computational methods for management and economics carla gomes module 7b duality and sensitivity...

Computational Methods forManagement and Economics

Carla Gomes

Module 7bDuality and Sensitivity Analysis

Economic Interpretation of Duality

(slides adapted from: M. Hillier’s, J. Orlin’s, and H. Sarper’s)

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

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

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)

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


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

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)?

Graphical Analysis of Dual variables – Variation in RHS Increasing level of resource 2 (b2)

0 2 4 6 8

8

6

4

10

2

Feasible

region

Production rate for doorsD

W

2 W =12

D = 4

3 D + 2 W = 18

Production rate for windows

(2,6)

Z=3(2)+5(6)=36

2w=13 Z=3(5/3)+5(13/2)=37.5

(5/3,13/2)

∆ Z=1.5= y2*

Why is y1*=0?

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.

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.

Does Z always increase at the same rate if we keep increasing the amount of resource 2?

0 2 4 6 8

8

6

4

10

2

Feasible

region


W

2 W =12

D = 4

3 D + 2 W = 18


(2,6)

Z=3(2)+5(6)=36

2w=13 Z=3(5/3)+5(13/2)=37.5(5/3,13/2)

∆ Z=1.5= y2*

What if b2 > 18 (i.e., 2W>18)?

the optimal solution will stay at (0,9) for b2>=18

b2=18(0,9)

If b2 < 6 the solution will no longer vary proportionally. The optimalsolution varies proportionally to the variation in b2 only if 6 <= b2 <=18.In other words, the current basis remainsoptimal for 6 ≤ b2 ≤ 18, but the decision variable values and z-value will change.

Does Z always decrease at the same rate if we decrease resource 2?

(2,6)

(5/3,13/2)

0 2 4 6 8

8

6

4

10

2

Feasible

region


W

2 W =12

D = 4

3 D + 2 W = 18


Z=3(2)+5(6)=36

2w=13 Z=3(5/3)+5(13/2)=37.5∆ Z=1.5= y2*

b2=6

• 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.

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)

How do we interpret the intervals?

• If we change one coefficient in the RHS, say capacity of plant 2, by 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. If is big enough or small enough the basis will change.

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.

Sensitivity analysis for c1

How much can we vary c1 without changing

the current basic optimal solution?

Sensitivity analysis for c1

Dc

5

1

0 2 4 6 8

8

6

4

2

Production rate

for windows

Production rate for doors

Feasible

region

(2, 6)

Optimal solution

10

W

D

P = 3600 = 300D + 500W

P = 3000 = 300D + 500W

P = 1500 = 300D + 500W

Our objective function is: Z= c1 D+5W=k

slope of iso-profit line is:

isoprofit line

How much can c1 vary until the slope of the iso-profit line equals the slope of constraint 2 and constraint 3?

• 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

5.7102

15123

51

01051

c

cDc

cDc

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.

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

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 x1 + 4.5 x2 + 6 x3 ($100s)

s.t 6 x1 + 5 x2 + 8 x3 60 (prod. cap. in hrs)

10 x1 + 20 x2 + 10 x3 150 (wareh. cap. in ft2)

x1 8 (6-0z. glass dem.)

x1 0, x2 0, x3 0

(from AMP and slides from James Orlin)

• Z* = 51.4286Decision Variables• x1 = 6.4286 (# of cases of 6-oz juice glasses (in 100s))• x2 = 4.2857 (# 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* = 1.5714Dual Variables• y1* = 0.7857• y2* = 0.0286• y3* = 0

Complementary optimal slackness

conditions

• 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 11/14

63 53 11/14 11/14

The dual /shadow Price is 11/14.

More changes in the RHS

Production

hours

Optimal obj. value

difference

64 54 4/7 11/14

65 55 5/14 11/14

66 56 1/11 *

67 56 17/22 15/22

The shadow Price is 11/14 until production = 65.5

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.

The verbal description for the optimum basis for the glass problem:

1. Produce Juice Glasses and cocktail glasses only

2. Fully utilize production and warehouse capacity

z = 5 x1 + 4.5 x2

6 x1 + 5 x2 = 60

10 x1 + 20 x2 = 150

x1 = 6 3/7 (6.4286)

x2 = 4 2/7 (4.2857)

z = 51 3/7 (51.4286)

The verbal description for the optimum basis for the glass problem:

1. Produce Juice Glasses and cocktail glasses only

2. Fully utilize production and warehouse capacity

z = 5 x1 + 4.5x2

6 x1 + 5 x2 = 60 +

10 x1 + 20 x2 = 150

x1 = 6 3/7 + 2/7

x2 = 4 2/7 – /7

z = 51 3/7 + 11/14

For = 5.5, x1 = 8, and the constraint x1 8 becomes binding.

