Modelling the interdependent relationships among epidemic antecedents using fuzzy multiple attribute decision making (F-MADM) approaches

2.1 Overview of the dengue epidemic

An epidemic is an event described by aspects such as fear and sudden widespread death. Its episodic quality elicits an immediate and widespread response [91]. The deadliest epidemics that affected the population at a global scale include Covid-19, Acquired Immune Deficiency Syndrome (AIDS), Severe Acute Respiratory Syndrome (SARS), Black Death, Small Pox, and Tuberculosis, among others [15, 130]. Likewise, Dengue Fever is among such epidemics that shares similar implications on societal fear of widespread death. Dengue is an acute systemic viral disease that has established itself globally in both endemic and epidemic transmission cycles [14, 68]. It was first described in the 1780s and is said to flourish in less economically productive urban areas, suburbs, and the countryside [119]. In that respect, it also affects affluent neighborhoods especially in tropical and subtropical countries [68, 120]. As reported by the World Health Organization (WHO), dengue is endemic to more than 100 countries in Africa, the Americas, the Eastern Mediterranean, South-East Asia, and the Western Pacific. As a consequence of its potential as a threat to public health, dengue is considered a critical contributor that increases systemic social risks. Scholars estimate dengue infections per year to be at 390 million using cartographic approaches [14]. Moreover, in the Americas alone 2.35 million cases of dengue were reported in 2015, of which 10,200 cases were diagnosed as severe causing 1,181 death cases [120].

In the Philippines, a developing country known to have the highest occurrences of dengue incidents in Southeast Asia, the dengue epidemic is one that significantly affects its stature concerning economic, health, and social performance [24, 26, 116]. A recent report in the period of January 1 – March 11, 2018 indicates that a total of 20,108 dengue cases were detected nationwide. Although such statistics are 25.59% lower compared to the same time period in 2017 (i.e. with a trend declining from 2010), the morbidity rate still exhibits an alarming level which has contributed to out-of-pocket spending of the government as well as some political and social concerns that are derived as a result of the country's aim to mitigate the severity of the epidemic [27]. One of the most controversial issues associated with initiatives in mitigating the epidemic is the halting of a dengue immunization campaign of the world's first approved dengue vaccine, Sanofi Pasteurs Dengvaxia, after post-market testing confirmed that it could increase the risk of severe dengue in people who have never been exposed to the virus [29]. To this end, the vaccine has become the center of a significant social and political issue in the Philippines that has still been debated up to the present. Moreover, the issue has affected not only the Philippines but also other nations across the globe after the recommendation of the WHO for additional testing of the vaccine due to safety concerns [74].

2.2 Current works in the study of dengue

As a critical factor affecting society at a global scale, it is crucial that the dengue epidemic be addressed in both domain and relevant literature, as well as in practice. Such a problem is widely tackled by several scholars in the extant literature from different perspectives. For instance, [25] generated a panel of human monoclonal antibodies to dengue virus and proposed implications that may be used for future vaccine design. Similarly, [108] reviewed current techniques and future strategies of diagnostic methods for confirming dengue infection and argued that newly established methods must be standardized to maintain high-quality laboratory performance. Moreover, [119] discussed some of the crucial lessons learned in the literature and practice about dengue incidents, as well as its history since 265–420 AD during the Jin dynasty of China up to the incident that occurred in the Philippines in 2015. Also, [70] investigated the potential effects of global climate change on human health, particularly on vector-borne diseases such as dengue fever. With such published scholarly articles, it can be safely inferred that the problem of the dengue epidemic is widely embodied in the literature particularly in the scientific, medical, and health perspectives.

Several scholars in the literature investigated factors that potentially influence the occurrence of dengue incidents. For instance, [28] explored characterization of mosquito breeding in urban hotspot areas and found that larvae density is related to factors such as frequency, intensity, and duration of epidemic variation which pertain to the mosquito habitat's environmental characteristics. Moreover, [64] pointed out the role of socio-economic status as one of the main antecedents of arboviral diseases (e.g. dengue). The paper argues that poverty may lead to other antecedents of dengue incidents such as inadequate vector control and prevention programs, sub-optimal health infrastructures, and the creation of environmental conditions that promote mosquito breeding. Furthermore, the paper laid out several other antecedents that may potentially interrelate with other antecedents such as climatic conditions and climate change, and man-related activities, among others. Likewise, [30] and [127] pointed out the role of community involvement and other social antecedents as highly influential to the prevention of dengue outbreaks in a community. A summary of previous works is presented in Table 1.

