共D兲
Total crew interruption days 共R兲
Selected interruption vectors 共v j兲 Inter3,j
Inter2,j Excavation 共i = 1兲
1 106.8 15 2 107.0 14 3 107.6 13 4 108.5 11 5 109.0 10 6 109.9 9 7 110.9 8 8 111.4 7 9 112.3 6 10 113.3 5 11 114.3 4 12 115.3 3 13 116.3 2 14 116.8 1 15 117.9 0 Interi,j ⫽ interruption days in activity i
Foundation 共i = 2兲
Inter4,j
Columns 共i = 3兲
Beams 共i = 4兲
Slabs 共i = 5兲
j=1
j=2
j=3
j=1
j=2
j=3
j=1
j=2
j=3
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
1 1 1 2 2 2 2 3 3 3 3 3 3 3 3
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
5 5 5 5 5 5 5 5 5 4 3 2 1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
6 6 5 4 4 3 3 1 1 1 1 1 1 1 0
3 2 2 1 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 after repetitive unit j.
Application Example An example of a three-span concrete bridge is analyzed in order to illustrate the use of the present model and demonstrate its capabilities in generating and evaluating optimal tradeoffs between project duration and crew work continuity. The project consists of five construction activities: excavation, foundations, columns, beams, and slabs that are repeated in four sections of the project, as shown in Fig. 1. The precedence relationships among these five successive activities are finish to start with no lag time. Table 1 summarizes the quantities of work for each activity in the four repetitive sections as well its available crew formation options and their daily productivity rates. The example was previously analyzed in the literature by Selinger 共1980兲, Russell and
Caselton 共1988兲, and El-Rayes and Moselhi 共2001兲, producing three different solutions as shown in Fig. 1 and Table 2. The present model was utilized to analyze the same example in order to: 共1兲 enable a comparison between its results and those generated by available models in the literature; and 共2兲 illustrate its capability of evaluating and ranking the obtained tradeoff solutions between project duration and crew work continuity. First, the model was used to generate a set of 15 optimal and nondominated construction plans, where each represents an optimal and unique tradeoff between project duration 共D兲 and crew work continuity 共R兲, as shown in Fig. 2 and Table 3. The results generated by the present model were compared to those produced by previous models 共see Fig. 2 and Table 3兲. The results of this comparative analysis illustrate the capabilities of the present
Table 4. Ranking of Generated Optimal Solutions Overall score in % 共ck兲
Performance in project objectives Solution 共g兲 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Duration in days 共Dg兲
Crew interruptions in days 共Rg兲
Duration performance in % 共PDg兲
Work continuity performance in % 共PRg兲
Scenario 1 WtD = 80% WtR = 20%
Scenario 2 WtD = 20% WtR = 80%
Scenario 3 WtD = 60% WtR = 40%
106.8 107.0 107.6 108.5 109.0 109.9 110.9 111.4 112.3 113.3 114.3 115.3 116.3 116.8 117.9
15 14 13 11 10 9 8 7 6 5 4 3 2 1 0
1.00 0.98 0.94 0.88 0.84 0.77 0.71 0.67 0.60 0.53 0.46 0.39 0.31 0.28 0.20
0 0.067 0.133 0.267 0.333 0.400 0.467 0.533 0.600 0.667 0.733 0.800 0.867 0.933 1.000
80.2 80.0 77.9 75.7 73.8 69.9 65.8 64.3 60.1 55.7 51.3 46.8 42.4 41.0 36.0
20.0 25.0 29.5 38.9 43.5 47.5 51.4 56.1 60.0 63.9 67.8 71.7 75.6 80.3 84.0
60.1 61.7 61.7 63.5 63.7 62.4 61.0 61.5 60.1 58.4 56.8 55.1 53.5 54.1 52.0
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model in: 共1兲 generating identical optimal solutions to those produced by the two existing models of Selinger 共1980兲 and El-Rayes and Moselhi 共2001兲, as shown in solutions A and B in Fig. 2; 共2兲 producing a wide range of optimal tradeoff solutions that cannot be developed using existing models, as shown in Fig. 2; and 共3兲 obtaining optimal tradeoff solutions that dominate and provide superior performance in both optimization objectives simultaneously than that obtained by the Russell and Caselton model 共1988兲, as highlighted by group D and solution C in Fig. 2, respectively. Second, the present model was used to evaluate and rank the 15 generated tradeoff solutions to facilitate the selection and execution of the best overall plan for the project. As stated earlier, construction planners need to specify: 共1兲 their degree of satisfaction with the maximum duration solution and the maximum interruption solution, as shown in Fig. 5; and 共2兲 the relative importance of minimizing the project duration and maximizing crew work continuity in the form of two relative weights 共WtD and WtR兲. In this example, it is assumed that the planner expressed a satisfaction level of 共P