Hi, I'm Noah Gans, I'm a Professor in the Department of Operations, Information and Decisions at the Wharton School. I'll be your guide for week four of our, Operations Analytics course. Before we get started though, I wanted to remind you of where we are in the course. In week one, Sentil introduced you to the news vendor problem, which is a fundamental operations problem of matching your supply with uncertain future demand. Sentil also offered you a look at the first steps of the business analytic cycle. The problem of characterizing data with descriptive analytics. Then in weeks two and three, Sergei introduced predictive and prescriptive tools, that are helpful when deciding the best course of action when faced with uncertainty. In week two, you saw how you can use optimization methods to find the best course of action, when there's little uncertainty regarding the data. In week three, you then saw how you can use simulation to evaluate any single course of action, when you're faced with more significant uncertainty. And along the way, Sergei introduced you to production and distribution problems, often faced in operations. Because of their complexity, Sergei introduced optimization and simulation one at a time. Sometimes though you need to optimize in settings with significant uncertainty. This is often called decision making under uncertainty. And that's where we pick up in week four. In session one, we'll begin by introducing a new tool that provides a useful way to think about and evaluate decisions made under uncertainty. They're called decision trees. Then in sessions two and three, we'll look at how simulation, optimization, and decision trees can be used together, to solve more complex problems of decision making under uncertainty. To keep the sessions focused on how the tools can complement each other, we'll use the same example in sessions one to three. That of a Scandinavian furniture retailer that we call IDEA. Finally, in session four. We'll go back to the news vendor problem, Sentil introduced in week one. And we'll see that we can use the simulation and optimization framework we've developed to tackle the problem. That's our agenda for the week. In session one, we'll introduce decision trees, and we'll introduce them through the example of IDEA, a Scandinavian furniture retailer. IDEA plans to introduce a gazebo tent that it calls, Krusbar. The tent's to be sold next summer in Scandinavia, for a price that's about 150 euros. It will be manufactured in advance, next spring and summer. At that 150 euro price, there's uncertainty regarding the demand for Krusbar though. There's a 50 50 chance that sales will be weak or strong. If sales are strong, IDEA will be able to sell 10,000 units. If sales are weak, IDEA will only be able to sell 5,000 units. This demand distribution is quite simple. It only has two points, 5,000 units of demand or 10,000 units. But that simplicity let's us focus on the analysis. We'll consider more complex demand distributions later this week. IDEAs considering two potential suppliers to manufacture the Krusbar. One's in Sweden, and the other is in Poland. We'll call the Swedish supplier, supplier S and the Polish supplier, supplier P. Each of the suppliers requires that IDEA uses all of its capacity. And IDEA will contract with at most one of the two suppliers. That is, IDEA may decide to use supplier S, or to use supplier P, or IDEA may decide not to use a supplier at all and not sell the Krusbar tent. Here are the statistics for the two suppliers. They have different capacities. You can see that supplier S has 5,000 units of capacity, and if IDEA contracts with it, it will order 5,000 units of the tent. Supplier P has 10,000 units of capacit, and if IDEAs contracts with it, it will order 10,000 units of the tent. The two suppliers also have different up-front charges to IDEA. If IDEA contracts with supplier S there is no up-front charge. Where as, if IDEA contracts with Supplier P, there's a 50,000 euro upfront charge. Unit costs are also different. You can see that Supplier S has a unit cost to IDEA of 120 euros per tent. Whereas Supplier P, has a unit cost of a hundred euros per tent. So, we can compare Supplier S has no upfront charge, but it has a higher unit cost, whereas Supplier P has an upfront charge, but it has a lower unit cost. We can use a decision tree to represent IDEA's choices. Before I get in to it, I wanna just review quickly for you, what the elements are of a decision tree. The structures of a decision trees made up of three building blocks. They're decision nodes, which we'll represent with squares. Decision nodes are points at which a decision maker has to decide on an action. For IDEA, those actions are which supplier to select, if any. Event nodes are represented with circles. Event nodes are points of uncertainty at which the outcome is random. For IDEA, that's whether the market will be weak or strong. Finally, outcomes are represented as triangles. Outcomes are payouts that occur, that are due to specific sequences of decisions and events. For IDEA, the outcomes are its profits. They depend on which supplier IDEA chooses, S, P, or none. And it depends on whether the market is weak or strong. Now that we've defined the decision nodes, the event nodes, and the outcomes we'll build the tree. The first thing that we need to do to build the tree, is include a decision. With whom will IDEA contract? If IDEA contracts with no one, it earns no revenue and pays no costs. And you can see we have put a cash flow of zero euros. And that's the end of that part of the tree. The outcome for IDEA in that case, would be to earn zero euros. A second possible decision would be to contract with Supplier