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Introduction to Management Science
(8th Edition, Bernard W. Taylor III)
Chapter 11
Chapter 11 - Nonlinear Programming 1
Nonlinear Programming
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Nonlinear Profit Analysis
Constrained Optimization
Solution of Nonlinear Programming Problems with Excel
Chapter Topics
Solution of Nonlinear Programming Problems with Excel
A Nonlinear Programming Model with Multiple Constraints
Nonlinear Model Examples
Chapter 11 - Nonlinear Programming 2
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Many business problems can be modeled only with nonlinear functions.
Problems that fit the general linear programming format but
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.
Often difficult, if not impossible, to determine optimal solution.
Chapter 11 - Nonlinear Programming 3
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.
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Profit function, Z, with volume independent of price:
Z = vp - cf - vcv
Optimal Value of a Single Nonlinear FunctionBasic Model
where v = sales volume
p = price
cf = fixed cost
cv = variable cost
Add volume/price relationship:
Chapter 11 - Nonlinear Programming 4
v = 1,500 - 24.6p
Figure 11.1Linear Relationship of Volume to Price
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With fixed cost (cf = $10,000) and variable cost (cv = $8):
Z = 1,696.8p - 24.6p2 - 22,000
Optimal Value of a Single Nonlinear FunctionExpanding the Basic Model to a Nonlinear Model
Figure 11.2The NonlinearProfit Function
Chapter 11 - Nonlinear Programming 5
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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.
Optimal Value of a Single Nonlinear FunctionMaximum Point on a Curve
The slope of a curve at its highest point equals zero.
Figure 11.3Maximum profit
Chapter 11 - Nonlinear Programming 6
for the profit function
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Z = 1,696.8p - 24.6p2 - 22,000
dZ/dp = 1,696.8 - 49.2p
Optimal Value of a Single Nonlinear FunctionSolution Using Calculus
nonlinearity!
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
Chapter 11 - Nonlinear Programming 7
Z = $7,259.45
Figure 11.4Maximum Profit, Optimal Price, and Optimal Volume
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If a nonlinear problem contains one or more constraints it becomes a constrained optimization model or a nonlinearprogramming model.
Constrained Optimization in Nonlinear ProblemsDefinition
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 andno guaranteed procedure exists.
Chapter 11 - Nonlinear Programming 8
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Effect of adding constraints to nonlinear problem:
Constrained Optimization in Nonlinear ProblemsGraphical Interpretation (1 of 3)
Chapter 11 - Nonlinear Programming 9
Figure 11.5Nonlinear Profit Curve for the Profit Analysis Model
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Constrained Optimization in Nonlinear ProblemsGraphical Interpretation (2 of 3)
Chapter 11 - Nonlinear Programming 10
Figure 11.6A Constrained Optimization Model
![Page 11: Nonlinear Programming - WordPress.com · A nonlinear programming model has the same general form as the linear programming model except that the objective function and/or the constraint(s)](https://reader035.vdocument.in/reader035/viewer/2022062603/5f0da8ae7e708231d43b7273/html5/thumbnails/11.jpg)
Constrained Optimization in Nonlinear ProblemsGraphical Interpretation (3 of 3)
Chapter 11 - Nonlinear Programming 11
Figure 11.7A Constrained Optimization Model with a Solution
Point Not on the Constraint Boundary
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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
Constrained Optimization in Nonlinear ProblemsCharacteristics
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 12
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Western Clothing ProblemSolution Using Excel (1 of 3)
Exhibit 11.1
Chapter 11 - Nonlinear Programming 13
Exhibit 11.1
![Page 14: Nonlinear Programming - WordPress.com · A nonlinear programming model has the same general form as the linear programming model except that the objective function and/or the constraint(s)](https://reader035.vdocument.in/reader035/viewer/2022062603/5f0da8ae7e708231d43b7273/html5/thumbnails/14.jpg)
Western Clothing ProblemSolution Using Excel (2 of 3)
Chapter 11 - Nonlinear Programming 14
Exhibit 11.2
![Page 15: Nonlinear Programming - WordPress.com · A nonlinear programming model has the same general form as the linear programming model except that the objective function and/or the constraint(s)](https://reader035.vdocument.in/reader035/viewer/2022062603/5f0da8ae7e708231d43b7273/html5/thumbnails/15.jpg)
Western Clothing ProblemSolution Using Excel (3 of 3)
