Decision Analysis

1
DecisionAnalysis

• Decision MakingUnder Uncertainty

2
Elements of a Decision Analysis

• Bidding for a Government Contract at SciTools

3
Background InformationBidding for a Government
Contract at SciTools

• SciTools Incorporated specializes in scientific
  instruments and has been invited to make a bid on
  a government contract to provide these
  instruments this coming year
• SciTools estimates that it will cost 5000 to
  prepare a bid and 95,000 to supply the
  instruments
• On the basis of past contracts, SciTools
  estimated the probabilities of the low bid from
  competitors at a certain dollar level
• In addition, they believe there is a 30 chance
  that there will be no competing bids

4
Decision Making Elements

• Although there is a wide variety of contexts in
  decision making, all decision making problems
  have three elements
• the set of decisions (or strategies) available to
  the decision maker
• the set of possible outcomes and the
  probabilities of these outcome
• a value model that prescribes results, usually
  monetary values, for the various combinations of
  decisions and outcomes
• Once these elements are known, the decision
  maker can find an optimal decision

5
SciTools ProblemBidding for a Government
Contract at SciTools

• SciTools decision is whether to submit a bid and
  how much they should bid (the bid must be greater
  than 100,000 for SciTools to make a profit)
• Based on the estimated probabilities, SciTools
  should bid either 115,000, 120,000, 125,000
  (well assume that they will never bid less than
  115,000 or more than 125,000 due to the small
  profit margin or low chances of winning the bid)
• The primary source of uncertainty is the behavior
  of the competitors - will they bid and, if so,
  how much?
• The behavior of the competitors depends on how
  many competitors are likely to bid and how the
  competitors assess their costs of supplying the
  instruments

6
SciTools ProblemBidding for a Government
Contract at SciTools

• The value model in this example is
  straightforward but in other examples it is often
  complex
• If SciTools decides right now not to bid, then
  its monetary values is 0 - no gain, no loss
• If they make a bid and are underbid by a
  competitor, then they lose 5000, the cost of
  preparing the bid
• If they bid B dollars and win the contract, then
  they make a profit of B - 100,000 that is, B
  dollars for winning the bid, less 5000 for
  preparing the bid, less 95,000 for supplying the
  instruments
• It is often convenient to list the monetary
  values in a payoff table

7
SciTools Payoff TablesBidding for a Government
Contract at SciTools
Payoff Table for SciTools Bidding Example Payoff Table for SciTools Bidding Example Payoff Table for SciTools Bidding Example Payoff Table for SciTools Bidding Example Payoff Table for SciTools Bidding Example Payoff Table for SciTools Bidding Example
Competitors Low Bid (1000s) Competitors Low Bid (1000s) Competitors Low Bid (1000s) Competitors Low Bid (1000s) Competitors Low Bid (1000s) Competitors Low Bid (1000s)
Bid level No Bid lt115 gt115, lt120 gt120, lt125 gt125 gt125
No Bid 0 0 0 0 0 0
115 15 -5 15 15 15 15
120 20 -5 -5 20 20 20
125 25 -5 -5 -5 25 25

Probability 0.3 0.7(0.2) 0.7(0.4) 0.7(0.3) 0.7(0.1) 0.7(0.1)
0.3 0.14 0.28 0.21 0.07 0.07
8
SciTools Payoff TablesBidding for a Government
Contract at SciTools
Alternative Payoff Table for SciTools Bidding Example Alternative Payoff Table for SciTools Bidding Example Alternative Payoff Table for SciTools Bidding Example Alternative Payoff Table for SciTools Bidding Example
Monetary Value Monetary Value
SciTools Bid SciTools Wins SciTools Loses Probability That SciTools Wins
No Bid NA 0 0.00
115 15 -5 0.86
120 20 -5 0.58
125 25 -5 0.37
9
Risk Profiles for SciToolsBidding for a
Government Contract at SciTools

• A risk profile simply lists all possible monetary
  values and their corresponding probabilities
• From the alternate payoff table we can obtain
  risk profiles for SciTools
• For example, if SciTools bids 120,000 there are
  two possibly monetary values, a profit of 20,000
  or a loss of 5000, and their probabilities are
  0.58 and 0.42, respectively
• Risk profiles can be illustrated on a bar chart
  (the bars above each possible monetary value
  measure the probability of that value occurring)

10
SciTools Expected Monetary Values (EMV)Bidding
for a Government Contract at SciTools
Alternative EMV Calculation EMV
No Bid 0(1) 0
Bid 115,000 15,000(0.86) (-5000)(0.14) 12,200
Bid 120,00 20,000 (0.58) (-5000)(0.42) 9500
Bid 125,000 25,000(0.37) (-5000)(0.63) 6100

• EMV is a weighted average of the possible
  monetary values, weighted by their probabilities
• What exactly does the EMV mean?
• It means that if SciTools were to enter many
  gambles like this, where on each gamble the
  gains, losses and probabilities were the same,
  then on average it would win 12,200 per gamble

11
Decision Tree ConventionsBidding for a
Government Contract at SciTools

• A decision tree is a graphical tool that can
  represent a decision problem with probabilities
• Decision trees are composed of nodes (circles,
  squares and triangles) and branches (lines)
• The nodes represent points in time
• A decision node (a square) is a time when the
  decision maker makes a decision
• A probability node (a circle) is a time when the
  result of an uncertain event becomes known
• An end node (a triangle) indicates that the
  problem is completed - all decisions have been
  made, all uncertainty have been resolved and all
  payoffs have been incurred

12
Decision Tree ConventionsBidding for a
Government Contract at SciTools

• Time proceeds from left to right
• Branches leading out of a decision node represent
  the possible decisions
• Branches leading out of probability nodes
  represent the possible outcomes of uncertain
  events
• Probabilities are listed on probability branches
  (these probabilities are conditional on the
  events that have already been observed )
• Individual monetary values are shown on the
  branches where they occur, and cumulative
  monetary values are shown to the right of the end
  nodes
• Two values are often found to the right of each
  end node the top one is the probability of
  getting to that end node, and the bottom one is
  the associated monetary value

