Applications and Overviews

**Chapter 4**

**Abstract**

Overview of Multi-Objective

Optimization Approaches in

*Ibraheem Alothaimeen and David Arditi*

Construction Project Management

The difficulties that are met in construction projects include budget issues, contractual time constraints, complying with sustainability rating systems, meeting local building codes, and achieving the desired quality level, to name but a few. Construction researchers have proposed and construction practitioners have used optimization strategies to meet various objectives over the years. They started out by optimizing one objective at a time (e.g., minimizing construction cost) while disregarding others. Because the objectives of construction projects often conflict with each other, single-objective optimization does not offer practical solutions as optimizing one objective would often adversely affect the other objectives that are not being optimized. They then experimented with multi-objective optimization. The many multi-objective optimization approaches that they used have their own advantages and drawbacks when used in some scenarios with different sets of objectives. In this chapter, a review is presented of 16 multi-objective optimization approaches used in 55 research studies performed in the construction industry and

that were published in the period 2012–2016. The discussion highlights the strengths and weaknesses of these approaches when used in different scenarios.

**Keywords:** construction project management, multi-objective optimization, evolutionary algorithms, swarm intelligence algorithms, analytic network process,

The main objective of the construction industry is to directly and indirectly provide people's daily needs. Mostly, a construction project involves the use of different resources (e.g., machinery, materials, manpower, etc.) to produce the final product (e.g., a building, a bridge, a water distribution system, etc.) that serves the targeted users' needs. The difficulties that are met in construction projects include budget limitations, contractual time constraints, safety and health issues, sustainability ratings, local building codes, the desired level of quality, to name but a few. Consequently, a construction project has multiple objectives including maximum productivity, minimum cost, minimum duration, specified quality, safety, and sustainability. Making decisions is difficult when one wants to

nature-based algorithms, Hungarian algorithm, mixed-integer nonlinear

reach the optimal solution for a combination of objectives.

programming, hybrid approaches

**1. Introduction**

**47**

#### **Chapter 4**

## Overview of Multi-Objective Optimization Approaches in Construction Project Management

*Ibraheem Alothaimeen and David Arditi*

#### **Abstract**

The difficulties that are met in construction projects include budget issues, contractual time constraints, complying with sustainability rating systems, meeting local building codes, and achieving the desired quality level, to name but a few. Construction researchers have proposed and construction practitioners have used optimization strategies to meet various objectives over the years. They started out by optimizing one objective at a time (e.g., minimizing construction cost) while disregarding others. Because the objectives of construction projects often conflict with each other, single-objective optimization does not offer practical solutions as optimizing one objective would often adversely affect the other objectives that are not being optimized. They then experimented with multi-objective optimization. The many multi-objective optimization approaches that they used have their own advantages and drawbacks when used in some scenarios with different sets of objectives. In this chapter, a review is presented of 16 multi-objective optimization approaches used in 55 research studies performed in the construction industry and that were published in the period 2012–2016. The discussion highlights the strengths and weaknesses of these approaches when used in different scenarios.

**Keywords:** construction project management, multi-objective optimization, evolutionary algorithms, swarm intelligence algorithms, analytic network process, nature-based algorithms, Hungarian algorithm, mixed-integer nonlinear programming, hybrid approaches

#### **1. Introduction**

The main objective of the construction industry is to directly and indirectly provide people's daily needs. Mostly, a construction project involves the use of different resources (e.g., machinery, materials, manpower, etc.) to produce the final product (e.g., a building, a bridge, a water distribution system, etc.) that serves the targeted users' needs. The difficulties that are met in construction projects include budget limitations, contractual time constraints, safety and health issues, sustainability ratings, local building codes, the desired level of quality, to name but a few. Consequently, a construction project has multiple objectives including maximum productivity, minimum cost, minimum duration, specified quality, safety, and sustainability. Making decisions is difficult when one wants to reach the optimal solution for a combination of objectives.

Construction practitioners have been using single-objective optimization strategies to meet the desired level of construction objectives. However, because the multiple objectives of construction projects often conflict with each other, singleobjective optimization does not offer practical solutions, as optimizing one objective would often adversely affect the other objectives that are not being optimized. As a result, some projects fail to meet some of the objectives. In order to avoid such failures, researchers have developed tools that can help efficiently manage construction projects and achieve the required objectives. These tools include many multi-objective optimization approaches, each of which has its own advantages and drawbacks when used in some scenarios with different sets of objectives.

A review is presented in this chapter of the various multi-objective optimization approaches used in recent studies in the construction industry to highlight the strengths and weaknesses of these approaches when used in different scenarios.

#### **2. Overview**

A total of 55 studies that applied multi-objective optimization methods in the construction industry are reviewed in this chapter. To avoid overlapping and redundancy of reviews with Evins' work [1], the review in this chapter includes only the recent studies which were published in the period late 2012 to early 2016. Evins [1] covered the period of 1990 to late 2012 and conducted a review of the studies that applied optimization methods in sustainable building design.

The 55 studies are reviewed relative to (1) the optimization method, (2) the project phase, (3) the optimization problem, (4) the type and number of targeted objectives, (5) the example used to test a model, and (6) the comparison with other methods when applicable.

These optimization methods were used to tackle different numbers of objectives

As expected, the large majority of the studies optimized two or three objectives that concern most practitioners. The number of times the objectives were used is presented in **Table 2**. Among the objectives used in the 55 papers, cost was the mostly optimized, accounting for 93% (51 times) of the total number of studies, duration was the second most optimized objective accounting for 42% (23 times), and the energy and environment category was the third most optimized with 31% (17 times). The rating system score was used only 3 times, i.e., in only 5% of the

GA is one of the popular evolutionary algorithms used by researchers. GA uses the concept of chromosomes to present the possible solutions in these chromosomes'strings [2]. The different aspects of each solution are positioned into the slots

studies, which represents the least optimized objective.

**construction-related studies**

**3.1 Genetic algorithms (GA)**

**49**

**3. Multi-objective optimization methods used in recent**

at a time. The number of objectives that was simultaneously optimized ranged between 2 and 7. The most common number of objectives in a study was 2 or 3 objectives (27 and 24 times, respectively) distributed by methods as shown in **Table 1**. The least common number of objectives considered in a study was 4, 6, and 7 (one time each). It should be noted that one of the 55 papers used two optimization methods, i.e., NSGA-II and PSO. Therefore, the total number of methods used

**Optimization method Number of objectives**

*Overview of Multi-Objective Optimization Approaches in Construction Project Management*

Genetic algorithms (GA) 2 3 ———— Differential evolution (DE) 1 3 ———— Strength Pareto evolutionary algorithm (SPEA) — 1 ———— Non-dominated sorting genetic algorithm-II (NSGA-II) 8 6 ———— Niched Pareto genetic algorithm (NPGA) — 1 ———— Multi-objective genetic algorithm (MOGA) 1 ——— 1 1 Particle swarm optimization (PSO) 3 3 — 2 — — Ant colony optimization (ACO) 1 ————— Analytic network process (ANP) — — 1 ——— Shuffled frog-leaping algorithm (SFLA) — 1 ———— Simulated annealing algorithm (SA) 1 ————— Plant growth simulation algorithm (PGSA) 1 ————— Hungarian algorithm (HA) 1 ————— Mixed-integer nonlinear programming (MINLP) 2 ————— Hybrid methods 6 6 ———— Total (56 methods) 27 24 1 2 1 1

**2 34567**

in the 55 papers is 56.

*Number of objectives used in the literature.*

*DOI: http://dx.doi.org/10.5772/intechopen.88185*

**Table 1.**

The number of optimization methods found in the review of the 55 papers was 16. These 16 methods and their usage frequency are presented in **Figure 1**, which shows that NSGA-II is the most used method (14 times) followed by a hybrid method (12 times) which pairs two or more methods for the optimization process. The acronyms in this figure are spelled out in **Table 1**.

**Figure 1.** *Frequency of methods used in literature.*


*Overview of Multi-Objective Optimization Approaches in Construction Project Management DOI: http://dx.doi.org/10.5772/intechopen.88185*

#### **Table 1.**

Construction practitioners have been using single-objective optimization strate-

A review is presented in this chapter of the various multi-objective optimization

A total of 55 studies that applied multi-objective optimization methods in the construction industry are reviewed in this chapter. To avoid overlapping and redundancy of reviews with Evins' work [1], the review in this chapter includes only the recent studies which were published in the period late 2012 to early 2016. Evins [1] covered the period of 1990 to late 2012 and conducted a review of the studies that applied optimization methods in sustainable building design.

The 55 studies are reviewed relative to (1) the optimization method, (2) the project phase, (3) the optimization problem, (4) the type and number of targeted objectives, (5) the example used to test a model, and (6) the comparison with other

The number of optimization methods found in the review of the 55 papers was 16. These 16 methods and their usage frequency are presented in **Figure 1**, which shows that NSGA-II is the most used method (14 times) followed by a hybrid method (12 times) which pairs two or more methods for the optimization process.

gies to meet the desired level of construction objectives. However, because the multiple objectives of construction projects often conflict with each other, singleobjective optimization does not offer practical solutions, as optimizing one objective would often adversely affect the other objectives that are not being optimized. As a result, some projects fail to meet some of the objectives. In order to avoid such failures, researchers have developed tools that can help efficiently manage construction projects and achieve the required objectives. These tools include many multi-objective optimization approaches, each of which has its own advantages and

*Multicriteria Optimization - Pareto-Optimality and Threshold-Optimality*

drawbacks when used in some scenarios with different sets of objectives.

**2. Overview**

methods when applicable.

**Figure 1.**

**48**

*Frequency of methods used in literature.*

The acronyms in this figure are spelled out in **Table 1**.

approaches used in recent studies in the construction industry to highlight the strengths and weaknesses of these approaches when used in different scenarios.

*Number of objectives used in the literature.*

These optimization methods were used to tackle different numbers of objectives at a time. The number of objectives that was simultaneously optimized ranged between 2 and 7. The most common number of objectives in a study was 2 or 3 objectives (27 and 24 times, respectively) distributed by methods as shown in **Table 1**. The least common number of objectives considered in a study was 4, 6, and 7 (one time each). It should be noted that one of the 55 papers used two optimization methods, i.e., NSGA-II and PSO. Therefore, the total number of methods used in the 55 papers is 56.

As expected, the large majority of the studies optimized two or three objectives that concern most practitioners. The number of times the objectives were used is presented in **Table 2**. Among the objectives used in the 55 papers, cost was the mostly optimized, accounting for 93% (51 times) of the total number of studies, duration was the second most optimized objective accounting for 42% (23 times), and the energy and environment category was the third most optimized with 31% (17 times). The rating system score was used only 3 times, i.e., in only 5% of the studies, which represents the least optimized objective.

#### **3. Multi-objective optimization methods used in recent construction-related studies**

#### **3.1 Genetic algorithms (GA)**

GA is one of the popular evolutionary algorithms used by researchers. GA uses the concept of chromosomes to present the possible solutions in these chromosomes'strings [2]. The different aspects of each solution are positioned into the slots

#### *Multicriteria Optimization - Pareto-Optimality and Threshold-Optimality*


*Min TLC* ¼ *OC* þ *FC* þ *SC* þ *LC* (1)

*SSi* � *ESi TFi*

(2)

*i*¼1

*<sup>N</sup>*<sup>0</sup> �<sup>X</sup> *N*0

*Min SCI* <sup>¼</sup> <sup>1</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.88185*

standard GA.

processes [12].

For example:

NSGA-II.

**51**

**3.2 Differential evolution (DE)**

*<sup>N</sup>*<sup>0</sup> �<sup>X</sup> *N*0

*i*¼1

*Overview of Multi-Objective Optimization Approaches in Construction Project Management*

where,*TLC* = total logistics costs; *OC* = ordering cost; *FC* = financing cost; *SC* = stock-out cost; *LC* = layout cost; *SCI* = schedule criticality index; *N*<sup>0</sup> = number of noncritical activities; *CIi* = criticality index of activity *i*; *SSi* = scheduled start time of activity *i*; *ESi* = early start time of activity *i*; and *TFi* = total float of activity *i*. Because the search space is large and the problem is complex, the authors justified the use of a GA model that involves 152 decision variables and 462 constraints. The model generated 361 optimal solutions. For equipment management problems, Xu et al. [9] proposed dynamic programming-based GA because they believed it would be capable of solving this type of problem more efficiently than traditional methods. The goal of the method was to minimize the project's total cost and maximize equipment operations such that in case of equipment failure there would be an equipment available. Moreover, to make the method more reliable, the failure rate of the equipment was considered a fuzzy variable. An actual hydropower project in China was selected to test the model. Under the same environment, the proposed algorithm performed better in searching than the

In summary, there is evidence that GA can optimize different objectives in the construction industry in the field of scheduling, sustainability, and site operation.

The DE approach is efficient and has low algorithmic complexity. There is also some evidence of its effectiveness in tackling problems of continuous optimization with different types of constraints and functions [10]. The members of the population in DE use floating-points which identify each member's direction and distance [11]. Therefore, the main concept behind the DE approach is that it creates a new population member with a vector that has the difference between two members' vectors; that process is done by the mutation and crossover

DE has proved its effectiveness in complex planning and scheduling problems by optimizing cost and time in addition to quality, environmental impact, or resources.

• Narayanan and Suribabu [13] applied DE to assist contractors in optimizing their plans for subcontracting in terms of cost, time and quality. To examine the model, they used a 7-activity and an 18-activity project. By comparison, the DE model generated better solutions than ant colony optimization (ACO) for

• Alternatively, Cheng and Tran [14] used a two-phase DE model on a 37 activity warehouse project to minimize total project cost and duration, while accounting for resource constraints. In the first phase, a multiple objective DE model was used to find the optimal tradeoff between time and cost in construction activities. Based on the solution obtained in the first phase, the best schedule was found within resource constraints in the second phase. A comparison of the results showed that the developed model outperformed three evolutionary algorithms: DE, particle swarm optimization (PSO) and

cost in the first case, and for cost and time in the second case.

*CIi* <sup>¼</sup> <sup>1</sup>

#### **Table 2.**

*Number of times the objectives were used in the 55 studies.*

which form the string [3]. A new set of solutions are found by the crossover between two strings (parent strings), and the new strings (children) will inherit the best features of the parent strings.

In construction-related fields, GA has been applied in many multi-objective optimization problems. For example:


*Overview of Multi-Objective Optimization Approaches in Construction Project Management DOI: http://dx.doi.org/10.5772/intechopen.88185*

$$\text{Min TLC} = \text{OC} + \text{FC} + \text{SC} + \text{LC} \tag{1}$$

$$Min\text{ SCI} = \frac{1}{N'} \times \sum\_{i=1}^{N'} \text{CI}\_i = \frac{1}{N'} \times \sum\_{i=1}^{N} \frac{\text{SS}\_i - \text{ES}\_i}{\text{TF}\_i} \tag{2}$$

where,*TLC* = total logistics costs; *OC* = ordering cost; *FC* = financing cost; *SC* = stock-out cost; *LC* = layout cost; *SCI* = schedule criticality index; *N*<sup>0</sup> = number of noncritical activities; *CIi* = criticality index of activity *i*; *SSi* = scheduled start time of activity *i*; *ESi* = early start time of activity *i*; and *TFi* = total float of activity *i*.

Because the search space is large and the problem is complex, the authors justified the use of a GA model that involves 152 decision variables and 462 constraints. The model generated 361 optimal solutions. For equipment management problems, Xu et al. [9] proposed dynamic programming-based GA because they believed it would be capable of solving this type of problem more efficiently than traditional methods. The goal of the method was to minimize the project's total cost and maximize equipment operations such that in case of equipment failure there would be an equipment available. Moreover, to make the method more reliable, the failure rate of the equipment was considered a fuzzy variable. An actual hydropower project in China was selected to test the model. Under the same environment, the proposed algorithm performed better in searching than the standard GA.

In summary, there is evidence that GA can optimize different objectives in the construction industry in the field of scheduling, sustainability, and site operation.

#### **3.2 Differential evolution (DE)**

which form the string [3]. A new set of solutions are found by the crossover between two strings (parent strings), and the new strings (children) will inherit the

[4] used GA in a single-family housing unit to optimize operational

In construction-related fields, GA has been applied in many multi-objective

**Objective Number of times objective used in studies**

Cost 51 Duration 23 Quality 7 Resources 7 Energy and environment 17 Thermal 13 Safety 6 Rating system score 3 Other 23

*Multicriteria Optimization - Pareto-Optimality and Threshold-Optimality*

• GA was used to improve sustainability in housing units. Karatas and El-Rayes

• GA was used to solve conflicting objectives in construction scheduling. For instance, Agrama [5] used GA to optimize building schedules. The author analyzed a 5-storey building and used nine scenarios for the weights of three objectives: project duration, total actual crews, and total interruptions for all activities. The model was implemented in Excel (Evolver) and solved by GA. In addition, it was found that the model performs consistently and can be used with both the critical path and line of balance methods. Moreover, the results obtained were identical to those in the literature but required less time and effort. Alternatively, Aziz et al. [6] introduced a method that combines CPM with GA to optimize the utilization of resources for mega construction projects in terms of time, cost, and quality. An 18-activity schedule was tested using the proposed method. To avoid complexity, the five decision variables which were construction materials, crew formation, crew overtime policy, machinery efficiency, and contractor class were all combined into a single decision variable called resource utilization. In this test, 305 optimal solutions were identified. Additionally, the results showed that the model outperformed the

approach used by Feng et al. [7] with the same case example.

• GA was used in managing site operations. For example, in material logistics, Said and El-Rayes [8] presented an example of a 10-storey building consisting of 107 activities with four temporary facilities. The aim of the model was to minimize total construction logistics costs (Eq. (1)) and minimize project

environmental performance, social quality of life, and life cycle cost. They used 33 decision variables in the model and computed in 47.5 hours 210 nearoptimal solutions within a large search space of configurations and decisions

best features of the parent strings.

**Table 2.**

optimization problems. For example:

*Number of times the objectives were used in the 55 studies.*

(more than 2.6 quadrillion).

schedule criticality (Eq. (2)).

**50**

The DE approach is efficient and has low algorithmic complexity. There is also some evidence of its effectiveness in tackling problems of continuous optimization with different types of constraints and functions [10]. The members of the population in DE use floating-points which identify each member's direction and distance [11]. Therefore, the main concept behind the DE approach is that it creates a new population member with a vector that has the difference between two members' vectors; that process is done by the mutation and crossover processes [12].

DE has proved its effectiveness in complex planning and scheduling problems by optimizing cost and time in addition to quality, environmental impact, or resources. For example:


$$\text{Min } T = \sum\_{n=1}^{l} T\_n^{\text{S}\_n} = \text{Max}\_{\mathbb{W}n}(\text{ES}\_{\mathbb{n}} + D\_n) \tag{3}$$

$$\text{Min } \mathbf{C} = \sum\_{n=1}^{N} \mathbf{Cost}\_n^{\mathbf{S}\_n} \tag{4}$$

Four solutions (maximum profit, minimum finance, minimum resource idle days, and the top compromised solution) were drawn from the 48 solutions obtained in the Pareto-optimal front. Clustering the Pareto solutions set was used to keep it within a manageable size. Nevertheless, because of the clustering, this method may

*Overview of Multi-Objective Optimization Approaches in Construction Project Management*

By optimizing the construction objectives of profit and resources, SPEA has verified its efficiency in the scheduling field. However, the clustering method proposed by Elazouni and Abido [18] should be avoided when using SPEA in order

One of the most powerful tools of genetic algorithms is NSGA-II. It uses the nondominated sorting for the solutions in the population. The non-dominated solutions are ranked at every iteration, and are excluded from the population in every iteration afterwards. In addition, in each ranked-solution set, the solutions are compared to each other by their crowding formation. In the crowding step, the position of a single solution is measured by its distance from the adjacent solutions' points, and based on its distance, the solution is assigned with a rank, as the best ranks start

NSGA-II has been used to solve multi-objective problems aimed at the optimal-

• Eliades et al. [19] used NSGA-II to optimally select the installation locations for indoor air quality sensors, in terms of number of sensors, and average and worst-case impact damage while considering the building's usage in the parameters. A simple 5-room building and a 14-room house were studied to

contamination scenarios, respectively. Grid and random sampling were used to construct the contamination scenarios, and the multi-zone building program

• In zero-energy-building (ZEB), Hamdy et al. [20] used a modified version of NSGA-II to find solutions for the optimal cost and nearly zero energy building performance with respect to the guidelines of European directives for the energy performance of buildings. Due to the large number of combinations, the solution space was divided into three stages. The total number of combinations (179, 712) in the first stage were searched in 800 runs.

• Huws and Jankovic [21] took into account future weather changes that could affect retrofitting strategies. These weather changes may eventually unsettle the performance of zero-carbon buildings by increasing the carbon emissions or cost, or in some cases a combination of these may create thermal discomfort. For that reason and to achieve optimal solutions for retrofit, environmental, social, and economic constraints were considered in optimizing the objectives of minimizing cost, CO2, and thermal discomfort. A simple 60 m<sup>2</sup> box model was created using the DesignBuilder program. DesignBuilder and JEPlus were used to perform the optimization process. NSGA-II within JEPlus was used for its capability of searching a large solutions space, and to avoid being stuck in a local suboptimum. The results indicated that there is an applicable alternative

ity of energy consumption and sustainability in buildings. For instance:

illustrate the performance of the proposed model, with 5 and 2310

to avoid the elimination of some extreme Pareto solutions. New clustering

result in the loss of some extreme Pareto solutions.

*DOI: http://dx.doi.org/10.5772/intechopen.88185*

approaches should be explored in upcoming studies.

from the shortest distance to the longest one [10].

CONTAM simulated them.

for both current and future weather.

**53**

**3.4 Non-dominated sorting genetic algorithm-II (NSGA-II)**

*Min LHEN* ¼ *LHE* þ *LHN 1*ð Þ þ *W if* SS ¼ 3 ð Þ *three* � *shift system* (5)

$$\text{Min LHNE} = \text{LHE if SS} = \text{2 (two} - \text{shift system)} \tag{6}$$

where in Eq. (3), *TSn <sup>n</sup>* is the duration of the activity *n{n = 1, 2, …, l}* on the critical path for a specific option of resources (*Sn*); *l* is the total number of critical activities on a specific critical path; *ESn* is the earliest start of activity *n; Dn* is the duration of activity *n.* In Eq. (4), *CostSn <sup>n</sup>* is the total cost which includes direct and indirect cost of activity *n* for a specific option of resources (*Sn*); *N* is the total number of activities. In Eqs. (5) and (6), *LHE* is the total number of evening shift work hours and *LHN* is the total number of night shift work hours. Because risks faced in night shiftwork are typically higher than in other shifts, *W* is the defined weight that represents the relative importance of minimizing *LHN*.

A 15-activity and a 60-activity project were used to test the model. In just one run, the model was capable of finding Pareto-optimal solutions to solve the objectives of the problem.

It can be concluded that the DE algorithm is capable of optimizing several objectives of time, cost, resource utilization, and environmental impact. Moreover, as DE and its variations successfully optimized those objectives, they also surpassed ACO, PSO, and NSGA-II in construction scheduling optimization.

#### **3.3 Strength Pareto evolutionary algorithm (SPEA)**

SPEA works by archiving the non-dominated solutions found in the Pareto-front at every iteration. Then, based on the number of solutions it dominates, each solution in the archive is ranked with a strength rate [10, 17].

