Fuzzy Similar Priority Method for Mixed Attributes

Fuzzy similar priority ratio method is to select the most suitable one to the specific object from feasible alternatives. However, this method considering only the index values of real number has its disadvantages of inaccuracy in result and complexity in calculation. So, this method was extended to handle mixed attributes including real number, interval number, triangular fuzzy number, and trapezoidal fuzzy number. The proposed method decides the optimal alternative by the minimum of integrated nearness degrees calculated by all index vectors and the fixed index vectors based on the theory of similarity. The improved method can not only address mixed attributes but also simplify the calculation and improve the accuracy of result. A case study illustrated this method.


Introduction
Knowing some index values of the specific object, the most suitable one can be determined from some feasible alternatives using the method of fuzzy similar priority ratio, which is of fuzzy mathematic method [1].Being of simple principle and clear meaning, this method has been widely used in many fields.However, the classical method only concerns the indices of real numbers.Furthermore, it has its disadvantages of inaccuracy in result and complexity in calculation.In decision making, the indices of an alternative may be characterized as interval number, triangular fuzzy number, and trapezoid fuzzy number.But the traditional method fails to deal with the mixed types of numbers.Article [2] presented a novel method in grey target decision making for handling mixed attributes.The method in [2] introduced the binary connection number provided in set pair analysis (SPA) [3,4] to transform all types of indices into connection number vectors regarded as in the determinacy-uncertainty space [5,6].Then all the nearness degrees of index vectors and the given index vectors could be calculated based on the theory of similarity.And the decision making was decided by the nearness degrees.The novel method addressed some alternatives for mixed attribute values, but its accuracy in solving the uncertain terms of index vectors needs improving.
Thus, an improved fuzzy similarity priority method is proposed to handle the mixed attributes based on the previous study of decision making by the nearness degrees.The improved fuzzy similar priority method has some features.First, it can deal with mixed attributes.Second, its calculation is simple compared with the traditional method.Third, it improves the accuracy of the nearness degrees provided in [2] through optimizing the uncertain terms of the index vectors.
The remainder of the organization is as follows.Section 2 presents some basic concepts.Section 3 discusses the proposed method.A case study is given in Section 4. And Section 5 concludes this work.

