Welcome to Module 2, Fundamentals of Quantitative Modeling. And we're going to talk about deterministic models and some optimization. In particular, we'll start with a discussion of the most fundamental and commonly used models is our linear models. We're then going to go on and discuss growth and decay. The idea there being that one of the most fundamental processes that a business is interested in is in growth, growth of customers, growth of profits, these sorts of things. And models that address growth directly are going to be very, very helpful. We are going to see, though, models for growth and decay that work in both, one set of models with discrete time and another for continuous time. Once we have some models in place, I'm also going to talk about some optimization. And when I say classical optimization, I mean optimization using calculus. So those are the topics that we're going to discuss. As a reminder, what's a deterministic model as compared to a probabilistic or stochastic model? Deterministic models don't have any random components, either inputs or outputs. And that means that there's nothing random going on, then you can be sure that if the same input goes in, you're going to get exactly the same output every single time. And so, that's what we mean by a deterministic model. A deterministic model is a frequently used practice, but there is a downside to them. And the downside to them is that, because we don't have any random uncertain components, it's very hard to assess the uncertainty in the outputs. Remember, all models are wrong, but some are useful. We would like sometimes to be able to talk about the precision of our forecasts and the output in the model. Well, that's really not a construct that works well in terms of a deterministic model because everything is fixed, there's no uncertainty by definition. But anyway, today is deterministic models. We will see the stacastical probabilistic models in another module. Let's start off by talking about linear models. Now we've discussed already linear functions, the straight line function. Remember the equation that we used to write down the straight line, it's y equals mx plus b. And, from the point of view of this course, what you need to understand about the line is its essential characteristic. And its essential characteristic is that the slope is constant. It doesn't matter what the value of x is. If x goes up by one unit, then y is always going to go up by m units, regardless of the value of x. Now, you have to ask yourself when you're modeling whether or not that idea makes sense in your context. So that's our definition of the line. If you feel that the constant slope assumption is not reasonable, that your process doesn't evolve in such a fashion, then you're probably saying you shouldn't be using a line as a model. So, these linear models aren't going to work everywhere, but they are a very important building block, and they are characterized through this constant slope idea. So those are our linear models, y equals m x plus b. I'm going to give you a couple of illustrations of linear models in practice. And the first one that I'm going to show you is a linear cost function. So, costs are an attribute that a business is often trying to get a handle on. Typically, get a handle on means model in some fashion. And a linear cost function is not a bad starting place for modeling costs. So, introducing some notation, let's call the number of units produced q, naturally q for quantity. And we'll call the total cost of producing those units C, capital C. Now I'm presenting to you an example of a linear cost function now. Let's say that the cost C is equal to 100 plus 30 times q. So there's a formula, it's a linear formula. What does it tell us about this cost process? I'd always get started by calculating some illustrative values. So if q is equal to 0, then you'd put 0 in the equation, and you're going to get a 100 plus 30 times 0 which is just 100. Working down through the table, if you were to put q equal to 10 in, you going to get 100 plus 300, give you 400. Q equal 20 will give you 700, so there's some illustrative values associated with this cost model. A picture is certainly worth a thousand words. So, here's a picture of this linear function, the cost model. And you can see, it's a straight line model. We have quantity on the horizontal axis and total cost, the variable that we are trying to understand on the vertical axis. And that's pretty much how it's always going to happen. The inputs on the x-axis, the outputs from the model on the y-axis. I've written the equation here where C equals 100 plus 30q onto the graph. And you should confirm as you look at this graph that the intercept, so that means follow quantity, all the way down to zero and eyeball what the value is, it's about 100 there. And you could also by choosing a couple values, say, q equals 10 and 20, look to see how much the graph is gone up by. And it should go up by x is going by 10 units, then y goes up by 30 units if it's a linear function here. And so you could confirm the coefficients simply by, the reasonableness of the coefficients simply by looking at the graph. Now, the two coefficients in the equation, the intercept and the slope, which we write as b and m in general, 100 and 30 in this particular instance, have interpretations. And one of the activities that one typically goes through with, in terms of a quantitative model, is to try and interpret features of that model. What are they capturing? What aspect of the business process are they encoding? So, interpretation is, in fact, a critical skill when it comes to modeling. And it's important, because at some point, remember the end point of the modeling is implementation. Some people say that implementation is the sincerest form of flattery. So you would like your models to be implemented. But for them to be implemented, you need to convince other people that they are useful and helpful. Now that process of convincing other people tends not to happen by you showing them the formula behind the model, because most people don't understand formula, they don't do math. What it involves is if you're discussing in language that they can understand what the model is capturing, and that language is all about interpretation. So I believe that interpretation is absolutely critical when it comes to modelling. If you want to convince other people that your model is reasonable, and ultimately, to get it implemented. So let's do some interpretation for this example. So let's look at the intercept, which is b. Now formally, you can say that the intercept b is a value of y when x is equal to zero, the cost of producing zero units. But it doesn't really make a lot of sense that there's some cost in producing zero units. A better understanding of that coefficient is to think of it as the part of total cost that doesn't depend on the quantity produced, and that's the definition of fixed cost. So every time you produce some of this particular product, there's a cost that is independent of the number of units that you are producing. And we call that one of the fixed costs. So the intercept has the interpretation of fixed cost. And m, the slope of the line, well, that's as quantity goes up by one unit, we anticipate the total cost to go up by m units. That is known as the variable cost. So the equation in this particular instance has a nice interpretations of the intercept and the slope as fixed and variable costs. All right, so that's our first linear function. Let's have a look at a second linear function. 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 a 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 certainly answers some practical questions, the time to produce function. So in the example that I'm looking at, we're given some information. The information as it takes two 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 modeling 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 it's 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 two hours. There's a two-hour setup time. And then, once we've set the machine up, it's quarter of an hour, 0.25 of an hour to produce each additional unit. And so in this example, the interpretation of b is the setup 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 had the linear cost function. Our 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'm 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, and the output, the time to produce, on the y-axis. We've called them T and q here. We look at the line and look to see where it intercepts the point X equal to zero. By just looking at the scale, we can say, yes, that's about two. 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 10 unit change in X. For 10 unit change in x, we're getting 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 a 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 work horses 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 in linear, 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 to 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 modeling 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 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 linear programming implemented in spreadsheets sometimes with add-ins. And so, Excel has a solver 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.