Читать книгу Optimizations and Programming - Bouchaib Radi, Ghias Kharmanda, Michel Ledoux - Страница 39
Case of a non-basic variable
ОглавлениеWrite ei for the canonical basis vector of ℝn, i.e. ei = (0, 0, . . . , 1, 0, . . . , 0)T with 1 at position i. Then:
The ith component of the vector is
where the ri are the components of the final row of the simplex tableau. In other words, we have all the information that we need to compute the stability intervals of the coefficients ci.
EXAMPLE 1.13.– Again, consider the previous example. The final simplex tableau is:
The last row gives us the vector of reduced costs The basic variables are B = {x1, x2}, and the non-basic variables are N = {x3, x4, x5}. The objective function is of the form z = c1x1 + c2x2 + c3x3 with and c3 = 3. It does not make sense to perturb the coefficients c4 and c5. Accordingly, we can only perturb x3. Let us compute the stability interval around c3 associated with the non-basic variable x3. The stability condition is
The optimal solution x = (2, 4, 0, 0, 0)T is the same for every value Furthermore, the value of the objective function (z = 8) remains unchanged because the variable is non-basic.