Basic Concepts
In this section, some concepts are presented.Definition 1.Let  be a set of real numbers; then x is called a fuzzy number if it has one of the following forms [6][7][8][9][10][11]: where   ,   ∈ ,   <   ,   , and   are the interval number's lower limits and upper limits, respectively; (2) if x = [  ,   ,   ], then x is called triangular fuzzy number, where   ,   ,   ∈ , 0 <   <   <   ,   , and   are the triangular fuzzy number's lower limits and upper limits, respectively, while   is medium value; (3) if x = [  ,   ,   ,   ], then x is called trapezoidal fuzzy number, where 0 <   <   <   <   ∈ ,   and   are the trapezoidal fuzzy number's lower limits and upper limits, respectively, while [  ,   ] is medium interval.
Definition 2. Assume x is a fuzzy number with its parameters which can be seen as  ( ⩾ 2) observed values [2], and then the fuzzy number's average value, standard deviation, and maximum deviation are as follows: where   ( = 1, . . ., ) is the parameter of the fuzzy number,  is the average value of the parameters,  is the standard deviation of the parameters,  is the maximum deviation of the parameters, and   and   are the fuzzy number's lower limits and upper limits, respectively.Definition 3. Let  be a set of real numbers;  +  is called binary connection number, where ,  ∈ ,  ∈ [−1, 1],  denotes the deterministic term,  denotes the uncertain term, and  is a variable term unifying the determinacy and uncertainty of a set pair.So the determinacy-uncertainty vector based on connection number can be constructed as follows [2]: where  and V are the concentrated value and the diverse value of  parameter values of the same fuzzy number x, respectively, x can be calculated using (1), and V can be obtained by the following equation: Article [12] provided the idea of using maximum deviation as the uncertain term of an index vector only involving the upper limits in trapezoidal number.However, considering both the lower limits and upper limits of a fuzzy number is more reasonable.Remark 4. A real number can be regarded as the special form of a fuzzy number; thus the real number can also be converted into a binary connection number with the form (, V) = + 0, where  is the real number itself and the uncertain term is 0. deviation) V of the connection number (, V) can be mapped to the binary determinacy-uncertainty space (D-U space) based on SPA.If (, ) =  + V represents the vector of D-U space, then the "" only denotes the signal of the uncertain term without representing the changeable value [2].
Figure 1 is a D-U space.The  axis and the  axis represent the relative uncertain measure and the relative deterministic measure, respectively.In Figure 1, the  and the V interact with each other, and the space reflection is the vector  from  to , and the degree of interaction means the module of the vector  denoted by  [6].Measures between vectors can use distance method [13] such as Hamming distance and Euclidean distance or similarity method [14] such as cosine value and projection value.Definition 6.Let  = ( 1 ,  2 , . . .,   ) be a vector; then is called the module of .Definition 7. Let  = ( 1 ,  2 , . . .,   ) and  = ( 1 ,  2 , . . .,   ) be two vectors; then is the projection of vector  on vector .Generally speaking, the bigger the value of Prj  () the more the similarity between  and  [15].
Definition 8. Assume  is the module of vector  and   is the projection value of vector  on vector ; then is the distance of projection value   and the module  is called nearness degree.When the two index vectors are all transformed from real numbers, the  0 in ( 8) is actually degraded to Hamming distance [2].
Journal of Applied Mathematics 3 Remark 9. Measuring the similarity of the two index vectors with nearness degree has its advantages over cosine value and projection value.First, it can specifically reflect the similarity of the two vectors whether they transformed from real number or fuzzy number.Second, it is the generalized distance especially for that it is degraded to Hamming distance when the two index vectors are converted from real numbers.The detailed discussion can be seen in the paper [2].(1)   = 0,  = 1, 2, . . ., ; (2)   +   = 1 ( ̸ = , ,  = 1, 2, . . ., ).

Fuzzy Similar Priority Method for Mixed Attributes
Equation (1) indicates that   has indifference with itself, so   = 0.And (2) indicates that if   is preferred to   , denoted by   , then   is preferred to   , denoted by    = 1 −   , especially, and   = 1 indicates that   is absolutely preferred to   with respect to  0 [16][17][18].
Step 1.With respect to the attribute   , calculate the Hamming distances of   and   with  0 , respectively, Step 2. Construct the fuzzy preference relation so the fuzzy similar matrix for attribute   is Step 3. A series of -sectional sets can be determined through operating   with the  ascending in [0, 1].To get the sectional set, the elements of   must be compared with the .
The elements of   will be rewritten as 1 if they are no less than the  value.Under attribute   , the most similarity alternative to the specific object is the one whose row first gets all "1" except the diagonal elements and the alternative acquires the order value "1." Then, the impact of the alternative must be eliminated by removing the row corresponding to the alternative and the same column in   .Repeating the same work, a series of similarity alternatives can be obtained and assigned the order values 2, 3, . . ., .And the order value that the alternative gains under attribute   is denoted by   ( = 1, 2, . . ., ,  = 1, 2, . . ., ).
Step 4. Integrate the order values under different attributes for every alternative denoted by TC  with the following equation: Step 5. Rank the alternatives with TC  in ascending order, and the optimal alternative to the specific object is the one with the minimum value TC  .
Step 1. Calculate the average values, standard deviations, and maximum deviations of all indices (  and  0 ) by using Equations from (1) to (3).
Step 4. Normalize the nearness degrees of all index vectors and the fixed index vector by where   is the normalized nearness degree.
Step 5. Determine the weights of all attributes denoted as   ( = 1, . . ., ) by objective method or subjective method.
Step 6. Integrate the normalized nearness degrees considering weights of all attributes using the following equation: Step 7. Rank the feasible alternatives by the integrated normalized nearness degrees   ( = 1, . . ., ).

