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Modeling and Solving the Deadline Satisfaction Problem in
Line-of-Balance Scheduling
Downloaded from ascelibrary.org by University Of Florida on 10/28/17. Copyright ASCE. For personal use only; all rights reserved.
Xin Zou1; Qian Zhang2; and Lihui Zhang3
Abstract: This study investigates the deadline satisfaction problem in line-of-balance (LOB) scheduling, where each activity can be done
concurrently in several units by hiring additional crews, and all employed crews are allocated to implement an activity in sequential units
smoothly with no interruption. Furthermore, the project must be completed within a given deadline. In previous studies, the objective was to
minimize the total number of crews under the assumption that the cost of hiring a crew is the same for all activities. In fact, this cost usually differs among activities because of the differences in type of work, crew size, and technical level of workers. This paper presents a mixed-integer
linear programming model with an extended objective of minimizing the crew employment cost of all activities. The model is equipped with
an exact procedure to validate the feasibility of a prespecified deadline. A highway project was used to show the application of the proposed
method, and then extensive computational experiments were conducted to investigate the performance of the proposed method by comparing
the obtained results with those obtained by the existing methods. The results highlight the significance of integrating the time-cost trade-off
analysis, and indicate that the proposed method can find optimal solutions for large-scale projects within a reasonable amount of time. DOI:
10.1061/(ASCE)ME.1943-5479.0000565. © 2017 American Society of Civil Engineers.
Author Keywords: Repetitive projects; Line-of-balance; Time-cost trade-off; Deadline constraint; Mixed-integer linear programming.
Introduction
Repetitive projects are those that consist of a number of units to be
finished and a set of activities that need to be repeated for each unit
in the length of the job. In real life, most multistage projects (e.g.,
high-rise buildings and multiple similar houses) and infrastructure
projects (e.g., highways, railways, and pipelines) fall into this category. Repetitive projects represent a large portion of the construction industry. According to China’s 13th Five-Year Plan, regional
road networks will be expanded to 5,000,000 km, with an increase
of 420,000 km; approximately 30,000 km of railways will be built
and start operation by 2020. Thus, efficient planning of this type of
project is an important determinant of the success of construction
companies.
In repetitive projects, each crew usually needs to perform the
same or similar activities in different units by moving from one unit
to the next. Therefore, the needless waste of resources owing to
waiting for preceding activities to finish should be eliminated by
maintaining work continuity of crews (Long and Ohsato 2009).
Moreover, maintaining work continuity is beneficial for the crews,
as it allows them to develop the so-called learning effect that will
1
Lecturer, Dept. of Economic Management, North China Electric
Power Univ., Hebei 071003, People’s Republic of China. E-mail:
zoux788@126.com
2
Ph.D. Degree Candidate, School of Economics and Management,
North China Electric Power Univ., Beijing 102206, People’s Republic of
China (corresponding author). E-mail: zhangqian881111@163.com
3
Professor, School of Economics and Management, North China
Electric Power Univ., Beijing 102206, People’s Republic of China. E-mail:
zlh6699@126.com
Note. This manuscript was submitted on February 13, 2017; approved
on July 3, 2017; published online on October 25, 2017. Discussion period
open until March 25, 2018; separate discussions must be submitted for
individual papers. This paper is part of the Journal of Management in
Engineering, © ASCE, ISSN 0742-597X.
© ASCE
likely reduce the time and money required for completing later units
(Arditi et al. 2001a).
The critical-path method (CPM) was widely implemented for
construction project planning in the late 50s (Mohammadipour and
Sadjadi 2016). When used to schedule repetitive projects, however,
CPM becomes less practical because it is not oriented toward providing work continuity for crews. Since the 1960s, many techniques
with the capability of eliminating CPM deficiencies in scheduling
repetitive projects have been developed (Arditi and Albulak 1986;
Al Sarraj 1990; Reda 1990; Harmelink and Rowings 1998; Harris
and Ioannou 1998; Lucko 2009). As shown in Table 1, although
these techniques differ somewhat in their focus, they all plan a
schedule in a space-time, two-dimensional coordinate system, and
have the ability to maintain work continuity of crews.
In the literature, several studies have been conducted to improve
existing methods at different levels of detail. For instance, Tang et
al. (2014) developed a two-stage scheduling model for resource leveling based on the linear-scheduling model (LSM), where optimal
or near-optimal schedules are obtained using constraint programming techniques. Hyari and El-Raye (2006) presented a multiobjective optimization model to minimize the project duration and maximize crew work continuity. Hsie et al. (2009) constructed a LSMbased resource-constrained scheduling model with the objective of
minimizing the project duration by searching for the optimal set of
production rates for activities in different time periods. Recently,
Huang et al. (2016) designed a genetic algorithm to solve the multimode discrete time-cost trade-off problem considering soft logic.
The objective was to find a set of activity modes, start times, and
work sequences between units such that the total cost was minimized while meeting a given deadline. A notable assumption
adopted by the aforementioned studies is that only one crew could
be used for each activity.
Fan et al. (2012) highlighted that, if the units in an activity are
technically independent (i.e., there is no precedence relation
between them), in theory, these units can be performed at the same
time by hiring additional crews. Considering multiple crews can
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Table 1. Review of Techniques for Scheduling Repetitive Projects
Method
LOB
LSM
Repetitive scheduling
method
Productivity scheduling
method
Repetitive project modeling
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a
Work
Typical
activitya interruption
Author(s)
Multiple crew
assignment
Deadline constraint
satisfaction
Graphic-based
method
Arditi and Albulak (1986); Al Sarraj (1990)
Harmelink and Rowings (1998)
Harris and Ioannou (1998)
Yes
No
No
No
No
allowed
Yes
No
No
No
No
No
Yes
Yes
Yes
Lucko (2009)
No
No
No
No
No
Reda (1990)
Yes
No
No
Yes
Yes
A typical activity means that the duration of the activity is constant along all units (locations).
