0:58

Once we have some models in place,

Â 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 a deterministic model as compared to a probabilistic or

Â stochastic model?

Â Deterministic models don't have any random components, either inputs or outputs.

Â 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 And so, that's what we mean by a deterministic model.

Â A deterministic model is a frequently used practice, but there is a downside to them.

Â And the downside to them is that, because we don't have any random uncertain

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

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

Â We would like sometimes to be able to talk about the precision of our forecasts and

Â the output in 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 stacastical probabilistic models in another module.

Â 3:25

If you feel that the constant slope assumption is not reasonable,

Â that your process doesn't evolve in such a fashion,

Â then you're probably saying you shouldn't be using a line as a model.

Â So, these linear models aren't going to work everywhere, but

Â they are a very important building block, and

Â they are characterized through this constant slope idea.

Â So those are our linear models, y equals m x plus b.

Â I'm going to give you a couple of

Â illustrations of linear models in practice.

Â And the first one that I'm going to show you is a linear cost function.

Â 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.

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

Â So, introducing some notation,

Â let's call the number of units produced q, naturally 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 4:56

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

Â So if q is equal to 0, then you'd put 0 in the equation, and

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

Â Working down through the table, if you were to put q equal to 10 in,

Â you going to get 100 plus 300, give you 400.

Â Q equal 20 will give you 700, so

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

Â A picture is certainly worth a thousand words.

Â So, here's a picture of this linear function, the cost model.

Â 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 And that's pretty much how it's always going to happen.

Â The inputs on the x-axis, the outputs from the model on the y-axis.

Â I've written the equation here where C equals 100 plus 30q onto the graph.

Â And you should confirm as you look at this graph that the intercept, so

Â that means follow quantity, all the way down to zero and

Â eyeball what the value is, it's about 100 there.

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

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

Â And it should go up by x is going by 10 units,

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

Â And so you could confirm the coefficients simply by,

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

Â Now, the two coefficients in the equation, the intercept and

Â the slope, which we write as b and m in general, 100 and

Â 30 in this particular instance, have interpretations.

Â And one of the activities that one typically goes through with,

Â in terms of a quantitative model, is to try and interpret features of that model.

Â What are they capturing?

Â What aspect of the business process are they encoding?

Â So, interpretation is, in fact, a critical skill when it comes to modeling.

Â And it's important, because at some point,

Â remember the end point of the modeling is implementation.

Â Some people say that implementation is the sincerest form of flattery.

Â So you would like your models to be implemented.

Â But for them to be implemented,

Â you need to convince other people that they are useful and helpful.

Â Now that process of convincing other people tends not to happen by you showing

Â them the formula behind the model, because most people don't understand formula,

Â they don't do math.

Â What it involves is if you're discussing in language that they can understand

Â what the model is capturing, and that language is all about interpretation.

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

Â to modelling.

Â If you want to convince other people that your model is reasonable, and ultimately,

Â 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 y when x is

Â equal to zero, the cost of producing zero units.

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

Â zero units.

Â A better understanding of that coefficient is to think of it as the part

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

Â that's the definition of fixed cost.

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

Â there's a cost that is independent of the number of units that you are producing.

Â And we call that one of the fixed costs.

Â So the intercept has the interpretation of fixed cost.

Â And m, the slope of the line, 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 So the equation in this particular instance has a nice interpretations of

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

Â 9:27

All right, so that's our first linear function.

Â Let's have a look at a second linear function.

Â Again, talk about interpretation of coefficients.

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

Â I'm interested in modeling the time to produce as a function

Â 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 a customer is going to say to you is when

Â 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 certainly answers 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 as it takes two hours to set up a production run.

Â And each incremental unit produced, every extra unit,

Â always takes an additional 15 minutes.

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

Â Now, in terms of modeling 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 15 minutes,

Â regardless of the number of units being produced.

Â So that's the constant slope statement coming in that is associated

Â with the linear function or straight line function.

Â So it's that always there that it's 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

Â starts off with two hours.

Â There's a two-hour setup time.

Â And then, once we've set the machine up, it's quarter of an hour,

Â 0.25 of an hour to produce each additional unit.

Â And so in this example, the interpretation of b is the setup time,

Â and 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.

Â The second bullet point is the articulation of the process

Â 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 had

Â the linear cost function.

Â Our 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'm 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, and once again,

Â we've got the input to the model, that's the quantity on the x-axis,

Â and the output, the time to produce, on the y-axis.

Â We've called them T and q here.

Â We look at the line and look to see where it intercepts the point X equal to zero.

Â By just looking at the scale, we can say, yes, that's about two.

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

Â the graph goes up between 20 and 30, that's a 10 unit change in X.

Â For 10 unit change in x, we're getting 2.5 extra hours to produce.

Â So I'm just eyeballing this graph to confirm that it is consistent with

Â the equation that I've written down.

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

Â And 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, so

Â a model for a time-to-produce.

Â I want to briefly talk about a topic that uses

Â 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 work horses of operations research,

Â it often goes by the acronym LP.

Â And 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 in linear, with a linear construct, basically lots of lines.

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

Â they explicitly incorporate what we term as constraints.

Â So when we try to 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 achieve,

Â 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

Â in our modeling process to try and make sure that our models really do

Â 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 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 linear programming implemented in

Â spreadsheets sometimes with add-ins.

Â And so, Excel has a solver 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