Prioritizing healthcare waste disposal methods considering environmental health using an enhanced multi-criteria decision-making method

ABSTRACT

The Healthcare Waste Disposal Method Selection (HCWDMS) is a complicated problem due to multiple and often contradictory criteria with different importance degrees. Thus, decision-makers are restored to multi-criteria decision-making (MCDM) methods to prioritize and select the best HCW disposal methods. This study introduces an enhanced MCDM method to deal with the HCWDMS problem. To address the problem, a comprehensive list of criteria and HCW disposal methods are identified. All the criteria are categorized into four main criteria, and Fuzzy Analysis Hierarchy Process is used to determine the weights of considered criteria and sub-criteria. The study results show that environmental, economic, technical, and social criteria are the most important in selecting disposal methods, respectively. Moreover, the sub-criteria of ‘Health Risk’, ‘Release with health effects’, and ‘Capital cost’ have the highest importance, respectively. Additionally, the methods of ‘Microwave’, ‘Sterilization by autoclave’, and ‘Reverse polymerization’ have the highest priority, respectively.

KEYWORDS:

• Healthcare waste disposal
• MCDM
• fuzzy theory
• analysis hierarchy process
• VIKOR

1. Introduction

One of the critical issues that significantly impact the health of humans and animals is the disposal of large amounts of healthcare waste (HCW). Facilities such as hospitals, doctors’ offices, blood banks, medical laboratories, and research centers are the main sources of HCW production. Different types of HCW include blood, human or animal tissues, infectious diseases, waste from rooms of patients with contagious diseases, and discarded vaccines. If these HCWs are not properly disposed of, they threaten humans’ and animals’ health [1,2]. HCW is considered the second most hazardous waste after radioactive waste. Therefore, using a proper method in HCW disposal can prevent environmental and health disasters [3]. Contact with HCW may cause various human diseases, such as bacterial, viral, parasitic, or fungal [4]. Studies show that 2.5 million people, mainly children, die from HCW-related diseases yearly [5]. Mismanagement of HCWs has dangerous consequences for the environment and human health.

There are various methods for disposing of HCW, such as sanitary landfill, incineration, microwave, sterilization by autoclave, chemical disinfection, radiation treatment by NEWater process, encapsulation, compaction, reverse polymerization, and plasma pyrolysis. To prevent environmental pollution and many harmful diseases, choosing an effective and safe HCW disposal method is essential for human health and the environment [6]. Each HCW disposal method has various economic, environmental, technical, and social effects, and selecting the most suitable method is a challenging task that should be carefully done considering several factors. This led to the definition of a problem known as the Healthcare Waste Disposal Method Selection (HCWDMS).

The HCWDMS is a complex issue due to multiple and often contradictory criteria with different importance degrees in selecting the best disposal method. In this case, Multi-Criteria Decision Making (MCDM) methods are often used to choose the best HCW disposal method. In general, using MCDM to deal with a problem, some alternatives are prioritized by considering several criteria with different importance degrees. In the HCWDMS problem, the HCW disposal methods are considered alternatives that should be evaluated and ranked by MCDM methods considering some related criteria. Each HCW disposal method may be better than other methods in some criteria, but a method is selected as the best that has a relatively good situation in all the considered criteria.

This study addresses the HCWDMS problem in a case study and aims to evaluate and prioritize the HCW disposal methods considering a set of criteria. The possible disposal methods and the relevant criteria for the studied case are identified through the literature review and experts’ opinions. In total, 11 alternatives for HCW disposal methods and 30 criteria are determined, in which the experts propose one alternative and five criteria, and the rest are extracted from the existing literature. The identified criteria are categorized into four main criteria, i.e. economic, environmental, technical, and social. To prioritize the alternative HCW disposal methods, a revised version of the well-known ‘Vlse Kriterijumsk Optimizacija Kompromisno Resenje (VIKOR)’ method, named (RVIKOR), is proposed. Moreover, the fuzzy theory is used to deal with the ambiguity of the parameters of the studied case.

The main contributions of this study can be summarized as follows:

• Considering a comprehensive list of criteria for ranking HCW disposal methods, composed of 30 sub-criteria, including five new criteria that have not been considered in previous studies.

• Considering a comprehensive list of HCW disposal methods as alternatives containing one alternative that has not been discussed in previous research.

• Introducing a new MCDM method called fuzzy RVIKOR inspired by the well-known VIKOR method to address the HCWDMS problem.

The remainder of the paper is structured as follows. Section 2 reviews the literature on the HCWDMS where an MCDM method is used. Section 3 discusses the proposed research method. The case study is briefly presented in Section 4. The results of the implementation of the research method are presented in Section 5. Concluding remarks and suggestions for future research are presented in Section 6.

2. Literature review

The MCDM methods are used in various scopes, such as supplier selection (Chatterjee and Kar [7], project portfolio selection [8], and Green supply chain problem [9]. In the case of waste management, various researchers used MCDM methods in different scopes to determine the best disposal method. Chadderton et al. [10] used multi-criteria decision analysis to solve technology selection for food waste management in U.S. Army. The authors also listed the strengths and weaknesses of each technology in different situations. Rani et al. [11] considered 13 criteria for ranking four food waste treatment methods, namely Composting, Anaerobic Digestion, Incineration, and Landfill. They applied the single-valued neutrosophic-CRITIC-MULTIMOORA approach to determine the most suitable treatment method. Their study showed that Anaerobic Digestion is the best food waste treatment method. Mao et al. [12] used a method-based fuzzy DEMATEL to determine the best plastic solid waste treatment method. This study found that combining recycling and incineration was the best solid plastic waste treatment method. Zhou and Dan [13] applied a combination of AHP and TOPSIS to solve municipal solid waste treatment problems in the Tibet plateau area of China. The study results showed that the incineration method is better than the landfill method. Salamirad et al. [14] used the best – worst method (BWM) and behavioral TOPSIS methods to solve the industrial wastewater treatment technologies problem. The authors considered 12 criteria for ranking seven industrial wastewater treatment methods. The results show that using an integrated fixed-film activated sludge is the best industrial wastewater treatment method. A comprehensive overview of the application of MCDM methods for waste management can be found in recent review studies published by Garcia-Garcia [15] and Torkayesh et al. [16].

2.1. Ranking HCW disposal methods

This section reviews the relevant studies that addressed the HCWDMS problem using an MCDM method. The review aims to identify the HCW disposal methods and the essential criteria for prioritizing them.

Dursun et al. [17] used fuzzy multi-criteria group decision-making (MCGDM) to solve the HCWDMS problem in Turkey. They considered 4 HCW disposal methods: incineration, steam sterilization, microwave, and landfill. The results showed that the best method for HCW disposal is steam sterilization. Dursun et al. [18] used fuzzy MCGDM based on fuzzy measures and integrals to solve the HCWDMS problem in Istanbul, Turkey. They used the Ordered Weighted Averaging (OWA) method to properly carry out the process of collecting the opinions of the decision-makers. The results showed that steam sterilization is the best HCW disposal method in the considered case. Liu et al. [19] addressed the HCWDMS problem in Shanghai, China using the fuzzy VIKOR method. The considered alternatives were incineration, steam sterilization, microwave, and landfill. The results showed that incineration is not suitable for hospital solid waste. The results of this study also suggested that although landfill is an economical method, its harmful effects on the environment make it an improper method for HCW disposal. The study concluded that steam sterilization is the best method of HCW disposal method. Liu et al. [20] used the interval 2-tuple linguistic MULTIMOORA (ITL-MULMOORA) method to solve the HCWDMS problem in Shanghai, China. This study found that steam sterilization was the best method of HCW disposal. Liu et al. [21] proposed a new MCDM method by merging the fuzzy MULTIMOORA and 2-Tuple DEMATEL methods to solve the HCWDMS problem. This study identified the steam sterilization method as the best HCW disposal method among the four alternatives: incineration, steam sterilization, microwave, and landfill.

Kalhor et al. [22] used Analytical Hierarchy Process (AHP) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) to solve the HCWDMS problem in Qazvin, Iran. They found that the most important criterion in choosing the appropriate method of HCW disposal is the environmental criterion and the least important criterion is the financial criterion. Also, the microwave method was identified as the best method of HCW disposal. Lu et al. [23] discussed the HCWDMS problem in Shanghai, China. They proposed the Interval 2-Tuple Induced TOPSIS (ITI-TOPSIS) method to solve the problem. The study concluded that the environmental aspect is the most important criterion and the irradiation method is the best method of HCW disposal. Shi et al. [24] studied the HCWDMS problem and proposed Multi-Attributive Border Approximation Area Comparison (MABAC) to identify the best HCW disposal method in Shanghai, China. The results showed that the steam sterilization method is the best HCW disposal method among the four methods of incineration, steam sterilization, microwave, and landfill. Xiao [25] addressed the HCWDMS problem using a new MCDM method based on the D-number method. After evaluating the four forms of incineration (i.e. steam sterilization, microwave, and landfill), steam sterilization was chosen as the best method. Hinduja and Pandey [26] used Decision-Making Trial and Evaluation Laboratory (DEMATEL), fuzzy AHP, and fuzzy ANP methods to deal with the HCWDMS problem in India. The authors considered 12 criteria and 6 HCW disposal methods. Finally, steam sterilization was chosen as the best HCW disposal method. Liu et al. [27] presented a new Multi-Attribute Group Decision Making (MAGDM) method based on Intuitionistic Uncertain Linguistic (IUL) to solve the HCWDMS problem in Shanghai, China. The result of this study suggested that steam sterilization is the best HCW disposal method.

Geetha et al. [28] employed Intuitionistic Hesitant Fuzzy MULTIMOORA (IHF- MULTIMOORA) to solve the HCWDMS problem. They considered several alternatives, such as deep burial, incineration, autoclave, and chemical disinfection. The study concluded that the ‘deep burial’ method is better than other HCW disposal methods. Mishra et al. [29] proposed a new method called Evaluation Based on Distance from Average Solution (EDAS) in fuzzy mode to cope with the HCWDMS problem. They identified steam sterilization as the best method between the four methods of incineration, steam sterilization, microwave, and landfill.

Narayanamoorthy et al. [30] used Hesitant Fuzzy Subjective and Objective Weight Integrated Approach (HF-SOWIA) and Hesitant Fuzzy Multi-Objective Optimization based on Simple Ratio Analysis (HF-MOOSRA) to deal with the HCWDMS problem. The authors considered 6 criteria and 5 alternatives for selecting the HCW disposal method. The study concluded that using an autoclave is the best HCW disposal method. Ghram and Frikha [31] investigated the HCWDMS problem in Tunis city. They considered 8 alternatives of sterilization by autoclave, microwave disinfection, landfill, chemical disinfection in situ, encapsulation, physio-chemical disinfection off situ, treatment by NEWater Process, and incineration. Moreover, 9 criteria are considered: cost, waste residuals, noise, health effects, reliability, effectiveness, occupational hazards, and public acceptance. Using the fuzzy ARAS-H (F-ARAS-H) method, microwave disinfection was identified as the best HCW disposal method. Li et al. [32] used combinations of DEMATEL and TOPSIS methods to select the best HCW disposal method in Beijing, China, considering fuzzy interval numbers. They studied 4 HCW disposal methods, including incineration, steam sterilization, microwave, and landfill. Considering five criteria for sustainability, they showed steam sterilization as the best HCW disposal method.

