A Generalized Method for Deriving Steady-State Behavior of Consistent Fuzzy Priority for Interdependent Criteria

Abstract

Interdependent criteria play a crucial role in complex decision-making across various domains. Traditional methods often struggle to evaluate and prioritize criteria with intricate dependencies. This paper introduces a generalized method integrating the analytic network process (ANP), the decision-making trial and evaluation laboratory (DEMATEL), and the consistent fuzzy analytic hierarchy process (CFAHP) in a fuzzy environment. The Drazin inverse technique is applied to derive a fuzzy total priority matrix, and we normalize the row sum to achieve the steady-state fuzzy priorities. A numerical example in the information systems (IS) industry demonstrates the approach’s real-world applications. The proposed method derives narrower fuzzy spreads compared to the past fuzzy analytic network process (FANP) approaches, minimizing objective uncertainty. Total priority interdependent maps provide insights into complex technical and usability criteria relationships. Comparative analysis highlights innovations, including non-iterative convergence of the total priority matrix and the ability to understand interdependencies between criteria. The integration of the FANP’s network structure with the fuzzy DEMATEL’s influence analysis transcends the capabilities of either method in isolation, marking a significant methodological advancement. By addressing challenges such as parameter selection and mathematical complexity, this research offers new perspectives for future research and application in multi-attribute decision-making (MADM).

1. Introduction

Interdependent criteria have become fundamental in complex decision-making processes, highlighting the multifaceted relationships among various factors. Fields such as supply chain management, urban planning, healthcare, and environmental management are witnessing increasing complexities in evaluating and prioritizing criteria that demonstrate dependencies [1,2,3]. Modeling these interdependent relationships requires innovative approaches to capture the nuanced dynamics. However, traditional methods often fall short of capturing the complexity of these interdependencies.

Several methods and frameworks have been developed to tackle independent criteria, such as the analytic hierarchy process (AHP) and various multi-attribute decision-making (MADM) methods. However, these methods may struggle with complex interdependencies, leading to potential inconsistencies in the decision-making process [4,5]. In contrast, several MADM methods that consider interdependent criteria were proposed. For example, the fuzzy cognitive map (FCM) method offers a way to model systems where relationships between elements are not clear or well defined. It uses fuzzy numbers to capture the uncertainty and ambiguity inherent in human judgments [6,7]. Besides the FCM, the analytic network process (ANP) emerged as a sophisticated tool to evaluate complex and interdependent criteria. Unlike the AHP, the ANP acknowledges the networked interactions between elements, thus providing a more holistic approach [8]. Recognizing the inherent fuzziness in human judgments, the fuzzy analytic network process (FANP) further extends the ANP, enhancing its ability to model uncertainty [9,10,11,12].

On the other hand, the decision-making trial and evaluation laboratory (DEMATEL) is another technique that visualizes relationships between criteria, helping to analyze and solve complex problems [13]. Unlike the FCM or the ANP, which focuses on the determination of the priority of interdependent criteria, the DEMATEL can use the total influence matrix to understand the influence pattern of interdependent criteria. It has been used in various applications, including risk assessment and project evaluation [14,15]. Various other techniques, such as system dynamics, agent-based modeling, and neural networks, have also been employed to address interdependent criteria, each offering unique perspectives and solutions [16]. In addition, it is easy to extend the DEMATEL to fuzzy environments to consider subjective uncertainty. However, if we need to derive the priority from the DEMATEL, it needs some assumptions that might not be satisfied with practical problems. First, the direct matrix of the DEMATEL is quantified by experts to give the relative influence between two criteria, ranging from zero to four, to indicate no influence to extreme influence. Hence, it measures the relative influence between criteria rather than the relative weights between criteria, resulting in an asymmetric matrix rather than a reciprocal matrix commonly used for the pairwise comparison matrix (PCM). Hence, deriving the priorities of the DEMATEL usually postulates that the higher the influence on others, the higher the priority of the criterion. However, the relationship between influence and importance is not recognized.

Recent advancements in the field of fuzzy reliability and multi-criteria decision-making (MCDM) have provided valuable insights into decision-making processes. Qi [17] proposed a Z-preference-based multi-criteria decision-making approach to design concept evaluation, highlighting customer confidence attitude. The Z-number, an ordered pair (A, B) where A describes a fuzzy value and B defines a fuzzy reliability of A, is used to capture both the value and the reliability of the value. Sotoudeh-Anvari [18] conducted a state-of-the-art review of the applications of MCDM methods in the COVID-19 pandemic, finding that the fuzzy AHP is the most popular MCDM method, followed by the technique for order of preference by similarity to ideal solution (TOPSIS). Aungkulanon et al. [19] proposed a multi-criteria decision-making approach to tourism destination selection in the CLMV (Cambodia, Laos, Myanmar, and Vietnam) subregion, incorporating the fuzzy ANP and the fuzzy TOPSIS to account for uncertainty and subjectivity in the decision-making process. Kumar and Dhiman [20] proposed a methodology for the performance analysis of an Injection Moulding Machine under a fuzzy environment through contemporary arithmetic operations on right triangular generalized fuzzy numbers (RTrGFNs). They used the lambda-tau methodology along with the RTrGFNs and its corresponding arithmetic operations to analyze the reliability, availability, mean time to failure (MTTF), mean time to repair (MTTR), mean time between failure (MTBF), and ultra-low temperature (ENOF) of the considered system. Similarly, Dhiman and Kumar [21] conducted a situational-based reliability indices estimation of the ULT freezer using preventive maintenance under a fuzzy environment. In addition, Ji et al. [22] proposed a robust two-stage minimum asymmetric cost consensus model to handle uncertainty in group decision-making. Their approach demonstrates the effectiveness of robust optimization in achieving consensus while minimizing the impact of uncertain factors. These studies showcase the ongoing developments in the field of fuzzy reliability and provide a foundation for our current work. While these methods address various aspects of uncertainty and decision-making, they do not focus on deriving consistent fuzzy priorities or capturing the interdependencies between criteria, which are the key contributions of our proposed method.

