A new approach is supplied for locating efficiency frontiers and evaluating the efficiency of Decision Making Units (DMU's). This is accomplished from observational data by means of an envelopment procedure called DEA (Data Envelopment Analysis) originally developed by Charnes, Cooper and Rhodes (Charnes, A., W. W. Cooper, E. Rhodes, 1978. Measuring the efficiency of decision making units. Eur. J. Oper. Res. 2 (6). See also, Short communication. Eur. J. Oper. Res. 3 (1979) 339.) in connection with their ratio formulation for relative efficiency measurement. The current variant employs a bi-extremal principle which, though nonlinear, is subsequently shown to be reducible to a finite sequence of linear programming problems. The development is illustrated by means of multiple output functions which are piecewise of Cobb-Douglas or general log linear type and which also allow for increasing, decreasing and constant returns to scale. More than one production function for the DMU's is also allowed. The reduction of the bi-extremal principle to linear programming equivalence is also accomplished for much more general classes of functions. A numerical example illustrates some of these ideas and also provides a springboard for a new theorem which relates these efficiency measures to ones which were supplied earlier in the Charnes, Cooper and Rhodes's work (Charnes, A., W. W. Cooper, E. Rhodes, 1978. Measuring the efficiency of decision making units. Eur. J. Oper. Res. 2 (6). See also, Short communication. Eur. J. Oper. Res. 3 (1979) 339.).