Www.ischool.drexel.edu INFO 631 Prof. Glenn Booker Week 9 – Chapters 24-26 1INFO631 Week 9

www.ischool.drexel.edu

INFO 631 Prof. Glenn Booker

Week 9 – Chapters 24-26

1INFO631 Week 9


Decisions Under Risk

Ch. 24

INFO631 Week 9 2



Outline

• Introducing decisions under risk

• Different techniques– Expected value decision making– Expectation variance– Monte Carlo analysis– Decision trees– Expected value of perfect information

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• When you know the probabilities of the different outcomes and will incorporate them– Expected value decision making– Expectation variance– Monte Carlo analysis– Decision trees– Expected value of perfect information

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Expected Value Decision Making

• The value of an alternative with multiple outcomes can be thought of as the average of the random individual outcomes that would occur if that alternative were repeated a large number of times– Can use PW(i), FW(i), or AE(i)

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Expected Value of a Single Alternative

• Denali project at Mountain Systems

• Imagine 1000 parallel universes where the Denali project could be run at the same time– Should expect most favorable outcome would happen in 15% or

150 of those universes– Fair outcome would happen in 650– Least favorable outcome would happen in 200

Least Most favorable Fair favorablePW(MARR) -$1234 $5678 $9012Probability 0.20 0.65 0.15

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• Total PW(i) income generated

• Average PW(i) income in each universe

• Notice

200 * -$1234 = -246,800650 * $5678 = $3,690,700150 * $9012 = $1,351,800 $4,795,700

$4,795,700 / 1000 = $4795.70

(0.20 * -$1234) + (0.65 * $5678) + (0.15 * $9012) = $4795.70

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• General formula

• Can be used to help decide between multiple alternatives

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Expected Value of Multiple Alternatives

• Same probability• Several projects at Mountain Systems

• Expected values

– Choose Shasta, it has the highest expected value

Least Most favorable Fair favorableAlternative 20% 65% 15%Denali -$1234 $5678 $9012Shasta -2101 6601 9282Washington -3724 4104 9804

Denali (0.20 * -$1234) + (0.65 * $5678) + (0.15 * $9012) = $4795.70Shasta (0.20 * -$1201) + (0.65 * $6601) + (0.15 * $9282) = $5262.75Washington (0.20 * -$3724) + (0.65 * $4104) + (0.15 * $9804) = $3393.40

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Expectation Variance

• What if probabilities were different for each alternative?

• Comparing projects– Lassen has higher expected value but win big-lose big– Moana Loa has lower expected value but more probability of

profit

Outcome Probability AE(i)Least favorable 45% -$3494Nominal 10% 728Most favorable 45% 4811

Expected value = $665

Outcome Probability AE(i)Least favorable 10% -$200Low nominal 20% 108High nominal 30% 378Most favorable 40% 877 Expected value = $466

Lassen Moana Loa

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Monte Carlo Analysis

• Randomly generate combinations of input values and look at distribution of outcomes– Named after gambling resort in Monaco

• Use [a variant of] Zymurgenics project (different data)

Least favorable Fair Most favorable estimate estimate estimateInitial investment $500,000 $400,000 $360,000Operating & maintenance $1500 $1000 $800Development staff cost / month $49,000 $35,000 $24,500Development project duration 15 months 10 months 7 monthsIncome / month $24,000 $40,000 $56,000

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• Simulation run results

Income range Number of occurrences-$75,000 to -$50,001 3-$50,000 to -$25,001 32-$25,000 to -$1 76$0 to $24,999 258$25,000 to $49,999 655$50,000 to $74,999 921$75,000 to $99,999 1044$100,000 to $124,999 865$125,000 to $149,999 586$150,000 to $174,999 329$175,000 to $199,999 159$200,000 to $224,999 53$225,000 to $249,999 17$250,000 to $274,999 5

