An Intuitionistic Fuzzy Decision-Making for Developing Cause and Effect Criteria of Subcontractors Selection

Lazim Abdullah; Zheeching Ong; Nuraini Rahim

doi:10.2991/ijcis.d.210222.001

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Volume 14, Issue 1, 2021, Pages 991 - 1002

An Intuitionistic Fuzzy Decision-Making for Developing Cause and Effect Criteria of Subcontractors Selection

Authors

Lazim Abdullah¹^{, *}^,, Zheeching Ong², Nuraini Rahim³

¹Management Science Research Group, Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, Kuala Nerus, 21030, Malaysia

²Public Bank Berhad, Jalan Sultan Yahya Petra, Kota Bharu, 15200, Malaysia

³Management Science Research Group, Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, Kuala Nerus, 21030, Malaysia

^*Corresponding author. Email: lazim_m@umt.edu.my

Corresponding Author

Lazim Abdullah

Received 30 September 2020, Accepted 2 February 2021, Available Online 12 March 2021.

DOI: 10.2991/ijcis.d.210222.001 How to use a DOI?
Keywords: Intuitionistic fuzzy set; Decision-making; DEMATEL; Causal diagram; Subcontractor selection.
Abstract: The decision-making trial and evaluation laboratory (DEMATEL) method has been applied to solve numerous multi-criteria decision-making (MCDM) problems where crisp numbers are utilized in defining linguistic evaluation. Previous literature suggests that the intuitionistic fuzzy DEMATEL (IF-DEMATEL) can offer a new decision-making method in solving MCDM problems where intuitionistic fuzzy sets (IFSs) are utilized in defining linguistic evaluation. This paper aims to develop a cause–effect diagram of subcontractor selection using a modified IF-DEMATEL method. In this paper, three modifications are made to the IF-DEMATEL method. Two memberships of IFSs, relative weights of experts, and a transformation equation are the elements introduced to the IF-DEMATEL. The linguistic variables that defined in IFSs are meant to capture wide arrays of uncertain and fuzzy information in solving MCDM problems. Furthermore, the modified IF-DEMATEL is applied it to a subcontractors' selection problem where groups of cause and effect criteria are segregated. A group of experts' opinions were sought to provide linguistic evaluations regarding the degree of influence between criteria in subcontractors' selection. The results show that four criteria are identified as cause criteria while six other criteria are identified as effect criteria. The results also suggest that the criteria “experience” is the main cause that influence the selection of subcontractors. The identification of cause and effect criteria would be a great significance for practical implementation of subcontractors' selection.
Copyright: © 2021 The Authors. Published by Atlantis Press B.V.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

The decision-making trial and evaluation laboratory (DEMATEL) method is one of the many multi-criteria decision-making (MCDM) methods that available in literature. The DEMATEL was initially developed by the Science and Human Affairs Program of the Battelle Memorial Institute of Geneva between 1972 and 1976 to study and resolve the complicated and intertwined problems [1,2]. Compared with other traditional MCDM techniques such as the analytic hierarchy process where criteria are independent, this method is one of the structural modeling techniques that can identify the interdependencies of system elements through causality graphs. The causal diagram uses digraphs rather than directionless graphs to portray the basic concept of contextual relationships and the strengths of influence among the elements [3]. This method has been applied in analyzing and developing the relationship of cause and effect among evaluation criteria [4]. In other words, the DEMATEL is used to derive inter-relationship among factors or criteria [5]. The DEMATEL is a comprehensive method for developing a basic model that contains causal connections between complex criteria. As to provide more evidence on the state-of-the-art of the DEMATEL, Si et al. [6], made a systematic review of the DEMATEL and its applications. In the DEMATEL, all criteria of the MCDM problems are partitioned into classifications, that is the cause and effect groups.

Despite all these advantages, the linguistic evaluation in DEMATEL suffers from several limitations. The DEMATEL is insufficient to provide a good decision tool in this age because most of the information are regularly exorbitant and huge. In addition, elicitation of experts' opinion could give evaluation in the situation where information is restricted or incomplete. In fact, the fuzziness in experts' opinions or insufficient knowledge about an issue could make the decision-making process more complicated [6,7]. In response to these limitations, the integration of fuzzy sets and DEMATEL (fuzzy DEMATEL) was proposed. The fuzzy DEMATEL basically works similar with the DEMATEL frameworks except for the linguistic used in evaluation. In the fuzzy DEMATEL, the seven-step computational procedures are embedded with fuzzy numbers where incomplete and vague information could be dealt with accordingly. The fuzzy DEMATEL has been used extensively in solving various decision problems [8–16]. Despite its success, previous literature suggests that intuitionistic fuzzy numbers (IFNs) are better options in dealing with incomplete and vague information in decision-making. These numbers are characterized by membership and nonmembership of intuitionistic fuzzy sets (IFS). Along a similar line, the fuzzy DEMATEL has been extended to the intuitionistic fuzzy DEMATEL (IF-DEMATEL) where single membership of fuzzy numbers is replaced by IFNs in defining linguistic variables. In particular, the linguistic variables used in IF-DEMATEL are defined using the concept of two or three numbers of IFNs.

