1. Computational Methods for Management and Economics Carla Gomes
Module 7b
Duality and Sensitivity Analysis
Economic Interpretation of Duality
(slides adapted from: M. Hillier’s, J. Orlin’s, and H. Sarper’s)

2. Post-optimality Analysis
Post-optimality – very important phase of modeling.
Duality plays and important role in post-optimality analysis
Simplex provides several tools to perform post-optimality analysis

3. Post-optimality analysis for LP
Task
Purpose
Technique
Model Debugging
Model Validation
Final Managerial on
resource allocations
Evaluate estimates of
model parameters
Evaluate parameter
trade-offs
Find errors and weaknesses in the model
Demonstrate validity of final model
Allocation of organizational resources
Determine if changes in parameters change optimal solution
Determine best trade-offs between model
parameters
Re-optimization
Analysis results
Dual (shadow)
prices
Sensitivity Analysis
Parametric Linear Programming

4. Economic Interpretation of Duality
LP problems – quite often can be interpreted as allocating resources to activities.
Let’s consider the standard form:
xi >= 0 , (i =1,2,…,n)

5. What if we change our resources – can we improve our optimal solution?
Resources – m (plants)
Activities – n (2 products)
Wyndor Glass problem optimal product mix --- allocation of resources to activities i.e., choose the levels of the activities that achieve best overall measure of performance
What if we change our resources – can we improve our optimal solution?

6. Sensitivity Analysis
How would changes in the problem’s objective function coefficients or right-hand side values change the optimal solution?

7. Dual Variables (Shadow Prices)
y1*= 0  dual variable (shadow price) for resource 1
y2*= 1.5  dual variable (shadow price) for resource 2
y3*= 1  dual variable (shadow price) for resource 3
How much does Z increase if we increase resource 2 by
1 unit (i.e., b2 = 12  b2=13)?

8. Graphical Analysis of Dual variables – Variation in RHS Increasing level of resource 2 (b2)
(5/3,13/2)
2w=13  Z=3(5/3)+5(13/2)=37.5
∆ Z=1.5
= y2*
Z=3(2)+5(6)=36
(2,6)
Figure Graph showing how the feasible region is formed by the constraint boundary lines, where the arrows indicate which side of each line is permitted by the corresponding constraint.
Why is y1*=0?

9. Economic Interpretation of Dual Variables
The dual variable associated with resource i
(also called shadow price), denoted by yi*, measures
the marginal value of this resource, i.e., the rate at
which Z could be increased by (slightly) increasing
the amount of this resource (bi), assuming everything
else stays the same. The dual variable yi* is identified
by the simplex method as the coefficient of the ith slack
variable in row 0 of the final simplex tableau.

10. Dual Variables: binding and non-binding constraints
The shadow prices (dual variables) associated with non-binding constraints are necessarily 0 (complementary optimal slackness)  there is a surplus of non-binding resource and therefore increasing it will not increase the optimal solution. Economist refer to such resources as free resources (shadow price =0)
Binding constraints on the other hand correspond to scarce resources – there is no surplus. In general they have a positive shadow price.

11. Does Z always increase at the same rate if we keep increasing the amount of resource 2?
(0,9)
b2=18
(5/3,13/2)
2w=13  Z=3(5/3)+5(13/2)=37.5
∆ Z=1.5
= y2*
Z=3(2)+5(6)=36
(2,6)
Figure Graph showing how the feasible region is formed by the constraint boundary lines, where the arrows indicate which side of each line is permitted by the corresponding constraint.
What if b2 > 18 (i.e., 2W>18)?
 the optimal solution will stay at (0,9) for b2>=18

12. Does Z always decrease at the same rate if we decrease resource 2?
2w=13  Z=3(5/3)+5(13/2)=37.5
∆ Z=1.5
= y2*
b2=6
(5/3,13/2)
(2,6)
If b2 < 6 the solution will no longer
vary proportionally. The optimal
solution varies proportionally to the
variation in b2 only if 6 <= b2 <=18.
In other words, the current basis remains
optimal for 6 ≤ b2 ≤ 18, but the decision
variable values and z-value will change.
Figure Graph showing how the feasible region is formed by the constraint boundary lines, where the arrows indicate which side of each line is permitted by the corresponding constraint.

