Optimized Reinforcement Detailing Design for Sustainable Construction: Slab Case Study

Abstract

Reinforced steel rebar is commonly supplied in one-dimensional stocks and typically designed for and installed in various structural components in civil and industrial construction. Surplus reinforcement constitutes a major fraction of construction generated waste. Cutting one-dimensional stocks to suit construction project requirements results in cutting losses. Therefore, reducing steel waste (or minimizing cutting losses) has long been the focus of academic research in one-dimensional stock design and cutting problems. Previous research developed mathematical models in an attempt to analytically minimize cutting losses based on preliminary engineering designs, but little insight has been provided on how to integrate minimization of cutting losses and engineering design into an integrated optimization problem, let alone considering minimizing total steel rebar installation cost as a parallel objective. The sustainability issue in regard to balancing reinforcement waste and crew installation costs on the basis of optimized engineering design has yet to be addressed. This study introduces a Mixed Integer Programming (MIP) approach to generate optimal cutting patterns, minimum cutting losses and associated total installation cost. A reinforced concrete slab case is adopted as a test to show that the proposed methodology is capable of producing optimal tradeoff solutions in slab reinforcement detailing design, resulting in less wastage and lower crew installation cost.

Full text

Procedia Engineering 145 ( 2016 ) 1478 – 1485
1877-7058 © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-review under responsibility of the organizing committee of ICSDEC 2016
doi: 10.1016/j.proeng.2016.04.186
ScienceDirect
Available online at www.sciencedirect.com
International Conference on Sustainable Design, Engineering and Construction
Optimized Reinforcement Detailing Design for Sustainable
Construction: Slab Case Study
Chaoyu Zhenga, Ming Lub
*
aDepartment of Civil and Envrionmental Engineering, University of Alberta, Edmonton,Canada
bDepartment of Civil and Envrionmental Engineering, University of Alberta, Edmonton,Canada
Abstract
Reinforced steel rebar is commonly supplied in one-dimensional stocks and typically designed for and installed in various structural
components in civil and industrial construction. Surplus reinforcement constitutes a major fraction of construction generated waste.
Cutting one-dimensional stocks to suit construction project requirements results in cutting losses. Therefore, reducing steel waste
(or minimizing cutting losses) has long been the focus of academic research in one-dimensional stock design and cutting problems.
Previous research developed mathematical models in an attempt to analytically minimize cutting losses based on preliminary
engineering designs, but little insight has been provided on how to integrate minimization of cutting losses and engineering design
into an integrated optimization problem, let alone considering minimizing total steel rebar installation cost as a parallel objective.
The sustainability issue in regard to balancing reinforcement waste and crew installation costs on the basis of optimized engineering
design has yet to be addressed. This study introduces a Mixed Integer Programming (MIP) approach to generate optimal cutting
patterns, minimum cutting losses and associated total installation cost. A reinforced concrete slab case is adopted as a test to show
that the proposed methodology is capable of producing optimal tradeoff solutions in slab reinforcement detailing design, resulting
in less wastage and lower crew installation cost.
© 2015 Zheng and Lu. Published by Elsevier Ltd.
Peer-review under responsibility of organizing committee of the International Conference on Sustainable Design, Engineering
and Construction 2015.
Keywords: Steel Rebar; Cutting Loss; Mixed Integer Programming; Multi Objective Optimization
* Corresponding author. Tel.: +1-780-492-5110; fax: +1-780-492-0249.
E-mail address: mlu6@ualberta.ca
© 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-review under responsibility of the organizing committee of ICSDEC 2016

