Math Models of OR: Sensitivity Analysis John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 8 USA October 8 Mitchell Sensitivity Analysis / 9
Optimal tableau and pivot matrix Outline Optimal tableau and pivot matrix Increase nonbasic variable Increase resource available Change selling price 5 New product available 6 Additional constraints violated by current solution Mitchell Sensitivity Analysis / 9
Optimal tableau and pivot matrix An example problem Consider the linear programming problem min x x s.t. x + x x + x 6 (P) x x 6 x i i =,. How does the optimal solution change if the problem changes? How sensitive is the solution to the parameters? Mitchell Sensitivity Analysis / 9
Optimal tableau and pivot matrix Graphing the problem x x x = x + x = 6, x = x x = 6, x 5 = (,) x + x =, x = x Mitchell Sensitivity Analysis / 9
Optimal tableau and pivot matrix Tableaus The initial tableau for this problem with slack variables x, x, x 5 is M = x x x x x 5 6 6 The optimal tableau for this problem is M = x x x x x 5 7 Mitchell Sensitivity Analysis 5 / 9
Optimal tableau and pivot matrix Pivot matrix The pivot matrix Q satisfying M = QM is Q = 7. Mitchell Sensitivity Analysis 6 / 9
Increase nonbasic variable Outline Optimal tableau and pivot matrix Increase nonbasic variable Increase resource available Change selling price 5 New product available 6 Additional constraints violated by current solution Mitchell Sensitivity Analysis 7 / 9
Increase nonbasic variable Increase a nonbasic variable Want x = a >, with a fixed. How to get new optimal solution? What if a >? Approach: place the x column into the right hand side column. Reoptimize using dual simplex if necessary. Modified optimal tableau is: M = x x x x 5 x + 7 x x x The modified tableau is in optimal form if 6 7 x, with optimal value + x. The same variables are basic, but their values have changed. We now have x = x, x = x, x = + 7 x. Mitchell Sensitivity Analysis 8 / 9
Increase nonbasic variable Graphing the modified problem x x + x = 6, x = x x = + a x x = 6, x 5 = x + x = 6 a, x = a (,) x + x =, x = ( a, a) x Mitchell Sensitivity Analysis 9 / 9
Increase nonbasic variable Larger changes in x If x = a >, then we need to use dual simplex to reoptimize. x x x x 5 x + 7 x x x x x x x 5 x 6 + x 6 + x 6 x The variable x is now nonbasic, and x 5 is basic. The updated objective function value is + x, so the shadow price has increased to as the resource is consumed. This updated tableau is in optimal form provided x 6. If x increases beyond 6 then further reoptimization using dual simplex is required (the problem becomes infeasible). Mitchell Sensitivity Analysis / 9
Increase nonbasic variable Graphing the modified problem x x + x = 6, x = x x = 6, x 5 = x + x =, x = x x = + a x + x = 6 a, x = a (,) (6 a, ) x Mitchell Sensitivity Analysis / 9
Increase resource available Outline Optimal tableau and pivot matrix Increase nonbasic variable Increase resource available Change selling price 5 New product available 6 Additional constraints violated by current solution Mitchell Sensitivity Analysis / 9
Increase resource available Increase resource available Change second constraint to x + x 6 + h with h >. How does the solution change? Approach: Change M to M h, then calculate M h = QM h. Reoptimize using dual simplex if necessary. We have M h = x x x x x 5 6 + h 6 Mitchell Sensitivity Analysis / 9
Increase resource available Calculate the updated tableau QM h = 7 6 + h 6 = + h 7 h 7 + h + h Notice that only the b-column of the tableau has changed. This tableau is in optimal form provided h 6 7. Mitchell Sensitivity Analysis / 9
Increase resource available Graphing the modified problem x x + x = 6, x = x x = h x + x = 6 + h, x = h x x = 6, x 5 = (,) ( + h, + h) x + x =, x = Mitchell Sensitivity Analysis 5 / 9 x
