1. Sensitivity analysis BSAD 30 Dave Novak
Source: Anderson et al., 2013 Quantitative Methods for Business 12th edition – some slides are directly from J. Loucks © 2013 Cengage Learning

2. Overview Introduction to sensitivity analysis
Changes in objective function coefficients
Changes in RHS values
Interpretation of shadow prices
Sunk costs versus relevant costs
Example

3. Sensitivity analysis
Also called post-optimality analysis considers how changes in the coefficients of a linear programming problem (within specified ranges) affect the optimal solution
Keep in mind that an LP is solved in terms of a specific set of objective function coefficients and constraints
It is not a generalized formula that you can automatically use to consider all kinds of different coefficient values

4. Sensitivity analysis
Sensitivity analysis is important because decision-makers operate in dynamic environments with imperfect information
The coefficients in the OF, the LHS of the constraints, and the RHS of the constraints are often estimated, and are typically imprecise
Allows decision-maker to ask what-if questions about the problem

5. Sensitivity analysis
Prices of raw material change, demand for products change, production capabilities change, stock prices change, etc.
If you are using an LP, you can generally expect some type of change over time
How do these changes affect the optimal solution, and at what point do you need to completely revise (and re-solve) the LP?

6. Sensitivity analysis
In sensitivity analysis, we consider only ONE change at a time
Change ONE OF coefficient
Change ONE RHS constraint value
We cannot use this approach to examine multiple simultaneous changes

7. Sensitivity analysis Consider the problem from Lecture 4 Max 5x1 + 7x2
Objective
Function
Max x1 + 7x2
s.t x < 6
2x1 + 3x2 < 19
x1 + x2 < 8
x1 > 0 and x2 > 0
“Regular”
Constraints
Non-negativity
Constraints

8. Sensitivity report
Download Lecture9 example.xlsx from the class website
The problem is already solved

9. Sensitivity report
The Sensitivity Report is:

10. Sensitivity analysis
First, let’s consider how changes in the objective function coefficients might affect the optimal solution
We want to find the range of OF coefficient values (for each coefficient) over which the current solution will remain optimal
Managers should focus on objective coefficients that have a narrow range of optimality and coefficients near the endpoints of the range

11. Sensitivity analysis
Note that any change in the OF coefficients will NOT impact the feasible region, or the extreme points within the feasible region, as the constraints are not changed in any way
Sensitivity focused on changes in OF coefficient values address the question of “at what point does a change in the profit or cost associated with a particular decision variable result in a different extreme point becoming the “new” optimal solution”?

12. Sensitivity analysis
We want to know how much each one of the coefficients in the OF can change before the optimal solution changes
Max 5x1 + 7x2
What if (c1) the profit associated with x1 were to change – would the solution of (5,3) still be optimal?
Currently, we want to produce 5 units of x1 and 3 units of x2 at profit of $5 and $7 per unit respectively

13. Sensitivity report
The “profit” coefficient associated with x1 (referred to as c1) can range
between $4.67 and $7 before The optimal solution mix will change from
(5, 3) to something else (another extreme point value in our feasible region)

14. Sensitivity analysis
If the profit associated with x1 were to increase by $1 per unit, we would still produce 5 units of x1, although our total profit would now be $51 instead of $46
6(5) + 7(3) = 51
As long as c1 (the profit contribution associated with x1) is between $4.67 and $7 per unit, the optimal solution mix of 5 units of x1 and 3 units of x2 will not change

15. Sensitivity report
Likewise, (c2) the profit coefficient associated with x2 can range between
$5 and $7.50 before the optimal solution mix will change from (5, 3) to
something else (another extreme point value in our feasible region)

16. Sensitivity analysis
If (c2) the profit associated with x2 were to decrease by $2 per unit, we would still produce 3 units of x2, although our total profit would now be $35 instead of $46
5(5) + 5(3) = 35
As long as c2 (the profit contribution associated with x2) is between $5 and $7.50 per unit, the optimal solution mix of 5 units of x1 and 3 units of x2 will not change

17. Sensitivity analysis
The range of values c1 and c2 can take on without changing the optimal solution is referred to as the range of optimality for the objective function coefficient
Range of optimality for c1: $4.67 – $7
Range of optimality for c2: $5 – $7.50

18. Graphical solution x2
Con 3: x1 + x2 < 8 8 7 6 5 4 3 2 1
OF: Max 5x1 + 7x2
Con 1: x1 < 6
Optimal Solution:
x1 = 5, x2 = 3
Con 2: 2x1 + 3x2 < 19 x1

19. Graphical solution x2
Con 3: x1 + x2 < 8
OF: Max 5x1 + 7x2 8 7 6 5 4 3 2 1
Con 1: x1 < 6
Optimal Solution:
x1 = 5, x2 = 3
Con 2: 2x1 + 3x2 < 19 x1