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Chaoyu Zheng and Ming Lu / Procedia Engineering 145 ( 2016 ) 1478 – 1485
1. Introduction
Materials management is a vital function for improving productivity, safety, quality and sustainability in
construction projects. The management of materials is one crucial part of the construction planning process (i.e.
material planning) as poor materials management can often affect overall construction time, quality and budget. In
construction, material management has become a special activity concerning whether materials are procured,
processed and installed economically and sustainably. It is important to ensure that material waste, crew installation
cost and engineering design are optimally balanced in both reinforcement design and construction.
With the penetration of the Integrated Project Delivery (IPD) concept and the application of BIM technologies,
construction performance is improved by developing a project team that focuses on work processes and decisions
benefitting the entire project rather than individual team members [1]. The integration allows evaluation of numerous
alternatives for design and construction, benefiting reinforcement detailing design, which is indispensably aligned
with pivotal construction issues (i.e. constructability and sustainability). Through interaction of functional activities
between reinforcement management and engineering design by involving experienced engineering designers and
sophisticated field superintendent and foremen, it is anticipated that both material procurement cost and material waste,
along with crew installation cost, can be further reduced while being adherent to design code.
Reinforcing steel installation, as the main component of foundation and superstructure construction, is significant
to construction material management and cost control. Usually, the budget for reinforcing steel accounts for a very
large proportion and can reach as high as 26% in terms of the total project cost [2]. And for some steel structure
dominated buildings, the cost of reinforcing steel can take up to 60% of the entire project cost [3]. On one hand, rebar
waste has a direct effect on the project cost. The generation of waste is inevitable when rebar is supplied in market
available lengths for on-site fabrication [4]. On the other hand, the reinforcing steel layout arrangement ready for crew
installation at the workface (i.e. rebar schedule detailing design) impacts the total material consumption. Standard
lengths for rebar available in the Canadian market are 6 m (20 ft), 9 m (30 ft), 12 m (40 ft), and 18 m (60 ft) [3],
whereas in remote areas where steel manufacturing is undeveloped and transportation is inconvenient, the supply of
reinforcing steel is constrained by limited stock size alternatives (only one or two stock sizes available to order).
Therefore, (1) how to purchase the minimum quantity of reinforcing steel stocks in order to save in the budget, (2)
how to reduce the generated material wastes, and (3) how to detail the reinforcing steel installation for workface
execution, in the context of material management constraints, become significant problems for applied research in
construction engineering.
2. Literature Review
The rebar cutting problem is a typical one-dimensional material cutting optimization problem [4]. The one-
dimensional cutting stock problem (CSP) is known for achieving the best cutting pattern (i.e. how to cut stock rebar)
so as to meet particular construction project requirements, with rebar cutting losses being the major cause of the
construction material waste [5]. The main objective for the classical CSP is minimizing the material waste, when the
order quantity (i.e. required rebar type and length as per design drawings) has been predetermined [2], [4], [6–9],
resulting in the optimized cutting pattern and corresponding waste. Though significant contributions have been made,
few previous research endeavors have integrated engineering design at the workface level into the cutting optimization
problem.
To minimize either the material waste or the total procurement cost subject to certain market stock sizes is not an
easy task since it is a combinatorial optimization problem under complicated practical constraints. Applying
optimization algorithms on computers is one of the most effective ways to solve those problems. Gilmore and Gomory
[7] introduced an ingenious column generation technique to generate the cutting patterns and solve the cutting
optimization problem. However, the solution using LP to obtain relaxed non-integer solutions would normally depart
from optimality, giving rise to unnecessary waste. Navon et al [10] introduced the benefits of computer-aided design
and computer-aided manufacturing (CAD/CAM) systems for concrete reinforcement; they developed a model for
rebar constructability diagnosis and correction in an object-oriented programming environment. With the swift
development of integer programming techniques, this useful technique has been widely applied in various areas, until

1480 Chaoyu Zheng and Ming Lu / Procedia Engineering 145 ( 2016 ) 1478 – 1485
recently; Salem et al [4] adopted an integer programming approach to minimize cutting wastes of reinforcing steel
rebars.
In a nutshell, different from the traditional rebar cutting optimization problem, the problem defined in this research
is a material optimization problem related to temporary engineering design of detailed rebar schedules focusing on
two objectives: (1) to purchase less reinforcing steel (purchasing less reinforcing steel means lower material cost), and
(2) to reduce waste. Thus, there is an urgent need to propose a scientific approach to deal with this new problem in
order to answer the questions: What is the most cost-efficient reinforcing steel detailed arrangement? What is the most
cost-efficient reinforcing steel procurement method? And what is the most environmentally-friendly cutting pattern
to apply to the procured reinforcing steel stocks?
3. Methodology
For different construction components (e.g. pillar, beam, slab, wall, etc.), reinforcement layout design may vary
considerably. Construction components in which rebars are positioned in a two-way arrangement in both vertical and
horizontal directions (e.g. slab and wall) entail high reinforcing steel consumption. In this research, we considered
slab as the typical case in order to illustrate how the rebar stock is cut and how the cut rebar is arranged on slab in
order to meet sustainability requirements (i.e. reducing steel waste and lowering crew installation cost) through
optimization analyses. This research aims to optimize both the rebar detailed arrangement (i.e. lap length, rebar cutting
length, cutting rebar quantities on both directions) on the slab and the cutting pattern for the reinforcing steel stock,
attempting to achieve the objectives of least steel waste and lowest procurement cost. The flowchart of methodology
is shown in figure 1.
Start
Input:
1. Component Configuration
2. Stock Size
3. Lap Length Range
4. Rebar Spacing
5. Unit cost of material of labor
and equipment
6. Other practical constraints
Calculate all the
Layout Arrangement
Possibilities
Output Cutting
Patterns, Total Cost
and Total Waste
Mixed Integer
Programming
Tradeoff Analysis
Have all the Layout
Arrangements been
selected?
Pick one Layout
Arrangement
Alternative
No
Yes
Finish
Drawing Engineering
Sketches for
Records
Fig.1. Flowchart of Methodology