Unit
Buffer time
6
Crew 1
5
4
Crew 2
Crew 2
Crew 1
Crew 1
Crew 1
Crew 1
2
1
Crew 1
Activity A Crew 1
3
Crew 3
Crew 2
Crew 1
Activity C
Crew 3
Crew 2
Crew 2
Activity B
Crew 1
Crew 1
Crew 1
Time
Buffer time
Fig. 1. Sample LOB schedule
provide more flexibility in scheduling repetitive projects, this
allows the project to be rescheduled by adjusting the number of
crews assigned to each activity. As a result, a more competitive
schedule in terms of time, cost, and/or resource distribution may be
obtained. In the literature, the concept of multiple crews has been
integrated into many project scheduling problems [e.g., cost optimization (Hegazy and Wassef 2001), time-cost trade-off (Hegazy
et al. 2004), deadline satisfaction (Tokdemir et al. 2006), profit
maximization considering cash flow (Ali and Elazouni 2009),
resource leveling (Damci et al. 2013), and schedule compression
(Bakry et al. 2014)]. It is worth noting that most studies involving
multiple crews prefer to construct their optimal schedules in a lineof-balance (LOB) diagram, because LOB permits more than one
crew to be assigned to an activity concurrently compared with other
scheduling techniques.
This paper focuses on the classic deadline satisfaction problem
in LOB scheduling. The objective is to construct a LOB schedule
by determining the optimal number of crews for each activity, such
that each crew has a continuous workflow and the project is completed within a given deadline. LOB is typically represented by a
two-dimensional diagram similar to the one illustrated in Fig. 1, in
which each activity is represented by a sloped bar and crews are
allocated to implement this activity in sequential units smoothly
with no interruption.
To the best of the authors’ knowledge, the integrated CPM and
LOB method (CPM/LOB) proposed by Suhail and Neale (1994)
may be the first approach to put forward a formulation for determining crews needed to meet a given deadline. This method integrates
the capabilities of CPM, as an analytical method, into the LOB technique to schedule repetitive projects in a nongraphical way. An
improved version of CPM/LOB can be found in the work by
© ASCE
Ammar (2013). Other alternative methods include the repetitive
unit scheduling system (RUSS) proposed by Arditi et al. (2001b),
and the advanced linear scheduling system (ALISS) developed by
Tokdemir et al. (2006). Both RUSS and ALISS start with an initial
LOB schedule in which each activity is performed by only one crew
without interruption. Then, the schedule is updated iteratively by
increasing the number of crews by one for the selected activity. The
process continues either until the given deadline is met or until the
project cannot be compressed by accelerating any one activity.
Dolabi et al. (2014) pointed out that these two methods cannot guarantee optimal/near-optimal crew distribution, because the selection
of consecutive activities with identical progress rates is prohibited.
Therefore, the authors presented a heuristic iterative algorithm
called search-based heuristic line of balance (SHLOB). In each iteration, a blocking process is executed such that parallel and consecutive activities are grouped together as a block. Then, SHLOB identifies a decision domain including blocks that, when increased, their
progress rates may decrease the project completion time. Next, the
better selection within the decision domain is determined based on
several heuristic rules, and the number of crews for each individual
activity in the selected block is increased by one.
Although several methods have been developed for solving the
deadline satisfaction problem in LOB scheduling, they suffer from
the following limitations. First, they may fail to obtain a LOB
schedule that satisfies the deadline constraint. Although CPM/LOB
allocates crews in light of the stated deadline, the final schedule
may still be infeasible if the calculated number of crews is not an integer. ALISS and SHLOB also violate the deadline constraint if no
activity is selected by their heuristic rules in an iteration. Second,
SHLOB is valid only for serial projects in which the precedence
relations between activities are finish to start and each activity has
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just one successor. Third, as heuristic procedures, CPM/LOB,
ALISS, and SHLOB provide a way to obtain good solutions but do
not guarantee optimality. Therefore, extensive experiments should
be conducted to validate the effectiveness of these methods by comparing their results with some benchmark results. Unfortunately, no
such experiments are found in existing research works. Fourth, their
objective is to minimize the total number of crews under the
assumption that the cost of hiring a crew is the same for all activities. In fact, this cost is usually different between activities owing
to the differences in type of work, crew size, and technical level of
workers. Therefore, an extended objective of minimizing the crew
employment cost of all activities should be considered. Finally,
when an infeasible schedule is generated, existing methods are
unable to provide any useful information for planners to make decisions. Any infeasible solution of the problem under study is caused
by either the problem itself, which includes an infeasible deadline
constraint, or the method itself, which fails to provide a feasible solution. It is important to tell planners the real reason, because if the
deadline constraint is infeasible, they have the right to ask for a
looser deadline; otherwise, using other methods is recommended.
Thus, this study aimed to eliminate the limitations of existing
methods by presenting a novel method with time-cost trade-off consideration. The proposed method consists of two modules: feasibility test and scheduling optimization. In the feasibility test module,
an exact procedure is developed to evaluate the shortest project duration (SPD), and then, the rationality of a prespecified deadline can
be checked by comparing it with the SPD. Once the deadline passes
the feasibility test, the scheduling optimization module is implemented to minimize the crew employment cost of all activities without exceeding a given deadline. Repetitive projects usually have a
smaller number of activities, which makes exact optimization techniques a feasible option (Bakry et al. 2014). Based on this, the
approach of mixed-integer linear programming was adopted to
guarantee optimal crew distribution, where the integer variables
relate to the crew assignment, whereas the continuous variables
involve the start times of activities in the first and last units. Finally,
a highway project was used to show the application of the proposed
method, and extensive computational experiments were conducted
to investigate the performance of the proposed method, with the
obtained results compared against the existing methods.
The rest of the paper is organized as follows. The next section
provides a capsule description for CPM/LOB, ALISS, and SHLOB.
The third section describes how the proposed method is developed.