13. A dual variable yi* gives us the rate at which Z could be increased by increasing the amount of resource i slightly.
However this is only true for a small increase in the amount of the resource. I.e., this definition applies only if the change in the RHS of constraint i leaves the current basis optimal. It also assumes everything else stays the same.
Another interpretation of yi* is: if a premium price must be paid for the resource i in the market place, yi* is the maximum premium (excess over the regular price) that would be worth paying.

14. Optimal Basis in the Wyndor Glass Problem
How can we characterize (verbally) the optimal basis of the Wyndor Glass problem?
Plant 1 – unutilized capacity (non-binding constraint)
Plant 2 – fully utilized capacity (binding constraint)
Plant 3 - fully utilized capacity (binding constraint)

15. How do we interpret the intervals?
If we change one coefficient in the RHS, say capacity of plant 2, by D the “basis” remains optimal, that is, the same equations remain binding.
So long as the basis remains optimal, the shadow prices are unchanged.
The basic feasible solution varies linearly with D. If D is big enough or small enough the basis will change.

16. The dual price or shadow price for the i th constraint
of an LP is the amount by which the optimal z-value
is improved (increased in a max problem or
decreased in a min problem) if the rhs of the i th
constraint is increased by one. This definition
applies only if the change in the rhs of constraint i
leaves the current basis optimal.
The dual variables or shadow prices are valid in a
given interval.

17. Sensitivity analysis for c1
How much can we vary c1 without changing
the current basic optimal solution?

18. Sensitivity analysis for c1
Our objective function is:
Z= c1 D+5W=k
slope of iso-profit line is:
Figure Graph showing three objective function lines for the Wyndor Glass Co. product-mix problem, where the top one passes through the optimal solution.
isoprofit line
How much can c1 vary until the slope of the iso-profit line
equals the slope of constraint 2 and constraint 3?

19. How much can c1 vary until the slope of the iso-profit line equals the slope of constraint 2 and constraint 3?
Slope of constraint 2 0
Slope of constraint 3  -3/2

20. Importance of Sensitivity Analysis
Sensitivity analysis is important for several reasons:
Values of LP parameters might change. If a parameter changes, sensitivity analysis shows it is unnecessary to solve the problem again. For example in the Wyndor problem, if the profit contribution of product 1 changes to 5, sensitivity analysis shows the current solution remains optimal.
Uncertainty about LP parameters. In the Wyndor problem for example, if the capacity of plant 1 decreases to 2, the optimal solution remains a weekly rate of 2 doors and 6 windows. Thus, even if availability of capacity of plant 1 uncertain, the company can be fairly confident that it is still optimal to produce a weekly rate of 2 doors and 6 windows.

21. Does the shadow price always have an economic interpretation?
Not necessarily
For example,there is no economic interpretation for dual variables associated with ratio constraints

22. Glass Example x1 = # of cases of 6-oz juice glasses (in 100s)
x2 = # of cases of 10-oz cocktail glasses (in 100s)
x3 = # of cases of champagne glasses (in 100s)
max 5 x x x3 ($100s)
s.t x x x3  (prod. cap. in hrs)
10 x x x3  (wareh. cap. in ft2)
x  (6-0z. glass dem.)
x1  0, x2  0, x3  0
(from AMP and slides from James Orlin)

23. Complementary optimal slackness conditions Z* = 51.4286
Decision Variables
x1 = (# of cases of 6-oz juice glasses (in 100s))
x2 = (# of cases of 10-oz cocktail glasses (in 100s))
x3 = 0 (# of cases of champagne glasses (in 100s))
Slack Variables
s1* = 0
s2* = 0
s3* =
Dual Variables
y1* =
y2* =
y3* = 0
Complementary
optimal slackness
conditions

24. Consider constraint 1. 6 x1 + 5 x2 + 8 x3  60 (prod. cap. in hrs)
Let’s look at the objective function if we change the production time from 60 and keep all other values the same.
Production
hours
Optimal obj. value
difference 60
51 3/7 61
52 3/14
11/14 62 53 63
53 11/14
The dual /shadow Price is 11/14.

