A Decision Making Method By Combining Topsis and Grey Relation Method Under Fuzzy Soft Sets

Abstract

Keywords: Soft sets; fuzzy sets; fuzzy soft sets; TOPSIS; multi-criteria decision making; grey relational analysis; drug selection.

1. Introduction

There are a lot of uncertainties in many areas such as business, service, management, military, economy, engineering, etc. To deal with this uncertainty, there some mathematical models. One of them is fuzzy set theory is suggested by Zadeh [34] in 1965. The other one is soft set theory is proposed by Molodtsov [24] in 1999. The Molodtsov’s theory was a new approach different from the Zadeh’s theory for modeling uncertainty. Some operations of soft sets and their properties were defined by Maji et al. [22]. Then some modifications of operations of soft sets and their properties were given by some researchers such as Ali et al. [1], Çağman and Enginoğlu [7], Zhu and Wen [36], Çağman [8]. In 2001, Maji et al. [21] defined the concept of fuzzy soft set and fuzzy soft set operations that are a generalization of Molodtsov’s soft set. Then Roy and Maji [29] applied the fuzzy soft sets to a decision making problem. Majumdar and Samanta [23] defined the generalized fuzzy soft sets and investigated their properties. Zhou et al. [35] suggested the generalized interval valued fuzzy soft sets and investigated their properties.

Corresponding Author/Sorumlu Yazar: e-mail/e-ileti: seraslan4@yahoo.com, tel: (318) 357 42 42 53

S. Eraslan, N. Çağman / Sigma J Eng & Nat Sci 8 (1), 53-64, 2017

TOPSIS being one of classical MADM methods such as ELECTRE [28], VIKOR [27], PROMETHEE [2], developed by Hwang and Yoon [17]. Chen et al. [4] extended the TOPSIS method for solving Multi Criteria Decision Making (MCDM) problems in fuzzy environment. Boran et al. [3] developed TOPSIS method for MCDM problems based on intuitionistic fuzzy sets. Chi and Liu [6] extended TOPSIS to interval neutrosophic sets, and with respect to the MADM problems in which the attribute weights were unknown and the attribute values take the form of interval neutrosophic sets. Eraslan [13] gave a decision making method by using TOPSIS on soft set theory. The Grey theory, proposed by Deng [9] in 1982, similar to fuzzy set theory, is an effective mathematical means to deal with systems analysis characterized by incomplete information. Grey theory is widely applied in fields such as systems analysis, data processing, modeling and prediction, as well as control and decision-making [10], [11], [12], [31]. The GRA is part of grey system theory, which is suitable for solving problems with complicated interrelationships between multiple factors and variables [25]. The GRA has been successfully applied in solving a variety of MADM problems, such as the hiring decision [26], the restoration planning for power distribution systems [5], the inspection of integrated-circuit marking process [18], the modeling of quality function deployment [37], the detection of silicon wafer slicing defects [20], etc. The GRA has been widely used to solve the uncertainty problems under the discrete data and incomplete information [11], [12]. In addition, the GRA method is one of the very popular methods to analyze various relationships among the discrete data sets and make decisions in multiple attribute situations. The major advantages of the GRA method are that the results are based on the original data, the calculations are simple and straightforward and it is one of the best methods to make decisions under business environment [16]. The GRA and TOPSIS [17], [19], [33], both use the idea of minimizing a distance function. Feng and Wang [15], applied grey relation analysis to select representative criteria among a large set of available choices, and then used TOPSIS for outranking. The aim of this paper is to extended the concept of the grey relation based on the concepts of TOPSIS to solve the multiple criteria decision making problem for selection of drugs problem and it appears to be more appropriate. The rest of the paper is organized as follows: in the Section 2, we present briefly some definitions and properties required in the other sections of study. In Section 3, we give an group decision making method combining TOPSIS and GRA method under fuzzy soft environment. In Section 4, we present an application of suggested method to a real life problem containing drug selection. In Section 5, conclusions are presented. This work is presented in The 4th International Fuzzy Systems Symposium (FUZZYSS15), 5-6 November 2015.