Computational experiments are reported in the fourth section, and
concluding remarks are given in the last section.
the prespecified deadline and completion time of the unit network, respectively; Rj is the desired progress rate of activity j;
and TFj is the corresponding total float calculated from the previous step
Rj ¼
m1
Td þ TFj T1
(1)
3. Calculate the number of crews (Cj) required to maintain the
desired progress rate of activity j using Eq. (2). If Cj is not an
integer, it should be rounded using Eq. (3) to determine the
actual number of crews (Caj). Eq. (3) also requires that Caj be
j ) of activless than or equal to the maximal crew availability (C
ity j
C j ¼ Rj d j
(2)
jg
Caj ¼ minfdCj e; C
(3)
4. Draw the final LOB schedule by further considering precedence constraints between activities.
ALISS
This method first initializes a LOB schedule by assigning only one
crew for each activity. If the deadline constraint is satisfied, then the
procedure stops and an optimal solution is found; otherwise, the following procedure is executed to improve the initial LOB schedule:
1. List the activities that have more than one crew available, and
sort them in descending order of durations along all units. If no
such activity exists, the procedure stops and it is reported that
no feasible solution is found; otherwise, the first activity in the
list is selected.
2. Run the LOB analysis to see if the project duration will be
reduced by assigning an additional crew to the selected activity.
If such a project compression happens, update the schedule by
increasing the number of crews by one for the selected activity
and go to the next step; otherwise, select the next activity in the
list and repeat this step. If the deadline constraint is still not satisfied after all listed activities have been investigated, the procedure fails to find a feasible schedule.
3. If the deadline constraint is met, the procedure stops and a local
optimum is found; otherwise, return to Step 1 in the ALISS
method.
SHLOB
Existing Solution Methods
As mentioned earlier, the available methods for solving the deadline
satisfaction problem in LOB scheduling include, but are not limited
to, CPM/LOB, ALISS, and SHLOB. These methods are designed
to minimize the total number of crews without exceeding a predetermined deadline.
CPM/LOB
The integrated CPM/LOB method proposed by Ammar (2013) was
considered in this study. Generally, this method can be decomposed
into the following four steps:
1. Perform the CPM calculation to obtain the completion time of
a unit network and the float times of activities.
2. Evaluate the desired progress rates of all activities using Eq.
(1), where m denotes the number of units; Td and T1 represent
© ASCE
This method is designed for repetitive projects with serial activities.
It first constructs an initial schedule using CPM/LOB. If the deadline constraint is met, the procedure terminates and an optimal solution is found; otherwise, the following procedure is implemented:
1. Classify parallel and consecutive activities into the same block.
2. Determine the set of permitted blocks that may lead to a
reduced project duration when their progress rates increase. If
the set of permitted blocks is empty, the procedure stops and
provides feedback that no feasible solution is found.
3. Calculate the potential reduction time (RTj) for each permitted
block j, and determine the corresponding reduction time index
using RTIj = RTj/nj, where nj is the number of activities existing in block j. For more detailed information regarding the calculation of RTj, please refer to Dolabi et al. (2014).
4. Generate a random number (RN 2 ½0; 1) and compare it with
the prespecified constant C 2 ½0; 1. If RN is greater than C, the
04017044-3
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block with the highest reduction time index is selected, and the
number of crews of each activity in this block is increased by
one; otherwise, a block from the set of permitted blocks is
selected randomly for schedule modification.
5. Update the schedule according to the modified crew distribution. If the deadline constraint is satisfied, then the procedure stops and a local optimum is found; otherwise, return
to Step 1.
Because SHLOB introduces a random search operator, the
authors decided to equip this algorithm with an iterative construct
to gain better results. In each iteration, an improvement on the initial LOB schedule is made, which may be different from previous
solutions owing to the randomness of parameter RN. When the
maximum number of iterations is reached, the feasible solution
with the minimal number of crews is treated as the final LOB
schedule.
Proposed Method
This section presents a mathematical programming-based optimization method for the deadline satisfaction problem in LOB scheduling. As mentioned earlier, the proposed method consists of two
modules: feasibility test and scheduling optimization. For a given
problem, the feasibility test module is designed to calculate the SPD
and then to check the feasibility of the predetermined deadline by
comparing it with the SPD. If it fails to pass the test, the procedure
reports a message that the current problem is infeasible due to a tight
deadline and shows the corresponding SPD to planners; otherwise,
the scheduling optimization module is used to search for a feasible
solution with the lowest project cost. To facilitate the understanding
of the proposed method, the authors first introduce the scheduling
optimization module.
min
k j
XX
© ASCE
(4)
where binary variables xjk need to meet the following constraints to
guarantee that the number of crews assigned to each activity is
definite:
k j
X
xjk ¼ 1; j 2 A
(5)
k¼1
In LOB scheduling, the precedence constraints between each
pair of activities [ð p; jÞ 2 E] are satisfied in all units only if they
are preserved in the first and last units. More precisely, if the progress rate of Activity j is less than that of its predecessor (p; see
Activities A and B in Fig. 1), the precedence constraints between
these two activities can be met by preserving that in the first unit;
otherwise, the last unit has to be considered (see Activities B and
C in Fig. 1). Based on the assumption that the buffer time is equal
to zero for each activity pair in E, the precedence constraints read
as follows:
sp1 þ dp sj1 ; ð p; jÞ 2 E
(6)
spm þ dp sjm ; ð p; jÞ 2 E
(7)
The start times of each Activity j in the first and last
units are
linked, with reference to Fig. 2, by sj1 þ ðm 1Þ=Rj ¼ sjm . In
contrast, the progress rate (Rj) of Activity j must equal to the ratio
of the actual number of crews employed by this activity and its
unit P
duration
to maintain work continuity of crews (i.e.,
k j
kxjk =dj ). By combining these two equations, the folRj ¼ k¼1
lowing linear constraints are obtained:
Scheduling Optimization Module
In this module, an exact mixed-integer linear programming (MILP)
model is developed for solving the problem. This idea is inspired by
the conclusion that constructing a model with linear constraints and
a linear objective function is a key factor for obtaining exact solutions to combinatorial optimization problems in a reasonable time
(Rieck et al. 2012).
The parameters used for model development are described.
