A fuzzy DEMATEL approach for evaluating the risk factors

Many risk factors are dependent on each other and taking this into account can be helpful in managing appropriate decisions. The poor evaluation of these factors will impose high costs in many real applications especially for a supply chain. There are different methods for risk evaluating and their effects for ranking them. For instant, in fuzzy DEMATEL method, the experts’ linguistic opinions and preferences on the agent effects are used as the input. The important point is how these opinions are aggregated to produce less computational error. In this regard, this paper proposes a new method based on statistical inferences for a fuzzy DEMATEL approach for evaluating the factors of a supply chain. This method was applied to a case study and the results showed that the proposed method was better than other methods for integrating the experts’ opinions in a supply chain. However, this method was proposed for the evaluation of supply chain factors, it can be applied to other systems as well.


Introduction
Many factors such as international competitors, demanding customers and rapid technological change profoundly impact the markets. Therefore, successful competition in this environment requires reducing operational costs and enlarging profit margins. For most industrial firms, the purchasing of raw material and component parts from suppliers constitutes a major expense. Hence, among the various strategic activities involved in the supply chain management, the purchase decision has profound impacts on the overall system [1]. When supply chain is faced with risk events, selecting the right suppliers becomes more essential than ever for the business. Several factors such as unquantifiable information, incomplete information, unobtainable information and partial ignorance cause the imprecision in decision making. Since conventional MADM methods cannot effectively handle problems with such imprecise information, therefore fuzzy multiple attribute decision-making methods have been developed owing to the imprecision in assessing the relative importance of attributes and the performance ratings of alternatives with respect to attributes [2]. Decision-Making Trial and Evaluation Laboratory (DEMATEL) is a widely used method to analyze and visualize the structure of complex systems through matrices and digraphs. The method typically requires dealing with substantial uncertainties and subjectivities inherent in the judgment process.
A review of the literature shows that several extensions of DEMATEL have been suggested so far dealing with a variety of sources of uncertainty. However, the uncertainty originating from the human doubt that might arise in the assignment of membership degrees during the assessments is partly or entirely ignored in these studies [3]. The decision-making trial and evaluation laboratory (DEMATEL) method is a hot issue in industrial engineering field for it can help determine critical factors in complex system. Although lots of efforts have been spent on improving the DEMATEL, they are just the extensions from the subjective perspective but lack of the objective perspective [4]. However, the crisp values sometimes cannot reflect human thinking comprehensively [5]. To address this problem, the -maker's DMs preferences were measured using fuzzy membership degrees [6]. However, in general, the estimation of experts' opinions using exact numerical values, especially in terms of uncertainty, is highly challenging. This is because decision making results are severely dependent upon inaccurate and ambiguous subjective judgments. This factor has led to the need to fuzzy logic DEMATEL. As a result, in DEMATEL, type-2 fuzzy linguistic variables are used, and this can facilitate decision-making under environmental uncertainty conditions [7]. 2 The literature shows that in reflecting the linguistic preferences fuzzy numbers are more powerful than crisp numbers. However, it is not clear that which method can make less computational error in aggregating the experts' opinions. In this paper, a mathematical and statistical based method is proposed such that we have a more suitable method for aggregating the experts' opinions and preferences in fuzzy DEMATEL method.
In this paper, first the literature on supply chain, DEMATEL method, the theory of fuzzy systems, and Fuzzy DEMATEL are investigated. Then in the following, the research method and proposed method for transforming the experts' opinions in fuzzy DEMATEL method has been explained. Moreover, the proposed method is applied to a case study. The paper is concluded in last section.

Subject Literature
From the literature, the following can be referred to:

Supply chain and its influential and influenced factors
The complexities in present day supply chain are numerous and are evolving due to globalization, customization, innovation, flexibility, sustainability and uncertainties. The growing supply chain complexity results in negative consequences on cost, customer service and reputation. Managing supply chain complexity without compromising the profitability is a challenging task. Supply chain complexity (SCC) management involves identifying, prioritizing, measuring, analyzing and controlling/eliminating the drivers of complexity. The SCC drivers denote number and variety of suppliers, customers, products, processes and uncertainties which are highly interdependent. Firms need to prioritize the drivers in order to manage and simplify SCC. [8]. Thus identifying all the complexity drivers and their interrelations that lead to unpredictable outcomes in supply chain is the first step in managing the complexity. Firms within supply chain are interested to address the dominant drivers rather than addressing all the drivers [9].

