Ranking Indices for Fuzzy Numbers

Fuzzy set theory has been studied extensively over the past 30 years. Most of the early interest in fuzzy set theory pertained to representing uncertainty in human cognitive processes (see for example Zadeh (1965)). Fuzzy set theory is now applied to problems in engineering, business, medical and related health sciences, and the natural sciences. In an effort to gain a better understanding of the use of fuzzy set theory in production management research and to provide a basis for future research, a literature review of fuzzy set theory in production management has been conducted. While similar survey efforts have been undertaken for other topical areas, there is a need in production management for the same. Over the years there have been successful applications and implementations of fuzzy set theory in production management. Fuzzy set theory is being recognized as an important problem modeling and solution technique.


Introduction
Fuzzy set theory has been studied extensively over the past 30 years.Most of the early interest in fuzzy set theory pertained to representing uncertainty in human cognitive processes (see for example Zadeh (1965)).Fuzzy set theory is now applied to problems in engineering, business, medical and related health sciences, and the natural sciences.In an effort to gain a better understanding of the use of fuzzy set theory in production management research and to provide a basis for future research, a literature review of fuzzy set theory in production management has been conducted.While similar survey efforts have been undertaken for other topical areas, there is a need in production management for the same.Over the years there have been successful applications and implementations of fuzzy set theory in production management.Fuzzy set theory is being recognized as an important problem modeling and solution technique.Kaufmann and Gupta (1988) report that over 7,000 research papers, reports, monographs, and books on fuzzy set theory and applications have been published since 1965.
As evidenced by the large number of citations found, fuzzy set theory is an established and growing research discipline.The use of fuzzy set theory as a methodology for modeling and analyzing decision systems is of particular interest to researchers in production management due to fuzzy set theory's ability to quantitatively and qualitatively model problems which involve vagueness and imprecision.Karwowski and Evans (1986) identify the potential applications of fuzzy set theory to the following areas of production management: new product development, facilities location and layout, production scheduling and control, inventory management, quality and cost benefit analysis.Karwowski and Evans identify three key reasons why fuzzy set theory is relevant to production management research.First, imprecision and vagueness are inherent to the decision maker's mental model of the problem under study.Thus, the decision maker's experience and judgment may be used to complement established theories to foster a better understanding of the problem.Second, in the production management environment, the information required to formulate a model's objective, decision variables, constraints and parameters may be vague or not precisely measurable.Third, imprecision and vagueness as a result of personal bias and subjective opinion may further dampen the quality and quantity of available information.Hence, fuzzy set theory can be used to bridge modeling gaps in descriptive and prescriptive decision models in production management research.
We often face difficultly in selecting appropriate defuzzification, which is mainly based on intuition and there is no explicit decision making for these parameters.For more comparison details on most of these methods, in this chapter we review some of ranking methods.

Definition
First, In general, a generalized fuzzy number A is membership () A x  can be defined as (Dubios & Prade, 1978) where 0 1   is a constant, and normal fuzzy number; otherwise, it is a trapezoidal fuzzy number and is usually denoted by (,,, , ) In particular, when bc  , the trapezoidal fuzzy number is reduced to a triangular fuzzy number denoted by (,, , ) The set of all elements that have a nonzero degree of membership in A , it is called the support of A , i.e.

 
The set of elements having the largest degree of membership in A , it is called the core of A , i.e.
() | () s u p () In the following, we will always assume that A is continuous and bounded support () SA .
The strong support of A should be   () , SA ad  .

Definition
The addition and scalar multiplication of fuzzy numbers are defined by the extension principle and can be equivalent represented in (Zadeh, 1965;Ma et al., 1999;Dubois & Prade, 1980) as follows.
For arbitrary .
To emphasis, the collection of all fuzzy numbers with addition and multiplication as defined by ( 8) is denoted by E, which is a convex cone.The image (opposite) of (,,, ) (Zadeh, L.A, 1965;Dubois, D. and H. Prade, 1980).

Definition
and (1) 1 f  .We say that s is a regular function if () 1 / 2 frd r .

Definition
If A is a fuzzy number with r-cut representation,   11 () , ()

AA LrR r
 and s is a reducing function, then the value of A (with respect to s); it is defined by

Definition
If A is a fuzzy number with r-cut representation   11 () , () , and s is a reducing function then the ambiguity of A (with respect to s) is defined by Let also recall that the expected interval () EI A of a fuzzy number A is given by Another parameter is utilized for representing the typical value of the fuzzy number is the middle of the expected interval of a fuzzy number and it is called the expected value of a fuzzy number A i.e. number A is given by (Bodjanova, 2005) 11

Definition
The first of maxima (FOM) is the smallest element of () .core A i.e.

Definition
The last of maxima (LOM) is the greatest element of () .core A i.e.

