Fuzzy Malmquist Productivity Index

The Malmquist productivity index (MPI) has been widely used in measuring the change in performance of a unit between two periods. This paper extends the conventional MPI from an environment of certainty to uncertainty, where the imprecise observations are represented by fuzzy numbers. An example explains how to derive the fuzzy MPI analytically from the fuzzy observations for very simple problems. For general problems, the extension principle is applied to formulate the problem of finding the a-cut of the fuzzy MPI as a pair of two-level mathematical programs. The twolevel program is converted to a bound-constrained nonlinear program, and a quasi-Newton method is devised for a solution. The fuzzy MPI is derived numerically from the a-cuts at various a values. The case of Taiwanese forests after reorganization is used to illustrate the validity of the proposed model and the solution method devised for obtaining the fuzzy MPI. The results also demonstrate that the fuzzy approach provides more information than the conventional approach, which simplifies the problem by assuming the imprecise observations to have precise values.