Introduction

The decision-making trial and evaluation laboratory (DEMATEL) is one of the multi-criteria decision making (MCDM) methods that purposely used for building and analysing a structural model. Developing the causal relationships between complex criteria in MCDM problems are the ultimate aim behind the structural model. According to Chang et al. [8] the DEMATEL was developed to explore and solve complex and interrelated criteria in groups of MCDM problems. Fragmented and antagonistic phenomena of MCDM problems are solved using the DEMATEL in an integrated manner. The DEMATEL method can convert the relationship between the causes and effects of criteria in MCDM into an intelligible structural mode. The strength of DEMATEL method lies on the applications of decision matrices and digraphs to consider the structure of complicated causal relationships [24]. The DEMATEL is particularly practical and useful for visualizing the structure of complicated causal relationships between criteria using digraphs. Using the DEMATEL, a visual representation is finally constructed to unravel the relationships between criteria of MCDM problems. In addition, the DEMATEL method can improve understanding of the specific problem antique, the cluster of intertwined problems, and contribute to identification of workable solutions by a hierarchical structure [33]. The causal diagram uses digraphs rather than directionless graphs to portray the basic concept of contextual relationships and the strengths of influence among the elements. Owing to these advantages, the DAMATEL has been applied in many recent MCDM problems (see [10, 21, 34]).

The DEMATEL also has been successfully combined with fuzzy sets as to handle the uncertainties and vagueness in MCDM problems, and also incomplete information about the data of the MCDM problems. The concept of fuzzy set theory was firstly proposed by Zadeh [46] and since then, the sets have been wisely fused into many MCDM methods with no exceptional of the DEMATEL. Likewise the DEMATEL, the fuzzy DEMATEL also has been applied in many MCDM problems in diverse areas. In supply chain research, for example, the fuzzy DEMATEL method was made to identify the critical success criteria [7, 25]. The DEMATEL was applied in marketing resources [3], knowledge management [29], and information support management [23]. Very recently, Chakraborty, et al. [6] developed a causal model to evaluate the critical issues in reverse supply chain implementation using fuzzy DEMATEL. Also in business related research, Mavi and Standing [26] analysed cause and effect of business intelligence benefits with fuzzy DEMATEL. As an extension to fuzzy sets, intuitionistic fuzzy set (IFS) was proposed by Atanassov [4] as to handle the issues of membership, non-memberships and hesitation degrees of decision problems. The IFS was intended to be an extension of single membership of fuzzy set theory. With the ultimate aim to solve the relationships between criteria in MCDM, the fusion of IFS and the DEMATEL was proposed (see [28, 32]). This fusion is normally written as intuitionistic fuzzy DEMATEL (IF-DEMATEL), and has been recently utilised by Zhou et al. [49]. The IF-DEMATEL method is a potent method that helps in gathering group knowledge for forming a structural model under uncertain and incomplete information. The IF-DEMATEL is introduced to represent the correlation among criteria in an intuitionistic fuzzy environment. The proposed IF-DEMATEL uses triangular intuitionistic fuzzy numbers to find weights [32]. In real case applications, the IF-DEMATEL method was used in prioritising the components in insurance industry [28], developed green practices and performances in a green supply chain [22], ranking the risk of construction projects [39], risk analysis of coal combustion [47]. The vagueness of human’s subjective judgments are conquered by embracing the two memberships in judgements of the IF- DEMATEL.

