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Minimizing duration and crew work interruptions of repetitive construction projects
Automation in Construction 88 (2018) 59–72
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Automation in Construction journal homepage: www.elsevier.com/locate/autcon
Minimizing duration and crew work interruptions of repetitive construction projects Ayman Altuwaima,b, Khaled El-Rayesa, a b
T
⁎
Department of Civil and Environmental Engineering, University of Illinois at Urbana–Champaign, Urbana, IL 61801, United States Department of Civil Engineering, King Saud University, Riyadh, Saudi Arabia
A R T I C L E I N F O
A B S T R A C T
Keywords: Repetitive construction Linear scheduling Resource-driven scheduling Crew work continuity Scheduling algorithms
This paper presents the development of a novel scheduling model for minimizing the duration and crew work interruptions of repetitive construction projects. The main contributions of the developed model are its ability to (1) generate early and late start schedules that minimize the duration of repetitive construction projects while keeping the total work interruptions of their utilized crews to a minimum; (2) calculate novel types of crew work-continuity floats that consider the impact of delaying the early start of repetitive activities on crew work continuity; (3) develop a wide range of intermediate schedules between the early and late start schedules that maintain the least project duration and minimum total crew work interruptions; and (4) compare shortest duration schedules with and without interruptions to identify the best schedule that fits the specific project needs. The model performance was evaluated using an application example of a repetitive construction project.
1. Introduction Repetitive construction projects require construction crews to repeat their work in a number of locations in the same project, moving from one location to the next. This continuous movement of crews on site is often encountered during the construction of high-rise buildings, housing projects, highways, pipeline networks, and bridges. Traditional scheduling methods such as bar charts and critical path method are ineffective in scheduling this class of projects due to their inability to consider and maximize work continuity for the construction crews on these projects [1–12]. Maximizing work continuity improves construction productivity by minimizing crew idle and non-productive times and by maximizing the benefits of the learning curve effect for working crews [1,8,13,14]. To realize these benefits, a number of scheduling models were developed for repetitive construction projects that are capable of considering and maximizing crew work continuity. Existing scheduling models for repetitive construction projects that consider crew work continuity can be classified in two main categories: (1) models that strictly enforce work continuity for all construction crews without allowing any interruptions [2,4,6,9,11,15–25]; and (2) models that maximize work continuity for all construction crews while allowing selected interruptions to minimize project duration, as shown in Fig. 1[5,7,8,18,26–36]. The scheduling models in the first category are capable of maintaining full work continuity for all crews, however their generated schedule often leads to longer project duration due to
⁎
the strict enforcement of the crew work continuity constraint, as shown in Fig. 1. The scheduling models in the second category provide shorter project duration than those in the first category because they provide the flexibility of enabling selected interruptions only when needed to minimize the project duration [2,5,8,29]. Scheduling models in the second category can be further classified into optimization models and heuristic models. First, optimization models in this category utilized various optimization techniques such as linear programing, dynamic programing or genetic algorithms to minimize project duration, cost, and/or work interruptions [5,8,14,18,26–32,35,36]. For example, linear programing was utilized by Ipsilandis [32] to minimize crew work interruptions in scheduling repetitive projects under a specified project duration. Dynamic programming was also used by Russell and Caselton [5], El-Rayes and Moselhi [8], and El-Rayes [14,26] to optimize the scheduling of repetitive construction projects to minimize project duration, work interruptions, project cost, and/or total combined bid price for A + B highway contracts. Similarly, genetic algorithms were utilized by Nasser [30], Hegazy and Wassef [18], Hegazy et al. [27], Hyari and ElRayes [28,29], Long and Ohsato [35], and Hyari and El-Rayes [36] to either minimize project duration and work interruptions, minimize project cost, trade-offs between project duration and work interruptions, or trade-offs between project duration and cost. Second, heuristic scheduling models in the second category utilized various algorithms and heuristics to schedule repetitive projects with selected work
Corresponding author. E-mail address: [email protected] (K. El-Rayes).
https://doi.org/10.1016/j.autcon.2017.12.024 Received 23 November 2016; Received in revised form 9 November 2017; Accepted 7 December 2017 Available online 03 January 2018 0926-5805/ © 2017 Elsevier B.V. All rights reserved.
Automation in Construction 88 (2018) 59–72
A. Altuwaim, K. El-Rayes
Fig. 1. Impact of allowing work interruptions on project duration.
3. Phase 1: early schedule computation
interruptions [7,35]. An example of heuristic scheduling models that allow interruptions for repetitive construction projects is the resourcedriven scheduling model developed by El-Rayes and Moselhi [7] that enables decision makers to assign multiple crew formations for each repetitive activity, and to specify work interruptions between the repetitive units in order to minimize the project duration. Another heuristic scheduling model was developed by Long and Ohsato [35] for repetitive projects which is capable of identifying work interruptions between repetitive units in order to minimize the project duration and keep work interruptions to a minimum. Despite the contributions of the aforementioned scheduling models for repetitive construction projects that allow selected work interruptions, they have a number of limitations as they are incapable of (1) generating an early start schedule that minimizes both the duration project and its total work interruptions; (2) calculating important types of floats for repetitive construction projects that can be used to analyze the impact of delaying the early start of repetitive construction activities on the work continuity of construction crews; and (3) providing planners with the early start and late start schedules for each repetitive activity which provides them the flexibility to generate a wide range of intermediate schedules. Accordingly, there is a pressing need for a novel scheduling model for repetitive construction projects that is capable of circumventing these three main limitations of existing heuristic models.
This phase of the model focuses on developing a novel scheduling algorithm for repetitive construction projects that seeks to minimize project duration by allowing selected work interruptions while maximizing work continuity. The scheduling algorithm is designed to (1) calculate early start (ESi,j), late start (LSi,j), early finish (EFi,j), late finish (LFi,j) and work interruptions (Interj, j − 1i) for each section (j) in the repetitive activities (i), as shown in Fig. 2; (2) compute the project duration (D) and total work interruptions (TR), as shown in Fig. 2; (3) comply with crew availability and precedence relationships constraints; (4) maximize work continuity; and (5) enable the scheduling of activities with varying durations in its repetitive units (see Fig. 1). The required input data for this schedule computation phase include: (a) repetitive activities data that include the total number of activities (I) and their logical precedence relationships; (b) number of the repetitive units (J); (c) the quantity of work (Qi,j) for each repetitive section (j) in each activity (i); and (d) daily productivity rates for the specified crew in each activity (Pi), as shown in Fig. 2. The computations of this phase are performed using the following two stages. 3.1. Stage 1 This stage creates an initial project schedule that provides the shortest possible duration for the project while complying with the job logic/precedence relationships and crew availability constraints. This is achieved in two main steps that are designed to calculate: (1.1) the duration of each activity (i) at each repetitive unit (j); and (1.2) the early start and finish dates of construction for each activity (i) at each repetitive unit (j), as shown in Fig. 3.
2. Objective The goal of this paper is to develop a novel scheduling model for repetitive construction projects that overcomes the aforementioned three main limitations of existing heuristic models. To accomplish this goal, the three main research objectives of this paper are to (1) develop an innovative heuristic methodology for generating an early start schedule that simultaneously minimizes project duration and total work interruptions; (2) create novel types of floats to calculate and analyze the impact of delaying the early start of repetitive construction activities on the work continuity of construction crews; and (3) generate a wide range of intermediate schedules to provide planners with alternative schedules that minimize both project duration and work interruptions. These three objectives are accomplished in four main phases: early schedule computation phase, work-continuity float calculation phase, strict work continuity phase, and performance evaluation phase, as shown in Fig. 2. The following sections of the paper describe these four development phases of the scheduling model.
3.2. Stage 2 This stage is designed to revise the initial schedule generated in Stage 1 to maximize compliance with the crew work continuity constraint. This is achieved by identifying for each repetitive activity (i) the required shift for the start date of its earlier repetitive sections (j = 1 to J-1) while maintaining the initial/earliest possible start and finish dates for its last repetitive section (j = J) to minimize work interruptions for all assigned crews while maintaining the shortest project duration, as shown in Fig. 4. The revised final schedules generated in this stage identifies the early start, early finish, late start, and late finish for each activity (i) at all its repetitive unit (j). The scheduling computations in 60
Automation in Construction 88 (2018) 59–72
A. Altuwaim, K. El-Rayes
Fig. 2. Model development phases.
3.2.2. Step 2.2: forward pass The purpose of this step is to calculate the early start schedule that minimizes the project duration and work interruptions simultaneously. This early start schedule for each activity (i) is calculated based on the early start of its predecessor activity (i-1) and the availability of its assigned crews, as shown in Fig. 7. The forward pass computations start from activity 1 to I and from repetitive unit 1 to J (see Fig. 6) and they are designed to calculate the early start and finish dates of each activity (i) at each repetitive unit (j) as shown in Eqs. (4) to (6), and the minimum project duration (D), as shown in Eq. (7). It should be noted that Eq. (4) in this step identifies the early start date of all activities at the first repetitive section (ESi,1) to be identical to its late start (LSi,1) to identify the earliest start date for the remaining repetitive sections (j = 2 to J) while maintaining the minimum work interruptions achieved in Step 2.1 and the shortest project duration, as shown in Fig. 7.
this stage are performed in three steps: backward pass, forward pass, and work interruption computations. 3.2.1. Step 2.1: backward pass The goal of the backward pass is to determine the latest start schedule that minimizes the project duration and work interruptions simultaneously. For each activity (i), this latest start schedule is calculated based on the late start of its successor activity (i + 1) and the availability of its assigned crews, as shown in Fig. 5. The backward pass computations start from activity I to 1 and from repetitive unit J to 1 (see Fig. 6) and they are designed to calculate the late finish and late start date of each activity (i) at each repetitive unit (j) as shown in Eqs. (1) to (3) and Fig. 5. It should be noted that Eq. (1) in this step identifies the late finish date of the last repetitive section (LFi,J) of all activities to be identical to its early finish (EFi,J) to minimize work interruptions for all assigned crews while maintaining the shortest project duration, as shown in Fig. 5.
LFi, J = EFi, J LFi, j = Min (LSi + 1, j − lagi + 1, i , LSi, j + 1)
LSi, j = LFi, j − Di, j
j
ESi,1 = LSi,1
(4)
(1)
ESi, j = Max (EFi, j − 1 + lagi, i − 1 , EFi − 1, j )
(5)
(2)
EFi, j = ESi, j + Di, j
(6)
(3)
D = EFI , J
(7)
where: ESi,1: Early start date of activity (i) at first repetitive unit (1). LSi,1: Late start date of activity (i) at first repetitive unit (1). ESi,j: Early start date of activity (i) at repetitive unit (j). EFi,j-1: Early finish date of activity (i) at previous repetitive unit (j1). Lagi,i-1: Lag time between activity (i) and activity (i-1). EFi-1,j: Early finish date of previous activity (i-1) at repetitive unit (j). EFi,j: Early finish date of activity (i) at repetitive unit (j).
where: LFi,J: Late finish date of activity (i) at repetitive unit (J). EFi,J: Early finish date of activity (i) at repetitive unit (J). LFi,j: Late finish date of activity (i) at repetitive unit (j). LSi + 1,j: Late start date of successor activity (i + 1) at repetitive unit (j). Lagi + 1,i: Lag time between activity (i + 1) and activity (i). LSi,j + 1: Late start date of activity (i) at next repetitive unit (j + 1). LSi,j: Late start date of activity (i) at repetitive unit (j). Di,j: Duration of activity (i) at repetitive unit (j). 61
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A. Altuwaim, K. El-Rayes
Step 1.2: Calculate Initial ES schedule for current activity (i)
4
3
50.3
2 43.4 Step 2.1: Calculate LS schedule for current activity (i)
1
: Precedence relationship constraint and direction of computations. Fig. 5. Calculation Steps 1.2 and 2.1.
for the early start schedule and Eq. (9) for the late start schedule; total work interruptions for the project based on either early or late start schedule as shown in Eq. (10); and number of interruptions for the project for both schedules as shown in Eq. (11).
Inter ij, j − 1 = ESi, j − EFi, j − 1
j=2→J
(8)
Inter ij, j − 1 = LSi, j − LFi, j − 1
j=2→J
(9)
I
TR =
J
∑ ∑ Inter ij,j−1 (10)
i=1 j=2
NR =
I
J
∑i =1 ∑ j=2
Ri, j
where Ri, j =
i ⎧ 1 Inter j, j − 1 > 0 ⎨ 0 Inter ij, j − 1 = 0 ⎩
(11)
where: Interij, j − 1: Work interruptions of activity (i) between repetitive units (j) and (j-1). TR: Total work interruptions in the project. NR: Number of crew work interruptions in the project for the considered schedule. Ri,j: Binary variable representing the presence of work interruptions in activity (i) between repetitive units (j) and (j-1), where Ri,j = 1 represents the presence of interruption and Ri,j = 0 indicates lack of interruption.
Fig. 3. Flowchart of Stage 1 of the early schedule computation phase.
4. Phase 2: work-continuity float calculation This phase of the model creates and utilizes novel types of floats for repetitive construction projects to calculate and analyze the impact of delaying the early start of repetitive construction activities on the work continuity of construction crews. These newly developed floats are named ‘work-continuity total float’ (CTFi,j) and ‘work-continuity free float’ (CFFi,j). Fig. 8 illustrates the limitation of using traditional floats such as total float for repetitive construction projects due to its inability to consider the impact of delaying the early start of repetitive construction activities on the total work interruption. For example, delaying the early start of activity B by its traditional total float (see Fig. 8) will not extend the project duration, however it will increase the total work interruption for all construction crews from 57 to 93 days. To overcome this limitation, the newly developed ‘work-continuity total float’ (CTFi,j) is designed to calculate the total number of days that an activity can be delayed beyond its early start without increasing the total work interruption days for the project nor extending its duration. For example, the early start of activity B in Fig. 9 can be delayed by its work-continuity total float without increasing the total project work interruption days and without extending the project duration. These novel types of floats are calculated in this model to enable construction managers to develop intermediate schedules between the
Fig. 4. Maximizing work continuity constraint while maintaining shortest project duration.
