to accompany quantitative analysis for management, 9e by render/stair/hanna 3-1 © 2006 by prentice...

To accompany Quantitative Analysis for Management, 9e by Render/Stair/Hanna

Prepared by Lee Revere and John LargePrepared by Lee Revere and John Large

Chapter 3Chapter 3

Decision AnalysisDecision Analysis

Learning ObjectivesLearning ObjectivesStudents will be able to:

1. List the steps of the decision-making process.

2. Describe the types of decision-making environments.

3. Make decisions under uncertainty.

4. Use probability values to make decisions under risk.

5. Develop accurate and useful decision trees.

6. Revise probabilities using Bayesian analysis.

7. Use computers to solve basic decision-making problems.

8. Understand the importance and use of utility theory in decision theory.

Chapter OutlineChapter Outline

3.1 Introduction

3.2 The Six Steps in Decision Theory

3.3 Types of Decision-Making Environments

3.4 Decision Making under Uncertainty

3.5 Decision Making under Risk

3.6 Decision Trees

3.7 How Probability Values Are Estimated by Bayesian Analysis

3.8 Utility Theory

IntroductionIntroduction

Decision theory is an analytical and systematic way to tackle problems.

A good decision is based on logic.

The Six Steps in The Six Steps in Decision TheoryDecision Theory

1. Clearly define the problem at hand.

2. List the possible alternatives.

3. Identify the possible outcomes.

4. List the payoff or profit of each combination of alternatives and outcomes.

5. Select one of the mathematical decision theory models.

6. Apply the model and make your decision.

John Thompson’s John Thompson’s Backyard Storage Backyard Storage

ShedsShedsDefine problem To manufacture or market

backyard storage sheds

List alternatives 1. Construct a large new plant

2. A small plant

3. No plant at all

Identify outcomes The market could be favorable or unfavorable for storage sheds

List payoffs List the payoff for each state of nature/decision alternative combination

Select a model Decision tables and/or trees can be used to solve the problem

Apply model and make decision

Solutions can be obtained and a sensitivity analysis used to make a decision

Decision Table Decision Table for Thompson Lumberfor Thompson Lumber

Alternative

State of Nature

Favorable Market ($)

Unfavorable Market ($)

Construct a large plant

200,000 -180,000

Construct a small plant

100,000 -20,000

Do nothing 0 0

Types of Decision-Types of Decision-Making EnvironmentsMaking Environments

Type 1: Decision making under certainty.Decision maker knows with certainty

the consequences of every alternative or decision choice.

Type 2: Decision making under risk.The decision maker does know the

probabilities of the various outcomes.

Decision making under uncertainty.The decision maker does not know the

probabilities of the various outcomes.

Decision MakingDecision Making under Uncertainty under Uncertainty

Maximax

Maximin

Equally likely (Laplace)

Criterion of realism

Minimax

Decision Table for Decision Table for Thompson LumberThompson Lumber

Alternative

State of Nature

200,000 -180,000

100,000 -20,000

Do nothing 0 0

Maximax: Optimistic Approach Find the alternative that maximizes the maximum

outcome for every alternative.

Thompson Lumber: Thompson Lumber: Maximax SolutionMaximax Solution

Alternative

State of Nature

Maximax

200,000 -180,000 200,000

100,000 -20,000 100,000

Do nothing 0 0 0

Decision Table for Decision Table for Thompson LumberThompson Lumber

Alternative

State of Nature

200,000 -180,000

100,000 -20,000

Do nothing 0 0

Maximin: Pessimistic Approach Choose the alternative with maximum

minimum output.

Thompson Lumber: Thompson Lumber: Maximin SolutionMaximin Solution

Alternative

State of Nature

MaximinFavorable Market ($)

200,000 -180,000 -180,000

100,000 -20,000 -20,000

Do nothing 0 0 0

Thompson Lumber: Thompson Lumber: HurwiczHurwicz

Criterion of Realism (Hurwicz) Decision maker uses a weighted average based

on optimism of the future.

Alternative

State of Nature

200,000 -180,000

100,000 -20,000

Do nothing 0 0

Thompson Lumber: Thompson Lumber: Hurwicz SolutionHurwicz Solution

CR = α*(row max)+(1- α)*(row min)

Alternative

State of NatureCriterion

of Realism or

Weighted Average (α = 0.8) ($)

200,000 -180,000 124,000

100,000 -20,000 76,000

Do nothing 0 0 0

Equally likely (Laplace)Assume all states of nature to be

equally likely, choose maximum Average.

