## 1 Introduction

*et al.*, 2021). After introducing fuzzy sets by Zadeh (1965), they have been prevalent in almost all branches of science (Kutlu Gündoğdu and Kahraman, 2019a). Many researchers (Zadeh, 1975; Grattan-Guinness, 1976; Atanassov, 1986; Yager, 1986; Atanassov, 1999; Smarandache, 1998; Torra, 2010; Cu’ò’ng, 2014; Yager, 2017; Gündoğdu and Kahraman, 2019a) have introduced many extensions of ordinary fuzzy sets in the literature. Numerous researchers have utilized these extensions in recent years in the solution of multi-attribute decision-making problems (Gündoğdu and Kahraman, 2019b). These extensions are presented in a chronological order, as given in Fig. 1.

##### Fig. 1

*et al.*, 2018). Spherical fuzzy sets are located between picture fuzzy sets and t-spherical fuzzy sets.

*et al.*, 2019). Ashraf

*et al.*(2019a,2019b) proposed spherical fuzzy sets with some operational rules and aggregation operations based on Archimedean t-norm and t-conorms.

*et al.*, 2019). MADM deals with problems where there are discrete attributes and more alternatives than one for evaluation (Kahraman

*et al.*, 2019b). MADM problems include numerous attributes, which are tangible or intangible (Karasan

*et al.*, 2019). Group decision making is a type of MADM problems that is usually understood as aggregating different discrete preferences on a given set of alternatives to a single collective preference. Group decision making involves multiple decision-makers, each with different skills, experience, and knowledge related to the problem’s different attributes. This type of problems are known as Multiple-Attribute Group Decision-Making (MAGDM) (Robinson and Amirtharaj, 2015).

*et al.*(2019) investigated the medical diagnostics and decision-making problem in the spherical fuzzy environment as a practical application. Zeng

*et al.*(2019), using the SF-TOPSIS methodology, developed a multi-attribute decision-making problem in an SF environment. They adopted a new approach of covering-based spherical fuzzy rough set (CSFRS) models utilizing spherical fuzzy

*β*-neighbourhoods to hybrid spherical fuzzy sets with notions of covering the rough set. In another research, Kutlu Gündoğdu and Kahraman (2019c), Kutlu Gündoğdu

*et al.*(2020) extended the classical (VIKOR) method to the spherical fuzzy VIKOR (SF-VIKOR) method and showed its applicability and validity through a waste management problem. Kahraman

*et al.*(2019b) developed and used the spherical fuzzy TOPSIS method in a hospital location selection problem. The extension of the classical analytic hierarchy process (AHP) to spherical fuzzy AHP (SF-AHP) method and its application to renewable energy location selection is proposed by Kutlu Gündoğdu and Kahraman (2019b). The application of the spherical fuzzy analytic hierarchy process (AHP) method in an industrial robot selection problem has been researched by Kutlu Gündoğdu and Kahraman (2020). The multi-criteria decision-making method TOPSIS is extended to spherical fuzzy TOPSIS in Kutlu Gündoğdu and Kahraman’s (2019c) research. The extensions of traditional WASPAS and CODAS methods to spherical fuzzy WASPAS and CODAS have also been developed by Kutlu Gündoğdu and Kahraman (2019a; 2019b). Kutlu Gündoğdu (2020) developed the spherical fuzzy MULTIMOORA method to efficiently solve complex problems, which require assessment and estimation under an unstable data environment. The interval-valued spherical fuzzy sets are employed in developing the extension of TOPSIS under fuzziness by Kutlu Gündoğdu and Kahraman (2019a). They used the proposed method in solving a multiple criteria selection problems among 3D printers. Liu

*et al.*(2020) proposed an approach based on linguistic spherical fuzzy sets for public evaluation of shared bicycles in China. Kahraman

*et al.*(2020) developed a performance measurement method in order to rank the firms using a spherical fuzzy multi-attribute decision making approach. A new approach to the fuzzy TOPSIS method based on entropy measures using spherical fuzzy information-based decision-making techniques for MAGDM problems was presented by Barukab

*et al.*(2019). Finally, some novel similarity measures in the spherical fuzzy environment have been proposed by Seyfi Shishavan

*et al.*(2020).