While works in the literature have explored different factors affecting dengue incidents, they focused on a very contained view of the factors affecting the epidemic. For instance, [127] only considered social antecedents of the dengue epidemic while [57] considered only climate and mosquito density as factors of the disease transmission. As such, no work is given to understanding how the different factors interplay with one another in a more holistic perspective. In other words, very limited attention is given to system dynamics or cybernetics approach in modelling the antecedents of the dengue epidemic. While studying the epidemic in a reductionist approach is important to develop a more fundamental understanding of it, viewing the problem in a holistic perspective is also as important to take into account the complexity of the system when making predictions. [98] pointed out in their review that with the growing interest of the literature in predicting disease dynamics, understanding the interplay of the different factors (e.g. climate and socioeconomic antecedents) would be crucial. Several scholars (e.g. [6, 64, 98, 126]) collectively agree on the importance of the development of an integrated approach in modelling the dynamics of infectious diseases. Similarly, [73] reiterated the WHO's movement for an Integrated Vector Management (IVM), which implies the development of an integrated approach in tackling the disease by incorporating key elements such as socio-economic, environmental, and other factors. Despite the importance of such an integrated approach, limited attention is given by scholars in the literature to analyzing the interplay between the different antecedents of dengue in a holistic perspective. As it currently stands, the problem remains a significant gap in the literature.

2.3 Antecedents of the dengue epidemic

In the literature, factors that influence the occurrence of an event/phenomenon/process are termed as antecedents, drivers, or causes. Throughout the paper, no distinction between the terms is made and they are used interchangeably. Studies on the dengue epidemic have provided results that emphasize the roles of different antecedents in influencing the transmission and spread of dengue in a community [6]. For instance, [57] investigated the role of climate and mosquito density as antecedents of dengue fever incidents in China. [105] studied how socio-demographic factors, socio-cultural factors, knowledge, attitudes, practices, environmental characteristics, and caregiver characteristics affect the risk factors of dengue hemorrhagic fever incidents. Likewise, [109] investigated how community empowerment influences the success for developing dengue vector control. Moreover, [64] explored the effect of socio-economic factors (e.g. poverty, resources) to the occurrence of arboviral diseases such as dengue. Clearly, the antecedents of the dengue epidemic play a crucial role in developing useful models and frameworks. A list of antecedents of the dengue epidemic extracted from peer-reviewed sources is presented in Table 2.

With the existence of numerous dengue epidemic antecedents studied in the literature, an overarching categorization that would span a set of antecedent would be desirable. For example, poverty and limited resources would be classified as economic antecedents. Likewise, Formation of open stagnant clean water in household proximity, and climatic factors would be classified as environmental antecedents. Although no formal consensus has been arrived at in the literature on how these antecedents are categorized, some scholars have termed a classification for some of the antecedents. For instance, [64] classified poverty, inadequate vector control, suboptimal healthcare infrastructure, differences in sovereignty between the various islands and archipelagos, under-developed water distribution system, and under-developed waste management system as socio-economic antecedents. Moreover, [49] used the classifications surveillance factors, environmental and socio-economic factors, and climatic factors in grouping the drivers used in analyzing autochthonous transmission in Southern France. Despite the adoption of such classification by some scholars, without an agreed framework on the classification of the antecedent, a gap in the literature remains concerning the characterization of the dengue epidemic antecedents.

2.4 Modelling the interdependence of antecedents in a structural modelling framework

Aside from the literature of dengue epidemic, the study involving antecedents has been invaluable in other fields as well. Similar to the field of dengue epidemic, antecedents are considered as concepts that influence the occurrence of a particular phenomenon. For instance, [66] studied the antecedents of secondhand clothing purchasing behavior of millennials in a marketing perspective. [79] studied the antecedents of using green technologies in an innovation point of view. [75] studied the adoption antecedents to Industry 4.0 in a technology management perspective. Moreover, [100] studied antecedents in a sustainable supply chain management application. Clearly, the study of antecedents is a crucial undertaking in many domains.

