0:00

Hi, I'm Noah Gans, I'm a Professor in the Department of Operations,

Â Information and 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

Â operations problem of matching your supply with uncertain future demand.

Â Sentil also offered you a look at the first steps

Â of the business analytic 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 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.

Â And that's where we pick 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,

Â to solve more complex problems of decision making under uncertainty.

Â To keep the sessions focused on how the tools can complement 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

IDEAs considering two potential suppliers to manufacture the Krusbar.

Â One's 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 a supplier at all and not sell the Krusbar tent.

Â Here are the statistics for the two suppliers.

Â 3:52

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 capacit, and

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

Â The two suppliers also have different up-front charges to IDEA.

Â If IDEA contracts with supplier S there is no up-front charge.

Â Where as, if IDEA contracts with Supplier P, there's a 50,000 euro upfront charge.

Â 4:27

Unit costs are also different.

Â You can see that Supplier S has a unit cost to IDEA of 120 euros per tent.

Â Whereas Supplier P, has a unit cost of a hundred euros per tent.

Â So, we can compare Supplier S has no upfront charge, but

Â it has a higher unit cost, whereas Supplier P has an upfront charge,

Â but it has a lower unit cost.

Â 4:53

We can use a decision tree to represent IDEA's choices.

Â Before I get in to it, I wanna just review quickly for

Â you, what the elements are of a decision tree.

Â The structures of a decision trees made up of three building blocks.

Â They're 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 Event nodes are represented with circles.

Â Event nodes are points of uncertainty at which the outcome is random.

Â For IDEA, that's whether the market will be weak or strong.

Â 5:35

Finally, outcomes are represented as triangles.

Â Outcomes are payouts that occur, that are due to specific sequences of decisions and

Â 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

Â the outcomes 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 costs.

Â And you can see we have 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 zero 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 zero euros there.

Â 6:35

If IDEA contracts with Supplier P however, it will pay an upfront fixed

Â fee of 50,000 euros, and you can see we've written it on that choice as well.

Â Those are the three choices that IDEA has.

Â Now that we have defined the decision node,

Â the next thing that happens is an event, whether the market is weak or strong.

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 5000 units of capacity.

Â And 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 it would earn 5,000

Â units times 150 euros per unit, but pay unit costs of 120 euros

Â on all 5,000 units for a gross profit of 150,000 euros.

Â 7:45

If on the other hand, the market is weak, that happens with a probability of 0.05.

Â And in that case, again, we can calculate the gross profit.

Â Remember, if the market's weak, demand is only 5,000 units.

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

Â So again, IDEA earns 150 euros per tent,

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

Â times 5,000 tents, unit cost to supplier S for a gross profit of 150,000.

Â And again, now we can calculate the profits for IDEA's outcome,

Â if it orders from supplier S and the market is weak, it's going to earn

Â 150,000 euros, that's no money upfront and 150,000 euros in gross profit.

Â 8:35

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

Â One is if the market is strong, that happens with probability .05.

Â And here the gross profit is calculated knowing that demand

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

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

Â minus 10,000 units times the 100 euro unit cost, 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 up front fixed cost with

Â the 500,000 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, that happens

Â with a probability of .5, and we can calculate these gross profits as well.

Â Remember, when IDEA orders from supplier P,

Â it orders 10,000 units, 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 negative 250,000 Euros.

Â 10:02

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

Â That's the 50,000 Euro up front cost, fixed cost,

Â plus the 250,000 euro negative gross profit.

Â So that's the entire decision tree.

Â Before I move on,

Â I wanna 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 the branches leading to

Â an outcome, and add up all of the cash flows associated with those branches.

Â You can see for this bottom branch it's -50,000 euros,

Â minus 250,000 euros, gives idea -300,000 euros gross profit.

Â The second point that 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 one.

Â As always, probabilities are always greater than or

Â equal to zero, and they ought to add up to one.

