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Objective-driven and Pareto Front Analysis: Optimizing ١
Time, Cost, and Job-site Movements ٢
Vahid Faghihi, Ph.D.1 ٣
Kenneth F. Reinschmidt, Ph.D.2 ٤
Julian H. Kang, Ph.D.3 ٥
Abstract ٦
Finding the optimized trade-off relationship between cost and time, two important objectives of ٧
construction projects, helps project managers and their teams select a more suitable schedule for ٨
a given project. This trade-off relationship can roughly be estimated using past and cumulative ٩
knowledge, but since the early 1970s, researchers have been working on a systematic and ١٠
mathematical solution to define this relationship more accurately. These researchers have used ١١
different optimization techniques such as the genetic algorithm (GA), ant colony, and fuzzy logic ١٢
to further explore the relationship. ١٣
In the present paper, the authors have used their previously introduced construction schedule ١٤
generator algorithm to present graphical relationships between pre-defined objectives of schedule ١٥
optimizations. The process starts with developing construction schedules from the project’s ١٦
Building Information Model (BIM) as part of the input along with resource data. Then the ١٧
process continues with optimization of all developed construction schedules according to the two ١٨
mentioned objectives along with the introduced job-site movement objective, which ١٩
mathematically helps the sequence of installation be more logical and practical. Finally ٢٠
1 Assistant Professor, Dept. of Civil & Environmental Engineering, Amirkabir University of Technology, Tehran,
Iran. E-mail: svfaghihi@gmail.com
2 Professor of Civil Engineering and J. L. Frank/Marathon Ashland Petroleum LLC Chair in Engineering Project
Management, Zachry Dept. of Civil Engineering, Texas A&M Univ., College Station, TX 77843-3136. E-mail:
kreinschmidt@civil.tamu.edu
3 Associate Professor, Dept. of Construction Science, Texas A&M Univ., 3137 TAMU, College Station, TX E-mail:
juliankang@tamu.edu
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generation of a 3D space for all the created and calculated construction schedules in the form of ٢١
a 3D solution cloud point. These 3D construction schedules show solution cloud points and three ٢٢
Pareto Fronts for the given project. ٢٣
Keywords ٢٤
Pareto Front, Optimization, Genetic Algorithm, Construction Project Scheduling, Building ٢٥
Information Model ٢٦
1 Introduction ٢٧
Extending or shortening a construction project’s duration clearly affects the total construction ٢٨
cost. The most important aspect is how project time and cost are related, and how much a single ٢٩
change in either of them, can effect and change the other one. This means the in-between ٣٠
relationship needs to be formulated and shown graphically in order to bring a better ٣١
understanding of the effects. Several successful attempts have been conducted to show this ٣٢
relationship. Different optimization tools have been applied to find the time-cost relationship of ٣٣
projects (Faghihi, Nejat, Reinschmidt, & Kang, 2015). In most cases, optimization tools that can ٣٤
produce numerous outputs while optimizing the solutions (e.g., genetic algorithm) are selected ٣٥
for this type of research. This feature of having numerous outputs can result in a Pareto Front ٣٦
graph representing the relationship between the defined objectives. Therefore, for each ٣٧
optimization output (project schedule in this context), multiple objective scores are needed. A ٣٨
common problem is whether the original project schedules are comprehensive enough to cover ٣٩
all project elements and needed tasks. It is important to make sure the initial project schedule ٤٠
represents the project well so that the optimization makes sense. The Building Information ٤١
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Model (BIM), on the other hand, contains all the project information in a 3D representation view. ٤٢
This source of project data can be and possibly should be used for the mentioned optimization ٤٣
purposes and to generate project specific time-cost optimization (TCO) graphs and reports. ٤٤
The contribution of the current research is the ability to use the inherent data of a construction ٤٥
project from its BIM to generate the project schedule initially and then find and calculate the ٤٦
relationship between predefined objectives for the given project. Thus, the main purpose of this ٤٧
paper is to use the outputs of the previously developed algorithm to find the relationship between ٤٨
the defined objectives. These objectives are “cost,” “time,” and “job-site movements.” As the ٤٩
first step toward conducting this research of finding objective relationships, the authors extracted ٥٠
and calculated a matrix of constructability relationships between all the elements directly from ٥١
the BIM of the project and called our calculations a matrix of constructability constraints ٥٢
(MoCC). ٥٣
Using the GA and the MoCC as the primary calculation basis for the GA fitness function, the ٥٤
authors developed a method that was able to generate valid construction sequencing of the ٥٥
building structure for the given 3D model (Faghihi, Reinschmidt, & Kang, 2014). By “a valid ٥٦
construction sequence,” the authors imply that all the project elements are scheduled for ٥٧
installation in a way that the structural stability requirements of the building are preserved ٥٨
throughout the construction process. To make the algorithm more mature and complete, the ٥٩
authors defined a new objective as “job-site movements.” This new objective is supposed to ٦٠
make element installation patterns more logical and acceptable by minimizing the distance ٦١
between installation groups of each type of element. By minimizing the distances, the installation ٦٢
