0:00

Hi, I'm Noah Gans.

Â I'm a professor in the Department of Operations, Information, and

Â Decisions at the Wharton School.

Â I'll be your guide for Week Four of our Operations Analytics course.

Â Before we get started though, I wanted to remind you of where we are in the course.

Â In week one,

Â Sentil introduced you to the news vendor problem, which is a fundamental operations

Â problem of matching your supply with uncertain future demand.

Â Sentil also offered you a look at the first steps

Â of the business analytics cycle.

Â The problem of characterizing data with descriptive analytics.

Â Then in weeks two and three, Sergei introduced predictive and

Â prescriptive tools that are helpful when deciding the best course of action

Â when faced with uncertainty.

Â In week two, you saw how you can use optimization methods to find the best

Â course of action when there's little uncertainty regarding the data.

Â 0:55

In week three, you then saw how you can use simulation to evaluate

Â any single course of action when you're faced with more significant uncertainty.

Â And along the way, Sergei introduced you to production and

Â distribution problems often faced in operations.

Â Because of their complexity, Sergei introduced optimization and

Â simulation one at a time.

Â Sometimes, though, you need to optimize in settings with significant uncertainty.

Â This is often called decision making under uncertainty And

Â that's where we picked up in week four.

Â In session one, we'll begin by introducing a new tool that provides a useful way to

Â think about and evaluate decisions made under uncertainty.

Â They're called decision trees.

Â Then in sessions two and three We'll look at how simulation, optimization and

Â decision trees can be used together to solve more complex problems of

Â decision making under uncertainty.

Â To keep the sessions focused on how the tools can compliment each other,

Â we'll use the same example in sessions one to three,

Â that of a Scandinavian furniture retailer that we call Idea.

Â Finally, in session four,

Â we'll go back to the news vendor problem Sentil introduced in week one.

Â And we'll see that we can use the simulation and

Â optimization framework we've developed to tackle the problem.

Â That's our agenda for the week.

Â 3:17

IDEA's considering two potential suppliers to manufacture the Krusbar.

Â One is in Sweden and the other is in Poland.

Â We'll call the Swedish supplier, supplier S and the Polish supplier, supplier P.

Â Each of the suppliers requires that IDEA uses all of its capacity.

Â And IDEA will contract with at most one of the two suppliers.

Â That is, IDEA may decide to use supplier S or to use supplier P.

Â Or, IDEA may decide not to use the supplier at all and

Â not sell the Krusbar tent.

Â Here are the statistics for the two suppliers, they have different capacities.

Â You can see that supplier S has 5,000 units of capacity and

Â if IDEA contracts with it, it will order 5,000 units of the tent.

Â Supplier P has 10,000 units of capacity, and

Â if IDEA contracts with it, it will order 10,000 units of the tent.

Â The two suppliers also have different upfront charges to IDEA.

Â If IDEA contracts with supplier S, there is no upfront charge.

Â Whereas if IDEA contracts with supplier P, there's a 50,000 euro upfront charge.

Â 4:53

We can use a decision tree to represent ideas, choices.

Â Before I get into it, I want to just review quickly for

Â you what the elements are of a decision tree.

Â The structure of a decision tree is made up of three building blocks.

Â There are decision nodes, which we'll represent with squares.

Â Decision nodes are points at which a decision-maker has to decide on an action.

Â For IDEA, those actions are which supplier to select, if any.

Â 5:35

Finally, outcomes are represented as triangles.

Â Outcomes are payouts that occur,

Â that are due to specific sequences of decisions and events.

Â For IDEA, the outcomes are its profits,

Â they depend on which supplier IDEA chooses, S, P, or none.

Â And it depends on whether the market is weak or strong.

Â Now that we've defined the decision nodes, the event nodes, and the outcomes,

Â we'll build the tree.

Â The first thing that we need to do to build the tree is include a decision.

Â With whom will IDEA contract?