In dealing with scheduling problems, SPEA was proposed by Elazouni and Abido [18] to optimize the three conflicting objectives of maximizing profit and minimizing required finance and resource idle days. The study used two examples from the literature to test the efficiency and scalability of the model. In the first example, the model was tested for its effectiveness in solving a 9-activity project. The model confirmed its robustness by achieving 50 identical solutions. By searching these solutions using a fuzzy based method, the top ones were selected. In the second example, an 18-activity project was used to assess the model's scalability.

*Overview of Multi-Objective Optimization Approaches in Construction Project Management DOI: http://dx.doi.org/10.5772/intechopen.88185*

Four solutions (maximum profit, minimum finance, minimum resource idle days, and the top compromised solution) were drawn from the 48 solutions obtained in the Pareto-optimal front. Clustering the Pareto solutions set was used to keep it within a manageable size. Nevertheless, because of the clustering, this method may result in the loss of some extreme Pareto solutions.

By optimizing the construction objectives of profit and resources, SPEA has verified its efficiency in the scheduling field. However, the clustering method proposed by Elazouni and Abido [18] should be avoided when using SPEA in order to avoid the elimination of some extreme Pareto solutions. New clustering approaches should be explored in upcoming studies.

#### **3.4 Non-dominated sorting genetic algorithm-II (NSGA-II)**

One of the most powerful tools of genetic algorithms is NSGA-II. It uses the nondominated sorting for the solutions in the population. The non-dominated solutions are ranked at every iteration, and are excluded from the population in every iteration afterwards. In addition, in each ranked-solution set, the solutions are compared to each other by their crowding formation. In the crowding step, the position of a single solution is measured by its distance from the adjacent solutions' points, and based on its distance, the solution is assigned with a rank, as the best ranks start from the shortest distance to the longest one [10].

NSGA-II has been used to solve multi-objective problems aimed at the optimality of energy consumption and sustainability in buildings. For instance:


• Subsequently, Cheng and Tran [15] proposed opposition-based multi-objective DE. The aim was to optimize construction products in terms of cost, time and environmental impact. The model used opposition-based learning to increase precision and convergence speed. A tunnel project consisting of 25 activities was used to test the model. The proposed model was superior compared to NSGA-II, PSO, and DE algorithms. The exact approach also outperformed these algorithms in a similar study conducted by Cheng and Tran [16].

• The goal of the Cheng and Tran [16] study was to minimize project time (Eq. (3)), project cost (Eq. (4)), and the utilization of resources (Eqs. (5) and

*Min T* <sup>¼</sup> <sup>X</sup>

represents the relative importance of minimizing *LHN*.

**3.3 Strength Pareto evolutionary algorithm (SPEA)**

solution in the archive is ranked with a strength rate [10, 17].

*l*

*Multicriteria Optimization - Pareto-Optimality and Threshold-Optimality*

*TSn*

*Min* <sup>C</sup> <sup>¼</sup> <sup>X</sup>

*N*

*CostSn*

*Min LHNE* ¼ *LHE if* SS ¼ 2 two ð Þ � *shift system* (6)

*<sup>n</sup>* is the duration of the activity *n{n = 1, 2, …, l}* on the critical

*<sup>n</sup>* is the total cost which includes direct and indirect cost

*n*¼1

*Min LHEN* ¼ *LHE* þ *LHN 1*ð Þ þ *W if* SS ¼ 3 ð Þ *three* � *shift system* (5)

path for a specific option of resources (*Sn*); *l* is the total number of critical activities on a specific critical path; *ESn* is the earliest start of activity *n; Dn* is the duration of

A 15-activity and a 60-activity project were used to test the model. In just one run, the model was capable of finding Pareto-optimal solutions to solve the objec-

SPEA works by archiving the non-dominated solutions found in the Pareto-front

In dealing with scheduling problems, SPEA was proposed by Elazouni and Abido [18] to optimize the three conflicting objectives of maximizing profit and minimizing required finance and resource idle days. The study used two examples from the literature to test the efficiency and scalability of the model. In the first example, the model was tested for its effectiveness in solving a 9-activity project. The model confirmed its robustness by achieving 50 identical solutions. By searching these solutions using a fuzzy based method, the top ones were selected. In the second example, an 18-activity project was used to assess the model's scalability.

at every iteration. Then, based on the number of solutions it dominates, each

It can be concluded that the DE algorithm is capable of optimizing several objectives of time, cost, resource utilization, and environmental impact. Moreover, as DE and its variations successfully optimized those objectives, they also surpassed

ACO, PSO, and NSGA-II in construction scheduling optimization.

of activity *n* for a specific option of resources (*Sn*); *N* is the total number of activities. In Eqs. (5) and (6), *LHE* is the total number of evening shift work hours and *LHN* is the total number of night shift work hours. Because risks faced in night shiftwork are typically higher than in other shifts, *W* is the defined weight that

*<sup>n</sup>* ¼ Max*<sup>Ɐ</sup>n*ð Þ ESn þ *Dn* (3)

*<sup>n</sup>* (4)

*n*¼1

(6)) in overtime shifts.

where in Eq. (3), *TSn*

activity *n.* In Eq. (4), *CostSn*

tives of the problem.

**52**

• In sustainability for low-income housing, Marzouk and Metawie [22] incorporated NSGA-II with BIM to assist the Egyptian government find solutions that best optimize those objectives. The BIM model was created using Revit. The model was defined based on the quantities and properties of the materials extracted from the BIM model. These quantities helped to find the different solutions in terms of project cost, duration and maximum LEED points. Construction productivity and cost were determined using a 44 activitiy low-income housing building. Moreover, LEED points were calculated through five credits chosen from the materials and resources category.

Site operations and planning problems were also tackled using NSGA-II.

*Overview of Multi-Objective Optimization Approaches in Construction Project Management*

II was superior to multi-objective PSO.

*DOI: http://dx.doi.org/10.5772/intechopen.88185*

algorithm as well as manual allocation.

water networks and slum areas. For example:

comparison was about the worst-case scenario.

**55**

• Fallah-Mehdipour et al. [26] applied NSGA-II to solve two tradeoff problems, time-cost and time-cost-quality, respectively. To validate the proposed method, an 18-activity and a 7-activity work schedule were utilized.

Additionally, multi-objective PSO was applied. The results showed that NSGA-

• In managing and storing materials in a construction site, Said and El-Rayes [27] presented an automated module, which imports its data from BIM files and historical schedule data. A module in the system was named construction logistics planning (CLP) and aimed to minimize the cost of logistics and the criticality of the schedule. These objectives were optimized by tackling four decision variables using NSGA-II. An application model of a 10-storey building project was used to apply the optimization process. The automated system generated better results compared to using CLP alone. A total of 361 optimal solutions were produced within 65 hours. Unlike CLP, which considered the utilization of exterior site space and disregarded the interior one, the system generated the solutions accounting

• In site operations, Parente et al. [28] proposed NSGA-II to optimize the allocation of compaction equipment within the criteria of cost and time associated with earthworks in large infrastructure projects. Additionally, linear programming was used for the allocation of the remaining equipment such as trucks and excavators. The proposed method which uses an actual construction site as a case study proved to be superior to the S-metric selection evolutionary

NSGA-II was used to find solutions in problems involving upgrade plans for

• Creaco et al. [29] divided the construction phases of a water network upgrade into four phases, considering the different phases of upgrades to the water network in a 100-year plan of possible upgrades. NSGA-II was used with a model of six network nodes and eight pipe laying locations to find the optimal solutions within the two objective functions: maximizing the minimum pressure and minimizing the cost, while the pipe diameters are acting as the decision variables. The proposed approach provided better results than the studies that used single phasing, by giving the optimal solution for maintaining the water distribution and pressure quality through the time of upgrade phases. In a similar study, Creaco et al. [30] proposed the use of NSGA-II while considering an additional factor to the study, which was the uncertainty of water demand. The authors determined the uncertainty using a probabilistic approach. Based on an example with 26 network nodes and 31 pipe laying locations. The probabilistic approach was compared with the deterministic approach used by Creaco et al. [29]. The results revealed that the solutions obtained by the probabilistic approach had higher costs than the solutions of the deterministic approach, especially in the first phase. However, the probabilistic solutions generated better results in terms of costs when the

For instance:

for both spaces.


$$\text{Min Cost}\_{total} = \text{Cost}\_{cub} + \text{Cost}\_{elec\\_n} \tag{7}$$

$$\text{Min }Imp\_{\text{total}} = Imp\_{\text{cub}} + Imp\_{\text{elec}} \tag{8}$$

where *Costtotal* is the total cost, *Costcub* is the cost of the materials used; *Costelec\_n* is the cost of the electricity consumed over the operational phase (*n* years); *Imptotal* is the total environmental impact; *Impcub* is the total impact of the materials used; and *Impelec* is the impact of the consumed electricity over the operational phase.

The authors used the example of a cubicle without insulation to compare the different results collected from using two cases of insulation. In the first case, similar thicknesses were used over the cubicle, while in the second case, different thicknesses were considered. Three materials were considered in the insulation selection process (polyurethane, mineral wool, and polystyrene). From the results, the polyurethane insulation was the least costly solution, whereas the optimal environmental impact solution was mineral wool insulation. The proposed methodology could improve the costs and environmental impacts by almost 40% when compared to a non-insulated cubicle.

*Overview of Multi-Objective Optimization Approaches in Construction Project Management DOI: http://dx.doi.org/10.5772/intechopen.88185*

Site operations and planning problems were also tackled using NSGA-II. For instance:


NSGA-II was used to find solutions in problems involving upgrade plans for water networks and slum areas. For example:

• Creaco et al. [29] divided the construction phases of a water network upgrade into four phases, considering the different phases of upgrades to the water network in a 100-year plan of possible upgrades. NSGA-II was used with a model of six network nodes and eight pipe laying locations to find the optimal solutions within the two objective functions: maximizing the minimum pressure and minimizing the cost, while the pipe diameters are acting as the decision variables. The proposed approach provided better results than the studies that used single phasing, by giving the optimal solution for maintaining the water distribution and pressure quality through the time of upgrade phases. In a similar study, Creaco et al. [30] proposed the use of NSGA-II while considering an additional factor to the study, which was the uncertainty of water demand. The authors determined the uncertainty using a probabilistic approach. Based on an example with 26 network nodes and 31 pipe laying locations. The probabilistic approach was compared with the deterministic approach used by Creaco et al. [29]. The results revealed that the solutions obtained by the probabilistic approach had higher costs than the solutions of the deterministic approach, especially in the first phase. However, the probabilistic solutions generated better results in terms of costs when the comparison was about the worst-case scenario.

• In sustainability for low-income housing, Marzouk and Metawie [22] incorporated NSGA-II with BIM to assist the Egyptian government find solutions that best optimize those objectives. The BIM model was created using Revit. The model was defined based on the quantities and properties of the materials extracted from the BIM model. These quantities helped to find the different solutions in terms of project cost, duration and maximum LEED points. Construction productivity and cost were determined using a 44 activitiy low-income housing building. Moreover, LEED points were calculated

*Multicriteria Optimization - Pareto-Optimality and Threshold-Optimality*

through five credits chosen from the materials and resources category.

• Kasinalis et al. [23] studied the improvement of indoor environment while reducing the energy consumption in climate adaptive building shells, and quantified the impact of using seasonal adaptation façade on those objectives. The example of an office zone model was used to evaluate the approach. The combination of daylight and energy simulations were utilized with NSGA-II to perform multi-objective optimization on that example. The optimization process considered six design parameters for the façade. The results showed that using a seasonal adaptation façade with these parameters is more efficient than a non-adaptive façade, since it can save up to 18% of energy consumption

• Inyim et al. [24] approached the problem of building components and material selection by using (SimulEICon) a BIM tool that simulates the environmental impact in buildings. The optimization process of time, cost and CO2 emissions was performed by NSGA-II. The case study was an actual zero net energy house. The model considered 16 activities and 185 building components. It was found that some of the combinations of components suggested by SimulEICon matched the original component combinations used in the existing house. However, SimulEICon lacked the ability to account for more than three

• Carreras et al. [25] introduced an approach for selecting the thickness of

insulation material for building shells. The objective of the study was to select the best option for the insulation that optimizes the costs (Eq. (7)) and environmental impacts (Eq. (8)) associated with energy consumption.

where *Costtotal* is the total cost, *Costcub* is the cost of the materials used; *Costelec\_n* is the cost of the electricity consumed over the operational phase (*n* years); *Imptotal* is the total environmental impact; *Impcub* is the total impact of the materials used; and *Impelec* is the impact of the consumed electricity over the operational phase. The authors used the example of a cubicle without insulation to compare the different results collected from using two cases of insulation. In the first case, similar thicknesses were used over the cubicle, while in the second case, different thicknesses were considered. Three materials were considered in the insulation selection process (polyurethane, mineral wool, and polystyrene). From the results, the polyurethane insulation was the least costly solution, whereas the optimal environmental impact solution was mineral wool insulation. The proposed methodology could improve the costs and environmental impacts by almost 40% when compared

*Min Costtotal* ¼ *Costcub* þ *Costelec*\_*<sup>n</sup>* (7) *Min Imptotal* ¼ *Impcub* þ *Impelec* (8)

and enhance the quality of the indoor environment.

objectives.

to a non-insulated cubicle.

**54**

• In uneven ground levels of slum areas, El-Anwar and Abdel Aziz [31] used an example of nine-zone slum area with a population of 2770 families to select the optimal upgrade plan. The optimization process involved three objectives: maximization of benefit of proposed upgrading projects, minimization of costs and socioeconomic disruption for the families. Due to its superiority over other GAs in solving multi-objective problems, NSGA-II was selected to solve the problem in which it generated 2000 solutions in less than 1 minute. Nevertheless, the time schedules module was not included in the model hence affecting its robustness.

by another individual in the population then it is assigned with the rank of one. But if an individual is dominated by other individuals then it is assigned with a rank corresponding to the total number of dominating individuals plus one [36].

*Overview of Multi-Objective Optimization Approaches in Construction Project Management*

• MOGA has shown its capabilities in achieving optimal structural design. For example, Richardson et al. [37] tackled the design problem of an x-bracing structural system for a building façade. Minimizing the cost of the bracing connections and the effectiveness of the bracings were the objectives under the

min*<sup>x</sup> f x*ð Þ¼ *<sup>f</sup>* <sup>1</sup>*; <sup>f</sup>* <sup>2</sup>

where *f1* is the cost objective function expressed in Eq. (10), *x* is the variable vector of length *n*, and *f2* is the relative tier deflection objective function expressed

*Min* <sup>f</sup> <sup>1</sup> <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*

∣*d*1∣ *h*1 *;*

of tier *j;* and *dj* is the measured deflection of tier *j* from rest position.

*i*¼1

where *ai* is a weighting coefficient related to the grouping of components based on symmetry; *xi* is the topology variable associated with bracing(s) *i; hj* is the height

While the constraints change as the design progresses, the proposed approach dynamically adapts to those constraints. Museum façades were picked to test the

• In reducing the energy consumed and environmental impact in buildings, Baglivo et al. [38] have used an improved version of MOGA (MOGA-II) on combinations of sustainable building materials for external walls of zero energy buildings, to achieve the best optimal solutions in balancing the life cycle cost and energy consumption. The materials were tested according to their thermal characteristics based on the Mediterranean climate. The assessment of material

combinations was carried on six thermal-related objectives. The study

concluded that the best selection of materials for external walls was by placing the insulation coating on the external side of the wall, while placing the high internal capacity material on the interior side. Similarly, Baglivo et al. [39] have conducted a study that added one more objective to the same six objectives.

The pattern of flocking birds and fish was the inspiration of PSO. In PSO, a set of solutions is called swarm, while a solution is called particle [26]. The particles are positioned in a D-dimensional search space. In each step, every particle changes its velocity to move toward the best solution and toward the global best solution [40]. Different issues of construction engineering and management were tackled by PSO. Some studies proposed PSO to solve site planning problems. For instance:

• Xu and Li [41] proposed permutation-based PSO to solve the planning problem of a dynamic construction site layout, in which ordinal numbers assigned to the

∣*d*<sup>2</sup> � *d*1∣ *h*2

*;*

∣*d*<sup>3</sup> � *d*2∣ *h*3 � � (11)

� � (9)

*ai:xi* (10)

multi-objective topology optimization process (Eq. (9)).

*Min f <sup>2</sup>* ¼ max

performance of the optimization method.

*DOI: http://dx.doi.org/10.5772/intechopen.88185*

**3.7 Particle swarm optimization (PSO)**

**57**

in Eq. (11).

• Brownlee and Wright [32] analyzed the relationship between design objectives and the effectiveness of design variables on the design objectives by using NSGA-II. They sorted the objectives by simple ranking. The approach was performed on a five-zone building with only two design objectives. The objectives to be minimized were total annual energy use and capital cost, and the design variables were 52 in total. Forty-nine solutions were generated using NSGA-II. However, the proposed approach failed to discriminate the distance variables which are the variables that measure the sets from the true Paretooptimal set from the floating variables which are the variables that have no effect on the objective function.

As the above-cited studies show, the NSGA-II proved its capability in optimizing for scheduling, urban planning, infrastructure, sustainability, energy and environmental design, and resource management. In addition to its superiority over other GAs, NSGA-II has also outperformed other methods in some fields. One of those is the multi-objective PSO applied to scheduling problems.

#### **3.5 Niched Pareto genetic algorithm (NPGA)**

The tournament selection among a population's individuals and Pareto dominance are the two basic ideas of NPGA's process. The selection process is based on the dominance of two randomly selected individuals from the population. To determine which individual of these two is dominant over the other, another set of individuals are picked and used to go against the two competing individuals, to examine the level of the two competing individuals in dominating each individual of the set. The winning criterion is defined by Pareto-front dominance. Therefore, one of the two competing individuals is selected if the other is dominated by one of the individuals in the set [33, 34].

Kim et al. [35] used NPGA to optimize cost, time and resource utilization. They optimized the three objectives at the same time. To test the performance of the method, they conducted two case studies. The first case had 11 activities, and measured the method's efficiency in solving the tradeoff problem between cost and time. In addition to the objectives in the first case, the second case extended the examination of the approach by including the resource-leveling index as an objective. The results showed that this method could provide decision makers with different solutions to enable them selecting the one that meets their preferences.

#### **3.6 Multi-objective genetic algorithm (MOGA)**

MOGA is an advanced version of traditional GA. The difference between MOGA and GA is the individual fitness assignment, while the remaining steps are followed as in GA. In MOGA, ranking is assigned for each individual in the population. The rank is assigned based on individual's dominance, if the individual is not dominated *Overview of Multi-Objective Optimization Approaches in Construction Project Management DOI: http://dx.doi.org/10.5772/intechopen.88185*

by another individual in the population then it is assigned with the rank of one. But if an individual is dominated by other individuals then it is assigned with a rank corresponding to the total number of dominating individuals plus one [36].

• MOGA has shown its capabilities in achieving optimal structural design. For example, Richardson et al. [37] tackled the design problem of an x-bracing structural system for a building façade. Minimizing the cost of the bracing connections and the effectiveness of the bracings were the objectives under the multi-objective topology optimization process (Eq. (9)).

$$\min\_{\mathbf{x}} f(\mathbf{x}) = \begin{pmatrix} f\_1, f\_2 \end{pmatrix} \tag{9}$$

where *f1* is the cost objective function expressed in Eq. (10), *x* is the variable vector of length *n*, and *f2* is the relative tier deflection objective function expressed in Eq. (11).

$$\text{Min } \mathbf{f}\_1 = \sum\_{i=1}^n a\_i.x\_i \tag{10}$$

$$\dim f\_2 = \max\left\{ \frac{|d\_1|}{h\_1}, \frac{|d\_2 - d\_1|}{h\_2}, \frac{|d\_3 - d\_2|}{h\_3} \right\} \tag{11}$$

where *ai* is a weighting coefficient related to the grouping of components based on symmetry; *xi* is the topology variable associated with bracing(s) *i; hj* is the height of tier *j;* and *dj* is the measured deflection of tier *j* from rest position.

While the constraints change as the design progresses, the proposed approach dynamically adapts to those constraints. Museum façades were picked to test the performance of the optimization method.

• In reducing the energy consumed and environmental impact in buildings, Baglivo et al. [38] have used an improved version of MOGA (MOGA-II) on combinations of sustainable building materials for external walls of zero energy buildings, to achieve the best optimal solutions in balancing the life cycle cost and energy consumption. The materials were tested according to their thermal characteristics based on the Mediterranean climate. The assessment of material combinations was carried on six thermal-related objectives. The study concluded that the best selection of materials for external walls was by placing the insulation coating on the external side of the wall, while placing the high internal capacity material on the interior side. Similarly, Baglivo et al. [39] have conducted a study that added one more objective to the same six objectives.

#### **3.7 Particle swarm optimization (PSO)**

The pattern of flocking birds and fish was the inspiration of PSO. In PSO, a set of solutions is called swarm, while a solution is called particle [26]. The particles are positioned in a D-dimensional search space. In each step, every particle changes its velocity to move toward the best solution and toward the global best solution [40].

Different issues of construction engineering and management were tackled by PSO. Some studies proposed PSO to solve site planning problems. For instance:

• Xu and Li [41] proposed permutation-based PSO to solve the planning problem of a dynamic construction site layout, in which ordinal numbers assigned to the

• In uneven ground levels of slum areas, El-Anwar and Abdel Aziz [31] used an example of nine-zone slum area with a population of 2770 families to select the optimal upgrade plan. The optimization process involved three objectives: maximization of benefit of proposed upgrading projects, minimization of costs and socioeconomic disruption for the families. Due to its superiority over other GAs in solving multi-objective problems, NSGA-II was selected to solve the

Nevertheless, the time schedules module was not included in the model hence

• Brownlee and Wright [32] analyzed the relationship between design objectives and the effectiveness of design variables on the design objectives by using NSGA-II. They sorted the objectives by simple ranking. The approach was performed on a five-zone building with only two design objectives. The objectives to be minimized were total annual energy use and capital cost, and the design variables were 52 in total. Forty-nine solutions were generated using NSGA-II. However, the proposed approach failed to discriminate the distance variables which are the variables that measure the sets from the true Paretooptimal set from the floating variables which are the variables that have no

As the above-cited studies show, the NSGA-II proved its capability in optimizing for scheduling, urban planning, infrastructure, sustainability, energy and environmental design, and resource management. In addition to its superiority over other GAs, NSGA-II has also outperformed other methods in some fields. One of those is

The tournament selection among a population's individuals and Pareto dominance are the two basic ideas of NPGA's process. The selection process is based on the dominance of two randomly selected individuals from the population. To determine which individual of these two is dominant over the other, another set of individuals are picked and used to go against the two competing individuals, to examine the level of the two competing individuals in dominating each individual of the set. The winning criterion is defined by Pareto-front dominance. Therefore, one of the two competing individuals is selected if the other is dominated by one of

Kim et al. [35] used NPGA to optimize cost, time and resource utilization. They optimized the three objectives at the same time. To test the performance of the method, they conducted two case studies. The first case had 11 activities, and measured the method's efficiency in solving the tradeoff problem between cost and time. In addition to the objectives in the first case, the second case extended the examination of the approach by including the resource-leveling index as an objective. The results showed that this method could provide decision makers with different solutions to enable them selecting the one that meets their preferences.