Definition
For arbitrary fuzzy numbers is the distance between A and B. The function (,)  DAB is a metric in E and (, ) ED is a complete metric space.
The ordering indices are organized into three categories by Wang and Kerre (Wang & Kerre, 2001) as follows:


Defuzzification method: Each index is associated with a mapping from the set of fuzzy quantities to the real line.In this case, fuzzy quantities are compared according to the corresponding real numbers.


Reference set method: in this case, a fuzzy set as a reference set is set up and all the fuzzy quantities to be ranked are compared with the reference set. Fuzzy relation method: In this case, a fuzzy relation is constructed to make pair wise comparisons between the fuzzy quantities involved.
Let M be an ordering method on E. The statement two elements 1 A and 2 A in E satisfy that 1 A has a higher ranking than 2 A when M is applied will be written as 11

Ranking indices a. Methods of centroid point
In order to determine the centroid points 00 (,) xy of a fuzzy number A , Cheng (Cheng, 1998) provided a formula then Wang et al. (Y. M. Wang et al., 2006) found from the point of view of analytical geometry and showed the corrected centroid points as follows: Since non-normal triangular fuzzy numbers are, special cases of normal trapezoidal fuzzy numbers with bc  , formulas (12) can be simplified as In this case, normal triangular fuzzy numbers could be compared or ranked directly in terms of their centroid coordinates on horizontal axis.
Cheng (Cheng, 1998) formulated his idea as follows: To overcome the drawback of Cheng's distance Chu and Tsao's (Chu & Tsao, 2002) computed the area between the centroid and original points to rank fuzzy numbers as: Then Wang and Lee (Y.J. Wang, 2008) ranked the fuzzy numbers based on their 0 x 's values if they are different.In the case that they are equal, they further compare their 0 y 's values to form their ranks.
Further, for two fuzzy numbers A and B if 00 () () By shortcoming of the mentioned methods finally, Abbasbandy and Hajjari (Abbasbandy & Hajjari 2010) improved Cheng's distance centroid as follows: Where However, there are some problems on the centroid point methods.In next section, we will present a new index for ranking fuzzy numbers.The proposed index will be constructed by fuzzy distance and centroid point.

b. Method of D-distance (Ma et al. 2000)
Let all of fuzzy numbers are positive or negative.Without less of generality, assume that all of them are positive.The membership function of aR  is the distance between A and B .

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Ranking Indices for Fuzzy Numbers 57

Definition
Let () : {1 , 1 } AE    be a function that is defined as follows:

Definition
For A and BE  define the ranking order of A and B by p d on E .i.e.


In fact, a reducing has the reflection of weighting the influence of the different r-cuts and diminishes the contribution of the lower r-levels.This is reasonable since these levels arise from values of membership function for which there is a considerable amount of uncertainty.For example, we can use () .
sr r 

Definition
For A and BE  we define H-distance of A and B by where H d is the Housdorf metric between intervals and  1 . is the 1-cut representation of a fuzzy number.

Definition
For A and BE  we define source distance of A and B by

where [, ]
A A mt and [, ] BB mt are the cores of fuzzy numbers A and B respectively.

Property
The In the following, we use an example to illustrate the ranking process of the proposed method.
Moreover, for normal fuzzy numbers we have www.intechopen.com

h. Methods of deviation degree
Ranking L-R fuzzy numbers based on deviation degree (Z.X.Wang et al., 2009)

Definition
For any groups of fuzzy numbers 12 , ,.

Definition
For any groups of fuzzy numbers , 1,2,..., . 1 Now, by using ( 15), for any two fuzzy numbers A i and Aj the ranking order is based on the following rules.
1.  Consider two fuzzy numbers A and B the ranking order is based on the following situations: where Ranking fuzzy numbers based on the left and the right sides of fuzzy numbers (Nejad & Mashinchi, 2011) Recently Nejad and Mashinchi (Nejad & Mashinchi, 2011) ,  , ( Nevertheless, the new ranking method has drawback.
In the next section, we discuss on those methods that based on deviation degree by a number numerical counter examples.

