Abstract
A Pythagorean fuzzy set is the superset of fuzzy and intuitionistic fuzzy sets, respectively. Yager proposed the concept of Pythagorean fuzzy sets in which he relaxed the condition that sum of square of both membership degree and nonmembership degree of an element of a set must not be greater than 1. This paper introduces two new techniques to solve -type fully Pythagorean fuzzy linear programming problems with mixed constraints having unrestricted -type Pythagorean fuzzy numbers as variables and parameters by introducing unknown variables and using a ranking function. Furthermore, we show the equivalence of both the proposed methods and compare the solutions obtained by the two techniques. Besides this, we solve an already existing practical model using proposed techniques and compare the result.
1. Introduction
The origin of linear programming is the 1940s (World War II). Linear programming is a technique in which a function (called objective function) is an optimized subject to a given set of restrictions (called constraints). It is mostly used in a situation where there is some quantity to be optimized within available resources. The nature of the linear programming model is trivial and easily applicable to various real-life applications, including transportation problems, supplier selections, assignment problems, production planning problems, and supply chain management. Linear programming in a fuzzy environment is a very interesting field in which many researchers showed interest around the globe. It can be used very effectively in the situations, where the data is fuzzy, vague, or uncertain, where crisp theory fails to cope with. Hence, in these situations, fuzzy linear programming technique is very effective in making decisions.
Zadeh [1, 2] introduced the concepts of fuzzy sets and fuzzy numbers. Bellman and Zadeh [3] first introduced the concept of decision-making in a fuzzy environment. Zimmermann [4] studied the fuzzy programming technique to solve the multiobjective linear programming problem under a fuzzy environment. The fuzzy optimization technique is based on the maximization of the marginal satisfaction (membership functions and degree of belongingness) of each element into the fuzzy decision set. Tanaka et al. [5] also discussed mathematical linear programming in a fuzzy environment. Allahviranloo [6] presented the Adomian decomposition method for a fuzzy system of linear equations. Allahviranloo et al. [7] discovered a method for solving fully fuzzy linear programming problems by the ranking function. Lotfi et al. [8] presented a method for solving a full fuzzy linear programming using lexicography method and fuzzy approximate solution. Kumar et al. [9] proposed a new method for solving fully fuzzy linear programming problems. Kumar and Kaur [10] studied a method for exact fuzzy optimal solution of fully fuzzy linear programming problems with unrestricted fuzzy variables. Kaur and Kumar [11] presented Mehar’s method for solving fully fuzzy linear programming problems with fuzzy parameters. Moloudzadeh et al. [12] introduced a new method for solving an arbitrary fully fuzzy linear system. Behera et al. [13] studied new methods for solving imprecisely defined linear programming problems under trapezoidal fuzzy uncertainty. Najafi and Edalatpanah [14] introduced a new method for solving fully fuzzy linear programming problems. Prez-Caedo et al. [15] gave a revised version of a lexicographical-based method for solving fully fuzzy linear programming problems with inequality constraints.
Later on, it was realized that only the membership degrees are not well enough to represent the marginal attainment of the element into the fuzzy decision set. To extend or explore the fuzzy set, Atanassov [16] introduced the concept of fuzzy set to intuitionistic fuzzy set in which there is a nonmembership function along with the membership function. In an intuitionistic fuzzy set, the sum of membership degree and nonmembership degree of an element should not be greater than 1. Angelov [17] first considered the intuitionistic fuzzy optimization techniques based on intuitionistic fuzzy decision set in decision-making problems. Dubey and Mehra [18] presented linear programming with triangular intuitionistic fuzzy numbers. Parvathi and Malathi [19] proposed a method to solve intuitionistic fuzzy linear programming problems. Nagoorgani and Ponnalagu [20] revealed a new approach on solving intuitionistic fuzzy linear programming problems. Parvathi and Malathi [21] studied intuitionistic fuzzy linear optimization. Parvathi et al. [22] gave intuitionistic fuzzy linear regression analysis. Garg et al. [23] presented an intuitionistic fuzzy optimization technique for solving multiobjective reliability optimization problems in an interval environment. Suresh et al. [24] gave a method of solving intuitionistic fuzzy linear programming problems by ranking function. Nagoorgani et al. [25] presented the knowledge of expert opinion in intuitionistic fuzzy linear programming problems. Singh and Yadav [26] proposed optimization of unrestricted -type intuitionistic fuzzy mathematical programming problems. Singh and Yadav [27] proposed intuitionistic fuzzy multiobjective linear programming problems with various membership functions. Malathi and Umadevi [28] gave a new procedure for solving linear programming problems in an intuitionistic fuzzy environment. Abhishekh and Nishad [29] proposed a novel ranking approach for solving fully -intuitionistic fuzzy transportation problem. Bharati and Singh [30] studied a method for the solution of multiobjective linear programming problems in interval-valued intuitionistic fuzzy environments. Kabiraj et al. [31] proposed another method for intuitionistic fuzzy linear programming problems. Perez-Canedo and Concepcion-Morales [15] presented a method for unique optimal values of -type fully intuitionistic fuzzy linear programming with inequality constraints.
