SmartISM 2.0: A Roadmap and System to Implement Fuzzy ISM and Fuzzy MICMAC

Abstract

Interpretive structural modeling (ISM) is a widely used technique to establish hierarchical relationships among a set of variables in diverse domains, including sustainability. This technique is generally coupled with MICMAC (Matrice d’Impacts Croisés Multiplication Appliquée á un Classement (cross-impact matrix multiplication applied to classification)) to classify variables in four clusters, although the manual application of the technique is complex and prone to error. In one of the previous works, a novel concept of reduced conical matrix was introduced, and the SmartISM software was developed for the user-friendly implementation of ISM and MICMAC. The web-based SmartISM software has been used more than 48,123 times in 87 countries to generate ISM models and MICMAC diagrams. This work attempts to identify existing approaches to fuzzy ISM and fuzzy MICMAC and upscale the SmartISM to incorporate fuzzy approaches. The fuzzy set theory proposed by Zadeh 1965 and Goguen 1969 helps the decision makers to provide their input with the consideration of vagueness in the real environment. The systematic review of 32 studies identified five significant approaches that have used different linguistic scales, fuzzy numbers, and defuzzification methods. Further, the approaches have differences in either using single or double defuzzification, and the aggregation of inputs of decision makers either before or after defuzzification, as well as the incorporation of transitivity either before or after defuzzification. A roadmap was devised to aggregate and generalize different approaches. Further, two of the identified approaches have been implemented in SmartISM 2.0 and the results have been reported. Finally, the comparative analysis of different approaches using SmartISM 2.0 in the area of digital transformation shows that, with a wide flexibility of fuzzy scales, the results converge and improve the confidence in the final model. The roadmap and SmartISM 2.0 will help in the implementation of fuzzy ISM and fuzzy MICMAC in a more robust and informed way.

1. Introduction

The interpretive structural modeling (ISM) developed by Warfield [1,2,3,4] is a method to create a model that shows the interrelationships among a set of variables in a particular domain with the help of domain experts. ISM “helps in representing partial, fragmented, and distributed knowledge into integrated, interactive, and actionable knowledge” [5]. Further, ISM “is a process that transforms unclear and poorly articulated mental models of systems into visible, well-defined models” [6]. The ISM technique provides practitioners with a comprehensive view of key concepts in an organized format that aids in problem solving. For its simple and modular approach, and significant value addition in solving the complex system of variables, it has been applied in various domains such as technology-assisted learning, sustainability, energy, industrial engineering, operations, information systems, and supply chain management, among others. Similarly, another important technique known as MICMAC (Matrice d’Impacts Croisés Multiplication Appliquée á un Classement (cross-impact matrix multiplication applied to classification)) was devised by J. C. Duperrin and M. Godet [7]. MICMAC classifies variables into four classes such as dependent, independent, linkage, and autonomous variables. ISM and MICMAC together provide a robust systematic framework to understand the nature of variables and the relationships among them.

In previous research [5], it was identified that many articles implementing ISM had errors at various steps due to the complex calculations involved in it, and an end-to-end automation, SmartISM, was developed to implement all steps of ISM. The software was originally developed using Microsoft Excel and VBA (Visual Basic for Applications). Later it was ported into the ASP.NET and VB.NET programming environment to make it available online (http://smartism.sgetm.com, accessed on 30 July 2024). Since then, it has been used more than 48,127 times to generate ISM models and MICMAC diagrams. Looking at the statistics, so far it has been used in 87 countries and 921 cities globally based on the users’ IP address analysis. The top users’ countries can be seen in Figure 1, and the number of variables being handled have ranged between 3 and 50. As discussed with users, the most frequent problem was the clustering of all variables into one point on the MICMAC diagram and the positioning of all variables on a single level in the ISM model. This happens when decision makers assign inconsistent relationships to the pair of variables. Further, many users made feedback to upscale the software into more advanced approaches of ISM such as fuzzy ISM or total ISM (TISM).

The comments from two YouTube videos and emails, and diverse approaches of fuzzy ISM in the literature, have prompted the upscaling of the software in newer dimensions. There is the possibility to upscale ISM in various directions such as fuzzy interpretive structural modeling (fuzzy ISM) [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27] and total interpretive structural modeling (TISM) [6]. The fuzzy ISM is a more primitive advancement [28] as it modifies the technique to incorporate the vagueness in the decisions of the experts using fuzzy set theory [29] and builds a more robust model from the inception. Conversely, TISM adds more value to the ISM models by incorporating the knowledge of the interpretation of relations, and the TISM can also be made as fuzzy TISM [6,30]. However, this additional knowledge may also be incorporated manually in the final model. Further, similar to fuzzy set theory, there are other theories such as interval-valued optimization theory that may also be incorporated into the ISM to represent uncertainties in the real world [31,32], although this has currently not been greatly explored by the researchers. Therefore, it warrants first to simplify the implementation of fuzzy ISM and fuzzy MICMAC.

The fuzzy ISM and fuzzy MICMAC integrate fuzzy logic in defining the relationships between the variables. It allows for the flexibility of the decision makers to assign relationships with vagueness using a linguistic scale as opposed to a binary scale in traditional ISM. And, there is a wide range of options to choose linguistic scales and fuzzy numbers such as singleton fuzzy number [13,14,17,25], triangular fuzzy number [8,9,24,30,33,34], and trapezoidal fuzzy number [26,35]. Also, the aggregation of binary inputs from multiple decision makers was mostly performed by taking a mode in the traditional approach that completely omits the less frequent inputs, whereas, in a fuzzy environment in addition to mode [30], various techniques are used for aggregation such as average [15], geometric mean [10], and 2-tuple average operator [23]. Further, the writers have used different techniques for defuzzification such as the simplified individual conversion of fuzzy to crisp such as the centroid method [16]. Conversely, some studies have used more comprehensive approaches that normalize the set of fuzzy numbers such as CFCS (converting the fuzzy data into crisp score) [36] and the technique illustrated in [15]. Hence, there is a need to formulate a roadmap illustrating the different techniques of fuzzy ISM and fuzzy MICMAC.