Alternative

State of Nature

200,000 -180,000

100,000 -20,000

Do nothing 0 0

Alternative

State of Nature

Avg.Favorable Market ($)

200,000 -180,000 10,000

100,000 -20,000 40,000

Do nothing 0 0 0

Thompson Lumber;Thompson Lumber;Minimax RegretMinimax Regret

Minimax Regret: Choose the alternative that minimizes the

maximum opportunity loss .

Alternative

State of Nature

200,000 -180,000

100,000 -20,000

Do nothing 0 0

Thompson Lumber:Thompson Lumber:Opportunity Loss Opportunity Loss

TableTable

Alternative

State of Nature

200,000 – 200,000 = 0

0- (-180,000) = 180,000

200,000 - 100,000 = 100,000

0- (-20,000) = 20,000

Do nothing 200,000 – 0 = 0 0 – 0 = 0

Thompson Lumber:Thompson Lumber:Minimax Regret Minimax Regret

SolutionSolution

Alternative

State of NatureMaximum

Opportunity LossFavorable

Market ($)Unfavorable Market ($)

0 180,000 180,000

100,000 20,000 100,000

Do nothing 200,000 0 200,000

In-Class Example 1In-Class Example 1

Let’s practice what we’ve learned. Use the decision table below to compute (1) Mazimax (2) Maximin (3) Minimax regret

Alternative

State of Nature

Market

Average

Market

75,000 25,000 -40,000

100,000 35,000 -60,000

Do nothing 0 0 0

In-Class Example 1:In-Class Example 1:MaximaxMaximax

Alternative

State of Nature

MaximaxGood

Market

Average

Market

75,000 25,000 -40,000 75,000

100,000 35,000 -60,000 100,000

Do nothing 0 0 0 0

In-Class Example 1:In-Class Example 1:MaximinMaximin

Alternative

State of Nature

MaximinGood

Market

Average

Market

75,000 25,000 -40,000 -40,000

100,000 35,000 -60,000 -60,000

Do nothing 0 0 0 0

In-Class Example 1:In-Class Example 1:Minimax Regret Minimax Regret

Opportunity Loss TableOpportunity Loss Table

Alternative

State of Nature

Maximum Opp. Loss

Market

Average

Market

25,000 75,000 40,000 40,000

0 0 60,000 60,000

Do nothing 100,000 35,000 0 100,000

Decision Making under Decision Making under RiskRisk

Expected Monetary Value:

In other words:

EMVAlternative n = Payoff 1 * PAlt. 1 + Payoff 2 * PAlt. 2 + … + Payoff n * PAlt.

nature. of stagesof numbern where

SP SPayoffative)EMV(Altern j

Thompson Lumber:Thompson Lumber:EMVEMV

Alternative

State of Nature

EMVFavorable Market ($)

200,000 -180,000200,000*0.5 +

(-180,000)*0.5 = 10,000

100,000 -20,000100,000*0.5 +

(-20,000)*0.5 = 40,000

Do nothing 0 0 0*0.5 + 0*0.5 = 0

Probabilities 0.50 0.50

Thompson Lumber:Thompson Lumber: EV|PI and EMV EV|PI and EMV

SolutionSolution

Alternative

State of Nature

EMVFavorable Market

Unfavorable Market

200,000 -180,000 10,000

100,000 -20,000 40,000

Do nothing 0 0 0

EV׀PI200,000*

0.5 = 100,000

0*0.5 = 0

Expected Value of Expected Value of Perfect Information Perfect Information

((EVPIEVPI)) EVPI places an upper bound on what

one would pay for additional information.

EVPI is the expected value with perfect information minus the maximum EMV.

Expected Value with Expected Value with Perfect Information (Perfect Information (EV|EV|

PIPI))

In other words

EV׀PI = Best Outcome of Alt 1 * PAlt. 1 + Best Outcome of Alt 2 * PAlt. 2 +… + Best Outcome of Alt n * PAlt. n

nature. of states ofnumber n

)P(S*nature) of statefor outcome(Best PI|EVn

Expected Value of Expected Value of Perfect InformationPerfect Information

EVPI = EV|PI - maximum EMV

Expected value with perfect information

Expected value with no additional

information

Thompson Lumber:Thompson Lumber:EVPI SolutionEVPI Solution

EVPIEVPI = expected value with perfect

information - max(EMVEMV)

= $200,000*0.50 + 0*0.50 - $40,000

= $60,000 From previous slide

Let’s practice what we’ve learned. Using the table below compute EMV, EV׀PI, and EVPI.

