Integrating Quality and Safety in Construction
Scheduling Time-Cost Trade-Off Model
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Abhilasha Panwar 1 and Kumar Neeraj Jha 2
Abstract: Quality and safety are important and are the leading concerns in a construction project. Planning a project without properly
incorporating these two performance parameters propagates multifarious defects and accidents in construction projects and this ultimately
leads to cost overrun and time delay. In order to deal with quality and safety at the planning stage (along with time and cost), a decisionmaking model considering time, cost, quality, and safety (TCQS) is developed on the basis of a many-objective evolutionary algorithm
(nondominated sorting genetic algorithm III) and presented in this study. An example is illustrated to demonstrate the capabilities of
the proposed model for generating and visualizing the optimal trade-off among TCQS. The proposed model shows better results compared
to earlier reported scheduling models. Additionally, the study has also conducted a correlation analysis to identify the interrelationship, if any,
among the TCQS components. Furthermore, the solution obtained from the model was put through sensitivity analysis. The developed model
was also checked for two large-scale scheduling cases and found to work efficiently. The inclusion of objectives like quality and safety will
provide a holistic scheduling model for the construction professionals. This study is expected to provide an alternative planning strategy
and assist project managers in selecting optimal TCQS trade-offs while constructing a facility. DOI: 10.1061/(ASCE)CO.1943-7862.0001979.
© 2020 American Society of Civil Engineers.
Author keywords: Trade-off; Time; Cost; Coordinate plot; Many-objectives; Sensitivity analysis.
Introduction
Quality and safety are essential parameters for the successful execution of a construction project (Ogwueleka 2013). Nonadherence
to contractual specifications on quality and safety measures often
leads to remedial work and accidents, respectively. Rework shortens the time available for execution of the project and has an adverse effect on the completion schedule. This may create a situation
in which safety has to be compromised to achieve the planned
schedule of the project (Wanberg et al. 2013). Lack of quality
and safety measures may also lead to a reduction in the motivation
of the workers and this can ultimately affect productivity and quality of workmanship (Li et al. 2012). Compromise in these two
parameters has the potential to increase the time and cost of projects
(Doloi et al. 2012), which is undesirable and affects a project’s success. Therefore, these two parameters cannot be treated in isolation
and attention being paid to one at the cost of the other is not desirable (Ogwueleka 2013). Although quality and safety are referred to
in qualitative models, they are often found to be overlooked in
quantitative models (Wanberg et al. 2013).
It is essential for project managers to ensure that a project should
be done right the first time. Further, there should be no major accidents and rework during the project. To do so, it is important to
engage quality and safety parameters in the planning stage along
with the primary objectives such as time and cost in quantitative
modeling. There are a number of planning and scheduling models
1
Formerly, Ph.D. Student, Dept. of Civil Engineering, Indian Institute of
Technology Delhi, Delhi 110016, India (corresponding author). ORCID:
https://orcid.org/0000-0002-4293-5061. Email: apanwar38@gmail.com
2
Professor, Dept. of Civil Engineering, Indian Institute of Technology
Delhi, Delhi 110016, India. Email: knjha@civil.iitd.ac.in
Note. This manuscript was submitted on March 11, 2020; approved on
August 28, 2020; published online on November 22, 2020. Discussion period open until April 22, 2021; separate discussions must be submitted for
individual papers. This paper is part of the Journal of Construction Engineering and Management, © ASCE, ISSN 0733-9364.
© ASCE
available in the literature and in practice. Some of these practical
models give a project schedule similar to one generated by software
such as Microsoft Project and some are more advanced and can
build an optimized scenario-based project management model considering multiple-objectives such as the one generated by the software Project Team Builder.
From the literature, it is evident that significant numbers of
researchers have developed optimization models to solve construction management problems, ranging from layout planning to schedule preparation. Several attempts have also been made to solve
multiobjective scheduling problems using optimization models,
in which genetic algorithms (GA) are most commonly used. In
the past, various researchers have also made efforts to develop different optimization algorithms, such as particle swarm optimization
(PSO), ant colony optimization, and so forth. Although adequate
work has been carried out in the field of multiobjective scheduling
problems (MOSP) that consider quality as one of the parameters,
only a few studies have been reported on safety, while also solving
MOSP. Afshar and Dolabi (2014) were the first to work with a
GA-based optimization model for time-cost-safety (TCS) tradeoff. Although safety and quality play a crucial role in improving
performance, profitability, and productivity during project implementation (Ogwueleka 2013), notable research that uses these two
parameters together in an optimization model for MOSP has not yet
been undertaken.
Further, most of the scheduling problems have concentrated on
two or three objectives in trade-off problems, the results of which
are straightforward to visualize in low dimensions. However, in
real-life scenarios, there are several objectives that affect a project
and should be included in a construction schedule, which makes it a
many-objective problem. For a higher-dimension problem, most of
the existing scheduling models lack effective research in terms of
dealing with large populations, computational complexities, and
visualization (Jain and Deb 2014).
This study takes the aforementioned two issues into consideration for setting the research ground and considers (1) integration
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of quality and safety into scheduling model, and (2) developing a
many-objective scheduling model. Hence, this study makes an attempt to develop many-objective scheduling model by considering
quality and safety together along with essential scheduling parameters, time and cost, in a construction scheduling problem.
The literature review is presented subsequently, followed by the
methodology adopted for instant research and model development.
This follows the illustration of the model by using a case study.
After a detailed analysis of the results, discussions, and sensitivity
analysis, important conclusions have been drawn and presented at
the end of this study.
Literature Review
The literature review is presented under two broad and related
headings. The first pertains to the schedule trade-off problem
and the second pertains to the many-objective optimization models.
Review of Trade-Off Problem
In the past, while solving the MOSP, the main focus of research has
been time and cost (Alavipour and Arditi 2019; Feng et al. 1997).
Due to rapid industrialization, technological advancements, and the
growth of the construction industry, the focus has shifted toward
other competing objectives, such as resource utilization, safety,
and quality, along with the prime objectives of time and cost. This
has led to the development of various multiobjective, metaheuristic optimization models that help in solving MOSP. Initially, biobjective models involving time-cost trade-off problems were
solved (Kandil and El-Rayes 2006; Tiwari and Johari 2015). Furthermore, these models were extended to achieve multiobjective
optimization by considering other competing objectives in these
models.
In the literature, theoretical qualitative relationships were established between construction safety and quality with the help of expert opinion (Wanberg et al. 2013), but the same was not evident
while modeling quantitatively. With regard to the trade-off problem, both quality and safety parameters were dealt with independently, e.g., time-cost-quality (TCQ) trade-off (El-Rayes and Kandil
2005; Tareghian and Taheri 2006; Zhang and Xing 2010) and timecost-safety trade-off (Afshar and Dolabi 2014).
Time-Cost-Quality Trade-Off Models
Babu and Suresh (1996) opined that quality might get affected during the crashing of critical activities in time and cost trade-off, and
therefore quality should also be considered along with time-cost
trade-off. To quantify the quality, this study measured the quality
of each activity on a continuous scale, varying from one to zero. In
this scale, zero represents fully crashed and one represents normal
activity. The quality of the overall project was then assessed by the
quality levels achieved in each activity. Three linear programming
models were developed, which optimized one parameter at a time
by considering the other two parameters as desired boundary conditions. To check the practical applicability of this model, Khang
and Myint (1999) used it in the case of a cement factory construction project and underlined the emerging managerial benefits. This
class of problem is categorized as a continuous TCQ trade-off problem (Ghodsi and Skandari 2009).
El-Rayes and Kandil (2005) introduced a three-dimensional,
discrete TCQ problem and used a modified genetic algorithm to
optimize all three objectives simultaneously. The element of simultaneity had been missing in previously developed TCQ trade-off
models (Babu and Suresh 1996; Khang and Myint 1999). The major concern of their study was to quantify the quality of executed
© ASCE
work. In order to solve this, three indicators (of measurable quality
and with assigned weightages) were used by the latter researchers
for each construction activity. For example, for the activity “concrete pavement” the suggested measurable quality indicators were
“compressive strength,” “flexural strength,” and “ride quality.” The
results gave a measure of the quality performance for the activity
concrete pavement in quantitative terms. Similarly, different quality
indicators were assigned for each construction activity for the quantification of each quality performance.