Further, there is a need to develop a software system for the user-friendly and error-free implementation of fuzzy ISM and fuzzy MICMAC: a flexible system that adapts to different approaches easily and facilitates collaboration among researchers and domain experts. Therefore, the objectives of the present study are as follows:

• The identification of the existing approaches for the implementation of fuzzy ISM and fuzzy MICMAC.
• The formulation of a roadmap to implement fuzzy ISM and fuzzy MICMAC.
• The development of SmartISM 2.0 to incorporate fuzzy ISM and fuzzy MICMAC approaches.
• The comparative analysis of different approaches using SmartISM 2.0.

The remainder of the article is organized as background, research methods, existing approaches for fuzzy ISM and fuzzy MICMAC, a roadmap to adopt fuzzy ISM and fuzzy MICMAC, the development of SmartISM 2.0, the case of digital transformation, and conclusion.

2. Background

2.1. Traditional ISM

The interpretive structural modeling (ISM) technique is applied in certain steps such as variable identification, the selection of domain experts, the formation of the structural self-interaction matrix (SSIM), the reachability matrix (RM) and final reachability matrix (FRM), level partitioning, the conical matrix (CM) and Digraph, the reduced conical matrix (RCM), and the final ISM model [5]. The concept of the reduced conical matrix was introduced in article [5] to remove some of the transitive links to produce the final ISM model in the SmartISM software. The SSIM is generated with the input of domain experts who define the relationships among the variables through pairwise comparisons. Each pair of variables ‘a’ and ‘b’ are assigned one of the four possible relationships such as ‘a’ influences ‘b’, ‘b’ influences ‘a’, ‘a’ and ‘b’ mutually influence each other, or ‘a’ and ‘b’ do not mutually influence each other, generally denoted with ‘V’, ‘A’, ‘X’, or ‘O’, respectively. Thereafter, influences are converted in binary forms with specified rules to generate RM.

2.2. Fuzzy ISM

These binary numbers indicate whether influence exists or does not exist. Although, in the real environment, it is not that crisp; rather, there exists vagueness in decisions of domain experts. This vagueness can be captured through the fuzzy set theory proposed by Zadeh 1965 [29] and Goguen 1969 [37]. The first adaptation of ISM to fuzzy ISM was proposed by Ragade 1976 [28]. However, there is an increased adoption of the fuzzy ISM approaches in recent times [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]. In the fuzzy ISM approach, there are some additional steps that need to be taken care of, such as the identification of a linguistic scale, fuzzy numbers to be used, method of aggregation of fuzzy numbers for multiple decision makers, method of defuzzification, and method of binary conversion that results in RM, and the remaining steps may follow from the traditional ISM approach.

2.3. Fuzzy Basics

The fuzzy set theory defines fuzzy numbers in different forms with the help of membership functions [38]. In one of the definitions, fuzzy number $\tilde{M}$ is the subset of X, a set of real numbers, and is characterized by a membership function ${\mathrm{\mu }}_{\tilde{M}}\left(x\right)$, where ${\mathrm{\mu }}_{\tilde{M}} :X\to \left[\mathrm{0,1}\right]$ [19,30]. The singleton fuzzy number is represented by a single value at which the membership function returns 1 and at all other values it returns zero [39]. A triangular fuzzy number (TFN) $\tilde{M}$ defined by a triplet (a, b, c), or a trapezoidal fuzzy number (TzFN) $\tilde{M}$ defined by a quadruplet (a, b, c, d), and their associated membership functions are illustrated in Figure 2 and Equation (1) [19,30]. Additionally, certain helpful mathematical operations like addition, multiplication, scalar multiplication, and inverse on TFNs or TzFNs ${\tilde{M}}_1=\left(a_1,b_1,c_1\right)\mathrm{o}\mathrm{r}\left(a_1,b_1,c_1, d_1\right)$ and ${\tilde{M}}_2=\left(a_2,b_2,c_2\right)\mathrm{o}\mathrm{r}\left(a_2,b_2,c_2, d_2\right)$ are demonstrated in Equations (2)–(5), respectively [40].

Table 1. Five-level linguistic scale for TFN and TzFN.

Linguistic Description	TFN [8,24,30,33]	TFN [9,34]	TFN	TzFN [26,35]	Influence Scope	Notation
Very good (high) influence/completely related	(0.75, 1.0, 1.0)	(0.75, 1.00, 1.00)	(0.7, 0.9, 1.0)	(0.75, 0.85, 0.95, 1.00)	4	VG
Good (high) influence/strongly related	(0.5, 0.75, 1.0)	(0.50, 0.75, 1.00)	(0.5, 0.7, 0.9)	(0.55, 0.65, 0.75, 0.85)	3	G
Fair influence/fairly related	(0.25, 0.5, 0.75)	(0.25, 0.50, 0.75)	(0.3, 0.5, 0.7)	(0.35, 0.45, 0.55, 0.65)	2	F
Poor (very low) influence/low related	(0, 0.25, 0.5)	(0.01, 0.25, 0.50)	(0.1, 0.3, 0.5)	(0.15, 0.25, 0.35, 0.45)	1	P
Very poor (no) influence/unrelated	(0, 0, 0.25)	(0.01, 0.01, 0.01)	(0, 0.1, 0.3)	(0.00, 0.05, 0.15, 0.25)	0	VP

and Figure 3 shows some sample fuzzy numbers along with linguistic scales that are used in the sample studies.

$${\mathrm{\mu }}_{\tilde{M}}\left(x\right)=\left\{\begin{array}{c}0, x\le 0\\ {\displaystyle \frac{x-a}{b-a}}, a\le x\le b\\ {\displaystyle \frac{x-c}{b-c}}, b\le x \le c\\ 0, x\ge c \end{array} \mathrm{o}\mathrm{r} \right.\left\{\begin{array}{c}0, x\le 0\\ {\displaystyle \frac{x-a}{b-a}}, a\le x\le b\\ 1, b\le x\le c\\ {\displaystyle \frac{x-d}{c-d}}, c\le x\le d\\ 0, x\ge d \end{array} \right.$$

(1)

$${\tilde{M}}_1\oplus {\tilde{M}}_2=\left(a_1+a_2,b_1+b_2,c_1+c_2\right) \mathrm{o}\mathrm{r} \left(a_1+a_2,b_1+b_2,c_1+c_2, d_1+d_2\right)$$