Max Dg = 20% 兲 and 共P Max Rg = 0 % 兲 with the maximum duration and the maximum interruption solutions, respectively, as shown in Fig. 5. Furthermore, three different combinations of weights are evaluated in this example to illustrate the impact of the weighting process on the ranking of the tradeoff solutions, as shown in Table 4. As expected, the first 共WtD = 80% and WtR = 20%兲 and second 共WtD = 20% and WtR = 80%兲 combinations of weights that favored the performance in project duration and crew work continuity produced the highest overall score for the least duration solution 共g = 1兲 and the least interruption solution 共g = 15兲, respectively 共see Table 4兲. On the other hand, the third combination of weights 共WtD = 60% and WtR = 40%兲 produced an intermediate tradeoff solution 共g = 5兲 as shown in Table 4. This illustrates the capability of the present model in performing instant sensitivity analysis in order to study the effects of varying the weights on the ranked solutions, without the need to rerun the computationally demanding genetic algorithms each time these weights are changed, as shown in Table 4.
Conclusions A robust multiobjective optimization model was developed to support the planning and scheduling of repetitive construction projects. The model enables construction planners to generate and evaluate optimal construction plans that establish optimal tradeoffs between project duration and crew work continuity. Each of these plans identifies, from a set of feasible alternatives, an optimal level of resource utilization for each activity in the project. To accomplish this, the model incorporates: 共1兲 a scheduling module that calculates the project duration and level of crew work continuity for repetitive construction projects; 共2兲 a multiobjective optimization module that searches for and identifies near optimal construction plans; and 共3兲 a ranking and selection module that ranks the generated solutions according to the specific requirements of the project being considered. An application example was analyzed to enable a comparison between the present model and those available in the literature. The results of this analysis illustrates the new capabilities of the developed model in: 共1兲 generating all optimal tradeoff solutions between project duration and crew work continuity in a single run, where each provides the least project duration that can be achieved at a given level of crew work continuity; and 共2兲 ranking these
generated tradeoff solutions to facilitate the selection of the best overall plan for constructing the project. These new and unique capabilities should prove useful to construction planners and is expected to advance existing planning and scheduling practices for repetitive construction projects.
Acknowledgment The writers gratefully acknowledge the financial support provided by the National Science Foundation for this research project under NSF CAREER Award No. CMS-0238470.
Notation The following symbols are used in this paper: Cg ⫽ overall score that represents combined performance of solution g in both objectives; D ⫽ project duration; Dg ⫽ project duration of solution g; di,j ⫽ construction duration of activity i in repetitive unit j; FCrew关i,j兴 ⫽ earliest finish times for activity i in unit j that satisfy crew availability and crew work continuity constraints; Fi,j ⫽ scheduled finish time for activity i in repetitive unit j; Interij ⫽ interruption days in activity i after repetitive unit j; Max Dg ⫽ maximum duration solution; Max Rg ⫽ maximum interruption solution; Min Dg ⫽ minimum duration solution; Min Rg ⫽ minimum interruption solution; Pi,n ⫽ productivity rate of crew n in activity i; PDg ⫽ project duration performance level; PRg ⫽ crew work continuity performance level; Qi,j ⫽ quantity of work of activity i in repetitive unit j; R ⫽ total number of crew interruption days; Rg ⫽ total interruption days in solution g; SCrew关i,j兴 ⫽ earliest start time for activity i in unit j that satisfy crew availability and crew work continuity constraints; Si,j ⫽ scheduled start time for activity i in repetitive unit j; and SLogic关i,j兴 ⫽ earliest start time that satisfies job logic and precedence relationships between activity i and all its predecessors. Subscripts and Superscripts G ⫽ number of nondominated optimal solutions 共g = 1 – G兲; I ⫽ number of construction activities 共i = 1 – I兲; J ⫽ number of repetitive units 共from j = 1–J兲. K ⫽ population size 共k = 1 – K兲; and T ⫽ number of generations 共t = 1 – T兲.
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