*et al.*, 2018). The combination of the criteria-wise rankings into an overall preference ranking that produces an optimal compromise among the several component rankings is the LAM’s basic idea (Liang

*et al.*, 2019). In the classical linear assignment method, usually crisp values of decision matrix are used. Considering the inevitable uncertainty of decision-making problems and real-life, LAM is extended under different fuzzy extensions by scholars using different modelling and solving approaches.

*et al.*(2018) proposed a method based on a linear assignment method to solve the group decision-making problems using hesitant fuzzy linguistic term sets. The results are compared with other methods to outline the model’s efficiency. An approach to solve the multiple criteria decision making (MCDM) problems under the hesitant fuzzy environment was developed by Wei

*et al.*(2017). The information about the criteria weights are correlative and based on the

*λ*-fuzzy measure. Based on signed distances, Chen (2013) developed a new linear assignment method within the interval type-2 trapezoidal fuzzy numbers framework to produce an optimal preference ranking of the alternatives. In another research, Yang

*et al.*(2018) developed a new multiple attribute decision-making method based on the interval neutrosophic sets and linear assignment. Developing an extended linear assignment method to solve multi-criteria decision-making (MCDM) problems under Pythagorean fuzzy environment was the aim of Liang

*et al.*(2019). A new approach based on a linear assignment method was presented by Bashiri

*et al.*(2011) for selecting the optimum maintenance strategy using qualitative and quantitative data through interaction with the maintenance experts. By extending the traditional linear assignment method, Chen (2014) proposed an efficient method for solving MCDM problems in the interval-valued intuitionistic fuzzy environment. Finally, Liang

*et al.*(2018) developed the linear assignment method for interval-valued Pythagorean fuzzy sets.

## 2 Preliminaries

##### Definition 1 *(See,* Gündoğdu and Kahraman, 2019c*).*

*X*is given by:

##### (1)

\[ {\tilde{A}_{s}}=\big\{x,{\mu _{{\tilde{A}_{s}}}}(x),{\vartheta _{{\tilde{A}_{s}}}}(x),{I_{{\tilde{A}_{s}}}}(x)\hspace{0.1667em}\big|\hspace{0.1667em}x\in X\big\},\]*x*to ${\tilde{A}_{S}}$, respectively, and:

##### (2)

\[ 0\leqslant {\mu _{{\tilde{A}_{s}}}^{2}}(x)+{\vartheta _{{\tilde{A}_{s}}}^{2}}(x)+{I_{{\tilde{A}_{s}}}^{2}}(x)\leqslant 1.\]*x*in

*X*.

##### Definition 2 *(See,* Gündoğdu and Kahraman, 2019c*).*

*Addition*

##### (3)

\[\begin{aligned}{}{\tilde{A}_{s}}\oplus {\tilde{B}_{s}}=& \Big\{\sqrt{{\mu _{{\tilde{A}_{s}}}^{2}}+{\mu _{{\tilde{B}_{s}}}^{2}}-{\mu _{{\tilde{A}_{s}}}^{2}}{\mu _{{\tilde{B}_{s}}}^{2}}},{\vartheta _{{\tilde{A}_{s}}}}{\vartheta _{{\tilde{B}_{s}}}},\\ {} & \sqrt{\big(1-{\mu _{{\tilde{B}_{s}}}^{2}}\big){I_{{\tilde{A}_{s}}}^{2}}+\big(1-{\mu _{{\tilde{A}_{s}}}^{2}}\big){I_{{\tilde{B}_{s}}}^{2}}-{I_{{\tilde{A}_{s}}}^{2}}{I_{{\tilde{B}_{s}}}^{2}}}\hspace{0.1667em}\Big\}.\end{aligned}\]*Multiplication*