It has been found in several results in the literature that antecedents exhibit an interdependent structure. In other words, antecedents usually influence one another in causing a phenomenon/process/event to occur. Such results have been observed in many areas such as in green supply chain [62], environmental sustainability [40], green lean practices [103], mobile banking [102], green workforce [69], and technology transfer [86], among others. The interdependence arising from the interplay of the antecedents are modelled using structural modelling techniques such as the interpretative structural model (ISM), decision making trial and evaluation laboratory (DEMATEL), and fuzzy cognitive mapping (FCM), among others. For instance, [66] used ISM in analyzing the hierarchical interdependence of the secondhand clothing buying behavior antecedents. [125] used DEMATEL in finding cause-effect chains between antecedents affecting college students’ usage intention of green public welfare activity platforms. [60] used FCM in analyzing the impact of social factors on homelessness.

While ISM, DEMATEL, and FCM can model interdependence between antecedents, the decision concerning which of the structural models to use depends on the insights that decision-makers want to obtain from the model. ISM is suitable for modelling the hierarchical structure of the antecedents [66]. As such, it can be used to determine which of the antecedents are the primary drivers or enablers [2]. With this, [2] used ISM to find the root cause of a quality problem due to its capability to reveal the hierarchical nature of antecedents. On the contrary, DEMATEL is suitable for problems in which decision-makers are interested in characterizing the antecedents into either a cause or an effect [125]. Since antecedents exhibit interdependence, DEMATEL is capable of analyzing the cause and effect chains that exist between them [125]. Furthermore, [10] showed how the DEMATEL is used to characterize barriers to implementing industrial symbiosis networks into a cause group and effect group as well as showing which of the antecedents are the critical factors. Finally, FCM is suitable for modelling problems in which decision-makers are interested in knowing how two antecedents (i.e. how strongly one concept causes another concept and if the antecedents are directly or inversely related) [80, 86]. As such, [81] adopted FCM to simulate how a system evolves over time in a terrorism and insurgency context. Therefore, the choice of which structural model technique to use depends on the insights that decision-makers are interested in obtaining.

Figure 1

Flow of the research process. The rectangular blocks represent a process to be executed. The circular blocks represent actors for a process. The unboxed words on the right represent the outputs produced by the preceding process. Such outputs will be used by their succeeding process.

3.1 Case background

In tropical and sub-tropical countries, mosquito breeding is found to be abundant and significant due to favorable environmental characteristics such as temperature, and wind velocity, among others [18, 28]. With this, mosquito-borne diseases such as dengue warrant serious health issues in the global landscape [23]. Dengue incidents are prevalent in many Asian countries, particularly the Philippines [18, 23]. In the Philippines, dengue is one of the more severe health issues, which causes exasperating morbidity and mortality to its population. The country maintains its position of having the highest occurrences of dengue incidents in Southeast Asia [26, 27, 116]. Despite the country's alarming health position, dengue contains to be less studied in the country's epidemiological, environmental, social, and even public health domains, making it one of the country's less understood health issues [18]. [18] maintained that only a few published works tackle the issue in a Philippine context and that most of these works are based on only one community (i.e. Manila). Hence, they argued that more studies would have to be conducted in different knowledge domains as well as across different regions of the country. Such approach would enable the development of a holistic perspective both for scientific understanding and policy-making.

Cebu, a province located in the Visayas, is the second largest urban area of the Philippines, next to Manila (i.e. the capital and principal city of the Philippines). It is the second most populated area in the country with a population count of 4.633 million as of the year 2015 [84]. Metropolitan Cebu or Metro Cebu, Cebus main urban center, accounts for the 20% of the land area and 61.5% of the population resulting in a high population density of 2700 per square kilometer. Because high population density is related to increasing dengue incidents [18, 23, 28] Metro Cebu may be highly susceptible to the epidemic. The Department of Health in Central Visayas (DOH-7) recorded a total of 6097 dengue cases with 37 deaths from January 1 to March 30 of 2019 [37]. These statistics are enough to manifest the issue of dengue incidents in the area. There are several perspectives upon which such issues can be looked at: (1) epidemiological [18], (2) sociological [107], and (3) ecological and environmental [106], among others. This paper explores the issue with a mix of (2) and (3). The research process used in this study is presented in Figure 1.

