Simple LP Problems

Dr Yamini S.
Faculty of Operations and
Analytics

24 November 2020 Yamini | Operations Research 1

1. Holiday Mean Turkey Ranch is considering buying two
different types of turkey feed. Each feed contains, in varying
proportions, some or all of the three nutritional ingredients
essential for fattening turkeys. Brand Y feed costs the
ranch $0.2 per pound. Brand Z costs $.03 per pound. The
rancher would like to determine the lowest-cost diet that meets
the minimum monthly intake requirement for each nutritional
ingredient. The following table contains relevant information
about the composition of brand Y and brand Z feeds, as well as
the minimum monthly requirement for each nutritional
ingredient per turkey.
24 November 2020 Yamini | Operations Research 2
Composition of Each Pound of Feed

Minimum
Ingredient Brand Y Feed Brand Z Feed Monthly
Requirement
A 5 oz 10 oz 90 oz
B 4 oz 3 oz 48 oz
C .5 oz 0 1.5 oz
Cost/lb $0.2 $.03

24 November 2020 Yamini | Operations Research 3

2. The New England Cheese Company produces two
cheese spreads by blending mild cheddar cheese with
extra sharp cheddar cheese. The cheese spreads are
packaged in 12-ounce containers, which are then sold to
distributors throughout the Northeast. The Regular blend
contains 80% mild cheddar and 20% extra sharp, and the
Zesty blend contains 60% mild cheddar and 40% extra
sharp. This year, a local dairy cooperative offered to
provide up to 8100 pounds of mild cheddar cheese for
$1.20 per pound and up to 3000 pounds of extra sharp
cheddar cheese for $1.40 per pound.
24 November 2020 Yamini | Operations Research 4
The cost to blend and package the cheese spreads,
excluding the cost of the cheese, is $0.20 per container. If
each container of Regular is sold for $1.95 and each
container of Zesty is sold for $2.20, how many containers
of Regular and Zesty should New England Cheese
produce?

24 November 2020 Yamini | Operations Research 5

3. Creative sports design (CSD) manufactures a standard size
racket and an over sized racket. The firm's rackets are
extremely light due to the use of a magnesium- graphite
alloy that was invented by the firm's founder. Each standard
size racket uses 0.125 kilograms of the alloy and each
oversize racket uses 0.4 kilograms; over the next two week
production period only 80 kilograms of the alloy are
available. Each standard size racket uses 10 min. of
manufacturing time and each oversize racket uses 12 min.

24 November 2020 Yamini | Operations Research 6

The profit contributions are $10 for each standard size
racket and $15 for each oversized racket, and 40 hours of
manufacturing time are available each week. Management
specified that at least 20% of the total production must3 be
the standard size racket. How many rackets of each type
should CSD manufacture over the next 2 weeks to maximize
the total profit contributions? Assume that because of the
unique nature of their products, CSD can sell as many
rackets as they can produce?

24 November 2020 Yamini | Operations Research 7

4. English Motors, Ltd. (EML), developed a new all-wheel-
drive sports utility vehicle. As part of the marketing
campaign, EML produced a video tape sales presentation to
send to both owners of current EML four-wheel-drive
vehicles as well as to owners of four-wheel-drive sports
utility vehicles offered by competitors; EML refers to these
two target markets as the current customer market and the
new customer market. Individuals who receive the new
promotion video will also receive a coupon for a test drive
of the new EML model for one weekend.

24 November 2020 Yamini | Operations Research 8

A key factor in the success of the new promotion is the
response rate, the percentage of individuals who receive
the new promotion and test drive the new model. EML
estimates that the response rate for the current customer
market is 25% and the response rate for the new customer
market is 20%. For the customers who test drive the new
model, the sales rate is the percentage of individuals that
makes a purchase. Marketing research studies indicate that
the sales rate is 12% for the current customer market and
20% for the new customer market.
24 November 2020 Yamini | Operations Research 9
The cost for each promotion, excluding the test drive costs, is $4
for each promotion sent to the current customer market and $6 for
each promotion sent to the new customer market. Management
also specified that a minimum of 30,000 current customers should
test drive the new model and a minimum of 10,000 new customers
should test drive the new model. In addition, the numbers of
current customers who test drive the new vehicle must be at least
twice the number of new customers who test drive the new
vehicle. If the marketing budget, excluding test drive costs, is $1.2
million, how many promotions should be sent to each group of
customers in order to maximize total sales?