D: Total project duration. EFI,J: Early finish date of activity (I) at repetitive unit (J). 3.2.3. Step 2.3: work interruptions computation Work interruptions among all repetitive units are calculated in this step based on both the early and late start schedules developed in Step 2.1 and Step 2.2. The computations in this step as shown in Fig. 6 are designed to calculate: the work interruptions between each repetitive unit (j) and its predecessor unit (j-1) for each activity (i) using Eq. (8) 62
Automation in Construction 88 (2018) 59–72
A. Altuwaim, K. El-Rayes
Fig. 6. Flowchart of Stage 2 of the early schedule computation phase.
CTFi, j = LSi, j − ESi, j CTFi, J = 0
4
j = 1 to J − 1
(12)
j=J
(13)
where: CTFi,j: Work-continuity total float of activity (i) at repetitive unit (j). CTFi,J: Work-continuity total float of activity (i) at last repetitive unit (J).
3
2
4.2. Step 2: work-continuity free float calculation 1
ES is always equal to LS for first unit.
The work-continuity free float is defined in this model as the maximum amount of time that the repetitive unit may be delayed from its early start date without delaying the early start of its successor activities while ensuring that the project duration and total work interruptions are not increased by that delay. The work-continuity free float is calculated in this step for each activity (i) at each repetitive unit (j) as shown in Eqs. (14) and (15). It should be noted that Eq. (15) identifies the work-continuity free float of all activities at their last repetitive section (J) to be zero since work-continuity total floats for last repetitive section (J) of all activities are zero as discussed earlier (See Eq. (13)).
Fig. 7. Calculation of early start schedule in Step 2.2.
aforementioned early and late start schedules while maintaining the shortest project duration and least work interruptions achieved in Phase 1 of this model. The computations in this phase are performed in three main steps, as shown in Fig. 10.
CFFi, j = Min (ESi, j + 1 , ESi + 1, j − lagi + 1, i ) − EFi, j 4.1. Step 1: work-continuity total float calculation
CFFi, J = 0
The work-continuity total float (CTF) is defined in this model as the maximum amount of time that a repetitive unit may be delayed from its early start date calculated in Step 2.2 of Phase 1 without delaying the project completion time nor increasing the total work interruptions. The work-continuity total float is determined for each activity (i) at each repetitive unit (j) by computing the difference between its late start date and its early start date, as shown in Eqs. (12) and (13). It should be noted that Eq. (13) identifies the work-continuity total float of all activities at their last repetitive section (J) to be zero since all activities have identical early and late start dates for their last repetitive section (J) as described earlier (see Fig. 4).
j=J
j = 1 to J − 1
(14) (15)
where: CFFi,j: Work-continuity free float of activity (i) at repetitive unit (j). CFFi,J: Work-continuity free float of activity (i) at last repetitive unit (J). 4.3. Step 3: intermediate schedule calculation The purpose of this step is to generate an intermediate schedule between the early and late start schedules generated in Phase 1 that maintains the minimum project duration and the minimum total work interruptions achieved in Phase 1. The model is designed to enable 63
Automation in Construction 88 (2018) 59–72
A. Altuwaim, K. El-Rayes
Fig. 8. Impact of delaying early start by traditional total float on total work interruptions of construction crews.
construction planners to analyze the calculated work-continuity total float (CTFi,j) in the previous step and specify their suggested shifts (shifti,j) for delaying the early start date of repetitive activities within these floats, as shown in Fig. 11 and Eq. (16). The intermediate schedule for each activity (i) is calculated based on the planned start of its predecessor activity (i-1) and the availability of its assigned crews (see Fig. 11). The computations in this step are performed in a forward pass starting from activity 1 to I and from repetitive unit 1 to J (see Fig. 10) and they are designed to calculate (1) the planned start and finish dates of each activity (i) at each repetitive unit (j), as shown in Eqs. (16) and (17); (2) the work interruptions between each repetitive unit (j) and its predecessor unit (j-1) for each activity (i), as shown in Eq. (18); and (3) the number of interruptions for the entire intermediate schedule as shown in Eq. (11). It should be noted that the computations in this step enable planners to specify a set of shifts (shifti,j) for each activity (i) at each repetitive unit (j), and accordingly develop a unique intermediate project schedule for each of these specified sets of shifts.
PSi, j = Max (ESi, j + Shifti, j , PFi, j − 1 + lagi, i − 1 , PFi − 1, j ) Shifti, j ≤ CTFi, j
(16)
PFi, j = PSi, j + Di, j
(17)
Inter ij, j − 1 = PSi, j − PFi, j − 1
j=2→J
(18)
where: PSi,j: Planned start date of activity (i) at repetitive unit (j). Shifti,j: Planner-specified shift for delaying the start of activity (i) at repetitive unit (j) beyond its early start date (ESi,j). PFi,j-1: Planned finish date of activity (i) at previous repetitive unit (j-1). PFi-1,j: Planned finish date of previous activity (i-1) at repetitive unit (j). PFi,j: Planned finish date of activity (i) at repetitive unit (j).
Fig. 9. Impact of delaying early start by proposed work-continuity total float on total work interruptions of construction crews.
64
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A. Altuwaim, K. El-Rayes
Fig. 10. Flowchart of work-continuity float calculation phase.
Suggested shifti,j for activity (i): Shifti,1= 0 Shifti,2= 4.8 Shifti,3= 0 Shifti,4= 0
project schedules with and without interruptions to identify the best schedule that fits the specific project needs by analyzing the impacts of interruptions on resource utilization, interruption cost, and total project cost. The computations in this phase are performed in two main stages that are designed to: (1) generate a project schedule that strictly enforces crew work continuity for all repetitive activities; and (2) calculate the total project cost for schedules that allow interruptions and for schedules that strictly enforces work continuity. The first stage computations are designed to: (1) identify the initial schedule for activity (i) at each of its repetitive unit (j) that complies with specified activity precedence relationships and crew availability constraints using Eqs. (19) to (23); and (2) revise the initial schedule generated in step (1) to further comply with the crew work continuity constraint for all activities using Eqs. (24) to (27). These computations are performed in a forward pass starting from the first activity through the last.
28.6
Shifti,2 31.6
Fig. 11. Calculation of intermediate schedule in Step 3 of Phase 2.
S1,1 = 0 5. Phase 3: strict work continuity
j = 1, i = 1
Si,1 = Max (ESi,1 , Fi∗− 1,1 + lagi, i − 1)
This phase of the model is designed to develop a schedule for repetitive construction projects that strictly maintains work continuity for all construction crews and therefore does not allow any work interruptions, as shown in Fig. 1. This schedule enables planners to compare
Si, j = Max (Fi, j − 1 , Fi∗− 1, j + lagi, i − 1) Fi, j = Si, j + Di, j 65
(19)
j = 1, i > 1
j>1
(20) (21) (22)
Automation in Construction 88 (2018) 59–72
A. Altuwaim, K. El-Rayes
Table 1 Application example data. Crew daily output rate (Pi) (m3/day)
Repetitive activity (i)
Repetitive unit (j)
Quantity of work (Qij) (m3)
545 650 602 602 633 675
80
Activity E (i = 5)
1 2 3 4 5 6
88 110 118 84 139 84
8.6
1 2 3 4 5 6
700 650 450 500 525 475
100
Activity F (i = 6)
1 2 3 4 5 6
107 144 107 169 107 180
9.9
Activity C (i = 3)
1 2 3 4 5 6
1090 1816 1204 1204 1266 1681
92
Activity G (i = 7)
1 2 3 4 5 6
126 177 158 153 179 97
8.7
Activity D (i = 4)
1 2 3 4 5 6
917 983 1163 1109 714 1189
94.2
Activity H (i = 8)
1 2 3 4 5 6
250 200 300 250 150 220
50
Repetitive activity (i)
Repetitive unit (j)
Quantity of work (Qij) (m3)
Activity A (i = 1)
1 2 3 4 5 6
Activity B (i = 2)
Inter ij, j − 1 = Si, j − Fi, j − 1
Shifti =
J
∑2
Inter ij, j − 1
Si∗,1 = Si,1 + Shifti Si∗, j Fi∗, j
=
Fi∗, j − 1
=
Si∗, j
j=2→J
j=1
j>1
+ Di, j
I
J
∑2 ∑2
(Inter ij, j − 1 × DRCi )
Crew daily output rate (Pi) (m3/day)
(23)
RC =
(24)
C = (Ds − D) × DEC
(31)
(25)
LC = (D − Ds ) × DLC
(32)
(26)
where: TC: Project total cost. DC: Project direct cost without interruption cost. IC: Project indirect cost. RC: Project interruption cost. EC: Project early completion incentives. LC: Project liquidated damages. DIC: Daily project indirect cost in dollars. DRCi: Daily interruption cost of activity (i) in dollars. Ds: Specified project deadline or duration in days. DEC: Early completion incentives in dollars per day. DLC: Liquidated damages in dollars per day. The scope of the present model is limited to repetitive construction projects that allow interruptions for construction crews to enable minimizing the project duration while maximizing crew work continuity. The present scheduling model is developed based on a number of assumptions and limitations that include (1) each repetitive activity is constructed by a single crew formation that moves sequentially from the first repetitive unit to the last, and each of these crew formations may consist of one or more construction crews that have constant productivity rate throughout the duration of the activity; (2) the precedence relationships among successor repetitive activities are assumed to be finish to start with or without lag time; (3) the interruption of work continuity for construction crews can cause additional costs resulting from keeping the crew idle or under-utilized on site, and these daily crew interruption costs can be specified by the planner; and (4) the total cost of repetitive construction projects can be quantified by calculating its direct cost, indirect cost, interruption cost, early completion incentives, and/or late completion penalties.
(27)
where: S1,1: Start date of first activity (1) at first repetitive unit (1). Si,1: Initial start date of activity (i) at first repetitive unit (1). Fi-1,1: Initial finish date of previous activity (i-1) at first repetitive unit (1). Si,j: Initial start date of activity (i) at repetitive unit (j). Fi,j-1: Initial finish date of activity (i) at previous repetitive unit (j-1). Fi-1,j: Initial finish date of previous activity (i-1) at repetitive unit (j). Fi,j: Initial finish date of activity (i) at repetitive unit (j). Shifti: Shift for delaying the start of activity (i). S⁎i,1: Start date of activity (i) at first repetitive unit (1). F⁎i-1,1: Finish date of previous activity (i-1) at first repetitive unit (1). S⁎i,j: Start date of activity (i) at repetitive unit (j). F⁎i,j-1: Finish date of activity (i) at previous repetitive unit (j-1). F⁎i-1,j: Finish date of previous activity (i-1) at repetitive unit (j). F*i,j: Finish date of activity (i) at repetitive unit (j). The computations of the second stage are designed to calculate the total project cost (TC) for the generated schedules in Phase 1 and 3 that considers project direct cost without interruptions (DC), project indirect cost (IC), project interruptions cost (RC), project early completion incentives (EC), and project liquidated damages (LC) using Eqs (28) to (32).
TC = DC + IC + RC − EC + LC
(28)
IC = (D × DIC )
(29) 66
(30)
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A. Altuwaim, K. El-Rayes
6. Phase 4: performance evaluation
generated schedule in Stage 1 to maximize crew work continuity. The computations in the second stage of the first phase are performed in three main steps. The first step calculates for each activity at each repetitive unit its late finish date (LFi,j) using Eqs. (1) and (2), and its late start date (LSi,j) using Eq. (3), as shown in columns 5 and 4 in Tables 2 and 3. These computations are performed using a backward pass that starts from the last repetitive unit to the first repetitive and progresses sequentially from the last activity (H) to the first (A). The second step of the computations in this stage is performed using a forward pass to re-calculate for each activity at each repetitive unit its early start date (ESi,j) using Eqs. (4) and (5), and its early finish date (EFi,j) using Eq. (6) that maximize crew work continuity, as shown in columns 6 and 7 in Tables 2 and 3. The last step of computations in the second stage calculates the total project duration (D) using Eq. (7), work interruptions among all repetitive units (Interij, j − 1) using Eq. (8), total work interruptions (TR) using Eq. (10), and number of interruptions (NR) for early start schedule using Eq. (11), as shown in columns 8 and 9 in Tables 2 and 3. The generated schedules in this stage are designed to (a) comply with the precedence relationship and crew availability constraints, and (b) minimize crew work interruptions, as shown in Fig. 13. Compared to the earlier schedule of Stage 1 (see Fig. 12), the generated early start (ES) and late start (LS) schedules in Stage 2 (see Fig. 13) was able to minimize crew work interruptions from 161.2 days to 39.4 days while maintaining a minimum project duration of 175.4 days. The purpose of the second computation phase is to calculate workcontinuity floats and to generate a number of feasible intermediate schedules. The second phase of this model calculates for each activity at each repetitive unit its work-continuity total float (CTFi,j) using Eqs. (12) and (13), and its work-continuity free float (CFFi,j) using Eqs. (14) and (15), as shown in columns 10 and 11 in Tables 2 and 3,
The main purpose of this phase of the model is to illustrate the use of the present model and demonstrate its capability by analyzing an application example of a repetitive construction project that consists of eight activities (A, B, C, D, E, F, G, H) and each activity is repeated in six units in the project. The example input data are summarized in Table 1, including the quantity of work of all activities at each repetitive unit, and the daily output rate of each construction crew. In this example, the logical relationship between any two successive activities is finish to start with no lag time. A single crew is assigned to perform each repetitive activity moving from the first repetitive unit to the sixth unit sequentially. The total project direct cost and daily indirect costs of this example are estimated to be $1 million and $2000 per day, respectively. Furthermore, the daily interruption cost for all construction crews is specified by the planner to be $1000 per day. The application example is scheduled by the present model using the aforementioned three computation phases. The purpose of first computation phase is to identify the early and late start schedules that minimize project duration while keeping total work interruptions to a minimum, and its computations are performed in two stages. The first stage calculates for each activity at each repetitive unit its duration (Di,j), its early start date (ESi,j), and its early finish date (EFi,j) using the computation procedure in Fig. 3, as shown in columns 1, 2 and 3 in Tables 2 and 3, respectively. The computations in this stage are performed using a forward pass that starts from the first repetitive unit to the sixth and progresses sequentially from the first activity (A) to the last (H). The generated schedule in this stage complies with the precedence relationship and crew availability constraints; however, it does not seek to minimize crew work interruptions, shown in Fig. 12. To overcome this limitation, the second stage of the first phase revises the Table 2 Model computations for application example. Activity (i)
Rept. units (j)
Phase 1
Phase 2
Stage 1 Duration (days)
Activity (A) (i = 1)
Activity (B) (i = 2)
Activity (C) (i = 3)
Activity (D) (i = 4)
Stage 2
Forward pass
Backward pass
Forward pass
Inter.