There is a handful of newly published research papers discussing the IF-DEMATEL. Govindan et al. [17], for example, conducted a research in green supply chain management where IF-DEMATEL method was used to handle the importance and causal relationships between green supply management practices. A research of municipal solid waste management was conducted by Zulkifli and Abdullah [18] to visualize the most viable solid waste treatment using IF-DEMATEL. In a business study, Vafadarnikjoo et al. [19], applied IF-DEMATEL in prioritizing the selection criteria of bank branch locations. There was also a study of location selection of freight villages where IF-DEMATEL was integrated with two other decision-making methods for determining the most effective criteria [20]. Recently, IF-DEMATEL integrated with an elimination method to obtain the importance-weights of criteria in personnel selection [21]. Most of these related researches introduced IFS in DEMATEL using the 7-point linguistic scale from 0 to 6. This scale was written in IFSs with the inclusion of the third membership of hesitation degree. It seems that the authors used three numbers of IFSs where hesitation degree is written independently.

In contrast to the three numbers, this paper proposed a modified IF-DEMATEL where two numbers of IFNs are used as the third number of hesitation degree is already considered in the two numbers. In addition, the proposed IF-DEMATEL method also includes transformation equation proposed by [22] instead of expected value [23] for IFN values to crisp values. In the transformation equation, the deviations between membership and nonmembership numbers are predetermined to ensure true crisp value that can reflect the initial evaluation. Furthermore, the proposed IF-DEMATEL also considers the importance weight of experts. It was noticed that there are no weights of experts assumed in most of the DEMATEL based methods [24–27]. In response to this issue, this proposed method introduced relative weights of experts based on the three numbers of IFNs of five linguistic terms. Proportion equation proposed by [28] is adopted to the proposed IF-DEMATEL method. Differently from the fuzzy DEMATEL, this paper proposes an IF-DEMATEL method where dual memberships of IFS, relative weights of experts, and transformation equation are among the highlights. The proposed method is applied to unravel the multiple conflicting criteria of subcontractors' selection in which the cause and effect criteria of subcontractors' selection will be identified.

The contributions of this paper are threefold: (1) we propose to use relative weights of experts instead of considering equal weights among the experts. The proposed method introduces a proportion equation which makes the weights of experts more suitable for real-life application, (2) we propose to use a weighted averaging operator to find aggregated direct relation matrix where series of multiplications of assessment scales and relative weights of experts are involved, (3) we propose to introduce a transformation equation instead of typical averaged defuzzification method to transform two numbers of IFSs to single real numbers. These three contributions are embedded in the proposed IF-DEMATEL where the total relation matrix is used to obtain two groups of criteria. The rest of this paper is arranged as follows. Section 2 briefly reviews definitions of IFS, IFN, and its memberships. Section 3 presents the proposed method that is to be applied to a case study. In Section 4, detailed description of subcontractors' selection including its computational procedures and results are presented. Section 5 provides a brief conclusion.

2. PRELIMINARY

The IFS developed by [29] is a capable device for managing equivocalness. An extraordinary element of IFSs is that it is a degree of membership and a degree of nonmembership to every component. The IFS component is an expansion of [7] which just allocates one degree of membership to each element. In the past two decades, numerous authors have been acknowledged that the IFS theory is effectively used to connect various applications. IFSs have been applied to many areas such as medical diagnosis [30,31], decision-making problems [32,33], and pattern recognition [34,35]. The IFSs are one of the generalization of nonstandard fuzzy sets and are particularly useful in dealing with uncertainty [29].

Atanassov [29] has stated that let X be a fixed set, an IFS A in X is given as A=x,μA(x),νA(x),x∈X where μA(x):X→[0,1] and νA(x):X→[0,1] are membership function and nonmembership function, respectively, satisfying 0≤μA(x)+νA(x)≤1,∀x∈X. The numbers μA(x) and νA(x) denote the degrees of membership and nonmembership, respectively, of the element x to A, for all x∈X.

In addition, πA(x)=1−μA(x)−νA(x) is called the hesitation degree of x∈A, representing the degree of indeterminacy or the degree of hesitancy of x to A. It is obvious that 0≤μA(x)≤1,∀x∈X. If πA(x) is small, then the value of x is more certain; if πA(x) is greater, then knowledge about x is more uncertain. Obviously, when πA(x)=1−νA(x) for all elements of the universe, then IFS is reduced to a traditional fuzzy set. Therefore, fuzzy sets can be considered as a special case of IFS.