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332

76

258

655

921

1044

865

586

329

159

5017 5

Income Range

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Decision Trees

• Maps out possible results when there are sequences of decisions and future random events– Useful when decisions can be made in stages

• Basic Elements– Decision nodes – points in time where a decision maker makes

a decision (square)– Chance nodes – points in time where the outcome is outside the

control of the decision maker (circles)– Node sequencing

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Sample Decision Tree

Do A

Do B

-$6000

-$4000

$1200/yr

$800/yr

$2848/yr

$1437/yr

$1100/yr

$1000/yr

Do D

Do C

-$5490

$835/yr

$3615/yr

$851/yr

$1526/yr

$1037/yr

$1214/yr

Period 1 2 Years

Period 2 6 Years

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Decision Tree Analysis, Part 1

1. Add the financial consequences for each arc (PW(i), FW(i), or AE(i))– Properly adjust for time periods as required

2. Sum financial consequences from the root node to all leaf nodes

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Do A

Do B

-$6000

-$4000

$2000

$1335

$8917

$4500

$1835

$1668

Do D

Do C

-$4300

$2615

$11,315

$2665

$4778

$3245

$3800

$4917

$500

-$2050

$4850

-$3800

$2613

$1080

$1467

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Decision Tree Analysis, Part 2

3. Write probabilities for each arc out of each chance node– Probabilities out of a chance node must = 1.0

4. Roll back values from leaf nodes to root– If node is chance node, calculate expected

value at that node based on values on all nodes to its right

– If node is decision node, select the maximum profit (or minimum cost) from nodes to its right

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Do A

Do B

$1200

$3150

$1800

Do D

Do C

$2000

$1390

$4917

$500

-$2050

$4850

-$3800

$2613

$1080

$1467

3/5

2/5

5/8

3/8

3/5

2/5

2/5

3/55/8

3/8$2000

$1800

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Expected Value of Perfect Information

• Value at root node is expected value of decision tree based on current information– Current information is known to be imperfect

• Reasonable follow-on question

– Research, experimentation, prototyping, …– Might even be able to eliminate one or more paths through the tree

because you may discover them to be impossible

• Analyzed decision tree provides information that will help answer that question

“Would there be any value in taking actions that would reducethe probability of ending up in an undesirable future state?”

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• If we had a crystal ball and knew outcomes for chance nodes, we could find which path would be best– Finding best path can be repeated for all possible combinations of

random variables

• Probabilities for random variables are known– Can calculate probability for each combination of outcomes

• For each combination of outcomes, multiply its best value by probability of that combination

• Sum the results of (value * probability) for all combinations of outcomes– Sum is expected value given perfect information– Difference between sum and expected value given current information is

expected value of perfect information

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• EVPI is upper limit on how much to spend to gain further knowledge– Probably impossible to actually get perfect

information, organization should plan on spending less

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Key Points• Value of an alternative with multiple outcomes is the average of the random individual

outcomes that would occur if that alternative were repeated a large number of times (expected value)

– The alternative with the highest expected value is best• With expectation variance, differing probabilities could influence the decision

– Alternative with lower expected value might be a better choice if it also has a much lower probability of a negative outcome

• Monte Carlo analysis generates random combinations of the input variables and calculates results under those conditions

– Repeated many times and statistical distribution of outcomes is analyzed• Decision trees map out possible results when there are sequences of decisions

together with a set of future random events that have known probabilities– Useful with many possible future states and decisions can be made in stages

• The Expected value of perfect information provides answer to, “Would there be any value in taking actions that would reduce the probability of ending up in an undesirable future state?”