In line with discrete TCQ, Tareghian and Taheri (2006) developed three interrelated integer programming models with the aim of
optimizing one of the given entities, simultaneously assigning the
required and desired boundary conditions for the other two. The
mathematical modeling involved the arithmetic mean, geometric
mean, and the minimum value of measurements. However, in successive research studies, arithmetic mean was the one most used by
researchers for assessing the quality (Fu and Zhang 2016). Furthermore, Tareghian and Taheri (2007) proposed a metaheuristic approach relying on electromagnetic scatter search space with the
aim of optimizing all three objectives comprising TCQ. Afshar et al.
(2007) used the quality indicators from the study carried out by
El-Rayes and Kandil (2005) and developed a metaheuristic model,
using the ant colony optimization model. Their results exhibited a
better capability of generating Pareto optimal solutions in a new
search space with less iteration. Still, these models were found incapable of determining the global optimal solution for time, cost, and
quality performance.
Ghodsi and Skandari (2009) developed a model for dealing with
similar problems by considering a more realistic relationship between the TCQ of activities in a project. The researchers used
an ε-constraint model where the general characteristics of a relationship function were presented as six axioms to simulate the real
world. Zhang and Xing (2010) introduced another evolutionary algorithm (PSO) to deal with TCQ trade-off in fuzzy environments.
The research work considered imprecise data and used fuzzy multiobjective PSO to solve the problem. Tavana et al. (2014) introduced
a pre-emptive, generalized precedence, discrete TCQ trade-off model
based on a nondominated sorting genetic algorithm (NSGA-II),
for scheduling optimization problems. The algorithm had a unique
chromosome that could handle both pre-emptive and non-preemptive activities concurrently. This model showed dominance in
the results of the model by Ghodsi and Skandari (2009). KhaliliDamghani et al. (2015) presented a trade-off model for the generalized precedence relationship by using mixed-integer linear
programming and also presented some deterministic and heuristic
procedures to solve the same. Wood (2017) applied integrated stochastic and fuzzy multiobjective optimization for the TCQ trade-off
problem in gas and oil projects. Mrad et al. (2019) developed a
simulation-based integer linear programming tool, which was able
to assess the impact of the stochastic behavior of time and quality.
The developed model was simple and less computationally expensive for large-scale networks, which made it capable of dealing with
real-life problems.
In past studies, quality was treated as a linear function of duration
(Babu and Suresh 1996; Khang and Myint 1999). Furthermore,
researchers have expressed the nonlinear quality function in their
studies (e.g., Tran et al. 2015). Zhang et al. (2014) expressed the
function in the quadratic term of duration. Later, researchers also
expressed quality in terms of a nonlinear function of time and cost
(Fu and Zhang 2016; Liberatore and Pollack-Johnson 2013).
Time-Cost-Safety Trade-Off
Only a few studies were found in the literature pertaining to TCS
trade-off. For instance, Afshar and Dolabi (2014) have considered
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the importance of safety and implemented a safety risk score with
time-cost trade-off. This study portrays the strong need to incorporate safety assessment into construction planning and further developed a TCS trade-off model using NSGA-II. Afshar and Dolabi
(2014) identified that safety risk assessment methods form two categories, the first being job-based and the other being activity-based.
They suggested that as the discrete time-cost trade-off problem is
activity-based, safety risk assessment should follow an activitybased methodology. They further argued that a qualitative safety
risk assessment should be more reasonable than quantitative, as accurate safety statistics are rare. With these ideas, researchers developed a qualitative activity-based safety risk method. A few other
studies have considered safety along with time/cost; for example,
El-Rayes and Khalafallah (2005) developed an optimization model
for site layout planning, incorporating safety issues with cost.
Similarly, Ning and Lam (2013) developed a trade-off model for
layout planning, considering safety along with the cost.
As previously indicated, quality and safety work together to
make a project successful, whereas from the literature it is clear
that past studies did not consider these parameters together to make
a holistic model in any of the scheduling models. The study makes
a maiden attempt to bring together all the primary success measures
such as time, cost, quality, and safety through developing a manyobjective scheduling model.
Review of Many-Objective Optimization
Many-objective optimization has been becoming a focus area
for the study of evolutionary optimization due to its ability to
solve real-life problems. Traditional multiobjective (two or three
objectives) algorithms such as NSGA-II, and strength Pareto evolutionary have proved to be ineffective in solving many-objective
(i.e., more than three objectives) optimization problems (Chand
and Wagner 2015). While solving many-objective optimization,
a number of challenges have been identified in the literature, such
as an exponential increase in the number of nondominated solutions, computationally expensive performance metrics, difficulties in visualization, and so forth (Elarbi et al. 2018; Jain and
Deb 2014; Von Lücken et al. 2014). To overcome these difficulties, researchers have proposed various many-objective evolutionary algorithms, and generally categorized them into six classes
(Li and Tang 2016).
Köppen and Yoshida (2007) argued that NSGA-II is inefficient
when dealing with many-objective optimization problems. To increase its efficiency, this study introduced a number of matrices
to replace the crowding distance operator in the original NSGAII. Bader and Zitzler (2011) suggested a hypervolume estimation
(HypE) algorithm, where the exact value of the hypervolume is replaced with an estimated value, obtained from Monte Carlo simulations, while solving many-objective problems. Furthermore,
Jain and Deb (2014) proposed NSGA-III based on Pareto and aggregation, where the Pareto dominance–based, nondominated sorting is used to move the population towards the Pareto front. In this
study, reference points are used to maintain diversity. The experimental results showed that NSGA-III had good convergence
and satisfactory diversity but were critically dependent on the reference points provided, either in advance or during the running of the
algorithm. In a different study, Wang et al. (2015) developed a lowcomplexity, two archive-2 algorithm. This algorithm deals with
convergence and diversity separately, with two archives being
maintained by the algorithm, i.e., the convergence archive and
the diversity archive. A hybrid, many-objective algorithm was developed using the preference-based and decomposition-based concepts in NSGA-II by Elarbi et al. (2018). Hybridization brings
© ASCE
forward the benefits of both concepts, and therefore the results outperform the other algorithms. Zou et al. (2018) further suggested
that all the multiobjective algorithms can provide only a limited
insight into dealing with many objectives. However, a boundless
domain can be explored to deal with many-objective optimization.
As many-objective problems are ever-evolving, new and more
advanced algorithms will continue to be developed (Tran et al.
2016).
Past studies have successfully incorporated quality and safety
separately in trade-off models. However, no significant literature
is available where these two parameters have been included simultaneously in the model. Razavi-Hajiagha et al. (2015) have
suggested that the trade-off model should be developed by
appending risk issues with TCQ. Also, the available models can
work with only two to three objectives at a time and their applicability for use in higher dimensions is limited. This may be attributed to the fact that a number of challenges have to be faced by
researchers in the development of many-objective optimization
models (consisting of more than three objectives). There have been
very few studies pertaining to construction scheduling that account
for four objectives (Panwar and Jha 2019). Elbeltagi et al. (2016)
had developed an optimization model taking four objectives (time,
cost, resource utilization, and cash flow) simultaneously. The researchers had used a PSO, with a new evolutionary strategy based
on the Pareto compromise solution. Although this research optimized four objectives using a PSO model, the researchers did
not come up with either the details or appropriateness of applying
a PSO model in a many-objective optimization problem. It was
stated that with the increase in the number of objectives, the probability of finding a nondominated solution decreased in search
spaces, which left the multiobjective PSOs with little or no use
in many-objective optimization (Köppen and Yoshida 2007).
In a study by Zheng (2017), a time-cost-quality-environment
trade-off scheduling problem was taken into account. The researcher had used an a priori approach to deliver the final solution
for the trade-off problems. The methodology incorporated steps for
calculating the weighted sum to assess the combined effect of all
the objectives. This was followed by the development of a GA
based model for the analysis of the problem. Because this study
ultimately considered a single objective by combining the effect
of four objectives, it was of little use for solving many-objective
optimization scheduling problems. In a recent study, Tran et al.
