Evaluation of Investment Opportunities With Interval-Valued Fuzzy Topsis Method

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 ].

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,..., A_n and m criterias, C₁, C₂,..., C_m. Each alternative is estimated regarding the m criteria. All the values/ratings are determined to alternatives with respect to decision matrix define by X(x_ij)_n×m. The criteria’s weight vector is w = (w₁, w₂, ..., w_m) that ∑j=1mwj=1\sum\nolimits_{j = 1}^m w_j = 1 . TOPSIS method includes a process consisting of 6-steps as follows:

Normalize the decision matrix using the following evolution for each r_ij. (1)rij=aij∑i=1maij2 i=1,2,…,m j=1,2,…,nr_{ij} = \frac{{a_{ij} }}{{\sqrt {\sum\limits_{i = 1}^m a_{ij}^2 } }}\;\;\;\;\;i = 1,2, \ldots ,m\;\;\;j = 1,2, \ldots ,n

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: (6)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∈(−∞,+∞)\begin{array}{l} A = \left\{ {x,\left[ {\mu _A^L \left( x \right),\mu _A^U \left( x \right)} \right]} \right\}\mu _A^L \left( x \right),\mu _A^U \left( x \right):X \to \left[ {0,1} \right]\;\;\forall x \in X\;,\mu _A^L \left( x \right) \le \mu _A^U \left( x \right) \\ \bar \mu _A \left( x \right) = \left[ {\mu _A^L \left( x \right),\mu _A^U \left( x \right)} \right]A = \left\{ {(x,\bar \mu _A^{\left( x \right)} )} \right\}\;,x \in ( - \infty , + \infty ) \\ \end{array} That μAL(x)\mu _A^L \left( x \right) is the lower limit of degree of membership and μAU(x)\mu _A^U \left( x \right) 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)\mu _A^L \left( x \right),\mu _A^U \left( x \right) respectively.

Fig. 1

Here are two interval-valued fuzzy numbers Px=[Px−;Px+]P_x = \left[ {P_x^ - ;P_x^ + } \right and Qx=[Qx−;Qx+]Q_x = \left[ {Q_x^ - ;Q_x^ + } \right due to the [5], we have:

P.Q(x.y)=[Px−.Qx−;Px+.Qx+]P.Q\left( {x.y} \right) = \left[ {P_x^ - .Q_x^ - ;P_x^ + .Q_x^ + } \right if . ∈ (+, −, ×, ÷).

3.2. Definition

The Normalized Euclidean distance between P⌣ andQ⌣\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over P} \;{\rm{and}}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over Q} is as: (7)D(P⌣,Q⌣)=16∑i=13[(Pxi−−Qxi−)2+(Pxi+−Qxi+)2]D\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over P} ,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over Q} } \right) = \sqrt {\frac{1}{6}\sum\limits_{i = 1}^3 \left[ {(P_{x_i }^ - - Q_{x_i }^ - )^2 + (P_{x_i }^ + - Q_{x_i }^ + )^2 } \right]}

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_i with notice to the jth attribute, x_j. 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\tilde X = \left[ {\tilde x_{ij} } \right]_{n \times m} be a fuzzy decision matrix for a multi criteria decision making problem where A₁, A₂,...,A_n are n possible alternatives and C₁, C₂,...,C_m are m criteria. So x˜ij\tilde x_{ij} is the performance of alternative A_i with notice to criterion C_j. Figure 2 represents x˜ij\tilde x_{ij} and w˜j\tilde w_j as triangular interval valued fuzzy numbers [10] x˜={(x1,x2,x3)(x′1,x2,x′3)\tilde x = \left\{ {\begin{array}{*{20}c} {\left( {x_1 ,x_2 ,x_3 } \right)} \hfill \\ {\left( {x'_1 ,x_2 ,x'_3 } \right)} \hfill \\\end{array}} \right.