Unfortunately, intuitionistic fuzzy sets fail to deal with the situations where the sum of membership degree and nonmembership degree of an element exceeds 1. To overcome this difficulty, Yager [32] introduced the concept of Pythagorean fuzzy set in which he relaxed the condition that sum of square of both membership degree and nonmembership degree of an element of a set must not be greater than 1. Yager and Abbasov [33] presented Pythagorean membership grades, complex numbers, and decision-making. Yager [34] introduced Pythagorean membership grades in multicriteria decision-making. Zhang and Xu [35] extended the TOPSIS to multiple-criteria decision-making with Pythagorean fuzzy sets. Peng et al. [36] studied Pythagorean fuzzy information measures and their applications. Wan et al. [37] gave a Pythagorean fuzzy mathematical programming method for multiattribute group decision-making with Pythagorean fuzzy truth degrees. Kumar et al. [38] proposed a Pythagorean fuzzy approach to the transportation problem. Luqman et al. [39] presented a digraph and matrix approach for risk evaluations under Pythagorean fuzzy information. Wan et al. [40] gave a new order relation for Pythagorean fuzzy numbers and its application to multiattribute group decision-making. On the contrary, Ahmad et al. [41] studied spherical fuzzy linear programming problems and Akram et al. [42] developed methods to solve fully Pythagorean fuzzy linear programming problems with equality constraints. Wei et al. [43] studied green supplier selection based on the CODAS method in a probabilistic uncertain linguistic environment. Wei et al. [44] extended COPRAS model for multiple attribute group decision-making based on single-valued neutrosophic 2-tuple linguistic environment. Zhang et al. [45] studied the TODIM method based on cumulative prospect theory for multiple attribute group decision-making under a 2-tuple linguistic Pythagorean fuzzy environment. Recently, Akram et al. [46] have introduced a method to solve linear programming problems in which all the variables and parameters are -type PFNs having equality constraints. As a continuation of this work, we study two methods for solving FPFLPPs with mixed constraints in which all the variables and parameters are unrestricted -type Pythagorean fuzzy numbers (PFNs).
The main contribution of this research paper is as follows:(1)The first method is presented to solve FPFLPPs with mixed constraints, in which all the variables and parameters are -type PFNs(2)The second method is presented to handle inequality constraints in FPFLPPs using a ranking function(3)A comparison of the proposed methods with the existing methods is given(4)The results of the methods are shown graphically
The article is organized as follows. Section 2 presents some preliminaries. Section 3 explains the proposed methods to solve FPFLPPs with unrestricted -type PFNs with mixed constraints. Section 4 presents the equivalence of the proposed techniques. Section 5 is devoted for numerical examples to explain the proposed methods. Section 6 includes comparative analysis and some discussions. In Section 7, merits of the proposed methods are given. In Section 8, we conclude the paper.
For further information, the reader can refer to [47–57].
2. Preliminaries
In this section, we review elementary concepts that are useful for this article.
Definition 1. (see [34]). A PFS on is an object of the form:where and are membership function and nonmembership function of , respectively, such that . Moreover, for all , is called a Pythagorean fuzzy index or degree of hesitancy of in . For computational convenience, is called a PFN [35].
Definition 2 (see [46]). A PFN is defined as an -type PFN if its membership and nonmembership functions are given aswhere and . and are continuous, decreasing functions on and and are continuous increasing functions on such that(1)(2)(3)(4) is called the mean value of , and are the left and right spreads of , and and are left and right spreads of , respectively.
Remark 1 (see [46]). (1)If we set and in Definition 2, then becomes -type intuitionistic fuzzy number [26].(2)If we takein Definition 2, then becomes triangular PFN.