Mi et al. [33] presented a hybrid MCDM method based on likelihood function and D-numbers to handle the HCWDMS problem. The authors considered 4 methods of incineration, steam sterilization, microwave, and landfill. The methods were ranked considering 8 criteria. The results showed that the best HCW disposal method is steam sterilization. Chen et al. [34] solved the HCWDMS problem using a new MCGDM method based on TODIM and Z-number methods. They considered 4 alternatives of landfill, autoclave, incineration, and microwave. The alternatives were ranked based on six criteria of cost per ton, waste residuals, health effects, technical reliability, treatment effectiveness, and public acceptance. The results showed that the autoclave is the best HCW disposal method. Liu et al. [35] discussed the HCWDMS problem using 5 criteria of technology, social acceptance, environmental protection, total cost, and health hazards. They used the Pythagorean Fuzzy Combined Compromise Solution (PF- COCOS) method to rank the alternatives and concluded that the autoclave is the best HCW disposal method. Chaurasiya and Jain [36] employed the Pythagorean Fuzzy Entropy Measure-Based Complex Proportional Assessment (PF-COPRAS) method to deal with the HCWDMS problem considering 5 HCW disposal methods of steam sterilization, microwave disinfection, plasma pyrolysis, chemical disinfection, and incineration. Considering 8 different criteria, they concluded steam sterilization is the best HCW disposal method. Saha et al. [37] addressed the HCWDMS problem by considering social, technical, economic, and environmental criteria. They determined the weight of the criteria through the Full Consistency Method (FUCOM). Then, 5 HCW disposal methods of chemical disinfection, microwave disinfection, incineration, autoclaving, and reverse polymerization were ranked using Double-Normalization-Based Multi-Aggregation (DNMA) and q-Rung Orthopair Fuzzy Information (q-ROFI). The results show that autoclaving is the most suitable HCW disposal method. Salimian and Mousavi [38] proposed a new MCGDM method based on Intuitionistic Fuzzy Sets (IFs) to cope with the HCWDMS problem considering 5 HCW disposal methods of incineration, steam sterilization, microwave, and landfill. The criteria used were cost of net per ton, waste residuals, health effect release, reliability, treatment effectiveness, and society admission. The weights of the considered criteria were calculated based on the Shannon entropy method with uncertain information. Finally, incineration was chosen as the best HCW disposal method. Puška et al. [39] proposed a full consistency method (FUCOM) and compromised the ranking of alternatives from distance to ideal solution (CRADIS) to rank healthcare waste incinerators in Bosnia and Herzegovina. They only considered the incineration method and ranked various types of healthcare waste incinerator machines using 16 sub-criteria. Wang and Wang [40] employed a combined compromise solution (CoCoSo) method to determine the best HCW methods in Shenyang, China, using the linguistic term with the weakened hedge. They used 8 criteria to select the best HCW disposal methods among 4 alternatives of incineration, landfill, steam sterilization, and microwave. The results showed that Steam sterilization is the best HCW disposal method.

Pamučar et al. [41] applied fuzzy rough numbers and Aczel – Alsina Function to select HCW treatment in Brčko District of Bosnia. The authors considered 12 criteria and 6 HCW disposal methods. Finally, the microwave was chosen as the best HCW disposal method, while the landfill method obtained the worst rank. Chen et al. [42] used Fermatean Fuzzy IWP-TOPSIS-GRA Multi-Criteria Group Analysis to rank the HCW disposal methods. They considered six criteria for ranking 4 HCW disposal methods of incineration, steam sterilization, microwave, and landfill. The results showed that steam sterilization was the best HCW disposal method.

2.2. Review findings and highlights

The literature review shows various criteria and alternatives for HCW disposal methods are considered in the previous studies. Moreover, various MCDM methods are used to address the HCWDMS problem. The review also shows that all the studies contain a real case, and the best HCW disposal method may differ in each region based on the region’s conditions. Table 1 summarizes the criteria and MCDM method used for ranking HCW disposal methods in the reviewed studies. Moreover, the considered HCW disposal methods in each research are reported in Table 2.

Table 1. Summary of effective criteria and solution methods for HCW disposal.

	Economic				Environmental														Technical							Social
Author(s)	Disposal cost	Operation and maintenance cost	Capital cost	The affordability of technology	Release with health effects	Compliance with national environmental laws	Consistency with WHO policies	Liquid residuals and environmental impacts	Solid residuals and environmental impacts	Air residuals and environmental impacts	Microbial inactivation	Health Risk	Noise	Odor	Water consumption	Energy consumption	toxic gas emissions	Volume reduction	Need for skilled operators	Level of automation	Treatment effectiveness	Reliability	Occupational hazards occurrence impact	Meets capacity requirements	Ability to treat wide range of infectious waste	Public Acceptance	Land Requirement	Technology acceptance	Adaptability to environmental policy	Cost acceptance	Method	Using fuzzy theory
Dursun et al. [17],		*	*		*			*	*	*			*	*				*	*	*	*	*	*			*	*		*		MCGDM	*
Dursun et al. [18]		*	*		*			*	*	*			*	*				*	*	*	*	*	*			*	*		*		MCGDM	*
Liu et al. [19]					*			*	*	*											*	*				*					VIKOR	*
Liu et al. [20]					*			*	*	*			*								*	*	*			*					ITL-MULTIMOORA
Liu et al. [21]					*			*	*	*			*								*	*	*			*					DEMATEL. MULTIMOORA	*
Kalhor et al. [22]			*		*			*	*	*				*						*	*	*				*	*				AHP. TOPSIS
Lu et al. [23]					*			*	*	*											*	*				*					2-Tuple Induced TOPSIS
Shi et al. [24]					*			*	*	*											*	*				*					MABAC	*
Xiao [25]					*			*	*	*			*								*	*	*			*					D number	*
Hinduja and Pandey [26]	*	*	*					*	*	*						*		*	*	*	*		*			*	*				DEMATEL. ANP. AHP	*
Liu et al. [27]								*	*	*							*				*	*				*					IUL-MAGDM
Geetha et al. [28]								*	*	*			*	*					*	*							*				MULTIMOORA	*
Mishra et al. [29]					*			*	*	*											*	*				*					EDAS	*
Narayanamoorthy et al. [30]					*							*	*													*					SOWIA. MOOSRA	*
Ghram and Frikha [31]					*			*	*	*			*									*	*								ARAS-H	*
Li et al. [32]			*		*			*	*	*		*	*	*					*		*	*	*			*	*			*	TOPSIS, DEMATEL	*
Mi et al. [33]					*			*	*	*			*								*	*	*			*					D-numbers
Chen et al. [34]					*			*	*	*											*	*				*					TODIM – Z number
Liu et al. [35]	*	*	*					*	*	*	*				*	*	*	*	*	*	*							*		*	PF-CoCoSo	*
Chaurasiya and Jain [36]	*							*	*								*		*	*	*					*	*				PF-COPRAS	*
Puška et al. [39]	*		*		*					*			*	*		*											*	*		*	FUCOM, CRADIS
Wang and Wang [40]	*	*	*		*			*	*	*			*								*	*	*			*					CoCoSo	*
Saha et al. [37]	*	*	*								*				*		*	*	*	*						*		*		*	FUCOM. DNMA	*
Salimian and Mousavi [38]					*			*	*	*											*	*				*					MCGDM	*
Pamučar et al. [41]	*	*	*	*	*	*		*	*	*		*				*				*	*					*		*			Rough numbers	*
Chen et al. [42]	*	*	*		*			*	*	*											*	*				*					TOPSIS	*
Current research	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	*	New method	*

Table 2. Summary of the identified HCW disposal methods in the reviewed literature.

Author(s)	Methods
	Sanitary landfill	Incineration	Microwave	Sterilization by autoclave	Chemical disinfection	Radiation	Treatment by NEWater process	Encapsulation	Compaction	Reverse polymerization	Plasma pyrolysis
Dursun et al. [17],	*	*	*	*
Dursun et al. [18]	*	*	*	*
Liu et al. [19]	*	*	*	*
Liu et al. [20]	*	*	*	*
Liu et al. [21]	*	*	*	*
Kalhor et al. [22]	*	*	*	*	*	*
Lu et al. [23]	*	*	*	*
Shi et al. [24]	*	*	*	*
Xiao [25]	*	*	*	*
Hinduja and Pandey [26]		*	*	*	*					*	*
Liu et al. [27]	*	*	*	*
Geetha et al. [28]	*	*	*	*	*
Mishra et al. [29]	*	*	*	*
Narayanamoorthy et al. [30]	*	*	*	*	*
Ghram and Frikha [31]		*	*	*	*		*	*
Li et al. [32]	*	*	*	*
Mi et al. [33]	*	*	*	*
Chen et al. [34]	*	*	*	*
Liu et al. [35]	*	*	*	*						*
Chaurasiya and Jain [36]		*	*	*	*						*
Puška et al. [39]		*
Wang and Wang [40]	*	*	*	*
Saha et al. [37]	*	*	*	*						*
Salimian and Mousavi [38]	*	*	*	*
Pamučar et al. [41]	*	*	*	*	*
Chen et al. [42]	*	*	*	*
Current research	*	*	*	*	*	*	*	*	*	*	*

In light of the performed literature review, in this research, a comprehensive set of criteria composed of 30 sub-criteria classified into four main criteria (i.e. economic, environmental, technical, and social) are considered for evaluating and prioritizing the HCW disposal methods. Moreover, this study benefits from a comprehensive set of alternative HCW disposal methods (i.e. 11 methods). Considering more criteria when choosing the best HCW disposal method means including various aspects in the decision-making and, consequently, a more reliable decision. Moreover, considering more alternatives gives the decision-makers a higher chance of finding a more suitable solution for the specific case.

In this study, 25 out of 30 sub-criteria are identified through the literature review, while 5 of the sub-criteria are new and suggested by the expert in the considered case study. The new sub-criteria are: ‘the affordability of technology’, ‘consistency with WHO policies’, ‘compliance with national environmental laws’, ‘meets capacity requirements’, and ‘ability to treat a wide range of infectious waste’. Concerning the HCW disposal methods, 10 out of 11 alternatives already exist in the literature, while one alternative (i.e. ‘compaction’) is proposed by the experts. In terms of the applied method, an enhanced version of the well-known VIKOR method, i.e. named RVIKOR, is proposed to rank the alternatives.

