# Developing Shuffled Frog-Leaping Algorithm (SFLA) Method to Solve Power Load-Constrained TCRTO Problems in Civil Engineering

Authors: Xingyu Tao; Heng Li; Chao Mao; Chen Wang; Jeffrey Boon Hui Yap; Samad Sepasgozar; Sara Shirowzhan; Timothy Rose
Journal: Advances in Civil Engineering
 (2019)

---

## Abstract

It is extensively acknowledged that excessive on-site electricity power load often causes power failure across a construction site and surrounding residential zones and can result in unforeseen schedule delay, construction quality problems, life inconvenience, and even property loss. However, energy management, such as power load optimization, has long been ignored in construction scheduling. This study aims to develop a modified shuffled frog-leaping algorithm (SFLA) approach in project scheduling to aid decision-makers in identifying the best Pareto solution for time-cost-resource trade-off (TCRTO) problems under the constraint of precedence, resource availability, and on-site peak electricity power load. A mathematical model including three objective functions and five constraints was established followed by the application of the modified SLFA on real-case multiobjective optimization problems in construction scheduling. The performance of SLFA was compared with that of the nondominated sorting genetic algorithm (NSGA II). The results showed that the developed new approach was superior in identifying optimal project planning solutions, which could essentially assist on-site power load-oriented schedule decision-making for construction teams.


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## Body

## 1. Introduction

The objectives of sustainable construction have resulted in increasing demand to adopt strategies to facilitate energy management throughout the entire lifecycle of buildings [1]. Multipronged research efforts, related to technology and policy aspects, have been applied to scheduling energy supply and consumption in the building operation phase [2, 3]. However, very few previous studies have proposed methods to enable energy optimization in the construction stage, given a lack of awareness in scheduling energy supply and multiple on-site restrictions. Energy management, especially electricity power load optimization, has long been ignored in construction scheduling. With the rapid development of new construction techniques, an increased amount of electric-powered construction equipment has been adopted on-site, placing a significant burden on the electric power supply. Especially, this occurs when more equipment is used simultaneously to push for target deadlines, resulting in a phenomenon called “electricity snatching.” This results in an increased risk of circuit breaker tripping due to power overload. An on-site power outage can generate negative project impacts, including mechanical breakdowns, schedule delays, and possible quality problems. For instance, the sudden stop of concrete pumps can have a negative impact on the efficiency of pouring and setting time, which is crucial to structural quality. Further, temporary electricity supply on construction sites is often sourced from surrounding residential zones. Peak loading or overloading on construction sites can result in power failure of residential areas, which can result in inconvenience for residents and property loss. The control of on-site power load has become a major issue to both the construction team and the public. Therefore, the need to integrate power load constraints into construction planning poses an obvious and urgent challenge in the construction industry.Many previous studies have attempted to find solutions to facilitate multiobjective optimization, such as minimization of construction duration, cost, and resource fluctuation [4, 5]. Meanwhile, various methods have been introduced to optimize construction scheduling while considering constraints such as the workforce, resource availability, or construction site layout that have been considered in earlier studies [6–8]. Nevertheless, optimizing and balancing on-site peaking power load has received little attention. Given the advances in electrical technology, on-site power supply reliability has been enhanced. But excessive power load and power failure occur occasionally because contractors usually manage project scheduling and power load requirements separately. The on-site power supply cannot be easily matched with actual construction demand without integrating power load into the construction schedule. A method that can be implemented to simultaneously optimize total duration, total project cost, and resource variation, while considering issues related to resource and power load constraints, is necessary and worth of developing.To address this research gap, this paper developed a novel shuffled frog-leaping algorithm (SFLA) to solve time-cost-resource trade-off (TCRTO) problems under the constraint of precedence, resource availability, and on-site peak electricity power load. SFLA is a promising and reliable approach for improving project planning solutions since it integrates the strengths of two evolutionary algorithms (i.e., particle swarm optimization and shuffling complex evolution algorithm) simultaneously [9, 10]. By optimizing the combination of on-site electric devices, the power load at any time instant could be automatically controlled within a safety range. The frequency of on-site power outage that led to mechanical breakdown, project quality problems, and inconvenience for surrounding communities was significantly reduced. The developed SFLA could help construction teams to explore better and more effective planning solutions with shorter duration, lower cost, and less resource fluctuation. In specific, this study proposed a mathematical model to optimize and to integrate power load constraints and developed a SFLA method to solve the power load-constrained TCRTO problems in construction planning. The modified SFLA model was implemented in a real case scenario compared with a typical existing method to assess its optimization capability.


## 2. Construction Scheduling Optimization and Shuffled Frog-Leaping Algorithm

Generally, objectives that need to be optimized usually contain projected duration and financial and resource aspects. Balancing these three parameters is crucial to ensure the achievement of project objectives, namely, minimizing the project duration (TD), project cost (TC), and resource fluctuation (RF). Another aspect of optimization is constraint parameters. Determining appropriate constraints is a critical factor for solving optimization problems. For example, precedence relationship requires an activity can start only when all the previous work has been completed. Further, the start time of each activity must be earlier later than its latest start time [11]. Other limitations, such as resource constraint, space constraint, information constraint, site layout, and safety, have also been considered as factors in recent years. Optimization parameters defined in previous studies are shown in Table 1. Unfortunately, energy management, such as on-site power load scheduling, has not been considered as an optimization parameter in previous studies, despite that uncontrolled on-site electricity power load often causes power failure on construction sites and surrounding residential zones, resulting in inconvenience, property loss, and even safety problems. This is the gap in knowledge that this research is focused on.Table 1
Optimization parameters and methods for the project scheduling problem.
Research studies
Optimization parameters
Optimization method
Time
Cost
Resource fluctuation
Quality
Cash flow
Resource availability
Site layout
Productivity
Safety
[11]
√
√
PSO
[12]
√
√
ACA
[4]
√
√
√
Commercial planning software
[13]
√
√
√
GA
[14]
√
√
GA
[15]
√
√
√
GA (NSGA II)
[10]
√
√
√
SFLA
[16]
√
√
√
IP
[17]
√
BCA
[18]
√
√
√
√
GA and evidential reasoning
[19]
√
√
Evolutionary algorithm
[5]
√
√
√
GA (NPGA)
[8]
√
√
√
√
√
GA
[20]
√
√
√
Metaheuristics
[6]
√
√
√
GA
[21]
√
√
√
√
√
NSGA II
[22]
√
√
√
√
GA


### 2.1. Methods for Construction Scheduling Optimization

#### 2.1.1. Mathematical Methods

Two mainstream mathematical approaches were used in optimizing construction schedules. (1) Critical path method (CPM): although CPM is widely used in scheduling in the construction industry [23], a major limitation of CPM is that this method cannot deal with optimization problems with multiple objectives [24]. (2) Integer programming (IP) and linear programming (LP): IP, LP, and hybrid IP/LP are analytical algorithms that have been commonly used to solve construction scheduling optimization problems due to their innate efficiency and accuracy [25, 26]. In most cases, IP or LP approaches focus on a single target. A few studies have applied this method in multiobjective optimization problems by providing different weights to different objectives to convert this problem to a single objective problem [27]. However, how to set convincing weights prevents the usage of LP/IP methods in multiobjective optimization problems.


#### 2.1.2. Metaheuristic Methods

(1) GA and NSGA II. GA is a method to search for finding the optimal solution by simulating the natural evolutionary process [28]. Previous studies have attempted to solve the multimode resource-constrained time-cost trade-off problem using GA or improved GA method [13, 29]. In multiobjective optimization problems, nondominated sorting genetic algorithm (NSGA II) is often used to achieve the Pareto front and a set of optimal solutions can be obtained. The results showed that this method can help construction planners achieve all the projects’ multiple objectives under certain limits [21, 30]. GA is an efficient method to obtain an optimal or a set of optimal solutions in construction scheduling. However, improperly selected fitness function and parameters may cause the algorithm to be trapped in local optimal, where global optimal cannot be guaranteed. In addition, it is difficult to determine the stopping criterion of a GA algorithm [31].(2) Ant Colony Optimization (ACO). Ant colony algorithm uses the ant’s walking path to represent the feasible solution of the problem to be optimized, and all the paths of the whole ant colony constitute the solution space of the problem to be optimized. Many studies have used this method to solve single and multiple objective problems [32–34]. Although ACO is a powerful tool, several limitations, including the premature convergence phenomenon, the stopping criterion, and the parameter determining the method, need to be addressed to improve the application of the ACO algorithm.(3) Particle Swarm Optimization (PSO). PSO is a population-based search process. Due to the poor efficiency of traditional PSO, many researchers have proposed modified PSO for solving the construction scheduling problem by introducing new crossover operators that operate on those coupled particles selected from half of the particle population [11, 35]. However, the selection of the PSO parameters, which are mainly selected based on empirical results, could cause convergence, divergence, and oscillation of the particles.


