Choose Best Criteria for Decision Making Via Fuzzy Topsis Method

: Multi Criteria Decision Making (MCDM) uses different techniques to find a best alternative from multi-alternative and multi-criteria conditions. TOPSIS is an important practical technique for ranking and selection of different alternatives by using distance measures. Classical TOPSIS uses crisp techniques for the linguistic assessments, but due to imprecise and fuzziness nature of the linguistic assessments, we faced some problems to find out the solution of these problems. We proposed a Fuzzy TOPSIS with its example in this work and use this method for decision making.


Introduction
Hwang and Yoon invented the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) in order to solve MCDM problem with many alternatives [1]. From crisp to fuzzy data, Chen & Hwang remodelled TOPSIS [2]. Furthermore, Chen widened the TOPSIS for Group Decision Making in fuzzy atmosphere [3]. Awasthi et al. used fuzzy TOPSIS for the evaluation and selection of the best location planning for urban distribution centres [4]. Chu [5] and Yong [6] applied fuzzy TOPSIS for choosing plant location with minimum costs and maximum use of resources. The relative closeness coefficients were obtained as fuzzy numbers and after defuzzification, alternatives were ranked [7]. Wang and Elhang found that their method is much closer to the fuzzy weighted approach presented by Dong and Wong [8]. Mahmoodzadeh et al. incorporating fuzzy AHP and TOPSIS method gave a new procedure for the project selection problem. Improved fuzzy AHP was used to compute the weights of each criterion at first and then TOPSIS algorithm was engaged for ranking the projects to be selected [9].
Yong Tsao et al. utilized fuzzy TOPSIS for supporting contractors to choose suitable project for bidding with the help of MAGDM. Triangular fuzzy numbers (TFNs) were assigned to each linguistic variable for alternative ratings and criteria weights [10]. Zadeh proposed Type-2 FS to encompass uncertainty about the membership function in fuzzy set theory (FST) [11]. Saremi and Montazer, to cope with the decision-making problems having data with large uncertainty, used a Type-2 fuzzy TOPSIS based method. A Study of TOPSIS in Classical, Fuzzy, Intuitionistic Fuzzy and Neutrosophic Environments Interval-valued fuzzy (IVF) set is a special type of Type-2 FS. Saremi and Montazer applied IVF sets having lower and upper triangular membership functions; they ranked the alternatives [12]. Chen and Tsao broadened IVF-TOPSIS to solve MADM problem [13].
Zulqarnain. M. and Saeed. M. proposed and proved the credibility of interval valued fuzzy soft matrix (IVFSM) in decision making. They discussed its different properties [14]. They also studied fuzzy soft matrix (FSM) and IVFSM and redefined the product of IVFSM. Later they used IVFSM and FSM in decision making problem with examples and compare the results. They observed that FSM method is more appropriate for decision making [15]. They proposed a new decision making method on IVFSM named as "interval valued fuzzy soft max-min decision making method" with the help of interval valued fuzzy soft max-min decision making function and used this method for decision making [16].

Prelimnires
In this section, we discussed about FS and fuzzy TOPSIS which is proposed by (Zadeh 1965) with examples and some important propositions.

Definition 1 [22]
If A be a set and U be a universal set. Set A contains different elements of set U then we assign all elements of A any membership value between the interval [0, 1] is called FS.

Definition 2 [18]
A fuzzy number is a convex and normal fuzzy subset of the universe of discourse X.

Definition 3 [22]
If A and B are two FS than A is called fuzzy subset of B if µ A (t) ≤ µ B (t) for all t ∈ U. It can be represented as A ≤ B.

Definition 3 [22]
A FS in which µ A (t) = 0 for all t ∈ U is called empty FS.

Definition 4 [22]
A FS in which µ A (t) = 1 for all t ∈ U is called universal FS.

Definition 5 [22]
If A and B are two FS than they are called equal FS if µ A (t) = µ B (t) for all t ∈ U. It can be represented as A = B.

Definition 6 [22]
Let A be a FS than its complement is defined as f A c (t) = 1-f A (t) for all t ∈ U.

Definition 7 [22]
If A and B are two FS whose membership functions are µ A (t) and µ B (t) respectively, than their union is also a FS, i.e. C = A U B which is defined as f C (t) = max (µ A (t), µ B (t)) for all t ∈ U.

