Selection of plastic solid waste treatment technology based on cumulative prospect theory and fuzzy DEMATEL

Abstract

Under the global implementation of a low-carbon economy, the treatment of municipal plastic solid waste (PSW) has become an important task to be solved urgently. In the actual decision-making process of PSW treatment, the evaluation information is usually fuzzy, and the decision-makers (DMs) are bounded rational. For selecting the most appropriate PSW treatment technology, we propose a multi-criteria decision-making (MCDM) method based on cumulative prospect theory and fuzzy decision-making trail and evaluation laboratory (DEMATEL). Firstly, we construct the criteria system of PSW treatment that consists of 9 sub-criteria from the perspectives of environment, economy, society, and technology. Then, considering the interdependences and interactions between these evaluation criteria and allowing multiple stakeholders to participate in decision-making, we propose a fuzzy DEMATEL method to deal with the fuzziness of evaluation in the decision-making process and determine the weights of the evaluation criteria. Subsequently, taking into account the different opinions of different stakeholders and psychological factors such as risk preference and loss aversion of stakeholders, we aggregate the evaluation information of different stakeholders and develop the PSW treatment alternatives to rank the orders by using the proposed multi-actor cumulative prospect theory (CPT) method. We study seven alternative processes for PSW treatment by the developed model, including landfill, recycling, pyrolysis, incineration, and the combination of landfilling and recycling, landfill and incineration, and recycling and pyrolysis. According to the ranking results, we find the combination of recycling and incineration is the best treatment alternative. We take the seven PSW treatment technologies in Shanghai as the case study to verify the effectiveness and feasibility of the proposed method. Through the sensitivity analysis and comparison analysis with fuzzy similarity to ideal solution (FTOPSIS) method and an acronym in Portuguese of the interactive and multi-criteria decision-making (TODIM) method, we illustrate the effectiveness and superiority of the proposed method. This research provides significant references for the PSW treatment technology selection problems under uncertain environments and extends the methods in the decision-making field.

Introduction

The comprehensive green transformation of economic and social development requires the harmless and resource-based and efficient treatment of municipal solid waste (MSW) to solve the problems of health, resources, environment, and economy associated with the improper disposal of waste (Zhao et al. 2016; Ali et al. 2020; Colpo et al. 2022). The compositions of MSW are diverse and the treatment is extremely complicated. The commonly used treatment technologies are landfill, recycling, pyrolysis, and gasification (Singh et al. 2011). As an important component of MSW, the treatment of PSW has attracted more and more attention recently, and become an urgent problem to be solved. With the improvement of people’s living standards, the demand for plastics in daily life is increasing rapidly (Al-Salem et al. 2014). Economic growth and changes in consumption and production patterns result in a significant increase in the generation of PSW. The global plastic production increased from 359 million tonnes in 2018 to 368 million tonnes in 2019, and with an estimation to be augmented to 500 million tonnes in 2025. About 60% of plastics enter the environment as plastic waste, and the plastic waste increased from 24.5 million tonnes in 2018 to 29.1 million tonnes in 2019 (Hasanzadeh et al. 2022). Plastic waste not only contains many potentially harmful chemicals, which pollute water, air, and soil, but also brings adverse factors such as health problems to organisms and humans. So it is extraordinarily significant to choose the most appropriate PSW treatment technology. There are many technologies for the treatment of PSW, i.e., land landfill, recycling, incineration and pyrolysis, etc. (Alhazmi et al. 2021). These PSW treatment technologies perform differently in environmental, economic, social, and technical aspects (Verma et al. 2022). In addition, Zhao (2021) pointed out that the process of waste management requires multiple stakeholders and summarized the stakeholder-associated factors influencing construction and demolition waste demolition management. Then, various stakeholders are usually involved in the selection process of municipal PSW, including local residents, engineers, scholars, managers, etc. Therefore, it is necessary to construct a multi-criteria decision-making (MCDM) model for the optimal selection of PSW treatment alternatives, and multiple stakeholders need to participate in the decision-making.

PSW is known as an environmental threat (Foschi et al. 2020). Extensive use of plastics causes environmental hazards, thereby endangering marine organisms, reducing soil fertility, and polluting groundwater (Mourshed et al. 2021; Singh et al. 2020). At the same time, plastic waste management has also become a problem. At present, due to pollution and technical constraints, only a small part of plastic waste has been recycled, and a large amount of plastic waste has accumulated in the environment. As a material, plastics are generating greenhouse gas emissions in the production process (Mumbach et al. 2019). It is challenging for the whole society to manage these huge plastic wastes. In recent years, people have been committed to improving the management of PSW, but their purpose is often to identify and develop the method of PSW recovery, ignoring the research on the selection of PSW treatment technology (Ragaert et al. 2020). In fact, in the decision-making process, paying full attention to the selection of PSW treatment technology has a significantly positive and far-reaching impact on society. On the one hand, the selection of appropriate PSW treatment technology can effectively achieve the economic, social, and environmental sustainability of waste management, reduce the waste of resources and reduce the environmental pollution caused by plastic waste. On the other hand, appropriate PSW treatment technology can reduce the pollution in the treatment process to a minimum and the cost is also low. At the same time, the social citizens have a high degree of acceptability. So, the selection process of PSW treatment technology has a positive impact on society and the environment. PSW treatment technologies, such as landfill, incineration, and pyrolysis, have different economic, social, and technical performances in terms of treatment costs, transportation costs, etc. And the economic and environmental impacts on society in the process of treatment are also different (Altieri et al. 2021). Therefore, it is necessary to establish an evaluation criteria system to measure the quality of PSW treatment technology, and it is also extremely meaningful to establish a multi-attribute decision-making model to evaluate and select PSW treatment technology.

In real life, the optimal selection of PSW treatment is a complex and multidisciplinary MCDM problem, which should be considered from the aspects of economy, society, environment and technology. MCDM is a branch of a general model, which uses multiple criteria to optimize decision-making problems (Kahraman 2008). Many waste management problems have been solved by applying these types of MCDM models, which include urban solid waste management systems (Hokkanen et al. 1995; Ojeda-Benítez et al. 2003) and ocean disposal sites (Leschine et al. 1992). However, in the actual MCDM decision process, a number of criteria cannot be accurately expressed in numbers. Mishra et al. (2022) combined Fermatean fuzzy sets (FFSs) with CRITIC and EDAS methods to solve the sustainable third-party reverse logistics providers selection problem. Afrane et al. (2022) used the fuzzy set theory and AHP-TOPSIS method to rank and select the waste-to-energy technology. Liu et al. (2022) extended the VIKOR method to the intuitionistic fuzzy set (IFS) environment for assessing the capacity of COVID-19 medical waste (CMW) recycling channels. In general, due to the uncertainty of information and the vagueness of human feeling and recognition, MCDM problems related to the environment in the real world should be regarded as essentially fuzzy problems, including attributes and alternatives (Aslan et al. n.d). The evaluation data of the applicability of alternatives to various subjective criteria and the weight of the criteria are often expressed by the decision-makers (DMs) in linguistic terms (Wang et al. 2009). Therefore, linguistic data are represented by fuzzy sets, and the fuzzy set theory is applied to PSW treatment decision problems for mathematical operations, so as to deal with the uncertainty and fuzziness in the decision-making process. The fuzzy set theory proposed by Zadeh (1965) provides convenience for dealing with the uncertainty of decision-making problems. The Fuzzy set theory has advantages in dealing with the fuzziness and uncertainty of human judgments, as well as inaccurate or insufficient information about quantitative and qualitative data. Therefore, in the selection process of PSW treatment alternative, the fuzzy set theory is significantly suitable to be combined with other MCDM methods to solve the problem.

The selection of a PSW treatment alternative is a MCDM problem involving multi-stakeholders. There are usually various MCDM methods for the selection of PSW treatment technology and MSW treatment. Vinodh et al. (2014) developed the MCDM method of fuzzy AHP (analytic hierarchy process) and TOPSIS (technique for order of preference by similarity to ideal solution) to determine the best plastic recycling technology. Soltani et al. (2016) extended a life cycle sustainability assessment method to select the optimal MSW treatment process by considering the social, environmental and economic impacts of different solutions. Mir et al. (2016) determined a method based on technique for order of preference by similarity to ideal solution (TOPSIS) and extended Viekriterijumsko Kompromisno Rangiranje (VIKOR) method to evaluate and select the best MSW treatment technology, which contributed to the DMs to optimize the sequencing and selection of MSW treatment technology. Ekmekçioğlu et al. (2010) used the improved fuzzy TOPSIS method for the optimal selection of a MSW treatment alternative and determined refuse-derived fuel (RDF) combustion as the optimal alternative. Yap and Nixon (2015) proposed the AHP method to evaluate the trade-offs among the benefits, opportunities, costs, and risks of alternative energy from waste technologies, considering technologies i.e., large-scale incineration, RDF combustion, gasification, anaerobic digestion and landfill recycling, etc. Karmperis et al. (2012) put forward a risk-based MCDM method to select the best alternative for waste management. Khan and Faisal (2008) constructed a hierarchical network decision-making structure and applied the analytic network process (ANP) method to help DMs select the optimal MSW treatment solution. Oyoo et al. (2013) proposed a new comprehensive urban waste flow model to determine the optimal waste management scenario. Shahnazari et al. (2020) applied the AHP method and TOPSIS method to select the best thermochemical technology by the technical, economic, and environmental criteria. Alao et al. (2020) utilized the TOPSIS method with the entropy-weighted method to select the optimal technology among the waste-to-energy technological options using the waste stream of Lagos, Nigeria. Badi et al. (2019) used the AHP method for waste management in respect to four main criteria and twenty-two sub-criteria. Antelava et al. (2019) evaluated the environmental burdens of chemical processes and material cycles from an environmental context by applying life cycle assessment (LCA). In order to select an adaptable recycling method for plastic materials, Geetha et al. (2021) presented the hesitant Pythagorean fuzzy ELECTRE III method. Liu et al. (2022) proposed a novel MCGDM approach based on intuitionistic fuzzy sets (IFSs) and the VIKOR method for assessing the capacity of COVID-19 medical waste (CMW) recycling channels. Kabirifar et al. (2020) developed a framework based on the theory of planned behavior (TPB) to study and assess the effectiveness of construction and demolition waste management (CDWM). Wu et al. (2020) assessed the incineration power plant performance from a refuse classification perspective with the fuzzy synthetic evaluation (FSE) method and the AHP method. Brans and Vincke (1985) proposed a new PROMETHEE method to solve the MCDM problems of several industries, such as waste treatment, wastewater management, and so on. Zhou et al. (2022) reviewed the quality function deployment (QFD) technique used in various areas of real-world applications which can provide new ideas for the field of waste management. Altan and Karasu (2019) proposed a support vector machine (SVM) method which provides a positive contribution to the MCDM decision process in the MSW and waste management fields. These methods can help DMs to select the most suitable solution among multiple PSW technologies by considering a variety of evaluation criteria. However, these methods which assume the DMs are completely rational exist some shortcomings in the decision-making of PSW alternatives. In the actual decision-making process, the DMs are usually bounded rational. It is necessary to consider the impact of behavioral psychologies of DMs, such as loss aversion, reference dependency and subjective judgment bias, etc., to select the best PSW treatment technology under environments of risk and uncertainty. In recent years, many scholars have combined behavioral decision-making theories with MCDM methods to study the decision-making problems, such as prospect theory, cumulative prospect theory (CPT) (Tversky and Kahneman 1992), regret theory (Bell 1982), and disappointment theory (Bell 1985). Among them, CPT has been deemed as the most popular and practical theory since it can well describe the behavior characteristics of DMs and give the calculation formulas of the value and weights of potential results.

CPT has been widely used to solve a variety of decision-making problems considering DMs’ behaviors. Ying et al. (2018) established a mixed multi-attribute decision-making (MADM) model based on CPT to select a new product development concept approach. In order to make the most appropriate selection for the third-party reverse logistics suppliers, Li et al. (2018) used CPT to rank and select the optimal third-party reverse logistics provider by considering various criteria about reverse logistics. Liu et al. (2014) proposed a risk decision analysis method based on CPT for risk decision-making in emergency response. Wilton et al. (2014) applied CPT to study the risk tendency of DMs with different capacity credits of wind power generation. Klein and Deissenroth (2017) solved the residential photovoltaic system by applying CPT and came to the conclusion that the system depended not only on the profitability but also on the change in profitability relative to the system. Zhao et al. (2021) established the risk assessment model of science and technology projects and verified its feasibility based on CPT and the Pythagorean fuzzy sets (PFSs). In order to solve the optimal selection problems of renewable power sources (RPS) used in China, Wu et al. (2018) utilized a fuzzy MCDM technique based on CPT method under an uncertain environment. Xu et al. (2021) proposed a new method based on CPT and cellular automata which was applied to establish the travel route choice model and analyze the characteristics of travelers with bounded rational travel behavior. For the site selection problems of photovoltaic power plants, Liu et al. (2017) employed the gray CPT method to study and analyze from the perspective of sustainability. Zhang et al. (2022) proposed a new method based on CPT and hybrid information to assess the commercially used photovoltaic technologies from a sustainable perspective. To select the best material alternative for the production of automotive parts produced through the high-pressure die-casting process, Zindani et al. (2021) put forward the complex interval-valued intuitionistic fuzzy TODIM approach based on the CPT method. However, CPT is rarely applied to the selection of PSW treatment technology. On the one hand, the risks in the process of PSW technology selection are significant because of the uncertainty and dynamics of the PSW technology selection process. On the other hand, the risk preferences of DMs are various, i.e., risk neutral, risk aversion and risk seeking, etc., and thus, different risk attitudes will lead to different selection results in final. Thus, it is noteworthy and necessary to incorporate the CPT into the selection process of PSW technology.

In MCDM problems, the determination of criteria weights is the key and critical step. There exist many methods to determine the criteria weights. Huang et al. (2021) used the AHP method to compute the weights of photovoltaic poverty alleviation (PVPA)’s social impact factors. Zhou et al. (2020) utilized the best–worst method (BWM) to determine the weights of criteria for the location decision of the photovoltaic charging station. Krstić et al. (2022) used the best–worst method (BWM) method and comprehensive distance based ranking (COBRA) method to study the problems of the applicability of Industry 4.0 technologies in treverse logistics. Matin et al. (2020) analyzed and determined the weights and prioritization determinants of heat stress control in workers (environment, worker, work) using the analytic network process (ANP) method. Although we can use these methods to determine the criteria weights, they still have some weaknesses. The BWM and AHP methods do not consider the correlation among evaluation criteria in the process of calculating weights, while the ANP method has the disadvantage of a large amount of calculation and will produce certain errors. And these methods lack the consideration of the interaction and independence of evaluation criteria and are difficult to reflect the causal relationship among evaluation factors, and there generally exists a certain interdependence and independence among environmental, economic, social, and technical evaluation criteria in the selection process of PSW treatment alternatives. In order to overcome and solve these problems, we employed the DEMATEL method to calculate the criteria weights with the consideration of the indicators’ interaction. In recent years, many scholars have used the DEMATEL method to determine the criteria weights and conduct decision-making research. Tzeng and Huang (2012) considered the independence and dependence between the evaluation criteria and combined DEMATEL and ANP to determine the weight of the evaluation criteria. Baykasoglu and Gölcü (2017) combined DEMATEL and TOPSIS to solve the MADM problems. Soroudi et al. (2018) integrated DEMATEL with ANP to determine the weight of rainwater management criteria, so as to study the site selection of landfill. Seleem et al. (2020) adopted the fuzzy- DEMATEL method to determine the priority and criteria weights for selecting suitable lean manufacturing tools. Wang et al. (2021) applied the DEMATEL method to analyze the relationship between the criteria affecting the economic benefits of photovoltaic power generation. Lin et al. (2018) used the DEMATEL method and D numbers to analyze the cause-effect relations and the significance degree of risk elements for a new energy power system in China. However, the influence matrix in the classical DEMATEL method is obtained by quantifying the language of experts with real numbers, which cannot well express the fuzziness and hesitation of experts’ thinking. Therefore, in the decision-making process of PSW treatment technology selection, DEMATEL is extended to the fuzzy environment, and the triangular fuzzy number quantization influence matrix is used to better deal with the uncertainty and inconsistency in the decision-making process and improve the accuracy of decision-making.