(2)

$${\tilde{M}}_1\otimes {\tilde{M}}_2=\left(a_1{a}_2,b_1{b}_2,c_1{c}_2\right) \mathrm{o}\mathrm{r} \left(a_1{a}_2,b_1{b}_2,c_1{c}_2, d_1{d}_2\right)$$

(3)

$$k\otimes {\tilde{M}}_1=\left(k{a}_1,k{b}_1,k{c}_1\right)\mathrm{o}\mathrm{r}\left(k{a}_1,k{b}_1,k{c}_1, k{d}_1\right), \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} k>0$$

(4)

$${{\tilde{M}}_1}^{-1}=\left({\displaystyle \frac{1}{c_1}},{\displaystyle \frac{1}{b_1}},{\displaystyle \frac{1}{a_1}}\right) \mathrm{o}\mathrm{r} \left({\displaystyle \frac{1}{d_1}},{\displaystyle \frac{1}{c_1}},{\displaystyle \frac{1}{b_1}}, {\displaystyle \frac{1}{a_1}}\right)$$

(5)

3. Research Methods

This research attempts to identify the existing approaches to implement fuzzy ISM. Therefore, one of the extensive databases, Google Scholar, was searched for the selection of studies as shown in the PRISMA flow diagram, Figure 4. For a more focused search, a title-level advanced search was performed twice with the keyword “fuzzy ism” and “fuzzy interpretive structural modeling”. It resulted in a total of 75 records. Out of these 75 records, 5 were duplicate, 10 were not relevant, 15 were in non-English languages, and 8 records were inaccessible. Out of the remaining 37 reports, 16 had applied only fuzzy MICMAC and were therefore excluded. In essence the studies that had applied fuzzy logic for both ISM and MICMAC, relevant, accessible in full length, and written in the English language, were included, and others were excluded. Also, all relevant articles that made the basis of the identified 21 articles were reviewed in the process of citation searching that further resulted in an additional 11 records being retrieved. The selected 32 studies were published between the year of 1996 and 2023; 25 belong to journal articles, five to conference papers, one book, and one PhD thesis (

Table 2. Bibliographic information of selected studies.

Publication Sources	1976	1999	2003	2007	2010	2011	2013	2015	2016	2017	2018	2019	2020	2021	2022	2023	Total
Journals:	1	1	1	1		1	1	2	2	1	3		2	7	1	1	25
Adv. Fuzzy Math.										1							1
Adv. Fuzzy Syst.									1								1
Ann. Oper. Res.														1			1
Appl. Soft Comput.								1									1
Comput. Math. Appl.		1															1
Expert Syst. Appl.				1													1
Flex. Serv. Manuf. J.						1											1
Fuzzy Sets Syst.													1				1
Glob. J. Flex. Syst. Manag.								1									1
Ind. Manag. Data Syst.											1						1
Int. J. Circ. Syst. Signal Process.														1			1
Int. J. Energy Sect. Manag.																1	1
Int. J. Ind. Syst. Eng.														1			1
Int. J. Ind. Ergon.											1						1
Int. J. Inf. Syst. Supply Chain Manag.							1										1
Int. J. Product. Qual. Manag.											1						1
Int. J. Uncertainty, Fuzziness Knowledge-Based Syst.			1														1
Jindal J. Bus. Res.													1				1
J. Cybern.	1																1
J. Enterp. Inf. Manag.														1			1
J. Model. Manag.									1								1
Mathematics														1			1
PLoS ONE														1			1
Secur. Commun. Netw.														1			1
Soc. Responsib. J.															1		1
Conferences:				2	1							1	1				5
CSCE Annual Conference 2019, Laval (Greater Montreal), Canada												1					1
Fuzzy Information and Engineering 2010, Huludao, China					1												1
Information Sciences 2007, Salt Lake City, Utah, USA				1													1
ICMET 2019, India													1				1
IIH-MSP 2007: Kaohsiung, Taiwan				1													1
Book: Automaton, Los Angeles, USA				1													1
PhD Thesis: Bahir Dar University, Ethiopia															1		1
Total	1	1	1	4	1	1	1	2	2	1	3	1	3	7	2	1	32

). Finally, these 32 studies were analyzed to identify the existing approaches for the fuzzy ISM and fuzzy MICMAC. The studies were clustered based on the similarity of the approaches adopted for fuzzy ISM and fuzzy MICMAC analysis. Five prominent approaches have been identified through analyzing these clusters. These five approaches are presented in the next section. Also, this article presents the development of the SmartISM 2.0 software system for the fully automated implementation of fuzzy ISM and fuzzy MICMAC for selected approaches, as per the availability of the data. Finally, the comparative analysis is performed on different approaches using SmartISM 2.0 on a case study of digital transformation with two experiments. The list of critical success factors for digital transformation and the relationships between them were identified in one of the workshops conducted before a group of faculty members belonging to the college of computer science. In the first experiment, highest intensity linguistic scales were used in all approaches to compare the results of fuzzy approaches with the traditional approach. In the second experiment, the inputs were simulated to four decision makers by changing the intensities to represent four values of the linguistic scale without changing the nature of the relationships.

4. Existing Approaches for Fuzzy ISM and Fuzzy MICMAC

There are five approaches that have been identified to have demonstrated the well-structured application of fuzzy ISM and fuzzy MICMAC. These five approaches are mentioned in the following sections. Studies that have used other miscellaneous approaches have been summarized in the final paragraph.

4.1. The First Basic Approach

The studies [16,19,22] falling in this category use 4, 5, or 7 scale fuzzy numbers, either TFN or TzFN, to establish the relationships between pairs of variables. These studies use more simpler approaches of defuzzification to convert the individual fuzzy numbers into crisp numbers such as the centroid method [16] and alpha-cut [22,41]. Thereafter, the crisp numbers are converted into binary numbers using some chosen threshold values to obtain the initial RM. Afterwards, traditional ISM and MICMAC steps are applied to develop a hierarchical final ISM model. [16] took a mode for the aggregation of responses and considered strong and moderate influence as ‘1’ and weak and no influence as ‘0’. Conversely, the crisp values that were obtained through the defuzzification method are used for MICMCA analysis. This dual defuzzification used in [16] is similar to Khatwani 2015 [30], as explained in the next section.

Let there be a TzFN ${\tilde{M}}_k=\left(a_k,b_k,c_k, d_k\right);k=1, 2, \cdots , n$ and $M_k$ be the crisp value.

