A Method For Solving Interval-valued Fuzzy Linear Programming Problems And Its Application

Fuzzy optimization is a combination of fuzzy mathematics and optimization theory,among which fuzzy linear programming is an important branch of fuzzy optimization.As the promotion of fuzzy linear programming,interval-valued fuzzy linear programming has been widely used in the fields of water resources management,investment portfolio,transportation management,project scheduling and so on.The search for the solution of interval-valued fuzzy linear programming is one of the hot research topics.The effective methods in the literature include the signed distance ranking method and the infinite α-cut set method.In this thesis,the interval-valued level cut set method is discussed to solve this kind of problem,and the method is applied to the actual problem solving.By comparing with the signed distance ranking method,the effectiveness of the method is proved.The specific research results are as follows:In Chapter 1,the research status of fuzzy linear programming and interval-valued fuzzy linear programming is described,and the relevant theoretical knowledge of fuzzy set,interval-valued fuzzy set and cut set is introduced.In Chapter 2,based on the three-valued fuzzy sets,the interval-valued level cut sets of the interval-valued fuzzy sets and the interval-valued triangular fuzzy numbers are defined respectively,and the corresponding properties are proved.The properties are exactly the same as the Zadeh cut sets,which shows that the definition is reasonable.In Chapter 3,the theoretical reasoning of the signed distance ranking method is introduced for the whole coefficient fuzzy interval-valued fuzzy linear programming problem,and the theoretical framework of the interval-valued level cut set method for solving this problem is constructed: Firstly,for any interval-valued level α,the corresponding new levelα’ is constructed,and the original objective function is transformed into membership function about interval-valued level α ’ by extension principle;Then,based on the membership function,a pair of two-level mathematical programming is constructed to determine the upper and lower bounds of the objective function of the interval-valued fuzzy linear programming.Based on the dual theory of linear programming,the two-level mathematical programming is transformed into a traditional linear programming model,so as to determine the interval of the objective function of the interval-valued fuzzy linear programming;Finally,the program of interval-valued level cut set method is complied by Matlab.The effectiveness of the interval-valued level cut set method for solving interval-valued fuzzy linear programming problem is verified by a concrete example.In Chapter 4,an interval-valued fuzzy linear programming model is established for the actual production planning problem,which is solved by signed distance ranking method and the interval-valued level cut set method respectively.The numerical results show that: The optimal value obtained by the signed distance ranking method belongs to the optimal value interval obtained by the interval-valued level cut set method.When the interval-valued fuzzy linear programming model degenerates into the traditional linear programming model,the optimal value also belongs to the optimal value interval.