TM 732
Engr. Economics for Managers
Decision Analysis
GoferBroke
Alternative
Drill fer Oil
Sell Land
Chance
Oil
700
90
0.25
Dry
-100
90
0.75
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.
Crane 1
Crane 2
Crane 3
Low Gr
43,000
37,000
30,000
Med. Gr.
60,000
52,000
57,000
High Gr.
68,000
75,000
80,000
Digger Construction
Alternative
Crane 1
Crane 2
Crane 3
Prob
Low Gr
43,000
37,000
30,000
0.2
Payoff
Med. Gr.
60,000
52,000
57,000
0.5
High Gr.
68,000
75,000
80,000
0.3
Decision Matrix
Decision Model for Lift Truck
Alternatives
A1 - Lease 4
A2 - Lease 5
A3 - Lease 6
A4 - Buy 4
A5 - Buy 5
A6 - Buy 6
Probability
EUAW
No. Trucks Required
4
5
6
18,000
20,000
22,000
20,000
20,000
22,000
21,000
21,000
21,000
12,000
14,500
15,000
14,000
14,000
16,000
14,000
15,500
18,000
0.30
0.40
0.30
Matrix Decision Model
A1
A2
:
:
Aj
:
:
An
p1
S1
V(11)
V(1)
p2
S2
V(12)
V(22)
:
:
V(j1)
:
:
V(n1)
:
:
V(j2)
:
:
V(n2)
-----
--
--
pk
Sk
V(1k)
V(2k)
:
:
V(jk)
:
:
V(nk)
-----
pm
Sm
V(1m)
V(2m)
--
:
:
V(jm)
--
:
:
V(nm)
Aj = alternative strategy j under decision makers control
Sk = a state or possible future that can occur given Aj
pk = the probability state Sk will occur
Matrix Decision Model
A1
A2
:
:
Aj
:
:
An
p1
S1
V(11)
V(1)
p2
S2
V(12)
V(22)
:
:
V(j1)
:
:
V(n1)
:
:
V(j2)
:
:
V(n2)
-----
--
--
pk
Sk
V(1k)
V(2k)
:
:
V(jk)
:
:
V(nk)
-----
pm
Sm
V(1m)
V(2m)
--
:
:
V(jm)
--
:
:
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
A1
A2
:
:
Aj
:
:
An
p= 1
S
V(1)
V()
:
:
V(j)
:
:
V(n)
Decisions Under Certainty
A1
A2
:
:
Aj
:
:
An
p= 1
S
V(1)
V()
:
:
V(j)
:
:
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
Alternative
Crane 1
Crane 2
Crane 3
Prob
Low Gr
43,000
37,000
30,000
0.2
Payoff
Med. Gr.
60,000
52,000
57,000
0.5
High Gr.
68,000
75,000
80,000
0.3
Select Aj: maxjminkV(jk)
e.g., Find the min payoff for each alternative.
Maximin
Alternative
Crane 1
Crane 2
Crane 3
Prob
Low Gr
43,000
37,000
30,000
0.2
Payoff
Med. Gr.
60,000
52,000
57,000
0.5
High Gr.
68,000
75,000
80,000
0.3
Select Aj: maxjminkV(jk)
e.g., Find the min payoff for each alternative.
Find the maximum of minimums
Select Crane 1
Choose best alternative when comparing worst
possible outcomes for each alternative.
Maximin
Alternative
Drill fer Oil
Sell Land
Chance
Oil
700
90
0.25
Dry
-100
90
0.75
Select Aj: maxjminkV(jk)
e.g., Find the min payoff for each alternative.
Find the maximum of minimums
Sell Land
Choose best alternative when comparing worst
possible outcomes for each alternative.
MiniMax
Alternative
Crane 1
Crane 2
Crane 3
Prob
Low Gr
43,000
37,000
30,000
0.2
Payoff
Med. Gr.
60,000
52,000
57,000
0.5
High Gr.