MOGA is an advanced version of traditional GA. The difference between MOGA and GA is the individual fitness assignment, while the remaining steps are followed as in GA. In MOGA, ranking is assigned for each individual in the population. The rank is assigned based on individual's dominance, if the individual is not dominated

problem in which it generated 2000 solutions in less than 1 minute.

*Multicriteria Optimization - Pareto-Optimality and Threshold-Optimality*

affecting its robustness.

effect on the objective function.

the multi-objective PSO applied to scheduling problems.

**3.5 Niched Pareto genetic algorithm (NPGA)**

**3.6 Multi-objective genetic algorithm (MOGA)**

the individuals in the set [33, 34].

**56**

particles were used to present the potential solutions. The objectives considered in the problem were the layout cost and the environmental and safety accidents. Since the study accounted for uncertainty, fuzzy random variables were included in the model. The model used the example of 14 temporary facilities in a hydropower project to evaluate its efficiency. The proposed approach proved to be more realistic than existing traditional approaches.

Some researchers used PSO to tackle different design objectives and constraints

• Decker et al. [46] have proposed a PSO algorithm to reach better design solutions in timber buildings. In addition to architectural, energy and environmental constraints, the study added structural constraints. The optimization process was in terms of energy needs, thermal discomfort, floor vibration, CO2 emissions, and embodied energy. To minimize computing time, the simulation model was transformed using a metamodeling procedure. A three-story office building was used as a case study to validate the proposed

*Overview of Multi-Objective Optimization Approaches in Construction Project Management*

• Chou and Le [47] used PSO in combination with MCS to attain the optimal solutions for building designs in terms of minimizing duration (Eq. (12)), cost

*Min Fdur* <sup>¼</sup> ESfin <sup>þ</sup>X*<sup>n</sup>*

*i*¼1

where *Fdur*, *Fcost*, and *FCO2* represent project duration, project cost, and CO2 emissions, respectively; *ESfin* is the early start of the finish activity; *ESi* is the early start of activity *i*; *COSTi* is the unit cost of activity *i; n* is the number of activities; and *FCi* is the amount of CO2 emitted to complete a unit of work of

In addition to PSO, a probabilistic method was applied to handle the uncertainties associated with the objectives of the study. The case study of a 12-activity roadway pavement project was used to evaluate the performance of the proposed

In sum, PSO proved its effectiveness in tackling the multi-objective optimization problems in different construction engineering and management areas such as site planning, maintenance of a structure, and sustainability issues. It was found that PSO's performance was superior compared to traditional approaches such as GA

The stimulus in discovering the ACO algorithm was the movement of ants and

The proximity and number of construction facilities and other resources on a construction site could contribute to an increase in cost and safety issues. Ning and Lam [49] developed a modified ACO model to tackle safety and cost problems within a site layout of irregular shape. The model was aimed to minimizing safety/

their trails of pheromones when searching for food. In the ACO process, each solution is connected to a route that is searched by an ant. Each solution's quality is evaluated by the quantity of pheromones that were deposited on the route by ants. The amount of pheromone left on a route indicates the closeness to the optimal solution. The chance of finding the shortest route increases for an ant as the amount

*i*¼1

*Min Fcost* <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*

*Min FCO*<sup>2</sup> <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*

*i*¼1

*ESi* (12)

*Wi:COSTi* (13)

*Wi:FCi* (14)

to achieve optimal sustainable design solutions. For instance:

*DOI: http://dx.doi.org/10.5772/intechopen.88185*

(Eq. (13)), and CO2 emissions (Eq. (14)).

and advanced approaches such as NSGA-II.

**3.8 Ant colony optimization (ACO)**

of pheromone on a route increases [48].

approach.

activity *i*.

method.

**59**


PSO has also been used in tackling different objectives in the maintenance of deteriorating structures. For example:


*Overview of Multi-Objective Optimization Approaches in Construction Project Management DOI: http://dx.doi.org/10.5772/intechopen.88185*

Some researchers used PSO to tackle different design objectives and constraints to achieve optimal sustainable design solutions. For instance:


$$\text{Min } F\_{dur} = \text{ES}\_{\text{fin}} + \sum\_{i=1}^{n} \text{ES}\_{i} \tag{12}$$

$$\text{Min } F\_{\text{cost}} = \sum\_{i=1}^{n} W\_i. \text{COST}\_i \tag{13}$$

$$\text{Min } F\_{\text{CO}\_2} = \sum\_{i=1}^{n} W\_i.F\text{C}\_i \tag{14}$$

where *Fdur*, *Fcost*, and *FCO2* represent project duration, project cost, and CO2 emissions, respectively; *ESfin* is the early start of the finish activity; *ESi* is the early start of activity *i*; *COSTi* is the unit cost of activity *i; n* is the number of activities; and *FCi* is the amount of CO2 emitted to complete a unit of work of activity *i*.

In addition to PSO, a probabilistic method was applied to handle the uncertainties associated with the objectives of the study. The case study of a 12-activity roadway pavement project was used to evaluate the performance of the proposed method.

In sum, PSO proved its effectiveness in tackling the multi-objective optimization problems in different construction engineering and management areas such as site planning, maintenance of a structure, and sustainability issues. It was found that PSO's performance was superior compared to traditional approaches such as GA and advanced approaches such as NSGA-II.

#### **3.8 Ant colony optimization (ACO)**

The stimulus in discovering the ACO algorithm was the movement of ants and their trails of pheromones when searching for food. In the ACO process, each solution is connected to a route that is searched by an ant. Each solution's quality is evaluated by the quantity of pheromones that were deposited on the route by ants. The amount of pheromone left on a route indicates the closeness to the optimal solution. The chance of finding the shortest route increases for an ant as the amount of pheromone on a route increases [48].

The proximity and number of construction facilities and other resources on a construction site could contribute to an increase in cost and safety issues. Ning and Lam [49] developed a modified ACO model to tackle safety and cost problems within a site layout of irregular shape. The model was aimed to minimizing safety/

particles were used to present the potential solutions. The objectives considered in the problem were the layout cost and the environmental and safety accidents. Since the study accounted for uncertainty, fuzzy random variables were included in the model. The model used the example of 14 temporary facilities in a hydropower project to evaluate its efficiency. The proposed approach proved to be more realistic than existing traditional

*Multicriteria Optimization - Pareto-Optimality and Threshold-Optimality*

• Xu and Song [42] approached the problem of unequal-area departments in dynamic temporary facility layout using position-based adaptive PSO. By using the facilities' coordinates as base for its model, the optimization process aimed at minimizing the total distance between adjacent facilities and the resulting costs associated with rearrangement and transportation, in which the

transportation costs were considered as fuzzy random variables. The modified

• Li et al. [43] proposed a modified PSO to achieve optimal solutions for dynamic construction site layout and security planning. The study approached the problem using the Stackelberg Game theory, in which the construction manager (the leader) must set up the layout and secure the facilities, then the attacker (the follower) has to create the maximum possible economic damage to the facilities. Bi-level multi-objective PSO was proposed to solve the problem. The method was implemented in a hydropower construction project to test its performance. The proposed method outperformed GA in achieving

PSO has also been used in tackling different objectives in the maintenance of

• Yang et al. [44] approached the problem of life cycle maintenance planning for deteriorating bridges using PSO with Monte Carlo simulation (MCS). Cost, safety and condition levels were the main objectives in the maintenance problem. Uncertainties in the maintenance cost, work effects of maintenance, and the structure' deterioration rate were also accounted for in the study. Parallel programming was used to minimize the computing time to solve the problem. Yang et al. [44] considered three paradigms in the programming process, namely master-slave, island, and diffusion. In each paradigm, the computers have a different set up to run MCS in parallel. From the analysis, the island paradigm surpassed the other two in terms of solution quality. By comparison, the multi-objective PSO algorithm outperformed NSGA-II.

• Chiu and Lin [45] proposed PSO to achieve the optimal strategies in maintaining reinforced concrete buildings. The authors considered five objectives in the study, which are life cycle cost, failure possibility, spalling possibility, maintenance rationality, and maintenance times. Assessment models of probabilistic effects were employed to observe the effects of maintenance strategies on the damage index. The four processes of analysis of deterioration, assessment of seismic performance, forming maintenance strategies, and multi-objective optimization were performed in the proposed maintenance strategy. The evaluation was completed using a case study of a

four-story reinforced concrete school building.

PSO was evaluated through a case study of a large-scale hydropower construction project. The proposed method showed better performance in obtaining optimal solutions when compared to standard PSO and GA.

approaches.

optimal solutions.

**58**

deteriorating structures. For example:

environmental concerns by reducing the occurrence of accidents (Eq. (15)) as well as minimizing the total handling cost between facilities by reducing the cost associated with resource exchanges among facilities (Eq. (16)).

$$\dim f\_1 = \min \sum\_{i=1}^{n} \sum\_{j=1}^{n} \sum\_{l=1}^{n} \sum\_{k=1}^{n} S\_{ij} d\_{kl} \mathbf{x}\_{ik} \mathbf{x}\_{jl} \tag{15}$$

*Maximize project ranking* <sup>¼</sup> *Max* <sup>X</sup>*<sup>n</sup>*

*Overview of Multi-Objective Optimization Approaches in Construction Project Management*

where *n* is the number of projects; *m* is the number of scenarios; *y*<sup>þ</sup>

the number of project managers needed to complete project *i*.

memeplexes to ultimately improve their quality of search [53, 54, 56].

*Min SSR* ð Þ¼ *min* <sup>X</sup>

solutions to the problem at hand, the model accounts for the reallocation of

for use in day *d* with *d* = 1, 2, … ,*TD* of the project duration.

better solutions than other algorithms used prior to the study.

**3.10 Shuffled frog-leaping algorithm (SFLA)**

*DOI: http://dx.doi.org/10.5772/intechopen.88185*

allocation variations expressed in Eq. (21).

**3.11 Simulated annealing algorithm (SA)**

**61**

*Maximize number of completed projects* <sup>¼</sup> *Max* <sup>X</sup>*<sup>n</sup>*

positive deviation of the cost of the scenario from the expected cost of the project; *RANKi* is the ranking given to project *i* based on the ANP computation; *xi* is a binary variable which has a value of 1 if project *i* is selected, and 0 otherwise; and *PMNOi* is

The SFLA idea is based on frogs' behavior in their search to locate the largest quantities of food [53]. A single solution is represented by one frog [54, 55]. The frogs are divided into groups (memeplexes). Each memeplex of frogs performs a local search, and every frog has an idea which is affected by other frogs' ideas to improve the quality of the local search [56]. A shuffling process is performed to allow the memeplexes in exchanging information between them and create new

Improving the quality of the final product with limited resources is the ultimate goal of construction managers and planners. Time, cost, and resources play important roles in achieving this goal. Ashuri and Tavakolan [57] concurrently optimized three objectives: the duration expressed by sum of the durations of activities on the critical path, the project cost including direct and indirect costs, and resource

*TD*

X *S*

!

*n*¼1 *R*2 *n,d*

*k*¼1

where *Rn,d* is the number of the *n*th resource with *n* = 1, 2, … , *S* that is planned

To solve these problems, they used the SFLA model. In order to find feasible

resources and for activity interruptions and splitting. In addition, the authors made use of the advantages of PSO and the shuffling complex evolution algorithm, which helped the model achieve better solutions and converge more rapidly. A 7-activity and an 18-activity project were utilized to assess the efficiency of the model. Delphi was the coding environment for the model. Due to the complexity of the problem, the solutions obtained were near-optimal. However, the proposed model generated

SA inherits its method from the movements of atoms within a material during the process of heating and then slowly cooling down [58]. In the optimization problem, the physical system's characteristics resemble the actual annealing process

corresponding characteristics of the optimization problem. In physical annealing, temperature and speed of cooling down play important roles on the strength of metals. Deficiencies (metastable states) occur when cooling down speed is fast or

[10]. Talbi [10] listed the characteristics of physical annealing with their

*i*¼1

*RANKixi* (19)

*xi* (20)

*ij* is the

(21)

*i*¼1

$$\dim f\_2 = \min \sum\_{i=1}^{n} \sum\_{j=1}^{n} \sum\_{l=1}^{n} \sum\_{k=1}^{n} C\_{ij} d\_{kl} \varkappa\_{ik} \varkappa\_{jl} \tag{16}$$

where *Sij* is the closeness relationship value for safety/environmental concerns between facilities *i* and *j; Cij* is the total closeness relationship value for the total handling cost between facilities *i* and *j*; *dkl* is the distance between facilities *k* and *l; xik* means when facility *i* is assigned to location *k;* and *xjl* means when facility *j* is assigned to location *l.*

The optimization process started by dividing the site layout into a grid. The grid units were selected based upon the dimensions of the facilities. Then, the ACO model was used to assign the different facilities on the site grid. To test the soundness of the model, a residential project composed of four buildings was selected. The proposed grid strategy reduced the complexity of the computational process.

#### **3.9 Analytic network process (ANP)**

Like the analytic hierarchy process, decision makers utilize ANP to solve multicriteria decision problems. The AHP uses a one-way top-down hierarchal process for its components such as goals, criteria, and alternatives [50]. The ANP which is a generalized version of AHP uses a network for some problems when their components have interdependencies between them. The flow in the ANP's network is open and allows any component to interact with another regardless of their levels, which is not possible in AHP [51].

Liang and Wey [52] proposed an ANP model to optimally select government projects by accounting for the limitation of resources along with uncertainties and socioeconomic factors. In order to test the model's effectiveness, seven projects in a nation-wide highway improvement project were used as an example. In the example, construction costs were determined by probability distributions and seven criteria were used to evaluate the projects. Moreover, since the model involves the use of multiple criteria, ANP was combined with MCS to make the selection of projects based on the solutions achieved by solving the multi-objective problems. ANP ranking was used to rank each project based on its value of priority among other projects. A cost-benefit approach was used to optimize the selection of projects based on the existing budget plan and the allocation of remaining budget to fund a project in full. The four objectives within these two problems were minimization of cost (Eq. (17)) and the number of project managers (Eq. (18)), and the maximization of project ranking (Eq. (19)) and the number of completed projects (Eq. (20)).

$$\text{Minimize modified mean absolute deviation of cost} = \text{Min } \frac{\sum\_{i=1}^{n} \sum\_{j=1}^{m} \mathcal{Y}\_{ij}^{+}}{nm} \quad \text{(17)}$$

$$\text{Minimize number of project managers} = \text{Min} \sum\_{i=1}^{n} \text{PMNO}\_i \mathbf{x}\_i \tag{18}$$

*Overview of Multi-Objective Optimization Approaches in Construction Project Management DOI: http://dx.doi.org/10.5772/intechopen.88185*

$$\text{Maximize project ranking} = \text{Max} \sum\_{i=1}^{n} \text{RANK}\_{i} \mathbf{x}\_{i} \tag{19}$$

$$\text{Maximize number of completed projects} = \text{Max} \sum\_{i=1}^{n} \mathbf{x}\_i \tag{20}$$

where *n* is the number of projects; *m* is the number of scenarios; *y*<sup>þ</sup> *ij* is the positive deviation of the cost of the scenario from the expected cost of the project; *RANKi* is the ranking given to project *i* based on the ANP computation; *xi* is a binary variable which has a value of 1 if project *i* is selected, and 0 otherwise; and *PMNOi* is the number of project managers needed to complete project *i*.

#### **3.10 Shuffled frog-leaping algorithm (SFLA)**

environmental concerns by reducing the occurrence of accidents (Eq. (15)) as well as minimizing the total handling cost between facilities by reducing the cost associ-

> X*n j*¼1

X*n j*¼1

where *Sij* is the closeness relationship value for safety/environmental concerns between facilities *i* and *j; Cij* is the total closeness relationship value for the total handling cost between facilities *i* and *j*; *dkl* is the distance between facilities *k* and *l; xik* means when facility *i* is assigned to location *k;* and *xjl* means when facility *j* is

The optimization process started by dividing the site layout into a grid. The grid

Like the analytic hierarchy process, decision makers utilize ANP to solve multicriteria decision problems. The AHP uses a one-way top-down hierarchal process for its components such as goals, criteria, and alternatives [50]. The ANP which is a generalized version of AHP uses a network for some problems when their components have interdependencies between them. The flow in the ANP's network is open and allows any component to interact with another regardless of their levels, which

Liang and Wey [52] proposed an ANP model to optimally select government projects by accounting for the limitation of resources along with uncertainties and socioeconomic factors. In order to test the model's effectiveness, seven projects in a nation-wide highway improvement project were used as an example. In the example, construction costs were determined by probability distributions and seven criteria were used to evaluate the projects. Moreover, since the model involves the use of multiple criteria, ANP was combined with MCS to make the selection of projects based on the solutions achieved by solving the multi-objective problems. ANP ranking was used to rank each project based on its value of priority among other projects. A cost-benefit approach was used to optimize the selection of projects based on the existing budget plan and the allocation of remaining budget to fund a project in full. The four objectives within these two problems were minimization of cost (Eq. (17)) and the number of project managers (Eq. (18)), and the maximization of project ranking (Eq. (19)) and the number of completed projects

*Minimize modified mean absolute deviation of cost* ¼ *Min*

*Minimize number of project managers* <sup>¼</sup> *Min* <sup>X</sup>*<sup>n</sup>*

units were selected based upon the dimensions of the facilities. Then, the ACO model was used to assign the different facilities on the site grid. To test the soundness of the model, a residential project composed of four buildings was selected. The proposed grid strategy reduced the complexity of the computational process.

X*n l*¼1

X*n l*¼1

X*n k*¼1

X*n k*¼1

*Sijdklxikxjl* (15)

*Cijdklxikxjl* (16)

P*<sup>n</sup> i*¼1 P*<sup>m</sup> <sup>j</sup>*¼<sup>1</sup> *<sup>y</sup>*<sup>þ</sup> *ij*

*i*¼1

*nm*

*PMNOixi* (18)

(17)

*i*¼1

*i*¼1

ated with resource exchanges among facilities (Eq. (16)).

*Min f <sup>1</sup>* <sup>¼</sup> minX*<sup>n</sup>*

*Multicriteria Optimization - Pareto-Optimality and Threshold-Optimality*

*Min f <sup>2</sup>* <sup>¼</sup> minX*<sup>n</sup>*

assigned to location *l.*

**3.9 Analytic network process (ANP)**

is not possible in AHP [51].

(Eq. (20)).

**60**

The SFLA idea is based on frogs' behavior in their search to locate the largest quantities of food [53]. A single solution is represented by one frog [54, 55]. The frogs are divided into groups (memeplexes). Each memeplex of frogs performs a local search, and every frog has an idea which is affected by other frogs' ideas to improve the quality of the local search [56]. A shuffling process is performed to allow the memeplexes in exchanging information between them and create new memeplexes to ultimately improve their quality of search [53, 54, 56].

Improving the quality of the final product with limited resources is the ultimate goal of construction managers and planners. Time, cost, and resources play important roles in achieving this goal. Ashuri and Tavakolan [57] concurrently optimized three objectives: the duration expressed by sum of the durations of activities on the critical path, the project cost including direct and indirect costs, and resource allocation variations expressed in Eq. (21).

$$\text{Min } (\text{SSR}) = \min \left( \sum\_{k=1}^{\text{TD}} \sum\_{n=1}^{\text{S}} R\_{n,d}^2 \right) \tag{21}$$

where *Rn,d* is the number of the *n*th resource with *n* = 1, 2, … , *S* that is planned for use in day *d* with *d* = 1, 2, … ,*TD* of the project duration.

To solve these problems, they used the SFLA model. In order to find feasible solutions to the problem at hand, the model accounts for the reallocation of resources and for activity interruptions and splitting. In addition, the authors made use of the advantages of PSO and the shuffling complex evolution algorithm, which helped the model achieve better solutions and converge more rapidly. A 7-activity and an 18-activity project were utilized to assess the efficiency of the model. Delphi was the coding environment for the model. Due to the complexity of the problem, the solutions obtained were near-optimal. However, the proposed model generated better solutions than other algorithms used prior to the study.

#### **3.11 Simulated annealing algorithm (SA)**

SA inherits its method from the movements of atoms within a material during the process of heating and then slowly cooling down [58]. In the optimization problem, the physical system's characteristics resemble the actual annealing process [10]. Talbi [10] listed the characteristics of physical annealing with their corresponding characteristics of the optimization problem. In physical annealing, temperature and speed of cooling down play important roles on the strength of metals. Deficiencies (metastable states) occur when cooling down speed is fast or

the temperature at the start is not high enough [59]. That means carefully setting up the temperature and cooling down speed is essential in escaping the local optimum —metastable state in physical annealing—and reaching the global optimum. A solution that is generated after an iteration is used, if feasible, to generate a new solution, but if the solution is infeasible, it is accepted only if it meets the probability criterion [10, 60]. The probability increases in obtaining an optimal or nearoptimal solution when the annealing is slowed down [61].

where *D* is the resource supply rate; *zj* is 1 when facility *j* is secured and 0 otherwise; *sj* is 1 when facility *j* is attacked and 0 otherwise; *θij* is the weight of demand's importance; 0 ≤ *θij* ≤ 1; *dij* is 1 when demand of facility *i* is served by facility *j* and 0 otherwise; *pk* is the occurrence probabilities of the *k*th degree attack,

*Overview of Multi-Objective Optimization Approaches in Construction Project Management*

X 5

*pkμ<sup>c</sup>*

where *C* is the economic loss of defender; *Mj* is the cost of securing facility *j; and*

Because integer programming made the problem complicated, the authors proposed PGSA. The model was applied on an actual hydropower project. Fifty runs were executed to achieve the optimal solution in less than 4 minutes. Even though the proposed model efficiently solved the problem, it did not top the list of algorithms. This study was the first study to apply PGSA on the problem of construction

The Hungarian algorithm is a modified form of the primal-dual algorithm that is

used to solve network flows. In assignment problems, the Hungarian algorithm changes the weights in a matrix to locate the optimal assignment. Eventually, a new

El-Anwar and Chen [68] proposed a modified Hungarian algorithm to solve post-disaster temporary housing problems. They considered the problem as an integer problem. An earthquake simulation example was used to examine the model's efficiency. The number of decision variables was determined by multiplying the housing alternatives (178) with the number displaced families (5000). Throughout the 13 temporary housing problems, a varying number of decision variables were considered. In terms of the running time, the Hungarian algorithm has shown superiority over integer programming. In the example, the running time for integer programming increased exponentially as the number of decision variables increased, and ran out of memory in case more than 24,000 decision variables were used. The Hungarian algorithm, on the other hand, solved all the problems

MINLP is an optimization problem in which the variables are constrained to continuous (e.g., costs, dimensions, mass, etc.) and integer values (typically binary 0 and 1), and the solution space and the objective functions are represented by nonlinear functions [69–71]. To solve complex problems that involve nonlinearity and mixed-integers, MINLP utilizes the combination of mixed-integer programming (MIP) and nonlinear programming (NLP) [72]. Thus, in solving MINLP problems, the approach is not considered a direct problem solver. The methods used to solve MINLP optimization problems include: branch and bound method, benders

• Fan and Xia [74] used MINLP to reduce energy consumption in residential buildings. The objectives of the study were to increase the energy savings and

matrix is obtained in which the optimal assignment is identified [67].

with the maximum number of decision variables (890,000).

**3.14 Mixed-integer nonlinear programming (MINLP)**

decomposition, and outer approximation algorithm [73].