patterns of the elements tend to get more logical and doable in the construction process. The ٦٣
authors implemented this new objective along with cost and time in the GA optimization ٦٤
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process. By developing this three-objective GA (cost, time, and minimal distance), the entire ٦٥
proposed method is able to generate constructible and optimized construction schedules only ٦٦
from the BIM of a project. ٦٧
This section is followed by a comprehensive literature review on multi-objective genetic ٦٨
algorithm as the chosen methodology in this research. Then, Section 4 will provide a descriptive ٦٩
section on how the 3D model input is handled along with the process of generating MoCC. Next, ٧٠
Section 5 gives a definition of the genetic algorithm and lists the objectives. After that, input ٧١
parameters of the entire algorithm are elaborated upon in Section 6, followed by Section 7 with ٧٢
tests and their results. Finally, the conclusion section and future works are summarized in ٧٣
Section 8. ٧٤
2 Literature Review ٧٥
This research is mainly focused on using the genetic algorithm for the calculation and ٧٦
optimization of defined objectives. The genetic algorithm is selected as the optimization tool for ٧٧
this research, not because it is the only or the best optimization tool for multi-objective ٧٨
optimization problems, but because of the nature of the extracted data from the BIM of the ٧٩
project and how easily data can be translated to GA genomes. Obviously, other optimization ٨٠
methods, as long as they can use existing data from BIM, can replace GA in further research for ٨١
evaluation and comparison purposes. The other optimization methods that have already been ٨٢
used for the time-cost optimization problem in the construction industry and are known by their ٨٣
ant colony optimization (e.g. Xiong & Kuang, 2008; Liu, Ni, & Qiu, 2015; Li & Zhang, 2013), ٨٤
particle swarm optimization (e.g. Yang, 2007; Zhang, Li, Li, & Huang, 2005; Liu, Al-Hussein, & ٨٥
Lu, 2015), linear and integer programming (e.g. Liu, Burns, & Feng, 1995), artificial intelligent ٨٦
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system (e.g. Chen, Griffis, Chen, & Chang, 2012), and mathematical modeling (e.g. Lucko, Said, ٨٧
& Bouferguene, 2014). These methods can be evaluated later and compared to GA when using ٨٨
BIM data to do time-cost optimization, which can also be referred to as ant colony optimization. ٨٩
In current research, a 3D model along with other GA variables are imported into the genetic ٩٠
algorithm, after which, the algorithm yields numerous logical and constructible schedules for the ٩١
given 3D model. These results from GA are used to generate 3D cloud point and Pareto front ٩٢
graphs. Therefore, the literature review for this research is limited to multi-objective genetic ٩٣
algorithms used for the optimization of construction projects. ٩٤
For multi-objective optimization of construction schedules, the GA has been used successfully ٩٥
among researchers solving engineering problems (Feng, Liu, & Burns, 1997). In 1997, Feng et ٩٦
al. (1997) introduced a GA methodology for optimizing time-cost relationships in construction ٩٧
projects. They also produced a computer application based on their methodology, which could ٩٨
run the algorithm. Zheng et al. (2002) also showed their interests in using GA for time-cost ٩٩
trade-off optimization problems in construction projects. By comparing GA with other ١٠٠
techniques, they showed that GA is capable of generating the most optimum results for the time-١٠١
cost optimization (TCO) problems in large construction projects. They also presented their own ١٠٢
multi-objective GA using the adaptive weight approach, which was able to point out an optimal ١٠٣
total project cost and duration (Zheng, Ng, & Kumaraswamy, 2004). In their next step, they ١٠٤
showed that using niche formation, Pareto ranking, and the adaptive weighting approach in ١٠٥
multi-objective GA could result in more robust time-cost optimization results (Zheng, Ng, & ١٠٦
Kumaraswamy, 2005). ١٠٧
In 2005, Azaron et al. (2005) introduced their multi-objective GA for solving time-cost ١٠٨
relationship problems, specifically in PERT networks. In their research they defined four ١٠٩
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objectives as minimizing project direct cost, minimizing mean of project duration, minimizing ١١٠
variance of project duration, and maximizing probability of reaching project duration limit. ١١١
Another group of researchers developed their own multi-objective GA to reach set of project ١١٢
schedules with near optimum duration, cost, and resource allocation and embedded their ١١٣
algorithm as a MS Project macro (Dawood & Sriprasert, 2006). In 2008, a multi-objective GA ١١٤
was introduced for scheduling linear construction projects and focused on optimizing both ١١٥
project cost and time as its objectives (Senouci & Al-Derham, 2008). Hooshyar et al. (2008) ١١٦
presented their GA time-cost tradeoff problem solver with higher calculation speed than made ١١٧
possible in the highly efficient Siemens’ algorithm (Siemens, 1971). ١١٨
Abd El Razek et al. (2010) developed an algorithm that used the line-of-balance technique and ١١٩
critical path method concepts in a multi-objective GA. This proposed algorithm was designed to ١٢٠
help project planners in optimizing resource usage. This resource usage optimization was ١٢١
conducted by minimizing cost and time while maximizing the project quality by increasing the ١٢٢
resource usage efficiency. Late in 2011, Mohammadi (2011) introduced his MOGA (multi-١٢٣
objective genetic algorithm) that generated Pareto front in its approach toward solving the time ١٢٤
cost optimization (TCO) problem in industrial environment. In 2012, Lin et al. (2012) designed ١٢٥
