tm 732 engr. economics for managers decision analysis

TM 732Engr. Economics for

ManagersDecision AnalysisDecision Analysis

GoferBrokeAlternative Oil DryDrill fer Oil 700 -100Sell Land 90 90Chance 0.25 0.75

Prototype Ex. 2

Low Gr Med. Gr. High Gr.Crane 1 43,000 60,000 68,000Crane 2 37,000 52,000 75,000Crane 3 30,000 57,000 80,000

Digger Construction is interested in purchasing 1 of 3 cranes. The cranes differ in capacity, age, and mechanical condition, but each is fully capable of performing the jobs expected. The firm anticipates a growing market and that there will be sufficient demand to justify each of the cranes. However, low, medium, and high growth estimates result in different cash flow profiles for each crane. Based on ATCF at 15%, the analyst estimates the following NPWs for each of the cranes for each of the growth market conditions.

Digger ConstructionPayoff

Alternative Low Gr Med. Gr. High Gr.Crane 1 43,000 60,000 68,000Crane 2 37,000 52,000 75,000Crane 3 30,000 57,000 80,000Prob 0.2 0.5 0.3

Decision Matrix

Decision Model for Lift TruckNo. Trucks Required

Alternatives 4 5 6 A1 - Lease 4 18,000 20,000 22,000 A2 - Lease 5 20,000 20,000 22,000 A3 - Lease 6 21,000 21,000 21,000 A4 - Buy 4 12,000 14,500 15,000 A5 - Buy 5 14,000 14,000 16,000 A6 - Buy 6 14,000 15,500 18,000

Probability 0.30 0.40 0.30

EUAW

Matrix Decision Model

p1 p2 -- pk -- pmS1 S2 -- Sk -- Sm

A1 V(11) V(12) -- V(1k) -- V(1m)A2 V(1) V(22) -- V(2k) -- V(2m) : : : : : : : : : :Aj V(j1) V(j2) -- V(jk) -- V(jm) : : : : : : : : : :An V(n1) V(n2) -- V(nk) -- V(nm)

Aj = alternative strategy j under decision makers controlSk = a state or possible future that can occur given Aj

pk = the probability state Sk will occur

Matrix Decision Model

p1 p2 -- pk -- pmS1 S2 -- Sk -- Sm

A1 V(11) V(12) -- V(1k) -- V(1m)A2 V(1) V(22) -- V(2k) -- V(2m) : : : : : : : : : :Aj V(j1) V(j2) -- V(jk) -- V(jm) : : : : : : : : : :An V(n1) V(n2) -- V(nk) -- V(nm)

V(jk) = the value of outcome jk (terms of $, time, distance, . . )jk = the outcome of choosing Aj and having state Sk occur

Decisions Under Certainty

p=1S

A1 V(1)A2 V() : : : :Aj V(j) : : : :An V(n)

Decisions Under Certainty

p=1S

A1 V(1)A2 V() : : : :Aj V(j) : : : :An V(n)

Investor wishes to invest $10,000 in one of five govt. securities. Effective yields are:

A1 = 8.0%A2 = 7.3%A3 = 8.7%A4 = 6.0%A5 = 6.5%

choose A3.

Maximin

Select Aj: maxjminkV(jk)e.g., Find the min payoff for each alternative.

PayoffAlternative Low Gr Med. Gr. High Gr.

Crane 1 43,000 60,000 68,000Crane 2 37,000 52,000 75,000Crane 3 30,000 57,000 80,000Prob 0.2 0.5 0.3

Maximin

Select Aj: maxjminkV(jk)e.g., Find the min payoff for each alternative.

Find the maximum of minimums Select Crane 1

PayoffAlternative Low Gr Med. Gr. High Gr.

Crane 1 43,000 60,000 68,000Crane 2 37,000 52,000 75,000Crane 3 30,000 57,000 80,000Prob 0.2 0.5 0.3

Choose best alternative when comparing worst possible outcomes for each alternative.