*k*¼1

Min*zj*

*DOI: http://dx.doi.org/10.5772/intechopen.88185*

*<sup>C</sup>* <sup>¼</sup> <sup>X</sup> *N*

*jk* is the mean value of the economic loss when facility *j* is attacked.

*j*¼1

*ijk* is the mean value of the fill rate of facility *j* to facility *i* when

*jksj* <sup>þ</sup><sup>X</sup> *N*

*j*¼1

*Mjzj* (24)

*k* ∈ {1, … , 5}; and *μ<sup>r</sup>*

facility *j* is attacked.

*μc*

**63**

site security.

**3.13 Hungarian algorithm (HA)**

To optimally design and construct a water distribution network, Marques et al. [62] proposed a model that used the SA algorithm combined with the EPANET hydraulic simulator. The objective was to minimize the cost of construction and operation including the initial and future costs, and to minimize violations in pressure as expressed in Eq. (22).

$$\text{Min TPV} = \sum\_{s=1}^{\text{NS}} \sum\_{t=1}^{\text{NTI}} \sum\_{d=1}^{\text{NDC}} \sum\_{n=1}^{\text{NN}} \max\{\mathbf{0}; (Pds\_{\text{min},n,d} - P\_{n,d,t,s})\}\tag{22}$$

where *TPV* is the total pressure violations; *NS* is the number of scenarios; *NTI* is the number of periods into which the planning horizon is subdivided; *NDC* is the number of demand conditions considered for the design; *NN* is the number of nodes; *Pdesmin,n,d* is the minimum desirable pressure at node *n* for demand condition *d;* and *Pn,d,t,s* is the pressure at node *n* at demand condition *d* for time interval *t* and in scenario *s*.

Eight scenarios were accounted for varying between three possible patterns of growth in the area: expansion, no expansion, and depopulation in a 60-year period. They split the 60-year duration of the plan into 320-year stages, and structured them into a decision tree to show the probability of the paths in each scenario. They used a 17-node distribution network to illustrate the model's efficiency. The decision variables included cost, diameters of pipes (eight diameters were considered), and carbon emissions produced during construction and operation (in terms of tons). The value of the objective function was not noticeably affected by the decision variable of carbon emission costs.

#### **3.12 Plant growth simulation algorithm (PGSA)**

The PGSA imitates the growth process of trees. The model's formulation for the optimization process in PGSA is based on the growth of plants [63]. It begins at the root then moves toward the light source (global optimum solution) to grow the branches [64]. A probability model is employed to form new branches which are used to guide the objective function toward the optimal solution [65].

To better minimize the losses and costs caused by an attack to the construction site and to increase the safety precautions to counter these attacks, Li et al. [66] used a bi-level model. The objectives of reducing attack-related cost and increasing facility productivity were considered at the upper level, in which the secured facilities were constrained by cost. The attacker, on the other hand, has the objective of reducing facility productivity, which is considered in the lower level. The formulation of the objective functions is as follows:

$$\text{Max}\_{\mathbf{z}\_{\parallel}} \mathbf{D} = \sum\_{j=1}^{N} \sum\_{\mathbf{i}=\mathbf{l}\_{\text{i}}}^{N} \sum\_{\mathbf{i}\neq\mathbf{j}}^{N} \Theta\_{\overline{\mathbf{i}}\overline{\mathbf{p}}} \mathbf{h}\_{\overline{\mathbf{i}}\mathbf{k}}^{\mathbf{r}} \mathbf{d}\_{\overline{\mathbf{i}}\mathbf{k}} \mathbf{s}\_{\overline{\mathbf{j}}} + \sum\_{j=1}^{N} \sum\_{\mathbf{i}=\mathbf{l}\_{\text{i}}}^{N} \Theta\_{\overline{\mathbf{i}}\overline{\mathbf{l}}} \mathbf{d}\_{\overline{\mathbf{i}}\mathbf{l}} \mathbf{z}\_{\overline{\mathbf{j}}} + \sum\_{j=1}^{N} \sum\_{\mathbf{i}=\mathbf{l}\_{\text{i}}}^{N} \Theta\_{\overline{\mathbf{i}}\overline{\mathbf{l}}} d\_{\overline{\mathbf{i}}\mathbf{l}} (\mathbf{1} - \mathbf{s}\_{\overline{\mathbf{i}}}) (\mathbf{1} - \mathbf{z}\_{\overline{\mathbf{i}}}) \tag{23}$$

*Overview of Multi-Objective Optimization Approaches in Construction Project Management DOI: http://dx.doi.org/10.5772/intechopen.88185*

where *D* is the resource supply rate; *zj* is 1 when facility *j* is secured and 0 otherwise; *sj* is 1 when facility *j* is attacked and 0 otherwise; *θij* is the weight of demand's importance; 0 ≤ *θij* ≤ 1; *dij* is 1 when demand of facility *i* is served by facility *j* and 0 otherwise; *pk* is the occurrence probabilities of the *k*th degree attack, *k* ∈ {1, … , 5}; and *μ<sup>r</sup> ijk* is the mean value of the fill rate of facility *j* to facility *i* when facility *j* is attacked.

$$\mathbf{Min}\_{\mathbf{z}\_{j}}\mathbf{C} = \sum\_{j=1}^{N} \sum\_{k=1}^{5} p\_{k} \mu\_{jk}^{\varepsilon} \mathbf{s}\_{j} + \sum\_{j=1}^{N} \mathbf{M}\_{j} \mathbf{z}\_{j} \tag{24}$$

where *C* is the economic loss of defender; *Mj* is the cost of securing facility *j; and μc jk* is the mean value of the economic loss when facility *j* is attacked.

Because integer programming made the problem complicated, the authors proposed PGSA. The model was applied on an actual hydropower project. Fifty runs were executed to achieve the optimal solution in less than 4 minutes. Even though the proposed model efficiently solved the problem, it did not top the list of algorithms. This study was the first study to apply PGSA on the problem of construction site security.

#### **3.13 Hungarian algorithm (HA)**

the temperature at the start is not high enough [59]. That means carefully setting up the temperature and cooling down speed is essential in escaping the local optimum —metastable state in physical annealing—and reaching the global optimum. A solution that is generated after an iteration is used, if feasible, to generate a new solution, but if the solution is infeasible, it is accepted only if it meets the probability criterion [10, 60]. The probability increases in obtaining an optimal or near-

To optimally design and construct a water distribution network, Marques et al. [62] proposed a model that used the SA algorithm combined with the EPANET hydraulic simulator. The objective was to minimize the cost of construction and operation including the initial and future costs, and to minimize violations in

where *TPV* is the total pressure violations; *NS* is the number of scenarios; *NTI* is the number of periods into which the planning horizon is subdivided; *NDC* is the number of demand conditions considered for the design; *NN* is the number of nodes; *Pdesmin,n,d* is the minimum desirable pressure at node *n* for demand condition *d;* and *Pn,d,t,s* is the pressure at node *n* at demand condition *d* for time interval *t* and

Eight scenarios were accounted for varying between three possible patterns of growth in the area: expansion, no expansion, and depopulation in a 60-year period. They split the 60-year duration of the plan into 320-year stages, and structured them into a decision tree to show the probability of the paths in each scenario. They used a 17-node distribution network to illustrate the model's efficiency. The decision variables included cost, diameters of pipes (eight diameters were considered), and carbon emissions produced during construction and operation (in terms of tons). The value of the objective function was not noticeably affected

The PGSA imitates the growth process of trees. The model's formulation for the optimization process in PGSA is based on the growth of plants [63]. It begins at the root then moves toward the light source (global optimum solution) to grow the branches [64]. A probability model is employed to form new branches which are

To better minimize the losses and costs caused by an attack to the construction site and to increase the safety precautions to counter these attacks, Li et al. [66] used a bi-level model. The objectives of reducing attack-related cost and increasing facility productivity were considered at the upper level, in which the secured facilities were constrained by cost. The attacker, on the other hand, has the objective of reducing facility productivity, which is considered in the lower level. The

used to guide the objective function toward the optimal solution [65].

ijkdijsj <sup>þ</sup><sup>X</sup>

N

X N

<sup>θ</sup>ijdijzj <sup>þ</sup><sup>X</sup>

*N*

X *N*

*θijdij* 1 � *sj*

� � <sup>1</sup> � *zj* � �

(23)

*<sup>i</sup>*¼1*,i*6¼*<sup>j</sup>*

*j*¼1

<sup>i</sup>¼1*,*i6¼<sup>j</sup>

j¼1

max 0f g ;ð Þ *Pdes*min*,n,d* � *Pn,d,t,s* (22)

optimal solution when the annealing is slowed down [61].

*Multicriteria Optimization - Pareto-Optimality and Threshold-Optimality*

pressure as expressed in Eq. (22).

*Min TPV* <sup>¼</sup> <sup>X</sup>

in scenario *s*.

*Max*zj

**62**

<sup>D</sup> <sup>¼</sup> <sup>X</sup> N

j¼1

X N

X 5

k¼1

<sup>i</sup>¼1*,*i6¼<sup>j</sup>

*NS*

X *NTI*

*NDC* X *d*¼1

X *NN*

*n*¼1

*t*¼1

*s*¼1

by the decision variable of carbon emission costs.

**3.12 Plant growth simulation algorithm (PGSA)**

formulation of the objective functions is as follows:

θijpkμ<sup>r</sup>

The Hungarian algorithm is a modified form of the primal-dual algorithm that is used to solve network flows. In assignment problems, the Hungarian algorithm changes the weights in a matrix to locate the optimal assignment. Eventually, a new matrix is obtained in which the optimal assignment is identified [67].

El-Anwar and Chen [68] proposed a modified Hungarian algorithm to solve post-disaster temporary housing problems. They considered the problem as an integer problem. An earthquake simulation example was used to examine the model's efficiency. The number of decision variables was determined by multiplying the housing alternatives (178) with the number displaced families (5000). Throughout the 13 temporary housing problems, a varying number of decision variables were considered. In terms of the running time, the Hungarian algorithm has shown superiority over integer programming. In the example, the running time for integer programming increased exponentially as the number of decision variables increased, and ran out of memory in case more than 24,000 decision variables were used. The Hungarian algorithm, on the other hand, solved all the problems with the maximum number of decision variables (890,000).

#### **3.14 Mixed-integer nonlinear programming (MINLP)**

MINLP is an optimization problem in which the variables are constrained to continuous (e.g., costs, dimensions, mass, etc.) and integer values (typically binary 0 and 1), and the solution space and the objective functions are represented by nonlinear functions [69–71]. To solve complex problems that involve nonlinearity and mixed-integers, MINLP utilizes the combination of mixed-integer programming (MIP) and nonlinear programming (NLP) [72]. Thus, in solving MINLP problems, the approach is not considered a direct problem solver. The methods used to solve MINLP optimization problems include: branch and bound method, benders decomposition, and outer approximation algorithm [73].

• Fan and Xia [74] used MINLP to reduce energy consumption in residential buildings. The objectives of the study were to increase the energy savings and economic benefits within budget limitations. The example of a 69-year old house was used to test the model, in which the retrofitting plan included the insulation materials for the roof and external walls, windows, and the installation of solar panels. The model proved to be effective in minimizing the energy consumed by the building; from the results obtained, in a 10-year period, the house could save around 288.44 MWh.

performing the evidential reasoning method, which assist decision makers in

• Xu et al. [79] proposed a multi-objective bi-level PSO (MOBLPSO) to optimize the minimum cost network flow of construction material transportation in terms of duration and cost. In the upper level of the model, the time to transport materials in addition to direct costs were minimized by the

contractor by selecting the most convenient routes for transporting materials.

transportation manager to reduce the costs of transportation. Because of the complexity of the problem the PSO approach was hybridized with two other methods, one in each level. In the upper level, PSO was integrated with Pareto Archived Evolution Strategy (PAES) to keep the best position for the solutions up to date. In the lower level, PSO used passive congregation to prevent the convergence from happening too early. The case of an actual hydropower construction project was utilized to examine the model's soundness. The model outperformed multi-objective bi-level genetic algorithms (MOBLGA) and the

• Xu et al. [80] conducted a similar study to the one mentioned above, but in this study the cost and duration of transportation were considered as fuzzy random variables. A fuzzy random simulation-based constraint checking procedure was coupled with MOBLPSO to solve the transportation assignment problem which was used to control the flow of materials within a given period. The road network of an existing hydropower project was used for the evaluation of the model. With accounting for uncertainties, the model showed its efficiency and

Depending on the decisions made in the model's upper level, every transporter's flow of material in those routes were considered by the

multi-objective bi-level simulated annealing algorithm (MOBLSA).

• Zhang et al. [81] proposed immune genetic PSO (IGPSO) which couples immune genetic algorithm with PSO. The approach was used to tackle the trade-off problem of time-cost-quality in construction, and accounting for

characteristics from three methods: (1) from the immune algorithm, whereby the hybrid method inherits the immune selection and the memory recognition; (2) from the genetic algorithm, which implements mutation and crossover; and (3) by limiting the particles' maximum velocity using the constriction

bonus and penalty. The hybrid method in the research obtained its

capability of solving the transportation problem.

**65**

• Brownlee and Wright [78] proposed modified approaches of NSGA-II on a simulation-based optimization problem for building energy. The minimization

of energy usage and construction cost were the two objectives in the optimization process. The aim of the study was to find an approach that surpasses the basic NSGA-II in terms of convergence rate and solution quality. The study used a middle floor from a commercial office building in three different cities to test the proposed model. The authors merged NSGA-II with two other approaches, namely radial basis function networks and deterministic infeasibility sorting. These approaches enabled the model to prevent the elimination of infeasible solutions and to keep them in the population. The objectives were represented by 50 decision variables (30 continuous, 8 integers and 12 categorical) and 18 inequality constraints. Moreover, the optimization runs were limited to 5000 completed within almost a day by six parallel simulations. The model was found superior to the basic NSGA-II in two of the

*Overview of Multi-Objective Optimization Approaches in Construction Project Management*

assessing each solution in terms of performance.

*DOI: http://dx.doi.org/10.5772/intechopen.88185*

three building examples.

• Karmellos et al. [75] also used MINLP to optimize the energy used by a building. The minimization of energy consumption every year and the cost of investments were the two main objectives in the optimization process. To test the model's soundness, the energy consumption in two houses was investigated. The first case involved a new house located in the UK while the second case was an existing house located in Greece. Fifty-four decision variables were accounted for in the model, which represented different building components including electrical appliances, building envelope, and lighting and energy systems. The model was effective in solving the optimization problem of and building energy. It was found that energy consumption goes down when investments in energy efficiency are increased.

#### **3.15 Hybrid approaches**

One way in approaching complex optimization problems is to combine two or more techniques together in order to overcome the deficiencies that one or some of them may possess. This approach could affect the overall quality of the solution in an optimization problem. The hybridization of methods has shown its efficacy in accomplishing optimization quality in construction. Hybrid methods have different operational characteristics in tackling optimization problems. While some hybrid methods work by carrying the entire solution process as a single novel technique, others work in tandem whereby one method works on some steps of the solution process and the other steps are completed by another method.

NSGA-II was hybridized with other approaches to solve optimization problems in construction planning, scheduling, energy conservation, transportation, and environmental design. For example:


performing the evidential reasoning method, which assist decision makers in assessing each solution in terms of performance.


economic benefits within budget limitations. The example of a 69-year old house was used to test the model, in which the retrofitting plan included the insulation materials for the roof and external walls, windows, and the

• Karmellos et al. [75] also used MINLP to optimize the energy used by a

the model's soundness, the energy consumption in two houses was

lighting and energy systems. The model was effective in solving the optimization problem of and building energy. It was found that energy consumption goes down when investments in energy efficiency are increased.

period, the house could save around 288.44 MWh.

*Multicriteria Optimization - Pareto-Optimality and Threshold-Optimality*

process and the other steps are completed by another method.

**3.15 Hybrid approaches**

environmental design. For example:

degree of convergence.

**64**

installation of solar panels. The model proved to be effective in minimizing the energy consumed by the building; from the results obtained, in a 10-year

building. The minimization of energy consumption every year and the cost of investments were the two main objectives in the optimization process. To test

investigated. The first case involved a new house located in the UK while the second case was an existing house located in Greece. Fifty-four decision variables were accounted for in the model, which represented different building components including electrical appliances, building envelope, and

One way in approaching complex optimization problems is to combine two or more techniques together in order to overcome the deficiencies that one or some of them may possess. This approach could affect the overall quality of the solution in an optimization problem. The hybridization of methods has shown its efficacy in accomplishing optimization quality in construction. Hybrid methods have different operational characteristics in tackling optimization problems. While some hybrid methods work by carrying the entire solution process as a single novel technique, others work in tandem whereby one method works on some steps of the solution

NSGA-II was hybridized with other approaches to solve optimization problems in construction planning, scheduling, energy conservation, transportation, and

• Mungle et al. [76] used fuzzy clustering-based genetic algorithm (FCGA) to find optimal solutions for the trade-off problem of time, cost and quality within the construction tasks. The method hybridized the fuzzy clustering approach with NSGA-II. In addition, AHP was utilized to measure the construction activities' quality. To evaluate the model's efficiency, a highway construction project was selected as an example. The authors used the example in three cases with different number of activities, i.e., eighteen, twelve, and seven-activity networks, in which the proposed approach was compared to other methods. The results of the comparison showed that FCGA surpassed MOPSO, MOGA and SPEA-II in terms of diversity as well as the speed and

• Monghasemi et al. [77] proposed an approach that combines NSGA-II with MOGA to solve a discrete problem of cost, time, and quality in construction project scheduling. An 18-activity highway construction project was used to examine the proposed model. MOGA was utilized to search the large size of 3.6 billion solutions and obtain near true optimal solutions. Shannon's entropy method was used to assign normalized weights to the three objectives in the obtained solutions. These weights were used to rank the solutions by

factor in PSO, which speeds up the convergence in initial steps. In addition, the study used a PERT network instead of CPM for the schedule. The model was applied on the 19 activities of a three-floor office building, and proved its effectiveness in solving the trade-off problem.

• Marzouk et al. [87] presented a hybrid approach that combined ACO with system dynamics to optimize the selection of sustainable materials. The aim of the study was the maximization of LEED credits and the minimization of cost. The authors employed a study case of a two-floor residential building to validate the efficacy of the model. From the achieved results, the model proved

*Overview of Multi-Objective Optimization Approaches in Construction Project Management*

its capability in accomplishing the two objectives of the problem.

in finding the optimal maintenance plan.

*DOI: http://dx.doi.org/10.5772/intechopen.88185*

a method until it is compared with another method.

**4. Conclusion**

**67**

• In building maintenance planning, Wang and Xia [88] used a predictive control model and DE algorithm to achieve the optimal retrofitting plan that lowers energy consumption. The study's first objective aimed at increasing a project's internal rate of return. The study's second objective was to increase energy savings while accounting for the sustainability period. The authors tackled the optimization of the maintenance plan in two instances. They started by solving the optimization problem without the assumption of uncertainties. They then solved the problem with uncertainties, in which the predictive control model was utilized. To check the approach's validity, a case study that involved the retrofitting of an office building consisting of 50 stories was considered. The results showed that the proposed approach was effective

The complexity of the problems in construction projects makes objective optimization usually difficult using a single approach. Hybrid techniques are effective and useful in generating optimal solutions in complex optimization problems. In some studies, these hybrid methods have outperformed some methods in their basic and variant forms. In scheduling for example, they were superior to multi-objective PSO, multi-objective ABC, multi-objective DE, MOGA, SPEA-II, and NSGA-II. In material logistics, they surpassed multi-objective bi-level GA and multi-objective bi-level SA. In site planning, they outperformed the basic form of ABC and one of the ACO variants. Finally, in sustainability, they were superior to NSGA-II.

This review included 55 papers that were published in refereed journals and conference proceedings published in the years 2012–2016. The authors of these papers conducted studies using various multi-objective optimization methods in the construction industry. There were 16 methods used in these studies in which some of the authors justify their picks on multiple factors (e.g., construction project type, project size, number of objectives, number of constraints, convergence rate, problem complexity such as constraints' nonlinearity with discontinuity and continuity, etc.). Moreover, some methods were found to be more efficient than others in some studies. For example, in water network planning, Creaco et al. [30] showed that their NSGA-II using a probabilistic approach was superior to NSGA-II used by Creaco et al. [29] in an earlier study in which they used a deterministic approach. The GA proposed by Aziz et al. [6] in a scheduling problem outperformed the GA utilized by Feng et al. [7] for the same case study. Fallah-Mehdipour et al. [26] concluded that NSGA-II has performed better than multi-objective PSO in solving a scheduling problem. Most of the time, it is difficult to guarantee the performance of

The most common number of objectives used in the literature is two and three.

As expected, cost and duration were the most targeted objectives as cost and duration are important objectives for all construction practitioners. The quality objective has also drawn the interest of researchers as they sometimes include it in


*Overview of Multi-Objective Optimization Approaches in Construction Project Management DOI: http://dx.doi.org/10.5772/intechopen.88185*


The complexity of the problems in construction projects makes objective optimization usually difficult using a single approach. Hybrid techniques are effective and useful in generating optimal solutions in complex optimization problems. In some studies, these hybrid methods have outperformed some methods in their basic and variant forms. In scheduling for example, they were superior to multi-objective PSO, multi-objective ABC, multi-objective DE, MOGA, SPEA-II, and NSGA-II. In material logistics, they surpassed multi-objective bi-level GA and multi-objective bi-level SA. In site planning, they outperformed the basic form of ABC and one of the ACO variants. Finally, in sustainability, they were superior to NSGA-II.

#### **4. Conclusion**

factor in PSO, which speeds up the convergence in initial steps. In addition, the study used a PERT network instead of CPM for the schedule. The model was applied on the 19 activities of a three-floor office building, and proved its

• In the trade-off problem, some researchers used the double-loop technique, in which the internal loop executes the simulation, while the external loop carries out the optimization process. However, this technique uses MCS and can sometimes take days to finish the process. Therefore, Yang et al. [82] proposed a procedure that combines the double-loop into one, and used MCS and support vector regression (SVR) with PSO to expedite the process of obtaining optimal solutions for the time-cost trade-off problem. MCS was set to assess the initial solutions' objective values, and a decision function gained by SVR promptly assesses the solutions obtained by PSO for their objective values. SVR's direct assessment contributed in shortening the search time of MCS. To test the model, an 18-activity project was utilized as an example. The results obtained showed that the proposed method was superior compared to the

• Futrell et al. [83] used PSO coupled with Hooke Jeeves and the generic

significant conflict between those objectives in the case of windows

optimization program (GenOpt) to optimize the performance of daylighting and thermal systems in buildings. Hooke Jeeves was utilized to fine-tune the best solution found in the PSO algorithm by locally searching it. The case study of a classroom design was utilized to evaluate the proposed approach. The classroom was tested on windows facing north, south, west, and east. It was found that there was no significant conflict between the optimization

objectives when the windows were facing south, west, or east, but there was a

• Yahya and Saka [84] used multi-objective artificial bee colony (ABC) with the Levy flights algorithm to find the ideal layout for a construction site. Levy flight which uses a random walk pattern searches food locations found by ABC to locate new solutions. The objective functions of the study were the reduction of the facilities' total handling cost, and minimization of environmental and safety risks. Two practical study cases were used to assess the proposed model. The first case was a residential project consisting of four high-rise buildings, and the second case was a three-floor private hospital. The first case which was a dynamic site layout was taken from Ning et al.'s [85] study, in which they applied a modified ACO. From the results, the model succeeded in optimizing the site layout problems. By comparison, the method proposed by Yahya and Saka [84] surpassed the plain-ABC and the modified ACO used by

• Tran et al. [86] tackled the trade-off problem of time, cost, and quality using the combination of multi-objective ABC with DE. DE was included to use its crossover mutation operators to optimize the stages of exploration and exploitation. A study case of a construction project consisting of 60 activities was used to test the model. The result proved the model's efficacy in the tradeoff problem. The approach was also compared against four other approaches

that were used to solve the trade-off problem. The proposed method

outperformed multi-objective ABC, multi-objective DE, multi-objective PSO,

effectiveness in solving the trade-off problem.

