Chapter 2 Problems (HW # 2)
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OPR320
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Industrial Engineering
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Dec 6, 2023
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SELFzest
A"
SELFzest
Problems
63
13.
14.
15.
16.
17.
Consider
the
following
linear
program:
Max
1A
+
2B
s.t.
1A
=5
1B=4
2A
+2B
=12
A,
B=0
a.
Show
the
feasible
region.
b.
What
are
the
extreme
points
of
the
feasible
region?
c.
Find
the
optimal
solution
using
the
graphical
procedure.
RMC,
Inc.,
is
a
small
firm
that
produces
a
variety
of
chemical
products.
In
a
particular
pro-
duction
process,
three
raw
materials
are
blended
(mixed
together)
to
produce
two
products:
a
fuel
additive
and
a
solvent
base.
Each
ton
of
fuel
additive
is
a
mixture
of
%
ton
of
material
1
and
%
of
material
3.
A
ton
of
solvent
base
is
a
mixture
of
¥2
ton
of
material
1,
¥
ton
of
material
2,
and
%o
ton
of
material
3.
After
deducting
relevant
costs,
the
profit
contribution
is
$40
for
every
ton
of
fuel
additive
produced
and
$30
for
every
ton
of
solvent
base
produced.
RMC’s
production
is
constrained
by
a
limited
availability
of
the
three
raw
materials.
For
the
current
production
period,
RMC
has
available
the
following
quantities
of
each
raw
material:
Raw
Material
Amount
Available
for
Production
Material
1
20
tons
Material
2
5
tons
Material
3
21
tons
Assuming
that
RMC
is
interested
in
maximizing
the
total
profit
contribution,
answer
the
following:
a.
What
is
the
linear
programming
model
for
this
problem?
b.
Find
the
optimal
solution
using
the
graphical
solution
procedure.
How
many
tons
of
each
product
should
be
produced,
and
what
is
the
projected
total
profit
contribution?
c.
Isthere
any
unused
material?
If
so,
how
much?
d.
Are
any
of
the
constraints
redundant?
If
so,
which
ones?
Refer
to
the
Par,
Inc.,
problem
described
in
Section
2.1.
Suppose
that
Par,
Inc.,
management
encounters
the
following
situations:
a.
The
accounting
department
revises
its
estimate
of
the
profit
contribution
fer
the
deluxe
bag
to
$18
per
bag.
b.
A
new
low-cost
material
is
available
for
the
standard
bag,
and
the
profit
contribution
per
standard
bag
can
be
increased
to
$20
per
bag.
(Assume
that
the
profit
contribution
of
the
deluxe
bag
is
the
original
$9
value.)
c.
New
sewing
equipment
is
available
that
would
increase
the
sewing
operation
capacity
to
750
hours.
(Assume
that
104
+
98B
is
the
appropriate
objective
function.)
If
each
of
these
situations
is
encountered
separately,
what
is
the
optimal
solution
and
the
total
profit
contribution?
Refer
to
the
feasible
region
for
Par,
Inc.,
problem
in
Figure
2.13.
a.
Develop
an
objective
function
that
will
make
extreme
point
(3the
optimal
extreme
point.
b.
What
is
the
optimal
solution
for
the
objective
function
you
selected
in
part
(a)?
¢.
What
are the
values
of
the
slack
variables
associated
with
this
solution?
Write
the
following
linear
program
in
standard
form:
Max
5A
+
2B
S.t.
1A
—
2B
=420
2A
+
3B
=
610
6A
—
1B
=
125
A,B=0
Problems
67
25.
26.
27.
Assuming
that
the
company
is
interested
in
maximizing
the
total
profit
contribution,
answer
the
following:
a.
What
is
the
linear
programming
model
for
this
problem?
b.
Find
the
optimal
solution
using
the
graphical
solution
procedure.
How
many
gloves
of
each
model
should
Kelson
manufacture?
c¢.
What
is
the
total
profit
contribution
Kelson
can
earn
with
the
given
production
quantities?
d.
How
many
hours
of
production
time
will
be
scheduled
in
each
department?
e.
What
is
the
slack
time
in
each
department?
George
Johnson
recently
inherited
a
large
sum
of
money;
he
wants
to
use
a
portion
of
this
money
to
set
up
a
trust
fund
for
his
two
children.
The
trust
fund
has
two
investment
options:
(1)
a
bond
fund
and
(2)
a
stock
fund.
The
projected
returns
over
the
life
of
the
investments
are
6%
for
the
bond
fund
and
10%
for
the
stock
fund.
Whatever
portion
of
the
inheritance
he
finally
decides
to
commit
to
the
trust
fund,
he
wants
to
invest
at
least
30%
of
that
amount
in
the
bond
fund.