(2018) considered TCQ with work continuity using opposition
multiple objective symbiotic organisms search. This method is still
in the development stage and needs more experimental work in the
field of many-objective optimizations. The lacunae in the existing
model pertaining to many-objective optimizations mentioned previously motivated the authors to use NSGA-III, an algorithm developed exclusively for many-objective optimization problems.
Furthermore, NSGA is one of the most recognized metaheuristic
algorithms (Tavana et al. 2014) and found to be one of the best
algorithms to deal with construction scheduling trade-off problems
(Panwar et al. 2019). NSGA-III is the latest version of the algorithm
that deals with many-objective optimization problems (Jain and
Deb 2014). Preliminary studies indicate that NSGA-III is a viable,
evolutionary, many-objective optimization algorithm (for more
than three objectives) for handling constrained and unconstrained
problems. Because the NSGA-III is a relatively new algorithm, the
proficiency of the algorithm in solving MOSP considering timecost-quality-safety (TCQS) trade-offs will be challenging but interesting while it is being explored. The next section illustrates the
development of many-objective construction scheduling model using NSGA-III.
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Construction Project Scheduling Optimization Model
The current study has developed a many-objective schedule model
based on NSGA-III. The following section initially explains the
process of NSGA-III and then gives the formulation of the construction scheduling problem.
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Nondominating Sorting Genetic-III
NSGA-III is an extended, many-objective, evolutionary version of
NSGA-II. This is an improvement when dealing with highdimensional, objective space. Both algorithms differ in their selection process. The crowding distance of NSGA-II is replaced with a
new reference point-based method, where each individual is associated with the closest reference point, which can be supplied manually or generated uniformly within NSGA-III (Ramírez et al.
2015). This methodology helps in ensuring a well-diversified solution set. Apart from belonging to a many-objective approach,
NSGA-III has shown promising results when coping with complex
real-world combinatorial problems (Mkaouer et al. 2014). To solve
MOSP (discrete search space combinatorial problems) widespread
adoption of NSGA can be seen in the literature because of its simplistic implementation (Panwar et al. 2019) and its efficiency in
finding the optimal solutions. Some of the applications can be seen
in the studies by Zheng et al. (2005), Tavana et al. (2014), and
Monghasemi et al. (2015). However, although the use of NSGAIII is limited in MOSP (Panwar and Jha 2019), the authors explored
its applicability further in this study. The pseudocode for NSGA-III
is given in Fig. 1.
Formulation of a Construction Project Scheduling
Problem
The objective functions and associated constraints of a construction
scheduling problem are discussed in this section. It has become
evident, from both theory and practice, that the project TCQS plays
a crucial role in construction project success (Wanberg et al. 2013).
Thus, to assess the applicability of quality and safety, along with
time and cost, all four crucial objectives are considered in the proposed model. The formulation of each objective is explained
subsequently.
Time
The precedence diagram method (PDM) is a representation technique that depicts the project activities involved in a network
and it has been used to determine the total time of a project.
The PDM is widely used, based on the activity of nodes, to prepare
project schedules. The time function is defined as the sum of the
duration taken by all the activities in the critical path of the project,
while maintaining relationships between the predecessor and successor activities.
Objective 1: Minimization of total time (T min )
X j
ð1Þ
T¼
ticp
where T = total project duration; and tjicp = duration associated with
the jth execution mode of ith activity on the critical path (cp).
Cost
The total project cost is accounted for as the sum of the costs incurred to execute the individual activities. The minimization of the
total cost is considered as the second objective for the optimization model.
Objective 2: Minimization of total cost (Cmin )
X j
C¼
DCi þ I jCi
ð2Þ
A
DjCi ¼
Cji
ð3Þ
I jCi ¼ Cic × T
ð4Þ
While (Termination criteria not met)
{
2.
Genetic operations (crossover + mutation)
3.
Non-dominating sorting of front
4.
Selection based on reference point mechanism
5.
Normalization of population
6.
Find reference points and associated members
7.
8.
Applied the niche preservation
Update population
}
Fig. 1. Pseudocode for NSGA-III.
© ASCE
i¼1
where C = total project cost; DjCi = total direct cost; I jCi = total
indirect cost associated with the jth execution mode of ith activity;
Cji = cost associated with the jth execution mode of ith activity;
Cic = indirect cost per unit of time; and T = total project duration.
Quality
Adequate implementation of quality measures during the construction process are essential. A low-quality execution leads to defects
or failures in constructed facilities, which increases the costs and
delays of a project. In the worst case, failures can cause personal
injuries or fatalities, which further trigger time and cost overrun in
an unsafe working environment. To integrate quality in the schedule, El-Rayes and Kandil (2005) suggested the aggregation of quality performance as one of the parameters in MOSP.
The quality function had been formulated by taking into account
the weighted sum of the quality of each activity in a project.
Objective 3: Maximization of total quality (Qmax )
Q¼
X
A
1. Initialize population ()
n
X
wi
P
X
p¼1
wji;p × qji;p
ð5Þ
where Q = quality of the project; wi = weight of ith activity (indicative of the relative importance and contribution of the individual
activity quality to overall project quality); wji;p = weight of jth execution mode of ith activity for the pth quality indicator (indicative
of the relative importance and contribution of the quality indicator
over and above the other activity indicator measures); and qji;p denotes the performance of the quality indicator value of the jth execution mode of the ith activity for the pth quality indicator.
El-Rayes and Kandil (2005) allowed consideration of a number
of quality indicators for each activity in the measurement and quantification of construction quality [Eq. (5)]. The quality indicators
considered had been identified by considering the long-term performance of each activity with regard to the individual quality
indicators (Anderson and Russell 2001; Minchin and Smith 2001).
For example, for the structural concreting activity, quality indicators
include density, rebar cover, water-cement ratio, curing, air content,
and compressive/tensile strength (Minchin and Smith 2001). These
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indicators were recommended to be selected in such a way that allowed for the quantification of each one.
As quality indicators have different units of measurement, note
that all the indicators of an activity must be converted into a normalized scale. The adopted model normalizes the quality indicator
values between 0% and 100% (El-Rayes and Kandil 2005). The
value 0 represents poor quality performance, while 100 represents
excellent quality performance.
Safety
The construction industry is recognized as being the most hazardous (Patel et al. 2016). Traditionally, time, cost, and quality were
the three prominent parameters required to ensure the success of a
project. However, health and safety measures were not given due
importance in the time-cost-quality trade-off models but have
slowly gained appropriate roles in the trade-off models. Afshar
and Dolabi (2014) incorporated safety measures, considering
the risk-based health and safety analytical model developed by
Hallowell and Gambatese (2009). Afshar and Dolabi (2014) quantified the safety risk on the basis of an activity-based safety risk
method. This method involved three basic steps: (1) identification
of significant safety risks, (2) determination of the likelihood of
the occurrence and evaluation of the severity of safety risks, and
(3) determination of overall safety risk score.
In the first step, significant safety risks for all the activities were
compiled, based on the data available from different government
reports and literature (Afshar and Dolabi 2014). After identification
of the safety risks, the method evaluated the probable likelihood
and severity of each identified safety risk. The method sought comments and opinions from domain experts. The process involved
seeking opinions on a 1–6 scale for both likelihood and severity.
In the case of likelihood, 1 denoted remote likelihood occurrence
and 6 denoted a highly probable occurrence, with the intermediate
gradations lying in between. Similarly, a severity of 1 represented
minor injury and 6 represented fatality. After getting all the scores
from the experts, the overall safety risk score was calculated using Eq. (7).
Objective 4: Minimization of project safety risk score (SRSmin )
X j
SRS ¼
S Ri
ð6Þ
SjRi ¼
K
X
k¼1
ðLjk × Sjk Þi
ð7Þ
where SRS = total safety risk score of the project; SjRi = safety risk
associated with the jth execution mode of ith activity; K = total
applicable safety risk in the ith activity; Ljk = likelihood of kth
safety risk occurring in jth execution mode; and Sjk = severity index
of the kth safety risk in jth execution mode.