##### (4)

\[\begin{aligned}{}{\tilde{A}_{s}}\otimes {\tilde{B}_{s}}=& \Big\{{\mu _{{\tilde{A}_{s}}}}{\mu _{{\tilde{B}_{s}}}},\sqrt{{\vartheta _{{\tilde{A}_{s}}}^{2}}+{\vartheta _{{\tilde{B}_{s}}}^{2}}-{\vartheta _{{\tilde{A}_{s}}}^{2}}{\vartheta _{{\tilde{B}_{s}}}^{2}}},\\ {} & \sqrt{\big(1-{\vartheta _{{\tilde{B}_{s}}}^{2}}\big){I_{{\tilde{A}_{s}}}^{2}}+\big(1-{\vartheta _{{\tilde{A}_{s}}}^{2}}\big){I_{{\tilde{B}_{s}}}^{2}}-{I_{{\tilde{A}_{s}}}^{2}}{I_{{\tilde{B}_{s}}}^{2}}}\hspace{0.1667em}\Big\}.\end{aligned}\]*Multiplication by a scalar*; $k>0$

##### (5)

\[ k{\tilde{A}_{s}}=\Big\{\sqrt{1-{\big(1-{\mu _{{\tilde{A}_{s}}}^{2}}\big)^{k}},}{\vartheta _{{\tilde{A}_{s}}}^{k}},\sqrt{{\big(1-{\mu _{{\tilde{A}_{s}}}^{2}}\big)^{k}}-{\big(1-{\mu _{{\tilde{A}_{s}}}^{2}}-{I_{{\tilde{A}_{s}}}^{2}}\big)^{k}}}\hspace{0.1667em}\Big\}.\]*Power of*${\tilde{A}_{s}}$; $k>0$

##### Definition 3 *(See,* Gündoğdu and Kahraman, 2019c*).*

*k*, ${k_{1}}$ and ${k_{2}}\geqslant 0$.

##### (7)

\[\begin{aligned}{}\text{I.}& \hspace{1em}{\tilde{A}_{s}}\oplus {\tilde{B}_{s}}={\tilde{B}_{s}}\oplus {\tilde{A}_{s}},\end{aligned}\]##### (8)

\[\begin{aligned}{}\text{II.}& \hspace{1em}{\tilde{A}_{s}}\otimes {\tilde{B}_{s}}={\tilde{B}_{s}}\otimes {\tilde{A}_{s}},\end{aligned}\]##### (9)

\[\begin{aligned}{}\text{III.}& \hspace{1em}k({\tilde{A}_{s}}\oplus {\tilde{B}_{s}})=k{\tilde{A}_{s}}\oplus k{\tilde{B}_{ss}},\end{aligned}\]##### (10)

\[\begin{aligned}{}\text{IV.}& \hspace{1em}{k_{1}}{\tilde{A}_{s}}\oplus {k_{2}}{\tilde{A}_{s}}=({k_{1}}+{k_{2}}){\tilde{A}_{s}},\end{aligned}\]##### Definition 4 *(See,* Gündoğdu and Kahraman, 2019c*).*

##### Definition 5 *(See,* Gündoğdu and Kahraman, 2019c*).*

##### (15)

\[\begin{aligned}{}& {\mathit{SWAM}_{w}}({\tilde{A}_{S1}},{\tilde{A}_{S2}},\dots ,{\tilde{A}_{Sn}})={w_{1}}{\tilde{A}_{S1}}+{w_{2}}{\tilde{A}_{S2}}+\cdots +{w_{n}}{\tilde{A}_{Sn}}\\ {} & \hspace{1em}=\Bigg\{\hspace{-0.1667em}\sqrt{1-\hspace{-0.1667em}{\prod \limits_{i=1}^{n}}{\big(1-{\mu _{{A_{s}}}^{2}}\big)^{{w_{i}}}}},{\prod \limits_{i=1}^{n}}{\vartheta _{{A_{s}}}^{{w_{i}}}},\sqrt{{\prod \limits_{i=1}^{n}}{\big(1-{\mu _{{A_{s}}}^{2}}\big)^{{w_{i}}}}\hspace{-0.1667em}-\hspace{-0.1667em}{\prod \limits_{i=1}^{n}}{\big(1-{\mu _{{A_{s}}}^{2}}-{I_{{A_{s}}}^{2}}\big)^{{w_{i}}}}}\hspace{0.1667em}\Bigg\}.\end{aligned}\]##### Definition 6 *(See* Gündoğdu and Kahraman (2019c)*).*

