0:58

Once we have the models in place, I'm also going to talk about some optimization.

Â And when I say classical optimization, I mean optimization using calculus.

Â So those are the topics that we're going to discuss.

Â As a reminder, what's the deterministic model as compared to a probabilistic or

Â stochastic model?

Â Deterministic models don't have any random components,

Â either inputs or outputs, and that means that there's nothing random going on.

Â Then you can be sure that if the same input goes in,

Â you're going to get exactly the same output every single time.

Â So that's what we mean by a deterministic model.

Â Deterministic models are frequently used in practice but

Â there is a downside to them.

Â The downside to them is that because we don't have any random

Â uncertain components, it's very hard to assess the uncertainty in the outputs.

Â Remember, old models are wrong but some are useful.

Â We would like, sometimes,

Â to be able to talk about the precision of our forecast and the output of the model.

Â Well, that's really not a construct that works well in terms of a deterministic

Â model because everything is fixed, there's no uncertainty by definition.

Â But anyway, today is deterministic models,

Â we will see the stocastical probabilistic models in another module.

Â 4:05

So costs are an attribute that a business is often trying to get a handle on.

Â Typically, get a handle on means model in some fashion, and

Â a linear cost function is not a bad starting place for modeling cost.

Â So, introducing some notation,

Â let's call the number of units produced q, not really q for quantity.

Â And we'll call the total cost of producing those units C, capital C.

Â Now, I'm presenting to you an example of a linear cost function now.

Â Let's say that the cost C is equal to 100, plus 30 times Q.

Â So, there's a formula, it's a Linear formula.

Â What does it tell us about this cost process?

Â I'd always get started by calculating some illustrative values.

Â So if Q is equal to 0, then you'd put 0 into your equation and

Â you're going to get 100 + 30 times 0 which is just 100.

Â Working down through the table, if you were to put Q=10n,

Â you're going to get 100 plus 300 getting 400.

Â Q=20 will give you 700.

Â So there's some illustrative values associated with this cost model.

Â 5:30

function, the cost model, and you can see, it's a straight line model.

Â We have quantity on the horizontal axis and total cost,

Â the variable that we are trying to understand, on the vertical axis.

Â That's pretty much how it's always going to happen, the inputs on the x-axis,

Â the outputs from the model on the y-axis.

Â I've written the equation here, C = 100 + 30q onto the graph.

Â And you should confirm, as you look at this graph,

Â that the intercept, so that means follow quality all the way down to zero and

Â eyeball what the value is, it's about a hundred there.

Â And you could also, by choosing a couple of values, say q=10 and

Â 20, look to see how much the graph has gone up by.

Â And it should go up by, if X is going by ten units,

Â then y goes up by 30 units if it's a linear function here.

Â And so you could confirm the coefficient simply

Â by the reasonableness of the coefficient simply by looking at the graph.

Â 7:51

the model is capturing.

Â And that language is all about interpretation.

Â So I believe that interpretation is absolutely critical when it comes

Â to modelling if you want to convince other people that your model is reasonable and

Â ultimately to get it implemented.

Â So, let's do some interpretation for this example.

Â So, let's look at the intercept which is B.

Â Now formally, you can say that the intercept B is a value of

Â y when x is zero, the cost of producing zero units.

Â But it doesn't really make a lot of sense that there's some cost in

Â producing 0 units.

Â A better understanding of that coefficient is to think of it

Â as the part of total cost that doesn't depend on the quantity produced, and

Â that's the definition of a fixed cost.

Â So every time you produce some of this particular product,

Â there's a cost that is independent of the number

Â of units that you are producing, and we call that one the fixed cost.

Â So the intercept has the interpretation of fixed cost and m, the slope of the line,

Â well, that's as quantity goes up by one unit,

Â we anticipate the total cost to go up by m units.

Â That is known as the variable cost.

Â And so the equation in this particular instance has nice interpretations of

Â the intercept and the slope as fixed and variable costs.

Â 9:27

All right, that's our first linear function.

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

Â again, talk about interpretation of coefficients.

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

Â And I'm interested in modeling the time to produce

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

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

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

Â when is it going to be ready?

Â Well, how long does it take to produce?

Â That's the idea here.

Â And so it would certainly answer some practical questions,

Â the time-to-produce function.

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

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

Â And each incremental unit produced,

Â every extra unit, always takes an additional 15 minutes.

Â 15 minutes is a quarter, 0.25 of an hour.

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

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

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

Â So that's the constant slope statement coming in

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

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

Â a straight line function.

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

Â then we need to start defining variables.

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

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

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

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

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

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

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

Â because a slope is a rate of change.

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

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

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

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

Â So there's a second example.

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

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

Â This time around in the time to produce function,

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

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

Â it takes to produce a job of any particular size.

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

Â We should confirm by looking at the axis.

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

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

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

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

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

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

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

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

Â So I'm just eyeballing this graph

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

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

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

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

Â So a model for time to produce.

Â I want to briefly talk about a topic that

Â uses linear functions as an essential input.

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

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

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

Â And that technique is known as Linear Programming.

Â It's one of the workhorses of operations research.

Â It often goes by the acronym LP and

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

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

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

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

Â explicitly incorporate what we term as constraints.

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

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

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

Â achieve because we don't have enough workers or

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

Â And so constraints are ideas that we can incorporate in our

Â modelling process to try and make sure that our models

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

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

Â incorporating those constraints.

Â They just happen to be linear constraints in linear programming.

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

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

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

Â And you can often find the linear programming implemented in

Â spreadsheets, sometimes with add-ins.

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

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

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

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

Â