Uncertain Dynamical Systems: Analysis and ApplicationsView this Special Issue
General Solutions of Fully Fuzzy Linear Systems
We propose a method to approximate the solutions of fully fuzzy linear system (FFLS), the so-called general solutions. So, we firstly solve the 1-cut position of a system, then some unknown spreads are allocated to each row of an FFLS. Using this methodology, we obtain some general solutions which are placed in the well-known solution sets like Tolerable solution set (TSS) and Controllable solution set (CSS). Finally, we solved two examples in order to demonstrate the ability of the proposed method.
Systems of simulations linear equations play major role in various areas such as mathematics, statistics, and social sciences. Since in many applications, at least some of thesystem’s parameters and measurements are represented by fuzzy rather than crisp numbers, therefore, it is important to develop mathematical models and numerical procedures that would appropriately treat general fuzzy linear systems and solve them.
The system of linear equations where the coefficient matrix is crisp, while is a fuzzy number vector, is called a fuzzy system of linear equations (FSLE). Fuzzy linear systems have been studied by many authors. The first person who suggested the solution for solving FSLE was Fridman. Friedman et al.  proposed a general model to solve FSLE by using an embedding approach. Following Friedman et al. , Ma et al.  analyzed the solution of the duality of fuzzy systems. Allahviranloo et al. suggested some famous numerical methods for solving an FSLE [3–8]. Also, in [9, 10], Abbasbandy et al. proposed the LU-decomposition method and the Steepest descent method to solve system, respectively. For more research see [11–22].
The linear system , where the elements, , of the matrix and the elements, , of the vector are fuzzy numbers, is called a fully fuzzy linear system (FFLS).
Buckley and Qu in their consecutive works [23–25] proposed different solutions for FFLSs. Also, they found relation between these solutions. Based on their works, Muzzioli and Reynaerts in  studied FFLS of the form , while for implementing their method crisp systems should be solved.
Consequently, Dehghan et al. have studied some methods for solving FFLS. They have represented Cramer's rule, Gaussian elimination, fuzzy LU decomposition (Doolittle algorithm), and its simplification; they also have showed the applicability of linear programming approach for overdetermined FFLS in [27–29]. Also, in , Allahviranloo and Mikaeilvand proposed an analytical method to obtain solution of FFLS by an embedding method. Their method is constructed based on obtaining a nonzero solution of the FFLS.
Vroman et al. in their continuous works [31–33] suggested two methods for solving system. In , they have proposed Cramer's rule to solve FFLS approximately, then they proved that their solution is better than Buckley and Qu’s approximate solution vector. Furthermore, they have proposed an algorithm to improve their method to solve FFLS by parametric functions .
Recently, Allahviranloo et al.  have proposed a new practical method to solve an FFLS based on the 1-cut expansion. In their method, some spreads and then some new solutions have been derived that belong to TSS or CSS. Note that they have obtained some spreads which are symmetric.
We show that, using the proposed method in the present paper, we can obtain better solutions. On the other hand, the created errors in some certain cases with respect to the proposed distance are less than the errors that are obtained via Allahviranloo et al.'s method .
The structure of this paper is organized as follows.
In Section 2, we discuss concisely some important basic concepts and definitions which will be used later. In Section 3, we present our new method and concentrate on how we could derive the linear general spreads of fuzzy vector solution corresponding to TSS or CSS. The proposed method is illustrated by solving some examples in Section 4, and conclusion is drawn in Section 5.
Let denote the family of all nonempty compact convex subset of .
Definition 1. A fuzzy number is a function such as satisfying the following properties:(1)is normal, that is, there exists an such that ;(2) is fuzzy convex, that is, for any ;(3) is upper semicontinuous;(4) is compact.The set of all fuzzy real numbers is denoted by .
An alternative definition of fuzzy number is as follows.
Definition 2. A fuzzy number in parametric form is a pair of functions , , , which satisfies the following requirements:(1) is a bounded monotonic increasing left continuous function;(2) is a bounded monotonic decreasing left continuous function;(3), .A popular fuzzy number is the symmetric triangular fuzzy number centered at with basic : We define arbitrary , , addition, subtraction, and multiplication:(1), ,(2), ,(3), ?.
Definition 3. The Hausdorff distance between fuzzy numbers given by ,
where , , is utilized in . Then it is easy to see that is a metric in and has the following properties (see ):(1),? for??all?,(2), for??all ,(3), for??all ,(4) is a complete metric space.
Also, we define the distance between two fuzzy vectors (each vector with fuzzy components) and as follows: where is the th row of FFLS and is the th compnent of fuzzy vector .
Definition 4. The linear systems of equations where the elements, , of the coefficient matrix , , and the elements, , of the vector are fuzzy numbers, are called fully fuzzy linear systems (FFLSs).
Definition 5. A fuzzy vector , given by , is called the solution of (10) if
Definition 7. A fuzzy vector , given by , , is called the minimal symmetric solution of (5) which is placed in CSS, if for any arbitrary symmetric solution which is placed in CSS and , we have where and are symmetric spreads of and , respectively. See [34, 39].
Definition 8. A fuzzy vector , given by , , is called the maximal symmetric solution of (5) which is placed in TSS, if for any arbitrary symmetric solution which is placed in TSS and , we have where and are symmetric spreads of and , respectively. See [34, 39].
