I am having a hard time understanding reduced costs and was wondering if anyone could help me out with this problem?
Consider a linear programming problem in standard form and suppose that x∗ is an optimal basic feasible solution. Consider an optimal basis associated with x∗. Let B and N be the set of basic and nonbasic indices, respectively. Let I be the set of nonbasic indices i for which the corresponding reduced costs c ̄i are zero.
(a) Show that if I is empty, then x∗ is the only optimal solution
(b) Show that x∗ is the unique optimal solution if and only if the following linear programming problem has an optimal value of zero:
maximize (sum of xi over i within I) subject to Ax=b xi=0 , i within N/I xi>=0 , i within B union I