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Introduction to Management Science 8th Edition by Bernard W. Taylor III. Chapter 11 Nonlinear Programming. Chapter Topics. Nonlinear Profit Analysis Constrained Optimization Solution of Nonlinear Programming Problems with Excel A Nonlinear Programming Model with Multiple Constraints - PowerPoint PPT Presentation
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Chapter 11 - Nonlinear Programming 1
Chapter 11
Nonlinear Programming
Introduction to Management Science
8th Edition
by
Bernard W. Taylor III
Chapter 11 - 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
Chapter 11 - 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
Chapter 11 - 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 11.1Linear Relationship of Volume to Price
Chapter 11 - 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 11.2The NonlinearProfit Function
Chapter 11 - 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 11.3Maximum profit
for the profit function
Optimal Value of a Single Nonlinear FunctionMaximum Point on a Curve
Chapter 11 - 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 11.4Maximum Profit, Optimal Price, and Optimal Volume
Optimal Value of a Single Nonlinear FunctionSolution Using Calculus
nonlinearity!
Chapter 11 - 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 andno guaranteed procedure exists.
Constrained Optimization in Nonlinear ProblemsDefinition
Chapter 11 - Nonlinear Programming 9
Effect of adding constraints to nonlinear problem:
Figure 11.5Nonlinear Profit Curve for the Profit Analysis Model
Constrained Optimization in Nonlinear ProblemsGraphical Interpretation (1 of 3)
Chapter 11 - Nonlinear Programming 10
Figure 11.6A Constrained Optimization Model
Constrained Optimization in Nonlinear ProblemsGraphical Interpretation (2 of 3)
Chapter 11 - Nonlinear Programming 11
Figure 11.7A Constrained Optimization Model with a Solution
Point Not on the Constraint Boundary
Constrained Optimization in Nonlinear ProblemsGraphical Interpretation (3 of 3)
Chapter 11 - 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
Chapter 11 - Nonlinear Programming 13
Exhibit 11.1
Western Clothing ProblemSolution Using Excel (1 of 3)
Chapter 11 - Nonlinear Programming 14
Exhibit 11.2
Western Clothing ProblemSolution Using Excel (2 of 3)
Chapter 11 - Nonlinear Programming 15
Exhibit 11.3
Western Clothing ProblemSolution Using Excel (3 of 3)
Chapter 11 - 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)
Chapter 11 - Nonlinear Programming 17
Exhibit 11.4
Beaver Creek Pottery Company ProblemSolution Using Excel (2 of 6)
Chapter 11 - Nonlinear Programming 18
Exhibit 11.5
Beaver Creek Pottery Company ProblemSolution Using Excel (3 of 6)
Chapter 11 - Nonlinear Programming 19
Exhibit 11.6
Beaver Creek Pottery Company ProblemSolution Using Excel (4 of 6)
Chapter 11 - Nonlinear Programming 20
Exhibit 11.7
Beaver Creek Pottery Company ProblemSolution Using Excel (5 of 6)
Chapter 11 - Nonlinear Programming 21
Exhibit 11.8
Beaver Creek Pottery Company ProblemSolution Using Excel (6 of 6)
Chapter 11 - 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)
Chapter 11 - Nonlinear Programming 23
Exhibit 11.9
Western Clothing Company ProblemSolution Using Excel (2 of 4)
Chapter 11 - Nonlinear Programming 24
Exhibit 11.10
Western Clothing Company ProblemSolution Using Excel (3 of 4)
Chapter 11 - Nonlinear Programming 25
Exhibit 11.11
Western Clothing Company ProblemSolution Using Excel (4 of 4)
Chapter 11 - 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) (= 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 iti =annual trips to town i
Facility Location Example ProblemProblem Definition and Data (1 of 2)
Chapter 11 - 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)
Chapter 11 - Nonlinear Programming 28
Exhibit 11.12
Facility Location Example ProblemSolution Using Excel
Chapter 11 - Nonlinear Programming 29
Figure 11.8Rescue Squad Facility Location
Facility Location Example ProblemSolution Map
Chapter 11 - 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.
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 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)
“straight-line distance”
Chapter 11 - 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)
Chapter 11 - Nonlinear Programming 33
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 (2 of 5)
Chapter 11 - Nonlinear Programming 34
Exhibit 11.13
Investment Portfolio Selection Example ProblemSolution Using Excel (3 of 5)
Chapter 11 - Nonlinear Programming 35
Exhibit 11.14
Investment Portfolio Selection Example ProblemSolution Using Excel (4 of 5)
Chapter 11 - Nonlinear Programming 36
Exhibit 11.15
Investment Portfolio Selection Example ProblemSolution Using Excel (5 of 5)
Chapter 11 - 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)
Chapter 11 - Nonlinear Programming 38
Hickory Cabinet and Furniture CompanyExample Problem and Solution (2 of 2)
Chapter 11 - Nonlinear Programming 39