Meanwhile, the determination of the DMs’ weights is as important as the determination of the criteria, which also needs to be paid attention to. Because we cannot simply assume that the weights of DMs are identical, reasonable decision-making methods should be used to determine. The methods for determining the criteria weight, such as the BWM method, the eigenvector method, weighted least square method, entropy method, AHP, LINMAP (linear programming techniques for multidimensional analysis preference), fuzzy preference programming (FPP), logarithmic fuzzy preference (LFPP), and modified LFPP, are also applicable to determine the DMs’ weights. We need to select the most appropriate DMs weights determination method to better make decisions on PSW treatment technology selection. The selection of the method depends on the nature of the problems. The selection of PSW treatment technology is a complex and extensive issue and needs the most inclusive and flexible approach. The AHP proposed by Saaty (Saaty 1977, 1980) can deal with the MCDM problem well and has been applied in many waste disposal decision fields successfully (Eskandari et al. 2016; Khoshand et al. 2019; Zhang et al. 2020). However, the traditional AHP method does not consider the uncertainty and vagueness of human judgment, and the results are rather imprecise. In the selection process of PSW treatment technology, experts’ evaluation information is usually uncertain and fuzzy, so it is more humanistic and convenient to assess evaluation information by linguistic terms than exact values. Therefore, integrating fuzzy theory with AHP is a vital way to deal with the uncertainty and vagueness of experts’ judgments. Buckley (1985) extended Satty’s AHP method to the FAHP method, the fuzzy ratios are applied to replace the exact ratios in order to deal with the difficulty of allocating accurate ratios when comparing two criteria, and the fuzzy weights of criteria are derived by the geometric mean method. And thus, we use the FAHP method proposed by Buckley to determine the weights of the stakeholder of PSW treatment.

Compared with traditional decision-making tools, the proposed method in this paper can not only reflect the decision results of the reference dependence of stakeholders' psychological factors but also consider the interaction between criteria when determining the criteria weights, so that the alternatives can be ranked according to it. We construct the evaluation criteria system of PSW treatment technology, which comprehensively includes the sustainability dimensions, i.e., environmental criteria, economic criteria, social criteria, and technical criteria, which can help us better analyze the factors affecting the treatment technology of PSW and make more accurate decisions. Moreover, in the decision-making process of PSW technology selection, we consider multiple groups of stakeholders on waste treatment, so as to make the decision-making process more objective and reliable. The difference between this research and other waste treatment researches is that in the decision-making process, we not only consider the interaction and dependence between the evaluation criteria, so as to determine the more objective and accurate criteria weights and the causal relationship between the criteria, but also construct a decision-making model considering the behavior characteristics of DMs, which reduces subjective biased judgment or information loss, so as to achieve more rational decision results. However, the established criteria system for the treatment of PSW may not be comprehensive, but it still has certain reference significance for the assessment of waste treatment.

When we examine the researches on waste management, the following shortcomings seems to emerge:

1. The psychological factors of DMs are not considered in the decision-making process.

2. Ignoring the fuzziness and uncertainty of human judgments and not considering incomplete, corrupted or insufficient data.

3. When calculating the criteria weights, most studies did not consider the interdependence and interaction between the criteria.

4. In the decision-making process of waste management, the DMs are not considered comprehensively or multiple groups of stakeholders are not considered, and the appropriate decision method is not selected to calculate the weights of the DMs.

5. The evaluation criteria system of PSW treatment technology selection is not sound enough. Due to the particularity and complexity of PSW treatment, PSW should not be disposed of as general waste. There is no suitable and targeted evaluation criteria system of PSW treatment technology selection in existing studies.

Motivated by the previous researches, assuming that the DMs are bounded rational and considering the interaction and dependence between evaluation criteria, we propose a method based on CPT and fuzzy DEMATEL to sort and select the best PSW treatment technology. The fuzzy set theory is applied to the decision framework and combined with the decision method, so as to better deal with fuzzy information, solve the uncertainty in the decision-making process, and reduce the loss of information. It is a meaningful and significant research on the selection of PSW treatment technology, which combines the criteria of four sustainable dimensions with the DMs’ psychological and behavioral characteristics. The main contributions of our work are summarized into two aspects, i.e., the theoretical aspect and practical aspect, which are shown as follows:

1. The theoretical aspect

   1. A MCDM framework for PSW treatment based on CPT and fuzzy DEMATEL is proposed, in which fuzzy set theory is applied to transform linguistic information into triangular fuzzy numbers for decision analysis.
   2. Compared with the traditional single method, the hybrid method proposed in this paper considers the psychological factors of DMs, and is a reasonable method to choose PSW treatment technology.
   3. Fuzzy set theory is introduced into decision-making model, which has advantages in dealing with the fuzziness and uncertainty of human judgments, as well as inaccurate or insufficient information about quantitative and qualitative data.
2. The practical aspect

   • (i)
     The complete criteria system about PSW treatment technology selection problems that consists of 9 sub-criteria from the perspectives of environment, economy, society and technology is established.
   • (ii)
     According to the proposed feasible decision framework, seven typical PSW treatment technologies in Shanghai are evaluated and selected, and the results can provide reference for managers.
   • (iii)
     By analyzing the results obtained by the proposed method, some managerial implications on environmental, economic, social and technological aspects are put forward to realize the sustainable development of the environment and reduce the waste of resources.

Besides the introduction, the remainder of this article is organized as follows. In “Materials and methods,” the used materials and methods are presented. “Case study” applies the method proposed in this paper to the case study. In “Results and discussion,” the results and discussions are performed. Finally, “Conclusions” gives the conclusions of the study and the directions for future research.

Materials and methods

Establishment of evaluation criteria system for PSW treatment technology

The selected evaluation criteria should minimize uncertainty and measure the performance of PSW treatment technology according to specific objectives. As a result, through the reading and reference of relevant literature, the understanding of the actual situation and the measurement under the sustainable low-carbon goal, this paper focuses on studying and determining the evaluation criteria of PSW treatment technology. The evaluation criteria system of PSW treatment technology is finally established and classified into four categories, i.e., environment, economy, society, and technology aspects, as shown in Fig. 1.

Fig. 1.

The evaluation criteria system of PSW treatment technology

Environmental criteria—C₁

The environmental criteria of PSW treatment technology is related to the goal of low-carbon sustainable environmental governance, including recycling raw materials and minimizing environmental pollution. The treatment of PSW may lead to air pollution (C₁₁), water pollution (C₁₂), and land pollution (C₁₃), and thus, these criteria are regarded as environmental criteria. PSW reduces the water permeability of the soil, even affects the fertility of the soil, and often causes serious blockage of sewage ditches and drainage ditches, resulting in extremely serious consequences (Saikia and Brito 2012).

Economic criteria—C₂

The treatment of PSW usually involves the cost of garbage collection, waste transportation and the cost of plastic solid waste treatment (C₂₁). At the same time, the cost in the process of waste treatment can be used as the economic criterion for evaluating PSW treatment technology (Bhagat et al. 2016). In some cases, the treatment of PSW may produce some economic benefits, such as the economic benefits brought by the sale of recycled items under the recycled treatment technology, i.e., the economic benefits of by-products (Vučijak et al. 2016) (C₂₂) can be used as a sub-criteria of the economic criterion to evaluate and select the PSW treatment technology.

Social criteria—C₃

Social acceptance (C₃₁) is the key criterion for evaluating PSW treatment technology. The acceptance of the public and residents should be considered when dealing with plastic waste in a certain area. The public and local residents are the stakeholders of PSW treatment (Deshpande et al. 2020). Social acceptance directly affects the difficulty and feasibility of the implementation of treatment technology, which is an important factor in the evaluation of treatment technology. Jobs creation (C₃₂) aims to rank the alternatives according to the ability of the PSW treatment technology to create new jobs for the public (Hopewell et al. 2009), which can be used as the criterion for evaluating the PSW treatment technology.

Technical criteria—C4

Technical feasibility (C₄₁) and technical reliability (C₄₂) are sub-criteria of technical criteria and evaluation indicators to measure the technologies of PSW treatment. Technical feasibility refers to the availability of the implementation of waste treatment technology whether the treatment technology is easy to implement, available and easy to use in the process of waste treatment at a certain place and time (Antelava et al. 2019). Technical feasibility can be used as an evaluation criterion to measure the performance of alternatives. Technical reliability refers to whether the PSW treatment technology is reliable and implementable (Coelho et al. 2017). When evaluating the treatment technologies of PSW, the reliability of the technology is one of its evaluation attributes, and the performance of the alternative can be evaluated by ranking.

Linguistic variable

IT is difficult for traditional quantitative methods to reasonably express those situations that are obviously complex or difficult to define, so, in this case, the notion of a language variable is necessary. Professor L.A. Zadeh, the founder of fuzzy set theory, first introduced the concept of linguistic variable in 1975 (Zadeh 1975). The notion of linguistic variable is used to explain occurrences that are too vague to be expressed in standard quantitative terms, so as to eliminate the vagueness and subjectivity of human judgment (Wang and Chang 2007).

When making qualitative judgments, DMs need to set appropriate language evaluation scales in advance according to their own experience, and express their preferences through the language scale value of language variables. Let $B=\left\{\frac{b_a}{a},=,0,1,,,\cdots ,,,\frac{M}{2},-,1,,,\frac{M}{2},,,\frac{M}{2},+,1,\cdots ,M\right\}$ be a linguistic term set, where $b_a$ denotes the a + 1th possible value for a linguistic variable in set B, and M is an even number. For example, let M = 6, then the set B can be defined as $B=\{b_0=\text{extremely poor},b_1=\text{very poor},b_2=\text{poor},b_3=\text{moderate},b_3=\text{moderate},b_4=\text{good},b_5=\text{very good},b_6=\text{extremely good}\}$. Generally speaking, any two linguistic terms satisfy the following three operational laws (Herrera et al. 2005):

1. The set is ordered: if a > k, then $b_a>b_k$ (i.e., $b_a$ is not inferior to $b_k$).

2. Negation operator: $neg\left(b_k\right)=b_a$, if $k=M-a$.

3. Max operator and min operator: there are $ax\left\{b_a,,,b_k\right\}=b_a$ and $min\left\{b_a,,,b_k\right\}=b_k$, if $b_a>b_k$.

Cumulative prospect theory method

In 1992, Tversky and Kahneman (1992) proposed an extended and improved version of prospect theory based on prospect theory, i.e., cumulative prospect theory (CPT). CPT is a descriptive theory that can better explain human decision behavior under risk and uncertainty, considering the psychological and behavioral characteristics of DMs, such as risk preference, loss aversion and so on. CPT that is more in line with the decision-making in actual uncertainty cases considers the bounded rational of DMs, which has a certain practicability and reference value in the actual decision-making process. Prospect value V is mainly determined by the value function $v\left(\mathrm{\Delta },x_i\right)$ and the weight function $\mathrm{\Pi }\left(p_i\right)$, calculated by Eq. (1).

$$V=\sum _{i=1}^{k}v(\mathrm{\Delta }{x}_i)\mathrm{\Pi }(p_i)$$ 1

The value function

The value function which represents the value compared with the reference point is determined by the subjective feeling of the DMs. The specific expression of the value function is as follows:

$$v(\mathrm{\Delta }{x}_i)\text{=}\left\{\begin{array}{c}\begin{array}{cc}{\left(\mathrm{\Delta },x\right)}^{\alpha },& \mathrm{\Delta }x\ge 0;\end{array}\hfill \\ -\lambda ,\mathrm{\Delta }x<0.\hfill \end{array}\right)$$ 2

where $\alpha$ and $\beta$ denote exponential parameters relative to gains and losses, respectively, $0\le \alpha ,\beta \le 1$. $\mathrm{\Delta }x$ is the gain value or loss value of the decision alternative related to the reference point, where $\mathrm{\Delta }x>0$ represents the gains and $\mathrm{\Delta }x<0$ represents the losses. $\lambda$ is the loss aversion parameter, which is widely recognized that $\lambda$ should be greater than 1 because the DMs are more sensitive to losses than gains. A graphical expression of the value function is shown in Fig. 2.

Fig. 2.

The value function of CPT

The weight function

The main improvement of CPT based on prospect theory is that the weight function used is not a linear function, but an inverse S-shaped curve, indicating that DMs usually overestimate small probability events while underestimating the possibility of medium and high probability events. Accordingly, the weight function formula of losses and gains is as follows:

$$\mathrm{\Pi }(p_i)=\left\{\begin{array}{c}\frac{p_i^{\eta }}{{\left[p_i^{\eta },+,{\left(1,-,p_i\right)}^{\eta }\right]}^{1/\eta }},\mathrm{\Delta }x>0;\hfill \\ \frac{p_i^{\gamma }}{{\left[p_i^{\gamma },+,{\left(1,-,p_i\right)}^{\gamma }\right]}^{1/\gamma }},\mathrm{\Delta }\mathrm{x}<0.\hfill \end{array}\right)$$ 3

where $\eta$ and $\gamma$ respectively represent the risk gains attitude coefficient and risk losses model coefficient of the DMs, $0<\eta ,\gamma <1$. Generally speaking, there is usually $\eta <\gamma$, which means that DMs often underestimate high probability events and overestimate low probability events. The shape of the weight function is shown in Fig. 3.

Fig. 3.

The weight function of CPT

Generally, the values of these parameters in the theoretical formula of CPT are defined as $\alpha =\beta =0.88$, $\lambda =2.25$, $\eta =0.61$, $\gamma =0.69$, which are given through Tversky and Kahneman's empirical experiments (Tversky and Kahneman 1992).

The decision-making trial and evaluation laboratory (DEMATEL) method

The decision-making trial and evaluation laboratory method, i.e., DEMATEL method, is an effective and comprehensive method for establishing and identifying the structural model of the causal-effect relation diagram between key factors in the system. Scholars Gabus and Fontela 1972, 1973) of Battelle laboratory in the USA carried out the DEMATEL method project through the Geneva Research Center. In a complex system in real life, all evaluation criteria are interdependent, whether direct or indirect relationship, which makes it difficult for us to define criteria in isolation. DEMATEL method mainly solves the practical MADM, MCDM or group decision-making problems from the perspective of identifying the key factors that have the greatest impact on the real complex system. As a MCDM method, the DEMATEL method can not only determine the interdependence among variables or criteria and deal with the internal correlation between criteria in a feasible way, but also convert the relations between factors into a visual structural model (Wu 2008). As a key tool in DEMATEL method, the initial direct-relation (IDR) matrix is used to describe the direct influence degree between various factors in the system.

