1. Sensitivity of the Right Hand Side Coefficients
Linear Programming
Sensitivity of the Right Hand Side Coefficients

2. Sensitivity of RHS Coefficients
RHS coefficients usually give some maximum limit for a resource or some minimum requirement that must be met.
Changes to the RHS can happen when extra units of the resource become available or when some of the original resource becomes unavailable.
Or the minimum requirement is loosened (made less) or strengthened (made greater).
Extra units may be available for a price.
The question becomes how much would an extra unit add to the value of the objective function, that is, “what is the most we would be willing to pay for extra units of the resource?”

3. Finding the Optimal Point - ReviewX2
1000
900
800
700
600
500
400
300
200
100 X1
Max 8X1 + 5X2
s.t.
OPTIMAL POINT (320,360)
= 4360
2X1 + 1X2 ≤ (Plastic)
3X1 + 4X2 ≤ (Time)
1X1 + 1X2 ≤
(Limit)
1X1 - 1X2 ≤ (Mix)
X1, X2 ≥ 0

4. Optimal Point With One Extra Unit of Plastic
1000
900
800
700
600
500
400
300
200
100 X1
Shadow Price (for Plastic)
– 4360 =
(new profit) (old profit)
$3.40
Max 8X1 + 5X2
s.t.
New OPTIMAL POINT (320.8,359.4) =
1001
2X1 + 1X2 ≤ (Plastic)
3X1 + 4X2 ≤ (Time)
1X1 + 1X2 ≤
(Limit)
1X1 - 1X2 ≤ (Mix)
X1, X2 ≥ 0
Still determined by Plastic and Time constraints

5. Shadow Prices
The shadow price for a constraint is the amount the objective function value will change given:
1 additional unit on the RHS of the constraint
No other changes
This shadow price is valid as long as the same constraints (including the one whose RHS is changing) determine the optimal point.
In this case plastic and production time
It can be shown that if the RHS for plastic were 1002 the profit would increase another $3.40 to $
It can also be shown that if the RHS for plastic were 999 the profit would decrease by $3.40 to $

6. Allowable Increase and Allowable Decrease of a RHS Value
The shadow prices remain valid as long as the same constraints (called the binding constraints) determine the optimal point.
When the RHS of the constraint is increased or decreased to the point that another constraint replaces one of the binding constraints to determine the optimal point a new shadow price becomes valid for the constraint.
The amount the RHS can increase or decrease before another constraint becomes one of the binding constraints is what Excel calls the Allowable Increase and the Allowable Decrease respectively.

7. Increasing the Right Handside for PlasticX2
1000
900
800
700
600
500
400
300
200
100 X1
Max 8X1 + 5X2
s.t.
1030
1060
1100
2X1 + 1X2 ≤ (Plastic)
3X1 + 4X2 ≤ (Time)
1X1 + 1X2 ≤
(Limit)
1X1 - 1X2 ≤ (Mix)
X1, X2 ≥ 0
Plastic and Time constraints still determine the optimal solution to this point.

8. Further Increasing the Right Hand Side for PlasticX2
1000
900
800
700
600
500
400
300
200
100 X1
Max 8X1 + 5X2
s.t.
1030
1060
1100
1102
2X1 + 1X2 ≤ (Plastic)
3X1 + 4X2 ≤ (Time)
1X1 + 1X2 ≤
(Limit)
1X1 - 1X2 ≤ (Mix)
X1, X2 ≥ 0
Further increases to the RHS side of plastic have now made the plastic and the Limit constraints as the ones that determine the optimal point.
The shadow prices will now CHANGE

9. Decreasing the RHS for PlasticX2
1000
900
800
700
600
500
400
300
200
100 X1
Max 8X1 + 5X2
s.t.
600
700
850
1000
2X1 + 1X2 ≤ (Plastic)
3X1 + 4X2 ≤ (Time)
1X1 + 1X2 ≤
(Limit) Redundant
1X1 - 1X2 ≤ (Mix)
X1, X2 ≥ 0
Optimal solution determined by Plastic and Time Constraints
and by X2 axis!

10. Further Decreasing the RHS for PlasticX2
1000
900
800
700
600
500
400
300
200
100 X1
Max 8X1 + 5X2
s.t.
590
600
700
850
1000
2X1 + 1X2 ≤ (Plastic)
3X1 + 4X2 ≤ (Time)
1X1 + 1X2 ≤
(Limit) Redundant
1X1 - 1X2 ≤ (Mix)
X1, X2 ≥ 0
Optimal Point now determined by the plastic constraint and the X2-axis
The shadow prices will now CHANGE

11. Comparison With Excel
Here is the printout out of the sensitivity analysis dealing with the objective RHS coefficients for the original Galaxy Industries problem.
Shadow Price for each constraint
Range of Feasibility for RHS1
1000 – 400  
Range of Feasibility for RHS2  
Range of Feasibility for RHS3
700 – 20  ∞
 ∞
Range of Feasibility for RHS4
350 – 390  ∞
 ∞
Range of Feasibility is the range of values that an RHS coefficient can assume without changing the shadow prices as long as no other changes are made.

12. Exact Meaning of Shadow Prices
A shadow price always means the amount the objective function will change given a one unit increase in the RHS value of a constraint.
But does this mean that this is the value (the most you would be willing to pay) for an extra unit? The answer depends on how the objective function coefficients were calculated.
If the objective function coefficients did not take the value of the resource into consideration, these are sunk costs.
Shadow price = the value of an extra unit of the resource.
If the objective function coefficients did take the value of the resource into consideration, these are included costs.
Shadow price = a premium above the current price of the item that one would be willing to pay for an extra unit.

13. Plastic is an included cost
EXAMPLE
Suppose the $8 objective function coefficient for dozens of Space Rays and the $5 objective function coefficient for dozens of Zappers were calculated as follows:
DOZ DOZ.
SPACE RAYS ZAPPERS
Selling Price $ $26
Costs
Plastic ($3/lb) $ 6 (2 lbs.) $ 3 (1 lb.)
Other Variable Costs $ $18
=========== ==========
Total Profit Per Dozen $ $ 5
Plastic is an included cost
Production time is a sunk cost
The $3.40 shadow price for plastic
means we would be willing to pay
up to $3.40 more than the current
price of $3 per pound (that is up to
$6.40/ lb.) for extra plastic.
It is not included in the objective
function coefficient calculation.
The $0.40 shadow price is the
value of an extra minute of
production time.

14. Complementary Slackness
Complementary slackness also holds for RHS values. This property for RHS values states:
Again, it can happen, that both are 0.
Complementary Slackness
For RHS Coefficients
For each constraint, either the slack
(difference between RHS – LHS) is 0 or its shadow price will be 0.
Plastic: Shadow Price ≠ 0; Slack = = 0
Time: Shadow Price ≠ 0; Slack = = 0
Prod. Limit: Slack = ≠ 0; Shadow Price = 0
Prod. Mix: Slack = 350-(-40) ≠ 0; Shadow Price = 0

15. Review Shadow price Range of Feasibility Complementary Slackness
Found by subtracting the original objective function value from the objective function value with one more unit of the resource on the RHS
Meaning
Included Cost
Sunk Cost
Range of Feasibility
Range of RHS value in which shadow price does not change
The same constraints determine the optimal solution in the range of feasibility
Complementary Slackness
Either the slack is 0 or the shadow price is 0