Introduction to Management Science
8th Edition
by
Bernard W. Taylor III
Chapter 11
Nonlinear Programming
Chapter 11 - Nonlinear Programming
1
Chapter Topics
Nonlinear Profit Analysis
Constrained Optimization
Solution of Nonlinear Programming Problems with Excel
A Nonlinear Programming Model with Multiple Constraints
Nonlinear Model Examples
Chapter 11 - Nonlinear Programming
2
Overview
Many business problems can be modeled only with nonlinear
functions.
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.
Often difficult, if not impossible, to determine optimal solution.
Solution techniques generally involve searching a solution
surface for high or low points requiring the use of advanced
mathematics.
Computer techniques (Excel) are used in this chapter.
Chapter 11 - Nonlinear Programming
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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 = fixed cost
cv = variable cost
Add volume/price relationship:
v = 1,500 - 24.6p
Figure 11.1
Linear Relationship of Volume to Price
Chapter 11 - Nonlinear Programming
4
Optimal Value of a Single Nonlinear Function
Expanding the Basic Model to a Nonlinear Model
With fixed cost (cf = $10,000) and variable cost (cv = $8):
Z = 1,696.8p - 24.6p2 - 22,000
Figure 11.2
The Nonlinear
Profit Function
Chapter 11 - Nonlinear Programming
5
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 11.3
Maximum profit
for the
profit function
Chapter 11 - Nonlinear Programming
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Optimal Value of a Single Nonlinear Function
Solution Using Calculus
nonlinearity!
Z = 1,696.8p - 24.6p2 - 22,000
dZ/dp = 1,696.8 - 49.2p
p = $34.49
v = 1,500 - 24.6p
v = 651.6 pairs of jeans
Z = $7,259.45
Figure 11.4
Maximum Profit, Optimal Price, and Optimal Volume
Chapter 11 - Nonlinear Programming
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Constrained Optimization in Nonlinear Problems
Definition
If a nonlinear problem contains one or more constraints it
becomes a constrained optimization model or a nonlinear
programming 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.
Chapter 11 - Nonlinear Programming
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Constrained Optimization in Nonlinear Problems
Graphical Interpretation (1 of 3)
Effect of adding constraints to nonlinear problem:
Figure 11.5
Nonlinear Profit Curve for the Profit Analysis Model
Chapter 11 - Nonlinear Programming
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Constrained Optimization in Nonlinear Problems
Graphical Interpretation (2 of 3)
Figure 11.6
A Constrained Optimization Model
Chapter 11 - Nonlinear Programming
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Constrained Optimization in Nonlinear Problems
Graphical Interpretation (3 of 3)
Figure 11.7
A Constrained Optimization Model with a Solution
Point Not on the Constraint Boundary
Chapter 11 - Nonlinear Programming
11
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.
Chapter 11 - Nonlinear Programming
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Western Clothing Problem
Solution Using Excel (1 of 3)
Exhibit 11.1
Chapter 11 - Nonlinear Programming
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Western Clothing Problem
Solution Using Excel (2 of 3)
Exhibit 11.2
Chapter 11 - Nonlinear Programming
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Western Clothing Problem
Solution Using Excel (3 of 3)
Exhibit 11.3
Chapter 11 - Nonlinear Programming
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Beaver Creek Pottery Company Problem
Solution Using Excel (1 of 6)
Maximize Z = $(4 - 0.1x1)x1 + (5 - 0.2x2)x2
subject to:
x1 + x2 = 40
Where:
x1 = number of bowls produced
x2 = number of mugs produced
Chapter 11 - Nonlinear Programming
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Beaver Creek Pottery Company Problem
Solution Using Excel (2 of 6)
Exhibit 11.4
Chapter 11 - Nonlinear Programming
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Beaver Creek Pottery Company Problem
Solution Using Excel (3 of 6)
Exhibit 11.5
Chapter 11 - Nonlinear Programming
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Beaver Creek Pottery Company Problem
Solution Using Excel (4 of 6)
Exhibit 11.6
Chapter 11 - Nonlinear Programming
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Beaver Creek Pottery Company Problem
Solution Using Excel (5 of 6)
