SIMILARITY BASED FUZZY INFERIOR RATIO FOR SOLVING MULTICRITERIA DECISION MAKING PROBLEMS

The method of Fuzzy Inferior Ratio (FIR) has been recognized as one of advantageous methods in multi criteria decision-making under fuzzy environment as it considers the element of compromise solution between the positive and negative aspect of the evaluation simultaneously. It is considered as an improvised version of Fuzzy TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) method for solving decision-making problems. However, the FIR utilizes the distance approach in the evaluation of obtaining the compromise solution. A defuzzification process is carried out to transform the fuzzy values into a crisp form. Hence, loss of information may occur in the computation. In this paper, we proposed a similarity-based FIR that overcomes the above-mentioned problem. A new compromise solution for the proposed FIR is developed and an improvised procedure of FIR is suggested using the similarity measure approach. A comparative analysis between the distance based and the similarity-based FIR is carried out using a case study of preferred client selection for a loan application. The proposed method is found to be effective in solving decision-making problems as the utilization of similarity measure will sufficiently preserve the data information in the computational process of evaluation.


Introduction
Decision making is an essential process of obtaining a desired result based on a set of criteria under consideration from a set of alternatives. There are many techniques and approaches that have been introduced to provide a systematic way in finding the solutions. A multi criteria decision making method (MCDM) is one of the techniques that can be employed when conflicting benefits and cost criteria are included. According to Mulliner et al. (2016), the objective of MCDM is to deliver a ranking, classification, sorting, description, and choices of alternatives. The MCDM approaches have been successfully implemented in various applications such as in operational research, engineering system, management science and decision theory (Kumar et al., 2017). For instance, some specific applications such as in selection problem of a new hub airport (Janic & Reggiani, 2002), supplier selection (Nourianfar & Montazer, 2013), water resources planning (Opricovic, 2011), transportation planning (Ramani et al., 2010) and network selection process in heterogeneous wireless (Obayiuwana & Falowo, 2015).
Since the introduction of fuzzy set theory by Zadeh (1965), the emergence of decisionmaking methods under fuzzy environment is timely, in order to cater for subjective and vague evaluation process. Instead of using crisp values to determine criteria weight and the rating of alternatives, the evaluation is now in the form of linguistic which is closer to human perception and thought. The first discussion on the decision making under fuzzy environment was discussed by Bellman & Zadeh (1970) and later, many MCDM methods have incorporated fuzziness in their techniques such as Fuzzy TOPSIS (Chen, 2000), Fuzzy VIKOR (Opricovic, 2011) and Fuzzy AHP (Bozbura et al., 2007). Fuzzy TOPSIS is a well-known method that utilizes the position of positive and negative solutions in the computation. Thus, the result takes into consideration the compromise solution between the two. Nevertheless, in obtaining the rating index, they only consider the negative solution in calculating the final index of preference. To overcome this problem, Hadi-Vencheh & Mirjaberi (2014) proposed the Fuzzy Inferior Ratio (FIR) method as an improvement to the fuzzy TOPSIS method which now considers the positive and negative solution simultaneously in the calculation of the rating index. Since the best solution does not always indicate the most remote from the negative solution, the balance between the shortest distance to the positive ideal solution and the farthest distance from the negative ideal solution is desired. The method employs the distance approach in calculating difference. This has become a major setback since vital information may loss due to the simplification process of fuzzy values into crisp values.
Similarity measure (SM) between two fuzzy values is an important tool to compare similarity from various perspectives of fuzzy values such as distance, shape, size, geometrical properties, etc. Chen (1996) was the first to propose an SM based on the distance between fuzzy numbers for trapezoidal and triangular fuzzy numbers, in order to aggregate the decision makers' opinions. Lee (1999) applied SM to deal with fuzzy opinions using the distance between fuzzy numbers based on metric for group decision making and the difference between minimum and maximum universe of discourse. Chen & Chen (2003) proposed a new SM to calculate the degree between two generalized trapezoidal fuzzy numbers based on the centre of gravity (COG) points meanwhile, Yong et al. (2004) introduced a new SM based on radius of gyration to cater the drawback of the method of Chen & Chen (2003). Some variants of new SM have been proposed recently such as by Ahmad et al. (2018), Mohamad et al. (2019) and Wu et al. (2020). Some applications of SM in solving real problems are discussed. Hejazi et al. (2011) and Wei & Chen (2009) used SM in determining risk analysis in manufacturing. Zuo et al. (2013) implemented similarity measure-based method in diagnosing rotor fault and Niyigena et al. (2012) offered a similarity based procedure to solve supplier selection problem.
In this paper, a similarity-based FIR is proposed for solving fuzzy decision making problem. The SM used is adapted from Ahmad et al. (2018) that includes geometric distance, center of gravity, the Hausdorff distance, and Dice similarity index. Each component in the SM has its role to capture some specific features of the fuzzy numbers.

