This document provides three examples of using linear programming to optimize objectives like minimizing costs or maximizing profits. The first example involves determining the optimal number of artisans and apprentices a furniture factory should employ to minimize wages while delivering at least 90 units per week. The second example involves determining the optimal amounts of two product mixtures a sweet factory should make to maximize daily profit. The third example involves determining the optimal number of passengers a company should transport by bus and minibus each day to maximize daily profit.

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LINEAR
PROGRAMMING
Compare Par 7.6
Linear Programming is often
useful in making real-life
economic decisions. For example
to determine the optimal cost of
a product or maximum profit
INFORMATION - WEDNESDAY 26 AUGUST 2015
1. Library Projects still outstanding? Please
submit to Ms. Durandt before the reses.
Late submission = less marks.
2. Lecture on Thursday 27 August on par. 8.1.
3. Find all memos for homework tasks &
worksheets on uLink.
4. Semester tests can be collected from the
Collection Facility at the Department of
Mathematics from Friday 28 August.
Compiled by R Durandt & Material
form Cengage Learning
Example 1
A furniture factory has a contract to deliver at
least 90 units of a certain piece of furniture per
week. There may not be more than 18
employees. An artisan, who earns R600 per
week, can produce 7 units per week, while an
apprentice, who earns R300 per week, can
produce only 4 units per week.
The labour laws specify that at least one
apprentice should be employed for every five
artisans. The labour union, however, insists that
the ration of apprentices to artisans should not
exceed 1:2.
Example 1 - Questions
1. Let the number of artisans be x and the number
of apprentices y. Write down the constraints in
terms of x and y.
2. Represent all the constraints graphically and
clearly indicate the feasible region.
3. Express the amount (L) representing weekly
wages in terms of x and y.
4. Show the optimal position of the curve of the
objective function which will minimise the wage
bill.
5. How many artisans and apprentices should be
employed so that the wage bill is kept to a
minimum but the largest number of units are
delivered?
Example 2
The daily production of a sweet factory consists
of at most 100kg chocolate covered nuts and at
most 125kg chocolate covered raisins, which are
then sold in two different mixtures. Mixture A
consists of equal amounts of nuts and raisins,
and is sold at a profit of R5 per kilogram.
Mixture B consists of one third nuts and two
thirds raisins and is sold at a profit of R4 a
kilogram. Let there be x kg of mixture A and y
kg of mixture B.

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Example 2 - Questions
1. Express the mathematical constraints which
have to be adhered to.
2. Write down the objective function which can
be used to determine maximum profit.
3. Represent the constraints graphically and
clearly show the feasible region.
4. Use the graph and determine maximum
profit obtained.
Example 3
A company uses buses and minibuses to
transport a minimum of 800 and a
maximum of 1200 passengers per day. The
number of passengers, x, transported by
bus, must be a minimum of 400 per day,
but cannot be more than three times the
number of passengers y, transported daily
by minibus.
Example 3 - Questions
1. Represent the data above as a set of
inequalities.
2. Represent these inequalities graphically and
clearly indicate the feasible region.
3. The daily profit per passenger travelling by
bus, is R1, and the daily profit per passenger
travelling by minibus is 80c. Use your graph
to determine the values of x and y which will
give a maximum profit.
4. Hence determine the maximum daily profit.