Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management...
-
Upload
megan-bennett -
Category
Documents
-
view
255 -
download
13
Transcript of Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management...
![Page 1: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/1.jpg)
Chapter 10 - Nonlinear Programming 1
Chapter 10
Nonlinear Programming
Introduction to Management Science
8th Edition
by
Bernard W. Taylor III
![Page 2: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/2.jpg)
Chapter 10 - Nonlinear Programming 2
Nonlinear Profit Analysis
Constrained Optimization
Solution of Nonlinear Programming Problems with Excel
A Nonlinear Programming Model with Multiple Constraints
Nonlinear Model Examples
Chapter Topics
![Page 3: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/3.jpg)
Chapter 10 - Nonlinear Programming 3
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.
Overview
![Page 4: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/4.jpg)
Chapter 10 - Nonlinear Programming 4
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
Optimal Value of a Single Nonlinear FunctionBasic Model
Figure 10.1Linear Relationship of Volume to Price
![Page 5: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/5.jpg)
Chapter 10 - Nonlinear Programming 5
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 10.2The NonlinearProfit Function
![Page 6: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/6.jpg)
Chapter 10 - Nonlinear Programming 6
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 10.3Maximum profit
for the profit function
Optimal Value of a Single Nonlinear FunctionMaximum Point on a Curve
![Page 7: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/7.jpg)
Chapter 10 - Nonlinear Programming 7
Z = 1,696.8p - 24.6p2 - 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 10.4Maximum Profit, Optimal Price, and Optimal Volume
Optimal Value of a Single Nonlinear FunctionSolution Using Calculus
![Page 8: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/8.jpg)
Chapter 10 - Nonlinear Programming 8
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.
Constrained Optimization in Nonlinear ProblemsDefinition
![Page 9: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/9.jpg)
Chapter 10 - Nonlinear Programming 9
Effect of adding constraints to nonlinear problem:
Figure 10.5Nonlinear Profit Curve for the Profit Analysis Model
Constrained Optimization in Nonlinear ProblemsGraphical Interpretation (1 of 3)
![Page 10: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/10.jpg)
Chapter 10 - Nonlinear Programming 10
Figure 10.6A Constrained Optimization Model
Constrained Optimization in Nonlinear ProblemsGraphical Interpretation (2 of 3)
![Page 11: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/11.jpg)
Chapter 10 - Nonlinear Programming 11
Figure 10.7A Constrained Optimization Model with a Solution
Point Not on the Constraint Boundary
Constrained Optimization in Nonlinear ProblemsGraphical Interpretation (3 of 3)
![Page 12: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/12.jpg)
Chapter 10 - Nonlinear Programming 12
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.
Constrained Optimization in Nonlinear ProblemsCharacteristics
![Page 13: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/13.jpg)
Chapter 10 - Nonlinear Programming 13
Exhibit 10.1
Western Clothing ProblemSolution Using Excel (1 of 3)
![Page 14: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/14.jpg)
Chapter 10 - Nonlinear Programming 14
Exhibit 10.2
Western Clothing ProblemSolution Using Excel (2 of 3)
![Page 15: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/15.jpg)
Chapter 10 - Nonlinear Programming 15
Exhibit 10.3
Western Clothing ProblemSolution Using Excel (3 of 3)
![Page 16: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/16.jpg)
Chapter 10 - Nonlinear Programming 16
Maximize Z = $(4 - 0.1x1)x1 + (5 - 0.2x2)x2
subject to: x1 + x2 = 40
Where: x1 = number of bowls producedx2 = number of mugs produced
Beaver Creek Pottery Company ProblemSolution Using Excel (1 of 6)
![Page 17: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/17.jpg)
Chapter 10 - Nonlinear Programming 17
Exhibit 10.4
Beaver Creek Pottery Company ProblemSolution Using Excel (2 of 6)
![Page 18: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/18.jpg)
Chapter 10 - Nonlinear Programming 18
Exhibit 10.5