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Chaoyu Zheng and Ming Lu / Procedia Engineering 145 ( 2016 ) 1478 – 1485
To formulate the optimization problem, the following assumptions were made:
(1) There is only one type of rebar stock size available on the market;
(2) The slab size is longer than the length of rebar stock available on the market, so when we place rebar into the
slab, rebar needs to be spliced up with laps;
(3) Concrete cover depth is neglected in this study because its impact on optimization results is minimal;
(4) Steel rebar is cut in particular lengths at the laydown area on site by experienced ironmen. To meet workface
management requirements, the procured rebar stock needs to be cut into identical lengths along the long direction
or short direction, respectively. However, the cut length for both directions of the slab do not need to be equal.
Given permanent and temporary rebar design specifications, the procured rebar stock size, and other practical
constraints, all the possible combinations of rebar detailing schedules on the slab can be sorted out. By applying mixed
integer programming techniques, minimum cutting losses, optimal cutting patterns and minimum total material cost
can be identified for each alternative. The optimized outputs of each rebar schedule alternative are compared and
tradeoff analyses are conducted between steel waste and total procurement cost in order to identify the best solution.
4. Optimization process
The methodology used to minimize cutting wastes in the industry is adapted from the steel waste reduction method
of one-dimensional stocks in the construction industry [11]. The solution to the problem is divided into the following
two steps.
4.1. Generating rebar detailing schedules
The first step to solve this problem is to generate all feasible rebar detailing schedules. The procedure as adapted
from Pierce [12] can be used to generate all the efficient feasible solutions. Figure 2 and Figure 3 are instrumental in
understanding the basics of slab rebar placement temporary design.
The rebar detailing schedule is determined by two integer parameters ݊௫ and ݊௬. Note that ሺȀሻ൑
௫൑
௫ି௠௔௫ and ሺܾȀܮሻ൑
௬൑
௬ି௠௔௫, and ݊௫, ݊௫ି௠௔௫, ݊௬, ݊௬ି௠௔௫belong to positive integers setכ. So the
number of feasible combinations of rebar detailing schedules ܰ௟௔௬௢௨௧can be denoted by the following equation:
ܰ௟௔௬௢௨௧ ൌ(݊௫ି௠௔௫ െሺȀሻ൅ͳሻሺ݊௬ି௠௔௫ െሺȀሻ൅ͳሻ
(1)
where ݊௫ is the cutting rebar quantity in one row of long direction while ݊௬ is the cutting rebar
quantity in one row of short direction; ݊௫has a lower bound equal to round up (a/L) and a upper
bound equal to ௫ି௠௔௫ (determined by the superintendent for practical concerns); ݊௬has a lower
bound equal to round up (b/L) and a upper bound equal to݊௬ି௠௔௫ (same as௫ି௠௔௫); L is stock
length; a is the width of the slab; b is the length of the slab; and c is the rebar spacing along both
long and short direction.
b
a
c
c
Fig.2 Rebar Layout Sample