Definition 3. (see [46]). An -type PFN is nonnegative (respectively, nonpositive), denoted as (respectively, ), if (respectively ) and is unrestricted if belongs to real numbers.
Definition 4. (see [46]). An -type PFN is positive if and negative if .
Definition 5. (see [46]). An -type PFN is zero if and only if , , , , and .
Definition 6. (see [46]). Two -type PFNs and are equal if , , , .
Theorem 1 (see [46]). Let and be two -type PFNs; then, .
Theorem 2 (see [46]). Let and be two -type PFNs; then, .
Theorem 3 (see [46]). Let be an -type PFN and c be any real number; then,
Theorem 4 (see [46]). Let and be two nonnegative -type PFNs; then, .
Theorem 5 (see [46]). Let be nonnegative -type PFN and be nonpositive -type PFN; then, .
Theorem 6 (see [46]). Let be an -type PFN in which , and be an unrestricted -type PFN; then, , where :
Theorem 7 (see [46]). Let be an -type PFN in which , and be an unrestricted -type PFN; then, , where :
Theorem 8 (see [46]). Let be an -type PFN in which , and be an unrestricted -type PFN; then, , where :
Theorem 9 (see [46]). Let be an -type PFN in which , and be an unrestricted -type PFN; then, , where :
Theorem 10 (see [46]). Let be an -type PFN in which and be an unrestricted -type PFN; then, , where :
Theorem 11 (see [46]). Let be an -type PFN in which and be an unrestricted -type PFN; then, , where :
Definition 7. (see [46]). Let be an -type PFN; then, ranking of , denoted , can be defined asLet and be two -type PFN; then, we see that(i) if (ii) if (iii) if
Remark 2. (see [46]). Ranking function, as defined in Definition 7, is a linear function.
Proof. Let and be two -type PFNs; then, .
Now, if ,Similarly, for some scalar , .
Hence, ranking function, as defined in Definition 7, is a linear function.
3. Methodology to Solve -Type Fully Pythagorean Fuzzy Linear Programming Problems
We state here our proposed FPFLPP with -type PFNs aswhich subject towhere , and are -type PFNs.
3.1. Method 1: FPFLPP Using Unknown Variables
Here, we state a criterion for the optimal solution of FPFLPP (13).
Definition 8. An -type Pythagorean fuzzy optimal solution of FPFLPP (13) with -type PFNs will be -type PFNs if(1) are -type PFNs(2), , such that for some PFNs and satisfying ; , , such that for some PFNs and satisfying ; and , , such that (3)If there exist any -type PFNs satisfying step 2, then in maximization problem and in minimization problem
The statement of our proposed problem is given in equation (13). We now present steps to solve proposed FPFLPP (13). Step 1: separating all the constraints into three categories, , and , the FPFLPP (13) can be rewritten as which subject to where are -type PFNs and , , and . Step 2: introduce the variable on left side and on right side of the inequality constraint , to convert it into equality constraint as below: where . Introduce the variable on left side and on right side of the inequality constraint , to convert it into equality constraint as below: where . The FPFLPP (15) can be written as which subject to where , and are -type PFNs. Step 3: by assuming , , , , , and , the FPFLPP (19) can be rewritten as which subject to where , , , , and are -type PFN. Step 4: by using the product as discussed in Section 2 and taking , the FPFLPP (21) can be written as where , , , , and are -type PFN. Step 5: using arithmetic operations as discussed in Section 2 and using Definition 6, the FPFLPP (23) takes the form which subject to where , , , , and are -type PFN. Step 6: now, we have to find -type Pythagorean fuzzy feasible solution out of all -type Pythagorean fuzzy feasible solutions corresponding to which the ranking of the objective is optimum. By applying ranking, the FPFLPP (24) can be written as which subject to , , , , , , , and , and . Step 7: by taking , problem (26) can be written as which subject to , , , , , , , and , and . Step 8: by using the linearity property of ranking function, problem (28) takes the form which subject to , , , , , , , and , and . Step 9: by using Definition 7, problem (30) can be converted into which subject to , , , , , , , and , and . Step 10: now, solve the crisp linear/nonlinear programming problem (32) by any existing method to find the optimal solution . Step 11: find the -type Pythagorean fuzzy optimal solution of the FPFLPP (13) by substituting the values of , , , , and in . Step 12: find the -type Pythagorean fuzzy optimal value of the FPFLPP (13) by substituting the values of , as calculated in Step (11), in .
