## 1 Introduction

Decision making is one of the most complicated administrative processes in management. Over the years, various methods have been designed to simplify the process as well as developing new methods. Since, there are many imprecise concepts all around us that routinely expressed in different terms. In fact, the human brain works with considering various factors and based on inferential thinking and value of sentences that modeling of them with mathematical formulas if not impossible would be a complex task.

Because crisp data are inexpressive to model real life situations Zadeh in 1965, has suggested fuzzy logic that is closer to human thinking and Chen [3] developed the TOPSIS method to fuzzy decision-making situations. The purpose of fuzzy logic as a decision-making technique is to improve decision making process in vague and unclear circumstances. Fuzzy management science, while creating the flexibility in the model, with entering some data such as knowledge, experience and human judgment in the model also offers fully functional responses to it [5] . However, if a decision is not possible for linguistic variables based on fuzzy sets, Interval-valued fuzzy set theory can provide a more detailed modeling. In this paper, interval-valued fuzzy TOPSIS method is proposed to solve MCDM (Multi-Criteria Decision Making) problems, where the weight of the criterias are unequal [ 2 , 6 , 7 , 10 , 11 , 12 ].

## 2 TOPSIS Method

As mentioned, this method was developed by Hwang and Yoon (1981) in which the best alternative should have the shortest distance from an ideal solution and the worst alternative is the furthest from an ideal solution [2, 6].

Assume a multi criteria decision making problem has *n* alternatives, *A*_{1}, *A*_{2},..., *A _{n}* and

*m*criterias,

*C*

_{1},

*C*

_{2},...,

*C*. Each alternative is estimated regarding the m criteria. All the values/ratings are determined to alternatives with respect to decision matrix define by

_{m}*X*(

*x*)

_{ij}

_{n}_{×}

*. The criteria’s weight vector is*

_{m}*w*= (

*w*

_{1},

*w*

_{2}, ...,

*w*) that

_{m}- Normalize the decision matrix using the following evolution for each
*r*._{ij}${r}_{\mathrm{ij}}=\frac{{a}_{\mathrm{ij}}}{\sqrt{\sum _{i=1}^{m}{a}_{\mathrm{ij}}^{2}}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1,2,\dots ,m\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,2,\dots ,n$ -
Multiply the columns of the normalized decision matrix by the connected weights. The weighted and normalized decision matrix is come as:
Which
${V}_{\mathrm{ij}}={w}_{j}\times {r}_{\mathrm{ij}};\hspace{0.17em}i=1,2,\dots ,m\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=1,2,\dots ,n$ *w*is the weight of the_{j}*jth*criteria. - specify the ideal and negative ideal alternatives respectively as follows:
Where
$\begin{array}{c}{A}^{+}=\left\{{v}_{1}^{+},{v}_{2}^{+},\dots ,{v}_{n}^{+}\right\}=\left\{\left(\mathrm{ma}{x}_{i}{v}_{\mathrm{ij}}j\in {J}_{1}\right),\left(\mathrm{mi}{n}_{i}{v}_{\mathrm{ij}}j\in {J}_{2}\right)i=1,2,\dots ,m\right\}\\ {A}^{-}=\left\{{v}_{1}^{-},{v}_{2}^{-},\dots ,{v}_{n}^{-}\right\}=\left\{\left(\mathrm{mi}{n}_{i}{v}_{\mathrm{ij}}j\in {J}_{1}\right),\left(\mathrm{ma}{x}_{i}{v}_{\mathrm{ij}}j\in {J}_{2}\right)i=1,2,\dots ,m\right\}\end{array}$ *J*_{1}is the set of benefit criterias and*J*_{2}is the set of cost criterias. - With using of the two Euclidean distances to calculate the distance of the existing alternatives from ideal and negative ideal alternatives as:
$\begin{array}{c}{S}_{i}^{+}=\sqrt{\sum _{j=1}^{n}{\left({v}_{\mathrm{ij}}-{v}_{j}^{+}\right)}^{2}}i=1,2,\dots ,m\\ {S}_{i}^{-}=\sqrt{\sum _{j=1}^{n}{\left({v}_{\mathrm{ij}}-{v}_{j}^{-}\right)}^{2}}\hspace{0.17em}\hspace{0.17em}i=1,2,\dots ,m\end{array}$ -
The relevant closeness to the ideal alternatives can be defined as:
Where
${C}_{i}^{+}=\frac{{S}_{i}^{-}}{{S}_{i}^{-}+{S}_{i}^{+}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1,2,\dots ,m$ .$0\le {C}_{i}^{+}\le 1$ -
According to the relative closeness to the ideal alternatives rank the alternatives the bigger
is related to better alternative${C}_{i}^{+}$ *A*[1] ._{i}

