The TOPSIS (Technique for Order Performance by Similarity to Ideal Solution) is a widely applied technique to solve MCDM problems that aim to rank the alternatives and select the best solution based on criteria and using a simple and intuitive algorithm (Montanari et al., 2021; Yadav et al., 2018; Onat et al., 2016; Wanke et al., 2015). The development of the TOPSIS algorithm is based on the idea to achieve a solution that is both as close as possible to the positive ideal solution and as far as possible from the negative ideal solution (Haleem et al., 2021; Raut et al., 2017; Shukla et al., 2017). Since its proposition by Hwang & Yoon (1981), the TOPSIS technique has become one of the most used MCDM methods with the development of different extension proposals (Çelikbilek & Tüysüz, 2020). To deal with different experts’ subjective judgments, which can be imprecise and uncertain, Chen (2000) proposed the fuzzy TOPSIS. The fuzzy TOPSIS also became a very popular and successfully applied technique in several current real-world challenges from many research fields (Palczewski & Sałabun, 2019).

In the fuzzy TOPSIS, linguistic variables are used by the decision-makers to evaluate the alternatives. These linguistic variables are represented by fuzzy numbers that are used in the following steps of the TOPSIS algorithm (Nădăban et al., 2016).

Step 1: aggregate the linguistic variables provided by the decision-makers on the performance of each alternative and the weight of the criteria through Equations 20 and 21.

Where ${\tilde{x}}_{ij}^k$ is a fuzzy element that represents the judgment of decision-maker $k=\left(\mathrm{1,} \mathrm{2,} \dots , K\right)$ for the performance of the alternative $i$ = (1, 2, …, n) related to the criterion $j=\left(\mathrm{1,} \mathrm{2,} \dots ,m\right)$. ${\tilde{w}}_j^k$ is the fuzzy element that represents the judgment of decision-maker $k$ for the weight of the criterion $j$.

1. Step 2

   create a decision matrix $\tilde{D}$ to represent all the performances ${\tilde{p}}_{ij}\left(\text{i}=\mathrm{1,}2\dots ,\text{n};\text{j}=\mathrm{1,}\mathrm{2,}\dots ,\text{m}\right)$ of the alternative $A_i \left(i= \mathrm{1,} 2 \dots , n\right)$ according to each criterion $C_j \left(j = \mathrm{1,} \mathrm{2,} \dots , m\right)$ as represented in Equation 22. In addition, a vector of criterion weights $\tilde{W}$ =${\tilde{w}}_j\left(j=\mathrm{1,} \mathrm{2,} \dots m\right)$ is defined by Equation 23.

   $\begin{array}{l}\begin{array}{cc}\begin{array}{ccc}{C}_1 & C_2 & \dots \end{array}& C_m\end{array}\\ \tilde{D}=\begin{array}{c}\begin{array}{c}{A}_1\\ A_2\end{array}\\ ⋮\\ A_n\end{array} \left(\begin{array}{cc}\begin{array}{c} \begin{array}{ccc}{\tilde{p}}_{11}& {\tilde{p}}_{12}& \dots \\ {\tilde{p}}_{21}& {\tilde{p}}_{22}& \dots \\ ⋮& ⋮& ⋮\end{array}\\ \begin{array}{ccc}{\tilde{p}}_{n1}& {\tilde{p}}_{n2}& \dots \end{array}\end{array}& \begin{array}{c}{\tilde{p}}_{1m}\\ {\tilde{p}}_{2m}\\ \begin{array}{c}⋮\\ {\tilde{p}}_{nm}\end{array}\end{array}\end{array} \right)\end{array}$(22)
   $\tilde{W}=\left[{\tilde{w}}_1, {\tilde{w}}_2, \dots , {\tilde{w}}_m\right]$(23)

Step 3: normalize the matrix $\tilde{D}$ using a linear transformation scale represented by the Equations 24 to 26 to get the matrix $\tilde{R}$.

Step 4: multiply the elements ${\tilde{r}}_{ij}$ in matrix $\tilde{R}$ by the weights ${\tilde{w}}_j$ to obtain the matrix $\tilde{V}$ as shown in Equations 27 and 28.

