# Evaluation of Investment Opportunities With Interval-Valued Fuzzy Topsis Method

Naiyer Mohammadi Lanbaran 1 , Ercan Celik 2  and Muhammed Yiğider 3
• 1 Atatürk University Faculty of Science
• 2 Atatürk University Faculty of Science
• 3 Department of Mathematics, Faculty of Science
, Ercan Celik
and Muhammed Yiğider

## Abstract

The purpose of this study is extended the TOPSIS method based on interval-valued fuzzy set in decision analysis. After the introduction of TOPSIS method by Hwang and Yoon in 1981, this method has been extensively used in decision-making, rankings also in optimal choice. Due to this fact that uncertainty in decision-making and linguistic variables has been caused to develop some new approaches based on fuzzy-logic theory. Indeed, it is difficult to achieve the numerical measures of the relative importance of attributes and the effects of alternatives on the attributes in some cases. In this paper to reduce the estimation error due to any uncertainty, a method has been developed based on interval-valued fuzzy set. In the suggested TOPSIS method, we use Shannon entropy for weighting the criteria and apply the Euclid distance to calculate the separation measures of each alternative from the positive and negative ideal solutions to determine the relative closeness coefficients. According to the values of the closeness coefficients, the alternatives can be ranked and the most desirable one(s) can be selected in the decision-making process.

## 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, A1, A2,..., An and m criterias, C1, C2,..., Cm. Each alternative is estimated regarding the m criteria. All the values/ratings are determined to alternatives with respect to decision matrix define by X(xij)n×m. The criteria’s weight vector is w = (w1, w2, ..., wm) that $∑j=1mwj=1$. TOPSIS method includes a process consisting of 6-steps as follows:

• Normalize the decision matrix using the following evolution for each rij.
$rij=aij∑i=1maij2 i=1,2,…,m j=1,2,…,n$
• Multiply the columns of the normalized decision matrix by the connected weights. The weighted and normalized decision matrix is come as:
$Vij=wj×rij; i=1,2,…,m j=1,2,…,n$
Which wj is the weight of the jth criteria.
• specify the ideal and negative ideal alternatives respectively as follows:
$A+={v1+,v2+,…,vn+}={(maxivijj∈J1),(minivijj∈J2)i=1,2,…,m}A−={v1−,v2−,…,vn−}={(minivijj∈J1),(maxivijj∈J2)i=1,2,…,m}$
Where J1 is the set of benefit criterias and J2 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:
$Si+=∑j=1n(vij−vj+)2i=1,2,…,mSi−=∑j=1n(vij−vj−)2 i=1,2,…,m$
• The relevant closeness to the ideal alternatives can be defined as:
$Ci+=Si−Si−+Si+ i=1,2,…,m$
Where $0≤Ci+≤1$.
• According to the relative closeness to the ideal alternatives rank the alternatives the bigger $Ci+$ is related to better alternative Ai [1] .

## 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:

$A={x,[μAL(x),μAU(x)]}μAL(x),μAU(x):X→[0,1]∀x∈X,μAL(x)≤μAU(x)μ¯A(x)=[μAL(x),μAU(x)]A={(x,μ¯A(x))},x∈(−∞,+∞)$
That $μAL(x)$ is the lower limit of degree of membership and $μAU(x)$ is the upper limit of degree of membership.

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 $μAL(x),μAU(x)$ respectively.

Here are two interval-valued fuzzy numbers $Px=[Px−;Px+]$ and $Qx=[Qx−;Qx+]$ due to the [5], we have: 3.1. Definition

$P.Q(x.y)=[Px−.Qx−;Px+.Qx+]$ if . ∈ (+, −, ×, ÷).

3.2. Definition

The Normalized Euclidean distance between $P⌣ andQ⌣$ is as:

$D(P⌣,Q⌣)=16∑i=13[(Pxi−−Qxi−)2+(Pxi+−Qxi+)2]$

A standard MCDM (Multi-Criteria Decision Making) problem can be briefly demonstrated in a decision matrix that xij represents value of the ith alternative of Ai with notice to the jth attribute, xj. 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.