Let set A ¼ f0; 1; :::; n þ 1g consist of all activities and set U ¼
f1; 2; :::; mg consist of all units. Without loss of generality, the
authors assume that Activity 0 is the single dummy start activity, and Activity n þ 1 is the single dummy end activity. For
each activity i = 1, ..., n, denote by dj the unit duration, k j , the
maximal number of available crews, and l jk the cost for
employing k =1, ..., k j , crews. For dummy Activities 0 and
n þ 1, set d0 ¼ dnþ1 ¼ 0; λ01 ¼ λnþ1;1 ¼ 0, and k 0 ¼ k nþ1 ¼ 1.
Pre-cedence constraints between activities are represented by a
set E A A in which each pair of activities [ð p; jÞ 2 E] indicates that Activity p is a predecessor of Activity j. Let parameter
d be the given deadline.
The present objective is to determine the optimal number of
crews for each activity and then construct a feasible LOB schedule
such that the project cost is minimized. Based on this, the authors
define two sets of decision variables: (1) sj1 ; sjm 2 Rþ , which represents the start times of Activity j in the first and last units, respectively; and (2) xjk, which takes a value of 1 if k crews are employed
for Activity j and zero otherwise. Then, the objective function can
be mathematically described as follows:
λjk xjk
j2A k¼1
sj1 þ ðm 1Þdj
k j
X
xjk
k¼1
k
¼ sjm ; j 2 A
(8)
The deadline constraint forces the project to be completed before
the given deadline (d ). Because Activity n þ 1 is the single dummy
end activity, the deadline constraint can be mathematically
described by
snþ1;m d
(9)
Unit
Activity j
5
unit duration (d j ) = 3
number of crews (C j ) = 2
4
progress rate ( R j ) = 2 3
3
2
1
Crew 2
Cj
Crew 1
Rj
di
Time
Fig. 2. Synchronization and work continuity of crews
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Finally, the exact MILP model for solving the deadline satisfaction problem in LOB scheduling is stated as follows:
k j
XX
min
λjk xjk
ESm
jkj can be iteratively calculated for each j 2 A and kj ¼ 1; :::; k j ,
then the value of the SPD is obtained by Eq. (11), because Activity
n þ 1 is the single dummy end activity with only one available crew
SPD ¼ ESm
nþ1;1
(11)
j2A k¼1
k j
X
xjk ¼ 1; j 2 A
k¼1
sp1 þ dp sj1 ; ð p; jÞ 2 E
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spm þ dp sjm ; ð p; jÞ 2 E
sj1 þ ðm 1Þdj
k j
X
xjk
k¼1
k
¼ sjm ; j 2 A
snþ1; m d
sj1 ; sjm 0; xjk 2 f0; 1g
For the aforementioned model, if l jk = k is assumed for each
j and k, the reduced model, with the objective to minimize the total
number of crews, represents the exact version of existing LOB scheduling methods, such as CPM/LOB, ALISS, and SHLOB.
In fact, the value of l jk is the sum of all costs incurred in performing Activity j with k crews. Generally, it can be divided into
material cost (MC), labor cost (LC), and equipment cost (EC). In
addition, the costs that cannot be associated with a certain activity
but may exist in the contract can also be integrated in the objective
function. These cost elements may include project indirect cost
(IC), early completion incentives (ECI), and liquidated damage cost
(LDC). IC is a function of the time needed to complete the project
(i.e., IC ¼ snþ1;m ICR, where ICR is the indirect cost rate per day
of the project, and snþi,m is used to evaluate the project duration).
ECI is the monetary incentive provided by the owner to the contractor if the project is completed before the given deadline. This cost is
calculated by ECI ¼ I p , where I is the daily ECI paid to the contractor, and p – is the time when project completion is early. LDC is
the financial penalty that needs to be paid by the contractor to the
owner if the project is not completed by the desired deadline. This
cost is given by LDC ¼ DLD p þ , where DLD is the daily liquidated damage paid by the contractor, and p þ is the time when project completion is late. Therefore, if necessary, anyone can easily
replace the objective function [Eq. (4)] by a refined one [see Eq.
(10)]. In doing so, additional constraints should be further considered to determine the values of p – and p þ
min
k j
XX
Eqs. (12)–(17) demonstrate formulas for calculating parameters
ES1jkj and ESm
jkj . In these equations, Pj is the set of predecessors of
Activity j. Rjkj is the progress rate of Activity j when kj crews are
hired and therefore can be calculated by Rjkj ¼ kj =dj . In particular,
Rjkj is assumed to be infinite for Activities 0 and n þ 1, as their unit
duration is zero. On the condition that kj crews are assigned to
Activity j and kp crews are employed by Activity p, parameters
1
m
and tj;k
denote the earliest start times of Activity j in
tj;k
j ð p;kp Þ
j ð p;kp Þ
the first and last units, respectively, such that the precedence constraint between Activity j and its predecessor p is satisfied. Because
the dummy Activity 0 has no predecessor, parameters ES101 and
ESm
01 are calculated as zero [see Eq. (12)]. For each j 2 A f0g and
1
kj ¼ 1; :::; k j ; ES1jkj is determined by the minimum of those tj;k
j ð p;kp Þ
that can preserve precedence constraints between Activity j and all
its predecessors [see Eq. (13)]. When the value of ES1jkj is determined, ESm
jkj can be calculated using Eq. (16)
ES101 ¼ ESm
01 ¼ 0
ES1jkj ¼ max
p2Pj
n
1
min ftj;k
g
j ð p;kp Þ
(12)
o
(13)
1kp k p
ðMCjk þ LCjk þ ECjk Þxjk þ IC ECI þ LDC
j2A k¼1
(10)
Feasibility Test Module
This module aims to calculate the SPD of the project and then to analyze the feasibility of the given deadline (d ) by comparing it with
the SPD. Obviously, the deadline (d ) is feasible if and only if the inequality d ≥ SPD is true. Note that, owing to the presence of work
continuity, finishing a project at the SPD does not mean that the
number of crews employed by each activity can reach its maximal
value—that is, it is impossible to determine the value of the SPD
directly by assigning k j crews to each Activity j.
