Taylor Introms10 Ppt 10

5/27/2018 Taylor Introms10 Ppt 10

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10-1Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall

Nonlinear Programming

Chapter 10


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10-2

Nonlinear Profit Analysis

Constrained Optimization

Solution of Nonlinear Programming Problems with Excel

Nonlinear Programming Model with Multiple Constraints

Nonlinear Model Examples

Chapter Topics

Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall


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Problems that fit the general linear programming format butcontain nonlinear functions are termed nonlinear programming(NLP) problems.

Solution methods are more complex than linear programmingmethods.

Determining an optimal solution is often difficult, if notimpossible.

Solution techniques generally involve searching a solution surface

for high or low points requiring the use of advanced mathematics.

Overview



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Profit function, Z, with volumeindependent of price:

Z = vp - cf- vcv

where v = sales volume

p = pricecf= unit fixed costcv= unit variable cost

Add volume/price relationship:v = 1,500 - 24.6p

Optimal Value of a Single Nonlinear Function

Basic Model

Figure 10.1 Linear Relationship of Volume to PriceCopyright 2010 Pearson Education, Inc. Publishing as Prentice Hall


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With fixed cost (cf = $10,000) and variable cost (cv= $8):

Profit, Z = 1,696.8p - 24.6p2- 22,000


Figure 10.2 The Nonlinear Profit FunctionCopyright 2010 Pearson Education, Inc. Publishing as Prentice Hall


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The slope of a curve at any point is equal to the derivative of the

curves function.

The slope of a curve at its highest point equals zero.

Figure 10.3 Maximum profit for the profit function


Maximum Point on a Curve



7/3910-7Figure 10.4


Solution Using Calculus


Z = 1,696.8p - 24.6p2

- 2,000

dZ/dp = 1,696.8 - 49.2p

= 0

p = 1696.8/49.2

= $34.49

v = 1,500 - 24.6p

v = 651.6 pairs of jeans

Z = $7,259.45


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A nonlinear problem containing one or more constraints becomes a

constrained optimization model or a nonlinear programming

(NLP) model.

A nonlinear programming model has the same general form as the

linearprogramming model except that the objective function and/or

the constraint(s) are nonlinear.

Solution procedures are much more complex and no guaranteedprocedure exists for all NLP models.

Constrained Optimization in Nonlinear Problems

Definition



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Effect of adding constraints to nonlinear problem:

Figure 10.5 Nonlinear Profit Curve for the Profit Analysis Model


Graphical Interpretation (1 of 3)



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Figure 10.6 A Constrained Optimization Model





11/3910-11Figure 10.7





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Unlike linear programming, solution is often not on the

boundary of the feasible solution space.

Cannot simply look at points on the solution space boundary but

must consider other points on the surface of the objective

function.

This greatly complicates solution approaches.

Solution techniques can be very complex.


Characteristics



13/3910-13Exhibit 10.1

Western Clothing Problem

Solution Using Excel (1 of 3)



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Exhibit 10.2





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Exhibit 10.3





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Maximize Z = $(4 - 0.1x1)x1+ (5 - 0.2x2)x2

subject to:

x1+ 2x2= 40

Where:x1= number of bowls produced

x2= number of mugs produced

40.1X1= profit ($) per bowl50.2X2= profit ($) per mug

Beaver Creek Pottery Company Problem




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Exhibit 10.5





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Exhibit 10.6





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Exhibit 10.7




C C


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Exhibit 10.8




W Cl hi C P bl


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Maximize Z = (p1- 12)x1+ (p2- 9)x2

subject to:

2x1+ 2.7x2 6,000

3.6x1+ 2.9x2 8,5007.2x1+ 8.5x2 15,000

where:

x1= 1,500 - 24.6p1

x2= 2,700 - 63.8p2p1= price of designer jeans

p2= price of straight jeans

Western Clothing Company Problem



W Cl hi C P bl


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Exhibit 10.9




W Cl hi C P bl


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Exhibit 10.10




W Cl hi C P bl


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Exhibit 10.11




F ili L i E l P bl


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Centrally locate a facility that serves several customers or otherfacilities in order to minimize distance or miles traveled (d) betweenfacility and customers.

