Modeling data envelopment analysis by chance method in hybrid uncertain environments

Abstract

This article first presents several formulas of chance distributions for trapezoidal fuzzy random variables and their functions, then develops a new class of chance model (C-model for short) about data envelopment analysis (DEA) in fuzzy random environments, in which the inputs and outputs are assumed to be characterized by fuzzy random variables with known possibility and probability distributions. Since the objective and constraint functions contain the chance of fuzzy random events, for general fuzzy random inputs and outputs, we suggest an approximation method to compute the chance. When the inputs and outputs are mutually independent trapezoidal fuzzy random variables, we can turn the chance constraints and the chance objective into their equivalent stochastic ones by applying the established formulas for the chance distributions. In the case when the inputs and the outputs are mutually independent trapezoidal fuzzy random vectors, the proposed C-model can be transformed to its equivalent stochastic programming one, in which the objective and the constraint functions include a number of standard normal distribution functions. To solve such an equivalent stochastic programming, we design a hybrid algorithm by integrating Monte Carlo (MC) simulation and genetic algorithm (GA), in which MC simulation is used to calculate standard normal distribution functions, and GA is used to solve the optimization problems. Finally, one numerical example is presented to demonstrate the proposed modeling idea and the efficiency in the proposed model.