Beaver Creek Pottery Company ProblemSolution Using Excel (3 of 6)
![Page 19: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/19.jpg)
Chapter 10 - Nonlinear Programming 19
Exhibit 10.6
Beaver Creek Pottery Company ProblemSolution Using Excel (4 of 6)
![Page 20: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/20.jpg)
Chapter 10 - Nonlinear Programming 20
Exhibit 10.7
Beaver Creek Pottery Company ProblemSolution Using Excel (5 of 6)
![Page 21: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/21.jpg)
Chapter 10 - Nonlinear Programming 21
Exhibit 10.8
Beaver Creek Pottery Company ProblemSolution Using Excel (6 of 6)
![Page 22: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/22.jpg)
Chapter 10 - Nonlinear Programming 22
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 - 63.8pp1 = price of designer jeansp2 = price of straight jeans
Western Clothing Company ProblemSolution Using Excel (1 of 4)
![Page 23: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/23.jpg)
Chapter 10 - Nonlinear Programming 23
Exhibit 10.9
Western Clothing Company ProblemSolution Using Excel (2 of 4)
![Page 24: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/24.jpg)
Chapter 10 - Nonlinear Programming 24
Exhibit 10.10
Western Clothing Company ProblemSolution Using Excel (3 of 4)
![Page 25: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/25.jpg)
Chapter 10 - Nonlinear Programming 25
Exhibit 10.11
Western Clothing Company ProblemSolution Using Excel (4 of 4)
![Page 26: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/26.jpg)
Chapter 10 - Nonlinear Programming 26
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 = diti
Where:di = distance to town iti =annual trips to town i
Facility Location Example ProblemProblem Definition and Data (1 of 2)
![Page 27: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/27.jpg)
Chapter 10 - Nonlinear Programming 27
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)
![Page 28: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/28.jpg)
Chapter 10 - Nonlinear Programming 28
Exhibit 10.12
Facility Location Example ProblemSolution Using Excel
![Page 29: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/29.jpg)
Chapter 10 - Nonlinear Programming 29
Figure 10.8Rescue Squad Facility Location
Facility Location Example ProblemSolution Map
![Page 30: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/30.jpg)
Chapter 10 - Nonlinear Programming 30
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.
![Page 31: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/31.jpg)
Chapter 10 - Nonlinear Programming 31
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 j
subject 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)
![Page 32: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/32.jpg)
Chapter 10 - Nonlinear Programming 32
Four stocks, desired annual return of at least 0.11.
MinimizeZ = S = xA
2(.009) + xB2(.015) + xC
2(.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
Investment Portfolio Selection Example ProblemSolution Using Excel (1 of 5)
![Page 33: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/33.jpg)
Chapter 10 - Nonlinear Programming 33
Stock (xi) Annual Return (ri) Variance (s i)
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 (2 of 5)
![Page 34: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/34.jpg)
Chapter 10 - Nonlinear Programming 34
Exhibit 10.13
Investment Portfolio Selection Example ProblemSolution Using Excel (3 of 5)
![Page 35: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/35.jpg)
Chapter 10 - Nonlinear Programming 35
Exhibit 10.14
Investment Portfolio Selection Example ProblemSolution Using Excel (4 of 5)
![Page 36: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/36.jpg)
Chapter 10 - Nonlinear Programming 36
Exhibit 10.15
Investment Portfolio Selection Example ProblemSolution Using Excel (5 of 5)
![Page 37: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/37.jpg)
Chapter 10 - Nonlinear Programming 37
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)
![Page 38: Chapter 10 - Nonlinear Programming 1 Chapter 10 Nonlinear Programming Introduction to Management Science 8th Edition by Bernard W. Taylor III.](https://reader036.fdocuments.net/reader036/viewer/2022081420/56649cf75503460f949c7ffd/html5/thumbnails/38.jpg)
Chapter 10 - Nonlinear Programming 38
Hickory Cabinet and Furniture CompanyExample Problem and Solution (2 of 2)