Triangular fuzzy number

Let $\tilde{r}=\left(r^L,,,r^M,,,r^U\right)$ denotes a triangular number. Then the membership degree function of triangular fuzzy number is defined mathematically as follows (Wu et al. 2018; Sun 2010):

$$\mu (x)=\left\{\begin{array}{c}0,x<r^L\hfill \\ \frac{x-r^L}{r^M-r^L},r^L\le x<r^M\hfill \\ \begin{array}{c}\frac{r^U-x}{r^U-r^M},r^M\le x<r^U\hfill \\ 0,x\ge r^U\hfill \end{array}\hfill \end{array}\right)$$ 4

where $r^L$, $r^M$, and $r^U$ are the lower value, middle value, and upper value of the triangular fuzzy number, respectively. In particular, when$r^L=r^M=r^U$, triangular fuzzy numbers degenerate into real numbers.

Let $\tilde{r}=\left(r^L,,,r^M,,,r^U\right)$ and $\tilde{s}=\left(s^L,,,s^M,,,s^U\right)$ be two triangular fuzzy numbers, then basic arithmetic operations are given as follows (Senthil et al. 2014):

$$\tilde{r}(+)\tilde{s}=(r^L,r^M,r^U)(+)(s^L,s^M,s^U)=(r^L+s^L,r^M+s^M,r^U+s^U)$$ 5

$$\tilde{r}(-)\tilde{s}=(r^L,r^M,r^U)(-)(s^L,s^M,s^U)=(r^L-s^L,r^M-s^M,r^U-s^U)$$ 6

$$\tilde{r}(\times )\tilde{s}=(r^L,r^M,r^U)(\times )(s^L,s^M,s^U)=(r^L·s^L,r^M·s^M,r^U·s^U)$$ 7

$$\tilde{r}(÷)\tilde{s}=(r^L,r^M,r^U)(÷)(s^L,s^M,s^U)=(r^L/s^L,r^M/s^M,r^U/s^U)$$ 8

$$k\tilde{r}=(k{r}^L,k{r}^M,k{r}^U)$$ 9

$${(\tilde{r})}^{\delta }=({(r^L)}^{\delta },{(r^M)}^{\delta },{(r^U)}^{\delta })$$ 10

The Euclidean distance between $\tilde{r}$ and $\tilde{s}$ can be calculated as:

$$d(\tilde{r},\tilde{s})=\sqrt{\frac{{(r^L-s^L)}^2+{(r^M-s^M)}^2+{(r^U-s^U)}^2}{3}}$$ 11

Definition 1

Suppose that $\tilde{r}=\left(r^L,,,r^M,,,r^U\right)$ is a triangular fuzzy number, then it can be defuzzified by the following Eq. (12) into a crisp number:

$$S\left(\tilde{r}\right)=\frac{r^L+4r^M+r^U}{6}$$ 12

The transformation of linguistic terms

For the convenience of calculation, the decision information in the form of language terms is ordinarily transformed into triangular fuzzy numbers. Let $B=\{b_a|a=0,1,\cdots ,\frac{M}{2}-1,\frac{M}{2},\frac{M}{2}+1,\cdots ,M\}$ be a linguistic term set, which can solve the problem that the evaluation criteria cannot be measured accurately. The conversion formula for transforming linguistic terms into triangular fuzzy numbers based on membership function is as follows (Ma et al. 2019):

$$\tilde{r}\text{=}(r^L,r^M,r^U)=(\max \{\frac{a-1}{M},0\},\frac{a}{M},\min \{\frac{a+1}{M},1\}),a=0,1,2,\cdots ,M.$$ 13

The graphical expression of these linguistic terms relative to membership function is shown in Fig. 4. The transformation relationship between triangular fuzzy numbers and linguistics terms is depicted in Table 1.

Fig. 4.

The graphical expression of these linguistic terms relative to membership function

Table 1.

The transformation relationship between triangular fuzzy numbers and linguistics terms

Linguistics terms	Corresponding triangular fuzzy numbers
Very poor (VP)	(0,0,0.25)
Poor (P)	(0,0.25,0.5)
Moderate (M)	(0.25,0.5,0.75)
Good (G)	(0.5,0.75,1)
Very good (VG)	(0.75,1,1)

Fuzzy DEMATEL method

The DEMATEL method can reveal the complex relationship among elements through matrix and digraph. Due to the fuzziness and uncertainty of human language, it is generally difficult for DMs to quantify the impact intensity among elements with accurate numbers (Yuan et al. 2022). Hence, the causal relationship between the causes and effects of criteria can be converted into an intelligible system structure model by the fuzzy DEMATEL method, which is analyzed by combining the DEMATEL method with fuzzy set theory.

The detailed steps of the fuzzy DEMATEL are as follows (Chen et al. 2007; Patil and Kant 2014):

• Step 1. Define the evaluation criteria and use the fuzzy linguistic scale to transform linguistic terms into triangular fuzzy numbers. Firstly, a set of evaluation criteria for the selection of PSW treatment technology is established, which is shown in Fig. 1. Because the evaluation criteria are not completely independent, but have certain dependence, and usually comprise various complicated aspects, the fuzzy DEMATEL method is used to divide the relevant criteria into causal structural models. To deal with the ambiguities of human assessments, linguistic terms are used to describe the influence relationship among evaluation criteria. The scale that transform linguistic terms into triangular fuzzy numbers is commonly used in the literature, and authors such as Valmohammadi and Sofiyabadi (2015) recommend it in their methodological proposals. By using the linguistic variables shown in Table 2, determine the relative degree of one factor affecting another.

• Step 2. Obtain the initial direct-influence fuzzy matrix. The decision team composed of $p$ experts needs to make sets of pair-wise comparisons in terms of linguistic terms to measure the relationship among evaluation criteria $C=\left\{C_i,|i=1|2|\cdots |n\right\}$. Then the initial direct-influence fuzzy matrix $\tilde{Z}\left(k\right)$ is established as follows:

  $$\tilde{Z}\left(k\right)=\begin{array}{c}{C}_1\\ C_2\\ \begin{array}{c}⋮\\ C_n\end{array}\end{array}\left[\begin{array}{cccc}0& {\tilde{Z}}_{12}^{\left(k\right)}& \cdots & {\tilde{Z}}_{1n}^{\left(k\right)}\\ {\tilde{Z}}_{21}^{\left(k\right)}& 0& \cdots & {\tilde{Z}}_{2n}^{\left(k\right)}\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{Z}}_{n1}^{\left(k\right)}& {\tilde{Z}}_{n2}^{\left(k\right)}& \cdots & 0\end{array}\right],k=1,2,\cdots ,p$$ 14

where ${\tilde{Z}}_{\mathit{ij}}^{\left(k\right)}=\left(L_{\mathit{ij}}^{\left(k\right)},,,M_{\mathit{ij}}^{\left(k\right)},,,U_{\mathit{ij}}^{\left(k\right)}\right)$ are triangular fuzzy numbers, and ${\tilde{Z}}_{\mathit{ij}}^{\left(k\right)}=\left(L_{\mathit{ij}}^{\left(k\right)},,,M_{\mathit{ij}}^{\left(k\right)},,,U_{\mathit{ij}}^{\left(k\right)}\right)$, $M$, $U$ are the lower value, middle value, and upper value of the triangular fuzzy number, respectively.

• Step 3. Calculate the normalized direct-influence fuzzy matrix. The linear scale transformation is used as a normalization formula to convert the evaluation criteria scale into comparable scales. Let ${\tilde{a}}_i^{\left(k\right)}$ and $U_i^{\left(k\right)}$ be two triangular fuzzy numbers and satisfy the following Eq. (15).

$${\tilde{a}}_i^{(k)}=\sum _{j=1}^n{\tilde{Z}}_{\mathit{ij}}^{(k)}=\left(\sum _{j=1}^n,L_{\mathit{ij}}^{(k)},,,\sum _{j=1}^n,M_{\mathit{ij}}^{(k)},,,\sum _{j=1}^n,U_{\mathit{ij}}^{(k)}\right),U_i^{(k)}=\underset{1\le i\le n}{\max }\left(\sum _{j=1}^n,U_{\mathit{ij}}^{(k)}\right)$$ 15

Table 2.

The correspondence of linguistic terms and triangular fuzzy numbers

Linguistic terms	Triangular fuzzy numbers
No influence (N)	(0,0,0)
Very low influence (VL)	(0,0,0.25)
Low influence (L)	(0,0.25,0.5)
High influence (H)	(0.25,0.5,0.75)
Very high influence (VH)	(0.5,0.75,1)

Then normalize the direct-influence fuzzy matrix to obtain the normalized direct-influence fuzzy matrix $\tilde{Y}\left(k\right)$, which is expressed as follows:

$$\tilde{Y}\left(k\right)=\begin{array}{c}\begin{array}{c}{C}_1\\ C_2\end{array}\\ ⋮\\ C_n\end{array}\left[\begin{array}{cccc}0& {\tilde{Y}}_{12}^{\left(k\right)}& \cdots & {\tilde{Y}}_{1n}^{\left(k\right)}\\ {\tilde{Y}}_{21}^{\left(k\right)}& 0& \cdots & {\tilde{Y}}_{2n}^{\left(k\right)}\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{Y}}_{n1}^{\left(k\right)}& {\tilde{Y}}_{n2}^{\left(k\right)}& \cdots & 0\end{array}\right],k=1,2,\cdots ,p$$ 16

where ${\tilde{Y}}_{\mathit{ij}}^{(k)}\text{=}\frac{{\tilde{Z}}_{\mathit{ij}}^{(k)}}{U_i^{(k)}}\text{=}\left(\frac{L_{\mathit{ij}}^{(k)}}{U_i^{(k)}},,,\frac{M_{\mathit{ij}}^{(k)}}{U_i^{(k)}},,,\frac{U_{\mathit{ij}}^{(k)}}{U_i^{(k)}}\right)$, and we assume that at least one $i$ such that $\sum _{j=1}^n{U}_{\mathit{ij}}^{\left(k\right)}<U_i^{\left(k\right)}$, furthermore, $\underset{k\to \infty }{\mathit{lim}}\tilde{Y}\left(k\right)={\left[0\right]}_{n\times n}$.

• Step 4. Derive the total-influence fuzzy matrix. Once the normalized direct-influence fuzzy matrix $\tilde{Y}\left(k\right)$ is obtained, total-influence fuzzy matrix $\tilde{T}$ can be computed. The total-influence fuzzy matrix $\tilde{T}$ can be obtained by the following Eq. (17).

$$\tilde{T}=\underset{k\to \infty }{\lim }({\tilde{Y}}^1+{\tilde{Y}}^2+\cdots +{\tilde{Y}}^k)=Y\times {(I-Y)}^{-1}$$ 17

The total-influence fuzzy matrix $\tilde{T}$ is expressed as follows:

$$\tilde{T}=\left[\begin{array}{cccc}{\tilde{T}}_{11}& {\tilde{T}}_{12}& \cdots & {\tilde{T}}_{1n}\\ {\tilde{T}}_{21}& {\tilde{T}}_{22}& \cdots & {\tilde{T}}_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{T}}_{n1}& {\tilde{T}}_{n2}& \cdots & {\tilde{T}}_{\mathit{nn}}\end{array}\right]$$ 18

where ${\tilde{T}}_{\mathit{ij}}=\left(L_{\mathit{ij}}^{{}^{\prime }},,,M_{\mathit{ij}}^{{}^{\prime }},,,U_{\mathit{ij}}^{{}^{\prime }}\right)$, and Matrix $\left[L_{\mathit{ij}}^{{}^{\prime }}\right]=Y_L\times {\left(I_L,-,Y_L\right)}^{-1}$, Matrix $\left[M_{\mathit{ij}}^{{}^{\prime }}\right]=Y_M\times {\left(I_M,-,Y_M\right)}^{-1}$, Matrix $\left[U_{\mathit{ij}}^{{}^{\prime }}\right]=Y_U\times {\left(I_U,-,Y_U\right)}^{-1}$.

• Step 5. Defuzzify the matrix. The purpose of defuzzifying the matrix is to turn the fuzzy triangular values into the score of each causal relationship, so as to facilitate comparison and calculation. The minimum, intermediate and maximum numerical values of triangular fuzzy numbers are weighted average, which can be used as a method of defuzzification (Acuña-Carvajal et al. 2019). Thus, the final total-influence fuzzy matrix after final defuzzification is obtained. Each value ${\tilde{F}}_{\mathit{ij}}$ of the matrix represents the total influence degree of a given objective i on the realization of objective j.

$${\tilde{F}}_{\mathit{ij}}=\frac{1}{3}\left(L_{\mathit{ij}}^{{}^{\prime }},+,M_{\mathit{ij}}^{{}^{\prime }},+,U_{\mathit{ij}}^{{}^{\prime }}\right)$$ 19

$$\tilde{F}=\left[\begin{array}{cccc}{\tilde{F}}_{11}& {\tilde{F}}_{12}& \cdots & {\tilde{F}}_{1n}\\ {\tilde{F}}_{21}& {\tilde{F}}_{22}& \cdots & {\tilde{F}}_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{F}}_{n1}& {\tilde{F}}_{n2}& \cdots & {\tilde{F}}_{\mathit{nn}}\end{array}\right]$$ 20

• Step 6. Compute the influence degree $\tilde{D}$ and the influenced degree $\tilde{R}$

$${\tilde{D}}_i=\sum _{j=1}^n{\tilde{F}}_{\mathit{ij}},{\tilde{R}}_j=\sum _{i=1}^n{\tilde{F}}_{\mathit{ij}}$$ 21

${\tilde{D}}_i$ shows the total effect dispatching from factor i to the other factors both directly and indirectly, i.e., the synthesis of the influence of factor i on all other factors. ${\tilde{D}}_i+{\tilde{R}}_i$ represents the total effect of the ith factor, i.e., the relative importance of the ith factor in the whole complex system and ${\tilde{D}}_i-{\tilde{R}}_i$ denotes the net effect of the ith factor to the complex system. If ${\tilde{D}}_i-{\tilde{R}}_i>0$, then the evaluation criterion $C_i$ is a causal factor and has a strong impact on other factors; if ${\tilde{D}}_i-{\tilde{R}}_i<0$, then evaluation criterion $C_i$ is resulting factor and largely influenced by other factors.