13
SciTools Decision Tree
14
Decision Tree Folding Back Procedure Bidding for
a Government Contract at SciTools

• The solution for the decision tree is on the next
  slide
• The solution procedure used to develop this
  result is called folding back on the tree
• Starting at the right on the tree and working
  back to the left, the procedure consists of two
  types of calculations
• At each probability node we calculate EMV and
  write it below the name of the node
• At each decision node we find the maximum of the
  EMVs and write it below the node name
• After folding back is completed we have
  calculated EMVs for all nodes

15
Decision Tree Results
16
The PrecisionTree Add-In

• This add-in enables us to build and label a
  decision tree, but it performs the folding-back
  procedure automatically and then allows us to
  perform sensitivity analysis on key input
  parameters
• There are three options to run PrecisionTree
• If Excel is not currently running , you can
  launch Excel and PrecisionTree by clicking on the
  Windows Start button and selecting the
  PrecisionTree item
• If Excel is currently running, the procedure in
  the previous bullet will launch PrecisionTree on
  top of Excel
• If Excel is already running and the Desktop Tools
  toolbar is showing, you can start PrecisionTree
  by clicking on its icon

17
Using PrecisionTreeBidding for a Government
Contract at SciTools

• Inputs Enter the cost and probability inputs
• New tree Click on the new tree button (the far
  left button) on the PrecisionTree toolbar, and
  then click on any cell below the input section
  to start a new tree. Click on the name box of
  this new tree to open a dialog box. Type in a
  descriptive name for the tree.
• Decision nodes and branches To obtain decision
  nodes and branches, click on the (only) triangle
  end node to open the dialog box shown here

18
The PrecisionTree Add-InBidding for a Government
Contract at SciTools

• Were calling this decision Bid? and specifying
  that there are two possible decisions. The tree
  expands as shown here.
• The boxes that say branch show the default
  labels for these branches. Click on either of
  them to open another dialog box where you can
  provide a more descriptive name for the branch.
  Do this to label the two branches No and Yes.
  Also, you can enter the immediate payoff/cost for
  either branch right below it. Since there is a
  5000 cost of bidding, enter the formula BidCost
  right below the Yes branch in cell B19.

19
The PrecisionTree Add-InBidding for a Government
Contract at SciTools

• More decision branches The top branch is
  completed if SciTools does not bid, there is
  nothing left to do. So click on the bottom end
  node, following SciTools decision to bid, and
  proceed as in the previous step to add and label
  the decision node and three decision branches
  for the amount to bid. Note that there are no
  monetary values below these decision branches
  because no immediate payoffs or costs are
  associated with the bid amount decision.

20
The PrecisionTree Add-InBidding for a Government
Contract at SciTools

• Probability nodes and branches We now need a
  probability node and branches from the rightmost
  end nodes to capture the competition bids
• Click on the top one of these end nodes to bring
  up the dialog box
• Click on the red circle box to indicate that this
  is a probability node. Label it Any competing
  bid?, specify two branches, and click on OK.
• Label the two branches No and Yes.
• Repeat this procedure to form another probability
  node following the Yes branch, call it Win
  bid?, and label its branches as shown on the
  next slide.

21
The PrecisionTree Add-InBidding for a Government
Contract at SciTools

• Copying probability nodes and branches You
  could build the next node and branches or take
  advantage of PrecisionTrees copy and paste
  function. Decision trees can be very bushy, but
  this copy and paste feature can make them much
  less tedious to construct.

22

• Labeling probability branches
• First enter the probability of no competing bid
  in cell D18 with the formula PrNoBid and enter
  its complement in cell D24 with the formula
  1-D18
• Next, enter the probability that SciTools wins
  the bid in cell E22 with the formula
  SUM(B10B12) and enter its complement in cell
  E26 with the formula 1-E22

23
The PrecisionTree Add-InBidding for a Government
Contract at SciTools

• For the monetary values, enter the formula
  115000-ProdCost in the two cells, D19 and E23,
  where SciTools wins the contract
• Enter the other formulas on probability branches
• Using the previous step and the final decision
  tree as a guide, enter formulas for the
  probabilities and monetary values on the other
  probability branches, that is, those following
  the decision to bid 120,000 or 125,000
• Were finished! The completed tree shows the best
  strategy and its associated EMV.
• Once the decision tree is completed,
  PrecisionTree has several tools we can use to
  gain more information about the decision analysis

24
Risk Profile of Optimal StrategyBidding for a
Government Contract at SciTools

• Click on the fourth button from the left on the
  PrecisionTree toolbar (Decision Analysis) and
  fill in the resulting dialog box (the Policy
  Suggestion option allows us to see only that part
  of the tree that corresponds to the best
  decision)
• The Risk Profile option allows us to see a
  graphical risk of the optimal decision (there are
  only two possible monetary outcomes if SciTools
  bids 115,000)

25
Sensitivity AnalysisBidding for a Government
Contract at SciTools

• We can enter any values not the input cells and
  watch how the tree changes
• To obtain more systematic information, click on
  the PrecisionTree sensitivity button
• The dialog box requires an EMV cell to analyze at
  the top and one or more input cells in the middle
• The cell to analyze is usually the EMV cell at
  the far left of the decision tree but it can be
  any EMV cell
• For any input cell (such as the production cost
  cell) we enter a minimum value, a maximum value,
  a base value, and a step size
• When we click Run Analysis, PrecisionTree varies
  each of the specified inputs and presents the
  results in several ways in a new Excel file with
  Sensitivity, Tornado, and Spider Graph sheets

26
Sensitivity Analysis Sensitivity ChartBidding
for a Government Contract at SciTools

• The Sensitivity sheet includes several charts, a
  typical one of which appears here
• This shows how the EMV varies with the production
  cost for both of the original decisions
• This type of graph is useful for seeing whether
  the optimal decision changes over the range of
  input variable