S. If IDEA contracts with Supplier S, it pays no upfront fixed fee, so we've written zero euros there. If IDEA contracts with Supplier P however, it will pay an upfront fixed fee of 50,000 euros, and you can see we've written it on that choice as well. Those are the three choices that IDEA has. Now that we have defined the decision node, the next thing that happens is an event, whether the market is weak or strong. If the market is strong and IDEA contracts with Supplier S, that happens with a probability of 0.5. In that case, we can calculate the gross profit. Remember that Supplier S has 5000 units of capacity. And a strong market has 10,000 units of demand. So IDEA would be capacity limited. It would sell everything that it had ordered, and it would earn 5,000 units times 150 euros per unit, but pay unit costs of 120 euros on all 5,000 units for a gross profit of 150,000 euros. That gives us an outcome of 150 thousand euros. And we calculated it by adding the zero upfront cost, with 150 thousand euros of gross profit. If on the other hand, the market is weak, that happens with a probability of 0.05. And in that case, again, we can calculate the gross profit. Remember, if the market's weak, demand is only 5,000 units. At the same time, supplier S only supplies 5,000 units to IDEA. So again, IDEA earns 150 euros per tent, times 5,000 tents revenue, and it pays 120 euros per tent times 5,000 tents, unit cost to supplier S for a gross profit of 150,000. And again, now we can calculate the profits for IDEA's outcome, if it orders from supplier S and the market is weak, it's going to earn 150,000 euros, that's no money upfront and 150,000 euros in gross profit. Finally, if IDEA contracts with supplier P, there are again, two outcomes. One is if the market is strong, that happens with probability .05. And here the gross profit is calculated knowing that demand is 10,000 units and IDEA has ordered 10,000 units from supplier P. So the gross profit is 10,000 units times the 150 euro revenue, minus 10,000 units times the 100 euro unit cost, for a gross profit of 500,000 euros. Now that we've got the gross profit, we can calculate the total profits for idea. If it orders from supplier P and the market is strong, by adding the -50,000 up front fixed cost with the 500,000 euro gross profit to get a net profit of 450,000 euros. If IDEA orders from supplier P and the market is weak, that happens with a probability of .5, and we can calculate these gross profits as well. Remember, when IDEA orders from supplier P, it orders 10,000 units, but when the market is weak, it only sells 5,000 units. So the gross profit is 5,000 units sold, times 150 Euros per unit, minus 10,000 units ordered, times 100 Euros per unit cost, for a gross profit of negative 250,000 Euros. We can finally calculate this last outcome as -300,000 Euros. That's the 50,000 Euro up front cost, fixed cost, plus the 250,000 euro negative gross profit. So that's the entire decision tree. Before I move on, I wanna point out two important facts about the construction of decision trees. The first is that, to calculate the profit for IDEA or the outcome, we always move from the root of the tree all the way along the branches leading to an outcome, and add up all of the cash flows associated with those branches. You can see for this bottom branch it's -50,000 euros, minus 250,000 euros, gives idea -300,000 euros gross profit. The second point that I would like to make is that when we look at event nodes, we always want to make sure that the sums of the probabilities add up to one. As always, probabilities are always greater than or equal to zero, and they ought to add up to one. Just looking at the finished tree provides us with some interesting information. If IDEA contracts with no one, it earns no revenue or pays no costs, and it has a net profit of zero for certain. If IDEA orders from supplier S, with a probability of 0.5, it earns a net profit of 150,000 euros when the market is strong. And with a probability of 0.5, it earns a net profit again of 150,000 euros when the market is weak. So, even though there are two outcomes and they're random, both yield the same net profit. Finally, if IDEA orders from supplier P, if the market's strong, IDEA earns a net profit of 450,000 euros, with probably 0.5, and with probability 0.5 the market's weak, and IDEA would lose 300,00 euros. So you can see that ordering from supplier P has a chance of making the most money, but it also has the chance of the largest loss. If we compare the outcomes with supplier S or with ordering from no one, we can see that both provide sure profits. Ordering from no one provides a sure profit of zero, while ordering from supplier S provides a sure profit of 150,000. In that sense, looking at the tree gives us a sense of the risk as [INAUDIBLE] had defined it in week three. But that's just looking at a small decision tree. It turns out that there are systematic ways of evaluating the risks of decisions and the rewards from decisions. And that's what we're gonna turn to next. There are three common approaches for evaluating these options, and they bound the risk posture of the decision maker. First, there's the Maxi-min strategy. What's that? That chooses the action always that maximizes the minimum outcome. By maximizing the minimum outcome, the decision maker is minimizing his or her losses. So that's a risk adverse strategy. It avoids bad outcomes. But notice it doesn't say anything about good outcomes. It ignores the possibility of good outcomes. At the other end of the spectrum, there are maxi-max strategies. Those are actions that maximize the maximum outcome. They seek good outcomes, but then again they completely ignore bad