For an IFS, the pair μA(x),νA(x) is called an IFN and each IFN can be simply denoted as α=(μα,να) where μα∈[0,1],να∈[0,1] and 0≤μα+να≤1. For an IFN α=(μα,να), if the value μα gets bigger and the value να gets smaller, then the IFN α=(μα,να) gets greater. Obviously, α+=(1,0) and α−=(0,1) are the largest and the smallest IFNs, respectively. In addition, S(α)=μα−να and H(α)=μα+να are called the score and accuracy degrees of α, respectively.

3. PROPOSED IF DEMATEL

The DEMATEL is a classical method for solving criteria of MCDM problems. It can be used to visualize the relationship between criteria. The DEMATEL also can be used to separate all criteria into two groups which are known as cause and effect groups. As an extension of DEMATEL, the fuzzy DEMATEL was proposed by many researchers and applied to various fields [4,5,8,28,36]. With further extension of fuzzy DEMATEL, this section presents the IF-DEMATEL method in which the fuzzy DEMATEL is modified by introducing the IFS into the fuzzy DEMATEL. As a result of this fusion, the proposed IF-DEMATEL has several innovations. Firstly, instead of considering equal weights among the experts, this proposed method introduced relative weights of experts based on the three numbers of IFSs of five linguistic terms. Proportion equation proposed by [28] is adopted to the proposed method. Secondly, individual assessment of experts is aggregated using the concept of summation of weighted average. This is an averaging operator where relative weight of experts is multiplied with decision assessment. Finally, in this proposed method, we introduce a transformation equation instead of typical defuzzification method to transform two numbers of IFSs to single crisp numbers. These three innovations are embedded into the general framework of DEMATEL where total relation matrix is used to derive causal and effect groups of criteria. In this section, we present an improvement to the DEMATEL where the IFS is the prominent element in the proposed method. It is illustrated in Figure A1 (See Appendix).

Detailed computational procedures of the proposed method are presented as follows:

Step 1: Construct Direct Relation Matrix

Suppose an MCDM problem has n criteria c1,c2,c3,…,cn. Data are collected from kth experts. Expert's assessments are collected, and the data is arranged in a direct relation matrix. This matrix represents an assessment of interrelationship between criteria using the five linguistic rating scale of DEMATEL. With linguistic variable of influence, one criterion is compared with other criteria in terms of degree of influence. The assessments are written in memberships of IFSs. Table 1 shows five linguistic terms and its respective IFS.

Linguistic variable	IFSs
No influence	(0.10, 0.90)
Low influence	(0.20, 0.65)
Fairly low influence	(0.30, 0.55)
Medium influence	(0.50, 0.50)
Fairly high influence	(0.65, 0.25)
High influence	(0.80, 0.05)
Absolutely high influence	(0.90, 0.10)

Table 1

Linguistic variable of influence and its intuitionistic fuzzy sets (IFSs) [8].

The assessments are made to collect the first-hand data on the influence of one criterion as compared to other criteria of MCDM problem. The assessments collected from experts are arranged in form of matrix to construct initial matrix or direct relation matrix.

The direct relation matrix is arranged in the matrix A^k where elements in A^k are the influential rating one criterion with respect to other criteria.

Ak=aijkn×n= c1 c2 … cn0a12k…a1nkan1k0…a2nk…………an1kan2k…0(1)

where c1,c2,…,cn are the criteria and aijk is the rating by kth expert in evaluation of inter-relationship between criteria.

Step 2: Priority Values of Experts

Each expert's opinion has a specific weight that must be accounted as they have varied in working experience and knowledge. So, the importance or weights of these experts are obtained by utilizing the IFS linguistic variable of importance. Table 2 presents the linguistic terms and its respective three memberships of IFS.

Linguistic variable	IFSs
Very important	(0.9, 0.05, 0.05)
Important	(0.75, 0.20, 0.05)
Medium	(0.50, 0.50, 0.00)
Unimportant	(0.25, 0.70, 0.05)
Very unimportant	(0.10, 0.85, 0.05)

Table 2

Linguistic variable for experts' relative importance or weight [28].

Assume that λk=(μk,νk,πk) is the memberships of IFS for relative importance weights of kth expert. The priority value of kth expert could be computed using Eq. (2).

λ=μk+πkμkμk+νk∑k=1lμk+πkμkμk+νk(2)

where ∑k=1lλk=1. In this paper, the hesitation degree is omitted.

Step 3: Construct Aggregated Weighted Direct Relation

Each individual expert assessment is aggregated in order to construct a collective intuitionistic fuzzy decision matrix. By using the intuitionistic fuzzy weighted averaging (IFWA) operator Eq. (3) [37], the aggregated intuitionistic fuzzy ratings xij represent the influence level of criterion i on j could be acquired. It is good to note that this step is indeed series of multiplications of assessment scales and weights of experts.

xij=1−∏k=1l1−μijkλk,∏i=1lνijkλk,i,j∈{1,2.3,…,n}(3)

where λk is the importance weight of kth expert aijk is is the corresponding of IFN of kth expert's opinion when comparing i to j.