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Decisions Under Uncertainty

Ch. 25

INFO631 Week 9

Slides adapted from Steve Tockey – Return on Software

24



Outline

• Introducing decisions under uncertainty

• Different Techniques– Payoff matrix– Laplace Rule– Maximin Rule– Maximax Rule– Hurwicz Rule– Minimax Regret Rule

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• Used when impossible to assign probabilities to outcomes– Can also be used when you don’t want to put probabilities on

outcomes, e.g., safety-critical software system where a failure could threaten human life

• People may not react well to an assigned probability of fatality

• If probabilities can be assigned, Decision Making under Risk should be used

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Payoff Matrix

• Shows all possible outcomes to consider– One axis lists mutually exclusive alternatives– Other axis lists different states of nature

• Each state of nature is a future outcome the decision maker doesn’t have control over

– Cells have PW(i), FW(i), AE(i), …

Alternative State1 State2 State3 A1 -4010 1002 2001 A2 948 1101 4021 A3 -2005 1516 6004 A4 0 2020 5104 A5 1005 3014 2008

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Reduced Payoff Matrix

• One alternative may be “dominated” by another– Another alternative has equal or better payoff under every state

of nature

• Reduced payoff matrix has no dominated alternatives– Less work if dominated alternatives are removed

Alternative State1 State2 State3 A1 -4010 1002 2001 A2 948 1101 4021 A3 -2005 1516 6004 A4 0 2020 5104 A5 1005 3014 2008

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Laplace Rule

• Assumes each state of nature is equally likely– Sometimes called “principle of insufficient

reason”

• Calculate average payoff for each alternative across all states of nature– Same as expected value analysis for multiple

alternatives with equal probabilities

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Laplace Rule

• Example

– Alternative A4 is chosen; the highest payoff always wins!

Alternative State1 State2 State3 Average payoff A2 948 1101 4021 1933 A3 -2005 1516 6004 1838 A4 0 2020 5104 2374 A5 1005 3014 2008 2009

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Maximin Rule

• Assumes worst state of nature will happen– Most pessimistic technique– Pick alternative that has best payoff from all

worst payoffs

• Formula

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Maximin Rule

• Example

– Alternative A5 is chosen

Alternative State1 State2 State3 Worst payoff A2 948 1101 4021 948 A3 -2005 1516 6004 -2005 A4 0 2020 5104 0 A5 1005 3014 2008 1005

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Maximax Rule

• Assumes best state of nature will happen– Most optimistic technique– Pick alternative that has best payoff from all

best payoffs

• Formula

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Maximax Rule

• Example


Alternative State1 State2 State3 Best payoff A2 948 1101 4021 4021 A3 -2005 1516 6004 6004 A4 0 2020 5104 5104 A5 1005 3014 2008 3014

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Hurwicz Rule

• Assumes that without guidance people will tend to focus on extremes– Blends optimism and pessimism using a

selected ratio

• Index of optimism, , between 0 and 1 = 0.2 means more pessimism than optimism = 0.1 means more pessimism than = 0.2 = 0.85 means lots of optimism but a small

amount of pessimism (15%) remains

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Hurwicz Rule

• Formula

• Example = 0.2


Alternative State1 State2 State3 Blended payoff A2 948 1101 4021 (0.2 * 4021) + (0.8 * 948) = 1563 A3 -2005 1516 6004 (0.2 * 6004) + (0.8 * -2005) = -403 A4 0 2020 5104 (0.2 * 5104) + (0.8 * 0) = 1021 A5 1005 3014 2008 (0.2 * 3014) + (0.8 * 1005) = 1407

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Hurwicz Rule6000

4000

2000

0

-2000

6000

4000

2000

0

-2000

0.5

A2

A3

A4

A5

.25

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Minimax Regret Rule

• Minimize regret you would have if you chose wrong alternative under each state of nature– If you selected A1 and state of nature happened where A1 had

the best payoff then you would have no regrets– If you selected A1 and state of nature happened where another

alternative was better, you can quantify regret as difference between payoff you chose and best payoff under that state of nature

• Regret matrix– Need to calculate– Difference between payoff you chose and best payoff under that

state of nature

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Minimax Regret Rule – Calculate Regret matrix