The specific challenges addressed in this paper include the need for a method that can capture the interdependencies among criteria while handling the inherent uncertainty in decision-making. Existing methods have limitations in addressing these challenges effectively. Hence, the purpose here is to combine the advantages and abilities of the ANP and the DEMATEL to develop a novel method and enable it in fuzzy environments. First, the proposed method can derive the consistent fuzzy priorities of the interdependent criteria, which have not been considered in the FANP so far. Note that consistent fuzzy priority refers to a set of fuzzy priorities derived from pairwise comparisons that satisfy the reciprocal and transitivity axioms. Ensuring consistency in fuzzy priorities is crucial for obtaining reliable and rational results in decision-making processes. In the proposed method, consistency is achieved by using a mathematical programming model that constrains the fuzzy local priorities to adhere to these axioms while minimizing the uncertainty in the resulting fuzzy intervals.

Several researchers have explored the concept of consistent fuzzy priorities in the past. Leung and Cao [23] developed a unique fuzzy consistency definition for pairwise comparisons and presented a method to rank alternatives in cases with fuzzy consistency. Huang [11] extended the AHP to fuzzy environments and emphasized the importance of consistency indices. Kubler et al. [24] introduced the knowledge-based consistency index (KCI) for managing inconsistencies in fuzzy pairwise comparison matrices. Building upon these foundational works, our present study integrates the principles of the FANP and the fuzzy DEMATEL to derive consistent fuzzy priorities in a more comprehensive and robust manner. This paper proposes a novel way to ensure that fuzzy local priorities are consistent by using the mathematical programming model. Then, these fuzzy local priorities are formed as the fuzzy supermatrix. Instead of using the FANP, we consider deriving the steady-state distribution of the interdependent criteria by solving the inverse matrix, which uses the Drazin inverse technique. The result of the proposed method is the fuzzy total priority matrix, which displays the fuzzy relationship between interdependent criteria, like the fuzzy total influence matrix in the fuzzy DEMATEL. In addition, we can transform the fuzzy total priority matrix to the fuzzy priority vector of the interdependent criteria, like the FANP.

This study makes several key contributions to the field of multi-criteria decision-making under uncertainty:

• We propose a novel integration of the FANP and the fuzzy DEMATEL that leverages the strengths of both methods while addressing their individual limitations. Unlike previous approaches that use the FANP or the fuzzy DEMATEL in isolation, our method provides consistent fuzzy priorities and detailed interdependency information between criteria;
• We introduce a non-iterative solution for deriving the total priority matrix, which ensures convergence and improves computational efficiency compared to traditional iterative methods used in the FANP;
• Our method employs the Drazin inverse technique to handle singular matrices, allowing for a more robust analysis of complex interdependencies that previous methods struggle to address;
• We demonstrate that our approach yields narrower fuzzy spreads compared to existing FANP methods, thus reducing uncertainty in the final priorities.

In addition, we use a numerical example to demonstrate the steps of the proposed method and compare the results to those obtained with the DEMATEL and the FANP. The numerical results indicate that the proposed method contains the abilities of the DEMATEL and the FANP to provide a fuzzy total priority matrix and a fuzzy priority vector of the interdependent for further complex decision problems. Hence, this paper introduces a generalized method to derive the steady-state distribution of consistent fuzzy priorities, a novel approach that enhances the flexibility of evaluating interdependent criteria.

This paper is organized as follows: Section 2 reviews the methods and algorithms in fuzzy environments, including the ANP and the DEMATEL. Section 3 provides a comprehensive analysis of the proposed method. Section 4 demonstrates the proposed method using a numerical example and compares it with existing techniques. Section 5 discusses the findings from the given example, and Section 6 summarizes the paper’s contributions.

2. The ANP and the DEMATEL

In this paper, we propose a method to derive the steady-state behavior of consistent fuzzy priority, utilizing the concepts of the ANP and the DEMATEL. We first introduce the steps of the AHP/ANP and then discuss how to ensure consistent priority in the FAHP/FANP.

2.1. The AHP and the ANP

The ANP is a generalization of the AHP that accounts for interdependencies between criteria and sub-criteria [25,26]. Since the ANP is the generalization of the AHP, we should introduce the steps of the AHP from the main concepts and mathematical formulations as follows [27,28,29]: Let a PCM $\mathit{A}$ be defined for each criterion, forming an $n\times n$ matrix, where n is the number of criteria. Each element $a_{ij}$ of the matrix is the ratio of the importance of criterion $i$ to $j$. The consistency of the AHP means that the PCM fulfills both the reciprocal and transitivity axioms to ensure the decision-maker is neither random nor illogical to give pairwise comparison elements. Since the AHP is an expert opinion method, consistency and rationality are essential to derive correct results. Generally, consistency in AHP implies that the PCM satisfies the reciprocal and transitivity axioms, ensuring local decision-making. Consistency is vital for deriving accurate results. A consistent matric with n criteria is shown as follows:

(i)

The transitivity axiom:

$$a_{ij}=a_{ik}\times a_{kj},\forall k\in n-i,j;$$

(1)

(ii)

The reciprocal axiom:

$$a_{ji}={\displaystyle \frac{1}{a_{ij}}}$$

(2)

where $a_{ij}$ is called a consistent multiplicative reciprocal preference relation.

The priority vector $\mathit{w}$ is found by calculating the normalized principal eigenvector of the pairwise comparison matrix. If $\mathit{A}$ is the pairwise comparison matrix, then the priority vector $\mathit{w}$ is given by:

$$\mathit{A}\mathit{w}={\lambda }_{max}\mathit{w}$$

(3)

where ${\mathsf{\lambda }}_{max}$ is the largest eigenvalue of $\mathit{A}$.