Step 4: Construct Normalized Aggregated Weighted Direct Relation Matrix

In order to construct normalized direct relation matrix, summation of rows and summation of columns are computed first.

The maximum number from summation of rows and columns are identified. With these maximum numbers, is m calculated using Eq. (4).

m=min1max∑j=1naij,1max∑i=1naij,i,j∈{1,2.3,…,n}(4)

The direct relation matrix in Table 2 is normalized by multiplying with m.

The normalized direct relation matrix of membership and nonmembership is transformed into a crisp number p using transformation Eq. (5) [22].

p=(μ−ν+1)2(5)

where p is a crisp number

μ is the membership values in total relation matrix

ν is the nonmembership values in total relation matrix.

Step 5: Construct Total Relation Matrix [5]

In order to construct Total relation matrix T, summation of rows and summation of columns of normalized aggregated weighted direct relation matrix are computed first.

The maximum numbers from summation of rows and columns are used to construct total relation matrix.

T=X(I−X)−1(6)

where I is the identity matrix.

and X=m×A(7)

where m is defined in Eq. (4).

From the total relation matrix, we are able to find sum of columns and rows.

Step 6: Sum of Rows and Sum of Columns of Total Relation Matrix.

T=[ty]n×ni,j=1,2,…,nR=∑i=1nty1×n=[tj]1×nD=∑j=1ntyn×1=[ti]n×1

R = total of row m for matrix,

D= total of column n for matrix

Summation of rows and columns of the total relation matrix produced vectors D and R, respectively. R+D is the horizontal axis vector named “Prominence” showed the importance of a criterion while R−D which is the vertical axis called “Relation.”

A criterion is considered as a member of the cause category if R−D is positive, and in the case of its negative value, the criterion attributes it to the effect group. The R+D and R−D values are computed using MATLAB software.

Step 7: Construct the Causal Diagram

Hence, the causal diagram was depicted by mapping the data set of the R+D,R−D which provides the discernment about the recognition of the entire system. Therefore, the causal diagram can visualize the complicated causal relationships between elements and provide valuable insights for decision-making.

Based on the coordinate positions of R+D and R−D, attributes can be divided into the following 4 types:

R−D is positive and R+D is large:

This indicates that the attributes are causes, which are also driving factors for solving problems.
R−D is positive and R+D is small:

This indicates that the attributes are independent and can influence only a few other attributes.
R−D is negative and R+D is large:

This indicates that the attributes are the core problems that must be solved. However, these are effect-type attributes, which cannot be directly improved.
R−D is negative and R+D is small:

This indicates that the attributes are independent and can be influenced by only a few other attributes.

Finally, the causal diagram is illustrated by mapping the data set (R+D,R−D) onto two-dimensional planes. This mapping enables us to understand about the recognition of the entire system of criteria of decision-making problems. The entire system of criteria of subcontractors' selections is elucidated in the following section. Summarily, this proposed method has advantages in several ways. First is about the weights of experts. This paper proposed relative weights for experts using a proportion equation and this surely has an edge over the typical DEMATEL where weights of experts are assumed to be equal. The second advantage lies on the aggregation method used. In the proposed method, an aggregated membership and nonmembership of entries in direct relation matrix is simply obtained by summing the multiplication of individual judgment with relative weight of experts. This aggregation method is straightforward as compared to other aggregation operators where several parameters are involved. The third advantage of the proposed method is the use of transformation equation. To transform the membership and nonmembership of IFS to single real numbers, a simple average equation is applied. Instead of using three memberships, this equation is restricted to two memberships, thereby reduces the computational complexity.

Criteria	Descriptions
Price (C₁)	Choosing subcontractors on a premise of most reduced cost regularly brings about cases for expansions of time, claims for extra charges, higher capital expenses of development and task, and a diminished quality in workmanship
Completing on Time (C₂)	It is normal to experience delays in development ventures. Deferral isn't generally caused by a cataclysmic occasion. Deferrals may make significant harm the proprietor. The date of finishing may change over the span of the agreement, The genuine finishing must be confirmed by the as of late concurred culmination date.
Experience (C₃)	The execution of the preliminary endeavor is considered by the legally binding laborer, subcontractor, and customer gathers as the most basic. While assessing who is welcome to exhibit a subcontract quote, the essential impermanent laborer will either consider the people who had past working associations or fuse subcontractors who have in advance viably completed made by various legally binding specialists of tantamount nature, scale, and capriciousness.
Financial Stability (C₄)	Compared with larger subcontractors, subcontractors trust that a superior money-related foundation is more imperative. In the construction industry, it is troublesome for little subcontractors to get budgetary help from banks. Without the necessary financial background, small subcontractors may not be eligible for a certain amount of engineering.
Compliance with Regulations (C₅)	Some subcontractors may themselves be a contractual worker of little development projects. In this manner, they ought to know about the significance of consistence with directions. They might not want to be denied from any offering openings, nor would they like to acquire a terrible notoriety and demolish their altruism.
Quality (C₆)	On the client/consultant side, they need to have quality facilities as well as for the duration of their life cycle keeping in mind the end goal to limit progressing upkeep costs, which is a typical marvel in construction subcontracting.
Performance History (C₇)	Linking contract continuation to contractor performance can serve as an incentive for performance excellence. Thorough review of past execution contributes important information to the department's decision-making regarding the exercise of options. Past performance assessment should include program and technical issues, cost control, business management, socio-economic programs, and other relevant contract obligations and requirements.
Safety Management (C₈)	Safety issues have increased vital importance throughout the construction industry. Numerous construction companies around the global are implementing safety, wellbeing, and environmental management framework to reduce injuries, eliminate illness, and to provide a safe work environment in their construction sites
Timely Payment to Labor (C₉)	Subcontractors are under the legally binding commitments to pay their worker on time.
Length of Time in Industry (C₁₀)	Flexibility and cooperation when resolving delays is one of the criteria that need to be considered. They have to resolve the delay when there is a delay where the subcontractor and the main contractor face it. If the subcontractor is accustomed to time delays in the past projects. Years in the industry can affect the selection for subcontractors because they have to solve a lot of problems.