Definition 8 [22]
If A and B are two FS whose membership functions are µ A (t) and µ B (t) respectively, than their intersection is also a FS, i.e. C = A ∩ B which is defined as

Fuzzy Topsis Algorithm
In order to deal with data and information containing nonstatistical uncertainties, the best tool is the theory of FS which was proposed by Zadeh. It is a matter of fact that we are not always in a position to express our viewpoint exactly. Many opinions are only unclear and uncertain. To model such scenarios more accurately, in 1965 Zadeh proposed a new theory, named as FST [19]. As far as fuzzy TOPSIS is concerned, Chen extended TOPSIS to Fuzzy-TOPSIS using TFNs to replace the numeric linguistic scales for rating the alternatives and to assign the weights to the criteria [20].
One of the major differences between Fuzzy-TOPSIS and Classical-TOPSIS is that the later utilizes precisely known ratings and weights of the criteria [21]. Fuzzy-TOPSIS is an application of Fuzzy logic and FS. The Algorithm of Fuzzy TOPSIS is given below.
Step 1: Selection of a fuzzy rating scale for linguistic variables, since alternatives as well as criteria both are in the form of linguistic variables.
Step 2: Criteria weightage and fuzzy linguistic ratings for the alternatives given by DM's Taking as fuzzy ratings by k th decision maker for the i th alternative w. r. t the j th criterion, represented by the following equation.

= ( , , )
And is the weight allocated by the k th decision maker for the j th criterion, represented by the equation given below.
Step 3: Compute the aggregated fuzzy ratings for the alternatives, taking as aggregated fuzzy ratings for the i th alternative w. r. t the j th criterion, and w j is the aggregated fuzzy weights for the j th criterion.

Step 4: Construction of AFDM and Aggregated Fuzzy Weight Matrix (AFWM).
Now the Fuzzy MCDM problem will be converted to an AFDM as given below where is aggregated fuzzy rating for the i th alternative w. r. t the j th criterion. Moreover, AFWM is defined as where ́ , is the aggregated fuzzy weight of j th criterion.
Step 5: Normalization of the FDM: The NFDM is given below The Linear Scale Transformation (LST) is used to normalize the Decision Matrix. The important point of this normalization method is that the normalized TFN are within the interval [0, 1].
Step 6: WNFDM The WNFDM 6 ( = is given below Step 7 Step 8: Calculation of distances E * and E 4 : The distances E * and E 4 of each weighted alternative 7́ ; 1, 2,..., m from FPIS and FNIS are calculated by using the distance formulas. where E is the distance between two fuzzy numbers.
Step 9: Determination of Closeness Coefficient CC i : In order to rank the alternatives closeness coefficient is computed by the following equation. Step 10: Ranking the alternatives: The alternatives are ranked according to the fact that "an alternative with CCi closer to 1 indicates that the alternative is close to the FPIS and distant form FNIS. A large value of closeness index indicates a good performance of the alternative [24].

Problem Scenario
A company Alpha has to choose a best one from two alternatives A 1 and A 2 . For the selection of best alternative, the company hire a team of decision makers which consist on three members, D = {DM 1 , DM 2 , DM 3 }. Four evaluation criteria's are mentioned for the selection of alternative which are given below.
Four Evaluation Criteria (n=4) represented by

Solution by Fuzzy Topsis
Step

1: Selection of a Fuzzy Ratings Scale for Linguistic Variables
Rating scale for linguistic variables is given below in table 1. Very Good (VG) (7,9,9) Step 2: Criteria weightage and fuzzy linguistic ratings for the alternatives given by DM's In order to integrate the opinions of all the DMs, each DM allocates some weights to the criteria, given in the table 2.

Conclusion
Classical TOPSIS uses crisp techniques for the linguistic assessments, but due to imprecise and fuzziness nature of the linguistic assessments, we faced some problems to find out the solution of these problems we proposed the Fuzzy TOPSIS with its example. It is evident that fuzzy TOPSIS has the ability to deal situations where ambiguity occurs due to the presence of linguistic variables, whereas Classical TOPSIS is lacking this property.