Research Article | Open Access

Lakshmana Gomathi Nayagam Velu, Jeevaraj Selvaraj, Dhanasekaran Ponnialagan, "A New Ranking Principle For Ordering Trapezoidal Intuitionistic Fuzzy Numbers", *Complexity*, vol. 2017, Article ID 3049041, 24 pages, 2017. https://doi.org/10.1155/2017/3049041

# A New Ranking Principle For Ordering Trapezoidal Intuitionistic Fuzzy Numbers

**Academic Editor:**Jose Egea

#### Abstract

Modelling real life (industrial) problems using intuitionistic fuzzy numbers is inevitable in the present scenario due to their efficiency in solving problems and their accuracy in the results. Particularly, trapezoidal intuitionistic fuzzy numbers (TrIFNs) are widely used in describing impreciseness and incompleteness of a data. Any intuitionistic fuzzy decision-making problem requires the ranking procedure for intuitionistic fuzzy numbers. Ranking trapezoidal intuitionistic fuzzy numbers play an important role in problems involving incomplete and uncertain information. The available intuitionistic fuzzy decision-making methods cannot perform well in all types of problems, due to the partial ordering on the set of intuitionistic fuzzy numbers. In this paper, a new total ordering on the class of TrIFNs using eight different score functions, namely, imprecise score, nonvague score, incomplete score, accuracy score, spread score, nonaccuracy score, left area score, and right area score, is achieved and our proposed method is validated using illustrative examples. Significance of our proposed method with familiar existing methods is discussed.

#### 1. Introduction

Classical set theory cannot be the better choice for modelling problems involving qualitative or imprecise information. To model such problems, fuzzy number was introduced by Jain [1] and some operations on fuzzy numbers are defined in [2, 3]. Intuitionistic fuzzy numbers are comparatively better in modelling real life problems involving uncertainties and imprecise information. Particularly, trapezoidal intuitionistic fuzzy numbers are more effective in describing impreciseness and incompleteness of a data. To resolve the task of comparing trapezoidal intuitionistic fuzzy numbers, many authors have proposed different ranking methods but none of them yield a total order on the class of TrIFNs with finite number of scores. Different ranking methodologies on the class of intuitionistic fuzzy numbers are discussed in [4, 5].

Nehi and Maleki [6] generalised the idea of natural ordering on real numbers to the triangular intuitionistic fuzzy numbers (TIFNs) by adopting a statistical view point. Nehi [7] compared TIFNs using lexicographic technique. Li [2] developed the idea of value and ambiguity of a triangular intuitionistic fuzzy number and introduced the new ranking method using the concept of the ratio of the value index to the ambiguity index. Ye [8] presented the new ranking method using expected value of a trapezoidal intuitionistic fuzzy number and solved the decision-making problem using weighted expected value of TrIFN. Dubey and Mehara [9] extended the concept of value and ambiguity to the slightly modified TIFN and proposed a new approach to solve intuitionistic fuzzy linear programing problem. Nehi [7] introduced the concept of characteristic values of membership and nonmembership functions of TrIFN and proposed a new ranking method for trapezoidal intuitionistic fuzzy numbers by using it. Zhang and Nan [10] developed a compromise ratio ranking method for fuzzy multiattribute decision-making (MADM) problem based on the concept that larger TIFN among other TIFNs will be closer to the maximum value index and it will be far away from the minimum ambiguity index simultaneously. Kumar and Kaur [11] proposed the ranking method for TrIFNs by modifying Nehiâ€™s [10] method. Zeng et al. [12] introduced a new ranking method for TrIFNs by extending the concept value and ambiguity of TIFN defined in Li [2]. Wan and Dong [13] introduced the concept of lower and upper weighted possibility mean and possibility mean for a trapezoidal intuitionistic fuzzy numbers and proposed the new ranking method by use of it. Different ranking methods and their applications on multicriteria decision-making problem and other domains are studied in ([14â€“20]). Lakshmana Gomathi Nayagam et al. [19, 21, 22] have introduced a complete ranking procedure on the class of intuitionistic fuzzy numbers using countable number of parameter. In this paper, a new total ordering on the class of TrIFNs using finite number of score functions is achieved. Also the limitations and drawbacks of all the abovementioned methods are discussed and the efficiency of our proposed method is shown by comparing all existing methods.