and introduced their multi-section GA model for scheduling problems. They combined that ١٢٦
model with their proposed network modeling technique to perform automatic scheduling in the ١٢٧
manufacturing system. ١٢٨
In recent years also, researchers have shown interest in new ways to solve the time-cost ١٢٩
optimization problem. Amiri et al. (2013) added quality to the TCO problem and used the Non-١٣٠
dominated Sorting Genetic Algorithm-II (NSGA-II) for time-cost-quality trade-off project ١٣١
scheduling problems (GPDTCQTP). Ke (2014) considered the indeterminacy of the environment ١٣٢
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in the proposed model and used the genetic algorithm to solve uncertain time-cost trade-off ١٣٣
problems. Using Line of Balance (LoB) technique, Agrama (2014) used her MOGA for ١٣٤
scheduling multistory buildings, in which project duration, total crews, and total interruptions ١٣٥
were defined as conflicting objectives. Cheng et al. (2014) proposed a novel approach by ١٣٦
introducing their two-phase differential evolution (DE) model which was able to successfully ١٣٧
reflect both time-cost effects and resource constraints. They (2015) included opposition ١٣٨
technique to their multi-objective DE, introducing the Opposition-based Multiple Objective ١٣٩
Differential Evolution (OMODE), to solve the time-cost-utilization work shift tradeoff (TCUT) ١٤٠
problem. They continued their work (2015) on the TCO problem by proposing a new hybrid ١٤١
multiple objective evolutionary algorithm that is based on the hybridization of an artificial bee ١٤٢
colony and DE (MOABCDE-TCQT). Later on Tran et al. (2016) showed the benefits of using ١٤٣
their novel approach named “Multiple Objective Symbiotic Organisms Search” (MOSOS) to ١٤٤
solve multiple work shifts problem in the context of TCO problems by adding labor utilization. ١٤٥
Lee et al. (2015) used the existing data from the project schedules for each individual task to find ١٤٦
optimal set of parameters for GA as an advanced stochastic time-cost tradeoff (ASTCT) method ١٤٧
to solve the TCO problem. Zhang (2015) applied genetic algorithm in repetitive construction ١٤٨
projects, such as bridges, to solve discrete time/cost trade-off problem (DTCTP) adding soft ١٤٩
logic to make it more complex. ١٥٠
All of these researchers successfully tackled the time-cost trade-off problem in construction ١٥١
schedules, but the a research information gap exists due the lack of a way to ensure the complete ١٥٢
and automated coverage of all the project elements in the calculations and scheduling. The ١٥٣
techniques mentioned herein were able to calculate and retrieve enough data from the existing ١٥٤
project schedules to solve TCO problem. However, the project schedules used had some inherent ١٥٥
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scheduling problems such as incompleteness in not covering the entire scope of the project and a ١٥٦
lack of logic in not satisfying proper relations. It is obvious that problematic input will result in ١٥٧
wrong and useless output in this type of optimization problem. Therefore, enriching the existing ١٥٨
approach with automated project scheduling technique is a needed step toward eliminating the ١٥٩
above mentioned problems. ١٦٠
In the current research, since the key input to the algorithm is the BIM of the project and the ١٦١
algorithm uses all the inherent geometry data from this input, full coverage of the project ١٦٢
elements as well as well-defined processor relationships between elements is guaranteed. ١٦٣
Considering all the project elements in both schedule development and later analysis, the end ١٦٤
result of the new algorithm is more realistic and has proven accurate. ١٦٥
3 Methodology ١٦٦
The entire methodology is summarized in Figure 1. As shown in this figure, the user inputs the ١٦٧
BIM along with resource-related information on the project. Then, the matrix of constructability ١٦٨
constraints (MoCC) algorithm detects the geometric information of the 3D elements and creates ١٦٩
the matrix. This matrix will be the basis for the genetic algorithm to ensure all the generated ١٧٠
construction schedules are structurally stable. Using the resource-related data, initially provided ١٧١
by the user, GA optimizes the resulting schedules toward multiple project objectives. This ١٧٢
calculation can be repeated whenever there is any change in designs or other project parameters ١٧٣
to obtain updated project schedules. In the proposed MoCC algorithm, in addition to beams and ١٧٤
columns (Faghihi, Reinschmidt, & Kang, 2014), other common building components are ١٧٥
included. These other building components are slabs (floors), roofs, walls, doors, and windows. ١٧٦
Adding this support for more element types makes it possible to input a more realistic 3D model ١٧٧
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into the algorithm. In addition, three more objective functions were defined to help generate
١٧٨
more workable and reasonable project schedules by optimizing time, cost, and required job-site
١٧٩
movements.
١٨٠
١٨١
Figure 1- Schematic view of the methodology
١٨٢
4 Reading the Geometry
١٨٣
In this research, the 3D model input to the algorithm is in a standard data format, called Industry
١٨٤
Foundation Classes (IFC). The IFC file format, as an open and neutral specification, is an object-
١٨٥
based file format. The data model of this neutral file format is developed by buildingSMART
١٨٦
with the main goal of facilitating interoperability between the AEC companies, as a commonly
١٨٧
used file format for BIM (buildingSMART, 2013). The IFC model specification is listed as an
١٨٨
official International Standard ISO 16739:2013 (International Organization for Standardization,
١٨٩
2013).