27
Sensitivity Analysis Tornado ChartBidding for
a Government Contract at SciTools

• The Tornado sheet shows how sensitive the EMV of
  the optimal decision is to each of the selected
  inputs over the ranges selected
• The production cost has the largest effect on
  EMV, and bid cost has the smallest effect

28
Sensitivity Analysis Spider ChartBidding for a
Government Contract at SciTools

• Finally, the Spider Chart shows how much the
  optimal EMV varies in magnitude for various
  percentage changes in the input variables
• Again, the production cost has a relatively large
  effect, whereas the other two inputs have
  relatively small effects

29
Sensitivity Analysis Two-Way AnalysisBidding
for a Government Contract at SciTools

• Use a two-way analysis to see how the selected
  EMV varies as each pair of inputs vary
  simultaneously
• For each of the possible values of production
  cost and probability of no competitor bid, this
  chart indicates which bid amount is optimal
• The optimal bid amount remains 115,000 unless
  the production cost and the probability of no
  competing bid are both large. Then it becomes
  optimal to bid 125,000

30
Multistage Decision Trees

• Marketing a New Product at Acme

31
Background Information

• Acme Company is trying to decide whether to
  market a new product.
• As in many new-product situations, there is much
  uncertainty about whether the product will
  catch-on.
• Acme believes that it would be prudent to
  introduce the product to a test market first.
• Thus the first decision is whether to conduct the
  test market.

32
Background Information -- continued

• Acme has determined that the fixed cost of the
  test market is 3 million.
• If they proceed with the test, they must then
  wait for the results to decide if they will
  market the product nationally at a fixed cost of
  90 million.
• If the decision is not to conduct the test
  market, then the product can be marketed
  nationally with no delay.
• Acmes unit margin, the difference between its
  selling price and its unit variable cost, is 18
  in both markets.

33
Background Information -- continued

• Acme classifies the results in either market as
  great, fair or awful.
• Each of these has a forecasted total units sold
  as (in 1000s of units) 200, 100 and 30 in the
  test market and 6000, 3000 and 900 for the
  national market.
• Based on previous test markets for similar
  products, it estimates that probabilities of the
  three test market outcomes are 0.3, 0.6 and 0.1.

34
Background Information -- continued

• Then based on historical data on products that
  were tested then marketed nationally, it assesses
  the probabilities of the national market outcomes
  given each test market outcome.
• If the test market is great, the probabilities
  for the national market are 0.8, 0.15, and 0.05.
• If the test market is fair. then the
  probabilities are 0.3, 0.5, 0.2.
• If the test market is awful, then the
  probabilities are 0.05, 0.25, and 0.7.
• Note how the probabilities of the national market
  mirrors those of the test market.

35
Elements of Decision Problem

• The three basic elements of this decision problem
  are
• the possible strategies
• the possible outcomes and their probabilities
• the value model
• The possible strategies are clear
• Acme must first decide whether to conduct the
  test market.
• Then it must decide whether to introduce the
  product nationally.

36
Contingency Plan

• If Acme decides to conduct a test market they
  will base the decision to market nationally on
  the test market results.
• In this case its final strategy will be a
  contingency plan, where it conducts the test
  market, then introduces the product nationally if
  it receives sufficiently positive test market
  results and abandons the product if it receives
  negative test market results.
• The optimal strategies from many multistage
  decision problems involve similar contingency
  plans.

37
Conditional Probabilities - continued

• The probabilities of the test market outcomes and
  conditional probabilities of national market
  outcomes given the test market outcomes are
  exactly the ones we need in the decision tree.
• However, suppose Acme decides not to run a test
  market and then decides to market nationally.
  Then what are the probabilities of the national
  market outcomes? We cannot simply assess three
  new probabilities.

38
Conditional Probabilities - continued

• These probabilities are implied by the given
  probabilities. This follows from the rule of
  conditional probability.
• If we let T1, T2, and T3 be the test market
  outcomes and N be any national market outcomes,
  then by the addition rule of probability and the
  conditional probability formulaP(N) P(N and
  T1) P(N and T2) and P(N and T3)
  P(NT1)P(T1) P(NT2)P(T2) P(NT3)P(T3)

39
Conditional Probabilities - continued

• This is sometimes called the law of
  probabilities.
• We determine the probabilities as 0.425 for a
  great market, 0.37 for a fair market and 0.205
  for an awful market.
• The monetary values are the fixed costs of test
  marketing or marketing nationally and these are
  incurred as soon as the go ahead decisions are
  made.

40
ACME.XLS

• This file contains the inputs for the decision
  tree.
• The only calculated values in this spreadsheet
  are in row 28, which follow from the equation.
  The formula in cell B28 is
  SUMPRODUCT(B22B24,B16B18)then it is
  copied across row 28.
• The creation of the tree is then straightforward
  to build and label.

41
Inputs
42
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43
Interpreting the Tree

• The interpretation of the tree is fairly
  straightforward if we realize that each value
  just below each node name is an EMV.
• Each of these EMVs have been calculated with the
  folding back procedure.
• We can also see Acmes optimal strategy by
  following the TRUE branches from left to right.

44
Optimal Strategy

• Acme should first run a test market and if the
  results are great then they should market it
  nationally.
• If the test results are fair or awful they should
  abandon the product.
• The risk profile for the optimal strategy can be
  seen on the next slide.
• The risk profile (created by clicking on
  PreceisionTrees staircase button and selecting
  Statistics and Risk Profile options) that there
  is a small chance of two possible large losses,
  there is a 70 chance of a moderate loss and
  there is a 24 chance of a nice profit.

45
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46
Optimal Strategy -- continued

• One might argue that the large potential (70)
  chance of some loss should persuade Acme to
  abandon the product right away - without a test
  market.
• This is what playing the averages with EMV is
  all about.
• Since the EMV of this optimal strategy is greater
  than 0, Acme should go ahead with this strategy
  if it is an EMV maximizer.

