This study deals with evaluating and prioritizing HCWDCLs considering economic, social, environmental, technical, and geological criteria. A hybrid MCDM method named PROMSIS is introduced to tackle the problem. Moreover, fuzzy theory is used to describe the problem parameters’ uncertainty. In this research, fuzzy triangular numbers are used to determine the values of linguistic terms. Each triangular number \(\tilde{A}\) = (*l,m,u*) is proposed by three elements of *l, m,* and *u* that show the lowest, most likely, and highest value for the number, respectively. The used mathematical calculation between two fuzzy numbers of \(\tilde{L} = \left( {l_{1} ,m_{1} ,u_{1} } \right)\) and \(\tilde{M} = \left( {l_{2} ,m_{2} ,u_{2} } \right)\) are presented in Eqs. (1, 2, 3, 4, 5)^{28,29}.

$$\tilde{L} + \tilde{M} = \left( {l_{1} + l_{2} ,m_{1} + m_{2} ,u_{1} + u_{2} } \right)$$

(1)

$$\tilde{L} – \tilde{M} = \left( {l_{1} – u_{2} ,m_{1} – m_{2} ,u_{1} – l_{2} } \right)$$

(2)

$$\tilde{L} \times \tilde{M} = \left( {\min \left( {l_{1} l_{2} ,l_{1} u_{2} ,l_{2} u_{1} ,u_{1} u_{2} } \right),m_{1} m_{2} ,\max \left( {l_{1} l_{2} ,l_{1} u_{2} ,l_{2} u_{1} ,u_{1} u_{2} } \right)} \right)$$

(3)

$$\tilde{L}/\tilde{M} = \left( {\min \left( {l_{1} /l_{2} ,l_{1} /u_{2} ,u_{1} /l_{2} ,u_{1} /u_{2} } \right),m_{1} /m_{2} ,\max \left( {l_{1} /l_{2} ,l_{1} /u_{2} ,u_{1} /l_{2} ,u_{1} /u_{2} } \right)} \right)$$

(4)

$${\text{Distance}} (\tilde{L},\tilde{M}) = \sqrt {\frac{1}{3} \times \left\{ {(l_{1} – l_{2} )^{2} + (m_{1} – m_{2} )^{2} + (u_{1} – u_{2} )^{2} } \right\}}$$

(5)

Moreover, defuzzification of fuzzy number \(\tilde{A}\) = (*l,m,u*) is obtained using Eq. (6).

$${\text{Defuzzify}} \;\left( {\tilde{A}} \right) = \frac{l + 4m + u}{6}$$

(6)

### Research steps

The following five steps are taken to evaluate and prioritize the HCWDCLs in this study. It is worth mentioning this study benefits from two questionnaires. The first questionnaire is a pairwise comparisons matrix (used in the AHP method). In the pairwise comparisons matrix questionnaire, each respondent uses a number between 1 (Just equal) to 9 (Extremely Preferred) to determine the privilege of two criteria against each other.

The second is a decision matrix questionnaire. In the decision matrix questionnaire, each respondent uses the Likert scale to determine the score of each candidate site (alternative) in each criterion. The linguistic terms used in the Likert scale and their corresponding fuzzy numbers are as follows: (1) Very low with the value of (0, 0, 0.75), (2) Low with the value of (0.5, 1.25, 2), (3) Average with the value of (1.75, 2.5, 3.25), (4) High with the value of (3, 3.75, 4.5), and (5) Very high with the value of (4.25, 5, 5).

The questionnaires used in this study are standard, and their validity is confirmed by previous studies^{21,30}. Both questionnaires are filled out by ten experts. Table 2 shows the experts’ information.

** Step 1.** Determine the effective criteria and sub-criteria in evaluating and prioritizing HCWDCLs. These criteria are obtained by reviewing the literature and considering experts’ opinions.

** Step 2.** Determine the weight of the considered criteria and sub-criteria using the AHP method. A pairwise comparison questionnaire is used to obtain the input matrix needed for the AHP method

^{31}. This questionnaire compares the effective criteria for deciding on the HCWDCL. In comparing the two criteria, the respondents were asked to choose one of the following alternatives: (1) Very slightly preferred, (2) Slightly preferred, (3) Preferred, (4) Preferred, and (5) Very highly preferred

^{32}. These linguistic terms are converted to a number to obtain the criteria weights. The considered values for the first to fifth linguistic terms are 1, 3, 5, 7, and 9, respectively .