2. Preliminary

In this section, we summarize the preliminary TOPSIS method and definitions which are fuzzy set [34], soft set [24], [7], fuzzy soft set and their results. Definition II.1. [34] Let U be an initial universe set. A fuzzy set X over U is a set defined by a function  X representing a mapping

 X : U  0, 1 Here,  X called membership function of X, and the value  X (u ) is called the grade of

membership of u  U. The value represents the degree of u belonging to the fuzzy set X. Thus, a fuzzy set X over U can be represented as follows,

Note that the set of all the fuzzy sets over U will be denoted by F(U) . 54

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Definition II.2. [24] Let U and X be two non empty set and P(U) is the power set of U. Then, a soft set f over U is a function defined by f : X  P(U) where U refer to an initial universe and X is a set of parameters. In other words, the soft set is a parametrized family of subsets of the set U. A soft set over U can be represented by the set of ordered pairs f = {(x; f(x)) : x  X} U

Note that the set of all soft sets defined from X to P(U) will be denoted by S X . Definition II.3. [7] Let f , g  S X . Then, U

f is called an empty soft set, denoted by X, if f(x) = , for all x  X.

f is called a universal soft set, denoted by f X~ , if f ( x)  U , for all x  X .

The set Im(f) = {f(x) : x  X} is called image of . ~ f is a soft subset of g, denoted by f  g , if f ( x )  g ( x ) , for all x  X .

f and g are soft equal, denoted by f = g, if and only if f(x) = g(x) for all x  X.

the set ( f  g )( x)  f(x)  g(x) for all x  X is called union of f and g.

~ ~ the set ( f  g )( x)  f(x)  g(x) for all x  X is called intersection of f and g.

 the set fc(x) = U \ f(x) for all x  X is called complement of f. Definition II.4. [21] Let U be an initial universe set, X be a set of all parameters, µ be a fuzzy set over U for every x  X and F(U) denote the set of all fuzzy sets in U. Then, a fuzzy soft set  over U is defined by a function  representing a mapping  : X  F(U) such that  (x) = θ if x  X Here, for every x  X,  (x) is a fuzzy set over U and it is called fuzzy value set of parameter xelement of the fs-set. Thus, an fs-set  over U can be represented by the set of ordered pairs  = {(x;  (x)) : x  X;  (x)  F(U)} Note that from now on the sets of all fs-sets over U will be denoted by FS(U). TOPSIS Method The operations within the TOPSIS process include: decision matrix normalization, distance measures, and aggregation operators [30]. For more detail of TOPSIS, we refer to the earlier studies [17], [32], [13]. The TOPSIS process is carried out as Step 1. Constructing of decision matrix D. Step 2. Creating of standard (normalized) decision matrix R. Step 3. Creating the weighted normalized decision matrix V . Step 4. Determining of positive ideal solution (PIS), A+ and negative ideal solution (NIS), A. 

Step 5. Calculating of separation measurements of positive ideal ( S i ) and the negative 

Step 6. Calculating of relative closeness (C i ) of alternatives to the ideal solution. Step 7. Ranking the preference order.

3. Combining Topsis And GREY Relation Method

In this section, we combine TOPSIS and grey relation method to determine separation measurements for each parameter. Therefore, we calculate the separation measurements in step 7 by using formula in [31] defined by 55

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here, r(A (j);Ai(j)) and r(A (j);Ai(j)) are the grey relation coefficients of each alternative to ideal and negative ideal solutions, respectively. For simplicity of representation, A+(j)  Ai(j) and 

A (j)  Ai (j) will be presented  i and  i , respectively.

After calculating r(A+(j), Ai(j)) and r(A (j), Ai(j)), separation measurements will be calculated according to the following Formula

(4) Now we will give the operations of proposed method. The main procedure of this method is presented with the following steps: Step 1. Defining of problem. Step 2. Constructing of weighed fuzzy parameter matrix D with choosing linguistic rating from Table 1. Table 1. Linguistic Terms for Evaluation of Parameters.

Step 3. Constructing of weighted normalized fuzzy parameter matrix R and forming weighed vector W = (W1, W2, …, Wn). Step 4. Constructing fuzzy decision matrices Dk for each decision makers and building of fuzzy average decision matrix V . Step 5. Constructing of weighed fuzzy decision matrix V. Step 6. Finding of fuzzy valued positive ideal solution (FVPIS)and fuzzy valued negative ideal solution (FVNIS). 