Discussion and counter examples 6.1 Example
Let two fuzzy numbers A= (3,6,9) and B= (5,6,7) from (Z.-X.Wang et al., 2009) as shown in Fig. 1.Fig. 1. Fuzzy numbers A=(3,6,9) and B= (5,6,7) Through the approaches in this paper, the ranking index can be obtained as Mag From the obtained results we have A  B, for two triangular fuzzy numbers A= (3,6,9) and B= (5,6,7).Now we review the ranking approaches by promoter operator.Since A and B have the same ranking order and the same centroid points we then compute their ambiguities.Hence, from (Deng et al., 2006) it will be obtained amb(A) = 1 and 1 () 3 amb B  .Consequently, by using promoter operator we have   (Cheng, 1998), Chu and Tsao's method (Chu & Tsao, 2002), Chen and Chen's method (S.J. Chen et al. 2009) get the same ranking order: BA  , whereas the ranking order by Mag-method (Abbasbandy & Hajjari, 2009) is AB  .By comparing the ranking result of Magmethod with other methods with respect to Set 7 of Fig. 2, we can see that Mag-method considers the fact that defuzzified value of a fuzzy number is more important than the spread of a fuzzy number.10.For the fuzzy numbers A and B shown in Set 8 of Fig. 2, Cheng's method (Cheng, 1998), Chu and Tsao's method (Chu & Tsao, 2002), Chen and Chen's method (S.J. Chen et al., 2009) and Mag-method (Abbasbandy & Hajjari, 2009) get the same ranking order: ABC , whereas the ranking order by Chen and Chen's method is AC B .
By comparing the ranking result of mentioned method with other methods with respect to Set 8 of Fig. 4, we can see that Chen's method considers the fact that the spread of a fuzzy number is more important than defuzzified value of a fuzzy number.
idea of ranking fuzzy numbers by deviation degree is useful, but a significant approaches should be reserved the important properties such that BC AC Now we give some numerical example to show the drawback of the aforementioned methods.
On the other hand, ranking order for A and B and their images by Wang et al.'s method and Asady's revised are ,    AB A B respectively.This example could be indicated that all methods are reasonable.However, we will show that functions of all three methods are not the same in different conditions.
Utilizing Nejad and Mashinchi's method the ranking order is ABC  and the ranking order of their images will be obtained -C-A-B, which is illogical.By using Wang et al.'s method the ranking order is B  A  C and for their images is -A  -C  -B, which is unreasonable too.
From point of revised deviation degree method (Asady, 2010) the ranking orders are B AC, -C-A-B, respectively.
From this example, it seems the revised method can rank correctly.
In the next example, we will indicate that none of the methods based on deviation degree can rank correctly in all situations.

Example
Let the triangular fuzzy number (1,2,3) A  and the fuzzy number (1,2,4) B  with the membership function (See Fig. 5) From mentioned examples, we can theorize that ranking fuzzy numbers based on deviation degree cannot rank fuzzy numbers in all situations.

Conclusion
With the increasing development of fuzzy set theory in various scientific fields and the need to compare fuzzy numbers in different areas.Therefore, Ranking of fuzzy numbers plays a very important role in linguistic decision making, neural network and some other fuzzy application systems .Several strategies have been proposed for ranking of fuzzy numbers.Each of these techniques has been shown to produce non-intuitive results in certain case.In this chapter, we reviewed some recent ranking methods, which will be useful for the researchers who are interested in this area.

Acknowledgment
This work was partly supported by Islamic Azad University, FiroozKooh Branch.
triangular fuzzy numbers are special cases of trapezoidal fuzzy numbers.Since A L and A R are both strictly monotonical and continuous functions, their inverse functions exist and should be continuous and strictly monotonical.Let


left fuzziness and right fuzziness are 0, so for each A E  www.intechopen.com of ranking L-R fizzy number based on deviation degree(Asady, 2010) Asady(Asady, 2010) revised Wang et al. (Z.X.Wang et al. 2009) method and suggested (.) D operator for ranking of fuzzy numbers as follows: (A)=Mag(B)=12 and EV(A) = EV(B) = 6.Then the ranking order of fuzzy numbers is A  B. Because fuzzy numbers A and B have the same mode and symmetric spread, most of existing approaches have the identical results.For instance, by Abbasbandy and Asady's approach (Abbasbandy & Asady, 2006), different ranking orders are obtained when different index values p are taken.When p = 1 and p = 2 the ranking order is the same, i.e., A  B Nevertheless, the same results produced when distance index, CV index of Cheng's approach and Chu and Tsao's area are respectively used, i.e., x A = x B = 6 and 1 3 AB yy  then from Cheng's distance and Chau and Tsao's area we get that R(A) = R(B) = 2.2608, S(A) = S(B) = 1.4142 respectively.

.2 Property
If AB  it is not necessary that AB  .

Method of sign distance (Abbasbandy & Asady 2006) 5.4 Definition
(5)emark 3.12 part(5)we can logically infer ranking order of the image of the fuzzy numbers.
p dA 

Method of promoter operator (Hajjari & Abbasbandy 2011)
Then we formulate the order  and  as AB  if and only if AB  or AB  , AB  if and only if AB  or AB  .In other words, this method is placed in the first class of Kerre's categories(X.Wang & Kerre 2001).g.AA A LrR r Wang et al. , 2009)wback of Wang et al. (Z.X.Wang et al. , 2009)hen they presented a novel ranking method as follows.