A Plmf-Based Decomposition And Coordination Methodology For Fuzzy Programming

To deal with linear programming problems with soft constraints, nonlinear membership functions (such as the Gaussian type) are always employed taking advantage of their objective descriptions for satisfaction degrees. However, nonlinearity of the problems may become more severe due to involvements of the nonlinear membership functions, increasing solving difficulties, even leading to local optimum. In this sense, seeking a methodology which can both retain the characteristics of nonlinear membership functions and linearize the programming model has practical significance.Since nonlinear membership functions can be approximated by piecewise linear ones efficiently, therefore, focusing on fuzzy programing with soft constraints which described by nonlinear membership functions, a decomposition and coordination methodology based on piecewise linear membership functions(PLMFs) has been proposed, which highlights key issues related with the approximations, constructions and solution associated with the corresponding mathematical programming problems. The main research contents are addressed as follows.(1) Aiming at replacing the nonlinear membership functions in fuzzy linear programming with soft constraints, a PLMF construction methodology based on sensitivity analysis, formulation metrics and subjective preferences has been introduced. Thus, the fuzzy programming can be constructed as the model with PLMFs instead of nonlinear membership functions;(2) An improved binary variable representation has been studied for the configuration of PLMFs. By means of this methodology, the programming models with PLMFs can be initially transformed into a decomposable form and then decomposed into several linear sub-models for sake of model linearization.(3) A parallel big M algorithm has been proposed for coordinating the linear sub-models. By the algorithm implementation, the solving process for each sub-model has been carried out simultaneously in order to improve the calculation efficiency, and the overall optimum solutions can be achieved eventually; meanwhile, an error elimination iteration algorithm has been provided for eliminating the error associate to the employment of PLMF, which can remove the error gradually.(4) The feasibility and validity of the proposed methodology has been illustrated employing a production planning optimization problem in a workshop of an oil refinery, and the results are satisfactory.The features of this contribution can be summarized as follows:the nonlinearity in the fuzzy programming is removed by means of the decomposition, thereupon the local optimum problem is avoided completely; subsequently, the models are solved efficiently through the coordination process which can also remove the replacement error to a certain extent. The proposed methodology provides a new way for the utilization of PLMFs in fuzzy programming.