代写留学生作业-决策树之产品决策研究-DecisionTrees

代写留学生作业，决策树之产品决策研究（DecisionTrees），决策树分析法是企业管理中常用的一种利于展示结构的重要方法。决策树很擅长处理非数值型数据，这与神经网络只能处理数值型数据比起来，就免去了很多数据预处理工作。甚至有些决策树算法专为处理非数值型数据而设计，因此当采用此种方法建立决策树同时又要处理数值型数据时，反而要做把数值型数据映射到非数值型数据的预处理。DecisionTrees
The analysis of complex decisions with signi¯ cant uncertainty can be confusingbecause 1) the consequence that will result from selecting any speci¯ ed decisionalternative cannot be predicted with certainty, 2) there are often a large numberof di® erent factors that must be taken into account when making the decision,3) it may be useful to consider the possibility of reducing the uncertainty in thedecision by collecting additional information, and 4) a decision maker's attitudetoward risk taking can impact the relative desirability of di® erent alternatives.This chapter reviews decision tree analysis procedures for addressing such complexities.
1.1 Decision TreesTo illustrate the analysis approach, a decision tree is used in the following exampleto help make a decision.
Example 1.1
Product decision. To absorb some short-term excess production capacityat its Arizona plant, Special Instrument Products is considering a short manufacturingrun for either of two new products, a temperature sensor or a pressuresensor. The market for each product is known if the products can be successfullydeveloped. However, there is some chance that it will not be possible to
successfully develop them.Revenue of $1,000,000 would be realized from selling the temperature sensor
and revenue of $400,000 would be realized from selling the pressure sensor. Bothof these amounts are net of production cost but do not include development cost.If development is unsuccessful for a product, then there will be no sales, and thedevelopment cost will be totally lost. Development cost would be $100,000 forthe temperature sensor and $10,000 for the pressure sensor.
2 CHAPTER 1 DECISION TREES
Figure 1.1 Special Instrument Products decision
Question 1.1: Which, if either, of these products should Special
Instrument Products attempt to develop?
To answer Question 1.1 it is useful to represent the decision as shown in Figure
1.1. The tree-like diagram in this ¯ gure is read from left to right. At the left,indicated with a small square, is the decision to select among the three availablealternatives, which are 1) the temperature sensor, 2) the pressure sensor, or 3) neither.
The development costs for the \develop temperature sensor" and \developpressure sensor" alternatives are shown on the branches for those alternatives.At the right of the development costs are small circles which represent the uncertaintyabout whether the development outcome will be a success or a failure. Thebranches to the right of each circle show the possible development outcomes. Onhe branch representing each possible development outcome, the sales revenue isshown for the alternative, assuming either success or failure for the development.
http://www.1daixie.com/liuxueshengzuoye/Finally, the net pro¯ t is shown at the far right of the tree for each possible combinationof development alternative and development outcome. For example, thetopmost result of $900,000 is calculated as $1; 000; 000 ¡ $100; 000 = $900; 000.
(Pro¯ ts with negative signs indicate losses.)The notation used in Figure 1.1 will be discussed in more detail shortly, but
for now concentrate on determining the alternative Special Instrument Productsshould select. We can see from Figure 1.1 that developing the temperature sensorcould yield the largest net pro¯ t ($900,000), but it could also yield the largestloss ($100,000). Developing the pressure sensor could only yield a net pro¯ t
1.1 DECISION TREES 3
of $390,000, but the possible loss is limited to $10,000. On the other hand,not developing either of the sensors is risk free in the sense that there is nopossibility of a loss. However, if Special Instrument Products decides not toattempt to develop one of the sensors, then the company is giving up the potentialopportunity to make either $900,000 or $390,000. Question 1.1 will be answeredin a following example after we discuss the criterion for making such a decision.You may be thinking that the decision about which alternative is preferred dependson the probabilities that development will be successful for the temperatureor pressure sensors. This is indeed the case, although knowing the probabilitieswill not by itself always make the best alternative in a decision immediately clear.However, if the outcomes are the same for the di® erent alternatives, and only theprobabilities di® er, then probabilities alone are su± cient to determine the best
alternative, as illustrated by Example 1.2.
