In week two you have learned how to use the optimization analytics tool kit, how to setup an optimization model, how to create a spreadsheet implementation of your model, and how to identify the best decision using software. All the examples we looked at in week two assumed that the data you have is certain, and that the outcomes of any course of action you choose are also certain. In particular, for every decision we considered, we knew exactly what its financial impact would be and whether it would satisfy all business requirements. This week, we'll look at the settings where the business environment contains a substantial degree of uncertainty. In environments like that, a particular course of action may lead to uncertain outcomes. For example, uncertain values of profit or cost. How do we value it and compare decisions when we do not know exactly how they impact our profits or costs? This is what we will focus on in week three. In particular, we will learn how to use another important analytics tool kit, simulation. Mustering the optimization and the simulation analytics tool kits will prepare you for tackling complex business decisions that involve choosing the best alternative and uncertainty. In week three we will have three class sessions. In session one we will discuss how different high uncertainty settings are from low uncertainty settings when it comes to charting the best course of action. We will look at a specific example that focuses on evaluating a wireless data plan, and discuss the notions of reward and risk that play a central role in decision making and uncertainty. In session two, we will introduce a simulation tool kit that we will use to estimate measures of reward and risk when making choices in uncertain environments. We will look at how to set up an algebraic model for a simulation, how to implement it in a spreadsheet form, and how to run a simulation model. Finally, in session three we'll focus on interpretation and visualization of simulation results. Let's start with session one. One important feature of low uncertainty settings, is that each particular course of action results in certain outcomes, single values, rather than multi value probability distributions. Take Zooter example, if a company chooses to produce 500 units of each scooter, it knows exactly how much profit it will make, and how much of each resource it will consume. Now, in high uncertainty settings, the situation could be different. Let's look again at the news vendor example from week one. Consider a product that sells for 12 talers per unit. It costs three talers to produce one unit of this product, and all units that are not sold cannot be salvaged. In this case, as in many other settings, decision on how many product q to put on the market, must be made before the demand for the product is known. So, the overall profit, which is the combination of decision q and random demand d, may also become random. Let's start with random demand. We can model the demand using scenarios approach. For example, we can take historical demand values and use those as possible scenarios for the future, attaching the same probability to each scenario, or we can find a continuous probability distribution that provides the best fit for our historical data. Whichever way we choose to model the future demand, its random nature may translate into random profit values. Let's look at an example. Let's choose three data points from our historical demand values, he first one, 29, the one from the middle, 63 and the last one observed, 41. Let's say we put on the market q equals 50 units of a product. Let's calculate the profit we will generate from each of those three demand values. If we have 50 products on the market, and the demand happens to be 29, our profit is $198, as we produce 50 and sell only 29 units of our product. On the other hand, if the demand turns out to be 63, we sell all 50 units we produce, and get a profit of 450 talers. Now, if the demand turns out to be 41, we generate a profit of 342 talers, since we sell only 41 units out of 50 we produce. So, a certain decision today results in an uncertain outcome tomorrow. One q value can generate three profit values depending on what demand value is realized. So, let's compare the process of selecting the best option in low and in high uncertainty settings. In low uncertainty settings, for each combination of decision variables we must calculate the value of the objective, and determine the feasibility of the solution. In other words, determine whether it satisfies all problem constraints or not. After that, one must find the best feasible solution. In contrast, in high uncertainty settings, we must understand how to calculate the distribution for any key performance indicator, profit, costs, etcetera. So, if we want to choose the best option, we must understand how to choose one probability distribution, such as the distribution of profit, over another. We will look at how to generate probability distributions of key performance indicators such as profit, cost, etcetera, in week three. In week 4 you will learn how to use these probability distributions to make the best decisions. Let's start our analysis of how to make decisions in high uncertainty environments by looking at an example. Consider a business analytics consultant that explores new options on the wireless data plan for her family. She's currently subscribed to a plan that charges $10 for each gigabyte of data. Now, she found an alternative plan, and in this plan her family gets 20 gigabytes of data every month for $160. In addition, for data usage amounts above 20 gigabytes, the new plan will charge $15 for