In
addition,
he
wants
to
select
a
mix
that
will
enable
him
to
obtain
a
total
return
of
at
least
7.5%.
a.
Formulate
a
linear
programming
model
that
can
be
used
to
determine
the
percentage
that
should
be
allocated
to
each
of
the
possible
investment
alternatives.
b.
Solve
the
problem
using
the
graphical
solution
procedure.
The
Sea
Wharf
Restaurant
would
like
to
determine
the
best
way
to
allocate
a
monthly
ad-
vertising
budget
of
$1000
between
digital
advertising
and
radio
advertising.
Management
decided
that
at
least
25%
of
the
budget
must
be
spent
on
each
type
of
media,
and
that
the
amount
of
money
spent
on
digital
advertising
must
be
at
least
twice
the
amount
spent
on
radio
advertising.
A
marketing
consultant
developed
an
index
that
measures
audience
exposure
per
dollar
of
advertising
on
a
scale
from
0
to
100,
with
higher
values
implying
greater
audience
exposure.
If
the
value
of
the
index
for
digital
advertising
is
50
and
the
value
of
the
index
for
spot
radio
advertising
is
80,
how
should
the
restaurant
allocate
its
advertising
budget
in
order
to
maximize
the
value
of
total
audience
exposure?
a.
Formulate
a
linear
programming
model
that
can
be
used
to
determine
how
the
restau-
rant
should
allocate
its
advertising
budget
in
order
to
maximize
the
value
of
total
audi-
ence
exposure.
b.
Solve
the
problem
using
the
graphical
solution
procedure.
Blair
&
Rosen,
Inc.
(B&R),
is
a
brokerage
firm
that
specializes
in
investment
portfolios
de-
signed
to
meet
the
specific
risk
tolerances
of
its
clients.
A
client
who
contacted
B&R
this
past
week
has
a
maximum
of
$50,000
to
invest.
B&R’s
investment
advisor
decides
to
recommend
a
portfolio
consisting
of
two
investment
funds:
an
Internet
fund
and
a
Blue
Chip
fund.
The
In-
ternet
fund
has
a
projected
annual
return
of
12%,
whereas
the
Blue
Chip
fund
has
a
projected
annual
return
of
9%.
The
investment
advisor
requires
that
at
most
$35,000
of
the
client’s
funds
should
be
invested
in
the
Internet
fund.
B&R
services
include
a
risk
rating
for
each
investment
alternative.
The
Internet
fund,
which
is
the
more
risky
of
the
two
investment
alternatives,
has
a
risk
rating
of
6
per
thousand
dollars
invested.
The
Blue
Chip
fund
has
a
risk
rating
of
4
per
thousand
dollars
invested.
For
example,
if
$10,000
is
invested
in
each
of
the
two
investment
funds,
B&R'’s
risk
rating
for
the
portfolio
would
be
6(10)
+
4(10)
=
100.
Finally,
B&R
de-
veloped
a
questionnaire
to
measure
each
client’s
risk
tolerance.
Based
on
the
responses,
each
client
is
classified
as
a
conservative,
moderate,
or
aggressive
investor.
Suppose
that
the
ques-
tionnaire
results
classified
the
current
client
as
a
moderate
investor.
B&R
recommends
that
a
client
who
is
a
moderate
investor
limit
his
or
her
portfolio
to
a
maximum
risk
rating
of
240.
a.
What
is
the
recommended
investment
portfolio
for
this
client?
What
is
the
annual
return
for the
portfolio?
b.
Suppose
that
a
second
client
with
$50,000
to
invest
has
been
classified
as
an
aggressive
investor.
B&R
recommends
that
the
maximum
portfolio
risk
rating
for
an
aggressive
investor
is
320.
What
is
the
recommended
investment
portfolio
for
this
aggressive
in-
vestor?
Discuss
what
happens
to
the
portfolio
under
the
aggressive
investor
strategy.
c.
Suppose
that
a
third
client
with
$50,000
to
invest
has
been
classified
as
a
conservative
in-
vestor.
B&R
recommends
that
the
maximum
portfolio
risk
rating
for
a
conservative
inves-
tor
is
160.
Develop
the
recommended
investment
portfolio
for the
conservative
investor.
Discuss
the
interpretation
of
the
slack
variable
for
the
total
investment
fund
constraint.
68
L"
SELFrest
Chapter
2
An
Introduction
fo
Linear
Programming
28.
29.
30.
31.
Tom’s,
Inc.,
produces
various
Mexican
food
products
and
sells
them
to
Western
Foods,
a
chain
of
grocery
stores
located
in
Texas
and
New
Mexico.
Tom’s,
Inc.,
makes
two
salsa
products:
Western
Foods
Salsa and
Mexico
City
Salsa.
Essentially,
the
two
products
have
different
blends
of
whole
tomatoes,
tomato
sauce,
and
tomato
paste.