Step 5: Define the Fuzzy Positive Ideal Solution - FPIS ($F^{+}$), and the Fuzzy Negative Ideal Solution – FNIS ($F^{-}$) as presented in Equations 29 and 30, in which ${\tilde{v}}_j^{+}=\underset{i}{\max }\left\{{\tilde{v}}_{ij}\right\}$ and ${\tilde{v}}_j^{-}=\underset{i}{\min }\left\{{\tilde{v}}_{ij}\right\}$, $j=\mathrm{1,} \mathrm{2,} \dots , m$.

Step 6: Calculate de distance $S_i^{+}$ between the values of FPIS ($F^{+}$) and the scores of the alternative $A_i$ in the $\tilde{V}$ matrix using Equations 31 and 32. The same procedure should be performed for the calculus of the distance $S_i^{-}$ between the values of FNIS ($F^{-}$) and the scores of the $\tilde{V}$ matrix. For this purpose, the Euclidian distance between two triangular fuzzy numbers presented in Equation 2 is used.

Step 7: Compute the closeness coefficient $C{C}_i$ for each alternative $A_i$ using Equation 33. Then, based on the $C{C}_i$ results, a ranking of the alternatives $A_i$ is created in decreasing order. The closer the coefficient is to 1, the better is the overall performance of the alternative.

Several approaches can be found in literature that proposes the hybridization of different MCDM techniques, since each MCDM is appropriate for different purposes or requires different types of information (Büyüközkan & Göçer, 2021; Büyüközkan & Güler, 2021; Ortiz-Barrios & Alfaro-Saiz, 2020; Ortiz-Barrios et al., 2020). The proposed model uses the hybridization of fuzzy AHP and fuzzy TOPSIS techniques, since both techniques are necessary for different purposes. The fuzzy AHP is commonly used to determine the relative weights of decision criteria, as can be seen in several studies (Büyüközkan & Güler, 2021; Ortiz-Barrios & Alfaro-Saiz, 2020; Ortiz-Barrios et al., 2020). In addition, the fuzzy TOPSIS is an adequate technique to find an alternative’s priority rank (Montanari et al., 2021). Despite of fuzzy AHP is also often used to rank alternatives, it was not appropriate to use it to rank the alternatives, since the available inputs of relationship between CE indicator and CBM (Table 1, Table 2 and Table 3) are not pairwise comparisons.

As can be seen in Figure 1, the proposed decision making model is composed of four steps. Step 1 consists of the parametrization of the fuzzy numbers as detailed in section 3 and the selection of the CE principles. In this way, it is necessary to define the linguistic terms and the correspondent fuzzy sets to the judgments related to the pairwise comparisons between the preference orders of the CE principles. In addition, it is necessary to define the fuzzy sets related to the linguistic term which represents the intensity level of the relationship between the CE principles, the CE indicators, and CBM. An example of linguistic terms and the correspondent fuzzy sets can be seen in Table 6 and 7.

Figure 1
Proposed model for CBM prioritization.

Table 6
Fuzzy linguistic terms for the pairwise comparisons.

Table 7
Fuzzy linguistic terms to represent the intensity level of the relationship.

In Step 2, the experts should evaluate the preference relations between the CE principles, presented in section 2, using the linguistic terms defined in Step 1. Note that, since the experts’ assessments are independent, the consistency ratio (CR) should be checked for each one of them. Therefore, it is necessary to apply Equations 14 to 19 to calculate the CR of each decision maker set of assessments. If CR < 0.1, the assessments are considered consistent, otherwise, the decision makers with a CR ≥ 0.1 must review their assessments. Then, the expert’s assessments can be aggregated using the arithmetic mean (Lima-Junior et al., 2014):

Following the fuzzy AHP procedure, it is necessary to compute the values of the fuzzy synthetic extent, by applying Equations 5 and 7. Next, the degrees of possibility of the fuzzy values must be calculated with Equations 9 and 10. Finally, the CE principles weight vector $W\text{'}$ can be calculated using Equation 12, which should be normalized.

To begin Step 3, firstly it is necessary to select the adequate CE indicators for the organization since not all of them can be applied for all organizations. Next, is necessary to calculate the weights of the CE indicators, by using the fuzzy weighted averaging (FWA) aggregation operator, to aggregate the intensity level of the relationship of each CE principle for all CE indicators. Therefore, by applying Equation 3, using the weights of the CE principles, computed in Step 2, and the level of the relationship between the CE principles and the CE indicators, the CE indicators weights can be calculated. Note that the intensity level of the relationship of each CE principle for all CE indicators should be assessed by the DMs.