25. More changes in the RHS Production hours Optimal obj. value difference64
54 4/7
11/14 65
55 5/14 66
56 1/11 * 67
56 17/22
15/22
The shadow Price is 11/14 until production = 65.5

26. What is the intuition for the shadow price staying constant, and then changing?
Recall from the simplex method that the simplex method produces a “basic feasible solution.” The basis can often be described easily in terms of a brief verbal description.

27. The verbal description for the optimum basis for the glass problem:
Produce Juice Glasses and cocktail glasses only
Fully utilize production and warehouse capacity
z = 5 x x2
6 x x2 = 60
10 x x2 = 150
x1 = 6 3/7 (6.4286)
x2 = 4 2/7 (4.2857)
z = 51 3/7 ( )

28. The verbal description for the optimum basis for the glass problem:
Produce Juice Glasses and cocktail glasses only
Fully utilize production and warehouse capacity
z = 5 x x2
6 x x2 = D
10 x x2 = 150
For D = 5.5, x1 = 8, and the constraint x1  8 becomes binding.
x1 = 6 3/7 + 2D/7
x2 = 4 2/7 – D/7
z = 51 3/7 + 11/14 D

29. How do we interpret the intervals?
If we change one coefficient in the RHS, say production capacity, by D the “basis” remains optimal, that is, the same equations remain binding.
So long as the basis remains optimal, the shadow prices are unchanged.
The basic feasible solution varies linearly with D. If D is big enough or small enough the basis will change.

30. Illustration with the glass example:
max 5 x x x3 ($100s)
s.t x x x3  (prod. cap. in hrs)
10 x x x3  (wareh. cap. in ft2)
x  (6-0z. glass dem.)
x1  0, x2  0, x3  0
The shadow price is the “increase” in the optimal value per unit increase in the RHS.
If an increase in RHS coefficient leads to an increase in optimal objective value, then the shadow price is positive.
If an increase in RHS coefficient leads to a decrease in optimal objective value, then the shadow price is negative.

31. Illustration with the glass example:
max 5 x x x3 ($100s)
s.t x x x3  (prod. cap. in hrs)
10 x x x3  (wareh. cap. in ft2)
x  (6-0z. glass dem.)
x1  0, x2  0, x3  0
Claim: the shadow price of the production capacity constraint cannot be negative.
Reason: any feasible solution for this problem remains feasible after the production capacity increases. So, the increase in production capacity cannot cause the optimum objective value to go down.

32. Illustration with the glass example:
max 5 x x x3 ($100s)
s.t x x x3  (prod. cap. in hrs)
10 x x x3  (wareh. cap. in ft2)
x  (6-0z. glass dem.)
x1  0, x2  0, x3  0
Claim: the shadow price of the “x1  0” constraint cannot be positive.
Reason: Let x* be the solution if we replace the constraint “x1  0” with the constraint “x1  1”. Then x* is feasible for the original problem, and thus the original problem has at least as high an objective value.

33. Signs of Shadow Prices for maximization problems
“  constraint” . The shadow price is non-negative.
“  constraint” . The shadow price is non-positive.
“ = constraint”. The shadow price could be zero or positive or negative.

34. Signs of Shadow Prices for minimization problems
The shadow price for a minimization problem is the “increase” in the objective function per unit increase in the RHS.
“  constraint” . The shadow price is non-positive.
“  constraint” . The shadow price is non-negative
“ = constraint”. The shadow price could be zero or positive or negative.
Please answer with your partner.

35. s.t 6 x1 + 5 x2 + 8 x3  60 (prod. cap. in hrs)
The shadow price of a non-binding constraint is 0. “Complementary Slackness.”
max 5 x x x3 ($100s)
s.t x x x3  (prod. cap. in hrs)
10 x x x3  (wareh. cap. in ft2)
x  (6-0z. glass dem.)
x1  0, x2  0, x3  0
In the optimal solution, x1 = 6 3/7.
Claim: The shadow price for the constraint “x1  8” is zero.
Intuitive Reason: If your optimum solution has x1 < 8, one does not get a better solution by permitting x1 > 8.

36. Is the shadow price the change in the optimal objective value if the RHS increases by 1 unit.
That is an excellent rule of thumb! It is true so long as the shadow price is valid in an interval that includes an increase of 1 unit.