1482 Chaoyu Zheng and Ming Lu / Procedia Engineering 145 ( 2016 ) 1478 – 1485
Fig.3 Rebar Lap
4.2. Mixed Integer Programming
The second step after generating the rebar detailing schedule is to formulate the mixed integer programming (MIP)
model, as follows:
1. Decision variables: to assign decision variables for each pattern (i.e. a particular rebar detailing schedule). At
the end of the solution, the final values of decision variables would inform the detailed rebar configurations ready for
cutting leading to the minimum possible waste. The decision variables are (1) cutting length of the rebar arranged in
the long direction of the slab, denoted asݔ; (2) cutting length of the rebar arranged in the short direction of the slab,
denoted asݕ; (3) the quantity of ݔ long cutting rebar associated with cutting pattern i, denoted as ݎ௜; and (4) the
quantity of ݕ long cutting rebar associated with cutting pattern i, denoted as ݏ௜. i is the sequence number of a cutting
pattern in connection with a particular rebar detailing schedule.
Variables ݔ and ݕ are fractional numbers while ݎ௜ and ݏ௜ are integer numbers. Thus, the defined problem is a mixed
integer programming (MIP) problem. These four variables are solved for each cutting pattern of a certain rebar
detailing schedule by iteration. The values of the MIP solution are further used to calculate total waste and total cost.
2. Objective function: the objective is to minimize total cutting losses, which can be written as:
Minimize σݓ௜ݖ௜൅ሺσݖ௜ݎ௜െ݉
ଵכ݊௫ሻכݔ൅൫σݖ௜ݏ௜െ݉ଶכ݊௬൯כݕ
(2)
where ݓ௜ is the total cutting losses of cutting pattern i; ݖ௜ is the quantity of cutting pattern i; ݉ଵ is
the rows of rebar in short direction of the slab and ݉ଵൌݎ݋ݑ݊݀ݑ݌ሺܾȀܿሻ ൅ͳ; m2 is the rows of
rebar in long direction of the slab and ݉ଶൌݎ݋ݑ݊݀ݑ݌ሺܽȀܿሻ ൅ ͳ.
The objective function considered both cutting losses σݓ௜ݖ௜ǡand surplus cutting lengths in both directions denoted
asሺσݖ௜ݎ௜െ݉
ଵכ݊௫ሻכݔ൅൫σݖ௜ݏ௜െ݉ଶכ݊௬൯כݕ. Note surplus cutting lengths are redundant for cutting rebar,
which is cut from rebar stock by default once cutting patterns are defined (typically for spare use).
3. Constraints: after setting up the objective function, some constraints must be fulfilled. The constraints are simply
to satisfy the demand of the constant cutting length (i.e. ݔ and ݕ) along both directions, which can be given as Eq.(3):
෍ݖ௜ݎ௜൒݉
ଵכ݊௫
෍ݖ௜ݏ௜൒݉
ଶכ݊௬
(3)
Additional constraints should be set up to ensure that the lap length is within the range given by design code. The
lap length along the long direction can be formulated as ሺݔൈ݊
௫െܽሻȀሺ݊௫െͳሻ, while the lap length along the short
direction can be formulated as ൫ݕ ൈ ݊௬െܾ൯Ȁ൫݊௬െͳ൯. Then the lap constraint can be given as Eq.(4):
݈ଵ൏ሺݔൈ݊
௫െܽሻȀሺ݊௫െͳሻ൏݈
ଶ
݈ଵ൏൫ݕൈ݊
௬െܾ൯Ȁ൫݊௬െͳ൯൏݈ଶ
(4)
where ݈ଵ is the lower bound of lap length range; ݈ଶ is the higher bound of lap length range.

