RATES OF SUBSTITUTION

Rates of substitution - Arises when equations are split into dependent (or basic ) variables and independent (or nonbasic) variables. In linear programming , we rewrite Ax=b as Bu + Nv = b, where u is the vector of basic variables and v is the vector of nonbasics. Then, the original equations are equivalent to u = b' + Mv, where b' = B^–1b and M = -B^–1N. This implies that the rate of substitution between u_i and v_j is M(i, j) because it describes the marginal rate at which u_i must change in response to a change in v_j to maintain the equations with all other v's held fixed.