Example 1.2
Tossing a die. Suppose you are o® ered two alternatives, each of whichconsists of a single toss of a fair die. With the ¯ rst alternative, you will win
$10 if any number greater than 4 is thrown, and lose $5 otherwise. With thesecond alternative, you will win $10 if any number greater than 3 is thrown, andlose $5 otherwise. In this case, since there are 6 faces on a die, the probability
of winning is 2=6 = 1=3 for the ¯ rst alternative and 3=6 = 1=2 for the secondalternative. Since the consequences are the same for both alternatives and theprobability of winning is greater for the second alternative, you should select the
second alternative.However, the possible outcomes are often not the same in realistic business
decisions and this causes additional complexities, as illustrated by further considerationof the Special Instruments Product decision in Example 1.3.Example 1.3
Product decision. Suppose that in Example 1.1 the probability of developmentsuccess is 0.5 for the temperature sensor and 0.8 for the pressure sensor.Figure 1.2 is a diagram with these probabilities shown in parentheses on the
branches representing the possible outcomes for each sensor development e® ort.(While only the probability of success is speci¯ ed for each development e® ort,the probability of failure is determined by the rules of probability since the probabilities
of development success and development failure must add up to one.)A study of Figure 1.2 shows that having the probabilities does not resolve thisdecision for us. The ¯ gure shows that although the probability of developmentsuccess is considerably lower for the temperature sensor than it is for the pressuresensor (0.5 versus 0.8), the net pro¯ t from successful development of the temperaturesensor is considerably higher than the net pro¯ t from successful development
4 CHAPTER 1 DECISION TREES
Success
Failure
Success
Failure
Figure 1.2 Special Instrument Products decision tree
代写留学生作业of the pressure sensor ($900,000 versus $390,000). That is, the alternative withthe higher potential payo® has a lower probability that thispayo® will actuallybe realized.
The resolution of this decision dilemma is addressed in the next section, butbefore doing this, De¯ nition 1.1 clari¯ es the notation in Figures 1.1 and 1.2.
De¯ nition 1.1: Decision tree notation
A diagram of a decision, as illustrated in Figure 1.2, is called a decisiontree. This diagram is read from left to right. The leftmost node in adecision tree is called the root node. In Figure 1.2, this is a smallsquare called a decision node. The branches emanating to the rightfrom a decision node represent the set of decision alternatives that areavailable. One, and only one, of these alternatives can be selected. Thesmall circles in the tree are called chance nodes. The number shownin parentheses on each branch of a chance node is the probability thatthe outcome shown on that branch will occur at the chance node. Theright end of each path through the tree is called an endpoint, and eachendpoint represents the ¯ nal outcome of following a path from the rootnode of the decision tree to that endpoint.
1.2 EXPECTED VALUE 5

Figure 1.3 Die toss decision tree
1.2 Expected Value
In order to decide which alternative to select in a decision problem, we needa decision criterion; that is, a rule for making a decision. Expected value is acriterion for making a decision that takes into account both the possible outcomesfor each decision alternative and the probability that each outcome will occur.To illustrate the concept of expected value, we consider a simpler decision withlower stakes than the Special Instrument Products decision.
Example 1.4
Rolling a die. A friend proposes a wager: You will pay her $9.00, and then afair die will be rolled. If the die comes up a 3, 4, 5, or 6, then your friend will payyou $15.00. If the die comes up 1 or 2, she will pay you nothing. Furthermore,
your friend agrees to repeat this game as many times as you wish to play.Question 1.2: Should you agree to play this game?
If a six-sided die is fair, there is a 1/6 probability that any speci¯ ed side willcome up on a roll. Therefore there is a 4/6 (= 2=3) probability that a 3, 4, 5, or
6 will come up and you will win. Figure 1.3 shows the decision tree for one playof this game.
At ¯ rst glance, this may not look like a good bet since you can lose $9.00,while you can only win $6.00. However, the probability of winning the $6.00 is2/3, while the probability of losing the $9.00 is only 1/3. Perhaps this isn't sucha bad bet after all since the probability of winning is greater than the probabilityof losing.
The key to logically analyzing this decision is to realize that your friend will letyou play this game as many times as you want. For example, how often would youexpect to win if you play the game 1,500 times? Based on what you have learned
6 CHAPTER 1 DECISION TREESabout probability, you know that the proportion of games in which you will winover the long run is approximately equal to the probability of winning a singlegame. Thus, out of the 1,500 games, you would expect to win approximately
(2=3)£1; 500 = 1; 000 times. Therefore, over the 1,500 games, you would expectto win a total of approximately 1; 000£$6+500£(¡ $9) = $1; 500. So this gamelooks like a good deal!
Based on this logic, what is each play of the game worth? Well, if 1,500plays of the game are worth $1,500, then one play of the game should be worth
$1; 500=1; 500 = $1:00. Putting this another way, you will make an average of$1.00 each time you play the game.A little thought about the logic of these calculations shows that you can directlydetermine the average payo® from one play of the game by multiplyingeach possible payo® from the game by the probability of that payo® , and thenadding up the results. For the die tossing game, this calculation is (2=3)£$6 +(1=3)£(¡ $9) = $1.