### 2.2. Shuffled Frog-Leaping Algorithm

The emergence of shuffled frog-leaping algorithm (SFLA) offers a promising and effective solution for multiobjective and combinatorial optimization problems. Two main steps are involved in SFLA, namely, local search and global search [9]. Specific optimization process of SFLA is illustrated in Section 4.The performance of SFLA has been compared with that of the genetic algorithm (GA) by testing 11 theoretical test functions and two applications. The result indicates that the SFLA is superior to, or at least, comparable with the GA. Also, SFLA and other existing optimal algorithms were employed in four realistic engineering problems, and the results of SFLA were extremely promising in terms of robustness in finding global solutions and solution speed [36, 37]. The advantages of SFLA have been highlighted within different industrial contexts in recent years. For example, Samuel and Rajan proposed a hybrid particle swarm optimization-based SFLA for solving long-term generation maintenance scheduling problems, and the results demonstrated great superiority of SFLA in searching solutions for combinatorial optimization problems [38]. Luo et al. introduced a novel hybrid SFLA for vehicle routing problems (VRP). The experimental results show that the presented algorithm is very effective for handling VRP when compared with other state-of-the-art heuristics [39]. Kaur et al. presented an augmented SFLA-based technique for resource provisioning and workflow scheduling. The efficacy of ASFLA has been assessed over some well-known scientific workflows, and the simulation results show a marked improvement in the performance criteria of achieving minimum execution cost and meeting schedule deadlines [40]. Compared with other existing optimal algorithms, three main advantages of SFLA can be highlighted: (1) SFLA is more powerful in solving complex combinatorial optimization problems; (2) SFLA has a faster search capability; and most importantly, (3) SFLA is robust in determining the global solution because every memeplex evolution and shuffle process can enhance the quality of individuals, thus a better set of solutions is generated.Despite these advantages, very few studies have adopted this novel method in the context of the construction schedule. Ashuri and Tavakolan established a construction project planning model using SFLA, which allows the interruption of project activities, optimizing three objectives [10]. However, no research has been identified that uses SFLA to enable planners to simultaneously achieve reduced total project duration, less total project cost, and less resource variation under the constraint of precedence relationships, resource availability, and on-site power load.


## 2.1. Methods for Construction Scheduling Optimization

### 2.1.1. Mathematical Methods

Two mainstream mathematical approaches were used in optimizing construction schedules. (1) Critical path method (CPM): although CPM is widely used in scheduling in the construction industry [23], a major limitation of CPM is that this method cannot deal with optimization problems with multiple objectives [24]. (2) Integer programming (IP) and linear programming (LP): IP, LP, and hybrid IP/LP are analytical algorithms that have been commonly used to solve construction scheduling optimization problems due to their innate efficiency and accuracy [25, 26]. In most cases, IP or LP approaches focus on a single target. A few studies have applied this method in multiobjective optimization problems by providing different weights to different objectives to convert this problem to a single objective problem [27]. However, how to set convincing weights prevents the usage of LP/IP methods in multiobjective optimization problems.


### 2.1.2. Metaheuristic Methods

(1) GA and NSGA II. GA is a method to search for finding the optimal solution by simulating the natural evolutionary process [28]. Previous studies have attempted to solve the multimode resource-constrained time-cost trade-off problem using GA or improved GA method [13, 29]. In multiobjective optimization problems, nondominated sorting genetic algorithm (NSGA II) is often used to achieve the Pareto front and a set of optimal solutions can be obtained. The results showed that this method can help construction planners achieve all the projects’ multiple objectives under certain limits [21, 30]. GA is an efficient method to obtain an optimal or a set of optimal solutions in construction scheduling. However, improperly selected fitness function and parameters may cause the algorithm to be trapped in local optimal, where global optimal cannot be guaranteed. In addition, it is difficult to determine the stopping criterion of a GA algorithm [31].(2) Ant Colony Optimization (ACO). Ant colony algorithm uses the ant’s walking path to represent the feasible solution of the problem to be optimized, and all the paths of the whole ant colony constitute the solution space of the problem to be optimized. Many studies have used this method to solve single and multiple objective problems [32–34]. Although ACO is a powerful tool, several limitations, including the premature convergence phenomenon, the stopping criterion, and the parameter determining the method, need to be addressed to improve the application of the ACO algorithm.(3) Particle Swarm Optimization (PSO). PSO is a population-based search process. Due to the poor efficiency of traditional PSO, many researchers have proposed modified PSO for solving the construction scheduling problem by introducing new crossover operators that operate on those coupled particles selected from half of the particle population [11, 35]. However, the selection of the PSO parameters, which are mainly selected based on empirical results, could cause convergence, divergence, and oscillation of the particles.


## 2.1.1. Mathematical Methods

Two mainstream mathematical approaches were used in optimizing construction schedules. (1) Critical path method (CPM): although CPM is widely used in scheduling in the construction industry [23], a major limitation of CPM is that this method cannot deal with optimization problems with multiple objectives [24]. (2) Integer programming (IP) and linear programming (LP): IP, LP, and hybrid IP/LP are analytical algorithms that have been commonly used to solve construction scheduling optimization problems due to their innate efficiency and accuracy [25, 26]. In most cases, IP or LP approaches focus on a single target. A few studies have applied this method in multiobjective optimization problems by providing different weights to different objectives to convert this problem to a single objective problem [27]. However, how to set convincing weights prevents the usage of LP/IP methods in multiobjective optimization problems.


## 2.1.2. Metaheuristic Methods

(1) GA and NSGA II. GA is a method to search for finding the optimal solution by simulating the natural evolutionary process [28]. Previous studies have attempted to solve the multimode resource-constrained time-cost trade-off problem using GA or improved GA method [13, 29]. In multiobjective optimization problems, nondominated sorting genetic algorithm (NSGA II) is often used to achieve the Pareto front and a set of optimal solutions can be obtained. The results showed that this method can help construction planners achieve all the projects’ multiple objectives under certain limits [21, 30]. GA is an efficient method to obtain an optimal or a set of optimal solutions in construction scheduling. However, improperly selected fitness function and parameters may cause the algorithm to be trapped in local optimal, where global optimal cannot be guaranteed. In addition, it is difficult to determine the stopping criterion of a GA algorithm [31].(2) Ant Colony Optimization (ACO). Ant colony algorithm uses the ant’s walking path to represent the feasible solution of the problem to be optimized, and all the paths of the whole ant colony constitute the solution space of the problem to be optimized. Many studies have used this method to solve single and multiple objective problems [32–34]. Although ACO is a powerful tool, several limitations, including the premature convergence phenomenon, the stopping criterion, and the parameter determining the method, need to be addressed to improve the application of the ACO algorithm.(3) Particle Swarm Optimization (PSO). PSO is a population-based search process. Due to the poor efficiency of traditional PSO, many researchers have proposed modified PSO for solving the construction scheduling problem by introducing new crossover operators that operate on those coupled particles selected from half of the particle population [11, 35]. However, the selection of the PSO parameters, which are mainly selected based on empirical results, could cause convergence, divergence, and oscillation of the particles.