47
Expected Value of Sample Information

• The role of the test market in this example is to
  provide Acme with information.
• Information usually costs, in this case its fixed
  cost is 3 million.
• From the decision tree we can see that the EMV
  from test marketing is 807,000 better than the
  decision not to test market. The most Acme would
  be willing to pay for the test marketing is
  3.807 million.
• This value is called the expected value of sample
  information or EVSI.

48
Expected Value of Sample Information -- continued

• In general we can write the following
  expressionEVSIEMV with free information - EMV
  without information
• For the Acme example the EVSI is 3.807 million.
• The reason for the term sample is that the
  information does not remove all uncertainty about
  the future.
• That is, even after the test market results are
  in, there is still uncertainty about the national
  market.

49
Expected Value of Perfect Information

• We can go one step further and ask how much
  perfect information is worth.
• Perfect could be imagined as an envelope
  containing the final outcome.
• No such envelope exists but if it did how much
  would Acme be willing to pay for it?
• This question can be answered with the decision
  tree shown on the next slide.
• Folding back produces an EMV of 7.65 million.

50
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51
Expected Value of Perfect Information -- continued

• This value is called the expected value of
  perfect information, or EVPI.
• The EVPI may appear to be irrelevant because
  perfect information is almost never available -
  at any price.
• However, it is often used as an upper bound on
  EVSI for any potential sample information.
• That is, no sample information can ever be worth
  more than the EVPI.

52
Bayes Rule

• Drug Testing College Athletes

53
Background Information

• If an athlete is tested for a certain type of
  drug use, the test will come out either positive
  or negative.
• However, these tests are never perfect. Some
  athletes who are drug-free test positive (false
  positives) and some who are drug users test
  negative (false negatives). We will assume that
• 5 of all athletes use drugs
• 3 of all tests on drug-free athletes yield false
  positives
• 7 of all tests on drug users yield false
  negatives.
• The question then is what we can conclude from a
  positive or negative test result.

54
Solution

• Let D and ND denote that a randomly chosen
  athlete is or is not a drug user, and let T and
  T- indicate a positive or negative test result.
• We know the following probabilities
• First, since 5 of all athletes are drug users,
  we know that P(D) 0.05 and P(ND) 0.95. These
  are called prior probabilities because they
  represent the chance that an athlete is or is not
  a drug user prior to the results of a drug test.

55
Solution -- continued

• Second, from the information on drug test
  accuracy, we know the conditional probabilities
  P(TND) 0.08 and P(T-D) 0.03.
• But a drug-free athlete either tests positive or
  negative, and the same is true for a drug user.
  Therefore, P(T-ND) 0.92 and P(TD) 0.97.
• These four conditional probabilities of test
  results given drug user status are often called
  the likelihoods of the test results.
• Given these priors and likelihoods we want
  posterior probabilities such as P(DT) or
  P(NDT-).

56
Solution -- continued

• These are called posterior probabilities because
  they are assessed after the drug test results.
  This is where Bayes rule enters.
• Bayes Rule says that a typical posterior
  probability is a ratio. The numerator is a
  likelihood times a prior, and the denominator is
  the sum of likelihoods times priors.

57
DRUGBAYES.XLS

• This file shows how easy it is to implement
  Bayes rule in a spreadsheet.
• The given priors and likelihoods are listed in
  the ranges B5C5 and B9C10.

58
Calculations

• We calculate the products of likelihoods and
  priors in the range B15C16. The formula in cell
  B15 isB5B9 and it is copied to the rest of
  B15C16 range.
• Their row sums are calculated in the range
  D15D16. These represent the unconditional
  probabilities of the two possible results. They
  are also the denominator of Bayes rule.
• Finally we calculate the posterior probabilities
  in the range B21C22. The formula in B21 is
  B15/D15 and it is copied to the rest of the
  range B21C22.

59
Resulting Probabilities

• A negative test result leaves little doubt that
  the athlete is drug-free this probability is
  0.996.
• A positive test result leaves a lot of doubt of
  whether the athlete is drug-free. The probability
  that the athlete uses drugs is 0.620.
• Since only 5 of athletes use drugs it takes a
  lot of evidence to convince us otherwise. This
  plus the fact that the test produces false
  positives means the athletes that test positive
  still have a decent chance of being drug-free.

60
Background Information

• The administrators at State University are trying
  to decide whether to institute mandatory drug
  testing for the athletes.
• They have all the same information of priors and
  likelihoods as the previous example.
• They want to use a decision tree approach to see
  whether the benefits outweigh the costs

61
Decision Alternatives

• We will assume that there are only two
  alternatives
• perform drug testing on all athletes
• dont perform any drug testing.
• In the former case we assume that if an athlete
  tests positive, this athlete is barred from
  sports.

62
Monetary Values

• The monetary values are more difficult to
  assess. They include
• the benefit B from correctly identifying a drug
  user and barring him or her from sports
• the cost C1 of the test itself for a single
  athlete (materials and labor)
• the cost C2 of falsely accusing a nonuser (and
  barring him or her from sports)
• the cost C3 of not identifying a drug user
  (either by not testing at all or by obtaining a
  false negative)
• the cost C4 of violating a nonusers privacy by
  performing the test

63
Monetary Values -- continued

• Only C1 is a direct monetary cost that is easy to
  measure.
• The other costs and the benefit are real, and
  they must be compared on some scale to enable
  administrators to make a rational decision.
• We will do this by comparing everything to C1 to
  which we will assign a value 1.
• There is a lot of subjectivity so sensitivity
  analysis on the final decision is a must.

64
Benefit-Cost Table

• Before constructing the decision tree it is
  useful to form a benefit-cost table for both
  alternatives and all possible outcomes.
• All benefits in this table have a positive sign
  and all costs have a negative sign.

