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Copyright 2007 by Oxford University Press, Inc.PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 11


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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 andselling an optimal quantity of goods

Firms minimize their cost of production by

using an optimal combination of inputs


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50

00

50

200

250

300

0

2 3 4 5 6 7

Q

TR

Expressing Economic

Relationships

Equations: TR = 100Q - 10Q2

Tables:

Graphs:

Q 0 1 2 3 4 5 6

TR 0 90 160 210 240 250 240


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Total, Average, and Marginal

Revenue

TR = PQ

AR = TR/Q

MR = (TR/(Q

Q T A M

0 0 - -

1 90 90 90

2 160 80 70

3 210 70 50

4 240 60 305 250 50 10

6 240 40 -10


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0

50

100

150

200

250

300

0 1 2 3 4 5 6 7

Q

TR

-40

-20

0

20

40

60

80

100

120

0 1 2 3 4 5 6 7

Q

AR, R

Total Revenue

Average and

Marginal Revenue


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Total, Average, and

Marginal Cost

Q TC AC MC

4 4

6 8

3 8 6

4 4 6 6

5 48 96 4

AC = TC/Q

MC = (TC/(Q


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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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Geometric Relationships A marginal value is positive, zero, and

negative, respectively, when a total

curve slopes upward, is horizontal, andslopes downward

A marginal value is above, equal to, and

below an average value, respectively,when the slope of the average curve is

positive, zero, and negative


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Profit Maximization

Profit


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Steps in Optimization

Define an objective mathematically as a

function of one or more choice variables

Define one or more constraints on thevalues of the objective function and/or

the choice variables

Determine the values of the choice

variables that maximize or minimize the

objective function while satisfying all of

the constraints


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New Management Tools

Benchmarking

Total Quality Management

Reengineering

The Learning Organization


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Other Management Tools Broadbanding

Direct Business Model

Networking

Performance Management


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Other Management Tools Pricing Power

Small-World Model

Strategic Development

Virtual Integration

Virtual Management


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Chapter 2 Appendix


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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.

0limX

dY Y

dX X( p

(

!(


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Rules of DifferentiationConstant Function Rule: The derivative

of a constant, Y = f(X) = a, is zero for all

values of a (the constant).

( )Y f a! !

0dY

dX!


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Rules of DifferentiationPower Function Rule: The derivative of

a power function, where a and b are

constants, is defined as follows.

( ) bY f a! !

1bdY

b aXdX

!


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Rules of DifferentiationSum-and-Differences Rule: The derivative

of the sum or difference of two functions,

U and V, is defined as follows.

( )U g X! ( )V h X!

dY dU dV

dX dX dX ! s

Y U V! s


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Rules of DifferentiationProduct Rule: The derivative of the

product of two functions, U and V, is

defined as follows.

( )U g X! ( )V h X!

dY dV dUU V

dX dX dX!

Y U V!


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Rules of DifferentiationQuotient Rule: The derivative of the

ratio of two functions, U and V, is

defined as follows.

( )U g X! ( )V h X!U

YV

!

2

dU dVV UdY dX dX

dX V

!


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Rules of DifferentiationChain Rule: The derivative of a function

that is a function of X is defined as follows.

( )U g X!( )Y f U!

dY dY d U

dX dU dX!


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Optimization with Calculus

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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Univariate OptimizationGiven objective function Y = f(X)

Find X such that dY/dX = 0Second derivative rules:

If d2Y/dX2 > 0, then X is a minimum.

If d2Y/dX2 < 0, then X is a maximum.


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Example1

Given the following total revenue (TR)

function, determine the quantity of

output (Q) that will maximize totalrevenue:

TR = 100Q 10Q2

dTR/dQ = 100 20Q = 0

Q* = and d2TR/dQ2 = -20 < 0


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Example 2 Given the following total revenue (TR)

function, determine the quantity of

output (Q) that will maximize totalrevenue:

TR = Q 0. Q2

dTR/dQ = Q = 0

Q* = and d2TR/dQ2 = -1 < 0


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Example Given the following marginal cost

function (MC), determine the quantity of

output that will minimize MC:

MC = Q2 1 Q + 7

dMC/dQ = Q - 1 = 0

Q* = 2. 7 and d2MC/dQ2 = > 0


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Example Given

TR = Q 0. Q2

TC = Q Q2 + 7Q + 2

Determine Q that maximizes profit ():

= Q 0. Q2 (Q Q2 + 7Q + 2)


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Example : Solution Method 1

d/dQ = Q - Q2 + 1 Q 7 = 0

-12 + 1 Q - Q2 = 0

Method 2

MR = dTR/dQ = Q

MC = dTC/dQ = Q2 - 1 Q + 7

Set MR = MC: Q = Q2 - 1 Q + 7

Use quadratic formula: Q* =


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Quadratic Formula

Write the equation in the following form:

aX2 + bX + c = 0

The solutions have the following form:2

b b 4ac

2a

s


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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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Example Determine the values of X and Y that

maximize the following profit function:

= 0X 2X2 XY Y 2 + 100Y

Solution

/X = 0 X Y = 0

/Y = -X Y + 100 = 0

Solve simultaneously

X = 1 . 2 and Y = 1 . 2


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Constrained Optimization Substitution Method

Substitute constraints into the objective

function and then maximize the objectivefunction

Lagrangian Method

Form the Lagrangian function by addingthe Lagrangian variables and constraints to

the objective function and then maximize

the Lagrangian function


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Example Use the substitution method to


= 0X 2X2 XY Y 2 + 100Y

Subject to the following constraint:

X + Y = 12


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Example : Solution Substitute X = 12 Y into profit:

= 0(12 Y) 2(12 Y)2 (12 Y)Y Y2 + 100Y

= Y2 + Y + 72

Solve as univariate function:

d/dY = Y + = 0

Y = 7 and X =


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Example 7 Use the Lagrangian method to


= 0X 2X2 XY Y 2 + 100Y

Subject to the following constraint:

X + Y = 12


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Example 7: Solution

Form the Lagrangian function L = 0X 2X2 XY Y 2 + 100Y + P(X + Y 12)

Find the partial derivatives and solvesimultaneously

dL/dX = 0 X Y + P = 0

dL/dY = X Y + 100 + P = 0 dL/dP = X + Y 12 = 0

Solution: X = , Y = 7, and P = -


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Interpretation of the

Lagrangian Multiplier, P Lambda, P, is the derivative of the

optimal value of the objective function

with respect to the constraint In Example 7, P = - , so a one-unit

increase in the value of the constraint (from

-12 to -11) will cause profit to decrease by

approximately units

Actual decrease is . units

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