Let ES1jkj and ESm
jkj represent the earliest possible start times of
Activity j in the first and last units, respectively, when kj crews are
employed by this activity. It is not difficult to see that if ES1jkj and
© ASCE
Fig. 3. Computational procedure of proposed method
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1
tj;k
j ð p;kp Þ
8
1
>
>
< ESpkp þ dp ;
¼
Rpkp Rjkj ;
dj
m
>
>
: tj;kj ð p;kp Þ ðm 1Þ k ; otherwise:
(14)
j
m
tj;k
j ð p;kp Þ
8
>
dj
>
1
< tj;k
þ ðm 1Þ ; Rpkp Rjkj ;
j ð p;kp Þ
kj
¼
>
>
: ESm
otherwise:
pkp þ dp ;
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1
ESm
jkj ¼ ESjkj þ
Rjkj
m1
Rjkj
(15)
(16)
8
>
< kj ;
j 2 A; f0; n þ 1g;
¼ dj
>
:
þ1; j ¼ 0 or n þ 1
(17)
Fig. 3 introduces the computational procedure of the proposed
method. Because MILP problems are nondeterministic polynomial
(NP)-hard, the CPLEX MIP Solver may not terminate in a reasonable amount of time. Hence, a maximum running time (e.g., 1 min, 1
h, or 1 day) is enforced. If the procedure stops before the maximum
running time, an optimal solution will be found; otherwise, a nearoptimal solution is reported.
Illustrative Example
To show an application of the proposed method, a highway project
that was first presented by Dolabi et al. (2014) is considered. From
the practitioners’ point of view, the project is assumed to consist of
24 serial activities with no buffers and is divided into 10 units.
Activity descriptions, staff-hour estimates to finish a unit, crew
sizes, daily working hours, and unit durations are given in Table 2.
The unit duration of an activity (Column 6) is calculated by dividing
the required staff-hours to finish a unit (Column 3) by the product
of the number of workers in a crew (Column 4) and the daily working hours (Column 5). Considering that the cost of an activity for
employing a crew is usually proportional to the crew size, the
authors adopt a simple but realistic setting of λjk ¼ kNWj , in which
NWj represents the number of workers in a crew of Activity j
(Column 4). For the illustrative example, the maximal number of
available crews is 10 for each activity, and the deadline is set to 238
days.
The proposed method is used to schedule the project considering two cases. In Case 1, the authors do not consider the timecost trade-off analysis and adopt the objective of minimizing the
totalPnumber
Pk j of crews as existing methods have done (i.e.,
kxjk ). In Case 2, the proposed objective function
min j2A k¼1
[Eq. (4)] is considered. The optimal crew distribution for each case
is presented in the last two columns in Table 2. Then, the project
cost for each case can be calculated by the sum of the crew employment costs of all activities, where the crew employment cost of an
activity is the product of the crew size (Column 4) and the calculated
number of crews (Column 7 or 8).
The results show that, in total, 66 crews should be employed for
project completion in Case 1 with a project cost of 373. In comparison, several activities in Case 2 have to be rescheduled by adjusting
their numbers of crews under pressure to reduce the project cost.
Specifically, the numbers of crews are reduced by one for Activities
3, 22, and 23; Activities 5 and 6 get an additional crew. The aforementioned adjustment results in a total number of crews of 65, and
Table 2. Project Information and Computational Results of Different Methods
Act
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Description
Mobilization
Surveying
Access road construction
Clearing grubbing
Earthmoving
Ditch excavation
Culvert installation
Peat excavation and
swamp backfill
Embankment
Utility work
Fine grading
Granular base (bulk)
Barrier walls
Granular base (fine)
Signage (foundations)
Paving (base)
Signage (installation)
High mast lighting
Paving (wearing course)
Shoulder granular
Lane marking
Landscaping
Checkout and acceptance
Demobilization
Number of crews
Required staff-hours
to finish a unit
Number of workers
in a crew
Daily working
hours
Unit duration
(days)
Case1
Case 2
192
64
576
576
1,040
400
80
192
4
2
6
8
10
5
5
6
8
8
8
8
8
8
8
8
6
4
12
9
13
10
2
4
3
2
5
3
4
3
1
2
3
2
4
3
5
4
1
2
432
160
256
336
576
384
1,600
336
288
224
288
320
16
72
80
160
6
4
4
6
8
6
10
6
4
4
6
5
2
3
2
5
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
9
5
8
7
9
8
20
7
9
7
6
8
1
3
5
4
4
2
3
2
2
2
5
2
3
2
2
3
1
3
4
3
4
2
3
2
2
2
5
2
3
2
2
3
1
2
3
3
Note: The boldface means that the number of crews assigned to the activity is different in the two cases.
© ASCE
04017044-6
J. Manage. Eng., 2018, 34(1): 04017044
J. Manage. Eng.
Computational Experiments
This section reports the results of computational experiments that
were used to analyze the performance of the proposed method in
solving large-size problems, with the obtained results compared
against three existing LOB scheduling methods, including CPM/
LOB, ALISS, and SHLOB. All of these methods are coded with
MATLAB (R2015b) and run on a PC with an i7-4700MQ CPU, an
8-GB RAM memory, and Windows 7 (64 bit) operating system.
The CPLEX MIP Solver (Version 12.6) is used for solving the
MILP model presented in this paper.
Test Problems
All test problems are generated randomly based on the following
four parameters: number of real activities (n), maximal number of
available crews (k j ), coefficient of network complexity (CNC), and
deadline tightness (u ). Table 3 presents the settings for these parameters. Furthermore, the number of units (m) is fixed at 30. Because
Computational Results and Discussions
An intuitive report on the results of the computational experiments
is shown in Fig. 4, where the symbol %svd is the percentage of
instances that can be solved, and %opt is the percentage of instances
solved to optimality. Fig. 4 shows that the proposed method is able
to obtain optimal solutions for all 2,700 instances. In comparison,
SHLOB provides each instance with a feasible solution, but only
11.93% (161 out of 1,350) are found to be optimal. Of the 2,700
instances, 57.59% can be solved by ALISS, of which only 1.7% (46
out of 2,700) are solved to optimality. CPM/LOB fails to solve any
instance (i.e., %svd = 0). The results highlight the good performance of the proposed method in deadline constraint satisfaction and
crew number minimization.