Â If IDEA contracts with no one, it earns no revenue and pays no cost.

Â And you can see we've put a cash flow of zero euros.

Â And that's the end of that part of the tree.

Â The outcome for IDEA in that case, would be to earn 0 Euros.

Â A second possible decision, would be to contract with supplier S.

Â If IDEA contracts with supplier S, it pays no upfront fixed fee,

Â so we've written 0 Euros there.

Â 6:55

If the market is strong, and

Â IDEA contracts with Supplier S, that happens with a probability of 0.5.

Â In that case, we can calculate the gross profit.

Â Remember that Supplier S has 5,000 units of capacity and

Â a strong market has 10,000 units of demand.

Â So IDEA would be capacity limited.

Â It would sell everything that it had ordered and

Â it would earn 5,000 units times 150 Euros per unit.

Â But pay unit cost of a 120 Euros on all 5,000 units

Â with gross profit of 150,000 Euros.

Â 7:59

At the same time, supplier S only supplies 5,000 units to IDEA.

Â So again, IDEA earns 150 euros per tent times 5,000 tents revenue.

Â And it pays 120 euros per tent times 5,000 tents

Â unit cost to supplier S for a gross profit of 150,000.

Â Again, now we can calculate the profits for IDEA's outcome.

Â If it orders from supplier S and the market is weak,

Â it's going to earn 150,000 euros.

Â That's no money up front, and 150,000 euros of gross profit.

Â 8:36

Finally, if IDEA contracts with supplier P, there are again two outcomes.

Â One is if the market is strong, that happens with probability 0.5.

Â And here the gross profit is calculated knowing that demand is 10,000 units.

Â And IDEA has ordered 10,000 units from supplier P.

Â So the gross profit is 10,000 units times the 150 euro

Â revenue minus 10,000 units times the 100 euro unit cost.

Â For a gross profit of 500,000 euros.

Â Now that we've got the gross profit, we can calculate the total profits for IDEA.

Â If it orders from supplier P and the market is strong.

Â By adding the -50,000 upfront fixed cost with the 500,000

Â euro gross profit to get a net profit of 450,000 euros.

Â 9:29

If IDEA orders from supplier P and the market is weak.

Â That happens with a probability of 0.5.

Â And we can calculate these gross profits as well.

Â Remember, when IDEA orders from supplier P, it orders 10,000 units.

Â But when the market is weak, it only sells 5,000 units.

Â So the gross profit is 5,000 units sold times 150 euros per unit

Â minus 10,000 units ordered times 100 euros per unit cost.

Â For a gross profit of -250,000 euros.

Â 10:02

We can finally calculate this last outcome as -300,000 euros.

Â That's the 50,000 euro upfront fixed cost plus

Â the 250,000 euro negative gross profit.

Â So that's the entire decision tree.

Â Before I move on,

Â I want to point out two important facts about the construction of decision trees.

Â The first is that to calculate the profit for IDEA, or the outcome.

Â We always move from the root of the tree all the way along

Â the branches leading to an outcome.

Â And add up all of the cash flows associated with those branches.

Â And you can see, for this bottom branch, it's -50,000 euros minus

Â 250,000 euros gives IDEA -300,000 euros gross profit.

Â The second point I would like to make is that when we look at event nodes.

Â We always want to make sure that the sums of the probabilities add up to 1.

Â As always, probabilities are all greater than or equal to 0 and

Â they all add up to 1.

Â Just looking at the finished tree provides us with some interesting information.

Â If IDEA contracts with no one, it earns no revenue, it pays no costs.

Â And it has a net profit of 0 for certain.

Â 11:23

If IDEA orders from supplier S with a probability of 0.5.

Â It earns a net profit of 150,000 euros when the market is strong.

Â And with a probability of 0.5,

Â it earns a net profit again of 150,000 euros when the market is weak.

Â So even though there are two outcomes in the random,

Â both yield the same net profit.