## 2.2. Shuffled Frog-Leaping Algorithm

The emergence of shuffled frog-leaping algorithm (SFLA) offers a promising and effective solution for multiobjective and combinatorial optimization problems. Two main steps are involved in SFLA, namely, local search and global search [9]. Specific optimization process of SFLA is illustrated in Section 4.The performance of SFLA has been compared with that of the genetic algorithm (GA) by testing 11 theoretical test functions and two applications. The result indicates that the SFLA is superior to, or at least, comparable with the GA. Also, SFLA and other existing optimal algorithms were employed in four realistic engineering problems, and the results of SFLA were extremely promising in terms of robustness in finding global solutions and solution speed [36, 37]. The advantages of SFLA have been highlighted within different industrial contexts in recent years. For example, Samuel and Rajan proposed a hybrid particle swarm optimization-based SFLA for solving long-term generation maintenance scheduling problems, and the results demonstrated great superiority of SFLA in searching solutions for combinatorial optimization problems [38]. Luo et al. introduced a novel hybrid SFLA for vehicle routing problems (VRP). The experimental results show that the presented algorithm is very effective for handling VRP when compared with other state-of-the-art heuristics [39]. Kaur et al. presented an augmented SFLA-based technique for resource provisioning and workflow scheduling. The efficacy of ASFLA has been assessed over some well-known scientific workflows, and the simulation results show a marked improvement in the performance criteria of achieving minimum execution cost and meeting schedule deadlines [40]. Compared with other existing optimal algorithms, three main advantages of SFLA can be highlighted: (1) SFLA is more powerful in solving complex combinatorial optimization problems; (2) SFLA has a faster search capability; and most importantly, (3) SFLA is robust in determining the global solution because every memeplex evolution and shuffle process can enhance the quality of individuals, thus a better set of solutions is generated.Despite these advantages, very few studies have adopted this novel method in the context of the construction schedule. Ashuri and Tavakolan established a construction project planning model using SFLA, which allows the interruption of project activities, optimizing three objectives [10]. However, no research has been identified that uses SFLA to enable planners to simultaneously achieve reduced total project duration, less total project cost, and less resource variation under the constraint of precedence relationships, resource availability, and on-site power load.


## 3. Mathematical Model for TCRTO Optimization

### 3.1. Duration, Cost, and Resource Model

The multiple variables to be optimized were the total duration (TD), the total cost (TC), and resource fluctuation (RF), under the constraint of on-site power load, precedence relationships, and resource availability. In this study, the number of construction activities was J (1, 2, …,j, …, J). Each activity j could be performed in different methods (modes) Mj Mj=1,2,…,m,…,Mj, and only one mode m could be chosen as the execution mode for j. Once the execution mode of each activity was selected, the corresponding duration, cost, resources requirements, and power load generated by a set of equipment were determined [21]. Parts of the TD, TC, and RF model were based on the study of Ghoddousi et al. [15]:(1)TD=max:fj,fj−∑m∈Mjxjm×djm≥fi,TC=∑j∑m∈Mjxjm×cjm+TD×cind+y×cp×TD−Tcontract,RF=∑k=1K∑t=1TDrkt−rk¯2,rk¯=1TD∑t=1TDrkt,where fj is the finish time of activity j in mode m; xjm is a decision variable which equals to 1 when activity j is executed in mode m, otherwise xjm equals to 0; djm is the duration of j in mode m; i are the predecessors activities of j. cjm is the direct cost when activity j performs in mode m; cind is the indirect cost which is a constant amount; Tcontract is the project deadline that is defined in the contract; cp is a penalty if delay happens; and y is a decision variable which equals to 1 when TD>Tcontract, otherwise y equals to 0. The squared deviation is used to evaluate RF; k represents resource type; rkt is the resource usage of k in period t∈1,2,…,TD; and rk¯ is the average resource usage.


### 3.2. On-Site Power Load Constraints

The power supply system was composed of a set of loops (l = 1, 2, …, L), where different construction equipment was connected to different loops. Two constraint conditions were proposed: (1) for each time instant t and for each loop l, the total power load of loop l is not allowed to exceed Ppeakl, which is the threshold or peak load of loop l. (2) For each time instant t, the power load of all loops is not allowed to exceed Ppeak, which means the threshold or peak load of the on-site power supply system. For example, three activities named 1, 2, and 3 were currently processed at time instant t. Their optional execution mode and the corresponding electric construction equipment are listed in Table 2, respectively. Assuming these three activities are all executed in mode 1, the equipment set, as well as the loop each equipment classification belongs to, can be identified as follows: a-loop 1, b-loop 2, c-loop 1, e-loop 2, f-loop 3, g-loop 1, h-loop 3, and i-loop 1. According to the power load constraints, a set of functions are provided as(2)Pa1+Pc1+Pg1+Pi1=P1≤Ppeak1,Pb2+Pe2=P2≤Ppeak2,Pf3+Ph3=P3≤Ppeak3,∑l=13Pl+Pextra≤Ppeak,where Pa1 indicates the power load of equipment a in loop 1 and so on. In general, Pa1=Prate−a/ε, where Prate−a indicates rate power of a; ε indicates the loss factor (ε=0.9 for lighting equipment and ε=0.8 for construction equipment); Ppeak1 indicates the threshold or peak load of loop 1, and so on. Pextra indicates the extra power supply by the on-site electric generator.Table 2
Example of on-site power load constraint.
Act ID
Execution mode
Equipment
Loop
Rate power
Loss factorε
1
1
a
1
P
rate
−
a
0.8
b
2
P
rate
−
b
0.8
c
1
P
rate
−
c
0.8
2
d
1
P
rate
−
d
0.8
c
1
P
rate
−
c
0.8
2
1
e
2
P
rate
−
e
0.8
f
2
P
rate
−
f
0.8
3
1
g
1
P
rate
−
g
0.8
h
3
P
rate
−
h
0.8
i
1
P
rate
−
i
0.8
2
g
1
P
rate
−
g
0.8
j
3
P
rate
−
j
0.8
Two power load constraints were presented as follows:(3)
∑
j
∈
A
t
∑
m
∈
M
j
x
j
m
×
P
j
m
l
≤
P
peak
l
,
l
=
1
,
2
,
…
,
L
,
A
t
=
j
∣
f
j
−
d
j
<
t
≤
f
j
,
(4)
∑
l
=
1
L
∑
j
∈
A
t
∑
m
∈
M
j
x
j
m
×
P
j
m
l
+
P
extra
≤
P
peak
,
l
=
1,2
,
…
,
L
,
A
t
=
j
∣
f
j
−
d
j
<
t
≤
f
j
.Constraint set (3) indicated that for each time instant t and for each loop l, the power load of the activities which were currently processed (i.e., At) and connected to loop l could not exceed the threshold or peak load of loop l, where Pjml was the power load of loop l generated by activity j, if it was executed in mode m. Constraint set (4) indicated that for each time instant t, the total power load of the activities which were currently processed (i.e., At) could not exceed the threshold or peak load of on-site supply.


## 3.1. Duration, Cost, and Resource Model

The multiple variables to be optimized were the total duration (TD), the total cost (TC), and resource fluctuation (RF), under the constraint of on-site power load, precedence relationships, and resource availability. In this study, the number of construction activities was J (1, 2, …,j, …, J). Each activity j could be performed in different methods (modes) Mj Mj=1,2,…,m,…,Mj, and only one mode m could be chosen as the execution mode for j. Once the execution mode of each activity was selected, the corresponding duration, cost, resources requirements, and power load generated by a set of equipment were determined [21]. Parts of the TD, TC, and RF model were based on the study of Ghoddousi et al. [15]:(1)TD=max:fj,fj−∑m∈Mjxjm×djm≥fi,TC=∑j∑m∈Mjxjm×cjm+TD×cind+y×cp×TD−Tcontract,RF=∑k=1K∑t=1TDrkt−rk¯2,rk¯=1TD∑t=1TDrkt,where fj is the finish time of activity j in mode m; xjm is a decision variable which equals to 1 when activity j is executed in mode m, otherwise xjm equals to 0; djm is the duration of j in mode m; i are the predecessors activities of j. cjm is the direct cost when activity j performs in mode m; cind is the indirect cost which is a constant amount; Tcontract is the project deadline that is defined in the contract; cp is a penalty if delay happens; and y is a decision variable which equals to 1 when TD>Tcontract, otherwise y equals to 0. The squared deviation is used to evaluate RF; k represents resource type; rkt is the resource usage of k in period t∈1,2,…,TD; and rk¯ is the average resource usage.