The following constraints were considered in the instant model:
(1) all the activities represented in the activity network are executed,
(2) each activity must be executed using only one of the available
executing modes, (3) decision variables must be positive integers
subject to the boundaries of upper and lower limit (which varies with
each activity execution mode), and (4) the project schedule must
maintain the relationships between the activities.
Proposed Many-Objective NSGA-III for
Time-Cost-Quality-Safety Trade-Off Model
Construction projects consist of several activities and can be
executed by one or more methods. These methods depend on
the resources utilized (material, equipment, labor) to perform
© ASCE
different activities. Different sets of resource combinations take different time durations to achieve a particular activity with the same
cost and similarly, quality, and safety environment. For example, excavation can be done manually or mechanically. Both execution
modes involve different sets of equipment, labor, crew, and material.
Based on these requirements, both methods entail varying TCQS
measures for individual activity. It has been observed that, in general,
the time taken by the manual model will be more than the time taken
by employing mechanical means. However, the cost is higher in the
case of the latter (DSR 2016). A similar situation is likely for all
project activities. The selection of an execution mode for an activity
should be such that an optimal balance between these competing
parameters can be achieved. The selection of the combination of
execution modes in a project is a tedious task, due to the fact that
planning and/or scheduling involves numerous activities and their
respective execution modes. The developed model provided an optimal combination of all execution modes for the activities for all of
the considered objectives simultaneously. A Pareto-optimal solution,
considering the best-suited alternatives for the overall project activities, is found to be the result by solving MOSP using NSGA-III.
To make it effective and hands on, a TCQS trade-off model is developed based on NSGA-III. The process for the development of the
NSGA-III model is explained as follows.
Initialization: The process begins by setting the initial population (Pt ) of size N with a set of individuals. These individuals are
generated with the help of project input data. These data consisted
of activity numbers, activity relationships, number of execution
modes, and the value of set objectives. Each individual in the population denotes a solution to the MOSP problem. An individual is
characterized by a set of variables known as genes. In the case of
the scheduling problem, these variables are execution modes.
Genes are joined together in the form of a string to represent a
chromosome (solution). The total number of activities taken together symbolizes a chromosome in MOSP. The length of the
chromosome is equal to the total number of decision variables
(Dv) existing in the problem. In the case of MOSP, the total number
of activities represented the number of Dv in the model. For different genes, an individual cell of a chromosome gave information
regarding the decision variables. This information suggested the
execution mode that could be used for an activity. In order to understand the representation of a solution chromosome, a project having
n activities was considered. Each of these activities could be
executed by three possible alternatives. The chromosome representation of one of the possible solutions to this MOSP problem is
shown in Fig. 2. For this solution, activities from 1 to n were
planned to be executed using execution modes 3, 2, 3, 1, 2, and
2, respectively.
Evaluation of objective function: The execution mode of each
activity was obtained from the chromosome representation and respective values for the objectives were summed up with respect to
Eqs. (1), (2), (5), and (6). By this process, the fitness value of the
objective function was calculated for a set of N number of the initial
(parent) population. The fitness value of these objectives helped in
sorting out the population for the next step.
Genetic operations: Generally, there are three main genetic operations, namely selection, crossover, and mutation. The selection
process ensures the preservation of the solutions from the population based on their fitness values. In this operation, the initial solutions were ranked on the basis of nondominated sorting (NDS)
from the parent population. NDS suggests that an individual A is
said to be nondominated from individual B if A is no worse than B
in all objectives or A is strictly better than B in at least one objective. From the parent population, N pairs were randomly selected
and subsequently, a tournament selection was carried out. Based on
04020160-5
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J. Constr. Eng. Manage.
Activities
Chromosome
Start
Algorithm inputs: Population size (N); and
H number of structured reference point Z r
Act1 Act2 Act3 Act4 Act5 … Actn
EM3 EM2 EM3 EM1 EM2 … EM2
Chromosome representation
Project details
Activities (Act)
Initialize population (Pt ) and evaluate
objective function
Execution modes (EM)
Generation of offspring population
(Ot ) = (recombination + mutation) of Pt
Combined population (Rt) = Pt + Ot
Objective values
T, C, Q, S
Generation of non-dominated front
(F1, F2……….Fn) from Rt
Intermediate population (St) =0; i=1
St = S t
Fi and i=i+1;
till last front Fl =Fi
Yes
Normalization
Computation of ideal point (zmin )
Computation of extreme point (z max )
Normalization of objective:
St=N
No
Mapping each reference point on the
normalized hyper-plane and saving in Zr
Pt+1=
Point chosen from Fl : K = N -
Pt+1=S t
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Optimization model
Normalization of objectives followed by
generation of reference point Z r
Association of each members of St with
reference point
Computation of niche count of reference points
Reference point association
Computation of reference line for
each point
Calculation of r distance of each
St from reference line
Associate the population member as
per their proximity from reference
point
Selection of K members to construct Pt+1
End
No
Met stopping criteria
Yes
Final set of solutions
Visualization
Co-ordinate plot
Fig. 2. NSGA-III modeling for TCQS trade-off flowchart.
the NDS ranking, the winning solution for each tournament was
identified. Thus, a pool of N solutions was determined after tournament selection. Subsequently, these N solutions had to undergo the
process of crossover and mutation. The purpose of these operations
is to maintain diversity within the population. In NSGA-III, in order
to generate the child population (Qt ), simulated binary crossover
(SBX) and polynomial mutation (PM) were used. Both SBX and
PM have been extensively used in the evolutionary algorithm for
solving multiobjective optimization problems (Lim et al. 2017).
The SBX was designed on the basis of the one-point crossover
property in binary coded GA. The PM reproduces the offspring
(child) distribution of binary-encoded bit-flip mutation on real values of the decision variables (Deb and Goyal 1996). By using both
the SBX and PM operator, the offspring were created next to the
parent population. The distribution index of SBX generally controlled the shape of the distribution of the child population from
© ASCE
the parent population. The larger value provided a “near-parent”
child solution, while the lower value generated the child population
far from the parent population. Commonly, the mutation probability had been previously recommended as 1=Dv (Deb and Goyal
1996). The detailed description of the genetic operation can be
found in Deb and Deb (2014) and Lim et al. (2017).
Generation of nondominated front: A pool of a 2N population
was generated by combining child and parent populations. With the
help of the NDS ranking given to the population, the solutions were
grouped into different fronts (F1 ; F2 ; : : : ; Fn ) and a new population set (St ) is alienated from the generated fronts until St ≥ N.
Selection-reference point: If the number of solutions of St
equals N, then St will be the new population (Pt þ 1) for the next
generation. Whereas, if the number of solutions in St > N then the
mandated solutions were selected based on the maximum diversity
in the solutions from the front (FL ). In NSGA-III, diversity is
04020160-6
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J. Constr. Eng. Manage.
obtained by the reference point (Zs )–based approach. This approach initially normalizes the objective function and then generates an H number of reference points (Zr ) in a hyperplane. These
reference points (H) are calculated by Eq. (8)
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H¼
mþp−1
p
ð8Þ
where m = number of objectives; and p = number of divisions desired in each objective axis in the hyperplane.
Subsequently, the distance between each solution and the reference lines are calculated. The reference line connects the reference
points and the origin of the search space of the normalized objectives. With the help of the calculated distance, each solution of the
St population is associated with the generated reference points
based on the minimum perpendicular distance between the solutions and the reference line. Then, with the help of niche count
strategy, the necessary solutions are chosen from FL . The selected
solutions are those which are associated with the least reference
points in the population (Pt þ 1). A solution is more preferred
if it represents the hitherto underrepresented or unrepresented reference points. Furthermore, to complete the population size considering nondominated solutions, the whole process was found to have
at least one individual corresponding to each generated reference
point, which is close to the Pareto-optimal front. The niching strategy puts an emphasis on selecting a population member from as
many reference points as possible so that a well-diversified solution
set can be obtained in the search space. Based on the process mentioned previously, K number of members are selected from FL in
such a way that the new generation (Pt þ 1) equals the number of
the initial population (N).