The quantity calculated in the manner illustrated in Example 1.4 is called theexpected value for an alternative, as shown in De¯ nition 1.2. Expected valueis often a good measure of the value of an alternative since over the long run thisis the average amount that you expect to make from selecting the alternative.
De¯ nition 1.2: Expected ValueThe expected value for an uncertain alternative is calculated by multiplyingeach possible outcome of the uncertain alternative by its probability,and summing the results. The expected value decision criterion
selects the alternative that has the best expected value. In situationsinvolving pro¯ ts where \more is better," the alternative with the highestexpected value is best, and in situations involving costs, where \lessis better," the alternative with the lowest expected value is best.
Example 1.5
Product decision. The expected values for the Special Instrument Productsdecision are designated by \EV" in Figure 1.4. These are determinedas follows: 1) For the temperature sensor alternative, 0:5 £$900; 000 + 0:5 £ (¡ $100; 000) = $400; 000, 2) for the pressure sensor alternative, 0:8£$390; 000+
0:2 £(¡ $10; 000) = $310; 000, and 3) for doing neither of these $0. Thus, the
alternative with the highest expected value is developing the temperature sensor,
and if the expected value criterion is applied, then the temperature sensor should
be developed.
1.2 EXPECTED VALUE 7
Success
Failure
Success
Failure
$100,000
$10,000
$1,000,000
$400,000
$0
$0
Neither
-$100,000
$900,000
$390,000
-$10,000
$0
Development
Cost
Development
Outcome
Sales
Revenue
Net
Profit
(0.5)
(0.5)
(0.8)
(0.2)
EV=$400,000
EV=$310,000
EV=$400,000
EV=$0
Temperature
Sensor
Pressure
Sensor
Figure 1.4 Special Instrument Products decision tree, with expected values
Figure 1.4 illustrates some additional notation that is often used in decisiontrees. The branches representing the two alternatives that are less preferred areshown with crosshatching (//) on their branches. The expected value for each
chance node is designated by \EV". Finally, the expected value at the decisionnode on the left is shown as equal to the expected value of the selected alternative.Xanadu Traders
We conclude this section by analyzing a decision involving international commerce.This example will be extended in the remainder of this chapter
Example 1.6Xanadu Traders. Xanadu Traders, a privately held U.S. metals broker, hasacquired an option to purchase one million kilograms of partially re¯ ned molyzirconiumore from the Zeldavian government for $5.00 per kilogram. Molyzirconiumcan be processed into several di® erent products which are used in semiconductormanufacturing, and George Xanadu, the owner of Xanadu Traders,estimates that he would be able to sell the ore for $8.00 per kilogram after importingit. However, the U.S. government is currently negotiating with Zeldaviaover alleged \dumping" of certain manufactured goods which that country exportsto the United States. As part of these negotiations, the U.S. governmenthas threatened to ban the import from Zeldavia of a class of materials that includesmolyzirconium. If the U.S. government refuses to issue an import license8 CHAPTER 1 DECISION TREES
for the molyzirconium after Xanadu has purchased it, then Xanadu will have to
pay a penalty of $1.00 per kilogram to the Zeldavian government to annul the
purchase of the molyzirconium.Xanadu has used the services of Daniel A. Analyst, a decision analyst, to helpin making decisions of this type in the past, and George Xanadu calls on him toassist with this analysis. From prior analyses, George Xanadu is well-versed indecision analysis terminology, and he is able to use decision analysis terms in hisdiscussion with Analyst.
Analyst: As I understand it, you can buy the one million kilograms of molyzirconiumore for $5.00 a kilogram and sell it for $8.00, which gives a pro¯ t of($8:00 ¡ $5:00)£1; 000; 000 = $3; 000; 000. However, there is some chance that
you cannot obtain an import license, in which case you will have to pay $1.00per kilogram to annul the purchase contract. In that case, you will not haveto actually take the molyzirconium and pay Zeldavia for it, but you will lose
$1:00£1; 000; 000 = $1; 000; 000 due to the cost of annulling the contract.Xanadu: Actually, \some chance" may be an understatement. The internalpolitics of Zeldavia make it hard for their government to agree to stop selling
their manufactured goods at very low prices here in the United States. Thechances are only ¯ fty-¯ fty that I will be able to obtain the import license. As you
know, Xanadu Traders is not a very large company. The $1,000,000 loss wouldbe serious, although certainly not fatal. On the other hand, making $3,000,000would help the balance sheet: : :
Question 1.3: Which alternative should Xanadu select? Assumethat Xanadu uses expected value as his decision criterion.
