Introduction to linear programming (1/3) METAL film 4.01

Now we’re going to look at two businesses
and see how they make decisions in order to help them maximize profits using this technique that we call linear programming. Belgian chocolates are famous the world
over but there are many different kinds of
chocolate that can be made. How can a producer pick the best
combinations within the various constraints imposed
on the business? We see how the mathematics of linear
programming can help to give an insight into this
question. Our tomato grower has focused entirely on different varieties of tomato. Is it time to diversify and produce
another product entirely? This will depend upon a whole series
of factors including: the time it will take to care for a different crop, the limits of greenhouse space, and the commitment he already has to supply tomatoes to retail outlets. Does it make sense for him to give up
some tomato production? The use of mathematical principles of
linear programming will enable us to understand the answer
to this. In the second part of the film, we begin to examine non-linear
mathematical techniques. We begin with competitive industries and
we’ll see that many industries have supply-and-demand curves that can’t realistically be thought of as linear. So how can we forecast the price and output of such industries? Then we turn to labour markets.
It’s often the case here too that we’ll need nonlinear functions to
find employment levels and wage rates. The mathematics of nonlinear equations
will help us to arrive at a better understanding of these and related issues that are raised in this film. In the last section of the film, we
introduce a problem for firms that we solve by differential calculus. We look here at industries where firms have more power to set prices. We then see that calculus can help
understand macro issues also. In this case, we examine consumption and saving behaviour. Belgian chocolate is famous throughout
the world and the Belgians have a reputation for fine craftsmanship
with chocolate. What makes it special? Why… What is it about Belgian chocolate that gives it its reputation? Well, Belgian chocolate is not always, everywhere, the same. I mean, you have Belgian chocolate that is made industrially, but you also have Belgian chocolate that is hand made. [Right.] and that means that the quality of
the chocolate is higher and they have a regulation in Belgium: if we make chocolate for the Belgian market, we are obligated from the government only to use pure cocoa butter in our chocolates. 100% pure cocoa butter. [Right.] Other countries, Scandanavia, almost every other country
in Europe have another regulation that means that if cocoa butter was very expensive, it puts other vegetable oil in its chocolate. Chocolatier Burie is one of many
chocolate factories in Belgium. There are supermarkets that sell chocolate
and they sell it a lot cheaper than you do. [Yeah] So why are people prepared to pay
so much more for these? It’s all fresh: you really taste it. We made it all by hand so it’s
a lot of work and you taste it. All the chocolates produced here are created by hand using fresh ingredients. This makes their chocolates more costly
to produce than machine-made chocolates. What makes the quality difference is also the filling that we use in our candies. We use good butter, cream, eggs; no preservatives. We have here, orange peel. It’s a speciality: sugared orange peel. We still sugar our orange peel ourselves as you can see. And this takes for about three or four
weeks. This is no preservatives the old fashion way of conserving fruit. That gives us a very high cost, but this is the only way we can say “This is quality”. The price is also, of course, higher. [Sure.] Let’s assume that they’d like to increase production by making some chocolates by machine and some by hand. What’s the best
amount to produce of each type? In other words, how do they make sure
that the amounts of each sort they make are optimal? They cannot simply make any
amount they choose because they’re constrained in all kinds of
ways. It’s very difficult now to find people , very specialized, trained people for a
long time commitment. Let’s assume the following constraints:
let’s assume that there are thirty-six people who work in the
factory and they work a 36-hour week. That represents a maximum of 1,296 hours labor time each week. Overtime is
clearly possible but it’s very expensive so the company
doesn’t regard this is a meaningful option. They have 1,824 hours of machine time
each week and a batch of handmade chocolates requires 18 hours of labour time and 6 hours of machine time. A batch of machine-made chocolates requires only 4 hours of labour time but 12
hours of machine time. Chocolatier Burie wishes to use its labor
and machines as much as possible. Furthermore, the deluxe handmade
chocolates make 89 euros a batch in profit but the standard chocolates make only 55
euros a batch profit. What does the mathematics of linear programming tell us about the best way for the chocolatier to organize
production so as to maximize profits? There are seven steps but we’ll focus for a time on just the
first five. These are: first, identify the unknowns and label them in some way: X&Y or whatever. Second, write down an expression for
the objective function – the thing you wish to maximize or
minimize – an express it in terms of X&Y. In this case we’ve got something to maximize: profit. Other times it might be something to minimize such as the cost to doing something. Third, write down all constraints on the
variables X&Y. Fourth, write down any obvious constraints such as which you may have forgotten about. Often these are non-negativity constraints. Here, for example, the output of batches of chocolate can’t be less than zero. Fifth, graph, the constraints in order to find the feasible area that is, all the options that are possible. Just what choices does our chocolate manufacturer have that he can consider? So let’s work through these 5 steps then afterwards we can see how to finish
the problem by picking the best combination of chocolate batches for maximizing profits. Here we follow a series of steps in
linear programming to determine how a Belgian chocolate producer should allocate his resources to
maximize profits. In this case, we want to find the optimum
number of batches of machine-made chocolates, that we’ll call M, and handmade chocolates that we’ll call H. Since a batch of M makes 55 euros profit and a batch of H makes 89 euros profit our objective function can be written as maximize 55 M plus 89 H. But he has to do this within a series of constraints. First, there’s a labor constraint: the
handmade chocolates, H, needs 18 hours for each batch and M needs four hours for each batch and the total hours
available is 1296. So the constraint is 4 M +18 H is equal to or less than 1296. We also have a machine time
constraint. Each batch of M needs 12 machine hours and each batch of H needs 6. And the number of machines available limits us to a total of 1824 hours. So our machine time constraint is 12 M + 6 H has to be equal to or less than 1824. But there are also some non-negativity constraints: batches of
each type can’t be less than 0. So we also require that H is equal to or greater than 0 and M is equal to or greater than 0. Now we haven’t yet found the best
combination of handmade and machine-made chocolates for
maximizing profit but what we have done is to establish the nature of the problem. We now have an objective function and we know what the constraints are within which we have to operate. So we now have the problem in a form which is soluble.

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