## 3 Interval-Valued Fuzzy Sets

Since the theory of fuzzy sets by Zadeh can be used in vague and imprecise terms, many studies, have developed TOPSIS method in the interval-fuzzy environment.Because of the complexity of the socio-economic environment in many practical decision problems that option often would arrange shady by decision-makers [8, 9]. An interval-valued fuzzy set *A* defined on (−∞, +∞) is given by:

Figure 1 Shows the value of membership at *x′* of interval-valued fuzzy set A. Thus, the minimum and maximum value of the membership *x′* are

Here are two interval-valued fuzzy numbers

The Normalized Euclidean distance between

A standard MCDM (Multi-Criteria Decision Making) problem can be briefly demonstrated in a decision matrix that *x _{ij}* represents value of the ith alternative of

*A*with notice to the jth attribute,

_{i}*x*. In this article, we develop the canonical matrix to interval-valued fuzzy decision matrix. The value and weighing of criteria, have been considered as linguistic variables. By using of Tables 1 and 2, these linguistic variables can be turned to interval-valued fuzzy triangular numbers.

_{j}Definition of linguistic variables for the ratings

Very Poor (VP) | [(0,0);0;(1,1.5)] |

Poor (P) | [(0,0.5);1;(2.5,3.5)] |

Moderately Poor (MP) | [(0,1.5);3;(4.5,5.5)] |

Fair (F) | [(2.5,3.5),5,(6.5,7.5)] |

Moderately Good (MG) | [(4.5,5.5),7,(8.9.5)] |

Good (G) | [(5.5,7.5),9,(9.5,10)] |

Very Good (VG) | [(8.5,9.5),10,(10,10)] |

Definition of linguistic variables for the importance of each criterion

Very low (VL) | [(0,0);0;(0.1,0.15)] |

Low (L) | [(0,0.05);0.1;(0.25,0.35)] |

Medium low (ML) | [(0,0.15);0.3;(0.45,0.55)] |

Medium (M) | [(0.25,0.35),0.5,(0.65,0.75)] |

Medium high (MH) | [(0.45,0.55),0.7,(0,8,0.95)] |

High (H) | [(0.55,0.75),0.9,(0.95,1)] |

Very high (VH) | [(0.85,0.95),1,(1,1)] |

Suppose that *A*_{1}, *A*_{2},...,*A _{n}* are n possible alternatives and

*C*

_{1},

*C*

_{2},...,

*C*are m criteria. So

_{m}*A*with notice to criterion

_{i}*C*. Figure 2 represents

_{j}Here

Therefore, the normalized matrix

Here the suggested technique for building up the TOPSIS to interval-valued fuzzy TOPSIS can be described as follows:

- Make the weighted normalized fuzzy decree matrix with notice that each criterias has own importance as:
that$\tilde{V}={\left[{\tilde{v}}_{\mathrm{ij}}\right]}_{n\times m}$ . Now from Defintion 3.1:${\tilde{v}}_{\mathrm{ij}}={\tilde{r}}_{\mathrm{ij}}\times {\tilde{w}}_{j}$ ${\tilde{v}}_{\mathrm{ij}}=\left[({\tilde{r}}_{\mathrm{ij}}\times {\tilde{w}}_{{1}_{j}},{{\tilde{r}}^{\prime}}_{{1}_{\mathrm{ij}}}\times {{\tilde{w}}^{\prime}}_{{1}_{j}});{\tilde{r}}_{{2}_{\mathrm{ij}}}\times {\tilde{w}}_{{2}_{j}};({\tilde{r}}_{{3}_{\mathrm{ij}}}\times {\tilde{w}}_{{3}_{j}},{\tilde{r}}_{{3}_{\mathrm{ij}}}\times {\tilde{w}}_{{3}_{j}})\right]=\left[\left({g}_{\mathrm{ij}},{{g}^{\prime}}_{\mathrm{ij}}\right);{h}_{\mathrm{ij}};({{l}^{\prime}}_{\mathrm{ij}},{l}_{\mathrm{ij}})\right]$ - Defined the optimal and negative optimal solution as:
$\hspace{0.17em}{A}^{+}=\left[\left(1,1\right);1;\left(1,1\right)\right]\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j\in {\Omega}_{b}\hspace{0.17em}{A}^{-}=\left[\left(0,0\right);0;\left(0,0\right)\right],\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j\in {\Omega}_{c\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}$ -
Normalized Euclidean distance can be figured out using Definition 3.2 as follows:
Where
$\begin{array}{c}{D}^{-}(\tilde{N},\tilde{M})=\sqrt{\frac{1}{3}\sum _{i=1}^{3}\left[{\left({N}_{{x}_{i}}^{-}-{M}_{{y}_{i}}^{-}\right)}^{2}\right]}\\ {D}^{+}(\tilde{N},\tilde{M})=\sqrt{\frac{1}{3}\sum _{i=1}^{3}\left[{\left({N}_{{x}_{i}}^{+}-{M}_{{y}_{i}}^{+}\right)}^{2}\right]}\end{array}$ *D*^{−},*D*^{+}are the initial and secondary distance measure, respectively. Hence, we can calculate distance from the ideal alternative for each alternative as follows:As the same way, calculate gap of the negative ideal solution by:$\begin{array}{c}{D}_{i1}^{+}=\sum _{j=1}^{m}\sqrt{\frac{1}{3}\left[{({g}_{\mathrm{ij}}-1)}^{2}+{({h}_{\mathrm{ij}}-1)}^{2}+{({l}_{\mathrm{ij}}-1)}^{2}\right]}\\ {D}_{i2}^{+}=\sum _{j=1}^{m}\sqrt{\frac{1}{3}\left[{({{g}^{\prime}}_{\mathrm{ij}}-1)}^{2}+{({h}_{\mathrm{ij}}-1)}^{2}+{({{l}^{\prime}}_{\mathrm{ij}}-1)}^{2}\right]}\end{array}$ Eqs. (12) and (13) are used to specify the distance from ideal and negative ideal alternatives in interval values.$\begin{array}{c}{D}_{i1}^{-}=\sum _{j=1}^{m}\sqrt{\frac{1}{3}\left[{({g}_{\mathrm{ij}}-0)}^{2}+{({h}_{\mathrm{ij}}-0)}^{2}+{({l}_{\mathrm{ij}}-0)}^{2}\right]}\\ {D}_{i2}^{-}=\sum _{j=1}^{m}\sqrt{\frac{1}{3}\left[{({{g}^{\prime}}_{\mathrm{ij}}-0)}^{2}+{({h}_{\mathrm{ij}}-0)}^{2}+{({{l}^{\prime}}_{\mathrm{ij}}-0)}^{2}\right]}\end{array}$ -
The involved sepreation can be calculated by:
The latest worths of
$R{C}_{1}=\frac{{D}_{i2}^{-}}{{D}_{i2}^{+}+{D}_{i2}^{-}}$ are identified as:$R{C}_{i}^{*}$ $R{c}_{i}^{*}=\frac{R{C}_{1}+R{C}_{2}}{2}$

## 4 The Implementation ofthe Extended Technique toSolveProblems

Suppose aninvestment corporation plans to allocate its limited resources toinvest on four projectsaccording toimportance and profitability of each project respectively. In thispaper, a model is presented for prioritizing investments in various industrial fields. In this case, committee of company’s decision makers intend to evaluate and ultimately rank the possible company’s investment options. Desired options for investment are given in the following table.

