Quantitative Methods MAT 540 Decision Analysis
Objectives• When you complete this lesson, you will be able to:
• List the components of a decision-making situation.• Make decisions without probabilities using the maximax, maximin,
minimax regret, Hurwicz, and equal likelihood criteria. • Compute the expected value of a decision when probabilities are
known.• Compute the expected value of perfect information.• Construct decision trees to calculate the expected value of a
decision. • Construct decision trees to calculate the expected value of a
sequence of decisions.• Conduct a decision analysis with additional information.• Compute the expected value and efficiency of sample information.
The Minimax Regret Criterion
• Minimizes the maximum regretGood Economic Conditions Poor Economic Conditions
$100,000 - 50,000 = $50,000 $30,000 - 30,000 = $0
$100,000 - 100,000 = $0 $30,000 – (- 40,000) = $70,000
$100,000 - 30,000 = $70,000 $30,000 - 10,000 = $20,000
The Hurwicz Criterion
• Compromise between maximax and maximin• Coefficient of optimism
• α = 0.4
• 1 – α = 0.6
Decision Values
Apartment building $50,000(.4) + 30,000(.6) = $38,000
Office building $100,000(.4) – 40,000(.6) = $16,000
Warehouse $30,000(.4) + 10,000(.6) = $18,000
The Equal Likelihood Criterion
• Weights each state of nature equally
Decision Values
Apartment building $50,000(.5) + 30,000(.5) = $40,000
Office building $100,000(.5) – 40,000(.5) = $30,000
Warehouse $30,000(.5) + 10,000(.5) = $20,000
Summary of Criteria Results
• Warehouse is dominated by apartment building
Criterion Decision (Purchase)
Maximax Office building
Maximin Apartment building
Minimax regret Apartment building
Hurwicz Apartment building
Equal likelihood Apartment building
Decision Making with Probabilities
• Expected value• Estimate the probability of occurrence of each
state of nature• Compute
n
iii xPxxE
1
Expected Value
EV(Apartment) = $50,000(.6) + 30,000(.4) = 42,000
EV(Office) = $100,000(.6) - 40,000(.4) = 44,000
EV(Warehouse) = $30,000(.6) + 10,000(.4) = 22,000
Expected Opportunity Loss
EOL(Apartment) = $50,000(.6) + 0(.4) = 30,000
EOL(Office) = $0(.6) + 70,000(.4) = 28,000
EOL(Warehouse) = $70,000(.6) + 20,000(.4) = 50,000
Expected Value of Perfect Information
$100,000(.60) + 30,000(.40) = $72,000
EV(office) = $100,000(.60) - 40,000(.40) = $44,000
EVPI = $72,000 - 44,000 = $28,000
Decision Trees, continued
• Compute expected values• EV(node 2) = .60($50,000) + .40(30,000) = $42,000
• EV(node 3) = .60($100,000) + .40(-40,000) = $44,000
• EV(node 4) = .60($30,000) + .40(10,000) = $22,000
Sequential Decision Trees, continued
• Apartment building: $1,290,000 – 800,000 = $490,000
• Land: $1,360,000 – 200,000 = $1,160,000
Decision Analysis with Additional Information, continued
• Conditional probabilities:
g = good economic conditions, P(g) = .6
p = poor economic conditions, P(p) = .4
P = positive economic report
N = negative economic report
P(Pg) = .80 P(NG) = .20
P(Pp) = .10 P(Np) = .90
Decision Analysis with Additional Information, continued
• Posterior probabilities
• P(g|N) = .25• P(p|P) = 0.77• P(p|N)=.75
923.
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6.8.
ppPggP
ggPPg
PPPP
PPP
Decision Trees with Posterior Probabilities, continued
• Probability of two dependent events• P(AB) = P(A|B)P(B)
• P(Pg) = P(P|g)P(g)
• P(Pp) = P(P|p)P(p)
Decision Trees with Posterior Probabilities, continued
• Mutually exclusive events• P(P) = P(Pg) + P(Pp) = P(P|g)P(g) + P(P|p)P(p) = (.8)(.6) + (.1)(.4) = .52
• P(N) = P(N|g)P(g) + P(N|p)P(p) = (.2)(.6) + (.9)(.4) = .48
Decision Trees with Posterior Probabilities, continued
• EV(apartment) = $50,000(.923) + 30,000(.077) = $48,460
The Expected Value of Sample Information
• EVSI = EVwith information – EVwithout information
= $63,194 – 44,000
= $19,194
• Efficiency = EVSI ÷ EVPI
= $19,194 / 28,000
= .68