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

Â If IDEA contracts with no one, it earns no revenue or

Â pays no costs, and it has a net profit of zero 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 and

Â they're random, 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 probably 0.5, and

Â with probability 0.5 the market's weak, and IDEA would lose 300,00 euros.

Â So you can see that ordering from supplier P has a chance of

Â making the most money, but it also has the chance of the largest loss.

Â If we compare the outcomes with supplier S or with ordering from no one,

Â we can see that both provide sure profits.

Â Ordering from no one provides a sure profit of zero,

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 [INAUDIBLE] had defined it in week three.

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 gonna turn to next.

Â There are three common approaches for evaluating these options, and

Â 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 adverse 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 13:40

In the middle are strategies that maximum the expected values of the outcomes.

Â They give equal weight to good and bad outcomes by calculating the expected

Â value, and we'll come back to that calculation in a moment.

Â There are risk neutral strategies that lie somewhere in between the natural

Â extremes of a maxi-min strategy and maxi-max strategy.

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

Â that's what we're gonna do next.

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 notes.

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

Â We wanna see how bad things can get.

Â If we look down at the bottom, we can see when the market is weak or

Â strong for supplier P.

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

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

Â Moving up now to supplier S, if the market is weak or

Â 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 wanna find the action that maximizes the associated value.

Â That decision's 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 those two

Â little red lines that 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 16:24

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

Â Remember, maxi-max set of 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 a maxi-max decision tree,

Â we again 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 IDEAS 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.

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

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

Â the outcome with the maximum value.

Â 17:28

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

Â So we'll replace that event node.

Â And now 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 value 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,

Â for 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 the 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 we'll then choose the action that maximizes

Â the associated value.

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

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

Â 18:55

The first set of nodes we see are event nodes, and

Â first we'll evaluate the event node associated with supplier P.

Â And we'll calculate the expected value.

Â The outcomes are four hundred fifty thousand, and

Â negative three hundred thousand.

Â And we need to weight them by the probabilities that they occur.

Â Point five each.

Â The calculation is that we take .05*(450,000)+ .05*(-300,000) to

Â 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

Â outcomes by 0.5 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 highest is 150,000.

Â That's contracting with supplier S.

Â 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.

Â Maximin, maximax, and expected value maximizing strategies.

Â The maxi-min strategy was to chose 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 chose supplier P,

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

Â The maxi-max strategy ensures

Â that idea has 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 the maxi-min strategy and

Â the risk-neutral strategy have the same value.

Â That is the expected value maximizing strategy is also

Â a very safe strategy because it's also maximizing the minimum payout.

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 analyse decision trees.

Â First we construct a decision tree.

Â 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 and rewards leading up to each leaf of the tree.

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

Â And looking at the range of outcomes and the probabilities,

Â itself can be instructive.

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

Â a more systematic way of taking a look at the range of possibilities.

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

Â to look at 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 notes, we then calculated either the minimum,

Â the maximum value or the expected value, and that differed with the max/min,

Â max/max, or expected value maximizing strategy.

Â And finally,

Â at decision nodes, we cut away the decision that did not maximize the value.

Â 22:34

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

Â avoiding max/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.

Â Specifically so that it all could 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, 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 tents similar to the Krusbar in previous years.

Â Sometimes it comes from "expert judgment".

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

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

Â 23:58

Finally, it's easiest to use a fence that have just a few discreet 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.

Â And that's what we're gonna look at in session two this week.

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

Â practice.

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

Â But in practise,

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

Â And I'm gonna list just a few of them that you can find published in Interfaces, but

Â they are many many more out there.

Â One example would be in research and development licensing.

Â For example, there's an interfaces article on Phytopharm, which was deciding whether

Â to keep developing its products or to license them.

Â Eventually Phytopharm was actually bought out by another Pxharma company.

Â Another nice example is credit scoring.

Â There's an Interfaces 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 Is 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 decision trees

Â to figure out what's the best course of action is.

Â There also exists different software packages to help analyse 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 look to the 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, see you at session two.

Â