Abstract
Corresponding to chance constraints, real-life possibility, necessity, and credibility measures on intuitionistic fuzzy set are defined. For the first time the mathematical and graphical representations of different types of measures in trapezoidal intuitionistic fuzzy environment are defined in this paper. We have developed intuitionistic fuzzy chance constraints model (CCM) based on possibility and necessity measures. We have also proposed a new method for solving an intuitionistic fuzzy CCM using chance operators. To validate the proposed method, we have discussed three different approaches to solve the intuitionistic fuzzy linear programming (IFLPP) using possibility, necessity and credibility measures. Numerical and graphical representations of optimal solutions of the given example at different possibility and necessity, levels have been discussed.
1. Introduction
In the real world some data often provide imprecision and vagueness at certain level. Such vagueness has been represented through fuzzy sets. Zadeh [1] first introduced the fuzzy sets. The perception of intuitionistic fuzzy set (IFS) can be analysed as an unconventional approach to define a fuzzy set where available information is not adequate for the definition of an imprecise concept by means of a usual fuzzy set. This IFS was first introduced by Atanassov [2]. Many researchers have shown their interest in the study of intuitionistic fuzzy sets/numbers [3–7]. Fuzzy sets are defined by the membership function in all its entirety (c.f. Pramanik et al. [8, 9]), but IFS is characterized by a membership function and a nonmembership function so that the sum of both values lies between zero and one [10]. Esmailzadeh and Esmailzadeh [11] provided new distance between triangular intuitionistic fuzzy numbers.
Recently, the IFN has also found its application in fuzzy optimization. Angelov [12] proposed the optimization in an intuitionistic fuzzy environment. Dubey and Mehra [13] solved linear programming with triangular intuitionistic fuzzy number. Parvathi and Malathi [14] developed intuitionistic fuzzy simplex method. Hussain and Kumar [15] and Nagoor Gani and Abbas [16] proposed a method for solving intuitionistic fuzzy transportation problem. Ye [17] discussed expected value method for intuitionistic trapezoidal fuzzy multicriteria decision-making problems. Wan and Dong [18] used possibility degree method for interval-valued intuitionistic fuzzy for decision making.
Possibility, necessity, and credibility measures have a significant role in fuzzy and intuitionistic fuzzy optimization. Buckley [19] introduced possibility and necessity in optimization and Jamison and Lodwick [20] developed the construction of consistent possibility and necessity measures. Duality in fuzzy linear programming with possibility and necessity relations has been developed by Ramík [21]. Iskander [22] suggested an approach for possibility and necessity dominance indices in stochastic fuzzy linear programming. Sakawa et al. [23] used possibility and necessity to solve fuzzy random bilevel linear programming. Pathak et al. [24] discussed a possibility and necessity approach to solve fuzzy production inventory model for deteriorating items with shortages under the effect of time dependent learning and forgetting. Maity [25] established possibility and necessity representations of fuzzy inequality and its application to two warehouse production-inventory problem. Wu [26] presented possibility and necessity measures fuzzy optimization problems based on the embedding theorem. Xu and Zhou [27] discussed possibility, necessity, and credibility measures for fuzzy optimization. Maity and Maiti [28] developed the possibility and necessity constraints and their defuzzification for multiitem production-inventory scenario via optimal control theory. Das et al. [29] presented a two-warehouse supply-chain model under possibility, necessity, and credibility measures. Panda et al. [30] proposed a single period inventory model with imperfect production and stochastic demand under chance and imprecise constraints. Intuitionistic fuzzy-valued possibility and necessity measures have been devolved by Ban [31] using measure theory. With our best knowledge, however, none of them introduced chance constraints model based on possibility, necessity, and credibility measures on intuitionistic fuzzy set for membership and nonmembership functions.
The rest of this paper is organized into different section as follows demonstrating the deduction of our theory and its application. In Section 2, we recall some preliminary knowledge about intuitionistic fuzzy and its arithmetic operation. Section 3 has provided possibility, necessity, and credibility measures in trapezoidal intuitionistic fuzzy number and its graphical representation. In Section 4, we have proposed intuitionistic fuzzy chance constraint models based on possibility, necessity, and credibility measures. The solution methodology of the proposed models using chance operator has been discussed in Section 5. In Section 6, a numerical example is presented to validate the proposed method. The numerical and graphical results at different possibility and necessity levels of the given problems have also been discussed here. Section 7 summarizes the paper and also discusses about the scope of future work.
