Reference no: EM131467508

In one of the papers (to be cited along with solutions) published in Operations Research journal, the author considers the following problem and its LP model:

Suppose that there are n decision-making units to be evaluated. Each unit consumes m inputs to produce s outputs. In particular, the jth unit consumes amounts X_j = {x_ij} of inputs (I = 1, . . . ,m) and produces amounts Y_j = {y_rj} of outputs (r = 1, . . . ,s). Without loss of generality, it is assumed that all the input and output data x_ij and y_rj (i = 1, . . . ,m; r = 1, . . . , s; j = 1, . . . , n) cannot be exactly obtained due to the existence of uncertainty. They are only known to lie within the upper and lower bounds represented by the intervals [x_ij^L, x_ij^U] and [y_rj^L, y_rj^U], where x_ij^L > 0 and y_rj^L > 0.

Here j_o is the unit under evaluation (usually denoted by subscript 0); u_r and v_i are the weights assigned to the outputs and inputs; θ_jo^U stands for the best possible relative efficiency achieved by the unit under evaluation when all the units are in the state of best production activity, while θ_jo^L stands for the lower bound of the best possible relative efficiency of the unit under observation.

They constitute a possible best relative efficiency interval [θ_jo^L, θ_jo^U].

To analyze the problems the author uses the concept of duality. Formulate the dual problems to problems (1) and (2).