Exhibit 11.7
Chapter 11 - Nonlinear Programming
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Beaver Creek Pottery Company Problem
Solution Using Excel (6 of 6)
Exhibit 11.8
Chapter 11 - Nonlinear Programming
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Western Clothing Company Problem
Solution Using Excel (1 of 4)
Maximize Z = (p1 - 12)x1 + (p2 - 9)x2
subject to:
2x1 + 2.7x2  6,00
3.6x1 + 2.9x2  8,500
7.2x1 + 8.5x2  15,000
where:
x1 = 1,500 - 24.6p1
x2 = 2,700 - 63.8p
p1 = price of designer jeans
p2 = price of straight jeans
Chapter 11 - Nonlinear Programming
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Western Clothing Company Problem
Solution Using Excel (2 of 4)
Exhibit 11.9
Chapter 11 - Nonlinear Programming
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Western Clothing Company Problem
Solution Using Excel (3 of 4)
Exhibit 11.10
Chapter 11 - Nonlinear Programming
24
Western Clothing Company Problem
Solution Using Excel (4 of 4)
Exhibit 11.11
Chapter 11 - Nonlinear Programming
25
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) (= straight-line distance)
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 i
ti =annual trips to town i
Chapter 11 - Nonlinear Programming
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Facility Location Example Problem
Problem Definition and Data (2 of 2)
Town
Abbeville
Benton
Clayton
Dunnig
Eden
Coordinates
x
y
20
20
10
35
25
9
32
15
10
8
Chapter 11 - Nonlinear Programming
Annual Trips
75
105
135
60
90
27
Facility Location Example Problem
Solution Using Excel
Exhibit 11.12
Chapter 11 - Nonlinear Programming
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Facility Location Example Problem
Solution Map
Figure 11.8
Rescue Squad Facility Location
Chapter 11 - Nonlinear Programming
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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.
Since variance is the sum of squares of differences, it is
mathematically identical to the “straight-line distance”!
Thus, it is possible to visualize variances as such
distances, and minimizing the overall variance is then
mathematically identical to minimizing such distances.
Chapter 11 - Nonlinear Programming
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Investment Portfolio Selection Example Problem
Definition and Model Formulation (2 of 2)
Minimize S = x12s12 + x22s22 + … +xn2sn2 + xixjrijsisj
“straight-line distance”
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
Chapter 11 - Nonlinear Programming
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Investment Portfolio Selection Example Problem
Solution Using Excel (1 of 5)
Four stocks, desired annual return of at least 0.11.
Minimize
Z = S = xA2(.009) + xB2(.015) + xC2(.040) + XD2(.023)
+ xAxB (.4)(.009)1/2(0.015)1/2 + xAxC(.3)(.009)1/2(.040)1/2 +
xAxD(.6)(.009)1/2(.023)1/2 + xBxC(.2)(.015)1/2(.040)1/2 +
xBxD(.7)(.015)1/2(.023)1/2 + xCxD(.4)(.040)1/2(.023)1/2 +
xBxA(.4)(.015)1/2(.009)1/2 + xCxA(.3)(.040)1/2 + (.009)1/2 +
xDxA(.6)(.023)1/2(.009)1/2 + xCxB(.2)(.040)1/2(.015)1/2 +
xDxB(.7)(.023)1/2(.015)1/2 + xDxC (.4)(.023)1/2(.040)1/2
subject to:
.08x1 + .09x2 + .16x3 + .12x4  0.11
x1 + x2 + x3 + x4 = 1.00
xi  0
Chapter 11 - Nonlinear Programming
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Investment Portfolio Selection Example Problem
Solution Using Excel (2 of 5)
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
Chapter 11 - Nonlinear Programming
.4
.3
.6
.2
.7
.4
33
Investment Portfolio Selection Example Problem
Solution Using Excel (3 of 5)
Exhibit 11.13
Chapter 11 - Nonlinear Programming
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Investment Portfolio Selection Example Problem
Solution Using Excel (4 of 5)
Exhibit 11.14
Chapter 11 - Nonlinear Programming
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Investment Portfolio Selection Example Problem
Solution Using Excel (5 of 5)
Exhibit 11.15
Chapter 11 - Nonlinear Programming
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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
Chapter 11 - Nonlinear Programming
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Hickory Cabinet and Furniture Company
Example Problem and Solution (2 of 2)
Chapter 11 - Nonlinear Programming
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Chapter 11 - Nonlinear Programming
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