3. General Solutions
In this section, we suggest a novel and practical method to obtain general solutions of FFLS. To this end, we solve the 1-cut system (5), which is a crisp system. So, we solve the following crisp system: where and , are unknown crisp variables which determine by solving system (10). Therefore, we fuzzify, the obtained solution from the crisp system (10), by allocating some unknown general spreads (asymmetric spreads) to each row of the system (10).
Then, crisp system (10) is converted to the following system linear equations: In the above system, , are of the obtained of crisp system (11) and , , are unknown spreads. However, for obtaining general solutions of the FFLS, first we have to solve the above equation system, which requires finding the general spreads solution. Now, to solve system (11), we suppose the following one type for the components of fuzzy matrix:
Remark 9. Without loss of generality, we explain our method with the assumption that, in interval , function is positive. We just remove some types which elements of fuzzy matrixes are negative and positive-negative, and also zero does exists in the support of elements of fuzzy matrixes and fuzzy solution. Moreover, the ordering means that if and only if and if and only if . Because of positivity of elements of fuzzy matrix , the th row of system (11) is the supposed like the following: Since, we considered is positive, the compact form of the above equations are calculated as follows: in which (15) and (16) are rendered, respectively, to We offer 4 ways, to determine the spreads of solutions of the FFLS, which are gained as follows: in which the obtained , by using (19) or (20), are as follows: Since, spreads are not linear functions, they may be piece-wise linear functions. Thus, for obtaining linear spreads, we have to make some changes in the structure of the obtained spreads. So, linear forms of spreads are as follows: Let , where supp , then Similarly, consider , then Therefore, we get So, we consider some situations on linear asymmetric spreads of solutions as follows: Hence, the fuzzy vector solution of FFLS, by using general spreads, will be obtained like the following: However, the obtained spreads of the FFLS in crisp manner should be zero, which we will talk about in the following part. Have in mind this feature is applicable for the mentioned type.
Proof. Based on the proposed method, we have assumed that 1-cut position is crisp (since the fuzzy values are triangular). So, all spreads are zero, that is, and then we deduce that the solutions in this case coincide with the 1-cut solution, that is, which completes the proof.
Theorem 11. The solution of fully fuzzy linear system (10) is a fuzzy vector.
Proof. We state the proof for the solution, , and the proof for the other solution, , is similar. So, we omit it. It is easy to verify that satisfies for each . Also, let us consider , then we have So, we deduce that is a fuzzy vector which completes the proof.
Proof. Let us consider , then based on the proposed approach, we have which shows that . Similarly, using the proposed method and definition of , one has which indicates that .
Theorem 13. Consider and as defined in Theorem 12, then one has the following:(1)is maximal general solution in TSS,(2) is minimal general solution in CSS.
Proof. Based on the definition of maximal and minimal general solutions, the proof is straightforward.
Theorem 14. Assuming that , then one has:
The following results show that, under certain conditions, the approach suggested in the present paper has less errors than the Allahviranloo et al.’s method .
Theorem 15. Assuming that , then one has: Also, let , then we have:
Proof. By comparing obtained results in , the proof is straightforward.
Theorem 16. If , then the following property holds:
Proof. Based on definitions of and , we have
Clearly, in the mentioned fuzzy vector solutions, the lower functions of each component, , , are equivalent. So, we can state that .
Now, we discuss the upper functions of the fuzzy vector solutions and . Since , we define the following positive function: Then, after simple calculations we obtain: for all and . It is easy to verify that is a positive function: Consequently, we have Thus, which completes the proof.
In this section, we take an example which has been solved in [28, 30, 34] with their proposed methods. We obtain the maximal-minimal general solutions, which are placed in a TSS and CSS, respectively.
Example 17. Consider the following FFLS: Then, the crisp solution of 1-cut of the FFLS is . So, we fuzzify the crisp system as follows: Now, we ought to solve the equations to find the spreads: which, the obtained spreads from (46)–(49), are as follows, respectively: After, the obtain spreads, by using (26), we determine linear unsymmetric spreads of solutions of the FFLS: In this way, unsymmetric solutions, in which the use of the above mention spreads, will be obtained as follows: We insert the obtained solutions (52), into FFLS, in order to compare the differences between the values of row 1 with and the values of row 2 with (see Figures 1 and 2).
Furthermore, the solutions of the suggested method are plotted to compare with Dehghan's method () which is offered in  and Allahviranloo's methods () which are proposed in [30, 34] (see Figures 3 and 4).
Example 18. Consider the following fully fuzzy linear system:
The 1-cut solution of system is .
So, we have Then, asymmetric linear spreads are derived as the following: Using asymmetric spreads, we obtain We depict all the solutions via Figures 5 and 6.
In this paper, we presented a practicl method for determining the general solutions of a fully fuzzy linear system. To do so, we firstly solved the system in 1-cut form, then we fuzzify 1-cut solution of the FFLS by devoting general spreads. Therefore, the crisp system was changed into a new system that we should have obtained its spreads.
Moreover, we have discussed the obtained result which was placed in the TSS and CSS. Furthermore, we have established that, under certain conditions, proposed method has less errors than the previously reported symmetric solutions. This method is a new approach to find the general solutions of the fully fuzzy linear systems. Also, the presented method always gives a fuzzy vector solution.
The reviewers' comments, which have improved the quality of this paper, are greatly appreciated.
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