Chapter 11 - Nonlinear Programming 15
Exhibit 11.3
![Page 16: Nonlinear Programming - WordPress.com · A nonlinear programming model has the same general form as the linear programming model except that the objective function and/or the constraint(s)](https://reader035.vdocument.in/reader035/viewer/2022062603/5f0da8ae7e708231d43b7273/html5/thumbnails/16.jpg)
Maximize Z = $(4 - 0.1x1)x1 + (5 - 0.2x2)x2
subject to: x + x = 40
Beaver Creek Pottery Company ProblemSolution Using Excel (1 of 6)
x1 + x2 = 40
Where: x1 = number of bowls producedx2 = number of mugs produced
Chapter 11 - Nonlinear Programming 16
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Beaver Creek Pottery Company ProblemSolution Using Excel (2 of 6)
Chapter 11 - Nonlinear Programming 17
Exhibit 11.4
![Page 18: Nonlinear Programming - WordPress.com · A nonlinear programming model has the same general form as the linear programming model except that the objective function and/or the constraint(s)](https://reader035.vdocument.in/reader035/viewer/2022062603/5f0da8ae7e708231d43b7273/html5/thumbnails/18.jpg)
Beaver Creek Pottery Company ProblemSolution Using Excel (3 of 6)
Chapter 11 - Nonlinear Programming 18
Exhibit 11.5
![Page 19: Nonlinear Programming - WordPress.com · A nonlinear programming model has the same general form as the linear programming model except that the objective function and/or the constraint(s)](https://reader035.vdocument.in/reader035/viewer/2022062603/5f0da8ae7e708231d43b7273/html5/thumbnails/19.jpg)
Beaver Creek Pottery Company ProblemSolution Using Excel (4 of 6)
Chapter 11 - Nonlinear Programming 19
Exhibit 11.6
![Page 20: Nonlinear Programming - WordPress.com · A nonlinear programming model has the same general form as the linear programming model except that the objective function and/or the constraint(s)](https://reader035.vdocument.in/reader035/viewer/2022062603/5f0da8ae7e708231d43b7273/html5/thumbnails/20.jpg)
Beaver Creek Pottery Company ProblemSolution Using Excel (5 of 6)
Chapter 11 - Nonlinear Programming 20
Exhibit 11.7
![Page 21: Nonlinear Programming - WordPress.com · A nonlinear programming model has the same general form as the linear programming model except that the objective function and/or the constraint(s)](https://reader035.vdocument.in/reader035/viewer/2022062603/5f0da8ae7e708231d43b7273/html5/thumbnails/21.jpg)
Beaver Creek Pottery Company ProblemSolution Using Excel (6 of 6)
Chapter 11 - Nonlinear Programming 21
Exhibit 11.8
![Page 22: Nonlinear Programming - WordPress.com · A nonlinear programming model has the same general form as the linear programming model except that the objective function and/or the constraint(s)](https://reader035.vdocument.in/reader035/viewer/2022062603/5f0da8ae7e708231d43b7273/html5/thumbnails/22.jpg)
Maximize Z = (p1 - 12)x1 + (p2 - 9)x2
subject to:2x + 2.7x 6,00
Western Clothing Company ProblemSolution Using Excel (1 of 4)
2x1 + 2.7x2 6,00 3.6x1 + 2.9x2 8,5007.2x1 + 8.5x2 15,000
where:x1 = 1,500 - 24.6p1
x2 = 2,700 - 63.8p
Chapter 11 - Nonlinear Programming 22
2
p1 = price of designer jeansp2 = price of straight jeans
![Page 23: Nonlinear Programming - WordPress.com · A nonlinear programming model has the same general form as the linear programming model except that the objective function and/or the constraint(s)](https://reader035.vdocument.in/reader035/viewer/2022062603/5f0da8ae7e708231d43b7273/html5/thumbnails/23.jpg)
Western Clothing Company ProblemSolution Using Excel (2 of 4)
Chapter 11 - Nonlinear Programming 23
Exhibit 11.9
![Page 24: Nonlinear Programming - WordPress.com · A nonlinear programming model has the same general form as the linear programming model except that the objective function and/or the constraint(s)](https://reader035.vdocument.in/reader035/viewer/2022062603/5f0da8ae7e708231d43b7273/html5/thumbnails/24.jpg)
Western Clothing Company ProblemSolution Using Excel (3 of 4)
Chapter 11 - Nonlinear Programming 24
Exhibit 11.10
![Page 25: Nonlinear Programming - WordPress.com · A nonlinear programming model has the same general form as the linear programming model except that the objective function and/or the constraint(s)](https://reader035.vdocument.in/reader035/viewer/2022062603/5f0da8ae7e708231d43b7273/html5/thumbnails/25.jpg)
Western Clothing Company ProblemSolution Using Excel (4 of 4)
Chapter 11 - Nonlinear Programming 25
Exhibit 11.11
![Page 26: Nonlinear Programming - WordPress.com · A nonlinear programming model has the same general form as the linear programming model except that the objective function and/or the constraint(s)](https://reader035.vdocument.in/reader035/viewer/2022062603/5f0da8ae7e708231d43b7273/html5/thumbnails/26.jpg)
Centrally locate a facility that serves several customers or other facilities in order to minimize distance or miles traveled (d) between facility and customers.