A pairwise comparisons matrix is formed as presented in Eq. (7), where *n* and* a*_{kj} are the numbers of criteria, and the privilege of criterion *k* against criterion *j*, respectively^{33}.

$${\text{A}} = \left[ {a_{kj} } \right]_{n \times n} = \left[ {\begin{array}{*{20}c} {a_{11} } & {a_{12} } & {a_{13} } & \ldots & {a_{1n} } \\ {a_{21} } & {a_{22} } & {a_{23} } & \ldots & {a_{12} } \\ {a_{31} } & {a_{32} } & {a_{33} } & \ldots & {a_{13} } \\ \ldots & \ldots & \ldots & \cdots & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ {a_{n1} } & {a_{n2} } & {a_{n3} } & \cdots & {a_{nn} } \\ \end{array} } \right] \;\;\;\; k, j = 1, 2, \ldots , n$$

(7)

Then, the pairwise comparisons matrix is normalized using Eq. (8), where *H*_{kj} is the normalized value of *a*_{kj}.

$$H_{kj} = {{a_{kj} } \mathord{\left/ {\vphantom {{a_{kj} } {\mathop \sum \limits_{k = 1}^{n} a_{kj} }}} \right. \kern-0pt} {\mathop \sum \limits_{k = 1}^{n} a_{kj} }}\;\; \;k, j = 1, 2, \ldots , n$$

(8)

The final weight of each criterion is obtained by Eq. (9), where *w*_{j} is the final weight of criterion *i*.

$$w_{j} = \mathop \sum \limits_{k = 1}^{n} \frac{{H_{kj} }}{n}\;\;\;\;j = 1, 2, \ldots , n$$

(9)

A pairwise comparison matrix should be established to compare the main criteria. Moreover, a pairwise comparisons matrix should be established for each main criterion to compare its related sub-criteria. The final weight of each sub-criterion is calculated by multiplying its weight by the weight of its corresponding main criterion.

**S**** tep 3.** Determine candidate locations (alternative) for the HCWDCL. Experts usually determine the alternatives based on some factors.

** Step 4.** Create the decision matrix (determine the score of each candidate for the HCWDCL in each criterion) using the decision matrix questionnaire. This questionnaire is used to determine the score of each HCW disposal candidate site in each of the considered criteria and form the decision matrix.

** Step 5.** Rank the alternatives using the fuzzy PROMSIS method. Fuzzy PROMSIS is a combination of fuzzy TOPSIS and fuzzy PROMETHEE methods. The details of the fuzzy PROMSIS are described in section “The proposed fuzzy PROMSIS”.

### The proposed fuzzy PROMSIS

In this section, the proposed fuzzy PROMSIS method is presented. As stated before, PROMSIS is a product of hybridizing TOPSIS and PROMETHEE. Each MCDM method has a different viewpoint in prioritizing the alternatives, resulting in a different solution. In the TOPSIS method, an alternative is preferable if its distance from the positive ideal solution (PIS) is low and from the negative ideal solution (NIS) is high. On the other hand, in the PROMETHEE method, an alternative is preferable if its net preference flow is high. The proposed PROMSIS method tries to integrate these two viewpoints.

In the PROMSIS method, an alternative is preferable if its distance from NIS is high, its distance from PIS is low, and its net preference flow value is simultaneously high. Having various viewpoints in ranking the alternatives caused the decision-making process to have a comprehensive attitude and reliable results. Before presenting the steps of the fuzzy PROMSIS method, the PROMETHEE and TOPSIS methods are briefly described.

#### PROMETHEE

The PROMETHEE method is an MCDM method that is popular for its simplicity, clarity, and reliability of the results^{34}. This method is suitable for evaluating a limited set of alternatives in the form of a partial or complete ranking.

This method gives the decision matrix (i.e., the score of each alternative in each criterion) and the weight of the criteria as its input. Suppose the decision matrix is as presented in Eq. (10), the main steps of the PROMETHEE method can be explained as follows.

$$X = \left[ {x_{ij} } \right]_{m \times n} = \left[ {\begin{array}{*{20}c} {x_{11} } & {x_{12} } & {x_{13} } & \ldots & {x_{1n} } \\ {x_{21} } & {x_{22} } & {x_{23} } & \ldots & {x_{2n} } \\ {x_{31} } & {x_{32} } & {x_{33} } & \ldots & {x_{3n} } \\ \ldots & \ldots & \ldots & \cdots & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ {x_{m1} } & {x_{m2} } & {x_{m3} } & \cdots & {x_{mn} } \\ \end{array} } \right] \;\;\;i = 1, 2, \ldots , m;\;\; j = 1, 2, \ldots , n$$

(10)

where \(x_{ij}\), *n* and *m* are the scores of alternative *i* in criterion *j*, number of criteria, and number of alternatives, respectively.