Table 1

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)]
Table 2

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 $X˜=[x˜ij]n×m$ be a fuzzy decision matrix for a multi criteria decision making problem where A1, A2,...,An are n possible alternatives and C1, C2,...,Cm are m criteria. So $x˜ij$ is the performance of alternative Ai with notice to criterion Cj. Figure 2 represents $x˜ij$ and $w˜j$ as triangular interval valued fuzzy numbers [10]

$x˜={(x1,x2,x3)(x′1,x2,x′3)$

Here $x˜$ can be indicated by $x˜=[(x1,x′1);x2;(x′3;x3)]$. The normalized performance of rating as an expansion to Chen [3] for $x˜=[(aij,a′ij);bij;(c′ij,cij)]$ can be calculated as:

$r˜ij=[(aijcj+,a′ijcj+);bijcj+;(c′ijcj+;cijcj+)],i=1,2,…,n j∈Ωbr˜ij=[(aj−a′ij,aj−aij);aj−bij;(aj−cij;aj−cij)],i=1,2,…,n j∈Ωccj+=maxicij,j∈Ωbaj−=minia′ij,j∈Ωc$

Therefore, the normalized matrix $R˜=[r˜ij]n×m$ can be obtained.

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: $V˜=[v˜ij]n×m$ that $v˜ij=r˜ij×w˜j$. Now from Defintion 3.1:
$v˜ij=[(r˜ij×w˜1j,r˜′1ij×w˜′1j);r˜2ij×w˜2j;(r˜3ij×w˜3j,r˜3ij×w˜3j)]=[(gij,g′ij);hij;(l′ij,lij)]$
• Defined the optimal and negative optimal solution as:
$A+=[(1,1);1;(1,1)] , j∈Ωb A−=[(0,0);0;(0,0)], j∈Ωc$
• Normalized Euclidean distance can be figured out using Definition 3.2 as follows:
$D−(N˜,M˜)=13∑i=13[(Nxi−−Myi−)2]D+(N˜,M˜)=13∑i=13[(Nxi+−Myi+)2]$
Where D , D+ are the initial and secondary distance measure, respectively.
Hence, we can calculate distance from the ideal alternative for each alternative as follows:
$Di1+=∑j=1m13[(gij−1)2+(hij−1)2+(lij−1)2]Di2+=∑j=1m13[(g′ij−1)2+(hij−1)2+(l′ij−1)2]$
As the same way, calculate gap of the negative ideal solution by:
$Di1−=∑j=1m13[(gij−0)2+(hij−0)2+(lij−0)2]Di2−=∑j=1m13[(g′ij−0)2+(hij−0)2+(l′ij−0)2]$
Eqs. (12) and (13) are used to specify the distance from ideal and negative ideal alternatives in interval values.
• The involved sepreation can be calculated by:
$RC1=Di2−Di2++Di2−$
The latest worths of $RCi*$ are identified as:
$Rci*=RC1+RC22$

## 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.

Table 3

Desired options for investment

CodeTitle
A1Project 1
A2Project 2
A3Project 3
A4Project 4
Table 4

The hierarchy of criteria and sub-criteria

C1-Industrial efficiencyC2-Compliance withcompany’sstrategyC3-Industrial experience
C11Increasing demand for industrial products (P)C21Ability to attract foreign investors (P)C31Receivables in the subordinate (C)
C12Alternative Products (C)C22Entrepreneurship (P)C32Implementation process of industrial projects (P)
C13government intervention in product pricing (P)C23Technology transfer Capacity (P)
C14Current value of the industry on the exchange (P)C24Ability to reduce dependency on foreign products (P)
C15Average process for the delivery of industrial projects (C)C25Amount of dependency on foreign raw material (C)
C26Exportamount (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 $[Di1+,Di2+]$, given in Table 11.
• 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 $cj+$ and $aj−$ as followes:

$x˜ij=[(aij,a′ij),bij,(c′ij,cij)]cj+=maxicij, j∈Ωb, aj−=miniaij′ ,j∈Ωc$
104.835.5104.83109.8310105.59.835.510
$c1+$$a2−$$a3−$$C4+$$a5−$$C6+$$C7+$$C8+$$C9+$$a10−$$C11+$$a12−$$C13+$
Now with using of:
$r˜ij=[(aijcj+,aij′cj+),bijcj+,(cij′cj+,cijcj+)], i=1,2,…,n , j∈Ωbr˜ij=[(aj−aij′,aj−aij),aj−bij,(aj−cij,aj−cij′)], i=1,2,…,n , j∈Ωc$
Make the $R˜=[r˜ij]n×m$ .