3. Research methodology

This section provides an overview of the applied methodology and research questions to address the HCWDMS problem.

3.1. Research questions

To reach the aim of the study and prioritize the HCW disposal methods in the case study, the primary research question of the study is formulated as follows.

• How can the HCW disposal methods in the considered case study be prioritized according to economic, environmental, technical, and social criteria?

To answer the main research question, finding an answer to the following sub-questions is necessary.

• What criteria and sub-criteria should be considered to choose the best HCW disposal methods?

• What is the important degree of each considered criterion?

• What HCW disposal methods could be considered for the case study?

• What is the score of each considered alternative in each considered sub-criterion?

• What is the ranking of the alternative HCW disposal methods using the RVIKOR method?

3.2. Steps of the applied methodology

In this study, the following five steps are performed to answer the research questions.

Step 1. Determining effective criteria and sub-criteria in evaluating and ranking HCW disposal methods. The criteria and sub-criteria are identified and defined through the literature review and experts’ opinions.

Step 2. Determining HCW disposal methods (alternatives). The alternatives are determined by experts based on the region’s conditions and the types of waste.

Step 3. Determining the weight of the considered criteria. Fuzzy Analysis Hierarchy Process (FAHP) is used for determining the criteria weight. In this study, a pairwise comparison matrix is also obtained through a questionnaire used in the FAHP. This questionnaire compares the importance of the considered criteria against each other, and each respondent chooses one of the answers shown in Table 3 in each comparison [43].

Table 3. Linguistic terms in the pairwise comparison matrix.

Linguistic variable	Fuzzy number
Just equal	(1,1,1)
Equally important	(1,1,3)
Weakly important	(1,3,5)
Essential or strongly important	(3,5,7)
Very strongly important	(5,7,9)
Extremely Preferred	(7,9,9)

Step 4. Determining the score of the considered alternatives in the considered criteria. In this step, a questionnaire is used to determine the score of each waste disposal method in each considered criterion (formation of the decision matrix) called decision matrix questionnaire. In this questionnaire, each respondent determines the score of each alternative in each criterion by selecting one of the 7-point Likert spectrum choices shown in Table 4 [44].

Table 4. The number of preferences in the decision matrix.

Score	Fuzzy number
Very weak	(0,1,2)
Weak	(1,2,3)
Moderately weak	(2,3.5,5)
Normal	(4,5,6)
Moderately good	(5,6.5,8)
Good	(7,8,9)
Very Good	(8,9,10)

Step 5. Ranking of HCW disposal methods using the RVIKOR method. The details of the proposed RVIKOR method are presented in section 3.4.

It is worth mentioning that the paired comparison questionnaire used in Step 3 and the decision matrix questionnaire used in Step 4 were completed by 12 experts composed of 6 university professors and 6 hospital staff with 10 years of experience in the field of HCW disposal management. The questionnaires are both standard questionnaires, and their validity and reliability have been confirmed by previous research [45].

3.3. Preliminaries

This section presents the methods and theories underlying the proposed methodology.

3.3.1. Fuzzy theory

Fuzzy theory is commonly used in MCDM studies to deal with the ambiguines of input parameters [46]. Triangular fuzzy numbers are used in this study, as described below.

Considering Ḿ₁ = (l₁, m₁, u₁) and Ḿ₂ = (l₂, m₂, u₂) as two triangular fuzzy numbers, the mathematical operations of these fuzzy numbers are presented as Equations (1)(1) $\mathrm{M}{ }_1+\mathrm{M}{ }_2=\left({\mathrm{l}}_1+{\mathrm{l}}_2,\hspace{0.17em}{\mathrm{m}}_1+{\mathrm{m}}_2,\hspace{0.17em}{\mathrm{u}}_1+{\mathrm{u}}_2\right)$(1) to (4(4) $\frac{{\tilde{\mathrm{M}}}_{1\hspace{0.17em}}}{{\tilde{\mathrm{M}}}_2}=\left(\mathrm{M}\mathrm{i}\mathrm{n}\left\{\frac{{\mathrm{l}}_1}{{\mathrm{l}}_2}\hspace{0.17em},\hspace{0.17em}\frac{{\mathrm{l}}_1}{{\mathrm{u}}_2},\frac{{\mathrm{u}}_1}{l_2}\hspace{0.17em},\hspace{0.17em}\frac{{\mathrm{u}}_1}{{\mathrm{u}}_2}\right\},\hspace{0.17em}\frac{{\mathrm{m}}_1}{{\mathrm{m}}_2}\hspace{0.17em},\mathrm{M}\mathrm{a}\mathrm{x}\left\{\frac{{\mathrm{l}}_1}{{\mathrm{l}}_2}\hspace{0.17em},\hspace{0.17em}\frac{{\mathrm{l}}_1}{{\mathrm{u}}_2},\frac{{\mathrm{u}}_1}{l_2}\hspace{0.17em},\hspace{0.17em}\frac{{\mathrm{u}}_1}{{\mathrm{u}}_2}\right\}\right)$(4) ).

(1) $\mathrm{M}{ }_1+\mathrm{M}{ }_2=\left({\mathrm{l}}_1+{\mathrm{l}}_2,\hspace{0.17em}{\mathrm{m}}_1+{\mathrm{m}}_2,\hspace{0.17em}{\mathrm{u}}_1+{\mathrm{u}}_2\right)$(1)

(2) $\mathrm{M}{ }_1-\mathrm{M}{ }_2=\left({\mathrm{l}}_1-u_2,\hspace{0.17em}{\mathrm{m}}_1-{\mathrm{m}}_2,\hspace{0.17em}{\mathrm{u}}_1-{\mathrm{l}}_2\right)$(2)

(3) ${\mathrm{M}}_1\times {\mathrm{M}}_2=\left(\begin{array}{c}\mathrm{M}\mathrm{i}\mathrm{n}\left\{{\mathrm{l}}_1\times {\mathrm{l}}_2,{\mathrm{u}}_1\times {\mathrm{u}}_2,{\mathrm{l}}_1\times {\mathrm{u}}_2,{\mathrm{u}}_1\times l_2\right\},\\ {\mathrm{m}}_1\times {\mathrm{m}}_2,\mathrm{M}\mathrm{a}\mathrm{x}\left\{{\mathrm{l}}_1\times {\mathrm{l}}_2,{\mathrm{u}}_1\times {\mathrm{u}}_2,{\mathrm{l}}_1\times {\mathrm{u}}_2,{\mathrm{u}}_1\times l_2\right\}\end{array}\right)$(3)

(4) $\frac{{\tilde{\mathrm{M}}}_{1\hspace{0.17em}}}{{\tilde{\mathrm{M}}}_2}=\left(\mathrm{M}\mathrm{i}\mathrm{n}\left\{\frac{{\mathrm{l}}_1}{{\mathrm{l}}_2}\hspace{0.17em},\hspace{0.17em}\frac{{\mathrm{l}}_1}{{\mathrm{u}}_2},\frac{{\mathrm{u}}_1}{l_2}\hspace{0.17em},\hspace{0.17em}\frac{{\mathrm{u}}_1}{{\mathrm{u}}_2}\right\},\hspace{0.17em}\frac{{\mathrm{m}}_1}{{\mathrm{m}}_2}\hspace{0.17em},\mathrm{M}\mathrm{a}\mathrm{x}\left\{\frac{{\mathrm{l}}_1}{{\mathrm{l}}_2}\hspace{0.17em},\hspace{0.17em}\frac{{\mathrm{l}}_1}{{\mathrm{u}}_2},\frac{{\mathrm{u}}_1}{l_2}\hspace{0.17em},\hspace{0.17em}\frac{{\mathrm{u}}_1}{{\mathrm{u}}_2}\right\}\right)$(4)

The distance of two fuzzy numbers of Ḿ₁ and Ḿ₂ is obtained by Equation (5)(5) $D\left({\tilde{\mathrm{M}}}_1,{\tilde{\mathrm{M}}}_2\right)=\sqrt{\frac{1}{3}\{{({\mathrm{l}}_1-{\mathrm{l}}_2)}^2+{({\mathrm{m}}_1-{\mathrm{m}}_2)}^2+{({\mathrm{u}}_1-{\mathrm{u}}_2)}^2\}}$(5) .

(5) $D\left({\tilde{\mathrm{M}}}_1,{\tilde{\mathrm{M}}}_2\right)=\sqrt{\frac{1}{3}\{{({\mathrm{l}}_1-{\mathrm{l}}_2)}^2+{({\mathrm{m}}_1-{\mathrm{m}}_2)}^2+{({\mathrm{u}}_1-{\mathrm{u}}_2)}^2\}}$(5)

Moreover, defuzzification of a fuzzy number is obtained using Equation (6)(6) $Defuzzy\left({\tilde{\mathrm{M}}}_1\right)=\frac{{\mathrm{l}}_1+4{\mathrm{m}}_1+{\mathrm{u}}_1}{6}$(6) .

(6) $Defuzzy\left({\tilde{\mathrm{M}}}_1\right)=\frac{{\mathrm{l}}_1+4{\mathrm{m}}_1+{\mathrm{u}}_1}{6}$(6)

3.3.2. FAHP method

AHP is a well-known MCDM method used in different studies for ranking alternatives [47]. In this study, fuzzy AHP is used to determine the weight of the criteria. The main steps of the FAHP method are as follows [48].

Step 1. Create a pairwise comparison matrix: suppose ${\tilde{p}}_{ij}$ is the value of comparing criterion i (ith row in pairwise comparison matrix) against criterion j (jth column in pairwise comparison matrix).

Step 2. Calculate fuzzy synthetic extent (${\tilde{S}}_i$) for each criterion using Equation (7)(7) ${\tilde{S}}_i=\sum _{j=1}^n{\tilde{p}}_{ij}\ast {\left[\sum _{i=1}^n\sum _{j=1}^n{\tilde{p}}_{ij}\right]}^{-1}$(7) , where n is the number of criteria.

(7) ${\tilde{S}}_i=\sum _{j=1}^n{\tilde{p}}_{ij}\ast {\left[\sum _{i=1}^n\sum _{j=1}^n{\tilde{p}}_{ij}\right]}^{-1}$(7)

Step 3. Calculate the magnitude of criterion i $\left({\tilde{S}}_{i\hspace{0.17em}}\right)$ against other criteria, using Equation (8)(8) $V=\left(S_i\ge S_k\right)=\left\{\begin{array}{c}1;m_i\ge m_k\\ 0;l_k\ge u_i\\ \frac{l_k-u_i}{\left(M_i-u_i\right)-\left(M_k-l_k\right)}otherwise\end{array}\right.$(8) , where $S_i=\left(l_i,m_i,u_i\right)$ and $S_k=\left(l_k,m_k,u_k\right)$.