Step 1: Defuzzification to obtain crisp values used in [19].

$$\begin{array}{c}{M}_k={\displaystyle \frac{\left({c_k}^2+{d_k}^2+c_k{d}_k-{a_k}^2-{b_k}^2-b_k{c}_k\right)}{3\left(c_k+d_k-a_k-b_k\right)}};\\ k=1, 2, \cdots , n\end{array}$$

Defuzzification to obtain crisp values used in [22].

$$M_k=\left(a_k+{2b}_k+2c_k+d_k\right)/6;k=1, 2, \cdots , n$$

Step 2: Binary conversion to obtain initial reachability matrix while the threshold value τ may be 0.35 [19] or 0.45 [22].

$$R_{ij}=\left\{\begin{array}{c}1 M_{ij}>\tau \\ 0 M_{ij}\le \tau \end{array}\right.$$

4.2. The Second Approach Using Singleton Fuzzy Numbers and Fuzzy Transitive Closure

There are some articles [13,14,17,25] that have used singleton fuzzy numbers (SFNs). The linguistic scales and SFNs help in developing fuzzy structural self-interaction matrix. And, fuzzy transitive closure may be used to incorporate transitivity in a fuzzy reachability matrix [42]. The fuzzy transitive closure is estimated by executing the power operation on the matrix A + I, where A is the adjacency matrix and I is the identity matrix [25,42]. That is, to increase the power on the A + I matrix until the result converges, meaning that the current power is equal to the previous power. The concept uses the max min approach of matrix multiplication [43]. This operator actually helps in calculating transitive relations. The singleton fuzzy numbers between 0 and 1 (SFN) ${\tilde{M}}_k=a_k;k=1, 2, \cdots , n;a_k\in \left[\mathrm{0,1}\right]$ are used to quantify the relationships between two elements. The approach may be illustrated in the following steps.

Step 1: Firstly, the fuzzy adjacency matrix $\tilde{A}$ is created by assigning the aggregated linguistic term and associated SFN on the basis of data gathered from decision makers for the pairwise comparisons of n variables.

$$\tilde{A}=\left\{\begin{array}{c}{a}_{ij};i,j=\mathrm{1,2}, \cdots ,n;i\ne j;a_{ij}\in \left[\mathrm{0,1}\right]\\ 0;i,j=\mathrm{1,2},\cdots ,n;i=j\end{array}\right.$$

Step 2: The fuzzy reachability matrix $\tilde{R}$ is obtained by adding identity matrix I to $\tilde{A}$, and the fuzzy transitive closure [42] is estimated by executing the power operation on $\tilde{B}=\tilde{A}+I$ until the results converge in kth power, utilizing the max min fuzzy matrix multiplication [43], as shown below.

$$\begin{array}{c}{\tilde{B}·\tilde{B}=b}_{ij}=\max \left(\min \left(i^{th} row of \tilde{B},{ j}^{th} column of \widetilde{B }transposed \right)\right);\\ i,j=\mathrm{1,2}, \cdots ,n\\ \tilde{R}={\left(\tilde{B}\right)}^{k+1}={\left(\tilde{B}\right)}^k\ne {\left(\tilde{B}\right)}^{k-1}\ne \cdots \ne {\left(\tilde{B}\right)}^3\ne {\left(\tilde{B}\right)}^2\ne \tilde{B}\end{array}$$

Step 3: Then, the binary reachability matrix is derived from the fuzzy reachability matrix $\tilde{R}$ by applying the cut-off coefficient α with the convenient value of 0.5.

$$R_{\alpha =0.5}=\left\{\begin{array}{c}1, r_{ij}\ge 0.5 \\ 0, r_{ij}<0.5\end{array}\right.$$

4.3. The Third Approach Using 25 Different Relations and the CFCS Defuzzification Method

Some studies [8,24,26] have adopted the fuzzy ISM based on the approach of Khatwani 2015 [30]. In this approach, the TFNs [8,24,30] or TzFNs [26] are used and the aggregation is performed on the basis of mode [30], which is the most frequent decision given by the experts. Further, a five-level linguistic scale for the TFN [33] and TzFN [35] is used, along with the compatibility functions (the mapping of linguistic terms to fuzzy numbers), as defined in Figure 3, Table 1. These linguistic scales increase four possible relationships V, A, X, and O in traditional ISM to 25 possible relationships [8,24,26,30] in the fuzzy ISM. Therefore, the mode approach for aggregation may cause deadlock, as each expert might choose a different relationship for the same pair of variables. Further, the linguistic terms are defuzzified to obtain the binary initial RM on the basis of the significance of the relationship; VG and G values are converted to 1 and F, P, and VP is converted to 0 [8,24,26,30]. Once the binary initial RM is identified, traditional ISM may follow.

The second defuzzification in Khatwani’s 2015 [30] approach is based on the concept of CFCS (converting the fuzzy data into crisp score) [36]. It is performed in the following four steps for TFNs. Let there be a TFN ${\tilde{M}}_k=\left(a_k,b_k,c_k\right);k=1, 2, \cdots , n$, where k represents the number of variables and $M_k$ is the crisp value. These crisp values are used in MICMAC analysis. This process helps in finding the driving and dependence values of each variable. The following steps are repeated twice for rows and columns.

Step 1: Normalization.

$$Compute L=\min \left(a_k\right); R=\max \left(c_k\right); ∆=L-R$$

$$\begin{gathered} {\widehat{a}}_k=(a_k-L)/∆; \\ {\widehat{b}}_k=(b_k-L)/∆; \\ {\widehat{c}}_k=(c_k-L)/∆; \end{gathered}$$

$$k=1, 2, \cdots , n$$

Step 2: Calculate left-hand-side and right-hand-side normalized values.

$${\widehat{l}}_k={\widehat{b}}_k/(1+{\widehat{b}}_k-{\widehat{a}}_k); {\widehat{r}}_k={\widehat{c}}_k/(1+{\widehat{c}}_k-{\widehat{b}}_k);k=1, 2, \cdots , n$$

Step 3: Calculate total normalized crisp value.

$$\begin{array}{c}{\widehat{M}}_k={\displaystyle \frac{\left[ {\widehat{l}}_k\left(1-{\widehat{l}}_k\right)+{\widehat{r}}_k\times {\widehat{r}}_k\right]}{1-{\widehat{l}}_k+{\widehat{r}}_k}};\\ k=1, 2, \cdots , n\end{array}$$