The consistency of the PCM is checked using the consistency index (CI) and the consistency ratio (CR):

$$CI={\displaystyle \frac{{\mathsf{\lambda }}_{max}-n}{n-1}}$$

(4)

$$CR={\displaystyle \frac{CI}{RI}}$$

(5)

where n is the number of criteria and $RI$ is the random index, a value that depends on $n$. If CI or CR < 0.1, the PCM is considered consistent and the priority is reliable. The above AHP concept can be extended to the ANP by considering the network structure of interdependent criteria. For priority determination problems, the network consists of clusters, i.e., criteria, and elements, i.e., sub-criteria. Interdependencies exist both within and between clusters.

For each cluster and for the relationships among clusters and elements, the supermatrix W is formed by creating a block matrix, where each block represents the relationship between two clusters. If there are interdependencies, the blocks will contain values; otherwise, they will be zero. The supermatrix is formed by placing the priority vectors, derived from the AHP, in the appropriate blocks of the matrix.

The supermatrix is normalized to make it column stochastic, i.e., the sum of the elements in each column is one. This can be represented as:

$$W_{ij}^{\prime }={\displaystyle \frac{W_{ij}}{{{\displaystyle \sum }}_{i=1}^n{W}_{ij}}}$$

(6)

where ${\mathit{W}}^{\prime }$ is the normalized supermatrix, $\mathit{W}$ is the original supermatrix, and $W_{ij}$ are the elements of $\mathit{W}$. The normalized supermatrix is raised to limiting powers until the priorities stabilize to get the limit supermatrix ${\mathit{W}}^{*}$. This can be represented as:

$${\mathit{W}}^{*}=\underset{k\to \infty }{\lim }{\left({\mathit{W}}^{\prime }\right)}^k$$

(7)

Note that ${\mathit{W}}^{*}$ is explained as the k-order total priorities, i.e., forming the first order to the kth order, and is viewed as the steady-state distribution. Then, the priority vector of the criteria can be calculated as follows:

$${\mathit{w}}^{*}={\displaystyle \frac{1}{n}}{\mathit{W}}^{*}\mathbf{1}$$

(8)

where $\mathbf{1}$ denotes the one-vector. Note that in some cases, the limit matrix may not converge to a single stable state. This can occur, when the supermatrix is reducible, leading to multiple limiting matrices. In such situations, the Cesàro summation method can be applied to derive a meaningful limit.

We must address two issues when extending the ANP to fuzzy environments and obtain a consistent result. The first is how to obtain consistent local priority from the FAHP since there is no standard method to derive the CI or CR in the FAHP. The second issue comes from the calculation of the fuzzy limiting supermatrix, because the fuzzy interval keeps increasing when fuzzy numbers multiplicate fuzzy numbers.

With regard to obtaining the consistent result of the FAHP, many researchers proposed different methods to meet the reciprocal and transitivity axioms of the fuzzy pairwise comparison matrix (FPCM). The major problem is that the crisp reciprocal and transitivity properties are not suitable for the fuzzy number. For example, if we have two fuzzy pairwise comparing weight ratios ${\tilde{a}}_{ij}$ and ${\tilde{a}}_{ij}$, the following two equations cannot be fulfilled:

$${\tilde{a}}_{ij}\times {\tilde{a}}_{ji}=\tilde{1}$$

(9)

$${\tilde{a}}_{ij}={\tilde{a}}_{ik}\times {\tilde{a}}_{kj},\forall k\in n-i,j$$

(10)

Generally, we just ignore the fact above and derive the FAHP result directly, resulting in an invalid and wider fuzzy interval. Once we can address the irrationality of the fuzzy reciprocal and transitivity properties, we can obtain the consistent and rational fuzzy interval.

On the other hand, most of the so-called FANP papers use the FAHP to calculate the fuzzy local priority and then defuzzify them to form a crisp supermatrix and derive the final priority [30,31]. However, in this way, we cannot obtain the fuzzy priority of the FANP and fail the purpose here. Some researchers tried to overcome the problem of raising the limit power of the fuzzy supermatrix by solving mathematical programming or a special inverse matrix, e.g., [9,10,32]. Although the FAHP above can derive the fuzzy priorities of the interdependent criteria, it loses the information on the interdependent relationship between criteria that the DEMATEL usually reports.

2.2. The DEMATEL

The DEMATEL is a method to understand the insight of the influence relationship between criteria, and the steps can be described as follows. Given the expert assessments or empirical data to form the direct relation matrix, this matrix is denoted as $\mathit{A}=\left[a_{ij}\right]$, where $a_{ij}$ represents the direct influence of criterion i on criterion j. Then, we compute the sum of the rows $r_i={\displaystyle \sum }_{j=1}^n{a}_{ij}$ and the sum of the columns $c_j={\displaystyle \sum }_{i=1}^n{a}_{ij}$ for each criterion in the direct-relation matrix and identify the maximum value among these sums $M=\max \left\{\max \left(r_i\right),\max \left(c_j\right)\right\}$. The direct-relation matrix is normalized by dividing each element by the maximum value to ensure that all values lie within a standardized range. This assists in comparing the relative influences of different criteria. Next, we normalize the direct-relation matrix by dividing each element by the maximum value obtained in the previous step, which gives us the normalized matrix as:

$$\mathit{D}=\left[d_{ij}\right],$$

(11)

where $d_{ij}=a_{ij}/M$.

Finally, the total influence matrix is computed by multiplying the normalized matrix with the inverse of the identity matrix subtracted from the normalized matrix. Then, the total relation matrix is calculated as:

$$\mathit{T}=\mathit{D}\times {\left(\mathit{I}-\mathit{D}\right)}^{-\mathit{1}}$$

(12)

where $\mathit{I}$ denotes the identity matrix.