١٩٠
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4.1 Detecting More 3D Elements ١٩١
The previously developed algorithm, which served to prove the usefulness of the methodology ١٩٢
(Faghihi, Reinschmidt, & Kang, 2014) supported the structural elements, columns (IfcColumn) ١٩٣
and beams (IfcBeam). The extended research supports more general 3D elements from the IFC ١٩٤
file format of the BIM of a project. These general 3D elements are geometric information of ١٩٥
basic element types, such as slabs (IfcSlab), roofs (IfcRoof), walls (IfcWall), doors (IfcDoor) and ١٩٦
windows (IfcWindow) to be detected, read, and calculated. For simplification of the geometric ١٩٧
calculations, these element types were assumed as either lines or plain squares with a boundary ١٩٨
box around them. The boundary boxes around elements are calculated with tolerances, different ١٩٩
for each type, that make them slightly larger than their actual size. These tolerances of the ٢٠٠
boundary boxes ensure that when two elements are connected, they will close enough to have ٢٠١
intersecting boundary boxes that are interpreted by the algorithm as the connection of the two ٢٠٢
elements. An example of how the boundary boxes of beams or columns are assumed is shown in ٢٠٣
Figure 2. ٢٠٤
(a) (b) (c)
Figure 2- 3D Element Simplification, a) actual element, b) element section and extrusion line, c) element ٢٠٥
boundary box ٢٠٦
In this research, whenever two elements are close enough to have intersecting boundary box ٢٠٧
regions, they are assumed to be physically connected. After locating these connections by ٢٠٨
performing geometric calculation on the data retrieved from IFC files and applying stability rules ٢٠٩
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as shown in Table 1, the MoCC can be generated. The information presented in this table simply ٢١٠
shows the construction prerequisites that must be constructed. For instance, to install a wall, the ٢١١
lower level beams in the same region must be installed prior to the wall as well as any columns ٢١٢
and/or beams that the wall is covering. ٢١٣
Table 1- Stability Prerequisites ٢١٤
Lower Level Same Level
Column Column -
Beam - Supporting Columns or Beams
Wall Beams Adjacent Columns and Beams
Slab Regional Beams -
Roof Regional Beams -
Door - Container Wall
Window - Container Wall
٢١٥
4.2 Generating MoCC ٢١٦
In the MoCC (shown in Equation 1), Ai represents elements as well as activities associated with ٢١٧
each element indicating installation of that activity. Values of si,j could be either one or zero, ٢١٨
indicating immediate prerequisite installation or no relationship, respectively. For instance, s2,6-٢١٩
=1 means that the element number 6, A6, should be installed prior to installation of element 2, A2. ٢٢٠
This matrix can be used in the genetic algorithm fitness function for checking the constructability ٢٢١
of genomes in each generation. The follow gives the MoCC calculation as ٢٢٢
Equation 1- Matrix of Constructability Constraints ٢٢٣
𝑀𝑎𝑡𝑟𝑖𝑥 𝑜𝑓 𝐶𝑜𝑛𝑠𝑡𝑟𝑢𝑐𝑡𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝐶𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡𝑠 𝑀𝑜𝐶𝐶 𝐴𝐴…𝐴
𝐴
𝐴
⋮
𝐴𝑠, 𝑠, ⋯𝑠
,
𝑠, 𝑠, ⋯𝑠
,
⋮⋮⋱⋮
𝑠, 𝑠, ⋯𝑠
, ٢٢٤
12
where: ٢٢٥
Ai: project tasks (geometric elements in the 3D model or the activities to be ٢٢٦ scheduled) ٢٢٧ Sj,i represents dependencies between elements (that could be either 0 or 1 ٢٢٨ showing not dependent or dependent respectively). ٢٢٩
These relationships represent parent-child dependencies in the model. This means if s2,6 =1 then ٢٣٠
element A6 has a child relation to element A2. For example, element 6 is a door and element 2 is a ٢٣١
wall with that door installed in it. ٢٣٢
Having the MoCC generated and ready, the next step would be defining the GA that can generate ٢٣٣
optimized construction schedules and produce 2D and 3D Pareto fronts. The GA calculations are ٢٣٤
restricted by the constructability rules and constraints detected and defined in MoCC as well as ٢٣٥
resource limitations defined by the user. The resource limitations are representing the availability ٢٣٦
of materials and project crew to perform project tasks. ٢٣٧
5 Genetic Algorithm ٢٣٨
The genetic algorithm is an optimization tool that uses a heuristic search and mimics the natural ٢٣٩
evolutionary process (Mitchell, 1996). Using a well-defined fitness function as the objective ٢٤٠
function or core metric randomly generates initial genomes that can evolve into optimized ٢٤١
solution(s) for a given problem, considering objectives that are mathematically defined by the ٢٤٢
fitness function. The GA has become a practical optimization tool in construction-related fields ٢٤٣
of research due to its inherent features and characteristics. In this paper, each GA genome matrix ٢٤٤
(MoG), to be used as a construction schedule in this case, is defined as shown in Equation 2 to be ٢٤٥
used in the GA. ٢٤٦
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Equation 2- Genome Matrix (MoG) ٢٤٧
𝐺𝑒𝑛𝑜𝑚𝑒 𝑀𝑎𝑡𝑟𝑖𝑥 𝑀𝑜𝐺 𝐷𝐷…𝐷
𝐴
𝐴
⋮
𝐴𝑔, 𝑔, ⋯𝑔
,
𝑔, 𝑔, ⋯𝑔
,
⋮⋮⋱⋮
𝑔, 𝑔, ⋯𝑔
,
٢٤٨
OR٢٤٩
𝑔𝑒𝑛𝑜𝑚𝑒𝑔,,…,𝑔,,𝑔,,…,𝑔,,…,𝑔,,…,𝑔,
where: ٢٥٠ Ai: project tasks i (geometric elements in the 3D model, also interpreted as ٢٥١ the activities to be scheduled) ٢٥٢ Dj: duration (time span) number j ٢٥٣
gi,j: indicates if the installation of element i is occurring in time span j or not, ٢٥٤ with 1 or 0, respectively. ٢٥٥ n: number of project tasks (geometric elements in the 3D model or the ٢٥٦ number of activities to be scheduled) ٢٥٧ k: total project time-unit (e.g., days, weeks, or months) ٢٥٨
For instance, if g2,6=1, it means that element number 2 is scheduled to be installed in the 6th ٢٥٩
timespan. ٢٦٠
5.1 Objective Definition ٢٦١
The following three fitness function scores, which are calculated for each member (genome) of ٢٦٢
GA populations, should be summed up to a single value to treat the multi-objective GA as a ٢٦٣
single objective GA. The summation approach is described later in this section. ٢٦٤