Increase resource available Graphing the modified problem x x + x = 6, x = x x = h x + x = 6 + h, x = h x x = 6, x 5 = (,) ( 7, 7 ) x + x =, x = Mitchell Sensitivity Analysis 5 / 9 x
Change selling price Outline Optimal tableau and pivot matrix Increase nonbasic variable Increase resource available Change selling price 5 New product available 6 Additional constraints violated by current solution Mitchell Sensitivity Analysis 6 / 9
Change selling price Change selling price Change objective function to min x ( + q)x. How does the solution change? Approach: Change M to M q, then calculate M q = QM q. Recover canonical form. Reoptimize using simplex if necessary. We have M q = x x x x x 5 q 6 6 Mitchell Sensitivity Analysis 7 / 9
Change selling price Calculate new tableau QM q = 7 q 6 6 = q 7 Notice that the only change in the optimal tableau is to the reduced cost for x. The same result would hold even if a nonbasic cost coefficient was changed instead. Mitchell Sensitivity Analysis 8 / 9
Change selling price Recover canonical form Since x was basic in M, the new tableau is no longer in canonical form. Pivoting on the x entry in R gives the updated tableau x x x x x 5 + q + q q 7 This is still in optimal form provided q. If q >, no longer in optimal form, and x 5 enters the basis. Mitchell Sensitivity Analysis 9 / 9
Change selling price Reoptimize From the minimum ratio test, x leaves the basis and the updated tableau is x x x x x 5 9 + q + q 5 q 7 This tableau is in optimal form provided q 5. Mitchell Sensitivity Analysis / 9
Change selling price Graphing the modified problem x x + x = 6, x = x x = 6, x 5 = (,) (,) x ( + q)x = 9 q x + x =, x = x Mitchell Sensitivity Analysis / 9
New product available Outline Optimal tableau and pivot matrix Increase nonbasic variable Increase resource available Change selling price 5 New product available 6 Additional constraints violated by current solution Mitchell Sensitivity Analysis / 9
New product available New product available Have new product x 6, and change the LP to min x x x 6 s.t. x + x + x 6 x + x + x 6 6 (P) x x + x 6 6 x i i =,, 6. How does the solution change? Approach: Change M to M p, then calculate M p = QM p. Reoptimize using simplex if necessary. M p = x x x x x 5 x 6 6 6 Mitchell Sensitivity Analysis / 9
New product available Calculate updated optimal tableau QM p = 7 6 6 = 7 Notice that the only change is in the x 6 column. No longer in optimal form, so need to reoptimize using simplex: x 6 enters the basis; from the minimum ratio test, x leaves the basis. Mitchell Sensitivity Analysis / 9
Outline Additional constraints violated by current solution Optimal tableau and pivot matrix Increase nonbasic variable Increase resource available Change selling price 5 New product available 6 Additional constraints violated by current solution Mitchell Sensitivity Analysis 5 / 9
Additional constraints violated by current solution Add additional constraints Approach: Recover canonical form. Reoptimize using dual simplex. For example: add the constraint x x. Adding this constraint to M with slack variable x 6 gives the tableau M = x x x x x 5 x 6 7 Mitchell Sensitivity Analysis 6 / 9
Additional constraints violated by current solution Graphing the modified problem x x x = 6 (.5,.5) x x =, x 6 = x x = 6, x 5 = (,) x + x = 6, x = x + x =, x = x Mitchell Sensitivity Analysis 7 / 9
Additional constraints violated by current solution Get (dual) canonical form M = x x x x x 5 x 6 7 R R, R + R x x x x x 5 x 6 7 Need to reoptimize using dual simplex. x 6 replaced by x 5 in basis. Mitchell Sensitivity Analysis 8 / 9
Additional constraints violated by current solution Pivot to optimality x x x x x 5 x 6 8-6 This is in optimal form. x x x x x 5 x 6 6 5 6 7 Mitchell Sensitivity Analysis 9 / 9