##### (16)

\[\begin{aligned}{}& {\mathit{SWGM}_{w}}({\tilde{A}_{S1}},{\tilde{A}_{S2}},\dots ,{\tilde{A}_{Sn}})={\tilde{A}_{S1}^{{w_{i}}}}+{\tilde{A}_{S2}^{{w_{i}}}}+\cdots +{\tilde{A}_{Sn}^{{w_{i}}}}\\ {} & \hspace{1em}=\Bigg\{{\prod \limits_{i=1}^{n}}{\mu _{{A_{s}}}^{{w_{i}}}},\hspace{-0.1667em}\sqrt{1-\hspace{-0.1667em}{\prod \limits_{i=1}^{n}}{\big(1-{\vartheta _{{A_{s}}}^{2}}\big)^{{w_{i}}}}},\hspace{-0.1667em}\sqrt{{\prod \limits_{i=1}^{n}}{\big(1-{\vartheta _{{A_{s}}}^{2}}\big)^{{w_{i}}}}-{\prod \limits_{i=1}^{n}}{\big(1-{\vartheta _{{A_{s}}}^{2}}-{I_{{A_{s}}}^{2}}\big)^{{w_{i}}}}}\hspace{0.1667em}\Bigg\}.\end{aligned}\]## 3 Spherical Fuzzy Linear Assignment Method (SF-LAM)

**Step 1.**Collect the decision-makers’ judgments by using Table 1. Consider a group of

*K*decision-makers, $D=\{{D_{1}},{D_{2}},\dots ,{D_{k}}\}$ participated in a group decision-making problem, where a finite set of alternatives, $A=\{{A_{1}},{A_{2}},\dots ,{A_{m}}\}$ are evaluated based on a finite set of criteria, $C=\{{C_{1}},{C_{2}},\dots ,{C_{n}}\}$, with corresponding weight vector ${w_{i}}=\{{w_{1}},{w_{2}},\dots ,{w_{n}}\}$ where ${\textstyle\sum _{i=1}^{n}}{w_{i}}=1$, ${w_{i}}\geqslant 0$. The weight of each criterion calculated using a pairwise comparison matrix based on decision-makers’ preference. Judgments of decision-makers are stated in a linguistic term based on Table 1. Each decision-maker

*k*expresses his opinion about the performance of alternative ${A_{m}}$ with regard to criterion ${c_{n}}$ using ${\mathit{SFS}_{k}^{mn}}$, so that ${\mathit{SFS}_{mn}^{k}}=({\mu _{mn}^{k}},{\vartheta _{mn}^{k}},{I_{mn}^{k}})$; therefore, the individual decision matrices are obtained as in Table 2.

##### Table 1

Linguistic terms | Spherical fuzzy numbers $(\mu ,\vartheta ,I)$ |

Absolutely Low Importance (ALI) | $(0.1,0.9,0.0)$ |

Very Low Importance (VLI) | $(0.2,0.8,0.1)$ |

Low Importance (LI) | $(0.3,0.7,0.2)$ |

Slightly Low Importance (SLI) | $(0.4,0.6,0.3)$ |

Equal Importance (EI) | $(0.5,0.4,0.4)$ |

Slightly More Importance (SMI) | $(0.6,0.4,0.3)$ |

More Importance (MI) | $(0.7,0.3,0.2)$ |

Very More Importance (VMI) | $(0.8,0.2,0.1)$ |

Absolutely More Importance (AMI) | $(0.9,0.1,0.0)$ |

**Step 2.**Aggregate the individual decision matrices based on aggregation operators. Naturally, decision-makers have different judgments about elements of the decision matrix. Therefore, the aggregation operators must be used in order to get the unified matrix. Hence, in this step, an aggregated decision matrix is composed, as in Table 3.