*Multicriteria Optimization - Pareto-Optimality and Threshold-Optimality*

methods that used the double loop.

facing north.

Ning et al. [85].

and NSGA-II.

**66**

This review included 55 papers that were published in refereed journals and conference proceedings published in the years 2012–2016. The authors of these papers conducted studies using various multi-objective optimization methods in the construction industry. There were 16 methods used in these studies in which some of the authors justify their picks on multiple factors (e.g., construction project type, project size, number of objectives, number of constraints, convergence rate, problem complexity such as constraints' nonlinearity with discontinuity and continuity, etc.). Moreover, some methods were found to be more efficient than others in some studies. For example, in water network planning, Creaco et al. [30] showed that their NSGA-II using a probabilistic approach was superior to NSGA-II used by Creaco et al. [29] in an earlier study in which they used a deterministic approach. The GA proposed by Aziz et al. [6] in a scheduling problem outperformed the GA utilized by Feng et al. [7] for the same case study. Fallah-Mehdipour et al. [26] concluded that NSGA-II has performed better than multi-objective PSO in solving a scheduling problem. Most of the time, it is difficult to guarantee the performance of a method until it is compared with another method.

The most common number of objectives used in the literature is two and three. As expected, cost and duration were the most targeted objectives as cost and duration are important objectives for all construction practitioners. The quality objective has also drawn the interest of researchers as they sometimes include it in trade-off problems with cost and/or duration. However, quality has not been optimized in any other set of objectives than three-objective optimization problems. The energy and environment category is an important candidate in the optimization process, as it came after cost and duration objectives based on the number of times it was optimized. That may be the result of efforts to optimally construct sustainable buildings and lower the depletion of natural resources.

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*DOI: http://dx.doi.org/10.5772/intechopen.88185*

*Overview of Multi-Objective Optimization Approaches in Construction Project Management*

[9] Xu J, Meng J, Zeng Z, Wu S, Shen M. Resource sharing-based multi objective multistage construction equipment allocation under fuzzy environment. Journal of Construction Engineering and Management. 2013;**139**(2):161-173. DOI: 10.1061/(ASCE)CO.1943-7862.0000593

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multiobjective optimization of time-cost tradeoffs in resource-constrained construction projects. IEEE Transactions on Engineering

Management. 2014;**61**(3):450-461. DOI:

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autcon.2015.02.010

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Among the multi-objective methods used in the literature, NSGA-II was the most used method. NSGA-II has proven its capability in solving optimization problems in different fields of construction. In addition to its popularity among researchers, NSGA-II has many advantages that make it suitable for many types of optimization problems such as obtaining diverse solutions in Pareto-front, low computational complexity, solving problems that involve nonlinearity and discontinuity.

#### **Author details**

Ibraheem Alothaimeen and David Arditi\* Illinois Institute of Technology, Chicago, IL, USA

\*Address all correspondence to: arditi@iit.edu

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Overview of Multi-Objective Optimization Approaches in Construction Project Management DOI: http://dx.doi.org/10.5772/intechopen.88185*

### **References**

trade-off problems with cost and/or duration. However, quality has not been optimized in any other set of objectives than three-objective optimization problems. The energy and environment category is an important candidate in the optimization process, as it came after cost and duration objectives based on the number of times it was optimized. That may be the result of efforts to optimally construct sustain-

Among the multi-objective methods used in the literature, NSGA-II was the most used method. NSGA-II has proven its capability in solving optimization prob-

researchers, NSGA-II has many advantages that make it suitable for many types of optimization problems such as obtaining diverse solutions in Pareto-front, low computational complexity, solving problems that involve nonlinearity and

lems in different fields of construction. In addition to its popularity among

able buildings and lower the depletion of natural resources.

*Multicriteria Optimization - Pareto-Optimality and Threshold-Optimality*

discontinuity.

**Author details**

**68**

Ibraheem Alothaimeen and David Arditi\*

provided the original work is properly cited.

Illinois Institute of Technology, Chicago, IL, USA

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: arditi@iit.edu

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0254535

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scheduling. In: Proceedings of the Third International Conference on Soft Computing for Problem Solving; 26-28 December 2013; Roorkee, India. New Delhi, India: Springer; 2014. pp. 11-24. DOI: 10.1007/978-81-322-1771-8\_2

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[55] Lu K, Ting L, Keming W, Hanbing Z, Makoto T, Bin Y. An improved shuffled frog-leaping algorithm for flexible job shop scheduling problem. Algorithms. 2015;**8**(1):19-31. DOI:

[56] Eusuff M, Lansey K, Pasha F. Shuffled frog-leaping algorithm: A memetic meta-heuristic for discrete

Optimization. 2006;**38**(2):129-154. DOI:

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optimization. Engineering

10.1080/03052150500384759

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reinforced concrete buildings.

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[47] Chou JS, Le TS. Probabilistic multiobjective optimization of sustainable engineering design. KSCE Journal of Civil Engineering. 2014; **18**(4):853-864. DOI: 10.1007/

[48] Parpinelli RS, Lopes HS, Freitas AA.

Computation. 2002;**6**(4):321-332. DOI:

[49] Ning X, Lam KC. Cost-safety tradeoff in unequal-area construction site layout planning. Automation in Construction. 2013;**32**:96-103. DOI: 10.1016/j.autcon.2013.01.011

[50] Saaty TL. Decision making with the analytic hierarchy process. International Journal of Services Sciences. 2008;**1**(1): 83-98. DOI: 10.1504/IJSSCI.2008.

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017590

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[60] Basu M. A simulated annealingbased goal-attainment method for economic emission load dispatch of fixed head hydrothermal power systems. International Journal of Electrical Power & Energy Systems. 2005;**27**(2):147-153. DOI: 10.1016/j. ijepes.2004.09.004

[61] Zhao X. Simulated annealing algorithm with adaptive neighborhood. Applied Soft Computing. 2011;**11**(2): 1827-1836. DOI: 10.1016/j. asoc.2010.05.029

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[63] Wang C, Cheng HZ, Hu ZC, Wang Y. Distribution system optimization planning based on plant growth simulation algorithm. Journal of Shanghai Jiaotong University (Science). 2008;**13**(4):462-467. DOI: 10.1007/ s12204-008-0462-4

[64] Rao PVVR, Sivanagaraju S. Radial distribution network reconfiguration for loss reduction and load balancing using plant growth simulation algorithm. International Journal on Electrical Engineering and Informatics. 2010; **2**(4):266-277. DOI: 10.15676/ijeei.2010. 2.4.2

[65] Guney K, Durmus A, Basbug S. A plant growth simulation algorithm for pattern nulling of linear antenna arrays by amplitude control. Progress in Electromagnetics Research B. 2009;**17**: 69-84. DOI: 10.2528/PIERB09061709

[66] Li Z, Xu J, Shen W, Lev B, Lei X. Bilevel multi-objective construction site security planning with twofold random phenomenon. Journal of Industrial and Management Optimization. 2015;**11**(2): 595-617. DOI: 10.3934/jimo.2015.11.595

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[68] El-Anwar O, Chen L. Maximizing the computational efficiency of temporary housing decision support following disasters. Journal of Computing in Civil Engineering. 2014; **28**(1):113-123. DOI: 10.1061/(ASCE) CP.1943-5487.0000244

[69] Bonami P, Kilinç M, Linderoth J. Algorithms and software for convex mixed integer nonlinear programs. In: Lee J, Leyffer S, editors. Mixed Integer Nonlinear Programming. New York, NY: Springer; 2012. pp. 1-39. DOI: 10.1007/978-1-4614-1927-3\_1

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[71] Bussieck MR, Pruessner A. Mixedinteger nonlinear programming. SIAG/ OPT Views and News. 2003;**14**(1):19-22 [72] Lu Y, Wang S, Sun Y, Yan C. Optimal scheduling of buildings with energy generation and thermal energy storage under dynamic electricity pricing using mixed-integer nonlinear programming. Applied Energy. 2015; **147**:49-58. DOI: 10.1016/j. apenergy.2015.02.060

[73] Shabani N, Sowlati T. A mixed integer non-linear programming model for tactical value chain optimization of a wood biomass power plant. Applied Energy. 2013;**104**:353-361. DOI: 10.1016/j.apenergy.2012.11.01

[74] Fan Y, Xia X. A multi-objective optimization model for building envelope retrofit planning. Energy Procedia. 2015;**75**:1299-1304. DOI: 10.1016/j.egypro.2015.07.193

[75] Karmellos M, Kiprakis A, Mavrotas G. A multi-objective approach for optimal prioritization of energy efficiency measures in buildings: Model, software and case studies. Applied Energy. 2015;**139**:131-150. DOI: 10.1016/ j.apenergy.2014.11.023

[76] Mungle S, Benyoucef L, Son YJ, Tiwari MK. A fuzzy clustering-based genetic algorithm approach for timecost-quality trade-off problems: A case study of highway construction project. Engineering Applications of Artificial Intelligence. 2013;**26**(8):1953-1966. DOI: 10.1016/j.engappai.2013.05.006

[77] Monghasemi S, Nikoo MR, Khaksar Fasaee MA, Adamowski J. A novel multi criteria decision making model for optimizing time-cost-quality trade-off problems in construction projects. Expert Systems With Applications. 2015;**42**(6):3089-3104. DOI: 10.1016/j. eswa.2014.11.032

[78] Brownlee AEI, Wright JA. Constrained, mixed-integer and multiobjective optimisation of building designs by NSGA-II with fitness approximation. Applied Soft

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*Overview of Multi-Objective Optimization Approaches in Construction Project Management*

[86] Tran DH, Cheng MY, Cao MT. Hybrid multiple objective artificial bee colony with differential evolution for the time-cost-quality tradeoff problem. Knowledge-Based Systems. 2015;**74**:

[87] Marzouk M, Abdelhamid M, Elsheikh M. Selecting sustainable building materials using system dynamics and ant colony optimization. Journal of Environmental Engineering and Landscape Management. 2013; **21**(4):237-247. DOI: 10.3846/ 16486897.2013.788506

[88] Wang B, Xia X. Optimal maintenance planning for building energy efficiency retrofitting from optimization and control system perspectives. Energy and Buildings. 2015;**96**:299-308. DOI: 10.1016/j.

enbuild.2015.03.032

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autcon.2009.09.002

176-186. DOI: 10.1016/j. knosys.2014.11.018

[79] Xu J, Tu Y, Zeng Z. A nonlinear multiobjective bilevel model for minimum cost network flow problem in a large-scale construction project. Mathematical Problems in Engineering. 2012. 40 pages. DOI: 10.1155/2012/ 463976. Article ID: 463976

[80] Xu J, Tu Y, Lei X. Applying multiobjective bilevel optimization under fuzzy random environment to traffic assignment problem: Case study of a large-scale construction project. Journal of Infrastructure Systems. 2014; **20**(3):05014003. DOI: 10.1061/(ASCE) IS.1943-555X.0000147

[81] Zhang L, Du J, Zhang S. Solution to the time-cost-quality trade-off problem in construction projects based on immune genetic particle swarm optimization. Journal of Management in Engineering. 2014;**30**(2):163-172. DOI: 10.1061/(ASCE)ME.1943-5479. 0000189

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[84] Yahya M, Saka MP. Construction site layout planning using multiobjective artificial bee colony algorithm with levy flights. Automation in Construction. 2014;**38**(3):14-29. DOI: 10.1016/j.autcon.2013.11.001

[85] Ning X, Lam KC, Lam MCK. Dynamic construction site layout *Overview of Multi-Objective Optimization Approaches in Construction Project Management DOI: http://dx.doi.org/10.5772/intechopen.88185*

planning using max-min ant system. Automation in Construction. 2010; **19**(1):55-65. DOI: 10.1016/j. autcon.2009.09.002

[72] Lu Y, Wang S, Sun Y, Yan C. Optimal scheduling of buildings with energy generation and thermal energy storage under dynamic electricity pricing using mixed-integer nonlinear programming. Applied Energy. 2015;

*Multicriteria Optimization - Pareto-Optimality and Threshold-Optimality*

Computing. 2015;**33**:114-126. DOI: 10.1016/j.asoc.2015.04.010

[79] Xu J, Tu Y, Zeng Z. A nonlinear multiobjective bilevel model for

463976. Article ID: 463976

IS.1943-555X.0000147

0000189

0000784

[80] Xu J, Tu Y, Lei X. Applying multiobjective bilevel optimization under fuzzy random environment to traffic assignment problem: Case study of a large-scale construction project. Journal of Infrastructure Systems. 2014; **20**(3):05014003. DOI: 10.1061/(ASCE)

minimum cost network flow problem in a large-scale construction project. Mathematical Problems in Engineering. 2012. 40 pages. DOI: 10.1155/2012/

[81] Zhang L, Du J, Zhang S. Solution to the time-cost-quality trade-off problem in construction projects based on immune genetic particle swarm

optimization. Journal of Management in Engineering. 2014;**30**(2):163-172. DOI: 10.1061/(ASCE)ME.1943-5479.

[82] Yang IT, Lin YC, Lee HY. Use of support vector regression to improve computational efficiency of stochastic

Management. 2014;**140**(1):04013036. DOI: 10.1061/(ASCE)CO.1943-7862.

[83] Futrell BJ, Ozelkan EC, Brentrup D. Bi-objective optimization of building enclosure design for thermal and lighting performance. Building and Environment. 2015;**2**:591-602. DOI: 10.1016/j.buildenv.2015.03.039

[84] Yahya M, Saka MP. Construction site layout planning using multiobjective artificial bee colony algorithm

with levy flights. Automation in Construction. 2014;**38**(3):14-29. DOI:

[85] Ning X, Lam KC, Lam MCK. Dynamic construction site layout

10.1016/j.autcon.2013.11.001

time-cost trade-off. Journal of Construction Engineering and

[73] Shabani N, Sowlati T. A mixed integer non-linear programming model for tactical value chain optimization of a wood biomass power plant. Applied Energy. 2013;**104**:353-361. DOI: 10.1016/j.apenergy.2012.11.01

[74] Fan Y, Xia X. A multi-objective optimization model for building envelope retrofit planning. Energy Procedia. 2015;**75**:1299-1304. DOI: 10.1016/j.egypro.2015.07.193

[75] Karmellos M, Kiprakis A, Mavrotas G. A multi-objective approach for optimal prioritization of energy

efficiency measures in buildings: Model, software and case studies. Applied Energy. 2015;**139**:131-150. DOI: 10.1016/

[76] Mungle S, Benyoucef L, Son YJ, Tiwari MK. A fuzzy clustering-based genetic algorithm approach for timecost-quality trade-off problems: A case study of highway construction project. Engineering Applications of Artificial Intelligence. 2013;**26**(8):1953-1966. DOI: 10.1016/j.engappai.2013.05.006

[77] Monghasemi S, Nikoo MR, Khaksar Fasaee MA, Adamowski J. A novel multi criteria decision making model for optimizing time-cost-quality trade-off problems in construction projects. Expert Systems With Applications. 2015;**42**(6):3089-3104. DOI: 10.1016/j.

j.apenergy.2014.11.023

eswa.2014.11.032

**74**

[78] Brownlee AEI, Wright JA.

Constrained, mixed-integer and multiobjective optimisation of building designs by NSGA-II with fitness approximation. Applied Soft

**147**:49-58. DOI: 10.1016/j. apenergy.2015.02.060

[86] Tran DH, Cheng MY, Cao MT. Hybrid multiple objective artificial bee colony with differential evolution for the time-cost-quality tradeoff problem. Knowledge-Based Systems. 2015;**74**: 176-186. DOI: 10.1016/j. knosys.2014.11.018

[87] Marzouk M, Abdelhamid M, Elsheikh M. Selecting sustainable building materials using system dynamics and ant colony optimization. Journal of Environmental Engineering and Landscape Management. 2013; **21**(4):237-247. DOI: 10.3846/ 16486897.2013.788506

[88] Wang B, Xia X. Optimal maintenance planning for building energy efficiency retrofitting from optimization and control system perspectives. Energy and Buildings. 2015;**96**:299-308. DOI: 10.1016/j. enbuild.2015.03.032

**Chapter 5**

Overview

**Abstract**

**1. Introduction**

**77**

*Maznah Mat Kasim*

On the Practical Consideration of

Evaluating Relative Importance of

A multicriteria (MC) problem usually consists of a set of predetermined alternatives or subjects to be analyzed, which is prescribed under a finite number of criteria. MC problems are found in various applications to solve various area problems. There are three goals in solving the problems: ranking, sorting or grouping the alternatives according to their overall scores. Most of MC methods require the criteria weights to be combined mathematically with the quality of the criteria in finding the overall score of each alternative. This chapter provides an overview on the practical consideration of evaluators' credibility or superiority in calculating the criteria weights and overall scores of the alternatives. In order to show how the degree of credibility of evaluators can be practically considered in solving a real problem, a numerical example of evaluation of students' academic performance is available in the Appendix at the end of the chapter. The degree of credibility of teachers who participated in weighting the academic subjects was determined objectively, and the rank-based criteria weighting methods were used in the example. Inclusion of the degree of credibility of evaluators who participated in solving

multicriteria problems would make the results more realistic and accurate.

**Keywords:** multicriteria problem, credibility, weights, subjective, aggregation

Multicriteria decision-making (MCDM) is now considered as one discipline of knowledge, which has been expanding very fast in its own domain. Basically, it is about how to make decision when the undertaken issue is surrounded with a multiple number of criteria. The MC problem consists of two main components, alternatives and criteria. In real-life situations, the alternatives are options, organizations, people, or units to be analyzed which are prescribed under a set of finite

Evaluators' Credibility in

Criteria for Some Real-Life

Multicriteria Problems: An

### **Chapter 5**

On the Practical Consideration of Evaluators' Credibility in Evaluating Relative Importance of Criteria for Some Real-Life Multicriteria Problems: An Overview

*Maznah Mat Kasim*

### **Abstract**

A multicriteria (MC) problem usually consists of a set of predetermined alternatives or subjects to be analyzed, which is prescribed under a finite number of criteria. MC problems are found in various applications to solve various area problems. There are three goals in solving the problems: ranking, sorting or grouping the alternatives according to their overall scores. Most of MC methods require the criteria weights to be combined mathematically with the quality of the criteria in finding the overall score of each alternative. This chapter provides an overview on the practical consideration of evaluators' credibility or superiority in calculating the criteria weights and overall scores of the alternatives. In order to show how the degree of credibility of evaluators can be practically considered in solving a real problem, a numerical example of evaluation of students' academic performance is available in the Appendix at the end of the chapter. The degree of credibility of teachers who participated in weighting the academic subjects was determined objectively, and the rank-based criteria weighting methods were used in the example. Inclusion of the degree of credibility of evaluators who participated in solving multicriteria problems would make the results more realistic and accurate.

**Keywords:** multicriteria problem, credibility, weights, subjective, aggregation

### **1. Introduction**

Multicriteria decision-making (MCDM) is now considered as one discipline of knowledge, which has been expanding very fast in its own domain. Basically, it is about how to make decision when the undertaken issue is surrounded with a multiple number of criteria. The MC problem consists of two main components, alternatives and criteria. In real-life situations, the alternatives are options, organizations, people, or units to be analyzed which are prescribed under a set of finite

criteria or attributes. If the number of alternatives is finite and known, the task is to select the best or the optimal alternative, to rank the alternatives according to their overall quality or performance, or to sort or group the alternatives based on certain measurements or values. In this case, the MC problem is usually called as a multiattribute decision-making (MADM) problem, and the alternatives are prescribed under a finite number of criteria or attributes [1]. The MADM methods are utilized to handle discrete MCDM problems [2]. This chapter focuses on MADM problems or more generally MCDM problems, where this type of problem has a finite number of predetermined alternatives, which is described by several criteria or attributes. MCDM problems can be found in various sectors.

consideration of the credibility of the evaluators, the problem of evaluation of students' academic performance is extended by considering the credibility of the teachers who participated in weighting the academic subjects. The detailed discus-

*On the Practical Consideration of Evaluators' Credibility in Evaluating Relative…*

(s) are involved in many stages of the evaluation process in searching for the optimal solution. As all MCDM problems have two main components, the alternatives and the criteria or attributes, the decision-maker(s) or the evaluator(s) would involve in at least two situations: deciding the quality of each alternative based on each of the criteria and also finding the relative importance of the criteria toward the overall performance of the alternatives. As what is usually arose in solving MCDM problems, criteria are contributing at different level of importance and should become a concern to the decision-maker(s) or evaluator(s). The criteria or attributes of the units to be analyzed should not be assumed to have same contri-

Besides having a challenge in finding the suitable evaluator(s) or decisionmaker(s), since they might come with different background and experience, they also come with different levels of superiority or credibility that should be taken into consideration. This issue should be thought seriously because the results may be misleading if those who are involved in doing the evaluation or judgment do not have enough experience or less credible to give judgment regarding the MCDM problem under study. Moreover, the results may differ among the evaluators if the evaluators are at different levels of superiority [9]. Therefore, the credibility of expert(s) or evaluator(s) or decision-maker(s) who are involved in assessing quality of the alternatives or relative importance of attributes should be taken into

Webster's New World College Dictionary defines credibility as the quality of being trustworthy or believable. Credibility is also interpreted by good reputation, reputation, honor, and the presence of someone who stands out in the professional

expected of a professional. In other words, a professional is someone who is skilled, reliable, and entirely responsible for carrying out their duties and profession [11]. This definition of professionalism has a resemblance to the term of credibility so that the two are like two sides of a coin that cannot be separated. For the purpose of assessment or evaluation, professionalism and credibility are the competencies of assessors in carrying out their functions and roles well, full of commitment, trust-

It is normal that the assessors have different levels of credibility, and their credibility should be considered together with their assessments or evaluations. This chapter provides an overview of the current work on how the credibility of the decision-maker(s) or evaluator(s) could be considered especially on evaluating the importance of the criteria or attributes of any MCDM under investigation, how to quantify the credibility of those people, and how that quantitative values could be incorporated in finding the overall score of the alternatives. This issue falls under the concept of group decision-making and extends it with the consideration of the degree of superiority or credibility of the decision-maker(s) or evaluator(s). By deliberation of different relative importance of the attributes plus the different level of credibility or superiority of those who are involved in finding the optimal solution of the MCDM problem, the solution of the problem would be more realistic,

accurate, and representative of the true setting of the problem.

community [10]. Meanwhile, professionalism refers to competence or skill

Referring to those examples of MCDM scenarios, decision-maker(s) or evaluator

sion is available in the Appendix at the end of the chapter.

bution toward the overall quality of the alternatives.