Merits of the Proposed Method.
The improved fuzzy similar method has its advantages over the traditional method.Above all, the proposed method can address the mixed numbers based on the similarity of the improved determinacy-uncertainty index vectors through optimizing the uncertain terms of them.Moreover, it can simplify the calculation by avoiding the traditional tedious computing procedure such that Steps 2 and 3 in the traditional method are reduced or avoided.Nevertheless, it determines the optimal alternative by the minimum of integrated nearness degrees instead of the minimum order values and can improve the accuracy of results.Finally, it can also reflect the decision makers' preferences by considering attribute weights.indices can be calculated using Equations from (1) to ( 3) and the results are shown in Table 2.

Translate All Index Values into Connection Number Vectors.
Having obtained the deterministic terms and uncertain terms of the connection numbers above, all index values can be transformed into index vectors using (4).Table 3 shows the connection numbers converted from all indices.

Calculate Nearness Degrees of All Index Vectors with the
Fixed Index Vectors.All nearness degrees of all index vectors with the fixed index vectors can be calculated as shown in Table 4 using Equations from ( 6) to (8).

Normalize the Nearness Degrees.
The normalized nearness degrees shown in Table 5 can be calculated using (14).

Discussion.
A comparison between the order value method and the nearness degree method is performed to illustrate the advantage of the proposed method to determine the optimal alternative.Table 6 shows the order values of all indices near the fixed indices determined by nearness degrees.The order values of all alternatives under some attribute are given by their index nearness degrees in terms of the principle of the smaller the better.
So the total integrated order values can be calculated as W = (16,16,21,26,34,27,39,19) using (12).And the alternatives ranking is   Table 7 is the comparison of two types of calculation method: one by integrated order values and one by integrated nearness degrees.However, decision making by integrated nearness degrees includes considering the attribute weights or not.
Seen from Table 7, alternatives ranking by order values is rough with the alternatives  1 and  2 in the same position.While the results by nearness degrees are accuracy with the alternatives  1 and  2 in different position.Furthermore, the proposed method can also use the attribute weights to reflect the decision makers' preferences.There is somewhat difference between the decision making considering attribute weights or not.

Conclusions
An improved fuzzy similar priority method was proposed to handle alternatives for mixed attributes.The proposed method solves the mixed attributes using the nearness degrees as derived from the similarity of the index vectors.And it improves the uncertain term of the index vector by selecting the minimum of the standard deviation or the minimum deviation of the parameters of the index and will contribute to the accuracy in result.Furthermore, the new method can reflect the decision maker's preferences through considering attribute weights.Another feature of this method is that its procedure and calculation are simple.A case study illustrated that it has advantages over the classical method.

Table 1 :
Safety data from coal mines.
1 to  6 respectively.Thus some coal enterprises with different index values can be regarded as different alternatives.Here  0 represents the specific coal enterprise wants to learn advanced safety management experience.And the eight feasible alternatives (coal enterprises) are denoted as  1 to  8 respectively.Besides the index values are given as mixed attribute values including real numbers, interval numbers, triangle fuzzy numbers and trapezoid fuzzy numbers.The aim of the decision makers is to seek for the most suitable enterprise whose safety management experiece can be learnt.So the proposed method can address this problem easily.The data is shown in Table1.
4.2.Process of Decision Making4.2.1.Calculate the Parameters of Index Vectors.All average values, standard deviations, and maximum deviations of all

Table 3 :
Index vectors transformed from index values.

Table 4 :
Nearness degrees of all index vectors with the fixed index vectors.

Table 5 :
Normalized nearness degrees of all index vectors with the fixed index vectors.

Table 6 :
Order values of index vectors with fixed index vectors.

Table 7 :
Alternatives ranking with two types of method.