##### Table 2

Alternatives | ${C_{1}}$ | ${C_{2}}$ | $\cdots \hspace{0.1667em}$ | ${C_{n}}$ |

${A_{1}}$ | ${\mathit{SFS}_{11}^{k}}$ | ${\mathit{SFS}_{12}^{k}}$ | $\cdots \hspace{0.1667em}$ | ${\mathit{SFS}_{1n}^{k}}$ |

${A_{2}}$ | ${\mathit{SFS}_{21}^{k}}$ | ${\mathit{SFS}_{22}^{k}}$ | $\cdots \hspace{0.1667em}$ | ${\mathit{SFS}_{2n}^{k}}$ |

⋮ | ⋮ | ⋮ | ⋱ | ⋮ |

${A_{m}}$ | ${\mathit{SFS}_{m1}^{k}}$ | ${\mathit{SFS}_{m2}^{k}}$ | $\cdots \hspace{0.1667em}$ | ${\mathit{SFS}_{mn}^{k}}$ |

**Step 3.**Compute the elements of the scored decision matrix by utilizing the spherical fuzzy score function (Eq. (13)). The obtained defuzzified (scored) decision matrix is presented in Table 4.

##### Table 3

Alternatives | ${C_{1}}$ | ${C_{2}}$ | $\cdots \hspace{0.1667em}$ | ${C_{n}}$ |

${A_{1}}$ | ${\mathit{SFS}_{11}}$ | ${\mathit{SFS}_{12}}$ | $\cdots \hspace{0.1667em}$ | ${\mathit{SFS}_{1n}}$ |

${A_{2}}$ | ${\mathit{SFS}_{21}}$ | ${\mathit{SFS}_{22}}$ | $\cdots \hspace{0.1667em}$ | ${\mathit{SFS}_{2n}}$ |

⋮ | ⋮ | ⋮ | ⋱ | ⋮ |

${A_{m}}$ | ${\mathit{SFS}_{m1}}$ | ${\mathit{SFS}_{m2}}$ | $\cdots \hspace{0.1667em}$ | ${\mathit{SFS}_{mn}}$ |

**Step 4.**Establish the rank frequency non-negative matrix ${\lambda _{ik}}$ with elements that represent the frequency that ${A_{m}}$ is ranked as the

*m*th criterion-wise ranking. By comparing the ${\mathit{SC}_{mn}}$ value of each column in the scored decision matrix (see Table 3), the

*m*alternatives can be ranked with respect to each criterion ${C_{n}}\in C$ according to the decreasing order of $S{C_{mn}}$ for all ${A_{m}}\in A$. The results of the rank frequency matrix are shown in Table 5.

##### Table 4

Alternatives | ${C_{1}}$ | ${C_{2}}$ | $\cdots \hspace{0.1667em}$ | ${C_{n}}$ |

${A_{1}}$ | ${\mathit{SC}_{11}}$ | ${\mathit{SC}_{12}}$ | $\cdots \hspace{0.1667em}$ | ${\mathit{SC}_{1n}}$ |

${A_{2}}$ | ${\mathit{SC}_{21}}$ | ${\mathit{SC}_{22}}$ | $\cdots \hspace{0.1667em}$ | ${\mathit{SC}_{2n}}$ |

⋮ | ⋮ | ⋮ | ⋱ | ⋮ |

${A_{m}}$ | ${\mathit{SC}_{m1}}$ | ${\mathit{SC}_{m2}}$ | $\cdots \hspace{0.1667em}$ | ${\mathit{SC}_{mn}}$ |

**Step 5.**Calculate and establish the weighted rank frequency matrix , where the measures the contribution of ${A_{m}}$ to the overall ranking. Note that each entry of the weighted rank frequency matrix is a measure of the concordance among all criteria in ranking the

*m*th alternative

*k*th (Table 6). Where

##### Table 5

*λ*.

Alternatives | 1st | 2nd | $\cdots \hspace{0.1667em}$ |
mth |

${A_{1}}$ | ${\lambda _{11}}$ | ${\lambda _{12}}$ | $\cdots \hspace{0.1667em}$ | ${\lambda _{1m}}$ |

${A_{2}}$ | ${\lambda _{21}}$ | ${\lambda _{22}}$ | $\cdots \hspace{0.1667em}$ | ${\lambda _{2m}}$ |

⋮ | ⋮ | ⋮ | ⋱ | ⋮ |

${A_{m}}$ | ${\lambda _{m1}}$ | ${\lambda _{m2}}$ | $\cdots \hspace{0.1667em}$ | ${\lambda _{mm}}$ |

**Step 6.**Define the permutation matrix

*P*as a square ($m\times m$) matrix and set up the following linear assignment model according to the value. The linear assignment model can be written in the following linear programming format:

**Step 7.**Solve the linear assignment model, and obtain the optimal permutation matrix ${P^{\ast }}$ for all

*i*and

*k*.