3.2. Method 2: FPFLPP Using Ranking Function
Now, we present another method to solve FPFLPP (13). We present a criterion for the optimal solution.
Definition 9. An -type Pythagorean fuzzy optimal solution of FPFLPP (13) with -type PFNs will be -type PFNs if(1) are -type PFNs(2), for all (3)If there exist any -type PFNs satisfying step 2, then in maximization problem and in minimization problem Step 1: by assuming , , , and , the FPFLPP (15) can be rewritten as which subject to where is an -type PFN. Step 2: by using the product as discussed in Section 2 and taking , the FPFLPP (34) can be written as where is an -type PFN. Step 3: using arithmetic operations as discussed in Section 2 and using Definition 6, the FPFLPP (36) can be rewritten as which subject to where is an -type PFN. Step 4: by applying ranking, the FPFLPP (37) takes the form which subject to , , , and , . Step 5: by taking , problem (39) can be rewritten as which subject to , , , and , . Step 6: by using the linearity of ranking function, problem (41) can be written as which subject to , , , and , . Step 7: by using Definition 7, problem (43) can be converted into problem (45): which subject to , , , and , . Step 8: solve the crisp linear/nonlinear programming problem (45) by any existing method to find the optimal solution . Step 9: find the -type Pythagorean fuzzy optimal solution of the FPFLPP (13) by substituting the values of , , , , and in . Step 10: find the -type Pythagorean fuzzy optimal value of the FPFLPP (13) by substituting the values of , as calculated in Step (9), in .
4. Equivalence of the Proposed Methods
Here, we confirm that the two techniques proposed in Section 3.1 and Section 3.2 give the same solution.
If and are any two PFNs such that , then . Thus, the and constraints [, ] of Problem (19) can be written as
Since the ranking function as discussed in Definition 7 is linear, so equations (47) and (48) can be written as
Thus, equations (49) and (50) can be written as
Now, by using the and constraints [] of problem (19), equations (51) and (52) convert toor
Hence, the proposed techniques (method 1 and method 2) are equivalent. Both the techniques give almost the same solution. However, there is a little bit of difference that when solving the translated crisp problem, one of them may give an answer more faster than the other one. So, depending on the initial guess for the solver, technique which gives faster optimal solution is not known in advance.
5. Numerical Examples
Example 1. A farmer has square-feet land. He wants to grow two types of plants, namely, X and Y. Each X plant needs square-feet of land and man-per-hour labor. Each Y plant needs square-feet of land and man-per-hour labor. Maximum labor which is available is man-per-hour. Profit for each X plant is and for each Y plant is . Farmer wants to maximize his profit subject to give available resources with and .
Let be the number of X plants and be the number of Y plants that farmer should grow. Then, the problem converts to the following -type FPFLPP: which subject towhere are -type PFNs for and and .
Now, we solve Example 1 by using method 1 as discussed in Section 3.1. Step 1: by applying Step 2 of the presented method 1 in Section 3.1, the problem becomes which subject to , where , and are -type PFNs for and and . Step 2: let , , , and , and the problem obtained in Step (1) can be written as which subject to where , , , and are -type PFNs. Step 3: using the product as discussed in Section 2, the FPFLPP, obtained in Step 2, can be written as which subject to where , , , and are -type PFNs. Step 4: by using arithmetic operations discussed in Section 2 and using Definition 6, the FPFLPP, obtained in Step 3, can be rewritten as which subject to , where , , , and are -type PFNs. Step 5: using Step 6 of the proposed method 1, the FPFLPP, obtained in Step 4, can be rewritten as which subject to ,,. Step 6: using and and Steps 8 and 9 of method 1, presented in Section 3.1, the FPFLPP, obtained in Step 5, can be written as which subject to , . Step 7: the optimal solution of the crisp nonlinear programming problem, obtained in Step 5(using: MATLAB R2014a, solver “fmincon,” algorithm “interior point,” TolFun = eps, TolX = eps, TolCon = 0.000001) is , , , , , , , , , and . Step 8: substituting the values of , and in and , the exact -type Pythagorean fuzzy optimal solution is , and . Step 9: substituting the values of and , obtained in Step 8, into the objective function, the -type Pythagorean fuzzy optimal value is . So, the farmer should grow number of plants of X and number of plants of Y to get a maximum profit of .We now solve Example 1 by using method 2 as discussed in Section 3.2: Step 1: let and ; then, problem of Example 1 can be written as which subject to where and are -type PFNs. Step 2: using the product as discussed in Section 2, the FPFLPP, obtained in Step 1, can be written as which subject to where and are -type PFNs. Step 3: by using arithmetic operations discussed in Section 2 and Definition 6, the FPFLPP, obtained in Step 2, can be rewritten as which subject to , where and are -type PFNs. Step 4: using Step 4 of the proposed method 2, the FPFLPP, obtained in Step 3, can be rewritten as which subject to ,. Step 5: using and and Steps 6 and 7 of the presented method 2, the FPFLPP, obtained in Step 4, can be written as which subject to . Step 6: the optimal solution of the crisp nonlinear programming problem, obtained in Step 5(using MATLAB R2014a, solver “fmincon,” algorithm “interior point,” TolFun = 1, TolX = eps, TolCon = 1), is , , , , , , , , , and . Step 7: substituting the values of , and in and , the exact -type Pythagorean fuzzy optimal solution is and . Step 8: substituting the values of and , obtained in Step 7 into the objective function, the -type Pythagorean fuzzy optimal value is .So, according to this technique, the farmer should grow number of plants of X and number of plants of Y to get a maximum profit of .
