Extended FTOPSIS with Distance and Set Theoretic-Based Similarity Measure

ABSTRACT


INTRODUCTION
Decision making (DM) in fuzzy environment requires extensive use of fuzzy numbers. Rating of alternatives and criteria weights determination are commonly expressed in terms of fuzzy linguistic values defined mathematically by fuzzy numbers. Due to this representation, comparison of fuzzy numbers become a crucial element in solving a DM problem. Similarity measure is a very useful means for the purpose of comparing fuzzy numbers as it has the advantage of minimizing the loss of information in the computational process [1]. Early works on fuzzy similarity measures can be found in [2]and [3]. Various fuzzy similarity measures which take into consideration factors like distance, center of gravity, spread, Jaccard index, Dice similarity index, geometric mean and geometric shape characteristics like height, area and parameter have been introduced in the literature [4][5][6][7][8][9][10]. Recently [11] introduced a generalized similarity measure that can measure most types of fuzzy numbers, meanwhile [12] proposed a similarity with multiple features to overcome shortcomings of some existing similarity measures.
Similarity measure has been applied to solve problems in various fields. Similarity measure has been incorporated in the development of a fuzzy knowledge based system [13]. Ref [14] introduced and utilized similarity measure in matching fingerprint image. Risk analysis problems have been solved by

PRELIMINARIES
In this section, a new decision making (DM) procedure is proposed whereby the similarity measure by [10] is integrated into the extended FTOPSIS procedure [1]. Some definitions related to the generalized trapezoidal fuzzy numbers (GTFNs) are given as follows.

Definition 1 [23]
A generalized trapezoidal fuzzy number (GTFN)  Figure 1 shows a graphical representation of a GTFN.

Definition 3[10]
Given a continuous universe U = [0,1] and a set of generalized fuzzy numbers over U, FS(U). Let be two generalized trapezoidal fuzzy numbers in The similarity measure between Ã and B is defined as The above similarity measure embeds four elements in the formula which are the geometric distance, the center of gravity, Hausdorff distance, and Dice similarity index that are important and favorable in similarity measurement. The measure has the advantage of discriminating two similar shape fuzzy numbers effectively with two different locations [10].

EXTENDED FTOPSIS USING DISTANCE AND SET THEORETIC-BASED SIMILARITY MEASURE
An extended FTOPSIS procedure incorporating a similarity measure by [10] particularly in calculating the closeness coefficients of the decision alternatives is presented as follows. Step 1: Set up a committee of K decision makers to determine the importance weights of n criteria and to rate m alternatives based on the criteria. Linguistic terms and the corresponding trapezoidal fuzzy numbers used for these purposes are as shown in  (8,9,10,10) Step 2: Obtain the aggregated criteria weight Step 3: Obtain the aggregated fuzzy ratings, Step 4: Construct a fuzzy decision matrix, Step Step 7: Using Definition 3, calculate the similarity values Step 8: Calculate two types of closeness coefficients for the i-th alternative, Step 9: Determine the ranking position of the i-th alternatives according to the

RESULTS AND ANALYSIS
In this section, the performance of the similarity measure by [10] in the context of decision making (DM) is investigated by implementing the proposed DM procedure in solving a supplier selection problem adopted from [1] in which the data are presented in Table 2 and Table 3. The weights of criteria and rating of alternatives by the decision makers are presented in Table 3 and 4, respectively.  D   C1  C2  C3  C4  C5  D1  D2  D3  D1  D2  D3  D1  D2  D3  D1  D2  D3  D1  D2  D3  A1  MG  MG  MG  MG  MG  VG  G  G  G  G  G  G  G  G  G  A2  G  G  G  VG  VG  VG  VG  VG  VG  G  VG  VG  VG  VG  VG  A3  VG  VG  G  VG  G  G  VG  VG  G  VG  VG  VG  G Table 5 shows the weighted normalized fuzzy decision matrix obtained by performing Step 3 to Step 5 in the procedure presented in Section 3. The  Table 6 and Table 7, respectively. Ranking orders of alternatives obtained based the calculated closeness coefficients with respect to the two types of FPIS and FNIS are also compared with the ranking orders by [1]. A . The ranking orders of alternatives are found to be consistent with [1].

CONCLUSION
In this paper, we have proposed an Extended FTOPSIS procedure incorporating the similarity measure proposed by [10]. The advantage of the similarity measure [10] which are composed of the geometric distance, the center of gravity, Hausdorf distance and the Dice similarity index lies in its ability in discriminating two similar shape fuzzy numbers effectively with two different locations. The implementation of the Extended FTOPSISin which the closeness coefficients of the decision alternatives are calculated using the similarity measure given by [10] is elucidated with an application in solving a supplier selection problem. Two different rules for determining the positive ideal solution and the negative ideal solution [1] are employed. The result shows that ranking orders of alternatives with respect to any type of ideal solutions being used as benchmarking fuzzy numbers in calculating the similarity degrees are consistent with one another. Comparison with [1] also indicates that ranking of alternatives are preserved as both approaches give similar orderings of alternatives. For future work, different similarity measures may be considered in determining the closeness coefficients in the Extended FTOPSIS and comparative analysis can be made.