Â Finally, if IDEA orders from supplier P, if the market's strong.

Â IDEA earns a net profit of 450,000 euros with probability 0.5.

Â And with probability 0.5, the market's weak, and IDEA would lose 300,000 euros.

Â So you can see that ordering from supplier P has a chance of making the most money.

Â But it also has the chance of the largest loss.

Â If we compare the outcomes with supplier S or with

Â ordering from no one, you can see that both provide sure profits.

Â Ordering from no one provides a sure profit of 0.

Â While ordering from supplier S provides a sure profit of 150,000.

Â In that sense, looking at the tree gives us a sense of the risk,

Â as Sergei had defined it in Week 3.

Â But that's just looking at a small decision tree.

Â It turns out that there are systematic ways of evaluating

Â the risks of decisions and the rewards from decisions.

Â And that's what we're going to turn to next.

Â There are three common approaches for evaluating these options.

Â And they bound the risk posture of the decision maker.

Â First, there's the Maxi-min strategy.

Â What's that?

Â That chooses the action always that maximizes the minimum outcome.

Â By maximizing the minimum outcome, the decision maker is minimizing his or

Â her losses.

Â So that's a risk averse strategy.

Â It avoids bad outcomes.

Â But notice it doesn't say anything about good outcomes.

Â It ignores the possibility of good outcomes.

Â At the other end of the spectrum, there are Maxi-max strategies.

Â Those are actions that maximize the maximum outcome.

Â 13:31

They seek good outcomes, but again, they completely ignore bad outcomes.

Â So those are risk seeking or gain seeking strategies.

Â In the middle are strategies that maximize the expected values of the outcomes.

Â They give equal weight to good and

Â bad outcomes by calculating the expected value.

Â And we'll come back to that calculation in a moment.

Â They are risk neutral strategies that lie somewhere in between

Â the natural extremes of a Maxi-min strategy and a Maxi-max strategy.

Â We can use IDEA's decision tree to determine each of those strategies.

Â And that's what we're going to do next.

Â So first, we'll start with Maxi-min strategies.

Â And I want to review for you how we're going to determine the strategy.

Â First, remember that Maxi-min decisions maximize the value of the minimum outcome.

Â We'll start at the tree's outcomes and work backwards towards its root.

Â 14:58

If we look at the outcomes and start working backwards.

Â We can see the first set of things we hit are event nodes.

Â Remember, at each event node, we want to find the outcome with the minimum value.

Â We want to see how bad things can get.

Â If we look down at the bottom,

Â we can see when the market is weak or strong for supplier P.

Â The worst outcome is when the market is weak, and IDEA loses 300,000 euros.

Â So we'll replace that event node with the 300,000.

Â Moving up now to supplier S.

Â If the market is weak or the market is strong, IDEA always earns 150,000 euros.

Â So, in this case, the minimum and the maximum value are both 150,000.

Â Finally, there's nothing to do if IDEA contracts with no one.

Â Because we already know that IDEA makes no money.

Â 15:49

The last step of determining the Maxi-min set of decisions is to go and

Â look at the decision nodes.

Â Here, there's one last decision node and

Â we want to find the action that maximizes the associated value.

Â That decisions who to contract with.

Â And we can see the value maximizing decision is to contract with supplier S.

Â So, what we'll do is, we'll cut away.

Â Those two little red lines are supposed to be cuts on branches of the tree

Â to indicate that we've chosen contracting with supplier S as the maxi-min strategy.

Â 16:25

The next strategy that we'll demonstrate is the maxi-max set of decisions.

Â Remember, maxi-max decisions maximize the value of the maximum outcome.

Â So, we want to see how good we can make the best possible outcome.

Â To roll back maxi-max decision tree, we again,

Â start with the tree's outcomes and work backwards towards its root.

Â At each event node, we now find the outcome with the maximum value, and

Â we replace the event node with that maximum value.