2. Preliminaries
Definition 1 (intuitionistic fuzzy set [2, 10]). Let be a given set and let be a set. An IFS in is given by , where and define the degree of membership and the degree of nonmembership of the element to satisfying the condition .
Definition 2 (intuitionistic fuzzy number [7]). An IFN is(i)an intuitionistic fuzzy subset on real line,(ii)there exist , such that , and .(iii)convex for the membership function ; that is, , , , .(iv)concave for the nonmembership function ; that is, , , , .
Definition 3 (trapezoidal intuitionistic fuzzy number (TIFN)). Let . A TIFN in written as has membership function (c.f. Figure 1) and nonmembership function
Definition 4 (triangular intuitionistic fuzzy number (TrIFN)). Let . A TrIFN in written as has membership function (c.f. Figure 2) and nonmembership function
Definition 5. A positive TIFN is denoted by , where all for all , , , , and for , .
Definition 6. Two TIFN and are said to be equal if and only if , , , , and .
Definition 7. Let and be two TIFN; then(i);(ii) if ;(iii) if ;(iv);(v).
3. Possibility, Necessity, and Credibility Measures of Intuitionistic Fuzzy Number
Definition 8. Let and be two IFN with membership function , and nonmembership function , , respectively, and is the set of real numbers. Then where the abbreviations and represent possibility of membership and nonmembership function, and and represent necessity of membership and nonmembership function. is any of the relations , , , , .
The dual relationship of possibility and necessity gives where represents complement of the event .
Definition 9. Let be a IFN. Then the intuitionistic fuzzy measures of for membership and nonmembership function are
where the abbreviation and represent measures of membership and nonmembership functions and is the optimistic-pessimistic parameter to determine the combined attitude of a decision maker.
If then , ; it means the decision maker is optimistic and maximum chance of holds.
If , then , ; it means the decision maker is pessimistic and minimal chance of holds.
If , then , , where Cr is the credibility measure; it means the decision maker takes compromise attitude.
3.1. Measures of Trapezoidal Intuitionistic Fuzzy Number
Let and be two TIFN. From Definition 8 the possibilities of for membership and nonmembership functions (c.f. Figures 3 and 4) are as follows:
From the Definition 8 the possibilities of for membership and nonmembership function (c.f. Figures 5 and 6) are as follows:
Now, by Definition 8, necessity of the event are as follows: By Definition 8 necessity of the event are as follows: By Definition 9 measures of the event are as follows: By Definition 9 measures of the event are as follows: For ,
Lemma 10. If and , then
Proof. Let us consider Now from (8)
Note. and and .
Lemma 11. If and , then
Proof. Let us consider Now from (10),
Note. and and .
Lemma 12. If and , then
Proof. Let us consider Now, from (14),
Note. , , and , .
4. Intuitionistic Fuzzy CCM
The chance operator is actually taken as possibility or necessity or credibility measures. We can use chance operator to transform the intuitionistic fuzzy problem into crisp problem, which is called as CCM [27]. A general single-objective mathematical programming problem with intuitionistic fuzzy parameter should have the following form: where is the decision vector, and are intuitionistic fuzzy parameters, is an imprecise objective function, and are constraints function for .
The general chance-constraints model for problem (24) is as follows: The abbreviations and represent chance operator (i.e., Pos or Nec measure) for membership and nonmembership functions. , , , and are the predetermined confidence levels such that and for .
4.1. Intuitionistic Fuzzy CCM Based on Possibility Measure
The CCM based on possibility measure is as follows: where , , , and are the predetermined confidence levels such that and for .
Definition 13. A solution of the problem (26) satisfies and for is called a feasible solution at possibility levels, .
Definition 14. A feasible solution at possibility levels, , is said to be efficient solution for problem (26) if and only if there exists no other feasible solution at possibility levels, such that and with .
4.2. Intuitionistic Fuzzy CCM Based on Necessity Measure
The CCM based on necessity measure is as follows: where , , , and are the predetermined confidence levels such that and for .