Firstly, criteria and sub-criteria were determined by applying the strategic documents of company. There are three main criteria as: “Industrial efficiency”, “Compliance with company’s strategy” and “The campany’s industrial experience”. The hierarchy of criteria and sub-criteria shows in Table 4.

Desired options for investment

Code | Title |
---|---|

A1 | Project 1 |

A2 | Project 2 |

A3 | Project 3 |

A4 | Project 4 |

The hierarchy of criteria and sub-criteria

C1-Industrial efficiency | C2-Compliance withcompany’sstrategy | C3-Industrial experience | |||
---|---|---|---|---|---|

C11 | Increasing demand for industrial products (P) | C21 | Ability to attract foreign investors (P) | C31 | Receivables in the subordinate (C) |

C12 | Alternative Products (C) | C22 | Entrepreneurship (P) | C32 | Implementation process of industrial projects (P) |

C13 | government intervention in product pricing (P) | C23 | Technology transfer Capacity (P) | ||

C14 | Current value of the industry on the exchange (P) | C24 | Ability to reduce dependency on foreign products (P) | ||

C15 | Average process for the delivery of industrial projects (C) | C25 | Amount of dependency on foreign raw material (C) | ||

C26 | Exportamount (P) |

### 4.1 Solution Steps

- After weighing the basic criteria by decision makers separately and unaware each other based on the target, then decision matrix is created by specified linguistic variables.As already mentioned, each linguistic variable has an interval fuzzy value. Table 6. gives these values as. So, the final decision matrix is given in Tables 7 with interval fuzzy numbers.
- In this step, decision matrix is normalized by the equation (8) and the results are expressed in Tables 8.
- As stated earlier, the weight of each criteria was previously determined by the decision makers (Shannon entropy) as given in Tables 9.
- Now we can make the weighted normalized fuzzy decision matrix by using the Eq. (9) given that each criterion has different importance. As in Table 10.
- By using Eq. (11,12,13) the distance of each alternative is calculated from the ideal alternative
, given in Table 11.$\left[{D}_{i1}^{+},{D}_{i2}^{+}\right]$ - At this step, the fuzzy relative closeness of each alternative is calculated by using the respective distinctions of each pair and the results are given in Table (11).
- In the last step alternatives are listed in Table (12) according to their relative closeness.

Now calculate

10 | 4.83 | 5.5 | 10 | 4.83 | 10 | 9.83 | 10 | 10 | 5.5 | 9.83 | 5.5 | 10 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Decision matrix according to linguistic variables

C11 | C12 | C13 | C14 | C15 | C21 | C22 | C23 | C24 | C25 | C26 | C31 | C32 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A1 | MG | P | MP | G | MP | MG | MG | M | M | MP | M | MP | MG |

A2 | VG | VP | P | MG | VP | G | G | G | G | P | MG | P | VG |

A3 | P | M | M | MP | P | M | MP | P | VP | MP | MP | M | M |

A4 | M | MP | MP | MG | P | MG | M | P | MG | MP | MG | MP | M |

Interval fuzzy value of linguistic variables

[(3.83,4.83);6.33;(7.5,8.83)] | VP |

[(4.5,5.5);6.67;(7.67,8.33)] | P |

[(5.17,6.17);7.33;(8.17,9)] | MP |

[(6.17,7.5);8.67;(,9.179.83)] | M |

[(7.17,8.17);9;(9.33,9.83)] | MG |

[(7.5,8.83);9.67;(9.83,10)] | G |

[(8.5,9.5);10;(10,10)] | VG |

Interval valued fuzzy decision matrix

C11 | C12 | C13 | C14 | C15 | |

A1 | [(7.17,8.17);9;(9.33,9.83)] | [(4.5,5.5);6.67;(7.67,8.33)] | [(5.17,6.17);7.33;(8.17,9)] | [(7.5,8.83);9.67;(9.83,10)] | [(5.17,6.17);7.33;(8.17,9)] |

A2 | [(8.5,9.5);10;(10,10)] | [(3.83,4.83);6.33;(7.5,8.83)] | [(4.5,5.5);6.67;(7.67,8.33)] | [(7.17,8.17);9;(9.33,9.83)] | [(3.83,4.83);6.33;(7.5,8.83)] |