• Step 7. Determine the relative importance of these factors. So far, the cause-effect relationship diagram can be obtained according to the coordinate values $\left({\tilde{D}}_i,+,{\tilde{R}}_i,,,{\tilde{D}}_i,-,{\tilde{R}}_i\right)$ with respect to all the factors. Calculate the relative importance of factors, that is, determine the relative weight of each criterion according to the values of ${\tilde{D}}_i+{\tilde{R}}_i$ and ${\tilde{D}}_i-{\tilde{R}}_i$, which is expressed by Eq. (22). The normalized weights of the relative importance of these factors can be determined by Eq. (23) (Wang et al. 2018).

$${\omega }_j=\sqrt{{\left({\tilde{D}}_j,+,{\tilde{R}}_j\right)}^2+{\left({\tilde{D}}_j,-,{\tilde{R}}_j\right)}^2}$$ 22

$${\overline{\omega }}_j=\frac{{\omega }_j}{\sum _{j=1}^n{\omega }_j}$$ 23

The framework of the proposed algorithm for the selection of PSW treatment technology based on CPT and fuzzy DEMATEL

Based on the above analysis and description, we depict a new MCDM problem of PSW treatment technology in this paper. Suppose that $S=\left\{S_1,,,S_2,,,\cdots ,,,S_m\right\}$ is the set of $m$ finite PSW treatment alternatives, where $S_i$denotes the ith alternative for assessment. Let $C=\left\{C_1,,,C_2,,,\cdots ,,,C_n\right\}$ be a finite evaluation criteria set, where $C_j$ represents the jth evaluation criterion. Assume that $\omega =\left\{{\omega }_1,,,{\omega }_2,,,\cdots ,,,{\omega }_n\right\}$ is the weight vector set for the evaluation criteria, where ${\omega }_j$ is the corresponding weight or importance degree relative to the criterion $C_j$. Let $k=\left\{1,,,2,,,\cdots ,,,p\right\}$ denote the set of $p$ stakeholders in the selection process of PSW treatment technology and then $\varpi =\left\{{\varpi }_1,,,{\varpi }_2,,,\cdots ,,,{\varpi }_p\right\}$ be the weight related to the kth stakeholder. The whole decision-making process of PSW treatment technology based on CPT and fuzzy DEMATEL can be summarized in the proposed algorithm, and Fig. 5 shows the framework of the proposed algorithm.

• Step 1. Determine the weights of the criteria by using the proposed fuzzy DEMATEL. The evaluation criteria of PSW treatment technology are not completely independent, but have a certain correlation. For example, a high degree of air pollution $C_{11}$, a high degree of water pollution $C_{12}$ and a high degree of land pollution $C_{13}$ will lead to low social acceptance ($C_{31}$). Considering that the DEMATEL method takes into account interdependences and interactions among the criteria, a fuzzy DEMATEL method is proposed to determine the weights of the evaluation criteria.

  • Sub-step 1.1: construct the initial direct-influence fuzzy matrix $\tilde{Z}\left(k\right)$.
  • Sub-step 1.2: Determine the weighted direct-influence fuzzy matrix $\tilde{F}$, which can be calculated by Eq. (24).

$$\tilde{F}={\left[{\tilde{f}}_{\mathit{ij}}\right]}_{n\times n}=\left[\begin{array}{cccc}0& {\tilde{f}}_{12}& \cdots & {\tilde{f}}_{1n}\\ {\tilde{f}}_{21}& 0& \cdots & {\tilde{f}}_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{f}}_{n1}& {\tilde{f}}_{n2}& \cdots & 0\end{array}\right]$$ 24

Fig. 5.

The selection framework of PSW treatment technology based on CPT and fuzzy DEMATEL

In the process of PSW treatment, there are multiple groups of stakeholders, so the weighted direct-influence fuzzy matrix should be determined by incorporating the relative importance of stakeholder roles. The relative importance of multiple groups of stakeholders can be determined by fuzzy analytic hierarchy process (FAHP) based on triangular fuzzy number (Chou et al. 2012).

$${\tilde{f}}_{\mathit{ij}}=\sum _{k=1}^p{\varpi }_k{\tilde{Z}}_{\mathit{ij}}^{(k)}=\sum _{k=1}^p{\varpi }_k\left(L_{\mathit{ij}}^{(k)},,,M_{\mathit{ij}}^{(k)},,,U_{\mathit{ij}}^{(k)}\right)=\left(\begin{array}{c}\sum _{k=1}^p{\varpi }_k{L}_{\mathit{ij}}^{(k)},\sum _{k=1}^p{\varpi }_k{M}_{\mathit{ij}}^{(k)},\sum _{k=1}^p{\varpi }_k\end{array},U_{\mathit{ij}}^{(k)}\right)$$ 25

• Step 1.2.1: Determine the weights of the stakeholders.

In this step, we use the FAHP method to determine the stakeholders’ weights which is proposed by Buckley (1985). The specific steps are summarized as follows.

1. Determine the comparison matrix by using linguistic variables.

The DMs determining the pair-wise comparison matrix include two representatives from group#1, a representative professor majoring in environmental engineering in universities and an engineer for municipal solid waste treatment from group#2, an urban environmentalist and a local resident who are highly concerned about the treatment of PSW in Shanghai from group#3, and the committee of three experts who have experience about this research issue for PSW treatment and importance assessment aspects. According to the interviews with nine representatives on the importance of stakeholders and their discussions, the pair-wise comparison matrix is obtained. The DMs use the linguistic variables to determine the pair-wise comparison matrix shown in Table 3 (Hsieh et al. 2004). Thus, a pair-wise comparison matrix composed of triangular fuzzy numbers can be obtained, as presented in Eq. (26).

$$\tilde{B}=\left[\begin{array}{cccc}1& {\tilde{b}}_{12}& \cdots & {\tilde{b}}_{1n}\\ {\tilde{b}}_{21}& 1& \cdots & {\tilde{b}}_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{b}}_{n1}& {\tilde{b}}_{n2}& \cdots & 1\end{array}\right]=\left[\begin{array}{cccc}1& {\tilde{b}}_{12}& \cdots & {\tilde{b}}_{1n}\\ 1/{\tilde{b}}_{12}& 1& \cdots & {\tilde{b}}_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ 1/{\tilde{b}}_{1n}& 1/{\tilde{b}}_{2n}& \cdots & 1\end{array}\right]$$ 26

where ${\tilde{b}}_{\mathit{ij}}=\left(b_{\mathit{ij}}^L,,,b_{\mathit{ij}}^M,,,b_{\mathit{ij}}^U\right)$ denotes the relative importance of the ith stakeholder relative to the jth stakeholder.

• (2)Calculate the fuzzy weights of the stakeholders.

Table 3.

Linguistic variables and corresponding triangular fuzzy numbers

Linguistic variables	Triangular fuzzy numbers
Equal importance (E)	(1,1,1)
Weak importance (W)	(1,2,3)
Moderate importance (M)	(2,3,4)
Strong importance (S)	(3,4,5)
Very importance (V)	(4,5,6)

The fuzzy geometric mean and fuzzy weight of each stakeholder can be defined by the geometric mean technique (Buckley 1985), as shown in Eq. (27).

$$\begin{array}{c}{\tilde{r}}_i={\left[{\tilde{b}}_{i1},(x),\cdots ,(x),{\tilde{b}}_{\mathit{ij}},(x),\cdots ,(x),{\tilde{b}}_{\mathit{in}}\right]}^{1/n}\hfill \\ {\varpi }_i={\tilde{r}}_i(\times ){\left[{\tilde{r}}_1,(+),\cdots ,(+),{\tilde{r}}_i,(+),\cdots ,(+),{\tilde{r}}_n\right]}^{-1}\hfill \end{array}$$ 27

where ${\tilde{r}}_i$ represents the geometric average of the fuzzy comparison value of the jth stakeholder, and ${\varpi }_k$ represents the weight of the kth stakeholder.

• Sub-step 1.3: normalize the direct-influence fuzzy matrix by Eq. (15) to obtain the normalized direct-influence fuzzy matrix $\tilde{Y}\left(k\right)$ in the format of Eq. (16).

• Sub-step 1.4: Determine the total-influence fuzzy matrix $\tilde{T}$ by Eq. (17).

• Sub-step 1.5: Use Eq. (19) to defuzzify the total-influence matrix $\tilde{F}$ after defuzzification in the format of Eq. (20) is obtained.

• Sub-step 1.6: Use Eq. (21) to calculate the influence degree $\tilde{D}$ and the influenced degree $\tilde{R}$ of each element.

• Sub-step 1.7: Determine the weights of these evaluation criteria by Eqs. (22) and (23).

• Step 2. Firstly, the DMs give the description for each evaluation criterion according to the DMs’ own experience and discussion and determine the performance of the alternatives relative to each evaluation criterion. The selection of PSW treatment technology usually involves multiple stakeholders, including managers, engineers and local residents. Considering the opinions of various stakeholders is significantly important for trade-off decision-making. Therefore, the DMs of PSW treatment technology often include multiple stakeholders.

• Step 3. Transform the corresponding expert description in the form of linguistic variables into triangular fuzzy numbers by Eq. (13) and Table 1. The results are presented in Eq. (28).

  $${\tilde{D}}^k=\left[\begin{array}{cccc}{\tilde{d}}_{11}^k& {\tilde{d}}_{12}^k& \cdots & {\tilde{d}}_{1n}^k\\ {\tilde{d}}_{21}^k& {\tilde{d}}_{22}^k& \cdots & {\tilde{d}}_{2n}^k\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{d}}_{m1}^k& {\tilde{d}}_{m2}^k& \cdots & {\tilde{d}}_{\mathit{mn}}^k\end{array}\right]$$ 28

  where ${\tilde{D}}^k$ represents the decision matrix of group kth stakeholders.

• Step 4. Aggregate expert information according to the information given by different stakeholders and determine the weighted fuzzy decision matrix by Eqs. (29) and (30).

  $$\tilde{D}=\left[\begin{array}{cccc}{\tilde{d}}_{11}& {\tilde{d}}_{12}& \cdots & {\tilde{d}}_{1n}\\ {\tilde{d}}_{21}& {\tilde{d}}_{22}& \cdots & {\tilde{d}}_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{d}}_{m1}& {\tilde{d}}_{m2}& \cdots & {\tilde{d}}_{\mathit{mn}}\end{array}\right]$$ 29

  $${\tilde{d}}_{\mathit{ij}}=\sum _{k=1}^p{\varpi }_k{\tilde{d}}_{\mathit{ij}}^k=\left(\begin{array}{c}\sum _{k=1}^p{\varpi }_k^L{d}_{\mathit{ij}}^L,\sum _{k=1}^p{\varpi }_k^M{d}_{\mathit{ij}}^M,\sum _{k=1}^p{\varpi }_k^U{d}_{\mathit{ij}}^U\end{array}\right)$$ 30

  where $\tilde{D}$ denotes the aggregated decision weight, and ${\varpi }_k$ represents the role weight of group kth stakeholders in the decision-making process.

• Step 5. Normalize the decision matrix. In order to eliminate the influence of different dimensions and measurements on the final decision, the decision matrix $D={\left[d_{\mathit{ij}}\right]}_{m\times n}$ needs to be normalized as $R={\left[r_{\mathit{ij}}\right]}_{m\times n}$. Generally, each attribute of the alternative can be divided into benefit attributes and cost attributes. The greater the benefit attribute value is, the better the performance will be. Conversely, for cost attributes, the smaller the value is, the better the performance will be.

  $$r_{\mathit{ij}}^L,r_{\mathit{ij}}^M{r}_{\mathit{ij}}^U\left\{\begin{array}{c}\left(\frac{x_{\mathit{ij}}^L}{x_{minj}^U,},,,\frac{x_{\mathit{ij}}^M}{x_{minj}^U},,,\frac{x_{\mathit{ij}}^U}{x_{minj}^U}\right),\mathrm{the}\mathrm{evaluation}\mathrm{criteria}\mathrm{belongs}\mathrm{to}\mathrm{benefit}\mathrm{criteria}\hfill \\ \left(\frac{x_{minj}^L}{x_{\mathit{ij}}^U},,,\frac{x_{minj}^L}{x_{\mathit{ij}}^M},,,\frac{x_{minj}^L}{x_{\mathit{ij}}^L}\right),\mathrm{the}\mathrm{evaluation}\mathrm{criteria}\mathrm{belongs}\mathrm{to}\mathrm{cost}\mathrm{criteria}\hfill \end{array}\right)$$ 31

  where $x_{\max j}^U=\max \{x_{\mathit{ij}}^U|i=1,2,\cdots ,m\}$ and $x_{\min j}^L=\min \{x_{\mathit{ij}}^L|i=1,2,\cdots ,m\}$.

• Step 6. Calculate the PIS and NIS for each criterion. When employing CPT, the reference point plays an important role in decision-making. Based on the maximum value and minimum value, this paper draws lessons from the concept of the TOPSIS method, and takes the positive ideal solution (PIS) and negative ideal solution (NIS) as the reference points to embody the risk attitude of DMs, i.e., stakeholders. When the reference point is PIS, DMs tend to seek risk because they face losses while when the reference point is NIS, the benefits make the DMs tend to avoid risk.

First of all, the normalized decision matrix is defuzzified by Eq. (12). Then, the triangular fuzzy numbers of the alternatives are sorted according to the defuzzification value of each criterion. Thus, PIs and NIS of each criterion are obtained, which are denoted as and $N_j$ ($j=1,2,\cdots ,n$), respectively.

• Step 7. Compute the gain and loss values of each alternative, i.e., the distance between each alternative and PIS/NIS, and then obtain the gain $G={\left[g_{\mathit{ij}}\right]}_{m\times n}$ matrix and loss matrix $L={\left[l_{\mathit{ij}}\right]}_{m\times n}$.

• Step 8. Calculate prospect values and determine the positive value matrix $V_{\mathit{ij}}^{+}$ and negative prospect value matrix $V_{\mathit{ij}}^{-}$, respectively.

• Step 9. Figure the cumulative prospect weights and gain the positive decision weight matrix ${\mathrm{\Pi }}^{+}\left({\omega }_j\right)$ and negative weight matrix ${\mathrm{\Pi }}^{-}\left({\omega }_j\right)$, respectively.

• Step 10. Calculate the comprehensive prospect value $V$ of each alternative and make a ranking.

$$V_i=\sum _{j=1}^n(V_{\mathit{ij}}^{+}{\mathrm{\Pi }}^{+}({\omega }_j)+V_{\mathit{ij}}^{-}{\mathrm{\Pi }}^{-}({\omega }_j))$$ 32

Case study

In order to illustrate the feasibility and superiority of the established MCDM method in selecting the optimal PSW treatment technology among multiple treatment technologies, a selection problem of plastic solid waste treatment technology in Shanghai is studied. Seven typical PSW treatment alternatives are investigated, and they are landfill ($S_1$), recycling ($S_2$), pyrolysis ($S_3$), incineration ($S_4$), combination of recycling and landfill ($S_5$), combination of landfill and incineration ($S_6$), and combination of recycling and incineration ($S_7$). Landfill is considered the traditional method of MSW treatment, and recycling, pyrolysis, and incineration are representative PSW treatment technologies. Generally speaking, a better treatment effect can be achieved through the combination of a variety of technologies and methods. Therefore, the combination of recycling and landfill, landfill and incineration, and recovery and pyrolysis are considered other alternatives in the selection process of PSW treatment technology.

• Landfill ($S_1$): Landfill is the process of accumulating waste in a certain land area, resulting in the decline of soil quality. It is a traditional method of waste treatment and a commonly used waste treatment option in China. The purpose of landfill is to reduce the environmental pollution caused by waste. Typically, most solid waste and plastic solid waste are buried in various regions of the world.

• Recycling ($S_2$): The meaning of recycling and treatment of PSW is to reprocess the plastic waste to produce new usable products, so as to reduce waste, reduce environmental pollution, and improve production efficiency. Plastic recycling includes four categories: primary (mechanical reprocessing into a product with equivalent properties), secondary (mechanical reprocessing into products requiring lower properties), tertiary (recovery of chemical constituents), and quaternary (recovery of energy).