*Step 1***.** Calculate the difference of alternatives in different criteria with pairwise comparisons of alternatives in each criterion using Eq. (11).

$$d_{j} \left( {A,B} \right) = x_{Aj} – x_{Bj}$$

(11)

where \(d_{j} \left( {A,B} \right)\) is the difference between the score of alternative *A* against alternative *B* in criterion *j*. This difference represents the privilege of alternative *A* against *B* if \(x_{Aj} \ge x_{Bj}\) for the profit criteria, or \(x_{Aj} \le x_{Bj}\) for the cost criteria.

*Step 2***.** Calculate the superiority of the alternatives against each other according to the superiority function *P*. There are six superiority functions, each of which takes the value of \(d_{j} \left( {A,B} \right)\) as the input and gives a value between 0 and 1 as the output^{35}. In this research, a Gaussian preference function is used, as shown in Eq. (12).

$$P_{j} (A,B) = P\left[ {d_{j} (A,B)} \right] = \left\{ {\begin{array}{*{20}l} {1 – e^{{\frac{{ – (d_{j} (A,B))^{2} }}{{2\sigma_{j}^{2} }}}} } \hfill & {if\,d_{j} (A,B) > 0} \hfill \\ 0 \hfill & {if\,d_{j} (A,B) \le 0} \hfill \\ \end{array} } \right.$$

(12)

The value of \(\sigma\) represents the threshold value between the indifferent and strict preference areas. In the other preference functions, if \(d_{j} \left( {A,B} \right)\) is bigger than a threshold value, the function returns value of 1. In this case, the differences between \(d_{j} \left( {A,B} \right)\) values are neglected. For example, if the threshold *p* is equal to 1, then there is no difference between \(d_{j} \left( {A,B} \right) = 1\) and \(d_{j} \left( {A,B} \right) = 5\). But the Gaussian method considers any differences between \(d_{j} \left( {A,B} \right)\) values.

** Step 3.** Calculate the multi-criteria preference degree of alternative

*A*against alternative

*B*using Eq. (13).

$${\uppi }\left( {A,B} \right) = \mathop \sum \limits_{{}} P_{j} \left( {A,B} \right) \times w_{j}$$

(13)

*Step 4***.** Calculate the input preference flow (*Φ*^{in}) and the output preference flow (*Φ*^{out}) of each alternative using Eqs. (14) and (15). The input preference flow indicates how much an alternative like *A* is superior to other alternatives. The higher this value is, the better this alternative is. The output preference flow indicates how much other alternatives are superior to alternative *A*. The lower this value is, the better this alternative is.

$${\Phi }^{in} \left( A \right) = \frac{{\mathop \sum \nolimits_{x = 1}^{m} \pi \left( {A,x} \right)}}{{\left( {n – 1} \right)}}$$

(14)

$${\Phi }^{out} \left( A \right) = \frac{{\mathop \sum \nolimits_{x = 1}^{m} \pi \left( {x,A} \right)}}{{\left( {n – 1} \right)}}$$

(15)

*Step 5***.** Calculate the net preference flow (*Φ*) for each alternative using Eq. (16). The higher the net preference flow of an alternative, the better it is.

$$\Phi_{{\text{A}}} = \Phi^{in} \left( A \right){-} \Phi^{out} \left( A \right)$$

(16)

#### TOPSIS

The TOPSIS is a well-known MCDM method. The TOPSIS method identifies the PIS and NIS. This method prefers an alternative whose sum of its distance from PIS is low and, simultaneously, its distance from NIS is high^{36}. The main steps of the TOPSIS are summarized below.