R
CTF
ES
EF
LS
LF
ES
EF
1
2
3
4
5
6
7
8
9
10
1 2 3 4 5 6
6.81 8.13 7.53 7.53 7.91 8.44
0 6.81 14.94 22.46 29.99 37.90
6.81 14.94 22.46 29.99 37.90 46.34
0.00 6.81 14.94 22.46 29.99 37.90
6.81 14.94 22.46 29.99 37.90 46.34
0 6.81 14.94 22.46 29.99 37.90
6.81 14.94 22.46 29.99 37.90 46.34
– 0 0 0 0 0
– 0 0 0 0 0
1 2 3 4 5 6
7.00 6.50 4.50 5.00 5.25 4.75
6.81 14.94 22.46 29.99 37.90 46.34
13.81 21.44 26.96 34.99 43.15 51.09
6.81 19.16 31.59 36.09 41.09 46.34
13.81 25.66 36.09 41.09 46.34 51.09
6.81 14.94 22.46 29.99 37.90 46.34
13.81 21.44 26.96 34.99 43.15 51.09
– 1.13 1.03 3.03 2.91 3.19
1 2 3 4 5 6
11.85 19.74 13.09 13.09 13.76 18.27
13.81 25.66 45.40 58.49 71.57 85.33
25.66 45.40 58.49 71.57 85.33 103.61
13.81 25.66 45.40 58.49 71.57 85.33
25.66 45.40 58.49 71.57 85.33 103.61
13.81 25.66 45.40 58.49 71.57 85.33
25.66 45.40 58.49 71.57 85.33 103.61
1 2 3 4 5 6
9.73 10.44 12.35 11.77 7.58 12.62
25.66 45.40 58.49 71.57 85.33 103.61
35.39 55.83 70.83 83.35 92.91 116.23
35.66 45.40 61.24 76.58 91.45 103.61
45.40 55.83 73.59 88.35 99.02 116.23
35.66 45.40 58.49 71.57 85.33 103.61
45.40 55.83 70.83 83.35 92.91 116.23
67
CFF
Intermediate schedule C Shift
PS
PF
Inter.
R
11
12
13
14
15
16
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 6.81 14.94 22.46 29.99 37.90
6.81 14.94 22.46 29.99 37.90 46.34
– 0 0 0 0 0
– 0 0 0 0 0
– 1 1 1 1 1
0 4.22 9.13 6.10 3.19 0
0 1.03 3.03 2.91 3.19 0
0 0 9.13 0 0 0
6.81 14.94 31.59 36.09 41.09 46.34
13.81 21.44 36.09 41.09 46.34 51.09
– 1.13 10.15 0 0 0
– 1 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
13.81 25.66 45.40 58.49 71.57 85.33
25.66 45.40 58.49 71.57 85.33 103.61
– 0 0 0 0 0
– 0 0 0 0 0
– 0 2.65 0.74 1.99 10.69
– 0 1 1 1 1
0 0 2.76 5.00 6.11 0
0 0 0 1.21 1.41 0
0 0 2 0 0 0
35.66 45.40 60.49 72.83 85.33 103.61
45.40 55.83 72.83 84.61 92.91 116.23
– 0 4.65 0 0.73 10.69
– 0 1 0 1 1
Automation in Construction 88 (2018) 59–72
A. Altuwaim, K. El-Rayes
Table 3 Model computations for application example (cont'd). Activity (i)
Rept. units (j)
Phase 1 Stage 1 Duration (days)
Activity (E) (i = 5)
Activity (F) (i = 6)
Activity (G) (i = 7)
Activity (H) (i = 8)
Phase 2 Stage 2
Forward pass
Backward pass
Forward pass
Inter.
R
CTF
ES
EF
LS
LF
ES
EF
1
2
3
4
5
6
7
8
9
10
1 2 3 4 5 6
10.23 12.79 13.72 9.77 16.16 9.77
35.39 55.83 70.83 84.55 94.32 116.23
45.63 68.63 84.55 94.32 110.48 126.00
45.60 55.83 73.59 88.35 99.02 116.23
55.83 68.63 87.31 98.12 115.19 126.00
45.60 55.83 70.83 84.55 94.32 116.23
55.83 68.63 84.55 94.32 110.48 126.00
0 2.21 0 0 5.74
0 1 0 0 1
1 2 3 4 5 6
10.81 14.55 10.81 17.07 10.81 18.18
45.63 68.63 84.55 95.36 112.43 126.00
56.44 83.17 95.36 112.43 123.24 144.18
57.82 68.63 87.31 98.12 115.19 126.00
68.63 83.17 98.12 115.19 126.00 144.18
57.82 68.63 84.55 95.36 112.43 126.00
68.63 83.17 95.36 112.43 123.24 144.18
– 0 1.38 0 0 2.76
1 2 3 4 5 6
14.48 20.34 18.16 17.59 20.57 11.15
56.44 83.17 103.52 121.68 139.26 159.84
70.92 103.52 121.68 139.26 159.84 170.99
68.69 83.17 103.52 121.68 139.26 159.84
83.17 103.52 121.68 139.26 159.84 170.99
68.69 83.17 103.52 121.68 139.26 159.84
83.17 103.52 121.68 139.26 159.84 170.99
1 2 3 4 5 6
5.00 4.00 6.00 5.00 3.00 4.40
70.92 103.52 121.68 139.26 159.84 170.99
75.92 107.52 127.68 144.26 162.84 175.39
147.99 152.99 156.99 162.99 167.99 170.99
152.99 156.99 162.99 167.99 170.99 175.39
147.99 152.99 156.99 162.99 167.99 170.99
152.99 156.99 162.99 167.99 170.99 175.39 Total
CFF
Intermediate schedule C Shift
PS
PF
Inter.
R
11
12
13
14
15
16
0 0 2.76 3.80 4.70 0
0 0 0 0 1.95 0
0 0 2.76 0 0 0
45.60 55.83 73.59 87.31 97.08 116.23
55.83 68.63 87.31 97.08 113.24 126.00
– 0 4.96 0 0 2.99
– 0 1 0 0 1
– 0 1 0 0 1
0 0 2.76 2.76 2.76 0
0 0 0 0 2.76 0
0 0 2 0 0 0
57.82 68.63 87.31 98.12 115.19 126.00
68.63 83.17 98.12 115.19 126.00 144.18
– 0 4.14 0 0 0
– 0 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
68.69 83.17 103.52 121.68 139.26 159.84
83.17 103.52 121.68 139.26 159.84 170.99
– 0 0 0 0 0
– 0 0 0 0 0
– 0 0 0 0 0 39.44
– 0 0 0 0 0 13
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
147.99 152.99 156.99 162.99 167.99 170.99
152.99 156.99 162.99 167.99 170.99 175.39 Total
– 0 0 0 0 0 39.44
– 0 0 0 0 0 8
The bold data in the table is to highlight the result of the computation of the project duration that was mentioned and highlighted in the text.
Fig. 12. Generated schedule in Stage 1 of Phase 1. Fig. 13. Generated ES and LS schedules in Stage 2 of phase 1.
respectively. The proposed model is capable of generating a wide range of intermediate schedules using the proposed novel work-continuity floats compared to existing models that are capable of generating only a single solution. As stated earlier, planners can analyze the generated floats in Tables 2 and 3 and provide user-specified shifts (shifti,j) for each activity (i) at each repetitive unit (j) that are less than or equal its work-continuity total float (CTFi,j). These user-specified shifts (shifti,j) can then be used to generate and analyze a number of intermediate schedules that maintains the least project duration (175.4 days) and minimum total work interruptions (39.4 days), as shown in Fig. 14. The number of interruptions for each of these generated intermediate schedules is identified and provided as an output to enable planners to consider its impacts on learning curve, productivity, and cost. In this example, three intermediate schedules are generated based on three
sets of planner-specified shifts: (1) intermediate schedule A based on the first set of user-specified shifts of Shift2,4 = 1.5 days, Shift4,4 = 1 days, Shift4,5 = 2 days, Shift5,4 = 2 days, and Shift6,4 = 1.7; (2) intermediate schedule B based on the second set of user-specified shifts of Shift2,3 = 4 days, Shift2,5 = 3.19 days, Shift5,4 = 3.8 days, and Shift6,3 = 2 days; and (3) intermediate schedule C based on the third set of user-specified shifts of Shift2,3 = 9.13 days, Shift4,3 = 2 days, Shift5,3 = 2.76 days, and Shift6,3 = 2. These three intermediate schedules are shown in Fig. 14, and the planned start dates (PSi,j), finish dates (PFi,j), and number of interruptions of each of these intermediate schedules are calculated using Eqs. (16), (17) and (11), respectively. For example, the detailed computations of intermediate schedule C are summarized in columns 68
Automation in Construction 88 (2018) 59–72
A. Altuwaim, K. El-Rayes
Intermediate Schedule A
Schedule Type Early start schedule (ES) Late start schedule (LS) Intermediate schedule A Intermediate schedule B Intermediate schedule C
Project Total work Number of duration interruptions interruptions (days) (days) 175.4 39.4 11 175.4 39.4 13 175.4 39.4 15 175.4 39.4 12 175.4 39.4 8
Intermediate Schedule B
Intermediate Schedule C
Fig. 14. Intermediate schedules A, B and C.
work continuity, respectively, as shown in Table 4. In order to conduct a sensitivity analysis of the generated results to variations in the daily crew interruption cost, project early completion incentives, and project liquidated damages, two additional scenarios were analyzed for this application example. The first scenario illustrates the impact of doubling the daily interruption cost for all construction crews from $1000 to $2000. In this scenario, the least project cost can be achieved by strictly enforcing crew work continuity for all crews as shown in Table 4. The second scenario expands the assumptions of the first scenario to highlight the additional impact of including a contractspecified deadline of 195 days, an early completion incentive of $8000 per day, and liquidated damages of $ 5000 per day. In the second scenario, the least project cost can be achieved by allowing selected crew interruptions, as shown in Table 4. The novelty of the developed scheduling model and its new and
13 to 16 in Tables 2 and 3. As shown in Fig. 14, intermediate schedule C has the least number of interruptions compared to other generated schedules and accordingly can be selected by planners as one of the more practical schedules for the project. The purpose of the third computation phase is to generate a schedule that strictly enforces work continuity for all repetitive activities to enable a comparison of its performance to that of the schedule generated in Phase 1. The computations of this phase are performed using Eqs. (19) to (27) in order to generate a schedule for this application example without any work interruptions that has a duration of 202.7 days, as shown in Fig. 15. Additionally, the total project costs for both the generated schedules in Phases 1 and 3 are calculated using Eqs. (28) to (32). The total project cost was calculated to be $1.392 million and $1.405 million for the shortest duration schedule for this application example that allows crew work interruptions and strictly enforces
Fig. 15. Phase 3 generated schedule without interruptions.
69
Automation in Construction 88 (2018) 59–72
A. Altuwaim, K. El-Rayes
Table 4 Project total cost. Schedule type
Allow selected interruptions Maintain strictly work continuity
Project duration (days)
Total work interruptions (days)
Indirect cost ($)
Project performance
Scenario 1
Scenario 2
Interruption cost ($)
Total cost ($)
Interruption cost ($)
Total cost ($)
Completion incentives ($)
Liquidated damages ($)
Total cost ($)
175.4
39.4
350,800
39,400
1,390,200
78,800
1,429,600
156,800
0
1,272,800
202.7
0
405,400
0
1,405,400
0
1,405,400
0
38,500
1,443,900
The bold data in the table is to highlight the least project costs in the three different cases: (1) Project performance, (2) Scenario 1, and (3) Scenario.