Step 7. Calculating of the separation measurement ( S i , S i ) for each parameter according to formula 3, 4.

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Step 8. Calculating of the relative closeness (C i ) of alternative to the ideal solution. Step 9. Ranking the preference order.

4. Application

In this section, we have presented an application by using the algorithm of this new group decision making method. We can solve the following problem step by step as follows: Step 1. Defining the problem. Assume that a researcher group wants to decide a drug which is best for cancer. There are eight drugs which are the set of alternatives, U = {u1, u2, u3, u4,u5, u6, u7, u8}. The researchers take into consideration a set of parameters, X = {x1, x2, x3, x4, x5, x6, x7}. The parameters xi (i = 1, 2, 3, 4, 5, 6, 7) stand for “anti-inflammatory”, “degreaser”, “painkiller”, “increasing kidney load”, “increasing burden of liver”, “negatively affect pregnancy” and “weaken the heart muscle”, respectively. Then we can give the following examples. Suppose that four researchers (decision-makers) to chose the best drug. Firstly, each researcher has to consider their own set of parameters. Then, they can construct their fuzzy soft sets. Next, by using TOPSIS on fuzzy soft set theory decision making method we select a candidate on the basis for the sets of decision makers parameters. Finally, they applies the following steps: Assume that decision makers D1, D2, D3 and D4 construct fuzzy soft sets, respectively as follows,

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Step 3. Weighted normalized fuzzy parameter matrix and weighed vector W are obtained as follow,

Step 4. Fuzzy decision matrices can be constructed by decision makers as follows,

A Decision Making Method by Combining Topsis and … / Sigma J Eng & Nat Sci 8 (1), 53-64, 2017

S. Eraslan, N. Çağman / Sigma J Eng & Nat Sci 8 (1), 53-64, 2017

Step 5. Weighed fuzzy decision matrix V is constructed as below,

Step 6. Fuzzy valued positive ideal solution (FV-PIS) and fuzzy valued negative ideal solution (FV-NIS) can be obtained as below, + = {+ (1) = 0.12, + (2) = 0.10, + (3) = 0.12, + (4) = 0.04, + (5) = 0.03, + (6) = 0.03, + (7) = 0.04 }  = { (1) = 0.07,  (2) = 0.04,  (3) = 0.08,  (4) = 0.07,  (5) = 0.07,  (6) = 0.06,  (7) = 0.06 } Step 7. Grey relation coefficient of each alternative to the ideal r(A+(j);Ai(j)) and the negative 

ideal r(A (j), Ai(j)) solution can be obtained as below. We will briefly denote p = min  i , q = 

min  i , P = max  i , Q = max  i , p* = min{min  i }, q* = min{min  i }, P* = max{max  i } 

From the above chart, all grey relational coefficients can be calculated by Eq. (1). In the example,  = 0.5. For ideal solution, the entire results for the grey relational coefficient is shown in matrix as

A Decision Making Method by Combining Topsis and … / Sigma J Eng & Nat Sci 8 (1), 53-64, 2017

Similarly, for negative ideal solution, we have the entire results for the grey coefficient is shown in matrix as

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( j ), Ai ( j )) and r ( A  ( j ), Ai ( j )) , the connection between ith 

alternative and reference number sequence is calculated according to formula 3, 4. Then S i and

Step 8. Relative closeness of alternatives to the ideal solution is calculated as follows,

Step 9. Ranking the preference order is u 5  u 2  u 4  u 7  u 6  u 8  u 3  u1 .

5. Conclusion

In this paper, we have presented an application using new group decision making method [14]. Then, we combine the GRA based on the concepts of TOPSIS to evaluate and select the best alternative. The degree of grey relation between every alternative and FV-PIS, FV-NIS is calculated. An illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness. As can be seen from the examples given above, u1, u3 drugs seem to be more useful for cancer through eight different drugs. However, considering the side effects of the drug u1, u3 give the greatest harm to the patient. The drug u5 provide less benefit and more damage in terms of benefits, but u1, u3 drug is less damage and more benefit to the body. This method can be successfully worked. It can be applied to decision making problems of many fields that contain uncertainty. Finally, the approach should be more comprehensive in the future to solve the related problems and a large number of examples could be recommended for test in future studies.

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