Establishing an interval-valued fuzzy decision-making method for sustainable selection of healthcare waste treatment technologies in the emerging economies

Abstract

Healthcare services provided by hospitals and clinics inevitably produce waste that may hazardous to the environment and society. However, there is a lack of an effective and comprehensive evaluation framework that takes uncertainty and fuzziness into account to assess healthcare waste treatment technologies in the emerging economies. The objective of this paper is to present a new integrated multi-criteria decision-making method based on interval-valued fuzzy DEMATEL (Decision-Making Trial and Evaluation Laboratory) and interval-valued fuzzy TOPSIS for evaluating healthcare waste treatment technologies in the emerging economies from a sustainability perspective. In this study, the decision makers are allowed to determine the weights of the evaluation criteria and prioritize the alternatives using linguistic variables. The weights of the evaluation criteria are determined by the interval-valued fuzzy DEMATEL method, and the prioritization of the alternatives is determined by the interval-valued fuzzy TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) method. Four alternatives for healthcare waste treatment technologies including incineration, steam sterilization, microwave and landfill are studied, and the results show that our established method is effective to help the decision-makers to determine the prioritization of the alternatives for healthcare waste treatment technologies.

Appendix

The definition of the interval-valued fuzzy set are as follows [53]:

$$\tilde{A} = \left\{ {x,\left[ {\mathop \mu \nolimits_{{\tilde{A}}}^{L} (x)\mathop {,\mu }\nolimits_{{\tilde{A}}}^{U} (x)} \right]} \right\},x \in ( - \infty , + \infty ),\mathop \mu \nolimits_{{\tilde{A}}}^{L} (x)\mathop {,\mu }\nolimits_{{\tilde{A}}}^{U} (x):( - \infty , + \infty ) \to \left[ {0,1} \right]$$

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$$\mathop \mu \nolimits_{{\tilde{A}}} (x) = \left[ {\mu_{{\tilde{A}}}^{L} (x),\mu_{{\tilde{A}}}^{U} (x)} \right],\mu_{{\tilde{A}}}^{L} (x) \le \mu_{{\tilde{A}}}^{U} (x),\,\,\,\,\forall x \in ( - \infty , + \infty )$$

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where $\mu_{{\tilde{A}}}^{L}$ refers to the lower limit of the membership degree, $\mu_{{\tilde{A}}}^{U}$ refers to the upper limit of the membership degree, and $\tilde{A}$ is the interval-valued fuzzy number.

The arithmetic operations of the interval-valued fuzzy numbers are as follows:

(1) Addition of two interval-valued fuzzy numbers:

$$\begin{aligned} \tilde{a} + \tilde{d} & = \left[ {(a_{1} ,a^{\prime}_{1} );a_{2} ;(a^{\prime}_{3} ,a_{3} )} \right] \times \left[ {(d_{1} ,d^{\prime}_{1} );d_{2} ;(d^{\prime}_{3} ,d_{3} )} \right] \\ & = \left[ {(a_{1} + d_{1} ,a^{\prime}_{1} + d^{\prime}_{1} );a_{2} + d_{2} ;(a^{\prime}_{3} + d^{\prime}_{3} ,a_{3} + d_{3} )} \right] \\ \end{aligned}$$

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(2) Subtraction of two interval-valued fuzzy numbers:

$$\begin{aligned} \mathop a\limits^{ \sim } - \mathop d\limits^{ \sim } & = \left[ {(\mathop a\nolimits_{1} ,\mathop {a^{\prime}}\nolimits_{1} );\mathop a\nolimits_{2} ;(\mathop {a^{\prime}}\nolimits_{3} ,\mathop a\nolimits_{3} )} \right] - \left[ {(\mathop d\nolimits_{1} ,\mathop {d^{\prime}}\nolimits_{1} );\mathop d\nolimits_{2} ;(\mathop {d^{\prime}}\nolimits_{3} ,\mathop d\nolimits_{3} )} \right] \\ & = \left[ {(\mathop a\nolimits_{1} - \mathop d\nolimits_{3} ,\mathop {a^{\prime}}\nolimits_{1} - \mathop {d^{\prime}}\nolimits_{3} );\mathop a\nolimits_{2} - \mathop d\nolimits_{2} ;(\mathop {a^{\prime}}\nolimits_{3} - \mathop {d^{\prime}}\nolimits_{1} ,\mathop a\nolimits_{3} - \mathop d\nolimits_{1} )} \right] \\ \end{aligned}$$