Fig. 2

Here x˜\tilde x can be indicated by x˜=[(x1,x′1);x2;(x′3;x3)]\tilde x = \left[ {\left( {x_1 ,x^' _1 } \right);x_2 ;(x'_3 ;x_3 )} \right . The normalized performance of rating as an expansion to Chen [3] for x˜=[(aij,a′ij);bij;(c′ij,cij)]\tilde x = \left[ {\left( {a_{ij} ,a^' _{ij} } \right);b_{ij} ;(c^' _{ij} ,c_{ij} )} \right can be calculated as: (8)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\begin{array}{l} \tilde r_{ij} = \left[ {\left( {\frac{{a_{ij} }}{{c_j^ + }},\frac{{a^' _{ij} }}{{c_j^ + }}} \right);\frac{{b_{ij} }}{{c_j^ + }};\left( {\frac{{c^' _{ij} }}{{c_j^ + }};\frac{{c_{ij} }}{{c_j^ + }}} \right)} \right],\;\;\;\;\;\;\;\;\;\;\;\;\;i = 1,2, \ldots ,\;n\;\;\;\;j \in \Omega _b \\ \tilde r_{ij} = \left[ {\left( {\frac{{a_j^ - }}{{a^' _{ij} }},\frac{{a_j^ - }}{{a_{ij} }}} \right);\frac{{a_j^ - }}{{b_{ij} }};\left( {\frac{{a_j^ - }}{{c_{ij} }};\frac{{a_j^ - }}{{c^' _{ij} }}} \right)} \right],\;\;\;\;\;\;\;\;\;\;\;\;i = 1,2,\; \ldots ,\;n\;\;\;\;\;j \in \Omega _c \\ c_j^ + = \mathop {\max }\limits_i c_{ij} \;\;,\;\;j \in \Omega _b \\ a_j^ - = \mathop {\min }\limits_i a'_{ij} \;\;,\;\;j \in \Omega _c \\ \end{array}

Therefore, the normalized matrix R˜=[r˜ij]n×m\tilde R = \left[ {\tilde r_{ij} } \right]_{n \times 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\tilde V = \left[ {\tilde v_{ij} } \right]_{n \times m} that v˜ij=r˜ij×w˜j\tilde v_{ij} = \tilde r_{ij} \times \tilde w_j . Now from Defintion 3.1: (9)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)]\tilde v_{ij} = \left[ {\left( {\tilde r_{1_{ij} } \times \tilde w_{1_j } ,\tilde r'_{1_{ij} } \times \tilde w'_{1_j } } \right);\tilde r_{2_{ij} } \times \tilde w_{2_j } ;(\tilde r^' _{3_{ij} } \times \tilde w^' _{3_j } ,\tilde r_{3_{ij} } \times \tilde w_{3_j } )} \right] = \left[ {\left( {g_{ij} ,g^' _{ij} } \right);h_{ij} ;(l^' _{ij} ,l_{ij} )} \right

Defined the optimal and negative optimal solution as: (10)A+=[(1,1);1;(1,1)] , j∈Ωb A−=[(0,0);0;(0,0)], j∈Ωc\;A^ + = \left[ {\left( {1,1} \right);1;\left( {1,1} \right)} \right]\;,\;\;\;\;\;j \in \Omega _b \;A^ - = \left[ {\left( {0,0} \right);0;\left( {0,0} \right)} \right],\;\;\;\;\;\;j \in \Omega _{c\;\;\;\;\;\;\;\;\;}

Normalized Euclidean distance can be figured out using Definition 3.2 as follows: (11)D−(N˜,M˜)=13∑i=13[(Nxi−−Myi−)2]D+(N˜,M˜)=13∑i=13[(Nxi+−Myi+)2]\begin{array}{l} D^ - \left( {\tilde N,\tilde M} \right) = \sqrt {\frac{1}{3}\sum\limits_{i = 1}^3 \left[ {\left( {N_{x_i }^ - - M_{y_i }^ - } \right)^2 } \right]} \\ D_{i2}^ + = \sum\limits_{j = 1}^m \sqrt {\frac{1}{3}\left[ {(g'_{ij} - 1)^2 + (h_{ij} - 1)^2 + (l'_{ij} - 1)^2 } \right]} \\ \end{array}\ 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: (12)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]\begin{array}{l} D_{i1}^ + = \sum\limits_{j = 1}^m \sqrt {\frac{1}{3}\left[ {(g_{ij} - 1)^2 + (h_{ij} - 1)^2 + (l_{ij} - 1)^2 } \right]} \\ D_{i2}^ + = \sum\limits_{j = 1}^m \sqrt {\frac{1}{3}\left[ {(g'_{ij} - 1)^2 + (h_{ij} - 1)^2 + (l'_{ij} - 1)^2 } \right]} \\ \end{array}

As the same way, calculate gap of the negative ideal solution by: (13)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]\begin{array}{l} D_{i1}^ - = \sum\limits_{j = 1}^m \sqrt {\frac{1}{3}\left[ {(g_{ij} - 0)^2 + (h_{ij} - 0)^2 + (l_{ij} - 0)^2 } \right]} \\ D_{i2}^ - = \sum\limits_{j = 1}^m \sqrt {\frac{1}{3}\left[ {(g'_{ij} - 0)^2 + (h_{ij} - 0)^2 + (l'_{ij} - 0)^2 } \right]} \\ \end{array}

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: (14)RC1=Di2−Di2++Di2−RC_1 = \frac{{D_{i2}^ - }}{{D_{i2}^ + + D_{i2}^ - }}

The latest worths of RCi*RC_i^* are identified as: (15)Rci*=RC1+RC22Rc_i^* = \frac{{RC_1 + RC_2 }}{2}

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.

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.