To carry out the analysis, a literature review is first conducted to extract antecedents that potentially influence the severity of a dengue epidemic. Scopus is used as the database for searching relevant articles. As a result, 43 antecedents as presented in Table 2 in Section 2.3 and are extracted from 18 research articles to qualify as an initial list of antecedents influencing the severity of the dengue epidemic. To make the analysis more convenient and easy to understand, this paper proposes four classifications for grouping the antecedents – (i) social, (ii) economic, (iii) environmental, and (iv) institutional. Social antecedents pertain to antecedents brought about by human activities (individually or as a group) that influence the occurrence of dengue incidents [90]. Economic antecedents pertain to antecedents measured in terms of economic indices such as employment and monetary gains, among others [41]. Environmental antecedents pertain to environmental conditions influencing dengue incidents such as climate, weather, and temperature, among others [64]. Institutional antecedents pertain to administrative, political, governmental, and regulatory antecedents, among others, which influence the occurrence of dengue incidents.

A total of 15 expert decision-makers with at least 10 years of experience in studying or handling dengue cases are then adopted as respondents of the study. The expert decision-makers consist of 2 biologists, 3 medical technologists, 2 physicians, 3 environmentalists, and 5 public health practitioners. The experts’ judgment will be used to: (a) reduce the dimensionality of the initial set of drivers to a more comprehensive size, and (b) to rate the relationship between the antecedents. A supervised survey interview with each respondent is conducted to obtain the required data. To give respondents enough time and convenience in answering the survey, they were given one week before the data was retrieved.

3.2 Fuzzy Best-Worst Method (BWM)

In a decision-making environment where several involved criteria impose complexity to the decision-making process, multiple criteria decision-making (MCDM) approaches can be very suitable and useful. Scholars have adopted several MCDM approaches in the literature. The Analytic Hierarchy Process (AHP) developed by [93] can be considered as the most prominent among such approaches [87]. It has been widely applied in different domains. For instance, [52] used AHP in conjunction with a novel index-based methodology for determining flood-prone areas. Likewise, [111] used the AHP in investigating the lean-green implementation practices of Indian small and medium enterprises (SMEs). Moreover, [38] used the AHP in exploring the maintenance policy selection of naval ships. Aside from AHP, several MCDM methods are widely used in the literature such as the Analytic Network Process (ANP) [94], Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) [114], ELimination Et Choix Traduisant la REalité (ELECTRE) [92], VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) [78], and Preference Ranking Organization METhods for Enrichment Evaluations (PROMETHEE) [61]. The approaches above are primarily based on the pairwise comparison in obtaining relative preferences of actions. On the contrary, such pairwise comparison methods can be susceptible to lack of consistency of the pairwise comparison matrices as observed in practice due to the unstructured way comparisons are executed by pairwise comparison-based methods [45, 87].

To overcome such a concern, a new MCDM approach that uses a different execution of pairwise comparison called the Best-Worst Method (BWM) is developed in the literature [87]. The BWM is a novel MCDM approach developed by [87] for comparing several different alternatives by dividing the pairwise comparisons into two main categories: (1) reference comparisons, and (2) secondary comparisons. As such, it derives the weights based on the pairwise comparison of the best and the worst criteria (alternatives) with the other criteria (alternatives). Several scholars have found essential applications of the BWM. For instance, [89] applied the BWM to a supplier selection life cycle approach that integrates traditional and environmental criteria. Similarly, [43] adopted the BWM in conjunction with fuzzy TOPSIS in the supplier selection among SMEs based on their green innovation ability. Pursuing this further, [42] extended the BWM to the Fuzzy BWM utilizing the fuzzy reference comparisons between the best and worst criteria to generate fuzzy weights to treat the vagueness, ambiguity, and imprecision that frequently arise with the qualitative judgment of decision-makers. Interestingly, [42] found that aside from being able to obtain a reasonable preference ranking for criteria (alternatives) the fuzzy BWM (F-BWM) also exhibits a higher consistency ratio than the non-fuzzy BWM. Despite being relatively new compared to the non-fuzzy BWM, the applications of the F-BWM is growing. For instance, [44] applied the F-BWM for combining individual and group decision-making which enables trade-off between democratic and autocratic decision-making styles. Likewise, [77] adopted a combination of the Fuzzy TOPSIS and F-BWM in finding the optimal combination of power plants alternatives. Comparably, [112] adopted a combination of the F-BWM and other FMCDM methods in developing an approach for performance evaluation.