Alternative

State of Nature

Market

Average

Market

75,000 25,000 -40,000

100,000 35,000 -60,000

Do nothing 0 0 0

In-Class Example 2:In-Class Example 2: EMV and EV EMV and EV׀׀PIPI

SolutionSolution

Alternative

State of Nature

EMVGood

Market

Average

Market

75,000 25,000 -40,000 21,250

100,000 35,000 -60,000 27,500

Do nothing 0 0 0 0

In-Class Example 2:In-Class Example 2:EVPI SolutionEVPI Solution

EVPIEVPI = expected value with perfect

information - max(EMVEMV)

= $100,000*0.25 + 35,000*0.50 +0*0.25

= $ 42,500 - 27,500

= $ 15,000

Expected Opportunity Expected Opportunity LossLoss

EOL is the cost of not picking the best solution.EOL = Expected Regret

Thompson Lumber: EOLThompson Lumber: EOLThe Opportunity Loss TableThe Opportunity Loss Table

Alternative

State of Nature

200,000 – 200,000

0- (-180,000)

200,000 - 100,000

0 – (-20,000)

Do nothing 200,000 - 0 0-0

Thompson Lumber: Thompson Lumber: EOL TableEOL Table

Alternative

State of Nature

200,000 -180,000

100,000 -20,000

Do nothing 0 0

Thompson Lumber: Thompson Lumber: EOL SolutionEOL Solution

Alternative EOL

Large Plant (0.50)*$0 + (0.50)*($180,000)

$90,000

Small Plant (0.50)*($100,000)+ (0.50)(*$20,000)

$60,000

Do Nothing (0.50)*($200,000) + (0.50)*($0)

$100,000

Thompson Lumber:Thompson Lumber:Sensitivity AnalysisSensitivity Analysis

EMV(Large Plant):

= $200,000P - (1-P)$180,000

EMV(Small Plant):

= $100,000P - $20,000(1-P)

EMV(Do Nothing):

= $0P + 0(1-P)

Thompson Lumber:Thompson Lumber:Sensitivity AnalysisSensitivity Analysis (continued)(continued)

-200000

-150000

-100000

-50000

100000

150000

200000

250000

0 0.2 0.4 0.6 0.8 1

Values of P

Point 1 Point 2Small Plant

Large Plant EMV

Marginal AnalysisMarginal Analysis

P = probability that demand > a given supply.

1-P = probability that demand < supply. MP = marginal profit. ML = marginal loss. Optimal decision rule is: P*MP (1-P)*ML

MLMPML

Marginal Analysis -Marginal Analysis -Discrete DistributionsDiscrete Distributions

Steps using Discrete Distributions: Determine the value for P.P. Construct a probability table and add a

cumulative probability column. Keep ordering inventory as long as the

probability of selling at least one additional unit is greater than P.P.

Café du Donut:Café du Donut:Marginal AnalysisMarginal Analysis

Daily Sales

(Cartons)

Probability of Sales

at this Level

Probability that Sales Will

Be at this Level or Greater

4 0.05 1.00

5 0.15 0.95

6 0.15 0. 80

7 0.20 0.65

8 0.25 0.45

9 0.10 0.20

10 0.10 0.10

Café du Donut sells a dozen donuts for $6. It costs $4 to make each dozen. The following table shows the discrete distribution for Café du Donut sales.

Café du Donut: Café du Donut: Marginal Analysis SolutionMarginal Analysis Solution

Marginal profit = selling price - cost

= $6 - $4 = $2Marginal loss = cost

Therefore:

667.06

Café du Donut: Café du Donut: Marginal Analysis SolutionMarginal Analysis Solution

Daily Sales

(Cartons)

Probability of Sales

at this Level

Probability that Sales Will

Be at this Level or Greater

4 0.05 1.00 ≥ 0.66

5 0.15 0.95 ≥ 0.66

6 0.15 0. 80 ≥ 0.66

7 0.20 0.65

8 0.25 0.45

9 0.10 0.20

10 0.10 0.10

Let’s practice what we’ve learned. You sell cases of goods for $15/case, the raw materials cost you $4/case, and you pay $1/case commission.

