Parameterizing an Activity Vector in Linear Programming

This paper presents an algorithm that will parameterize an activity vector for a linear programming problem in the following manner. Assume that an optimum feasible basis for a parametric linear programming problem has been obtained for the low-bound value of the parameter θ, set to zero. The algorithm then determines a sequence of critical values θ1, …, θk, …, θT, where θk ≧ θk−1, and a series of bases B1, …, Bk, …, BT, in such a way that Bk is optimum feasible for θk ≦ θ ≦ θk+1 for 1 ≦ k ≦ T − 1 and BT is either optimum feasible for all θT ≦ θ ≦ γ or unbounded or infeasible for 0 ≦ θ − θT ≦ ϵ, where ϵ is an arbitrarily small positive number. The essence of the algorithm consists of a series of simple transformations from one optimum feasible basis to another. The algorithm is illustrated with a numerical example.