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5.1.1 ProjectDuration٢٦٥
As shown in Equation 2, the length of a genome is equal to the number of project elements
٢٦٦
multiplied by the number of time units (duration). For any given genome, the duration can be
٢٦٧
calculated by dividing the length of the genome by the number of 3D elements. With the same ٢٦٨
concept the total construction duration associated with a MoG is equal to the number of matrix ٢٦٩
columns. ٢٧٠
5.1.2 ProjectLaborCost٢٧١
As described earlier in this paper, the current research aims to introduce a framework that is ٢٧٢
capable of detecting the relationship between major construction schedule objectives, which are ٢٧٣
cost, time, and job-site movements. To build this framework, a few simplifications and ٢٧٤
assumptions are needed. For instance, a precise material take-off calculation is not in the scope ٢٧٥
of this research; therefore, the calculation of the project cost does not reflect the cost for material. ٢٧٦
The authors have calculated the project labor cost as follows: Initially, the algorithm collects ٢٧٧
resource-related data from the user. These user inputs contain the maximum number of each ٢٧٨
element type that could be installed in each day as well as the associated labor cost for those ٢٧٩
installations. The algorithm assigns the full labor cost for the days that the element installations ٢٨٠
do not exceed the maximum defined by the user. In case there is a day where the installation ٢٨١
number exceeds the maximum limit, the extra installations will be assigned a labor cost 1.5 times ٢٨٢
that of the regular installation, considering them as overtime work. This increase in cost is based ٢٨٣
on the U.S. Fair Labor Standards Act of 1938 (U.S. Department of Labor, 2009) that guarantees ٢٨٤
“time-and-a-half” for overtime in certain jobs. ٢٨٥
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5.1.3 Job‐siteMovements٢٨٦
Workability of construction processes can be increased by reducing the required movements of
٢٨٧
the crew and machinery installing the project elements. Having the location information for all
٢٨٨
the pieces of the project from its BIM in addition to the installation sequence of them from a ٢٨٩
given schedule (genome), the distances between all installations in a single time-unit and also ٢٩٠
between time-units can be calculated. A short mathematical description of the calculation is that ٢٩١
in each time-unit, the distances of all the scheduled installations for that time-unit to the central ٢٩٢
positioning point of those elements are calculated, and then the distances between the central ٢٩٣
positioning points of each time-unit are added to the sum. Minimizing the sum of total distances ٢٩٤
between installations of elements of each type will be another objective for the fitness function. ٢٩٥
5.1.4 CalculationExample٢٩٦
As an example of how to calculate the defined objectives, a simple structure and its represented ٢٩٧
MoG (Figure 3b) for the construction sequence of the 3D model (Figure 3a) is demonstrated. ٢٩٨
5.1.4.1 Calculating Duration Objective ٢٩٩
As mentioned earlier in the objective definition section, the construction duration for the given ٣٠٠
genome is equal to the number of the matrix columns, which is 5 unit-times. One unit-time will ٣٠١
be described and defined by the user. ٣٠٢
5.1.4.2 Calculating Cost Objective ٣٠٣
If a user entered the following inputs to the algorithm: ٣٠٤
Maximum number of columns per day: 4 Cost for this installation: $400/day ٣٠٥
Maximum number of beams per day: 5 Cost for this installation: $300/day ٣٠٦
16
Then, the calculation of the cost objective will be $400 for any project day that has 1 to 4 column ٣٠٧
installations and/or $300 for any day with 1 to 5 beam installations. If there is a day in which no ٣٠٨
column is scheduled to be installed, there would be no calculated cost associated to the column ٣٠٩
costs. On the other hand, if there are more than 4 columns (e.g. 6 columns) scheduled for a single ٣١٠
day to be installed, the algorithm will multiply the cost of the extra column installations by 1.5, ٣١١
simulating the cost for overtime work. Therefore, if there is a day with six columns scheduled to ٣١٢
be installed, the associated cost for that day for column installations would be: ٣١٣
𝐶𝑜𝑠𝑡$400$400
41.52$700
The total cost for the given genome shown in Figure 3b will be equal to a three-day cost of ٣١٤
column installations ($400) and a three-day cost of beam installations ($300): ٣١٥
𝐶𝑜𝑠𝑡$40033003$2100
5.1.4.3 Calculating Movement Objective ٣١٦
In the given genome, which has five unit-times as the total construction duration, the associated ٣١٧
installation duration for each element of the model is indicated as one time-unit. For instance, ٣١٨
elements number 1 and 3 are scheduled to be installed in the first time-unit, and element number ٣١٩
8 is scheduled for the second. For calculation purposes, the authors assume that the distances ٣٢٠
between column 1 and 3 is 10 feet, the height of the structure is also 10 feet, and it is symmetric ٣٢١
in all directions. ٣٢٢
17
𝑀𝑜𝐺
12345
1
2
3
4
5
6
7
8⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎡
10000
00100
10000
00010
00010
00001
00001
01000
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎤
(a)
(b)
Figure 3- (a) Sample structure and (b) a sample construction sequence
٣٢٣
To visualize the installation distances associated with the MoG and 3D model shown in Figure 3,
٣٢٤
a schematic top-view of the distances with consideration of the sequencing is drawn in Figure 4.
٣٢٥
In this figure when multiple elements are installed in a single time-span, the distances are shown
٣٢٦
with a dotted line with an arrow head on both ends, connecting all elements to be installed in that
٣٢٧
time span. The distance between installed elements in two time spans following each other is
٣٢٨
shown with a dotted line with arrow head on one end showing the direction of the steps, which
٣٢٩
connect the central positioning point of both sets of installing elements.