Definition 15. A solution of the problem (27) satisfies and for is called a feasible solution necessity levels, .
Definition 16. A feasible solution at necessity levels, , is said to be efficient solution for problem (27) if and only if there exists no other feasible solution at necessity levels, such that and with .
4.3. Intuitionistic Fuzzy CCM Based on Credibility Measure
The CCM based on credibility measure is as follows: where , , , and are the predetermined confidence levels such that and for .
Definition 17. A solution of the problem (28) satisfies and for is called a feasible solution credibility levels, .
Definition 18. A feasible solution at credibility levels, , is said to be efficient solution for problem (28) if and only if there exists no other feasible solution at credibility levels, such that and with .
5. Proposed Method to Solve IFLPP Using Chance Operator
To solve intuitionistic fuzzy CCM based on possibility or necessity or credibility measures we propose the following method.
Step 1. Apply chance operator possibility/necessity/credibility in intuitionistic fuzzy programming (24). Problem (24) can be converted into following problem: where and are the predefined confidence levels.
Step 2. Using Lemmas 10, 11, and/or Lemma 12, the above problem in Step 1 can also be written as where is obtained by applying Lemmas 10, 11, and/or Lemma 12 in (30) and (29).
Step 3. The above problem is equivalent to
Step 4. Crisp programming problem obtained in Step 2 can be solved using any well-known method to get the optimal solution.
6. Numerical Example
Let us consider the following intuitionistic fuzzy mathematical programming problem as: where , , , , , , , and .
6.1. Intuitionistic Fuzzy CCM Based on Possibility Measure
Now by using Step 2 of the method explained in Section 4 and Lemma 10, if we apply the possibility measure in intuitionistic fuzzy mathematical programming (37), problem (37) is converted into the following crisp programming problem: Solving the above crisp problem for efficient levels and different possibility levels, we get different optimal solutions. Optimal solution of (38) at different possibility levels (in Figure 7) are presented in Table 1. From Table 1, we can observe that maximum value () can be obtained at , and , possibility levels.
6.2. Intuitionistic Fuzzy CCM Based on Necessity Measure
Now by using Step 2 of the method explained in Section 4 and Lemma 11, if we apply the necessity measure in (37), problem (37) is converted into following crisp programming problem: Solving the above crisp linear programming problem for efficient levels and different necessity levels, we get different optimal solutions. Optimal solutions of (39) at different necessity levels (in Figures 8 and 9) are presented in Table 2. From Table 2, we can observed that at , and , the decision maker will get the maximum value .
7. Intuitionistic Fuzzy Transportation Problem Based on Possibility Measure
Let us consider the following intuitionistic fuzzy transportation problem (IFTP) (in Table 3).
Above transportation problem is a balanced transportation problem as . The above IFTP can be written as Now by using Step 2 of the method explained in Section 4 and Lemma 10, if we apply the possibility measure in intuitionistic fuzzy mathematical programming (40), problem (40) is converted into following crisp programming problem: Solving the above crisp problem for efficient levels , and possibility levels , for , using Lingo-11.0, we get , , , , , , , , and . Now the minimum intuitionstic fuzzy optimal cost is
8. Discussion
Intuitionistic fuzzy sets being a generalization of fuzzy sets give us an additional possibility to represent imperfect knowledge, making it possible to describe many real problems in a more adequate way. So in this paper, we have developed the possibility and necessity measures on intuitionistic fuzzy set. Here we have presented first time the mathematical representation of different types of measures in intuitionistic fuzzy environments and some graphical representations of them are depicted. We have also developed the theoretical calculation on possibility, necessity, and credibility measures for defuzzify intuitionistic fuzzy linear programming problem using chance operators. To validate the proposed method, we have discussed three different approaches to defuzzify the intuitionistic fuzzy relations using possibility, necessity, and credibility measures. Using chance operator we can convert a problem under imprecise models to corresponding crisp models. At different levels of possibility, necessity, and credibility, we have achieved different optimal solution. A numerical example is presented and solved using LINGO-11.0 to illustrate the proposed approaches. The proposed method can be applied for multiobjective, multiitem transportation problem. This method can be also extended to be applied into different types of optimization problem, namely, optimal control and solid transportation problems.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.