How do we interpret the intervals?

• If we change one coefficient in the RHS, say production capacity, by 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. If is big enough or small enough the basis will change.

Illustration with the glass example:

max 5 x1 + 4.5 x2 + 6 x3 ($100s)




x1 0, x2 0, x3 0The 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.

Illustration with the glass example:max 5 x1 + 4.5 x2 + 6 x3 ($100s)




x1 0, x2 0, x3 0Claim: 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.

Illustration with the glass example:max 5 x1 + 4.5 x2 + 6 x3 ($100s)




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.

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.

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.

The shadow price of a non-binding constraint is 0. “Complementary Slackness.”

max 5 x1 + 4.5 x2 + 6 x3 ($100s)




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.

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.

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.

Reduced Costs

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

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.

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.”

Pricing Out

max 5 x1 + 4.5 x2 + 6 x3 ($100s)

s.t 6 x1 + 5 x2 + 8 x3 60

10 x1 + 20 x2 + 10 x3 150

1 x1 8

x1 0, x2 0, x3 0

shadow price

……11/14

……1/35

…….0

Pricing out treats shadow prices as though they are real prices. The result is the “reduced costs.”

Pricing Out of x1

max 5 x1 + 4.5 x2 + 6 x3 ($100s)

s.t 6 x1 + 5 x2 + 8 x3 60

10 x1 + 20 x2 + 10 x3 150

1 x1 8

x1 0, x2 0, x3 0

shadow price

……11/14

……1/35

…….0

Reduced cost of x1 = 5

- 6 x 11/14

- 10 x 1/35

- 1 x 0

= 5 – 33/7 – 2/7 = 0

Pricing Out of x2

max 5 x1 + 4.5 x2 + 6 x3 ($100s)

s.t 6 x1 + 5 x2 + 8 x3 60

10 x1 + 20 x2 + 10 x3 150

1 x1 8

x1 0, x2 0, x3 0

shadow price

……11/14

……1/35

…….0

Reduced cost of x2 = 4.5

- 5 x 11/14

- 20 x 1/35

- 0 x 0

= 4.5 – 55/14 – 4/7 = 0

Pricing Out of x3

max 5 x1 + 4.5 x2 + 6 x3 ($100s)

s.t 6 x1 + 5 x2 + 8 x3 60

10 x1 + 20 x2 + 10 x3 150

1 x1 8

x1 0, x2 0, x3 0

shadow price

……11/14

……1/35

…….0


- 8 x 11/14

- 10 x 1/35

- 0 x 0

= 6 – 44/7 – 2/7 = -4/7

Can we use pricing out to figure out whether a new type of glass should be

produced?max 5 x1 + 4.5 x2 + 7 x4 ($100s)

s.t 6 x1 + 5 x2 + 8 x4 60

10 x1 + 20 x2 + 20 x4 150

1 x1 8

x1 0, x2 0, x4 0

shadow price

……11/14

……1/35

…….0


- 8 x 11/14

- 20 x 1/35

- 0 x 0

= 7 – 44/7 – 4/7 = 1/7

Pricing Out of xj

max 5 x1 + 4.5 x2 + cj xj ($100s)

s.t 6 x1 + 5 x2 + a1j xj 60

10 x1 + 20 x2 + a2j xj 150

………..

………. + amjxj = bm

x1 0, x2 0, x3 0

shadow price

……y1

……y2

………

……ym

Reduced cost of xj = ?

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).

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

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.

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.

Implications of Reduced Costs

• Implication 1: increasing the cost coefficient of a non-basic variable by leads to an increase of its reduced cost by .

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.

• Every tableau has “prices.” These are usually called simplex multipliers.

• The prices for the optimal tableau are the shadow prices.

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


Computer Analysis

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

MPL – Sensitivity analysis info

c1

b2

Dual variables

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).

MPL – Sensitivity analysis info

c1

b2

What about c2? And b1 and b3?

Allowable ranges (w/o changing basis) for the x1 coefficient (c1) is:

0 c1 7.5

Allowable range (w/o changing basis) for the rhs (b2) of the second constraint is:

6 b2 18

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

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

3 4 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.