Several works in BWM and FBWM have been discussed in domain and relevant literatures such as [88], [4], and [124], among others. These works have adopted the use of BWM and FBWM in a group decision-making environment where weights are established into concepts such as factors, antecedents, barriers, and indicators, among others. In this paper, an F-BWM is adopted to screen out relevant factors that qualify as socio-economic and socio-environmental antecedents of dengue incidents in the Philippine context using expert knowledge while treating the inherent ambiguity, imprecision, and vagueness of the human decision-making process. The algorithm used in this paper is adopted from [42] being the most cited work on F-BWM. As such, the steps in performing an F-BWM are as follows:

1. Step 1. Establish the antecedent set to be reduced. The antecedent set C is the set of antecedents to be analyzed. For n antecedents, we have C = {c₁, c₂, . . , c_n}.

2. Step 2. Determine the best (most important) antecedent and the worst (least important) antecedent. The decision-makers must select the best and the worst antecedents from the antecedent set C. We denote the best antecedent as c_B and the worst antecedent as c_W.

3. Step 3. Compare the best antecedent to the other antecedents. In this step, decision-makers compare the importance of the best antecedent c_B compared to antecedent c_j using the linguistic scale provided in Table 3. By repeating the comparison for each antecedent, we obtain the Best-to-Others vector in Equation (1) as follows.

   (1) A˜B=(a˜B1,a˜B2,…,a˜Bn),

   where Ã_B represents the fuzzy Best-to-Others vector; ã_Bj represents the fuzzy preference of the best antecedent c_B over antecedent c_j for j ∈ {1, 2, . . . , n}. We note that the preference of the best antecedent to itself is equally important, hence, ã_BB = (1, 1, 1).

4. Step 4. Compare the other antecedents to the worst antecedent. In this step, each antecedent c_j for j ∈ {1, 2, . . . , n} is compared to the worst antecedent c_W using the linguistic ratings in Table 3. As a result, we obtain the Others-to-Worst vector in Equation (2)

   (2) A˜W=(a˜1W,a˜2W,…,a˜nW),

   where Ã_W represents the Others-to-Worst vector, ã_jW represents the fuzzy preference of antecedent c_j for j ∈ {1, 2, . . . , n} over the worst antecedent c_W. As such, ã_WW = (1, 1, 1).

5. Step 5. Determine the optimal fuzzy weights. In this step, the importance of each antecedent c_j for j ∈ {1, 2, . . . , n} is calculated. Such importance value is represented using a fuzzy weight W˜ . In order to obtain an optimal weight assignment to each antecedent, a nonlinear optimization model presented in Equation (3), using the vectors Ã_B and Ã_W as parameters, is formulated. The optimal weights W˜∗=(w˜1∗,w˜2∗,…,w˜n∗) can then be obtained as a result. For a deeper discussion, the reader is referred to [42]. Considering l^ξ ≤ m^ξ ≤ u^ξ, it is supposed that ξ˜∗=(k∗,k∗,k∗) where k^* ≤ l^ξ. The nonlinearly constrained optimization problem is formulated as follows:

   (3) Min ξ˜∗subject to:|(lBw,mBw,uBw)(ljw,mjw,ujw)−(lBjw,mBjw,uBjw)|≤(k∗,k∗,k∗)|(ljw,mjw,ujw)(lWw,mWw,uWw)−(ljWw,mjWw,ujWw)|≤(k∗,k∗,k∗)∑j=1nR(w˜j)=1ljw≤mjw≤ujwljw≥0j=1,2,…,n

   The resulting optimal fuzzy weights (w˜1∗,w˜2∗,…,w˜n∗) with w˜j∗=(lj∗,mj∗,uj∗) will then be defuzzified to obtain the crisp weights W∗=(w1∗,w2∗,…,wn∗) of the antecedents [35]. In this paper, the defuzzification approach to be adopted is the graded mean integration representation (GMIR) [42].