** Step 1.** Calculate the normalized decision matrix

*R*= [

*r*

_{ij}]

_{m×n}using Eq. (17). Where \({x}_{ij}\) is the score of alternative

*i*in criterion

*j*.

$$r_{ij } = \frac{{x_{ij} }}{{\sqrt {\mathop \sum \nolimits_{i = 1}^{m} x_{ij}^{2} } }}$$

(17)

** Step 2.** Obtain the weighted normalized decision matrix

*V*= [

*v*

_{ij}]

_{m×n}using Eq. (18).

$$v_{ij} = w_{j} *r_{ij}$$

(18)

** Step 3.** Calculate \(PIS = \left[ {pis_{1}^{{}} ,pis_{2}^{{}} , \cdots ,pis_{n}^{{}} } \right]\) and \(NIS = \left[ {nis_{1}^{{}} ,nis_{2}^{{}} , \cdots ,nis_{n}^{{}} } \right]\) using Eqs. (19) and (20). Where

*J*

^{+}indicates the set of profit criteria, and

*J*

^{−}indicates the set of cost criteria.

$$pis_{i}^{{}} = \left\{ {\begin{array}{*{20}c} {\max v_{ij} \,\,\,\,\,\,if\,j \in J^{ + } } \\ {\min v_{ij} \,\,\,\,\,if\,j \in J^{ – } } \\ \end{array} } \right.\,\,\,\,\,\,\,\forall i = 1,2,…,m$$

(19)

$$nis_{i}^{{}} = \left\{ {\begin{array}{*{20}c} {\min v_{ij} \,\,\,\,\,\,if\,j \in J^{ + } } \\ {\max v_{ij} \,\,\,\,\,if\,j \in J^{ – } } \\ \end{array} } \right.\,\,\,\,\,\,\,\forall i = 1,2,…,m$$

(20)

** Step 4.** Calculate the distance of each alternative from the PIS (\(DPIS_{i}\)) and the NIS (\(DNIS_{i}\)) using Eqs. (21) and (22).

$$DPIS_{i} = \sqrt {\sum\limits_{j = 1}^{n} {\left( {\mathop v\nolimits_{ij} – \mathop {pis}\nolimits_{j}^{{}} } \right)^{2} } }$$

(21)

$$DNIS_{i} = \sqrt {\sum\limits_{j = 1}^{n} {\left( {\mathop v\nolimits_{ij} – \mathop {nis}\nolimits_{j}^{{}} } \right)^{2} } }$$

(22)

** Step 5.** Calculate the closeness coefficient for each alternative (

*CC*

_{i}) using Eq. (23).

$$CC_{i} = \frac{{DNIS_{i} }}{{DPIS_{i} + DNIS_{i} }}$$

(23)

** Step 6.** Sort the alternatives in descending order of closeness coefficient. Consider the alternative that has the highest value of closeness coefficient as the best alternative.

#### Fuzzy PROMSIS

The PROMSIS method and its steps are described in this section. In the fuzzy PROMSIS method, the decision matrix (i.e., the score of each alternative in each criterion) and the weight of the criteria are taken as input. Equation (24) shows the fuzzy decision matrix.

$$\begin{aligned} \tilde{X} & = \left[ {\tilde{x}_{ij} } \right]_{m \times n} = \left[ {\begin{array}{*{20}c} {\tilde{x}_{11} } & \cdots & {\tilde{x}_{1n} } \\ \vdots & \ddots & \vdots \\ {\tilde{x}_{m1} } & \cdots & {\tilde{x}_{mn} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\left( {x_{11}^{l} ,x_{11}^{m} ,x_{11}^{u} } \right)} & \cdots & {\left( {x_{1n}^{l} ,x_{1n}^{m} ,x_{1n}^{u} } \right)} \\ \vdots & \ddots & \vdots \\ {\left( {x_{m1}^{l} ,x_{m1}^{m} ,x_{m1}^{u} } \right)} & \cdots & {\left( {x_{mn}^{l} ,x_{mn}^{m} ,x_{mn}^{u} } \right)} \\ \end{array} } \right]\; \\ i & = 1, 2, \ldots , m;j = 1, 2, \ldots , n \\ \end{aligned}$$

(24)

where \(\tilde{x}_{ij}\), *n* and *m* are the scores of alternative *i* in criterion *j*, number of criteria, and number of alternatives, respectively.

The steps of the fuzzy PROMSIS method are as follows.

** Step 1.** Calculate the difference of alternatives in different criteria with pairwise comparisons of alternatives in each criterion. Equation (25) is used for profit criteria, and Eq. (26) for cost criteria. Where \(d_{j} \left( {A,B} \right)\) is the difference between alternatives A and B in criterion

*j*. This difference represents the privilege of alternative

*A*against

*B*. In this research, the fuzzy distance of these two numbers is obtained by Eq. (5).

