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 newsvendor problem. In order to successfully tackle a newsvendor 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 this tasks. First, in week two, we will focus on the optimization tool kit, and see how the best alternative is selected among many choices, incidents with low uncertainty. Then, in week three we will 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 this optimization and simulation tool kits 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 problem in terms of decisions the company must make, the objective 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 add in called, 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 through 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 this 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 number 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 spend 4 hours in manufacturing the frame, 1.5 hours in wheels and deck assembly, and 1 hour in Q and A and packaging. And, the corresponding numbers for Navajo scooter is 5, 2, and .8 hours. So, as you can see, Navajos are more profitable, take less time to do Q and A and packaging, but use quite a more of frame manufacturing and wheels and deck assembly hours. In this problem. Zooter wants to plan its production from 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-valued 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 justified in situations where a company, faced with a decision, exercises fairly strong control over its business environment, either because it considers the 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 uncertain factors influencing the outcomes of managerial actions are present. We will have a closer look at this 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 a 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 the particular choice of these decision variables. Such as 500 and 500. So, if Zooter sets its 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 produces R and the number of Navajo scooters produces N, the chief profit value will be 150*R + 160*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 user 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 our production plan of 500 of each mode fares in terms of other resources it requires. It is okay from the point of view of wheels and deck assembly hours. What it requires, 1750 hours, does not exceed what Zooter has available, 2200 hours. The same is true regarding Q and 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 the 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 our limit on the number of Q and A and packaging hours, 1100 required, as compared to 1200 available, but We still cannot implement it because of what it requires from other resources. We call such a solution infeasible. Note 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'll wanna say that the number of required frame manufacturing hours may not exceed the number of available hours. And, as a formula, we can write 4*R +5*N ≤ 5,610. What we have on the left hand side of this constraint, to the left of the less or equal than sign, is number of frame manufacturing hours required by any pair of R and N. Four hours for each unit of R and five hours of each unit of N. And, what we have on the right hand inside of this constraint, to the right of the less or equal than 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 and 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 want to choose the values of our decision variables R and N to make as much profit as possible. That's 150*R + 160*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 module can have many decision variables and constraints, but it can only have one objective. But, what if a company's interest in a number of so-called key performance indicators, such as profit, cost, 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 and treat 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 a 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 contained 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 their 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 you're 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 non-linear, or especially if you do both at the same time, you're making the model harder to optimize, whether you're 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 your 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 this 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 his 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 modelling 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.