Maximin

Select Aj: maxjminkV(jk)e.g., Find the min payoff for each alternative.

Find the maximum of minimums Sell LandChoose best alternative when comparing worst possible outcomes for each alternative.

Alternative Oil DryDrill fer Oil 700 -100Sell Land 90 90Chance 0.25 0.75

MiniMax

Select Aj: maxjminkV(jk)e.g., Find the max payoff for each alternative.

PayoffAlternative Low Gr Med. Gr. High Gr.

Crane 1 43,000 60,000 68,000Crane 2 37,000 52,000 75,000Crane 3 30,000 57,000 80,000Prob 0.2 0.5 0.3

MiniMax

Select Aj: maxjminkV(jk)e.g., Find the max payoff for each alternative.

Find the minimum of maximums Select Crane 1

Choose worst alternative when comparing bestpossible outcomes for each alternative.

PayoffAlternative Low Gr Med. Gr. High Gr.

Crane 1 43,000 60,000 68,000Crane 2 37,000 52,000 75,000Crane 3 30,000 57,000 80,000Prob 0.2 0.5 0.3

MiniMax

Select Aj: maxjminkV(jk)e.g., Find the max payoff for each alternative.

Find the minimum of maximums Sell Land

Choose worst alternative when comparing best possible outcomes for each alternative.

Alternative Oil DryDrill fer Oil 700 -100Sell Land 90 90Chance 0.25 0.75

Class ProblemProbability

Alternatives S1 S2 S3

A1 15,163 13,409 11,962A2 16,536 13,465 10,934A3 18,397 14,240 10,840

Choose best alternative usinga. Maximax criteria

b. Minimin criteria

Class ProblemProbability

Alternatives S1 S2 S3

A1 15,163 13,409 11,962A2 16,536 13,465 10,934A3 18,397 14,240 10,840

Choose best alternative usinga. Maximax criteria (best of the best)

maxj{15163, 16536, 18397} = 18,397

choose A3

ProbabilityAlternatives S1 S2 S3

A1 15,163 13,409 11,962A2 16,536 13,465 10,934A3 18,397 14,240 10,840

Class Problem

Choose best alternative usinga. Minimin criteria (worst of the worst)

minj{11,962 10,934 10,840} = 10,840

choose A3

Maximum LikelihoodPayoff

Alternative Low Gr Med. Gr. High Gr.Crane 1 43,000 60,000 68,000Crane 2 37,000 52,000 75,000Crane 3 30,000 57,000 80,000Prob 0.2 0.5 0.3

Assume S2 a certainty

Maximum LikelihoodPayoff

Alternative Low Gr Med. Gr. High Gr.Crane 1 43,000 60,000 68,000Crane 2 37,000 52,000 75,000Crane 3 30,000 57,000 80,000Prob 0.2 0.5 0.3

Assume S2 a certainty

max{PA1, PA2, PA3 | p2 =1.0}

choose A1

Most ProbablePayoff

Alternative Low Gr Med. Gr. High Gr.Crane 1 43,000 60,000 68,000Crane 2 37,000 52,000 75,000Crane 3 30,000 57,000 80,000Prob 0.2 0.5 0.3

Assume S2 a certainty

max{PA1, PA2, PA3 | p2 =1.0}

choose A1

Most ProbablePayoff

Alternative Low Gr Med. Gr. High Gr.Crane 1 43,000 60,000 68,000Crane 2 37,000 52,000 75,000Crane 3 30,000 57,000 80,000Prob 0.2 0.5 0.3

Assume S2 a certainty

max{PA1, PA2, PA3 | p2 =1.0}

choose A1

Most ProbablePayoff

Alternative Low Gr Med. Gr. High Gr.Crane 1 43,000 60,000 68,000Crane 2 37,000 52,000 75,000Crane 3 30,000 57,000 80,000Prob 0.2 0.5 0.3