**1.2 Credibility of the evaluators**

*DOI: http://dx.doi.org/10.5772/intechopen.92541*

consideration.

**79**

worthiness, and accountability.

#### **1.1 Examples of multicriteria decision-making problems**

Selection problems are really of an MCDM type, a simple problem that we are facing almost every day, for example, when we want to select a dress or a shirt to wear. A decision to choose which dress or shirt is based on certain attributes or factors, such as for what function (office, leisure, and business), color preference, and style or fashion. Here, the types of dress/cloth are the alternatives, while all factors that become the basis of evaluation are the attributes. Another example is when we want to choose the best location to set up projects such as housing, industrial, agricultural activities, recreation center, hoteling, and so on. Many factors or criteria that may be conflicting with each other should be considered by the decision-makers. Selecting the best candidate for various positions that can be conducted in many settings such as face-to-face interviews or online test is also an MCDM problem since the selection will be based on certain requirements. Selecting employees in different organizations with different scope of jobs with different requirements imposed by the related organization can also be categorized as an MCDM problem.

Another example is about selection of the best supplier of a manufacturing firm [3, 4], selection of the best personal computer [5], and selection of a suitable e-learning system [6] to be implemented in an educational institution. These studies focused on selecting the best alternative from a finite number of alternatives that were prescribed under a few evaluation criteria. These studies have the same main issue that is the relative importance or the weights of the evaluation criteria toward the overall performance of the alternatives under study. The studies provide ways to find weights subjectively and how to aggregate the weights when a group of decision-makers were involved in judging the importance of the criteria.

In addition, conducting an evaluation of a program, for example, is usually done after identifying the aspects of the program to evaluate. We may have many programs to be evaluated under several aspects of evaluations with the involvement of one evaluator or a group of evaluators. In a different situation, it may be only one program to be evaluated under several aspects and may be evaluated by one or many evaluators. Besides, many other evaluation situations are usually performed with the presence of many criteria such as evaluation of students, evaluation of employees' performance, evaluation of learning approaches [7], and evaluation of students' performance [8]. In relation to the study about the evaluation of students' academic performance in primary schools, five academic subjects were assumed to have different contribution toward the overall performance of the students. A few experienced teachers were asked to evaluate the degree of importance of the subjects. The resulting weights of the academic subjects were incorporated in finding the overall academic performance of the students in year six in one selected primary school in the northern part of Malaysia. For the purpose of illustrating the practical

consideration of the credibility of the evaluators, the problem of evaluation of students' academic performance is extended by considering the credibility of the teachers who participated in weighting the academic subjects. The detailed discussion is available in the Appendix at the end of the chapter.

### **1.2 Credibility of the evaluators**

criteria or attributes. If the number of alternatives is finite and known, the task is to select the best or the optimal alternative, to rank the alternatives according to their overall quality or performance, or to sort or group the alternatives based on certain

Selection problems are really of an MCDM type, a simple problem that we are facing almost every day, for example, when we want to select a dress or a shirt to wear. A decision to choose which dress or shirt is based on certain attributes or factors, such as for what function (office, leisure, and business), color preference, and style or fashion. Here, the types of dress/cloth are the alternatives, while all factors that become the basis of evaluation are the attributes. Another example is when we want to choose the best location to set up projects such as housing, industrial, agricultural activities, recreation center, hoteling, and so on. Many factors or criteria that may be conflicting with each other should be considered by the decision-makers. Selecting the best candidate for various positions that can be conducted in many settings such as face-to-face interviews or online test is also an MCDM problem since the selection will be based on certain requirements. Selecting employees in different organizations with different scope of jobs with different requirements imposed by the related organization can also be categorized as an

Another example is about selection of the best supplier of a manufacturing firm [3, 4], selection of the best personal computer [5], and selection of a suitable e-learning system [6] to be implemented in an educational institution. These studies focused on selecting the best alternative from a finite number of alternatives that were prescribed under a few evaluation criteria. These studies have the same main issue that is the relative importance or the weights of the evaluation criteria toward the overall performance of the alternatives under study. The studies provide ways to find weights subjectively and how to aggregate the weights when a group of decision-makers were involved in judging the importance

In addition, conducting an evaluation of a program, for example, is usually done after identifying the aspects of the program to evaluate. We may have many programs to be evaluated under several aspects of evaluations with the involvement of one evaluator or a group of evaluators. In a different situation, it may be only one program to be evaluated under several aspects and may be evaluated by one or many evaluators. Besides, many other evaluation situations are usually performed with the presence of many criteria such as evaluation of students, evaluation of employees' performance, evaluation of learning approaches [7], and evaluation of students' performance [8]. In relation to the study about the evaluation of students' academic performance in primary schools, five academic subjects were assumed to have different contribution toward the overall performance of the students. A few experienced teachers were asked to evaluate the degree of importance of the subjects. The resulting weights of the academic subjects were incorporated in finding the overall academic performance of the students in year six in one selected primary school in the northern part of Malaysia. For the purpose of illustrating the practical

measurements or values. In this case, the MC problem is usually called as a multiattribute decision-making (MADM) problem, and the alternatives are prescribed under a finite number of criteria or attributes [1]. The MADM methods are utilized to handle discrete MCDM problems [2]. This chapter focuses on MADM problems or more generally MCDM problems, where this type of problem has a finite number of predetermined alternatives, which is described by several criteria

or attributes. MCDM problems can be found in various sectors.

*Multicriteria Optimization - Pareto-Optimality and Threshold-Optimality*

**1.1 Examples of multicriteria decision-making problems**

MCDM problem.

of the criteria.

**78**

Referring to those examples of MCDM scenarios, decision-maker(s) or evaluator (s) are involved in many stages of the evaluation process in searching for the optimal solution. As all MCDM problems have two main components, the alternatives and the criteria or attributes, the decision-maker(s) or the evaluator(s) would involve in at least two situations: deciding the quality of each alternative based on each of the criteria and also finding the relative importance of the criteria toward the overall performance of the alternatives. As what is usually arose in solving MCDM problems, criteria are contributing at different level of importance and should become a concern to the decision-maker(s) or evaluator(s). The criteria or attributes of the units to be analyzed should not be assumed to have same contribution toward the overall quality of the alternatives.

Besides having a challenge in finding the suitable evaluator(s) or decisionmaker(s), since they might come with different background and experience, they also come with different levels of superiority or credibility that should be taken into consideration. This issue should be thought seriously because the results may be misleading if those who are involved in doing the evaluation or judgment do not have enough experience or less credible to give judgment regarding the MCDM problem under study. Moreover, the results may differ among the evaluators if the evaluators are at different levels of superiority [9]. Therefore, the credibility of expert(s) or evaluator(s) or decision-maker(s) who are involved in assessing quality of the alternatives or relative importance of attributes should be taken into consideration.

Webster's New World College Dictionary defines credibility as the quality of being trustworthy or believable. Credibility is also interpreted by good reputation, reputation, honor, and the presence of someone who stands out in the professional community [10]. Meanwhile, professionalism refers to competence or skill expected of a professional. In other words, a professional is someone who is skilled, reliable, and entirely responsible for carrying out their duties and profession [11]. This definition of professionalism has a resemblance to the term of credibility so that the two are like two sides of a coin that cannot be separated. For the purpose of assessment or evaluation, professionalism and credibility are the competencies of assessors in carrying out their functions and roles well, full of commitment, trustworthiness, and accountability.

It is normal that the assessors have different levels of credibility, and their credibility should be considered together with their assessments or evaluations. This chapter provides an overview of the current work on how the credibility of the decision-maker(s) or evaluator(s) could be considered especially on evaluating the importance of the criteria or attributes of any MCDM under investigation, how to quantify the credibility of those people, and how that quantitative values could be incorporated in finding the overall score of the alternatives. This issue falls under the concept of group decision-making and extends it with the consideration of the degree of superiority or credibility of the decision-maker(s) or evaluator(s). By deliberation of different relative importance of the attributes plus the different level of credibility or superiority of those who are involved in finding the optimal solution of the MCDM problem, the solution of the problem would be more realistic, accurate, and representative of the true setting of the problem.

In achieving the objective of the writing, the chapter is organized as follows. The next section describes the basic notations for this chapter. Section 3 discusses the concept of weights and the related methods, particularly the rank-based weighting method. Section 4 discusses on the aggregation of criteria weights and the values of criteria. Section 5 explains how to aggregate the credibility of the evaluators who are involved in weighting or finding weights or relative importance of the criteria. Furthermore, Section 5 also illustrates two approaches to aggregate the degree of credibility of evaluators in finding the relative importance in order to find the overall performance of the alternatives and their rankings. Section 6 suggests a few ways to quantify the credibility of the evaluators. The conclusion of the chapter is in Section 7, which is followed by a list of all references of the chapter. A numerical example is provided in the Appendix at the end of the chapter.

[12]. The objective methods are data-driven methods where quality values of the criteria should be available prior to the evaluation of criteria's relative importance. Based on the criteria's values, proxy measures such as standard deviation, correlation, variance, range, coefficient of variation, and entropy [13–17] would represent the criteria weights to be calculated. In relation to the concept of entropy, it was introduced in the communication theory, usually refers to uncertainty. The measure of entropy is often used to quantify the information or message. However, the entropy measure has become the proxy measures of criterion weights in MCDM domain. In other words, these objective methods produce weights of criteria based on the intrinsic information of the criteria. These methods do not require evaluators to do the criteria weighting. No further discussion is included in this chapter

*On the Practical Consideration of Evaluators' Credibility in Evaluating Relative…*

This subsection focuses on the discussion of rank-based weighting methods [18, 19] as these methods are used in this chapter in the illustration of practical consideration of evaluators' credibility in evaluating relative importance of criteria for some real-life multicriteria problems. These methods are very easy to use but have good impact [20]. Three popular rank-based methods are rank-sum (RS), rank reciprocal (RR), and rank order centroid (ROC). The mathematical representations

Suppose *rj* be a ranking of criterion *j* given by an evaluator where *rj* is an integer number with possible values from 1 to *m*. The smaller value of *rj* means that the ranking of that criterion is higher and more important than the other criteria. The value of *r* ¼ f g *r*1, … ,*rm* can be transformed into *w* ¼ f g *w*1, … , *wm* by using any of the following formula for RS, RR, and ROC, respectively. It should be noted that the

*<sup>w</sup> j rs* ð Þ <sup>¼</sup> <sup>2</sup> *<sup>m</sup>* <sup>þ</sup> <sup>1</sup> � *rj*

*<sup>w</sup> j rr* ð Þ <sup>¼</sup> <sup>P</sup>

*m* X*m k*¼1

Referring to the numerical example in the Appendix, there are five criteria representing five academic subjects; *rj* is a ranking of academic subject*j* where *rj* is

f g *r*1, … ,*r*<sup>5</sup> represents ranks of academic subjects 1 to 5 that can be transformed into

Many studies were conducted to study the performance of these rank-based methods as criteria weighting methods. For example, a simulation experiment was conducted on investigating the performance of the three rank-based weighting methods (RS, RR, RS) and equal weights (EW) where the data was generated on a random basis [16]. Three performance measures of the methods were "hit rate," "average value loss," and "average proportion of maximum value range achieved." The results show that the ROC was found to be the best technique in most cases an

*<sup>w</sup> j roc* ð Þ <sup>¼</sup> <sup>1</sup>

.

an integer number with possible values from 1 to 5, while the value of *r* ¼

weights of academic subjects 1 to 5, *w* ¼ f g *w*1, … , *w*<sup>5</sup> , respectively.

0 if*rk* <*rj*

� �

1*=r j m j*¼1 1*=r j*

1 *rk*

� *I rk* >*rj*

*m m*ð Þ � <sup>1</sup> (1)

� � (3)

(2)

because objective weights are not the focus of the chapter.

**3.1 Rank-based weighting methods**

*DOI: http://dx.doi.org/10.5772/intechopen.92541*

of the three methods are as follows.

where *I rk* > *rj*

**81**

sum of weights of the criteria is usually equal to one:

� � <sup>¼</sup> 1 if *rk* <sup>≥</sup> *rj*

�

#### **2. Basic notation**

Let *A* ¼ f g *A*1, … , *An* be a set of *n* alternatives that are prescribed under *m* criteria, *C* ¼ f g *C*1, … ,*Cm* , and *xij* be a value of alternative *i*, under criterion *j*, where *i* ¼ 1, … , *n* and *j* ¼ 1, … , *m.* Let *w* ¼ f g *w*1, … , *wm* be the weight of the criteria with conditions that 0≤ *w <sup>j</sup>* ≤1 and P*<sup>m</sup> <sup>j</sup>*¼<sup>1</sup>*<sup>w</sup> <sup>j</sup>* <sup>¼</sup> 1. This information can be illustrated as a decision matrix as shown in **Figure** 1.

In relation to the numerical example in the Appendix, the students are the alternatives, while the academic subjects are the criteria. So, *A* ¼ f g *A*1, … , *A*<sup>10</sup> represents a set of 10 students that are prescribed under five academic subjects, *C* ¼ f g *C*1, … ,*C*<sup>5</sup> , and *xij* is the score of student *i*, under academic subject *j*, where *i* ¼ 1, … , 10 and *j* ¼ 1, … , 5*.* The weights of the criteria, *w* ¼ f g *w*1, … , *w*<sup>5</sup> , obviously refer to the relative importance of the academic subjects toward the composite or final score of each student.


#### **Figure 1.**

*A multiattribute problem as a decision matrix.*

#### **3. Weights of criteria**

In finding the relative importance of the criteria or simply the weights of the criteria, *w* ¼ f g *w*1, … , *wm* , there are many methods available in literature which are classified into two main approaches, objective methods and subjective methods

*On the Practical Consideration of Evaluators' Credibility in Evaluating Relative… DOI: http://dx.doi.org/10.5772/intechopen.92541*

[12]. The objective methods are data-driven methods where quality values of the criteria should be available prior to the evaluation of criteria's relative importance. Based on the criteria's values, proxy measures such as standard deviation, correlation, variance, range, coefficient of variation, and entropy [13–17] would represent the criteria weights to be calculated. In relation to the concept of entropy, it was introduced in the communication theory, usually refers to uncertainty. The measure of entropy is often used to quantify the information or message. However, the entropy measure has become the proxy measures of criterion weights in MCDM domain. In other words, these objective methods produce weights of criteria based on the intrinsic information of the criteria. These methods do not require evaluators to do the criteria weighting. No further discussion is included in this chapter because objective weights are not the focus of the chapter.

#### **3.1 Rank-based weighting methods**

In achieving the objective of the writing, the chapter is organized as follows. The next section describes the basic notations for this chapter. Section 3 discusses the concept of weights and the related methods, particularly the rank-based weighting method. Section 4 discusses on the aggregation of criteria weights and the values of criteria. Section 5 explains how to aggregate the credibility of the evaluators who are involved in weighting or finding weights or relative importance of the criteria. Furthermore, Section 5 also illustrates two approaches to aggregate the degree of credibility of evaluators in finding the relative importance in order to find the overall performance of the alternatives and their rankings. Section 6 suggests a few ways to quantify the credibility of the evaluators. The conclusion of the chapter is in Section 7, which is followed by a list of all references of the chapter. A numerical

example is provided in the Appendix at the end of the chapter.

*Multicriteria Optimization - Pareto-Optimality and Threshold-Optimality*

Let *A* ¼ f g *A*1, … , *An* be a set of *n* alternatives that are prescribed under *m* criteria, *C* ¼ f g *C*1, … ,*Cm* , and *xij* be a value of alternative *i*, under criterion *j*, where *i* ¼ 1, … , *n* and *j* ¼ 1, … , *m.* Let *w* ¼ f g *w*1, … , *wm* be the weight of the criteria with

In relation to the numerical example in the Appendix, the students are the alternatives, while the academic subjects are the criteria. So, *A* ¼ f g *A*1, … , *A*<sup>10</sup> represents a set of 10 students that are prescribed under five academic subjects, *C* ¼ f g *C*1, … ,*C*<sup>5</sup> , and *xij* is the score of student *i*, under academic subject *j*, where *i* ¼ 1, … , 10 and *j* ¼ 1, … , 5*.* The weights of the criteria, *w* ¼ f g *w*1, … , *w*<sup>5</sup> , obviously refer to the relative importance of the academic subjects toward the composite or

In finding the relative importance of the criteria or simply the weights of the criteria, *w* ¼ f g *w*1, … , *wm* , there are many methods available in literature which are classified into two main approaches, objective methods and subjective methods

*<sup>j</sup>*¼<sup>1</sup>*<sup>w</sup> <sup>j</sup>* <sup>¼</sup> 1. This information can be illustrated as a

**2. Basic notation**

conditions that 0≤ *w <sup>j</sup>* ≤1 and P*<sup>m</sup>*

final score of each student.

**3. Weights of criteria**

*A multiattribute problem as a decision matrix.*

**Figure 1.**

**80**

decision matrix as shown in **Figure** 1.

This subsection focuses on the discussion of rank-based weighting methods [18, 19] as these methods are used in this chapter in the illustration of practical consideration of evaluators' credibility in evaluating relative importance of criteria for some real-life multicriteria problems. These methods are very easy to use but have good impact [20]. Three popular rank-based methods are rank-sum (RS), rank reciprocal (RR), and rank order centroid (ROC). The mathematical representations of the three methods are as follows.

Suppose *rj* be a ranking of criterion *j* given by an evaluator where *rj* is an integer number with possible values from 1 to *m*. The smaller value of *rj* means that the ranking of that criterion is higher and more important than the other criteria. The value of *r* ¼ f g *r*1, … ,*rm* can be transformed into *w* ¼ f g *w*1, … , *wm* by using any of the following formula for RS, RR, and ROC, respectively. It should be noted that the sum of weights of the criteria is usually equal to one:

$$\text{low}\_{j(\mathbf{r})} = \frac{2\left(m + \mathbf{1} - r\_j\right)}{m(m - \mathbf{1})} \tag{1}$$

$$\left(w\right)\_{j(r)} = \frac{\mathbb{1}\_{r\_j}}{\sum\_{j=1}^{m} \mathbb{1}\_{r\_j}}\tag{2}$$

$$\log\_{j(mc)} = \frac{1}{m} \sum\_{k=1}^{m} \frac{1}{r\_k} \times I(r\_k > r\_j) \tag{3}$$

where *I rk* > *rj* � � <sup>¼</sup> 1 if *rk* <sup>≥</sup> *rj* 0 if*rk* <*rj* � .

Referring to the numerical example in the Appendix, there are five criteria representing five academic subjects; *rj* is a ranking of academic subject*j* where *rj* is an integer number with possible values from 1 to 5, while the value of *r* ¼ f g *r*1, … ,*r*<sup>5</sup> represents ranks of academic subjects 1 to 5 that can be transformed into weights of academic subjects 1 to 5, *w* ¼ f g *w*1, … , *w*<sup>5</sup> , respectively.

Many studies were conducted to study the performance of these rank-based methods as criteria weighting methods. For example, a simulation experiment was conducted on investigating the performance of the three rank-based weighting methods (RS, RR, RS) and equal weights (EW) where the data was generated on a random basis [16]. Three performance measures of the methods were "hit rate," "average value loss," and "average proportion of maximum value range achieved." The results show that the ROC was found to be the best technique in most cases an in every measure. Another study on these three rank-based weighting techniques and EW concludes that the rank-based methods have higher correlations with the so-called true weights than EW [21].

For the swing method, the evaluator must identify an alternative with the worst consequences on all attribute. The evaluator(s) can change one of the criteria from the worst consequence to the best. Then, the evaluator(s) is asked to choose the criteria that he/she would most prefer to modify from its worst to its best level, the criterion with the most chosen swing is the most important, and 100 points is

*On the Practical Consideration of Evaluators' Credibility in Evaluating Relative…*

The GW method begins with a horizontal line that is marked with a series of number, such as (9-7-5-3-1-3-5-7-9). The evaluator is expected to place a mark that represents the relative importance of a criterion on the horizontal line with the basis that a criterion is either more, equally, or less important than another criterion by a factor of 1–9. Then, a decision matrix is built as a pairwise comparison matrix. A quantitative weight for a criterion can be calculated by taking the sum of each row, and then the scores are normalized to obtain an overall weight vector. The GW method enables the evaluators to express preferences in a purely visual way. However, GW is sometimes criticized, since it allows evaluator(s) to assign weights in a

A Delphi subjective weighting method [35] requires one focus group of evalua-

Finding the final score of each alternative is very important since the final scores of the alternatives are required to rank the alternatives. Basically, those alternatives with higher scores should be positioned at higher rankings and vice versa. In order to find the overall or composite or final values of each alternative, the criteria weights should be aggregated with each alternative's values of the corresponding criteria. There are many aggregation methods available in literature. The section focuses on simple additive weighted average (SAW) method as the chapter uses SAW in the numerical example (in the Appendix at the end of the chapter). Furthermore, SAW method is a very well-established method and very easy to use [16].

tors to evaluate the relative importance of the criteria. Each evaluator remains nameless to each other that can reduce the risk of personal effects or individual bias. The evaluation is conducted in more than one round until the group ends with a consensus of opinions on the relative importance of the criteria under study. The main advantage of this method is that the method avoids confrontation of the experts [36]. However, to pool up such a focus group is quite costly and timely.

**4. Aggregation of criteria weights and values of criteria**

**4.1 Simple additive weighted average (SAW) method**

score of student *i*, where *i* = 1, … , 10.

**83**

The mathematical equation for SAW is given as follows:

*Score Ai* <sup>¼</sup> <sup>X</sup>*<sup>m</sup>*

*j*¼1

*Score Ai* is the overall score of alternative *i*. Based on *Score Ai*, where *i* ¼ 1, … , *n*, the *n* alternatives could be ranked, selected, or sorted with the condition that the alternatives with the higher overall scores should be ranked at higher positions. Referring to the numerical example in the Appendix, *Score Ai* represents the overall

SAW is an old method, and MacCrimmon is one of the first researchers that summarized this method in 1968 [37]. As a well-established method, it is used widely [38] in solving MC problems, particularly for the evaluation of alternatives. Basically, this method is the same as the simple arithmetic average method, but

*w jxij* (4)

allocated to the most important criterion.

*DOI: http://dx.doi.org/10.5772/intechopen.92541*

more relaxed manner.

A study is also done where EW, RS, and ROC methods were compared to direct rating and ratio weight methods [22]. Basically, the direct rating method is a simple type of weighting approach in which the decision-maker or the evaluator must rate all the criteria according to their importance. The evaluator can directly quantify their preference of the criteria. The rating does not constrain the decision-maker's responses since it is possible for the evaluator to alter the importance of one criterion without adjusting the weight of another [23]. The comparison was conducted under a condition that the evaluators' judgments of the criteria weights are not certain and subject to random errors. The results show that the direct rating tends to give better quality of decision results when the uncertainty is set as small, while ROC provides comparable results to the ratio weights when a large degree of error is placed. Please note that the ratio weight method requires the evaluators firstly rank the related criteria based on their importance. The evaluators should allocate certain value such as 10 for the least important attribute, and the rest of attributes are judged as multiples of 10. The weight of a criterion is obtained by dividing the criterion's weight with the sum of all attributes' weights.