Project Work duration interruptions (days) (days)
Providing ES schedule
Identifying WorkContinuity Floats
Flexibility to Total generate Floats intermediate (days) schedules
Long and Ohsato Model [35]
175.4
39.4
No
No
0
No
Developed Model
175.4
39.4
Yes
Yes
33.4
Yes
Fig. 16. Generated solutions by developed model and Long and Ohsato model [35].
schedule for repetitive construction projects that minimizes both the project duration and its total work interruptions. The work-continuity float calculation phase presented original types of floats that can be used to calculate and analyze the impact of delaying the early start of repetitive construction activities on the work continuity of construction crews. The strict work continuity phase develops a schedule that enforces work continuity for all repetitive activities to enable a comparison of its performance to that of the schedule generated in Phase 1. The performance evaluation phase analyzed an application example of a repetitive construction project to illustrate the use of the model. The results of the performance evaluation phase highlighted the original contributions of the model and its novel capabilities that include (1) generating early and late start schedules for repetitive construction projects, (2) calculating two novel types of crew work-continuity floats for each repetitive unit in all activities to enable analyzing the impact of delaying the early start of repetitive construction activities on the work continuity of construction crews, (3) developing a wide range of intermediate schedules between the early and late start schedules that maintains the least project duration and minimum total work interruption, and (4) comparing shortest duration schedules with and without interruptions to identify the best schedule that fits the specific project needs. These novel and unique capabilities provide muchneeded support for construction planners and enable them to efficiently
unique capabilities can be highlighted by comparing its results to those generated by existing related models (Long and Ohsato [35]) for the same example, as shown in Fig. 16. The results of this comparison analysis confirms the original contributions of the developed model and its three new and unique capabilities of (1) generating early and late start schedules that minimize the project duration while keeping total work interruptions to a minimum; (2) identifying novel type of floats (CTF and CFF) that analyze the impact of delaying the early start of repetitive construction activities on the work continuity of construction crews; and (3) providing planners with the flexibility to generate a wide range of intermediate schedules between the ES and LS schedules that maintains the least project duration and minimum total work interruption as shown in Fig. 16. 7. Conclusion A novel scheduling model was developed for repetitive construction projects to minimize project duration by allowing selected work interruptions when needed while maximizing crew work continuity. The model was developed in four phases: early schedule computation phase, work-continuity float calculation phase, strict work continuity phase, and performance evaluation phase. The early schedule computation phase provided a novel methodology for generating an early start 70
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A. Altuwaim, K. El-Rayes
IC = Project indirect cost. RC = Project interruption cost. EC = Project early completion incentives. LC = Project liquidated damages. DIC = Daily project indirect cost in dollars. DRCi = Daily interruption cost of activity (i) in dollars. Ds = Specified project deadline or duration in days. DEC = Early completion incentives in dollars per day. DLC = Liquidated damages in dollars per day.
generate a wide range of least duration schedules for repetitive construction projects and analyze their impacts on the total project cost. Appendix A A.1. Notation Di,j = duration of repetitive unit (j) in activity (j). Qi,j = quantity of work in repetitive unit (j) in activity (i). Pi = daily output rate for construction crew performing activity (i). ESi,j = early start date of repetitive unit (j) in activity (i). ES1,1 = early start date of repetitive unit (1) in activity (1). ESi,1 = early start date in first repetitive unit (1) in activity (i). EFi,j = early finish date of repetitive unit (j) in activity (i). EFi-1,j = early finish date of repetitive unit (j) in previous activity (i-1). EFi,j-1 = early finish date of previous repetitive unit (j-1) in activity (i). EFi,J = early finish date of repetitive unit (J) in activity (i). EFI,J = early finish date of repetitive unit (J) in activity (I). LFi,j = late finish date of repetitive unit (j) in activity (i). LFi,J = late finish date of repetitive unit (J) in activity (i). LSi,j = late start date of repetitive unit (j) in activity (i). LSi + 1,j = late start date of repetitive unit (j) in successor activity (i + 1). LSi,j + 1 = late start date of next repetitive unit (j + 1) in activity (i). LSi,1 = late start date in first repetitive unit (1) in activity (i). Lagi,i-1 = Lag time between activity (i) and activity (i-1). D = total project duration. Interij, j − 1 = work interruptions between repetitive units (j) and (j-1) in activity (i). TR = total work interruptions in the project. Ri,j = binary variable representing the presence of work interruptions in activity (i) between repetitive units (j) and (j-1), where Ri,j = 1 represents the presence of interruption and Ri,j = 0 indicates lack of interruption. CTFi,j = work-continuity total float of repetitive unit (j) in activity (i). CTFi,J = work-continuity total float of last repetitive unit (J) in activity (i). CFFi,j = work-continuity free float of repetitive unit (j) in activity (i). CFFi,J = work-continuity free float of last repetitive unit (J) in activity (i). Shifti,j = planner-specified shift for delaying the start of repetitive unit (j) in activity (i) beyond its early start date (ESi,j). PSi,j = planned start date of repetitive unit (j) in activity (i). PSi,j-1 = planned finish date of previous repetitive unit (j-1) in activity (i). PFi-1,j = planned finish date of repetitive unit (j) in previous activity (i-1). PFi,j = planned finish date of repetitive unit (j) in activity (i). S1,1 = Start date of first activity (1) at first repetitive unit (1). Si,1 = Initial start date of activity (i) at first repetitive unit (1). Fi-1,1 = Initial finish date of previous activity (i-1) at first repetitive unit (1). Si,j = Initial start date of activity (i) at repetitive unit (j). Fi,j-1 = Initial finish date of activity (i) at previous repetitive unit (j-1). Fi-1,j = Initial finish date of previous activity (i-1) at repetitive unit (j). Fi,j = Initial finish date of activity (i) at repetitive unit (j). Shifti = Shift for delaying the start of activity (i). S⁎i,1 = Start date of activity (i) at first repetitive unit (1). F⁎i-1,1 = Finish date of previous activity (i-1) at first repetitive unit (1). S⁎i,j = Start date of activity (i) at repetitive unit (j). F⁎i,j-1 = Finish date of activity (i) at previous repetitive unit (j-1). F⁎i-1,j = Finish date of previous activity (i-1) at repetitive unit (j). F⁎i,j = Finish date of activity (i) at repetitive unit (j). TC = Project total cost. DC = Project direct cost without interruption cost.
A.2. Subscripts and superscripts i = construction activity (from i = 1 to I). j = repetitive unit (from j = 1 to J). References [1] G.S. Birrell, Construction planning—beyond the critical path, J. Constr. Div. 106 (3) (1980) 389–407 http://cedb.asce.org/CEDBsearch/record.jsp?dockey=0009725 (Accessed: November 1, 2017). [2] S. Selinger, Construction planning for linear projects, J. Constr. Div. 106 (2) (1980) 195–205 http://cedb.asce.org/CEDBsearch/record.jsp?dockey=0009573 , Accessed date: 2 November 2017. [3] D.P. Kavanagh, Siren: a repetitive construction simulation model, J. Constr. Eng. Manag. 111 (3) (1985) 308–323 https://doi.org/10.1061/(ASCE)0733-9364(1985) 111:3(308). [4] R.M. Reda, RPM: repetitive project modeling, J. Constr. Eng. Manag. 116 (2) (1990) 316–330 https://doi.org/10.1061/(ASCE)0733-9364(1990)116:2(316). [5] A.D. Russell, W.F. Caselton, Extensions to linear scheduling optimization, J. Constr. Eng. Manag. 114 (1) (1988) 36–52 https://doi.org/10.1061/(ASCE)07339364(1988)114:1(36. [6] H. Adeli, A. Karim, Scheduling/cost optimization and neural dynamics model for construction, J. Constr. Eng. Manag. 123 (4) (1997) 450–458 https://doi.org/10. 1061/(ASCE)0733-9364(1997)123:4(450). [7] K. El-Rayes, O. Moselhi, Resource-driven scheduling of repetitive activities, Constr. Manag. Econ. 16 (4) (1998) 433–446 https://doi.org/10.1080/014461998372213. [8] K. El-Rayes, O. Moselhi, Optimizing resource utilization for repetitive construction projects, J. Constr. Eng. Manag. 127 (1) (2001) 18–27 https://doi.org/10.1061/ (ASCE)0733-9364(2001)127:1(18). [9] A. Hassanein, O. Moselhi, Accelerating linear projects, Constr. Manag. Econ. 23 (4) (2005) 377–385 https://doi.org/10.1080/01446190410001673571. [10] S. Fan, H.P. Tserng, Object-oriented scheduling for repetitive projects with soft logics, J. Constr. Eng. Manag. 132 (1) (2006) 35–48 https://doi.org/10.1061/ (ASCE)0733-9364(2006)132:1(35. [11] M.E. Georgy, Evolutionary resource scheduler for linear projects, Autom. Constr. 17 (5) (2008) 573–583 https://doi.org/10.1016/j.autcon.2007.10.005. [12] H.R. Zolfaghar Dolabi, A. Afshar, R. Abbasnia, CPM/LOB scheduling method for project deadline constraint satisfaction, Autom. Constr. 48 (2014) 107–118 https:// doi.org/10.1016/j.autcon.2014.09.003. [13] D.B. Ashley, Simulation of repetitive-unit construction, J. Constr. Div. 106 (2) (1980) 185–194 http://cedb.asce.org/CEDBsearch/record.jsp?dockey=0009564 (Accessed: November 2, 2017). [14] K. El-Rayes, Optimum planning of highway construction under A + B bidding method, J. Constr. Eng. Manag. 127 (4) (2001) 261–269 https://doi.org/10.1061/ (ASCE)0733-9364(2001)127:4(261). [15] O. Moselhi, K. El-Rayes, Scheduling of repetitive projects with cost optimization, J. Constr. Eng. Manag. 119 (4) (1993) 681–697 https://doi.org/10.1061/(ASCE) 0733-9364(1993)119:4(681). [16] O. Moselhi, K. El-Rayes, Least cost scheduling for repetitive projects, Can. J. Civ. Eng. 20 (5) (1993) 834–843 https://doi.org/10.1139/l93-109. [17] A.B. Senouci, N.N. Eldin, Dynamic programming approach to scheduling of nonserial linear project, J. Comput. Civ. Eng. 10 (2) (1996) 106–114 https://doi.org/ 10.1061/(ASCE)0887-3801(1996)10:2(106). [18] T. Hegazy, N. Wassef, Cost optimization in projects with repetitive nonserial activities, J. Constr. Eng. Manag. 127 (3) (2001) 183–191 https://doi.org/10.1061/ (ASCE)0733-9364(2001)127:3(183). [19] G. Lucko, Mathematical Analysis of Linear Schedules, Proceedings of the 2007 Construction Research Congress, http://citeseerx.ist.psu.edu/viewdoc/summary? doi=10.1.1.554.4313, (2007) , Accessed date: 8 November 2017. [20] G. Lucko, A.a. Peña Orozco, Float types in linear schedule analysis with singularity functions, J. Constr. Eng. Manag. 135 (5) (2009) 368–377 https://doi.org/10.1061/ (ASCE)CO.1943-7862.0000007. [21] S.-L. Fan, K.-S. Sun, Y.-R. Wang, GA optimization model for repetitive projects with soft logic, Autom. Constr. 21 (2012) 253–261 https://doi.org/10.1016/j.autcon. 2011.06.009. [22] A. Damci, D. Arditi, G. Polat, Multiresource leveling in line-of-balance scheduling, J. Constr. Eng. Manag. 139 (9) (2013) 1108–1116 https://doi.org/10.1061/ (ASCE)CO.1943-7862.0000716. [23] A. Damci, D. Arditi, G. Polat, Resource leveling in line-of-balance scheduling, computer-aided civil and infrastructure, Engineering 28 (9) (2013) 679–692
71
Automation in Construction 88 (2018) 59–72
A. Altuwaim, K. El-Rayes
https://doi.org/10.1111/mice.12038. [24] I. Bakry, O. Moselhi, T. Zayed, Optimized acceleration of repetitive construction projects, Autom. Constr. 39 (2014) 145–151 https://doi.org/10.1016/j.autcon. 2013.07.003. [25] I. Bakry, Optimized Scheduling of Repetitive Construction Projects Under Uncertainty, Concordia University, PhD Dissertation, 2014http://spectrum.library. concordia.ca/978756/1/Bakry_PhD_S2014.pdf , Accessed date: 1 November 2017. [26] K. El-Rayes, Object-oriented model for repetitive construction scheduling, J. Constr. Eng. Manag. 127 (3) (2001) 199–205 https://doi.org/10.1061/(ASCE)07339364(2001)127:3(199). [27] T. Hegazy, A. Elhakeem, E. Elbeltagi, Distributed scheduling model for infrastructure networks, J. Constr. Eng. Manag. 130 (2) (2004) 160–167 https://doi.org/ 10.1061/(ASCE)0733-9364(2004)130:2(160). [28] K. Hyari, K. El-Rayes, A. Multi-objective Model, For optimizing construction planning of repetitive infrastructure projects, proceedings of the tenth international conference on computing in civil and building, Engineering (2004), https://doi. org/10.13140/2.1.5017.3128. [29] K. Hyari, K. El-Rayes, M. ASCE, Optimal planning and scheduling for repetitive construction projects, J. Manag. Eng. 22 (1) (2006) 11–19 https://doi.org/10. 1061/(ASCE)0742-597X(2006)22:1(11). [30] K. Nassar, Evolutionary optimization of resource allocation in repetitive
[31]
[32]
[33]
[34]
[35]
[36]
72
construction schedules, Electron. J. Inf. Technol. Constr. 10 (2005) 265–273 http:// www.itcon.org/2005/18 (Accessed November 2, 2017). S.-S. Liu, C.-J. Wang, Optimization model for resource assignment problems of linear construction projects, Autom. Constr. 16 (4) (2007) 460–473 https://doi.org/ 10.1016/j.autcon.2006.08.004. P.G. Ipsilandis, Multiobjective linear programming model for scheduling linear repetitive projects, J. Constr. Eng. Manag. 133 (6) (2007) 417–424 https://doi.org/ 10.1061/(ASCE)0733-9364(2007)133:6(417). G. Lucko, Productivity scheduling method compared to linear and repetitive project scheduling methods, J. Constr. Eng. Manag. 134 (9) (2008) 711–720 https://doi. org/10.1061/(ASCE)0733-9364(2008)134:9(711). G. Lucko, A.A.P. Orozco, Calculating float in linear schedules with singularity functions, Proceedings of the 2008 Winter Simulation Conference, 2008, pp. 2512–2518 https://doi.org/10.1109/WSC.2008.4736361. L.D. Long, A. Ohsato, A genetic algorithm-based method for scheduling repetitive construction projects, Autom. Constr. 18 (4) (2009) 499–511 https://doi.org/10. 1016/j.autcon.2008.11.005. K.H. Hyari, K. El-Rayes, M. El-Mashaleh, Automated trade-off between time and cost in planning repetitive construction projects, Constr. Manag. Econ. 27 (8) (2009) 749–761 https://doi.org/10.1080/01446190903117793.