Step 4: Calculate total crisp value.

$$M_k=L+{\widehat{M}}_k\times ∆;k=1, 2, \cdots , n$$

Similarly, for TzFNs, the following formula may be used to calculate the crisp values. Let there be a TzFN ${\tilde{M}}_{\mathrm{k}}=\left(a_k,b_k,c_k, d_k\right);k=1, 2, \cdots , n$ and $M_k$ be the crisp value [26]. The below formula is devised by the new approach of defuzzification based on the mean value of statistical beta distribution [44].

$$\begin{array}{c}{M}_k={\displaystyle \frac{2\times a_k+7\times b_k+7\times c_k+2\times d_k}{18}};\\ k=1, 2, \cdots , n\end{array}$$

One study [10] uses the same scale for relationship identification. While aggregation is performed by geometric mean, defuzzification is performed through the traditional center of gravity method, and the average value of the defuzzified matrix is used as the threshold value to derive a binary reachability matrix.

4.4. The Fourth Approach Applying Aggregation after the Defuzzification of RM

In this approach of Wang L. 2018 [15], a five-level linguistic scale is used to identify the relationship between pairs of variables forming the n × n matrix for n variables. Thereafter, linguistic variables are converted to triangular fuzzy numbers and defuzzified by the technique illustrated below. The aggregation of the input of different decision makers is performed by taking the average values of all the crisp values of a specific pair of variables. Subsequently, the threshold value is calculated by taking the average of the first column or row of the matrix representing these averaged crisp values. And, finally, the binary relationship matrix is obtained by comparing all values of the matrix with this threshold value. Once the binary relationship matrix is obtained, traditional ISM and MICMAC may be followed. The technique may be illustrated in the following steps. Let there be a TFN ${\tilde{M}}_{ijk}=\left(a_{ijk},b_{ijk},c_{ijk}\right);i,j=\mathrm{1,2}, \cdots ,n;k=1, 2, \cdots , m$; where k represents the number of decision makers and $M_{ijk}$ is the crisp value. This process will be repeated for each pair of variables forming a matrix of size n by n, where n is the number of variables.

Step 1: Normalize TFNs.

$$\begin{array}{c}\begin{array}{c}Compute ∆=\max \left(c_{\mathit{ijk}}\right)-\min \left(a_{ijk}\right); \\ {\widehat{a}}_{ijk}={\displaystyle \frac{\left(a_{ijk}-\min \left(a_{ijk}\right)\right)}{∆}};i\ne j\end{array}\\ {\widehat{b}}_{ijk}={\displaystyle \frac{\left(b_{ijk}-\mathit{min}\left(b_{ijk}\right)\right)}{∆}};i\ne j\\ {\widehat{c}}_{ijk}={\displaystyle \frac{\left(c_{ijk}-\mathit{min}\left(c_{ijk}\right)\right)}{∆}};i\ne j\end{array}$$

Step 2: Calculate left-hand-side and right-hand-side normalized values.

$$\begin{gathered} {\widehat{l}}_{ijk}={\displaystyle \frac{{\widehat{b}}_{ijk}}{(1+{\widehat{b}}_{ijk}-{\widehat{a}}_{ijk})}}; i\ne j \\ {\widehat{r}}_{ijk}={\displaystyle \frac{{\widehat{c}}_{ijk}}{(1+{\widehat{c}}_{ijk}-{\widehat{b}}_{ijk})}}; i\ne j \end{gathered}$$

Step 3: Calculate total normalized crisp value.

$${\widehat{M}}_{ijk}={\displaystyle \frac{\left[ {\widehat{l}}_{ijk}\left(1-{\widehat{l}}_{ijk}\right)+{\widehat{r}}_{ijk}\times {\widehat{r}}_{ijk}\right]}{1-{\widehat{l}}_{ijk}+{\widehat{r}}_{ijk}}}; i\ne j$$

Step 4: Calculate averaged normalized crisp value for m decision makers.

$${\widehat{M}}_{ij}=\left({\sum }_{k=1}^m{\widehat{M}}_{ijk}\right)/m ;i\ne j$$

Step 5: Derive the binary values ${\pi }_{ij}$ based on threshold value τ defined by the mean value of the first column of averaged normalized crisp value matrix ${\widehat{M}}_{ij}$.

$$\begin{gathered} \mathsf{\tau }=\left({\sum }_{i=1}^n{\widehat{M}}_{i1}\right)/n \\ {\pi }_{ij}=\left\{\begin{array}{c}1 {\widehat{M}}_{ij}\ge \tau \\ 0 {\widehat{M}}_{ij}<\tau \end{array}\right. \end{gathered}$$

4.5. The Fifth Approach Applying the Aggregation before the Defuzzification of RM

The study of Wang W. 2018 [9] took the α−cut approach of Kang and Lee 2007 [45] for the implementation of fuzzy ISM. Firstly, the TFNs are aggregated through the geometric mean, although the equation given in the article is for the arithmetic mean. Thereafter, the TFNs are characterized at the defined interval of confidence at α. Further, the defuzzified numbers or crisp numbers are derived with α−cuts and degree of optimism β. The higher the value of β, the higher the optimism. Finally, the binary conversion is carried out by identifying the threshold value by the decision makers [34] or the mean value of the defuzzified relationship matrix [9]. Once the binary relationship matrix is obtained, traditional ISM and MICMAC may be followed. The technique may be illustrated in the following steps. Let there be a TFN ${\tilde{M}}_{ijk}=\left(a_{ijk},b_{ijk},c_{ijk}\right);i,j=\mathrm{1,2}, \cdots ,n;k=1, 2, \cdots , m$; where k represents the number of decision makers and $M_{ij}$ is the crisp value. This process will be repeated for each pair of variables forming a matrix of size n by n, where n is the number of variables.