Net Benefit for Drug-Testing Example Net Benefit for Drug-Testing Example Net Benefit for Drug-Testing Example Net Benefit for Drug-Testing Example Net Benefit for Drug-Testing Example Net Benefit for Drug-Testing Example
Dont Test Dont Test Perform Test Perform Test Perform Test Perform Test
D ND D and T ND and T D and T- ND and T-
-C3 0 B C1 -(C1 C2 C4) - (C1 C3) - (C1 C4)
65
DRUG.XLS

• This file provides the data to create the
  decision tree with PrecisionTree.
• First, we enter the input data and then along
  with Bayes rule calculations from before we can
  create the tree and enter the links to the values
  and probabilities.

66
Timing

• Before interpreting the tree we need to discuss
  the timing (from left to right).
• If drug testing is performed, the result of the
  drug test is observed first (a probability node).
• Each result leads to an action (bar from sports
  or dont), and then the eventual benefit or cost
  depends on whether the athlete uses drugs (again
  a probability node).
• If no drug testing is performed, then there is no
  intermediate test result node or branches.

67
Interpretation

• First, we discuss the benefits and costs.
• The largest cost is falsely accusing (and
  barring) a nonuser.
• This is 50 times as large as the cost of the
  test.
• The benefit of identifying a drug user is only
  half this large, and the cost of not identifying
  a user is 40 as large as barring a nonuser.
• The violation of privacy of a nonuser is twice as
  large as the cost of the test.
• Based on these values, the decision tree implies
  that drug testing should not be performed. The
  EMVs are both negative thus costs outweigh
  benefits.

68
Interpretation -- continued

• What would it take to change this decision?
• Most people in society would agree that the costs
  of falsely accusing a nonuser should be the
  largest cost. In fact, with legal costs we might
  argue that it should be more than 50 times the
  cost of the test.
• On the other hand, if the benefit of identifying
  a user and/or the cost C3 for not identifying a
  user increase, the testing might be the preferred
  alternative.
• We can test this by varying the benefits and the
  costs.

69
Interpretation -- continued

• Other than benefits and costs, the only thing
  that we might vary is the accuracy of the test,
  measured by the error probabilities.
• Even when each error probability was decreased to
  0.01, the no-testing alternative was still
  optimal.
• In summary, based on a number of reasonable
  assumptions and parameter settings, this example
  has shown that it is difficult to make a case for
  mandatory drug testing.

70
DecisionAnalysis

• Attitudes Toward Risk andMulti-attribute
  Decision Making

71
Utility Functions

• A utility function is a mathematical function
  that transforms monetary values payoffs and
  costs into utility values.
• Most individuals are risk averse, which means
  intuitively that they are willing to sacrifice
  some EMV to avoid risky gambles.
• If a person is indifferent, then the expected
  utilities from the two options must be equal. We
  will call the resulting value the indifference
  value.

72
Example

• John Jacobs owns his own business.
• Because he is about to make an important decision
  where large losses or large gains are at stake,
  he wants to use the expected utility criterion to
  make his decision.
• He knows that he must first assess his own
  utility function, so he hires a decision analysis
  expert, Susan Schilling, to help him out.
• How might the session between John and Susan
  proceed?

73
Solution

• Susan first asks John for the largest loss and
  largest gain he can imagine.
• He answers with the values 200,000 and 300,000,
  so she assigns utility values U(-200,000) 0 and
  U(300,000) 1 as anchors for the utility
  function.
• Now she presents John with the choice between two
  options
• Option 1 Obtain a payoff of z (really a loss if
  z is negative).
• Option 2 Obtain a loss of 200,000 or a payoff
  of 300,000, depending on the flip of a fair coin.

74
Solution -- continued

• Susan reminds John that the EMV of option 2 is
  50,000.
• He realizes this, but because he is quite risk
  averse, he would far rather have 50,000 for
  certain than take the gamble for option 2.
• Therefore the indifference value of z must be
  less than 50,000.
• Susan then poses several values of z to John.

75
Solution -- continued

• Would he rather have 10,000 for sure or take
  option 2? He says he would rather take this
  10,000.
• Would he rather pay 5000 for sure or take option
  2. He says he would rather take option 2.
• By this time, we know the indifference value of z
  must be less than 10,000 and greater than
  -5000.
• With a few more questions of this type, John
  finally decides on z5000 as his indifference
  value. He is indifferent between obtaining 5000
  for sure and taking the gamble in option 2.

76
Solution -- continued

• We can substitute these values into the
  equationU(5000) 0.5U(-200,000)
  0.5U(300,000) 0.5(0) 0.5(1) 0.5
• Note that John is giving up 45,000 in EMV
  because of his risk aversion.
• The EMV of the gamble in option 2 is 50,000, and
  he is willing to accept a sure 5000 in its
  place.
• The process would then continue. For example,
  since she now knows U(5000) and U(300,000), Susan
  could ask John to choose between these options

77
Solution -- continued

• Option 1 Obtain a payoff of z.
• Option 2 Obtain a payoff of 5000 or a payoff of
  300,000, depending on the flip of a fair coin.
• If John decides that his indifference value is
  now z 130,000, then with the equation we know
  thatU(130,000) 0.5U(5000) 0.5U(300,000)
  0.5(0.5) 0.5(1) 0.75
• Note that John is now giving up 22,500 in EMV
  because the EMV of the gamble in option 2 is
  152,500. By continuing in this manner, Susan can
  help John assess enough utility values to
  approximate a continuous utility curve.

78
Incorporating Attitudes Toward Risk

• Deciding Whether to Enter Risky Ventures at
  Venture Limited

79
Background Information

• Venture Limited is a company with net sales of
  30 million. The company currently must decide
  whether to enter one of two risky ventures or do
  nothing.
• The possible outcomes of the less risky venture
  are 0.5 million loss, a 0.1 million gain, and a
  1 million gain.
• The probabilities of these outcomes are 0.25,
  0.50, and 0.25.
• The possible outcomes of the most risky venture
  is 1 million loss, a 1 million gain, and a 3
  million gain.

80
Background Information -- continued

• The probabilities of these outcomes are 0.35,
  0.60, and 0.05.
• If Venture Limited can enter at most one of the
  two risky ventures, what should it do?