Table 3

Criteria of subcontractors selection.

4. A CASE STUDY OF SUBCONTRACTORS' SELECTION

Subcontractors usually help main contractor to overcome problems that related to the need for special expertise, limitation in finances, and shortage in resources. The main contractor is still in charge and should administer contracts to guarantee the project is executed and finished as pointed out in contract. Specialist subcontractor can be utilized, when the main contractor acquires products or administrations, which the main contractor doesn't deliver or can't deliver by his own company. Therefore selecting the deliverable subcontractors is critical in making sure the implementation of the project is completed within the stipulated times. In this paper, subcontractors selection problem is presented where a decision- making approach is employed. In fulfilling this objective, a group of five experts are invited to provide assessment via scales of linguistic variables. Detailed descriptions of experts and criteria are presented in the next subsection.

4.1. Experts, Criteria, and Linguistic Data Collection

Five experts attached to a construction company are invited to offer a linguistic assessment on criteria in selecting subcontractor. The experts consist of two project managers, two engineers, and one site supervisor of which, all these experts have more than three years experiences in construction industry. Experts are requested to provide degree of influence of a criterion with respect to other criteria using linguistic variables (see Table 1). The assessments are made to collect the first-hand data on the influence of one criterion as compared to other criteria in subcontractors' selection.

In study, the criteria affecting the selection of subcontractors are retrieved from the relevant literature in subcontractors' selection [38–44]. Criteria and their brief descriptions are depicted in Table 3.

These lists of criteria are the main subjects in this analysis where their importance and unidirectional cause–effect relationship are determined through series of computational procedures. Detailed computations and results are presented in the following subsection.

4.2. Computation and Results

Using the information gathered, the implementation of the proposed method is performed. Detailed computations and results are presented as follows.

In this computation, priority values of experts are computed first using Eq. (2).

	Expert 1	Expert 2	Expert 3	Expert 4	Expert 5
Lambda, λk	0.2161	0.2138	0.2281	0.1900	0.1520

Table 4

Priority values of experts.

Criteria	C₁	C₂	C₃	…	C₉	C₁₀
C1	0.0000,1.0000	0.3886,0.6114	0.4734,0.4900	…	0.3917,0.5220	0.4143,0.5404
C2	0.3652,0.6348	0.0000,0.0000	0.6837,0.1825	…	0.2775,0.6312	0.6083,0.2903
C3	0.7040,0.2542	0.8313,0.0972	0.0000,0.0000	…	0.2908,0.6698	0.7873,0.1281
…	…	…	…	…	…	…
C10	0.5227,0.3238	0.7914,0.1565	0.7797,0.1569	…	0.3581,0.4500	0.0000,0.0000

Table 5

Aggregated direct relation matrix.

For example,

Expert 1:w1=0.90+0.050.900.90+0.05=0.9474

Expert 2:w2=0.75+0.200.750.75+0.05=0.9375

Expert 3:w3=0.50+0.500.500.50+0.00=1.0000

Expert 4:w4=0.25+0.700.250.25+0.05=0.8333

Expert 5:w5=0.10+0.850.100.10+0.05=0.6667

Then, λ1=0.94744.3849=0.2161

The priority values for four other experts are calculated similarly. Table 4 summarizes the priority values for experts.

These priority values of experts are then used in obtaining aggregated direct relation matrix. Direct relation matrices of expert's assessment are aggregated using Eq. (3).