This paper is organised in the following manner. After introduction, some important definitions on intuitionistic fuzzy numbers are given in Section 2. The different subclasses of TrIFNs are introduced and the new score functions on these subclasses are established in Section 3. In Section 4, a complete ranking on the class of trapezoidal intuitionistic fuzzy numbers by using score functions defined in Section 3 is explained. The ranking procedure is explained in detail with several examples and also our proposed method is compared with some other existing methods in the Section 5. Finally conclusions are given in Section 6.

#### 2. Preliminaries

Here we give a brief review of some preliminaries.

*Definition 1 (Atanassov [23]). *Let be a nonempty set. An intuitionistic fuzzy set (IFS) in is defined by , where and with the conditions . The numbers denote the degree of membership and nonmembership of to lie in , respectively. For each intuitionistic fuzzy subset in , is called hesitancy degree of to lie in .

*Definition 2 (Burillo et al., [24]). *An intuitionistic fuzzy number in the set of real numbers is defined aswhere and such that , and four functions are the legs of membership function and nonmembership function The functions and are nondecreasing continuous functions and the functions and are nonincreasing continuous functions.

An intuitionistic fuzzy number with is shown in Figure 1.

*Definition 3. *A trapezoidal intuitionistic fuzzy number with parameters is denoted as in the set of real numbers is an intuitionistic fuzzy number whose membership function and nonmembership function are given asIf (and in a trapezoidal intuitionistic fuzzy number , we have the triangular intuitionistic fuzzy numbers as special case of the trapezoidal intuitionistic fuzzy numbers.

A trapezoidal intuitionistic fuzzy number with , , , and is shown in Figure 2.

We note that the condition of the trapezoidal intuitionistic fuzzy number whose membership and nonmembership fuzzy numbers of are and implies , , , and on the legs of trapezoidal intuitionistic fuzzy number.

*Definition 4 (Atanassov and Gargov, [25]). *Let be the set of all closed subintervals of the interval . An interval valued intuitionistic fuzzy set on a set is an expression given by , where with the condition .

The intervals and denote, respectively, the degree of belongingness and nonbelongingness of the element to the set . Thus for each , and are closed intervals whose lower and upper end points are, respectively, denoted by , and , . We denote , where .

For each element , we can compute the unknown degree (hesitance degree) of belongingness to as . An intuitionistic fuzzy interval number (IFIN) is denoted by for convenience.

*Definition 5. *Let . An -cut of a trapezoidal intuitionistic fuzzy number, denoted by , is defined as , where and are -cut and -cut of membership and nonmembership trapezoidal fuzzy numbers.

The -cut of trapezoidal intuitionistic fuzzy number is given by = , , where and represents the lower and upper end points of the -cut of the membership function of and and represent the lower and upper end points of the -cut of the nonmembership function of , respectively.

#### 3. Different Classes of Trapezoidal Intuitionistic Fuzzy Numbers

In this section the entire class of TrIFNs is partitioned into eight subclasses and further score functions using different concepts are defined in order to give total order on the entire class of TrIFNs and some theorems related to these concepts are established.

In this paper, and always denote trapezoidal intuitionistic fuzzy number (TrIFN).

with conditions that and , and and with the condition that and , , and unless otherwise specifically stated.

##### 3.1. Imprecise Score of a Trapezoidal Intuitionistic Fuzzy Number

In this subsection, a new subclass of TrIFNs is introduced. The imprecise score function on this class of trapezoidal intuitionistic fuzzy numbers is defined and some of its properties are studied using illustrative examples.

*Definition 6. *Let and , be two TrIFNs.

A special subclass of the set of TrIFNs consist of TrIFNs for which every pair of and are with and . The imprecise relation on is denoted as and defined as follows: if such that then and .

If then one of the above inequalities becomes strict inequality.

*Note 1. *By Definition 6, we note that any pair of members of are related under .

The score function which measures the preciseness is defined as follows.

*Definition 7. *Let be a trapezoidal intuitionistic fuzzy number with . Then the imprecise score of a trapezoidal intuitionistic fuzzy number is defined by

The proofs of the following propositions are immediate from the above definition and hence they are omitted.