The superiority of ROC over other rank-based methods is also subsequently confirmed in different simulation conditions [24]. An investigation on RS, RR, and ROC weighting methods was also carried out by changing the number of criteria from two to seven [25]. It is found that ROC gives the largest gap between the weights of the most important criterion and the least. RS provides the flattest weight function in the linear form. For RR, the weight of the most important one descends most aggressively to that of the second highest weight value, and then, the function continues to move flatter. In relation to rank-based weighting methods, another rank-based method was proposed [26]. This new rank-based method is called as generalized sum of ranks (GRS). Further investigation was carried out where the performance of GRS was compared to RS, RR, and ROC using a simulation experiment. The result of the investigation shows that GRS has a similar performance to ROC.

Based on the previous discussion, it can be concluded that the three rank-based weighting methods, RS, RR, and ROC, are having good features especially the ROC method. Therefore, these rank-based methods are used in the current study to illustrate how to include the degree of credibility of the evaluators who are involved in ranking the importance of the criteria. Furthermore, converting the ranks into weight values is not difficult, and the related formula is given as in Equations (1), (2), and (3).

#### **3.2 Other subjective weighting methods**

Other subjective weighting methods are analytic hierarchy process (AHP) [4, 27, 28], swing methods [29, 30], graphical weighting (GW) method [31], and Delphi method [32]. The AHP technique was introduced in 1980 [33]. It is a very popular MC approach, and it is done by conducting pairwise comparison of the importance of each pair of criteria. A prioritization procedure is implemented to draw a corresponding priority vector, where this priority vector represents the criteria weights. Thus, if the judgments are consistent, all prioritization procedures would give the same results. At the same time, if the judgments are inconsistent, prioritization procedures will provide different priority vectors [34]. Nevertheless, AHP is widely criticized for being such a tedious process, especially when there are a significant number of criteria or alternatives.

*On the Practical Consideration of Evaluators' Credibility in Evaluating Relative… DOI: http://dx.doi.org/10.5772/intechopen.92541*

For the swing method, the evaluator must identify an alternative with the worst consequences on all attribute. The evaluator(s) can change one of the criteria from the worst consequence to the best. Then, the evaluator(s) is asked to choose the criteria that he/she would most prefer to modify from its worst to its best level, the criterion with the most chosen swing is the most important, and 100 points is allocated to the most important criterion.

The GW method begins with a horizontal line that is marked with a series of number, such as (9-7-5-3-1-3-5-7-9). The evaluator is expected to place a mark that represents the relative importance of a criterion on the horizontal line with the basis that a criterion is either more, equally, or less important than another criterion by a factor of 1–9. Then, a decision matrix is built as a pairwise comparison matrix. A quantitative weight for a criterion can be calculated by taking the sum of each row, and then the scores are normalized to obtain an overall weight vector. The GW method enables the evaluators to express preferences in a purely visual way. However, GW is sometimes criticized, since it allows evaluator(s) to assign weights in a more relaxed manner.

A Delphi subjective weighting method [35] requires one focus group of evaluators to evaluate the relative importance of the criteria. Each evaluator remains nameless to each other that can reduce the risk of personal effects or individual bias. The evaluation is conducted in more than one round until the group ends with a consensus of opinions on the relative importance of the criteria under study. The main advantage of this method is that the method avoids confrontation of the experts [36]. However, to pool up such a focus group is quite costly and timely.

#### **4. Aggregation of criteria weights and values of criteria**

Finding the final score of each alternative is very important since the final scores of the alternatives are required to rank the alternatives. Basically, those alternatives with higher scores should be positioned at higher rankings and vice versa. In order to find the overall or composite or final values of each alternative, the criteria weights should be aggregated with each alternative's values of the corresponding criteria. There are many aggregation methods available in literature. The section focuses on simple additive weighted average (SAW) method as the chapter uses SAW in the numerical example (in the Appendix at the end of the chapter). Furthermore, SAW method is a very well-established method and very easy to use [16].

#### **4.1 Simple additive weighted average (SAW) method**

The mathematical equation for SAW is given as follows:

$$\text{Score } A\_i = \sum\_{j=1}^{m} w\_j \mathbf{x}\_{ij} \tag{4}$$

*Score Ai* is the overall score of alternative *i*. Based on *Score Ai*, where *i* ¼ 1, … , *n*, the *n* alternatives could be ranked, selected, or sorted with the condition that the alternatives with the higher overall scores should be ranked at higher positions. Referring to the numerical example in the Appendix, *Score Ai* represents the overall score of student *i*, where *i* = 1, … , 10.

SAW is an old method, and MacCrimmon is one of the first researchers that summarized this method in 1968 [37]. As a well-established method, it is used widely [38] in solving MC problems, particularly for the evaluation of alternatives. Basically, this method is the same as the simple arithmetic average method, but

in every measure. Another study on these three rank-based weighting techniques and EW concludes that the rank-based methods have higher correlations with the

*Multicriteria Optimization - Pareto-Optimality and Threshold-Optimality*

A study is also done where EW, RS, and ROC methods were compared to direct rating and ratio weight methods [22]. Basically, the direct rating method is a simple type of weighting approach in which the decision-maker or the evaluator must rate all the criteria according to their importance. The evaluator can directly quantify their preference of the criteria. The rating does not constrain the decision-maker's responses since it is possible for the evaluator to alter the importance of one criterion without adjusting the weight of another [23]. The comparison was conducted under a condition that the evaluators' judgments of the criteria weights are not certain and subject to random errors. The results show that the direct rating tends to give better quality of decision results when the uncertainty is set as small, while ROC provides comparable results to the ratio weights when a large degree of error is placed. Please note that the ratio weight method requires the evaluators firstly rank the related criteria based on their importance. The evaluators should allocate certain value such as 10 for the least important attribute, and the rest of attributes are judged as multiples of 10. The weight of a criterion is obtained by dividing the

The superiority of ROC over other rank-based methods is also subsequently confirmed in different simulation conditions [24]. An investigation on RS, RR, and ROC weighting methods was also carried out by changing the number of criteria from two to seven [25]. It is found that ROC gives the largest gap between the weights of the most important criterion and the least. RS provides the flattest weight function in the linear form. For RR, the weight of the most important one descends most aggressively to that of the second highest weight value, and then, the function continues to move flatter. In relation to rank-based weighting methods, another rank-based method was proposed [26]. This new rank-based method is called as generalized sum of ranks (GRS). Further investigation was carried out where the performance of GRS was compared to RS, RR, and ROC using a simulation experiment. The result of the investigation shows that GRS has a similar

Based on the previous discussion, it can be concluded that the three rank-based weighting methods, RS, RR, and ROC, are having good features especially the ROC method. Therefore, these rank-based methods are used in the current study to illustrate how to include the degree of credibility of the evaluators who are involved in ranking the importance of the criteria. Furthermore, converting the ranks into weight values is not difficult, and the related formula is given as in Equations (1),

Other subjective weighting methods are analytic hierarchy process (AHP) [4, 27, 28], swing methods [29, 30], graphical weighting (GW) method [31], and Delphi method [32]. The AHP technique was introduced in 1980 [33]. It is a very popular MC approach, and it is done by conducting pairwise comparison of the importance of each pair of criteria. A prioritization procedure is implemented to draw a corresponding priority vector, where this priority vector represents the criteria weights. Thus, if the judgments are consistent, all prioritization procedures would give the same results. At the same time, if the judgments are inconsistent, prioritization procedures will provide different priority vectors [34]. Nevertheless, AHP is widely criticized for being such a tedious process, especially

when there are a significant number of criteria or alternatives.

so-called true weights than EW [21].

performance to ROC.

**3.2 Other subjective weighting methods**

(2), and (3).

**82**

criterion's weight with the sum of all attributes' weights.

instead of having the same weight values for the criteria, SAW method uses mostly distinct weights values of the criteria. As given in Eq. (4), the overall performance of each alternative is obtained by multiplying the rating of each alternative on each criterion by the weight assigned to the criterion and then summing these products over all criteria [15]. The best alternative is the one that obtained the highest score and will be selected or ranked at the first position. Many recent studies used the SAW method, for example, in [39–41], and a review on its applications is also available [42].

Besides SAW or also known as weighted sum method (WSM), there is another average technique, called weighted product model (WPM) or simple geometric weighted (SGW) or simple geometric average method. In WPM, the overall performance of each alternative is determined by raising the rating of the alternative to the power of the criterion weight and then multiplying these products over all criteria [15]. However, WPM is a little bit complex as compared to SAW since WPM involves power and multiplications.

of the criteria based on rank-based weighting methods as explained in Section 3.1.

*On the Practical Consideration of Evaluators' Credibility in Evaluating Relative…*

by evaluator *l*, where *l* ¼ 1, … , *p*. In order to include the credibility of the evaluators, let us introduce a new set of values that represents the different credibility of the evaluators. Let *u<sup>l</sup>* be the degree of credibility of evaluator *l*, where 0≤ *ul* ≤1, and

*<sup>l</sup>*¼<sup>1</sup>*ul* <sup>¼</sup> 1. There are two approaches [57] where the degree of credibility of the evaluators could be attached in finding the overall scores. The first approach is in calculating the final weight of criteria as given in **Figure 1**, and the second approach is in computing the overall performance of the alternatives as given in **Figure 2**. For the first approach as portrayed in **Figure 2**, the degree of credibility of the evaluators is attached to the resulted weights from the ranks of criteria by using any of the equations, Eq. (1), Eq. (2), or Eq. (3). So, here there are *p* sets of weights of the criteria, and the average of that p weights for each criterion is calculated by summing up all weights for that criteria and divide the sum with the total number of evaluators. So now, there is only one set of weights that can be aggregated with the values of alternatives for each corresponding criterion as given in Eq. (4). There

For the second approach, the criteria weights obtained from each evaluator are kept, and then each set of weights is aggregated with the quality values of each alternative. So, here there are *p* sets of overall values of the alternatives. In order to get the final overall score of the alternatives, the average of the *p* scores for each alternative should be calculated. The ranking or sorting of the alternatives or selecting the best alternative is done based on the average of that *p* overall scores of each individual alternative. The following section provides some suggestion on how

Referring to the numerical example in the Appendix, there were three evaluators involved in ranking the importance of the five academic subjects, and the number

Credibility is synonym to professionalism, integrity, trustworthiness, authority, and believability. A study focuses on how to assess the credibility of expert witnesses [58]. A 41-item measure was constructed based on the ratings by a panel of judges, and a factor analysis yielded that credibility is a product of four factors: likeability, trustworthiness, believability, and intelligence. Another study concerns about the credibility of information in digital era [59]. Credibility is said to have two main components: trustworthiness and expertise. However, the authors conclude that the relation among youth, digital media, and credibility today is sufficiently complex to resist simple explanations, and their study represents a first step toward mapping that complexity and providing a basis for future work that seeks to find

by evaluator *<sup>l</sup>*, where *<sup>l</sup>* <sup>¼</sup> 1, … , 3, and *<sup>n</sup>* = 10, while *<sup>u</sup><sup>l</sup>* represents the degree of

*<sup>j</sup>* is the rank of academic subject *j* , with *j =* 1, … ,5, evaluated

*<sup>l</sup>*¼<sup>1</sup>*u<sup>l</sup>* <sup>¼</sup> 1.

*<sup>j</sup>* be rank of criterion *j*, evaluated

Suppose there is a panel of *p* evaluators, and let *r<sup>l</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.92541*

is only one set of overall performance of all *n* alternatives.

to quantify the credibility of the evaluators.

credibility of evaluator *l*, where 0 ≤*u<sup>l</sup>* ≤ 1, and P<sup>3</sup>

**6. Quantification of credibility of evaluators**

of students is 10. So, *r<sup>l</sup>*

explanations.

**85**

P*<sup>p</sup>*

**Figure 2.** *Approach 1.*

#### **4.2 Other aggregation methods**

AHP [14], technique for order preference by similarity to ideal solution (TOPSIS), and *VlseKriterijumska Optimizacija Kompromisno Resenje* (VIKOR) [43] are also popular aggregation methods in solving MC problems. As previously mentioned in Section 3.2, AHP is built under the concept of pairwise comparison either in finding the criteria weights or criteria values of the alternatives. The aggregation of criteria weights and the criteria values obtained by AHP is sometimes done by using the SAW or SGW methods.

AHP and TOPSIS are two different aggregation methods. TOPSIS assigns the best alternative that relies on the concepts of compromise solution, where the best alternative is the one that has the shortest distance from the ideal solution and the farthest distance from the negative ideal solution [44]. In other words, alternatives are prioritized according to their distances from positive ideal solutions and negative ideal solutions, and the Euclidean distance approach is utilized to evaluate the relative closeness of the alternatives to the ideal solutions. There is a series of steps of TOPSIS, but this method starts with the weighted normalization of all performance values against each criterion. Some recent applications of the TOPSIS method are available [45–48].

VIKOR method [49] is quite similar to TOPSIS method, but there are some important differences, and one of the differences is about the normalization process. TOPSIS uses the vector linearization where the normalized value could be different for different evaluation unit of a certain criterion, while VIKOR uses linear normalization where the normalized value does not depend on the evaluation unit of a criterion. VIKOR has also been used in many real-world MCDM problems such as mobile banking services [50] digital music service platforms [51], military airport location selection [52], concrete bridge projects [53], risk evaluation of construction projects [54], maritime transportation [55], and energy management [56].

#### **5. Inclusion of credibility of evaluators in solving multicriteria problems**

This section discusses how credibility can be included practically in solving MC problems. Suppose the evaluators are requested to evaluate the relative importance *On the Practical Consideration of Evaluators' Credibility in Evaluating Relative… DOI: http://dx.doi.org/10.5772/intechopen.92541*

$$\begin{pmatrix} r\_1^1, \dots, r\_1^p \\ \vdots \\ r\_m^1, \dots, r\_m^p \end{pmatrix} \xrightarrow{r\_{\{w\}}^l} \begin{pmatrix} u^1 w\_1^1, \dots, u^1 w\_1^p \\ \vdots \\ u^p w\_m^p, \dots, u^p w\_m^p \end{pmatrix} \xrightarrow{\sum\_{i=1}^p u^1\_i} (w\_1^{av}, \dots, w\_m^{av}) \xrightarrow{\mathcal{S}coreA\_l} \begin{pmatrix} \mathcal{ScoreA\_l} \\ \vdots \\ \mathcal{ScoreA\_n} \end{pmatrix} \xrightarrow{} \mathcal{R}$$

**Figure 2.** *Approach 1.*

instead of having the same weight values for the criteria, SAW method uses mostly distinct weights values of the criteria. As given in Eq. (4), the overall performance of each alternative is obtained by multiplying the rating of each alternative on each criterion by the weight assigned to the criterion and then summing these products over all criteria [15]. The best alternative is the one that obtained the highest score and will be selected or ranked at the first position. Many recent studies used the SAW method, for example, in [39–41], and a review on its applications is also

*Multicriteria Optimization - Pareto-Optimality and Threshold-Optimality*

Besides SAW or also known as weighted sum method (WSM), there is another average technique, called weighted product model (WPM) or simple geometric weighted (SGW) or simple geometric average method. In WPM, the overall performance of each alternative is determined by raising the rating of the alternative to the power of the criterion weight and then multiplying these products over all criteria [15]. However, WPM is a little bit complex as compared to SAW since WPM

AHP [14], technique for order preference by similarity to ideal solution (TOPSIS), and *VlseKriterijumska Optimizacija Kompromisno Resenje* (VIKOR) [43] are also popular aggregation methods in solving MC problems. As previously mentioned in Section 3.2, AHP is built under the concept of pairwise comparison either in finding the criteria weights or criteria values of the alternatives. The aggregation of criteria weights and the criteria values obtained by AHP is sometimes done by

AHP and TOPSIS are two different aggregation methods. TOPSIS assigns the best alternative that relies on the concepts of compromise solution, where the best alternative is the one that has the shortest distance from the ideal solution and the farthest distance from the negative ideal solution [44]. In other words, alternatives are prioritized according to their distances from positive ideal solutions and negative ideal solutions, and the Euclidean distance approach is utilized to evaluate the relative closeness of the alternatives to the ideal solutions. There is a series of steps of TOPSIS, but this method starts with the weighted normalization of all performance values against each criterion. Some recent applications of the TOPSIS

VIKOR method [49] is quite similar to TOPSIS method, but there are some important differences, and one of the differences is about the normalization process. TOPSIS uses the vector linearization where the normalized value could be different for different evaluation unit of a certain criterion, while VIKOR uses linear normalization where the normalized value does not depend on the evaluation unit of a criterion. VIKOR has also been used in many real-world MCDM problems such as mobile banking services [50] digital music service platforms [51], military airport location selection [52], concrete bridge projects [53], risk evaluation of construction projects [54], maritime transportation [55], and

**5. Inclusion of credibility of evaluators in solving multicriteria**

This section discusses how credibility can be included practically in solving MC problems. Suppose the evaluators are requested to evaluate the relative importance

available [42].

involves power and multiplications.

**4.2 Other aggregation methods**

using the SAW or SGW methods.

method are available [45–48].

energy management [56].

**problems**

**84**

of the criteria based on rank-based weighting methods as explained in Section 3.1. Suppose there is a panel of *p* evaluators, and let *r<sup>l</sup> <sup>j</sup>* be rank of criterion *j*, evaluated by evaluator *l*, where *l* ¼ 1, … , *p*. In order to include the credibility of the evaluators, let us introduce a new set of values that represents the different credibility of the evaluators. Let *u<sup>l</sup>* be the degree of credibility of evaluator *l*, where 0≤ *ul* ≤1, and P*<sup>p</sup> <sup>l</sup>*¼<sup>1</sup>*ul* <sup>¼</sup> 1. There are two approaches [57] where the degree of credibility of the evaluators could be attached in finding the overall scores. The first approach is in calculating the final weight of criteria as given in **Figure 1**, and the second approach is in computing the overall performance of the alternatives as given in **Figure 2**.

For the first approach as portrayed in **Figure 2**, the degree of credibility of the evaluators is attached to the resulted weights from the ranks of criteria by using any of the equations, Eq. (1), Eq. (2), or Eq. (3). So, here there are *p* sets of weights of the criteria, and the average of that p weights for each criterion is calculated by summing up all weights for that criteria and divide the sum with the total number of evaluators. So now, there is only one set of weights that can be aggregated with the values of alternatives for each corresponding criterion as given in Eq. (4). There is only one set of overall performance of all *n* alternatives.

For the second approach, the criteria weights obtained from each evaluator are kept, and then each set of weights is aggregated with the quality values of each alternative. So, here there are *p* sets of overall values of the alternatives. In order to get the final overall score of the alternatives, the average of the *p* scores for each alternative should be calculated. The ranking or sorting of the alternatives or selecting the best alternative is done based on the average of that *p* overall scores of each individual alternative. The following section provides some suggestion on how to quantify the credibility of the evaluators.

Referring to the numerical example in the Appendix, there were three evaluators involved in ranking the importance of the five academic subjects, and the number of students is 10. So, *r<sup>l</sup> <sup>j</sup>* is the rank of academic subject *j* , with *j =* 1, … ,5, evaluated by evaluator *<sup>l</sup>*, where *<sup>l</sup>* <sup>¼</sup> 1, … , 3, and *<sup>n</sup>* = 10, while *ul* represents the degree of credibility of evaluator *l*, where 0 ≤*u<sup>l</sup>* ≤ 1, and P<sup>3</sup> *<sup>l</sup>*¼<sup>1</sup>*u<sup>l</sup>* <sup>¼</sup> 1.

#### **6. Quantification of credibility of evaluators**

Credibility is synonym to professionalism, integrity, trustworthiness, authority, and believability. A study focuses on how to assess the credibility of expert witnesses [58]. A 41-item measure was constructed based on the ratings by a panel of judges, and a factor analysis yielded that credibility is a product of four factors: likeability, trustworthiness, believability, and intelligence. Another study concerns about the credibility of information in digital era [59]. Credibility is said to have two main components: trustworthiness and expertise. However, the authors conclude that the relation among youth, digital media, and credibility today is sufficiently complex to resist simple explanations, and their study represents a first step toward mapping that complexity and providing a basis for future work that seeks to find explanations.

It can be argued that the degree of credibility of evaluators or judges or decisionmakers can be determined subjectively or objectively, where the former one can be done by using certain construct as proposed in [58] or can be determined based on certain objective or exact measures such as years of experience, salary scale, or amount of salary. The quantification of the degree of credibility opens a new potential area of research as there are very few researches done especially on finding the suitable objective proxy measures of the degree of credibility.

very limited. More researches should be conducted to find ways of measuring the credibility of evaluators or experts either subjectively or objectively. Inclusion of the credibility of evaluators in solving multicriteria problems is realistic since the evaluators come from different backgrounds and levels of experience. Quantification of the evaluators' credibility subjectively or objectively opens a new insight in group decision-making field. Furthermore, the credibility of the evaluators should also be considered in other multicriteria problems in other areas, so that the results

*On the Practical Consideration of Evaluators' Credibility in Evaluating Relative…*

Mr. Zachariah is a class teacher of 10 excellent students in one of the best primary schools of a country. The 10 students were already given the final marks of

Mr. Zachariah must rank the students according to their performance because these

Suppose three experienced teachers, Edward, Mary, and Foong, were asked to evaluate the relative importance of the five academic subjects with their degree of credibility as discussed in previous section, that is, the salary ratio of the three teachers is 0167: 0.333: 0.500. The rank-based technique is used to analyze the ranking of importance of the academic subjects given by these three teachers by

The results are given in **Table 2**. Column 2 displays the ranking of the criteria evaluated by teacher 1, and column 3 shows the corresponding criteria weights as analyzed by Eq. (1), while columns 4 and 5 and columns 6 and 7 show the respective results by teachers 2 and 3, respectively. The second last column of the table summarizes the criteria weights when the teachers are of same credibility. The values were computed as the simple arithmetic average of the corresponding criterion, while the last column has the final weights that were calculated as the simple arithmetic average as well but with consideration of the different degree of credibility according to Approach 1 as given in **Figure 2**. Please note that the both sets of final weights are already summed to one. So, the normalization process to guarantee

**Native language English language Mathematics Science History**

Student 1, *A*<sup>1</sup> 0.25 0.34 0.12 0.36 0.45 *A*<sup>2</sup> 0.33 0.54 0.22 0.44 0.76 *A*<sup>3</sup> 0.43 0.65 0.57 0.42 0.91 *A*<sup>4</sup> 0.55 0.32 0.37 0.67 0.53 *A*<sup>5</sup> 0.27 0.66 0.57 0.82 0.61 *A*<sup>6</sup> 0.67 0.56 0.46 0.46 0.31 *A*<sup>7</sup> 0.58 0.87 0.39 0.27 0.43 *A*<sup>8</sup> 0.32 0.76 0.41 0.37 0.51 *A*<sup>9</sup> 0.91 0.36 0.47 0.45 0.45 *A*<sup>10</sup> 0.12 0.33 0.81 0.75 0.32

five main academic subjects by their respective teachers as in **Table 1**.

students will be given awards and recognition on their graduation day.

are more practical and accurate.

using Eq. (1).

**Table 1.**

**87**

**Appendix: A numerical example**

*DOI: http://dx.doi.org/10.5772/intechopen.92541*

the sum of weights is one and is not necessary.