For example, membership and nonmembership of a11 are computed as:

1−∏k=1l1−μjλk=1−1−0.00^0.2161∗1−0.00^0.2138∗1−0.00^0.2281∗1−0.00^0.1900∗(1−0.00) ^0.1520=0.000

∏k=1lνjλk=1^0.2161∗1^0.2138∗1^0.2281∗1^0.1900∗1^(0.1520)=1.000

Similarly, membership and nonmembership for a21 are given as:

1−∏k=1l1−μjλk=1−1−0.1^0.2161∗1−0.5^0.2138∗1−0.5^0.2281∗1−0.1^0.1900∗(1−0.5)^0.1520=0.3652

∏k=1lνjλk=0.9^0.2161∗0.50^0.2138∗0.5^0.2281∗0.9^0.1900∗ 0.5^(0.1520)=0.6348

Similarly, other entries are calculated accordingly. The aggregated direct relation matrix is shown in Table 5.

Information in the aggregated direct relation matrix are given in IFNs. These numbers need to transform into a single value. Membership and nonmembership values are transformed into a single score (crisp value) using Eq. (5). Table 6 shows the aggregated crisp matrix.

In order to normalize the aggregated crisp matrix, summations of row and column of criteria are made. Table 7 shows these summations of criteria.

Criteria	C₁	C₂	C₃	C₄	C₅	…	C₈	C₉	C₁₀
C₁	0.5000	0.3886	0.4917	0.5636	0.2950	…	0.3443	0.4349	0.4369
C₂	0.3652	0.5000	0.7506	0.3804	0.3781	…	0.2733	0.3231	0.6590
C₃	0.7249	0.8670	0.5000	0.5543	0.6561	…	0.6410	0.3105	0.8296
C₄	0.6177	0.6953	0.2464	0.5000	0.4872	…	0.3427	0.8500	0.4764
C₅	0.1000	0.5960	0.3238	0.1297	0.5000	…	0.7775	0.4586	0.4519
C₆	0.8989	0.8672	0.7597	0.6754	0.4579	…	0.4357	0.2314	0.7026
C₇	0.8059	0.7641	0.8007	0.3641	0.5309	…	0.3818	0.3858	0.7115
C₈	0.5550	0.6950	0.2631	0.2573	0.8224	…	0.5000	0.1652	0.6286
C₉	0.3429	0.6347	0.3145	0.7714	0.4021	…	0.3407	0.5000	0.4601
C₁₀	0.5995	0.8175	0.8114	0.4995	0.3703	…	0.3613	0.4540	0.5000

Table 6

Aggregated crisp matrix.

Criteria	Summation of row	Summation of column
C₁	4.8128	5.5101
C₂	5.0850	6.8254
C₃	6.7730	5.2619
C₄	5.2839	4.6955
C₅	4.3742	4.9000
C₆	6.3383	6.4485
C₇	6.0495	6.3506
C₈	5.3256	4.3983
C₉	4.5046	4.1135
C₁₀	5.8137	5.8567
Maximum	6.7730	6.8254

Table 7

Summation of row and column.

The maximum numbers from summation of row and column will be chosen, respectively, Eq. (4).

m=min16.7730,16.8254 =min(0.1476,0.1465) =0.1465

The direct relation matrix is then normalized by multiplying with 0.1465 Eq. (7). Table 8 presents the normalized direct relation matrix.

Criteria	C₁	C₂	C₃	C₄	C₅	…	C₈	C₉	C₁₀
C₁	0.0733	0.0569	0.0720	0.0826	0.0432	…	0.0504	0.0637	0.0640
C₂	0.0535	0.0733	0.1100	0.0557	0.0554	…	0.0400	0.0473	0.0965
C₃	0.1062	0.1270	0.0733	0.0812	0.0961	…	0.0939	0.0455	0.1215
C₄	0.0905	0.1019	0.0361	0.0733	0.0714	…	0.0502	0.1245	0.0698
C₅	0.0147	0.0873	0.0474	0.0190	0.0733	…	0.1139	0.0672	0.0662
C₆	0.1317	0.1271	0.1113	0.0990	0.0671	…	0.0638	0.0339	0.1029
C₇	0.1181	0.1119	0.1173	0.0533	0.0778	…	0.0559	0.0565	0.1042
C₈	0.0813	0.1018	0.0385	0.0377	0.1205	…	0.0733	0.0242	0.0921
C₉	0.0502	0.0930	0.0461	0.1130	0.0589	…	0.0499	0.0733	0.0674
C₁₀	0.0878	0.1198	0.1189	0.0732	0.0543	…	0.0529	0.0665	0.0733

Table 8

Normalized direct relation matrix.

Using Eq. (6), we can obtain the total relation matrix. It is shown in Table 9.