Table 5

Decision matrix according to linguistic variables

C11C12C13C14C15C21C22C23C24C25C26C31C32
A1MGPMPGMPMGMGMMMPMMPMG
A2VGVPPMGVPGGGGPMGPVG
A3PMMMPPMMPPVPMPMPMM
A4MMPMPMGPMGMPMGMPMGMPM
Table 6

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
Table 7

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)]
Table 8

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)]
Table 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
Table 9

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
Table 10

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)]
Table 11

Distance of alternatives from ideal alternatives

A1$D11+$$D12+$$D11−$$D12−$A2$D11+$$D12+$$D11−$$D12−$
C110.2345210.140.8260350.890468C110.1616580.0577350.9162240.967815
C120.4434710.314590.5643290.697472C120.4224140.2783280.5874520.781537
C130.4242640.2880390.5884730.719097C130.3821870.2253890.6319810.798415
C140.8925620.8512930.1534060.209921C140.8983320.8534830.1425950.204369
C150.5859470.4911550.4159330.511631C150.5282050.38790.4750440.624873
C210.8981460.8512930.1425950.209921C210.8892880.8436030.1554560.212132
C220.7831560.7129280.2962540.312463C220.7924650.7169840.285190.356931
C230.6658330.5916080.3844480.471487C230.5763680.5037860.488740.554196
C240.8164970.7420920.2485290.321714C240.7783960.6984510.3042480.366742
C250.7916230.7767450.0346410.057735C250.9804080.9647280.0346410.063509
C260.6271630.5493630.4393940.525674C260.6078380.5415410.4576750.516333
C310.670820.5850640.3429290.420833C310.4858330.5430160.6653570.463537
C320.7895150.7108680.2882710.355387C320.7729810.6909170.312250.37063
Sum8.6235177.6050394.7252345.703802Sum8.2763717.3058625.4568536.281021
A3$D11+$$D12+$$D11−$$D12−$A4$D11+$$D12+$$D11−$$D12−$
C110.4266930.3503330.6288080.685128C110.3297470.2359380.7038470.784644
C120.5637380.4615190.4447470.541295C120.4943350.3769170.5141660.628808
C130.5052390.388930.5052390.614763C130.4238320.2894250.5884730.719097
C140.910860.8695210.1278020.189912C140.8983320.8534830.1425950.204369
C150.5482090.435890.4551920.568624C150.5282050.38790.4750440.624873
C210.910860.8658140.1278020.190526C210.8983320.8534830.1425950.204369
C220.8144120.7481980.2515290.326292C220.7924650.7169840.285190.356931
C230.7018310.6418980.3548240.422729C230.7018310.6418980.3548240.422729
C240.8350050.7727440.224870.300777C240.7893670.7115480.2882710.355387
C250.9804080.9678150.0346410.057735C250.8485280.9678150.346410.057735
C260.6715410.6024670.3885440.472193C260.6078380.5415410.4576750.516333
C310.7174960.6431690.2968730.365605C310.6715160.5863160.3429290.420833
C320.8175780.7417320.2485290.321714C320.8175780.7417320.2485290.321714
Sum9.4038698.4900314.0894015.057293Sum8.8019037.9049814.8905475.617823
Table 12

The final ranking of Options

RC1RC2RC*RANK
A10.4285720.3539830.3912782
A20.4622860.6026530.5324691
A30.3733060.303070.3381884
A40.4154330.3571710.3863023

## 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.

## References

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S. Ballı and S. Korukoğlu (2009), “Operating SystemSelection Using Fuzzy AHP and Topsis Methods”, Mathematical & Computational Applications, 14 (2), 119–130.

• [2]

C.L. Hwang, K. Yoon, Multiple Attributes Decision Making Methods and Applications, Springer, Berlin Heidelberg, 1981.

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C.T. Chen, Extensions of the TOPSIS for group decision-making under fuzzy environment, Fuzzy Sets and Systems 114 (2000) 1–9.