Step 1: Calculate the geometric mean [46].

$${\widehat{a}}_{ij}=\sqrt[m]{{\prod }_{k=1}^m{a}_{ijk}};i\ne j$$

$${\widehat{b}}_{ij}=\sqrt[m]{{\prod }_{k=1}^m{b}_{ijk}}; i\ne j$$

$${\widehat{c}}_{ij}=\sqrt[m]{{\prod }_{k=1}^m{c}_{ijk}} ;i\ne j$$

Step 2: Characterize the TFNs at the defined interval of confidence at α [45].

$$\begin{gathered} {\widehat{l}}_{ij}=\left({\widehat{b}}_{ij}-{\widehat{a}}_{ij}\right)\alpha +{\widehat{a}}_{ij}; \forall \alpha \in \left[\mathrm{0,1}\right]; i\ne j \\ {\widehat{r}}_{ij}=-\left({\widehat{c}}_{ij}-{\widehat{b}}_{ij}\right)\alpha +{\widehat{c}}_{ij}; \forall \alpha \in \left[\mathrm{0,1}\right]; i\ne j \end{gathered}$$

Step 3: Defuzzify the α−cut characterized numbers at β degree of optimism [45].

$$M_{ij}=\left(1-\beta \right){\widehat{l}}_{ij}+\beta {\widehat{r}}_{ij}; \forall \beta \in \left[\mathrm{0,1}\right]; i\ne j$$

Step 4: Derive the binary values ${\pi }_{ij}$ based on the threshold value τ defined by the decision makers [34] or the mean value of the defuzzified relationship matrix $M_{ij}$ [9].

$${\pi }_{ij}=\left\{\begin{array}{c}1 M_{ij}>\tau \\ 0 M_{ij}\le \tau \end{array}\right.$$

There were some other studies that implemented fuzzy ISM, such as [27], that used a 10-point Likert scale to quantify the intensity of the relationships and trapezoidal fuzzy numbers that were used for MICMAC analysis. Similarly, one study [23] used the concept of 2-tuple linguistic representation for fuzzy ISM. The aggregation of the inputs of decision makers was performed through a 2-tuple average operator. Also, the studies [11,12] integrated the fuzzy logic of model perception with ISM to analyze the concept structure that is useful in cognition diagnostics. Another study [21] used the fuzzy concept for fuzzy MICMAC analysis. Similarly, studies [13,14] used singleton fuzzy numbers between 0 and 1 to associate the strength of relationships in the traditional ISM approach and fuzzy MICMAC. The study [14] also applied singleton fuzzy numbers between 0 and 1, and for level partitioning, the approach of counting zeros in the fuzzy adjacency matrix was taken. The article [17] initially used a traditional approach to identify the relationships and identify the final reachability matrix after employing transitivity. Thereafter, the seven-level singleton fuzzy numbers are associated with all the relationships shown with digit ‘1’ with the help of decision makers. Subsequently, the obtained fuzzy final reachability matrix is used for the fuzzy ISM model and fuzzy MICMAC analysis. The study [18] used a survey technique to identify the relationships between variables on the basis of correlations and corroborated it with the decision makers for the direction of the relation. The strength of the correlation is mapped to the linguistic terms and these terms are fuzzified using TFNs. Thereafter, a basic fuzzy ISM approach is used to establish the final model. The study [20] did not clearly specify the method of defuzzification.

5. A Roadmap to Implement Fuzzy ISM and Fuzzy MICMAC

In the presence of a range of approaches for fuzzy ISM and fuzzy MICMAC, it is necessary to devise a roadmap, as in Figure 5, illustrating the key differences and similarities among them. Firstly, there are various linguistic scales and fuzzy numbers that are being used in different approaches, as illustrated in the previous section. Secondly, there is one primary distinction in all the studies, which is to use either the single defuzzification method in both fuzzy ISM and fuzzy MICMAC, or double defuzzification methods such as the conversion of linguistic symbols to binary numbers for fuzzy ISM and a more comprehensive defuzzification approach that normalizes the set of fuzzy numbers for the fuzzy MICMAC. As for the singleton fuzzy numbers, the defuzzification is simply the single value at which the membership function returns the value of 1. Thirdly, transitivity incorporation in some studies [25] is performed in the fuzzy RM itself before binary conversion, and in some studies [9] it is performed after binary conversion.

In the basic approach [19,22], generally, the individual fuzzy numbers are individually defuzzified with some simpler defuzzification methods. Thereafter, on the basis of some threshold value, these defuzzified values are converted into binary numbers and, thereafter, traditional ISM is applied. For fuzzy MICMAC analysis, these studies employ the defuzzified values for the computation of the driving and dependence values of individual variables. However, one study [16] follows a double defuzzification approach. The linguistic terms are defuzzified to binary numbers for fuzzy ISM, whereas, for fuzzy MICMAC, the linguistic terms are mapped to fuzzy numbers and the fuzzy numbers are translated to crisp values.

The second identified approach uses singleton fuzzy numbers. For singleton fuzzy numbers, the defuzzification is simply the single value at which the membership function returns the value of one, as at all other values it returns zero. The study [25] used fuzzy transitive closure [42] to incorporate transitivity in the fuzzy matrix of singleton fuzzy numbers. Further, it used an α-cut approach to drive a binary matrix to develop a fuzzy ISM model. There are several other studies [13,14,17] as well in the literature that use singleton fuzzy numbers to identify the strengths of relationships, and these strengths are depicted in the fuzzy ISM models in some studies. Further, these non-binary values are used in the fuzzy MICMAC analysis to calculate the driving and dependence powers of the variables.

The third identified approach devised by Khatwani 2015 [30] has been used in many articles [8,24,26,30]. In this approach, a five-scale linguistic scale is attached to the traditional 4 possible relationships that results in 25 different possible relationships. Thereafter, the aggregation of multiple decision makers is performed on the basis of mode. Further, the higher intensity scales are defuzzified to 1, whereas lower intensity scales are defuzzified to 0. Hence, the obtained binary reachability matrix is used in the traditional ISM steps to obtain the fuzzy ISM model. For MICMAC analysis, the driving and dependence powers are calculated using the CFCS (converting the fuzzy data into crisp score) [36] defuzzification method. There are also some variations to this approach in aggregation and defuzzification [10,26].

The fourth approach of Wang L. 2018 [15] first converted linguistic scales to the fuzzy numbers. Thereafter, these fuzzy numbers of different decision makers are defuzzified using some α-cut defuzzification approach. Further, the aggregated SSIM is generated by taking the average of these crisp values, whereas the fifth approach of Wang W. [9] first aggregated the fuzzy numbers using geometric mean, which were thereafter defuzzified using the α−cut approach of Kang and Lee 2007 [45]. Further, some threshold values are used in both approaches of Wang L. 2018 [15] and Wang W. [9] to convert these final crisp values to the binary numbers to obtain the RM. Finally, the obtained initial RM is used in the traditional ISM steps to produce a fuzzy ISM model and fuzzy MICMAC diagram.