Criteria	C₁	C₂	C₃	C₄	C₅	…	C₈	C₉	C₁₀
C₁	0.3812	0.4258	0.3719	0.3375	0.3013	…	0.2817	0.2806	0.3841
C₂	0.3856	0.4722	0.4367	0.3285	0.3336	…	0.2906	0.2780	0.4432
C₃	0.5322	0.6403	0.4953	0.4312	0.4568	…	0.4180	0.3452	0.5684
C₄	0.4083	0.4881	0.3506	0.3438	0.3416	…	0.2934	0.3550	0.4048
C₅	0.2852	0.4179	0.3157	0.2413	0.3085	…	0.3237	0.2556	0.3540
C₆	0.5325	0.6079	0.5068	0.4292	0.4051	…	0.3674	0.3181	0.5231
C₇	0.5012	0.5734	0.4965	0.3716	0.4009	…	0.3482	0.3254	0.5068
C₈	0.4106	0.5012	0.3684	0.3068	0.4019	…	0.3282	0.2563	0.4401
C₉	0.3238	0.4258	0.3148	0.3428	0.2927	…	0.2591	0.2740	0.3556
C₁₀	0.4585	0.5651	0.4842	0.3798	0.3670	…	0.3333	0.3272	0.4627

Table 9

Total relation matrix.

Causal diagram is obtained by calculating the summations of row and summation of column of the total relation matrix. The summations of row (R) summations of column (D), R+D and R−D of total relation matrix are shown in Table 10.

Criteria	Summation of row (R)	Summation of column (D)	R + D	R − D
C₁	3.6606	4.2192	7.8798	−0.5586
C₂	3.9370	5.1176	9.0546	−1.1806
C₃	5.1139	4.1409	9.2548	0.9731
C₄	3.8731	3.5125	7.3856	0.3606
C₅	3.2799	3.6095	6.8894	−0.3296
C₆	4.8040	4.8823	9.6864	−0.0783
C₇	4.5971	4.8271	9.4242	−0.2300
C₈	3.9867	3.2435	7.2302	0.7432
C₉	3.3266	3.0154	6.3419	0.3112
C₁₀	4.4321	4.4430	8.8750	−0.0109

Table 10

Summations of row and column.

The values of R+D and R−D are mapped onto Cartesian coordinate to construct the causal diagram. It is shown in Figure 1.

Based on the information from the causal diagram (see Figure 1), the criteria are divided into two groups by x-axis. The cause criteria are Experience, Safety Management, Financial Stability, and Timely Payment to Labor as the (R−D) is positive. On the other hand, the effect criteria are Length of Time in Industry, Quality, Performance History, Compliance with Regulations, Price, and Completing on Time. This result also indicates that “Experience” is the most important criteria in subcontractors selection owing to the largest value of (R−D).

5. COMPARATIVE ANALYSIS

It can be seen that the criteria of subcontractors' selection are segregated into two groups using the proposed IF-DEMATEL. However, these results are confined to the research frameworks and the evaluation model used. It is hypothesized that the results would no longer stable or unchanged if other computational method is used. Therefore, this section provides a comparative analysis in which the result obtained using the proposed method is compared to other methods. Specifically, the DEMATEL method [2] is used to compute real numbers input data and the results are compared to the results obtained using the proposed method. This comparative analysis is made from two perspectives. The first perspective is to investigate the criteria that belong to cause group and effect group. Table 11 presents the comparative results obtained from the use of two DEMATEL-based methods.

Methods	Cause criteria	Effect criteria
Proposed IF-DEMATEL	Experience (C₃) Safety Management (C₈) Financial Stability (C₄) Timely Payment to Labor (C₉)	Length of Time in Industry (C₁₀) Quality (C₆) Performance History (C₇) Compliance with Regulations (C₅) Price (C₁) Completing on Time (C₂)
DEMATEL	Experience (C₃) Quality (C₆) Compliance with Regulations (C₅) Safety Management (C₈) Length of Time in Industry (C₁₀)	Financial Stability (C₄) Performance History (C₇) Timely Payment to Labor (C₉) Price (C₁) Completing on Time (C₂)

Table 11

Comparative results of criteria using two different methods.

As shown in Table 11, some criteria are shared under the same group despite different methods used. For example, the criteria “Experience” and “Safety Management” are shared under the same cause criteria group. In contrast, the result is totally different for the criterion “Financial Stability.” This criterion is placed under cause group and effect group of the IF-DEMATEL and the DEMATEL, respectively. This difference could be attributed to the type of numbers employed in each method. Real numbers are employed in linguistic scales of DEMATEL. On the other hand, IFNs are employed in the proposed IF-DEMATEL.

This section also compares the results from the perspective of degree of importance. Using the proposed IF-DEMATEL and the DEMATEL, the importance of ten criteria can be prioritized by seeing to the values of (R-C). Table 12 shows the comparative results of ‘degree of importance’ of criteria.

Method	Degree of importance
Proposed IF-DEMATEL	C₃ > C₈ > C₄ > C₉ > C₁₀ > C₆ > C₇ > C₅ > C₁ > C₂
DEMATEL	C₃ > C₆ > C₅ > C₈ > C₁₀>C₄ > C₇ > C₉ > C₁ > C₂

Note: “>” denotes “more important than.”

Table 12

Comparative results of degree of importance using two different methods.