• Regret matrix– Difference between payoff you chose and best payoff under that

state of nature• For State 1 – A2

o 1005 – 948 = 57

• For State 1 – A3o 1005 – (-2005) = 3010

oEtc.o NOTE: use numbers from original matrix

Alternative State1 State2 State3 A2 57 2003 1983 A3 3010 1498 0 A4 1005 994 900 A5 0 0 3966

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Minimax Regret Rule

• Choose alternative with smallest maximum regret


Alternative State1 State2 State3 Maximum regret A2 57 2003 1983 2003 A3 3010 1498 0 3010 A4 1005 994 900 1005 A5 0 0 3966 3996

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Summary of Uncertainty Rules

Decision rule Alternative selected Optimism or pessimismLaplace A4 NeitherMaximin A5 PessimismMaximax A3 OptimismHurwicz (a=0.2) A2 BlendMinimax regret A4 Pessimism

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Key Points• Uncertainty techniques used when impossible, or impractical, to assign

probabilities to outcomes• Payoff matrix shows all possible outcomes to consider• Laplace rule assumes each state of nature is equally likely

– Essentially expected value with equal probabilities• Maximin rule is most pessimistic

– Pick alternative with best payoff from all worst payoffs• Maximax rule is most optimistic

– Pick alternative with best payoff from all best payoffs• Hurwicz Rule assumes that without guidance people will tend to focus on

the extremes– Blend optimism and pessimism using selected ratio

• Minimax Regret rule minimizes regret you would have if you chose the wrong alternative under each state of nature

– Choose alternative with smallest maximum regret

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Multiple Attribute Decisions

Ch. 26

INFO631 Week 9 43


Multiple Attribute Decisions

Outline

• Introducing multiple attribute decisions

• Case study: Fly-by-Night Air

• Different kinds of “value”

• Choosing attributes

• Measurement scales

• Non-compensatory techniques

• Compensatory techniques

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IntroducingMultiple Attribute Decisions

• Previous chapters explained how to make decisions using a single criterion, money– Alternative with best PW(i), AE(i), incremental IRR, incremental

benefit-cost ratio, etc. is selected

• Aside from technical feasibility, money is almost always the most important decision criterion– But not the only one– Often, other criteria (“attributes”) must be considered and can’t

be cast in terms of money

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Case Study: Fly-by-Night (FBN) Airlines

• 10-year old regional airline with above average growth• Moving into nationwide market as no-frills carrier• As part of strategic planning, IT department charged with

examining airline reservations systems– 10 year planning horizon, effective income tax rate=37%,

after-tax MARR=15%• Research has identified five technically-viable

alternatives – Keep existing software– Buy Jupiter commercial system– Buy Sword commercial system– Buy Guppy commercial system– Develop new software in-house– Develop new software offshore

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Different Kinds of “Value”

• Decision process is all about maximizing value– Choose from available alternatives the one that

maximizes value• When value is expressed as money, decision

process may be complex but is straightforward– Money isn’t the only kind of value– Money is really only a way to quantify value

• Two kinds of value– Use-value - the ability to get things done, the

properties of the object that cause it to perform– Esteem value - the properties that make it desirable

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Choosing Attributes

• Decisions should be based on appropriate attributes– Each attribute should capture a unique dimension of decision– Set of attributes should cover important aspects of decision– Differences in attribute values should be meaningful in

distinguishing among alternatives– Each attribute should distinguish at least two alternatives

• Selection of attributes may be subjective– Too many attributes is unwieldy– Too few attributes gives poor differentiation– Potential for better decisions needs to be balanced with extra

effort of more attributes

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FBN Air: Decision Attributes

• Total cost of ownership

• In-service availability

• Liffey performance index– From Liffey Consultancy, Ltd in Dublin,

Ireland

• Alignment with existing business processes

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Measurement Scales

• Each alternative will be evaluated on each attribute

• Many ways to measure things– In fact, different “classes” of measurements– Within a class, some manipulations make sense and others

don’t • So it’s important for you to know what the different classes of

measurements are, how to recognize them, and what can and can’t be done with them.