$$\begin{gathered} d_{j} (A,B) = {\text{Distance}} \left( {\tilde{x}_{Aj} ,\tilde{x}_{Bj} } \right) = \left\{ {\begin{array}{*{20}c} {\sqrt {\frac{1}{3} \times \left\{ {\left( {x_{Aj}^{l} – x_{Bj}^{l} } \right)^{2} + \left( {x_{Aj}^{m} – x_{Bj}^{m} } \right)^{2} + \left( {x_{Aj}^{u} – x_{Bj}^{u} } \right)^{2} } \right\}} } & {if\,\tilde{x}_{Aj} > \tilde{x}_{Bj} \,\,} \\ 0 & {otherwise} \\ \end{array} } \right. \hfill \\ \hfill \\ \end{gathered}$$

(25)

$$d_{j} (A,B) = {\text{Distance}} \left( {\tilde{x}_{Aj} ,\tilde{x}_{Bj} } \right) = \left\{ {\begin{array}{*{20}c} {\sqrt {\frac{1}{3} \times \left\{ {\left( {x_{Aj}^{l} – x_{Bj}^{l} } \right)^{2} + \left( {x_{Aj}^{m} – x_{Bj}^{m} } \right)^{2} + \left( {x_{Aj}^{u} – x_{Bj}^{u} } \right)^{2} } \right\}} } & {if\,\,\tilde{x}_{Aj} < \tilde{x}_{Bj} \,} \\ 0 & {otherwise} \\ \end{array} } \right.$$

(26)

where \(\tilde{x}_{Aj} = \left( {x_{Aj}^{l} ,x_{Aj}^{m} ,x_{Aj}^{u} } \right)\) and \(\tilde{x}_{Bj} = \left( {x_{Bj}^{l} ,x_{Bj}^{m} ,x_{Bj}^{u} } \right)\) are the score of alternatives A and B in criterion *j*, respectively.

** Step 2.** Calculate the superiority of the alternatives over each other in each criterion according to the superiority function using Eq. (27). In this research, the GAUSSIAN superiority function is used with \(\sigma =0.5\).

$$P_{j} (A,B) = P(d_{j} (A,B)) = 1 – e^{{\frac{{ – (d_{j} (A,B))^{2} }}{{2\sigma_{j}^{2} }}}} \,$$

(27)

** Step 3.** Calculate the multi-criteria preference degree of alternative

*A*against alternative

*B*using Eq. (28).

$${\uppi }\left( {A,B} \right) = \mathop \sum \limits_{{}} P_{j} \left( {A,B} \right) \times w_{j}$$

(28)

** Step 4.** For each alternative

*A*, calculate each alternative’s input preference flow (\({\Phi }^{in}\left(A\right)\)), output preference flow (\({\Phi }^{out}\left(A\right)\)) and net preference flow (

*Φ*

_{A}) using Eqs. (29, 30, 31), respectively.

$${\Phi }^{in} \left( A \right) = \frac{{\mathop \sum \nolimits_{x = 1}^{m} \pi \left( {A,x} \right)}}{{\left( {n – 1} \right)}}$$

(29)

$${\Phi }^{out} \left( A \right) = \frac{{\mathop \sum \nolimits_{x = 1}^{m} \pi \left( {x,A} \right)}}{{\left( {n – 1} \right)}}$$

(30)

$$\Phi_{A} = \Phi^{in} \left( A \right) \, {-} \Phi^{out} \left( A \right)$$

(31)

** Step 5.** Calculate the normalized decision matrix \(\tilde{R} = [\tilde{r}_{ij} ]_{m \times n}\) using Eq. (32).

$$\tilde{r}_{ij} = \left( {r_{ij}^{l} ,r_{ij}^{m} ,r_{ij}^{u} } \right) = \left\{ {\begin{array}{*{20}c} {\left( {\frac{{x_{ij}^{l} }}{{c_{j}^{*} }},\frac{{x_{ij}^{m} }}{{c_{j}^{*} }},\frac{{x_{ij}^{u} }}{{c_{j}^{*} }}} \right)} & {c_{j}^{*} = \max_{i} x_{ij}^{u} } & {For \, the \, positive \, criterion \, j} \\ {\left( {\frac{{a_{j}^{ \circ } }}{{x_{ij}^{u} }},\frac{{a_{j}^{ \circ } }}{{x_{ij}^{m} }},\frac{{a_{j}^{ \circ } }}{{x_{ij}^{l} }}} \right)} & {a_{j}^{^\circ } = \min_{i} x_{ij}^{l} } & {For \, the \, negative \, criterion \, j} \\ \end{array} } \right.$$

(32)

** Step 6.** Obtain the weighted normalized decision matrix \(\tilde{V} = [\tilde{v}_{ij} ]_{m \times n}\) using Eq. (33).