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Chaoyu Zheng and Ming Lu / Procedia Engineering 145 ( 2016 ) 1478 – 1485
4.3. Multi objective optimization
The main objective of this multi-objective optimization problem is to minimize the cutting losses (i.e. waste) of
rebar stocks and minimize total rebar installation cost (i.e. material, labor and equipment). For the multi objective
optimization problem, a Pareto optimal solution is commonly applied to simultaneously optimize each objective [13].
Once cutting pattern configuration variables are solved by applying MIP, total losses and total installation cost of
all the possible rebar detailing schedules can be calculated in Eq.(5) and Eq.(6), as follows:
Total Wastage (%) = [ σݓ௜ݖ௜൅ሺσݖ௜ݎ௜െ݉
ଵכ݊௫ሻכݔ൅൫σݖ௜ݏ௜െ݉ଶכ݊௬൯כݕሿȀσݖ௜כܮ
(5)
Total Cost =Ƚσݖ௜൅ߚ
ଵൣ݉ଵൈሺ݊௫Ȃͳሻ൅݉ଶൈ൫݊௬െͳ൯൧൅ߚ
ଶൣ݉ଵൈ݊௫൅݉ଶൈ݊௬൧൅
ߛൣ݉ଵൈ݊௫൅݉ଶൈ݊௬൧
(6)
whereȽ is the unit cost of reinforcing steel rebar stock in $/ea; ߚଵis unit labor cost to tie up one lap
in $/ea; ߚଶis unit labor cost to place one cutting rebar regardless of length in $/ea;ɀ is unit labor
and equipment cost to cut rebar stock once in $/ea.
Note σݖ௜ is the total number of rebar stock; ൣ݉ଵൈሺ݊௫Ȃͳሻ൅݉ଶൈ൫݊௬െͳ൯൧is the quantity of laps and stock
cuts;ൣ݉ଵൈ݊௫൅݉ଶൈ݊௬൧ is the quantity of cutting rebar.
After cutting losses and total installation cost of all the possible rebar detailing schedules are calculated, the
solutions are sorted by assigning a rank that represents the non-domination of each solution compared to the other
solutions. Note the non-domination is the number of times that the objective values of a certain solution is smaller
than all other possible solutions. The best solutions are then selected by comparing the ranks of all the possible
solutions. The optimization process is illustrated with a flowchart shown in Figure 4:
Population of Generated Solutions
Solution 1: Waste w1; Cost: c1
Solution 2: Waste w2; Cost: c2
Solution 3: Waste w3; Cost: c3
Solution 4: Waste w4; Cost: c4
Solution 5: Waste w5; Cost: c5
ĂĂ
Start
Assign ranks to solutions
Solution 1: Rank: 3;
Solution 2: Rank: 1;
Solution 3: Rank: 15
Solution 4: Rank: 9
Solution 5: Rank: 20
ĂĂ
Pareto optimal
solution Finish
Calculate non-domination
Solution 1: Waste 4; Cost Rank: 7
Solution 2: Waste 8; Cost Rank: 20
Solution 3: Waste 14; Cost Rank: 3
Solution 4: Waste 7; Cost Rank: 3
Solution 5: Waste 1; Cost Rank: 7
ĂĂ
Fig.4 Flowchart of Multi Objective Optimization Process
5. Case study
To illustrate and verify the proposed approach, a case study based on a reinforced concrete (RC) slab was chosen.
The RC slab is the work package of a one-story garage building construction in Alberta, Canada. The RC slab which
is 70 feet long and 55 feet wide is reinforced with steel rebar along both long and short directions. The steel rebar is
placed on the slab spaced at 1 foot along both directions. Confirmed by an experienced field engineer, a maximum of
6 steel rebars of identical length is allowed to be placed along the long direction while a maximum of 5 steel rebar of
identical length is allowed to be placed along the short direction. Lap length is from 35d to 45d (d is the diameter of
rebar). Due to resource constraints, the only available steel stock size is 20M rebar which is 30 feet long. The material
manager and the superintendent would both benefit from identifying the optimized solution for the rebar detailing
schedule on the slab in terms of minimized cutting losses and total installation cost.
In the case study, the parameters of the objective function are determined by empirical and historical data. Steel
cost for 20M rebar is $1/ft; diameter of 20M rebar is 0.064 feet; ironman hourly rate is $60/hr; cutting machine hourly
rate is $155/hr; average cutting time for one cut is 3 mins including loading, cutting and unloading; ground labor
hourly rate is $45/hr; moving and placing one cutting steel rebar (regardless of length) consumes an average of 5 mins;
tying one steel rebar lap consumes an average of 3 mins; working efficiency factor is 0.8; and 2 ironmen and 3 ground
laborers make up the crew.