DEMATEL Method
Decision-making trial and evaluation laboratory (DEMATEL) has been widely used in decision-making methods with the aim of discovering the relations among the criteria in complex and intertwined problems. DEMATEL method is capable of analyzing the total relations among sets of variables using the mathematical techniques adopted to obtain logical relationships and direct impact of relationships [10]. Decision-making trial and evaluation laboratory method) DEMATEL) is known as a powerful approach to sum up experts' opinions about a problem and utilize them to solve complex and multifaceted problems [11,12]. DEMATEL method, originally developed by Battelle Memorial Institute to search for integrated solutions [13], is one of the methods used to visualize the structure of causal relationships among factors in an understandable manner. It can transfer the cause-effect relations among factors into a comprehensive model to aid the process of decision making. DEMATEL is one the effective technique to find and analyses the inter-relationship among the system factors. This method transforms the causal relationships between the factors indicators into a tangible structural model. DEMATEL is a comprehensive method for the preparation and analysis of a structural model that includes causal relationships between complex factors. This technique acts on directional graphs, and these graphs are able to display directional relationships between sub-systems. The result of the DEMATEL technique is to divide the factors into two cause and effect groups [14]. A review of the literature shows that the most common theories used to model uncertainty inherent in the assessment process and analysis are ordinary fuzzy set theory [15], intuitionistic fuzzy set (IFS) theory [16], type-2 fuzzy set theory [7], evidence theory [17], grey theory [18,19] and 2tuple fuzzy linguistic representation [20,21]. According to the review of the literature, the most widely studied approach considering uncertainty in DEMATEL is based on ordinary fuzzy sets. In order to model uncertainty inherent in the assessment process, the fuzzy approach uses linguistic terms and corresponding predefined functions with varying membership grades (typically triangular or trapezoidal fuzzy numbers). This allows dealing with vague and/or ambiguous expressions and judgments of experts.

The Fuzzy System Theory
Implementing fuzzy sets in decision making problems represents an important and efficient application compared to classical set theory. Fuzzy logic is the one that replaces the conclusion methods in human brain to express ambiguity in the form of a number. It introduces a function for the membership in one group that 3 assigns a number between zero and one to every element. This number represents the level of the element's membership in the set. Zero indicates that the element is totally outside the set while one means the element is completely in the set [22]. In numerous real life situations, the judgments of decisionmakers are normally characterized by ambiguity. Fuzzy numbers are suggested to suitably express linguistic variables. The triangular and trapezoidal fuzzy numbers have identified most commonly used fuzzy numbers [23]. In this method, sharp numerical values are represented as bands of fuzzy numbers with an overlap [24]. In such cases, the use of a definitive scoring system can be criticized for two reasons; first, definitive methods ignore ambiguity resulting from people's judgment and may not be able to convert the changes in linguistic terms into numbers. Secondly, selecting priorities based on people's subjective judgments can largely influence the results [25] . Fuzzy logic is a useful tool for measuring ambiguous concepts related to individual subjective judgment and is a powerful method to overcome the above-mentioned problems [26].

Fuzzy DEMATEL
The main advantage of FDEMATEL is to consider the fuzziness and to provide flexibility in a fuzzy environment [27]. Therefore, prior to giving the details of FDEMATEL method, it requires to mention some preliminaries of fuzzy set theory and important notations. Fuzzy set theory reflects the uncertainties that result from vague and imprecise linguistic expressions [26]. Fuzzy DEMATEL method is currently applied in many areas such as management [28,29], emergency planning [30], health care systems [31,32], and safety management [11,33]. The overview of the literature show that for integrating the experts' opinions and preferences in DEAMTEL, arithmetic mean has been usually used [34,35]. Since arithmetic mean is sensitive to outliers, the integration of experts' opinions using this method has high error. Thus, a new method is proposed here through which different methods for the integration of opinions can be evaluated so that the most suitable one is selected.