Stopping criteria: A similar process was repeated to generate
future population until the stopping criterion was met. The developer can set the stopping criteria to be the maximum number of
generations or maximum number of function evaluations or a combination of the two. At the end of the optimization process, the
optimal solutions were obtained, known as the Pareto front. The
generated Pareto front is of great importance and aids planners
in assessing the pros and cons of every potential solution on the
basis of quantitative and experience-driven aspects.
with those obtained by previously developed models from the
literature.
The TCQ trade-off problem had been taken from the study of
El-Rayes and Kandil (2005). The problem consisted of a total of 18
activities having varying execution modes. The results obtained
after the application of the developed model are given in Table 1.
For comparison purposes, only those results obtained by the application of the existing model developed by El-Rayes and Kandil
(2005) and the minimum solution found from the developed model
are shown. The algorithm configuration was kept the same as in the
literature, in order to make the comparison.
As is evident from Table 1, the Pareto front solutions obtained
from the developed model out-performed the solutions obtained
from the model in the existing literature. The model not only provided a better solution but also gave the solution corresponding to
total project duration of 100 and 102 days, whereas the past model
did not provide the solution corresponding to the two previously
stated durations. With respect to the rest of the solutions, the average decrease in cost was found to be $180 ($127,876−$127,696)
and the increase in quality performance was 0.8% (81.8%−81.0%).
Hence, the obtained results justify the model’s applicability and
capability in solving MOSP.
In another example, TCS data taken from Afshar and Dolabi
(2014) were analyzed by using the developed model. The results
from the developed model were compared with the literature
findings (Table 2). It was evident that the developed manyobjective schedule model was capable of finding a better solution
in the search space when compared with the model developed
earlier.
The average decrease in time, cost, and safety risk score was
around 0.3 days (119.8−119.5 days), $22,858 ($139,154.30−
$116,296.30), and 0.8 (225.3−224.5), respectively. Thus, it was
obvious from the TCQ and TCS solution sets that the developed
many-objective schedule model was better at generating trade-off
solutions in comparison to the models used in the literature. The
results from the developed model were comparable in the case
of time and safety risk score with the previous model and also provided considerably improved solutions with respect to the project
cost in the instant case. Note that the developed model is probabilistic in nature as it uses a stochastic algorithm NSGA-III. Due to
the probabilistic nature, a global optimal solution cannot be guaranteed by the developed model.
Verification of the Model
To verify the applicability and efficiency of the developed model,
two problems from the literature had been taken (i.e., timecost-quality trade-off, and time-cost-safety trade-off) because no
example involving four objective MOSP was available. These
problems were then solved through the developed model. The respective results obtained for both problems were then compared
Comparison Based on Performance Metrics
The performance of an optimized model is evaluated using a different set of metrics. Degree of convergence, diversity, and speed of
convergence are three commonly used quantitative performance
metrics (Zhang and Li 2010; Zitzler and Thiele 1999). The degree
of convergence is analyzed by using the generational distance
Table 1. Results of TCQ example
Literature TCS model
Developed TCQS model
Solution
Time (days)
Cost ($)
Quality (%)
Execution modes
Time (days)
Cost ($)
Quality (%)
1
2
3
4
5
6
7
Average
—
—
104
109
114
115
124
113
—
—
166,320
121,350
105,470
141,620
104,620
127,876
—
—
95
77
71
90
72
81
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
1,1,2,1,1,1,1,1,1,1,1,1,2,1,1,1,2,1
1,2,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1
2,5,1,3,4,2,3,3,1,1,1,1,3,1,1,5,1,1
1,5,3,3,4,3,3,5,1,1,3,1,3,2,1,5,1,1
2,3,1,1,2,3,1,1,1,1,1,1,1,1,1,4,1,1
1,5,1,3,4,3,3,5,1,1,3,1,3,2,1,5,3,1
—
100
102
104
109
114
115
124
109.7
169,820
167,970
165,720
121,350
105,470
141,370
104,570
127,696
97
96
96
78
72
91
72
81.8
© ASCE
04020160-7
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Table 2. Results of TCS example
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Literature TCS model
Developed TCQS model
Solution
Time (days)
Cost ($)
Safety risk score
Execution mode
Time (days)
Cost ($)
Safety risk score
1
2
3
4
5
6
18
Average
100
102
105
108
112
116
126
119.8
153,320
148,470
141,070
140,870
128,170
127,970
127,770
139,154.3
254
255
248
255
254
259
243
225.3
1,5,3,3,3,1,3,5,1,1,1,1,1,2,1,2,1,1
2,5,3,3,4,1,3,5,1,1,1,1,1,2,1,2,1,1
2,5,3,3,3,2,3,5,1,1,1,1,1,2,1,2,1,1
2,5,3,3,3,2,3,5,2,1,1,1,1,2,1,2,1,1
1,5,3,3,4,2,3,5,1,1,1,4,1,2,1,2,1,1
1,2,5,3,3,4,2,3,5,1,1,3,1,1,3,1,5,3
1,1,5,3,3,4,3,3,5,1,1,3,3,1,3,1,2,3
—
100
102
105
108
112
116
126
119.5
136,820
131,570
128,570
128,510
122,820
120,020
106,770
116,296.3
239
240
229
227
226
222
220
224.47
Table 3. Performance metrics values
Metrics
Generational distance (GD)
Diversity
Computation time
Parameter
CSMOPSO
RSMOPSO
MOGA
NSGA-III
Average
Standard deviation
Average
Standard deviation
Average
Standard deviation
4.73
1.56
9.58
1.95
3.42
0.09
9.79
3.66
14.25
2.77
5.56
0.08
5.68
1.57
10.16
2.55
4.43
0.09
4.27
0.72
8.89
1.01
3.32
0.15
Note: CSMOPS = combined scheme based multiobjective particle swarm optimization; RSMOPS = random selection based multiobjective particle swarm
optimization; and MOGA = multiobjective genetic algorithm.
(GD). This metric measures the distance between the Pareto front
solution and the best Pareto front approximation (Garcia and Trinh
2019). A smaller value of the GD denotes a higher convergence
degree, and thus a zero value indicates complete convergence.
The diversity metric defines the range of variance of the solutions
(Zitzler and Thiele 1999). A lower value of the metric denotes
higher diversity within the solutions and a zero value indicates that
the solutions are uniformly distributed. The last metric (i.e., speed
of convergence) involves measuring the time required for computation in order to get the final results. A shorter computation time
indicates a faster convergence process.
To check the performance of the developed model based on the
previously mentioned three metrics, a benchmark problem involving time-cost trade-off has been chosen from the literature, consisting of 18 activities (Feng et al. 1997). The results are compared
with the results of those reported by Zhang and Li (2010). For the
sake of comparison, algorithmic parameters for the analysis are kept
similar to those considered by Zhang and Li (2010), i.e., the population size (100), crossover probability (0.4), mutation probability
0.02)), and run size (50). The exercise was carried out for 50 runs and
the average value of all three metrics and standard deviation for the
50 runs are depicted in Table 3.
The results show that the developed NSGA-III model performs
best out of all four models. The model has a better average value in
all three performance metrics, compared to the other reported models. Furthermore, a lower standard deviation value provides evidence of consistent performance by the developed model.
Case Study
This study numerically analyzed a four objective case, to demonstrate the effectiveness of the proposed many-objective schedule
model for the TCQS trade-off problem. The case data was initially
introduced by Feng et al. (1997) to illustrate construction time-cost
© ASCE
trade-off analysis. The quality data was taken from the study of ElRayes and Kandil (2005) and the data for safety was adopted from
the study of Afshar and Dolabi (2014). The case consisted of 18
activities, each activity being executed in two to five ways. Table 4
presents the project information data including the activity weight
(Wt) for each activity (Act) and its direct cost (C), duration (D),
safety risk (Sr ), and different quality indicator (K) with respective
indicator weight (K wt ) and their percentage quality performance
(Qp ) for each execution mode (EM). For example, in the case
of Activity 1, the number of execution modes was five. With respect to the first execution mode, the duration, direct cost, and
safety risk score of the activity were 14 days, $2,400, and 12, respectively. The indicator weight and quality performance for each
of the three quality indicators were 50, 100 (K wt , Qp ); 30, 96; and
20, 98, respectively. The quality indicators were identified by the
researchers in the literature, e.g., for concrete, the suggested quality
indicators were density, rebar cover, water cement (W/C) ratio, curing, air content, and strength (El-Rayes and Kandil 2005). The selection of the quality indicator was based on its ability to achieve
quantitative measures. For concrete, strength could be a quality indicator as it could be easily measured, in terms of the compressive
strength of the concrete. All of the indicators for the activities had
different units of measurement. To measure them under the same
unit, researchers had used a 0%–100% scale to represent the degree
of satisfaction of quality performance.