• Pyrolysis ($S_3$): Pyrolysis refers to the anaerobic combustion process at high temperatures. In the pyrolysis process of PSW, under the influence of temperature, residence time, catalyst, and other factors, macromolecular polymers are pyrolyzed into a mixture of CO, H2, CH4, and high hydrocarbons.

• Incineration ($S_4$): Incineration is the process of burning PSW. The incineration treatment process reduces the demand for landfill of plastic waste and recovers some energy from plastic waste for electricity generation, combined heat and power, or as solid refuse fuel for co-fuelling of blast furnaces or cement kilns, so as to reduce carbon dioxide emission. Using incineration as a method to treat PSW is now recommended to be applied in various places.

Based on the characteristics of PSW and the emerging technologies of waste treatment, the classical PSW treatment technologies are selected, but the typical four technologies of landfill, recycling, pyrolysis, and incineration have their own advantages and weak points. For instance, the landfill technology has the advantages of relatively lower cost and easier implementation, but it does not achieve the goal of reducing the volume of municipal solid waste and converting municipal solid waste into reusable fuel and requires a large area of occupied land (Cheng and Hu 2010). The advantage of pyrolysis is that a high fuel is produced in the treatment process, which can be easily sold on the market and used for heating and gas engine power generation (Demirbas 2004), but if specific products are required, the final fuel needs to be treated (Al-Salem et al. 2009). Therefore, it is highly laborious for the DMs to select the best PSW treatment technology among these seven PSW treatment options.

Four evaluation criteria, including environmental criteria ($C_1$), economic criteria ($C_2$), social criteria ($C_3$), and technical criteria ($C_4$), have been established to evaluate the PSW treatment technologies. Nine sub-criteria under the four major criteria are also employed in “Materials and methods” to evaluate and analyze the alternatives. There are three groups of stakeholders, i.e., DMs, who participate in the analysis and decision-making process of PSW technology. The first group is composed of three managers from the Environmental Protection Agency of a local government in China and two senior managers (group#1) from Shanghai MSW treatment-related companies. The second group is composed of four professors majoring in Environmental Engineering in universities and three engineers (group#2) for municipal solid waste treatment. The third group includes two urban environmentalists and three local residents who are highly concerned about the treatment of PSW in Shanghai (group#3). Three groups of stakeholders reached an agreement through internal coordination and discussion.

• Step 1: Determine the weights of the criteria by using the proposed fuzzy DEMATEL.

  • Step 1.1: Construct the initial direct-influence fuzzy matrix $\tilde{Z}\left(k\right)$, i.e., $\tilde{Z}\left(1\right)$, $\tilde{Z}\left(2\right)$ and $\tilde{Z}\left(3\right)$. The direct-influence fuzzy matrix determined by the three groups of stakeholders using linguistic variables is shown in Table 4. Then, the linguistic variables in Table 4 are transformed into triangular fuzzy numbers, and the results are represented in Table 5.
  • Step 1.2: Determine the weight of stakeholders, and accordingly obtain the weighted direct-influence fuzzy matrix $\tilde{F}$.

• Determine the comparison matrix by using linguistic variables $\tilde{B}$. The comparison matrix determined is displayed in Table 6.

• Calculate the fuzzy weights of the stakeholders by Eq. (27).

$$\begin{array}{c}{\tilde{r}}_1={\left[{\tilde{b}}_{11},\left(x\right),{\tilde{b}}_{12},\left(x\right),{\tilde{b}}_{13}\right]}^{1/3}={\left[\left(1,1,,,1\right),(x),\left(1,2,,,3\right),(x),\left(2,3,,,4\right)\right]}^{1/3}\hfill \\ =\left({\left(1,1,,,2\right)}^{1/3},,,{\left(1,2,,,3\right)}^{1/3},,,{\left(1,3,,,4\right)}^{1/3}\right)=(1.260,1.817,2.289)\hfill \end{array}$$

$$\begin{array}{c}{\tilde{r}}_2={\left[{\tilde{b}}_{21},\left(x\right),{\tilde{b}}_{22},\left(x\right),{\tilde{b}}_{23}\right]}^{1/3}={\left[(0.33,0.50,1.00),(x),\left(1,1,,,1\right),(x),\left(1,2,,,3\right)\right]}^{1/3}\hfill \\ =\left({\left(0.33,1,,,1\right)}^{1/3},,,{\left(0.50,1,,,2\right)}^{1/3},,,{\left(1.00,1,,,3\right)}^{1/3}\right)=(0.691,1.000,1.442)\hfill \end{array}$$

$$\begin{array}{c}{\tilde{r}}_3={\left[{\tilde{b}}_{31},\left(x\right),{\tilde{b}}_{32},\left(x\right),{\tilde{b}}_{33}\right]}^{1/3}={\left[(0.25,0.33,0.50),(x),\left(0.33,0.50,1.00\right),(x),\left(1,1,,,1\right)\right]}^{1/3}\hfill \\ =\left({\left(0.25,0.33,1\right)}^{1/3},,,{\left(0.33,0.50,1\right)}^{1/3},,,{\left(0.50,1.00,1\right)}^{1/3}\right)=(0.435,0.548,0.794)\hfill \end{array}$$

$$\begin{array}{c}{\varpi }_1={\tilde{r}}_1\left(x,\right){\left[{\tilde{r}}_1,\left(+\right),{\tilde{r}}_2,\left(+\right),{\tilde{r}}_3\right]}^{-1}=(1.260,1.817,2.289)(\mathrm{x})[(1.260,1.817,2.289)(+)\\ {(0.691,1.000,1.442)(+)(0.435,0.548,0.794)]}^{-1}=(1.260,1.817,2.289)(\mathrm{x})\\ [1/(2.289+1.442+0.794),1/(1.817+1.000+0.548),1/(1.260+0.691+0.435)]\\ =(0.278,0.540,0.959)\end{array}$$

$$\begin{array}{c}{\varpi }_2={\tilde{r}}_2\left(x,\right){\left[{\tilde{r}}_1,\left(+\right),{\tilde{r}}_2,\left(+\right),{\tilde{r}}_3\right]}^{-1}=(0.691,1.000,1.442)(\mathrm{x})[(1.260,1.817,2.289)(+)\\ {(0.691,1.000,1.442)(+)(0.435,0.548,0.794)]}^{-1}=(0.691,1.000,1.442)(\mathrm{x})\\ [1/(2.289+1.442+0.794),1/(1.817+1.000+0.548),1/(1.260+0.691+0.435)]\\ =(0.153,0.423,0.604)\end{array}$$

$$\begin{array}{c}{\varpi }_3={\tilde{r}}_3\left(\times \right){\left[{\tilde{r}}_1,\left(+\right),{\tilde{r}}_2,\left(+\right),{\tilde{r}}_3\right]}^{-1}=(0.435,0.548,0.794)(\times )[(1.260,1.817,2.289)(+)\hfill \\ (0.691,1.000,1.442)(+)(0.435,0.548,0.794){]}^{-1}=(0.435,0.548,0.794)(\times )\hfill \\ \begin{array}{c}[1/(2.289+1.442+0.794),1/(1.817+1.000+0.548),1/(1.260+0.691+0.435)]\hfill \\ (0.096,0.163,0.333)\hfill \end{array}\hfill \end{array}$$

Table 4.

The direct-influence fuzzy matrix determined by the three groups of stakeholders using linguistic variables

Group#1		C11	C12	C13	C21	C22	C31	C32	C41	C42
C1	C11	N	VL	VL	N	VL	VH	H	N	N
	C12	VL	N	VL	VL	VL	VH	VH	N	N
	C13	VL	VL	N	VL	VL	H	VH	N	N
C2	C21	N	N	N	N	VL	H	H	N	N
	C22	N	N	N	VL	N	VH	H	N	N
C3	C31	N	N	N	N	N	N	VL	N	N
	C32	N	N	N	N	N	VL	N	N	N
C4	C41	VH	VH	VH	H	VH	VH	H	N	L
	C42	H	VH	VH	VH	H	VH	VH	VL	N
Group#2		C11	C12	C13	C21	C22	C31	C32	C41	C42
C1	C11	N	VL	VL	N	N	VH	VH	N	N
	C12	VL	N	VL	N	VL	H	VH	N	N
	C13	N	N	N	VL	VL	VH	H	N	N
C2	C21	N	N	N	VL	VL	VH	VH	N	N
	C22	N	N	N	VL	VL	VH	H	N	N
C3	C31	N	N	N	N	N	VL	VL	N	N
	C32	N	N	N	N	N	N	N	N	N
C4	C41	H	H	VH	H	VH	VH	VH	N	L
	C42	VH	VH	H	VH	VH	VH	VH	N	N
Group#3		C11	C12	C13	C21	C22	C31	C32	C41	C42
C1	C11	N	VL	VL	N	N	VH	H	N	N
	C12	VL	N	VL	N	VL	VH	VH	N	N
	C13	VL	N	N	VL	VL	H	H	N	N
C2	C21	N	N	N	VL	VL	VH	H	N	N
	C22	N	N	N	N	VL	VH	H	N	N
C3	C31	N	N	N	N	N	VL	N	N	N
	C32	N	N	N	N	N	N	N	N	N
C4	C41	H	VH	H	H	VH	H	VH	N	VL
	C42	VH	VH	VH	H	VH	VH	H	N	N

Table 5.

The direct-influence fuzzy matrix determined by the three groups of stakeholders using triangular fuzzy numbers

Group#1		C11	C12	C13	C21	C22	C31	C32	C41	C42
C1	C11	(0,0,0)	(0,0,0.25)	(0,0,0.25)	(0,0,0)	(0,0,0.25)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0,0,0)	(0,0,0)
	C12	(0,0,0.25)	(0,0,0)	(0,0,0.25)	(0,0,0.25)	(0,0,0.25)	(0.5,0.75,1)	(0.5,0.75,1)	(0,0,0)	(0,0,0)
	C13	(0,0,0.25)	(0,0,0.25)	(0,0,0)	(0,0,0.25)	(0,0,0.25)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0,0,0)	(0,0,0)
C2	C21	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0.25)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0,0,0)	(0,0,0)
	C22	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0.25)	(0,0,0)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0,0,0)	(0,0,0)
C3	C31	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0.25)	(0,0,0)	(0,0,0)
	C32	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0.25)	(0,0,0)	(0,0,0)	(0,0,0)
C4	C41	(0.5,0.75,1)	(0.5,0.75,1)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0,0,0)	(0,0.25,0.5)
	C42	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.5,0.75,1)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0,0,0.25)	(0,0,0)
Group#2		C11	C12	C13	C21	C22	C31	C32	C41	C42
C1	C11	(0,0,0)	(0,0,0.25)	(0,0,0.25)	(0,0,0)	(0,0,0)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0,0,0)	(0,0,0)
	C12	(0,0,0.25)	(0,0,0)	(0,0,0.25)	(0,0,0)	(0,0,0.25)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0,0,0)	(0,0,0)
	C13	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0.25)	(0,0,0.25)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0,0,0)	(0,0,0)
C2	C21	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0.25)	(0,0,0.25)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0,0,0)	(0,0,0)
	C22	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0.25)	(0,0,0.25)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0,0,0)	(0,0,0)
C3	C31	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0.25)	(0,0,0.25)	(0,0,0)	(0,0,0)
	C32	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)
C4	C41	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0,0,0)	(0,0.25,0.5)
	C42	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0,0,0)	(0,0,0)
Group#3		C11	C12	C13	C21	C22	C31	C32	C41	C42
C1	C11	(0,0,0)	(0,0,0.25)	(0,0,0.25)	(0,0,0)	(0,0,0)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0,0,0)	(0,0,0)
	C12	(0,0,0.25)	(0,0,0)	(0,0,0.25)	(0,0,0)	(0,0,0.25)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0,0,0)	(0,0,0)
	C13	(0,0,0.25)	(0,0,0)	(0,0,0)	(0,0,0.25)	(0,0,0.25)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0,0,0)	(0,0,0)
C2	C21	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0.25)	(0,0,0.25)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0,0,0)	(0,0,0)
	C22	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0.25)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0,0,0)	(0,0,0)
C3	C31	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0.25)	(0,0,0)	(0,0,0)	(0,0,0)
	C32	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)
C4	C41	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0,0,0)	(0,0,0.25)
	C42	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0,0,0)	(0,0,0)

Table 6.

The comparison matrix for determining the role weights of the three groups of DMs

	Group#1	Group#2	Group#3
Group#1	E	W	M
Group#2	RW	E	W
Group#3	RM	RW	E
	Group#1	Group#2	Group#3
Group#1	(1,1,1)	(1,2,3)	(2,3,4)
Group#2	(0.33,0.50,1.00)	(1,1,1)	(1,2,3)
Group#3	(0.25,0.33,0.50)	(0.33,0.50,1.00)	(1,1,1)

RW and RM indicate the reciprocals of W and M

Then, the relative weights of three groups of stakeholders are $\left(0.278,,,0.540,,,0.959\right)$, $\left(0.153,,,0.423,,,0.604\right)$, and $\left(0.096,,,0.163,,,0.333\right)$, respectively.

The weighted direct-influence fuzzy matrix $\tilde{F}$ calculated by Eq. (24) is directed in Table 7.

• Step 1.3: Normalize the direct-influence fuzzy matrix by Eq. (15) to obtain the normalized direct-influence fuzzy matrix $\tilde{Y}\left(k\right)$in the format of Eq. (16), where $U_i^{\left(k\right)}$ = 12.721. The results of normalized direct-influence fuzzy matrix $\tilde{Y}\left(k\right)$ are shown in Table 8.

• Step 1.4: Determine the total-influence fuzzy matrix $\tilde{T}$ by Eq. (17), as Table 9.

• Step 1.5: Use Eq. (19) to defuzzify. The results after defuzzification are shown in Table 10.

• Step 1.6: Use Eq. (21) to calculate the influence degree $\tilde{D}$ and the influenced degree $\tilde{R}$ of each element, whose results are shown in Table 11.

• Step 1.7: Determine the weights of these evaluation criteria by Eqs. (22) and (23). The obtained evaluation criteria weights are $\omega =\left(0.06,,,0.07,,,0.07,,,0.08,,,0.08,,,0.17,,,0.16,,,0.16,,,0.15\right)$. The weights of criteria and sub-criteria are obtained, as shown in Fig. 6.

• Step 2: Firstly, the DMs give the description for each evaluation criterion according to the DMs’ own experience and discussion, and determine the performance of the alternatives relative to each evaluation criterion.

Table 7.