37. The shadow price is valid if only one right hand side changes
The shadow price is valid if only one right hand side changes. What if multiple right hand side coefficients change?
The shadow prices are valid if multiple RHS coefficients change, but the ranges are no longer valid.

38. Reduced Costs

39. Do the non-negativity constraints also have shadow prices?
Yes. They are very special and are called reduced costs?
Look at the reduced costs for
Juice glasses reduced cost = 0
Cocktail glasses reduced cost = 0
Champagne glasses red. cost = -4/7

40. What is the managerial interpretation of a reduced cost?
There are two interpretations. Here is one of them.
We are currently not producing champagne glasses. How much would the profit of champagne glasses need to go up for us to produce champagne glasses in an optimal solution?
The reduced cost for champagne classes is –4/7. If we increase the revenue for these glasses by 4/7 (from 6 to 6 4/7), then there will be an alternative optimum in which champagne glasses are produced.

41. Why are they called the reduced costs? Nothing appears to be “reduced”
The reduced costs can be obtained by treating the shadow prices are real costs. This operation is called “pricing out.”

42. Pricing Out shadow price max 5 x1 + 4.5 x2 + 6 x3 ($100s) ……11/14
……1/35
…….0
max 5 x x x3 ($100s)
s.t x x x3  60
10 x x x3  150
1 x 
x1  0, x2  0, x3  0
Pricing out treats shadow prices as though they are real prices. The result is the “reduced costs.”

43. Pricing Out of x1 shadow price max 5 x1 + 4.5 x2 + 6 x3 ($100s)
……11/14
……1/35
…….0
max 5 x x x3 ($100s)
s.t x x x3  60
10 x x x3  150
1 x 
x1  0, x2  0, x3  0 5
- 6 x 11/14
- 10 x 1/35
- 1 x 0
= 5 – 33/7 – 2/7 = 0
Reduced cost of x1 =

44. Pricing Out of x2 shadow price ……11/14
……1/35
…….0
max 5 x x x3 ($100s)
s.t x x x3  60
10 x x x3  150
1 x 
x1  0, x2  0, x3  0
4.5
- 5 x 11/14
- 20 x 1/35
- 0 x 0
= – 55/14 – 4/7 = 0
Reduced cost of x2 =

45. Pricing Out of x3 shadow price max 5 x1 + 4.5 x2 + 6 x3 ($100s)
……11/14
……1/35
…….0
max 5 x x x3 ($100s)
s.t x x x3  60
10 x x x3  150
1 x 
x1  0, x2  0, x3  0 6
- 8 x 11/14
- 10 x 1/35
- 0 x 0
= 6 – 44/7 – 2/7 = -4/7
Reduced cost of x3 =

46. Can we use pricing out to figure out whether a new type of glass should be produced?
shadow price
……11/14
……1/35
…….0
max 5 x x x4 ($100s)
s.t x x x4  60
10 x x x4  150
1 x 
x1  0, x2  0, x4  0 7
- 8 x 11/14
- 20 x 1/35
- 0 x 0
= 7 – 44/7 – 4/7 = 1/7
Reduced cost of x4 =

47. Pricing Out of xj shadow price ……y1 max 5 x1 + 4.5 x2 + cj xj ($100s)
………
……ym
max 5 x x2 + cj xj ($100s)
s.t x x2 + a1j xj  60
10 x x2 + a2j xj  150
………..
……… amjxj = bm
x1  0, x2  0, x3  0
Reduced cost of xj = ?

48. Brief summary on reduced costs
The reduced cost of a non-basic variable xj is the “increase” in the objective value of requiring that xj >= 1.
The reduced cost of a basic variable is 0.
The reduced cost can be computed by treating shadow prices as real prices. This operation is known as “pricing out.”
Pricing out can determine if a new variable would be of value (and would enter the basis).

49. Summary
The shadow price is the unit change in the optimal objective value per unit change in the RHS.
The shadow price for a “ 0” constraint is called the reduced cost.
Shadow prices usually but not always have economic interpretations that are managerially useful.
Non-binding constraints have a shadow price of 0.
The sign of a shadow price can often be determined by using the economic interpretation
Shadow prices are valid in an interval.
Reduced costs can be determined by pricing out

50. Reduced Costs
The reduced cost of a variable x is the shadow price of the “x  0” constraint. It is also the negative of cost coefficient for x in the final tableau.
Suppose in the previous example that we required that x3  1? What is the impact on the optimal objective value? What is the resulting solution?
By the previous slide, the impact is -4/7.

