Uncertain linear programming with cloud set constraints integrating fuzziness and randomness

Abstract

In reality, there is a significant need for processing methods of uncertain programming problems with fuzziness and randomness. Although there are many extensions of fuzzy sets to handle uncertain information, it is rare to organically integrate randomness and fuzziness, such as the cloud set which is the set extension of cloud model. To address the limitations of uncertain linear programming, which lacks consideration of fuzziness and randomness, a new approach called uncertain linear programming with cloud set constraints is proposed. This approach integrates both fuzziness and randomness properties. Firstly, the definition of uncertain linear programming model with cloud set constraints integrating fuzzy and randomness properties is given. Secondly, the construction method of cloud set membership function of uncertain linear programming model is studied. Thirdly, the solution method for uncertain linear programming model based on Ioperation and P operation in cloud set is argued. Finally, the effectiveness of the method is verified through a specific application example and a comparison case with other linear programs. This method of uncertain linear programming with cloud set constraints utilizes the normal cloud set to represent the objective and conditional constraints, effectively capturing the fuzziness and randomness inherent in these constraints. The contribution of this study is that it proposes the use of random membership degree to replace the subjective fixed single membership in the constraints of linear programming. The membership degree of normal cloud set is expressed by a normal random number with a stable tendency, which integrates fuzziness and randomness with a regular, discrete, sparse thickness instead of a fixed curve. It can more effectively express and adapt to the inherent uncertainty of the objective world because various variables in natural science and social science typically follow a normal distribution. So, it has good objectivity and universality.