٣٣٠
Figure 4- Top-view of the installation distances for sample MoG: (a) for the columns and (b) for the beams
٣٣١
Figure 4a shows a top-view for installation distances of the columns whereby column 1 and 3 are
٣٣٢
installed together in the first time unit, column number 2 is in the third time unit, and column
٣٣٣
number 4 is in the fourth time unit. In the first time unit, the installation distance is equal to the
٣٣٤
(a) (b)
3
1 2
4
7
5
68
18
distance between column 1 and 3, which is 10 feet. There are no columns scheduled for the ٣٣٥
second time unit, which is accordingly skipped. In the third time unit, there is only one column ٣٣٦
installed; therefore, the installation distance would be equal to the distance from column 2 to the ٣٣٧
central positioning point of the previous installations (i.e., columns 1 and 3). For the fourth time-٣٣٨
unit, the installation distance is simply calculated as the distance between column 4 and the last ٣٣٩
installation of the previous time span, column 2. With a similar concept and calculations, the ٣٤٠
total installation distances for beams shown in Figure 4b can be calculated. Calculated lengths of ٣٤١
installations for the given example are shown in Equation 3a and 3b. Notice that the total ٣٤٢
installation distances (or job-site movements required for installations) is equal to the sum of all ٣٤٣
the element types. In this example, the movement objective score is equal to the sum of total ٣٤٤
installation distances of columns and beams as shown in Equation 3c. ٣٤٥
Equation 3- the total installation distances in the sample presented as Figure 4 ٣٤٦
(a) 𝑇𝑜𝑡𝑎𝑙 𝑖𝑛𝑠𝑡𝑎𝑙𝑙𝑎𝑡𝑖𝑜𝑛 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒𝑠 𝑓𝑜𝑟 𝑐𝑜𝑙𝑢𝑚𝑛𝑠10 11.18 10 31.18 ٣٤٧ 𝑏 𝑇𝑜𝑡𝑎𝑙 𝑖𝑛𝑠𝑡𝑎𝑙𝑙𝑎𝑡𝑖𝑜𝑛 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒𝑠 𝑓𝑜𝑟 𝑏𝑒𝑎𝑚𝑠7.07 7.91 7.07 22.05
𝑐 𝑇𝑜𝑡𝑎𝑙 𝑖𝑛𝑠𝑡𝑎𝑙𝑙𝑎𝑡𝑖𝑜𝑛 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒𝑠 31.18 22.05 53.23
By defining this method in the fitness function of the GA in even more complex models, ٣٤٨
schedules with less job-site movement distance can be found in each generation cycle. By ٣٤٩
implementing this objective into the fitness function of the GA, the resulting construction ٣٥٠
schedules will have a more reasonable pattern of installation sequences as briefly shown in the ٣٥١
above example. ٣٥٢
5.1.5 ObjectiveSummation٣٥٣
Among the different approaches to calculate the fitness function score in multi-objective GAs ٣٥٤
(Konaka, Coitb, & Smith, 2006), the classic “weighted sum approach” is adopted as a very ٣٥٥
19
computationally efficient approach. In this approach, the user needs to simply assign weights for ٣٥٦
each of the objectives defined in the fitness function. For each objective, the score for each ٣٥٧
genome is divided by the average score of the entire population for the same score to normalize ٣٥٨
the scores as shown in (Equation 4). This normalization of objective scores forbids any ٣٥٩
individual objective to dominate the final score. The user-defined weights would be multiplied ٣٦٠
by the respective normalized scores, and then all three scores are summed up to form the final ٣٦١
score. The minimizing function is shown in Equation 5, where the weight factors (wd, we, and ٣٦٢
wm) are inputs received from user defining any desired priority for objectives. The following ٣٦٣
calculation shows how to obtain the normalized score for a genome objective ٣٦٤
Equation 4- Normalization of an objective score of a genome ٣٦٥ 𝑁𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑 𝑠𝑐𝑜𝑟𝑒 𝑓𝑜𝑟 𝑎𝑛 𝑜𝑏𝑗𝑒𝑐𝑡𝑖𝑣𝑒 𝑜𝑓 𝑎 𝑔𝑒𝑛𝑜𝑚𝑒٣٦٦
٣٦٧
Equation 5- Weighted sum approach ٣٦٨
min𝑍𝑤𝐹𝑥𝑤𝐹𝑥𝑤𝐹𝑥 ٣٦٩
6 Inputs and Variables ٣٧٠
The model shown in Figure 5a is a generic turbine building structural model with 42 columns ٣٧١
and 58 beams, adding up to 100 elements. This generic model of a turbine building was ٣٧٢
previously used for proving the usefulness of this algorithm in developing structurally stable ٣٧٣
construction schedules extracted from a real-world and complex structural 3D model (Faghihi, ٣٧٤
Reinschmidt, & Kang, 2014). To test the extended research, a typical 3D model of a two-story ٣٧٥
residential building was selected, as shown in Figure 5b. For the purpose of better internal ٣٧٦
viewing, the roof element is made invisible in the figure. This 3D model consists of 38 columns, ٣٧٧
20
56 beams, 1 roof, 1 floor, 18 walls, 24 windows, and 7 doors, summing up to 145 elements. The ٣٧٨
3D model shown in Figure 5b is used in this research for elaboration of the results and Pareto ٣٧٩
fronts. ٣٨٠
(a) (b)
Figure 5- 3D model input, (a) generic turbine building structural model, (b) typical 2-story residential ٣٨١
building ٣٨٢
The 3D model input along with the other input variables (shown in Table 2) was used to run the ٣٨٣
GA for testing the proposed algorithm with new extensions. ٣٨٤
Table 2- Variable inputs to the GA calculations ٣٨٥
Title Value
Population per generation 10
Elite rate 20%
Installment (maximum number
of installations in each time-unit) -
- Columns 2
- Beams
- Walls 1
- Slabs 1
- Roofs 1
- Doors 2
- Windows 4
Cost/Wage
(dollars per each time-unit) -
- Columns 500
21
- Beams 600
- Walls 300
- Slabs 400
- Roofs 450
- Doors 200
- Windows 320
Objective weights -
- Duration 1
- Cost 1
- Movement 1
7 Results ٣٨٦
Running the developed GA for 20,000 rounds of population generation with 10 genomes in each, ٣٨٧
for the 3D model, as shown in Figure 5b, produced 200,000 valid construction schedules. Each ٣٨٨
of these generated construction schedules has three fitness function scores for the three ٣٨٩
objectives defined earlier. ٣٩٠
7.1 2D Pareto Fronts ٣٩١
Plotting each of two objectives, calculated in the results on a single 2D coordinate system, ٣٩٢
shapes the following outputs. Figure 6 shows that by increasing the project duration, the labor ٣٩٣
cost for the project will be reduced before it starts to slightly increase. The reduction is due to ٣٩٤
decreasing the overtime work for the project. The latter increase of labor cost is caused by the ٣٩٥
growth in the number of working days. This interpretation matches the correct relationship ٣٩٦
between time and cost in real projects. ٣٩٧
22
٣٩٨
Figure 6- Cost-Time Pareto Front
٣٩٩
The time-movement Pareto Front graph as presented in Figure 7 shows a constant and
٤٠٠
exponential increase in the movements when the project duration increases. This constant
٤٠١
increase is a result of spreading out the element installations over the construction duration. Thus
٤٠٢
the distances between elements are not minimized by being installed as a group in a single day to
٤٠٣
have a shorter mid-point distance to the other set or setting up installation at the next time step.