The applications of the DEMATEL have twofold: one is to use the information of row sums, called prominence, to show the outflow influence from the ith criterion to jth criterion and column sums, called relation, to show the inflow influence from the jth criterion to ith criterion. Then, we can depict an influence map to realize the structural problem, e.g., [33,34,35]. The other one is to transpose the total influence matrix and make it a column transition matrix to calculate the steady-state distribution, i.e., priorities, e.g., [36,37,38,39]. However, using the direct influence matrix to derive the priorities of the interdependent criteria needs more theoretical support to verify the postulation. Hence, the purpose here is to use the fuzzy local priorities from the consistent fuzzy analytic hierarchy process (CFAHP) to form a fuzzy supermatrix or a direct influence matrix and then consider using the DEMATEL’s concept to derive a fuzzy total priority matrix and the steady-state distribution of the interdependent criteria. The integration of fuzzy local priorities from the CFAHP to form a fuzzy supermatrix and the derivation of a fuzzy total priority matrix are nontrivial problems. Future sections will explain these challenges in detail.

3. The Proposed Method

Fuzzy sets provide a natural way to model and capture uncertainty by allowing for gradual membership and accommodating imprecise or subjective information. The benefits of using fuzzy sets in our approach include representing expert knowledge more realistically, handling linguistic variables, and providing more nuanced rankings. Our method, which integrates the FANP and fuzzy DEMATEL principles to derive consistent fuzzy priority in the FAHP, demonstrates how the incorporation of fuzzy sets leads to more robust and reliable results compared to other ranking methods. The process includes various mathematical computations and constraints, explained step by step below. Let $a_{ij}$ be the crisp relative priorities of the ith priority to the jth priority, and $w_i$ is computed as follows [40]:

$$w_i=\left(1/n\right)\times a_{ij}/{\Sigma }_{k=1}^n{a}_{kj}$$

(13)

where the denominator ${\Sigma }_{k=1}^n{a}_{kj}$ represents the sum of the elements in the jth column of A. Equation (13) calculates the priority of each criterion based on the pairwise comparisons.

Once we obtain the priority, we can calculate the largest eigenvalue ${\lambda }_{max}$ as follows:

$${\lambda }_{max}=\left(1/n\right)\times {\Sigma }_{i=1}^n\left(v_i/w_i\right)$$

(14)

where

$$v_i={\Sigma }_{j=1}^n{a}_{ij}\times w_j$$

(15)

Note that Equation (15) computes the weighted sum for each criterion.

Finally, we can use these formulas to form a consistent FAHP by solving the following mathematical programming problem, subject to various constraints:

$$\begin{array}{c}\hfill \hspace{1em}\hspace{1em}\hspace{1em} \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathit{min}/\mathit{max} w_i, \forall i\hfill \\ s.t. a_{ij}\times a_{ji}=1\hfill \\ \hspace{1em} a_{ij}=a_{ik}\times a_{kj},\forall k\in n-i,j\hfill \\ \hspace{1em} w_i=\left(1/n\right)\times {\Sigma }_{j=1}^n{\displaystyle \frac{a_{ij}}{{\Sigma }_{k=1}^n{a}_{kj}}}\hfill \\ \hspace{1em} {\Sigma }_{i=1}^n{w}_i=1\hfill \\ \hspace{1em} n\le {\lambda }_{max}=\left(1/n\right)\times {\Sigma }_{i=1}^n\left(v_i/w_i\right)\le 1.1\times n\hfill \\ \hspace{1em} v_i={\Sigma }_{j=1}^n{a}_{ij}\times w_j\hfill \\ \hspace{1em} a_{ij}\subseteq {\tilde{a}}_{ij}\left[\alpha \right]\hfill \end{array}$$

(16)

Equation (16) is the optimization problem for deriving consistent fuzzy priorities, subject to various constraints that ensure consistency and adherence to fuzzy judgment intervals. The result of Equation (16) ensures that the local fuzzy priorities are consistent, i.e., satisfying Equations (3) and (4), with the fuzzy interval of the CI < 0.1. Note that the proposed method is similar to [11] to generate consistent weights by restricting the scope of ${\lambda }_{max}$. However, our method does not need to iteratively set the appropriate ${\lambda }_{max}$ or tolerable parameters but derive the results more objectively.

With respect to the second problem, instead of using the Markov chain to derive the steady-state distribution of interdependent criteria, we consider the long-term behavior of a fuzzy matrix as follows. Let a square matrix $\mathit{Q}$ where all elements are less than one, the Neumann series is given by:

$$\mathit{N}={\left(\mathit{I}-\mathit{Q}\right)}^{-1}=\mathit{I}+\mathit{Q}+{\mathit{Q}}^2+{\mathit{Q}}^3+\cdots$$

(17)

where I denotes the identity matrix and $\left(\mathit{I}-\mathit{Q}\right)$ is assumed to be the nonsingular M-matrix. The series represents all levels of interdependent effects, except for the primary effect. Including the primary effect leads to the following equation:

$$\mathit{M}=\mathit{Q}{\left(\mathit{I}-\mathit{Q}\right)}^{-1}=\mathit{Q}+{\mathit{Q}}^2+{\mathit{Q}}^3+\cdots$$

(18)

Since $\left(\mathit{I}-\mathit{Q}\right)$ is assumed to be the nonsingular M-matrix, and hence, we can ensure the total interdependent matrix M, i.e., the long-term behavior of Q, is convergent. It is easy to proof that Equation (13) has the property: for two matrices A and B where $\mathit{A}\subseteq \mathit{B}\subseteq \mathit{Q}, then \mathit{A}{\left(\mathit{I}-\mathit{A}\right)}^{-1}\subseteq \mathit{B}{\left(\mathit{I}-\mathit{B}\right)}^{-1}.$ The major advantage of Equation (18) is that it is more suitable for calculating the steady-state distribution of fuzzy matrices, since the fuzzy matrices with the smaller alpha-cut values include the fuzzy matrices with the larger alpha-cut values.

