In one of the papers to be cited along with solutions

Question

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).