1. Section 3.4
Sensitivity Analysis

2. 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.

3. 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

4. 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).

5. 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:

6. 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 = 30.

7. 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.

8. 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.

9. 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.