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(3) Multiplication of two interval-valued fuzzy numbers:

$$\begin{aligned} \tilde{a} \times \tilde{d} & = \left[ {(a_{1} ,a^{\prime}_{1} );a_{2} ;(a^{\prime}_{3} ,a_{3} )} \right] \times \left[ {(d_{1} ,d^{\prime}_{1} );d_{2} ;(d^{\prime}_{3} ,d_{3} )} \right] \\ & = \left[ {(a_{1} d_{1} ,a^{\prime}_{1} d^{\prime}_{1} );a_{2} d_{2} ;(a^{\prime}_{3} d^{\prime}_{3} ,a_{3} d_{3} )} \right] \\ \end{aligned}$$

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(4) Multiplication between a positive crisp number and an interval-valued fuzzy number:

$$\beta \cdot \tilde{a} = \beta \times [(a_{1} ,a^{\prime}_{1} );a_{2} ;(a^{\prime}_{3} ,a_{3} )] = [(\beta a_{1} ,\beta a^{\prime}_{1} );\beta a_{2} ;(\beta a^{\prime}_{3} ,\beta a_{3} )]$$

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(5) Division of two interval-valued fuzzy numbers:

$$\begin{aligned} \tilde{a} \div \tilde{d} = \left[ {(a_{1} ,a^{\prime}_{1} );a_{2} ;(a^{\prime}_{3} ,a_{3} )} \right] \div \left[ {(d_{1} ,d^{\prime}_{1} );d_{2} ;(d^{\prime}_{3} ,d_{3} )} \right] \hfill \\ = \left[ {(a_{1} /d_{3} ,a^{\prime}_{1} /d^{\prime}_{3} );a_{2} /d_{2} ;(a^{\prime}_{3} /d^{\prime}_{1} ,a_{3} /d_{1} )} \right] \hfill \\ \end{aligned}$$

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(6) Reciprocal of two interval-valued fuzzy numbers [12]:

$$(\tilde{a})^{ - 1} = \left[ {(a_{1} ,a^{\prime}_{1} );a_{2} ;(a^{\prime}_{3} ,a_{3} )} \right]^{ - 1} = \left[ {(1/a_{3} ,1/a^{\prime}_{3} );1/a_{2} ;(1/a^{\prime}_{1} ,1/a_{1} )} \right]$$

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(7) Exponentiation of two interval-valued fuzzy numbers [12]:

$$(\tilde{a})^{n} = \left[ {\left( {a_{1} ,\,\,a^{\prime}_{1} } \right);\,\,a_{2} ;\,\,\left( {a^{\prime}_{3} ,a_{3} } \right)} \right]^{n} = \left[ {\left( {(a_{1} )^{n} ,(a^{\prime}_{1} )^{n} } \right);\,\,(a_{2} )^{n} ;\,\,\left( {(a^{\prime}_{3} )^{n} ,(a_{3} )^{n} } \right)} \right]$$

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(8) Defuzzification of the interval-valued fuzzy number [32]:

$$f(\tilde{a}) = \frac{{a_{1} + a^{\prime}_{1} + 2a_{2} + a^{\prime}_{3} + a_{3} }}{6}$$

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See Tables 7, 8, 9, 10, 11, 12 and 13.

Table 7 The form of the interval-valued fuzzy numbers of the initial direct-influenced matrix

Table 8 The form of the interval-valued fuzzy numbers of the normalized direct-influenced matrix

Table 9 The form of the interval-valued fuzzy numbers of the total relation matrix

Table 10 The form of the interval-valued fuzzy numbers of the cause-effect relationship

Table 11 The form of the interval-valued fuzzy numbers of the initial decision-making matrix

Table 12 The form of the interval-valued fuzzy numbers of the normalized decision-making matrix

Table 13 The form of the interval-valued fuzzy numbers of the weighted decision-making matrix