The weighted direct-influence fuzzy matrix

		C11	C12	C13	C21	C22	C31	C32	C41	C42
C1	C11	(0,0,0)	(0,0,0.474)	(0,0,0.474)	(0,0,0)	(0,0,0.240)	(0.264,0.845,1.896)	(0.132,0.563,1.422)	(0,0,0)	(0,0,0)
	C12	(0,0,0.474)	(0,0,0)	(0,0,0.474)	(0,0,0.240)	(0,0,0.474)	(0.225,0.739,1.745)	(0.240,0.804,1.813)	(0,0,0)	(0,0,0)
	C13	(0,0,0.323)	(0,0,0.240)	(0,0,0)	(0,0,0.474)	(0,0,0.474)	(0.170,0.669,1.573)	(0.201,0.698,1.662)	(0,0,0)	(0,0,0)
C2	C21	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0.474)	(0.194,0.710,1.656)	(0.132,0.563,1.422)	(0,0,0)	(0,0,0)
	C22	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0.25)	(0,0,0)	(0.264,0.845,1.896)	(0.132,0.563,1.422)	(0,0,0)	(0,0,0)
C3	C31	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0.391)	(0,0,0)	(0,0,0)
	C32	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0.240)	(0,0,0)	(0,0,0)	(0,0,0)
C4	C41	(0.201,0.698,1.662)	(0.225,0.739,1.745)	(0.240,0.804,1.813)	(0.132,0.563,1.422)	(0.264,0.845,1.896)	(0.201,0.698,1.662)	(0.194,0.710,1.656)	(0,0,0)	(0,0.241,0.865)
	C42	(0.194,0.710,1.656)	(0.201,0.698,1.662)	(0.225,0.739,1.745)	(0.240,0.804,1.813)	(0.156,0.604,1.505)	(0.240,0.804,1.813)	(0.132,0.563,1.422)	(0,0,0.240)	(0,0,0)

Table 8.

Normalized direct-influence fuzzy matrix

		C11	C12	C13	C21	C22	C31	C32	C41	C42
C1	C11	(0,0,0)	(0,0,0.037)	(0,0,0.037)	(0,0,0.019)	(0,0,0.019)	(0.021,0.067,0.149)	(0.010,0.044,0.112)	(0,0,0)	(0,0,0)
	C12	(0,0,0.037)	(0,0,0)	(0,0,0.037)	(0,0,0.019)	(0,0,0.037)	(0.018,0.058,0.137)	(0.019,0.063,0.143)	(0,0,0)	(0,0,0)
	C13	(0,0,0.025)	(0,0,0.019)	(0,0,0)	(0,0,0.037)	(0,0,0.037)	(0.013,0.053,0.124)	(0.016,0.055,0.131)	(0,0,0)	(0,0,0)
C2	C21	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0.037)	(0.015,0.056,0.130)	(0.010,0.044,0.112)	(0,0,0)	(0,0,0)
	C22	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0.25)	(0,0,0)	(0.021,0.067,0.149)	(0.010,0.044,0.112)	(0,0,0)	(0,0,0)
C3	C31	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0.031)	(0,0,0)	(0,0,0)
	C32	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0.019)	(0,0,0)	(0,0,0)	(0,0,0)
C4	C41	(0.016,0.055,0.131)	(0.018,0.058,0.137)	(0.019,0.063,0.143)	(0.010,0.044,0.112)	(0.021,0.067,0.149)	(0.016,0.055,0.131)	(0.015,0.056,0.130)	(0,0,0)	(0,0.019,0.068)
	C42	(0.015,0.056,0.130)	(0.016,0.055,0.131)	(0.018,0.058,0.137)	(0.019,0.063,0.143)	(0.012,0.047,0.118)	(0.019,0.063,0.143)	(0.010,0.044,0.112)	(0,0,0.019)	(0,0,0)

Table 9.

The total-influence fuzzy matrix

		C11	C12	C13	C21	C22	C31	C32	C41	C42
C1	C11	(0,0,0.002)	(0,0,0.038)	(0,0,0.038)	(0,0,0.027)	(0,0,0.023)	(0.021,0.067,0.169)	(0.010,0.044,0.134)	(0,0,0)	(0,0,0)
	C12	(0,0,0.038)	(0,0,0.002)	(0,0,0.038)	(0,0,0.031)	(0,0,0.040)	(0.018,0.058,0.161)	(0.019,0.063,0.166)	(0,0,0)	(0,0,0)
	C13	(0,0,0.026)	(0,0,0.020)	(0,0,0.002)	(0,0,0.048)	(0,0,0.040)	(0.013,0.053,0.146)	(0.016,0.055,0.151)	(0,0,0)	(0,0,0)
C2	C21	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0.009)	(0,0,0.037)	(0.015,0.056,0.139)	(0.010,0.044,0.122)	(0,0,0)	(0,0,0)
	C22	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0.252)	(0,0,0.009)	(0.021,0.067,0.186)	(0.010,0.044,0.147)	(0,0,0)	(0,0,0)
C3	C31	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0.001)	(0,0,0.031)	(0,0,0)	(0,0,0)
	C32	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0.019)	(0,0,0)	(0,0,0)	(0,0,0)
C4	C41	(0.016,0.056,0.150)	(0.018,0.059,0.155)	(0.019,0.064,0.164)	(0.010,0.045,0.178)	(0.021,0.068,0.178)	(0.017,0.074,0.259)	(0.016,0.072,0.246)	(0,0,0)	(0,0.019,0.068)
	C42	(0.015,0.056,0.142)	(0.016,0.055,0.142)	(0.018,0.058,0.150)	(0.019,0.063,0.192)	(0.012,0.047,0.142)	(0.020,0.080,0.255)	(0.011,0.058,0.216)	(0,0,0.019)	(0,0,0.001)

Table 10.

The total-influence matrix after defuzzification

		C11	C12	C13	C21	C22	C31	C32	C41	C42
C1	C11	0.00067	0.01267	0.01267	0.00900	0.00767	0.08567	0.06267	0	0
	C12	0.01267	0.00067	0.01267	0.01033	0.01333	0.11233	0.08267	0	0
	C13	0.00867	0.00667	0.00067	0.01600	0.01333	0.07067	0.07400	0	0
C2	C21	0	0	0	0.00300	0.01233	0.07000	0.05867	0	0
	C22	0	0	0	0.08400	0.00300	0.09133	0.06700	0	0
C3	C31	0	0	0	0	0	0.00033	0.10333	0	0
	C32	0	0	0	0	0	0.00633	0	0	0
C4	C41	0.07400	0.07733	0.08233	0.07767	0.08900	0.11667	0.11133	0	0.02900
	C42	0.07100	0.07100	0.07533	0.09133	0.06700	0.11833	0.09500	0.00633	0.00033

Table 11.

The influence degree and the influenced degree of each element

		${\tilde{D}}_i$	${\tilde{R}}_j$	${\tilde{D}}_i+{\tilde{R}}_i$	${\tilde{D}}_i-{\tilde{R}}_i$	Criterion type
C1	C11	0.191	0.167	0.358	0.024	Causal factor
	C12	0.245	0.168	0.413	0.077	Causal factor
	C13	0.190	0.184	0.374	0.006	Causal factor
C2	C21	0.144	0.291	0.435	-0.147	Resulting factor
	C22	0.245	0.206	0.451	0.039	Causal factor
C3	C31	0.104	0.672	0.776	-0.568	Resulting factor
	C32	0.006	0.655	0.661	-0.649	Resulting factor
C4	C41	0.657	0.006	0.663	0.651	Causal factor
	C42	0.596	0.029	0.625	0.567	Causal factor

Fig. 6.

The weights of criteria and sub-criteria

$C_{11}$, $C_{12}$, $C_{13}$, and $C_{21}$ are cost criteria and while $C_{22}$, $C_{31}$, $C_{32}$, $C_{41}$, and $C_{42}$ are benefit criteria. For example, if a treatment technology is evaluated as "very good" in terms of “water pollution ($C_{12}$),” it means that the alternative has significantly little water pollution. If a waste treatment technology is evaluated as “very good” in terms of “social acceptance ($C_{31}$),” it means that the social acceptance of the alternative is extremely high. The decision matrix determined by the three groups of stakeholders by using linguistic terms is shown in Table 12.

• Step 3: Transform the corresponding expert description in the form of linguistic variables into triangular fuzzy numbers by Eq. (13) and Table 1. The transformed table is shown in Table 13.

• Step 4: Aggregate expert information according to the information given by different stakeholders and determine the weighted fuzzy decision matrix by Eqs. (29) and (30) as shown in Table 14.

• Step 5: Normalize the weighted fuzzy decision matrix $D={\left[d_{\mathit{ij}}\right]}_{m\times n}$ into $R={\left[r_{\mathit{ij}}\right]}_{m\times n}$. The normalized weighted fuzzy decision matrix is shown in Table 15.

• Step 6: Calculate the PIS and NIS for each criterion.

$$\begin{array}{c}P=\{P_1,P_2,\cdots ,P_n\}=\{(0.090,0.157,0.458),(0.094,0.220,0.804),(0.094,0.211,0.708),\\ (0.104,0.258,1),(0.147,0.480,0.956),(0.176,0.517,1),(0.172,0.523,1),\\ (0.090,0.353,0.830),(0.208,0.594,1)\},\end{array}$$

$$N=\{N_1,N_2,...,N_n\}=\{(0,0,0),(0,0,0),(0,0,0),(0,0,0),(0.022,0.222,0.630),(0.070,0.297,0.750),(0.044,0.301,0.831),(0.018,0.140,0.591),(0.028,0.289,0.821)\}.$$

• Step7: Compute the gain and loss values of each alternative, i.e., the distance between each alternative and PIS/NIS, and then obtain the gain $G={\left[g_{\mathit{ij}}\right]}_{m\times n}$ matrix and loss matrix $L={\left[l_{\mathit{ij}}\right]}_{m\times n}$.

$$\begin{array}{c}G=\left[\begin{array}{ccccccccc}0.000& 0.000& 0.000& 0.000& 0.199& 0.118& 0.112& 0.000& 0.000\\ 0.395& 0.484& 0.361& 0.599& 0.251& 0.202& 0.040& 0.161& 0.052\\ 0.092& 0.108& 0.084& 0.579& 0.000& 0.102& 0.130& 0.079& 0.037\\ 0.000& 0.000& 0.000& 0.000& 0.201& 0.156& 0.000& 0.206& 0.051\\ 0.339& 0.390& 0.584& 0.238& 0.259& 0.000& 0.059& 0.054& 0.048\\ 0.579& 0.108& 0.164& 0.084& 0.086& 0.173& 0.085& 0.032& 0.048\\ 0.284& 0.321& 0.430& 0.405& 0.173& 0.087& 0.177& 0.189& 0.229\end{array}\right]\hfill \\ L=-\left[\begin{array}{ccccccccc}0.284& 0.484& 0.430& 0.599& 0.092& 0.122& 0.093& 0.189& 0.229\\ 0.113& 0.000& 0.070& 0.000& 0.000& 0.000& 0.147& 0.029& 0.183\\ 0.193& 0.376& 0.346& 0.116& 0.251& 0.106& 0.063& 0.113& 0.233\\ 0.284& 0.484& 0.430& 0.599& 0.077& 0.085& 0.177& 0.041& 0.188\\ 0.075& 0.100& 0.175& 0.361& 0.032& 0.202& 0.120& 0.144& 0.238\\ 0.319& 0.376& 0.268& 0.515& 0.165& 0.046& 0.095& 0.201& 0.238\\ 0.000& 0.166& 0.000& 0.197& 0.083& 0.119& 0.000& 0.000& 0.000\end{array}\right]\hfill \end{array}$$

Table 12.

The decision matrix determined by the three groups of stakeholders by using linguistic terms

Group#1	C1			C2		C3		C4
	C11	C12	C13	C21	C22	C31	C32	C41	C42
S1	VP	P	VP	P	M	M	M	VP	P
S2	G	M	G	M	G	VG	M	M	M
S3	M	M	M	P	P	M	G	P	M
S4	P	P	VP	M	M	M	P	P	M
S5	M	M	P	M	G	M	M	P	M
S6	P	M	P	M	M	G	G	P	M
S7	VG	G	G	M	G	G	G	M	VG
Group#2	C1			C2		C3		C4
	C11	C12	C13	C21	C22	C31	C32	C41	C42
S1	VP	VP	P	P	P	P	M	P	M
S2	G	G	G	M	VG	G	G	M	G
S3	M	M	G	P	M	G	M	M	M
S4	VP	P	P	P	M	M	M	M	P
S5	M	P	M	M	G	M	G	M	M
S6	P	M	M	G	M	G	M	P	M
S7	VG	VG	G	VG	M	M	VG	G	VG
Group#3	C1			C2		C3		C4
	C11	C12	C13	C21	C22	C31	C32	C41	C42
S1	VP	P	P	VP	M	M	P	M	P
S2	M	G	G	G	M	G	M	G	M
S3	G	M	M	M	P	M	G	M	P
S4	P	P	VP	P	P	M	M	G	M
S5	P	M	M	G	G	M	G	M	M
S6	M	M	G	M	M	G	G	M	M
S7	G	VG	M	VG	M	M	VG	M	VG

Table 13.

The decision matrix determined by the three groups of stakeholders by using triangular fuzzy numbers

Group#1	C1			C2		C3		C4
	C11	C12	C13	C21	C22	C31	C32	C41	C42
S1	(0,0,0.25)	(0,0.25,0.5)	(0,0,0.25)	(0,0.25,0.5)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0,0,0.25)	(0,0.25,0.5)
S2	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.75,1,1)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0.25,0.5,0.75)
S3	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0,0.25,0.5)	(0,0.25,0.5)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0,0.25,0.5)	(0.25,0.5,0.75)
S4	(0,0.25,0.5)	(0,0.25,0.5)	(0,0,0.25)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0,0.25,0.5)	(0,0.25,0.5)	(0.25,0.5,0.75)
S5	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0,0.25,0.5)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0,0.25,0.5)	(0.25,0.5,0.75)
S6	(0,0.25,0.5)	(0.25,0.5,0.75)	(0,0.25,0.5)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.5,0.75,1)	(0,0.25,0.5)	(0.25,0.5,0.75)
S7	(0.75,1,1)	(0.5,0.75,1)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.5,0.75,1)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.75,1,1)
Group#2	C1			C2		C3		C4
	C11	C12	C13	C21	C22	C31	C32	C41	C42
S1	(0,0,0.25)	(0,0,0.25)	(0,0.25,0.5)	(0,0.25,0.5)	(0,0.25,0.5)	(0,0.25,0.5)	(0.25,0.5,0.75)	(0,0.25,0.5)	(0.25,0.5,0.75)
S2	(0.5,0.75,1)	(0.5,0.75,1)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.75,1,1)	(0.5,0.75,1)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.5,0.75,1)
S3	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0,0.25,0.5)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0.25,0.5,0.75)
S4	(0,0,0.25)	(0,0.25,0.5)	(0,0.25,0.5)	(0,0.25,0.5)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0,0.25,0.5)
S5	(0.25,0.5,0.75)	(0,0.25,0.5)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.25,0.5,0.75)
S6	(0,0.25,0.5)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0,0.25,0.5)	(0.25,0.5,0.75)
S7	(0.75,1,1)	(0.75,1,1)	(0.5,0.75,1)	(0.75,1,1)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0.75,1,1)	(0.5,0.75,1)	(0.75,1,1)
Group#3	C1			C2		C3		C4
	C11	C12	C13	C21	C22	C31	C32	C41	C42
S1	(0,0,0.25)	(0,0.25,0.5)	(0,0.25,0.5)	(0,0,0.25)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0,0.25,0.5)	(0.25,0.5,0.75)	(0,0.25,0.5)
S2	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.5,0.75,1)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.25,0.5,0.75)
S3	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0,0.25,0.5)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0,0.25,0.5)
S4	(0,0.25,0.5)	(0,0.25,0.5)	(0,0,0.25)	(0,0.25,0.5)	(0,0.25,0.5)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.25,0.5,0.75)
S5	(0,0.25,0.5)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.25,0.5,0.75)
S6	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0.5,0.75,1)	(0.5,0.75,1)	(0.25,0.5,0.75)	(0.25,0.5,0.75)
S7	(0.5,0.75,1)	(0.75,1,1)	(0.25,0.5,0.75)	(0.75,1,1)	(0.25,0.5,0.75)	(0.25,0.5,0.75)	(0.75,1,1)	(0.25,0.5,0.75)	(0.75,1,1)

Table 14.