each gigabyte above 20. For example, if monthly data usage turns out to be 22 gigabytes, then the plan will charge a fixed fee of $160 plus 15 times $230 for the two gigabytes of data above 20. So the total monthly charge in this case will be 160 plus 30, $190. Consider another example. Let's say that in a particular month, the data usage is 17 Gigabytes. Well, in that case, our consultant will still pay $160. The consultant is interested in the amount she will have to pay on monthly basis under this new plan. Since the amount of monthly payment is a function of monthly data usage, and monthly data usage cannot be predicted with certainty, her monthly payment cannot be predicted with certainty as well, but she can use predictive analytic methods, like the ones discussed in week one, to convert the past data usage values into a forecast for the future. Let's suppose that our consultant, after analyzing past monthly data usage values for her family, came up with a prediction model for the future data usage values. Let's say this prediction model tells her that her future monthly data usage values will be distributed as a normal random variable, with mean of 23 gigabytes and a standard deviation of five gigabytes. Then, since under the current plan she pays $10 per gigabyte, she knows that her future monthly payments under the old plan will be distributed as a normal random variable with mean of 230 and a standard deviation of $50. This is because when we multiple a normally distributed random variable, like the data usage u, by a constant like 10, the result is a new random variable with the mean equal to the product of that constant, and the mean value of the old normal random variable, and the standard deviation equal to the product of that constant, and the standard deviation of that old random variable. In our case, if we multiply random data usage u by 10, we get a normal random variable with mean equal to 23 times 10, 230, and the standard deviation of five times 10, 50. In other words, if our consultant keeps her old plan, she knows the distribution of her monthly payments. So, we have a distribution of outcomes, monthly payments, associated with the decision to keep the old plan. If we want to compare this decision to any other decisions, for example a decision to subscribe to some other plan, we must extract from this distribution of outcomes some key performance indicators. The idea is to then use one of these performance indicators as an objective and the rest is constraining factors when choosing between different decisions. So, what are those performance indicators? Let's start with the so called reward. Expected value of profit or cost Is often used as a measure of attractiveness of a particular decision. In our example an expected monthly payment is what consultant would pay per month on average if she stays with the plan forever. In general the lower is the expected payment under particular plan, the more attractive this blend would be, all other things being equal. Note that the term reward basically reflects the attractiveness of a particular decision. So, higher reward can mean higher expected profit, but it can also mean lower expected cost. Now, expected payment is what one get if calculates an average monthly payment over infinite number of months. In any particular month, the actual payment can be quite different from the expected value. So, for any decision we consider in high uncertainty settings we must, in addition to our reward measure, have a key performance indicator or indicators that tell us how risky our decision can get. One candidate for the measure of risk in our wireless data plan example is a standard deviation of actual monthly payments. The standard deviation tells us, roughly speaking, how far away from the expected value our actual payments in any particular month will be on average. A decision maker may prefer lower values of the standard deviation of monthly payments, all other things being equal. Now, a risk may mean different things to different decision makers. Some would like to control the value of the standard deviation of monthly payments, yet others may be concerned about the likelihood that their payment, in any given month, exceeds a particular threshold, for example $300. In other words, risk measures can come in all shapes and sizes. They can reflect multiple key performance indicators that a particular decision maker, a firm, a manager, wants to keep track of and control. So, here's a road map that we will be charting when making decisions in high uncertainty settings. First, we'll decide which key performance indicators we will select as measures of reward and risk. Then, we will use simulation to get the estimates for reward and risk measures, for all different courses of action we consider. Finally, we will use optimization tool kit, to select the best course of action using, for example, reward estimate as the objective function, and risk measures as constraints. Simulation tool kit is what we're going to focus on next, and the process of selecting the best alternative based on the results of simulation will be covered next week. >> In this session, we have looked at differences in the way a decision making process unfolds in low uncertainty versus high uncertainty business settings. We have looked at the data plan evaluation example and identified performance measures, broadly defined as reward and risk, that will help us to evaluate and compare decisions in high uncertainty settings. Next time, we're going to look at a simulation as a tool for estimating the values of reward and risk, for any course of action we choose. See you in session two.