Fuzzy sets
Definition 1 - Figure 1 shows a triangular fuzzy number (M ) and parameters ( , , u) such that is the smallest possible occurrence value, is the most probable value, and is the largest possible occurrence value. The left and right sides of each fuzzy number can be linearly determined by the following membership function:

Fuzzy DEMATEL
The fuzzy DEMATEL algorithm is executed in the following steps.
Step 1. The experts express their judgments in linguistic scale. Some linguistic scales of the literature that are changed to triangular fuzzy number (TFN) are shown in  ( 0.75 , 1 , 1 ) ( 0.7 , 0.9 , 1 ) ( 0.7 , 0.9 , 1 ) Step 2. The fuzzy initial direct-relation matrix is defined as follows: where ̃ shows the judgment of expert k on the impact of factor i on factor j. Therefore, the fuzzy initial direct-relation matrix is as follows.
Step 3. In this step, the judgments of k experts are aggregated. The review of the literature shows that most of the researchers have used average method for aggregating the judgments. [31,34,35,37,41,42]. This forms the average matrix as the following: Step 4. The fuzzy normalized direct-relation matrix is calculated. Step The elements of these matrices are extracted from as follows: Step 6. The fuzzy total relation matrix is obtained as follows: Step 7. The non-fuzzy total relation matrix is obtained as follows: Step 8. The experts should identify a threshold value ( ) to filter out some negligible casual relationships. If ≥ , then element has a significant causal relationship on element .
Step 9. The values of the matrices and , where is the sum of columns and is the sum of rows of matrix T, are obtained (they represents the influence on and the relationships with others): Some criteria have positive values of and therefore greatly influence other criteria. These criteria are called dispatchers; others have negative values of and thus are greatly influenced by other criteria. These are called receivers. The value of + indicates the degree of relationship between each criterion with other criteria. Criteria having greater values of + have stronger relationships with other criteria, while those having smaller values of + have less of a relationship with others.
Step 10: Using values of + and , make an impact relationship chart that shows the causal relationships of the elements.

ANOVA 4 -MANOVA 5
Hypothesis testing for a single variable, such as the t-test and Z-test, are used for testing one or two samples in a univariate case, while in a case with several variables (the multivariate case), Hotelling's test ( ) is used. These tests cannot be used for three or more samples. Thus a new technique should be considered for this purpose. Analysis of variance (ANOVA) for one dependent variable and a multivariate analysis of variance (MANOVA) for several dependent variables measured on each sampling unit will be used to test the equality of means for three or more samples. ANOVA can be used in the case of two means (two samples), and the results will be the same as the Z or t tests. Assumptions of ANOVA: 1-Normality: the samples must be selected from populations that are normally or approximately normally distributed. 2-Independence: the samples must be independent. 3-Homogeneity: the variances of the populations must be equal. One-way ANOVA It considers only one independent variable (X) (called a factor) at g levels (groups), and the objective is to study the effect of different levels of the factor on a continuous response (Y). In a one-way ANOVA, researchers are interested in testing the following hypotheses:

Two-way ANOVA
The idea of one-way analysis of variance can be extended to include two independent variables (factors); each variable has at least two levels and one response variable. The technique for analyzing the effect of two independent variables is called a two-way analysis of variance. Suppose there are two factors, factor A at g levels and factor B at q levels. The researchers are interested in testing the following hypotheses.
Hypothesis about the effect of the first factor (factor A): Hypothesis about the effect of the second factor (factor B): Hypothesis about the effect of interaction between the two factors A and B: : ( ) = 0 : ( ) ≠ 0

MULTIVARIATE ANALYSIS OF VARIANCE (MANOVA)
The idea of a multivariate analysis of variance (MANOVA) is the same as a univariate analysis of variance (ANOVA), because both methods are used to test the equality of means for three or more samples. However, the difference between ANOVA and MANOVA is that ANOVA is used when only one response variable (dependent variable) is measured for each experimental unit, while MANOVA is used when 7 several response variables (dependent variables) are measured for each experimental unit. Assumptions for MANOVA: 1-Normality: the samples must be selected from populations that are normally or approximately normally distributed. 2-Independence: the samples must be independent. 3-Homogeneity: all populations have a common variance-covariance matrix. 4-Multivariate normal: each population is multivariate normally distributed.

One-Way MANOVA
The concept of one-way MANOVA is the same as one-way ANOVA, and the difference between them is the number of response variables (dependent variables). In a multivariate case, there are g populations, and the objective is to compare these populations based on several measurements (dependent variables), i.e., whether the g samples' mean vectors are the same or not. In a one-way MANOVA, the researchers are interested in the following hypothesis (a sample with size of n is from k-variate normal populations with equal covariance matrices): Where, μ is a vector.
If the null hypothesis is not rejected, the g means are equal for each dependent variable, while rejecting the null hypothesis means that at least two means (groups) differ for at least one dependent variable.