The weighted fuzzy decision matrix

	C1			C2		C3		C4
	C11	C12	C13	C21	C22	C31	C32	C41	C42
S1	(0,0,0.474)	(0,0.176,0.797)	(0,0.147,0.708)	(0,0.241,0.865)	(0.094,0.457,1.271)	(0.094,0.457,1.271)	(0.108,0.522,1.339)	(0.024,0.187,0.792)	(0.038,0.387,1.099)
S2	(0.240,0.804,1.813)	(0.194,0.710,1.656)	(0.264,0.845,1.896)	(0.156,0.604,1.505)	(0.278,0.91,1.813)	(0.333,0.98,1.896)	(0.17,0.669,1.573)	(0.156,0.604,1.505)	(0.17,0.669,1.573)
S3	(0.156,0.604,1.505)	(0.132,0.563,1.422)	(0.170,0.669,1.573)	(0.024,0.322,1.031)	(0.038,0.387,1.099)	(0.17,0.669,1.573)	(0.225,0.739,1.745)	(0.062,0.428,1.182)	(0.108,0.522,1.339)
S4	(0,0.176,0.797)	(0,0.282,0.948)	(0,0.106,0.625)	(0.07,0.417,1.188)	(0.108,0.522,1.339)	(0.132,0.563,1.422)	(0.062,0.428,1.182)	(0.086,0.469,1.266)	(0.094,0.457,1.271)
S5	(0.108,0.522,1.339)	(0.094,0.457,1.271)	(0.062,0.428,1.182)	(0.156,0.604,1.505)	(0.264,0.845,1.896)	(0.132,0.563,1.422)	(0.194,0.71,1.656)	(0.062,0.428,1.182)	(0.132,0.563,1.422)
S6	(0.024,0.332,1.031)	(0.132,0.563,1.422)	(0.086,0.469,1.266)	(0.17,0.669,1.573)	(0.132,0.563,1.422)	(0.264,0.845,1.896)	(0.225,0.739,1.745)	(0.024,0.322,1.031)	(0.132,0.563,1.422)
S7	(0.371,1.085,1.896)	(0.326,0.991,1.896)	(0.240,0.804,1.813)	(0.256,0.856,1.656)	(0.201,0.698,1.662)	(0.201,0.698,1.662)	(0.326,0.991,1.896)	(0.17,0.669,1.573)	(0.395,1.126,1.896)

Table 15.

The weighted normalized decision matrix

	C1			C2		C3		C4
	C11	C12	C13	C21	C22	C31	C32	C41	C42
S1	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0.070,0.341,0.949)	(0.070,0.341,0.949)	(0.081,0.390,1)	(0.018,0.140,0.591)	(0.028,0.289,0.821)
S2	(0.086,0.194,0.650)	(0.094,0.220,0.804)	(0.082,0.185,0.591)	(0.104,0.258,1)	(0.147,0.480,0.956)	(0.176,0.517,1)	(0.090,0.353,0.830)	(0.082,0.319,0.794)	(0.090,0.353,0.830)
S3	(0.016,0.040,0.154)	(0.017,0.043,0.182)	(0.015,0.036,0.141)	(0.023,0.075,1)	(0.022,0.222,0.630)	(0.097,0.383,0.901)	(0.129,0.423,1)	(0.036,0.245,0.677)	(0.062,0.299,0.767)
S4	(0,0,0)	(0,0,0)	(0,0,0)	(0,0,0)	(0.076,0.367,0.942)	(0.093,0.396,1)	(0.044,0.301,0.831)	(0.060,0.330,0.890)	(0.066,0.321,0.894)
S5	(0.046,0.119,0.574)	(0.049,0.136,0.660)	(0.052,0.145,1)	(0.041,0.103,0.397)	(0.139,0.446,1)	(0.070,0.297,0.750)	(0.102,0.374,0.873)	(0.033,0.226,0.623)	(0.070,0.297,0.750)
S6	(0.023,0.075,1)	(0.017,0.043,0.182)	(0.019,0.051,0.279)	(0.015,0.036,0.141)	(0.070,0.297,0.750)	(0.139,0.446,1)	(0.119,0.390,0.920)	(0.013,0.170,0.544)	(0.070,0.297,0.750)
S7	(0.090,0.157,0.458)	(0.090,0.172,0.521)	(0.094,0.211,0.708)	(0.103,0.199,0.664)	(0.106,0.368,0.877)	(0.106,0.368,0.877)	(0.172,0.523,1)	(0.090,0.353,0.830)	(0.208,0.594,1)

• Step 8: Calculate prospect values and determine the positive value matrix $V_{\mathit{ij}}^{+}$ and negative prospect value matrix $V_{\mathit{ij}}^{-}$, respectively.

$$\begin{array}{c}{V}_{\mathit{ij}}^{+}=\left[\begin{array}{ccccccccc}0.000& 0.000& 0.000& 0.000& 0.242& 0.152& 0.146& 0.000& 0.000\\ 0.442& 0.528& 0.408& 0.637& 0.296& 0.245& 0.059& 0.200& 0.074\\ 0.122& 0.141& 0.113& 0.618& 0.000& 0.134& 0.166& 0.107& 0.055\\ 0.000& 0.000& 0.000& 0.000& 0.244& 0.195& 0.000& 0.249& 0.073\\ 0.386& 0.437& 0.623& 0.283& 0.305& 0.000& 0.083& 0.077& 0.069\\ 0.618& 0.141& 0.204& 0.113& 0.115& 0.214& 0.114& 0.048& 0.069\\ 0.330& 0.368& 0.476& 0.451& 0.214& 0.117& 0.218& 0.231& 0.273\end{array}\right]\hfill \\ V_{\mathit{ij}}^{-}=\left[\begin{array}{ccccccccc}-0.743& -1.188& -1.071& -1.433& -0.276& -0.353& -0.278& -0.519& -0.615\\ -0.330& 0.000& -0.217& 0.000& 0.000& 0.000& -0.416& -0.100& -0.505\\ -0.529& -0.951& -0.884& -0.338& -0.667& -0.312& -0.198& -0.330& -0.624\\ -0.743& -1.188& -1.071& -1.433& -0.236& -0.257& -0.490& -0.135& -0.517\\ -0.230& -0.297& -0.485& -0.918& -0.109& -0.551& -0.348& -0.409& -0.636\\ -0.823& -0.951& -0.706& -1.255& -0.461& -0.150& -0.284& -0.548& -0.636\\ 0.000& -0.463& 0.000& -0.539& -0.252& -0.346& 0.000& 0.000& 0.000\end{array}\right]\hfill \end{array}$$

• Step 9: Figure the cumulative prospect weights and gain the positive decision weight matrix ${\mathrm{\Pi }}^{+}\left({\omega }_j\right)$ and negative weight matrix ${\mathrm{\Pi }}^{-}\left({\omega }_j\right)$, respectively.

$$\begin{array}{c}{\mathrm{\Pi }}^{+}({\omega }_j)=\{0.144,0.156,0.156,0.167,0.167,0.241,0.234,0.234,0.227\},\hfill \\ {\mathrm{\Pi }}^{-}({\omega }_j)=\{0.125,0.137,0.137,0.149,0.149,0.233,0.225,0.225,0.217\}.\hfill \end{array}$$

• Step 10: Calculate the comprehensive prospect value $V$ of each alternative and make a ranking.

$$\begin{array}{ccc}{V}_1=-0.941,& V_2=0.206,& \begin{array}{ccc}{V}_3=-0.525,& V_4=-0.801,& \begin{array}{ccc}{V}_5=-0.353,& V_6=-0.659,& V_7=0.224\end{array}\end{array}\end{array}$$

Therefore, the seven alternatives are ranked as $S_7>S_2>S_5>S_3>S_6>S_4>S_1$. This means that the combination of recycling and incineration is the best treatment alternative, followed by recycling, the combination of recycling and landfill, pyrolysis, the combination of landfill and incineration, incineration and landfill. The values and ranking results of the comprehensive prospect values of the seven alternatives are shown in Fig. 7.

Fig. 7.

The values and ranking results of the comprehensive prospect values of the seven alternatives

The results show that the combination of recycling and incineration is the best treatment alternative, which means that the combination of recycling and incineration has two technical advantages of recycling and incineration, and can be better suited to handle the PSW. Landfill which is the worst PSW treatment technology obtained by the proposed method is mainly because some of the PSW are recyclable and cannot be simply landfilled. Moreover, landfill will cause certain pollution to the soil and air, and some wastes are not degradable. Therefore, the results obtained from the analysis with the method proposed in this paper are reasonable and have certain universality, and can also be applied to other environmental decision-making fields.

Moreover, we calculated and analyzed the performance of each alternative on the four criteria, and the results are shown in Fig. 8. As can be seen from Fig. 8, none of the PSW treatment technologies performed equally well on all four criteria considered. The combination of recycling and landfill which ranks first in environmental criteria performs best in the environmental criteria of the four criteria, while recycling has absolute advantages in economic criteria and the combination of landfill and incineration is inclined towards social criteria. Whereas, the combination of recycling and incineration performs best in technical criteria.

Fig. 8.

The performances of PSW treatment alternatives concerning with criteria

Results and discussions

Sensitivity analysis of$\lambda$

In practice, the psychological factors of DMs such as loss aversion and reference dependence will have a certain impact on the authenticity and effectiveness of the final decision. Therefore, since the loss aversion coefficient $\lambda$ reflect the risk attitude of stakeholders, in order to illustrate the feasibility of the model developed in this paper in the PSW treatment scenario, the sensitivity analysis is carried out by using the parameter $\lambda$, and its impact on the ranking of alternatives is discussed by fluctuating the parameter $\lambda$. The ranking of PSW treatment technologies obtained by sensitivity analysis of the value of parameter $\lambda$ fluctuating from 1 to 4 is shown in Table 16 and Fig. 9.

Table 16.

Ranking results with different values of λ

$\lambda$	Ranking
1	$S_2>S_7>S_5>S_3>S_6>S_4>S_1$
1.5	$S_7>S_2>S_5>S_3>S_6>S_4>S_1$
2	$S_7>S_2>S_5>S_3>S_6>S_4>S_1$
2.25	$S_7>S_2>S_5>S_3>S_6>S_4>S_1$
3	$S_7>S_2>S_5>S_3>S_6>S_4>S_1$
3.5	$S_7>S_2>S_5>S_3>S_6>S_4>S_1$
4	$S_7>S_2>S_5>S_3>S_6>S_4>S_1$

Fig. 9.

The ranking of PSW treatment technologies obtained by sensitivity analysis

When $\lambda$ = 1, recycling ($S_2$) is the best PSW treatment technology, and the order of the technologies is $S_2>S_7>S_5>S_3>S_6>S_4>S_1$, and when $\lambda$ > 1, the combination of recycling and incineration ($S_7$) is the best PSW treatment technology, and the alternative ranking is $S_7>S_2>S_5>S_3>S_6>S_4>S_1$, which shows that the alternative ranking calculated by the method proposed in this paper is robust and reliable. The main reason for the change of the optimal alternative is that the prospect values of the conceptual criteria of some loss types are underestimated in the case of low-level loss aversion. However, such losses are enlarged with the increase of loss aversion coefficients, resulting in the ranking of concepts with small loss. In practical fuzzy decision-making, there is a high degree of loss aversion, so the ranking results under the assumption of relatively high loss aversion have more meaning and implications.

Sensitivity analysis with criteria weight fluctuations

Sensitivity analysis of various criteria weights fluctuation is used to test the impact of the change of criteria weights on the ranking of the final PSW treatment technologies, that is, the perturbation method is used to analyze the sensitivity of the evaluation criteria weights to test the corresponding change of the ranking results of each PSW treatment technology when the evaluation criteria weights in the decision-making are slightly disturbed.

Let ${\omega }_j$ denote the initial evaluation criterion weight, and ${\omega }_j^{{}^{\prime }}={\upsilon \omega }_j$ represent the disturbed evaluation criterion weight, where $0\le {\omega }_j^{{}^{\prime }}\le 1$ and $0\le \upsilon \le 1/{\omega }_j$. Because the sum of the weights of the evaluation criteria is equal to 1, when ${\omega }_j$ changes, the weights of other evaluation criteria will also change, which are recorded as ${\omega }_k^{{}^{\prime }}={\tau \omega }_k$, $k\ne j$, $k=1,2,\cdots ,n$, and meet ${\upsilon \omega }_j+\tau {\sum }_{k\ne j,k=1}^n{\omega }_k=1$. Based on the above calculation and analysis, $\tau =\left(1,-,{\upsilon \omega }_j\right)/\left(1,-,{\omega }_j\right)$ can be obtained. In this paper, nine evaluation criteria are disturbed, respectively, and $\upsilon$ is taken as 1/4, 1/3, 1/2, 2, 3, and 4 in turn. A total of 54 experiments are carried out. The results of the sensitivity analysis are shown in Fig. 10.

Fig. 10.

The results of sensitivity analysis

As can be seen from Fig. 7, in 54 experiments, the comprehensive prospect values of $S_2$ and $S_7$ are always higher than other alternatives and the comprehensive prospect values of $S_2$ and $S_7$ are intersect at the highest. In the 54 experiments, 36 times of $S_7$ and 18 times of $S_2$ have the highest comprehensive prospect value. The comprehensive prospect value of $S_1$ is the smallest in 53 experiments, and the 42nd experiment is the second smallest. In addition to $S_2$ and $S_7$, the ranking results of other alternatives are $S_5>S_3>S_6>S_4>S_1$ in 48 out of 53 experiments. To sum up, $S_2$ and $S_7$ are sensitive to the fluctuations of evaluation criteria weights, while other alternatives are not sensitive to the fluctuations of criteria weights.

Comparison analysis

In order to illustrate the feasibility and superiority of the method proposed in this paper, the FTOPSIS (Kang et al. 2019; Chu 2002) and TODIM methods (Li and Cao 2019; Tosun and Akyüz 2015) will be used to study the case respectively, so as to compare with the method proposed in this paper. The criteria weights are still calculated by DEMATEL method in order to compare different methods. The FTOPSIS method is an efficient MCDM method, which ranks the alternatives according to their relative closeness to the ideal solution without considering the psychological factors of the DMs, while the TODIM method which is bounded rational considers the DMs’ psychological factors. The ranking results determined by different methods are shown in Table 17 and Fig. 11.

1. The comparison with the FTOPSIS method

Table 17.

_{The ranking results determined by different methods}

Methods	Ranking results
FTOPSIS	$S_2>S_7>S_5>S_3>S_6>S_4>S_1$
TODIM	$S_7>S_2>S_5>S_6>S_3>S_4>S_1$
The proposed method	$S_7>S_2>S_5>S_3>S_6>S_4>S_1$

Fig. 11.

