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 use models, these are 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 addressed 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 the 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 the 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. So that's what we mean by a deterministic model. Deterministic models are frequently used in practice but there is a downside to them. 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, old models are wrong but some are useful. We would like, sometimes, to be able to talk about the precision of our forecast and the output of 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 stocastical probabilistic models in another module. Let's start off by talking about linear models. Now, we've discussed already linear functions, the straight line functions. Remember the equation that we used to write down the straight line, it's y = mx + b. And from the point of view of this course, what you need to understand about the line is it's a central characteristic. And it's a central 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 modelling 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, using 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 all linear models, y = mx + b. I'm going to give you a couple 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 cost. So, introducing some notation, let's call the number of units produced q, not really 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 into your equation and you're going to get 100 + 30 times 0 which is just 100. Working down through the table, if you were to put Q=10n, you're going to get 100 plus 300 getting 400. Q=20 will give you 700. So there's some illustrative values associated with this cost model. A picture is certainly worth 1000 words, so here is 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. 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, C = 100 + 30q onto the graph. And you should confirm, as you look at this graph, that the intercept, so that means follow quality all the way down to zero and eyeball what the value is, it's about a hundred there. And you could also, by choosing a couple of values, say q=10 and 20, look to see how much the graph has gone up by. And it should go up by, if X is going by ten units, then y goes up by 30 units if it's a linear function here. And so you could confirm the coefficient simply by the reasonableness of the coefficient 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 modelling 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 you 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 zero, the cost of producing zero units. But it doesn't really make a lot of sense that there's some cost in producing 0 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 a 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 the fixed cost. 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. And so the equation in this particular instance has nice interpretations of the intercept and the slope as fixed and variable costs. 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.