Contents lists available at ScienceDirect
Automation in Construction journal homepage: www.elsevier.com/locate/autcon
Minimizing duration and crew work interruptions of repetitive construction projects Ayman Altuwaima,b, Khaled El-Rayesa, a b
T
⁎
Department of Civil and Environmental Engineering, University of Illinois at Urbana–Champaign, Urbana, IL 61801, United States Department of Civil Engineering, King Saud University, Riyadh, Saudi Arabia
A R T I C L E I N F O
A B S T R A C T
Keywords: Repetitive construction Linear scheduling Resource-driven scheduling Crew work continuity Scheduling algorithms
This paper presents the development of a novel scheduling model for minimizing the duration and crew work interruptions of repetitive construction projects. The main contributions of the developed model are its ability to (1) generate early and late start schedules that minimize the duration of repetitive construction projects while keeping the total work interruptions of their utilized crews to a minimum; (2) calculate novel types of crew work-continuity floats that consider the impact of delaying the early start of repetitive activities on crew work continuity; (3) develop a wide range of intermediate schedules between the early and late start schedules that maintain the least project duration and minimum total crew work interruptions; and (4) compare shortest duration schedules with and without interruptions to identify the best schedule that fits the specific project needs. The model performance was evaluated using an application example of a repetitive construction project.
1. Introduction Repetitive construction projects require construction crews to repeat their work in a number of locations in the same project, moving from one location to the next. This continuous movement of crews on site is often encountered during the construction of high-rise buildings, housing projects, highways, pipeline networks, and bridges. Traditional scheduling methods such as bar charts and critical path method are ineffective in scheduling this class of projects due to their inability to consider and maximize work continuity for the construction crews on these projects [1–12]. Maximizing work continuity improves construction productivity by minimizing crew idle and non-productive times and by maximizing the benefits of the learning curve effect for working crews [1,8,13,14]. To realize these benefits, a number of scheduling models were developed for repetitive construction projects that are capable of considering and maximizing crew work continuity. Existing scheduling models for repetitive construction projects that consider crew work continuity can be classified in two main categories: (1) models that strictly enforce work continuity for all construction crews without allowing any interruptions [2,4,6,9,11,15–25]; and (2) models that maximize work continuity for all construction crews while allowing selected interruptions to minimize project duration, as shown in Fig. 1[5,7,8,18,26–36]. The scheduling models in the first category are capable of maintaining full work continuity for all crews, however their generated schedule often leads to longer project duration due to
⁎
the strict enforcement of the crew work continuity constraint, as shown in Fig. 1. The scheduling models in the second category provide shorter project duration than those in the first category because they provide the flexibility of enabling selected interruptions only when needed to minimize the project duration [2,5,8,29]. Scheduling models in the second category can be further classified into optimization models and heuristic models. First, optimization models in this category utilized various optimization techniques such as linear programing, dynamic programing or genetic algorithms to minimize project duration, cost, and/or work interruptions [5,8,14,18,26–32,35,36]. For example, linear programing was utilized by Ipsilandis [32] to minimize crew work interruptions in scheduling repetitive projects under a specified project duration. Dynamic programming was also used by Russell and Caselton [5], El-Rayes and Moselhi [8], and El-Rayes [14,26] to optimize the scheduling of repetitive construction projects to minimize project duration, work interruptions, project cost, and/or total combined bid price for A + B highway contracts. Similarly, genetic algorithms were utilized by Nasser [30], Hegazy and Wassef [18], Hegazy et al. [27], Hyari and ElRayes [28,29], Long and Ohsato [35], and Hyari and El-Rayes [36] to either minimize project duration and work interruptions, minimize project cost, trade-offs between project duration and work interruptions, or trade-offs between project duration and cost. Second, heuristic scheduling models in the second category utilized various algorithms and heuristics to schedule repetitive projects with selected work
Corresponding author. E-mail address: [email protected] (K. El-Rayes).
https://doi.org/10.1016/j.autcon.2017.12.024 Received 23 November 2016; Received in revised form 9 November 2017; Accepted 7 December 2017 Available online 03 January 2018 0926-5805/ © 2017 Elsevier B.V. All rights reserved.
Automation in Construction 88 (2018) 59–72
A. Altuwaim, K. El-Rayes
Fig. 1. Impact of allowing work interruptions on project duration.
3. Phase 1: early schedule computation
interruptions [7,35]. An example of heuristic scheduling models that allow interruptions for repetitive construction projects is the resourcedriven scheduling model developed by El-Rayes and Moselhi [7] that enables decision makers to assign multiple crew formations for each repetitive activity, and to specify work interruptions between the repetitive units in order to minimize the project duration. Another heuristic scheduling model was developed by Long and Ohsato [35] for repetitive projects which is capable of identifying work interruptions between repetitive units in order to minimize the project duration and keep work interruptions to a minimum. Despite the contributions of the aforementioned scheduling models for repetitive construction projects that allow selected work interruptions, they have a number of limitations as they are incapable of (1) generating an early start schedule that minimizes both the duration project and its total work interruptions; (2) calculating important types of floats for repetitive construction projects that can be used to analyze the impact of delaying the early start of repetitive construction activities on the work continuity of construction crews; and (3) providing planners with the early start and late start schedules for each repetitive activity which provides them the flexibility to generate a wide range of intermediate schedules. Accordingly, there is a pressing need for a novel scheduling model for repetitive construction projects that is capable of circumventing these three main limitations of existing heuristic models.
This phase of the model focuses on developing a novel scheduling algorithm for repetitive construction projects that seeks to minimize project duration by allowing selected work interruptions while maximizing work continuity. The scheduling algorithm is designed to (1) calculate early start (ESi,j), late start (LSi,j), early finish (EFi,j), late finish (LFi,j) and work interruptions (Interj, j − 1i) for each section (j) in the repetitive activities (i), as shown in Fig. 2; (2) compute the project duration (D) and total work interruptions (TR), as shown in Fig. 2; (3) comply with crew availability and precedence relationships constraints; (4) maximize work continuity; and (5) enable the scheduling of activities with varying durations in its repetitive units (see Fig. 1). The required input data for this schedule computation phase include: (a) repetitive activities data that include the total number of activities (I) and their logical precedence relationships; (b) number of the repetitive units (J); (c) the quantity of work (Qi,j) for each repetitive section (j) in each activity (i); and (d) daily productivity rates for the specified crew in each activity (Pi), as shown in Fig. 2. The computations of this phase are performed using the following two stages. 3.1. Stage 1 This stage creates an initial project schedule that provides the shortest possible duration for the project while complying with the job logic/precedence relationships and crew availability constraints. This is achieved in two main steps that are designed to calculate: (1.1) the duration of each activity (i) at each repetitive unit (j); and (1.2) the early start and finish dates of construction for each activity (i) at each repetitive unit (j), as shown in Fig. 3.
2. Objective The goal of this paper is to develop a novel scheduling model for repetitive construction projects that overcomes the aforementioned three main limitations of existing heuristic models. To accomplish this goal, the three main research objectives of this paper are to (1) develop an innovative heuristic methodology for generating an early start schedule that simultaneously minimizes project duration and total work interruptions; (2) create novel types of floats to calculate and analyze the impact of delaying the early start of repetitive construction activities on the work continuity of construction crews; and (3) generate a wide range of intermediate schedules to provide planners with alternative schedules that minimize both project duration and work interruptions. These three objectives are accomplished in four main phases: early schedule computation phase, work-continuity float calculation phase, strict work continuity phase, and performance evaluation phase, as shown in Fig. 2. The following sections of the paper describe these four development phases of the scheduling model.
3.2. Stage 2 This stage is designed to revise the initial schedule generated in Stage 1 to maximize compliance with the crew work continuity constraint. This is achieved by identifying for each repetitive activity (i) the required shift for the start date of its earlier repetitive sections (j = 1 to J-1) while maintaining the initial/earliest possible start and finish dates for its last repetitive section (j = J) to minimize work interruptions for all assigned crews while maintaining the shortest project duration, as shown in Fig. 4. The revised final schedules generated in this stage identifies the early start, early finish, late start, and late finish for each activity (i) at all its repetitive unit (j). The scheduling computations in 60
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A. Altuwaim, K. El-Rayes
Fig. 2. Model development phases.
3.2.2. Step 2.2: forward pass The purpose of this step is to calculate the early start schedule that minimizes the project duration and work interruptions simultaneously. This early start schedule for each activity (i) is calculated based on the early start of its predecessor activity (i-1) and the availability of its assigned crews, as shown in Fig. 7. The forward pass computations start from activity 1 to I and from repetitive unit 1 to J (see Fig. 6) and they are designed to calculate the early start and finish dates of each activity (i) at each repetitive unit (j) as shown in Eqs. (4) to (6), and the minimum project duration (D), as shown in Eq. (7). It should be noted that Eq. (4) in this step identifies the early start date of all activities at the first repetitive section (ESi,1) to be identical to its late start (LSi,1) to identify the earliest start date for the remaining repetitive sections (j = 2 to J) while maintaining the minimum work interruptions achieved in Step 2.1 and the shortest project duration, as shown in Fig. 7.
this stage are performed in three steps: backward pass, forward pass, and work interruption computations. 3.2.1. Step 2.1: backward pass The goal of the backward pass is to determine the latest start schedule that minimizes the project duration and work interruptions simultaneously. For each activity (i), this latest start schedule is calculated based on the late start of its successor activity (i + 1) and the availability of its assigned crews, as shown in Fig. 5. The backward pass computations start from activity I to 1 and from repetitive unit J to 1 (see Fig. 6) and they are designed to calculate the late finish and late start date of each activity (i) at each repetitive unit (j) as shown in Eqs. (1) to (3) and Fig. 5. It should be noted that Eq. (1) in this step identifies the late finish date of the last repetitive section (LFi,J) of all activities to be identical to its early finish (EFi,J) to minimize work interruptions for all assigned crews while maintaining the shortest project duration, as shown in Fig. 5.
LFi, J = EFi, J LFi, j = Min (LSi + 1, j − lagi + 1, i , LSi, j + 1)
LSi, j = LFi, j − Di, j
j
ESi,1 = LSi,1
(4)
(1)
ESi, j = Max (EFi, j − 1 + lagi, i − 1 , EFi − 1, j )
(5)
(2)
EFi, j = ESi, j + Di, j
(6)
(3)
D = EFI , J
(7)
where: ESi,1: Early start date of activity (i) at first repetitive unit (1). LSi,1: Late start date of activity (i) at first repetitive unit (1). ESi,j: Early start date of activity (i) at repetitive unit (j). EFi,j-1: Early finish date of activity (i) at previous repetitive unit (j1). Lagi,i-1: Lag time between activity (i) and activity (i-1). EFi-1,j: Early finish date of previous activity (i-1) at repetitive unit (j). EFi,j: Early finish date of activity (i) at repetitive unit (j).
where: LFi,J: Late finish date of activity (i) at repetitive unit (J). EFi,J: Early finish date of activity (i) at repetitive unit (J). LFi,j: Late finish date of activity (i) at repetitive unit (j). LSi + 1,j: Late start date of successor activity (i + 1) at repetitive unit (j). Lagi + 1,i: Lag time between activity (i + 1) and activity (i). LSi,j + 1: Late start date of activity (i) at next repetitive unit (j + 1). LSi,j: Late start date of activity (i) at repetitive unit (j). Di,j: Duration of activity (i) at repetitive unit (j). 61
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Step 1.2: Calculate Initial ES schedule for current activity (i)
4
3
50.3
2 43.4 Step 2.1: Calculate LS schedule for current activity (i)
1
: Precedence relationship constraint and direction of computations. Fig. 5. Calculation Steps 1.2 and 2.1.
for the early start schedule and Eq. (9) for the late start schedule; total work interruptions for the project based on either early or late start schedule as shown in Eq. (10); and number of interruptions for the project for both schedules as shown in Eq. (11).
Inter ij, j − 1 = ESi, j − EFi, j − 1
j=2→J
(8)
Inter ij, j − 1 = LSi, j − LFi, j − 1
j=2→J
(9)
I
TR =
J
∑ ∑ Inter ij,j−1 (10)
i=1 j=2
NR =
I
J
∑i =1 ∑ j=2
Ri, j
where Ri, j =
i ⎧ 1 Inter j, j − 1 > 0 ⎨ 0 Inter ij, j − 1 = 0 ⎩
(11)
where: Interij, j − 1: Work interruptions of activity (i) between repetitive units (j) and (j-1). TR: Total work interruptions in the project. NR: Number of crew work interruptions in the project for the considered schedule. Ri,j: Binary variable representing the presence of work interruptions in activity (i) between repetitive units (j) and (j-1), where Ri,j = 1 represents the presence of interruption and Ri,j = 0 indicates lack of interruption.
Fig. 3. Flowchart of Stage 1 of the early schedule computation phase.
4. Phase 2: work-continuity float calculation This phase of the model creates and utilizes novel types of floats for repetitive construction projects to calculate and analyze the impact of delaying the early start of repetitive construction activities on the work continuity of construction crews. These newly developed floats are named ‘work-continuity total float’ (CTFi,j) and ‘work-continuity free float’ (CFFi,j). Fig. 8 illustrates the limitation of using traditional floats such as total float for repetitive construction projects due to its inability to consider the impact of delaying the early start of repetitive construction activities on the total work interruption. For example, delaying the early start of activity B by its traditional total float (see Fig. 8) will not extend the project duration, however it will increase the total work interruption for all construction crews from 57 to 93 days. To overcome this limitation, the newly developed ‘work-continuity total float’ (CTFi,j) is designed to calculate the total number of days that an activity can be delayed beyond its early start without increasing the total work interruption days for the project nor extending its duration. For example, the early start of activity B in Fig. 9 can be delayed by its work-continuity total float without increasing the total project work interruption days and without extending the project duration. These novel types of floats are calculated in this model to enable construction managers to develop intermediate schedules between the
Fig. 4. Maximizing work continuity constraint while maintaining shortest project duration.
