A Simple Class of Parametric Linear Programming Problems

To find changes in coefficients that can occur without affecting the choice of optimal feasible basic variables is a well known problem in linear programming sensitivity theory. This paper shows that there is a very simple solution to this problem when the m × m matrix A of optimal basis vectors is varied parametrically, provided the matrix B of variations is chosen so that the inverse of A + λ B is linear in λ. It gives a general expression for such matrices B, which allows a considerable degree of arbitrariness; in particular, there exist B's, that can have rank as high as m/2. Extensions to include variations in the nonbasic part of the coefficient matrix and in the objective function coefficients are briefly described.