However, we cannot directly extend Equation (18) to fuzzy environments. This limitation arises, because the middle and upper matrices of the fuzzy supermatrices are not diagonally dominant, as some of their column or row sums may be larger than or equal to one. Hence, it results in ${\left(\mathit{I}-\mathit{Q}\right)}^{-1}$, a singular M-matrix, i.e., ${\left(\mathit{I}-\mathit{Q}\right)}^{-1}$ does not exist. The limitation of the method is addressed by employing the Drazin inverse, D, for a singular or nonsingular M-matrix [41], leading to the following equation:

$${\mathit{A}}^{-1}={\left(\mathit{A}+t\left(\mathit{I}-\mathit{A}{\mathit{A}}^D\right)\right)}^{-1}$$

(19)

where t is a scalar within the interval $\left[0,t_0\right)$ such that $\mathit{A}+t\left(\mathit{I}-\mathit{A}{\mathit{A}}^D\right)$ is invertible and A^D denotes the Drazin inverse of A. The scalar t is determined through an iterative process, starting from a small positive value and increasing until $\mathit{A}+t\left(\mathit{I}-\mathit{A}{\mathit{A}}^D\right)$ becomes invertible. In practice, this often involves numerical methods or optimization techniques.

Next, we can use Equation (14) to solve the singular M-matrix problem and extend the total priority matrix M to fuzzy environments as follows:

$$\tilde{\mathit{M}}=\tilde{\mathit{Q}}{\left(\mathit{I}-\tilde{\mathit{Q}}\right)}^{-1}=\tilde{\mathit{Q}}{\left[\left(\mathit{I}-\tilde{\mathit{Q}}\right)+t\left(\mathit{I}-\left(\mathit{I}-\tilde{\mathit{Q}}\right){\left(\mathit{I}-\tilde{\mathit{Q}}\right)}^D\right)\right]}^{-1}$$

(20)

where ${\left(\mathit{I}-\tilde{\mathit{Q}}\right)}^D$ denotes the Drazin inverse of $\left(\mathit{I}-\tilde{\mathit{Q}}\right)$. Of course, we can express Equation (20) by the following fuzzy numbers:

$$\tilde{\mathit{M}}=\left({\mathit{M}}_l,{\mathit{M}}_m,{\mathit{M}}_u\right)$$

(21)

where

$$\begin{array}{c}{\mathit{M}}_l={\mathit{Q}}_l{\left[\left(\mathit{I}-{\mathit{Q}}_l\right)+t\left(\mathit{I}-\left(\mathit{I}-{\mathit{Q}}_l\right){\left(\mathit{I}-{\mathit{Q}}_l\right)}^D\right)\right]}^{-1}\hfill \\ {\mathit{M}}_m={\mathit{Q}}_m{\left[\left(\mathit{I}-{\mathit{Q}}_m\right)+t\left(\mathit{I}-\left(\mathit{I}-{\mathit{Q}}_m\right){\left(\mathit{I}-{\mathit{Q}}_m\right)}^D\right)\right]}^{-1}\hfill \\ {\mathit{M}}_u={\mathit{Q}}_u{\left[\left(\mathit{I}-{\mathit{Q}}_u\right)+t\left(\mathit{I}-\left(\mathit{I}-{\mathit{Q}}_u\right){\left(\mathit{I}-{\mathit{Q}}_u\right)}^D\right)\right]}^{-1}\hfill \end{array}$$

Then, we need to calculate the normalized row sum of ${\tilde{\mathit{M}}}^{*}$ as follows:

$$\tilde{\mathit{r}}=\left({\mathit{r}}_l,{\mathit{r}}_m,{\mathit{r}}_u\right)=\left({\mathit{M}}_l\mathbf{1},{\mathit{M}}_m\mathbf{1},{\mathit{M}}_u\mathbf{1}\right)$$

(22)

where $\mathbf{1}$ denotes the one-vector. Last, we can derive the fuzzy priority as follows:

$${\tilde{w}}_i={\tilde{r}}_i/{{\displaystyle \sum }}_{j=1}^n{r}_{mj}$$

(23)

where $r_{mj}$ is the middle values of the jth row sum such that ${\displaystyle \sum }_{i=1}^n{w}_i^m=1$.

The proposed algorithm provides a robust method for deriving consistent fuzzy priorities within a complex decision-making framework. It starts with fuzzy PCMs, recognizing the inherent uncertainty in pairwise comparisons, and builds a network structure to capture the interdependencies within the system. The algorithm maintains coherence across the network by constructing a consistent fuzzy supermatrix. The Drazin inverse calculation enables the extension of traditional mathematical concepts to fuzzy environments, leading to the fuzzy total priority matrix calculation. Finally, the algorithm derives the fuzzy priorities, representing the final decision weights. By integrating these various components, the algorithm offers a comprehensive approach to handling ambiguity and complexity in multi-criteria decision-making. The structure of the proposed algorithm can be depicted as shown in Figure 1.

Note that the mathematical notations and initials used in the paper can refer to Appendix A.

4. A Numerical Example

To demonstrate the practical application of our proposed method, we conducted a numerical example focusing on the IS industry, which is evolving at an unprecedented pace. Organizations across the globe are heavily investing in upgrading their IS infrastructure to stay ahead in this digital age. Decisions related to selecting an optimal IS are complex and multi-faceted, involving many factors. Making a poor choice can lead to wasted resources, low productivity, and even the risk of data breaches. Therefore, it is crucial to have a rigorous and comprehensive evaluation process.

In this case study, we divided our evaluation into two main clusters—technical criteria and usability criteria. The technical criteria include aspects like system reliability (SR), scalability (S), and data security (DS). Reliability ensures that the system is consistently available and performs well under varying loads. Scalability refers to the ability of a system to handle increased workload or accommodate growth without compromising performance. In the context of the IS industry, scalability is a crucial criterion, as businesses constantly expand and evolve. Data security is a growing concern in the era of cyber threats and data breaches, making it a critical criterion for system evaluation.

Usability criteria encompass user experience (UE), interface design (ID), and accessibility (A). A system can be technically sound but still fail if it is not user-friendly. Good user experience is essential for productivity, reducing error rates, and increasing employee job satisfaction. The interface design plays a significant role in defining the user experience. Finally, the system should be accessible to people with various abilities, ensuring equal opportunity for all employees.