3.3 Fuzzy decision-making trial and evaluation laboratory (DEMATEL)

The Decision-making Trial and Evaluation Laboratory (DEMATEL) is a structural modeling approach to multiple criteria decision making. It is developed by the Science and Human Affairs Program of the Battelle Memorial Institute of Geneva between 1972 and 1976 to solve and research the complicated, intertwined problem group [115]. The approach is widely adopted in the literature to develop structural models for causally interrelated factors or criteria using expert knowledge [7, 19, 123]. However, due to the vagueness, imprecision, and ambiguity of expert decisions resulting from the subjectivity of the human decision-making process, [123] proposed the Fuzzy DEMATEL approach, which is a fuzzified version of the DEMATEL, achieved through the integration of fuzzy set theory. Fuzzy set theory is a mathematical approach popularized by [129] capable of treating ambiguities, vagueness, and imprecision of the membership of elements in sets. As such, it becomes very suitable in treating the imprecision, ambiguities, and vagueness of the human decision-making process.

Several works have discussed and adopted the use of the Fuzzy DEMATEL approach in relevant domains such as:

1. Green supply chains [39]

2. Sustainable supply chain [56]

3. Sustainable recycling [58]

4. Environmental performance evaluation [113]

5. Solar power development [59]

6. Air transport management [16]

7. Human resource management [1]

8. Analyzing antecedents of organizational citizenship [76]

9. Knowledge transfer evaluation [97]

10. Operational hazards evaluation [5]

11. Supplier evaluation [53]

12. Project outcomes prediction [96]

13. Operational risk analysis [101]

14. Risk assessment of cargo ships [67]

15. Project management [63]

16. Sustainable hospitality management [122]

The steps for conducting the Fuzzy DEMATEL is provided as follows:

1. Step 1. Establish the linguistic relationships between the antecedents. First, let the decision-makers decide how antecedent c_i influences antecedent c_j for i, j ∈ {1, 2, . . . , n}. For decision-makers to conveniently express their judgment in a natural language, the linguistic scale presented in Table 4 is used to elicit judgment. By performing the pairwise comparison for each antecedent, an expert defined relationship matrix is formed. Since the elements of the matrix are still in linguistic value, they must be transformed to equivalent quantitative values. To do this, Table 4 is used to transform the linguistic values to triangular fuzzy numbers (TFNs). The result will be a fuzzy matrix, also known as the fuzzy direct relation matrix. For K decision-makers a fuzzy direct relation matrix is represented as Z^k=(zijk)n×n and z_ijk = (l_ijk, m_ijk, u_ijk) for i, j ∈ {1, 2, . . . , n}. Here, z^ijk is the equivalent TFN of the k^th decision-maker's elicited judgment for the influence of antecedent c_i on antecedent c_j.

2. Step 2. Aggregate the individual fuzzy direct relation matrices. In order to obtain a collective judgment of the K decision-makers, each fuzzy direct relation matrix Z^k can be averaged to form the average direct relation matrix Z^ as shown in Equation (4),

   (4) Z^=(z^ij)nxn=(z^ij1⊕z^ij2⊕…⊕z^ijKK)nxn.

3. Step 3. Normalize the average fuzzy direct relation matrix. In order to keep all elements in the same dimensions, the average fuzzy direct relation matrix must be normalized. Thus, the normalized fuzzy direct relation matrix X^ can be obtained using Equation (5),

   (5) (x^11x^12⋯x^1nx^21x^22⋯x^2n⋮⋮⋱⋮x^n1x^n2⋯x^nn),

   where x^ij=z^ijr=(l^ijr,m^ijr,u^ijr) ; r=max1≤i≤n(∑j=1nu^ij) .