Â Then, at each decision node as before,

Â we choose the action that maximizes the associated value.

Â 17:02

So, let's take a look and see what happens with IDEA's decision tree.

Â Again, we start at the tree's outcomes and we work backwards towards its root.

Â Starting at the outcomes and moving to the left,

Â we see that the first set of nodes are event nodes.

Â We'll start at the bottom and

Â we'll evaluate the decision to contract with supplier P.

Â Remember, at this event node, we're going to look for

Â the outcome with the maximum value.

Â And for IDEA, that maximum value is 450,000 euros when the market is strong.

Â So, we'll replace that event node and then we'll move to supplier S.

Â Again, in either case the outcome is 150,000 euros.

Â So by contracting with supplier S, the maximum is 150,000 euros.

Â And finally, we'll move to the decision node, which is whom to contract with,

Â and we'll choose the action that maximizes the associated value.

Â In this case, you can see the maximum value is 450,000 euros for

Â contracting with supplier P.

Â And so, we'll eliminate the other two choices, contracting with no one or

Â contracting with supplier S.

Â That is the maxi-max strategy is to contract with supplier P.

Â 18:18

Finally, we're going to determine what the expected value maximizing strategy is.

Â We'll start at the tree's outcomes and work backwards towards its root.

Â At each event node, we'll now calculate the expected value of the outcomes, and

Â we'll replace the event node with that expected value.

Â How do we calculate the expected value?

Â We take each of the outcomes and

Â we weight it by the estimate of the probability that it will occur.

Â At each decision node,

Â we'll then choose the action that maximizes the associated value.

Â So, let's take a look at how it works for IDEA.

Â Again, we'll start at the tree's outcomes and work backwards towards its root.

Â 18:55

The first set of nodes we see are event nodes.

Â And first, we'll evaluate the event node associated with supplier P, and

Â we'll calculate the expected value.

Â The outcomes are 450,000 and -300,000, and

Â we need to weight them by the probabilities that they occur, 0.5 each.

Â The calculation is that we take 0.5 * (+450,000) + 0.5

Â * (-300,000) to get an expected value of 75,000 for contracting with supplier P.

Â Well, take that expected value and substitute it for

Â the event node and move up to supplier S.

Â The same calculation for supplier S has 150,000 for both outcomes.

Â And it's not hard to see that when we weight both outcomes by 0.5,

Â the expected value is also 150,000.

Â We'll take that expected value and replace the event node by it.

Â And finally, we can decide who IDEA should contract with to maximize expected value.

Â Here, you can see the expected value that's highest is 150,000.

Â That's contracting with supplier S, and so we'll eliminate the two other options.

Â The expected value maximizing strategy for IDEA is to contract with supplier S.

Â 20:13

So those are the three strategies, maxi-min,

Â maxi-max and expected value maximizing strategies.

Â The maxi-min strategy was to choose supplier S, and

Â it had a maxi-min value of 150,000 euros.

Â That is, that was the strategy that would ensure that the worst that could happen

Â would be IDEA would earn 150,000 euros.

Â The maxi-max strategy was to choose supplier P, and

Â it had a maxi-max value of 450,000 euros.

Â The maxi-max strategy ensures that IDEA has

Â the chance of earning up to 450,000 euros.

Â Finally, the risk-neutral strategy was to choose supplier S.

Â And again, it had an expected value of 150,000 euros.

Â So, you can see maxi-min strategy and the risk-neutral strategy have the same value.

Â That is, the expected value maximizing strategy is also a very safe strategy,

Â because it's also maximizing the minimum path.

Â We've completed the building and the analysis of the decision tree, and

Â it's a good point to review the mechanics of what we do to analyze decision trees.

Â First, we construct a decision tree.

Â And a decision tree has three parts.

Â It has decision nodes, those are points at which you make choices among options.

Â It has event nodes, those are moments in time when there's a random occurrence.