Assume S2 a certainty

max{PA1, PA2, PA3 | p2 =1.0}

choose A1

Most ProbablePayoff

Alternative Low Gr Med. Gr. High Gr.Crane 1 43,000 60,000 68,000Crane 2 37,000 52,000 75,000Crane 3 30,000 57,000 80,000Prob 0.2 0.5 0.3

Assume S2 a certainty

max{PA1, PA2, PA3 | p2 =1.0}

choose A1

Most ProbablePayoff

Alternative Low Gr Med. Gr. High Gr.Crane 1 43,000 60,000 68,000Crane 2 37,000 52,000 75,000Crane 3 30,000 57,000 80,000Prob 0.2 0.5 0.3

Assume S2 a certainty

max{PA1, PA2, PA3 | p2 =1.0}

choose A1

Most ProbablePayoff

Alternative Low Gr Med. Gr. High Gr.Crane 1 43,000 60,000 68,000Crane 2 37,000 52,000 75,000Crane 3 30,000 57,000 80,000Prob 0.2 0.5 0.3

Assume S2 a certainty

max{PA1, PA2, PA3 | p2 =1.0}

choose A1

Most ProbablePayoff

Alternative Low Gr Med. Gr. High Gr.Crane 1 43,000 60,000 68,000Crane 2 37,000 52,000 75,000Crane 3 30,000 57,000 80,000Prob 0.2 0.5 0.3

Assume S2 a certainty

max{PA1, PA2, PA3 | p2 =1.0}

choose A1

Most ProbablePayoff

Alternative Low Gr Med. Gr. High Gr.Crane 1 43,000 60,000 68,000Crane 2 37,000 52,000 75,000Crane 3 30,000 57,000 80,000Prob 0.2 0.5 0.3

Assume S2 a certainty

max{PA1, PA2, PA3 | p2 =1.0}

choose A1

Assume S2 a certainty

max{PA1, PA2| p2 =1.0}

choose A2

Maximun Likelihood Most Probable

Alternative Oil DryDrill fer Oil 700 -100Sell Land 90 90Chance 0.25 0.75

Bayes’ Decision Rule

E[A1] > E[A2] > E[A3]

choose A1

PayoffAlternative Low Gr Med. Gr. High Gr. Expectation

Crane 1 43,000 60,000 68,000 59,000Crane 2 37,000 52,000 75,000 55,900Crane 3 30,000 57,000 80,000 58,500Prob 0.2 0.5 0.3

Bayes’ Decision Rule

E[A1] > E[A2]

choose A1

PayoffAlternative Oil Dry ExpectationDrill fer Oil 700 -100 100Sell Land 90 90 90Chance 0.25 0.75

Expectation

E[A1] > E[A2] > E[A3]

choose A1

PayoffAlternative Low Gr Med. Gr. High Gr. Expectation

Crane 1 43,000 60,000 68,000 59,000Crane 2 37,000 52,000 75,000 55,900Crane 3 30,000 57,000 80,000 58,500Prob 0.2 0.5 0.3

Expectation

E[A1] > E[A2] > E[A3]

choose A1

PayoffAlternative Low Gr Med. Gr. High Gr. Expectation

Crane 1 43,000 60,000 68,000 59,000Crane 2 37,000 52,000 75,000 55,900Crane 3 30,000 57,000 80,000 58,500Prob 0.2 0.5 0.3

Expectation

E[A1] > E[A2] > E[A3]

choose A1

PayoffAlternative Low Gr Med. Gr. High Gr. Expectation

Crane 1 43,000 60,000 68,000 59,000Crane 2 37,000 52,000 75,000 55,900Crane 3 30,000 57,000 80,000 58,500Prob 0.2 0.5 0.3

Expectation

E[A1] > E[A2] > E[A3]

choose A1

PayoffAlternative Low Gr Med. Gr. High Gr. Expectation

Crane 1 43,000 60,000 68,000 59,000Crane 2 37,000 52,000 75,000 55,900Crane 3 30,000 57,000 80,000 58,500Prob 0.2 0.5 0.3