(8) $V=\left(S_i\ge S_k\right)=\left\{\begin{array}{c}1;m_i\ge m_k\\ 0;l_k\ge u_i\\ \frac{l_k-u_i}{\left(M_i-u_i\right)-\left(M_k-l_k\right)}otherwise\end{array}\right.$(8)

Step 4. Compute the unnormalized weight of criterion i ($w_i^u$) using Equation (9)(9) $w_i^u=\underset{k=\mathrm{1.2.}\dots .n.k\ne i}{min}\left\{V\left(S_i\ge S_k\right)\right\}$(9) .

(9) $w_i^u=\underset{k=\mathrm{1.2.}\dots .n.k\ne i}{min}\left\{V\left(S_i\ge S_k\right)\right\}$(9)

Therefore, the unnormalized weight vector is given as Equation (10)(10) $W_{\hspace{0.17em}}^u={\left(\hspace{0.17em}{w}_1^u,\hspace{0.17em}{w}_2^u,\dots .w_n^u\hspace{0.17em}\right)}^{\hspace{0.17em}}\hspace{0.17em}$(10) .

(10) $W_{\hspace{0.17em}}^u={\left(\hspace{0.17em}{w}_1^u,\hspace{0.17em}{w}_2^u,\dots .w_n^u\hspace{0.17em}\right)}^{\hspace{0.17em}}\hspace{0.17em}$(10)

Step 5. Calculate the final weight vector by normalizing the weight vector as Equation (11)(11) $W={\left(\hspace{0.17em}{w}_1^{\hspace{0.17em}},\hspace{0.17em}{w}_2^{\hspace{0.17em}},\hspace{0.17em}\dots .,w_n^{\hspace{0.17em}}\hspace{0.17em}\right)}^{\hspace{0.17em}}$(11) .

(11) $W={\left(\hspace{0.17em}{w}_1^{\hspace{0.17em}},\hspace{0.17em}{w}_2^{\hspace{0.17em}},\hspace{0.17em}\dots .,w_n^{\hspace{0.17em}}\hspace{0.17em}\right)}^{\hspace{0.17em}}$(11)

3.3.3 Fuzzy VIKOR

The VIKOR is a popular MCDM method widely used in different field of study for evaluating and prioritizing alternatives [49]. The main steps of the fuzzy VIKOR method are presented below, where $\tilde{\mathrm{X}}$ is a fuzzy decision matrix with m alternative and n criteria as given in Equation (12)(12) $\tilde{\mathrm{X}}=\hspace{0.17em}{\left[{\tilde{x}}_{ij}\right]}_{m\times n}=\left[\begin{array}{ccc}{\tilde{x}}_{11}& \cdots & {\tilde{x}}_{1n}\\ ⋮& \ddots & ⋮\\ {\tilde{x}}_{m1}& \cdots & {\tilde{x}}_{mn}\end{array}\right]\mathrm{\forall }i.j$(12) .

(12) $\tilde{\mathrm{X}}=\hspace{0.17em}{\left[{\tilde{x}}_{ij}\right]}_{m\times n}=\left[\begin{array}{ccc}{\tilde{x}}_{11}& \cdots & {\tilde{x}}_{1n}\\ ⋮& \ddots & ⋮\\ {\tilde{x}}_{m1}& \cdots & {\tilde{x}}_{mn}\end{array}\right]\mathrm{\forall }i.j$(12)

Step 1. Determine the Fuzzy Positive Ideal Solution (FPIS) using Equation (13)(13) ${\widetilde{fpis}}_j=\left\{\begin{array}{cc}{max}_i{\tilde{x}}_{ij}& For\hspace{0.17em}the\hspace{0.17em}profit\hspace{0.17em}criterionj\\ {min}_i{\tilde{x}}_{ij}& For\hspace{0.17em}the\hspace{0.17em}cost\hspace{0.17em}criterionj\end{array}\right.\begin{array}{cc},& j=1,2,\dots ,n\end{array}$(13) , where${\widetilde{fpis}}_j=(l_j^{+},m_j^{+},u_j^{+})$ is the value of FPIS for jth criterion.

(13) ${\widetilde{fpis}}_j=\left\{\begin{array}{cc}{max}_i{\tilde{x}}_{ij}& For\hspace{0.17em}the\hspace{0.17em}profit\hspace{0.17em}criterionj\\ {min}_i{\tilde{x}}_{ij}& For\hspace{0.17em}the\hspace{0.17em}cost\hspace{0.17em}criterionj\end{array}\right.\begin{array}{cc},& j=1,2,\dots ,n\end{array}$(13)

Step 2. Obtain the normalized distance of each alternative from FPIS using Equation (14)(14) ${\tilde{d}}_{ij}=\left\{\begin{array}{cc}({\widetilde{fpis}}_j-{\tilde{x}}_{ij})/(u_j^{+}-l_j^{-})& For\hspace{0.17em}the\hspace{0.17em}profit\hspace{0.17em}criterionj\\ ({\tilde{x}}_{ij}-{\widetilde{fpis}}_j)/(u_j^{-}-l_j^{+})& For\hspace{0.17em}the\hspace{0.17em}cost\hspace{0.17em}criterionj\end{array}\right.$(14) , where ${\tilde{d}}_{ij}$ is the normalized distance of alternative i from FPIS in criterion j.

(14) ${\tilde{d}}_{ij}=\left\{\begin{array}{cc}({\widetilde{fpis}}_j-{\tilde{x}}_{ij})/(u_j^{+}-l_j^{-})& For\hspace{0.17em}the\hspace{0.17em}profit\hspace{0.17em}criterionj\\ ({\tilde{x}}_{ij}-{\widetilde{fpis}}_j)/(u_j^{-}-l_j^{+})& For\hspace{0.17em}the\hspace{0.17em}cost\hspace{0.17em}criterionj\end{array}\right.$(14)

Step 3. Obtain utility and regret indexes.

The utility index represents the total weighted distance of each alternative from FPIS (${\tilde{U}}_i$) and calculated using Equation (15)(15) ${\tilde{U}}_i=\sum _{j=1}^n(w_j\times {{\tilde{d}}^{+}}_{ij})$(15) .

(15) ${\tilde{U}}_i=\sum _{j=1}^n(w_j\times {{\tilde{d}}^{+}}_{ij})$(15)

The regret index represents the maximum distance of each alternative from FPIS in each criterion (${\tilde{R}}_i$) and calculated using Equation (16)(16) ${\tilde{R}}_i=\underset{j}{max}(w_j\times {{\tilde{d}}^{+}}_{ij})$(16) .

(16) ${\tilde{R}}_i=\underset{j}{max}(w_j\times {{\tilde{d}}^{+}}_{ij})$(16)

Step 4. Calculate the VIKOR index (${\tilde{Q}}_i$) for each alternative i using Equation (17)(17) ${\tilde{Q}}_i={\nu }_1\left[\frac{{\tilde{U}}_i-{\tilde{U}}^{+}}{U^u-U^l}\right]+{\nu }_2\left[\frac{{\tilde{R}}_i-{\tilde{R}}^{+}}{R^u-R^l}\right]$(17) .

(17) ${\tilde{Q}}_i={\nu }_1\left[\frac{{\tilde{U}}_i-{\tilde{U}}^{+}}{U^u-U^l}\right]+{\nu }_2\left[\frac{{\tilde{R}}_i-{\tilde{R}}^{+}}{R^u-R^l}\right]$(17)

$v_1+v_2=1$

${\tilde{U}}^{+}=\underset{i}{min}{\tilde{U}}_i^{+}$$U^u=\underset{i}{max}{U}_i^u$$U^l=\underset{i}{min}{U}_i^l$

${\tilde{R}}^{+}=\underset{i}{min}{\tilde{R}}_i^{+}$$R^u=\underset{i}{max}{R}_i^u$$R^l=\underset{i}{min}{R}_i^l$

Step 5. Prioritize the alternatives after obtaining the defuzzified value of the VIKOR index (Q_i). In this case, an alternative that has the lowest Q_i value has the highest priority.

3.4. The proposed Fuzzy RVIKOR

This research uses the fuzzy RVIKOR method to rank the HCW disposal methods. The RVIKOR method is an extension of the VIKOR method.

The VIKOR method only pays attention to Positive Ideal Solution (PIS). It gives a higher priority to alternatives with a low value in the parameters of the weighted sum of its distance from PIS (utility index) and the maximum distance from PIS in each criterion (regret index). However, the RVIKOR method simultaneously pays attention to PIS and Negative ideal solution (NIS). The RVIKOR method considers four parameters: two are related to PIS (utility and regret indexes), and two are associated with NIS (hate and escape) indexes. In this method, an alternative is better if the values of its utility and regret indexes are low and, at the same time, the weighted sum of the distances from NIS (hate index) and the maximum distance from NIS (escape index) are high. In other words, in the RVIKOR method, the closeness of an alternative to NIS reduces its attractiveness. On the other hand, attention to the regret and escape indexes simultaneously means an alternative receives a high priority if it has balanced scores in different criteria. The main steps of fuzzy RVIKOR are as follows.

Step 1. Obtain Fuzzy Positive Ideal Solution (FPIS) and Fuzzy Negative Ideal Solution (FNIS) using Equations (18)(18) ${\widetilde{fpis}}_j=\left\{\begin{array}{cc}{max}_i{\tilde{x}}_{ij}& For\hspace{0.17em}the\hspace{0.17em}profit\hspace{0.17em}criterionj\\ {min}_i{\tilde{x}}_{ij}& For\hspace{0.17em}the\hspace{0.17em}cost\hspace{0.17em}criterionj\end{array}\right.\begin{array}{cc},& j=1,2,\dots ,n\end{array}$(18) and (19(19) ${\widetilde{fnis}}_j=\left\{\begin{array}{cc}{min}_i{\tilde{x}}_{ij}& For\hspace{0.17em}the\hspace{0.17em}profit\hspace{0.17em}criterionj\\ {max}_i{\tilde{x}}_{ij}& For\hspace{0.17em}the\hspace{0.17em}cost\hspace{0.17em}criterionj\end{array}\right.\begin{array}{cc},& j=1,2,\dots ,n\end{array}$(19) ), where ${\widetilde{fpis}}_j=(l_j^{+},m_j^{+},u_j^{+})$ and ${\widetilde{fnis}}_j=(l_j^{-},m_j^{-},u_j^{-})$ are FPIS and FNIS in jth criteria, respectively.