1484 Chaoyu Zheng and Ming Lu / Procedia Engineering 145 ( 2016 ) 1478 – 1485
By inputting the parameters along with mathematical formulations and equations into the Excel Solver interface,
the optimized rebar detailing schedule can be obtained. By listing all the feasible layout arrangement patterns and
undergoing the MIP process one by one, rebar detailing schedules and corresponding total cutting losses and total cost
were obtained given certain settings of input data. A total of 16 alternatives of the slab rebar detailing schedule are
obtained from optimization analysis. Among the 16 options, Pattern No.14 has the minimum percentage of cutting
losses (i.e. actual lengths of cutting losses over total stock lengths), being the most environmentally-friendly and
sustainable solution; Pattern No.1 has the minimum total cost valued at $17,246, which is the most cost-effective
option. The parameters are calculated and tabulated in Table 1.
Table 1. Steel Layout Arrangement Pattern Comparison
Pattern No.
nx
ny
Cutting
Losses (%)
Non-Domi.
(1)
Total Cost ($)
Non-Domi
(2)
Non-Domi
(1)+(2)
Rank
1
3
2
21.72%
8
17,246
15
23
3
2
3
3
29.71%
4
20,651
14
18
6
3
3
4
14.49%
12
20,696
13
25
1
4
3
5
18.86%
9
23,261
10
19
5
5
4
2
34.07%
2
21,563
12
14
8
6
4
3
27.56%
7
24,968
8
15
7
7
4
4
28.14%
6
25,013
7
13
9
8
4
5
30.88%
3
27,578
3
6
10
9
5
2
42.67%
0
25,880
6
6
10
10
5
3
46.25%
1
29,285
2
3
11
11
5
4
17.96%
10
25,974
5
15
7
12
5
5
28.92%
5
36,095
0
5
10
13
6
2
15.57%
11
23,807
11
22
4
14
6
3
3.11%
15
24,692
9
24
2
15
6
4
8.34%
14
27,258
4
18
6
16
6
5
13.21%
13
29,823
1
14
8
In addition to the two extreme optimal solutions mentioned above, by applying Pareto optimal solution to integrate
the two objectives, tradeoff between the two optimization objectives being analyzed can be made. Planners can analyze
these solutions comprehensively and select a rebar detailing schedule that strikes the optimal balance between
reducing cutting losses and the total installation cost such as Pattern No.3. This solution provides savings of 25% of
total cost compared to Pattern No.14 with an increase of only 10% of the wastage. Similarly, Pattern No.3 provides a
reduction of 7% more wastage than Pattern No.1, which increases the total cost by 12.35%.
The results shown in Table 1 offer a good "sustainable" case: the design optimum selection cannot be based on crew
installation cost alone, or material waste alone; they need to be well balanced. The analysis of this case study
emphasizes the unique and practical traits of the presented approach. It also illustrates how the approach can be
effectively used to identify a wide range of optimal plans of slab steel rebar detailing schedules. Decision makers can
make the best tradeoff decision by selecting an optimal slab rebar detailing schedule that satisfies specific requirements
of the construction project being planned.
6. Conclusion
This research has introduced an optimization method for slab rebar detailing schedule design, giving rise to the
sustainable construction plan featuring the optimal tradeoff between reducing cutting losses and lowering the total
installation cost. In the particular case of a slab rebar design, the concepts of sustainability, integrated project delivery,
and workface engineering design have been materialized in the form of mathematical programming formulations,
resulting in analytical optimal solutions ready for workface execution. In detail, the problem has been formulated in
the form of Mixed Integer Programming and Multi Objective Optimization; the optimal tradeoff plan for selecting

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Chaoyu Zheng and Ming Lu / Procedia Engineering 145 ( 2016 ) 1478 – 1485
slab rebar detailing design is thus achievable. Excel Solver is utilized to conduct the optimization in the case study.
To some extent, the proposed methodology has converted an empirical rebar arrangement problem in construction
engineering into an analytical problem for optimization. Further, the approach used for slab in this study can be
replaced by other construction components. To assist in addressing a largely empirical designing problem in a
quantitative fashion, further improvements of the research reported in this paper are anticipated in the future, as
follows:
(1) As the optimization approach cannot be detached from the empirical cost data; it is foreseen that there is a need
to improve the reliability of empirical cost data in order to achieve better optimized solutions in designing the optimal
rebar detailing schedule.
(2) In practice, weighting in regard to relative importance of each objective can be added in order to conduct
weighted tradeoff analysis.
(3) The methodology can be generalized to deal with the rebar detailing schedule design on multiple structural
components in order to maximize the potential of optimization analysis.
(4) A real-world case study, combined with optimization result comparisons with other (conventional) approaches,
could be conducted, attempting to validate the method applicability and effectiveness.
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