$$\tilde{v}_{ij} = w_{j} \times \tilde{r}_{ij} = \left( {w_{j} \times r_{ij}^{l} ,w_{j} \times r_{ij}^{m} ,w_{j} \times r_{ij}^{u} } \right)$$

(33)

** Step 7.** Calculate Fuzzy Positive Ideal Solution (FPIS) and Fuzzy Negative Ideal Solution (FNIS) using Eqs. (34) and (35). Where

*J*

^{+}indicates the set of profit criteria, and

*J*

^{−}indicates the set of cost criteria \(\left( {FPIS} \right. = \left[ {\widetilde{fpis}_{1} ,\widetilde{fpis}_{2} , \ldots ,\widetilde{fpis}_{n} } \right]\) and \(FNIS = \left. {\left[ {\widetilde{fnis}_{1} ,\widetilde{fnis}_{2} , \ldots ,\widetilde{fnis}_{n} } \right]} \right)\).

$$\widetilde{fpis}_{i} = \left\{ {\begin{array}{*{20}c} {\max \tilde{v}_{ij} \,\,\,\,\,\,if\,j \in J^{ + } } \\ {\min \tilde{v}_{ij} \,\,\,\,\,if\,j \in J^{ – } } \\ \end{array} } \right.\,\,\,\,\,\,\,\forall i = 1,2,…,m$$

(34)

$$\widetilde{fnis}_{i} = \left\{ {\begin{array}{*{20}c} {\min \tilde{v}_{ij} \,\,\,\,\,\,if\,j \in J^{ + } } \\ {\max \tilde{v}_{ij} \,\,\,\,\,if\,j \in J^{ – } } \\ \end{array} } \right.\,\,\,\,\,\,\,\forall i = 1,2,…,m$$

(35)

** Step 8.** Calculate the distance of each alternative

*i*from the FPIS (

*DPIS*

_{i}) and the FNIS (

*DNIS*

_{i}) using Eqs. (36) and (37).

$$DNIS_{i} = \sum\limits_{j = 1}^{n} {{\text{Distance}}} \left( {\tilde{v}_{ij} ,\widetilde{fnis}_{j} } \right)\,\,\,\,\,\,\,i = 1,…,m$$

(36)

$$DPIS_{i} = \sum\limits_{j = 1}^{n} {\text{Distance}} \left( {\tilde{v}_{ij} ,\widetilde{fpis}_{j} } \right)\,\,\,\,\,\,\,i = 1,…,m$$

(37)

** Step 9.** Normalize the values of

*DPIS*

_{i}

*, DNIS*

_{i}and

*Φ*

_{i}using Eqs. (38) to (40).

$$NDNIS_{i} = \frac{{DNIS_{i} }}{{\sqrt {\sum\limits_{i = 1}^{m} {DNIS^{2}_{i} } } }}\,\,\,\,\,\,\,i = 1,…,m$$

(38)

$$NDNIS_{i} = \frac{{DPIS_{i} }}{{\sqrt {\sum\limits_{i = 1}^{m} {DPIS^{2}_{i} } } }}\,\,\,\,\,\,\,i = 1,…,m$$

(39)

$$N\Phi_{i} = \frac{{\Phi_{i} }}{{\sqrt {\sum\limits_{i = 1}^{m} {\Phi_{i}^{2} } } }}\,\,\,\,\,\,\,i = 1,…,m$$

(40)

** Step 10.** Calculate the PROMSIS index using Eq. (41).

$$Q_{i} = \frac{{\left( {NDNIS_{i} + N{\Phi }_{i} } \right)}}{{NDPIS_{i} + (NDNIS_{i} + N{\Phi }_{i} )}}$$

(41)

Reviewing TOPSIS and PROMETHEE shows that the lower value of \({NDPIS}_{i}\) and the higher values of \(NDNIS_{i}\) and \(N{\Phi }_{i}\) are favorable in ranking the alternatives. In calculating the closeness coefficient for each alternative, its distance from PIS appears in the numerator, and the summation of its distance from PIS and NIS appears in the denominator. In the calculation of PROMSIS index, the parameters that their higher values are favorable (\(NDNIS_{i}\) and \(N{\Phi }_{i}\)) appear in the numerator, and the summation of all the other parameters appears in the denominator.

** Step 11.** Prioritize the alternatives according to the PROMSIS index. In this case, an alternative with a higher value of PROMSIS index gets a higher priority.

The flowchart of the research steps is presented in Fig. S1 in Supplementary Material.