Evaluation of Investment Opportunities With Interval-Valued Fuzzy Topsis Method

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 decisionmaking, 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.


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].

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 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 i j ) n×m . The criteria's weight vector is w = (w 1 , w 2 , . . . , w m ) that ∑ m j=1 w j = 1. TOPSIS method includes a process consisting of 6-steps as follows: i Normalize the decision matrix using the following evolution for each r i j .
ii Multiply the columns of the normalized decision matrix by the connected weights. The weighted and normalized decision matrix is come as: Which w j is the weight of the jth criteria.
iii specify the ideal and negative ideal alternatives respectively as follows: . . , m Where J 1 is the set of benefit criterias and J 2 is the set of cost criterias.
iv With using of the two Euclidean distances to calculate the distance of the existing alternatives from ideal and negative ideal alternatives as: The relevant closeness to the ideal alternatives can be defined as: Where 0 ≤ C + i ≤ 1.
vi According to the relative closeness to the ideal alternatives rank the alternatives the bigger C + i is related to better alternative A i [1].

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: That µ L A (x) is the lower limit of degree of membership and µ U A (x) is the upper limit of degree of membership.  and Q x = [Q − x ; Q + x ] due to the [5], we have:

Definition
The Normalized Euclidean distance betweenP andQ is as: A standard MCDM (Multi-Criteria Decision Making) problem can be briefly demonstrated in a decision matrix that x i j 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.
Suppose thatX = [x i j ] n×m be a fuzzy decision matrix for a multi criteria decision making problem where A 1 , A 2 , . . . , A n are n possible alternatives and C 1 ,C 2 , . . . ,C m are m criteria. Sox i j is the performance of alternative A i with notice to criterion C j . Figure 2 representsx i j andw j as triangular interval valued fuzzy numbers [10]. Table 1 Definition of linguistic variables for the ratings Table 2 Definition of linguistic variables for the importance of each criterion The normalized performance of rating as an expan- can be calculated as: Therefore, the normalized matrixR = [r i j ] n×m can be obtained.
Here the suggested technique for building up the TOPSIS to interval-valued fuzzy TOPSIS can be described as follows: i Make the weighted normalized fuzzy decree matrix with notice that each criterias has own importance as: ii Defined the optimal and negative optimal solution as: iii Normalized Euclidean distance can be figured out using Definition 3.2 as follows: 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: As the same way, calculate gap of the negative ideal solution by: Eqs. (12) and (13) are used to specify the distance from ideal and negative ideal alternatives in interval values.
iv The involved sepreation can be calculated by: The latest worths of RC * i are identified as: 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.  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)

Solution Steps
i 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. ii In this step, decision matrix is normalized by the equation (8) and the results are expressed in Tables 8. iii As stated earlier, the weight of each criteria was previously determined by the decision makers (Shannon entropy) as given in Tables 9.   (8.5,9.5);10;(10,10)] VG iv 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.
v By using Eq. (11,12,13) the distance of each alternative is calculated from the ideal alternative D + i1 , D + i2 , given in Table 11.
vi 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).
vii In the last step alternatives are listed in Table (12) according to their relative closeness.
Now calculate c + j and a − j as followes: Now with using of:r

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.       T h i s p a g e i s i n t e n t i o n a l l y l e f t b l a n k