Example 2. Let us solve the practical model, discussed in [15], by method 1 as discussed in Section 3.1, using and :which subject towhere are -type PFNs for .
Let , , , , , and . By solving step by step as discussed in Section 3.1, we obtain the crisp problem as given underwhich subject to, , , , .
The optimal solution of this nonlinear problem (using MATLAB R2014a, solver “fmincon,” algorithm “interior point,” TolFun = 0.1, TolX = eps, TolCon = 0.1) is , , and . The optimal value of this problem is .
6. Comparison with Existing Linear Programming Model
Prez-Caedo et al. [15] developed a method to solve -type fully intuitionistic fuzzy linear programming model. We have proposed two methods to solve -type FPFLPP with mixed constraints. By applying the proposed methods to Example 1, we have obtained the solution. Results of Example 1 are given in Table 1 and are shown graphically in Figure 1. Furthermore, we have solved the practical model [15] by using and , and results are given in Table 2. Solution with existing method [15] with permutation and solution with method 1 as discussed in Section 3.1 are compared in Figure 2. We observe the following facts:(1)Our proposed methods are equivalent in terms of handling the inequality constraints of FPFLPP with -type PFNs as variables and parameters.(2)Example 1 is solved by using our proposed methods. We see from Table 1 and Figure 1 that both methods produce the optimal solution which is almost the same.(3)The solutions of both the methods (method 1 and method 2) are obtained by solving ultimately a crisp linear programming problem, which is mostly done using any software. The iterations needed for the solution of the crisp problem may vary problem to problem and may also depend on one of the methods used.(4)We compare the solutions of our proposed method 1 with the existing method [15]. We see from Table 2 and Figure 2 that both the solutions are consistent to a large extent.
7. Merits of the Proposed Methods
The proposed mathematical model is based on the Pythagorean fuzzy environment. The advantages of the proposed method as compared to the existing method are as follows:(1)There is no method to solve FPFLPP in which all the variables and parameters are unrestricted -type PFNs. Thus, this contribution is new and very helpful for the decision makers.(2)A Pythagorean fuzzy model is more powerful than an intuitionistic fuzzy model since the intuitionistic fuzzy model cannot handle the situation where sum of membership degree and nonmembership degree of an element exceed 1. So, these techniques are more general and can be used in an intuitionistic fuzzy environment or fuzzy environment.(3)The proposed techniques give almost the same results, so these techniques can be used depending on the interest of the decision maker.
Apart from all the benefits, the presented methods have some limitations. Our proposed methods fail where the condition .
8. Conclusions and Future Directions
In mathematical programming models, linear programming problems are the simplest and most extensively used model. The linear programming model is easily applicable to various real-life applications. In this article, we have studied two techniques to solve -type FPFLPP with mixed constraints. We have shown the equivalence of both the presented methods. We have compared the results obtained from both the proposed techniques which come out to be almost the same. Furthermore, we have compared with the existing method [15]. In the future, our research work can be extended for nonlinear programming problems, fractional programming problems, and transportation problems.
Data Availability
No data were used to support this study.
Ethical Approval
This article does not contain any studies with human participants or animals performed by any of the authors.
Conflicts of Interest
The authors declare that they have no conflicts of interest.