Past exam of the Probability, statistics and options II course of the University of Oxford / 2007 / T / 2 / c

In the usual linear programming sensitivity within a fixed optimal basis, retain the optimal basis $\{x_1,x_3\}$ while allowing its values to adjust. If the second resource amount is $b$, thenThe dual prices remain $(1,1)$, so optimality persists exactly while both basic variables are nonnegative:The optimal profit on that range is $33+b$. At the endpoints the basis is degenerate but remains optimal. If the wording is instead read as requiring the exact numerical production vector $(9,0,3)$ to remain unchanged and optimal, only $b=18$ does so; the standard allowable range concerns the basis and its adjusted solution.

Let the third profit coefficient be $c_3$. For the same basis, the resource dual prices solve $2y_1+y_2=3$ and $5y_1+3y_2=c_3$, givingDual feasibility requires $y_1,y_2\ge 0$ and $3y_1+2y_2\ge 4$. These inequalities are $c_3\le 9$, $c_3\ge 15/2$ and $c_3\ge 7$, respectively. HenceThe production vector is unchanged because the resource constraints are unchanged. Equality at a boundary can allow additional optimal solutions.

Changing only the second profit coefficient leaves the dual prices at $(1,1)$, because the basic products remain one and three. The nonbasic product has nonpositive reduced cost exactly when $c_2\le 3(1)+2(1)=5$. ThereforeThere is no mathematical lower bound; if profit coefficients are additionally restricted to be nonnegative, intersect this range with $[0,\infty )$. At $c_2=5$, alternate optimal solutions are possible.