Table 3. Parameters of Test Problems
Parameter
Number of real activities (n)
Maximal crew available
number (k j )
CNC
Deadline tightness (u )
Values
60, 90, or 120
Randomly chosen from the interval [2,5],
[6,10], or [11,20]
1 or 1.5
0.05, 0.2, 0.35, 0.5, and 0.65
%svd
Percentage (%)
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existing LOB scheduling methods are designed to minimize the
total number of crews and do not consider the time-cost trade-off, a
fair test environment should be considered by assuming l jk = k for
each j and k. When the values of parameters n, k j , CNC, and u are
determined, a random instance is obtained using the following four
steps:
1. Randomly construct an activity-on-node (AON) network (A, E)
using the problem generator (ProGen) developed by Kolisch et
al. (1995) based on parameters n and CNC. The parameter
CNC represents the average number of nonredundant arcs per
node in an AON network in which each node represents an activity and each arc indicates the precedence constraint between
the corresponding activity pair. If the value of CNC is set to 1,
then a serial network will be obtained in which each activity
has only one successor.
2. Randomly determine the unit duration (dj) for each Activity j
from a uniform distribution over the set of f1; 2; :::; 50g.
3. Calculate the SPD and longest project duration (LPD) of the
random instance. The SPD is determined according to the feasibility test module, whereas the LPD is equal to the makespan of
the LOB schedule in which each activity is assigned to only
one crew.
4. Set the deadline d ¼ SPD þ ðLPD SPDÞu .
For each parameter level combination, the authors generate 30
random instances, which gives a total of 3 3 2 5 30 = 2,700
instances. Each instance is solved separately by using CPM/LOB,
ALISS, and the proposed method. For SHLOB, only the instances
with CNC = 1 are considered. Besides that, a maximum running
time of 1 min is set for the proposed method.
reduces the project cost to 363 with a savings of 2.75%. Despite the
limited scope of the analysis, the results support the thesis that
choosing whether or not to implement the time-cost trade-off analysis may have a significant impact on crew distribution and project
cost. The authors discussed this study with some researchers and
practitioners, and they all agree that the deadline satisfaction problem in LOB scheduling should be extended to integrate the timecost trade-off analysis. Generally, the chief target of construction
companies is to maximize profits; however, this target may not be
achieved if the project cost cannot be evaluated accurately. If the
planners schedule the project according to the objective of minimizing the total number of crews rather than minimizing the project
cost, the final crew distribution may be far from optimal and bring
construction companies a loss.
100
90
80
70
60
50
40
30
20
10
0
100
%opt
100
57.59
11.93
1.7
Proposed method
SHLOB
ALISS
0
CPM/LOB
Fig. 4. Comparison of computational results for different methods
© ASCE
04017044-7
J. Manage. Eng., 2018, 34(1): 04017044
J. Manage. Eng.
Table 4. Computational Results for Instances with Serial Activities (CNC = 1)
CNC = 1
ALISS
Proposed method
k j
n
u
Nfea
Nopt
Avg dev (%)
Max dev (%)
Nfea
Nopt
Nfea
Nopt
Avg dev (%)
Max dev (%)
Avg time (s)
Max time (s)
[2,5]
60
0.05
0.2
0.35
0.5
0.65
0.05
0.2
0.35
0.5
0.65
0.05
0.2
0.35
0.5
0.65
0.05
0.2
0.35
0.5
0.65
0.05
0.2
0.35
0.5
0.65
0.05
0.2
0.35
0.5
0.65
0.05
0.2
0.35
0.5
0.65
0.05
0.2
0.35
0.5
0.65
0.05
0.2
0.35
0.5
0.65
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
1
1
9
18
26
0
1
0
3
16
0
0
0
1
5
0
0
0
4
19
0
0
0
0
15
0
0
0
0
5
0
0
1
4
18
0
0
0
1
8
0
0
0
1
4
3.23
2.79
1.34
0.54
0.19
4.77
3.82
2.77
1.28
0.49
5.63
3.78
2.55
1.47
0.77
7.31
5.04
3.15
2.06
0.52
8.75
6.05
3.85
2.51
0.63
10.27
7.02
4.65
2.69
1.03
8.44
4.70
3.96
1.63
0.66
10.18
6.41
4.17
2.39
0.85
11.07
7.71
5.18
2.76
1.23
7.02
6.45
3.70
1.43
1.49
10.34
7.97
5.13
2.75
1.96
9.25
5.91
4.88
3.97
1.50
12.28
9.02
6.98
5.19
1.43
15.50
10.49
6.43
6.78
2.83
15.93
12.24
8.42
6.25
2.16
20.00
10.00
9.89
4.82
2.78
19.23
10.23
8.97
4.24
2.75
19.95
12.02
8.52
4.94
3.57
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
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
14
30
30
30
0
14
28
30
30
0
20
30
30
30
0
5
25
29
30
0
2
25
30
30
0
3
25
29
30
0
2
20
30
30
0
1
17
30
30
0
2
19
30
30
0
0
0
2
3
0
0
0
0
0
0
0
1
0
0
0
0
1
1
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
—
4.39
2.80
2.58
2.71
—
5.30
3.65
3.30
3.75
—
5.24
3.22
3.31
3.48
—
7.00
6.64
3.52
3.39
—
7.35
7.45
2.91
2.88
—
12.88
7.24
3.44
3.18
—
7.44
6.38
4.02
3.47
—
9.82
5.68
3.50
3.62
—
6.81
8.87
3.60
3.28
—
7.95
9.76
5.71
5.88
—
9.29
8.47
6.42
6.73
—
9.34
6.92
6.16
5.88
—
13.51
23.66
8.75
7.04
—
9.70
24.09
7.50
4.76
—
22.37
17.88
8.44
7.19
—
7.48
20.45
13.75
8.45
—
9.82
9.63
7.32
7.62
—
7.02
24.32
12.43
5.00
0.09
0.05
0.03
0.03
0.03
0.18
0.09
0.06
0.05
0.04
0.29
0.12
0.08
0.06
0.06
1.05
0.14
0.07
0.05
0.04
2.42
0.27
0.11
0.07
0.06
4.44
0.50
0.17
0.10
0.08
2.08
0.21
0.09
0.07
0.06
4.85
0.40
0.14
0.10
0.08
9.12
0.75
0.22
0.13
0.11
0.21
0.10
0.06
0.04
0.04
0.37
0.16
0.11
0.07
0.06
0.56
0.18
0.14
0.11
0.09
2.50
0.31
0.19
0.14
0.06
4.01
0.70
0.23
0.11
0.11
12.72
2.14
0.50
0.20
0.12
3.52
0.49
0.20
0.10
0.08
11.51
0.99
0.26
0.13
0.13
18.80
1.65
0.40
0.24
0.16
90
Downloaded from ascelibrary.org by University Of Florida on 10/28/17. Copyright ASCE. For personal use only; all rights reserved.