However, the two memberships of IFS have some limitations particularly on the arithmetic addition of two memberships and also hesitation degree. To improve hesitation degree, author such as Zeng et al. [47, 48] proposed interval-valued hesitant fuzzy sets and its arithmetic operations. For the two memberships of IFS, the focal point is on its arithmetic addition. It is known that the sum of two memberships of IFS is limited to one. In response to this limitation, Yager [45] introduced Pythagorean fuzzy set (PFS) where the limitation in IFS has been modified. The sum of two memberships in IFS is now substituted with squares of each membership where the sum of these two squares is less or equal to one. In other words, the PFS is characterized by a membership degree and non-membership degree where the square sum of its membership degree and non-membership degree is less than or equal to one. The PFS is one of the most successful sets, in terms of representing comprehensively uncertain and vague information [11]. In this regards, the PFS has been fused to the many MCDM methods particularly in aggregation methods. For example, the new aggregation operators were proposed by combining PFS-Choquet–Frank aggregation operators [44], PFS-Einstein aggregation operator [14, 31], PFS-averaging and geometric operators with logarithmic laws [19], PFS- geometric-Einstein operators [20]. The PFS also was successfully integrated with the concepts of confidence level [15], and decision making with probabilities [17]. The new linguistic and exponential operational laws with PFS were also proposed [16, 18]. About similar with other sets, the PFS was also extended to interval-valued PFS and hesitant PFS. These two sets were successfully used in developing new aggregation operators such as Maclaurin Symmetric Mean Operator [13, 43], averaging and geometric aggregation operators [12].

The PFS fusion based models also has been applied in many real life problems. For example, a three-phase PFS-MCDM method has been applied to haze management [40]. Also very recently, the PFS based analysis model has been applied in solar power plants [9]. The PFS-analytic hierarchy process and PFS-similarity measures were proposed to solve MCDM problems [41, 42]. It can be seen that there were handful of research applied MCDM methods based on PFS. Moreover, so far, there has been little discussion about the applications of DEMATEL based on PFS to solid waste management (SWM). To bridge the literature gap between the DEMATEL based on PFS and other MCDM methods, this paper proposes a modified DEMATEL based on PFS and applies the proposed method to the case of SWM. Differently from the DEMATEL, the proposed method introduces new linguistic variable of influence, experts’ weights and score function based on PFS. In short, the objective of this research is to propose the DEMATEL method based on PFS and apply it to the case of SWM. This paper is organised as follows. “Preliminary” presents the definition that related to PFS. The proposed work is presented in “Proposed method”. “Empirical case to construct causal diagram” provides the application of the proposed method to the case of SWM. Finally, “Conclusions” concludes.

Preliminary

This section recalls the definitions of PFS and some its algebraic operations. As a basis to the proposed PFS-DEMATEL, the section also provides the basis of compuational steps in the DEMATEL.

Definition 1

PFS [45].

A PFS P in a finite universe of discourse is

$$ P = \{ < x,\mu_{P} (x),v_{P} (x) > \left| {x \in X} \right.\} > $$

where \( \mu_{P} ,v_{P} :X \to [0,1] \) with the condition that the square sum of its membership degree and non-membership degree is less than or equal to 1.

$$ \left[ {\mu_{P} (x)} \right]^{2} + \left[ {v_{P} (x)} \right]^{2} \le 1. $$

Definition 2

Degree of Indeterminacy [30].

The degree of indeterminacy of x to P is given by \( \pi_{P} \left( x \right) = \sqrt {1 - \mu_{P}^{2} \left( x \right) - v_{P}^{2} \left( x \right)} \). If \( B = P\left( {\mu_{B} ,v_{B} } \right) \) is a Pythagorean fuzzy number then, degree of indeterminacy of B is given as \( \pi_{B} = \sqrt {1 - \mu_{B}^{2} - v_{B}^{2} } \) where \( \mu_{B} ,v_{B} \in \left[ {0,1} \right] \) and \( \left( {\mu_{B} } \right)^{2} + \left( {v_{B} } \right)^{2} \le 1 \).

Definition 3

Algebraic Operations of Pythagorean fuzzy number (PFN).