Two-Way MANOVA
Suppose there are two factors: factor A has a levels and factor B has b levels. The objective is to investigate the effect of different levels of each factor on several responses measured on the same experimental unit. In a two-way multivariate analysis of variance (MANOVA), the researchers are interested in three hypotheses-the first two hypotheses are related to the main effect of each factor and the third hypothesis is related to the interaction between the different levels of factor A and factor B. The hypotheses for interaction and main effects can be tested by using the Wilk's test.

Proposed method
As was said in step 3 of section 3.2, in DEMATEL method, the aggregated matrix is obtained by combining the experts' opinions and preferences. In the literature, most of the studies have used arithmetic mean. However, this method is drastically sensitive to outliers. That is, the incorrect judgment of one or a few experts may affect the aggregated matrix of the judgments and increase the computational error. Taking this into account, this paper proposes a method that, among different aggregation methods, can identify the proper one for each case study.

The fuzzy initial direct-relation matrix
The steps 1 and 2 are the same as the general fuzzy DEMATEL explained in section 3.2. So, the fuzzy initial direct-relation matrix is shown as follows: where ̃ represents the judgment of expert k on the impact of factor i on factor j.  (Table 2). These two values depend on one subject (the method of aggregating experts' opinions). Thus, we use one-way MANOVA to evaluate the aggregation method. In One-way MANOVA, the hypothesis for Table 2 is expressed as follows: If the null hypothesis is rejected, it means that there is a significant difference between the aggregation methods. In this case, the most appropriate method can be selected using the other tests of one-way MANOVA.

Case Study
To evaluate the proposed method, a case study is presented here. The training supply chain of Iranian national oil company includes the training centers of the company, different companies, both internal and external, that provide training materials, training units of different oil companies covered by Iranian 9 national oil company as the distributor of the training courses, and all employees of Iranian national oil company as those who receive these trainings. Accordingly, Mahmoudabad oil training center was selected as our case study. To assess how influential these factors are and how they are influenced, a questionnaire was designed and sent for 50 experts of this field. They were asked to express their opinion in linguistic scale as no influence (NO), very low (VL), Low (L), high (H), and very high (VH) influence (Table 3). From these, 38 questionnaires were completed and returned. These linguistic scales are transformed to triangular fuzzy number (TFN) as in Table 4: Accordingly, the fuzzy initial direct-relation matrix for expert 1 is shown in Table 5.

Mahmoudabad oil training center
The fuzzy initial direct-relation matrix was calculated for all of the experts. For aggregating the experts' opinions, the conventional method (CM) and the proposed methods of this research have been used (Tables  6 to 9).
11 Table 5-the fuzzy initial direct-relation matrix for expert 1.

C5
The remaining of the calculations were done using fuzzy DEMATEL. The remarkable note for the case study is that in the proposed FM method (I-Xu) -1 is undefined, and consequently, " = × ( ) is not defined. Therefore, this method cannot be used in the case study. The values for D+R and D-R for each one Tables 6, 8, and 9 are reported in Table 10. Based on section 4.4 of the proposed method, to identify the better method for integrating the experts' opinions in fuzzy DEMATEL method, one-way MANOVA has been used. For this, the data of Table 10 was loaded into SPSS software. The results have been shown in Tables 11 to 13. According to Table 11, the hypothesis H0 is rejected and H1 is accepted, i.e., the different methods of integrating opinions have significant difference. Table 12 indicates that the dependent variable D+R has caused this deference. Table 13 shows that TM method is a better method for integrating the experts' opinions in this case study.

Conclusion
Making a suitable decision on critical factors of a supply chain is special interest for specialists. A nonproper evaluation of supply chain factors can incur huge cost to supply chain members. These factors have interactions with each other, and calculating the sum of influence given to and received by each factor can be a suitable measure for the evaluation and ranking of those critical factors. There are different methods for evaluating and ranking of the factors. In fuzzy DEMATEL approach, linguistic opinions and preferences are used, and usually, the arithmetic mean is applied for the integration of experts' opinions. However, this mean is drastically sensitive to outliers and makes considerable error. Thus, in this paper, we proposed a statistical-based method for identifying the appropriate method for the integration of experts' opinions. For this, the outputs of fuzzy DEMATEL (D+R and D-R) were evaluated considering the data input method. Since there are two dependent variables D+R and D-R in relation to the independent input method, we used one-way MANOVA for our analysis. As a case study, we considered the data of Mahmoudabad training center in Iran. The results showed that the TM method was better than other methods for integrating the experts' opinions.