CPM/
LOB
SHLOB
120
[6,10]
60
90
120
[11,20]
60
90
120
Tables 4 and 5 detail the computational results of all tested methods. In Tables 4 and 5, the symbol Nfea (or Nopt) denotes the number
of instances for which a feasible (or an exact) solution is found. The
column labeled Avg dev (or Max dev) displays the average (or maximal) percentage of deviation from optimality (only for the instances of column Nfea). The column labeled Avg time (or Max time)
indicates the average (or maximal) running time in seconds needed
to solve the instances. Considering that no feasible solution is found
by CPM/LOB for all instances, the corresponding columns Avg dev
and Max dev are ignored. Similarly, the columns Nfea, Nopt, Avg
dev, and Max dev are ignored for the proposed method because an
optimal solution can be given for each instance.
© ASCE
As shown in Tables 4 and 5, all instances with up to 120 activities can be solved by the proposed method within 20 s. Dolabi et al.
(2014) also developed an exact model in the form of mixed-integer
nonlinear programming, but their model is less practical because it
took approximately 31 h to deal with a project with 24 activities.
This means that the proposed method outperforms the existing exact
model and is capable of providing exact solutions for large problems in a reasonable amount of time. By comparison, the maximal
running times of SHLOB, ALISS, and CPM/LOB are less than 1 s
(the specific data are not listed). The result is foregone, because
these methods belong to heuristic algorithms that trade optimality,
accuracy, or precision for speed.
04017044-8
J. Manage. Eng., 2018, 34(1): 04017044
J. Manage. Eng.
Table 5. Computational Results for Instances with Nonserial Activities (CNC = 1.5)
CNC = 1.5
ALISS
Proposed method
k j
n
u
Nfea
Nopt
Nfea
Nopt
Avg dev (%)
Max dev (%)
Avg time (s)
Max time (s)
[2,5]
60
0.05
0.2
0.35
0.5
0.65
0.05
0.2
0.35
0.5
0.65
0.05
0.2
0.35
0.5
0.65
0.05
0.2
0.35
0.5
0.65
0.05
0.2
0.35
0.5
0.65
0.05
0.2
0.35
0.5
0.65
0.05
0.2
0.35
0.5
0.65
0.05
0.2
0.35
0.5
0.65
0.05
0.2
0.35
0.5
0.65
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
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
3
14
21
27
29
2
14
27
28
30
5
15
23
29
29
0
7
20
27
29
0
4
14
24
27
0
6
17
24
29
0
4
15
27
29
0
2
12
26
30
0
1
13
25
27
0
1
2
1
6
0
0
0
1
4
0
0
0
3
9
0
0
0
1
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
2
0
0
0
0
1
0
0
0
0
0
2.71
4.15
4.80
4.77
3.04
3.61
7.45
5.83
4.44
3.34
4.70
6.41
4.89
2.71
2.02
—
10.88
11.34
9.50
6.73
—
10.85
13.26
7.98
4.86
—
12.92
8.93
6.76
3.96
—
6.98
10.56
7.69
4.43
—
14.21
10.37
9.44
5.52
—
20.00
9.95
8.04
4.31
3.53
7.69
12.50
8.96
7.94
4.17
15.38
19.81
13.00
11.46
11.54
14.56
13.10
10.00
7.26
—
18.37
19.51
20.29
20.00
—
18.52
29.17
20.00
16.33
—
19.05
12.84
15.91
8.73
—
12.24
35.53
14.29
16.42
—
17.56
20.97
21.10
15.46
—
20.00
23.68
20.45
9.85
0.04
0.05
0.04
0.03
0.03
0.10
0.11
0.07
0.05
0.03
0.15
0.15
0.11
0.06
0.04
0.35
0.20
0.14
0.07
0.04
1.20
0.64
0.35
0.12
0.07
2.41
1.64
0.51
0.17
0.08
0.59
0.29
0.16
0.08
0.06
3.37
1.22
0.51
0.20
0.12
5.00
1.93
0.76
0.21
0.12
0.09
0.09
0.11
0.07
0.05
0.23
0.21
0.13
0.12
0.06
0.22
0.33
0.22
0.14
0.07
1.56
0.34
0.29
0.17
0.07
3.49
2.61
1.33
0.28
0.13
4.51
3.95
2.02
0.72
0.19
1.48
0.70
0.30
0.14
0.15
7.22
3.81
1.08
0.50
0.32
16.08
3.79
3.05
0.49
0.27
90
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CPM/LOB
120
[6,10]
60
90
120
[11,20]
60
90
120
Now, this paper further investigates the performance of SHLOB
and ALISS using the data listed in Table 4. It is not necessary to discuss CPM/LOB in more detail because the corresponding %svd and
%opt are equal to zero. The computational results of SHLOB and
ALISS for different n, k j , and u are shown in Figs. 5–7, respectively. The results conclude that SHLOB is superior to ALISS
owing to the following reasons: first, although their average deviations (Avg dev) are more or less the same, SHLOB can provide feasible solutions for more instances and keeps the maximal deviation
(Max dev) at a lower level for each parameter level combination;
and second, ALISS is effective only for the instances with a loose
deadline constraint (i.e., the value of u is larger). In particular, if the
© ASCE
value of u is set to 0.2, only 14% (63 out of 450) of the instances
can be solved by ALISS. This ratio will be reduced to zero if a
smaller deadline tightness (u ) of 0.05 is considered, as shown in
Fig. 7. For nonserial projects (i.e., the instances with CNC = 1.5),
SHLOB is invalid owing to its own flaws, and the computational
results listed in Table 5 suggest that ALISS outperforms CPM/
LOB.