To answer this question, construct a decision tree. There are two possiblealternatives, purchase the molyzirconium or don't purchase it. If the molyzirconiumis purchased, then there is uncertainty about whether the import license
will be issued or not. The decision tree is shown in Figure 1.5. Starting from theroot node for this tree, it costs $5 million to purchase the molyzirconium, and ifthe import license is issued, then the molyzirconium will be sold for $8 million,
yielding a net pro¯ t of $3 million. On the other hand, if the import license is notissued then Xanadu will recover $4 million of the $5 million that it invested, butwill lose the other $1 million due to the cost of annulling the contract.
The endpoint net pro¯ ts are shown in millions of dollars, and the expectedvalue for the \purchase" alternative is 0:5£$3 + 0:5 £(¡ $1) = $1, in millionsof dollars. Therefore, if expected value is used as the decision criterion, then thepreferred alternative is to purchase the molyzirconium.
1.3 DEPENDENT UNCERTAINTIES 9
$0
Cost
Yes
No
Purchase
Revenue
$5
(0.5)
(0.5)
$8 $3
$4 -$1
EV=$1
Don't
Purchase
Import License
Issued?
Net
Profit
Figure 1.5 Xanadu Traders initial decision tree (dollar amounts in millions)
1.3 Dependent Uncertainties
In this section, we consider an additional complexity that often occurs in businessdecisions: dependent uncertainties. Speci¯ cally, we will examine a case thatillustrates this complexity of real world decisions, and review a procedure foranalyzing decisions that include dependent uncertainties.
Example 1.7Xanadu Traders. This is a continuation of Example 1.6. We now consider
an expanded version of the decision that includes dependent uncertainties and
extend the analysis procedure to handle this new issue. We continue to follow the
discussion between Daniel Analyst and George Xanadu that started in Example
1.6.
Analyst: Maybe there is a way to reduce the risk. As I understand it, thereason you need to make a quick decision is that Zeldavia has also o® ered thisdeal to other brokers, and one of them may take it before you do. Is that reallyvery likely? Perhaps you can apply for the import license and wait until youknow whether it is approved before closing the deal with Zeldavia.Xanadu: That's not very likely. Some of those brokers are pretty big operators,
and dropping $1,000,000 wouldn't make them lose any sleep. I'd say there is a0.70 probability that someone else will take Zeldavia's o® er if I wait until theimport license comes through. Of course, it doesn't cost anything to apply foran import license, so maybe it is worth waiting to see what happens: : :
10 CHAPTER 1 DECISION TREES
$0
$5
Cost
(0.5)
(0.5)
EV=$1
Yes
No
Import License
Issued?
(0.5)
(0.5)
Yes
No
$3
Net
Profit
Don't
Purchase
-$1
(0.3)
(0.7)
$3
$0
$0
Purchase
EV=$0.45
Wait
$0
EV=$1
Still
Available?
Yes
No
Cost Revenue
$8
$4
$5 $8
EV=$0.9
Figure 1.6 Xanadu Traders revised decision tree, with expected values
Question 1.4: Should Xanadu Traders wait to see if an importlicense is issued before purchasing the molyzirconium?
The decision tree for this revised problem is shown in Figure 1.6. The twoalternatives at the top of this tree (\purchase" and \don't purchase") are the sameas the alternatives shown in Figure 1.5. The third alternative (\wait") considersthe situation where Xanadu waits to see whether it can obtain an import licensebefore purchasing the molyzirconium. This alternative introduces a new analysisissue that must be addressed before the expected value for this alternative canbe determined. This new issue concerns the fact that there are two stages ofuncertainty for this alternative. First, the issue of an import license is resolved,and then there is a further uncertainty about whether the molyzirconium willstill be available.Question 1.5: What is the expected value for the \wait" alternative?