Expectation

E[A1] > E[A2] > E[A3]

choose A1

PayoffAlternative Low Gr Med. Gr. High Gr. Expectation

Crane 1 43,000 60,000 68,000 59,000Crane 2 37,000 52,000 75,000 55,900Crane 3 30,000 57,000 80,000 58,500Prob 0.2 0.5 0.3

Expectation

E[A1] > E[A2] > E[A3]

choose A1

PayoffAlternative Low Gr Med. Gr. High Gr. Expectation

Crane 1 43,000 60,000 68,000 59,000Crane 2 37,000 52,000 75,000 55,900Crane 3 30,000 57,000 80,000 58,500Prob 0.2 0.5 0.3

Laplace PrincipleIf one can not assign probabilities, assume each state equally probable.

Max E[PAi] choose A1

PayoffAlternative Low Gr Med. Gr. High Gr. Expectation

Crane 1 43,000 60,000 68,000 56,943Crane 2 37,000 52,000 75,000 54,612Crane 3 30,000 57,000 80,000 55,611Prob 0.333 0.333 0.333

Expectation-Variance

If E[A1] = E[A2] = E[A3]

choose Aj with min. variance

PayoffAlternative Low Gr Med. Gr. High Gr. Expectation Variance

Crane 1 43,000 60,000 68,000 59,000 76,000,000Crane 2 37,000 52,000 75,000 55,900 188,490,000Crane 3 30,000 57,000 80,000 58,500 302,250,000Prob 0.2 0.5 0.3

Sensitivity Payoff

Alternative Oil DryDrill fer Oil 700 -100Sell Land 90 90Chance p 1-p

Suppose probability of finding oil (p) is somewherebetween 15 and 35 percent.

Sensitivity

Suppose probability of finding oil (p) is somewherebetween 15 and 35 percent.

PayoffAlternative Oil Dry ExpectationDrill fer Oil 700 -100 20Sell Land 90 90 90Chance 0.15 0.85

Sensitivity

Suppose probability of finding oil (p) is somewherebetween 15 and 35 percent.

Alternative Oil Dry ExpectationDrill fer Oil 700 -100 180Sell Land 90 90 90Chance 0.35 0.65

Sensitivityp Drill Sell

0.15 20 900.35 180 90

Sensitivityp Drill Sell

0.15 20 900.35 180 90

Sensitivity Plot

0

50

100

150

200

0 0.1 0.2 0.3 0.4

Prob. of Oil

Expe

cted

Val

ue

Drill

Sell

Sensitivity

We know E[payoff] = 700(p) -100(1-p) = 800p - 100

p Drill Sell0.15 20 900.35 180 90

Sensitivityp Drill Sell

0.15 20 900.35 180 90

Sensitivity Plot

0

50

100

150

200

0 0.1 0.2 0.3 0.4

Prob. of Oil

Expe

cted

Val

ue

DrillSell

Aspiration-Level

Aspiration: max probability that payoff > 60,000

PayoffAlternative Low Gr Med. Gr. High Gr.

Crane 1 43,000 60,000 68,000Crane 2 37,000 52,000 75,000Crane 3 30,000 57,000 80,000Prob 0.2 0.5 0.3

P{PA1 > 60,000} = 0.8P{PA2 > 60,000} = 0.3P{PA3 > 60,000} = 0.3

Choose A2 or A3

Aspiration-Level

Aspiration: max probability that payoff > 60,000

PayoffAlternative Low Gr Med. Gr. High Gr.

Crane 1 43,000 60,000 68,000Crane 2 37,000 52,000 75,000Crane 3 30,000 57,000 80,000Prob 0.2 0.5 0.3

P{PA1 > 60,000} = 0.8P{PA2 > 60,000} = 0.3P{PA3 > 60,000} = 0.3

Choose A2 or A3

Class ProblemAspiration LevelProbability 0.1 0.3 0.6

Alternatives S1 S2 S3

A1 15,163 13,409 11,962A2 16,536 13,465 10,934A3 18,397 14,240 10,840

Determine alternative Aj if aspiration level is NPW > $14,000.