## 3.2. On-Site Power Load Constraints

The power supply system was composed of a set of loops (l = 1, 2, …, L), where different construction equipment was connected to different loops. Two constraint conditions were proposed: (1) for each time instant t and for each loop l, the total power load of loop l is not allowed to exceed Ppeakl, which is the threshold or peak load of loop l. (2) For each time instant t, the power load of all loops is not allowed to exceed Ppeak, which means the threshold or peak load of the on-site power supply system. For example, three activities named 1, 2, and 3 were currently processed at time instant t. Their optional execution mode and the corresponding electric construction equipment are listed in Table 2, respectively. Assuming these three activities are all executed in mode 1, the equipment set, as well as the loop each equipment classification belongs to, can be identified as follows: a-loop 1, b-loop 2, c-loop 1, e-loop 2, f-loop 3, g-loop 1, h-loop 3, and i-loop 1. According to the power load constraints, a set of functions are provided as(2)Pa1+Pc1+Pg1+Pi1=P1≤Ppeak1,Pb2+Pe2=P2≤Ppeak2,Pf3+Ph3=P3≤Ppeak3,∑l=13Pl+Pextra≤Ppeak,where Pa1 indicates the power load of equipment a in loop 1 and so on. In general, Pa1=Prate−a/ε, where Prate−a indicates rate power of a; ε indicates the loss factor (ε=0.9 for lighting equipment and ε=0.8 for construction equipment); Ppeak1 indicates the threshold or peak load of loop 1, and so on. Pextra indicates the extra power supply by the on-site electric generator.Table 2
Example of on-site power load constraint.
Act ID
Execution mode
Equipment
Loop
Rate power
Loss factorε
1
1
a
1
P
rate
−
a
0.8
b
2
P
rate
−
b
0.8
c
1
P
rate
−
c
0.8
2
d
1
P
rate
−
d
0.8
c
1
P
rate
−
c
0.8
2
1
e
2
P
rate
−
e
0.8
f
2
P
rate
−
f
0.8
3
1
g
1
P
rate
−
g
0.8
h
3
P
rate
−
h
0.8
i
1
P
rate
−
i
0.8
2
g
1
P
rate
−
g
0.8
j
3
P
rate
−
j
0.8
Two power load constraints were presented as follows:(3)
∑
j
∈
A
t
∑
m
∈
M
j
x
j
m
×
P
j
m
l
≤
P
peak
l
,
l
=
1
,
2
,
…
,
L
,
A
t
=
j
∣
f
j
−
d
j
<
t
≤
f
j
,
(4)
∑
l
=
1
L
∑
j
∈
A
t
∑
m
∈
M
j
x
j
m
×
P
j
m
l
+
P
extra
≤
P
peak
,
l
=
1,2
,
…
,
L
,
A
t
=
j
∣
f
j
−
d
j
<
t
≤
f
j
.Constraint set (3) indicated that for each time instant t and for each loop l, the power load of the activities which were currently processed (i.e., At) and connected to loop l could not exceed the threshold or peak load of loop l, where Pjml was the power load of loop l generated by activity j, if it was executed in mode m. Constraint set (4) indicated that for each time instant t, the total power load of the activities which were currently processed (i.e., At) could not exceed the threshold or peak load of on-site supply.


## 4. Development of Shuffled Frog-Leaping Algorithm for TCRTO Problems

This effort was to optimize the time-cost-resource trade-off formulated as a multiobjective optimization problem that searches for nondominated schedules to minimize the TD, TC, and RF simultaneously under the constraints of precedence relationship, resource availability, and on-site power load. In this nondominated multiobjective scenario, some of the defined criteria were in conflict with each other, i.e., an improvement in one objective could only be achieved by sacrificing improvement in another. Moreover, some criteria were not commensurable to predefine preferences among the criteria. As a result, the fitness value, which was the key parameter in SFLA, could not be determined for each individual. Due to these reasons, this study developed a modified SFLA method by integrating the advantages of an elitist nondominated sorting genetic algorithm (NSGA II) to search a set of Pareto optimal solutions. To successfully use the developed SFLA method in TCRTO problem, eight key steps were followed: (a) data gathering and setting up preliminary requirements of the model; (b) selection of SFLA parameters; (c) solution encoding; (d) forming Pareto front based on domination relation; (e) partitioning frogs into memeplexes; (f) memetic evolution; (g) shuffled complex evolution (SCE); and (h) iteration. These steps were further specified as follows.


### 4.1. Data Gathering and Setting Up Model Preliminary Requirements

The setting up of project parameters included defining the number of activities, precedents and successors of activities, resources and their availability in each period during project execution, execution modes for each activity with corresponding durations, resource requirements and direct costs, the project deadline, and the penalty for the delay as defined in the contract. Further, the project parameters included a definition of the equipment used in each activity and their corresponding rate power, peak power load values of each loop, and the total circuit load.


### 4.2. Selection of SFLA Parameters

This stage comprised the setting of the size of population F and setting the iteration number for global exploration and local exploration.


### 4.3. Solution Encoding

The information reflected in each solution or “frog” representation should contain two aspects: (1) the activities’ order for the scheduling process and (2) the mode assignment for activities’ execution. With this in mind, this step considered a chromosome representation used by Ghoddousi [15]. Each individual was represented by a double list: an activity list (AL) and a mode assignment list (MSL). The AL was a precedence feasible list of the activities. The MSL showed the mode selected for each of the activities. Figure 1 shows an example of the structure of a chromosome for a project instance with seven activities.Figure 1
Structure of solution.


### 4.4. Forming Pareto Front Based on Domination Relation

TD, TC, and RF were calculated for every solution in the population F, according to the mathematical model developed by this study. The combined population F was sorted into nondominated fronts (F1,…,Fn) according to the fast nondominated sorting procedure, the first front being a completely nondominant set in the current population, the second front being dominated by the individuals in the first front only, and the fronts go on. Fitness values of these frogs were given based on the fronts they are allocated to (e.g., frogs in the first front were given a fitness value of 1).


### 4.5. Partitioning Frogs into Memeplexes

SolutionsF = m ∗ n were randomly partitioned into several memeplexes (m), each consisting of an equal number of solutions (n). Each memeplex evolved independently. Figure 2 shows the memeplex partition process.Figure 2
Schematic illustration of memeplex partitioning process [10].However, in the nondomination multiobjective optimization scenario, the best frog or solution was not unique, since solutions in the Pareto front 1 represented all the best solutions. The random assignment could cause the problem that the best frogs were assigned into the same memeplex and the worst frogs were assigned to another memeplex, and thus it was meaningless to continue the memetic evolution. Therefore, an improved allocation method was introduced as follows: (a) determining the number of memeplexes (m) by using the number of frogs in front X, which contains the largest number of frogs (the number of total fronts was the frogs in each memeplex (n)), and (b) sorting frogs. The first frog in front 1, the first frog in front 2, …, the first frog in front N were assigned to the first memeplex; the second frog in front 1, the second frog in front 2, …, the second frog in front N are assigned to the second memeplex, and so on. The partition process would not stop until the last frog is assigned in the Nth memeplex. If there were not enough frogs to fill the memeplex, new frogs would be generated randomly to facilitate memetic evolution.For example, F contains 12 frogs and forms 3 Pareto front based on domination relation, namely,F1 = {U1, U2, U3, U4, U5,}, F3 = {U6, U7, U8}, and F2 = {U9, U10, U11, U12}. F1 represents the best front and U1, U2, U3, U4, and U5 are nondominant frogs set in front 1, F2 represents the moderate front and U6, U7, and U8 are nondominant frogs set in front 2, and F3 represents the worst front and U9, U10, U11, and U12 are nondominant frogs set in front 3. Since F1 has the largest number of frogs among all fronts, the number of memeplex can be determined as 5. Each memeplex contains 3 frogs because there are 3 fronts. Thus, the allocation results (memeplexes) can be shown as: {U1, U6, U9}, {U2, U7, U10}, {U3, U8, U11}, {U4, Ui, U12}, and {U5, Uj, Uk}, where Ui, Uj, and Uk are new randomly generated solutions. This improved method solves the difficulty of parameter setting. Figure 3 presents the improved method for the memeplex partitioning process.Figure 3
The improved method for memeplex partitioning process.