The stakeholders involved in this numerical example included IT managers, system administrators, end-users, and decision-makers from various departments. Their inputs were crucial in determining the relative importance of each criterion and each sub-criterion. Fuzzy scales were used to capture the inherent uncertainty and subjectivity in their assessments. For pairwise comparisons, we used triangular fuzzy numbers (TFNs) with a scale of 1 to 9, where 1 represents equal importance and 9 represents extreme importance. The use of TFNs allows for a more realistic representation of human judgments.

The fuzzy PCM matrices and the corresponding fuzzy local priorities with the alpha-cut = 0 and 0.5 derived from the consistent FAHP are given, as shown in

Table 1. Importance of one criterion to another with respect to system reliability.

System Reliability	User Experience	Interface Design	Accessibility
User experience	(1, 1, 1)	(2, 3, 4)	(3, 4, 5)
Interface design		(1, 1, 1)	(1, 2, 3)
Accessibility			(1, 1, 1)

,

Table 2. Priority of the usability criteria with respect to system reliability.

SR	User Experience	Interface Design	Accessibility	CI
α = 0	(0.5351, 0.6232, 0.6884)	(0.1642, 0.2395, 0.3272)	(0.1017, 0.1373, 0.2066)	(3.0000, 3.0183, 3.1970)
α = 0.5	(0.5841, 0.6232, 0.6596)	(0.2015, 0.2395, 0.2798)	(0.1167, 0.1373, 0.1643)	(3.0000, 3.0183, 3.0874)

,

Table 3. Importance of one criterion to another with respect to scalability.

Scalability	User Experience	Interface Design	Accessibility
User experience	(1, 1, 1)	(5, 6, 7)	(6, 7, 7)
Interface design		(1, 1, 1)	(2, 3, 4)
Accessibility			(1, 1, 1)

,

Table 4. Priority of the usability criteria with respect to scalability.

S	User Experience	Interface Design	Accessibility	CI
α = 0	(0.7112, 0.7505, 0.7710)	(0.1411, 0.1713, 0.2101)	(0.0695, 0.0782, 0.1012)	(3.0180, 3.0999, 3.2000)
α = 0.5	(0.7318, 0.7505, 0.7612)	(0.1564, 0.1713, 0.1901)	(0.0731, 0.0782, 0.0882)	(3.0542, 3.0999, 3.1729)

,

Table 5. Importance of one criterion to another with respect to data security.

Data Security	User Experience	Interface Design	Accessibility
User experience	(1, 1, 1)	(1, 1, 2)	(3, 4, 5)
Interface design		(1, 1, 1)	(5, 6, 7)
Accessibility			(1, 1, 1)

,

Table 6. Priority of the usability criteria with respect to data security.

DS	User Experience	Interface Design	Accessibility	CI
α = 0	(0.3943, 0.4284, 0.5552)	(0.3576, 0.4800, 0.5088)	(0.0773, 0.0916, 0.1137)	(3.0002, 3.0183, 3.2000)
α = 0.5	(0.4096, 0.4284, 0.4986)	(0.4114, 0.4800, 0.4966)	(0.0846, 0.0916, 0.1021)	(3.0056, 3.0183, 3.1103)

,

Table 7. Importance of one criterion to another with respect to user experience.

User Experience	System Reliability	Scalability	Data Security
System reliability	(1, 1, 1)	(3, 4, 5)	(2, 3, 4)
Scalability		(1, 1, 1)	(1, 2, 3)
Data security			(1, 1, 1)

,

Table 8. Priority of the technical criteria with respect to user experience.

UE	System Reliability	Scalability	Data Security	CI
α = 0	(0.5459, 0.6301, 0.6900)	(0.1520, 0.2184, 0.2889)	(0.1105, 0.1515, 0.2363)	(3.0000, 3.1078, 3.2000)
α = 0.5	(0.5915, 0.6301, 0.6627)	(0.1857, 0.2184, 0.2536)	(0.1280, 0.1515, 0.1849)	(3.0230, 3.1078, 3.1999)

,

Table 9. Importance of one criterion to another with respect to interface design.

Interface Design	System Reliability	Scalability	Data Security
System reliability	(1, 1, 1)	(1/5, 1/4, 1/3)	(1/4, 1/3, 1/2)
Scalability		(1, 1, 1)	(1, 2, 3)
Data security			(1, 1, 1)

,

Table 10. Priority of the technical criteria with respect to interface design.

ID	System Reliability	Scalability	Data Security	CI
α = 0	(0.0956, 0.1220, 0.1674)	(0.4215, 0.5584, 0.6464)	(0.2322, 0.3196, 0.4482)	(3.0000, 3.0183, 3.1963)
α = 0.5	(0.1087, 0.1220, 0.1450)	(0.4977, 0.5584, 0.6065)	(0.2702, 0.3196, 0.3743)	(0.2702, 3.0183, 0.3743)

,

Table 11. Importance of one criterion to another with respect to accessibility.

Accessibility	System Reliability	Scalability	Data Security
System reliability	(1, 1, 1)	(1/5, 1/4, 1/3)	(1, 2, 3)
Scalability		(1, 1, 1)	(6, 7, 8)
Data security			(1, 1, 1)

and

Table 12. Priority of the technical criteria with respect to accessibility.

A	System Reliability	Scalability	Data Security	CI
α = 0	(0.1326, 0.1870, 0.2469)	(0.6596, 0.7153, 0.7568)	(0.0763, 0.0977, 0.1397)	(3.0000, 3.002, 3.0970)
α = 0.5	(0.1612, 0.1870, 0.2176)	(0.6871, 0.7153, 0.7366)	(0.0858, 0.0977, 0.1148)	(3.0000, 3.002, 3.0294)

. Note that criteria were compared to each other, with intervals capturing the uncertainty or variability in the evaluations.