Class ProblemAspiration LevelProbability 0.1 0.3 0.6

Alternatives S1 S2 S3

A1 15,163 13,409 11,962A2 16,536 13,465 10,934A3 18,397 14,240 10,840

Determine alternative Aj if aspiration level is Payoff > $14,000.

Class ProblemAspiration LevelProbability 0.1 0.3 0.6

Alternatives S1 S2 S3

A1 15,163 13,409 11,962A2 16,536 13,465 10,934A3 18,397 14,240 10,840

Determine alternative Aj if aspiration level is Payoff > $14,000.

P{PA1 > 14,000} = 0.1P{PA2 > 14,000} = 0.1P{PA3 > 14,000} = 0.4 Choose A3

Hurwicz Principle = 0.3

ProbabilityAlternatives S1 = 10% S2=15% S3=20% Hj

A1 15,163 13,409 11,962 12,922A2 16,536 13,465 10,934 12,615A3 18,397 14,240 10,840 13,107

Select j: maxj{Hj=maxk[V(jk)]+(1-)mink(V(jk)

max{12,922 12,615 13,107} = 13,107

choose A3

Hurwicz Principle = 0.3

ProbabilityAlternatives S1 = 10% S2=15% S3=20% Hj

A1 15,163 13,409 11,962 12,922A2 16,536 13,465 10,934 12,615A3 18,397 14,240 10,840 13,107

Select j: maxj{Hj=maxk[V(jk)]+(1-)mink(V(jk)

max{12,922 12,615 13,107} = 13,107

choose A3

Hurwicz Principle = 0.3

ProbabilityAlternatives S1 = 10% S2=15% S3=20% Hj

A1 15,163 13,409 11,962 12,922A2 16,536 13,465 10,934 12,615A3 18,397 14,240 10,840 13,107

Select j: maxj{Hj=maxk[V(jk)]+(1-)mink(V(jk)

max{12,922 12,615 13,107} = 13,107

choose A3

Hurwicz Principle = 0.3

ProbabilityAlternatives S1 = 10% S2=15% S3=20% Hj

A1 15,163 13,409 11,962 12,922A2 16,536 13,465 10,934 12,615A3 18,397 14,240 10,840 13,107

Select j: maxj{Hj=maxk[V(jk)]+(1-)mink(V(jk)

max{12,922 12,615 13,107} = 13,107

choose A3

Hurwicz Principle = 0.3

ProbabilityAlternatives S1 = 10% S2=15% S3=20% Hj

A1 15,163 13,409 11,962 12,922A2 16,536 13,465 10,934 12,615A3 18,397 14,240 10,840 13,107

Select j: maxj{Hj=maxk[V(jk)]+(1-)mink(V(jk)

max{12,922 12,615 13,107} = 13,107

choose A3

Hurwicz Principle = 0.3

ProbabilityAlternatives S1 = 10% S2=15% S3=20% Hj

A1 15,163 13,409 11,962 12,922A2 16,536 13,465 10,934 12,615A3 18,397 14,240 10,840 13,107

Select j: maxj{Hj=maxk[V(jk)]+(1-)mink(V(jk)

max{12,922 12,615 13,107} = 13,107

choose A3

Hurwicz Principle = 0.3

ProbabilityAlternatives S1 = 10% S2=15% S3=20% Hj

A1 15,163 13,409 11,962 12,922A2 16,536 13,465 10,934 12,615A3 18,397 14,240 10,840 13,107

Select j: maxj{Hj=maxk[V(jk)]+(1-)mink(V(jk)

max{12,922 12,615 13,107} = 13,107

choose A3

Hurwicz Principle = 0.3

ProbabilityAlternatives S1 = 10% S2=15% S3=20% Hj

A1 15,163 13,409 11,962 12,922A2 16,536 13,465 10,934 12,615A3 18,397 14,240 10,840 13,107

Select j: maxj{Hj=maxk[V(jk)]+(1-)mink(V(jk)