### 4.6. Memetic Evolution

The goal of this process was to improve the quality of feasible solutions through “leaping.” The distance that the worst frog “leaps” was(5)Si=rand0,1×Pb−Pw,where Si is the “leaping” distance and Pb and Pw are the best and worst frogs in this memeplex, respectively. Rand (0,1) is a random value between 0 and 1. The new position of the worst frog Pwk+1 is then computed by(6)Pwk+1=Pwk+Si.If this “leaping” does not improve the performance of the worst frog, the “leaping” distance will be recalculated by using the best frog of the whole populationPg, instead of Pb. If the position of Pw has not changed, a new frog will be generated randomly to replace the old Pw.As solutions for this TCROT optimization problem must be a set of integers, the “leaping” distance calculated by the method presented above may be nonintegers due to “rand (0,1).” In addition, the structure of the solution, as shown in Figure1, is not suitable for emetic evolution formulas, since the activities list, as well as the mode assignment list, cannot be pulsed and subtracted.Therefore, an adaptive evolution method was developed by integrating genetic operators and the one-point crossover method introduced by Hartmann considering the activity list concept, precedence feasibility, and the mode assignment [41]. Specifically, the best frog Pb and the worst frog Pw were selected in a memeplex and integer q1 with 1 ≤ q1 ≤ J is randomly chosen. The two “improved” frogs (i.e., Pimp1 and Pimp2) activity lists (ALs) and mode lists were generated. The q1 positions in Pimp1 structure were inherited from Pb. The positions from q1 + 1 to J were inherited from Pw by a left-to-right scan. In the same way, the q1 + 1 to J position of Pb and 1 to q1 position of Pw comprised Pimp2. These two “improved” frogs were then compared with Pw, and the best solution was selected as the new frog. If Pimp1 and Pimp2 were not superior to Pw, another best frog in the population (Pg) could be chosen to replace Pb to repeat the evolution process and another two “improved” frogs, Pimp3 and Pimp4, are generated. If Pimp3 and Pimp4 were still not superior to Pw, another new frog could be randomly generated to replace Pw. Figure 4 shows this adaptive memetic evolution method.Figure 4
The adaptive memetic evolution method.


### 4.7. Shuffled Complex Evolution (SCE)

This step mixed the memeplexes that completed memetic evolution and new memeplexes were generated through the shuffling process. New frogs were randomly generated in stepse and f; thus, the number of total frogs was beyond the initial population size F. Redundant frogs, which were identified as the worst frogs based on new Pareto front formation, could be eliminated to enable the shuffled evolution process. This shuffling enhanced the quality of the memes.


### 4.8. Iteration

This stage involved repeating Stepsd–g until no new project planning solutions were found (i.e., when the new population set of project planning options was equal to the current set of project planning options), and the final population set represented a Pareto optimal set of project planning solutions for the TCRTO problem. Figure 5 shows the flow chart of the modified SFLA.Figure 5
Flow chart of developed shuffled frog-leaping algorithm.


## 4.1. Data Gathering and Setting Up Model Preliminary Requirements

The setting up of project parameters included defining the number of activities, precedents and successors of activities, resources and their availability in each period during project execution, execution modes for each activity with corresponding durations, resource requirements and direct costs, the project deadline, and the penalty for the delay as defined in the contract. Further, the project parameters included a definition of the equipment used in each activity and their corresponding rate power, peak power load values of each loop, and the total circuit load.


## 4.2. Selection of SFLA Parameters

This stage comprised the setting of the size of population F and setting the iteration number for global exploration and local exploration.


## 4.3. Solution Encoding

The information reflected in each solution or “frog” representation should contain two aspects: (1) the activities’ order for the scheduling process and (2) the mode assignment for activities’ execution. With this in mind, this step considered a chromosome representation used by Ghoddousi [15]. Each individual was represented by a double list: an activity list (AL) and a mode assignment list (MSL). The AL was a precedence feasible list of the activities. The MSL showed the mode selected for each of the activities. Figure 1 shows an example of the structure of a chromosome for a project instance with seven activities.Figure 1
Structure of solution.


## 4.4. Forming Pareto Front Based on Domination Relation

TD, TC, and RF were calculated for every solution in the population F, according to the mathematical model developed by this study. The combined population F was sorted into nondominated fronts (F1,…,Fn) according to the fast nondominated sorting procedure, the first front being a completely nondominant set in the current population, the second front being dominated by the individuals in the first front only, and the fronts go on. Fitness values of these frogs were given based on the fronts they are allocated to (e.g., frogs in the first front were given a fitness value of 1).


## 4.5. Partitioning Frogs into Memeplexes

SolutionsF = m ∗ n were randomly partitioned into several memeplexes (m), each consisting of an equal number of solutions (n). Each memeplex evolved independently. Figure 2 shows the memeplex partition process.Figure 2
Schematic illustration of memeplex partitioning process [10].However, in the nondomination multiobjective optimization scenario, the best frog or solution was not unique, since solutions in the Pareto front 1 represented all the best solutions. The random assignment could cause the problem that the best frogs were assigned into the same memeplex and the worst frogs were assigned to another memeplex, and thus it was meaningless to continue the memetic evolution. Therefore, an improved allocation method was introduced as follows: (a) determining the number of memeplexes (m) by using the number of frogs in front X, which contains the largest number of frogs (the number of total fronts was the frogs in each memeplex (n)), and (b) sorting frogs. The first frog in front 1, the first frog in front 2, …, the first frog in front N were assigned to the first memeplex; the second frog in front 1, the second frog in front 2, …, the second frog in front N are assigned to the second memeplex, and so on. The partition process would not stop until the last frog is assigned in the Nth memeplex. If there were not enough frogs to fill the memeplex, new frogs would be generated randomly to facilitate memetic evolution.For example, F contains 12 frogs and forms 3 Pareto front based on domination relation, namely,F1 = {U1, U2, U3, U4, U5,}, F3 = {U6, U7, U8}, and F2 = {U9, U10, U11, U12}. F1 represents the best front and U1, U2, U3, U4, and U5 are nondominant frogs set in front 1, F2 represents the moderate front and U6, U7, and U8 are nondominant frogs set in front 2, and F3 represents the worst front and U9, U10, U11, and U12 are nondominant frogs set in front 3. Since F1 has the largest number of frogs among all fronts, the number of memeplex can be determined as 5. Each memeplex contains 3 frogs because there are 3 fronts. Thus, the allocation results (memeplexes) can be shown as: {U1, U6, U9}, {U2, U7, U10}, {U3, U8, U11}, {U4, Ui, U12}, and {U5, Uj, Uk}, where Ui, Uj, and Uk are new randomly generated solutions. This improved method solves the difficulty of parameter setting. Figure 3 presents the improved method for the memeplex partitioning process.Figure 3
The improved method for memeplex partitioning process.