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Measurement Scales

Scale type Description Example Operations

Nominal Two things are assigned the same symbol if they have thesame value

House style (Colonial,Contemporary, Ranch,Craftsman, Bungalow, …)

=, <>

Ordinal The order of the symbols reflectsan order defined on the attribute

Letter grades in school(A, B, C, ...)

=, <>,<, >, <=, =>

Interval Differences between the numbersreflect differences in the attribute

Temperature in degreesFahrenheit or Celsius,Calendar date

=, <>,<, >, <=, =>,

+, -

Ratio Differences and ratios between thenumbers reflect differences and

ratios of the attribute

Length in centimeters,Duration in seconds,Temperature in Kelvin

=, <>,<, >, <=, =>,

+, -, *, /

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FBN Air: Evaluation and Attribute Scales

Cost Availability Liffey index AlignmentAlternative PW(i) Months [65..135] [Ex, Vg,Ok,Pr, Vpr]Existing -$1.8M 3 99 ExcellentJupiter -$15.4M 6 115 PoorSword -$21.6M 5 128 OkGuppy -$16.7M 8 105 Very poorNew in-house -$30.3M 14 105 ExcellentNew off-shore -$17.5M 18 105 Very good

Attribute ScaleCost RatioAvailability RatioLiffey index IntervalAlignment Ordinal

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Dimensionality of Decision Techniques

• Two families of decision techniques– Differ in how attributes used

• Non-compensatory, or fully dimensioned, techniques– Each attribute treated as separate entity– No tradeoffs among attributes

• Compensatory, or single-dimensioned, techniques– Collapse attributes onto single figure of merit– Lower score in one attribute can be compensated by—or traded

off against—higher score in others

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Non-compensatory Decision Techniques

• Three will be described– Dominance– Satisficing– Lexicography

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Dominance• Compare each pair of alternatives on attribute-by-attribute basis

– Look for one alternative to be at least as good in every attribute and better in one or more

• When found, no problem deciding– One alternative is clearly superior to the other, inferior can be discarded

• May not lead to selecting one single alternative– Good for filtering alternatives and reducing work using other techniques

• In FBN Air, Jupiter dominates Guppy

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Satisficing

• Sometimes called “method of feasible ranges”– Establish acceptable ranges of attribute

values– Alternatives with any attributes outside

acceptable range are discarded

• May not lead to selecting one single alternative– Good for filtering alternatives and reducing

work using other techniques56INFO631 Week 9


Satisficing

• Can lead to selecting one alternative when used with an iterative propose-then-evaluate process

• Iterative version is appropriate when satisfactory performance, rather than optimal performance, is good enough– If optimal performance needed, always identify several

alternatives that meet satisficing criteria then do further decision analysis with one of other techniques

Repeat Propose a new solution Evaluate that solution against the decision attributesUntil the solution is within the acceptable range for all decision attributes

Note: Stops when 1st acceptable solution is proposed

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Lexicography• Two previous techniques assume attributes have equal importance

– If one attribute is far more important than others, final choice could be made on that one attribute alone

• If alternatives have identical values for most-important attribute, use next-most-important attribute to break tie– If still tied, compare next most important attribute, …– Continue until a single alternative chosen or all alternatives evaluated

• FBN Air– Alignment might be #1, eliminates all but Existing and In-house– Cost might be #2, eliminates in-house

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Compensatory Decision Techniques

• Attribute values converted into common “figure of merit”– Units for common scale are usually arbitrary– If common scale is at least interval scale then

scores can be compared meaningfully

• Two will be presented– Nondimensional scaling– Additive Weighting– Analytical Hierarchy Process (see text)

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Non-Dimensional Scaling• Convert attribute values into common scale so they can be added together

to make composite score for each alternative– Alternative with best composite score is selected– All attributes are defined to have equal importance

