Optimization Techniques

Optimization Techniques - Transcript

Optimization Techniques
Methods for maximizing or minimizing an objective function Examples
Consumers maximize utility by purchasing an optimal combination of goods Firms maximize profit by producing and selling an optimal quantity of goods Firms minimize their cost of production by using an optimal combination of inputs
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Slide 1

Concept of the Derivative
The derivative of Y with respect to X is equal to the limit of the ratio Y X as X approaches zero

dY Y lim dX X 0 X

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Slide 2

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Slide 3

Total Average and Marginal Revenue
TR PQ AR TR Q MR TR Q

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Slide 4

Total Revenue Equation
Equation Table Graph
Q TR
300 250 200 150 100 50 0 0 1 2 3 4 5 6 7 Q

TR 100Q 10Q2
0 0
TR

1 90

2 3 4 5 6 160 210 240 250 240

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Slide 5

Total Revenue Schedule of a Firm

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Slide 6

Total Revenue Curve of a Firm

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Slide 7

TR 300 250

Total Revenue

200 150 100 50 0 0 1 2 3 4 5 6 7 Q

AR MR 120

Average and Marginal Revenue

100 80 60 40 20 0 20 40 Q 0 1 2 3 4 5 6 7

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Slide 8

Total Average and Marginal Cost
AC TC Q MC TC Q Q 0 1 2 3 4 5 TC AC MC 20 140 140 120 160 80 20 180 60 20 240 60 60 480 96 240
Slide 9

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Total Average and Marginal Cost

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Slide 10

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Slide 11

Geometric Relationships
The slope of a tangent to a total curve at a point is equal to the marginal value at that point The slope of a ray from the origin to a point on a total curve is equal to the average value at that point

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Slide 12

Geometric Relationships
A marginal value is positive zero and negative respectively when a total curve slopes upward is horizontal and slopes downward A marginal value may be negative but an average value can never be negative

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Slide 13

Profit Maximization
Q 0 1 2 3 4 5
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TR 0 90 160 210 240 250

TC Profit 20 20 140 50 160 0 180 30 240 0 480 230
Slide 14

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Slide 15

Steps in Optimization
Define an objective function of one or more choice variables Define the constraint on the values of the objective function Determine the values of the choice variables that maximize or minimize the objective function while satisfying the constraint
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Slide 16

New Management Tools for Optimization
Benchmarking tool for improving productivity and quality Total Quality Management constantly improving the quality of products and the firm s processes to deliver more value to customers e g Six Sigma Reengineering radical redesign of all the firm s processes to achieve major gains Learning Organization values continuing learning both individual and collective
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Slide 17

Other Management Tools for Optimization
Broad banding elimination of multiple salary grades to foster movement among jobs within the firm and lower cost Direct Business Model eliminating the time and cost of third party distribution Networking forming of temporary strategic alliances among firms as per their core competence Performance Management holding executives and their subordinates accountable for delivering the desired results
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Slide 18

Pricing Power ability of a firm to raise prices faster than the rise in its costs and vice versa Small World Model linking well connected individuals from each level of the organization to one another to improve flow of information and the operational efficiency Strategic Development continuous review of strategic decisions Virtual Integration treating suppliers and customers as if they were part of the company which reduces the need for inventories Virtual Management ability of a manager to simulate consumer behavior using computer models
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Other Management Tools for Optimization

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Slide 19

Univariate Optimization
Given objective function Y f X Find X such that dY dX 0 Second derivative rules If d2Y dX2 0 then X is a minimum If d2Y dX2 0 then X is a maximum

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Slide 20

Example 1
Given the following total revenue TR function determine the quantity of output Q that will maximize total revenue TR 100Q 10Q2 dTR dQ 100 20Q 0 Q 5 and d2TR dQ2 20 0
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Slide 21

Example 2
Given the following total revenue TR function determine the quantity of output Q that will maximize total revenue TR 45Q 0 5Q2 dTR dQ 45 Q 0 Q 45 and d2TR dQ2 1 0
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Slide 22

Example 3
Given the following marginal cost function MC determine the quantity of output that will minimize MC MC 3Q2 16Q 57 dMC dQ 6Q 16 0 Q 2 67 and d2MC dQ2 6 0

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Slide 23

Example 4
Given
TR 45Q 0 5Q2 TC Q3 8Q2 57Q 2

Determine Q that maximizes profit
45Q 0 5Q2 Q3 8Q2 57Q 2

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Slide 24

Example 4 Solution
Method 1
d dQ 45 Q 3Q2 16Q 57 0 12 15Q 3Q2 0

Method 2
MR dTR dQ 45 Q MC dTC dQ 3Q2 16Q 57 Set MR MC 45 Q 3Q2 16Q 57

Use quadratic formula Q 4
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Slide 25

Quadratic Formula
Write the equation in the following form
aX2 bX c 0

The solutions have the following form

b

b 4ac 2a
2

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Slide 26

Multivariate Optimization
Objective function Y f X1 X2 Xk Find all Xi such that Y Xi 0 Partial derivative
Y Xi dY dXi while all Xj where j i are held constant

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Slide 27

Example 5
Determine the values of X and Y that maximize the following profit function
80X 2X2 XY 3Y2 100Y

Solution
X 80 4X Y 0 Y X 6Y 100 0 Solve simultaneously X 16 52 and Y 13 91
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Slide 28

Constrained Optimization
Substitution Method
Substitute constraints into the objective function and then maximize the objective function

Lagrangian Method
Form the Lagrangian function by adding the Lagrangian variable and constraint to the objective function and then maximize the Lagrangian function
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Slide 29

Example 6
Use the substitution method to maximize the following profit function
80X 2X2 XY 3Y2 100Y

Subject to the following constraint
X Y 12

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Slide 30

Example 6 Solution
Substitute X 12 Y into profit
80 12 Y 2 12 Y 2 12 Y Y 3Y2 100Y

4Y2 56Y 672

Solve as univariate function
d dY 8Y 56 0 Y 7 and X 5

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Slide 31

Example 7
Use the Lagrangian method to maximize the following profit function
80X 2X2 XY 3Y2 100Y

Subject to the following constraint
X Y 12

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Slide 32

Example 7 Solution
Form the Lagrangian function
L 80X 2X2 XY 3Y2 100Y X Y 12

Find the partial derivatives and solve simultaneously
dL dX 80 4X Y 0 dL dY X 6Y 100 0 dL d X Y 12 0

Solution X 5 Y 7 and 53
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Slide 33

Interpretation of the Lagrangian Multiplier
Lambda is the derivative of the optimal value of the objective function with respect to the constraint
In Example 7 53 so a one unit increase in the value of the constraint from 12 to 11 will cause profit to decrease by approximately 53 units

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Slide 34