Hi, I'm Sergei Savin. I'm an associate professor in the department of operations, information and decisions at the Wharton school. I will be guiding you through weeks two and three of operations analytics course. In week one, you have looked at the ways and means of describing and forecasting uncertain outcomes. You've also been introduced to a quintessential problem of matching demand with supply in uncertain settings, a news vendor problem. In order to successfully tackle a news vendor challenge, we must understand how to first evaluate any course of action when faced with uncertainty, and second, choose the best course of action out of all possible alternatives. In weeks two and three, we will learn how to use two basic business analytics tool kits to accomplish these tasks. First, in week two we will focus on the optimization toolkit, and see how the best alternative is selected among many choices in settings with low uncertainty. Then, in week three, we'll look at high uncertainty settings and we'll use the simulation toolkit to evaluate reward and risk associated with any possible course of action. All this will prepare you for week four, when the optimization and simulation toolkits will be used together to identify the optimal choice. Let's begin. In week two, we will look at how to identify the best decisions in settings with low uncertainty. We'll have three class sessions in week two. In the first session, we'll look at an example of a resource allocation problem faced by a manufacturing company. Our focus will be on describing this company in terms of the decisions the company must make, the objectives it wants to achieve, and the constraints it faces. We'll build an analytics model that expresses this problem using formulas that will help us later to conduct a spreadsheet based search for the best decision. In session two, we'll convert the analytics model built in session one into a spreadsheet form, and use an Excel [INAUDIBLE] Solver to find the best course of action. Finally in session three, we will model another decision frequently encountered in practice. A decision that involves shipping goods in a network of supply and demand locations. As in sessions one and two, we will build an analytics model describing the network management problem convert it into a spreadsheet formulation, and identify the best option using Solver. Let's start our session one. We start our analysis of how to make the best decisions in low uncertainty environments by looking at a small example. In this example, a scooter manufacturer called Zooter is faced with a problem of allocating its limited resources between its two main products, Razor and Navajo. Navajos are slightly more profitable than Razors. $160 of profit contribution per unit versus $150. The company is small, and it projects that this profit margins will not be affected by the numbers of scooters it can realistically produce and place on the market. Each scooter model requires the use of each of three resources. It must go through frame manufacturing, wheels and deck assembly, and quality assurance and packaging. Razor and Navajo scooters require different amounts of each resource. For example, in order to produce one Razor scooter, the company must spent 4 hours of manufacturing the frame, 1.5 hours in wheels and deck assembly, and 1 hour in Q &A and packaging. And the corresponding numbers for Navajo scooter are 5, 2, and 0.8 hours. So as you can see, Navajos are more profitable, take less time in Q & A packaging, but there's quite a bit more of frame manufacturing and wheels and decks assembly hours. In this problem, Zooter wants to plan its production for the coming week, and during that week, it estimates that its resources will have the following limits. It will have 5,610 hours of frame manufacturing available, 2,200 hours of wheels and deck assembly available and 1,200 hours of quality assurance and packaging available. So it wants to decide how many units of each scooter model should it plan to produce next week. So that its limited resources are allocated in the most profitable way. Note that all the data we use in this example are certain non-random quantities. In other words, each piece of data is a single number rather than a multi value probability distribution. For example, we assume that one Razor scooter will make a profit contribution of exactly $150. And that its frame will take exactly four hours to manufacture. What this implies is that any production plan Zooter chooses for the next week will result in certain non random outcomes in terms of profit as well as resource consumption. The absence of randomness is a very powerful help in the task of evaluating different courses of action and selecting the best. In the absence of uncertainty, even very large problems, in other words problems with very large numbers of products and resources, are easier to tackle. Assuming away uncertainty may be justified in situations where a company faced with a decision exercises fairly strong control over its business environment. Either because it considers a short-term planning or because it benefits from longer term contracts that allow it to confidently predict future data parameters. Naturally this may not be such a good assumption in settings where significant and certain factors influencing the outcomes of managerial actions are present. We will have a closer look, at these more complex settings in weeks three and four. Okay, let's return to the Zooter situation. The analysis of any problem focused on finding the best course of action, or using another word, on optimizing the course of action, must start by identifying the decision variables. Decision variables reflect actions that a manager or company must choose to achieve a desired outcome. In the Zooter example, the company must decide upon the numbers of each scooter model to produce. So the decision variables are R, the number of Razor scooters to produce in the coming week, and N, the number of Navajo scooters to produce in the coming week. A solution is a particular choice of these decision variables, such as 500 and 500. So if Zooter sets it decision variables to 500 and 500, it will earn $155,000 in profit. This calculation brings us to another important component of an optimization model, an objective. Objective is a criterion such as profit or cost that a company wants to make as big as possible or as small as possible. In the Zooter case, the objective is a profit and the company wants to maximize it. It is important to remember that once we decide upon values of the decision variables, we should be able to calculate the value of the objective. For example, if the number of Razor scooters produced is R, and the number of Navajo scooters produced is N, the chief profit value will be 150 times R plus 160 times N. The formula that expresses objective as a function of decision variables is called an objective function. And the objective function value is what we get if we plug in a particular combination of decision variable values into the objective function. So if we plug in 500 and 500 into that formula, we get the objective function value of $155,000. Constraints form the third building block of an optimization model. If Zooter makes 500 of each scooter model, how much of each limited resource will this require? Well, on