outcomes. So those are risk seeking or gain seeking strategies. In the middle are strategies that maximum the expected values of the outcomes. They give equal weight to good and bad outcomes by calculating the expected value, and we'll come back to that calculation in a moment. There are risk neutral strategies that lie somewhere in between the natural extremes of a maxi-min strategy and maxi-max strategy. We can use IDEA's decision tree to determine each of those strategies, and that's what we're gonna do next. So first, we'll start with maxi-min strategies. And I wanna review for you how we're going to determine the strategy. First, remember that maxi-min decisions maximize the value of the minimum outcome. We'll start at the tree's outcomes and work backwards towards it's root. At each event node, we'll look for the outcome with the minimum value, and we'll replace that event node with that minimum value. At each decision node, we'll choose the action that maximizes the associated values. So we'll be maximizing those minimum values. And we can illustrate that using IDEA's decision tree. Here's the decision tree and remember we're gonna start at the trees outcomes and work backwards towards its root. If we look at the outcomes and start working backwards, we can see the first set of things we hit are event notes. Remember, at each event node we wanna find the outcome with the minimum value. We wanna see how bad things can get. If we look down at the bottom, we can see when the market is weak or strong for supplier P. The worst outcome is when the market is weak and the idea loses 300,000 euros. So we'll replace that event node with the 300,000. Moving up now to supplier S, if the market is weak or the market is strong, IDEA always earns 150,000 euros. So, in this case, the minimum and the maximum value are both 150,000. Finally, there's nothing to do if IDEA contracts with no one. Because we already know that IDEA makes no money. The last step of determining the maxi-min set of decisions is to go and look at the decision nodes. Here there's one last decision node, and we wanna find the action that maximizes the associated value. That decision's who to contract with. And we can see the value maximizing decision is to contract with supplier s. So what we'll do is we'll cut away those two little red lines that are supposed to be cuts on branches of the tree, to indicate that we've chosen contracting with supplier s as the maxi min strategy. The next strategy that we'll demonstrate is the maxi-max set of decisions. Remember, maxi-max set of decisions maximize the value of the maximum outcome. So we want to see how good we can make the best possible outcome. To roll back a maxi-max decision tree, we again start with the tree's outcomes and work backwards towards its root. At each event node, we now find the outcome with the maximum value and we replace the event node with that maximum value. Then at each decision node as before, we choose the action that maximizes the associated value. So let's take a look and see what happens with IDEAS decision tree. Again, we start at the tree's outcomes and we work backwards towards its root. Starting at the outcomes and moving to the left, we see that the first set of nodes are event nodes. We'll start at the bottom. We'll evaluate the decision to contract with supplier P. Remember, at this event node, we going to look for the outcome with the maximum value. And for IDEA, that maximum value is 450,000 euros when the market is strong. So we'll replace that event node. And now we'll move to supplier S. Again, in either case, the outcome is 150,000 euros. So, by contracting with supplier S, the maximum value is 150,000 euros. And finally, we'll move to the decision node, which is whom to contract with, and we'll choose the action that maximizes the associated value. In this case, you can see the maximum value is 450,000 euros, for contracting with supplier P. And so, we'll eliminate the other two choices contracting with no one or contracting with supplier S. That is, the maxi-max strategy is to contract the supplier P. Finally, we're going to determine what the expected value maximizing strategy is. We'll start at the tree's outcomes and work backwards towards its root. At each event node, we'll now calculate the expected value of the outcomes, and we'll replace the event node with that expected value. How do we calculate the expected value? We take each of the outcomes and we weight it by the estimate of the probability that it will occur. At each decision node we'll then choose the action that maximizes the associated value. So let's take a look at how it works for idea. Again we'll start at the tree outcomes and work backwards towards it's root. The first set of nodes we see are event nodes, and first we'll evaluate the event node associated with supplier P. And we'll calculate the expected value. The outcomes are four hundred fifty thousand, and negative three hundred thousand. And we need to weight them by the probabilities that they occur. Point five each. The calculation is that we take .05*(450,000)+ .05*(-300,000) to get an expected value of 75,000 for contracting with supplier p. Well take that expected value and substitute it for the event node and move up to supplier s. The same calculation for supplier s has 150,000 for both outcomes. And it's not hard to see that when we weight both outcomes by 0.5 the expected value is also 150,000. We'll take that expected value and replace the event node by it. And finally we can decide who Idea should contract with to maximize expected value. Here you can see the expected value that highest is 150,000. That's contracting with supplier S. And so we'll eliminate the two other