51. More on reduced costs
In a pivot, multiples of constraints are added to the cost row.
We will use this fact to determine explicitly how the cost row in the final tableau is obtained.

52. Implications of Reduced Costs
Implication 1: increasing the cost coefficient of a non-basic variable by D leads to an increase of its reduced cost by D.

53. Implications of Reduced Costs
Implication 2: We can compute the reduced cost of any variable if we know the original column and if we know the “prices” for each constraint.
FACT: We can compute the reduced cost of a new variable. If the reduced cost is positive, it should be entered into the basis.

54. Every tableau has “prices
Every tableau has “prices.” These are usually called simplex multipliers.
The prices for the optimal tableau are the shadow prices.

55. Quick Summary
Connection between shadow prices and reduced cost. If xj is the slack variable for a constraint, then its reduced cost is the negative of the shadow price for the constraint.
The reduced cost for a variable is the negative of its cost coefficient in the final tableau

56. Sensitivity Analysis Computer Analysis

57. The Computer and Sensitivity Analysis
If an LP has more than two decision variables, the range of values for a rhs (or objective function coefficient) for which the basis remains optimal cannot be determined graphically.
These ranges can be computed by hand but this is often tedious, so they are usually determined by a packaged computer program. MPL and LINDO will be used and the interpretation of its sensitivity analysis discussed.
Note: sometimes Excel provides erroneous results

58. MPL – Sensitivity analysis infoc1
Reduced cost is the amount the objective function coefficient for variable i would have to be increased for there to be an alternative optimal solution.
More later…
Dual or Shadow prices are the amount the optimal z-value improves if the rhs of a constraint is increased by one unit (assuming no change in basis).
Dual variables b2

59. MPL – Sensitivity analysis info
Allowable ranges (w/o changing basis) for the x1 coefficient (c1) is:
0 £ c1 £ 7.5 c1
Allowable range (w/o changing basis) for the rhs (b2) of the second constraint is:
6 £ b2 £ 18 b2
What about c2? And b1 and b3?

60. Lindo Sensitivity Analysis
Allowable ranges – in terms of increase and decrease
(w/o changing basis) for the x1 coefficient (c1) is:
0 £ c1 £ 7.5

61. The Computer and Sensitivity Analysis
Consider the following maximization problem. Winco sells four types of products. The resources needed to produce one unit of each are:
Product 1
Product 2
Product 3
Product 4
Available
Raw material 2 3 4 7
4600
Hours of labor 5 6
5000
Sales price $4 $6 $7 $8
To meet customer demand, exactly 950 total units must be produced. Customers demand that at least 400 units of product 4 be produced. Formulate an LP to maximize profit.
Let xi = number of units of product i produced by Winco.

62. The Winco LP formulation:
max z = 4x1 + 6x2 +7x3 + 8x4
s.t. x1 + x x3 + x4 = 950
x4 ≥ 400
2x1 + 3x2 + 4x3 + 7x4 ≤ 4600
3x1 + 4x2 + 5x3 + 6x4 ≤ 5000
x1,x2,x3,x4 ≥ 0

63. LINDO output and sensitivity analysis example(s).
MAX X1 + 6 X2 + 7 X3 + 8 X4
SUBJECT TO
2) X1 + X2 + X3 + X4 =
3) X4 >= 400
4) 2 X1 + 3 X2 + 4 X3 + 7 X4 <=
5) 3 X1 + 4 X2 + 5 X3 + 6 X4 <=
END
LP OPTIMUM FOUND AT STEP
OBJECTIVE FUNCTION VALUE 1)
VARIABLE VALUE REDUCED COST X X X X
ROW SLACK OR SURPLUS DUAL PRICES 2) 3) 4) 5)
NO. ITERATIONS=
Reduced cost is the amount the objective function coefficient for variable i would have to be increased for there to be an alternative optimal solution.