These five approaches were aggregated and generalized, as illustrated in Figure 5. The first four steps include the identification of variables; the identification of linguistic scale and relationship scale; and identification of fuzzy number and the development of fuzzy SSIM using linguistic terms and relationship symbols, which are common in all approaches, although the outcome will depend on the choices being made. Thereafter, the first diversion happens in the aggregation of the inputs of different decision makers. The third approach generates the aggregated fuzzy SSIM and defuzzifies its linguistic terms into the binary form to obtain the initial RM to be used in traditional ISM. Similarly, the aggregated SSIM is converted into aggregated fuzzy RM and defuzzified to calculate crisp dependence and driving powers to make the fuzzy MICMAC diagram, whereas other approaches convert the fuzzy SSIM into fuzzy RM. The second diversion happens post fuzzy RM; in the fifth approach, the aggregated fuzzy RM is generated by taking the geometric mean and, thereafter, the defuzzification is performed to produce crisp initial RM; in the fourth approach, firstly, the defuzzification is performed to obtain crisp initial RM and, thereafter, by simple average the aggregated initial RM is computed; in the second approach, which uses singleton fuzzy numbers, fuzzy transitive closure is used to produce final fuzzy RM and, thereafter, defuzzification is performed to obtain crisp final RM as the transitivity is already incorporated in the fuzzy RM. These crisp initial or final RMs are translated into binary initial or final RMs and, thereafter, traditional ISM is followed to obtain the fuzzy ISM model and fuzzy MICMAC diagram.

6. The Development of SmartISM 2.0 to Incorporate Fuzzy ISM Approaches

For the development of the software, complete data and clarification on missing details are required to implement and test the identified approaches. The writers of these approaches were contacted for data collection and clarifications. As per the availability of complete data in the articles and responses received from the writers, two approaches, the second using data of [25] and third using data of [30], were chosen to be implemented and tested on Microsoft Excel using VBA functions. The first basic approach is difficult to standardize, as versatile configurations are used by different writers. The writers of the fourth and fifth approach did not provide sufficient response.

The second approach was relatively simpler to incorporate as it uses singleton fuzzy numbers (SFNs). Additional complexity was involved in implementing fuzzy transitive closure on the fuzzy matrix of SFNs. This operation helps in incorporating transitivity in the fuzzy reachability matrix. Also, this approach uses a very simple defuzzification method due to SFNs and cut-off value approach to drive binary FRM, as explained in Section 4.2. Moreover, post defuzzification, the binary FRM matrix is used for both final model and MICMAC analysis, as explained in [5]. Nonetheless, 22 VBA functions were used to implement this approach. The second approach was tested with the data of article [25]. The selected reproduced results from SmartISM 2.0 for this article are shown in Figure 6 and Figure 7. All the results of SmartISM 2.0 match with the original article [25].

For the third approach, the first function was written to populate a Microsoft Excel sheet with multiple fuzzy SSIM matrices, as per the number of decision makers with the provision of dropdown lists to choose one of the 25 relations defined in this approach for the couple of variables. Thereafter, functions were written to aggregate the inputs of DMs on the basis of mode, and drive fuzzy RM to show linguistic terms and binary RM from linguistic terms. Thereafter, traditional ISM was applied, as explained in [5]. Next, functions were written to obtain fuzzy RM in TFNs translated from linguistic terms and defuzzified RM, as explained in Section 4.3 using the CFCS defuzzification method. The obtained crisp dependence and driving values in the previous step were used in generating a fuzzy MICMAC diagram. A separate class TFN was defined in VBA to represent TFNs and associated functions such as defuzzification. In total, 23 VBA functions were written to implement this approach. The third approach was tested with the data of article [30]. The selected reproduced results from SmartISM 2.0 for this article are shown in Figure 8 and Figure 9. There is a slight discrepancy in the final model in Figure 8 due to the non-assignment of level 2 for element 10 in the level partitioning in the second iteration in the original article [30]. The rest of the results of SmartISM 2.0 match with the article [30].

7. The Case of Digital Transformation

This section presents the results of the application of fuzzy ISM and fuzzy MICMAC using SmartISM 2.0 in the case of digital transformation (DT). The critical success factors (CSFs) for DT and relationships were identified in one of the workshops conducted before a group of faculty members belonging to the college of computer science (

Table 3. List of critical success factors for digital transformation.

S. No.	Factor	Definition
1	Value creation	External orientation for the DT efforts
2	Strategic alignment	DT alignment with vision, mission, and goals
3	Organizational competencies	Competencies that result from DT
4	Steering team	The team for DT initiatives
5	Top management support	Commitment for DT
6	Change management	Efficient and effective change management for DT
7	Implementation framework	Iterative implementation methodologies
8	Technology selection	Big data, cloud, IoT, artificial intelligence, etc.
9	Vendor support	Due diligence of the vendor
10	Consultant support	Advising and guidance in the journey of DT
11	DT benefits realization	Identification, planning, and monitoring of DT benefits
12	DT initialization	Sound business case for DT

). In this workshop, about 50 faculty members were present, and through presentation and group discussion the CSFs of DT were verified and the pairwise relationships were defined. The relationships were defined in the form of ‘V’, ‘A’, ‘X’, and ‘O’ in order to establish the nature of relationships and develop a structural self-interactive matrix (SSIM). This SSIM was used for the traditional approach and the results were demonstrated before the group. Thereafter, in the first experiment, the SSIM was converted into fuzzy SSIM with the highest intensity linguistic scales. The highest intensity scales were chosen to compare the results of the second and third approaches with the traditional ISM approach. The highest intensity scales will lead to comparable input for fuzzy and traditional approaches. The SSIMs for different approaches are given in

Table 4. Structural self-interaction matrix for the traditional approach.