*Ten students assessed under five academic subjects.*

Finding the degree of credibility subjectively requires more time and much harder as it involves a construct or an instrument which would be used as a rating mechanism to obtain the degree of credibility. Meanwhile, finding the degree of credibility based on objective information is simpler and easier to do. As an illustration on how to quantify the credibility objectively, suppose there are three experts with their basic salaries in a simple ratio of 1:2:3. So, this ratio can be converted as 0.167:0.333:0.500, so that the sum of credibility of the evaluators is equal to 1. These values can be used to represent the degree of credibility of the evaluators or experts 1, 2, and 3, respectively. It should be noted that the sum of the degrees of credibility of the three evaluators is equal to one to make the future calculation simple while easier for interpretation of the values. Here, evaluator 3 is the most credible one since he/she has the highest salary among the three, and it is a usual practice that those who are higher in terms of expertise usually are paid higher. The same computation can be used for the years of experience or salary scale.

The numerical example in the Appendix extends the problem of evaluating students' academic performance which is discussed earlier in the Introduction. Here, the credibility of the teachers who were asked to assess the relative importance of the five subjects was considered. In order to incorporate the degree of credibility of the teachers, a new set of values is introduced to represent these different degrees of credibility. The example shows two ways of calculations on how the credibility values could be included in finding the overall scores of the alternatives. As expected, the overall scores and the overall ranking are different as compared to overall scores of not considering the different credibility of the teachers. The details and the step-by-step methodology are also included in the Appendix.

#### **7. Conclusion**

This chapter provides an overview on the practical consideration of evaluators' credibility in evaluating relative importance of criteria for some real-life multicriteria problems. Credibility of the evaluators who are involved in solving any multicriteria problem should be included in calculating the overall scores of the alternatives or the units of analysis. This chapter demonstrates how the credibility of evaluators who participated in finding the criteria weights can be combined with the criteria weights and the quality of the criteria of the alternatives. Rank-based criteria weighting methods are used as an illustration in a numerical example of evaluation of students' academic performance problem at the end of the chapter. However, other criteria subjective weighting methods are also possible to be used but with caution especially at the stage of aggregation of criteria weights and criteria values. It may exist only one approach to do the aggregation due to the underpinning concepts of the aggregation methods. The chapter uses simple additive weighted average method as the aggregation method since the method is very well established. The use of other aggregation techniques is also plausible. The chapter also suggests a few practical proxy measures of the credibility but is still

*On the Practical Consideration of Evaluators' Credibility in Evaluating Relative… DOI: http://dx.doi.org/10.5772/intechopen.92541*

very limited. More researches should be conducted to find ways of measuring the credibility of evaluators or experts either subjectively or objectively. Inclusion of the credibility of evaluators in solving multicriteria problems is realistic since the evaluators come from different backgrounds and levels of experience. Quantification of the evaluators' credibility subjectively or objectively opens a new insight in group decision-making field. Furthermore, the credibility of the evaluators should also be considered in other multicriteria problems in other areas, so that the results are more practical and accurate.

#### **Appendix: A numerical example**

It can be argued that the degree of credibility of evaluators or judges or decisionmakers can be determined subjectively or objectively, where the former one can be done by using certain construct as proposed in [58] or can be determined based on certain objective or exact measures such as years of experience, salary scale, or amount of salary. The quantification of the degree of credibility opens a new potential area of research as there are very few researches done especially on finding the suitable objective proxy measures of the degree of credibility.

*Multicriteria Optimization - Pareto-Optimality and Threshold-Optimality*

Finding the degree of credibility subjectively requires more time and much harder as it involves a construct or an instrument which would be used as a rating mechanism to obtain the degree of credibility. Meanwhile, finding the degree of credibility based on objective information is simpler and easier to do. As an illustration on how to quantify the credibility objectively, suppose there are three experts with their basic salaries in a simple ratio of 1:2:3. So, this ratio can be converted as 0.167:0.333:0.500, so that the sum of credibility of the evaluators is equal to 1. These values can be used to represent the degree of credibility of the evaluators or experts 1, 2, and 3, respectively. It should be noted that the sum of the degrees of credibility of the three evaluators is equal to one to make the future calculation simple while easier for interpretation of the values. Here, evaluator 3 is the most credible one since he/she has the highest salary among the three, and it is a usual practice that those who are higher in terms of expertise usually are paid higher. The same computation can be used for the years of experience or

The numerical example in the Appendix extends the problem of evaluating students' academic performance which is discussed earlier in the Introduction. Here, the credibility of the teachers who were asked to assess the relative importance of the five subjects was considered. In order to incorporate the degree of credibility of the teachers, a new set of values is introduced to represent these different degrees of credibility. The example shows two ways of calculations on how the credibility values could be included in finding the overall scores of the alternatives. As expected, the overall scores and the overall ranking are different as compared to overall scores of not considering the different credibility of the teachers. The details and the step-by-step methodology are also included in the

This chapter provides an overview on the practical consideration of evaluators'

multicriteria problems. Credibility of the evaluators who are involved in solving any multicriteria problem should be included in calculating the overall scores of the alternatives or the units of analysis. This chapter demonstrates how the credibility of evaluators who participated in finding the criteria weights can be combined with the criteria weights and the quality of the criteria of the alternatives. Rank-based criteria weighting methods are used as an illustration in a numerical example of evaluation of students' academic performance problem at the end of the chapter. However, other criteria subjective weighting methods are also possible to be used but with caution especially at the stage of aggregation of criteria weights and criteria values. It may exist only one approach to do the aggregation due to the underpinning concepts of the aggregation methods. The chapter uses simple additive weighted average method as the aggregation method since the method is very well established. The use of other aggregation techniques is also plausible. The chapter also suggests a few practical proxy measures of the credibility but is still

credibility in evaluating relative importance of criteria for some real-life

salary scale.

Appendix.

**86**

**7. Conclusion**

Mr. Zachariah is a class teacher of 10 excellent students in one of the best primary schools of a country. The 10 students were already given the final marks of five main academic subjects by their respective teachers as in **Table 1**. Mr. Zachariah must rank the students according to their performance because these students will be given awards and recognition on their graduation day.

Suppose three experienced teachers, Edward, Mary, and Foong, were asked to evaluate the relative importance of the five academic subjects with their degree of credibility as discussed in previous section, that is, the salary ratio of the three teachers is 0167: 0.333: 0.500. The rank-based technique is used to analyze the ranking of importance of the academic subjects given by these three teachers by using Eq. (1).

The results are given in **Table 2**. Column 2 displays the ranking of the criteria evaluated by teacher 1, and column 3 shows the corresponding criteria weights as analyzed by Eq. (1), while columns 4 and 5 and columns 6 and 7 show the respective results by teachers 2 and 3, respectively. The second last column of the table summarizes the criteria weights when the teachers are of same credibility. The values were computed as the simple arithmetic average of the corresponding criterion, while the last column has the final weights that were calculated as the simple arithmetic average as well but with consideration of the different degree of credibility according to Approach 1 as given in **Figure 2**. Please note that the both sets of final weights are already summed to one. So, the normalization process to guarantee the sum of weights is one and is not necessary.


#### **Table 1.**

*Ten students assessed under five academic subjects.*


**Table 2.**

*Criteria weights of five academic subjects evaluated by three teachers with the same and different credibility by using rank-sum weighting technique.*

Now, in order to find the overall performance of each student, for example, the overall performance of student 1 without consideration of credibility of teachers in evaluating the relative importance of the academic subjects, it is simply done by multiplying row 2 of **Table 1** with its corresponding criteria weights in the second last column of **Table 2** by using Eq. (4) as follows:

$$\begin{aligned} \text{Score } \mathbf{A}\_{1} &= \sum\_{j=1}^{5} \mathbf{w}\_{j} \mathbf{x}\_{1j} \\ &= (0.289)(0.25) + (0.2)(0.34) + (0.267)(0.12) + (0.067)(0.36) \\ &+ (0.178)(0.45) \\ &= 0.277 \end{aligned}$$

The same process is performed to find the overall scores of student 1, if the credibility of the teachers in finding weights of the criteria is considered but the weights in last column of **Table 2** is used, instead.

$$\begin{aligned} \text{Score } \mathbf{A}\_{1} &= \sum\_{j=1}^{5} \mathbf{w}\_{j} \mathbf{x}\_{1j} \\ &= (\mathbf{0.278})(\mathbf{0.25}) + (\mathbf{0.2})(\mathbf{0.34}) + (\mathbf{0.3})(\mathbf{0.12}) + (\mathbf{0.067})(\mathbf{0.36}) \\ &+ (\mathbf{0.156})(\mathbf{0.45}) \\ &= \mathbf{0.244} \end{aligned}$$

**Table 3** gives the overall scores and the corresponding final rankings of all students based on average criteria weights with the same (SC) and different (DC) credibility of the teachers. The overall scores are all different, while the rankings are different especially for ranks 8 and 9 and 4 and 5.

**Table 4** summarizes three individual overall score of the three different teachers without consideration of their credibility, while the second last column and the last column are the average overall scores of the three overall scores and its corresponding rankings, respectively.

**Table 5** shows the three overall scores by consideration of the credibility of teachers in finding the academic subjects' weights, and the average overall scores of the three overall scores. The ranking of the students is based on the average overall scores in column 5 of the table. Here, Approach 2 as in **Figure 3** is used to find the final overall scores of the students.

To make the comparison easier, **Table 6** summarizes the overall scores and their corresponding rankings of the students with SC and DC of the teachers when calculating the academic subjects' weights based on Approach 2.

As the two sets of the overall scores are different, all rankings based on both sets

*Different credibility: four different sets of overall scores and final ranking of the 10 students based on average*

*A***<sup>1</sup>** *A***<sup>2</sup>** *A***<sup>3</sup>** *A***<sup>4</sup>** *A***<sup>5</sup>** *A***<sup>6</sup>** *A***<sup>7</sup>** *A***<sup>8</sup>** *A***<sup>9</sup>** *A***<sup>10</sup>**

SC Score 0.277 0.427 0.597 0.461 0.526 0.514 0.540 0.470 0.571 0.424 Rank 10 8 1 7 4 5 3 6 2 9 DC Score 0.244 0.408 0.592 0.439 0.518 0.540 0.550 0.462 0.561 0.422 Rank 10 9 1 7 5 4 3 6 2 8

*On the Practical Consideration of Evaluators' Credibility in Evaluating Relative…*

*Overall scores and ranking of students with average criteria weights evaluated by teachers of the same and*

*<sup>i</sup> Score A***<sup>3</sup>**

*A*<sup>1</sup> 0.311 0.259 0.259 0.276 10 *A*<sup>2</sup> 0.479 0.400 0.400 0.426 8 *A*<sup>3</sup> 0.620 0.584 0.584 0.596 1 *A*<sup>4</sup> 0.483 0.449 0.449 0.460 7 *A*<sup>5</sup> 0.515 0.530 0.530 0.525 4 *A*<sup>6</sup> 0.510 0.516 0.516 0.514 5 *A*<sup>7</sup> 0.552 0.534 0.534 0.540 3 *A*<sup>8</sup> 0.474 0.467 0.467 0.469 6 *A*<sup>9</sup> 0.588 0.561 0.561 0.570 2 *A*<sup>10</sup> 0.349 0.461 0.461 0.424 9

*Same credibility: four different sets of overall scores and final ranking of the 10 students based on average*

*A*<sup>1</sup> 0.052 0.086 0.129 0.089 10 *A*<sup>2</sup> 0.080 0.133 0.200 0.138 9 *A*<sup>3</sup> 0.104 0.194 0.292 0.197 1 *A*<sup>4</sup> 0.081 0.150 0.225 0.152 7 *A*<sup>5</sup> 0.086 0.176 0.265 0.176 4 *A*<sup>6</sup> 0.085 0.172 0.258 0.172 5 *A*<sup>7</sup> 0.092 0.178 0.267 0.179 3 *A*<sup>8</sup> 0.079 0.155 0.233 0.156 6 *A*<sup>9</sup> 0.098 0.187 0.281 0.189 2 *A*<sup>10</sup> 0.058 0.153 0.230 0.147 8

*<sup>i</sup> u***<sup>3</sup>***Score A***<sup>3</sup>**

*<sup>i</sup> Score A*AV

*<sup>i</sup> Score A*AV

*<sup>i</sup>* **Ranking**

*<sup>i</sup>* **Ranking**

**Table 3.**

**Table 4.**

**Table 5.**

**89**

*overall scores.*

*overall scores.*

*different credibility based on Approach 1.*

*Score A***<sup>1</sup>**

*DOI: http://dx.doi.org/10.5772/intechopen.92541*

*u***<sup>1</sup>***ScoreA***<sup>1</sup>**

*<sup>i</sup> u***<sup>2</sup>***Score A***<sup>2</sup>**

*<sup>i</sup> Score A***<sup>2</sup>**

of the overall scores are the same except for ranks 8 and 9. There is not much different in the overall rankings since the MC problem that is considered here is

*On the Practical Consideration of Evaluators' Credibility in Evaluating Relative… DOI: http://dx.doi.org/10.5772/intechopen.92541*


**Table 3.**

Now, in order to find the overall performance of each student, for example, the overall performance of student 1 without consideration of credibility of teachers in evaluating the relative importance of the academic subjects, it is simply done by multiplying row 2 of **Table 1** with its corresponding criteria weights in the second

*Criteria weights of five academic subjects evaluated by three teachers with the same and different credibility by*

**Teacher 3 (0.500)**

Native language 1 0.333 2 0.267 2 0.267 0.289 0.278 English language 3 0.200 3 0.200 3 0.200 0.200 0.200 Mathematics 4 0.133 1 0.333 1 0.333 0.267 0.300 Science 5 0.067 5 0.067 5 0.067 0.067 0.067 History 2 0.267 4 0.133 4 0.133 0.178 0.156

**(DF)** *<sup>r</sup>***<sup>1</sup>** *<sup>w</sup>***<sup>1</sup>** *<sup>r</sup>***<sup>2</sup>** *<sup>w</sup>***<sup>2</sup>** *<sup>r</sup>***<sup>3</sup>** *<sup>w</sup>***<sup>3</sup>**

**Final weight same credibility (SC)**

**Final weight different credibility**

¼ ð Þ 0*:*289 ð Þþ 0*:*25 ð Þ 0*:*2 ð Þþ 0*:*34 ð Þ 0*:*267 ð Þþ 0*:*12 ð Þ 0*:*067 ð Þ 0*:*36

¼ ð Þ 0*:*278 ð Þþ 0*:*25 ð Þ 0*:*2 ð Þþ 0*:*34 ð Þ 0*:*3 ð Þþ 0*:*12 ð Þ 0*:*067 ð Þ 0*:*36

The same process is performed to find the overall scores of student 1, if the credibility of the teachers in finding weights of the criteria is considered but the

**Table 3** gives the overall scores and the corresponding final rankings of all students based on average criteria weights with the same (SC) and different (DC) credibility of the teachers. The overall scores are all different, while the rankings are

column are the average overall scores of the three overall scores and its

**Table 4** summarizes three individual overall score of the three different teachers without consideration of their credibility, while the second last column and the last

To make the comparison easier, **Table 6** summarizes the overall scores and their

corresponding rankings of the students with SC and DC of the teachers when

calculating the academic subjects' weights based on Approach 2.

**Table 5** shows the three overall scores by consideration of the credibility of teachers in finding the academic subjects' weights, and the average overall scores of the three overall scores. The ranking of the students is based on the average overall scores in column 5 of the table. Here, Approach 2 as in **Figure 3** is used to find the

last column of **Table 2** by using Eq. (4) as follows:

þ ð Þ 0*:*178 ð Þ 0*:*45

weights in last column of **Table 2** is used, instead.

wjx1j

different especially for ranks 8 and 9 and 4 and 5.

þ ð Þ 0*:*156 ð Þ 0*:*45

wjx1j

**Teacher 1 (0.167)**

**Teacher 2 (0.333)**

*Multicriteria Optimization - Pareto-Optimality and Threshold-Optimality*

Score A1 <sup>¼</sup> <sup>X</sup>

*using rank-sum weighting technique.*

**Table 2.**

Score A1 <sup>¼</sup> <sup>X</sup>

5

j¼1

¼ 0*:*277

5

j¼1

¼ 0*:*244

corresponding rankings, respectively.

final overall scores of the students.

**88**

*Overall scores and ranking of students with average criteria weights evaluated by teachers of the same and different credibility based on Approach 1.*


#### **Table 4.**

*Same credibility: four different sets of overall scores and final ranking of the 10 students based on average overall scores.*


#### **Table 5.**

*Different credibility: four different sets of overall scores and final ranking of the 10 students based on average overall scores.*

As the two sets of the overall scores are different, all rankings based on both sets of the overall scores are the same except for ranks 8 and 9. There is not much different in the overall rankings since the MC problem that is considered here is

*Multicriteria Optimization - Pareto-Optimality and Threshold-Optimality*

$$\begin{pmatrix} r\_1^1, \dots, r\_1^p \\ \vdots \\ r\_m^1, \dots, r\_m^p \end{pmatrix} \xrightarrow{r\_{\{tr\}}^1} \begin{pmatrix} w\_1^1, \dots, w\_1^p \\ \vdots \\ w\_m^p, \dots, w\_m^p \end{pmatrix} \xrightarrow{u^1 \mathcal{S}core \ A\_1^1} \begin{pmatrix} u^1 \mathcal{S}core A\_1^1, \dots, u^1 \mathcal{S}core A\_n^1 \\ \vdots \\ u^p \mathcal{S}core A\_1^p, \dots, u^p \mathcal{S}core A\_n^p \end{pmatrix} \rightarrow \begin{pmatrix} \mathcal{S}core A\_1^{\boxtimes \nu} \\ \vdots \\ \mathcal{S}core A\_n^{\boxtimes \nu} \end{pmatrix} \rightarrow R$$

**References**

1980;**7**:5-31

**53**:49-57

2016;**2**:61-71

**6**(4):289-293

p. 040019-1-6

**91**

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*DOI: http://dx.doi.org/10.5772/intechopen.92541*

average method. Malaysian Journal of Learning and Instructions. 2013;**10**:

[9] Metzger MJ. Making sense of credibility on the Web: Models for evaluating online information and recommendations for future research. Journal of the American Society for Information Science and Technology.

2007;**58**:2078-2091

**112**:397-404

[10] Zulfiqar S, Bin Tahir S. Professionalism and credibility of assessors in enhancing educational quality. 2019;**2**:162-175. Available from:

https://www.researchgate.net/ publication/334599511\_ PROFESSIONALISM\_AND\_

CREDIBILITY\_OF\_ASSESSORS\_IN\_ ENHANCING\_EDUCATIONAL\_ QUALITY [Accessed: 6 April 2020]

[11] Korten D, Alfonso F. Bureaucracy and the Poor: Closing the Gap. Singapore: McGraw-Hill; 1981

[12] Ma J, Fan Z, Huang L. A subjective and objective integrated approach to determine attribute weights. European Journal of Operational Research. 1999;

[13] Ray AM. On the measurement of certain aspects of social development. Social Indicators Research. 1989;**21**:35-92

[14] Diakoulaki D, Koumoutos N. Cardinal ranking of alternatives actions: Extension of the PROMETHEE method.

European Journal of Operational Research. 1991;**53**:337-347

[15] Hwang CL, Yoon K. Multiple Attribute Decision-Making: Methods and Applications. Berlin: Springer; 1981

[16] Triantaphyllou E. Multi Criteria Decision Making: A Comparative Study.

119-132

*On the Practical Consideration of Evaluators' Credibility in Evaluating Relative…*

[2] Rezaei J. Best-worst multi-criteria decision-making method. Omega. 2015;

[3] Ahmad N, Kasim MM, Rajoo SSK. Supplier selection using a fuzzy multi-criteria method. International Journal of Industrial Management.

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learning approach using TOPSIS technique with best ranked criteria weights. In: Proceedings of the 13th IMT-GT International Conference on Mathematics, Statistics and Their Applications (ICMSA 2017); 4-7 December 2017; Malaysia. Sintok: AIP Conference Proceedings 1905. 2017.

[7] Mohammed HJ, Kasim MM,

[8] Kasim MM, Abdullah SRG. Aggregating student academic achievement by simple weighted

Hamadi AK, Al-Dahneem E. Evaluating collaborative and cooperative learning using MCDM method. Advanced Science Letters. 2018;**24**(6):4084-4088

Shaharanee INM. Selection of suitable e-

**Figure 3.** *Approach 2.*


**Table 6.**

*Two different set of overall scores of the students by averaging overall performance of the students and their corresponding rankings based on Approach 2.*

only a small scale problem with only 10 alternatives and 5 criteria. However, the two sets of overall values are totally different. There may be much more differences in terms of rankings if a bigger MC problem with more alternatives and more criteria is considered. The final ranking of the students obtained by consideration of the different credibility of the teachers should be selected as the practical and valid results.

#### **Author details**

Maznah Mat Kasim School of Quantitative Sciences (SQS), UUM College of Arts and Sciences, Universiti Utara Malaysia, Sintok, Kedah, Malaysia

\*Address all correspondence to: maznah@uum.edu.my

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*On the Practical Consideration of Evaluators' Credibility in Evaluating Relative… DOI: http://dx.doi.org/10.5772/intechopen.92541*

#### **References**

only a small scale problem with only 10 alternatives and 5 criteria. However, the two sets of overall values are totally different. There may be much more differences in terms of rankings if a bigger MC problem with more alternatives and more criteria is considered. The final ranking of the students obtained by consideration of the different credibility of the teachers should be selected as the practical and valid

*Two different set of overall scores of the students by averaging overall performance of the students and their*

SC Score 0.276 0.426 0.596 0.460 0.525 0.514 0.540 0.469 0.570 0.424 Rank 10 8 1 7 4 5 3 6 2 9 DC Score 0.089 0.138 0.197 0.152 0.176 0.172 0.179 0.156 0.189 0.147 Rank 10 9 1 7 4 5 3 6 2 8

*Multicriteria Optimization - Pareto-Optimality and Threshold-Optimality*

*A***<sup>1</sup>** *A***<sup>2</sup>** *A***<sup>3</sup>** *A***<sup>4</sup>** *A***<sup>5</sup>** *A***<sup>6</sup>** *A***<sup>7</sup>** *A***<sup>8</sup>** *A***<sup>9</sup>** *A***<sup>10</sup>**

School of Quantitative Sciences (SQS), UUM College of Arts and Sciences,

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Universiti Utara Malaysia, Sintok, Kedah, Malaysia

provided the original work is properly cited.

\*Address all correspondence to: maznah@uum.edu.my

results.

**Table 6.**

*corresponding rankings based on Approach 2.*

**Figure 3.** *Approach 2.*

**Author details**

**90**

Maznah Mat Kasim

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2008;**35**(5):1660-1670

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2000

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100018

216-226

566-583

### *Edited by Nodari Vakhania and Frank Werner*

Multi-criteria optimization problems naturally arise in practice when there is no single criterion for measuring the quality of a feasible solution. Since different criteria are contradictory, it is difficult and often impossible to find a single feasible solution that is good for all the criteria. Hence, some compromise is needed. As such, this book examines the commonly accepted compromise of the traditional Pareto-optimality approach. It also proposes one new alternative approach for generating feasible solutions to multi-criteria optimization problems. Finally, the book presents two chapters on the existing solution methods for two real-life, multi-criteria optimization problems.

Published in London, UK © 2020 IntechOpen © sinemaslow / iStock

Multicriteria Optimization - Pareto-Optimality and Threshold-Optimality

Multicriteria Optimization

Pareto-Optimality and Threshold-Optimality

*Edited by Nodari Vakhania and Frank Werner*