Constructing and Solving Linear Models
Often students who have mastered mathematical techniques have
difficulty in knowing when and how to apply those techniques when
faced with problems that are not already expressed in familiar terms.
An important part of developing mathematical thinking is practice in
formulating appropriate mathematical models from the information
provided.
For each of the situations described below, decide if linear
programming is an appropriate approach to take to solve the problem.
What sort of things give you clues as to whether it is a linear
programming problem?
- A painter can use either Ludux or Bautman paint for a job that
involves painting 44 square metres of wall. Allowing for two coats,
the Ludux will cover 6 square metres per litre and the Bautman will
cover 7.5 square metres per litre. The paint can be bought in bulk for
$7 per litre (Ludux) and $8.50 per litre (Bautman). How much of each
paint should the painter buy to minimize cost, and how much will it cost?
- How many bags of flour should a bakery order so as to be able to
bake 200 loaves and 300 rolls a day for the next week, if it takes
500g of flour for each loaf and 150g for each roll and a bag of flour
contains 20kg?
- Nubbles, a small snack food company, has to decide how many of
each of two types of health bar to manufacture. They make $35 profit
on a carton of nutty bars, and $16 profit on a carton of
open sesame bars. The factory uses a 2-stage process to make
open sesame bars, and it takes two minutes for a carton of
bars to be processed by each stage. This means the maximum production
of open sesame bars is 15 cartons an hour. Nutty
bars, however, use a 3-stage process with each stage again taking two
minutes. This means the maximum production but nutty is 10
cartons per hour. The same equipment is used for both bars, so the
number of cartons of each type is restricted by the constraint
2 x the number of open sesame cartons + 3 x the number of
nutty may not exceed 30 per hour. In addition, management has
a policy that the number of cartons of nutty bars should be
no more than three times the number of open sesame bars.
Assuming that all bars made will be sold, what combination of bars
gives the most profit?
- In order to try to lose some weight I have decided to either walk
or swim. I'm going to spend at least 20 minutes a day on exercise, but
I'm not a good enough swimmer to swim for more than 45 minutes. If I
shed 2g of fat for each minute I spend swimming but only 1g of fat for
each minute I spend walking, how long should I walk and swim to lose
the most weight?
- A firm that manufactures metal components for the maritime
industry has a commission to cast a component that can be made of
either of two metals, angstide or beranium, or any combination of
them. There is 17kg of angstide available at a cost of $43.50 per
kilogram, while only 15.5kg of beranium is available at a cost of $32
per kilogram. The component must have a finished weight of at least
18kg. The foundry can handle volumes up to 100 decilitres.
1kg of angstide occupies 3.5 decilitres and 1kg of beranium occupies 6
decilitres. What amounts of the two metals should be used to minimize
cost? What is the minimum cost?
- A neighbour of mine keeps a sheep in her yard. If she wants to
make a rectangular pen using her fence as one side and some or all of
a 20 metre roll of wire mesh to make the other three sides, what
length sides should she use to give the maximum area for the pen. The
fence is only 18m long, and her yard only extends 6m away from the
fence.
Check your answers before continuing.
For those that are linear programming problems, write down the
constraints and objective function in algebraic form. Are you seeking
to maximize or minimize the objective?
Use the applet to solve the linear models you have constructed.
Note that the applet provides up to four constraints, but all are
"less than or equal to" type. This means that you will need to use the
same change of sign trick as was suggested on the last page. That is,
make use of
s ≥ t implies
-s ≤ -t.
(Of course, you won't have to worry about this if you're solving
problems like these by hand, and some software allows you to specify
what kind of constraint you are dealing with. But there is still some
simpler software around that, like this applet, assumes all constraints are
expressed in ≤ form.)
For any problems that require less than four constraints, set all
the parameters to 0 and the constraint line will disappear.