(18) ${\widetilde{fpis}}_j=\left\{\begin{array}{cc}{max}_i{\tilde{x}}_{ij}& For\hspace{0.17em}the\hspace{0.17em}profit\hspace{0.17em}criterionj\\ {min}_i{\tilde{x}}_{ij}& For\hspace{0.17em}the\hspace{0.17em}cost\hspace{0.17em}criterionj\end{array}\right.\begin{array}{cc},& j=1,2,\dots ,n\end{array}$(18)

(19) ${\widetilde{fnis}}_j=\left\{\begin{array}{cc}{min}_i{\tilde{x}}_{ij}& For\hspace{0.17em}the\hspace{0.17em}profit\hspace{0.17em}criterionj\\ {max}_i{\tilde{x}}_{ij}& For\hspace{0.17em}the\hspace{0.17em}cost\hspace{0.17em}criterionj\end{array}\right.\begin{array}{cc},& j=1,2,\dots ,n\end{array}$(19)

Step 2. Obtain the normalized distance of each alternative from FPIS and FNIS using Equations (20)(20) ${{\tilde{d}}^{+}}_{ij}=\left\{\begin{array}{cc}({\widetilde{fpis}}_j-{\tilde{x}}_{ij})/(u_j^{+}-l_j^{-})& For\hspace{0.17em}the\hspace{0.17em}profit\hspace{0.17em}criterionj\\ ({\tilde{x}}_{ij}-{\widetilde{fpis}}_j)/(u_j^{-}-l_j^{+})& For\hspace{0.17em}the\hspace{0.17em}cost\hspace{0.17em}criterionj\end{array}\right.$(20) and (21(21) ${{\tilde{d}}^{-}}_{ij}=\left\{\begin{array}{cc}({\tilde{x}}_{ij}-{\widetilde{fnis}}_j)/(u_j^{+}-l_j^{-})& For\hspace{0.17em}the\hspace{0.17em}profit\hspace{0.17em}criterionj\\ ({\widetilde{fnis}}_j-{\tilde{x}}_{ij})/(u_j^{-}-l_j^{+})& For\hspace{0.17em}the\hspace{0.17em}cost\hspace{0.17em}criterionj\end{array}\right.$(21) ), where ${{\tilde{d}}^{+}}_{ij}$ and ${{\tilde{d}}^{+}}_{ij}$ are the normalized distance of alternative i in criterion j from PIS and NIS, respectively.

(20) ${{\tilde{d}}^{+}}_{ij}=\left\{\begin{array}{cc}({\widetilde{fpis}}_j-{\tilde{x}}_{ij})/(u_j^{+}-l_j^{-})& For\hspace{0.17em}the\hspace{0.17em}profit\hspace{0.17em}criterionj\\ ({\tilde{x}}_{ij}-{\widetilde{fpis}}_j)/(u_j^{-}-l_j^{+})& For\hspace{0.17em}the\hspace{0.17em}cost\hspace{0.17em}criterionj\end{array}\right.$(20)

(21) ${{\tilde{d}}^{-}}_{ij}=\left\{\begin{array}{cc}({\tilde{x}}_{ij}-{\widetilde{fnis}}_j)/(u_j^{+}-l_j^{-})& For\hspace{0.17em}the\hspace{0.17em}profit\hspace{0.17em}criterionj\\ ({\widetilde{fnis}}_j-{\tilde{x}}_{ij})/(u_j^{-}-l_j^{+})& For\hspace{0.17em}the\hspace{0.17em}cost\hspace{0.17em}criterionj\end{array}\right.$(21)

Step 3. Obtain the utility, hate, regret, and escape indexes.

The utility index represents the total weighted distance of each alternative from FPIS (${\tilde{U}}_i$) using Equation (22)(22) ${\tilde{U}}_i=\sum _{j=1}^n(w_j\times {{\tilde{d}}^{+}}_{ij})$(22) .

(22) ${\tilde{U}}_i=\sum _{j=1}^n(w_j\times {{\tilde{d}}^{+}}_{ij})$(22)

The hate index represents the total weighted distance of each alternative from FNIS (${\tilde{H}}_i$) using Equation (23)(23) ${\tilde{H}}_i=\sum _{j=1}^n(w_j\times {{\tilde{d}}^{-}}_{ij})$(23) .

(23) ${\tilde{H}}_i=\sum _{j=1}^n(w_j\times {{\tilde{d}}^{-}}_{ij})$(23)

The regret index represents the maximum distance of each alternative from FPIS in each criterion (${\tilde{R}}_i$) using Equation (24)(24) ${\tilde{R}}_i=\underset{j}{max}(w_j\times {{\tilde{d}}^{+}}_{ij})$(24) .

(24) ${\tilde{R}}_i=\underset{j}{max}(w_j\times {{\tilde{d}}^{+}}_{ij})$(24)

The escape index represents the maximum distance of each alternative from FNIS in each criterion (${\tilde{E}}_i$) using Equation (25)(25) ${\tilde{E}}_i={max}_j(w_j\times {{\tilde{d}}^{-}}_{ij})$(25) .

(25) ${\tilde{E}}_i={max}_j(w_j\times {{\tilde{d}}^{-}}_{ij})$(25)

Step 4. Calculate the RVIKOR index (${\tilde{Q}}_i^r$) using Equation (26)(26) ${\tilde{Q}}_i^r={\nu }_1\left[\frac{{\tilde{U}}_i-{\tilde{U}}^{+}}{U^u-U^l}\right]+{\nu }_2\left[\frac{{\tilde{R}}_i-{\tilde{R}}^{+}}{R^u-R^l}\right]-{\nu }_3\left[\frac{{\tilde{H}}_i-{\tilde{H}}^{+}}{H^u-H^l}\right]-{\nu }_4\left[\frac{{\tilde{E}}_i-{\tilde{E}}^{+}}{E^u-E^l}\right]$(26) . In this Equation, $v_1$, $.v_2$, $v_3$ and $v_3$ are the weight of normalized utility, regret, hate and escape indexes in the calculation RVIKOR index.

(26) ${\tilde{Q}}_i^r={\nu }_1\left[\frac{{\tilde{U}}_i-{\tilde{U}}^{+}}{U^u-U^l}\right]+{\nu }_2\left[\frac{{\tilde{R}}_i-{\tilde{R}}^{+}}{R^u-R^l}\right]-{\nu }_3\left[\frac{{\tilde{H}}_i-{\tilde{H}}^{+}}{H^u-H^l}\right]-{\nu }_4\left[\frac{{\tilde{E}}_i-{\tilde{E}}^{+}}{E^u-E^l}\right]$(26)

$v_1+v_2+v_3+v_4=1$

${\tilde{U}}^{+}=\underset{i}{min}{\tilde{U}}_i^{+}$$U^u=\underset{i}{max}{U}_i^u$$U^l=\underset{i}{min}{U}_i^l$

${\tilde{R}}^{+}=\underset{i}{min}{\tilde{R}}_i^{+}$$E^u=\underset{i}{max}{E}_i^u$$E^l=\underset{i}{min}{E}_i^l$

${\tilde{H}}^{+}=\underset{i}{min}{\tilde{H}}_i^{+}$$H^u=\underset{i}{max}{H}_i^u$$H^l=\underset{i}{min}{H}_i^l$

${\tilde{E}}^{+}=\underset{i}{min}{\tilde{E}}_i^{+}$$R^u=\underset{i}{max}{R}_i^u$$R^l=\underset{i}{min}{R}_i^l$

Step 5. Rank the alternatives based on the defuzzified value of the RVIKOR index ($Q_i^r$). Between the two alternatives, the one with the lowest value for $Q_i^r$has the highest priority. It means that the lower values for utility and regret indexes and higher values for hate and escape indexes of an alternative make it more favorable.

4. Case study

Selecting the HCW disposal method depends on the types and amount of produced HCWs [50]. The amount and variety of waste are different in various religions. This study deals with a real case study conducted in Tehran, Iran. Tehran province is an area of about 13,688 square kilometers located in the central north of the country. Tehran is the most populous province of Iran, with a population of 13,267,637 people (i.e. 12452,230 in urban areas and 814,698 in rural areas). The capital of this province is the 34^th most populous city in the world. This province has 177 hospitals, 1012 healthcare laboratories, 552 clinics, and 172 animal clinics. Figure 1 shows the location of this province on the map. This city produces 11 thousand tons of normal waste and approximately 80 to 110 tons of HCW daily.

Figure 1. Location of Tehran Province on the map.

The large population and the high number of healthcare centers cause the creation of a considerable amount of HCWs. Determining a proper HCW disposal method is challenging for healthcare and municipality managers. Considering economic, environmental, technical, and social aspects when selecting the HCW disposal method makes it an even more complex decision-making problem. For example, an HCW disposal method may have a better economic effect than another method while having a worse environmental or social impact, and vice versa. Considering the complex nature of the HCWDMS, the decision-makers in the municipality are looking for a reliable decision-making approach to prioritize and choose the most suitable HCW disposal method among the possible alternatives while considering some important criteria.

5. Methodology implementation and results

This research follows the explained methodology and applies the proposed RVIKOR method to prioritize the alternative HCW disposal methods in the case study, considering a set of criteria. The implementation steps and the obtained results are presented in the following sections.

5.1. Identifying criteria affecting the HCWDMS

After reviewing the literature and consulting with experts, 30 sub-criteria were identified that were classified into 4 main criteria of ‘economic’, ‘environmental’, ‘technical’, and ‘social’. The criteria and sub-criteria and their notations are shown in Figure 2. It is worth noting that 25 of the sub-criteria are identified through the literature review, and five sub-criteria are suggested by the experts in the studied case. The newly introduced sub-criteria are ‘The affordability of technology’, ‘Consistency with WHO policies’, ‘Compliance with national environmental laws’, ‘Meets capacity requirements’, and ‘Ability to treat a wide range of infectious waste’.

Figure 2. The identified criteria and sub-criteria.

5.2. Identifying HCW disposal methods

After reviewing the literature and consulting with experts, 11 waste disposal methods as identified and considered decision alternatives. Table 5 shows these alternatives and their advantages and disadvantages. Ten disposal methods are identified from the literature, and the experts in the case study suggest one additional method (i.e. ‘compaction’).

Table 5. Main advantages and shortcomings of the different methods for HCW.