D: Total project duration. EFI,J: Early finish date of activity (I) at repetitive unit (J). 3.2.3. Step 2.3: work interruptions computation Work interruptions among all repetitive units are calculated in this step based on both the early and late start schedules developed in Step 2.1 and Step 2.2. The computations in this step as shown in Fig. 6 are designed to calculate: the work interruptions between each repetitive unit (j) and its predecessor unit (j-1) for each activity (i) using Eq. (8) 62
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A. Altuwaim, K. El-Rayes
Fig. 6. Flowchart of Stage 2 of the early schedule computation phase.
CTFi, j = LSi, j − ESi, j CTFi, J = 0
4
j = 1 to J − 1
(12)
j=J
(13)
where: CTFi,j: Work-continuity total float of activity (i) at repetitive unit (j). CTFi,J: Work-continuity total float of activity (i) at last repetitive unit (J).
3
2
4.2. Step 2: work-continuity free float calculation 1
ES is always equal to LS for first unit.
The work-continuity free float is defined in this model as the maximum amount of time that the repetitive unit may be delayed from its early start date without delaying the early start of its successor activities while ensuring that the project duration and total work interruptions are not increased by that delay. The work-continuity free float is calculated in this step for each activity (i) at each repetitive unit (j) as shown in Eqs. (14) and (15). It should be noted that Eq. (15) identifies the work-continuity free float of all activities at their last repetitive section (J) to be zero since work-continuity total floats for last repetitive section (J) of all activities are zero as discussed earlier (See Eq. (13)).
Fig. 7. Calculation of early start schedule in Step 2.2.
aforementioned early and late start schedules while maintaining the shortest project duration and least work interruptions achieved in Phase 1 of this model. The computations in this phase are performed in three main steps, as shown in Fig. 10.
CFFi, j = Min (ESi, j + 1 , ESi + 1, j − lagi + 1, i ) − EFi, j 4.1. Step 1: work-continuity total float calculation
CFFi, J = 0
The work-continuity total float (CTF) is defined in this model as the maximum amount of time that a repetitive unit may be delayed from its early start date calculated in Step 2.2 of Phase 1 without delaying the project completion time nor increasing the total work interruptions. The work-continuity total float is determined for each activity (i) at each repetitive unit (j) by computing the difference between its late start date and its early start date, as shown in Eqs. (12) and (13). It should be noted that Eq. (13) identifies the work-continuity total float of all activities at their last repetitive section (J) to be zero since all activities have identical early and late start dates for their last repetitive section (J) as described earlier (see Fig. 4).
j=J
j = 1 to J − 1
(14) (15)
where: CFFi,j: Work-continuity free float of activity (i) at repetitive unit (j). CFFi,J: Work-continuity free float of activity (i) at last repetitive unit (J). 4.3. Step 3: intermediate schedule calculation The purpose of this step is to generate an intermediate schedule between the early and late start schedules generated in Phase 1 that maintains the minimum project duration and the minimum total work interruptions achieved in Phase 1. The model is designed to enable 63
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A. Altuwaim, K. El-Rayes
Fig. 8. Impact of delaying early start by traditional total float on total work interruptions of construction crews.
construction planners to analyze the calculated work-continuity total float (CTFi,j) in the previous step and specify their suggested shifts (shifti,j) for delaying the early start date of repetitive activities within these floats, as shown in Fig. 11 and Eq. (16). The intermediate schedule for each activity (i) is calculated based on the planned start of its predecessor activity (i-1) and the availability of its assigned crews (see Fig. 11). The computations in this step are performed in a forward pass starting from activity 1 to I and from repetitive unit 1 to J (see Fig. 10) and they are designed to calculate (1) the planned start and finish dates of each activity (i) at each repetitive unit (j), as shown in Eqs. (16) and (17); (2) the work interruptions between each repetitive unit (j) and its predecessor unit (j-1) for each activity (i), as shown in Eq. (18); and (3) the number of interruptions for the entire intermediate schedule as shown in Eq. (11). It should be noted that the computations in this step enable planners to specify a set of shifts (shifti,j) for each activity (i) at each repetitive unit (j), and accordingly develop a unique intermediate project schedule for each of these specified sets of shifts.
PSi, j = Max (ESi, j + Shifti, j , PFi, j − 1 + lagi, i − 1 , PFi − 1, j ) Shifti, j ≤ CTFi, j
(16)
PFi, j = PSi, j + Di, j
(17)
Inter ij, j − 1 = PSi, j − PFi, j − 1
j=2→J
(18)
where: PSi,j: Planned start date of activity (i) at repetitive unit (j). Shifti,j: Planner-specified shift for delaying the start of activity (i) at repetitive unit (j) beyond its early start date (ESi,j). PFi,j-1: Planned finish date of activity (i) at previous repetitive unit (j-1). PFi-1,j: Planned finish date of previous activity (i-1) at repetitive unit (j). PFi,j: Planned finish date of activity (i) at repetitive unit (j).
Fig. 9. Impact of delaying early start by proposed work-continuity total float on total work interruptions of construction crews.
64
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A. Altuwaim, K. El-Rayes
Fig. 10. Flowchart of work-continuity float calculation phase.
Suggested shifti,j for activity (i): Shifti,1= 0 Shifti,2= 4.8 Shifti,3= 0 Shifti,4= 0
project schedules with and without interruptions to identify the best schedule that fits the specific project needs by analyzing the impacts of interruptions on resource utilization, interruption cost, and total project cost. The computations in this phase are performed in two main stages that are designed to: (1) generate a project schedule that strictly enforces crew work continuity for all repetitive activities; and (2) calculate the total project cost for schedules that allow interruptions and for schedules that strictly enforces work continuity. The first stage computations are designed to: (1) identify the initial schedule for activity (i) at each of its repetitive unit (j) that complies with specified activity precedence relationships and crew availability constraints using Eqs. (19) to (23); and (2) revise the initial schedule generated in step (1) to further comply with the crew work continuity constraint for all activities using Eqs. (24) to (27). These computations are performed in a forward pass starting from the first activity through the last.
28.6
Shifti,2 31.6
Fig. 11. Calculation of intermediate schedule in Step 3 of Phase 2.
S1,1 = 0 5. Phase 3: strict work continuity
j = 1, i = 1
Si,1 = Max (ESi,1 , Fi∗− 1,1 + lagi, i − 1)
This phase of the model is designed to develop a schedule for repetitive construction projects that strictly maintains work continuity for all construction crews and therefore does not allow any work interruptions, as shown in Fig. 1. This schedule enables planners to compare
Si, j = Max (Fi, j − 1 , Fi∗− 1, j + lagi, i − 1) Fi, j = Si, j + Di, j 65
(19)
j = 1, i > 1
j>1
(20) (21) (22)
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Table 1 Application example data. Crew daily output rate (Pi) (m3/day)
Repetitive activity (i)
Repetitive unit (j)
Quantity of work (Qij) (m3)
545 650 602 602 633 675
80
Activity E (i = 5)
1 2 3 4 5 6
88 110 118 84 139 84
8.6
1 2 3 4 5 6
700 650 450 500 525 475
100
Activity F (i = 6)
1 2 3 4 5 6
107 144 107 169 107 180
9.9
Activity C (i = 3)
1 2 3 4 5 6
1090 1816 1204 1204 1266 1681
92
Activity G (i = 7)
1 2 3 4 5 6
126 177 158 153 179 97
8.7
Activity D (i = 4)
1 2 3 4 5 6
917 983 1163 1109 714 1189
94.2
Activity H (i = 8)
1 2 3 4 5 6
250 200 300 250 150 220
50
Repetitive activity (i)
Repetitive unit (j)
Quantity of work (Qij) (m3)
Activity A (i = 1)
1 2 3 4 5 6
Activity B (i = 2)
Inter ij, j − 1 = Si, j − Fi, j − 1
Shifti =
J
∑2
Inter ij, j − 1
Si∗,1 = Si,1 + Shifti Si∗, j Fi∗, j
=
Fi∗, j − 1
=
Si∗, j
j=2→J
j=1
j>1
+ Di, j
I
J
∑2 ∑2
(Inter ij, j − 1 × DRCi )
Crew daily output rate (Pi) (m3/day)
(23)
RC =
(24)
C = (Ds − D) × DEC
(31)
(25)
LC = (D − Ds ) × DLC
(32)
(26)
where: TC: Project total cost. DC: Project direct cost without interruption cost. IC: Project indirect cost. RC: Project interruption cost. EC: Project early completion incentives. LC: Project liquidated damages. DIC: Daily project indirect cost in dollars. DRCi: Daily interruption cost of activity (i) in dollars. Ds: Specified project deadline or duration in days. DEC: Early completion incentives in dollars per day. DLC: Liquidated damages in dollars per day. The scope of the present model is limited to repetitive construction projects that allow interruptions for construction crews to enable minimizing the project duration while maximizing crew work continuity. The present scheduling model is developed based on a number of assumptions and limitations that include (1) each repetitive activity is constructed by a single crew formation that moves sequentially from the first repetitive unit to the last, and each of these crew formations may consist of one or more construction crews that have constant productivity rate throughout the duration of the activity; (2) the precedence relationships among successor repetitive activities are assumed to be finish to start with or without lag time; (3) the interruption of work continuity for construction crews can cause additional costs resulting from keeping the crew idle or under-utilized on site, and these daily crew interruption costs can be specified by the planner; and (4) the total cost of repetitive construction projects can be quantified by calculating its direct cost, indirect cost, interruption cost, early completion incentives, and/or late completion penalties.
(27)
where: S1,1: Start date of first activity (1) at first repetitive unit (1). Si,1: Initial start date of activity (i) at first repetitive unit (1). Fi-1,1: Initial finish date of previous activity (i-1) at first repetitive unit (1). Si,j: Initial start date of activity (i) at repetitive unit (j). Fi,j-1: Initial finish date of activity (i) at previous repetitive unit (j-1). Fi-1,j: Initial finish date of previous activity (i-1) at repetitive unit (j). Fi,j: Initial finish date of activity (i) at repetitive unit (j). Shifti: Shift for delaying the start of activity (i). S⁎i,1: Start date of activity (i) at first repetitive unit (1). F⁎i-1,1: Finish date of previous activity (i-1) at first repetitive unit (1). S⁎i,j: Start date of activity (i) at repetitive unit (j). F⁎i,j-1: Finish date of activity (i) at previous repetitive unit (j-1). F⁎i-1,j: Finish date of previous activity (i-1) at repetitive unit (j). F*i,j: Finish date of activity (i) at repetitive unit (j). The computations of the second stage are designed to calculate the total project cost (TC) for the generated schedules in Phase 1 and 3 that considers project direct cost without interruptions (DC), project indirect cost (IC), project interruptions cost (RC), project early completion incentives (EC), and project liquidated damages (LC) using Eqs (28) to (32).
TC = DC + IC + RC − EC + LC
(28)
IC = (D × DIC )
(29) 66
(30)
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6. Phase 4: performance evaluation
generated schedule in Stage 1 to maximize crew work continuity. The computations in the second stage of the first phase are performed in three main steps. The first step calculates for each activity at each repetitive unit its late finish date (LFi,j) using Eqs. (1) and (2), and its late start date (LSi,j) using Eq. (3), as shown in columns 5 and 4 in Tables 2 and 3. These computations are performed using a backward pass that starts from the last repetitive unit to the first repetitive and progresses sequentially from the last activity (H) to the first (A). The second step of the computations in this stage is performed using a forward pass to re-calculate for each activity at each repetitive unit its early start date (ESi,j) using Eqs. (4) and (5), and its early finish date (EFi,j) using Eq. (6) that maximize crew work continuity, as shown in columns 6 and 7 in Tables 2 and 3. The last step of computations in the second stage calculates the total project duration (D) using Eq. (7), work interruptions among all repetitive units (Interij, j − 1) using Eq. (8), total work interruptions (TR) using Eq. (10), and number of interruptions (NR) for early start schedule using Eq. (11), as shown in columns 8 and 9 in Tables 2 and 3. The generated schedules in this stage are designed to (a) comply with the precedence relationship and crew availability constraints, and (b) minimize crew work interruptions, as shown in Fig. 13. Compared to the earlier schedule of Stage 1 (see Fig. 12), the generated early start (ES) and late start (LS) schedules in Stage 2 (see Fig. 13) was able to minimize crew work interruptions from 161.2 days to 39.4 days while maintaining a minimum project duration of 175.4 days. The purpose of the second computation phase is to calculate workcontinuity floats and to generate a number of feasible intermediate schedules. The second phase of this model calculates for each activity at each repetitive unit its work-continuity total float (CTFi,j) using Eqs. (12) and (13), and its work-continuity free float (CFFi,j) using Eqs. (14) and (15), as shown in columns 10 and 11 in Tables 2 and 3,
The main purpose of this phase of the model is to illustrate the use of the present model and demonstrate its capability by analyzing an application example of a repetitive construction project that consists of eight activities (A, B, C, D, E, F, G, H) and each activity is repeated in six units in the project. The example input data are summarized in Table 1, including the quantity of work of all activities at each repetitive unit, and the daily output rate of each construction crew. In this example, the logical relationship between any two successive activities is finish to start with no lag time. A single crew is assigned to perform each repetitive activity moving from the first repetitive unit to the sixth unit sequentially. The total project direct cost and daily indirect costs of this example are estimated to be $1 million and $2000 per day, respectively. Furthermore, the daily interruption cost for all construction crews is specified by the planner to be $1000 per day. The application example is scheduled by the present model using the aforementioned three computation phases. The purpose of first computation phase is to identify the early and late start schedules that minimize project duration while keeping total work interruptions to a minimum, and its computations are performed in two stages. The first stage calculates for each activity at each repetitive unit its duration (Di,j), its early start date (ESi,j), and its early finish date (EFi,j) using the computation procedure in Fig. 3, as shown in columns 1, 2 and 3 in Tables 2 and 3, respectively. The computations in this stage are performed using a forward pass that starts from the first repetitive unit to the sixth and progresses sequentially from the first activity (A) to the last (H). The generated schedule in this stage complies with the precedence relationship and crew availability constraints; however, it does not seek to minimize crew work interruptions, shown in Fig. 12. To overcome this limitation, the second stage of the first phase revises the Table 2 Model computations for application example. Activity (i)
Rept. units (j)
Phase 1
Phase 2
Stage 1 Duration (days)
Activity (A) (i = 1)
Activity (B) (i = 2)
Activity (C) (i = 3)
Activity (D) (i = 4)
Stage 2
Forward pass
Backward pass
Forward pass
Inter.