Proposition 8. *For any real number , .*

Proposition 9. *If is a trapezoidal fuzzy number, then .*

Proposition 10. *Let be a triangular intuitionistic fuzzy number. Then .*

Proposition 11. *Let be an interval valued intuitionistic fuzzy number. Then .*

Theorem 12. *Let and . If then .*

*Proof. *Let . We claim that .

By Definition 6, we haveNow,From (4) it is very easy to see that all the terms in (5) are positive. Therefore their sum is also positive. From Definition 6, we know that at least one of the above inequalities in (4) becomes strict inequality and hence we get .

Theorem 13. *Let and such that ; then .*

*Proof. *Let . We claim that .

By Definition 6, without loss of generality, we haveNow,Therefore from (6) and (7), it is clear that all the terms in (7) are positive and their sum gives zero only when each term is equal to zero. Hence , , , and , hence the proof.

*Note 2. *The imprecise score can be calculated to any TrIFN but gives total order in .

*Definition 14. *If , then .

The following example explains the ranking procedure introduced in Definition 14.

*Example 15. *Let , and , . Now and .

Hence .

*Example 16. *Let and , but not in , where .

Now . But . Pictorial representation of this example is given in Figure 3.

Example 16 shows that is not enough cover the entire class of TrIFNs. Therefore it is needed for us to define another score function that can cover some subclass of TrIFNs which cannot be covered by .

##### 3.2. Nonvague Score of a Trapezoidal Intuitionistic Fuzzy Number

In this subsection, another subclass of TrIFNs is introduced using nonvague relation. The nonvague score function on this class of trapezoidal intuitionistic fuzzy numbers is defined and some of its properties are studied using illustrative examples.

*Definition 17. *Let and be two TrIFNs.

A special subclass of the set of TrIFNs consist of TrIFNs for which every pair of and are with and . The nonvague relation on is denoted as and defined as follows: if such that then and .

If then one of the above inequalities becomes strict inequality.

*Note 3. *By Definition 17, we note that any pair of members of are related under .

The score function which measures the nonvagueness is defined as follows.

*Definition 18. *Let be a trapezoidal intuitionistic fuzzy number. Then the nonvague score of a trapezoidal intuitionistic fuzzy number is defined by The proofs of the following propositions are immediate from Definition 18 and hence they are omitted.

Proposition 19. *For any real number , .*

Proposition 20. *If is a trapezoidal fuzzy number, then .*

Proposition 21. *Let be a triangular intuitionistic fuzzy number. Then .*

Proposition 22. *Let be an interval valued intuitionistic fuzzy number. Then .*

Theorem 23. *Let and . If then .*

*Proof. *Let us assume . We claim that .

By Definition 17, we haveNow,From (9) it is very easy to see that all the terms in (10) are positive. Therefore their sum is also positive. From Definition 17, we know that at least one of the above inequalities in (9) becomes strict inequality and hence we get .

Theorem 24. *Let and . If then .*

*Proof. *Let us assume . We claim that .

By Definition 17, without loss of generality, we haveNow,Therefore from (11) and (12), it is clear that all the terms in (12) are positive; therefore their sum gives zero only when each term is equal to zero. Hence and . Hence .

*Note 4. *The nonvague score can be calculated to any TrIFN. But gives total order on .

*Definition 25. *If , then .

Ranking relation defined above is explained in the following example.

*Example 26. *Let , and , . Now and .

Hence .

Example 27 shows the inefficiency of in comparing any two arbitrary TrIFNs and the importance of defining new score function .

*Example 27. *Let and , and .

NowButTherefore ; hence .

*Example 28. *Let and , and but not in . NowBut .

Pictorial representation of this example is given in Figure 4.

From Example 28 it is easy to see that and are not enough to cover the entire class of TrIFNs. Therefore in the next subsection we are introducing a new score function which covers some more subclasses of TrIFNs that cannot be covered by and .

##### 3.3. Incomplete Score of a Trapezoidal Intuitionistic Fuzzy Number

In this subsection, another class of TrIFNs is introduced. The incomplete score function on this class of trapezoidal intuitionistic fuzzy numbers is defined and some of its properties are studied using illustrative examples.

*Definition 29. *Let and be two TrIFNs.

A special subclass of the set of TrIFNs consist of TrIFNs for which every pair of and are with and . The incomplete relation on is denoted as and defined as follows: if such that then and .