In-Class Example 3:In-Class Example 3:SolutionSolution

Daily Sales Cases

Probability of Sales at this Level

Probability that Sales Will Be at this

Level or Greater 4 0.1 1.0 > .286 5 0.1 .9 > .286 6 0.4 .8 > .286 7 0.3 .4 > .286 8 0.1 .1 1.00

MP = $15-$4-$1 = $10 per case ML = $4P>= $4 / $10+$4 = .286

Marginal AnalysisMarginal AnalysisNormal DistributionNormal Distribution

= average or mean sales = standard deviation of sales MPMP = marginal profit MLML = Marginal loss

Marginal Analysis -Marginal Analysis -Discrete DistributionsDiscrete Distributions

• Steps using Normal Distributions: Determine the value for P.

Locate P on the normal distribution. For a given area under the curve, we find Z from the standard Normal table.

Using we can now solve for:

Marginal Analysis:Marginal Analysis:Normal Curve ReviewNormal Curve Review

Pcumulative

Marginal Analysis -Marginal Analysis -Normal Curve ReviewNormal Curve Review

area = .30

Use table to find Z

area = .70

Joe’s Newsstand Joe’s Newsstand ExampleExample

Joe sells newspapers for $1.00 each.

Papers cost him $.40 each. His average

daily demand is 50 papers with a standard

deviation of 10 papers. Assuming sales

follow a normal distribution, how many

papers should Joe stock?

MLML = $0.40 MPMP = $0.60 = Average demand = 50 papers per

day = Standard deviation of demand =

Joe’s Newsstand ExampleJoe’s Newsstand Example (continued)(continued)

Step 1:

MPMPMLML

MLMLPP

Joe’s Newsstand ExampleJoe’s Newsstand Example (continued)(continued)

Step 2: Look on the Normal table for

PP = 0.6 (i.e., 1 - .4) ZZ = 0.25,

XX** = 10 * 0.25 + 50 = 52.5 or 53 newspapers = 10 * 0.25 + 50 = 52.5 or 53 newspapers

Joe’s Newsstand Joe’s Newsstand Example BExample B

Joe also offers his clients the “Times” for $1.00. This paper is flown in from out of state, which greatly increases its costs. Joe pays $.80 for the “Times.” The “Times” has average daily sales of 100 papers with a standard deviation of 10. Assuming sales follow a normal distribution, how many “Times” papers should Joe stock?

MLML = $0.80 MPMP = $0.20 = Average demand = 100 papers per day = Standard deviation of demand = 10

Joe’s Newsstand Joe’s Newsstand Example BExample B (continued)(continued)

Step 1:

MPMPMLML

MLMLPP

Step 2:

Z = 0.80

= -0.84 for an area of 0.80

or: X=-8.4+100 or 92 newspapers

Joe’s Newsstand Joe’s Newsstand Example BExample B (continued)(continued)

Decision Making with Decision Making with Uncertainty: Using the Uncertainty: Using the

Decision TreesDecision Trees

Decision treesDecision trees enable one to look at

decisions:

With many alternativesalternatives and states states

of nature,of nature,

which must be made in sequence.

Five Steps toFive Steps toDecision Tree AnalysisDecision Tree Analysis

1. Define the problem.

2. Structure or draw the decision tree.

3. Assign probabilities to the states of nature.

4. Estimate payoffs for each possible combination of alternatives and states of nature.

5. Solve the problem by computing expected monetary values (EMVs) for each state of nature node.

Structure of Decision Structure of Decision TreesTrees

A graphical representation where:

A decision node from which one of several alternatives may be chosen.

A state-of-nature node out of which one state of nature will occur.

Thompson’s Decision Thompson’s Decision TreeTree

A A Decision Decision

NodeNode

A State of A State of Nature Nature NodeNode

Favorable Market

Unfavorable Market

Favorable Market

Unfavorable Market

Construct

Large Plant

Construct Small Plant

Do Nothing

Step 1: Define the problem

Lets re-look at John Thompson’s decision regarding storage sheds. This simple problem can be depicted using a decision tree.