Examining Figs. 5 and 6 again, the authors observe that the average deviation of SHLOB (ALISS) does not increase significantly as
the project size (measured in terms of n and k j ) grows. To be specific, when the number of activities (n) increases from 60 to 120, the
average deviation of SHLOB (ALISS) slightly increases from 2.36
04017044-9
J. Manage. Eng., 2018, 34(1): 04017044
J. Manage. Eng.
30
Avg. Dev. (SHLOB)
Avg. Dev. (ALISS)
Max. Dev. (SHLOB)
Max. Dev. (ALISS)
Percentage (%)
25
20
15
10
0
60
90
%svd
SHLOB
ALISS
n =60
100
61.11
120
n =90
100
59.33
n =120
100
61.78
Fig. 5. Average and maximal deviations of SHLOB and ALISS for different n
30
Avg. Dev. (SHLOB)
Avg. Dev. (ALISS)
Max. Dev. (SHLOB)
Max. Dev. (ALISS)
25
Percentage (%)
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5
20
15
10
5
0
U[2,5]
U[6,10]
%svd
SHLOB
ALISS
U[2,5]
100
70.22
U[6,10]
100
58.44
U[11,20]
U[11,20]
100
53.56
Fig. 6. Average and maximal deviations of SHLOB and ALISS for different k j
(3.47) to 4.76 (4.46%). Moreover, when the maximal number of
available crews (k j ) increases from U[2,5] to U[11,20], the average
deviation of SHLOB (ALISS) slightly increases from 3.04 (3.91) to
4.52 (4.35%). This phenomenon may indicate that SHLOB and
ALISS are robust to the activities and maximal crew available numbers. In contrast, Fig. 7 reveals that as the deadline tightness (u )
decreases, the average deviation of SHLOB (ALISS) rapidly
increases from 0.71 (3.72) to 7.74 (5.92%). Meanwhile, the maximal deviation of SHLOB (ALISS) rapidly increases from 3.57
(8.45) to 20 (24.32%). The reason lies in their limited ability in solving exact or high-quality solutions for the instances with a tight
deadline constraint.
The aforementioned experiment fully explains the limitations of
existing methods, and the computational results validate the
© ASCE
effectiveness of the method presented in this paper. However, the
proposed method also has some limitations. First, coding the objective function as well as the required constraints in machine language
may be time-consuming and prone to errors. Second, the proposed
method is based on MILP, and it is difficult for practitioners to
obtain the optimal LOB schedule if no commercial software is
available.
Concluding Remarks
This paper presents an exact MILP-based method for solving the
deadline satisfaction problem in LOB scheduling. The method may
be the first LOB scheduling method that considers the time-cost
04017044-10
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30
Avg. Dev. (SHLOB)
Avg. Dev. (ALISS)
Max. Dev. (SHLOB)
Max. Dev. (ALISS)
Percentage (%)
25
20
15
10
Downloaded from ascelibrary.org by University Of Florida on 10/28/17. Copyright ASCE. For personal use only; all rights reserved.
5
0
0.05
%svd
SHLOB
ALISS
0.2
0.05
100
0
0.35
0.2
100
14
0.5
0.35
100
48.67
0.5
100
59.56
0.65
0.65
100
60
Fig. 7. Average and maximal deviations of SHLOB and ALISS for different u
trade-off and deadline constraint satisfaction and is capable of providing optimal solutions for large projects in a reasonable amount of
time. More importantly, the method is easy for practitioners to use
and does not need them to master any optimization theory. The final
schedule constructed by the method is more competitive than ever.
As discussed in the previous studies, the proposed method can be
applied to repetitive projects with typical activities (e.g., highways,
high-rise buildings, and housing projects with identical units).
The main contributions of this study include the following three
aspects. First, the proposed method integrates the time-cost tradeoff analysis; it is thus capable of determining a crew distribution
that minimizes the project cost without exceeding the given deadline. Moreover, the proposed method is also valid for instances with
a tight deadline constraint. Second, this paper developed the first
procedure (i.e., the feasibility test module) for calculating the shortest duration of a project and judging the feasibility of the given
deadline, which enhances the value of this work for practical applications. Third, this paper conducted extensive computational
experiments, and the results will help practitioners fully understand
the performance of the proposed method and existing procedures.
Similar experiments have never been mentioned in the literature.
Moreover, the computational results presented in this paper may be
used as benchmarks to evaluate the performance of other future
heuristic methods.
In the future, a potential research direction lies in integrating
LOB scheduling with other realistic settings. These settings may
result from the needs of planners or the characteristics of the project
itself. A meaningful extension could be the deadline satisfaction
problem in LOB scheduling with nontypical projects (i.e., the activity duration may be variable for different units). Nontypical projects
are as common as typical projects in real life; however, the scheduling decision for nontypical projects is very different from that for
typical projects.
Acknowledgments
The authors acknowledgment the Natural Science Foundation of
China (Grants 71271081 and 71701069) and Fundamental
© ASCE
Research Funds for the Central Universities (Grant 2017MS175).
The authors are grateful to the anonymous reviewer and editor for a
careful scrutiny of details and for comments that helped improved
this paper.
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