Â And finally, there are outcomes.

Â They capture all the costs from awards leading up to each leaf of the tree.

Â Having built the decision tree, we can just take a look at it, and

Â looking at the range of outcomes and the probabilities itself can be constructive.

Â But for a very big tree, it's useful to have a more systematic way

Â of taking a look at the range of possibilities.

Â And to do that, we use three classic decision-making strategies to look at

Â risk-seeking, risk-avoiding and risk-neutral strategies.

Â For all three of them, we started at the end with the outcomes and

Â worked backwards to the root.

Â At event nodes, we then calculated either minimum,

Â the maximum value or the expected value.

Â And that differed with the maxi-min, maxi-max or

Â expected value maximizing strategy.

Â And finally, at decision nodes,

Â we cut away the decisions that did not maximize the value.

Â 22:34

This procedure identifies a range of risk-sensitive strategies from highly

Â risk-avoiding maxi-min strategies, to risk-seeking maxi-max strategies,

Â to expected value maximizing strategies that are somewhere in between.

Â When using decision trees, it's also worth keeping the following in mind.

Â The tree that we constructed in this session was quite small, specifically so

Â that it could all fit on one screen.

Â But in real life, decision trees can be very, very large and

Â have many branches and layers of decisions and events.

Â Cash flows in decision trees sometimes stream in over long periods of time,

Â like years.

Â In that case, you need to worry about the discounting of the cash flows.

Â Where do the cash flows and probabilities come from in the first place?

Â Sometimes, it comes from past data.

Â For example, maybe IDEA had sold ten similar to the in previous years.

Â Sometimes, it comes from expert judgement.

Â But, in either case, these are predictions about cash flows and probabilities.

Â And that's a form of predictive analytics that we've touched on already in week one.

Â You can even do sensitivity analysis to address shaky data.

Â For example, we might be interested in knowing at which probability we become

Â indifferent between contracting with supplier S and supplier P.

Â Rather than saying what the probability is,

Â we find out what the break-even probability is.

Â Finally, it's easiest to use events that have just a few discrete scenarios.

Â That's what we did this time.

Â There are only two scenarios, a weak market and a strong market.

Â But again, the reality can be more complex, and

Â that's what we're going to look at in Session 2 this week.

Â Finally, it's worth mentioning that decision trees are widely used in

Â practice.

Â IDEA's just a small example that we've designed to convey the essential ideas.

Â But in practice,

Â decision trees are used to evaluate a really wide range of complex problems.

Â And I'm going to list just a few of them that you can find published in interfaces,

Â but there are many, many more out there.

Â One example would be in research development licensing.

Â For example, this interfaces article on Phytopharm, which was deciding

Â whether to keep developing its products or to license them.

Â Eventually, Phytopharm actually bought another pharma company.

Â Another nice example is credit scoring.

Â There's an interface article on Bank One, which was subsequently bought by Chase.

Â And when people apply for credit cards and

Â other forms of credit, a common thing to do is to use decision trees

Â to figure out whether to accept that person or to not accept the person.

Â Finally, there's a nice article on the eradication of polio,

Â in which the Center for Disease Control in the United States is using

Â the decision trees to figure out what the best course of action is.

Â There also exists different software packages to help analyze and

Â manage large decision trees.

Â They range from single user products, such as TreePlan, to massive enterprise

Â wide products, such as those made by DecisionTools and Logical Decisions.

Â 25:36

Problems of decision-making under uncertainty are all around us.

Â We find them whenever we have to choose among competing actions, and

Â our choices lead to uncertain outcomes.

Â In this session, we looked at a simple one stage decision of which supplier to

Â select, along with a simple model of uncertainty in the market outcome.

Â But decisions made under uncertainty can have more complex decisions and outcomes.

Â And in the next two sessions, we'll extend our analysis to cover these cases.

Â That's it for week four and session one.

Â See you at session two.

Â