The process of determining the expected value for this alternative involves twostages of calculation. In particular, it is necessary to start at the right side of thedecision tree, and carry out successive calculations working toward the root nodeof the tree. Speci¯ cally, ¯ rst determine the expected value for the alternativeassuming that the import license is issued, and then use this result to calculate
1.4 SEQUENTIAL DECISIONS 11
the expected value for the \wait" alternative prior to learning whether the import
license is issued.Examine Figure 1.6 to see how this calculation process works. As this ¯ gureshows, if the import license is issued, then there is a 0.3 probability that themolyzirconium will still be available. In this case, Xanadu will pay $5 million
for the molyzirconium, and sell it for $8 million realizing $3 million in net pro¯ t.If the molyzirconium is not still available, then Xanadu will not have to payanything and will realize no net pro¯ t. Thus, the expected value for the situation
after the uncertainty about the import license has been resolved is 0:3£$3+0:7£ $0 = $0:9. This expected value is shown next to the lower right chance node onthe decision tree in Figure 1.6.From the discussion regarding expected value in Section 1.2, it follows that
this $0.9 million is the value of the alternative once the result of the importlicense application is known. Hence, this value should be used in the furtherexpected value calculation needed to determine the overall value of the \wait"
alternative. Thus, the expected value for the \wait" alternative is given by 0:5£ $0:9 + 0:5 £$0 = $0:45. This expected value is shown next to the lower left
chance node on the decision tree in Figure 1.6. Since the expected value for the
\wait" alternative is less than the $1 million expected value for purchasing the
molyzirconium right now, this alternative is less preferred than purchasing the
molyzirconium right now. Xanadu should not wait, assuming that expected value
is used as the decision criterion.
The process of sequentially determining expected values when there are dependent
uncertainties in a decision tree, as demonstrated in Example 1.6, is called
decision tree rollback. This term is de¯ ned in De¯ nition 1.3.
De¯ nition 1.3: Decision Tree Rollback
The process of successively calculating expected values from the endpoints
of the decision tree to the root node, as demonstrated in this
section, is called a decision tree rollback.
1.4 Sequential Decisions
In addition to dependent uncertainties, real business decisions often include sequential
decisions. This section considers an example that demonstrates how
to address sequential decisions.
12 CHAPTER 1 DECISION TREES
Example 1.8
ABC Computer Company. ABC Computer Company is considering submission
of a bid for a government contract to provide 10,000 specialized computers
for use in computer-aided design. There is only one other potential bidder for
this contract, Complex Computers, Inc., and the low bidder will receive the contract.
ABC's bidding decision is complicated by the fact that ABC is currently
working on a new process to manufacture the computers. If this process works as
hoped, then it may substantially lower the cost of making the computers. However,
there is some chance that the new process will actually be more expensive
than the current manufacturing process. Unfortunately, ABC will not be able to
determine the cost of the new process without actually using it to manufacture
the computers.
If ABC decides to bid, it will make one of three bids: $9,500 per computer,
$8,500 per computer, or $7,500 per computer. Complex Computers is certain
to bid, and it is equally likely that Complex will bid $10,000, $9,000, or $8,000
per computer. If ABC decides to bid, then it will cost $1,000,000 to prepare the
bid due to the requirement that a prototype computer be included with the bid.
This $1,000,000 will be totally lost regardless of whether ABC wins or loses the
bidding competition.
With ABC's current manufacturing process, it is certain to cost $8,000 per
computer to make each computer. With the proposed new manufacturing process,
there is a 0.25 probability that the manufacturing cost will be $5,000 per computer
and a 0.50 probability that the cost will be $7,500 per computer. Unfortunately,
there is also a 0.25 probability that the cost will be $8,500 per computer.
Question 1.6: Should ABC Computer Company submit a bid,
and if so, what should they bid per computer?
A decision tree for this situation is shown in Figure 1.7. First, ABC must
decide whether to bid and how much to bid. If ABC's bid is lower than Complex
Computer's, then ABC must decide which manufacturing process to use. If ABC
uses the proposed new manufacturing process, then the cost of manufacturing
the computers is uncertain. The net pro¯ t ¯ gures (in millions of dollars) shown
at the endpoints of the Figure 1.7 tree take into account the cost of preparing
the bid, the cost of manufacturing the computers, and the revenue that ABC will
receive for the computers. For example, examine the topmost endpoint value. It
costs $1 million to prepare the bid, and ABC bids $9,500, which is lower than
Complex Computers' bid of $10,000, and hence ABC wins the contract. Then the
proposed new manufacturing process is used, and it costs $8,500 per computer
to manufacture the 10,000 computers. Therefore, at this endpoint, ABC makes a
net pro¯ t of ¡ 1; 000; 000 ¡ 10; 000£$8; 500+10; 000£$9; 500 = $9; 000; 000 = $9
1.4 SEQUENTIAL DECISIONS 13
New
Current
New $7,500
$8,500
$5,000
Current
New $7,500
$8,500
$5,000
Current
$19
$44
$14
-$1
-$1
$9
$34
-$11
-$1
$24
-$6
$0
$9,500
$8,500
$7,500
No Bid
$10,000
$9,000/$8,000
$9
Cost Per
Computer
Manufacturing
Process
Complex
Bid
ABC
Bid
$8,500
$7,500
$5,000
(1/2)
(1/4)
(1/4)
$8,000
(1/2)
(1/4)
(1/4)
(1/2)
(1/4)
(1/4)
(1/3)
(2/3)
(2/3)
(1/3)
$4
-$1
Net
Profit
$10,000/$9,000
/$8,000
$8,000
$8,000
EV=$22.75
EV=$22.75
EV=$12.75
EV=$12.75
EV=$8.17
EV=$8.17
EV=$2.75
EV=$2.75
$10,000
/$9,000
EV=$6.92
$8,000
Figure 1.7 ABC Computer Company decision tree, with net pro¯ t in millions of dollars
million. Verify the net pro¯ ts shown at the other endpoints so that you better
understand this process.