64. LINDO sensitivity analysis example(s).
RANGES IN WHICH THE BASIS IS UNCHANGED:
OBJ COEFFICIENT RANGES
VARIABLE CURRENT ALLOWABLE ALLOWABLE
COEF INCREASE DECREASE
X INFINITY X X
X INFINITY
RIGHTHAND SIDE RANGES
ROW CURRENT ALLOWABLE ALLOWABLE
RHS INCREASE DECREASE
INFINITY
LINDO sensitivity analysis example(s).
Allowable range (w/o changing basis) for the x2 coefficient (c2) is:
5.50 £ c2 £ 6.667
Allowable range (w/o changing basis) for the rhs (b1) of the first constraint is:
850 £ b1 £ 1000

65. Shadow prices are shown in the Dual Prices section of LINDO output.
MAX X1 + 6 X2 + 7 X3 + 8 X4
SUBJECT TO
2) X1 + X2 + X3 + X4 =
3) X4 >= 400
4) 2 X1 + 3 X2 + 4 X3 + 7 X4 <=
5) 3 X1 + 4 X2 + 5 X3 + 6 X4 <=
END
LP OPTIMUM FOUND AT STEP
OBJECTIVE FUNCTION VALUE 1)
VARIABLE VALUE REDUCED COST X X X X
ROW SLACK OR SURPLUS DUAL PRICES 2) 3) 4) 5)
NO. ITERATIONS=
Shadow prices are shown in the Dual Prices section of LINDO output.
Shadow prices are the amount the optimal z-value improves if the rhs of a constraint is increased by one unit (assuming no change in basis).

66. Interpretation of shadow prices for the Winco LP
ROW SLACK OR SURPLUS DUAL PRICES
2) (overall demand)
3) (product 4 demand)
4) (raw material availability)
5) (labor availability)
Assuming the allowable range of the rhs is not violated, shadow (Dual) prices show: $3 for constraint 1 implies that each one-unit increase in total demand will increase net sales by $3. The -$2 for constraint 2 implies that each unit increase in the requirement for product 4 will decrease revenue by $2. The $1 shadow price for constraint 3 implies an additional unit of raw material (at no cost) increases total revenue by $1. Finally, constraint 4 implies any additional labor (at no cost) will not improve total revenue.

67. Shadow price signs
Constraints with ³ symbols will always have nonpositive shadow prices.
Constraints with £ will always have nonnegative shadow prices.
Equality constraints may have a positive, a negative, or a zero shadow price.

68. Managerial Use of Shadow Prices
The managerial significance of shadow prices is that they can often be used to determine the maximum amount a manager should be willing to pay for an additional unit of a resource. Reconsider the Winco to the right.
What is the most Winco should be willing to pay for additional units of raw material or labor?
MAX X1 + 6 X2 + 7 X3 + 8 X4
SUBJECT TO
2) X1 + X2 + X3 + X4 =
3) X4 >= 400
4) 2 X1 + 3 X2 + 4 X3 + 7 X4 <=
5) 3 X1 + 4 X2 + 5 X3 + 6 X4 <=
END
LP OPTIMUM FOUND AT STEP
OBJECTIVE FUNCTION VALUE 1)
VARIABLE VALUE REDUCED COST X X X X
ROW SLACK OR SURPLUS DUAL PRICES 2) 3) 4) 5)
NO. ITERATIONS=
raw material
labor

69. Managerial Use of Shadow Prices
The shadow price for raw material constraint (row 4) shows an extra unit of raw material would increase revenue $1. Winco could pay up to $1 for an extra unit of raw material and be as well off as it is now.
Labor constraint’s (row 5) shadow price is 0 meaning that an extra hour of labor will not increase revenue. So, Winco should not be willing to pay anything for an extra hour of labor.
MAX X1 + 6 X2 + 7 X3 + 8 X4
SUBJECT TO
2) X1 + X2 + X3 + X4 =
3) X4 >= 400
4) 2 X1 + 3 X2 + 4 X3 + 7 X4 <=
5) 3 X1 + 4 X2 + 5 X3 + 6 X4 <=
END
LP OPTIMUM FOUND AT STEP
OBJECTIVE FUNCTION VALUE 1)
VARIABLE VALUE REDUCED COST X X X X
ROW SLACK OR SURPLUS DUAL PRICES 2) 3) 4) 5)
NO. ITERATIONS=