the frame manufacturing front, Zooter will need 4,500 hours. No problem, it has many more hours available. Well, whatever combination of decision variables Zooter selects, the number of required manufacturing hours cannot be higher than the number of available hours. This is what we mean by constraint. Let's see how a production plan of 500 of each model varies in terms of other resources that requires. It is okay from the point of view of wheels and deck assembly hours, what it requires, 1,750 hours, does not exceed what Zooter has available, 2,200 hours. The same is true regarding Q&A and packaging hours. The required number, 900, does not exceed the availability, that is 1,200. We call a solution like that a feasible solution. What if Zooter decides to produce 500 Razor scooters and bump up the production of Navajo scooters to 750 units? Well, this production plan requires more frame manufacturing hours than Zooter has and more wheels and deck assembly hours than Zooter has. Well, it is still within a limit on the number of Q & A and packaging hours, 1,100 required as compared to 1,200 available, but we still cannot implement it because of what it requires from other resources. We call such a solution infeasible. Know that for the solution to be infeasible, it does not have to violate all constraints. Even violating one is enough. So if we want to write a constraint on the number of frame manufacturing hours as a formula that contains decision variables R and N, how do we do it? Well, in words, we want to say that the number of required frame manufacturing hours may not exceed the number of available hours. And as a formula, we can write four times R plus five times N should be less or equal than 5,610. What we have on the left hand side of this constraint to the left of less or equal than sign, is the number of firm manufacturing hours required by any pair of R and N. Four hours for each unit of R and five hours for each unit or N. And what we have on the right hand side of this constraint, to the right of the less or equal sign, is the number of frame manufacturing hours available. Now we can write similar expressions for the constraints on the other two resources. The first line is the expression for the constraint on the number of wheels and deck assembly hours and the second, for the constraint on the number of Q & A and packaging hours available. So, are there any other constraints we must have? Well, we must make sure that our R and N variables cannot be fractional. In other words, we cannot decide to produce, for example, 467.4 Navajo scooters since 0.4 scooters do not really sell well. So R and N must be integer numbers like 350 or 878. Finally, for obvious reasons, we cannot produce negative numbers of scooters. So let us put it all together. We wanted to choose the values of our decision variables R and N, to make as much profit as possible, that's 150 x R + 160 x N. While making sure that we do not exceed resource availabilities and that we manufacture integer, non negative scooter numbers. A model like that, in other words the model that uses formulas that express objective function and constraints in terms of decision variables, is called an algebraic model. Once we convert this algebraic formulation into a spreadsheet format, Solver will be the tool that will optimize the model. In other words, it will find for us the best combination of decision variables. A few things to keep in mind. An optimization model can have many decision variables and constraints, but it can only have one objective. But what if a company is interested in a number of so-called key performance indicators, such as profit, costs, customer service levels, etc. While in general, it is impossible to optimize all of the key performance indicators at the same time, you can always do this. You can choose one of them to be an objective. Enter the rest of them as constraining factors. For example, you can try to maximize profit while making sure that the resource utilization does not exceed some threshold level. Here a profit is chosen as an objective and resource utilization forms a constraint. Another thing to keep in mind is that some models are easier to tackle than others. Look at the Zooter model, it contains constants, products of decision variables and constants, and the addition, it can also contain subtraction of the resulting expressions. Models like that are called linear since they contain only linear functions of decision variables. Linear models, in general, are easier to optimize. In other words, optimization software like Solver or any commercial software will have an easier time in identifying the best decision. Then there are harder to optimize models. One example of harder models is nonlinear models. When a model contains products of decision variables, ratios of decision variables, powers of roots of decision variables, anything beyond linear function, such model is nonlinear. And those models are harder to optimize, especially if they have large numbers of variables and constraints. Even if your model is linear, adding integer requirements on the decision variables can also significantly complicate optimization process. So to summarize, the easiest class of models to deal with are linear models with variables that are allowed to take any fractional values. We will see such a model in session three of this week. If you add integer requirements to the variables of your model or make it nonlinear or especially if you do both at the same time, you're making the model harder to optimize, whether using Solver or some commercial optimization software. This distinction may not be that important for a small problem like. The problem that Zooter is trying to solve, two variables and three constraints. But it can become quite important if the numbers of variables and constraints in the model is large. If you would like to learn more about optimization and the types of optimization models, there's a number of books out there that can help you. Here are two examples. In the session we have looked at the task of selecting the best decision among many alternatives. In settings where each piece of data that goes into a decision making process is known with certainty. As an example, we looked at a small instance of a resource allocation problem in which two products are competing for limited resources. Using this example, we have identified three elements that any optimization model will have, decision variables, an objective function, and constraints. Even in low uncertainty settings, the task of identifying the best decision may become very challenging as the sheer number of possible decisions can be very large. So we may often need the help of an optimization tool such as Solver. Now before using software to find the best decision, it is very useful to express the problem using an algebraic language. In particular it can be much easier to identify modeling mistakes by looking at an algebraic formulation than by looking at a spreadsheet. And an algebraic formulation may also help you to create a well-structured easy to read spreadsheet formulation. In the next session, we'll focus on converting an algebraic formulation of the Zooter problem we created into a spreadsheet formulation. And on using Solver to identify the best production decision. See you soon.