4. Step 4. Decompose the normalized fuzzy direct relation matrix into l, m, u components. With X^=(x^ij)n×n=((lij′,mij′,uij′))n×n , let Xl=(lij′)nxn , Xm=(mij′)nxn , and Xu=(uij′)nxn be the decomposed components of the normalized fuzzy direct relation matrix X^ . The components are presented in Equation (6) as follows.

   (6) Xl=(0l12′⋯l1n′l21′0⋯l2n′⋮⋮⋱⋮ln1′ln2′⋯0),Xm=(0m12′⋯m1n′m21′0⋯m2n′⋮⋮⋱⋮mn1′mn2′⋯0),Xu=(0u12′⋯u1n′u21′0⋯u2n′⋮⋮⋱⋮un1′un2′⋯0).

5. Step 5. Obtain the fuzzy total relation matrix. Let Y represent any crisp total relation matrix, D represent any crisp direct relation matrix, and I be the identity matrix. By crisp, we mean a non-fuzzy version (or the “usual” numbers). The total relation matrix is then computed as

   (7) Y=D(I−D)−1.

   We apply Equation (7) to X_l, X_m and X_u to obtain three total relation matrices from each of the components of the decomposed fuzzy normal direct relation matrix. In similar terms, if we let t^ij be the elements of the fuzzy total relation matrix T^ , then we have

   (8) T^=(t^11t^12⋯t^1nt^21t^22⋯t^2n⋮⋮⋱⋮t^n1t^n2⋯t^nn)

   where t^ij=(lij″,mij″,uij″) is obtained from (lij″)n×n=Xl(I−Xl)−1 , (mij″)n×n=Xm(I−Xm)−1 , and (uij″)n×n=Xu(I−Xu)−1 .

6. Step 6. Obtain the crisp total relation matrix. While fuzzy numbers provide a means to handle ambiguities, vagueness, and imprecision, they are difficult to interpret in practice. Hence, it is desirable to obtain a crisp version of the fuzzy total relation matrix. To do this, a defuzzification process is performed. This would transform fuzzy numbers into crisp numbers. Let the resulting crisp total relation matrix be represented as T = (t_ij)_n×n, then each t_ij is obtained using the defuzzification process in Equation (9),

   (9) tij=(lij″+4mij″+uij″)6.

   The defuzzification method used in Equation (9) is known as the “Center of Gravity” method. There are many more defuzzification methods used in the literature and the interested reader is referred to [117] for a more in-depth discussion.

7. Step 7. Configure a threshold value to filter out insignificant relationships. A threshold value α is necessary to be established to filter out some negligible causal relationships. If t_ij ≥ α then antecedent c_i has a significant influence on antecedent c_j. In the current literature, the magnitude of the threshold value is determined through interviews with the respondents or judged by the researcher [54]. In this paper, the threshold value is agreed by the researchers to be the arithmetic mean of the elements of the defuzzified total relation matrix T. The same reasoning is also made by [22] and [128]. The threshold value can then be used in constructing an adjacency matrix where a value of 1 is assigned to links with significant relationships, and 0 otherwise.

8. Step 8. Develop the structural model or directed graph of the antecedents. Using the adjacency matrix, a directed graph of the antecedents can be constructed. If the value from c_i to c_j in the adjacency matrix is equal to 1, then a directed link from c_i to c_j must be drawn. Otherwise, no link can exist from c_i to c_j. The resulting structural model or directed graph provides a visual representation of the interdependence between the antecedents. Moreover, the graph can be described using the row sum (D) and column sum (R) of the total relation matrix as described in Equation (10). These measures provide a way to calculate two important structural characteristics for the DEMATEL approach known as: (i) relation (D − R), and (ii) prevalence (D + R). Relation enables the characterization of an antecedent as either a “net cause” or “net effect.” A negative relation (D − R) value implies that an antecedent is generally influenced by other antecedents, thus being a “net effect.” On the other hand, a positive relation (D − R) value implies that an antecedent generally influences other antecedents, thus being a “net cause.” Prevalence depicts an antecedent's degree of importance [76]. The higher the antecedent's prevalence, the more important it is in the system. These structural characteristics can then be used to partition the antecedents into cause and effect groups:

   (10) D=(∑j=1ntij)n×1=(ti)n×1,R=(∑i=1ntij)1×nT=(tj)n×1.