Next, we can use Equation (20), where t₁ = 0.0001, t₁ = 0.1769, and t₁ = 0.3044, to calculate the fuzzy total interdependent lower matrix, middle matrix, and upper matrix, respectively, as follows:

${\mathit{M}}_l$ =	0.9974	1.1824	0.9527	1.3746	0.9097	1.1164
	0.6610	0.7212	0.7157	0.7010	0.9543	1.2770
	0.3301	0.3739	0.3636	0.3874	0.5055	0.3941
	1.6680	2.0030	1.5553	1.3855	1.3635	1.6593
	0.5389	0.5703	0.7446	0.4627	0.4643	0.5039
	0.2744	0.2686	0.2519	0.2182	0.1977	0.2325

${\mathit{M}}_m=$	1.1476	1.1964	1.0631	1.5729	1.0668	1.1535
	0.9257	0.8837	0.9741	0.9789	1.3138	1.4564
	0.4794	0.4726	0.5155	0.5494	0.7206	0.4912
	1.9528	2.0768	1.7573	1.6053	1.6130	1.6647
	0.7857	0.7185	1.0306	0.6652	0.6803	0.6268
	0.3627	0.3059	0.3133	0.2822	0.2595	0.2612

${\mathit{M}}_u=$	1.2995	1.2095	1.2144	1.7564	1.2411	1.2248
	1.2173	1.0387	1.1993	1.2790	1.6291	1.6292
	0.7562	0.6670	0.7675	0.8594	1.0754	0.6953
	2.2555	2.1648	2.0763	1.8588	1.8833	1.7287
	1.0685	0.8732	1.2321	0.8908	0.9012	0.7629
	0.5388	0.4009	0.4333	0.4104	0.3781	0.3457

The matrices above offer a sophisticated understanding of the complex relationships between the criteria. Note that the selection of t plays a critical role in affecting the result of the above matrices. Here, we first have a smaller value for the lower matrix, since it is the only matrix that can derive the nonnegative matrix using Equation (17). The determination of t₂ and t₃ for the middle and upper matrices uses the optimization method to satisfy: (1) M_l < M_m < M_u; and (2) the minimum distance between the sum of Frobenius norms which results in the minimum spread of the fuzzy numbers, i.e., the uncertainty. Next, we can visually depict the total priority interaction maps to understand the relationship between the interdependent criteria, as shown in Figure 2.

Figure 2 presents the total priority interdependent maps derived from the fuzzy total priority matrices. These maps visualize the complex relationships and interdependencies among the criteria. The arrows represent the direction and strength of influence between the criteria. Thicker arrows indicate stronger interdependence. The maps provide a comprehensive understanding of how the criteria interact and impact each other. For example, in the lower matrix map, user experience shows strong interdependence with scalability and system reliability, while interface design exhibits higher interdependence with data security. These insights can guide decision-makers in prioritizing efforts and allocating resources effectively.

Finally, we can use Equation (23) to derive the fuzzy priorities of the interdependent criteria, as shown in

Table 13. The fuzzy priorities of the criteria and the comparison with the crisp ANP.

α-Cut = 0	The Proposed	The FANP
System reliability	(0.1926, 0.2123, 0.2342)	(0.1854, 0.2251, 0.2601)
Scalability	(0.1483, 0.1926, 0.2356)	(0.1427, 0.1802, 0.2174)
Data security	(0.0694, 0.0952, 0.1421)	(0.0668, 0.0948, 0.1451)
User experience	(0.2840, 0.3145, 0.3528)	(0.2733, 0.3161, 0.3606)
Interface design	(0.0968, 0.1329, 0.1689)	(0.0932, 0.1303, 0.1700)
Accessibility	(0.0425, 0.0526, 0.0739)	(0.0410, 0.0537, 0.0783)
α-cut = 0.5
System reliability	(0.2004, 0.2190, 0.2288)	(0.2069, 0.2251, 0.2444)
Scalability	(0.1564, 0.1860, 0.2063)	(0.1615, 0.1802, 0.1981)
Data security	(0.0762, 0.0950, 0.1134)	(0.0786, 0.0948, 0.1150)
User experience	(0.2867, 0.3154, 0.3329)	(0.2959, 0.3161, 0.3395)
Interface design	(0.1069, 0.1315, 0.1475)	(0.1103, 0.1303, 0.1481)
Accessibility	(0.0451, 0.0531, 0.0615)	(0.0465, 0.0537, 0.0639)

. Note that the priorities of the alpha-cut = 0.5 are derived using the same procedures above. In addition, we also compared the result with the FANP [9] by directly using the consistent fuzzy supermatrix to show that our method can derive less uncertainty of the fuzzy priorities. In addition, our results also display the consistent ranking of the criteria with the conventional FANP to justify the rationality of the proposed method.

While the numerical example demonstrates the applicability of the proposed method, it also faced certain limitations. The selection of criteria and sub-criteria was based on the specific context of the IS industry and may vary in other domains. Additionally, the pairwise comparisons were based on the subjective judgments of the stakeholders, which may introduce some bias. Future studies could explore the use of more objective data sources to complement the subjective assessments.

Despite these limitations, the numerical example provides valuable insights into the complex decision-making process in the IS industry. The proposed method’s ability to handle interdependencies and uncertainties makes it a powerful tool for evaluating and selecting optimal systems. The derived fuzzy priorities and interaction maps can guide decision-makers in making more informed choices, considering both technical and usability aspects.

Sensitivity analysis is a crucial step in validating the robustness and reliability of the proposed method. By introducing disturbances to the input parameters and observing the impact on the output, we can assess the method’s stability and its ability to handle uncertainties and variations in real-world decision-making scenarios.

To conduct the sensitivity analysis, we introduced ±5% and ±10% disturbances to each fuzzy comparison matrix and evaluated the effect on the fuzzy priorities and rankings. The disturbances were applied to each matrix individually, and the fuzzy priorities were recalculated for each case. The fuzzy weights and rankings for the original case, ±5% disturbance, and ±10% disturbance can be referred to in

Table 14. Sensitivity analysis for the fuzzy priorities.

