All right, that's our first linear function.

Let's have a look at a second linear function, and

again, talk about interpretation of coefficients.

So here, I'm thinking about a production process.

And I'm interested in modeling the time to produce

as a function of the number or the quantity of units that I'm producing.

So, obviously such a function would be very helpful if you had a customer who

gave you an order, one of the first things the customer is going to say to you is,

when is it going to be ready?

Well, how long does it take to produce?

That's the idea here.

And so it would certainly answer some practical questions,

the time-to-produce function.

So in the example that I'm looking at, we're given some information.

The information is it takes 2 hours to set up a production run.

And each incremental unit produced,

every extra unit, always takes an additional 15 minutes.

15 minutes is a quarter, 0.25 of an hour.

Now, in terms of modelling this, there's a key word here and that's the word always.

And what that is telling you is that the time to produce goes up by

15 minutes, regardless of the number of units being produced.

So that's the constant slope statement coming in

that is associated with the linear function or straight line function.

So it's that always there that is telling me that we're looking at

a straight line function.

So if we were to write down these words in terms of a quantitative model,

then we need to start defining variables.

So let's call T, the time to produce q units.

Then, what we're told is that the time to produce q units always starts

off with 2 hours as a 2 hour set up time, and then, once we've set the machine up,

it's quarter of an hour, or 0.25 of an hour to produce each additional unit.

And so, in this example, the interpretation of b is the set up time and

m, I might call the work rate, which is 15 minutes per additional item.

I certainly like to use the word rate here when we're talking about a slope,

because a slope is a rate of change.

And so, in this example, we were given the words associated with the process,

and it's really up to us to turn it into a mathematical or modelling formulation.

So the first bullet point is the description of the process, the second

bullet point is the articulation of the process in terms of a quantitative model.

So there's a second example.

So once again, we've got interpretations in the first example where we have

the linear cost function intercept and slope were fixed and variable cost.

This time around in the time to produce function,

they are setup time and, as I've termed it here, the work rate.

So with this function at hand, I am going to be able to predict how long

it takes to produce a job of any particular size.

And so, let's just check out the graph here quickly.

We should confirm by looking at the axis.

And once again, we've got the input to the model,

that's the quantity on the x axis on the output the time to produce on the y axis.

We've got them T and Q here, if we look at the line and

we look to see where it intercepts the point x equal to 0.

By just looking at the scale, we can see, yes, that's about 2 and

we could confirm for ourselves, for example, by looking to see how

much the graph goes up between 20 and 30, that's a ten unit change in x.

For ten unit change in x we're getting a core 2.5 extra hours to produce.

So I'm just eyeballing this graph

to confirm that it is consistent with the equation that I've written down.

And it's always a good idea to do that because mistakes happen and

it's good to have in place some kind of checks as we go along the way.

So there's our equation and the graphical representation of it.

So a model for time to produce.

I want to briefly talk about a topic that

uses linear functions as an essential input.

Now, in this particular course, I'm not going to show you the implementation but

I just want you to know that this technique is out there,

it solves a set of problems, and it is totally focused on linear functions.

And that technique is known as Linear Programming.

It's one of the workhorses of operations research.

It often goes by the acronym LP and

it is used to solve a certain set of optimization problems.

And those are optimization problems where all the features of the underlying

process can be captured with a linear construct, basically, lots of lines.

One of the interesting things about these linear programs is that they

explicitly incorporate what we term as constraints.

So when we try and optimize processes that really means doing the best that we can,

it's often important to recognize that we work within constraints.

So there's no point coming up with an optimal solution that we can't

achieve because we don't have enough workers or

we don't have enough of a certain product on hand to achieve that optimization.

And so constraints are ideas that we can incorporate in our

modelling process to try and make sure that our models

really do correspond to the world that we're trying to describe.

And, as I say, linear programming really does think carefully about

incorporating those constraints.

They just happen to be linear constraints in linear programming.

So, if you come across problems that are to do with optimization and

most of or all of the underlying features of the process can be captured through

a linear representation, then linear programming might be the thing for you.

And you can often find the linear programming implemented in

spreadsheets, sometimes with add-ins.

And so, Excel has a sorter, which can be used for doing linear programming.

So, this is one of the big uses of linear models for optimization.

Again, it's not a part of this particular course, but I want you to know that it's

out there, and it's one of the, as I say, big uses of linear models.