A3 | [(4.5,5.5);6.67;(7.67,8.33)] | [(6.17,7.5);8.67;(,9.179.83)] | [(6.17,7.5);8.67;(,9.179.83)] | [(5.17,6.17);7.33;(8.17,9)] | [(4.5,5.5);6.67;(7.67,8.33)] |

A4 | [(6.17,7.5);8.67;(,9.179.83)] | [(5.17,6.17);7.33;(8.17,9)] | [(5.17,6.17);7.33;(8.17,9)] | [(7.17,8.17);9;(9.33,9.83)] | [(4.5,5.5);6.67;(7.67,8.33)] |

C21 | C22 | C23 | C24 | C25 | |

A1 | [(7.17,8.17);9;(9.33,9.83)] | [(7.17,8.17);9;(9.33,9.83)] | [(6.17,7.5);8.67;(,9.179.83)] | [(6.17,7.5);8.67;(,9.179.83)] | [(5.17,6.17);7.33;(8.17,9)] |

A2 | [(8.5,9.5);10;(10,10)] | [(6.17,7.5);8.67;(,9.179.83)] | [(8.5,9.5);10;(10,10)] | [(7.5,8.83);9.67;(9.83,10)] | [(4.5,5.5);6.67;(7.67,8.33)] |

A3 | [(6.17,7.5);8.67;(,9.179.83)] | [(5.17,6.17);7.33;(8.17,9)] | [(4.5,5.5);6.67;(7.67,8.33)] | [(3.83,4.83);6.33;(7.5,8.83)] | [(5.17,6.17);7.33;(8.17,9)] |

A4 | [(7.17,8.17);9;(9.33,9.83)] | [(6.17,7.5);8.67;(,9.179.83)] | [(4.5,5.5);6.67;(7.67,8.33)] | [(7.17,8.17);9;(9.33,9.83)] | [(5.17,6.17);7.33;(8.17,9)] |

C26 | C31 | C32 | |||

A1 | [(6.17,7.5);8.67;(,9.179.83)] | [(5.17,6.17);7.33;(8.17,9)] | [(7.17,8.17);9;(9.33,9.83)] | ||

A2 | [(7.17,8.17);9;(9.33,9.83)] | [(4.5,5.5);6.67;(7.67,8.33)] | [(8.5,9.5);10;(10,10)] | ||

A3 | [(5.17,6.17);7.33;(8.17,9)] | [(6.17,7.5);8.67;(,9.179.83)] | [(6.17,7.5);8.67;(,9.179.83)] | ||

A4 | [(7.17,8.17);9;(9.33,9.83)] | [(5.17,6.17);7.33;(8.17,9)] | [(6.17,7.5);8.67;(,9.179.83)] |

Normalize Decision Matrix

C11 | C12 | C13 | C14 | C15 | |

A1 | [(0.72,0.82);0.9;(0.93,0.98)] | [(0.88,1.07);0.72;(0.58,0.63)] | [(0.89,1.06);0.75;(0.61,0.67)] | [(0.75,0.88);0.97;(0.98,1)] | [(0.78,0.93);0.66;(0.54,0.59)] |

A2 | [(0.85,0.95);1;(1,1)] | [(1,1.3);0.76;(0.55,0.64)] | [(1,1.22);0.82;(0.62,0.72)] | [(0.72,0.82);0.9;(0.93,0.98)] | [(1,1.3);0.76;(0.55,0.64)] |

A3 | [(0.45,0.55);0.67;(0.77,0.83)] | [(0.64,0.78);0.56;(0.49,0.53)] | [(0.73,0.89);0.63;(0.56,0.6)] | [(0.52,0.62);0.7;(0.82,0.9)] | [(0.88,1.07);0.72;(0.58,0.63)] |

A4 | [(0.62,0.75);0.73;(0.82,0.9)] | [(0.78,0.93);0.66;(0.54,0.59)] | [(0.89,1.06);0.75;(0.61,0.67)] | [(0.72,0.82);0.9;(0.93,0.98)] | [(0.88,1.07);0.72;(0.58,0.63)] |