Calculating the expected values shown on the Figure 1.7 decision tree requires
addressing a new issue, namely what to do when there are multiple decision nodes
in the tree. In this decision, the amount of the bid is the ¯ rst decision, and if
this is lower than the Complex Computers bid, then there is a second decision
involving the type of manufacturing process to use. The calculation procedure
for this situation is a straightforward extension of the calculation procedure that
was demonstrated in the preceding section for dependent uncertainties.
This procedure will be illustrated by considering the topmost set of nodes
in the Figure 1.7 tree. Start at the rightmost side of the tree, and calculate
the expected value for the top rightmost chance node. This is determined as
14 CHAPTER 1 DECISION TREES
(1=4)£$9 + (1=2)£$19 + (1=4)£$44 = $22:75. At the top rightmost decision
node, compare the expected values for the two branches. The expected value for
the top branch of this decision node is $22.75, and (since there is no uncertainty
regarding the cost of the current manufacturing process) the expected value for
the bottom branch is $14. Since the top branch has the higher expected value, it
is the preferred branch. That is, the proposed new manufacturing process should
be used. Hence, the expected value for the \manufacturing process" decision
node is equal to the expected value for the proposed new manufacturing process,
which is $22.75.
Now continue back toward the root of the decision tree by calculating the
expected value for the top leftmost chance node in the tree. Since the expected
value of the manufacturing process decision is $22.75, and there is no uncertainty
about the net pro¯ t if ABC loses the bid, the expected value for the top leftmost
chance node is (1=3)£$22:75 + (2=3)£(¡ $1) = $6:92.
A similar process is used to calculate the expected values for the other three
branches of the root node, and the results are shown in Figure 1.7. These calculations
show that an $8,500 bid has the highest expected value, which is $8.17
million. Hence, if ABC uses expected value as its decision criterion, then it
should bid $8,500. In addition, the calculations also show that ABC should use
the proposed new manufacturing process if it wins the contract. The less preferred
branches for each decision node have been indicated on the decision tree
with cross hatching.
The complete speci¯ cation of the alternatives that should be selected at all
decision nodes in a decision tree is called a decision strategy.
De¯ nition 1.5: Decision Strategy
The complete speci¯ cation of all the preferred decisions in a sequential
decision problem is called the decision strategy. The decision strategy
shown in Figure 1.7 can be summarized as follows: Bid $8,500, and if
you win the contract use the proposed new manufacturing process.
1.5 Exercises
1.1 Arthrodax Company has been approached by Ranger Sound with a rush order
o® er to purchase 100 units of a customized version of Arthrodax's SoundScreamer
audio mixer at $5,000 per unit, and Arthrodax needs to decide how to respond.
The electronic modi¯ cations of the standard SoundScreamer needed for this customized
version are straightforward, but there will be a ¯ xed cost of $100,000
to design the modi¯ cations and set up for assembly of the customized Sound-
Screamers, regardless of the number of units produced. It will cost $2,000 per
1.5 EXERCISES 15
unit to manufacture the circuit boards for the units. Since Arthrodax has some
short term spare manufacturing capacity, the Ranger o® er is potentially attractive.
However, the circuit boards for the customized units will not ¯ t into the
standard SoundScreamer case, and Arthrodax must decide what to do about acquiring
cases for the customized units as it decides whether to accept Ranger's
purchase o® er. An appropriate case can be purchased at $500 per case, but
Arthrodax could instead purchase an injection molder to make the cases. It will
cost $20,000 to purchase the molder, and there is a 0.6 probability that it will be
possible to successfully make the cases using the molder. If the molder does not
work, then the purchase price for the molder will be totally lost and Arthrodax
must still purchase the cases at $500 per case. If the molder works, then it will
cost $60 per case to make the cases using the molder. Regardless of which case is
used, the cost of assembling the SoundScreamer circuit boards into the case is $20
per unit. Unfortunately, there is no way to test the molder without purchasing
it. Assume that there is no other use for the molder except to make the cases for
the Ranger order.
