#### Transcript nonlinear programming

```Department of Business Administration
SPRING 2007-08
Management Science
by
Asst. Prof. Sami Fethi
Ch 7: Nonlinear programming
Chapter Topics
Nonlinear Profit Analysis
Constrained Optimization
Nonlinear Programming Model with Multiple Constraints
Nonlinear Model Examples
Operations Research
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Ch 7: Nonlinear programming
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) can be used in this NPL.
Operations Research
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Optimal Value of a Single Nonlinear Function
Basic Model
Ch 7: Nonlinear programming
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
relationship:
v = 1,500 - 24.6p
Operations Research
Figure 1 Linear Relationship of Volume to Price
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Optimal Value of a Single Nonlinear Function
Expanding the Basic Model to a Nonlinear Model
Ch 7: Nonlinear programming
With fixed cost (cf = \$10,000) and variable cost (cv = \$8):
Profit, Z = 1,696.8p - 24.6p2 - 22,000
Operations Research
Figure 2 The Nonlinear Profit Function
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Optimal Value of a Single Nonlinear Function
Maximum Point on a Curve
Ch 7: Nonlinear programming
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.
Operations Research
Figure 3 Maximum profit for the profit function
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Optimal Value of a Single Nonlinear Function Ch 7: Nonlinear programming
Solution Using Calculus
Z = 1,696.8p - 24.6p2
-22,000
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
Operations Research
Figure 4 Maximum Profit, Optimal Price, and Optimal Volume
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Ch 7: Nonlinear programming
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 (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.
Operations Research
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Constrained Optimization in Nonlinear Problems
Graphical Interpretation (1 of 3)
Ch 7: Nonlinear programming
Effect of adding constraints to nonlinear problem:
Figure 5 Nonlinear Profit Curve for the Profit Analysis Model
Operations Research
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Ch 7: Nonlinear programming
Constrained Optimization in Nonlinear Problems
Graphical Interpretation (2 of 3)- First constrained p<= 20
Operations Research
Figure 6 A Constrained Optimization Model
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Ch 7: Nonlinear programming
Constrained Optimization in Nonlinear Problems
Graphical Interpretation (3 of 3) Second constrained p<= 40
Figure 7 A Constrained Optimization Model with a Solution
Point Not on the Constraint Boundary
Operations Research
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Ch 7: Nonlinear programming
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.
Operations Research
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Furniture Company Problem
Solution (1 of 3)
Ch 7: Nonlinear programming
A company makes chairs.
FC per month of making chair \$ 7,500 and VC per chair is \$
40. Price is related to demand according to the following
linear equation: V= 400 -1.2 p.
(a) Develop the nonlinear profit function and
(b) Determine the price that will maximize profit, the optimal
volume and the maximum profit per month.
(c) Graphically illustrate the profit curve and indicate the
optimal price and maximum profit per month.
Operations Research
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Ch 7: Nonlinear programming
Furniture Company Problem
Solution (2 of 3)
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
v = 400 - 1.2p
Figure 8
Operations Research
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Ch 7: Nonlinear programming
Furniture Company Problem
Solution (3 of 3)
With fixed cost (cf = \$7,500) and variable cost (cv = \$40):
Profit, Z = 448p - 1.2p2 - 23,500
Z = 448p – 1.2p2
-23,500
dZ/dp = 448 – 2.4p
=0
p = 448/2.4
= \$186.66
v = 400 – 1.2p
v = 175.996 chairs
Z = \$18,313.87
Operations Research
Figure 9
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Ch 7: Nonlinear programming
Beaver Creek Pottery Company Problem
Solution (1 of 1)
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
Determine the optimal solution to this nonlinear programming model
Operations Research
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Beaver Creek Pottery Company Problem
Solution (2 of 2)
Ch 7: Nonlinear programming
Maximize Z = \$(4 - 0.1x1)x1 + (5 - 0.2x2)x2
Z = 4x1 - 0.1x21 + 5x2 - 0.2x22
Subs x1 = 40-2x2 into the eq above
Z = 13x2 - 0.6x22
dZ/dx2 = 13 – 1.2x2
= 0 →x2 =10.83 and x1 =18.34 subs these into
following profit fuction:
Z = 4x1 - 0.1x21 + 5x2 - 0.2x22
Z =\$ 70.42
Operations Research
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Ch 7: Nonlinear programming
Western Clothing Company Problem
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
Operations Research
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Western Clothing Company Problem
Solution Using Excel (2 of 4)
Operations Research
Ch 7: Nonlinear programming
Figure 10
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Western Clothing Company Problem
Solution Using Excel (3 of 4)
Operations Research
Ch 7: Nonlinear programming
Figure 11
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Western Clothing Company Problem
Solution Using Excel (4 of 4)
Operations Research
Ch 7: Nonlinear programming
Figure 12
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Ch 7: Nonlinear programming
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 =  diti
Where:
di = distance to town i
ti =annual trips to town i
Operations Research
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Ch 7: Nonlinear programming
Facility Location Example Problem
Problem Definition and Data (2 of 2)
Town
Abbeville
Benton
Clayton
Dunnig
Eden
Operations Research
Coordinates
y
x
20
20
35
10
9
25
15
32
8
10
Annual Trips
75
105
135
60
90
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Facility Location Example Problem
Solution Using Excel
Operations Research
Ch 7: Nonlinear programming
Figure 13
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Facility Location Example Problem
Solution Map
Operations Research
Ch 7: Nonlinear programming
Figure 14 Rescue Squad Facility Location
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Ch 7: Nonlinear programming
Facility Location Example Problem
Solution Map
di = sqrt[(xi - x)2 + (yi - y)2]
dA=sqrt[(20- 21.72)2 + (20- 14.14)2]
dA=6.11........................................ dE=6.22
d =  diti
d = 6.11(75)+................................+90(6.22)
d = 4432.53 total annual distance
Operations Research
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Ch 7: Nonlinear programming
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
Operations Research
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Hickory Cabinet and Furniture Company
Example Problem and Solution (2 of 2)
Operations Research
Ch 7: Nonlinear programming
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Ch 7: Nonlinear programming
End of chapter
Operations Research
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