C21 | C22 | C23 | C24 | C25 | |

A1 | [(0.72,0.82);0.9;(0.93,0.98)] | [(0.73,0.83);0.92;(0.95,0.95)] | [(0.62,0.75);0.73;(0.82,0.9)] | [(0.62,0.75);0.73;(0.82,0.9)] | [(0.89,1.06);0.75;(0.61,0.67)] |

A2 | [(0.85,0.95);1;(1,1)] | [(0.63,0.76);0.88;(0.93,1)] | [(0.85,0.95);1;(1,1)] | [(0.75,0.88);0.97;(0.98,1)] | [(1,1.22);0.82;(0.62,0.72)] |

A3 | [(0.62,0.75);0.73;(0.82,0.9)] | [(0.53,0.63);0.75;(0.83,0.92)] | [(0.45,0.55);0.67;(0.77,0.83)] | [(0.38,0.48);0.63;(0.75,0.88)] | [(0.89,1.06);0.75;(0.61,0.67)] |

A4 | [(0.72,0.82);0.9;(0.93,0.98)] | [(0.63,0.76);0.88;(0.93,1)] | [(0.45,0.55);0.67;(0.77,0.83)] | [(0.72,0.82);0.9;(0.93,0.98)] | [(0.89,1.06);0.75;(0.61,0.67)] |

C26 | C31 | C32 | |||

A1 | [(0.63,0.76);0.88;(0.93,1)] | [(0.89,1.06);0.75;(0.61,0.67)] | [(0.72,0.82);0.9;(0.93,0.98)] | ||

A2 | [(0.73,0.83);0.92;(0.95,0.95)] | [(1,1.22);0.82;(0.62,0.72)] | [(0.85,0.95);1;(1,1)] | ||

A3 | [(0.53,0.63);0.75;(0.83,0.92)] | [(0.73,0.89);0.63;(0.56,0.6)] | [(0.62,0.75);0.73;(0.82,0.9)] | ||

A4 | [(0.73,0.83);0.92;(0.95,0.95)] | [(0.89,1.06);0.75;(0.61,0.67)] | [(0.62,0.75);0.73;(0.82,0.9)] |

Weight values of criteria

[(0.85,0.95);1;(1,1)] | VH |

[(0.55,0.75);0.9;(0.95,1)] | H |

[(0.45,0.55);0.7;(0.8,0.95)] | MH |

[(0.25,0.35);0.5;(0.65,0.75)] | M |

[(0,0.15);0.3;(0.45,0.55)] | ML |

[(0,0.05);0.1;(0.25,0.35)] | L |

[(0,0);0;(0.1,0.15)] | VL |

Weight of criterias

C11 | VH | C21 | L | C31 | M |

C12 | H | C22 | ML | C32 | ML |

C13 | H | C23 | M | ||

C14 | L | C24 | ML | ||

C15 | MH | C25 | VL | ||

C26 | M |

Weighted normalize fuzzy decision matrix

C11 | C12 | C13 | C14 | C15 | |

A1 | [(0.61,0.78);0.9;(0.93,0.98)] | [(0.48,0.80);0.65;(0.55,0.63)] | [(0.49,0.8);0.68;(0.58,0.67)] | [(0,0.04);0.09;(0.25,0.35)] | [(0.35,0.51);0.46;(0.43,0.56)] |

A2 | [(0.72,0.9);1;(1,1)] | [(0.55,0.98);0.68;(0.52,0.64)] | [(0.55,0.92);0.74;(0.59,0.72)] | [(0,0.04);0.09;(0.23,0.34)] | [(0.45,0.72);0.53;(0.44,0.61)] |

A3 | [(0.38,0.52);0.67;(0.77,0.83)] | [(0.35,0.59);0.50;(0.47,0.53)] | [(0.40,0.67);0.57;(0.53,0.6)] | [(0,0.03);0.07;(0.21,0.32)] | [(0.4,0.6);0.5;(0.46,0.6)] |