R
CTF
ES
EF
LS
LF
ES
EF
1
2
3
4
5
6
7
8
9
10
1 2 3 4 5 6
6.81 8.13 7.53 7.53 7.91 8.44
0 6.81 14.94 22.46 29.99 37.90
6.81 14.94 22.46 29.99 37.90 46.34
0.00 6.81 14.94 22.46 29.99 37.90
6.81 14.94 22.46 29.99 37.90 46.34
0 6.81 14.94 22.46 29.99 37.90
6.81 14.94 22.46 29.99 37.90 46.34
– 0 0 0 0 0
– 0 0 0 0 0
1 2 3 4 5 6
7.00 6.50 4.50 5.00 5.25 4.75
6.81 14.94 22.46 29.99 37.90 46.34
13.81 21.44 26.96 34.99 43.15 51.09
6.81 19.16 31.59 36.09 41.09 46.34
13.81 25.66 36.09 41.09 46.34 51.09
6.81 14.94 22.46 29.99 37.90 46.34
13.81 21.44 26.96 34.99 43.15 51.09
– 1.13 1.03 3.03 2.91 3.19
1 2 3 4 5 6
11.85 19.74 13.09 13.09 13.76 18.27
13.81 25.66 45.40 58.49 71.57 85.33
25.66 45.40 58.49 71.57 85.33 103.61
13.81 25.66 45.40 58.49 71.57 85.33
25.66 45.40 58.49 71.57 85.33 103.61
13.81 25.66 45.40 58.49 71.57 85.33
25.66 45.40 58.49 71.57 85.33 103.61
1 2 3 4 5 6
9.73 10.44 12.35 11.77 7.58 12.62
25.66 45.40 58.49 71.57 85.33 103.61
35.39 55.83 70.83 83.35 92.91 116.23
35.66 45.40 61.24 76.58 91.45 103.61
45.40 55.83 73.59 88.35 99.02 116.23
35.66 45.40 58.49 71.57 85.33 103.61
45.40 55.83 70.83 83.35 92.91 116.23
67
CFF
Intermediate schedule C Shift
PS
PF
Inter.
R
11
12
13
14
15
16
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 6.81 14.94 22.46 29.99 37.90
6.81 14.94 22.46 29.99 37.90 46.34
– 0 0 0 0 0
– 0 0 0 0 0
– 1 1 1 1 1
0 4.22 9.13 6.10 3.19 0
0 1.03 3.03 2.91 3.19 0
0 0 9.13 0 0 0
6.81 14.94 31.59 36.09 41.09 46.34
13.81 21.44 36.09 41.09 46.34 51.09
– 1.13 10.15 0 0 0
– 1 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
13.81 25.66 45.40 58.49 71.57 85.33
25.66 45.40 58.49 71.57 85.33 103.61
– 0 0 0 0 0
– 0 0 0 0 0
– 0 2.65 0.74 1.99 10.69
– 0 1 1 1 1
0 0 2.76 5.00 6.11 0
0 0 0 1.21 1.41 0
0 0 2 0 0 0
35.66 45.40 60.49 72.83 85.33 103.61
45.40 55.83 72.83 84.61 92.91 116.23
– 0 4.65 0 0.73 10.69
– 0 1 0 1 1
Automation in Construction 88 (2018) 59–72
A. Altuwaim, K. El-Rayes
Table 3 Model computations for application example (cont'd). Activity (i)
Rept. units (j)
Phase 1 Stage 1 Duration (days)
Activity (E) (i = 5)
Activity (F) (i = 6)
Activity (G) (i = 7)
Activity (H) (i = 8)
Phase 2 Stage 2
Forward pass
Backward pass
Forward pass
Inter.
R
CTF
ES
EF
LS
LF
ES
EF
1
2
3
4
5
6
7
8
9
10
1 2 3 4 5 6
10.23 12.79 13.72 9.77 16.16 9.77
35.39 55.83 70.83 84.55 94.32 116.23
45.63 68.63 84.55 94.32 110.48 126.00
45.60 55.83 73.59 88.35 99.02 116.23
55.83 68.63 87.31 98.12 115.19 126.00
45.60 55.83 70.83 84.55 94.32 116.23
55.83 68.63 84.55 94.32 110.48 126.00
0 2.21 0 0 5.74
0 1 0 0 1
1 2 3 4 5 6
10.81 14.55 10.81 17.07 10.81 18.18
45.63 68.63 84.55 95.36 112.43 126.00
56.44 83.17 95.36 112.43 123.24 144.18
57.82 68.63 87.31 98.12 115.19 126.00
68.63 83.17 98.12 115.19 126.00 144.18
57.82 68.63 84.55 95.36 112.43 126.00
68.63 83.17 95.36 112.43 123.24 144.18
– 0 1.38 0 0 2.76
1 2 3 4 5 6
14.48 20.34 18.16 17.59 20.57 11.15
56.44 83.17 103.52 121.68 139.26 159.84
70.92 103.52 121.68 139.26 159.84 170.99
68.69 83.17 103.52 121.68 139.26 159.84
83.17 103.52 121.68 139.26 159.84 170.99
68.69 83.17 103.52 121.68 139.26 159.84
83.17 103.52 121.68 139.26 159.84 170.99
1 2 3 4 5 6
5.00 4.00 6.00 5.00 3.00 4.40
70.92 103.52 121.68 139.26 159.84 170.99
75.92 107.52 127.68 144.26 162.84 175.39
147.99 152.99 156.99 162.99 167.99 170.99
152.99 156.99 162.99 167.99 170.99 175.39
147.99 152.99 156.99 162.99 167.99 170.99
152.99 156.99 162.99 167.99 170.99 175.39 Total
CFF
Intermediate schedule C Shift
PS
PF
Inter.
R
11
12
13
14
15
16
0 0 2.76 3.80 4.70 0
0 0 0 0 1.95 0
0 0 2.76 0 0 0
45.60 55.83 73.59 87.31 97.08 116.23
55.83 68.63 87.31 97.08 113.24 126.00
– 0 4.96 0 0 2.99
– 0 1 0 0 1
– 0 1 0 0 1
0 0 2.76 2.76 2.76 0
0 0 0 0 2.76 0
0 0 2 0 0 0
57.82 68.63 87.31 98.12 115.19 126.00
68.63 83.17 98.12 115.19 126.00 144.18
– 0 4.14 0 0 0
– 0 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
68.69 83.17 103.52 121.68 139.26 159.84
83.17 103.52 121.68 139.26 159.84 170.99
– 0 0 0 0 0
– 0 0 0 0 0
– 0 0 0 0 0 39.44
– 0 0 0 0 0 13
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
147.99 152.99 156.99 162.99 167.99 170.99
152.99 156.99 162.99 167.99 170.99 175.39 Total
– 0 0 0 0 0 39.44
– 0 0 0 0 0 8
The bold data in the table is to highlight the result of the computation of the project duration that was mentioned and highlighted in the text.
Fig. 12. Generated schedule in Stage 1 of Phase 1. Fig. 13. Generated ES and LS schedules in Stage 2 of phase 1.
respectively. The proposed model is capable of generating a wide range of intermediate schedules using the proposed novel work-continuity floats compared to existing models that are capable of generating only a single solution. As stated earlier, planners can analyze the generated floats in Tables 2 and 3 and provide user-specified shifts (shifti,j) for each activity (i) at each repetitive unit (j) that are less than or equal its work-continuity total float (CTFi,j). These user-specified shifts (shifti,j) can then be used to generate and analyze a number of intermediate schedules that maintains the least project duration (175.4 days) and minimum total work interruptions (39.4 days), as shown in Fig. 14. The number of interruptions for each of these generated intermediate schedules is identified and provided as an output to enable planners to consider its impacts on learning curve, productivity, and cost. In this example, three intermediate schedules are generated based on three
sets of planner-specified shifts: (1) intermediate schedule A based on the first set of user-specified shifts of Shift2,4 = 1.5 days, Shift4,4 = 1 days, Shift4,5 = 2 days, Shift5,4 = 2 days, and Shift6,4 = 1.7; (2) intermediate schedule B based on the second set of user-specified shifts of Shift2,3 = 4 days, Shift2,5 = 3.19 days, Shift5,4 = 3.8 days, and Shift6,3 = 2 days; and (3) intermediate schedule C based on the third set of user-specified shifts of Shift2,3 = 9.13 days, Shift4,3 = 2 days, Shift5,3 = 2.76 days, and Shift6,3 = 2. These three intermediate schedules are shown in Fig. 14, and the planned start dates (PSi,j), finish dates (PFi,j), and number of interruptions of each of these intermediate schedules are calculated using Eqs. (16), (17) and (11), respectively. For example, the detailed computations of intermediate schedule C are summarized in columns 68
Automation in Construction 88 (2018) 59–72
A. Altuwaim, K. El-Rayes
Intermediate Schedule A
Schedule Type Early start schedule (ES) Late start schedule (LS) Intermediate schedule A Intermediate schedule B Intermediate schedule C
Project Total work Number of duration interruptions interruptions (days) (days) 175.4 39.4 11 175.4 39.4 13 175.4 39.4 15 175.4 39.4 12 175.4 39.4 8
Intermediate Schedule B
Intermediate Schedule C
Fig. 14. Intermediate schedules A, B and C.
work continuity, respectively, as shown in Table 4. In order to conduct a sensitivity analysis of the generated results to variations in the daily crew interruption cost, project early completion incentives, and project liquidated damages, two additional scenarios were analyzed for this application example. The first scenario illustrates the impact of doubling the daily interruption cost for all construction crews from $1000 to $2000. In this scenario, the least project cost can be achieved by strictly enforcing crew work continuity for all crews as shown in Table 4. The second scenario expands the assumptions of the first scenario to highlight the additional impact of including a contractspecified deadline of 195 days, an early completion incentive of $8000 per day, and liquidated damages of $ 5000 per day. In the second scenario, the least project cost can be achieved by allowing selected crew interruptions, as shown in Table 4. The novelty of the developed scheduling model and its new and
13 to 16 in Tables 2 and 3. As shown in Fig. 14, intermediate schedule C has the least number of interruptions compared to other generated schedules and accordingly can be selected by planners as one of the more practical schedules for the project. The purpose of the third computation phase is to generate a schedule that strictly enforces work continuity for all repetitive activities to enable a comparison of its performance to that of the schedule generated in Phase 1. The computations of this phase are performed using Eqs. (19) to (27) in order to generate a schedule for this application example without any work interruptions that has a duration of 202.7 days, as shown in Fig. 15. Additionally, the total project costs for both the generated schedules in Phases 1 and 3 are calculated using Eqs. (28) to (32). The total project cost was calculated to be $1.392 million and $1.405 million for the shortest duration schedule for this application example that allows crew work interruptions and strictly enforces
Fig. 15. Phase 3 generated schedule without interruptions.
69
Automation in Construction 88 (2018) 59–72
A. Altuwaim, K. El-Rayes
Table 4 Project total cost. Schedule type
Allow selected interruptions Maintain strictly work continuity
Project duration (days)
Total work interruptions (days)
Indirect cost ($)
Project performance
Scenario 1
Scenario 2
Interruption cost ($)
Total cost ($)
Interruption cost ($)
Total cost ($)
Completion incentives ($)
Liquidated damages ($)
Total cost ($)
175.4
39.4
350,800
39,400
1,390,200
78,800
1,429,600
156,800
0
1,272,800
202.7
0
405,400
0
1,405,400
0
1,405,400
0
38,500
1,443,900
The bold data in the table is to highlight the least project costs in the three different cases: (1) Project performance, (2) Scenario 1, and (3) Scenario.