Given two PFNs, \( A = P(\mu_{A} ,\upsilon_{A} ) \) and \( B = P\left( {\mu_{B} ,v_{B} } \right) \), where \( \mu_{B} ,v_{B} \in \left[ {0,1} \right] \),\( \mu_{A} ,\upsilon_{a} \in [0,1] \), then some arithmetic operations can be described as follows:

  1. 1.
    $$ A \cup B = P\left( {\hbox{max} \left\{ {\mu_{A} ,\mu_{B} } \right\},\hbox{min} \left\{ {v_{A} ,v_{B} } \right\}} \right) $$
  2. 2.
    $$ A \cap B = P\left( {\hbox{min} \left\{ {\mu_{A} ,\mu_{B} } \right\},\hbox{max} \left\{ {v_{A} ,v_{B} } \right\}} \right) $$
  3. 3.
    $$ A^{C} = P\left( {v_{A} ,\mu_{A} } \right) $$
  4. 4.
    $$ A \oplus B = P\left( {\sqrt {\mu_{A}^{2} + \mu_{B}^{2} - \mu_{A}^{2} \mu_{B}^{2} } ,v_{A} v_{B} } \right) $$
  5. 5.
    $$ A \otimes B = P\left( {\mu_{A} \mu_{B} ,\sqrt {v_{A}^{2} + v_{B}^{2} - v_{A}^{2} v_{B}^{2} } } \right) $$
  6. 6.
    $$ \lambda A = P\left( {\sqrt {1 - \left( {1 - \mu_{A}^{2} } \right)^{\lambda } } ,\left( {v_{A} } \right)^{\lambda } } \right),\lambda > 0 $$
  7. 7.
    $$ A^{\lambda } = P\left( {\left( {\mu_{A} } \right)^{\lambda } ,\sqrt {1 - \left( {1 - v_{A}^{2} } \right)^{\lambda } } } \right),\lambda > 0 $$
  8. 8.
    $$ \lambda (A \oplus B) = \lambda A \oplus \lambda B,\lambda > 0 $$

These definitions are relevant and necessitated in the proposed method.

Proposed method

Most studies in the field of decision making, have only focused on specific methods of MCDM such as the DEMATEL method and combined with fuzzy sets and its extensions. For example, the DEMATEL method has been combined with interval type-2 fuzzy sets [1, 5, 27]. To the best of authors knowledge, previous studies of DEMATEL have not been combined with PFS. In this paper, the proposed method is a fusion of the PFS and the DEMATEL method. The PFS is used as a linguistic judgment and substitute it into the decision-making method DEMATEL. In this section, we provide acronym for this combination as PF-DEMATEL and will be used throughout this text. In other words, the PF-DEMATEL is a decision-making method that worked in the framework of DEMATEL with the use of PFS in linguistic judgement. It is anticipated that the proposed PF-DEMATEL method is a potent method that helps in gathering group knowledge for developing a structural model under Pythagorean fuzzy condition. The proposed framework is divided into three phases. In phase 1, a new linguistic variable based on PFS is developed. The integration of PFS with DEMATEL is made in the phase 2. Multiplication with the weights of DMs, aggregation operators and defuzzification are the main mathematical operations in this phase. Finally, the causal diagram is illustrated is phase 3. Figure 1 presents a schematic representation of the proposed method.

Fig. 1
figure 1

Schematic representation of the PF-DEMATEL

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Based on the framework, the algorithm of PF-DEMATEL is proposed as follows.

Step 1: Define linguistic variables.

With the linguistic variable ‘influence’, seven linguistic terms are defined based on the rating scales of DEMATEL. The pair-wise comparison scales are made in seven terms, where the scores of 0, 1, 2, 3, 4, 5 and 6 represent “Very low influence”, “Low influence”, “Medium low influence”, “Medium influence”, “Medium high influence”, “High influence”, and “Very high influence”, respectively. These scales are now being introduced in PFS instead of crisp number Table 1 shows the linguistic terms used in judgment and its respective scales in PFS.

Table 1 Linguistic variable in Pythagorean fuzzy sets
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Step 2: Obtain a n × n matrix as initial direct-relation matrix Z by pair-wise comparisons in terms of influences and directions between criteria, in which is denoted as the degree to which the criterion i affects the criterion j, i.e., \( Z = \left[ {z_{ij} } \right]_{n \times n} \). The value of Zij is written in PFS.

