14.
Digital Controls, Inc. (DCI), manufactures two
models of a radar gun used by police to monitor the speed of automobiles. Model
A has an accuracy of plus or minus 1 mile per hour, whereas the smaller model B
has an accuracy of plus or minus 3 miles per hour. For the next week, the
company has orders for 100 units of model A and 150 units of model B. Although
DCI purchases all the electronic components used in both models, the plastic
cases for both models are manufactured at a DCI plant in Newark, New Jersey.
Each model A case requires 4 minutes of injectionmolding time and 6 minutes of
assembly time. Each model B case requires 3 minutes of injectionmolding time
and 8 minutes of assembly time. Each model B case requires 3 minutes of
injectionmolding time and 8 minutes of assembly time. For next week, the
Newark plant has 600 minutes of injectionmolding time available and 1080
minutes of assembly time available. The manufacturing cost is $10 per case for
model A and $6 per case for model B. Depending upon demand and the time
available at the Newark plant, DCI occasionally purchases cases for one or both
models from an outside supplier in order to fill customer orders that could not
be filled otherwise. The purchase cost is $14 for each model A case and $9 for
each model B case. Management wants to develop a minimum cost plan that will
determine how many cases of each model should be produced at the Newark plant
and how many cases of each model should be purchased. The following decision
variables were used to formulate a linear programming model for this problem:
AM=number of cases of model A manufactured
BM=number of cases of model B manufactured
AP=number of cases of model A purchased
BP=number of cases of model B purchased
The linear programming model that can be used
to solve this problem is as follows:


10
+ 6 + 14 + 9



. .

1
+


+
1 +

= 100




1
+

1 = 150




4
+ 3


? 600




6
+ 8


? 1080




, , ,
? 0




Quantitative
Analysis BA 452 Homework 3 Questions
.jpg">
The computer solution is shown in Figure 3.18.
a. What is the optimal solution and what is the
optimal value of the objective function?
b. Which constraints are binding?
c. What are the dual values? Interpret each.
d. If you could change the righthand side of one
constraint by one unit, which one would you choose? Why?
Quantitative
Analysis BA 452 Homework 3 Questions
15.
Refer to
the computer solution to Problem 14 in Figure 3.18.
a. Interpret the ranges of optimality for the
objective function coefficients.
b.
Suppose
that the manufacturing cost increases to $11.20 per case for model A. What is
the new optimal solution?
c.
Suppose
that the manufacturing cost increases to $11.20 per case for model A and the
manufacturing cost for model B decreases to $5 per unit. Would the optimal
solution change?
Quantitative
Analysis BA 452 Homework 3 Questions
16.
Tucker Inc.
produces highquality suits and sport coats for men. Each suit requires 1.2
hours of cutting time and 0.7 hours of sewing time, uses 6 yards of material,
and provides a profit contribution of $190. Each sport coat requires 0.8 hours
of cutting time and 0.6 hours of sewing time, uses 4 yards of material, and
provides a profit contribution of $150. For the coming week, 200 hours of
cutting time, 18 hours of sewing time, and 1200 yards of fabric are available.
Additional cutting and sewing time can be obtained by scheduling overtime for
these operations. Each hour of overtime for the cutting operation increase the
hourly cost by $15, and each hour of overtime for the sewing operation increase
the hourly cost by $10. A maximum of 100 hours of overtime can be scheduled.
Marketing requirements specify a minimum production of 100 suits and 75 sport
coats. Let
S=number of suits produced
SC=number of sport oats produced
D1=hours of overtime for the cutting operation
D2=hours of overtime for the sewing operation
The computer solution is shown in Figure 3.19.
a.
What is the
optimal solution, and what is the total profit? What is the plan for the use of
overtime?
b.
A price increase fir suits is being considered
that would result in a profit contribution of $210 per suit. If this price
increase is undertaken, how will the optimal solution change?
c.
Discuss the
need for additional material during the coming week. If a rush order for
material can be placed at the usual price plus an extra $8 per yard for
handling, would you recommend the company consider placing a rush order for
material? What is the maximum price Tucker would be willing to pay for an
additional yard of material? How many additional yards of material should
Tucker consider ordering?
d.
Suppose the
minimum production requirement for suits is lowered to 75. Would this change
help or hurt profit? Explain.
Quantitative
Analysis BA 452 Homework 3 Questions
.jpg">
Quantitative
Analysis BA 452 Homework 3 Questions
17.