No.	Alternative	Advantage	Disadvantage
$A_1$	Sanitary landfill	(1) Low cost, (2) Easy operation	(1) The need to access a wide place for HCW disposal, (2) Soil pollution, (3) Water pollution
$A_2$	Incineration	(1) Application for all types of waste, (2)Unrecognizability of all types of waste after treatment, (3) Volume reduction, (4) Energy recovery, (5) Fully sterilized waste	(1) Very expensive, (2) Finding a disposal site, (3)Emission of toxic gases into the air, (4) Heavy metals in the remaining ash, (5) Dioxin creation
$A_3$	Microwave	(1) Easy technology, (2) Volume reduction, (3) No liquid sewage, (4) Minimal emissions, (5) Compatible with the environment	(1) High cost, (2) Not suitable for all types of waste, (3) Use of press shredder, (4) Unpleasant smells
$A_4$	Sterilization by autoclave	(1) Long lifespan, (2) Compatible with the environment, (3) There is no risk of emission, (4) Low cost, (5)Convenient technology	(1) Need to dry, (2) Stink smell, (3) Not suitable for all types of waste, (4) Need to crush
$A_5$	Chemical disinfection	(1) It is suitable for disinfecting liquid waste such as blood, urine, feces, and hospital sewage, (2) Not reducing the amount of waste	(1) Environmental problems, (2) Only solid waste disinfects, (3) Need for trained personnel, (4) Dangerous disinfectants are required, (5) Possibility of penetration of chemical aerosols into the lower part of the respiratory system and causing respiratory problems
$A_6$	Radiation	(1) Low executive operation, (2) Silent technology, (3) Technology is automatic and requires a short time for an operator, (4) Nothing is added to waste.	(1) Personnel must be shielded from radiation, (2) If complete shielding is not part of the device design, the system requires concrete shielding that is several feet thick or an underground structure, either of which is costly, (3) Technology does not reduce the volume of waste, (4) The need for a shredder
$A_7$	Treatment by NEWater process	(1) Relatively low chamber temperature, (2) Volume reduction, (3) Short purification time	(1) High cost, (2) Cannot be used for all wastes, (3) Creating toxic chloride compounds, (4) Relatively low capacity
$A_8$	Encapsulation	(1) Simple and easy, (2) Low cost, (4) The Low risk of groundwater pollution	(1) High cost, (2) Not recommended for non-sharp infectious waste.
$A_9$	Compaction	(1) Volume reduction, (2) Low manpower required	(1) Not compatible with the environment, (2) It is proper for only solid wastes
$A_{10}$	Reverse polymerization	(1) Accurate process control, (2) The reaction occurs at a relatively low temperature in the chamber, (3) Oxidation of waste does not occur, (4) No production of toxic gases, (5) Reduction of waste volume, (6) Low energy consumption	(1) Very high cost, (2) leads to the formation of undesirable products such as dioxins or furans.
$A_{11}$	Plasma pyrolysis	(1) Suitable for all types of waste, (2) Requires less space, (3) Environmentally friendly, (4) No need for fume hoods, (5) Very low toxic residue, (6) No need for separation, (7) Energy recovery, (8) Volume reduction	(1) Need for technical people, (2) High cost

5.3. Obtaining the weight of the identified criteria

The FAHP method is used to obtain the weights of the criteria. First, the main criteria were compared with each other, and their weights were calculated using the FAHP method. Then, related sub-criteria in each main criterion were compared with each other, and their weights were calculated using the FAHP method. The weight of each sub-criterion is obtained by the product of the weight of the related main criterion, the weight of the sub-criterion in each main criterion and the number of sub-criteria in the related criterion. Table 6 shows the criteria and sub-criteria weights. The results show that the most important criteria are ‘Environmental’, ‘Economic’, ‘Technical’, and ‘Social’, respectively. Moreover, sub-criteria of ‘Health Risk’, ‘Release with health effects’, and ‘Capital cost’ have the highest importance among other sub-criteria, respectively.

Table 6. The weight of the identified criteria and sub-criteria.

Criteria.	Weight:	Sub-criteria	Weight	Final normalized Weight	Cost or benefit	Criteria.	Weight:	Sub-criteria	Weight	Final normalized Weight	Cost or benefit
C1	0.268	C1,1	0.183	0.0247	cost	C2	0.305	C2,12	0.064	0.0343	cost
		C1,2	0.281	0.0379	cost			C2,13	0.096	0.0515	cost
		C1,3	0.401	0.054	cost			C2,14	0.035	0.0188	benefit
		C1,4	0.135	0.0182	benefit	C3	0.239	C3,1	0.145	0.0305	cost
C2	0.305	C2,1	0.104	0.0558	cost			C3,2	0.101	0.0212	benefit
		C2,2	0.065	0.0349	benefit			C3,3	0.114	0.024	benefit
		C2,3	0.056	0.03	benefit			C3,4	0.144	0.0303	benefit
		C2,4	0.083	0.0445	cost			C3,5	0.135	0.0284	cost
		C2,5	0.083	0.0445	cost			C3,6	0.158	0.0332	benefit
		C2,6	0.079	0.0424	cost			C3,7	0.202	0.0425	benefit
		C2,7	0.085	0.0456	benefit	C4	0.189	C4,1	0.199	0.0236	benefit
		C2,8	0.109	0.0585	cost			C4,2	0.221	0.0262	cost
		C2,9	0.037	0.0199	cost			C4,3	0.126	0.015	benefit
		C2,10	0.04	0.0215	cost			C4,4	0.246	0.0292	benefit
		C2,11	0.064	0.0343	cost			C4,5	0.208	0.0247	benefit

5.4. Determining the score of each alternative in each criterion

The score of each alternative in each criterion (decision matrix) is shown in Table 7. The second questionnaire was used to collect the data required for the decision matrix. The arithmetic average of experts’ answers is considered the final decision matrix. The defuzzification of the decision matrix values shows that each HCW disposal method obtained the best score in one or more sub-criteria. However, finding an HCW disposal method with a good score in all sub-criteria is crucial and lies within the defined research question of the study. Figure 3 shows the HCW disposal methods with the best score in each sub-criterion. The results show that two HCW disposal methods obtained the best score in some sub-criteria simultaneously.

Figure 3. The best HCW disposal method in each sub-criterion.

Table 7. Decision matrix.

	A1	A2	A3	A4	A5	A6	A7	A8	A9	A10	A11
C1,1	(3,4.25,5.5)	(4.83,5.92,7)	(3.5,4.63,5.75)	(4.58,5.88,7.17)	(7,8.17,9.33)	(7.25,8.38,9.5)	(4.75,6.13,7.5)	(3.33,4.5,5.67)	(3.67,4.75,5.83)	(5.75,7.13,8.5)	(4.58,5.88,7.17)
C1,2	(1.75,3.13,4.5)	(1.08,2.13,3.17)	(3.33,4.5,5.67)	(7.5,8.58,9.67)	(8,9,10)	(7.5,8.58,9.67)	(7,8.17,9.33)	(1.83,3.25,4.67)	(1.83,3.25,4.67)	(6,7.33,8.67)	(7.25,8.38,9.5)
C1,3	(1.67,3,4.33)	(1.67,2.75,3.83)	(4.08,5.13,6.17)	(4.08,5.13,6.17)	(4.58,5.88,7.17)	(8,9,10)	(8,9,10)	(4.5,5.75,7)	(7.25,8.38,9.5)	(4.75,6.13,7.5)	(4.75,6.13,7.5)
C1,4	(5.25,6.71,8.17)	(1.75,3.13,4.5)	(1.25,2.38,3.5)	(6,7.33,8.67)	(1.25,2.38,3.5)	(1,2,3)	(1.33,2.5,3.67)	(1.42,2.63,3.83)	(2.67,4,5.33)	(1.92,3.38,4.83)	(1.17,2.25,3.33)
C2,1	(8,9,10)	(7.5,8.58,9.67)	(1.92,3.38,4.83)	(1.58,2.63,3.67)	(8,9,10)	(7.25,8.38,9.5)	(7.5,8.58,9.67)	(4.83,6.25,7.67)	(7.5,8.58,9.67)	(2.33,3.5,4.67)	(1.75,3.13,4.5)
C2,2	(1.33,2.5,3.67)	(1.25,2.38,3.5)	(5.5,6.92,8.33)	(7.42,8.46,9.5)	(1,2,3)	(1,2,3)	(4.33,5.5,6.67)	(1.42,2.63,3.83)	(2.33,3.75,5.17)	(5.5,6.92,8.33)	(4.33,5.5,6.67)
C2,3	(1,2,3)	(1.17,2.25,3.33)	(5.5,6.92,8.33)	(5.5,6.92,8.33)	(1,2,3)	(1.25,2.38,3.5)	(4.17,5.25,6.33)	(2.67,4,5.33)	(2.33,3.75,5.17)	(6,7.33,8.67)	(5.5,6.92,8.33)
C2,4	(8,9,10)	(4.92,6.38,7.83)	(1.67,3,4.33)	(1.17,2.25,3.33)	(8,9,10)	(3.5,4.63,5.75)	(3.67,4.75,5.83)	(4.83,6.25,7.67)	(3.5,4.63,5.75)	(1.75,3.13,4.5)	(1.83,3.25,4.67)
C2,5	(8,9,10)	(7.5,8.58,9.67)	(1.75,3.13,4.5)	(3.5,4.63,5.75)	(8,9,10)	(4.67,6,7.33)	(7.25,8.38,9.5)	(7.25,8.38,9.5)	(7,8.17,9.33)	(2.58,4,5.42)	(3.33,4.5,5.67)
C2,6	(7.5,8.58,9.67)	(5.75,7.13,8.5)	(1.83,2.88,3.92)	(1.67,3,4.33)	(4.75,6.13,7.5)	(1.92,3.38,4.83)	(8,9,10)	(3.67,4.75,5.83)	(4.75,6.13,7.5)	(1.67,3,4.33)	(1.83,3.25,4.67)
C2,7	(1,2,3)	(6.5,7.75,9)	(4.08,5.13,6.17)	(4.42,5.63,6.83)	(6,7.33,8.67)	(2.5,3.88,5.25)	(2.33,3.75,5.17)	(1.25,2.38,3.5)	(1.42,2.63,3.83)	(4.33,5.5,6.67)	(4.33,5.5,6.67)
C2,8	(8,9,10)	(7.75,8.79,9.83)	(3.33,4.5,5.67)	(3.67,4.75,5.83)	(7,8.17,9.33)	(4.83,6.25,7.67)	(4.5,5.75,7)	(7.25,8.38,9.5)	(8,9,10)	(3.5,4.63,5.75)	(3.17,4.38,5.58)
C2,9	(3.5,4.63,5.75)	(2.92,4.38,5.83)	(4.08,5.13,6.17)	(4.08,5.13,6.17)	(1.75,3.13,4.5)	(1.75,3.13,4.5)	(3.33,4.5,5.67)	(4.75,6.13,7.5)	(4.92,6.38,7.83)	(3.67,4.75,5.83)	(3.83,4.88,5.92)
C2,10	(7.25,8.38,9.5)	(5.5,6.92,8.33)	(3.67,4.75,5.83)	(1.67,2.75,3.83)	(7.25,8.38,9.5)	(3.33,4.5,5.67)	(4.75,6.13,7.5)	(4.75,6.13,7.5)	(7.25,8.38,9.5)	(1.67,3,4.33)	(3.5,4.63,5.75)
C2,11	(1.75,3.13,4.5)	(1.75,3.13,4.5)	(3.17,4.38,5.58)	(5.25,6.71,8.17)	(4.75,6.13,7.5)	(1.75,3.13,4.5)	(4.75,6.13,7.5)	(7,8.17,9.33)	(6.75,7.96,9.17)	(3.67,4.75,5.83)	(3.83,4.88,5.92)
C2,12	(1.83,3.25,4.67)	(3,4.38,5.75)	(2.17,3.63,5.08)	(4.08,5.13,6.17)	(4.67,6,7.33)	(8,9,10)	(3.5,4.63,5.75)	(7.25,8.38,9.5)	(8,9,10)	(1.33,2.38,3.42)	(1.92,3.38,4.83)
C2,13	(7,8.17,9.33)	(5.75,7.13,8.5)	(1.75,3.13,4.5)	(1.08,2.13,3.17)	(8,9,10)	(3.5,4.63,5.75)	(4.75,6.13,7.5)	(3.33,4.5,5.67)	(4.83,6.25,7.67)	(1.58,2.88,4.17)	(1.83,3.25,4.67)
C2,14	(1,2,3)	(6.25,7.54,8.83)	(7.08,8.13,9.17)	(3.58,4.63,5.67)	(1,2,3)	(1,2,3)	(4.33,5.5,6.67)	(2.33,3.75,5.17)	(4.08,5.13,6.17)	(4.92,6.38,7.83)	(5.75,7.13,8.5)
C3,1	(1.92,3.38,4.83)	(1.58,2.63,3.67)	(3.67,4.75,5.83)	(4.92,6.38,7.83)	(4.92,6.38,7.83)	(7.5,8.58,9.67)	(7,8.17,9.33)	(1.67,3,4.33)	(1.75,3.13,4.5)	(8,9,10)	(8,9,10)
C3,2	(6,7.33,8.67)	(5.5,6.92,8.33)	(4.08,5.13,6.17)	(7.17,8.25,9.33)	(1,2,3)	(1,2,3)	(2.67,4,5.33)	(1.25,2.38,3.5)	(1,2,3)	(5.75,7.13,8.5)	(4.42,5.63,6.83)
C3,3	(1,2,3)	(6.25,7.54,8.83)	(7.25,8.38,9.5)	(4.17,5.63,7.08)	(1,2,3)	(1.42,2.63,3.83)	(2.5,3.88,5.25)	(1.25,2.38,3.5)	(1.25,2.38,3.5)	(1.67,3,4.33)	(5.25,6.71,8.17)
C3,4	(2.5,3.88,5.25)	(5.75,7.13,8.5)	(4.25,5.38,6.5)	(4.25,5.38,6.5)	(1.25,2.38,3.5)	(4.33,5.5,6.67)	(1.17,2.25,3.33)	(1,2,3)	(1.25,2.38,3.5)	(4.5,5.75,7)	(4.17,5.25,6.33)
C3,5	(7.5,8.58,9.67)	(5.75,7.13,8.5)	(1.92,3.38,4.83)	(3.67,4.75,5.83)	(8,9,10)	(8,9,10)	(6.5,7.75,9)	(4.83,6.25,7.67)	(4.67,6,7.33)	(3.33,4.5,5.67)	(4.58,5.88,7.17)
C3,6	(1,2,3)	(2.67,4,5.33)	(5.5,6.92,8.33)	(6.75,7.79,8.83)	(1.17,2.25,3.33)	(1,2,3)	(1,2,3)	(1.5,2.75,4)	(1.33,2.5,3.67)	(7.75,8.79,9.83)	(5.75,7.13,8.5)
C3,7	(2.17,3.63,5.08)	(4.25,5.38,6.5)	(6.25,7.54,8.83)	(6.83,7.92,9)	(1,2,3)	(4.25,5.38,6.5)	(2.17,3.63,5.08)	(1,2,3)	(2.33,3.75,5.17)	(7.08,8.13,9.17)	(5.25,6.71,8.17)
C4,1	(1,2,3)	(1.25,2.38,3.5)	(4.25,5.38,6.5)	(5.5,6.92,8.33)	(1.33,2.5,3.67)	(4.17,5.25,6.33)	(5.75,7.13,8.5)	(2.33,3.75,5.17)	(1.33,2.5,3.67)	(4.17,5.25,6.33)	(4.33,5.5,6.67)
C4,2	(8,9,10)	(7,8.17,9.33)	(1.92,3,4.08)	(2.42,3.88,5.33)	(3.33,4.5,5.67)	(7.25,8.38,9.5)	(3.33,4.5,5.67)	(4.5,5.75,7)	(4.58,5.88,7.17)	(1.67,2.75,3.83)	(1.75,3.13,4.5)
C4,3	(1,2,3)	(1,2,3)	(5.5,6.92,8.33)	(4.25,5.38,6.5)	(1.33,2.5,3.67)	(2.33,3.75,5.17)	(2.33,3.75,5.17)	(1.33,2.5,3.67)	(1,2,3)	(4.17,5.25,6.33)	(2.33,3.75,5.17)
C4,4	(1,2,3)	(1.17,2.25,3.33)	(5.75,7.13,8.5)	(6.75,7.79,8.83)	(1.33,2.5,3.67)	(4.33,5.5,6.67)	(2.5,3.88,5.25)	(2.33,3.75,5.17)	(4.17,5.25,6.33)	(6.17,7.25,8.33)	(5.25,6.71,8.17)
C4,5	(1,2,3)	(1,2,3)	(6,7.33,8.67)	(4.17,5.25,6.33)	(1.5,2.75,4)	(2.5,3.88,5.25)	(2.33,3.75,5.17)	(1.25,2.38,3.5)	(1.5,2.75,4)	(4.17,5.25,6.33)	(4.42,5.63,6.83)