(i) Draw a decision tree for Arthrodax's decision about whether to accept the
Ranger o® er and how to acquire the cases for the customized SoundScreamers.
(ii) Using expected net pro¯ t as the decision criterion, determine the preferred
course of action for Arthrodax.
1.2 This is a continuation of Exercise 1.1. Assume that all information given in that
exercise is still valid, except as discussed in this exercise. Ranger now tells Arthrodax
that there is uncertainty about the number of customized SoundScreamers
that will be needed. Speci¯ cally, there is a 0.35 probability that it will need 100
units, and a 0.65 probability that it will need 50 units. If Arthrodax will agree
now to produce either number of units, then Ranger will pay $6,000 per unit if it
ultimately orders 50 units, and will pay $5,000 per unit if it ultimately orders 100
units. The timing is such on this rush order that Arthrodax will have to make a
decision about purchasing the injection molder before it knows how many units
Ranger will take. However, Arthrodax will only need to purchase or manufacture
the number of circuit boards and cases needed for the ¯ nal order of either 50 or
100 units.
(i) Draw a decision tree for Arthrodax's decision about whether to accept the
Ranger o® er and how to acquire the cases for the customized SoundScreamers.
Note that this is a situation with dependent uncertainties, as discussed
in Section 1.3.
(ii) Using expected net pro¯ t as the decision criterion, determine the preferred
course of action for Arthrodax.
1.3 This is a continuation of Exercise 1.2. Assume that all information given in
that exercise is still valid, except as discussed in this exercise. Assume now that
Arthrodax could delay the decision about purchasing the injection molder until
after it knows how many units Ranger will take.
16 CHAPTER 1 DECISION TREES
(i) Draw a decision tree for Arthrodax's decision about whether to accept the
Ranger o® er and how to acquire the cases for the customized SoundScreamers.
Note that this is a situation with sequential decisions, as discussed in
Section 1.4.
(ii) Using expected net pro¯ t as the decision criterion, determine the preferred
course of action for Arthrodax.
1.4 Aba Manufacturing has contracted to provide Zyz Electronics with printed circuit
(\PC") boards under the following terms: (1) 100,000 PC boards will be delivered
to Zyz in one month, and (2) Zyz has an option to take delivery of an additional
100,000 boards in three months by giving Aba 30 days notice. Zyz will pay $5.00
for each board that it purchases. Aba manufactures the PC boards using a batch
process, and manufacturing costs are as follows: (1) there is a ¯ xed setup cost
of $250,000 for any manufacturing batch run, regardless of the size of the run,
and (2) there is a marginal manufacturing cost of $2.00 per board regardless of
the size of the batch run. Aba must decide whether to manufacture all 200,000
PC boards now or whether to only manufacture 100,000 now and manufacture
the other 100,000 boards only if Zyz exercises its option to buy those boards. If
Aba manufactures 200,000 now and Zyz does not exercise its option, then the
manufacturing cost of the extra 100,000 boards will be totally lost. Aba believes
there is a 50% chance Zyz will exercise its option to buy the additional 100,000
PC boards.
(i) Explain why it might potentially be more pro¯ table to manufacture all
200,000 boards now.
(ii) Draw a decision tree for the decision that Aba faces.
(iii) Determine the preferred course of action for Aba assuming it uses expected
pro¯ t as its decision criterion.
1.5 Kezo Systems has agreed to supply 500,000 PC FAX systems to Tarja Stores in 90
days at a ¯ xed price. A key component in the FAX systems is a programmable
array logic integrated circuit chip (\PAL chip"), one of which is required in
each FAX system. Kezo has bought these chips in the past from an American
chip manufacturer AM Chips. However, Kezo has been approached by a Korean
manufacturer, KEC Electronics, which is o® ering a lower price on the chips. This
o® er is open for only 10 days, and Kezo must decide whether to buy some or all
of the PAL chips from KEC. Any chips that Kezo does not buy from KEC will be
bought from AM. AM Chips will sell PAL chips to Kezo for $3.00 per chip in any
quantity. KEC will accept orders only in multiples of 250,000 PAL chips, and is
o® ering to sell the chips for $2.00 per chip for 250,000 chips, and for $1.50 per chip
in quantities of 500,000 or more chips. However, the situation is complicated by
a dumping charge that has been ¯ led by AM Chips against KEC. If this charge
is upheld by the U. S. government, then the KEC chips will be subject to an
antidumping tax. This case will not be resolved until after the point in time
when Kezo must make the purchase decision. If Kezo buys the KEC chips, these
will not be shipped until after the antidumping tax would go into e® ect and the
chips would be subject to the tax. Under the terms o® ered by KEC, Kezo would
have to pay any antidumping tax that is imposed. Kezo believes there is a 60%
1.5 EXERCISES 17
chance the antidumping tax will be imposed. If it is imposed, then it is equally
likely that the tax will be 50%, 100%, or 200% of the sale price for each PAL
chip.