Finally, with fuzzy TOPSIS technique, it is possible to prioritize the mentioned CBMs in Section 2. In this way, Step 4 uses the CE indicators weights, calculated in Step 3, and the intensity level of the relationship between the CE indicators and the CBMs, which also have to be assessed by the DMs. The ${\tilde{p}}_{ij}$ index, from Equation 20, is composed by the level of the intensity of the relationship between the CE indicators and the CBMs. The ${\tilde{K}}_j$ value corresponds to the normalized CE indicators weights, which can be computed by defuzzifying the CE indicators weights computed in Step 3 with Equation 14 and normalizing them.

Following the fuzzy TOPSIS procedure, the intensity level of the relationship between the CE indicators and the CBMs converted in TFNs. Next, the decision matrix must be normalized with Equations 24 to 26. Using Equation 28, the normalized decision matrix has to be multiplied using the CE indicators weights.

Now, the Fuzzy Positive Ideal Solution FPIS ($F^{+}$) and the Fuzzy Negative Ideal Solution FNIS ($F^{-}$) must be computed with Equation 29 and Equation 30. The distance between the intensity values and the FPIS $\left(D_i^{+}\right)$ and the distance between the intensity values and Fuzzy Negative Ideal Solution FNIS ($D_i^{-}$) can be computed by Equation 31 and Equation 32. Finally, by applying Equation 33, it is possible to compute the closeness coefficient $C{C}_i$ of each CBM. The output is a rank of priority of CBMs, according to the organization's principles.

Table 7 presents the performance of the CBMs and the respective ranking considering the influence of all the three dimensions of indicators. The main purpose of this ranking is to show that to achieve greater performance in the proposed circularity indicators, CBMs should be prioritized as the presented classification. The best CBM ranked indicated that Recovery by-products is the most suitable CBM for improving the circularity of an organization when all the indicators of the three dimensions (economic, social, and material) are selected. Product life extension was classified as the second and Product as a service as the third. These three CBMs are closely related to each other.

Recovery by-products means that other products will be created from secondary raw materials and/or end-of-life materials (British Standards Institution, 2017; Ellen MacArthur Foundation, 2012). A lot of organizations emerged to apply this business model to deal with the existing waste. Applying this kind of CBM means that resources in the product remain available for consumption and that no virgin materials need to be mined (Prosman & Sacchi, 2018; Vegter et al., 2020). Moreover, the performance objective is to recover a product, so that is available for subsequent consumption, making it possible to use less materials, water, and energy (Vegter et al., 2020). Recovery by-products helps in the extension of product life and in closing the loop when Product as a Service CBM is applied (Arponen et al., 2018).

Product life extension seeks to design products to be more durable and to circulate in the system aiming to keep value. Product life extension is the key to reducing rapid and excessive consumption of Product as a service (Chen et al., 2020). It is also the main purpose to create durable products to slow material and energy flows (Milios, 2021). For companies and stakeholders, it is important to create and capture value via extending product lifetime in their business models (Hopkinson et al., 2020). Recovery by-products and Product life extension are directly connected to Product as a service, since the first two CBM are essential for the planning and implementation of successful Product as a service system.

Product as a service is based on the idea of a performance economy, where ownership-based business models are replaced with pay-per-use models to reduce production and fulfill CE requirements (British Standards Institution, 2017; Chen et al., 2020). Product as a service incorporates strategies such as leasing, renting, and pay-per-use which help manufacturers and product owners handle the whole life cycle of products and decide when they should be repaired, recycled, or remanufactured (Arponen et al., 2018; British Standards Institution, 2017; Chen et al., 2020).

Those CBM were classified as the third that most impacts on the achievement of a good circular performance because they incorporates the majority of the specific indicators of the three dimensions of CE indicators. Those CBM when correctly implemented in the automobile sector have the potential to reduce operating expenses, make companies participate in secondary sales, increase gross profits, generate and increase revenue, and reduce input material costs (Arponen et al., 2018).