Step 2: Draw the tree

A A Decision Decision

NodeNode

A State of A State of Nature NodeNature Node Favorable (0.5)

Market

Unfavorable (0.5) Market

Favorable (0.5) Market

Construct

Large Plant

Do Nothing

$200,000$200,000

-$180,000-$180,000

$100,000$100,000

-$20,000-$20,000

Step 3: Assign probabilities to the states of nature.

Step 4: Estimate payoffs.

A Decision A Decision NodeNode

A State A State of Nature of Nature NodeNode

Construct

Large Plant

Do Nothing

$200,000$200,000

-$180,000-$180,000

$100,000$100,000

-$20,000-$20,000

EMV EMV =$40,000=$40,000

EMVEMV=$10,000=$10,000

Step 5: Compute EMVs and make decision.

Thompson’s Decision:Thompson’s Decision:A More Complex A More Complex

ProblemProblem John Thompson has the opportunity of

obtaining a market survey that will give additional information on the probable state of nature. Results of the market survey will likely indicate there is a percent change of a favorable market. Historical data show market surveys accurately predict favorable markets 78 % of the time. Thus P(Fav. Mkt / Fav. Survey Results) = .78

Likewise, if the market survey predicts an unfavorable market, there is a 13 % chance of its occurring. P(Unfav. Mkt / Unfav. Survey Results) = .13

Now that we have redefined the problem (Step 1), let’s use this additional data and redraw Thompson’s decision tree (Step 2).

Step 3: Assign the new probabilities to the states of nature.

Step 4: Estimate the payoffs.

Step 5: Compute the EMVs and make decision.

John Thompson DilemmaJohn Thompson Dilemma

John Thompson is not sure how much value to place on market survey. He wants to determine the monetary worth of the survey. John Thompson is also interested in how sensitive his decision is to changes in the market survey results. What should he do?

Expected Value of Sample Information

Sensitivity Analysis

Expected Value of Expected Value of Sample InformationSample Information

Expected value of best decision withwith sample information, assuming no cost to gather it

Expected value of best decision withoutwithout sample information

EVSIEVSI =

EVSI for Thompson Lumber = $59,200 - $40,000 = $19,200Thompson could pay up to $19,200 and come out ahead.

Calculations for Thompson Calculations for Thompson Lumber Sensitivity Lumber Sensitivity

AnalysisAnalysis

2,400$104,000

($2,400)($106,400)1) EMV(node

Equating the EMVEMV(node 1) to the EMV of not conducting the survey, we have

0.36$104,000

$37,600

$37,600$104,000

$40,000$2,400$104,000

In-Class Problem 3In-Class Problem 3

Let’s practice what we’ve learned

Leo can purchase a historic home for $200,000 or land in a growing area for $50,000. There is a 60% chance the economy will grow and a 40% change it will not. If it grows, the historic home will appreciate in value by 15% yielding a $30,00 profit. If it does not grow, the profit is only $10,000. If Leo purchases the land he will hold it for 1 year to assess the economic growth. If the economy grew during the first year, there is an 80% chance it will continue to grow. If it did not grow during the first year, there is a 30% chance it will grow in the next 4 years. After a year, if the economy grew, Leo will decide either to build and sell a house or simply sell the land. It will cost Leo $75,000 to build a house that will sell for a profit of $55,000 if the economy grows, or $15,000 if it does not grow. Leo can sell the land for a profit of $15,000. If, after a year, the economy does not grow, Leo will either develop the land, which will cost $75,000, or sell the land for a profit of $5,000. If he develops the land and the economy begins to grow, he will make $45,000. If he develops the land and the economy does not grow, he will make $5,000.

In-Class Problem 3: In-Class Problem 3: SolutionSolution

Purchase historic home

Purchase land

Economy grows (.6)

No growth (.4)

Economy grows (.6)

No growth (.4)

Build house

Economy grows (.8)

No growth (.2)

Sell land

Develop land

Sell land

Economy grows (.3)

No growth (.7)

In-Class Problem 3: In-Class Problem 3: SolutionSolution

Purchase historic home

Purchase land

$35,000

$22,000 Economy grows (.6) $30,000

No growth (.4)

$10,000

Economy grows (.6)

No growth (.4)

$35,000

$47,000

Build house

$47,000

Economy grows (.8) $55,000

$15,000No growth (.2)

Sell land

$15,000

$17,000

Develop land

Sell land

$5,000

Economy grows (.3)