(i) Draw a decision tree for this decision.
(ii) Using expected value as the decision criterion, determine Kezo's preferred
ordering alternative for the PAL chips.
1.6 Intermodular Semiconductor Systems case. The Special Products Division
of Intermodular Semiconductor Systems has received a Request for Quotation
from Allied Intercontinental Corporation for 100 deep sea semiconductor
electrotransponders, a specialized instrument used in testing undersea engineered
structures. While Intermodular Semiconductor Systems has never produced deep
sea electrotransponders, they have manufactured subsurface towed transponders,
and it is clear that they could make an electrotransponder that meets Allied's
speci¯ cations. However, the production cost is uncertain due to their lack of
experience with this particular type of transponder. Furthermore, Allied has also
requested a quotation from the Undersea Systems Division of General Electrodevices.
Intermodular Semiconductor Systems and General Electrodevices are the
only companies capable of producing the electrotransponders within the time
frame required to meet the construction schedule for Allied's new undersea habitat
project.
Mack Reynolds, the Manager of the Special Products Division, must decide
whether to bid or not, and if Intermodular Semiconductor Systems does
submit a bid, what the quoted price should be. He has assembled a project team
consisting of Elizabeth Iron from manufacturing and John Traveler from marketing
to assist with the analysis. Daniel A. Analyst, a consulting decision analyst,
has also been called in to assist with the analysis.
Analyst: For this preliminary analysis, we have agreed to consider only a
small number of di® erent possible bids, production costs, and General Electrodevices
bids.
Reynolds: That's correct. We will look at possible per-unit bids of $3,000,
$5,000, and $7,000. We will look at possible production costs of $2,000, $4,000,
and $6,000 per unit, and possible per-unit bids by General Electrodevices of
$4,000, $6,000, and $8,000.
Iron: There is quite a bit of uncertainty about the cost of producing the
electrotransponders. I'd say there is a 50% chance we can produce them in a
volume of 100 units at $4,000 per unit. However, that still leaves a 50% chance
that they will either be $2,000 or $6,000 per unit.
Analyst: Is one of these more likely than the other?
Iron: No. It's equally likely to be either $2,000 or $6,000. We don't have
much experience with deep sea transponders. Our experience with subsurface
towed transponders is relevant, but it may take some e® ort to make units that
hold up to the pressure down deep. I'm sure we can do it, but it may be expensive.
Analyst: Could you do some type of cost-plus contract?
18 CHAPTER 1 DECISION TREES
Reynolds: No way! This isn't the defense business. Once we commit, we
have to produce at a ¯ xed price. Allied would take us to court otherwise. They're
tough cookies, but they pay their bills on time.
Iron: I want to emphasize that there is no problem making the electrotransponders
and meeting Allied's schedule. The real issue is what type of
material we have to use to take the pressure. We may be able to use molyaluminum
like we do in the subsurface towed units in which case the cost will be
lower. If we have to go to molyzirconium, then it will be more expensive. Most
likely, we will end up using some of each, which will put the price in the middle.
Analyst: What is General Electrodevices likely to bid?
Traveler: They have more experience than we do with this sort of product.
They have never made deep sea electrotransponders, but they have done a variety
of other deep sea products. I spent some time with Elizabeth discussing their
experience, and also reviewed what they did on a couple of recent bids. I'd say
there is a 50% chance they will bid $6,000 per unit. If not, they are more likely
to bid low than high|there is about a 35% percent chance they will bid $4,000
per unit.
Analyst: So that means there is 15% chance they will bid $8,000.
Traveler: Yes.
Reynolds: Suppose we had a better handle on our production costs.
Would that give us more of an idea what General Electrodevices would bid?
Iron: No. They use graphite-based materials to reinforce their transponders.
The cost structure for that type of production doesn't have any relationship
to our system using moly alloys.
(i) Draw a decision tree for the decision that Reynolds must make.
(ii) Determine the expected values for each of the alternatives, and specify
which alternative Reynolds should select if he uses expected value as a
decision criterion.