This paper is relevant for practitioners since, based on the analysis and tests conducted, it was possible to identify the CBMs that are most suitable for the improvement of performance regarding CE implementation. The ranking of the CBMs concerning the CE principles and the circularity indicators indicate that organizations have some paths to follow in the transition towards CE. The choice will depend on the main strategic goals of the organization. If the organization aims to have equal performance on the three dimensions of circularity indicators, it should choose Recovery by-products as its initial CBM. If the focus is on economic indicators, the organization could choose between Products as a Service or Recovery by-products. If the focus is on social indicators the decision is to implement Product as a service. In case of the intention is to improve the material indicators, the decision is to implement Recovery by-products. Hence, this research contributes to managers and practitioners by operationalizing forms to connect CBMs ans CE indicators systematically.

It is worth saying that a combination of different CBMs could be used to make the business stronger and to fulfill some aspects of the value proposition that sometimes could not be fulfilled using only one CBM. Also, it should be highlighted that the choice of the CE indicators can vary depending on the organizational context. Therefore, the model must be applied selecting only the relevant CE indicators, and then, the company can obtain the appropriate CBM ranking.

CE can support organizations to develop sustainable solutions, maximizing the collaboration between stakeholders to create positive social and environmental impacts. The CBMs are strategies used to achieve sustainability. However, it was not found in the literature quantitative approaches to indicate which CBMs can be prioritized towards CE implementation.

In contrast with similar papers, and in a wide search through the literature, to the best of our knowledge, just the study of Husain et al. (2021) similarly addresses CBMs, as it can be seen in Table 5. However, Husain et al. (2021) do not analyze the relationship between CE principles and circularity indicators as we propose in the present research. The authors only seek to rank the CBMs with pre-defined criteria. Hence, the use of CE principles and circularity indicators to rank CBMs remains innovative in this study. Performance indicators are critical for determining how well a business is performing in CE and where improvements may be made. In this regard, using circularity indicators to prioritize CBMs may assist firms in establishing the parameters like as technology, resources, people, cultural elements, regulations, and other things that are required for CE implementation.

Sensitivity analysis is usually performed to check the robustness of the current search structure by considering the sources of uncertainty within a multi-criteria analysis (Maliene et al., 2018). In the application of the Fuzzy TOPSIS technique, sensitivity analysis contributes to understanding the influence of varying input data on the final ranking of alternatives (Kumar & Barman, 2021). Thus, several studies dealing with a variety of criteria and applying Fuzzy TOPSIS seek to change the weights of the decision-making criteria to assess their influence on the final solution (Awasthi et al., 2011; Dwivedi et al., 2018; Kumar & Barman, 2021).

In this way tests were conducted to understand the influence of the indicators’ weights on the prioritization of the CBMs according to each dimension of indicators. To perform these tests, each dimension of indicator had its weight considered as integral while the other dimensions of indicators had their weights set to zero. Table 11 presents the performance of the CBMs and the respective ranking for each test carried out.

Table 11
Performance of each CBM when isolating each dimension of indicators.

The best CBM ranked according to each dimension of indicators indicates that Product as a service is the most suitable CBM in the economic and social dimension and Recovery by-products is the most suitable CBM for the material and economic dimension. Both CBM are complementary to each other since Recovery by-product is part of the Product as a service (Arponen et al., 2018). Moreover, Recovery by-products has a major focus on value optimization of materials while Product as a service has its major focus in creates new kinds of business structures.

In order to understand the influence of the principles' weights on the prioritization of the CBMs, tests were conducted by setting each principle with its weight considered as integral while the other principles had their weights set to zero. Table 12 presents the performance of the CBMs and the respective resulting ranking.

Table 12
Performance of each CBM when with integral weight in each criterion.

It’s relevant to note that although some variations in the rankings could be noted from the changes in the weights, the order of the CBMs remains unaltered. This happens because all CE indicators were selected in the performed test, which makes them all assume similar weights. This indicates the robustness of the proposed decision model and consistency in the execution of the soft computing techniques.

As it can also be observed in Table 12, the ranking of CBM consists of the Recovery by-products first, followed by Product life extension, Product as a service, sharing economy, and On-demand. In this direction, it should be emphasized that, in real case applications, each organization should select only the principles that apply to its reality, as commented in section 4. The company of the illustrative application did not especified which indicator’s dimension should be focused, which justifies the rank obtained in Table 10.