5.5. Ranking the HCW disposal methods

In this section, the results of the implementation of the RVIKOR method in ranking HCW disposal methods are presented. In calculating the RVIKOR index (${\tilde{Q}}_i^r$), the weight of all parameters of normalized utility, regret, hate, and escape indexes are considered equal ($v_1=v_2=v_3=v_4=0.25$). Table 8 shows the results of the implementation of the RVIKOR method. The results show that the alternatives of ‘Microwave’, ‘Sterilization by autoclave’, and ‘Reverse polymerization’ have the highest priority, respectively. In addition, the alternative of ‘Chemical disinfection’ and ‘Compaction’ have the lowest priority, respectively.

Table 8. Results of the RVIKOR method.

Alternative		Fuzzy RVIKOR
	${\tilde{U}}_i$	${\tilde{R}}_i$	${\tilde{H}}_i$	${\tilde{E}}_i$	${\tilde{Q}}_i^r$	Rank
Sanitary landfill	(0.233,0.523,0.813)	(0.029,0.042,0.058)	(−0.069,0.202,0.473)	(0.024,0.039,0.054)	(−0.431,0.128,0.662)	6
Incineration	(0.098,0.395,0.693)	(0.025,0.04,0.057)	(0.051,0.33,0.609)	(0.027,0.041,0.054)	(−0.513,0.043,0.579)	5
Microwave	(−0.158,0.141,0.442)	(0.006,0.015,0.029)	(0.302,0.584,0.865)	(0.021,0.039,0.057)	(−0.748,-0.186,0.354)	1
Sterilization by autoclave	(−0.123,0.168,0.46)	(0.018,0.027,0.036)	(0.284,0.557,0.829)	(0.029,0.042,0.056)	(−0.665,-0.135,0.357)	2
Chemical disinfection	(0.298,0.586,0.876)	(0.029,0.042,0.056)	(−0.132,0.139,0.409)	(0.017,0.03,0.044)	(−0.345,0.204,0.715)	11
Radiation	(0.185,0.478,0.772)	(0.027,0.041,0.054)	(−0.027,0.248,0.522)	(0.016,0.029,0.044)	(−0.412,0.151,0.66)	8
Treatment by NEWater process	(0.184,0.482,0.781)	(0.027,0.041,0.054)	(−0.037,0.243,0.523)	(0.011,0.028,0.047)	(−0.427,0.158,0.693)	9
Encapsulation	(0.189,0.488,0.789)	(0.021,0.034,0.054)	(−0.044,0.237,0.518)	(0.014,0.026,0.039)	(−0.407,0.141,0.68)	7
Compaction	(0.224,0.521,0.819)	(0.025,0.04,0.058)	(−0.075,0.204,0.482)	(0.014,0.024,0.035)	(−0.349,0.191,0.716)	10
Reverse polymerization	(−0.107,0.191,0.491)	(0.016,0.023,0.038)	(0.253,0.534,0.814)	(0.022,0.037,0.056)	(−0.669,-0.119,0.414)	3
Plasma pyrolysis	(−0.076,0.229,0.535)	(0.017,0.027,0.038)	(0.209,0.496,0.783)	(0.023,0.04,0.058)	(−0.661,-0.095,0.43)	4

5.6. Sensitivity analysis

In this section, a sensitivity analysis is performed to analyze the robustness of the proposed fuzzy RVIKOR method. In the sensitivity analysis, variations in the results are analyzed when the value of a parameter is changed. In the current sensitivity analysis, the change in the ranking of the alternatives is analyzed when the weight of the sub-criteria is changed. As mentioned previously, 30 sub-criteria are considered to assess the alternatives. In total, 61 scenarios are considered for the sensitivity analysis. In the 1^st to 30^th scenarios, the weight of one sub-criteria is increased by 5%, and the weight of the other sub-criteria is updated accordingly. Then, the ranking of the alternatives is obtained based on the new weights. For example, in the first scenario, the weight of the first sub-criterion is increased by 5%, and the alternatives are ranked based on the new weights. In the 31^st to 60^th scenarios, the weight of the first to 30^th sub-criteria is decreased by 5%. In the 61^st scenario, the weights of the sub-criteria are considered as those shown in Table 6.

Figure 4 shows the priorities of HCW disposal methods under 61 performed scenarios. In this figure k^th scenario is shown as S_k. The results show that the number of changes in the ranking of the alternatives is insignificant. The robustness of the proposed RVIKOR method can be better argued, knowing that the five top priorities in all scenarios remain unchanged.

Figure 4. Ranking HCW disposal methods under 61 scenarios.

5.7. Comparative analysis

As mentioned previously, the fuzzy RVIKOR originated from the fuzzy VIKOR method. Table 9 compares the results of the fuzzy VIKOR and fuzzy RVIKOR methods. This table shows the defuzzied values of utility (U_i), regret (R_i), hate (H_i), escape (E_i), RVIKOR ($Q_i^r$) and VIKOR (Q_i) indexes.

Table 9. Comparing the results of fuzzy RVIKOR and fuzzy VIKOR methods.