$$ Z_{m} = \left[ {z_{{ij}} } \right]_{{n \times n}} = \begin{array}{*{20}c} {C_{1} } \\ \vdots \\ {C_{n} } \\ \end{array} \left[ {\begin{array}{*{20}l} {C_{1} } \hfill & \ldots \hfill & {C_{n} } \hfill \\ {\left\langle {0,0} \right\rangle } \hfill & \cdots \hfill & {\left\langle {\mu _{{m_{{1n}} }} ,~v_{{m_{{1n}} }} } \right\rangle } \hfill \\ \vdots \hfill & \ddots \hfill & \vdots \hfill \\ {\left\langle {\mu _{{m_{{n1}} }} ,~v_{{m_{{n1}} }} } \right\rangle } \hfill & \cdots \hfill & {\left\langle {0,0} \right\rangle } \hfill \\ \end{array} } \right]. $$
(1)

Step 3: Calculate the weighted initial direct-relation matrix,

$$ \lambda_{m} Z_{m} = \left[ {\begin{array}{*{20}c} {\lambda_{m} \left\langle {0,0} \right\rangle } & \ldots & {\lambda_{m} P_{1n} } \\ \vdots & \ddots & \vdots \\ {\lambda_{m} P_{n1} } & \cdots & {\lambda_{m} \left\langle {0,0} \right\rangle } \\ \end{array} } \right] $$
(2)

where \( \lambda_{m} P_{ij} = \left\langle {\sqrt {1 - \left( {1 - \mu_{ij}^{2} } \right)^{\lambda } } ,\left( {v_{ij} } \right)^{\lambda } } \right\rangle \) is the weighted PFS element.

Step 4: Calculate the aggregated matrix using addition operation

$$ \begin{aligned} & \lambda_{{m_{1} }} Z_{{m_{1} }} \oplus \lambda_{{m_{2} }} Z_{{m_{2} }} \\ &\quad = \left[ {\begin{array}{*{20}c} {\lambda_{{m_{1} }} P_{{Z_{{m_{1} }} ,11}} \oplus \lambda_{{m_{2} }} P_{{Z_{{m_{2} }} ,11}} } & \ldots & {\lambda_{{m_{1} }} P_{{Z_{{m_{1} }} ,1n}} \oplus \lambda_{{m_{2} }} P_{{Z_{{m_{2} }} ,1n}} } \\ \vdots & \ddots & \vdots \\ {\lambda_{{m_{1} }} P_{{Z_{{m_{1} }} ,n1}} \oplus \lambda_{{m_{2} }} P_{{Z_{{m_{2} }} ,n1}} } & \cdots & {\lambda_{{m_{1} }} P_{{Z_{{m_{1} }} ,nn}} \oplus \lambda_{{m_{2} }} P_{{Z_{{m_{2} }} ,nn}} } \\ \end{array} } \right] \end{aligned} $$
(3)

where,

$$ \begin{aligned} & \lambda_{{m_{1} }} P_{{Z_{{m_{1} }} ,ij}} \oplus \lambda_{{m_{2} }} P_{{Z_{{m_{2} }} ,ij}} \\ &\quad = \left\langle {\sqrt {\mu_{{\lambda_{{m_{1} }} Z_{{m_{1} }} }}^{2} + \mu_{{\lambda_{{m_{2} }} Z_{{m_{2} }} }}^{2} - \mu_{{\lambda_{{m_{1} }} Z_{{m_{1} }} }}^{2} \mu_{{\lambda_{{m_{2} }} Z_{{m_{2} }} }}^{2} } ,v_{{\lambda_{{m_{1} }} Z_{{m_{1} }} }} v_{{\lambda_{{m_{2} }} Z_{{m_{2} }} }} } \right\rangle \end{aligned} $$

Step 5: Construct total average crisp matrix by using score function as a defuzzification function.

$$ A = \left[ {\begin{array}{*{20}c} {a_{11} } & \ldots & {a_{1n} } \\ \vdots & \ddots & \vdots \\ {a_{n1} } & \cdots & {a_{nn} } \\ \end{array} } \right] $$
(4)

where \( a_{ij} = \mu_{{P_{z} ,ij}}^{2} - v_{{P_{z} ,ij}}^{2} \)

Step 6: Construct the normalized average crisp matrix X.

