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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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