## 4.6. Memetic Evolution

The goal of this process was to improve the quality of feasible solutions through “leaping.” The distance that the worst frog “leaps” was(5)Si=rand0,1×Pb−Pw,where Si is the “leaping” distance and Pb and Pw are the best and worst frogs in this memeplex, respectively. Rand (0,1) is a random value between 0 and 1. The new position of the worst frog Pwk+1 is then computed by(6)Pwk+1=Pwk+Si.If this “leaping” does not improve the performance of the worst frog, the “leaping” distance will be recalculated by using the best frog of the whole populationPg, instead of Pb. If the position of Pw has not changed, a new frog will be generated randomly to replace the old Pw.As solutions for this TCROT optimization problem must be a set of integers, the “leaping” distance calculated by the method presented above may be nonintegers due to “rand (0,1).” In addition, the structure of the solution, as shown in Figure1, is not suitable for emetic evolution formulas, since the activities list, as well as the mode assignment list, cannot be pulsed and subtracted.Therefore, an adaptive evolution method was developed by integrating genetic operators and the one-point crossover method introduced by Hartmann considering the activity list concept, precedence feasibility, and the mode assignment [41]. Specifically, the best frog Pb and the worst frog Pw were selected in a memeplex and integer q1 with 1 ≤ q1 ≤ J is randomly chosen. The two “improved” frogs (i.e., Pimp1 and Pimp2) activity lists (ALs) and mode lists were generated. The q1 positions in Pimp1 structure were inherited from Pb. The positions from q1 + 1 to J were inherited from Pw by a left-to-right scan. In the same way, the q1 + 1 to J position of Pb and 1 to q1 position of Pw comprised Pimp2. These two “improved” frogs were then compared with Pw, and the best solution was selected as the new frog. If Pimp1 and Pimp2 were not superior to Pw, another best frog in the population (Pg) could be chosen to replace Pb to repeat the evolution process and another two “improved” frogs, Pimp3 and Pimp4, are generated. If Pimp3 and Pimp4 were still not superior to Pw, another new frog could be randomly generated to replace Pw. Figure 4 shows this adaptive memetic evolution method.Figure 4
The adaptive memetic evolution method.


## 4.7. Shuffled Complex Evolution (SCE)

This step mixed the memeplexes that completed memetic evolution and new memeplexes were generated through the shuffling process. New frogs were randomly generated in stepse and f; thus, the number of total frogs was beyond the initial population size F. Redundant frogs, which were identified as the worst frogs based on new Pareto front formation, could be eliminated to enable the shuffled evolution process. This shuffling enhanced the quality of the memes.


## 4.8. Iteration

This stage involved repeating Stepsd–g until no new project planning solutions were found (i.e., when the new population set of project planning options was equal to the current set of project planning options), and the final population set represented a Pareto optimal set of project planning solutions for the TCRTO problem. Figure 5 shows the flow chart of the modified SFLA.Figure 5
Flow chart of developed shuffled frog-leaping algorithm.


## 5. Case Study

To assess the effectiveness of the developed SFLA method, a simplified construction project case with 18 activities was selected to implement the model. In addition, a comparison study of optimization results between the SFLA method and a common existing optimization method, nondominated sorting genetic algorithm, was conducted. Both methods were used to solve project planning problems and to find Pareto optimal project planning solutions. The SFLA algorithm was coded in MATLAB R2016a. Results showed the superiority of the proposed method.The activity network of the project is shown in Figure6, and project information is shown in Table 3. In this case study, different activities had different execution modes from which to select, and each activity j could be executed in only one mode without splitting. For each mode, corresponding duration, cost and resource requirements, and power loads were presented. The workforce resource was included in the optimization problem and its per period availability was 75. The indirect cost of this project was $250. The contract project deadline was 250 days from commencement, and the contractor must pay another $550 per day when a delay occurred. The threshold of each electric loop was 60 kW and the peak power load value was 188 kW, with an extra generated power supply of 8 kW by an on-site power generator.Figure 6
Project network activities in the case study.Table 3
Activity data of case study.
Activity number
Preceding activities
Execution mode
Time (days)
Cost ($)
Resource (people)
Equipment type
Equipment number
Rated power (kW)
Electricity loop
1
1
14
2400
8
Eqpt. 1
1
5.5
1
Eqpt. 2
1
4
2
2
15
2150
10
Eqpt. 1
1
5.5
1
Eqpt. 3
1
3.5
2
3
16
1900
4
Eqpt. 4
2
6
3
2
1
15
3000
12
2
18
2400
14
3
1
15
4500
18
Eqpt. 5
1
7.5
2
Eqpt. 6
1
3
1
2
22
4000
15
Eqpt. 5
1
7.5
2
Eqpt. 7
1
4
3
4
1
12
45000
14
Eqpt. 8
1
32
1
2
16
35000
12
Eqpt. 8
1
32
1
5
1
1
22
20000
7
6
1
1
14
40000
21
Eqpt. 8
1
32
1
Eqpt. 9
2
6.5
2
2
18
32000
18
Eqpt. 8
1
32
1
Eqpt. 9
2
6.5
2
3
24
18000
16
Eqpt. 10
1
24.5
3
Eqpt. 9
2
6.5
2
7
5
1
9
30000
19
Eqpt. 11
1
6
2
2
15
24000
16
Eqpt. 11
1
6
2
8
6
1
14
220
8
2
15
215
7
3
16
200
5
9
6
1
15
300
11
2
18
240
9
10
2, 6
1
15
450
12
Eqpt. 8
1
32
1
Eqpt. 9
1
6.5
2
2
22
400
13
Eqpt. 8
1
32
1
Eqpt. 9
6.5
2
3
33
320
14
Eqpt. 8
1
32
1
Eqpt. 9
1
6.5
2
11
7, 8
1
12
450
4
12
5, 9, 10
1
22
2000
15
Eqpt. 4
1
6
3
Eqpt. 3
1
3.5
2
2
24
1750
14
Eqpt. 4
1
6
3
Eqpt. 3
1
3.5
2
13
3
1
14
4000
20
2
18
3200
22
3
24
1800
23
14
4, 10
1
9
3000
15
2
15
2400
17
3
18
2200
17
15
12
1
16
3500
9
Eqpt. 8
1
32
1
Eqpt. 2
1
4
2
16
13, 14
1
20
3000
10
Eqpt. 8
2
32
1
2
22
2000
11
Eqpt. 10
2
24.5
3
3
24
1750
14
Eqpt. 10
2
24.5
3
17
11, 14, 15
1
14
4000
12
2
18
3200
14
3
24
1800
17
18
16, 17
1
9
3000
13
Eqpt. 12
1
9.2
2
2
15
2400
15
Eqpt. 12
1
9.2
2