The crisp matrix X = xij where, 0 ≤ xij≤ 1 are obtained using the equation

$$ X = s \cdot A $$
(5)

where \( s = \frac{1}{{\max_{1 \le i \le n} \sum\nolimits_{j = 1}^{n} {a_{ij} } }}\quad \quad \quad i,j = 1,2, \cdots ,n. \)

Step 7: Construct the total-relation matrix T using the Eq. (6).

$$ T = X\left( {I - X} \right)^{ - 1} $$
(6)

where I is the identity matrix.

Step 8: The sum of rows and the sum of columns are separately denoted as D and R within the total-relation matrix T through \( T = t_{ij} , \) where i, j = 1, 2,…, n.

Sum of row,

$$ r = \sum\limits_{j = 1}^{n} {t_{ij} } , $$
(7)

Sum of Column,

$$ c = \sum\limits_{i = 1}^{n} {t_{ij} } $$
(8)

Step 9: Draw a causal diagram.

A causal diagram is obtained by mapping the dataset of (r + c, r − c), where the horizontal axis (r + c) is made by adding r to c, and the vertical axis (r − c) is made by subtracting r from c.

If (r − c) is positive, then the criteria is under the cause category.

If (r − c) negative, then the criteria is under the effect category.

The nine-step proposed algorithm is implemented to a case of SWM where two causal categories of criteria is determined.

Empirical case to construct causal diagram

One of the most significant current discussions in sustainability and environmental studies is solid waste management (SWM). Recent status of solid waste generation, trends and regulations was comprehensively reviewed by the researcher [37]. The relationship between SWM and fuzzy decision making was discussed in a study [36]. The fuzzy analytic hierarchy process were combined with the fuzzy ideal solution to achieve the integrated SWM decision. This current research mainly discusses about the evaluation of SWM by using the proposed PF-DEMATEL method. Specifically, the experiment attempts to visualise the cause criteria and effect criteria of SWM in Malaysia.

Experts and criteria

The evaluation criteria that influence solid waste management are retrieved from literature while the weight and priority of the criteria are provided by a group of experts in the field that related to solid waste management. Personal profiles of the experts are given in Table 2.

Table 2 Personal profiles of experts
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Interviews with experts were conducted to collect linguistic evaluation. The interviews mainly aimed at obtaining a pair-wise comparison between criteria based on the developed linguistic terms (see Table 1). The criteria selected for this study are Relative Cost (C1), Environmental Health (C2), Socio-culture (C3), Public Awareness(C4), Institutional(C5), Technical (C6), Operation and Maintenance Challenges (C7), Population Size (C8), Human Health (C9), and Consumption Habits (C10). A brief description about the criteria are summarised in Table 3.

Table 3 Description of criteria
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The linguistic evaluations provided by experts are then computed using the proposed PF-DEMATEL method (see “Proposed method”).

Computation

Computations are executed in accordance with the proposed method (see “Proposed method”). Linguistic judgements are transformed into a \( 10 \times 10 \) matrix as initial direct-relation matrix Z by pair-wise comparisons in the linguistic terms of influences between criteria. The elements of matrix Z represent the degree to which the criterion i affects the criterion j. Table 4. presents the initial direct matrix, \( Z_{m} = \left[ {z_{ij} } \right]_{10 \times 10} \) obtained from E1 (see Eq. (1)).

Table 4 Initial direct matrix (E1)
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The five other similar matrices are obtained from E2, E3, and E4 and E6. Details of the matrices are shown in “Appendix”.

In this empirical case, the weights of experts \( \lambda_{m} \) is determined using the following equation.

Expert weight, \( \lambda_{m} = \frac{{{\text{Expert}}\;{\text{weight}}\;{\text{score}}}}{{{\text{Sum}}\;{\text{of}}\;{\text{weight}}\;{\text{score}}}}. \)

The weights of experts are determined and summarised in Table 5.

Table 5 Experts weight
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The weights of experts are multiplied with the initial direct matrix using Eq. (2). The weighted initial direct matrix is calculated for every expert. Table 6 presents the weighted initial direct matrix of E1.