68,000
75,000
80,000
0.3
Select Aj: maxjminkV(jk)
e.g., Find the max payoff for each alternative.
MiniMax
Alternative
Crane 1
Crane 2
Crane 3
Prob
Low Gr
43,000
37,000
30,000
0.2
Payoff
Med. Gr.
60,000
52,000
57,000
0.5
High Gr.
68,000
75,000
80,000
0.3
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 best
possible outcomes for each alternative.
MiniMax
Alternative
Drill fer Oil
Sell Land
Chance
Oil
700
90
0.25
Dry
-100
90
0.75
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.
Class Problem
Probability
Alternatives
S1
S2
S3
A1
A2
A3
15,163
16,536
18,397
13,409
13,465
14,240
11,962
10,934
10,840
Choose best alternative using
a. Maximax criteria
b. Minimin criteria
Class Problem
Probability
Alternatives
S1
S2
S3
A1
A2
A3
15,163
16,536
18,397
13,409
13,465
14,240
11,962
10,934
10,840
Choose best alternative using
a. Maximax criteria (best of the best)
maxj{15163, 16536, 18397} = 18,397
choose A3
Class Problem
Probability
Alternatives
S1
S2
S3
A1
A2
A3
15,163
16,536
18,397
13,409
13,465
14,240
11,962
10,934
10,840
Choose best alternative using
a. Minimin criteria (worst of the worst)
minj{11,962 10,934 10,840} = 10,840
choose A3
Maximum Likelihood
Alternative
Crane 1
Crane 2
Crane 3
Prob
Low Gr
43,000
37,000
30,000
0.2
Payoff
Med. Gr.
60,000
52,000
57,000
0.5
Assume S2 a certainty
High Gr.
68,000
75,000
80,000
0.3
Maximum Likelihood
Alternative
Crane 1
Crane 2
Crane 3
Prob
Low Gr
43,000
37,000
30,000
0.2
Payoff
Med. Gr.
60,000
52,000
57,000
0.5
Assume S2 a certainty
max{PA1, PA2, PA3 | p2 =1.0}
choose A1
High Gr.
68,000
75,000
80,000
0.3
Most Probable
Alternative
Crane 1
Crane 2
Crane 3
Prob
Low Gr
43,000
37,000
30,000
0.2
Payoff
Med. Gr.
60,000
52,000
57,000
0.5
Assume S2 a certainty
max{PA1, PA2, PA3 | p2 =1.0}
choose A1
High Gr.
68,000
75,000
80,000
0.3
Most Probable
Alternative
Crane 1
Crane 2
Crane 3
Prob
Low Gr
43,000
37,000
30,000
0.2
Payoff
Med. Gr.
60,000
52,000
57,000
0.5
Assume S2 a certainty
max{PA1, PA2, PA3 | p2 =1.0}
choose A1
High Gr.
68,000
75,000
80,000
0.3
Most Probable
Alternative
Crane 1
Crane 2
Crane 3
Prob
Low Gr
43,000
37,000
30,000
0.2
Payoff
Med. Gr.
60,000
52,000
57,000
0.5
Assume S2 a certainty
max{PA1, PA2, PA3 | p2 =1.0}
choose A1
High Gr.
68,000
75,000
80,000
0.3
Most Probable
Alternative
Crane 1
Crane 2
Crane 3
Prob
Low Gr
43,000
37,000
30,000
0.2
Payoff
Med. Gr.
60,000
52,000
57,000
0.5
Assume S2 a certainty
max{PA1, PA2, PA3 | p2 =1.0}
choose A1
High Gr.
68,000
75,000
80,000
0.3
Most Probable
Alternative
Crane 1
Crane 2
Crane 3
Prob
Low Gr
43,000
37,000
30,000
0.2
Payoff
Med. Gr.
60,000
52,000
57,000
0.5
Assume S2 a certainty
max{PA1, PA2, PA3 | p2 =1.0}
choose A1
High Gr.
