tm 732 engr. economics for managers decision analysis
DESCRIPTION
Prototype Ex. 2 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.TRANSCRIPT
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