The project had an average of 3.61 execution modes for each
of the 18 activities. This could result in 10 billion (3.6118 ¼
10.8 × 109 ) possible combinations for completing the entire
project, which would make it an NP-hard problem of optimization.
Each possible solution had an exclusive effect on project performance. To find the potential optimal solution, which established
a balance among the objectives, the decision makers had to search
a large pool of possible solutions. The developed many-objective
optimization model was used to ease the effort of decision makers
and search an ample number of potential solutions.
04020160-8
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Table 4. Case study data
Table 4. (Continued.)
Quality performance (Qp ) and
quality indicator (K)
Quality performance (Qp ) and
quality indicator (K)
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K¼1
K¼2
K¼1
K¼3
K¼2
K¼3
Act EM D (days)
C ($)
Wt K wt Qp K wt
Qp
K wt
Qp
Sr
Act EM D (days)
C ($)
Wt K wt Qp K wt
Qp
K wt
Qp
Sr
1
2,400
2,150
1,900
1,500
1,200
3,000
2,400
1,800
1,500
1,000
4,500
4,000
3,200
45,000
35,000
30,000
20,000
17,500
15,000
10,000
40,000
32,000
18,000
30,000
24,000
22,000
220
215
200
208
120
300
240
180
150
100
450
400
320
450
350
300
2,000
1,750
1,500
1,000
4,000
3,200
1,800
3,000
2,400
2,200
3,500
4,500
3,000
2,000
1,750
1,500
1,000
4,000
3,200
1,800
3
96
89
77
72
60
94
94
92
72
66
97
82
60
95
71
63
97
89
71
64
95
74
62
99
73
62
N/A
N/A
N/A
N/A
N/A
99
92
88
75
64
97
83
69
95
75
66
98
85
71
61
96
71
62
95
82
67
98
98
96
85
79
73
60
97
75
65
20
20
20
20
20
20
20
20
20
20
15
15
15
15
15
15
20
20
20
20
25
25
25
40
40
40
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
15
15
15
15
20
20
20
10
10
20
0
0
40
40
40
40
40
10
10
10
98
89
84
73
65
99
95
85
70
59
98
81
63
94
76
64
99
89
72
61
100
79
68
93
71
67
N/A
N/A
N/A
N/A
N/A
0
0
0
0
0
0
0
0
0
0
0
95
87
79
63
97
76
63
98
81
66
N/A
N/A
98
87
78
74
62
99
72
61
12
9
12
8
5
30
24
20
20
18
20
24
14
5
5
4
12
8
5
9
12
5
9
24
20
12
0
N/A
N/A
N/A
N/A
6
4
8
3
4
9
12
8
3
5
4
30
36
24
20
15
18
24
16
15
16
30
25
10
3
6
8
6
36
36
20
18
3,000
2,400
2,200
5
99
77
66
25
25
25
94
71
67
20
18
12
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
1
2
3
4
5
1
2
3
4
5
1
2
3
1
2
3
1
2
3
4
1
2
3
1
2
3
1
2
3
4
5
1
2
3
4
5
1
2
3
1
2
3
1
2
3
4
1
2
3
1
2
3
1
2
1
2
3
4
5
1
2
3
© ASCE
14
15
16
21
24
15
18
20
23
25
15
22
33
12
16
20
22
24
28
30
14
18
24
9
15
18
14
15
16
21
24
15
18
20
23
25
15
22
33
12
16
20
22
24
28
30
14
18
24
9
15
18
16
12
20
22
24
28
30
14
18
24
5
8
11
10
11
10
1
1
1
1
2
3
1
7
3
6
50
50
50
50
50
40
40
40
40
40
70
70
70
50
50
50
60
60
60
60
50
50
50
30
30
30
100
100
100
100
100
50
50
50
50
50
60
60
60
70
70
70
50
50
50
50
40
40
40
80
80
80
70
70
30
30
30
30
30
70
70
70
100
90
86
75
63
98
87
81
77
60
100
80
62
99
74
59
100
93
77
61
95
76
59
97
70
61
95
83
75
68
61
100
97
81
71
63
94
79
63
96
72
61
99
89
70
62
99
73
60
100
79
63
100
100
97
89
81
72
67
98
73
62
30
30
30
30
30
40
40
40
40
40
15
15
15
35
35
35
20
20
20
20
25
25
25
30
30
30
0
0
0
0
—
50
50
50
50
50
40
40
40
30
30
30
35
35
35
35
40
40
40
10
10
10
30
30
30
30
30
30
30
20
20
20
1
2
3
9
15
18
30
30
30
98 45
75 45
63 45
Note: N/A = not applicable.
Results and Discussion
This section describes the simulated results for the considered example. To achieve this, fine tuning of the parameters of the optimization algorithm, such as population, number of generations,
crossover and mutation index, was performed. These parameters
were set based on the proposed values from the literature and trials
were carried out by altering the values of the previously mentioned
parameters. For the final analysis, the best possible combinations
selected were as follows:
• Number of generations = 200;
• Population size = 100; and
• Crossover and mutation distribution index = 2.0.
Optimal sets of different combinations of execution modes that
met the desired project objectives were obtained. Table 5 describes
the four nondominated best solutions with respect to TCQS. It can
be seen that Solution 1 generated the smallest project duration value
(100 days) for the project while Solutions 2 and 4 generated the
smallest values for cost ($100,865) and safety risk score (190)
for the project, respectively. The maximum quality performance
of the project was found to be 97.63%.
The obtained set of results can be used by project managers and
team members based on their priorities. A posteriori approach such
as weighted sum can be used for the selection of the most suitable
solution that best fits their priority amongst the available Pareto
optimal solutions. For example, if a project requires high-quality
product, the maximum weightage can be given to quality and other
parameters can have an equal share of weightage. Using weights, a
maximum quality Pareto solution can be selected from the pool of
optimal solutions provided by the model. Additionally, to choose
the best solution, one has to check the overall obtained solution by
using weighted sum, and thereafter the solution having optimal
value can be selected to achieve best objective values. This will
aid the decision makers with the required inputs in order to make
appropriate decisions during the course of project execution.
To get a better understanding of the objectives, with respect to one
another, trade-off graphs for the obtained results were plotted in three
dimensions. One was for TCQ trade-off and the second one was for
TCS trade-off, as depicted in Figs. 3 and 4, respectively. From these
figures, it can be observed that the direct cost increased with timecost trade-off, showing a similar pattern to those established in past
studies (Tiwari and Johari 2015; Yang 2007). This makes the model
results acceptable with regard to the other objectives.
The primary focus of this study was the integration of the quality
and safety measures in the schedule. In order to observe the behavior of both parameters with respect to each other, the trade-off graph
was analyzed further (Fig. 5). It can be seen that, for the initial
period carrying a higher safety risk score, the quality performance
was low. Therefore, it could be inferred that, due to the initial reluctance of workers toward the safety measures and lack of knowledge of safety norms, the quality of the workmanship decreases
04020160-9
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J. Constr. Eng. Manage.
Table 5. Best nondominated solution obtained by developed model
Objective
Time
Cost
Quality
Safety risk score
Execution mode
Time (days)
Cost ($)
Quality (%)
Safety risk score
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
4,5,3,3,4,3,3,2,1,1,3,4,3,2,2,5,3,2
1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1
5,5,3,1,3,2,3,2,2,1,1,4,1,2,1,2,3,3
100
145
104
144
169,820
100,865
168,820
138,655
97.63
67.63
97.63
77.49
285
226
290
190
0.6
0.4
0.2
0
Cost
Quality
Time
Safety risk
score
Fig. 6. Time-cost-quality-safety coordinate plot.