• [4]

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K. Yoon, System selection by multiple attribute decision making, Ph.D. dissertations, Kansas State University, Manhattan, Kansas, 1980.

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L. Zadeh, The concept of a linguistic variable and its application to approximate reasoning - I, Information Science 8 (1975) 199–249.

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L.A. Zadeh, the concept of a linguistic variable and its application to approximate reasoning—I, Inform. Sci. 8 (1975) 199–249.

• [9]

M.B. Gorzalczany, A method of inference in approximate reasoning based on interval-valued fuzzy sets, Fuzzy Sets and Systems 21 (1987) 1–17.

• [10]

P. Grzegorzewski, Distances between intuitionistic fuzzy sets and/or intervalvalued fuzzy sets based on the Hausdorff metric, Fuzzy Set and Systems 148 (2004) 319–328.

• [11]

Şenel, M., Şenel, B., Havle, C.A., (2018). Risk Analysis of Ports in Maritime Industry in Turkey Using FMEA Based Intuitionistic Fuzzy Topsis Approach, ITM Web of Conference, 01023 (8). DOI: https://doi.org/10.1051/itmconf/2018221023.

• [12]

M. Şenel, B. Şenel, Havle, C.A., (2018). Analysis of APSP Key Factors By Using Fuzzy Cognitive Map(FCM), Safety Science, (Yayınaşamasında).

• [13]

Şenel, B., Şenel, M., Aydemir, G., (2018). Use and Comparison of TOPSIS and Electre Methods in Personnel Selection. ITM Web of Conference, 01021 (10). DOI: https://doi.org/10.1015/itmconf/20182201021.

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• [1]

S. Ballı and S. Korukoğlu (2009), “Operating SystemSelection Using Fuzzy AHP and Topsis Methods”, Mathematical & Computational Applications, 14 (2), 119–130.

• [2]

C.L. Hwang, K. Yoon, Multiple Attributes Decision Making Methods and Applications, Springer, Berlin Heidelberg, 1981.

• [3]

C.T. Chen, Extensions of the TOPSIS for group decision-making under fuzzy environment, Fuzzy Sets and Systems 114 (2000) 1–9.

• [4]

D. Dubois, S. Gottwald, P. Hajek, J. Kacprzyk and H. Prade, Terminological difficulties in fuzzy set theory—the case of “intuitionistic fuzzy sets”, Fuzzy Sets and Systems 156 (2005) 485–491.

• [5]

D.H. Hong and S. Lee, Some algebraic properties and a distance measure for intervalvalued fuzzy numbers, Information Sciences 148 (2002) 1–10.

• [6]

K. Yoon, System selection by multiple attribute decision making, Ph.D. dissertations, Kansas State University, Manhattan, Kansas, 1980.

• [7]

L. Zadeh, The concept of a linguistic variable and its application to approximate reasoning - I, Information Science 8 (1975) 199–249.

• [8]

L.A. Zadeh, the concept of a linguistic variable and its application to approximate reasoning—I, Inform. Sci. 8 (1975) 199–249.

• [9]

M.B. Gorzalczany, A method of inference in approximate reasoning based on interval-valued fuzzy sets, Fuzzy Sets and Systems 21 (1987) 1–17.

• [10]

P. Grzegorzewski, Distances between intuitionistic fuzzy sets and/or intervalvalued fuzzy sets based on the Hausdorff metric, Fuzzy Set and Systems 148 (2004) 319–328.

• [11]

Şenel, M., Şenel, B., Havle, C.A., (2018). Risk Analysis of Ports in Maritime Industry in Turkey Using FMEA Based Intuitionistic Fuzzy Topsis Approach, ITM Web of Conference, 01023 (8). DOI: https://doi.org/10.1051/itmconf/2018221023.

• [12]

M. Şenel, B. Şenel, Havle, C.A., (2018). Analysis of APSP Key Factors By Using Fuzzy Cognitive Map(FCM), Safety Science, (Yayınaşamasında).

• [13]

Şenel, B., Şenel, M., Aydemir, G., (2018). Use and Comparison of TOPSIS and Electre Methods in Personnel Selection. ITM Web of Conference, 01021 (10). DOI: https://doi.org/10.1015/itmconf/20182201021.

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