Table 6 Weighted initial direct matrix (E1)
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The weighted initial direct matrices of E2, E3, E4, E5 and E6 are computed similarly.

The aggregated matrix to represent assessments made by six experts are calculated using Eq. (3). The aggregated matrix is shown in Table 7.

Table 7 Aggregated matrix
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The score function [see Eq. (4)] is used to obtain total average crisp matrix. The average crisp matrix is summarised in Table 8.

Table 8 Total average crisp matrix
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The average crisp matrix is then normalised using Eq. (5). Table 9 presents the normalised average crisp matrix.

Table 9 Normalized average crisp matrix
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Toward the end of this compuation, the total-relation matrix T is obtained using Eq. (6). Table 10 presents the total relation matrix.

Table 10 Total relation matrix
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The sum of rows and the sum of columns of total relation matrix are separately denoted as c and r within the total-relation matrix T using Eqs. (6) and (7). The values of c, r. c + r and c-r are shown in Table 11.

Table 11 Sum of rows, sum of columns and its substraction and addition
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Based on the information in Table 11, finally, the causal diagram is drawn. Date set of (r + c, r − c) are mapped out onto two-dimension plane. Figure 2 shows the coordinates of criteria in two dimesion plane.

Fig. 2
figure 2

Causal diagram

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It is apparent from this table that the criteria are separated into two categories based on the values of r − c. The first category is ‘cause group’ and the second category is denoted as ‘effect group’. The criteria in the ‘cause group’ are population size, consumption habits, human health and environmental health. On the other hand, the criteria in the ‘effect group’ are relative cost, socio-culture, public awareness, institutional, technical and operational and maintenance challenges. The causal diagram depicts the coordinates of all ten criteria of SWM where the four criteria in ‘cause group’ would influence the six criteria in ‘effect group’. The results indicate that consumer habit and population size are among the criteria in cause group where as Technical and operational maintainance are placed in effect group. The importance of criteria is determined based on the (r + c) values. It can be seen that ‘operational and maintenance challenges’ is the most important criteria in the effect group. All in all, the results of the empirical case provide evidence on the application of the proposed method.

The results obtained from the proposed method suggests that ‘operational and maintenance challenges’ is most influenced criteria. This result is not consistent with an empirical study conducted by Soroudi et al. [35] where ‘soil depth’ is the most influenced criteria. They utilised the DEMATEL and the analytical network process to determine interaction and weight of criteria of SWM. In another experiment, by using the DEMATEL method, Tseng [38] found that the criteria ‘natural resources’ is considered as the most influenced criteria. This result also differs from that of Abdullah et al. [2] who suggest that the criteria ‘political issues’ was selected as the most important criteria.

Conclusions

The DEMATEL provides a comprehensive tool and computationally feasible way in dealing with relationships between criteria of MCDM problems. The DEMATEL has been combined with IFS to address uncertain and vague information in solving MCDM problems. However, the combined IFS-DEMATEL fails to solve problems particularly on the cases where sum of two memberships is greater than one. To address this limitation, the PFS was proposed with the assumption that the linearity in two membership functions of IFS is now modified to the square functions. In this paper, the combination of DEMATEL and PFS has been proposed with three innovations. Firstly, two memberships of PHS were used to replace two linear membership functions of IFS. This combination entails newly defined linguistic variable, weights of experts and score function. The proposed PFS-DEMATEL shows some extent of advantages as the proposed MCDM method provides more obvious improvement in judgements where uncertainties are now expressed in squares of memberships. In other words, the PFS-DEMATEL creates a new perspective in solving MCDM problems where linguistic variables are defined by considering membership and non-membership of PFS. Another objective of this paper was to apply the proposed method in a SWM problem. Therefore, the proposed eleven-step DEMATEL method was computationally implemented to an empirical study in SWM. A group of experts were invited to assess the degree of influence among ten criteria of SWM using the seven-scale linguistic terms. The computational feasible method was successfully segregated the ten criteria into two categories of which four criteria included in cause group while six criteria are placed in effect group. The current study found that the criteria ‘operational and maintenance challenges’ is the most important criteria in solving SWM problem.