(Li et al. 2012). However, as soon as safety terms get familiar and
workers begin to use them, it can be seen that with time, there is
an improvement in quality, which leads to a higher safety risk score.
To avoid the initial drop in the quality measure, a proper quality and
safety training should be provided to the workers so that their reluctant attitude can be pursued toward a positive direction. For a
successful execution of a project, irrespective of quality and safety,
human perception has been identified as the critical element. However, human understanding on the perceptions of safety and quality
is not clear (Ramaswamy and Mosher 2017). Although this study
has tried to analyze these human perspectives in the quantitative
terms, there is a need for further research to understand the trend
of these qualitative measures.
As the human brain can generally visualize a maximum of
three dimensions, the visualization of a fourth dimension is a challenging task. Thus, to visualize all four objectives simultaneously,
authors have used the parallel coordinate plot system, as shown in
Fig. 6. The coordinate plot described the different objectives on
the x-axis and the normalized values of the objectives on the
y-axis. This technique has been one of the most widely adopted
methods in many-objective problems in order to visualize the conflicting nature of various objectives (Von Lücken et al. 2014).
From Fig. 6, it is shown that cost and quality follow an almost
similar pattern. At low cost, a low score of quality is obtained and at
a higher cost, higher quality performance is obtained. Conversely,
poor quality adversely affects a project’s timespan (Fu and Zhang
2016) and this can be analyzed easily by the given coordinate plot.
Furthermore, if the relation between time and safety risk score is
analyzed, for a minimum time duration, the safety risk score is high,
and similarly, for a higher duration, the safety risk score is low. The
pattern suggests that safety risk can increase in a project when the
duration assigned to an activity is less.
Fig. 3. Time-cost-quality trade-off.
Fig. 4. Time-cost-safety trade-off.
Quality-safety trade-off
290
280
Safety risk
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1
0.8
270
260
250
Correlation Analysis
240
To check the existence of correlations, if any, between all four of
the objectives, a Pearson correlation test was conducted. The Pearson correlation is a coefficient that provides statistical evidence for
interrelationships among the variables. The correlation indicates
whether there is a significant relationship between the two variables, how strong the relationship is, and the direction of the trends
(increasing or decreasing). Table 6 shows the coefficient of Pearson
230
70
75
80
85
90
Quality (%)
Fig. 5. Quality-safety trade-off.
© ASCE
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100
04020160-10
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J. Constr. Eng. Manage.
Table 6. Correlation among variables time, cost, quality, and safety risk
score
Variables
Time
Cost
Quality
Safety risk score
Cost
1
−0.837a
−0.875a
−0.849a
−0.837
1
0.964a
0.747a
a
Quality
Safety risk score
−0.875
0.964a
1
0.836a
−0.849a
0.747a
0.836a
1
a
Correlation is significant at the 0.01 level (2-tailed).
correlation among the four project objectives (time, cost, quality,
and safety risk score). It is evident from Table 6 that all four objective variables are significantly correlated with each other (statistically speaking) at a significance level of 0.01. Also, as all of the
coefficient values are greater than 0.5, it shows a strong interrelation between the variables and indicates that time is inversely proportional to the remaining three objectives: cost, quality, and safety
risk score. Cost, quality, and safety risk score have a positive correlation with each other. The results also show that cost, quality,
and safety risk scores are directly proportional to each other.
Sensitivity Analysis
First, to check the impact of the possible deviations in the input
parameters of the algorithm and that the outputs of the developed
model were the optimal values of the objectives, a sensitivity
analysis was carried out. Different studies have carried out sensitivity analyses with regard to objective weight, population size,
number of generations, and so forth (Fu and Zhang 2016; KhaliliDamghani et al. 2015; Taheri Amiri et al. 2017). In this study, a
sensitivity analysis was performed by altering the population
Gen 1
Cost
180000
Gen 100
(d)
140
Time
140
Time
(b)
140
Time
(e)
Gen 175
(g)
Cost
140000
120
140
Time
160
(f)
140
Time
180
160
180
Gen 200
140000
100000
100
180
180
140000
100000
100
180
140
Time
Gen 150
180000
180000
180000
100000
100
(c)
140000
100000
100
180
140000
100000
100
180
Gen 125
180000
140000
100000
100
140000
100000
100
180
Gen 75
180000
Cost
(a)
140
Time
Cost
100000
100
In order to ensure the robustness and the practical applicability of
the model, it is vital to test the developed model on a large-scale
problem with a greater number of activities in the network. However, owing to the scarcity of previous literature on many-objective
large-scale projects, the authors have attempted to verify the model
on a larger-scale scenario in two ways: (1) by replicating the
Cost
140000
Scalability Toward Large-Scale Problems
Gen 25
180000
Cost
Cost
180000
Cost
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a
Time
value and the number of generations in the optimization model
to examine the deviation in the optimal values of TCQS. The convergence of the population with the number of generations is
shown in Fig. 7. It can be seen that the population in Gen-1 was
randomly distributed in the search space. Furthermore, when the
authors increased the number of generations, the overall population followed the standard time-cost trade-off trend. At the final
Gen-200 a smooth, converged pattern of time-cost trade-off was
obtained, as presented in Fig. 7.
Second, the sensitivity analysis was carried out on the basis of
the alteration provided in the population and the number of generations. The results of the analysis are given in Table 7, with each
of the combination parameters of population and number of generations. When the sensitivity of all the four objectives was checked
with each other according to the parameters, it could be observed
that the time and cost are equally sensitive in the trade-off process.
Furthermore, it was observed that with a change in the population,
the percentage changes in time-cost were significantly lower. A
similar trend was noticed in the case of quality and safety except
that the percentage change was still a little less, compared to timecost. When sensitivity was analyzed with changing combinations
of time-quality, time-safety, cost-quality, and cost-safety, the percentage changes seemed to slightly increase.
(h)
120
140
Time
Fig. 7. Generationswise population convergence toward optimal solution.
© ASCE
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Table 7. Population and generationwise deviations in optimal solution
Pop
Gen
1
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25
Time
Objective (days)
Average
Min
Max
25
Average
Min
Max
50
Average
Min
Max
75
Average
Min
Max
100
Average
Min
Max
125
Average
Min
Max
150
Average
Min
Max
175
Average
Min
Max
200
Average
Min
Max
Average Average
Min
Max
% change
133
119
148
139
117
157
150
114
169
140
104
169
147
100
169
127
100
169
132
100
165
135
100
165
137
100
169
138
106
164
35
Cost
($)
50
Quality
Time
(%) Safety (days)
139,079
118,010
152,585
128,096
104,650
165,700
119,690
100,255
139,015
124,288
101,200
168,820
131,281
99,820
169,820
141,196
100,540
169,820
136,374
100,960
169,820
131,704
100,790
169,820
132,347
99,790
169,770
131,562
102,891
163,908
37
81
75
84
77
68
95
74
67
86
75
66
98
78
65
98
83
67
98
82
65
98
79
65
98
79
65
98
79
67
95
29
257
228
281
237
207
274
223
198
252
238
194
290
228
201
285
245
197
285
244
192
285
236
191
285
238
189
290
238
200
281
29
131
107
146
132
100
169
134
100
169
134
100
169
140
100
165
136
100
169
135
100
169
141
100
169
137
100
169
136
101
166
39
Cost
($)
75
Quality
Time
(%) Safety (days)
131,113
111,815
146,905
133,104
103,880
167,630
135,742
99,820
169,770
132,357
99,835
169,820
126,263
100,820
169,820
129,358
100,740
169,820
131,214
99,820
169,820
129,681
99,740
169,820
134,816
100,740
169,820
131,516
101,912
167,025
39
80
73
86
81
70
95
81
65
97
81
65
98
77
65
98
81
65
98
80
65
98
79
65
98
81
65
98
80
66
96
31
Table 8. Average percentage deviation and running time for large-scale
problems
Activities
108
504
1,080
Run Nos.
