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Linear Programming Applications in Marketing, Finance and Operations Management
Chapter 4
Linear Programming Applications in
Marketing, Finance and Operations
Management
2.
a.
Let
x1 = units of product 1 produced
x2 = units of product 2 produced
Max
s.t.
30×1
+
15×2
x1
0.30×1
0.20×1
+
+
+
0.35×2
0.20×2
0.50×2



100
36
50
Dept. A
Dept. B
Dept. C
x1 , x2  0
Solution: x1 = 77.89, x2 = 63.16 Profit = 3284.21
b.
The dual price for Dept. A is $15.79, for Dept. B it is $47.37, and for Dept. C it is $0.00. Therefore
we would attempt to schedule overtime in Departments A and B. Assuming the current labor
available is a sunk cost, we should be willing to pay up to $15.79 per hour in Department A and up
to $47.37 in Department B.
c.
Let
xA = hours of overtime in Dept. A
xB = hours of overtime in Dept. B
xC = hours of overtime in Dept. C
Max
s.t.
30×1
+
15×2

18xA
x1
0.30×1
0.20×1
+
+
+
0.35×2
0.20×2
0.50×2

xA


22.5xB

12xC
xB

xC
xA
xB
xC
x1, x2, xA, xB, xC  0






100
36
50
10
6
8
Chapter 4
x1 = 87.21
x2 = 65.12
Profit = $3341.34
Overtime
Dept. A
Dept. B
Dept. C
10 hrs.
3.186 hrs
0 hours
Increase in Profit from overtime = $3341.34 – 3284.21 = $57.13
17. a.
Let
FM
FP
SM
SP
TM
TP
Min
s.t.
38FM
3.5FM
2.2FM
3.1FM
FM
=
=
=
=
=
=
number of frames manufactured
number of frames purchased
number of supports manufactured
number of supports purchased
number of straps manufactured
number of straps purchased
+ 51FP
+
+ 11.5SM + 15SP
+ 6.5TM + 7.5TP
+
+
+
+ 0.8TM
1.3SM
1.7SM
2.6SM
+ 1.7TM
FP
SM

+
SP
TM +
FM, FP, SM, SP, TM, TP  0.
TP






21,000
25,200
40,800
5,000
10,000
5,000
Solution:
Frames
Supports
Straps
Manufacture
5000
2692
0
Purchase
0
7308
5000
b.
Total Cost = $368,076.91
c.
Subtract values of slack variables from minutes available to determine minutes used. Divide by 60
to determine hours of production time used.
Constraint
1
Cutting:
Slack = 0 350 hours used
Linear Programming Applications in Marketing, Finance and Operations Management
2
3
Milling:
Shaping:
(25200 – 9623) / 60 = 259.62 hours
(40800 – 18300) / 60 = 375 hours
d.
Nothing, there are already more hours available than are being used.
e.
Yes. The current purchase price is $51.00 and the reduced cost of 3.577 indicates that for a purchase
price below $47.423 the solution may improve. Resolving with the coefficient of FP = 45 shows
that 2714 frames should be purchased.
The optimal solution is as follows:
OPTIMAL SOLUTION
Objective Function Value =
361500.000
Variable
————-FM
FP
SM
SP
TM
TP
Value
————–2285.714
2714.286
10000.000
0.000
0.000
5000.000
Reduced Costs
—————–0.000
0.000
0.000
0.900
0.600
0.000
Constraint
————-1
2
3
4
5
6
Slack/Surplus
————–0.000
3171.429
7714.286
0.000
0.000
0.000
Dual Prices
—————–2.000
0.000
0.000
-45.000
-14.100
-7.500

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