What-If Analysis for Linear Programming
Sensitivity Analysis—“What if?”
Weekly supply of raw materials:
8 Small Bricks
6 large Bricks
Products:
Table
Chair
Profit = $20/Table
Profit = $15/Chair
Example: The Lego Production Problem (Revisited)
 Had 8 small bricks and 6 large bricks
to produce tables and chairs. The
optimal solution is to produce 2 tables
and 2 chairs.
 What if the profit per table really is
$25? Does the optimal solution
change?
 What if the profit per table really is
$35? Does the optimal solution
change?
 Thus, there is some range of values for
this profit over which the optimal
solution does not change, but beyond
which the optimal solution does
change.
 What if one additional large brick
becomes available? Does this enable
increasing the total profit?
 The amount of any increase is called
the shadow price for large bricks.
 What if a second additional large brick
becomes available? Does this enable
further increasing the total profit by the
amount of the shadow price?
 Thus, there is a certain range of values
(called the range of feasibility) for the
number of available large bricks over
which the shadow price is applicable.
Formulation
Maximize ($15)Chairs  ($20)Tables
subject to
Large Bricks:
Chairs  2Tables  6
Small Bricks:
2Chairs  2Tables  8
and
Chairs  0, Tables  0.
Generating the Sensitivity Report
Solve the problem using the Solver:
Then, choose “Sensitivity” under Reports.
Questions Answered by Excel
•What is the optimal solution?
•What is the profit for the optimal solution?
•If the net profit per table changes, will the solution change?
•If the net profit per chair changes, will the solution change?
•If more (or less) large bricks are available, how will this affect
profit?
•If more (or less) small bricks are available, how will this affect
profit?
The Sensitivity Report
Changing Cells
Cell
$B$3
$C$3
Name
Solution: Chairs
Solution: Tables
Final
Value
2
2
Reduced
Cost
0
0
Objective
Coefficient
15
20
Allowable
Increase
5
10
Allowable
Decrease
5
5
Name
Large Bricks LHS
Small Bricks LHS
Final
Value
6
8
Shadow
Price
5
5
Constraint
R.H. Side
6
8
Allowable
Increase
2
4
Allowable
Decrease
2
2
Constraints
Cell
$D$8
$D$9
Net Profit from Tables = $35
Changing Cells
Cell
$B$3
$C$3
Name
Solution: Chairs
Solution: Tables
Final
Value
0
3
Reduced Objective
Cost
Coefficient
-2.5
15
0
35
Allowable
Increase
2.5
1E+30
Allowable
Decrease
1E+30
5
Name
Large Bricks LHS
Small Bricks LHS
Final
Value
6
6
Shadow
Price
17.5
0
Allowable
Increase
2
1E+30
Allowable
Decrease
6
2
Constraints
Cell
$D$8
$D$9
Constraint
R.H. Side
6
8
Seven Large Bricks
Changing Cells
Cell
$B$3
$C$3
Name
Solution: Chairs
Solution: Tables
Final
Value
1
3
Reduced Objective
Cost
Coefficient
0
15
0
20
Allowable
Increase
5
10
Allowable
Decrease
5
5
Name
Large Bricks LHS
Small Bricks LHS
Final
Value
7
8
Shadow
Price
5
5
Allowable
Increase
1
6
Allowable
Decrease
3
1
Constraints
Cell
$D$8
$D$9
Constraint
R.H. Side
7
8
Nine Large Bricks
Changing Cells
Cell
$B$3
$C$3
Name
Solution: Chairs
Solution: Tables
Final
Value
0
4
Reduced Objective
Cost
Coefficient
-5
15
0
20
Allowable
Increase
5
1E+30
Allowable
Decrease
1E+30
5
Name
Large Bricks LHS
Small Bricks LHS
Final
Value
8
8
Shadow
Price
0
10
Allowable
Increase
1E+30
1
Allowable
Decrease
1
8
Constraints
Cell
$D$8
$D$9
Constraint
R.H. Side
9
8
Summary of Output from Computer Solution
Changing Cells:
Final Value
The value of the variable in the optimal solution
Reduced Cost Increase in the objective function value per unit
increase in the value of a zero-valued variable
(for small increases).
Allowable
Increase/
Decrease
Defines the range of the cost coefficients in the
objective function for which the current solution
(value of the variables in the optimal solution)
will not change.
Constraints:
Final Value
The usage of the resource in the optimal solution.
Shadow price
The change in the value of the objective function
per unit increase in the right hand side of the
constraint:
∆Z = (Shadow Price)(∆RHS)
(Note: only valid if change is within the allowable
range for RHS values—see below.)
Constraint
R.H. Side
The current value of the right hand side of the
constraint (the amount of the resource available).
Allowable
Increase/
Decrease
Defines the range of values of the RHS for which
the shadow price is valid and hence for which the
new objective function value can be calculated.
(NOT the range for which the current solution will
not change.)