If then one of the above inequalities becomes strict inequality.

*Note 5. *By Definition 29, we note that any pair of members of are related under .

The incomplete score function which measures the completeness is defined as follows.

*Definition 30. *Let be a trapezoidal intuitionistic fuzzy number. Then the incomplete score of is defined by

The proofs of the following propositions are immediate from the above definition and hence they are omitted.

Proposition 31. *For any real number , .*

Proposition 32. *If is a trapezoidal fuzzy number, then .*

Proposition 33. *Let be a triangular intuitionistic fuzzy number. Then*

Proposition 34. *Let be an interval valued intuitionistic fuzzy number. Then*

Theorem 35. *Let . If then .*

*Proof. *Let us assume that . We claim that .

By Definition 29, we haveNow,From (19) it is very easy to see that all the terms in (20) are positive. Therefore their sum is also positive. From Definition 29, we know that at least one of the above inequalities in (19) becomes strict inequality and hence we get , hence the proof.

Theorem 36. *Let . If then .*

*Proof. *Let us assume . We prove that .

By Definition 29, without loss of generality, we haveNow,Therefore from (21) and (22), it is clear that all the terms in (22) are positive and therefore their sum gives zero only when each term is equal to zero. Hence and . Hence .

*Note 6. *The incomplete score can be calculated to any TrIFN. But gives total order on .

*Definition 37. *If , then .

The following example is used to explain the ranking procedure defined in Definition 37.

*Example 38. *Let , and , . Now and .

Hence .

The inefficiency of the score functions and in discriminating any two arbitrary TrIFNs and the ability of in comparing arbitrary TrIFNs is shown in the following example.

*Example 39. *Let and , and .

NowButHence .

The inefficiency of the score functions to in the task of comparing TrIFNs is explained in the following example.

*Example 40. *Let and , and but not in .

NowBut .

Pictorial representation of this example is given in Figure 5.

Example 40 shows that , and cannot be sufficient to cover the entire class of TrIFNs and the class of TrIFNs in the above example excite us to define new score function that can fill the subclass of TrIFNs which cannot be covered by , , and .

##### 3.4. Accuracy Score of a Trapezoidal Intuitionistic Fuzzy Number

In this subsection, a new subclass of TrIFNs using accuracy relation is introduced and the accuracy score function on this class of trapezoidal intuitionistic fuzzy numbers is defined and some of its properties are studied using illustrative examples.

*Definition 41. *Let and be two TrIFNs.

A special subclass of the set of TrIFNs consist of TrIFNs for which every pair of and are with and . The accuracy relation on is denoted as and defined as follows: if such that then and .

If then one of the above inequalities becomes strict inequality.

*Note 7. *By Definition 41, we note that any pair of members of are related under .

The score function which measures the accuracy is defined as follows.

*Definition 42. *Let be a trapezoidal intuitionistic fuzzy number. Then the accuracy score of is defined by

The proofs of the following propositions are immediate from Definition 42 and hence they are omitted.

Proposition 43. *For any real number , .*

Proposition 44. *For any trapezoidal fuzzy number , .*

Proposition 45. *Let be a triangular intuitionistic fuzzy number. Then .*

Proposition 46. *Let be an interval valued intuitionistic fuzzy number. Then .*

*Note 8. *The accuracy score can be calculated to any TrIFN. But gives total order on which is proved in Theorems 47 and 48.

Theorem 47. *Let . If then .*

*Proof. *Let us assume that . We claim that .

By Definition 41, we haveNow,From (27) it is very easy to see that all the terms in (28) are positive. Therefore their sum is also positive. From Definition 41, we know that at least one of the above inequalities in (27) becomes strict inequality and hence we get .

Theorem 48. *Let such that ; then .*

*Proof. *Let us assume . We claim that .

By Definition 41, without loss of generality, we haveNow,Therefore from (29) and (30), it is clear that all the terms in (30) are positive and therefore their sum gives zero only when each term is equal to zero. Hence and , . Hence .

*Definition 49. *If then .

Ranking procedure defined in Definition 49 is illustrated in the following example.

*Example 50. *Let , and , . Now and .

Hence .

The relative importance of the score function in discriminating two different TrIFNs when , and fail to discriminate them is explained by Example 51.

*Example 51. *Let and , , and .

NowBut