68,000
75,000
80,000
0.3
Most Probable
Alternative
Crane 1
Crane 2
Crane 3
Prob
Low Gr
43,000
37,000
30,000
0.2
Payoff
Med. Gr.
60,000
52,000
57,000
0.5
Assume S2 a certainty
max{PA1, PA2, PA3 | p2 =1.0}
choose A1
High Gr.
68,000
75,000
80,000
0.3
Most Probable
Alternative
Crane 1
Crane 2
Crane 3
Prob
Low Gr
43,000
37,000
30,000
0.2
Payoff
Med. Gr.
60,000
52,000
57,000
0.5
Assume S2 a certainty
max{PA1, PA2, PA3 | p2 =1.0}
choose A1
High Gr.
68,000
75,000
80,000
0.3
Most Probable
Alternative
Crane 1
Crane 2
Crane 3
Prob
Low Gr
43,000
37,000
30,000
0.2
Payoff
Med. Gr.
60,000
52,000
57,000
0.5
Assume S2 a certainty
max{PA1, PA2, PA3 | p2 =1.0}
choose A1
High Gr.
68,000
75,000
80,000
0.3
Maximun Likelihood
Most Probable
Alternative
Drill fer Oil
Sell Land
Chance
Oil
700
90
0.25
Assume S2 a certainty
max{PA1, PA2| p2 =1.0}
choose A2
Dry
-100
90
0.75
Bayes’ Decision Rule
Alternative
Crane 1
Crane 2
Crane 3
Prob
Low Gr
43,000
37,000
30,000
0.2
Payoff
Med. Gr.
60,000
52,000
57,000
0.5
E[A1] > E[A2] > E[A3]
choose A1
High Gr. Expectation
68,000
59,000
75,000
55,900
80,000
58,500
0.3
Bayes’ Decision Rule
Payoff
Alternative
Drill fer Oil
Sell Land
Chance
Oil
700
90
0.25
E[A1] > E[A2]
choose A1
Dry
-100
90
0.75
Expectation
100
90
Expectation
Alternative
Crane 1
Crane 2
Crane 3
Prob
Low Gr
43,000
37,000
30,000
0.2
Payoff
Med. Gr.
60,000
52,000
57,000
0.5
E[A1] > E[A2] > E[A3]
choose A1
High Gr. Expectation
68,000
59,000
75,000
55,900
80,000
58,500
0.3
Expectation
Alternative
Crane 1
Crane 2
Crane 3
Prob
Low Gr
43,000
37,000
30,000
0.2
Payoff
Med. Gr.
60,000
52,000
57,000
0.5
E[A1] > E[A2] > E[A3]
choose A1
High Gr. Expectation
68,000
59,000
75,000
55,900
80,000
58,500
0.3
Expectation
Alternative
Crane 1
Crane 2
Crane 3
Prob
Low Gr
43,000
37,000
30,000
0.2
Payoff
Med. Gr.
60,000
52,000
57,000
0.5
E[A1] > E[A2] > E[A3]
choose A1
High Gr. Expectation
68,000
59,000
75,000
55,900
80,000
58,500
0.3
Expectation
Alternative
Crane 1
Crane 2
Crane 3
Prob
Low Gr
43,000
37,000
30,000
0.2
Payoff
Med. Gr.
60,000
52,000
57,000
0.5
E[A1] > E[A2] > E[A3]
choose A1
High Gr. Expectation
68,000
59,000
75,000
55,900
80,000
58,500
0.3
Expectation
Alternative
Crane 1
Crane 2
Crane 3
Prob
Low Gr
43,000
37,000
30,000
0.2
Payoff
Med. Gr.
60,000
52,000
57,000
0.5
E[A1] > E[A2] > E[A3]
choose A1
High Gr. Expectation
68,000
59,000
75,000
55,900
80,000
58,500
0.3
Expectation
Alternative
Crane 1
Crane 2
Crane 3
Prob
Low Gr
43,000
37,000
30,000
0.2
Payoff
Med. Gr.