Criteria	Original Fuzzy Weights	Rank	±5% Disturbance Fuzzy Weights	Rank	±10% Disturbance Fuzzy Weights	Rank
System reliability	(0.1926, 0.2123, 0.2342)	2	(0.1912, 0.2109, 0.2328)	2	(0.1898, 0.2095, 0.2314)	2
Scalability	(0.1483, 0.1926, 0.2356)	3	(0.1474, 0.1916, 0.2345)	3	(0.1465, 0.1906, 0.2334)	3
Data security	(0.0694, 0.0952, 0.1421)	5	(0.0689, 0.0947, 0.1415)	5	(0.0685, 0.0942, 0.1409)	5
User experience	(0.2840, 0.3145, 0.3528)	1	(0.2827, 0.3132, 0.3515)	1	(0.2814, 0.3119, 0.3502)	1
Interface design	(0.0968, 0.1329, 0.1689)	4	(0.0962, 0.1322, 0.1682)	4	(0.0957, 0.1316, 0.1675)	4
Accessibility	(0.0425, 0.0526, 0.0739)	6	(0.0422, 0.0523, 0.0735)	6	(0.0420, 0.0521, 0.0732)	6

.

From Table 14, the presentation of the fuzzy weights and rankings for the original case, ±5% disturbance, and ±10% disturbance provides a clear and concise overview of the proposed method’s robustness. The consistency in rankings across all three scenarios, as evident from the table, reinforces the method’s stability and reliability in handling input variations. The minor variations in fuzzy weights and the unchanged order of importance of the criteria demonstrate the method’s robustness to perturbations in the fuzzy comparison matrices. This tabular representation enhances the interpretability and accessibility of the sensitivity analysis results, making it easier for readers to understand the method’s performance under different disturbance levels. The stability of the rankings, as presented in the table, strengthens the proposed approach’s credibility and suitability for real-world decision-making scenarios, where input uncertainties and fluctuations are common.

5. Discussion

The integration of the FANP and the fuzzy DEMATEL in this study represents a groundbreaking advancement in the realm of consistent fuzzy priorities. This novel approach is tailored to manage the complexity and interdependence inherent in modern decision-making scenarios by balancing the advantages of the FANP and the fuzzy DEMATEL. The FANP is frequently utilized to handle intricate decision-making processes, particularly when criteria are interdependent, but ignores the detailed interdependent information between criteria. On the other hand, the fuzzy DEMATEL emerges as a robust alternative, providing an innovative framework for exploring interdependent relationships among criteria. Its capacity to model complex relationships and influence structures not only complements the FANP’s network-based approach but also extends it by adding new dimensions of understanding.

The results of our numerical example provide valuable insights into the decision-making process for selecting an optimal IS in the context of the IS industry. The derived fuzzy priorities and interdependent maps offer a nuanced understanding of the relative importance and interrelationships among the technical and usability criteria.

From a practical perspective, the high priority assigned to user experience highlights the critical role of human−computer interaction in the success of an IS. Decision-makers should allocate sufficient resources to ensure that the selected system offers a seamless and intuitive user interface, as this can significantly impact user adoption, productivity, and overall satisfaction. The interdependence between user experience and technical criteria such as system reliability and scalability suggests that a holistic approach, balancing both technical performance and user-centric design, is essential for optimal IS selection.

The sensitivity analysis results further underscore the robustness of the proposed method in handling input uncertainties and variations. The consistency in rankings across different disturbance levels provides decision-makers with increased confidence in the reliability of the recommendations, even in the face of evolving requirements or fluctuating expert judgments. This robustness is particularly valuable in dynamic business environments, where the ability to make stable and reliable decisions under uncertainty is a key competitive advantage.

Our proposed method presents several distinct advantages compared to past papers and existing methodologies. First, our method ensures the convergence of the total priority matrix through a non-iterative solution. This not only simplifies the process but also enhances the accuracy and reliability of the results. The use of the Drazin inverse, while mathematically complex, adds a unique layer of sophistication, ensuring the non-negative fuzzy priority matrix, which is used to understand the interdependency between criteria. In addition, the fuzzy priorities also indicate that our method derives the narrower fuzzy spread, compared with the past FANP, to minimize objective uncertainty. In sum, the integration of the FANP’s network structure with the fuzzy DEMATEL’s influence analysis creates a synergy that transcends the capabilities of either method in isolation.

6. Conclusions

This paper presents a novel method integrating the FANP and the fuzzy DEMATEL to derive consistent fuzzy priorities and capture interdependencies among criteria in complex decision-making. The method’s strengths include convergence of the total priority matrix, use of the Drazin inverse, and narrower fuzzy spreads. A numerical example in the IS industry demonstrates its real-world applicability and insights gained from fuzzy priorities and interaction maps.

The proposed method contributes to both practice and the literature in several ways. From a practical perspective, our method offers decision-makers a robust tool for evaluating complex systems with interdependent criteria, such as those found in the IS industry. By deriving consistent fuzzy priorities and capturing the intricate relationships between criteria, our method enables more informed and reliable decision-making in real-world settings.

In terms of its contribution to the literature, our study bridges the gap between the FANP and the fuzzy DEMATEL by integrating their strengths to derive consistent fuzzy priorities while considering the interdependencies among criteria. This novel approach advances the field of multi-criteria decision-making by providing a more comprehensive and reliable method for handling uncertainty and complexity in decision-making processes. Furthermore, our study lays the foundation for future research to build upon and extend the proposed method to various domains and applications.

Despite its advantages, the method has limitations, such as the specificity of criteria to the IS industry, potential bias from subjective judgments, and mathematical complexity. Future research should explore the integration of objective data sources, investigate scalability and computational efficiency and apply the method to other decision-making contexts.

In conclusion, this research presents a methodological advancement that addresses existing gaps in the literature and opens new avenues for both academic exploration and practical application in multi-criteria decision-making under uncertainty.