Note:= 1.0 MaxiMax

= 0.0 MaxiMin

Hurwicz PrincipleProbability

Alternatives S1 = 10% S2=15% S3=20% HjA1 15,163 13,409 11,962 15,163A2 16,536 13,465 10,934 16,536A3 18,397 14,240 10,840 18,397

= 1.0

MaxiMax = best of the best = max{maxkV(jk)}

max{15,163 16,536 18,397} = 18,397

choose A3

Hurwicz PrincipleProbabilityAlternatives S1 = 10% S2=15% S3=20% Hj

A1 15,163 13,409 11,962 11,962A2 16,536 13,465 10,934 10,934A3 18,397 14,240 10,840 10,840

= 0.0

MaxiMin = best of the worst = max{minkV(jk)} max{11,962 10,934 10,840} = 11,962 choose A1

Class ProblemLow Gr Med. Gr. High Gr.

Crane 1 43,000 60,000 68,000Crane 2 37,000 52,000 75,000Crane 3 30,000 57,000 80,000

You personally assess your boss’s risk level to beapproximately .3. Use Hurwicz’s principle to analyze the value matrix and determine the appropriate alternative.

Hurwicz Principle

Select j: maxj{Hj=maxk[V(jk)]+(1-)mink(V(jk)

= 0.3Payoff

Alternative Low Gr Med. Gr. High Gr. HiCrane 1 43,000 60,000 68,000 50,500Crane 2 37,000 52,000 75,000 48,400Crane 3 30,000 57,000 80,000 45,000Prob 0.333 0.333 0.333

Hurwicz Principle

Select j: maxj{Hj=maxk[V(jk)]+(1-)mink(V(jk)

max{50500, 48400, 45000} = 50,500

= 0.3Payoff

Alternative Low Gr Med. Gr. High Gr. HiCrane 1 43,000 60,000 68,000 50,500Crane 2 37,000 52,000 75,000 48,400Crane 3 30,000 57,000 80,000 45,000Prob 0.333 0.333 0.333

Hurwicz Principle

Select j: maxj{Hj=maxk[V(jk)]+(1-)mink(V(jk)

max{50500, 48400, 45000} = 50,500

choose A1

= 0.3Payoff

Alternative Low Gr Med. Gr. High Gr. HiCrane 1 43,000 60,000 68,000 50,500Crane 2 37,000 52,000 75,000 48,400Crane 3 30,000 57,000 80,000 45,000Prob 0.333 0.333 0.333

Savage Principle (Minimax Regret)Savage PrincipleProbabilityAlternatives S1 = 10% S2=15% S3=20%

A1 15,163 13,409 11,962A2 16,536 13,465 10,934A3 18,397 14,240 10,840

Build table of regrets: Rjk = maxj[V(jk)] - V(jk)(max in each column less cell value)

Savage Principle (Minimax Regret)Savage PrincipleProbabilityAlternatives S1 = 10% S2=15% S3=20%

A1 15,163 13,409 11,962A2 16,536 13,465 10,934A3 18,397 14,240 10,840

Table of RegretsProbabilityAlternatives S1 = 10% S2=15% S3=20%

A1 3,234 831 0A2 1,861 775 1,028A3 0 0 1,122

Savage Principle (Minimax Regret)Table of RegretsProbabilityAlternatives S1 = 10% S2=15% S3=20%

A1 3,234 831 0A2 1,861 775 1,028A3 0 0 1,122

Minimize the maximum regret Min {3,234 1,861 1,122} = 1,122 choose A3

Class Problem

Being somewhat of a pessimist, you constantly worry about lost opportunities. Compute a regret matrix and determine an alternative which minimizes the maximum regret.

Low Gr Med. Gr. High Gr.Crane 1 43,000 60,000 68,000Crane 2 37,000 52,000 75,000Crane 3 30,000 57,000 80,000