60,000
52,000
57,000
0.5
E[A1] > E[A2] > E[A3]
choose A1
High Gr. Expectation
68,000
59,000
75,000
55,900
80,000
58,500
0.3
Laplace Principle
If one can not assign probabilities, assume each state
equally probable.
Alternative
Crane 1
Crane 2
Crane 3
Prob
Max E[PAi]
Low Gr
43,000
37,000
30,000
0.333
Payoff
Med. Gr.
60,000
52,000
57,000
0.333
High Gr. Expectation
68,000
56,943
75,000
54,612
80,000
55,611
0.333
choose A1
Expectation-Variance
Alternative
Crane 1
Crane 2
Crane 3
Prob
Low Gr
43,000
37,000
30,000
0.2
Payoff
Med. Gr.
60,000
52,000
57,000
0.5
High Gr. Expectation Variance
68,000
59,000
76,000,000
75,000
55,900
188,490,000
80,000
58,500
302,250,000
0.3
If E[A1] = E[A2] = E[A3]
choose Aj with min. variance
Sensitivity
Payoff
Alternative
Drill fer Oil
Sell Land
Chance
Oil
700
90
p
Dry
-100
90
1-p
Suppose probability of finding oil (p) is somewhere
between 15 and 35 percent.
Sensitivity
Payoff
Alternative
Drill fer Oil
Sell Land
Chance
Oil
700
90
0.15
Dry
-100
90
0.85
Expectation
20
90
Suppose probability of finding oil (p) is somewhere
between 15 and 35 percent.
Sensitivity
Alternative
Drill fer Oil
Sell Land
Chance
Oil
700
90
0.35
Dry
-100
90
0.65
Expectation
180
90
Suppose probability of finding oil (p) is somewhere
between 15 and 35 percent.
Sensitivity
p
0.15
0.35
Drill
20
180
Sell
90
90
Sensitivity
Drill
20
180
p
0.15
0.35
Sell
90
90
Sensitivity Plot
Expected Value
200
150
Drill
100
Sell
50
0
0
0.1
0.2
Prob. of Oil
0.3
0.4
Sensitivity
p
0.15
0.35
Drill
20
180
Sell
90
90
We know
E[payoff] = 700(p) -100(1-p)
= 800p - 100
Sensitivity
Drill
20
180
p
0.15
0.35
Sell
90
90
Sensitivity Plot
Expected Value
200
150
Drill
100
Sell
50
0
0
0.1
0.2
Prob. of Oil
0.3
0.4
Aspiration-Level
Alternative
Crane 1
Crane 2
Crane 3
Prob
Low Gr
43,000
37,000
30,000
0.2
Payoff
Med. Gr.
60,000
52,000
57,000
0.5
High Gr.
68,000
75,000
80,000
0.3
Aspiration: max probability that payoff > 60,000
P{PA1 > 60,000} = 0.8
P{PA2 > 60,000} = 0.3
P{PA3 > 60,000} = 0.3
Choose A2 or A3
Aspiration-Level
Alternative
Crane 1
Crane 2
Crane 3
Prob
Low Gr
43,000
37,000
30,000
0.2
Payoff
Med. Gr.
60,000
52,000
57,000
0.5
High Gr.
68,000
75,000
80,000
0.3
Aspiration: max probability that payoff > 60,000
P{PA1 > 60,000} = 0.8
P{PA2 > 60,000} = 0.3
P{PA3 > 60,000} = 0.3
Choose A2 or A3
Class Problem
Aspiration Level
Probability
Alternatives
0.1
S1
0.3
S2
0.6
S3
A1
A2
A3
15,163
16,536
18,397
13,409
13,465
14,240
11,962
10,934
10,840
Determine alternative Aj if aspiration level is
NPW > $14,000.
Class Problem
Aspiration Level
Probability
Alternatives
0.1
S1
0.3
S2
0.6
S3
A1
A2
A3
15,163
16,536
18,397
13,409
13,465
14,240
11,962
10,934
10,840
Determine alternative Aj if aspiration level is
Payoff > $14,000.