### 5.1. Implementation of SFLA

By providing a “time attribute” and “optimal gap” to every activity, the developed SFLA method enabled decision-makers to quickly select effective solutions and to extend the solution range. The “time attribute” defined the early start time and duration of activityj in mode m. A traversal approach was adopted in the MATLAB coding process to calculating power load and resource usage at every time instance without the requirement of generating new network diagrams or resource usage histograms, which speeded up the optimization process. The traversal approach only needs to check the activities set at each time instance and calculate the corresponding value. This process is presented in Figure 7, where the broken lines represent the sequence of calculation. In a certain solution, j1 (from day 6 to 10), j2 (from day 10 to 20), j3 (from day 10 to 24), and j4 (from day 5 to 13) represent different activities with their “time attribute.” When the traversal process arrived at t = 8, only the activities j1 and j4 should be considered. When t = 12, the resource requirement value and power load of j2 and j3 and j4 should be calculated. Further, j2 and j3 should be calculated when t = 15. Instead of providing solutions that did not satisfy the constraints in TCRTO mathematic model (e.g., exceeding resource availability in a time instance) after initial traversal calculation, SFLA converted this invalid value to a valid value by automatically delaying the start time of some activities to balance the power load or resource usage. The delay value “optimal gap” was randomly generated by SFLA so that more effective solutions could be found. For these initial effective solutions (combinations), “optimal gap” was also used to identify better Pareto fronts. There were three advantages in seeking effective solutions using the SFLA: (a) a faster calculation process by using the traversal approach, (b) broader solution range by converting invalid values to valid values, and (c) better optimization solutions by using the “optimal gap” to facilitate time, cost, and resource fluctuation trade-offs.Figure 7
Process of seeking effective solutions in SFLA.Figures8(a)–8(e) show the process of fast convergence towards the Pareto optimal solutions set, while maintaining diversity in solutions through increasing iteration times. Seven nondominated solutions were identified after the 5th iteration; 9 solutions were identified after the 10th iteration; 13 solutions were identified after the 20th iteration; and 17 solutions were identified after the 50th iteration. It should be mentioned that all 19 solutions were found in the 100th iteration, where Pareto front solutions did not change and convergence criteria were reached. Only approximately 37% of nondominated solutions were achieved after the 5th iteration. After the 10th iteration, the percentage of the nondominated solution was enhanced to 47%, where dominated solutions were eliminated through iterations and replaced by new-found nondominated solutions. Respectively, 60% solutions and 89% solutions were found after 20th and 50th iterations. Finally, by reaching the maximum iteration number of 100 as a termination condition, all nondominated solutions were found and the optimal frog population was formed.Figure 8
Optimal solutions generated by SFLA. (a) Nondominated solutions after 5th generation. (b) Nondominated solutions after 10th generation. (c) Nondominated solutions after 20th generation. (d) Nondominated solutions after 50th generation. (e) Pareto front solutions of the TCRTO model after 100th generation.
(a)
(b)
(c)
(d)
(e)Schedules generated by the modified SFLA and their corresponding time, cost, and resource moment fluctuation are presented in Table4. The distribution of 19 nondominated points is shown in Figure 8(e), in a three-dimensional space after the 100th iteration. The range of time deviation in Pareto front solutions of the TCROTO model was 141 to 196. The range of cost deviation was $151540 to $158618, and the range of resource moment deviation was 3552.20 to 5156.86. In addition to the optimization results, information of these 19 best solutions in the Pareto front can also be obtained. For example, Table 5 shows the specific information of solution no. 1, including the execution mode combination and the start time of each activity.Table 4
Optimal solutions for case study.
Solution
Total duration
Total cost
Resource fluctuation
Solutions of SFLA
1
162
189665
4536.24
2
178
200640
3716.53
3
155
190965
5204.40
4
169
220420
3802.89
5
145
207850
4935.05
6
180
199640
3561.16
7
157
213350
4222.84
8
164
214605
4128.29
9
167
195800
4404.09
10
151
203850
4668.06
11
141
216100
5294.22
12
157
195300
4989.14
13
163
193510
4594.50
14
159
210190
4305.84
15
163
207140
4615.77
16
147
193350
5033.71
17
177
195020
4175.24
18
175
197225
4008.82
19
151
208305
4258.49
20
172
195700
4403.29
21
205
215760
3464.22
22
158
189120
4219.71
23
142
199600
4986.46
24
169
201510
4151.20
25
146
195005
5136.67
26
165
207405
4153.21
27
149
201580
4906.96
28
160
201080
4846.13
29
161
188840
4917.31
30
180
195700
4257.91
31
149
208190
4607.44
Solutions of NSGA II
1
183
201210
4033.26
2
155
188740
4806.04
3
179
223555
3761.24
4
143
224860
5170.60
5
145
225655
5114.27
6
143
224860
5245.04
7
163
207790
4725.61
8
170
196090
4638.32
9
170
220005
4156.89
10
152
214455
5001.21Table 5
Specific information of solution no. 1
Activity number
Start time
Execution mode
1
23
1
2
9
2
3
14
2
4
8
1
5
39
3
6
45
2
7
75
2
8
69
2
9
76
1
10
74
1
11
106
1
12
91
1
13
37
3
14
98
1
15
113
1
16
123
3
17
129
3
18
153
1Data analysis in two dimensions is presented in Figure9 including the relationship between “TD-TC,” “TD-RF,” and “TC-RF.” An interesting phenomenon occurs where the best solutions may contradict each other. For example, considering solutions no. 2 (178 days, $200,640) and no. 3 (155 days, $190,965), solution no. 2, with a large duration (178 days) and greater cost ($200,640), is still considered as the optimal solution. This is because the results cannot be interpreted by considering only time and cost, while ignoring the third dimension resource variation. Compared with no. 3 whose RF is 5,204.40, no. 2 remains in optimal population since it has a smaller resource moment deviation (3,716.53) and thus has a smoother resource usage histogram. That is also the reason why there is no obvious trend or regularity between any two objectives.Figure 9
Optimal solutions in two-dimensional space.
(a)
(b)
(c)


### 5.2. Comparison Study of Modified SFLA and NSGA II

A comparison study between the modified SFLA method and nondominated sorting genetic algorithm (NSGA II) was conducted with respect to the simultaneous minimization of TD, total TC, and RF. NSGA II was selected as a comparative method because it is a popular and widely used nondominate-based algorithm for multiobjective optimization [42]. The optimal scheduling solutions obtained by NSGA II and the solutions obtained by the SFLA are presented in the same three-dimensional space in Figure 10, where the number of solutions generated by SFLA are larger than that of NSGA II (i.e., 31 solutions were generated by the modified SFLA, while 10 solutions were generated by NSGA II). To demonstrate the advantage of the proposed algorithm, the data analysis is presented in Figure 11, where optimal project planning solutions derived by the two algorithms are displayed in the two-dimensional space of project planning objectives. The results show that the modified SFLA can find optimal solutions with the shortest TD (i.e., 141 days compared to 143 days obtained by the NSGA II algorithm) and less total variation of resources allocation (i.e., 3,464.22 compared to 3,761.24 obtained by the NSGA II algorithm). Although the slightly lower total project cost ($ 188,740) was obtained by NSGA II in comparison to the SFLA ($188,840), more solutions with lower cost (and less duration) were obtained by the SFLA. For example, in the “TD-TC” relationship in Figure 11, when the duration is less than 150 days, solutions obtained by SFLA were superior to that of NSGA II, due to the lower cost. In “TD-RF” and “TC-RF” relationships, it is clear that with the same, or near the same, value on the X-axis, the position of most “blue points” is beyond the lowest “red points.” This indicates that in relation to duration or cost solutions, the SFLA can always find solutions that have a lower resource fluctuation than that obtained by NSGA II. The two most significant contributions of the proposed algorithm over NSGA II are that the modified SFLA can find better solutions for multiobjective problems and is capable of finding additional optimal project scheduling solutions. This is because more possible frogs (solutions) have the chance to participate in memeplex evolution, due to the improved method to convert some invalid solutions into effective solutions by using the “optimal gap,” and as a result, more solutions are obtained. Instead of random evolution used in NSGA II, memetic evolution in SFLA ensures “worst solutions” can “leap” to a better position in every evolution process, which enables the identification of more optimal solutions.Figure 10
Optimal solutions generated by SFLA and NSGA II.Figure 11
Solution comparison in two-dimensional space.
(a)
(b)
(c)