٤٠٤
23
٤٠٥
Figure 7- Time-Movement Pareto Front
٤٠٦
As shown in Figure 8, when the movement score is at its minimum (meaning that elements are
٤٠٧
installed in groups in each time step), the associated labor cost is higher, which is due to
٤٠٨
overtime installations of the elements. When elements are installed in sets of groups in each time
٤٠٩
step (e.g., six columns in a day), the total distance between each of these elements and also the
٤١٠
distance between the mid-point of this installation group and the next one will be reduced based
٤١١
on the similar formulation and calculations shown in Equation 3. On the other hand, when there
٤١٢
are more elements to be installed in a single day than the user defined maximum limit (e.g.,
٤١٣
maximum of four columns per day), the surplus elements (e.g., two columns) have a labor cost
٤١٤
1.5 times higher than the regular installation. These two facts together make labor cost higher
٤١٥
when the movement score lowers.
٤١٦
24
٤١٧
Figure 8- Movement-Cost Pareto Front
٤١٨
When the movement score exceeds a certain number and continues to increase, it indicates that
٤١٩
the project elements installed are becoming much more scattered each day. Based on the authors’
٤٢٠
definition of labor cost, any number of installations per day can be equal to or less than the user-
٤٢١
defined maximum and will have the user-defined associated labor cost for the day. Putting these
٤٢٢
two factors together, we know that the more the element installations are spread out in each day,
٤٢٣
the higher the labor cost will be.
٤٢٤
Demonstrations of Pareto fronts in all three graphs mentioned in this section show that this
٤٢٥
methodology shows the correct behavior of the three objectives defined in this research. The
٤٢٦
time-cost-movement relations described in this section matches the same inter-relationships
٤٢٧
between these objectives as known in the construction field. These demonstrations and
٤٢٨
descriptions validate the use of this methodology.
٤٢٩
25
7.2 Solutions Cloud Point ٤٣٠
As described earlier in this paper, each of the generated project schedules has three objective
٤٣١
scores for time, cost, and movement objectives. Figure 6 through Figure 8 showed how all
٤٣٢
200,000 solutions can be represented in the 2D coordinate systems. Since there are three
٤٣٣
objectives in this paper, it is possible to show all the solution points in a single 3D scattered plot,
٤٣٤
as shown in Figure 9. Further development of this analysis can generate a 3D Pareto Front
٤٣٥
surface showing the optimum relationship between the defined three objectives of this research.
٤٣٦
٤٣٧
Figure 9- Solutions 3D Cloud Point
٤٣٨
7.3 Objective-driven Analysis ٤٣٩
Other than generating three previously shown Pareto Fronts and the solution cloud points, the
٤٤٠
proposed algorithm is capable of responding to the different project characteristics as being
٤٤١
objective driven. For instance, the solutions from this algorithm can properly demonstrate the
٤٤٢
nature of cost- or time-driven schedules. To receive these types of reactions, the user needs to
٤٤٣
26
input the desired behavior for the project schedules when defining the objective weights right ٤٤٤
before running the GA calculations. In the previously shown 200,000 results from the algorithm, ٤٤٥
the defined weights for the objectives were set as equal. This equality of objectives weights ٤٤٦
means that the GA calculations considered all three objectives with the same importance when ٤٤٧
the scores were summing up. Therefore, the changes in each objective had the same impact on ٤٤٨
the overall score for the genomes and thus on the chance for being selected as an elite member or ٤٤٩
for the crossover function of the GA. ٤٥٠
In this paper, the authors show how the solution cloud points will be changed reflecting different ٤٥١
objective weights. For this reason, three different runs with the same input 3D model and data ٤٥٢
have been conducted. In each of these three runs, one of the objectives received a weight of 100 ٤٥٣
while the other two were set to one. By inputting objective weights in this manner, in each of the ٤٥٤
new calculations, one of the objectives will be considered 100 times more important than the ٤٥٥
other two. This assumption will reflect the expectation of the user to have scheduling solutions ٤٥٦
driven toward a specific objective. For example, if the user set the cost objective weight at 100 ٤٥٧
times more than the duration and movement objective weights, the algorithm will understand that ٤٥٨
the cost object is much more important than time and distance. In other words, the cost-driven ٤٥٩
solutions are requested by the user. Then, the algorithm will use that input to produce the ٤٦٠
construction schedules for the project. ٤٦١
Figure 6 through Figure 8 show results from the calculation with the same objective weights for ٤٦٢
all three objectives. The following figures show how the cost-driven and time-driven calculations ٤٦٣
can differ. ٤٦٤
27
٤٦٥
Figure 10- Time-cost cloud points comparing objective-driven calculations
٤٦٦
As described earlier in this paper, by changing the weighting parameter for time and cost
٤٦٧
objectives in two different sets of runs from 1 to 100, while the other objective weights remain at
٤٦٨
one in that algorithm run, 200,000 new project schedules are generated for each run. Figure 10
٤٦٩
shows how the cloud points for the two new sets of calculations and construction schedules are
٤٧٠
different from the earlier run. As shown in the figure, when the calculation is set to be time-
٤٧١
driven (time objective has a weight 100 times greater than the other two objectives), the entire
٤٧٢
cloud point (red pluses in the figure) shifts to the left of the graph. This left-shift means that the
٤٧٣
entire solution cloud point has construction schedules shorter than the normal calculation (blue
٤٧٤
dots) in which the weighted objectives are equal. The calculated average of the cloud points are
٤٧٥
shown as big red, green, and blue dots indicating average values for time-driven, cost-driven, and
٤٧٦
normal calculations, respectively.