81
Solution

• We will assume that Venture Limited has an
  exponential utility function.
• An exponential utility function has only one
  adjustable numerical parameter, and there are
  straightforward ways to discover the most
  appropriate value of this parameter for a
  particular individual or company.
• Also, based on Howards guidelines, we will
  assume that the companys risk tolerance is 6.4
  of its net sales, or 1.92 million.

82
Solution -- continued

• We can substitute into the equation to find the
  utility of any monetary outcome.
• For example, the gain from doing nothing is 0,
  and its utility is U(0) 1 e-0/1.92 1-1 0.
  As another example, the utility of a 1 million
  loss is U(-1) 1 e-(-1)/1.92 1 1.683 -
  0.683.
• These are the values we use (instead of monetary
  values) in the decision tree.

83
Using PrecisionTree

• Fortunately, PrecisionTree takes care of all the
  details.
• After we build a decision tree and label it in
  the usual way, we click on the name of the tree
  to open the dialog box shown here.

84
Using PrecisionTree

• We then fill in the utility function information
  as shown in the upper right section of the dialog
  box.
• This says to use an exponential function with
  risk tolerance 1.92.
• It also indicates that we want the expected
  utilities (as opposed to EMVs) to appear in the
  decision tree.

85
VENTURE.XLS

• We build our tree exactly the same way and link
  probabilities and monetary values to its branches
  in the usual way.
• For example, there is a link in cell C22 to the
  monetary value in cell A10. However, the expected
  values shown in the tree are expected utilities,
  and the optimal decision is the one with the
  largest expected utility.
• In this case the expected utilities for doing
  nothing, investing in the less risky venture, and
  investing in the more risky venture are 0,0.0525,
  and 0.0439. Therefore, the optimal decision is to
  invest in the less risky venture.

86
Solution -- continued

• Note that the EMVs of the three decisions are 0,
  0.175 million, and 0.4 million. The later of
  these are calculated in row 14 as the usual
  sumproduct of monetary values and
  probabilities.
• How sensitive is the optimal decision to the key
  parameter, the risk tolerance?
• We can answer this by changing the risk tolerance
  and watching how the tree changes.
• So the optimal decision depends heavily on the
  attitudes toward risk of Venture Limiteds top
  management.

87
Certainty Equivalents

• Now suppose that Venture Limited has only two
  options.
• It can enter the less risky venture or receive a
  certain dollar amount x and avoid the gamble
  altogether.
• We want to find the dollar amount x such that the
  company is indifferent between these two options.
• If it enters the risky venture, its expected
  utility is 0.0525, calculated above. If he
  receives x dollars for certain, its (expected)
  utility is U(x) 1 e-x/1.92.

88
Certainty Equivalents -- continued

• To find the value x where it is indifferent
  between the two options, we set 1 e-x/1.92
  equal to 0.0525 or e-x/1.92 0.9475, and solve
  for x.
• Taking natural logarithms of both sides and
  multiplying by 1.92, we obtain x
  -1.92ln(0.9475) 0.104 million.
• This value is called the certainty equivalent of
  the risky venture.

89
Certainty Equivalents -- continued

• The company is indifferent between entering the
  less risky venture and receiving 0.104 million
  to avoid it.
• Although the EMV of the less risky venture is
  0.175 million, the company acts as if it is
  equivalent to a sure 0.104 million.
• In this sense, the company is willing to give up
  the difference in EMV, 71,000, to avoid a
  gamble.
• By a similar calculation, the certainty
  equivalent of the more risky venture is
  approximately 0.086 million, when in fact its
  EMV is a hefty 0.4 million!

90
Certainty Equivalents -- continued

• So in this case it is willing to give up the
  difference in EMV, 314,000, to avoid this
  particular gamble.
• Again, the reason is that the company dislikes
  risk.
• We can see these certainty equivalents in
  PrecisionTree by adjusting the Display box to
  show Certainty Equivalent.
• The tree then looks like the one on the next
  slide.
• The certainty equivalents we just discussed
  appear in cells C24 and C32.

91
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92
Multi-attribute Decision Making

• Using AHP to Select a Job

93
Background Information

• Jane is about to graduate from college and is
  trying to determine which job to accept from
  among three options.
• She plans to choose among the offers by
  determining how well each job offer meets the
  following four objectives
• Objective 1 High starting salary
• Objective 2 Quality of life in city where job is
  located
• Objective 3 Interest of work
• Objective 4 Nearness of job to family

94
Value structure propertiesHow do we determine
proper objectives?

• Completeness (covers all concerns)
• Non-redundant (no overlap)
• Decomposable (can consider each objective without
  considering others)
• Operability (must be understandable)
• Number of objectives (smaller generally better)

95
Objectives AssessmentDetermine attribute
measures for each objective

• Salary (measure annual salary in K assume that
  other benefits and cost of living in different
  locations are similar) Job 1 40K
  Job 2 36K Job 3 28K
• Quality of life (based on subjective 1-10 rating
  done by Jane using reference material)
  Job 1 5 Job 2 6 Job 3 8
• Work interest (based on subjective 1-10 rating
  using job description and interview)
  Job 1 3 Job 2 9 Job 3 6
• Nearness to family (measured in travel time)
  Job 1 3 hours Job 2 1 hour Job 3 20
  minutes

96
Additive Weighting Approach

• Develop a value function
• Transform attribute measures from
  quantitative/qualitative measures into a linear
  scaled measure
• Assess and normalize weights for each attribute
• Each alternative receives a value score (sum of
  attribute measures weighted by objective weights)
• Preferred alternative has the largest value score
• Simple Additive Weighting Template

97
Transforming Attribute Measures

• We want scales with the following properties
• Interval scale (difference between values has
  meaning)
• Consistent units of measurement
• Scales in the same direction
• We will use a linear proportional scale