## 5.1. Implementation of SFLA

By providing a “time attribute” and “optimal gap” to every activity, the developed SFLA method enabled decision-makers to quickly select effective solutions and to extend the solution range. The “time attribute” defined the early start time and duration of activityj in mode m. A traversal approach was adopted in the MATLAB coding process to calculating power load and resource usage at every time instance without the requirement of generating new network diagrams or resource usage histograms, which speeded up the optimization process. The traversal approach only needs to check the activities set at each time instance and calculate the corresponding value. This process is presented in Figure 7, where the broken lines represent the sequence of calculation. In a certain solution, j1 (from day 6 to 10), j2 (from day 10 to 20), j3 (from day 10 to 24), and j4 (from day 5 to 13) represent different activities with their “time attribute.” When the traversal process arrived at t = 8, only the activities j1 and j4 should be considered. When t = 12, the resource requirement value and power load of j2 and j3 and j4 should be calculated. Further, j2 and j3 should be calculated when t = 15. Instead of providing solutions that did not satisfy the constraints in TCRTO mathematic model (e.g., exceeding resource availability in a time instance) after initial traversal calculation, SFLA converted this invalid value to a valid value by automatically delaying the start time of some activities to balance the power load or resource usage. The delay value “optimal gap” was randomly generated by SFLA so that more effective solutions could be found. For these initial effective solutions (combinations), “optimal gap” was also used to identify better Pareto fronts. There were three advantages in seeking effective solutions using the SFLA: (a) a faster calculation process by using the traversal approach, (b) broader solution range by converting invalid values to valid values, and (c) better optimization solutions by using the “optimal gap” to facilitate time, cost, and resource fluctuation trade-offs.Figure 7
Process of seeking effective solutions in SFLA.Figures8(a)–8(e) show the process of fast convergence towards the Pareto optimal solutions set, while maintaining diversity in solutions through increasing iteration times. Seven nondominated solutions were identified after the 5th iteration; 9 solutions were identified after the 10th iteration; 13 solutions were identified after the 20th iteration; and 17 solutions were identified after the 50th iteration. It should be mentioned that all 19 solutions were found in the 100th iteration, where Pareto front solutions did not change and convergence criteria were reached. Only approximately 37% of nondominated solutions were achieved after the 5th iteration. After the 10th iteration, the percentage of the nondominated solution was enhanced to 47%, where dominated solutions were eliminated through iterations and replaced by new-found nondominated solutions. Respectively, 60% solutions and 89% solutions were found after 20th and 50th iterations. Finally, by reaching the maximum iteration number of 100 as a termination condition, all nondominated solutions were found and the optimal frog population was formed.Figure 8
Optimal solutions generated by SFLA. (a) Nondominated solutions after 5th generation. (b) Nondominated solutions after 10th generation. (c) Nondominated solutions after 20th generation. (d) Nondominated solutions after 50th generation. (e) Pareto front solutions of the TCRTO model after 100th generation.
(a)
(b)
(c)
(d)
(e)Schedules generated by the modified SFLA and their corresponding time, cost, and resource moment fluctuation are presented in Table4. The distribution of 19 nondominated points is shown in Figure 8(e), in a three-dimensional space after the 100th iteration. The range of time deviation in Pareto front solutions of the TCROTO model was 141 to 196. The range of cost deviation was $151540 to $158618, and the range of resource moment deviation was 3552.20 to 5156.86. In addition to the optimization results, information of these 19 best solutions in the Pareto front can also be obtained. For example, Table 5 shows the specific information of solution no. 1, including the execution mode combination and the start time of each activity.Table 4
Optimal solutions for case study.
Solution
Total duration
Total cost
Resource fluctuation
Solutions of SFLA
1
162
189665
4536.24
2
178
200640
3716.53
3
155
190965
5204.40
4
169
220420
3802.89
5
145
207850
4935.05
6
180
199640
3561.16
7
157
213350
4222.84
8
164
214605
4128.29
9
167
195800
4404.09
10
151
203850
4668.06
11
141
216100
5294.22
12
157
195300
4989.14
13
163
193510
4594.50
14
159
210190
4305.84
15
163
207140
4615.77
16
147
193350
5033.71
17
177
195020
4175.24
18
175
197225
4008.82
19
151
208305
4258.49
20
172
195700
4403.29
21
205
215760
3464.22
22
158
189120
4219.71
23
142
199600
4986.46
24
169
201510
4151.20
25
146
195005
5136.67
26
165
207405
4153.21
27
149
201580
4906.96
28
160
201080
4846.13
29
161
188840
4917.31
30
180
195700
4257.91
31
149
208190
4607.44
Solutions of NSGA II
1
183
201210
4033.26
2
155
188740
4806.04
3
179
223555
3761.24
4
143
224860
5170.60
5
145
225655
5114.27
6
143
224860
5245.04
7
163
207790
4725.61
8
170
196090
4638.32
9
170
220005
4156.89
10
152
214455
5001.21Table 5
Specific information of solution no. 1
Activity number
Start time
Execution mode
1
23
1
2
9
2
3
14
2
4
8
1
5
39
3
6
45
2
7
75
2
8
69
2
9
76
1
10
74
1
11
106
1
12
91
1
13
37
3
14
98
1
15
113
1
16
123
3
17
129
3
18
153
1Data analysis in two dimensions is presented in Figure9 including the relationship between “TD-TC,” “TD-RF,” and “TC-RF.” An interesting phenomenon occurs where the best solutions may contradict each other. For example, considering solutions no. 2 (178 days, $200,640) and no. 3 (155 days, $190,965), solution no. 2, with a large duration (178 days) and greater cost ($200,640), is still considered as the optimal solution. This is because the results cannot be interpreted by considering only time and cost, while ignoring the third dimension resource variation. Compared with no. 3 whose RF is 5,204.40, no. 2 remains in optimal population since it has a smaller resource moment deviation (3,716.53) and thus has a smoother resource usage histogram. That is also the reason why there is no obvious trend or regularity between any two objectives.Figure 9
Optimal solutions in two-dimensional space.
(a)
(b)
(c)


## 5.2. Comparison Study of Modified SFLA and NSGA II

A comparison study between the modified SFLA method and nondominated sorting genetic algorithm (NSGA II) was conducted with respect to the simultaneous minimization of TD, total TC, and RF. NSGA II was selected as a comparative method because it is a popular and widely used nondominate-based algorithm for multiobjective optimization [42]. The optimal scheduling solutions obtained by NSGA II and the solutions obtained by the SFLA are presented in the same three-dimensional space in Figure 10, where the number of solutions generated by SFLA are larger than that of NSGA II (i.e., 31 solutions were generated by the modified SFLA, while 10 solutions were generated by NSGA II). To demonstrate the advantage of the proposed algorithm, the data analysis is presented in Figure 11, where optimal project planning solutions derived by the two algorithms are displayed in the two-dimensional space of project planning objectives. The results show that the modified SFLA can find optimal solutions with the shortest TD (i.e., 141 days compared to 143 days obtained by the NSGA II algorithm) and less total variation of resources allocation (i.e., 3,464.22 compared to 3,761.24 obtained by the NSGA II algorithm). Although the slightly lower total project cost ($ 188,740) was obtained by NSGA II in comparison to the SFLA ($188,840), more solutions with lower cost (and less duration) were obtained by the SFLA. For example, in the “TD-TC” relationship in Figure 11, when the duration is less than 150 days, solutions obtained by SFLA were superior to that of NSGA II, due to the lower cost. In “TD-RF” and “TC-RF” relationships, it is clear that with the same, or near the same, value on the X-axis, the position of most “blue points” is beyond the lowest “red points.” This indicates that in relation to duration or cost solutions, the SFLA can always find solutions that have a lower resource fluctuation than that obtained by NSGA II. The two most significant contributions of the proposed algorithm over NSGA II are that the modified SFLA can find better solutions for multiobjective problems and is capable of finding additional optimal project scheduling solutions. This is because more possible frogs (solutions) have the chance to participate in memeplex evolution, due to the improved method to convert some invalid solutions into effective solutions by using the “optimal gap,” and as a result, more solutions are obtained. Instead of random evolution used in NSGA II, memetic evolution in SFLA ensures “worst solutions” can “leap” to a better position in every evolution process, which enables the identification of more optimal solutions.Figure 10
Optimal solutions generated by SFLA and NSGA II.Figure 11
Solution comparison in two-dimensional space.
(a)
(b)
(c)


## 6. Conclusion and Future Work

Uncontrolled on-site electricity power loading can often result in major site disruption including mechanical breakdown, schedule delays, and increased quality risks. As an exploratory effort to improve on-site power load scheduling, a shuffled frog-leaping algorithm (SFLA) approach that is commonly applied in solving complex combinatorial optimization problems was developed in this paper to aid decision-makers in identifying the best Pareto solution for time-cost-resource trade-off (TCRTO) problems under the constraints of precedence, resource availability, and on-site peak electricity power load. Improvements have been made to the SFLA method to suit the research context (i.e., an adaptive frog presentation, a new method for partitioning frogs into memeplexes and an improved memetic evolution process). The effectiveness of the developed SFLA method was verified through a real case and a comparison study between SFLA and a nondominated sorting genetic algorithm (NSGA II) indicating that the developed method could identify a larger amount and a wider range of optimal solutions, establishing the potential and value-addedness of this modified SFLA in delivering better power load optimization. The adoption of this novel SFLA in the planning of on-site power loading can provide strong theoretical and technical support through enhanced energy management solutions to guide construction teams to achieve both financial and environmental objectives.The developed SFLA has limitations. Only precedence, resource availability, and on-site peak electricity power load constraints have been considered. Future research should explore including a wider range of constraints such as materials, working areas, and cash flows to better reflect situational project constraints and improve robustness. Future work may also investigate the application of SFLA to real-time energy consumption optimization through the use of integrated wireless smart meters and wireless networks. Furthermore, extending this SFLA approach to household energy management is also important for more efficient energy consumption.



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*Source: 1404636-2019-05-02.xml*