Fuzzy Solution Of Fuzzy Equations And Fuzzy Programming Problems

The concept of fuzzy set was first put forward by Zadeh in 1965 and the research of fuzzy set beginning.When it comes,it brings to an wide affection.And applies to artificial intelligence,system evaluation,automatic control and so on.Along with Age of Big Data,how to get the optimum information resources is important in many information resources.That is why the fuzzy equation and fuzzy programming becoming.Its known that fuzzy equation and fuzzy programming problems are important research fields in mathematics,and have strong application background.In these research fields,many important results have been made.In [3],the paper discussed fuzzy equation?A ?x = ?b,where,the elements of fuzzy matrix A and the right hand b were trapezoidal fuzzy numbers.They argued that the previous method of solving for x,based on the extension principle and regular fuzzy arithmetic,should be abandoned since it too often fails to produce a solution.The matrix A was square and always non-singular.And proofed if A was singular,the solution x of fuzzy equation might be more complicated.In paper[1],the author discussed the basic arithmetic operations of the interval-valued trapezoidal fuzzy numbers and the necessary and sufficient condition for existence of solutions of linear equation AX = C.T.Allahviranloo,KH.Shamsolkotabi,N.A.Kiani and L.Alizadeh discussed fuzzy programming problems was reduced to two crisp integer linear programming problems,where,the right hand column b was fuzzy column,A,x were real numbers.And they managed to proof it by the core in [22].M.Duran Toksari and Y.Bilim presented a algorithm that is presented to solve multi-objective programming problems with single decision maker at the first level and multiple decision makers at the second level.And used Jacobian matrix to linearize membership functions associated with each objective of the problem in [23].We can see that there are two conditions in solving fuzzy linear equations of system and fuzzy linear programming problems.On the one hand,the unknown vector x is a real vector.Either A or b is fuzzy number,either A or b is real number.On the other hand,the unknown vector x is fuzzy vector.Either A or b is fuzzy number,either A or b is real number.In a fuzzy condition,A and b maybe fuzzy number.But the unsolved vector x maybe real number or maybe fuzzy number.In this paper,we give a condition that A and b are fuzzy number,but x is real number and we proof the solution of fuzzy linear equations and fuzzy linear programming are fuzzy number.At first,according to the general expression of fuzzy linear equation and the properties of fuzzy r-level set,the fuzzy linear equations is turned into fuzzy r-level set linear equations.Then,the r-level set solution of fuzzy linear equation is given by the general expression of fuzzy linear solution and properties of r-level set and the feasible and of solution is proofed.And then,by the feasible of solution of fuzzy programming,the definition of r-solution and arithmetic of fuzzy linear programming are defined and gotten.At last,r-best optimal solution and r-worst optimal solution and r-best optimal value and r-worst optimal value are defined,and the properties of solution are researched.The main work and results of this paper lie in the following:1.In the Chapter 1,we mainly introduce the research background,purpose and meaning.2.In the Chapter 2,some basic concepts of fuzzy set theory and fuzzy numbers are introduced.3.In the Chapter 3,we introduced the concept of fuzzy equations system and solution.The linear equations with fuzzy number coefficients are successfully translated into an equivalent linear equations with interval number coefficients.The solution of fuzzy interval linear equations system and fuzzy solution were defined.The theorem of fuzzy interval linear equations system was gave.At last,two examples to show the application of the method.4.In the Chapter 4,we introduced the concept of fuzzy programming problems.And the fuzzy programming problems were translated into an equivalent fuzzy programming problems with fuzzy interval by the fuzzy order.We gave a theory that the solution of Ax ? b is limited to r-minimum range and r-maximum range of two systems.The theory of solution of fuzzy programming problems was gave.If trapezoidal fuzzy number was given,the optimum solution of fuzzy programming problems was gotten.And then we proofed the r-level of Ax ? b was equal to r-minimum range and r-maximum range of Ax ? b.5.In the Chapter 5,we summarize the article and talk about the future research work.