Comparing the two results, it can be seen that the most important criterion and the least important criterion obtained using the DEMATEL method and IFS-DEMATEL method are consistent. However, the degree of importance of other criteria between the two methods are a bit inconsistent. It is understandable that the inconsistent results are due the different linguistic variables used. In DEMATEL method, the linguistic variables that are defined in real numbers are used. In contrast, the linguistic variables that are defined in dual memberships of IFSs are used in the proposed IFS-DEMATEL. It is believed that the IF-DEMATEL has an edge in dealing with a complex decision problem owing to the fact that IFSs are dealt with membership and nonmembership judgments. The proposed method in this study also contributes to the changes in the algorithm of the DEMATEL thereby would slightly affect the degree of importance.

Evidently the two methods suggest that the criterion “Experience” (C₃) is the most important criterion in subcontractors selection. It is also good to mention that the two methods also agree that the criterion “Completing on Time” (C₂) is the least important criterion. However, the remaining eight criteria show a slight inconsistency in their position of preference. These results are compared with the work of Ng et al. [43]. It is found that the results are slightly inconsistent where the criteria “financial strength” and “adequacy of experiences” are the two most important criteria in their research. With a statistical approach, Doloi [44] identified the criteria “technical expertise,” “past success,” “time in business,” “work methods,” and “working capital” are also important pre-qualification criteria in subcontractors selection.

6. CONCLUSION

The DEMATEL and fuzzy DEMATEL have been used extensively to solve various multi-criteria decision-making. By means of an extension of these two decision-making methods, IF-DEMATEL was introduced as to deal with wider range of vague and uncertain information. In group multi-criteria decision-making approaches, the weights of experts and aggregation operator are among the features that need particular attention. In this paper, several innovations to the IF-DEMATEL have been proposed. As to avoid the assumption of equal weights among experts, this paper proposed relative weights for experts using a proportion equation. To aggregate the evaluation made by experts, an averaging operator is used in the proposed method. Apart from these two innovations, this paper also proposed the use of a transformation equation as a technique to convert IFN to crisp numbers. This innovated IF-DEMATEL is then implemented to a case study of subcontractors' selection. The most interesting finding was about the two categories of criteria in subcontractors' selection. The ten most important criteria for scrutinizing subcontractors were identified and these criteria were assessed by five experts. The ultimate result of this research is segregating the criteria of subcontractors' selection into the cause and effect criteria. It is found that Experience, Safety Management, Financial Stability, and Timely Payment to Labor are the criteria in the cause group, while Length of Time in Industry, Quality, Performance History, Compliance with Regulations, Price, and Completing on Time are the criteria in the effect group. In a nutshell, the criteria “Experience” is identified as the main cause that affects the selection of subcontractors. It is shown that the proposed IF-DEMATEL method is a valuable instrument to decide the key success factors in subcontractors' selection. This finding has an important outlook of this study. The identification of two groups of criteria is not only significant to subcontractors' selection, but more importantly, this study has shown the superiority of the proposed method is suggesting the most important and the least important criteria. Nonetheless, the proposed mentioned method suffers from some limitations. The current study was limited by a group of experts where their evaluations are subjected to their experience and perceptions. The current study has only examined ten criteria in which these criteria are not exhaustive and could be added in future research. Considerably more work will need to be done to determine the correlation between criteria and also multi-person decision analysis among experts. A further study with more focus on DEMATEL and the new sets such as Fermatean fuzzy [45] and hesitant multi-fuzzy soft set [46] are therefore suggested.

CONFLICTS OF INTEREST

The authors declare no conflicts of interest.

AUTHORS' CONTRIBUTION

All authors contributed to the design and implementation of the research, to the analysis of the results and to the writing of the manuscript.

Funding Statement

This work was supported by the Fundamental Research Grant Scheme, Ministry of Higher Education Malaysia [grant number FRGS/1/2018/STG06/UMT/01/1].

ACKNOWLEDGMENTS

The authors would like to extend a deep appreciation to the Universiti Malaysia Terengganu and Ministry of Higher Education for providing financial support under the Fundamental Research Grant Scheme (FRGS), Malaysian Ministry of Education with vote number, FRGS/1/2018/STG06/UMT/01/1.

APPENDIX

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Journal: International Journal of Computational Intelligence Systems
Volume-Issue: 14 - 1
Pages: 991 - 1002
Publication Date: 2021/03/12
ISSN (Online): 1875-6883
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TY  - JOUR
AU  - Lazim Abdullah
AU  - Zheeching Ong
AU  - Nuraini Rahim
PY  - 2021
DA  - 2021/03/12
TI  - An Intuitionistic Fuzzy Decision-Making for Developing Cause and Effect Criteria of Subcontractors Selection
JO  - International Journal of Computational Intelligence Systems
SP  - 991
EP  - 1002
VL  - 14
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.210222.001
DO  - 10.2991/ijcis.d.210222.001
ID  - Abdullah2021
ER  -

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