Preliminaries
A fuzzy set A in a universal set U is defined as an ordered pair: is known as the membership degree to which x belongs to A. As opposed to crisp set where the characteristic function only gives the value 1 if the element is in the set and 0 when is not, the membership function of fuzzy set offers all values between 0 and 1 inclusively. A fuzzy number is a fuzzy set that is normal and convex; where normal refers to a fuzzy set with height 1 and convex is when the fuzzy set satisfies the membership inequality: (2) Ahmad et. al.,Malaysian Journal of Computing,5 (  ( . When w = 1, the generalized trapezoidal fuzzy number is known as the standardized trapezoidal fuzzy number. In addition, when a2 = a3, it becomes a triangular fuzzy number and it is a singleton when a1 = a2 = a3 = a4. Let ( ) be two trapezoidal fuzzy numbers. Then: i) Addition:

Similarity Measure (SM)
There are several SMs that have been proposed by researchers to cater some specific problems and issues under fuzzy environment. In this paper, the SM proposed by Ahmad et al. (2018) is be two trapezoidal fuzzy numbers. The SM between Ã and B is defined as: where Ã x and Ã ŷ are the horizontal center of gravity (COG) of Ã and B calculated as: The above SM consists of four components, which are geometric distance, center of gravity, Hausdorff distance, and Dice similarity index. Using the Hausdorff distance, the geometry information in evaluating the mismatch between two generalized fuzzy numbers can be obtained by considering the geometry shape of membership function of the fuzzy numbers. This SM has the advantage of discriminating two similar shape fuzzy numbers with two different locations effectively (Ahmad et al., 2018). The similarity measure ( ) The proof of the properties is given in Ahmad et al. (2018) and the SM has been validated by using some benchmark sets of fuzzy numbers. It was found that the SM to be comparable with other SM with some additional advantages for some specific cases of comparison.

Fuzzy Inferior Ratio
The method of the Fuzzy Inferior Ratio was introduced by Hadi-Vencheh & Mirjaberi (2014) to overcome weaknesses of some compensatory methods of decision making where most consider only one of the ideal solutions in obtaining the rating index. It was highlighted that the single remotest value does not guarantee the alternative to be the desired one. Hence, a compromise solution was suggested between the positive ideal solution and the negative ideal solution simultaneously.
be set of criteria under consideration in the evaluation. The decision matrix of evaluation is defined as is a trapezoidal fuzzy number. The pseudo steps of the FIR as given in Hadi-Vencheh & Mirjaberi (2014) are given as follows: • Evaluate each alternative in A with respect to each criterion in C to obtain the decision matrix D.
• Normalize the decision matrix D to eliminate the dimension of the attribute.
• Determine the positive ideal solution (PIS) and the negative ideal solution (NIS) • The difference between the alternative and the PIS/NIS is measured.
• The compromise solution is calculated to measure the extent to which each alternative is closest to PIS and is far away from NIS, simultaneously.
• The inferior ratio for both attributes of the shortest distance from PIS and farthest away from NIS is obtained.

Decision Making Procedure Using Similarity Based Fuzzy Inferior Ratio.
A decision making procedure using the FIR with the SM given earlier is proposed using the following steps. Some modification of the original FIR is necessary to accommodate the integration of the SM, in particular in evaluating the inferior ratio.
Step 1: Form a committee of K decision makers, k D to evaluate the importance of n criteria Cj and the rating of m alternatives Ai. The linguistic terms and the corresponding trapezoidal fuzzy numbers used in the evaluation of criteria weight and ratings of alternatives is shown in Table  1 and Table 2 respectively.  Step 2: Determine the aggregated fuzzy weight of the criteria, Step 3: Find the aggregated fuzzy rating Step 4: Construct the fuzzy decision matrix, (1, 3, 3, 5) Fair (F) (3, 5, 5, 7) Medium Good (MG) (5, 7, 7, 9) Good (G) (7, 9, 9, 10) Very Good (VG) (9, 10, 10, 10)  with B and C representing benefit and cost criteria, respectively.
Step 5 . Then, construct the weighted transformed normalized fuzzy decision matrix Ṽ by multiplying the weight, j w of evaluation criteria with the transformed normalized fuzzy decision matrix ij r as: (10) Step 6: Determine Fuzzy Positive Ideal Solution (FPIS), A + and Fuzzy Negative Ideal Solution Step 7 (13) Step 8: Calculate the similarity-based compromise solution for each alternative. The new compromise solution based on similarity measure, ( ) i A  is given as: where: The compromise solution, ( )  Ahmad et. al., Malaysian Journal of Computing, 5 (2): 597-608, 2020 603 Proof. Since ( ) ( ) This gives Equation (14) ( ) ( ) ( ) Step 9: Calculate the inferior ratio IR(Ai) for each alternative where: The ranking order of the alternatives is determined where the least value of ( ) i A IR will be placed in first ranking position.

Implementation
As an illustration, a decision making problem of a financial institution in evaluating some loan applications is solved using the proposed procedure. A committee of three experts, DM1, DM2 and DM3 evaluate six applicants (A1, A2, A3, A4, A5 and A6) based on four benefit criteria, which are credit history (C1), capacity (C2), capital (C3) and collateral (C4). The evaluations are made by the decision makers in determining the weight of criteria and the rating of the six applicants and are shown in Table 3 and Table 4 respectively.  The evaluations in Table 4 are converted into fuzzy numbers, which then are aggregated into a fuzzy decision matrix using Equation (7) as shown in Table 5. The elements in Table 5 are normalized using Equation (8) as in Table 6.

Conclusion
The emergence of new methods in solving decision making problem systematically has helped many sectors to improve their productivity. Nevertheless, the distance approach is usually employed in obtaining rating of alternative which may dissipate some of the information inside the fuzzy values. In addition, some methods such as fuzzy TOPSIS and fuzzy VIKOR use only one remotest value to obtain the preference index of alternatives which does not guarantee the ultimate result. The FIR method was suggested by Hadi-Vencheh & Mirjaberi (2014) based on compromise solution used the both positive and negative ideal solution simultaneously. In this paper, the method of FIR is further improved by integrating the similarity measure in the decision making procedure. This is to minimize the loss of information due to simplification. A decision making procedure was developed by introducing a new similarity based compromise solution and inferior ratio. An application was illustrated to validate its effectiveness. The similarity based approach has the potential to be extended to other fuzzy decision making methods that utilise distance method in the computation.