20. Sensitivity analysis
A change in an OF coefficient changes the slope of the OF line!
Graphically, the limits of a range of optimality are found by changing the slope of the objective function line within the limits of the slopes of the binding constraint lines
Slope of an objective function line, Max c1x1 + c2x2, is (-c1/c2), and the slope of a constraint, a1x1 + a2x2 = b, is (-a1/a2)

21. Sensitivity analysis
Mathematically find the range of optimality for OF coefficient c1 (the coefficient associated with x1, which is 5)
Slope of current OF line = (-c1/c2) = (-5 / 7)

22. Sensitivity analysis Two binding constraints: Con 2 and Con 3
Con 2: 2x1 + 3x2 = 19
Slope of Con 2 = (-a1/a2) = (-2 / 3)
Con 3: x1 + x2 = 8
Slope of Con 3 = (-a1/a2) = (-1 / 1) = -1

23. Sensitivity analysis
Find the range of values for c1 (while holding c2 constant, or staying at $7), such that the slope of the OF line stays between the 2 binding constraints
-1 < -c1 / 7 < -2 / 3
Slope of Con 2
Slope of Con 3

24. Sensitivity analysis
Find the range of optimality for c2 (the OF coefficient associated with x2 while holding c1 constant (fixed at 5)
-1 < -5 / c2 < -2 / 3

25. Sensitivity analysis x2 Feasible Region x1
Coincides with Con 3: x1 + x2 < 8 8 7 6 5 4 3 2 1
Objective function line for 5x1 + 7x2
Coincides with Con 2:
2x1 + 3x2 < 19
Feasible
Region x1

26. Sensitivity analysis
Second, let’s consider how changes in the constraint RHS coefficients might affect the feasible region as well as the optimal solution
Like changes in OF coefficients, we are considering one change at a time!
Changes in the RHS of a constraint are a more complicated sensitivity concept because changes in the RHS can change the entire feasible region for the problem

27. Sensitivity report
The number in the “Final Value” column associated with each constraint is
the amount of the resource that is used in the optimal solution. Note that the
difference between the values in the “Final Value” column and The
“Constraint RHS” is the value of the slack or surplus variables

28. Answer report

29. Shadow prices
The shadow price is a measure of the relative value of a resource, or the change in the OF value when increasing the resource by one unit
As the RHS of a constraint increases or decreases, other constraints may become binding and impact the optimal solution, so the shadow price interpretation is only applicable for SMALL changes in the RHS

30. Sensitivity report

31. Sensitivity report
Consider constraint #2: if we increase the RHS value from 19 to 20 units,
our OF value (or profit) will increase by $2 (from $46 to $48)
If we decrease the RHS value from 19 to 18 units, our OF value (or profit)
will decrease by $2 (from $46 to $44)

32. Graphical change in RHSx2
Coincides with Con 3: x1 + x2 < 8 8 7 6 5 4 3 2 1
Coincides with Con 1: x1 < 6
Coincides with NEW Con 2: 2x1 + 3x2 < 20
Feasible
Region x1

33. Shadow prices
The Sensitivity Report provides information (although it is limited) on how changes in the values of the RHS of the constraints impact the optimal solution as well as the feasible region
We don’t have to completely reformulate and resolve the LP

34. Shadow prices The shadow price for a non-binding constraint is 0
If a constraint is non-binding, it is not part of the optimal solution (and there is some slack or surplus associated with the constraint)
Logically, the marginal value of one additional unit of a resource associated with a non-binding constraint is 0

35. Shadow prices
The reduced cost associated with each decision variable is equal to the non-negativity constraint associated with the variable
This will become more of a focus as we investigate problems with more than 2 decision variables

36. Shadow prices
Some words of caution regarding interpretation of shadow prices
The value of a shadow prices is generally only applicable for small increases in the RHS
As more resources are available and the RHS value increases, different sets of constraints become binding and change the optimal solution mix (the optimal values for x1 and x2)

37. Shadow prices
Some words of caution regarding interpretation of shadow prices
Not all shadow prices can be interpreted the same way
While the shadow price is typically interpreted as “the maximum amount you should be willing to pay for one additional unit of a resource”, this interpretation is not always correct

38. Shadow prices
Understanding the difference between relevant and sunk costs
Relevant costs are affected by the decisions that are made
Sunk costs are NOT affected by the decisions that are made
Relevant costs will vary according to the values of the decision variables

39. Shadow prices
A resource cost is a relevant cost if the amount paid for that resource is dependent on the amount of the resource used by the decision variables
Relevant costs are reflected in the OF coefficients

40. Shadow prices
A resource cost is a sunk cost if it must be paid regardless of the amount of the resource that is actually used by the decision variables
Sunk costs are NOT reflected in the OF coefficients

41. Shadow price interpretation
Resource cost is relevant
The shadow price can be interpreted as the amount by which the value of the resource exceeds its cost (or the premium over the normal cost that you should be willing to pay for one unit of that resource)
Resource cost is sunk
The shadow price can be interpreted as the maximum that you should be willing to pay for one additional unit of that resource

