As indicated in the last column, the acreage to be cultivated is limited by the amount of money
available for cultivation costs and by the number of working days that can be put into this part of
the business. Find the maximum profit that the farmer can get from his land.
Each spot is expected to increase sales by 40000 cans, whereas each magazine ad will account for
95000 cans in sell. The company's gross profit on sales is $0.3 per can.
One television spot costs $1200; each magazine ad requires an expenditure of $3000.
To have a balanced marketing program, the advertising budget must involve no more than $36000
in magazine ads and no more than $60000 in television spots. What is the maximal net profit (gross
profit minus ads expenses) that the company can achieve making the right decision?
(Note: Number of units of ads bought can be fractional.)
A farmer has 340 acres of land available on which to grow some soybeans and some corn. The cost
of cultivation per acre, the workdays needed per acre, and the profit per acre are indicated in the
table:
Cultivation cost per
acre
A factory manufactures two products, each requiring the use of three machines.
The first machine can be used at most 70 hours; the second machine at most 60 hours; and the third
machine at most 80 hours.
One unit of the first product requires 4 hour(s) on machine 1, 4 hour(s) on machine 2, and 1 hour(s)
on machine 3.
One unit of the second product requires 1 hour(s) on machine 1, 3 hour(s) on machine 2, and 5
hour(s) on machine 3.
If the profit is £75 per unit for the first product and £45 per unit for the second product, what is the
maximum profit that can be achieved by manufacturing these products?
The marketing manager of a North America Brewery must determine how many television spots
and magazine ads to purchase within an advertising budget of $80000.
Problem 4
Problem 5
Votes/$
0.07
0.01
0.09
0.06
Problem 6
The company has £11000 to spend on producing and advertising applesauce and apple juice. The
company has a policy that at least 35% but no more than 55% of this budget should be used for
producing apple juice.
The amount allocated to job training must at least equal the amount spent on the sanitation.
What is the maximal expected number of votes the congressman can gain by distributing the money
among the programs? (The answer can be fractional)
A local company purchases apples from local growers and makes applesauce and apple juice. It
costs £0.7 to produce a kilogram of applesauce and £0.5 to produce a litre of apple juice.
The company wants to meet but not exceed the demand for each product.
The marketing manager estimates that the demand for applesauce is 3000 kilogram plus additional
2 kilograms for each £1 spent on advertising.
The demand for apple juice is estimated to be 4000 litres plus additional 4 litres for every £1 spent
to promote apple juice.
Parks
Sanitation
Mobile library
In order also to satisfy several local influential citizens who financed his election, he is obliged to
observe the following guidelines.
None of the programs can receive more than 45% and less than 8% of the total allocation.
The amount allocated to sanitation can not exceed the total allocated to the parks and mobile
library programs.
The remainder of each product consists of an inexpensive non-meat filler that the store has in
unlimited quantities.
What is the maximum amount of meat that can be used to make pasties?
A congressman of United States is responsible for allocation of $290000 for programs and projects
in his district. It is up to the congressman to decide how to distribute the money.
The congressman has decided to allocate the money to four ongoing programs because of their
importance to his district - a job training program, a parks project, a sanitation project, and a mobile
library. However, the congressman wants to distribute the money in a manner that will please the
most voters, or, in other words, gain him the most votes in the upcoming election. His staff's
estimates of the expected number of votes gained per dollar spent for the various programs are as
follows.
Program
Job training
The manager of the supermarket meat department finds that there are 200 kg of round steak, 220 kg
of chuck steak, and 130 kg of pork in stock on Saturday noon. This meat has to be ground into
hamburger pasties and picnic pasties for which there is a large weekend demand.
Each kilogram of hamburger pasties contains 25% ground round and 20% ground chuck.
Each kilogram of picnic pasties contains 40% ground pork and 25% ground chuck.
Problem 7
Problem 8
Problem 9
From 1 2 3 4
Plant 1 35 40 30 45
Plant 2 30 50 25 40
Plant 3 45 45 35 50
A firm producing a single product has three plants and four customers.
The three plants will produce 3400, 2200 and 5800 units, respectively, during the next time period.
The firm has made a commitment to sell 2800 units to customer 1, 4000 units to customer 2, and at
least 1150 units to customer 3. Both customers 3 and 4 also want to buy as many of the remaining
units as possible.
The profit associated with shipping a unit from plant i to customer j is given in the table below.
To Customer (£)
The cost of wool is £5 per square meter and the cost of nylon is £3 per square meter.
What is the maximum income that can be achieved by manufacturing the carpets?
Hint :_Income=revenue from sale-(production cost for material+labor)
The Archer family raises cattle on their farm in West Midlands. They also have a large garden in
which they grow ingredients for making two types of relish - SauceA and SauceB. These they sell at
local stores.
The profit per kilogram of SauceA is £3.25 and the profit per kilogram of SauceB is £3.5. The
ingredients in each relish are cabbage, tomatoes, onions, and oil. One kilogram of SauceA must
contain at least 65% but no more than 73% cabbage, and at least 4% onion, and at least 5% oil.
One kilogram of SauceB must contain at least 65% but no more than 75% tomatoes, and at least 5%
onion, and at least 7% oil. Both relishes contain no more than 12% onion and no more than 13%
oil.
The family has enough time to make no more than 760 kilograms of relish. They know also that
they will sell at least 25% more SauceA than SauceB. They will have this year 320 kilograms of
cabbage, 280 kilograms of tomatoes, and 140 kilograms of onion. They can use any amount of oil
needed.
What is the maximal profit that the family can gain by producing and selling the relish?
Given that the applesauce sells for £1.95 per kilogram, apple juice sells for £1.95 per litre, what is
the maximal profit that the company can achieve using the budget in optimal way?
Smart Rug Manufacturers has available1900 square meters of wool and 1400 square meters of
nylon for the manufacture of two grades of carpeting: high-grade, which sells for £800 per roll, and
low-grade, which sells for £650 per roll.
25 square meters of wool and 15 square meters of nylon are used in a roll of high-grade carpet.
30 square meters of wool and 40 square meters of nylon are used in a roll of low-grade carpet.
25 work-hours are required to manufacture each roll of the high-grade carpet, and 15 work-hours
are required for each roll of the low-grade carpet, at an average cost of £5 per work-hour. A
maximum of 900 work-hours are available.
Problem 10
Project Year 1 Year 2 Year 3 Year 4 Year 5 Year 6
1 -1 0.6 0.3 1.6 1.5 1.5
2 -1 0.5 0.3 1.8 1.6 1.9
3 0 -1 0.5 0.5 1.2 1.7
4 0 0 -1 1.4 1.7 1.7
An investor has four potential opportunities to invest a total of £70000. The following table gives
the cash flow for the four investments.
Cash flow at the start of
The information in the table can be interpreted as follows. For project 1, £1 invested at the
beginning of year 1 will yield £0.6 at the beginning of year 2, £0.3 at the beginning of year 3, £1.6
at the beginning of year 4, £1.5 at the beginning of year 5, and £1.5 at the beginning of year 6. The
remaining entry can be interpreted similarly. A 0 entry signifies no in or out cash flow.
The investor also has the additional option of investing in a bank account that earns 5.5% annually.
Any funds accumulated can immediately be reinvested in the options available.
What is the maximal profit that the investor can earn by the beginning of year 6?
What is the maximal profit which the company can achieve using an optimal transportation plan