##### Table 6

Alternatives | 1st | 2nd | $\cdots \hspace{0.1667em}$ |
mth |

${A_{1}}$ | $\cdots \hspace{0.1667em}$ | |||

${A_{2}}$ | $\cdots \hspace{0.1667em}$ | |||

⋮ | ⋮ | ⋮ | ⋱ | ⋮ |

${A_{m}}$ | $\cdots \hspace{0.1667em}$ |

**Step 8.**Calculate the internal multiplication of matrix ${P^{\ast }}.A={P^{\ast }}.\left[\begin{array}{c}{A_{1}}\\ {} {A_{2}}\\ {} \vdots \\ {} {A_{m}}\end{array}\right]$ and obtain the optimal order of alternatives.

## 4 An Application to Wind Power Farm Location Selection

*Step 1.*Construct the spherical fuzzy decision matrix based on evaluations of three decision-makers. The decision matrices are presented in Tables 7, 8, and 9.

##### Table 7

Alternatives | ${C_{1}}$ | ${C_{2}}$ | ${C_{3}}$ | ${C_{4}}$ |

${A_{1}}$ | $(0.6,0.4,0.3)$ | $(0.5,0.4,0.4)$ | $(0.3,0.7,0.2)$ | $(0.5,0.4,0.4)$ |

${A_{2}}$ | $(0.5,0.4,0.4)$ | $(0.3,0.7,0.2)$ | $(0.3,0.7,0.2)$ | $(0.4,0.6,0.3)$ |

${A_{3}}$ | $(0.6,0.4,0.3)$ | $(0.2,0.8,0.1)$ | $(0.5,0.4,0.4)$ | $(0.5,0.4,0.4)$ |

${A_{4}}$ | $(0.7,0.3,0.2)$ | $(0.4,0.6,0.3)$ | $(0.5,0.4,0.4)$ | $(0.7,0.3,0.2)$ |

##### Table 8

Alternatives | ${C_{1}}$ | ${C_{2}}$ | ${C_{3}}$ | ${C_{4}}$ |

${A_{1}}$ | $(0.5,0.4,0.4)$ | $(0.7,0.3,0.2)$ | $(0.3,0.7,0.2)$ | $(0.6,0.4,0.3)$ |

${A_{2}}$ | $(0.4,0.6,0.3)$ | $(0.5,0.4,0.4)$ | $(0.3,0.7,0.2)$ | $(0.5,0.4,0.4)$ |

${A_{3}}$ | $(0.6,0.4,0.3)$ | $(0.3,0.7,0.2)$ | $(0.6,0.4,0.3)$ | $(0.5,0.4,0.4)$ |

${A_{4}}$ | $(0.5,0.4,0.4)$ | $(0.4,0.6,0.3)$ | $(0.4,0.6,0.3)$ | $(0.8,0.2,0.1)$ |

*Step 2.*Aggregate the decision matrices using Eq. (15) into a single decision matrix, as given in Table 10.

##### Table 9

Alternatives | ${C_{1}}$ | ${C_{2}}$ | ${C_{3}}$ | ${C_{4}}$ |

${A_{1}}$ | $(0.7,0.3,0.2)$ | $(0.6,0.4,0.3)$ | $(0.5,0.4,0.4)$ | $(0.5,0.4,0.4)$ |

${A_{2}}$ | $(0.5,0.4,0.4)$ | $(0.4,0.6,0.3)$ | $(0.4,0.6,0.3)$ | $(0.6,0.4,0.3)$ |

${A_{3}}$ | $(0.6,0.4,0.3)$ | $(0.3,0.7,0.2)$ | $(0.7,0.3,0.2)$ | $(0.5,0.4,0.4)$ |

${A_{4}}$ | $(0.7,0.3,0.2)$ | $(0.7,0.3,0.2)$ | $(0.7,0.3,0.2)$ | $(0.9,0.1,0.0)$ |

*Step 3.*Calculate each alternative’s score value based on each criterion using Eq. (13). The results are shown in Table 11.