24 November 2020 Yamini | Operations Research 10

5. Applied Technology, Inc. (ATI) produces bicycle frames using two
fiberglass materials that improve the strength-to-weight ratio of
the frames. The cost of the standard grade material is $7.50 per
yard and the cost of the professional grade material is $9.00 per
yard. The standard and professional grade materials contain
different amounts of fiberglass, carbon fiber, and Kevlar as shown
in the following table.
Standard Grade Professional Grade
Fiber Glass 84% 58%
Carbon Fiber 10% 30%
Kevlon 6% 12%

24 November 2020 Yamini | Operations Research 11

ATI signed a contract with a bicycle manufacturer to produce
a new frame with a carbon fiber content of at least 20% and
a Kevlar content of not greater than 10%. To meet the
required weight specification, a total of 30 yards of material
must be used for each frame.
Formulate a linear program to determine the number of
yards of each grade of fiberglass material that ATI should
use in each frame in order to minimize total cost. Define the
decision variables and indicate the purpose of each
constraint.
24 November 2020 Yamini | Operations Research 12
6. Innis Investments manages funds for a number of companies
and wealthy clients. The investment strategy is tailored to each
client’s needs. For a new client, Innis has been authorized to
invest up to $1.2 million in two investment funds: a stock fund
and a money market fund. Each unit of the stock fund costs $50
and provides an annual rate of return of 10%; each unit of the
money market fund costs $100 and provides an annual rate of
return of 4%.
The client wants to minimize risk subject to the requirement
that the annual income from the investment be at least
$60,000.
24 November 2020 Yamini | Operations Research 13
According to Innis’s risk measurement system, each unit invested
in the stock fund has a risk index of 8, and each unit invested in the
money market fund has a risk index of 3; the higher risk index
associated with the stock fund simply indicates that it is the riskier
investment. Innis’s client also specifies that at least $300,000 be
invested in the money market fund.
a. Determine how many units in each fund Inns should purchase
for the client's minimize the total risk index of the portfolio.
b. How much annual income will this investment strategy
generate?
c. Suppose this client desires to maximize annual return. How
should the funds be invested?
24 November 2020 Yamini | Operations Research 14
7. Jackson Hole Manufacturing is a small manufacturer of plastic
products used in the automotive and computer industries. One
of its major contracts is with a large computer company and
involves the production of plastic printer cases for the computer
company’s portable printers. The printer cases are produced on
two injection moulding machines. The M-100 machine has a
production capacity of 25 printer cases per hour, and the M-200
machine has a production capacity of 40 cases per hour. Both
machines use the same chemical material to produce the
printer cases; the M-100 uses 40 pounds of the raw material per
hour and the M-200 uses 50 pounds per hour.
24 November 2020 Yamini | Operations Research 15
The computer company asked Jackson Hole to produce as many
of the cases during the upcoming week as possible; it will pay $18
for each case Jackson Hole can deliver. However, next week is a
regularly scheduled vacation period for most of Jackson Hole’s
production employees; during this time, annual maintenance is
performed for all equipment in the plant. Because of the
downtime for maintenance, the M-100 will be available for no
more than 15 hours, and the M-200 will be available for no more
than 10 hours. However, because of the high setup cost involved
with both machines, management requires that each machine
must be operated for at least 5 hours.

24 November 2020 Yamini | Operations Research 16

The supplier of the chemical material used in the
production process informed Jackson Hole that a maximum
of 1000 pounds of the chemical material will be available
for next week’s production; the cost for this raw material is
$6 per pound. In addition to the raw material cost, Jackson
Hole estimates that the hourly cost of operating the M-100
and the M-200 are $50 and $75, respectively.
a. Formulate a linear programming model that can be used
to maximize the contribution to profit.

24 November 2020 Yamini | Operations Research 17

8. Pharma Plus operates a chain of 30 pharmacies. The
pharmacies are staffed by licensed pharmacists and
pharmacy technicians. The company currently employs 105
full-time-equivalent pharmacists (combination of full time
and part time) and 175 full-time-equivalent technicians. Each
spring management reviews current staffing levels and
makes hiring plans for the year. A recent forecast of the
prescription load for the next year shows that at least 320
full-time-equivalent employees (pharmacists and
technicians) will be required to staff the pharmacies.

24 November 2020 Yamini | Operations Research 18

The personnel department expects 10 pharmacists and 30
technicians to leave over the next year. To accommodate the
expected attrition and prepare for future growth,
management states that at least 15 new pharmacists must be
hired. In addition, Pharma Plus’s new service quality
guidelines specify no more than two technicians per licensed
pharmacist. The average salary for licensed pharmacists is
$45 per hour and the average salary for technicians is $25 per
hour. Determine a minimum-cost staffing plan for Pharma
Plus. How many pharmacists and technicians are needed?

24 November 2020 Yamini | Operations Research 19