Alternative	Defuzzified parameters in fuzzy RVIKOR						Defuzzified parameters in fuzzy VIKOR
	Ui	Ri	Hi	Ei	$Q_i^r$	Rank	Ui	Ri	Qi	Rank
Sanitary landfill	0.523	0.0425	0.202	0.039	0.1238	6	0.523	0.0425	0.4375	10
Incineration	0.3952	0.0403	0.33	0.0408	0.0397	5	0.3952	0.0403	0.3507	5
Microwave	0.1413	0.0158	0.5838	0.039	−0.1897	1	0.1413	0.0158	0	1
Sterilization by autoclave	0.1682	0.027	0.5568	0.0422	−0.1413	2	0.1682	0.027	0.1203	3
Chemical disinfection	0.5863	0.0422	0.1388	0.0302	0.1977	11	0.5863	0.0422	0.464	11
Radiation	0.4782	0.0408	0.2478	0.0293	0.142	8	0.4782	0.0408	0.3943	7
Treatment by NEWater process	0.4822	0.0408	0.243	0.0283	0.1497	9	0.4822	0.0408	0.3972	8
Encapsulation	0.4883	0.0352	0.237	0.0262	0.1395	7	0.4883	0.0352	0.3512	6
Compaction	0.5212	0.0405	0.2038	0.0242	0.1885	10	0.5212	0.0405	0.4147	9
Reverse polymerization	0.1913	0.0243	0.5338	0.0377	−0.1218	3	0.1913	0.0243	0.1018	2
Plasma pyrolysis	0.2292	0.0272	0.496	0.0402	−0.1018	4	0.2292	0.0272	0.1447	4

According to Table 9, the ranking of alternatives by fuzzy VIKOR method shows that ‘Microwave’, ‘Reverse polymerization’, and ‘Sterilization by autoclave’ methods have the highest priority, respectively. Moreover, the alternative of ‘Chemical disinfection’ and ‘Sanitary landfill’ have the lowest priority, respectively. The results show that fuzzy RVIKOR and fuzzy VIKOR propose different alternative rankings.

To provide a more comprehensive comparison, the fuzzy RVIKOR is compared with fuzzy VIKOR [45], fuzzy TOPSIS [51], fuzzy ARAS [52], and fuzzy TOPKOR [44] methods. Table 10 shows the obtained rankings by each method for the case study. The comparative ranking of the mentioned MCDM methods is illustrated in Figure 5.

Figure 5. Comparative rankings based on different MCDM methods.

Table 10. Comparison of alternatives’ ranking obtained by different MCDM methods.

Alternative	Fuzzy RVIKOR	Fuzzy VIKOR	Fuzzy TOPSIS	Fuzzy ARAS	Fuzzy TOPKOR
Sanitary landfill	6	10	11	6	11
Incineration	5	5	5	5	6
Microwave	1	1	8	2	1
Sterilization by autoclave	2	3	2	1	2
Chemical disinfection	11	11	6	11	9
Radiation	8	7	9	7	7
Treatment by NEWater process	9	8	1	8	5
Encapsulation	7	6	10	9	10
Compaction	10	9	7	10	8
Reverse polymerization	3	2	3	3	3
Plasma pyrolysis	4	4	4	4	4

Each MCDM method has a unique viewpoint in ranking the alternatives that result in different rankings of alternatives. For example, fuzzy TOPSIS and fuzzy VIKOR proposed quite different ranks for the ‘Microwave’ method, as shown in Figure 5. This difference in ranking is caused because the TOPSIS method considers the total distance of the current value from PIS and NIS when ranking the alternative. This means that if an alternative obtains a bad score in a criterion, its good scores in other criteria could compensate for it. But the VIKOR method considers the total distance of each alternative from PIS (utility index) and the maximum distance of each alternative from PIS in each criterion (regret index). The regret index indicates that a bad score of an alternative in a criterion could not significantly compensate for its good scores in other criteria. In other words, an alternative with more balanced scores in all criteria has a better regret index and a better chance of obtaining a higher priority by the VIKOR method.

To evaluate the performance of the compared methods, an indicator called “Number of Change in Alternatives’ Ranking” (NCAR) is defined as presented in Equation (27)(27) $\begin{array}{rl}& \mathrm{N}\mathrm{C}\mathrm{A}\mathrm{R}(A,B)=\sum _{i=1}^m{X}_i\\ & X_i=\left\{\begin{array}{cc}1& if\mathrm{R}\mathrm{a}\mathrm{n}\mathrm{k}\mathrm{o}\mathrm{f}{i}^{\mathrm{t}\mathrm{h}}\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{i}\mathrm{n}A\ne \mathrm{R}\mathrm{a}\mathrm{n}\mathrm{k}\mathrm{o}\mathrm{f}{i}^{\mathrm{t}\mathrm{h}}\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{i}\mathrm{n}B\\ 0& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}\right.\end{array}$(27) . If A and B are two MCDM methods, NCAR(A,B) represents the number of alternatives whose priority differs in the A and B methods.

(27) $\begin{array}{rl}& \mathrm{N}\mathrm{C}\mathrm{A}\mathrm{R}(A,B)=\sum _{i=1}^m{X}_i\\ & X_i=\left\{\begin{array}{cc}1& if\mathrm{R}\mathrm{a}\mathrm{n}\mathrm{k}\mathrm{o}\mathrm{f}{i}^{\mathrm{t}\mathrm{h}}\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{i}\mathrm{n}A\ne \mathrm{R}\mathrm{a}\mathrm{n}\mathrm{k}\mathrm{o}\mathrm{f}{i}^{\mathrm{t}\mathrm{h}}\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{i}\mathrm{n}B\\ 0& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}\right.\end{array}$(27)

The lower value of NCAR(A,B) shows that A and B methods better justify each other. Figure 6 shows the NCAR index between the compared MCDM methods. The value mentioned in each edge shows the NCAR value between the connected MCDM methods. Moreover, the sum of the NCAR value of each MCDM method in all comparisons has been shown in the figure.

Figure 6. Comparing the results of MCDM methods.

The results show that the lowest value of the total NCAR indicator belongs to the RVIKOR method. It indicates that the fuzzy RVIKOR method is better justified by other MCDM methods. In other words, the ranking provided by the RVIKOR method has less deviation from the ranking proposed by the other methods.

5.8. Discussion

The results show that the environmental criterion has more weight than the economic criterion. This is because environmental pollutions of some HCW disposal methods have hazardous short-term and long-term effects on the life and health of citizens, and this criterion has more importance than the economic criterion.

Five new sub-criteria for prioritizing HCW disposal methods are introduced in this study. These sub-criteria are ‘affordability of technology’, ‘consistency with WHO policies’, ‘compliance with national environmental laws’, ‘meets capacity requirements’, and ‘ability to treat a wide range of infectious waste’. Considering these sub-criteria, besides the ones identified from the literature, allows the decision makers to make a more reliable decision by paying attention to more aspects affecting the HCW disposal. For example, in pandemic situations or natural disasters, the amount of HCW is increased and considering the sub-criterion ‘meets capacity requirements’ could be helpful to cope with these situations. Moreover, some HCW disposal methods may not be able to dispose of some HCW types. This indicates that other compensatory HCW disposal methods should be used, resulting in increased cost and resource usage. Considering the criteria of ‘ability to treat a wide range of infectious waste’ decrease the need for using other compensatory HCW disposal methods.

The RVIKOR method applied to the case study is an enhanced version of the VIKOR method. The results show that ‘Microwave’, ‘Sterilization by autoclave’, and ‘Reverse polymerization’ obtained the highest priorities in the fuzzy RVIKOR method, respectively. Comparing the ranking proposed by RVIKOR and VIKOR shows that these methods provide different alternatives’ ranking. The difference in ranking alternatives by fuzzy RVIKOR and fuzzy VIKOR can be explained as having roots in their viewpoints. The RVIKOR considered the distance of alternatives from NIS in ranking, but fuzzy VIKOR ignores this parameter. For example, the results show that the ‘Radiation’ method has better values of utility and regret indexes than the ‘Sanitary landfill’ method. Therefore, it obtained higher priority than the ‘Sanitary landfill’ method in the fuzzy VIKOR. However, the worse value of the ‘Radiation’ method in the escape index compared with the ‘Sanitary landfill’ method caused it to obtain a worse priority in the fuzzy RVIKOR.

Each MCDM method has a different viewpoint in ranking the alternatives. Comparing the fuzzy RVIKOR with some MCDM methods such as fuzzy VIKOR, fuzzy TOPSIS, fuzzy ARAS, and fuzzy TOPKOR methods show that the fuzzy RVIKOR has the lowest value in the “Number of Change in Alternatives’ Ranking (NCAR)” index. This could be interpreted that fuzzy RVIKOR has a comprehensive view in ranking the alternatives and considered utility, hate, regret, and escape indexes in ranking the alternatives.

6. Conclusion and future research directions

This research introduces an enhanced MCDM method named fuzzy RVIKOR to deal with the HCWDMS problem in Tehran, Iran. Effective criteria in ranking and evaluating HCW disposal methods were identified through a literary review and experts’ opinions. In total, 30 criteria were identified, of which five new these sub-criteria were introduced by experts. All the criteria are divided into four categories: social, technical, environmental, and economical. Moreover, this research provided a comprehensive list of criteria for HCWDMS (i.e. inkling 11 methods) where one of the methods is proposed by the experts and ten exist in the literature. The weights of the criteria were determined by the FAHP method. The score of each HCW disposal method in each criterion was obtained using a decision matrix. Finally, the considered HCW disposal methods were ranked with the fuzzy RVIKOR.

The proposed fuzzy RVIKOR provides a comprehensive view when ranking alternatives by considering utility, regret, hate, and escape indexes. Comparing the proposed method with some other MCDM methods, such as fuzzy VIKOR, fuzzy TOPSIS, fuzzy ARAS, and fuzzy TOPKOR, justified the fuzzy RVIKOR results. Moreover, the robustness of the fuzzy RVIKOR was investigated through a sensitivity analysis and considering several scenarios.

6.1. Research limitations and implications

The results obtained in this research are related to the gathered data from the studied case, such as pairwise comparisons and decision matrixes. Therefore, the results of this study might not be generalized to other cases. Nevertheless, the identified and used criteria are relevant to any similar studies on HCWDMS and could help the managers to consider various aspects in selecting the most suitable HCW disposal methods. Considering various aspects could be beneficial for managers of recycling disposal centers, healthcare centers, and hospitals to become more aware of their decisions’ impacts and make more reliable decisions. Moreover, the proposed fuzzy RVIKOR method is applicable to other similar decision-making problems and could support the managers in deciding to select the proper HCW disposal methods.

6.2. Future research scopes

As for future studies, identifying and adding new effective criteria for evaluating and ranking the HCW disposal methods could be a valuable research stream. Moreover, future studies can benefit from the proposed RVIKOR method to select the proper disposal method for other types of waste, such as urban waste. Using type-2 fuzzy sets in the proposed RVIKOR method and comparing its results with the current version could provide insightful findings. Applying the RVIKOR to problems in other domains and comparing its robustness with other MCDM methods could be a promising research avenue.

Disclosure statement

No potential conflict of interest was reported by the authors.

Data availability statement

The authors confirm that the data supporting the findings of this study are available within the article.