CH_10_NON_LINEAR_PROGRAMING

Nonlinear Programming

Chapter 10 10-1

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Chapter Topics

■ Nonlinear Profit Analysis
■ Constrained Optimization
■ Solution of Nonlinear Programming Problems with Excel
■ Nonlinear Programming Model with Multiple Constraints
■ Nonlinear Model Examples

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Overview

■ 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.

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Optimal Value of a Single Nonlinear Function
Basic Model

Profit function, Z, with volume
independent of price:
Z = vp - cf - vcv

where v = sales volume
p = price
cf = unit fixed cost
cv = unit variable cost

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

Figure 10.1 Linear Relationship of Volume to Price

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Optimal Value of a Single Nonlinear Function

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 Function 10-5

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Optimal Value of a Single Nonlinear Function
Maximum Point on a Curve

■ 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 10-6

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Optimal Value of a Single Nonlinear Function
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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10-7

Constrained Optimization in Nonlinear Problems
Definition

■ 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.

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Constrained Optimization in Nonlinear Problems
Graphical Interpretation (1 of 3)

Effect of adding constraints to nonlinear problem:

Figure 10.5 Nonlinear Profit Curve for the Profit Analysis Model 10-9

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Constrained Optimization in Nonlinear Problems
Graphical Interpretation (2 of 3)

Figure 10.6 A Constrained Optimization Model 10-10

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Constrained Optimization in Nonlinear Problems
Graphical Interpretation (3 of 3)

Figure 10.7

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Constrained Optimization in Nonlinear Problems
Characteristics

■ 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.

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Western Clothing Problem
Solution Using Excel (1 of 3)

Exhibit 10.1

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Western Clothing Problem
Solution Using Excel (2 of 3)

Exhibit 10.2

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Western Clothing Problem
Solution Using Excel (3 of 3)

Exhibit 10.3

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Beaver Creek Pottery Company Problem 10-16
Solution Using Excel (1 of 6)

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

4 – 0.1X1 = profit ($) per bowl
5 – 0.2X2 = profit ($) per mug

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Beaver Creek Pottery Company Problem
Solution Using Excel (2 of 6)

Exhibit 10.4

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Beaver Creek Pottery Company Problem
Solution Using Excel (3 of 6)

Exhibit 10.5

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Beaver Creek Pottery Company Problem
Solution Using Excel (4 of 6)

Exhibit 10.6

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Beaver Creek Pottery Company Problem
Solution Using Excel (5 of 6)

Exhibit 10.7

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Beaver Creek Pottery Company Problem
Solution Using Excel (6 of 6)

Exhibit 10.8

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Western Clothing Company Problem 10-22
Solution Using Excel (1 of 4)

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 jeans
p2 = price of straight jeans

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Western Clothing Company Problem
Solution Using Excel (2 of 4)

Exhibit 10.9

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Western Clothing Company Problem
Solution Using Excel (3 of 4)

Exhibit 10.10

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Western Clothing Company Problem
Solution Using Excel (4 of 4)

Exhibit 10.11

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Facility Location Example Problem
Problem Definition and Data (1 of 2)

Centrally locate a facility that serves several customers or other
facilities in order to minimize distance or miles traveled (d) between
facility 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 = Σ diti
Where:
di = distance to town i
ti =annual trips to town i

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

Town Coordinates Annual Trips

Abbeville xy 75
Benton 20 20 105
Clayton 10 35 135
Dunnig 25 9 60
Eden 32 15 90
10 8

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Facility Location Example Problem
Solution Using Excel

Exhibit 10.12

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Facility Location Example Problem
Solution Map

Figure 10.8 Rescue Squad Facility Location 10-29

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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 on

investment)
■ while achieving some specified minimum return on the total

portfolio investment.

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Investment Portfolio Selection Example Problem
Definition and Model Formulation (2 of 2)

Minimize S = x12s12 + x22s22 + … +xn2sn2 + Σxixjrijsisj
i≠j
where:

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 i

rij = the correlation between returns on investments i and j

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

subject to:

r1x1 + r2x2 + … + rnxn ≥ rm
x1 + x2 + …xn = 1.0

where:

ri = expected annual return on investment i
rm = the minimum desired annual return from the portfolio

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Investment Portfolio Selection Example Problem
Solution Using Excel (1 of 5)

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

Altacam .08 .009
Bestco .09 .015
Com.com .16 .040
Delphi .12 .023

Stock combination (i,j) Correlation (rij)

A,B .4
A,C .3
A,D .6
B,C .2
B,D .7
C,D .4

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Investment Portfolio Selection Example Problem
Solution Using Excel (2 of 5)

Four stocks, desired annual return of at least 0.11.

Minimize

Z = S = x12(.009) + x22(.015) + x32(.040) + X42(.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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Investment Portfolio Selection Example Problem
Solution Using Excel (3 of 5)

Exhibit 10.13

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Investment Portfolio Selection Example Problem
Solution Using Excel (4 of 5)

Exhibit 10.14

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Investment Portfolio Selection Example Problem
Solution Using Excel (5 of 5)

Exhibit 10.15

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Hickory Cabinet and Furniture Company
Example Problem and Solution (1 of 2)

Model:
Maximize Z = $280x1 - 6x12 + 160x2 - 3x22
subject to:

20x1 + 10x2 = 800 board ft.
Where:

x1 = number of chairs
x2 = number of tables

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Hickory Cabinet and Furniture Company
Example Problem and Solution (2 of 2)

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