98
Assess Normalize Weights

• Weights should be based on importance relative to
  the range of values for each attribute
• How much more/less important are the following
  attributes relative to each other?
• Salary (ranging from 28K to 40K)
• Quality of life (ranging from 5 to 8 on a 10
  point scale)
• Work interest (ranging from 3 to 9 on a 10 point
  scale)
• Nearness to family (ranging from 20 min to 3 hr)
• Rank order attributes, give most important
  attribute a top score, other scores relative to
  best
• Normalize divide each weight by the sum of
  weighted scores (sum of normalized weights equal
  1)

99
Determine Value Score

• Use weights of salary10, work interest5, near
  to family3 and quality of life2
• Final value scores Job 1.66, Job 2.82, Job
  3.77
• Preferred option is Job 2 (although Job 3 is
  close enough to examine these two jobs against
  each other)
• Sensitivity analysis is important due to
  subjectivity
• Always present and rationalize preferred option
  based on objectives (why was Job 2 preferred?
  best for work interest, overall strong on all
  objectives)
• Job 3 does well due to dominance in nearness to
  family (may consider rescaling attribute measure
  or importance if this does not seem reasonable)

100
Analytic Hierarchy Process (AHP)

• Transform objective measures from quantitative or
  qualitative measures using pairwise comparisons
  and an intensity scale of relative importance
• Assess weights using the same pairwise comparison
  method
• Each alternative receives a weighted score (sum
  of objective measures weighted by objective
  weights)
• Preferred alternative has the largest weighted
  score

101
Solution

• To illustrate how AHP works, suppose that Jane is
  facing three job offers and must determine which
  offer to accept.
• In this example there are four objectives, as
  listed earlier.
• For each objective, AHP generates a weight (by a
  method to be described shortly).
• By convention, the weights are always chosen so
  that they sum to 1.

102
Pairwise Comparison Matrices

• To obtain the weights for the various objectives,
  we begin by forming a matrix A, known as the
  pairwise comparison matrix.
• The entry in row i and column j of A, labeled
  aij, indicates how much more (or less) important
  objective i is than objective j.
• Importance is measured on an integer-valued 1-9
  scale with each number having the interpretation
  shown in the table.

103
Pairwise Comparison Matrices -- continued
Interpretation of Values in Pairwise Comparison Matrix Interpretation of Values in Pairwise Comparison Matrix
Value of aij Interpretation
1 Objective i and j are equally important.
3 Objective i are slightly more important than j.
5 Objective i are strongly more important than j.
7 Objective i are very strongly more important than j.
9 Objective i are absolutely more important than j.

• The phrases in this table, such as strongly more
  important than, are suggestive only.
• They simply indicate discrete points on a
  continuous scale that can be used to compare the
  relative importance of any two objectives.

104
Pairwise Comparison Matrices -- continued

• For example, if a13 3, then objective 1 is
  slightly more important than objective 3. If aij
  4, a value not in the table, then objective i
  is somewhere between slightly and strongly more
  important than objective j.
• If objective i is less important then objective
  j, we use the reciprocal of the appropriate
  index.
• For example, if objective i is slightly less
  important than objective j, the aij 1/3.
• Finally, for all objectives i, we use the
  convention that aij 1.

105
Pairwise Comparison Matrices -- continued

• For consistency, it is necessary to set aij
  1/aij.
• This simply states that if objective 1 is
  slightly more important than objective 3, then
  objective 3 is slightly less important than job
  1.
• It is usually easier to determine all aijs that
  are greater than 1 and then use the relationship
  aij 1/aij to determine the remaining entries in
  the pairwise comparison matrix.

106
Pairwise Comparison Matrices -- continued

• To illustrate, suppose that Jane has identified
  the following pairwise comparison matrix for her
  four objectives
• The rows and columns of A each correspond to
  Janes four objectives salary, quality of life,
  interest of work, and nearness to family.

107
Pairwise Comparison Matrices -- continued

• The entries in this matrix have built-in pairwise
  consistency because we require aij 1/aij for
  each i and j.
• However, they may not be consistent when three
  alternatives are considered simultaneously.
• For example, Jane claims that salary is strongly
  more important than quality of life (a12 5) and
  that salary is very slightly more important than
  interesting work (a13 2). But she also says
  that interesting work is very slightly more
  important than quality of life (a32 2).

108
Pairwise Comparison Matrices -- continued

• The question is whether these ratings are all
  consistent with one another.
• They are not, at least not exactly.
• It can be shown that some of Janes pairwise
  comparisons, slight inconsistencies are common
  and fortunately do not cause serious
  difficulties.
• An index that can be used to measure the
  consistency of Janes preferences will be
  discussed later in this section.

109
Determining the Weights

• Although the ideas behind AHP are fairly
  intuitive, the mathematical reasoning required to
  derive the weights for the objectives is quite
  advanced. Therefore, we simply describe how it is
  done.
• Starting with the pairwise comparison matrix A,
  we find the weights for Janes four objectives
  using the following two steps.

110
Determining the Weights -- continued

1. For each of the columns of A, divide each entry
   in the column by the sum of the entries in the
   column. This yields a new matrix (call it Anorm,
   for normalized) in which the sum of the entries
   in each column is 1.For Janes pairwise
   comparison matrix, this step yields

111
Determining the Weights -- continued

1. Estimate wi, the weight for objective i, as the
   average of the entries in row i of Anorm. For
   Janes matrix this yields

112
Determining the Weights -- continued

• Intuitively, why does wi approximate the weight
  for objective 1 (salary)? Here is the reasoning.
• The proportion of weight that salary is given in
  pairwise comparisons of each objective to salary
  is 0.5128. Similarly, 0.50 represents the
  proportion of total weight that salary is given
  in pairwise comparisons of each objective to
  quality of life.
• Therefore, we see that each of the four numbers
  averaged to obtain wi represents a measure of the
  total weight attached to salary. Averaging these
  numbers should give a good estimate of the
  proportion of the total weight given to salary.

113
Determining the Score of Each Decision
Alternative on Each Objective

• Now that we have determined the weights, we need
  to determine how well each job scores on each
  objective.
• To determine these scores, we use the same scale
  described in the table to construct a pairwise
  comparison matrix for each objective.
• Consider the salary objective, for example.
  Suppose that Jane assesses the following pairwise
  comparison matrix.