A4 | [(0.53,0.71);0.73;(0.82,0.9)] | [(0.43,0.70);0.59;(0.51,0.59)] | [(0.49,0.8);0.68;(0.58,0.67)] | [(0,0.04);0.09;(0.23,0.34)] | [(0.4,0.6);0.5;(0.46,0.6)] |

C21 | C22 | C23 | C24 | C25 | |

A1 | [(0,0.04);0.09;(0.23,0.35)] | [(0,0.12);0.28;(0.43,0.52)] | [(0.16,0.26);0.37;(0.53,0.68)] | [(0,0.11);0.22;(0.37,0.5)] | [(0,0);0;(0.06,0.1)] |

A2 | [(0,0.05);0.1;(0.25,0.35)] | [(0,0.11);0.26;(0.42,0.55)] | [(0.21,0.33);0.5;(0.65,0.75)] | [(0,0.13);0.29;(0.44,0.55)] | [(0,0);0;(0.06,0.11)] |

A3 | [(0,0.04);0.07;(0.21,0.32)] | [(0,0.08);0.23;(0.37,0.51)] | [(0.11,0.19);0.34;(0.5,0.62)] | [(0,0.07);0.19;(0.34,0.48)] | [(0,0);0;(0.06,0.1)] |

A4 | [(0,0.04);0.09;(0.23,0.34)] | [(0,0.11);0.26;(0.42,0.55)] | [(0.11,0.19);0.34;(0.5,0.62)] | [(0,0.12);0.27;(0.42,0.54)] | [(0,0);0;(0.06,0.1)] |

C26 | C31 | C32 | |||

A1 | [(0.16,0.27);0.44;(0.6,0.75)] | [(0.22,0.37);0.38;(0.4,0.5)] | [(0,0.12);0.27;(0.42,0.54)] | ||

A2 | [(0.18,0.29);0.46;(0.62,0.71)] | [(1,0.43);0.41;(0.4,0.54)] | [(0,0.14);0.3;(0.45,0.55)] | ||

A3 | [(0.13,0.22);0.38;(0.54,0.69)] | [(0.18,0.31);0.32;(0.36,0.45)] | [(0,0.11);0.22;(0.37,0.5)] | ||

A4 | [(0.18,0.29);0.46;(0.62,0.71)] | [(0.22,0.37);0.38;(0.4,0.5)] | [(0,0.11);0.22;(0.37,0.5)] |

Distance of alternatives from ideal alternatives

A1 | A2 | ||||||||
---|---|---|---|---|---|---|---|---|---|

C11 | 0.234521 | 0.14 | 0.826035 | 0.890468 | C11 | 0.161658 | 0.057735 | 0.916224 | 0.967815 |

C12 | 0.443471 | 0.31459 | 0.564329 | 0.697472 | C12 | 0.422414 | 0.278328 | 0.587452 | 0.781537 |

C13 | 0.424264 | 0.288039 | 0.588473 | 0.719097 | C13 | 0.382187 | 0.225389 | 0.631981 | 0.798415 |

C14 | 0.892562 | 0.851293 | 0.153406 | 0.209921 | C14 | 0.898332 | 0.853483 | 0.142595 | 0.204369 |

C15 | 0.585947 | 0.491155 | 0.415933 | 0.511631 | C15 | 0.528205 | 0.3879 | 0.475044 | 0.624873 |

C21 | 0.898146 | 0.851293 | 0.142595 | 0.209921 | C21 | 0.889288 | 0.843603 | 0.155456 | 0.212132 |

C22 | 0.783156 | 0.712928 | 0.296254 | 0.312463 | C22 | 0.792465 | 0.716984 | 0.28519 | 0.356931 |

C23 | 0.665833 | 0.591608 | 0.384448 | 0.471487 | C23 | 0.576368 | 0.503786 | 0.48874 | 0.554196 |

C24 | 0.816497 | 0.742092 | 0.248529 | 0.321714 | C24 | 0.778396 | 0.698451 | 0.304248 | 0.366742 |

C25 | 0.791623 | 0.776745 | 0.034641 | 0.057735 | C25 | 0.980408 | 0.964728 | 0.034641 | 0.063509 |

C26 | 0.627163 | 0.549363 | 0.439394 | 0.525674 | C26 | 0.607838 | 0.541541 | 0.457675 | 0.516333 |

C31 | 0.67082 | 0.585064 | 0.342929 | 0.420833 | C31 | 0.485833 | 0.543016 | 0.665357 | 0.463537 |

C32 | 0.789515 | 0.710868 | 0.288271 | 0.355387 | C32 | 0.772981 | 0.690917 | 0.31225 | 0.37063 |

Sum | 8.623517 | 7.605039 | 4.725234 | 5.703802 | Sum | 8.276371 | 7.305862 | 5.456853 | 6.281021 |