##### Table 10

Alternatives | ${C_{1}}$ | ${C_{2}}$ | ${C_{3}}$ | ${C_{4}}$ |

${A_{1}}$ | $(0.61,0.37,0.30)$ | $(0.61,0.37,0.30)$ | $(0.38,0.58,0.30)$ | $(0.53,0.40,0.37)$ |

${A_{2}}$ | $(0.47,0.46,0.37)$ | $(0.41,0.56,0.32)$ | $(0.34,0.67,0.24)$ | $(0.51,0.46,0.34)$ |

${A_{3}}$ | $(0.60,0.40,0.30)$ | $(0.27,0.73,0.17)$ | $(0.61,0.37,0.30)$ | $(0.50,0.40,0.40)$ |

${A_{4}}$ | $(0.65,0.33,0.27)$ | $(0.54,0.48,0.26)$ | $(0.56,0.42,0.30)$ | $(0.82,0.18,0.11)$ |

*Step 4.*Establish the rank frequency matrix based on the scored value matrix. First, we have to determine each alternative’s ranking based on each criterion, as shown in Table 12. Then, the rank frequency matrix

*λ*is established as in Table 13. For example, observe that $A1$ has the first rank once (on ${C_{2}}$), the second rank twice (on ${C_{1}}$ and ${C_{4}}$), the third rank once (on ${C_{3}}$) and none fourth rank. Thus, ${\lambda _{11}}=1$, ${\lambda _{12}}=2$, ${\lambda _{13}}=1$ and ${\lambda _{14}}=0$.

##### Table 11

Alternatives | ${C_{1}}$ | ${C_{2}}$ | ${C_{3}}$ | ${C_{4}}$ |

${A_{1}}$ | 0.16 | 0.16 | 0.05 | 0.08 |

${A_{2}}$ | 0.02 | 0.02 | 0.02 | 0.03 |

${A_{3}}$ | 0.14 | 0.00 | 0.16 | 0.05 |

${A_{4}}$ | 0.22 | 0.06 | 0.09 | 0.57 |

##### Table 12

Ranking | ${C_{1}}$ | ${C_{2}}$ | ${C_{3}}$ | ${C_{4}}$ |

1st | $A4$ | $A1$ | $A3$ | $A4$ |

2nd | $A1$ | $A4$ | $A4$ | $A1$ |

3rd | $A3$ | $A2$ | $A1$ | $A2$ |

4th | $A2$ | $A3$ | $A2$ | $A3$ |

*Step 5.*Compute and further establish the weighted rank frequency matrix , as shown in Table 14. For example, consider in the following:

##### Table 13

*λ*.

Alternatives | 1st | 2nd | 3rd | 4th |

$A1$ | 1 | 2 | 1 | 0 |

$A2$ | 0 | 0 | 2 | 2 |

$A3$ | 1 | 0 | 1 | 2 |

$A4$ | 2 | 2 | 0 | 0 |

*Step 6.*Construct the linear assignment model as follows. This binary mathematical model’s objective function tries to maximize the sum of the weights of alternatives by choosing the optimal order.

##### Table 14

Alternatives | 1st | 2nd | 3rd | 4th |

$A1$ | 0.158 | 0.673 | 0.169 | 0 |

$A2$ | 0 | 0 | 0.471 | 0.529 |

$A3$ | 0.169 | 0 | 0.36 | 0.471 |

$A4$ | 0.673 | 0.327 | 0 | 0 |

*Step 7.*Solve the above mathematical model by using GAMS 24.1.3 software, and the results are obtained as follows: After solving the model, the results are ${P_{12}}=1$, ${P_{23}}=1$, ${P_{34}}=1$ and ${P_{41}}=1$. Also, the objective function of the assignment model is $z=2.288$.

*Step 8.*Apply the permutation matrix ${P^{\ast }}$ to the matrix of alternatives (

*A*) to obtain the optimal order of alternatives.