٤٧٧
28
Figure 10 also shows that when the calculation is set to be time-driven (or cost-driven) the entire ٤٧٨
cloud point as well as the average point of the cloud shifts to the left for shorter construction ٤٧٩
durations (or shifts down for less construction labor costs in cost-driven runs). When the user ٤٨٠
intends to run the algorithm as time-driven, the algorithm produces more construction schedules ٤٨١
with shorter duration. Imagine this example project has a constraint of a construction plan for a ٤٨٢
building to be built in less than 23 days. To satisfy this constraint the user needs to put more ٤٨٣
weight on the time objective in the calculation (e.g., 100 for time and 1 for the other objectives). ٤٨٤
By running the algorithm with this setting, the normal run generated 11 construction schedules ٤٨٥
with a duration of less than 23 days while the time-driven run generated more than 38,000 ٤٨٦
different construction schedules satisfying the constraint. Similar results can be discussed with ٤٨٧
the cost-driven algorithm calculations. ٤٨٨
Also it is visible that in cost-driven calculation results, since the time objective had less weight ٤٨٩
(importance) set by the user, the average of the cloud point has been shifted to the right. This ٤٩٠
means that for cost-driven construction sequences, while the average cost has been reduced, the ٤٩١
average time has been increased due to less importance of the time objective. Similar ٤٩٢
descriptions can be explained for other objectives and calculations. ٤٩٣
29
٤٩٤
Figure 11- Time-movement cloud point comparing objective-driven calculations
٤٩٥
Similar to Figure 10, Figure 11 shows the objective-driven calculations versus the normal ones,
٤٩٦
which had equal weights assigned to all the objectives. As seen in this figure, in both cost and
٤٩٧
time driven calculations, the average value for the movement objective has been increased. As
٤٩٨
similarly described before, this behavior is due to less importance (objective weight) assigned to
٤٩٩
the movement objective for calculations by the user. Figure 12 shows the same behavior in its
٥٠٠
movement-cost graph.
٥٠١
30
٥٠٢
Figure 12- Movement-cost cloud point comparing objective-driven calculations
٥٠٣
The differences in the average values of the cloud points in all three runs are shown in Table 3.
٥٠٤
As described before and as visible in Table 3, the cost objective score is increased in time-driven
٥٠٥
runs and decreased in cost-driven runs as expected. Likewise, the time score was reduced when
٥٠٦
calculations were time-driven and enlarged when cost-driven. The movement objective score
٥٠٧
was increased in both runs since the user-defined objective weight of this objective was set to the
٥٠٨
minimum.
٥٠٩
Table 3- Average score difference for normal vs. objective driven runs
٥١٠
Normal Cost-driven Time-driven
Cost 35,214.6 33,718.4 35,255.3
Time 29.1 36.5 28.5
Movements 2,154 2,311.4 2,192.8
31
8 Conclusion and Future Work ٥١١
This article shows useful benefits of the proposed construction scheduling algorithm through the ٥١٢
Pareto front relationship between optimization objectives of a construction project. The entire ٥١٣
algorithm, as described in earlier papers, is able to read through the 3D model input in the form ٥١٤
of IFC, detect all the structural stability dependencies and relations, and form a structural ٥١٥
stability matrix called MoCC. Then, it uses that matrix as the basis for the GA fitness function to ٥١٦
validate the structural stability wellness of the populations and produces project schedules while ٥١٧
showing 4D construction animation (Faghihi, Reinschmidt, & Kang, 2014). The generated ٥١٨
populations that contain construction schedules are ordered and handled based on their objective ٥١٩
scores. Those construction schedules with better scores will have a better chance of reaching the ٥٢٠
next generations. ٥٢١
As mentioned in this paper, in addition to automatic construction schedule development, the ٥٢٢
proposed algorithm can also provide several managerial tools to help project managers and ٥٢٣
project management teams in their scheduling of projects. These managerial tools can provide ٥٢٤
two-by-two Pareto Front graphs for objectives, solution cloud points and a 3D Pareto Front ٥٢٥
surface (with later extensions). The algorithm’s tools can also be used to reflect the objective ٥٢٦
driven nature of the project in the provided solutions for construction project scheduling ٥٢٧
problems. ٥٢٨
Besides all of the mentioned benefits and advantages of this method, it still has some limitations, ٥٢٩
one of which is that it takes some time to produce the required number of rounds for GA ٥٣٠
calculations. When the project gets bigger, and its BIM model gets relatively more complex, the ٥٣١
number of elements will increase and the time needed to prepare will also increase if acceptable ٥٣٢
results are to increase exponentially. Another limitation to the current algorithm is its support for ٥٣٣
32
a building’s irregular shaped elements. If, for example, a building contains curved walls (similar ٥٣٤
to the Walt Disney Concert Hall) the existing algorithm cannot address that type of dimensions ٥٣٥
and connecting points for those walls and elements. Therefore, it cannot do the required ٥٣٦
calculations to generate a graph as shown in this paper. ٥٣٧
For further extension on this research, other multi-objective optimization methods can be ٥٣٨
evaluated and investigated to see if better representation of the objectives’ relationships can be ٥٣٩
determined. We also need to find a way to support more 3D element types so the input model ٥٤٠
can generate more realistic results. ٥٤١
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