10-1 copyright © 2013 pearson education, inc. publishing as prentice hall nonlinear programming...

10-1Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Nonlinear Programming

Chapter 10


■Nonlinear Profit Analysis

■Constrained Optimization

■Solution of Nonlinear Programming

Problems with Excel

■Nonlinear Programming Model with

Multiple Constraints

■Nonlinear Model Examples

Chapter Topics


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

■ Solution methods are more complex than linear programming methods.

■ Determining an optimal solution is often difficult, if not impossible.

■ Solution techniques generally involve searching a solution surface for high or low points requiring the use of advanced mathematics.

Overview


Profit function, Z, with volume independent of price:

Z = vp - cf - vcv

where v = sales volumep = pricecf = unit fixed

costcv = unit variable

cost

Add volume/price relationship:

v = 1,500 - 24.6p

Nonlinear Profit AnalysisOptimal Value of a Single Nonlinear Function

Figure 10.1 Linear relationship of volume to price


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

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

Optimal Value of a Single Nonlinear Function

Figure 10.2 The nonlinear profit function


■ The slope of a curve at any point is equal to the derivative of the curve’s function.

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

Figure 10.3 Maximum profit for the profit function

Optimal Value of a Single Nonlinear FunctionMaximum Point on a Curve


Figure 10.4 Maximum profit, optimal price and optimal volume

Optimal Value of a Single Nonlinear FunctionSolution 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


■ 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 linear programming model

except that the objective function and/or the

constraint(s) are nonlinear.

■ Solution procedures are much more complex and

no guaranteed procedure exists for all NLP models.

Constrained Optimization in Nonlinear ProblemsDefinition


Effect of adding constraints to nonlinear problem:

Figure 10.5 Nonlinear profit curve for the profit analysis model

Constrained Optimization in Nonlinear ProblemsGraphical Interpretation (1 of 3)

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Figure 10.6 A constrained optimization model


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Figure 10.7 A constrained optimization model with a solution point not on the constraint boundary


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■ Unlike linear programming, the 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.

Constrained Optimization in Nonlinear ProblemsCharacteristics

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Exhibit 10.1

Western Clothing ProblemSolution Using Excel (1 of 3)

Formula for profit

=1500-24.6*C5

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Exhibit 10.2


Click on “GRG Nonlinear”

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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 producedx2 = number of mugs produced

4 – 0.1X1 = profit ($) per bowl5 – 0.2X2 = profit ($) per mug

Beaver Creek Pottery Company ProblemSolution Using Excel (1 of 6)

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Exhibit 10.4


=C5+2*C6

=SUMPRODUCT (C5:C6,D5:D6)

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Exhibit 10.5


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Exhibit 10.6


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Exhibit 10.7


Select “Sensitivity”

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Exhibit 10.8


Lagrange multiplier for labor

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

subject to:2x1 + 2.7x2 6,000

3.6x1 + 2.9x2 8,500 7.2x1 + 8.5x2 15,000

where:x1 = 1,500 - 24.6p1

x2 = 2,700 - 63.8p2

p1 = price of designer jeansp2 = price of straight jeans

Western Clothing Company ProblemSolution Using Excel (1 of 4)

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Exhibit 10.9


=D5-12

=SUMPRODUCT (C5:C6,E5:E6)

=2*C5+2.7*C6

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Exhibit 10.10


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Exhibit 10.11


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

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 ProblemProblem Definition and Data (1 of 2)

2 2( ) ( )i id x x y y

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Coordinates Town x y Annual Trips Abbeville Benton Clayton Dunnig Eden

20 10 25 32 10

20 35 9

15 8

75 105 135 60 90

Facility Location Example ProblemProblem Definition and Data (2 of 2)

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Exhibit 10.12

Facility Location Example ProblemSolution Using Excel

=SQRT((B6-C14)^2+(C6-C15)^2)

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Figure 10.8 Rescue squad facility location

Facility Location Example ProblemSolution Map

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Investment Portfolio Selection Example ProblemDefinition and Model Formulation (1 of 2)Objective of the portfolio selection model is:

■ to minimize some measure of portfolio risk (variance in the return on investment)

■ while achieving some specified minimum return on the total portfolio investment.

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Minimize S = x12s1

2 + x22s2

2 + … +xn2sn

2 + xixjrijsisj

where:S = variance of annual return of the portfolioxi,xj = the proportion of money invested in

investments i or jsi

2 = the variance for investment irij = the correlation between returns on

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

and jsubject to:

r1x1 + r2x2 + … + rnxn rm

x1 + x2 + …xn = 1.0

where:ri = expected annual return on investment irm = the minimum desired annual return from

the portfolio

Investment Portfolio Selection Example ProblemDefinition and Model Formulation (2 of 2) i≠j

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

Altacam Bestco Com.com Delphi

.08

.09

.16

.12

.009

.015

.040

.023

Stock combination (i,j) Correlation (rij)

A,B A,C A,D B,C B,D C,D

.4

.3

.6

.2

.7

.4

Investment Portfolio Selection Example ProblemSolution Using Excel (1 of 5)

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Four stocks, desired annual return of at least 0.11.Minimize Z = S = x1

2(.009) + x22(.015) + x3

2(.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 + .12x4 0.11

x1 + x2 + x3 + x4 = 1.00 xi 0


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Exhibit 10.13


Doubling covariances will include all investment pairs

=SUMPRODUCT9B6:B9,E6:E9)

=SUM(E6:E9)

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Exhibit 10.14


All money is invested constraint

Investment return constraint

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Exhibit 10.15

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The Hickory Cabinet and Furniture Company makes chairs and tables. The company has developed the following nonlinear programming model to determine the optimal number of chairs and tables to produce each day to maximize profit. Determine the solution using Excel.

Model:

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

2

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

Where:x1 = number of chairsx2 = number of tables

Hickory Cabinet and Furniture CompanyExample Problem and Solution (1 of 2)

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Hickory Cabinet and Furniture CompanyExample Problem and Solution (2 of 2)

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