di= sqrt[(xi- x)2+ (yi- y)

2]

Where:(x,y) = coordinates of proposed facility(xi,yi) = coordinates of customer or location facility i

Minimize total miles d = diti

Where:di= distance to town iti=annual trips to town i

Facility Location Example Problem

Problem Definition and Data (1 of 2)


F ili L i E l P bl


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Coordinates

Town x y Annual Trips

AbbevilleBentonClaytonDunnig

Eden

20102532

10

2035915

8

7510513560

90


Problem Definition and Data (2 of 2)


F ili L i E l P bl


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Exhibit 10.12


Solution Using Excel


F ilit L ti E l P bl


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10-29Figure 10.8 Rescue Squad Facility Location


Solution Map


I t t P tf li S l ti E l P bl


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Investment Portfolio Selection Example Problem

Definition and Model Formulation (1 of 2)

Objective of the portfolio selection model is to:

minimize some measure of portfolio risk (variance in the return oninvestment)

while achieving some specified minimum return on the totalportfolio investment.


In t nt P rtf li S l ti n E pl Pr bl


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Minimize S = x12

s12

+ x22

s22

+ +x

n

2

sn2

+ xixjrijsisjwhere:

S = variance of annual return of the portfolio

xi,xj= the proportion of money invested in investments i or j

si2

= the variance for investment irij= the correlation between returns on investments i and j

si,sj= the std. dev. of returns for investments i and j

subject to:

r1

x1

+ r2

x2

+ + rn

xn

rm

x1+ x2+ xn= 1.0

where:

ri= expected annual return on investment i

rm= the minimum desired annual return from the portfolio


Definition and Model Formulation (2 of 2)

ij


In estment Portfolio Selection E mple Problem


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Stock (xi) Annual Return (ri) Variance (si)

AltacamBestcoCom.com

Delphi

.08

.09

.16

.12

.009

.015

.040

.023

Stock combination (i,j) Correlation (rij)

A,B

A,CA,DB,CB,DC,D

.4

.3

.6

.2

.7

.4






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Four stocks, desired annual return of at least 0.11.Minimize

Z = S = x12(.009) + x2

2(.015) + x32(.040) + X4

2(.023)

+ x1x2(.4)(.009)1/2(0.015)1/2+ x1x3(.3)(.009)

1/2(.040)1/2+

x1x4(.6)(.009)

1/2

(.023)

1/2

+ x2x3(.2)(.015)

1/2

(.040)

1/2

+x2x4(.7)(.015)1/2(.023)1/2+ x3x4(.4)(.040)

1/2(.023)1/2+

x2x1(.4)(.015)1/2(.009)1/2+ x3x1(.3)(.040)

1/2+ (.009)1/2+

x4x1(.6)(.023)1/2(.009)1/2+ x3x2(.2)(.040)

1/2(.015)1/2+

x4x2(.7)(.023)1/2(.015)1/2 + x4x3(.4)(.023)

1/2(.040)1/2

subject to:

.08x1+ .09x2+ .16x3+ .12x40.11x1+ x2+ x3+ x4= 1.00

xi0






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10-34Exhibit 10.13






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Exhibit 10.14






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Copyright 2010 Pearson Education, Inc. Publishing as Prentice HallExhibit 10.15

Hickory Cabinet and Furniture Company


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Model:

Maximize Z = $280x1- 6x12+ 160x2- 3x2

2

subject to:20x1+ 10x2= 800 board ft.

Where:

x1= number of chairs

x2= number of tables


Example Problem and Solution (1 of 2)




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Example Problem and Solution (2 of 2)



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