Section 3.4 Sensitivity Analysis

Sensitivity Analysis Investigating how changes in the parameters of a linear programming problem affect its optimal solution. Check Changes in: Coefficients of the Objective Function. Constants on the Right-Hand Side of Constraint Inequalities.

Ex. Changes in Coefficients of the Objective Function max is 25 at (5, 1) Maximize P = 4x + 5y Subject to Slope = –3/5 Slope = –1 Examine the first variable P = cx + 5y Which is Compare this slope to the slopes of the constraints

The optimal solution will remain unaffected as long as Slope = –3/5 Slope = –1 So if the value on x stays between 3 and 5 then the optimal solution is still at (5, 1).

Ex. Changes in Constants on the Right-Hand Side of the Constraints Maximize P = 4x + 5y Subject to max is 25 at (5, 1) Examine the first constraint This is parallel to the first constraint. Add h:

Find the intersection point of 3x + 5y = 20 + h and x + y = 6 Using So for a meaningful solution, the first constraint must be between 20 – 2 = 18 and 20 + 10 = 30.

For example, since If h = 2, then the constraint becomes P = 4(4) + 5(2)= 26 Therefore increasing the constraint from 20 to 22 results in an increase of P from 25 to 26.

Shadow Prices The shadow price of the ith resource is the amount by which the objective function is improved if the right-hand side of the ith constraint is increased by 1 unit.

From the last example, since Plugging into P = 4x + 5y we get Since the maximum for the original problem was 25, this shows that for each unit increase of the constant in first constraint there is a 0.5 increase in P.