The
Western
Foods
Salsa
is
a
blend
of
50%
whole
tomatoes,
30%
tomato
sauce,
and
20%
tomato
paste.
The
Mexico
City
Salsa,
which
has
a
thicker
and
chunkier
consistency,
consists
of
70%
whole
tomatoes,
10%
tomato
sauce,
and
20%
tomato
paste.
Each
jar
of
salsa
produced
weighs
10
ounces.
For
the
current
production
period,
Tom’s,
Inc.,
can
purchase
up
to
280
pounds
of
whole
tomatoes,
130
pounds
of
tomato
sauce,
and
100
pounds
of
tomato
paste;
the
price
per
pound
for
these
ingredients
is
$0.96,
$0.64,
and
$0.56,
respectively.
The
cost
of
the
spices
and
the
other
ingredients
is
approximately
$0.10
per
jar.
Tom’s,
Inc.,
buys
empty
glass
jars
for
$0.02
each,
and
labeling
and
filling
costs
are
estimated
to
be
$0.03
for
each
jar
of
salsa
produced.
Tom’s
contract
with
Western
Foods
results
in
sales
revenue
of
$1.64
for
each
jar
of
Western
Foods
Salsa
and
$1.93
for
each
jar
of
Mexico
City
Salsa.
a.
Develop
a
linear
programming
mode]
that will
enable
Tom’s
to
determine
the
mix
of
salsa
products
that
will
maximize
the
total
profit
contribution.
b.
Find
the
optimal
solution.
Autolgnite
produces
electronic
ignition
systems
for
automobiles
at
a
plant
in
Cleveland,
Ohio.
Each
ignition
system
is
assembled
from
two
components
produced
at
Autolgnite’s
plants
in
Buffalo,
New
York,
and
Dayton,
Ohio.
The
Buffalo
plant
can
produce
2000
units
of
component
1,
1000
units
of
component
2,
or
any
combination
of
the
two
components
each
day.
For
instance,
60%
of
Buffalo’s
production
time
could
be
used
to
produce
component
1
and
40%
of
Buffalo’s
production
time
could
be
used
to
produce
component
2;
in
this
case,
the
Buffalo
plant
would
be
able
to
produce
0.6(2000)
=
1200
units
of
component
1
each
day
and
0.4(1000)
=
400
units
of
component
2
each
day.
The
Dayton
plant
can
produce
600
units
of
component
1,
1400
units
of
component
2,
or
any
combination
of
the
two
components
each
day.
At
the
end
of
each
day,
the
component
production
at
Buffalo
and
Dayton
is
sent
to
Cleveland
for
assembly
of
the
ignition
systems
on
the
following
workday.
a.
Formulate
a
linear
programming
model
that
can
be
used
to
develop
a
daily
production
schedule
for
the
Buffalo
and
Dayton
plants
that
will
maximize
daily
production
of
ignition
systems
at
Cleveland.
b.
Find
the
optimal
solution.
A
financial
advisor
at
Diehl
Investments
identified
two
companies
that
are
likely
candidates
for
a
takeover
in
the
near
future.
Eastern
Cable
is
a
leading
manufacturer
of
flexible
cable
systems
used
in
the
construction
industry,
and
ComSwitch
is
a
new
firm
specializing
in
digital
switching
systems.
Eastern
Cable
is
currently
trading
for
$40
per
share,
and
ComSwitch
is
currently
trading
for
$25
per
share.
If
the
takeovers
occur,
the
financial
advisor
estimates
that
the
price
of
Eastern
Cable
will
go
to
$55
per
share
and
ComSwitch
will
go
to
$43
per
share.
At
this
point
in
time,
the
financial
advisor
has
identified
ComSwitch
as
the
higher
risk
alterna-
tive.
Assume
that
a
client
indicated
a
willingness
to
invest
a
maximum
of
$50,000
in
the
two
companies.
The
client
wants
to
invest
at
least
$15,000
in
Eastern
Cable
and
at
least
$10,000
in
ComSwitch.
Because
of
the
higher
risk
associated
with
ComSwitch,
the
financial
advisor
has
recommended
that
at
most
$25,000
should
be
invested
in
ComSwitch.
a.
Formulate
a
linear
programming
model
that
can
be
used
to
determine
the
number
of
shares
of
Eastern
Cable
and
the
number
of
shares
of
ComSwitch
that will
meet
the
in-
vestment
constraints
and
maximize
the
total
return
for
the
investment,
b.
Graph
the
feasible
region.
c.
Determine
the
coordinates
of
each
extreme
point.
,
d.
Find
the
optimal
solution.
Consider
the
following
linear
program:
Min
3A
+
4B
S.t.
1A+3B=6
IA+1B=4
A,
B=0
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