S. No.	12	11	10	9	8	7	6	5	4	3	2	1
1	A	A	A	O	O	O	A	O	A	A	A
2	X	O	O	O	V	O	O	A	A	V
3	A	A	A	A	A	O	A	O	A
4	V	V	V	V	V	V	V	A
5	V	O	O	O	O	O	V
6	A	V	A	A	A	X
7	A	V	A	A	O
8	A	V	X	X
9	A	V	A
10	A	V
11	A
12

,

Table 5. Structural self-interaction matrix for the second fuzzy approach.

S. No.	1	2	3	4	5	6	7	8	9	10	11	12
1	0	0	0	0	0	0	0	0	0	0	0	0
2	1	0	1	0	0	0	0	1	0	0	0	1
3	1	0	0	0	0	0	0	0	0	0	0	0
4	1	1	1	0	0	1	1	1	1	1	1	1
5	0	1	0	1	0	1	0	0	0	0	0	1
6	1	0	1	0	0	0	1	0	0	0	1	0
7	0	0	0	0	0	1	0	0	0	0	1	0
8	0	0	1	0	0	1	0	0	1	1	1	0
9	0	0	1	0	0	1	1	1	0	0	1	0
10	1	0	1	0	0	1	1	1	1	0	1	0
11	1	0	1	0	0	0	0	0	0	0	0	0
12	1	1	1	0	0	1	1	1	1	1	1	0

and

Table 6. Structural self-interaction matrix for the third fuzzy approach.

S. No.	12	11	10	9	8	7	6	5	4	3	2	1
1	A(VH)	A(VH)	A(VH)	O(No)	O(No)	O(No)	A(VH)	O(No)	A(VH)	A(VH)	A(VH)
2	X(VH)	O(No)	O(No)	O(No)	V(VH)	O(No)	O(No)	A(VH)	A(VH)	V(VH)
3	A(VH)	A(VH)	A(VH)	A(VH)	A(VH)	O(No)	A(VH)	O(No)	A(VH)
4	V(VH)	V(VH)	V(VH)	V(VH)	V(VH)	V(VH)	V(VH)	A(VH)
5	V(VH)	O(No)	O(No)	O(No)	O(No)	O(No)	V(VH)
6	A(VH)	V(VH)	A(VH)	A(VH)	A(VH)	X(VH)
7	A(VH)	V(VH)	A(VH)	A(VH)	O(No)
8	A(VH)	V(VH)	X(VH)	X(VH)
9	A(VH)	V(VH)	A(VH)
10	A(VH)	V(VH)
11	A(VH)
12

.

The final ISM models in the second and third fuzzy and traditional approaches were the same (Figure 10). The MICMAC diagrams in the second fuzzy and the traditional approaches were the same as both incorporated transitivity (Figure 11), whereas, in the third fuzzy approach, it is different, as this approach does not incorporate transitivity for fuzzy MICMAC (Figure 12).

The second experiment was conducted with the varied intensity input to compare the results between the second and third fuzzy approaches. Four decision makers (DMs) were simulated to include all scales of intensities such as very high, high, low, and very low for the third approach and, correspondingly, 1, 0.75, 0.5, and 0.25 for the second approach. The nature of relation was not changed, but intensities were reduced for the successive DMs. Therefore, SSIMs for second and third approaches resembled those of Table 5 and Table 6, except for intensities. The SmartISM 2.0 results in experiment 2 in both fuzzy approaches did not change from experiment 1. In the second fuzzy approach, it is observed that, if the nature of relation is not changed and there is equal distribution of all intensities of fuzzy scales from the decision makers, then the results will match with the traditional approach. Conversely, in the third fuzzy approach, it is identified that, due to the mode for aggregation, it is difficult to obtain conclusive results if there is equal distribution of all intensities of fuzzy scales from the decision makers. The SmartISM 2.0 gave similar results to that of experiment 1 for the third approach, since it picked up the first values of high intensity due to the mode and equal chance of each intensity.

Both experiments illustrate that the fuzzy scales give huge flexibility to the DMs and the results converge and improve the confidence in the final model. Moreover, the third approach can be improved by incorporating more advanced methods of aggregation and fuzzy transitive closure to include transitivity for fuzzy MICMAC.

8. Conclusions

As the nature of human decision making is generally vague, ISM and MICMAC applications must be performed utilizing the fuzzy set theory. Fuzzy ISM is an extension of the ISM technique by introducing the fuzzy set theory to incorporate fuzziness in the decision-making process in defining the relationships between the variables. It also allows for the variation of strengths in relationships instead of restricting them to a binary choice of presence and absence. In the traditional ISM approach, the aggregation of decision makers was achieved either through some group decision-making techniques or by taking the mode if the inputs were taken individually. This did not allow for the complete consideration of the input of all decision makers, whereas, in the fuzzy environment, various aggregation techniques allowed for the integration of inputs from all decision makers.

This article has illustrated five approaches of fuzzy ISM and fuzzy MICMAC identified through systematic review. Further, the approaches have been summarized in a roadmap to implement fuzzy ISM and fuzzy MICMAC. The variations in these approaches have also been summarized for making informed decisions to implement fuzzy ISM and fuzzy MICMAC. Further, Microsoft Excel-based SmartISM 2.0 has been developed to incorporate two fuzzy approaches and the results have been reported along with variations from those of the original articles. The system simplifies the implementation of fuzzy ISM and fuzzy MICMAC, as the computational efforts increase in a quadratic manner with the increase in the number of variables. Finally, the case of digital transformation has been studied to analyze different approaches using SmartISM 2.0. The results show that, with a wide flexibility of fuzzy scales, the results converge and improve the confidence in the final model. The findings of the case study will encourage researchers to use fuzzy approaches instead of a traditional approach. Also, the mode approach of aggregation may be modified with more advanced methods, and fuzzy transitive closure may be used to incorporate transitivity in the fuzzy environment.

Future work will develop web-based SmartISM 2.0 software using ASP.Net and VB.Net. Further, more approaches will be identified and incorporated into the system, and comparative analysis will be performed on different approaches. Further, similar to the fuzzy set theory, there are other theories such as interval-valued optimization theory that may also be incorporated into ISM to represent uncertainties in the real world. There is also a need to identify an index to establish consistency in the decision makers’ input in both traditional and fuzzy environments, such as the consistency ratio in the analytic hierarchy process (AHP). The acceptable value of this index will help in solving the problem of the clustering of variables on one point in the MICMAC diagram and the flat hierarchy in the final ISM model, as reported by many of the users.