Data Envelopment Analysis (DEA) is a widely used mathematical programming technique for comparing the inputs and outputs of a set of homogenous Decision Making Units (DMUs) by evaluating their relative efficiency. The conventional DEA methods assume deterministic and precise values for the input and output observations. However, the observed values of the input and output data in real-world problems can potentially be both random and fuzzy in nature. We introduce Random Fuzzy (Ra-Fu) variables in DEA where randomness and vagueness coexist in the same problem. In this paper, we propose three DEA models for measuring the radial efficiency of DMUs when the input and output data are Ra-Fu variables with Poisson, uniform and normal distributions. We then extend the formulation of the possibility–probability and the necessity–probability DEA models with Ra-Fu parameters for a production possibility set where the Ra-Fu inputs and outputs have normal distributions with fuzzy means and variances. We finally propose the general possibility–probability and necessity–probability DEA models with fuzzy thresholds. A set of numerical examples and a case study are presented to demonstrate the efficacy of the procedures and algorithms. 
Chance-constrained DEA models with random fuzzy inputs and outputs