Supplement 6
Linear Programming
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1
Examples of Successful LP Applications
 Scheduling school busses to minimize total distance traveled when
carrying students
 Allocating police patrol units to high crime areas in order to minimize
response time to 911 calls
 Scheduling tellers at banks so that needs are met during each hour of
the day while minimizing the total cost of labor
 Picking blends of raw materials in feed mills to produce finished feed
combinations at minimum costs
 Selecting the product mix in a factory to make best use of machine
and labor-hours available while maximizing the firm’s profit
 Allocating space for a tenant mix in a new shopping mall so as to
maximize revenues to the leasing company
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Simple Example and Solution
We make 2 products: Panels and Doors
Panel:
Labor:
2 hrs/unit
Material:
3 #/unit
Door:
Labor:
4 hrs/unit
Material:
1 #/unit
Available Resources:
Profit:
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Labor:
Material:
$10 per Panel
$ 8 per Door
3
80 hrs
60 #
Enumeration for Simple Example
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X2 - Doors
60
Add Paint Constraint (Resource)
8 X1  5.6 X 2  224Quarts
Material - wood
8 X1  5.6 X 2  176Quarts
40
31.43
20
10
Labor - hrs
0
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5
8
20 22
28
40
X1 - Panels
Example Solution Using Simplex
Let # of Colonial lots be
Let # of Western lots be
X1
X2
4) Budget:
20 X1  50 X 2  5,000
3X1  2 X 2  400
3 X1  4 X 2  500
50 X1  43.75 X 2  7,000
Max. profit
Z  80 X1  100 X 2
1) Wood:
2) Pressing Time:
3) Finishing Time:
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X2
250
Max(Z )  80 X1  100 X 2
3X1  2 X 2  400
200
50 X1  43.75 X 2  7000
20 X1  50 X 2  5000
3 X1  4 X 2  500
150
Optimal Solution:
X1 = 89.09
X2 = 58.18
Profit = $ 12,945.20
100
50
Z  8000
0
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50
100
150
7
200
250
X1
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Requirements of a Linear Programming Problem





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Must seek to maximize or minimize some quantity (the
objective function)
Objectives and constraints must be expressible as linear
equations or inequalities
Presence of restrictions or constraints - limits ability to
achieve objective
Must be willing to accept divisibility
Must have a convex feasible space
9
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Minimization Example
You’re an analyst for a division of
Kodak, which makes BW & color
chemicals. At least 30 tons of BW and
at least 20 tons of color must be made
each month. The total chemicals made
must be at least 60 tons. How many
tons of each chemical should be made
to minimize costs?
BW: $2,500
manufacturing cost
per ton per month
Color: $ 3,000
manufacturing cost
per ton per month
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Graphical Solution
80
X2
Find values for
X1 + X2  60
BW
 30
Tons, Color Chemical (X2)
X1
X2  20
60
Total
40
20
Color
0
X1
80
0
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20
40
Tons, BW Chemical (X1)
12
60