42. Shadow prices
Con 1: Since x1 < 6 is not a binding constraint, its shadow price is 0
The allowable increase is ∞
If the amount of the resource used in the optimal solution is 5, which is less than the amount available (which is 6), increasing the amount of the resource available will not impact the solution in any way
The allowable decrease is 1
Further reducing the amount of the resource available (below 5 units) will change the optimal solution as Con 1 now becomes binding

43. Shadow prices Con 2: Shadow price of Con 2 is 2
Adding one additional unit of resource (increasing the RHS from 19 to 20 units) will increase our total profit by $2
This interpretation of the shadow price is applicable within the range of feasibility
Allowable increase is 5
Allowable decrease is 1
The same set of binding constraints will remain binding within the RHS range of (18-24)

44. Shadow prices Con 3: Shadow price of Con 3 is 1
Adding one additional unit of resource (increasing the RHS from 8 to 9 units) will increase our total profit by $1
This interpretation of the shadow price is applicable within the range of feasibility
Allowable increase is 0.33
Allowable decrease is 1.66
The same set of binding constraints will remain binding within the RHS range of ( )

45. Shadow prices
Graphically, the range of feasibility is determined by finding the values of a RHS coefficient such that the same two lines that determined the original optimal solution continue to determine the optimal solution for the problem
To what extent can we make changes to the RHS of any one of the 3 constraints, so that the optimal solution still lies at the intersection of Con 2 and Con 3?

46. Shadow prices
This does not mean that the optimal solution won’t change within the range of feasibility, just that the same set of binding constraints (currently #2 and #3) will remain binding for changes to the RHS of either constraint – as long as those changes occur within the range of feasibility
Within the range of feasibility, the interpretation of the shadow price holds (the range where the shadow price is applicable)

47. Shadow prices
For changes to the RHS outside of the range of feasibility, the problem must be resolved to find the new shadow price
Only one RHS value can be changed at a time – does not apply to multiple simultaneous changes
Does not apply to changes in LHS constraint coefficients

48. Olympic bike example
Olympic Bike is introducing two new lightweight bicycle frames, the Deluxe and the Professional, to be made from special aluminum and steel alloys. The anticipated unit profits are $10 for the Deluxe and $15 for the Professional
A supplier delivers 100 lbs. of aluminum alloy and 80 lbs. of steel alloy per week. Each Deluxe requires 2lbs. of aluminum and 3 lbs. of steel. Each Professional requires 4 lbs. of aluminum and 2 lbs. of steel
How many of each type of frame should Olympic produce each week?

49. Olympic bike example

50. Olympic bike example

51. Olympic bike example
What happens if the unit profit for the Deluxe frame changes from $10 to $20 – is the current solution still optimal?

52. Olympic bike example
The current solution of 15 Deluxe and 17.5 Professional frames remains optimal
As long as the OF coefficient for x1 (c1) is between $7.50 and $22.50
So, increasing c1 by $10 (to $20) is within the range of optimality, and the solution
mix WILL NOT change – we will still produce 15 Deluxe and 17.5 Professional frames
The profit maximizing solution will change from $ to $ because we
Are gaining $10 for each Deluxe frame we produce (= 20(15)+15(17.5))

53. Olympic bike example
What happens if the unit profit for the Deluxe frame changes from $10 to $6 – is the current solution still optimal?

54. Olympic bike example
The current solution of 15 Deluxe and 17.5 Professional frames remains optimal
As long as the OF coefficient for x1 (c1) is between $7.50 and $22.50
So, decreasing c1 by $4 (to $6) is OUTSIDE the range of optimality, and the solution
mix WILL change – we have to reformulate and re-solve the problem

55. Olympic bike example Assume aluminum alloy is a sunk cost
What is the maximum amount Olympic should pay for 50 extra lbs. of aluminum alloy?

56. Olympic bike example
For a sunk cost, the shadow price is interpreted as the value of extra aluminum (we
already have paid for 100 lbs.) – what is the value of 50 additional lbs.?
Shadow price = $3.125 per lb. the allowable increase is 60, so the shadow price
interpretation holds for 50 additional lbs.
The value of 50 additional lbs. of aluminum alloy is 50 * $3.125 = $156.25

57. Olympic bike example Assume aluminum alloy is a relevant cost
What is the maximum amount Olympic should pay for 50 extra lbs. of aluminum alloy?

58. Olympic bike example
If aluminum alloy is a relevant cost, the shadow price represents the amount above the normal/initial price of aluminum the company should be willing to pay for additional aluminum (the premium Olympic is willing to pay for more of the resource)
If aluminum were initially $4 per lb., then additional units of aluminum (beyond the initial 100 lbs. and within the feasible range) would be worth $7.125 per lb. to Olympic

59. Summary Introduction to sensitivity analysis
Changes in objective function coefficients
What does this mean graphically?
How are these changes interpreted?
Changes in RHS values

60. Summary Interpretation of shadow prices Olympic bike example
Where to find these on the Sensitivity Report
Range of feasibility
Sunk costs versus relevant costs
Olympic bike example