A3 | A4 | ||||||||
---|---|---|---|---|---|---|---|---|---|

C11 | 0.426693 | 0.350333 | 0.628808 | 0.685128 | C11 | 0.329747 | 0.235938 | 0.703847 | 0.784644 |

C12 | 0.563738 | 0.461519 | 0.444747 | 0.541295 | C12 | 0.494335 | 0.376917 | 0.514166 | 0.628808 |

C13 | 0.505239 | 0.38893 | 0.505239 | 0.614763 | C13 | 0.423832 | 0.289425 | 0.588473 | 0.719097 |

C14 | 0.91086 | 0.869521 | 0.127802 | 0.189912 | C14 | 0.898332 | 0.853483 | 0.142595 | 0.204369 |

C15 | 0.548209 | 0.43589 | 0.455192 | 0.568624 | C15 | 0.528205 | 0.3879 | 0.475044 | 0.624873 |

C21 | 0.91086 | 0.865814 | 0.127802 | 0.190526 | C21 | 0.898332 | 0.853483 | 0.142595 | 0.204369 |

C22 | 0.814412 | 0.748198 | 0.251529 | 0.326292 | C22 | 0.792465 | 0.716984 | 0.28519 | 0.356931 |

C23 | 0.701831 | 0.641898 | 0.354824 | 0.422729 | C23 | 0.701831 | 0.641898 | 0.354824 | 0.422729 |

C24 | 0.835005 | 0.772744 | 0.22487 | 0.300777 | C24 | 0.789367 | 0.711548 | 0.288271 | 0.355387 |

C25 | 0.980408 | 0.967815 | 0.034641 | 0.057735 | C25 | 0.848528 | 0.967815 | 0.34641 | 0.057735 |

C26 | 0.671541 | 0.602467 | 0.388544 | 0.472193 | C26 | 0.607838 | 0.541541 | 0.457675 | 0.516333 |

C31 | 0.717496 | 0.643169 | 0.296873 | 0.365605 | C31 | 0.671516 | 0.586316 | 0.342929 | 0.420833 |

C32 | 0.817578 | 0.741732 | 0.248529 | 0.321714 | C32 | 0.817578 | 0.741732 | 0.248529 | 0.321714 |

Sum | 9.403869 | 8.490031 | 4.089401 | 5.057293 | Sum | 8.801903 | 7.904981 | 4.890547 | 5.617823 |

The final ranking of Options

RC1 | RC2 | RC* | RANK | |
---|---|---|---|---|

A1 | 0.428572 | 0.353983 | 0.391278 | 2 |

A2 | 0.462286 | 0.602653 | 0.532469 | 1 |

A3 | 0.373306 | 0.30307 | 0.338188 | 4 |

A4 | 0.415433 | 0.357171 | 0.386302 | 3 |

## 5 Conclusions

The increasing complexity of socio-economic communities causes the intricacy and ambiguity in the priorities of decision-makers; because decision-making is often done in some circumstances such as lack of information and knowledge, lack of decision-makers consensus, time limits. . . So, in such situation, Decision-making in an interval-valued fuzzy environment would be convenient. The main characteristic of using interval-valued fuzzy environment is that the membership functions would be an interval rather than an exact number. In fuzzy set theory, it is difficult to express a thought or linguistic variables entirely by an integer number in [0, 1]. Thus, expressing degree of certainty by an interval of [0, 1] would be more appropriate. It’s worth paying attention, the use of interval valuation numbers gives an occasion to proficients to define lower and upper bounds values as an interval for matrix elements and weights of criteria.

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