Running time (s)
APD (%)
50
50
50
1.05
10.35
25.28
0.59
0.60
0.62
249
218
274
255
210
290
243
196
288
236
188
290
230
193
285
239
192
285
243
189
290
235
195
285
240
188
290
241
197
286
31
132
113
154
134
106
169
130
100
161
133
100
169
132
100
165
130
100
165
131
100
169
132
100
165
131
100
165
132
102
165
38
Cost
($)
100
Quality
Time
(%) Safety (days)
133,730
113,608
153,400
126,368
100,170
168,040
131,399
100,180
169,760
136,222
99,820
169,820
131,406
100,870
169,820
129,214
100,820
169,820
129,954
99,820
169,820
130,688
100,740
169,820
135,599
100,740
169,820
131,620
101,863
167,791
39
80
73
88
79
66
95
80
66
98
81
65
98
81
65
98
81
65
98
80
65
98
80
65
98
82
65
98
80
66
97
32
256
233
288
243
199
281
238
196
283
242
193
285
240
188
290
243
189
285
241
188
285
237
187
285
244
193
290
243
196
286
31
132
109
155
133
100
169
133
100
169
132
100
169
134
100
169
132
100
165
130
100
165
132
100
169
134
100
169
132
101
167
40
Cost
($)
133,954
107,000
158,778
133,923
100,910
168,620
128,548
100,220
169,570
130,858
100,835
168,820
131,572
100,820
168,770
132,421
100,740
169,820
133,594
100,740
169,820
128,399
99,740
169,820
130,388
99,740
169,820
131,517
101,194
168,204
40
Quality
(%) Safety
81
72
87
81
67
97
80
65
98
80
65
98
81
65
97
81
65
98
81
65
98
80
65
98
81
65
98
81
66
97
32
254
220
281
242
193
286
240
189
291
242
195
290
240
190
294
237
188
287
235
188
285
235
188
287
239
193
290
240
194
288
33
activities of the available 18 activity many-objectives problem, and
(2) by using the available real-life, large-scale problem with twoobjectives. These mean models can be analyzed in terms of their
capability for solving complex, many-objective, large-scale, real-life
problems.
First, three many-objective hypothetical problems, consisting
of 108, 504, and 1,080 activities, were generated by replicating/
Table 9. Results of 63-activity project
GA
PSO
TLBO
Developed model
Number of runs
Time (days)
Cost ($)
Time (days)
Cost ($)
Time (days)
Cost ($)
Time (days)
Cost ($)
1
2
3
4
5
6
7
8
9
10
Average
Minimum
Maximum
Population size
No. of iterations
No. of evaluations
519
528
522
523
524
516
517
519
519
522
521
516
528
5,825,480
5,687,020
5,725,380
5,765,800
5,827,200
6,052,120
5,722,600
5,872,000
5,818,480
5,716,980
5,801,306
5,687,020
6,052,120
602
620
594
606
630
617
614
627
610
581
610
581
630
5,920,580
5,904,125
5,701,200
5,837,980
5,994,490
5,925,980
5,751,470
5,934,330
5,924,365
5,858,295
5,875,282
5,701,200
5,994,490
629
614
630
616
630
637
639
630
627
632
628
614
639
5,613,820
5,644,640
5,600,190
5,623,260
5,642,405
5,637,290
5,503,940
5,696,820
5,588,485
5,625,310
5,617,616
5,503,940
5,696,820
518
518
515
515
519
519
513
517
516
515
517
513
519
5,735,990
5,718,005
5,743,180
5,825,630
5,723,700
5,729,490
5,802,700
5,732,340
5,838,270
5,768,830
5,761,814
5,718,005
5,838,270
500
500
250,000
500
500
250,000
180
450
162,180
100
200
20,000
Note: Bold value indicates best value obtained for respective parameter. TLBO = teaching-learning-based optimization.
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repeating the existing 18 activity problem (Agdas et al. 2018) by
6, 28, and 60 times. Although, these are hypothetical examples,
given the complexity in terms of solving the 1,080 activity problem, it has approximately 4.5 × 10539 ways to schedule the project
and thus in a way, it also replicates a real-life example ensuring the
practical applicability of the model. By running all mentioned
hypothetical problems 50 times using the developed NSGA-III
model, their average percentage deviation (APD) and running time
per generation were measured and are presented in Table 8. The
measured APD and running time portrays the model’s robustness
in solving large-scale scheduling problems.
Second, an attempt was made to compare the optimal solutions
generated by Toğan and Eirgash (2019) for a real-life example consisting of a time-cost trade-off problem and the developed NSGA-III
model. The example problem consists of a real-life project having 63
activities. Each activity had an average of 4.69 modes of execution,
which lead to approximately 1.9 × 1042 ways to schedule the project.
The example checks the model for large-scale, real-life project complexity. The generated optimum solutions and the comparisons are
presented in Table 9.
From Table 9, it can be inferred that the developed model not
only successfully analyzed the large-scale problem successfully,
but also provided better solutions with less evaluation in comparison to the reported studies. Moreover, the lower descriptive statistical values strengthen the performance efficiency of the developed
model. The analysis outlined previously has established that the
developed model can successfully solve large-scale problems having real-life complexities. Furthermore, the developed model provides a number of optimal schedules, which help decision makers
to opt an appropriate schedule according to the project requirement.
interesting insight is that by examining sets of Pareto optimal solutions, additional quality and safety performances can be obtained
without putting additional cost. Further, the trade-off pattern shows
that the quality performance increases with an increase in project
cost and safety risk score decreases with increasing time duration.
Correlation analysis has also been conducted to verify the trade-off
pattern between the objectives. An optimal schedule with regard to
all these four essential construction objectives proves to be useful
for construction planners and can lead to significant improvements
in the safety and quality performance of constructed facilities at
the planning stage itself with time and cost. Sensitivity analysis
was performed to check the impact of these inputs on output
parameters.
It is observed that the many-objective schedule model works effectively in the simultaneous optimization of all four desired project
objectives. To see the applicability of the model in real-world construction problems, two large-scale scheduling examples were analyzed, which provided further evidence for the efficient performance
of the model. Although the results for these large-scale problems
need further validation through other optimization algorithms, the
authors believe that the proposed methodology using the evolutionary algorithm (NSGA-III) has capability to generate more practical
solutions in terms of many-objective project scheduling. Further, this
will encourage the decision makers to use the model in a holistic way
in the project planning stage itself. This would also ensure benefits to
the stakeholders of the construction projects, to guarantee a safe
work environment and deliver a quality end product. In the real
world, construction projects are full of uncertainties and ambiguities
and it would be interesting to see the further application of the developed model in real-life probabilistic scenario and fuzzy data sets
for the TCQS trade-off.
Conclusion
Quality and safety are the two performance parameters that directly
or indirectly affect the project time and cost. Incorporation of these
parameters into a MOSP provides decision makers with a holistic
construction schedule. Still, the integration of these two parameters
in trade-off models of MOSP was not observed in past studies. This
was mainly because the complexities and intricacies involved in
solving these problems increase exponentially with an increase in the
number of activities, their possible execution modes, and the number
of project objectives. To make it manageable, a many-objective optimization model was developed based on NSGA-III. The model carries out the trade-off between four key project objectives, namely
time, cost, quality, and safety risks. To indicate the comparability
and dominance of the developed model, a comparative study was
carried out on two multiobjective scheduling problems (TCQ and
TCS) taken from the literature. The obtained solution for both examples indicates that the model has better capabilities for dealing
with MOSPs. Further, the proposed model has the ability to deal with
more than three objectives and provides opportunity for the construction stakeholders to incorporate the parameters of their interest. The
generated Pareto front solution shows the superiority of the developed model when compared with the models available in the literature on all four objective TCQS.
The developed model is illustrated for many objectives with the
help of a case study example. The results obtained from the model
reveal the benefits of the inclusion of quality and safety with the
other project management objectives. To study the influence of
these two parameters, various trade-off graphs are analyzed and
discussed. The analysis results highlight the new and unique capabilities of the scheduling model in generating optimal tradeoffs between construction time, cost, quality, and safety risk. An
© ASCE
Data Availability Statement
Some or all data, models, or code generated or used during the
study are available from the corresponding author by request.
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