114
Determining the Score of Each Decision
Alternative on Each Objective -- continued

• We denote this matrix as A1 because it reflects
  her comparisons of the three jobs with respect to
  the first objective of salary.
• The rows and columns of this matrix correspond to
  the three jobs. For example, the first row means
  that Jane believes job 1 is superior to job 2
  (and even more superior than job 3) in terms of
  salary.

115
Determining the Score of Each Decision
Alternative on Each Objective -- continued

• To find the relative scores of the three jobs on
  salary, we now apply the same two-step procedure
  we did earlier to the salary pairwise comparison
  matrix A1. That is we first divide each column
  entry by the column sum to obtain

116
Determining the Score of Each Decision
Alternative on Each Objective -- continued

• Then we average the numbers in each row to obtain
  the vector of scores for the three jobs on
  salary, denoted by S1
• That is, the scores for jobs 1, 2, and 3 on
  salary are 0.5714, 0.2857, and 0.1429. In terms
  of salary, job 1 is clearly the favorite.

117
Determining the Score of Each Decision
Alternative on Each Objective -- continued

• Next we repeat these calculations for Jane's
  other objectives.
• Each of these objective requires a pairwise
  comparison matrix, which we will denote as A2,
  A3, A4.
• Suppose that Janes pairwise comparison matrix
  for quality of life is

118
Determining the Score of Each Decision
Alternative on Each Objective -- continued

• Then the corresponding normalized matrix
  isand by averaging, we obtain
• Here, Job 3 is the clear favorite. However this
  might not have much impact because Jane puts
  relatively little weight on quality of life.

119
Determining the Score of Each Decision
Alternative on Each Objective -- continued

• For interest of work, suppose the pairwise
  comparison matrix is
• Then the same type calculations show that the
  scores for jobs 1, 2, and 3 on interest of work
  are

120
Determining the Score of Each Decision
Alternative on Each Objective -- continued

• Finally, suppose the pairwise comparison matrix
  for nearness to family is
• In this case the scores for jobs 1, 2, and 3 on
  nearness to family are

121
Determining the Best Alternative

• Lets summarize what we determined so far.
• Jane first assesses a pairwise comparison matrix
  A that measures the relative importance of each
  of her objectives to one another.
• From this matrix we obtain a vector of weights
  that summarizes the relative importance of the
  objectives.
• Next, Jane assesses a pairwise comparison matrix
  Ai for each objective i. This matrix measures how
  well each job compares to other jobs with regard
  to this objective. For each matrix Ai we obtain a
  vector of scores Si that summarizes how the jobs
  compare in terms of achieving objective i.

122
Determining the Best Alternative -- continued

• The final step is to combine the scores in the Si
  vectors with the weights in the w vector.
• Actually we have already done this. Note that the
  columns of the Table are the Si vectors we just
  obtained.
• If we form a matrix S of these score vectors and
  multiply this matrix by w, we obtain a vector of
  overall scores for each job, as shown on the next
  slide.

123
Determining the Best Alternative -- continued

• These are the same overall scores that we
  obtained earlier. As before, the largest of these
  overall scores is for job 2, so AHP suggests Jane
  should accept this job. Job 1 follows closely
  behind, with job 3 somewhat farther behind.

124
Checking the Consistency

• As mentioned earlier, any pairwise comparison
  matrix can suffer from inconsistencies.
• We now describe a procedure to check for
  inconsistencies.
• We illustrate this on the A matrix and its
  associated vector of weights w. The same
  procedure can be used on any of the Ai matrices
  and their associated weights vector Si.

125
Checking the Consistency -- continued

1. Compare Aw. For the example, we obtain
2. Find the ratio of each element of Aw to the
   corresponding weight in w and average these
   ratios. For the example, this calculation is

126
Checking the Consistency -- continued

• Compute the consistency index (labeled CI)
  aswhere n is the number of objective. For the
  example this is
• Compare CI to the random index (labeled RI) in
  the table on the next slide for the appropriate
  value of n.

127
Checking the Consistency -- continued
Random Indices for Consistency Check for AHP Example Random Indices for Consistency Check for AHP Example Random Indices for Consistency Check for AHP Example Random Indices for Consistency Check for AHP Example Random Indices for Consistency Check for AHP Example Random Indices for Consistency Check for AHP Example Random Indices for Consistency Check for AHP Example Random Indices for Consistency Check for AHP Example Random Indices for Consistency Check for AHP Example Random Indices for Consistency Check for AHP Example
N 2 3 4 5 6 7 8 9 10
RI 0 0.58 0.90 1.12 1.24 1.32 1.41 1.45 1.51

• To be perfectly consistent decision maker, each
  ratio in Step 2 should equal n. This implies that
  a perfectly consistent decision maker has CI 0.
• The values of RI in the table give the average
  values of CI if the entries in A were chosen at
  random.

128
Checking the Consistency -- continued

• If the ratio of CI to RI is sufficiently small,
  then the decision makers comparisons are
  probably consistent enough to be useful.
• Saaty suggests that if CI/RI lt 0.10, then the
  degree of consistency is satisfactory, whereas if
  CI/RI gt 0.10, serious inconsistencies exist and
  AHP may not yield meaningful results.
• In Janes example, CI/RI 0.159/0.90 0.0177,
  which is much less than 0.10. Therefore Janes
  pairwise comparison matrix A does not exhibit any
  serious inconsistencies.

129
AHP Template

• The Template file will perform all of the
  calculations, including consistency indices and
  ratios.
• The next slide shows the pairwise judgments used
  by Jane
• Using Janes information, Job 2 has the highest
  preference score, followed by Job 1 and then Job
  3
• The results match the additive weighting approach
  for the highest preference, but the order of Job
  1 and Job 3 are changed (due to subjectivity in
  weighting scaling)
• Janes pairwise comparisons are reasonably
  consistent for all matrices

130
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