The ranking results determined by different methods

The key steps of the FTOPSIS method are as follows:

• Step 1: Determine the fuzzy positive ideal solution (FPIS) and fuzzy negative ideal solution (FNIS) according to the weighted normalized decision matrix $R={\left[r_{\mathit{ij}}\right]}_{m\times n}$ by Eqs. (33) and (34).

$$r_i^{+}=\left\{\begin{array}{c}\underset{1\le j\le n}{\max }\left\{r_{\mathit{ij}}\right\},j\in \mathrm{the}\mathrm{benifit}\mathrm{criteria}\hfill \\ \underset{1\le j\le n}{\text{min}}\left\{r_{\mathit{ij}}\right\},j\in \mathrm{the}\mathrm{cost}\mathrm{criteria}\hfill \end{array}\right)$$ 33

$$r_i^{-}=\left\{\begin{array}{c}{min}_{1\le j\le n}\{r_{\mathit{ij}}\},j\in \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{benifit}\mathrm{criteria}\hfill \\ {max}_{1\le j\le n}\{r_{\mathit{ij}}\},j\in \mathrm{th}\mathrm{e}\mathrm{c}\text{o}\mathrm{st}\mathrm{criteria}\hfill \end{array}\right)$$ 34

• Step 2: Calculate the distance of each alternative from FNIS and FPIS by the following equations

$$d_i^{+}=\sum _{j=1}^n\begin{array}{c}d\left(r_{\mathit{ij}},,,r_i^{+}\right),i=1,2,\cdots m\end{array}$$ 35

$$d_i^{-}=\sum _{j=1}^n\begin{array}{c}d\left(r_{\mathit{ij}},,,r_i^{-}\right),i=1,2,\cdots m\end{array}$$ 36

• Step 3: Compute the closeness coefficient of each alternative using Eq. (37).

$${\mathit{CC}}_i=\frac{d_i^{-}}{d_i^{+}+d_i^{-}}$$ 37

The results obtained by calculating the case with the above steps of the FTOPSIS method are shown in Table 18 and Fig. 12.

Table 18.

The results calculated by FTOPSIS method

	d+	d−	CCi	Ranking
S1	2.524	0.429	0.145	7
S2	0.541	2.544	0.824	1
S3	1.795	1.212	0.403	4
S4	2.366	0.614	0.206	6
S5	1.447	1.971	0.577	3
S6	2.223	1.360	0.380	5
S7	0.564	2.295	0.803	2

Fig. 12.

The calculation results calculated by the FTOPSIS method

The ranking result calculated by the FTOPSIS method is $S_2$ superior to $S_7$, conversely, while the result computed by the proposed method in this paper is $S_7$ superior to $S_2$. The differences between the ranking results obtained by these two methods are that the method proposed in this paper assumes that the stakeholders are bounded rational and take into account the psychological factors of the DMs, while theFTOPSIS method evaluates the alternatives rationally under the strict assumption of completely rational without considering of the psychological factors of DMs. At the same time, landfill is found as the worst treatment alternative in the two decision methods.

• (2)The comparison with the fuzzy TODIM method

The TODIM method (an acronym in Portuguese of interactive and multi-criteria decision-making) is a MCDM method based on the prospect theory. The critical steps of the fuzzy TODIM method are depicted as follows:

• Step 1: Compute the gains and losses of each alternative concerning the others. Then the gain and loss of alternatives S_i relative to S_k relative to the attribute C_j can be expressed by the following Equations.

For the benefit criterion:

$$G_{\mathit{ik}}^j=\left\{\begin{array}{c}d\left(r_{\mathit{ij}},,,r_{\mathit{kj}}\right),r_{\mathit{ij}}\ge r_{\mathit{kj}}\hfill \\ 0,r_{\mathit{ij}}<r_{\mathit{kj}}\hfill \end{array}\right)$$ 38

$$L_{\mathit{ik}}^j=\left\{\begin{array}{c}0,r_{\mathit{ij}}\ge r_{\mathit{kj}}\hfill \\ d\left(r_{\mathit{ij}},,,r_{\mathit{kj}}\right),r_{\mathit{ij}}<r_{\mathit{kj}}\hfill \end{array}\right)$$ 39

For the cost criterion:

$$G_{\mathit{ik}}^j=\left\{\begin{array}{c}0,r_{\mathit{ij}}\ge r_{\mathit{kj}}\hfill \\ d\left(r_{\mathit{ij}},,,r_{\mathit{kj}}\right),r_{\mathit{ij}}<r_{\mathit{kj}}\hfill \end{array}\right)$$ 40

$$L_{\mathit{ik}}^j=\left\{\begin{array}{c}-d\left(r_{\mathit{ij}},,,r_{\mathit{kj}}\right),r_{\mathit{ij}}\ge r_{\mathit{kj}}\hfill \\ 0,r_{\mathit{ij}}<r_{\mathit{kj}}\hfill \end{array}\right)$$ 41

• Step 2: Determine the relative weights of each criterion using Eq. (42).

$${\omega }_{\mathit{jr}}={\omega }_j/{\omega }_r,{\omega }_r=max\left\{{\omega }_j\right\}$$ 42

• Step 3: Calculate the dominance degree matrix for each criterion by Eqs. (43)–(45).

$${\varphi }_{\mathit{ik}}^{j+}=\sqrt{\frac{G_{\mathit{ik}}^j{\omega }_{\mathit{jr}}}{\sum _{j=1}^n{\omega }_{\mathit{jr}}}}$$ 43

$${\varphi }_{\mathit{ik}}^{j-}=-\frac{1}{\theta }\sqrt{\frac{-L_{\mathit{ik}}^j\sum _{j=1}^n{\omega }_{\mathit{jr}}}{{\omega }_{\mathit{jr}}}},$$ 44

$${\varphi }_{\mathit{ik}}^j={\varphi }_{\mathit{ik}}^{j+}+{\varphi }_{\mathit{ik}}^{j-}$$ 45

• Step 4: Calculate the overall dominance degree ${\eta }_{\mathit{ik}}$ and the comprehensive dominance value $\delta \left(S_i\right)$.

$${\eta }_{\mathit{ik}}=\sum _{j=1}^n{\varphi }_{\mathit{ik}}^j$$ 46

$$\delta (S_i)=\frac{\sum _{k=1}^m{\eta }_{\mathit{ik}}-min\left\{\sum _{k=1}^m,,{\eta }_{\mathit{ik}}\right\}}{max\left\{\sum _{k=1}^m,,{\eta }_{\mathit{ik}}\right\}-min\left\{\sum _{k=1}^m,,{\eta }_{\mathit{ik}}\right\}}$$ 47

The final overall dominance degree ${\eta }_{\mathit{ik}}$ and the comprehensive dominance value $\delta \left(S_i\right)$ determined by the fuzzy TODIM method are shown in Table 19.

Table 19.

The results determined by fuzzy TODIM method

	The overall dominance degree	The comprehensive dominance value	Ranking
S1	− 40.780	0	7
S2	− 1.703	0.888	2
S3	− 32.481	0.189	5
S4	− 35.743	0.114	6
S5	− 14.635	0.594	3
S6	− 26.165	0.332	4
S7	3.247	1	1

The method based on DEMATEL and CPT assumes that the DMs are bounded rational and more in line with the reality, so the calculated ranking result is more realistic. The optimal alternative calculated by the TODIM method is the same as the proposed method in this paper, but the difference is $S_3$ and $S_6$. $S_3$ is better than $S_6$ calculated by the proposed method, while $S_6$ is superior than $S_3$ gained by TODIM method. The reason for this difference is that the TODIM method obtains the ranking results without considering the reference point, in contrast, the method proposed in this paper applies the positive ideal point and negative ideal point of TOPSIS as the reference point, which is more objective and more in line with expectations. Compared with the abovementioned methods, the proposed method considers the psychological factors of the DMs and the relationship among evaluation sub-criteria, which effectively reduces the decision-making cost and improves the efficiency.

Discussions

This paper presents a MCDM method based on CPT and DEMATEL, which combines fuzzy set theory and uniquely considers the psychological factors of DMs in the process of PSW treatment. And then the case about the selection problem of PSW treatment technology in Shanghai is studied by the novel proposed method in order to verify the feasibility and effectiveness of the method. Finally, the comprehensive prospect values of the seven alternatives of the case are calculated based on the prospect values and cumulative prospect weights. The results imply that the alternative ${\mathit{S}}_7$ is the best technology of PSW treatment in the four sustainability perspectives. In terms of the current decision-making process based on 9 sub-criteria, the combination of recycling and incineration has become the most ideal and sustainable treatment process considering the stakeholders’ psychological factors. In addition, it is noted that when the criteria weights are determined by the fuzzy DEMATEL method which considers the interaction between the criteria, the social acceptance (C₃₁) that is calculated as the most effective criterion also plays a key role in the selection of risk-oriented PSW treatment technology in real life. Taking all environmental, economic, social and technological factors affecting PSW treatment into account, the proposed decision-making process reveals the realistic ranking of PSW treatment technology selection, rather than the ideal ranking. But the ranking results obtained can still provide a decision basis and reference value for the selection of PSW treatment technology in real life. Therefore, the decision framework proposed in this paper is feasible and has certain reference significance for similar solid waste technology selection issues.

Moreover, some potentially important results were also yielded by the analysis. First of all, relatively speaking, the results of the case calculation, that is, the combined treatment method of recycling and incineration is the best technology in terms of environment, economy, society, and technology aspects. However, other PSW treatment methods in the case are not good enough in some criteria, such as S₁ performs relatively poorly in economic and social aspects, S₂ has relatively bad performance in technical and social aspects, and unsatisfactory performance exists in S₃ and S₄ in all four aspects. Additionally, considering the influence of environment, economy, society, and technology is the key to choosing PSW treatment technology. This process helps to identify economically feasible, socially acceptable practices and technically mature, as well as key environmental factors that need to be considered, to ensure that the environment and human health are protected in a sustainable and coordinated manner. With population growth, environmental changes, and economic development, challenges related to the selection and implementation of the most feasible PSW treatment technology will continue to exist. Finally, although the decision framework proposed in this study is designed for the selection of PSW treatment technology, the versatility of its decision framework means that it can also be extended to other waste systems, such as wastewater treatment system, food waste treatment system, municipal solid waste, and industrial waste.

Managerial implications

The research on the selection of PSW treatment technology in this paper can provide a decision-making basis and reference value for the treatment of municipal plastic solid waste. Therefore, this paper provides some useful and valuable suggestions for the treatment of municipal plastic solid waste from the aspects of environment, economy, society, and technology to realize the sustainable development of the environment and reduce the waste of resources.

• (i)In the environmental aspect, the combined treatment of recycling and incineration can best reduce pollution and reduce the harm to the environment caused by PSW treatment. Thus, the government should effectively supervise and manage the treatment of municipal PSW and reprocess the recyclable plastic waste, so as to bring more economic benefits. The management and attention should also be strengthened by the government for the plastic waste that reduces land and water pollution through incineration.
• (ii)In the economic aspect, the treatment technology of the combination of recycling and incineration brings economic benefits and reduces the cost of treatment. Investors should strengthen the management of enterprises, improve the employment rate of residents and promote the circular development of economy.
• (iii)In the social aspect, the PSW technology of the combination of recycling and incineration should implement the people-oriented concept, not only considering public acceptability, taking effective measures to improve social acceptance, but also fully encouraging urban residents to perform waste classification so as to better deal with PSW and improve waste recycling ratio and energy recovery ratio.
• (iv)In technical aspects, the government and investors should fully guarantee the feasibility and reliability of the technology, enhance the maturity of the technology, improve the efficiency of the technology, and make the treatment of PSW more reasonable and reliable.

Conclusions

As the PSW management has attracted more and more social attention, people pay more and more attention to the applicability and reliability of municipal PSW treatment methods. In order to select the most suitable PSW treatment technology, a MCDM method based on CPT and fuzzy DEMATEL is developed in this study. Initially, the weights of evaluation criteria are determined by the fuzzy DEMATEL method, which considers the interaction among evaluation criteria. Subsequently, on the premise of considering different stakeholders, by aggregating the different opinions of various stakeholders and considering the psychological factors of stakeholders, i.e., risk preference and loss aversion, the CPT method is used to rank the PSW treatment alternatives and accordingly obtain the optimal treatment technology. The main advantages of the proposed model in this paper are that in the decision-making process of PSW treatment, the weights of evaluation criteria are determined, and the interaction relationship and complex independent relationship among evaluation criteria are considered. At the same time, the different opinions of different stakeholders and their psychological factors, i.e., risk preference and loss aversion, etc. are fully considered, and the proposed model deals with the fuzzy information caused by the uncertainty and fuzziness of people’s subjective judgment.

According to the results obtained by using the proposed method, the $S_7$ alternative of the combination of recycling and incineration has been found as the best alternative in the case study. Moreover, the TOPSIS method based on the completely rational and TODIM method based on bounded rational is used to compare and verify the effectiveness and superiority of the results calculated by the proposed method. Compared with the FTOPSIS method, the psychological factors of DMs play a vital role, whose influence on the final decision-making can be seen through the sensitivity analysis of $\lambda$. More interestingly, when assuming that the DMs are completely rational, i.e., $\lambda$ = 1, similar outcomes as FTOPSIS can be obtained, and when $\lambda$ > 1, the ranking order is robust. In addition, the sensitivity analysis with criteria weight fluctuations also shows the robustness of the ranking results obtained in this paper. Compared with the TODIM method, the model constructed in this paper takes positive ideal points and negative ideal points as reference points to sort the alternatives, which is more objective and closer to reality. The comparative analysis shows that the psychological factors and reference points of DMs are significant in the decision-making of PSW treatment and also shows the advantages of the proposed method compared with the other two methods.

The critical motivations and major contributions of the study are three-fold: (1) This paper establishes a comprehensive criteria system for the selection problems of PSW treatment technology that consists of 9 sub-criteria listed under the four main criteria, i.e., environmental, economic, social, and technological criteria, which provides an evaluation criteria reference for issues related to waste disposal. (2) The method takes both the DMs’ psychological factors into account that assumes the DMs are bounded rational and considers the interaction between the criteria when determining the weights of criteria, which can better simulate the real situation and make decisions more accurately. (3) The decision-making method in this paper can meaningfully handle the selection problems of PSW treatment technology under the uncertain environment, which can further be applied in other waste management problems, such as MSW treatment problems, the selection problems of wastewater treatment technology, health-care waste treatment, construction waste treatment, and so on.

In the future, we can conduct deeper research from the following three potential directions. First of all, in the actual decision-making process, there is not only decision information in the format of linguistic terms but also real numbers and interval numbers. So, in the future, we can expand a mixed MADM method that can deal with multiple decision information forms, so as to select the best PSW treatment technology. Secondly, because of the feasibility and practicability of the method proposed in this paper, it can also be applied in other industries. Finally, the treatment of PSW can be studied in an intuitionistic fuzzy environment to solve more complex and accurate problems.

Author contribution

All authors contributed to the study conception and design. Material preparation, methodology, formal analysis, and supervision were performed by Qinghua Mao. Jinjin Chen and Mengxin Guo contributed to the conceptualization, software, data analysis, methodology, original draft preparation, and visualization. The reviewing, English editing, and literature survey parts of the manuscript were written by Jian Lv. Pengzhen Xie contributed to the supervision and investigation. All authors have read and approved the final manuscript.

Funding

This research has been funded by the S&T Program of Hebei (215576116D) and the Key Research Base Project of Humanities and Social Sciences in Higher Education Institutions of Hebei Province (JJ2109).

Data availability

All data generated or analyzed during this study are included in this article.

Declarations

Ethics approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.