Project Work duration interruptions (days) (days)
Providing ES schedule
Identifying WorkContinuity Floats
Flexibility to Total generate Floats intermediate (days) schedules
Long and Ohsato Model [35]
175.4
39.4
No
No
0
No
Developed Model
175.4
39.4
Yes
Yes
33.4
Yes
Fig. 16. Generated solutions by developed model and Long and Ohsato model [35].
schedule for repetitive construction projects that minimizes both the project duration and its total work interruptions. The work-continuity float calculation phase presented original types of floats that can be used to calculate and analyze the impact of delaying the early start of repetitive construction activities on the work continuity of construction crews. The strict work continuity phase develops a schedule that enforces work continuity for all repetitive activities to enable a comparison of its performance to that of the schedule generated in Phase 1. The performance evaluation phase analyzed an application example of a repetitive construction project to illustrate the use of the model. The results of the performance evaluation phase highlighted the original contributions of the model and its novel capabilities that include (1) generating early and late start schedules for repetitive construction projects, (2) calculating two novel types of crew work-continuity floats for each repetitive unit in all activities to enable analyzing the impact of delaying the early start of repetitive construction activities on the work continuity of construction crews, (3) developing a wide range of intermediate schedules between the early and late start schedules that maintains the least project duration and minimum total work interruption, and (4) comparing shortest duration schedules with and without interruptions to identify the best schedule that fits the specific project needs. These novel and unique capabilities provide muchneeded support for construction planners and enable them to efficiently
unique capabilities can be highlighted by comparing its results to those generated by existing related models (Long and Ohsato [35]) for the same example, as shown in Fig. 16. The results of this comparison analysis confirms the original contributions of the developed model and its three new and unique capabilities of (1) generating early and late start schedules that minimize the project duration while keeping total work interruptions to a minimum; (2) identifying novel type of floats (CTF and CFF) that analyze the impact of delaying the early start of repetitive construction activities on the work continuity of construction crews; and (3) providing planners with the flexibility to generate a wide range of intermediate schedules between the ES and LS schedules that maintains the least project duration and minimum total work interruption as shown in Fig. 16. 7. Conclusion A novel scheduling model was developed for repetitive construction projects to minimize project duration by allowing selected work interruptions when needed while maximizing crew work continuity. The model was developed in four phases: early schedule computation phase, work-continuity float calculation phase, strict work continuity phase, and performance evaluation phase. The early schedule computation phase provided a novel methodology for generating an early start 70
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A. Altuwaim, K. El-Rayes
IC = Project indirect cost. RC = Project interruption cost. EC = Project early completion incentives. LC = Project liquidated damages. DIC = Daily project indirect cost in dollars. DRCi = Daily interruption cost of activity (i) in dollars. Ds = Specified project deadline or duration in days. DEC = Early completion incentives in dollars per day. DLC = Liquidated damages in dollars per day.
generate a wide range of least duration schedules for repetitive construction projects and analyze their impacts on the total project cost. Appendix A A.1. Notation Di,j = duration of repetitive unit (j) in activity (j). Qi,j = quantity of work in repetitive unit (j) in activity (i). Pi = daily output rate for construction crew performing activity (i). ESi,j = early start date of repetitive unit (j) in activity (i). ES1,1 = early start date of repetitive unit (1) in activity (1). ESi,1 = early start date in first repetitive unit (1) in activity (i). EFi,j = early finish date of repetitive unit (j) in activity (i). EFi-1,j = early finish date of repetitive unit (j) in previous activity (i-1). EFi,j-1 = early finish date of previous repetitive unit (j-1) in activity (i). EFi,J = early finish date of repetitive unit (J) in activity (i). EFI,J = early finish date of repetitive unit (J) in activity (I). LFi,j = late finish date of repetitive unit (j) in activity (i). LFi,J = late finish date of repetitive unit (J) in activity (i). LSi,j = late start date of repetitive unit (j) in activity (i). LSi + 1,j = late start date of repetitive unit (j) in successor activity (i + 1). LSi,j + 1 = late start date of next repetitive unit (j + 1) in activity (i). LSi,1 = late start date in first repetitive unit (1) in activity (i). Lagi,i-1 = Lag time between activity (i) and activity (i-1). D = total project duration. Interij, j − 1 = work interruptions between repetitive units (j) and (j-1) in activity (i). TR = total work interruptions in the project. Ri,j = binary variable representing the presence of work interruptions in activity (i) between repetitive units (j) and (j-1), where Ri,j = 1 represents the presence of interruption and Ri,j = 0 indicates lack of interruption. CTFi,j = work-continuity total float of repetitive unit (j) in activity (i). CTFi,J = work-continuity total float of last repetitive unit (J) in activity (i). CFFi,j = work-continuity free float of repetitive unit (j) in activity (i). CFFi,J = work-continuity free float of last repetitive unit (J) in activity (i). Shifti,j = planner-specified shift for delaying the start of repetitive unit (j) in activity (i) beyond its early start date (ESi,j). PSi,j = planned start date of repetitive unit (j) in activity (i). PSi,j-1 = planned finish date of previous repetitive unit (j-1) in activity (i). PFi-1,j = planned finish date of repetitive unit (j) in previous activity (i-1). PFi,j = planned finish date of repetitive unit (j) in activity (i). S1,1 = Start date of first activity (1) at first repetitive unit (1). Si,1 = Initial start date of activity (i) at first repetitive unit (1). Fi-1,1 = Initial finish date of previous activity (i-1) at first repetitive unit (1). Si,j = Initial start date of activity (i) at repetitive unit (j). Fi,j-1 = Initial finish date of activity (i) at previous repetitive unit (j-1). Fi-1,j = Initial finish date of previous activity (i-1) at repetitive unit (j). Fi,j = Initial finish date of activity (i) at repetitive unit (j). Shifti = Shift for delaying the start of activity (i). S⁎i,1 = Start date of activity (i) at first repetitive unit (1). F⁎i-1,1 = Finish date of previous activity (i-1) at first repetitive unit (1). S⁎i,j = Start date of activity (i) at repetitive unit (j). F⁎i,j-1 = Finish date of activity (i) at previous repetitive unit (j-1). F⁎i-1,j = Finish date of previous activity (i-1) at repetitive unit (j). F⁎i,j = Finish date of activity (i) at repetitive unit (j). TC = Project total cost. DC = Project direct cost without interruption cost.
A.2. Subscripts and superscripts i = construction activity (from i = 1 to I). j = repetitive unit (from j = 1 to J). References [1] G.S. Birrell, Construction planning—beyond the critical path, J. Constr. Div. 106 (3) (1980) 389–407 http://cedb.asce.org/CEDBsearch/record.jsp?dockey=0009725 (Accessed: November 1, 2017). [2] S. Selinger, Construction planning for linear projects, J. Constr. Div. 106 (2) (1980) 195–205 http://cedb.asce.org/CEDBsearch/record.jsp?dockey=0009573 , Accessed date: 2 November 2017. [3] D.P. Kavanagh, Siren: a repetitive construction simulation model, J. Constr. Eng. Manag. 111 (3) (1985) 308–323 https://doi.org/10.1061/(ASCE)0733-9364(1985) 111:3(308). [4] R.M. Reda, RPM: repetitive project modeling, J. Constr. Eng. Manag. 116 (2) (1990) 316–330 https://doi.org/10.1061/(ASCE)0733-9364(1990)116:2(316). [5] A.D. Russell, W.F. Caselton, Extensions to linear scheduling optimization, J. Constr. Eng. Manag. 114 (1) (1988) 36–52 https://doi.org/10.1061/(ASCE)07339364(1988)114:1(36. [6] H. Adeli, A. Karim, Scheduling/cost optimization and neural dynamics model for construction, J. Constr. Eng. Manag. 123 (4) (1997) 450–458 https://doi.org/10. 1061/(ASCE)0733-9364(1997)123:4(450). [7] K. El-Rayes, O. Moselhi, Resource-driven scheduling of repetitive activities, Constr. Manag. Econ. 16 (4) (1998) 433–446 https://doi.org/10.1080/014461998372213. [8] K. El-Rayes, O. Moselhi, Optimizing resource utilization for repetitive construction projects, J. Constr. Eng. Manag. 127 (1) (2001) 18–27 https://doi.org/10.1061/ (ASCE)0733-9364(2001)127:1(18). [9] A. Hassanein, O. Moselhi, Accelerating linear projects, Constr. Manag. Econ. 23 (4) (2005) 377–385 https://doi.org/10.1080/01446190410001673571. [10] S. Fan, H.P. Tserng, Object-oriented scheduling for repetitive projects with soft logics, J. Constr. Eng. Manag. 132 (1) (2006) 35–48 https://doi.org/10.1061/ (ASCE)0733-9364(2006)132:1(35. [11] M.E. Georgy, Evolutionary resource scheduler for linear projects, Autom. Constr. 17 (5) (2008) 573–583 https://doi.org/10.1016/j.autcon.2007.10.005. [12] H.R. Zolfaghar Dolabi, A. Afshar, R. Abbasnia, CPM/LOB scheduling method for project deadline constraint satisfaction, Autom. Constr. 48 (2014) 107–118 https:// doi.org/10.1016/j.autcon.2014.09.003. [13] D.B. Ashley, Simulation of repetitive-unit construction, J. Constr. Div. 106 (2) (1980) 185–194 http://cedb.asce.org/CEDBsearch/record.jsp?dockey=0009564 (Accessed: November 2, 2017). [14] K. El-Rayes, Optimum planning of highway construction under A + B bidding method, J. Constr. Eng. Manag. 127 (4) (2001) 261–269 https://doi.org/10.1061/ (ASCE)0733-9364(2001)127:4(261). [15] O. Moselhi, K. El-Rayes, Scheduling of repetitive projects with cost optimization, J. Constr. Eng. Manag. 119 (4) (1993) 681–697 https://doi.org/10.1061/(ASCE) 0733-9364(1993)119:4(681). [16] O. Moselhi, K. El-Rayes, Least cost scheduling for repetitive projects, Can. J. Civ. Eng. 20 (5) (1993) 834–843 https://doi.org/10.1139/l93-109. [17] A.B. Senouci, N.N. Eldin, Dynamic programming approach to scheduling of nonserial linear project, J. Comput. Civ. Eng. 10 (2) (1996) 106–114 https://doi.org/ 10.1061/(ASCE)0887-3801(1996)10:2(106). [18] T. Hegazy, N. Wassef, Cost optimization in projects with repetitive nonserial activities, J. Constr. Eng. Manag. 127 (3) (2001) 183–191 https://doi.org/10.1061/ (ASCE)0733-9364(2001)127:3(183). [19] G. Lucko, Mathematical Analysis of Linear Schedules, Proceedings of the 2007 Construction Research Congress, http://citeseerx.ist.psu.edu/viewdoc/summary? doi=10.1.1.554.4313, (2007) , Accessed date: 8 November 2017. [20] G. Lucko, A.a. Peña Orozco, Float types in linear schedule analysis with singularity functions, J. Constr. Eng. Manag. 135 (5) (2009) 368–377 https://doi.org/10.1061/ (ASCE)CO.1943-7862.0000007. [21] S.-L. Fan, K.-S. Sun, Y.-R. Wang, GA optimization model for repetitive projects with soft logic, Autom. Constr. 21 (2012) 253–261 https://doi.org/10.1016/j.autcon. 2011.06.009. [22] A. Damci, D. Arditi, G. Polat, Multiresource leveling in line-of-balance scheduling, J. Constr. Eng. Manag. 139 (9) (2013) 1108–1116 https://doi.org/10.1061/ (ASCE)CO.1943-7862.0000716. [23] A. Damci, D. Arditi, G. Polat, Resource leveling in line-of-balance scheduling, computer-aided civil and infrastructure, Engineering 28 (9) (2013) 679–692
71
Automation in Construction 88 (2018) 59–72
A. Altuwaim, K. El-Rayes
https://doi.org/10.1111/mice.12038. [24] I. Bakry, O. Moselhi, T. Zayed, Optimized acceleration of repetitive construction projects, Autom. Constr. 39 (2014) 145–151 https://doi.org/10.1016/j.autcon. 2013.07.003. [25] I. Bakry, Optimized Scheduling of Repetitive Construction Projects Under Uncertainty, Concordia University, PhD Dissertation, 2014http://spectrum.library. concordia.ca/978756/1/Bakry_PhD_S2014.pdf , Accessed date: 1 November 2017. [26] K. El-Rayes, Object-oriented model for repetitive construction scheduling, J. Constr. Eng. Manag. 127 (3) (2001) 199–205 https://doi.org/10.1061/(ASCE)07339364(2001)127:3(199). [27] T. Hegazy, A. Elhakeem, E. Elbeltagi, Distributed scheduling model for infrastructure networks, J. Constr. Eng. Manag. 130 (2) (2004) 160–167 https://doi.org/ 10.1061/(ASCE)0733-9364(2004)130:2(160). [28] K. Hyari, K. El-Rayes, A. Multi-objective Model, For optimizing construction planning of repetitive infrastructure projects, proceedings of the tenth international conference on computing in civil and building, Engineering (2004), https://doi. org/10.13140/2.1.5017.3128. [29] K. Hyari, K. El-Rayes, M. ASCE, Optimal planning and scheduling for repetitive construction projects, J. Manag. Eng. 22 (1) (2006) 11–19 https://doi.org/10. 1061/(ASCE)0742-597X(2006)22:1(11). [30] K. Nassar, Evolutionary optimization of resource allocation in repetitive
[31]
[32]
[33]
[34]
[35]
[36]
72
construction schedules, Electron. J. Inf. Technol. Constr. 10 (2005) 265–273 http:// www.itcon.org/2005/18 (Accessed November 2, 2017). S.-S. Liu, C.-J. Wang, Optimization model for resource assignment problems of linear construction projects, Autom. Constr. 16 (4) (2007) 460–473 https://doi.org/ 10.1016/j.autcon.2006.08.004. P.G. Ipsilandis, Multiobjective linear programming model for scheduling linear repetitive projects, J. Constr. Eng. Manag. 133 (6) (2007) 417–424 https://doi.org/ 10.1061/(ASCE)0733-9364(2007)133:6(417). G. Lucko, Productivity scheduling method compared to linear and repetitive project scheduling methods, J. Constr. Eng. Manag. 134 (9) (2008) 711–720 https://doi. org/10.1061/(ASCE)0733-9364(2008)134:9(711). G. Lucko, A.A.P. Orozco, Calculating float in linear schedules with singularity functions, Proceedings of the 2008 Winter Simulation Conference, 2008, pp. 2512–2518 https://doi.org/10.1109/WSC.2008.4736361. L.D. Long, A. Ohsato, A genetic algorithm-based method for scheduling repetitive construction projects, Autom. Constr. 18 (4) (2009) 499–511 https://doi.org/10. 1016/j.autcon.2008.11.005. K.H. Hyari, K. El-Rayes, M. El-Mashaleh, Automated trade-off between time and cost in planning repetitive construction projects, Constr. Manag. Econ. 27 (8) (2009) 749–761 https://doi.org/10.1080/01446190903117793.