The Porsche Club of America sponsors driver
education events that provide highperformance



driving instruction on actual race tracks.
Because safety is a primary consideration at such events,



many owners elect to install roll bars in their
cars. Deegan Industries manufactures two types of roll



bars for Porsches. Model DRB
is bolted to the car using existing holes in the car’s frame. Model DRW



is a heavier roll bar that must be welded to
the car’s frame. Model DRB requires 20 pounds of a



special high alloy steel, 40 minutes of
manufacturing time, and 60 minutes of assembly time. Model



DRW requires 25 pounds of the special high
alloy steel, 100 minutes of manufacturing time, and 40



minutes of assembly time.
Deegan’s steel supplier indicated that at most 40,000 pounds of the high



alloy steel will be available next quarter. In
addition, Deegan estimates that 20000 hours of



manufacturing time and 1600 hours of assembly
time will be available next quarter. The profit



contributions are $200 per unit for model DRB
and $280 per unit for model DRB. The linear



programming model for this problem is as
follows:





200+




. .

20+ 25? 40,000




40+ 100? 120,000




60+ 40? 96,000





,? 0



The computer solution is shown in Figure 3.20.
a. What are the optimal solution and the total
profit contribution/
b.
Another
supplier offered to provide Deegan Industries with an additional 500 pounds of
the steel alloy at $2 per pound. Should Deegan purchase the additional pounds
of the steel alloy? Explain.
c. Deegan is considering using overtime to increase
the available assembly time. What would you advise Deegan to do regarding this
option? Explain.
d.
Because of
increased competition, Deegan is considering reducing the price of model DRB
such that the new contribution to profit is $175 per unit. How would this
change in price affect the optimal solution? Explain.
e. If the available manufacturing time is increased
by 500 hours, will the dual value for the manufacturing time constraint change?
Explain.
Quantitative
Analysis BA 452 Homework 3 Questions
.jpg">
Quantitative
Analysis BA 452 Homework 3 Questions
18.
Davison
Electronics manufactures two LCD television monitors, identified as model A and
model B. Each model has its lowest possible production cost when produced on
Davison’s new production line. However, the new production line does not have
the capacity to handle the total production of both models. As a result, as
least some of the production must be routed to a highercost, old production
line. The following table shows the minimum production requirements for next
month, the production line table shows the minimum production requirements for
next month, the production line capacities in units per month, and the
production cost per unit for each production line:


Production Cost per




Unit


Model

New Line

Old Line

Minimum
Production




Requirements

A

$30

$50

50,000

B

$25

$40

70,000

Production Line

80,000

60,000


Capacity








Let:
AN= Units of model A produced on the new production line
AO= Units of model A produced on the old production line
BN = Units of model B produced on the new production line
BO= Units of model B produced on the old production line
Davison’s objective is to determine the minimum
cost production plan. The computer solution is shown below.
a. Formulate the linear programming model for this
problem using the following four constraints:
i. Constraint 1: Minimum production for model A
ii. Constraint 2: Minimum production for model B
iii. Constraint 3: Capacity of the new production line
iv. Constraint 4: Capacity of the old production line
b.
Using
computer solution in Figure 3.21, what is the optimal solution, and what is the
total production cost associated with this solution?
c. Which constraints are binding? Explain.
d.
The
production manager noted that the only constraint with a positive dual values
is the constraint on the capacity of the new production line. The manager’s
interpretation of the dual value was that a oneunit increase in the righthand
side of this constraint would actually increase the total production cost by
$15 per unit. Do you agree with this interpretation? Would an increase in
capacity for the new production line be desirable? Explain.
e. Would you recommend increasing the capacity of
the old production line? Explain.
Quantitative
Analysis BA 452 Homework 3 Questions
f.
The
production cost for model A on the old production line is $50 per unit. How
much would this cost have to change to make it worthwhile to produce model A on
the old production line? Explain.
g.
Suppose
that the minimum production requirement for model B is reduced from 70,000
units to 60,000 units. What effect would this change have on the total
production cost? Explain.
Optimal Objective Value


= 3850000.00000



Variable


Value

Reduced Cost


AN


50000.00000

0.00000


AO


0.0000

5.00000


BN


30000.00000

0.00000


BO


40000.00000

0.00000







Constraint


Slack/Surplus

Dual Value


1


0.00000

45.00000


2


0.00000

40.00000


3


0.00000

15.00000


4


20000.00000

0.00000






OBJECTIVE COEFFICIENT RANGES



Variable


Objective Coefficient

Allowable Increase

Allowable Decrease

AN


30.00000

5.00000

Infinite

AO


50.00000

Infinite

5.00000

BN


25.00000

15.00000

5.00000

BO


40.00000

5.00000

15.00000






RIGHT HAND SIDE RANGES




Constraint


RHS Value

Allowable Increase

Allowable Decrease

1


50000.00000

20000.00000

40000.00000

2


70000.00000

20000.00000

40000.00000

3


80000.00000

40000.00000

20000.00000

4


60000.00000

Infinite

20000.00000
