Aggregation of Objective (Criteria) Functions

Let’s have a multi criteria programming problem:

where

x = (x_j)_n … vector of model variables

z_k(x) … k-th criteria function

c_k = (c_kj)_{n …} matrix of cost coefficients of k-th criteria function

A = (a_ij)_{m.n …} matrix of coefficients of constraints

b = (b_i)_m … RHS vector

All criteria functions can be aggregated into a single one by the following way

where

Z(x) … aggregated criteria function

v_k … k-th criteria function weight