options. The expected value maximizing strategy for idea is to contract with supplier S. So those are the three strategies. Maximin, maximax, and expected value maximizing strategies. The maxi-min strategy was to chose supplier S, and it had a maxi-min value of 150,000 Euros. That is, that was the strategy that would ensure that the worst that could happen would be IDEA would earn 150,000 Euros. The maxi-max strategy was to chose supplier P, and it had a maxi-max value of 450,000 Euros. The maxi-max strategy ensures that idea has the chance of earning up to 450,000 euros. Finally the risk-neutral strategy was to choose supplier S. And again, it had an expected value of 150,000 euros. So you can see the maxi-min strategy and the risk-neutral strategy have the same value. That is the expected value maximizing strategy is also a very safe strategy because it's also maximizing the minimum payout. We've completed the building and the analysis of the decision tree, and it's a good point to review the mechanics of what we do to analyse decision trees. First we construct a decision tree. A decision tree has three parts. It has decision nodes, those are points at which you make choices among options. It has event nodes, those are moments in time when there's a random occurrence. And finally, there are outcomes. They capture all the costs and rewards leading up to each leaf of the tree. Having built the decision tree, we just take a look at it. And looking at the range of outcomes and the probabilities, itself can be instructive. But for a very big tree, it's useful to have a more systematic way of taking a look at the range of possibilities. And to do that we use three classic decision making strategies, to look at risk seeking, risk avoiding and risk neutral strategies. For all three of them, we started at the end with the outcomes, and worked backwards to the root. At event notes, we then calculated either the minimum, the maximum value or the expected value, and that differed with the max/min, max/max, or expected value maximizing strategy. And finally, at decision nodes, we cut away the decision that did not maximize the value. This procedure identifies a range of risk-sensitive strategies from highly risk avoiding max/min strategies, to risk-seeking maxi max strategies, to expected value maximizing strategies that are somewhere in between. When using decision trees, it's also worth keeping the following in mind. The tree that we constructed in this session was quite small. Specifically so that it all could fit on one screen. But in real life decision trees can be very, very large and have many branches and layers of decisions and events. Cash flows in decision trees sometimes stream in over long periods of time. Like years, In that case you need to worry about the discounting of the cash flows. Where do the cash flows and probabilities come from in the first place? Sometimes it comes from past data. For example, maybe IDEA had sold tents similar to the Krusbar in previous years. Sometimes it comes from "expert judgment". But in either case, these are predictions about cash flows and probabilities, and that's a form of "predictive" analytics that we've touched on already in week one. You can even do sensitivity analysis to address shaky data. For example, we might be interested in knowing at which probability we become indifferent between contracting with supplier S, and supplier P. Whether than saying what the probability is, we find out what the break-even probability is. Finally, it's easiest to use a fence that have just a few discreet scenarios. That's what we did this time. There are only two scenarios, a weak market and a strong market. But again, the reality can be more complex. And that's what we're gonna look at in session two this week. Finally, it's worth mentioning that decision trees are widely used in practice. IDEA is just a small example that we've designed to convey the essential ideas. But in practise, decision trees are used to evaluate a really wide range of complex problems. And I'm gonna list just a few of them that you can find published in Interfaces, but they are many many more out there. One example would be in research and development licensing. For example, there's an interfaces article on Phytopharm, which was deciding whether to keep developing its products or to license them. Eventually Phytopharm was actually bought out by another Pxharma company. Another nice example is credit scoring. There's an Interfaces article on Bank One, which was subsequently bought by Chase. And when people apply for credit cards and other forms of credit, a common thing to do is to use decision trees. Is to figure out, whether to accept that person or to not accept the person. Finally, there's a nice article on the eradication of Polio, in which the Center for Disease Control in the United States is using decision trees to figure out what's the best course of action is. There also exists different software packages to help analyse and manage large decision trees. They range from single user products, such as TreePlan, to massive enterprise wide products such as those made by DecisionTools and Logical Decisions. Problems of decision making under uncertainty are all around us. We find them whenever we have to choose among competing actions, and our choices lead to uncertain outcomes. In this session, we look to the simple one stage decision of which supplier to select, along with a simple model of uncertainty in the market outcome. But decisions made under uncertainty can have more complex decisions, and outcomes, and in the next two sessions we'll extend our analysis to cover these cases. That's it for week four and session one, see you at session two.