No growth (.7)

$45,000

$5,000

$17,000

Estimating Probability Estimating Probability Values with BayesianValues with Bayesian

Management experience or intuition History Existing data Need to be able to revise

probabilities based upon new data

Posteriorprobabilities

Priorprobabilities New data

Baye’s Theorem

Bayesian AnalysisBayesian Analysis

Market Survey Reliability in Predicting Actual States of Nature

Actual States of Nature

Result of Survey Favorable

Market (FM)

Unfavorable

Market (UM)

Positive (predicts

favorable market

for product)

P(survey positive|FM)

= 0.70

P(survey positive|UM)

= 0.20

Negative (predicts

unfavorable

market for

product)

P(survey

negative|FM) = 0.30

P(survey negative|UM)

= 0.80

The probabilities of a favorable / unfavorable state of nature can be obtained by analyzing the Market Survey Reliability in Predicting Actual States of Nature.

Bayesian Analysis Bayesian Analysis (continued):(continued): Favorable SurveyFavorable Survey

Probability Revisions Given a Favorable Survey

Conditional

Probability

Posterior

Probability

Nature

P(Survey positive|State of Nature

Prior ProbabilityJoint Probability

FM 0.70 * 0.50 0.350.45

0.35 = 0.78

UM 0.20 * 0.500.45

0.10 0.10 = 0.22

0.45 1.00

Bayesian Analysis Bayesian Analysis (continued):(continued): Unfavorable SurveyUnfavorable Survey

Probability Revisions Given an Unfavorable Survey

Conditional

Probability

Posterior

Probability

Nature

P(Survey

negative|State

of Nature)

Prior Probability

Joint Probability

FM 0.30 * 0.50 0.150.55

0.15 = 0.27

UM 0.80 * 0.50 0.400.55

0.40 = 0.73

0.55 1.00

Decision Making Using Decision Making Using Utility TheoryUtility Theory

Utility assessment assigns the worst outcome a utility of 0, and the best outcome, a utility of 1.

A standard gamble is used to determine utility values.

When you are indifferent, the utility values are equal.

Standard Gamble for Standard Gamble for Utility AssessmentUtility Assessment

Best outcomeUtility = 1

Worst outcomeUtility = 0

Other outcomeUtility = ??

(1-p)Alternative 1

Alternative 2

Simple Example: Utility Simple Example: Utility TheoryTheory

$5,000,000

$2,000,000

Accept Offer

Reject Offer

Heads(0.5)

Tails(0.5)

Let’s say you were offered $2,000,000 right now on a chance to win $5,000,000. The $5,000,000 is won only if you flip a coin and get tails. If you get heads you lose and get $0. What should you do?

Real Estate Example: Real Estate Example: Utility TheoryUtility Theory

Jane Dickson is considering a 3-year real estate investment. There is an 80 % chance the real estate market will soar and a 20 % chance it will bust. In a good market the real estate investment will pay $10,000, in an unfavorable market it is $0. Of course, she could leave her money in the bank and earn a $5,000 return. What should she do?

Real Estate Example: Real Estate Example: SolutionSolution

$10,000U($10,000) = 1.0

0U(0)=0

$5,000U($5,000)=p=0.80

p= 0.80

(1-p)= 0.20

Invest in

Real Estate

Invest in Bank

Utility Curve for Jane Utility Curve for Jane DicksonDickson

00.10.20.30.40.50.60.70.80.9

$- $2,000 $4,000 $6,000 $8,000 $10,000

Monetary Value

Utility

Preferences for RiskPreferences for Risk

Monetary Outcome

Avoider

Seeker

Risk In

differe

Decision Facing Mark Decision Facing Mark SimkinSimkin

Tack landspoint up (0.45)

Tack lands point down (0.55)

$10,000

-$10,000

Alternative 1

Mark plays

the game

Alternative 2

Mark does not play the game

Utility Curve for Mark Utility Curve for Mark SimkinSimkin

0.20.3

0.70.8

-$20,000 -$10,000 $0 $10,000 $20,000 $30,000

Thompson Decision Tree Thompson Decision Tree Problem Using QM for Problem Using QM for

WindowsWindows

Thompson Decision Tree Thompson Decision Tree Problem Using ExcelProblem Using Excel

to accompany quantitative analysis for management, 9e by render/stair/hanna 3-1 © 2006 by prentice...

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