Facility Location Example ProblemProblem Definition and Data (1 of 2)
(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
Chapter 11 - Nonlinear Programming 26
Minimize total miles d = ditiWhere:
di = distance to town iti =annual trips to town i
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Coordinates
Facility Location Example ProblemProblem Definition and Data (2 of 2)
CoordinatesTown x y Annual TripsAbbevilleBentonClaytonDunnig
20102532
20359
15
7510513560
Chapter 11 - Nonlinear Programming 27
DunnigEden
3210
158
6090
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Facility Location Example ProblemSolution Using Excel
Chapter 11 - Nonlinear Programming 28
Exhibit 11.12
![Page 29: Nonlinear Programming - WordPress.com · A nonlinear programming model has the same general form as the linear programming model except that the objective function and/or the constraint(s)](https://reader035.vdocument.in/reader035/viewer/2022062603/5f0da8ae7e708231d43b7273/html5/thumbnails/29.jpg)
Facility Location Example ProblemSolution Map
Chapter 11 - Nonlinear Programming 29
Figure 11.8Rescue Squad Facility Location
![Page 30: Nonlinear Programming - WordPress.com · A nonlinear programming model has the same general form as the linear programming model except that the objective function and/or the constraint(s)](https://reader035.vdocument.in/reader035/viewer/2022062603/5f0da8ae7e708231d43b7273/html5/thumbnails/30.jpg)
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 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 30
mathematically identical to minimizing such distances.
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Minimize S = x12s1
2 + x22s2
2 + … +xn2sn
2 + xixjrijsisj
where:S = variance of annual return of the portfolio
Investment Portfolio Selection Example ProblemDefinition and Model Formulation (2 of 2)
“straight-line distance”
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 j
subject to:r x + r x + … + r x r
Chapter 11 - Nonlinear Programming 31
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
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Four stocks, desired annual return of at least 0.11.
MinimizeZ = S = x 2(.009) + x 2(.015) + x 2(.040) + X 2(.023)
Investment Portfolio Selection Example ProblemSolution Using Excel (1 of 5)
Z = S = xA2(.009) + xB
2(.015) + xC2(.040) + XD
2(.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 + x x (.7)(.023)1/2(.015)1/2 + x x (.4)(.023)1/2(.040)1/2
Chapter 11 - Nonlinear Programming 32
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 = 1.00xi 0
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Stock (xi) Annual Return (ri) Variance (si)
Investment Portfolio Selection Example ProblemSolution Using Excel (2 of 5)
AltacamBestcoCom.comDelphi
.08
.09
.16
.12
.009
.015
.040
.023
Stock combination (i,j) Correlation (rij)
A,B .4
Chapter 11 - Nonlinear Programming 33
A,BA,CA,DB,CB,DC,D
.4
.3
.6
.2
.7
.4
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Investment Portfolio Selection Example ProblemSolution Using Excel (3 of 5)
Chapter 11 - Nonlinear Programming 34
Exhibit 11.13
![Page 35: Nonlinear Programming - WordPress.com · A nonlinear programming model has the same general form as the linear programming model except that the objective function and/or the constraint(s)](https://reader035.vdocument.in/reader035/viewer/2022062603/5f0da8ae7e708231d43b7273/html5/thumbnails/35.jpg)
Investment Portfolio Selection Example ProblemSolution Using Excel (4 of 5)
Chapter 11 - Nonlinear Programming 35
Exhibit 11.14
![Page 36: Nonlinear Programming - WordPress.com · A nonlinear programming model has the same general form as the linear programming model except that the objective function and/or the constraint(s)](https://reader035.vdocument.in/reader035/viewer/2022062603/5f0da8ae7e708231d43b7273/html5/thumbnails/36.jpg)
Investment Portfolio Selection Example ProblemSolution Using Excel (5 of 5)
Chapter 11 - Nonlinear Programming 36
Exhibit 11.15
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Model:
Maximize Z = $280x1 - 6x12 + 160x2 - 3x2
2
Hickory Cabinet and Furniture CompanyExample Problem and Solution (1 of 2)
subject to:20x1 + 10x2 = 800 board ft.
Where:x1 = number of chairsx2 = number of tables
Chapter 11 - Nonlinear Programming 37
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Hickory Cabinet and Furniture CompanyExample Problem and Solution (2 of 2)
Chapter 11 - Nonlinear Programming 38
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Chapter 11 - Nonlinear Programming 39