• The Winco LP formulation:

max z = 4x1 + 6x2 +7x3 + 8x4

s.t. x1 + x2 + x3 + x4 = 950

x4 ≥ 400

2x1 + 3x2 + 4x3 + 7x4 ≤ 4600

3x1 + 4x2 + 5x3 + 6x4 ≤ 5000

x1,x2,x3,x4 ≥ 0

MAX 4 X1 + 6 X2 + 7 X3 + 8 X4 SUBJECT TO 2) X1 + X2 + X3 + X4 = 950 3) X4 >= 400 4) 2 X1 + 3 X2 + 4 X3 + 7 X4 <= 4600 5) 3 X1 + 4 X2 + 5 X3 + 6 X4 <= 5000 END

LP OPTIMUM FOUND AT STEP 4

OBJECTIVE FUNCTION VALUE 1) 6650.000

VARIABLE VALUE REDUCED COST X1 0.000000 1.000000 X2 400.000000 0.000000 X3 150.000000 0.000000 X4 400.000000 0.000000

ROW SLACK OR SURPLUS DUAL PRICES 2) 0.000000 3.000000 3) 0.000000 -2.000000 4) 0.000000 1.000000 5) 250.000000 0.000000

NO. ITERATIONS= 4

LINDO output and sensitivity analysis example(s).

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.

RANGES IN WHICH THE BASIS IS UNCHANGED:

OBJ COEFFICIENT RANGES

VARIABLE CURRENT ALLOWABLE ALLOWABLE

COEF INCREASE DECREASE

X1 4.000000 1.000000 INFINITY

X2 6.000000 0.666667 0.500000

X3 7.000000 1.000000 0.500000

X4 8.000000 2.000000 INFINITY

RIGHTHAND SIDE RANGES

ROW CURRENT ALLOWABLE ALLOWABLE

RHS INCREASE DECREASE

2 950.000000 50.000000 100.000000

3 400.000000 37.500000 125.000000

4 4600.000000 250.000000 150.000000

5 5000.000000 INFINITY 250.000000

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

Shadow prices are shown in the Dual Prices section of LINDO output.

MAX 4 X1 + 6 X2 + 7 X3 + 8 X4 SUBJECT TO 2) X1 + X2 + X3 + X4 = 950 3) X4 >= 400 4) 2 X1 + 3 X2 + 4 X3 + 7 X4 <= 4600 5) 3 X1 + 4 X2 + 5 X3 + 6 X4 <= 5000 END



VARIABLE VALUE REDUCED COST X1 0.000000 1.000000 X2 400.000000 0.000000 X3 150.000000 0.000000 X4 400.000000 0.000000


NO. ITERATIONS= 4

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).

Interpretation of shadow prices for the Winco LP

ROW SLACK OR SURPLUS DUAL PRICES

2) 0.000000 3.000000 (overall demand)

3) 0.000000 -2.000000 (product 4 demand)

4) 0.000000 1.000000 (raw material availability)

5) 250.000000 0.000000 (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.

Shadow price signs

1. Constraints with symbols will always have nonpositive shadow prices.

2. Constraints with will always have nonnegative shadow prices.

3. Equality constraints may have a positive, a negative, or a zero shadow price.

Managerial Use of Shadow Prices MAX 4 X1 + 6 X2 + 7 X3 + 8 X4 SUBJECT TO 2) X1 + X2 + X3 + X4 = 950 3) X4 >= 400 4) 2 X1 + 3 X2 + 4 X3 + 7 X4 <= 4600 5) 3 X1 + 4 X2 + 5 X3 + 6 X4 <= 5000 END



VARIABLE VALUE REDUCED COST

X1 0.000000 1.000000 X2 400.000000 0.000000 X3 150.000000 0.000000 X4 400.000000 0.000000


NO. ITERATIONS= 4

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?

raw material

labor

Managerial Use of Shadow PricesMAX 4 X1 + 6 X2 + 7 X3 + 8 X4 SUBJECT TO 2) X1 + X2 + X3 + X4 = 950 3) X4 >= 400 4) 2 X1 + 3 X2 + 4 X3 + 7 X4 <= 4600 5) 3 X1 + 4 X2 + 5 X3 + 6 X4 <= 5000 END



VARIABLE VALUE REDUCED COST

X1 0.000000 1.000000 X2 400.000000 0.000000 X3 150.000000 0.000000 X4 400.000000 0.000000


NO. ITERATIONS= 4

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.

computational methods for management and economics carla gomes module 7b duality and sensitivity...

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