• Common scale needs same range for all attributes– Must also follow same trend on desirability; most-preferred value needs to

always be biggest or always be smallest common scale value

• Formula for converting attributes, as long as interval or ratio-scaled, into the common scale

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FBN Air: Scaled Attributes

Cost Availability Liffey indexAlternative [0..50] [0..50] [0..50] TotalExisting 50.0 50.0 0.0 100.0Jupiter 26.1 40.0 27.6 93.7Sword 15.3 43.3 50.0 108.6Guppy 23.9 33.3 10.3 67.5New in-house 0.0 13.3 10.3 23.6New off-shore 22.5 0.0 10.3 32.8

Note: Let’s entirely arbitrarily chose the common scale to be 0..50. In FBN’s case, • lower cost is better so lowest cost alternative highest common rating• higher Liffey Index (LI) is better so the highest LI alternative highest common rating.• Best = Sword

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Non-Dimensional Scaling and Ordinal Attributes

• When decision includes ordinal scaled attributes, you will need to:– Ignore ordinal-scaled attributes– Refine ordinal-scaled attributes to use interval or ratio scales and

include them in nondimensional scaling– Do nondimensional scaling for all interval- and ratio-scaled attributes

then finish using a non-compensatory technique

Alternative Total AlignmentExisting 100.0 ExcellentJupiter 93.7 PoorSword 108.6 OkGuppy 67.5 Very poorNew in-house 23.6 ExcellentNew off-shore 32.8 Very good

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Additive Weighting

• Identical to non-dimensional scaling except attributes have different “weights” or degrees of influence on the decision– An attribute that’s more important will have more influence on

outcome– Most popular

• Step 1: select common scale and convert all interval and ratio-scaled attribute values into that scale– Just like non-dimensional scaling

• Step 2: assign weights based on relative importance– Many different approaches to this– Recommended approach is

• Each attribute given “points” corresponding to importance• Weight for each attribute is its points divided by sum of points

across all attributes63INFO631 Week 9


FBN Air: Weighting the Attributes

• Suppose FBN Air gives point values as shown for ratio and interval-scaled attributes

Attribute Points WeightCost 50 50 / ( 50 + 10 + 25 ) = 0.588Availability 10 10 / ( 50 + 10 + 25 ) = 0.118Liffey index 25 25 / ( 50 + 10 + 25 ) = 0.294

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Additive Weighting

• Step 3: calculate each alternative’s total weighted score– Example Existing = (0.588*50)+(0.118*50)+(0.294*0) = 35.3

• Same as non-dimensional scaling, decision is made on total score if there are no relevant ordinal-scaled attributes

Cost Availability Liffey indexAlternative (0.588) (0.118) (0.294) TotalExisting 50.0 50.0 0.0 35.3Jupiter 26.1 40.0 27.6 28.2Sword 15.3 43.3 50.0 28.8Guppy 23.9 33.3 10.3 21.0New in-house 0.0 13.3 10.3 4.6New off-shore 22.5 0.0 10.3 16.3

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Key Points• Aside from technical feasibility, money is almost always the most important

decision criterion but it’s not always the only one• Use values can usually be quantified in terms of money• Esteem values can't be quantified in terms of money

– Decisions involving more than one attribute are almost inevitable• Choose decision attributes to cover all relevant use values and esteem

values• Several different classes of measurement

– Nominal, Ordinal, Interval, and Ratio– Within each class, some comparisons will make sense and others won’t

• Non-compensatory techniques treat each attribute as a separate entity– Dominance, Satisficing, Lexicography

• Compensatory techniques allow better performance on one attribute to compensate for poorer performance in another

– Nondimensional Scaling, Additive Weighting

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Www.ischool.drexel.edu INFO 631 Prof. Glenn Booker Week 9 – Chapters 24-26 1INFO631 Week 9