1. Linear Programming – Sensitivity Analysis
How much can the objective function coefficients change
before the values of the variables change?
How much can the right hand side of the constraints change
you obtain a different basic solution?
How much value is added/reduced to the objective function if I have
a larger/smaller quantity of a scarce resource?

2. Linear Programming – Sensitivity
Leo Coco Problem
Max 20x x2
s.t x x2 <= 1
3x x2 <= 7
x1, x2 >= 0
Solution:
x1 = 0, x2 = 7
Z = 70
Issue: How much can you change a cost coefficient without changing the solution?

3. Sensitivity – Change in cost coefficient
What if investment 2 only
pays $5000 per share?
Max 20x1 + 5 x2
s.t x x2 <= 1
3x x2 <= 7
x1, x2 >= 0
New objective function isobar
Optimal solution is now the point (2,1).
Issue: At what value of C2 does solution change?

4. Sensitivity – Change in cost coefficient
Issue: At what value of C2 does solution change?
Ans.: When objective isobar is parallel to the binding constraint.
Max 20x1 + C2* x2
s.t x x2 <= 7
3/1 = 20/C2 or C2 = 20/3 or 6.67

5. Sensitivity – Change in cost coefficient
Lindo Sensitivity Analysis Output – Leo Coco Problem
LP OPTIMUM FOUND AT STEP
OBJECTIVE FUNCTION VALUE 1)
VARIABLE VALUE REDUCED COST X X
ROW SLACK OR SURPLUS DUAL PRICES 1) 2)
NO. ITERATIONS=
RANGES IN WHICH THE BASIS IS UNCHANGED:
OBJ COEFFICIENT RANGES
VARIABLE CURRENT ALLOWABLE ALLOWABLE
COEF INCREASE DECREASE
X INFINITY
X INFINITY
RIGHTHAND SIDE RANGES
ROW CURRENT ALLOWABLE ALLOWABLE
RHS INCREASE DECREASE
INFINITY
INFINITY

6. Sensitivity – Change in right hand side
What if only 5 hours
available in time constraint?
Max 20x1 + 5 x2
s.t x x2 <= 1
3x1 + x2 <= 5
x1, x2 >= 0
New time constraint.
Optimal solution is now the point (5,0). But, basis does not change.
Issue: At what value of the r.h.s. does the basis change?

7. Sensitivity – Change in right hand side
Issue: At what value of the r.h.s.
does the basis change?
Max 20x1 + 5 x2
s.t x x2 <= 1
3x x2 <= ?
x1, x2 >= 0
Basis changes at this constraint.
x2 becomes non-basic at the origin. Or, when the constraint is:
3x x2 < 0

8. Sensitivity – Change in right hand side
Lindo Sensitivity Analysis Output – Leo Coco Problem
LP OPTIMUM FOUND AT STEP
OBJECTIVE FUNCTION VALUE 1)
VARIABLE VALUE REDUCED COST X X
ROW SLACK OR SURPLUS DUAL PRICES 1) 2)
NO. ITERATIONS=
RANGES IN WHICH THE BASIS IS UNCHANGED:
OBJ COEFFICIENT RANGES
VARIABLE CURRENT ALLOWABLE ALLOWABLE
COEF INCREASE DECREASE
X INFINITY
X INFINITY
RIGHTHAND SIDE RANGES
ROW CURRENT ALLOWABLE ALLOWABLE
RHS INCREASE DECREASE
INFINITY
INFINITY

9. Sensitivity – Shadow or Dual Prices
Issue, how much are you willing to pay for one additional unit of a limited resource?
Max 20x x2
s.t x x2 <= 1 (budget constraint
3x1 + x2 <= 7 (time constraint
x1, x2 >= 0
Knowing optimal solution is (0,7) and time constraint is binding:
Not willing to increase budget constraint (shadow price is $0).
If time constraint increase by one unit (to 8), solution will change to (0,8) and Z=80. Therefore should be willing to pay up to $10(000s) for each additional unit of time constraint.

10. Sensitivity – Change in right hand side
Lindo Sensitivity Analysis Output – Leo Coco Problem
LP OPTIMUM FOUND AT STEP
OBJECTIVE FUNCTION VALUE 1)
VARIABLE VALUE REDUCED COST X X
ROW SLACK OR SURPLUS DUAL PRICES 1) 2)
NO. ITERATIONS=
RANGES IN WHICH THE BASIS IS UNCHANGED:
OBJ COEFFICIENT RANGES
VARIABLE CURRENT ALLOWABLE ALLOWABLE
COEF INCREASE DECREASE
X INFINITY
X INFINITY
RIGHTHAND SIDE RANGES
ROW CURRENT ALLOWABLE ALLOWABLE
RHS INCREASE DECREASE
INFINITY
INFINITY

11. Linear Programming – Sensitivity Analysis
What if more than one coefficient is changed?:
100% Rule (for objective function coefficients):
if <= 1, the optimal solution will not change,
where is the actual increase (decrease) in the coefficient
and is the maximum allowable increase (decrease) from
the sensitivity analysis.

12. Linear Programming – Sensitivity Analysis
Example obj. function coefficient changes

13. Linear Programming – Sensitivity Analysis
Simultaneous variations in multiple coefficients:
100% Rule (for RHS constants):
if <= 1, the optimal basis and product mix will not change,
where is the actual increase (decrease) in the coefficient
and is the maximum allowable increase (decrease) from
the sensitivity analysis.

14. Linear Programming – Sensitivity Analysis
Example RHS constant changes