Class Problem
Aspiration Level
Probability
Alternatives
0.1
S1
0.3
S2
0.6
S3
A1
A2
A3
15,163
16,536
18,397
13,409
13,465
14,240
11,962
10,934
10,840
Determine alternative Aj if aspiration level is
Payoff > $14,000.
P{PA1 > 14,000} = 0.1
P{PA2 > 14,000} = 0.1
P{PA3 > 14,000} = 0.4
Choose A3
Hurwicz Principle
 = 0.3
Probability
Alternatives
A1
A2
A3
S1 = 10%
15,163
16,536
18,397
S2= 15%
13,409
13,465
14,240
S3= 20%
11,962
10,934
10,840
Hj
12,922
12,615
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
Probability
Alternatives
A1
A2
A3
S1 = 10%
15,163
16,536
18,397
S2= 15%
13,409
13,465
14,240
S3= 20%
11,962
10,934
10,840
Hj
12,922
12,615
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
Probability
Alternatives
A1
A2
A3
S1 = 10%
15,163
16,536
18,397
S2= 15%
13,409
13,465
14,240
S3= 20%
11,962
10,934
10,840
Hj
12,922
12,615
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
Probability
Alternatives
A1
A2
A3
S1 = 10%
15,163
16,536
18,397
S2= 15%
13,409
13,465
14,240
S3= 20%
11,962
10,934
10,840
Hj
12,922
12,615
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
Probability
Alternatives
A1
A2
A3
S1 = 10%
15,163
16,536
18,397
S2= 15%
13,409
13,465
14,240
S3= 20%
11,962
10,934
10,840
Hj
12,922
12,615
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
Probability
Alternatives
A1
A2
A3
S1 = 10%
15,163
16,536
18,397
S2= 15%
13,409
13,465
14,240
S3= 20%
11,962
10,934
10,840
Hj
12,922
12,615
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
Probability
Alternatives
A1
A2
A3
S1 = 10%
15,163
16,536
18,397
S2= 15%
13,409
13,465
14,240
S3= 20%
11,962
10,934
10,840
Hj
12,922
12,615
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
Probability
Alternatives
A1
A2
A3
S1 = 10%
15,163
16,536
18,397
S2= 15%
13,409
13,465
14,240
S3= 20%
11,962
10,934
10,840
Hj
12,922
12,615
13,107
Select j: maxj{Hj=maxk[V(jk)]+(1-)mink(V(jk)
Note:
= 1.0
MaxiMax
 = 0.0
MaxiMin
Hurwicz Principle
 = 1.0
Probability
Alternatives
A1
A2
A3
S1 = 10%
15,163
16,536
18,397
S2= 15%
13,409
13,465
14,240
S3= 20%
11,962
10,934
10,840
Hj
15,163
16,536
18,397
MaxiMax = best of the best = max{maxkV(jk)}
max{15,163 16,536 18,397} = 18,397
choose A3
Hurwicz Principle
 = 0.0
Probability
Alternatives S1 = 10%
A1
15,163
A2
16,536
A3
18,397
S2= 15%
13,409
13,465
14,240
S3= 20%
11,962
10,934
10,840
Hj
11,962
10,934
10,840
MaxiMin = best of the worst = max{minkV(jk)}
max{11,962 10,934 10,840} = 11,962
choose A1
Class Problem
Crane 1
Crane 2
Crane 3
Low Gr
43,000
37,000
30,000
Med. Gr.
60,000
52,000
57,000
High Gr.
68,000
75,000
80,000
You personally assess your boss’s risk level  to be
approximately .3. Use Hurwicz’s principle to analyze
the value matrix and determine the appropriate
alternative.
Hurwicz Principle
=
Alternative
Crane 1
Crane 2
Crane 3
Prob
0.3
Low Gr
43,000
37,000
30,000
0.333
Payoff
Med. Gr.
60,000
52,000
57,000
0.333
High Gr.
68,000
75,000
80,000
0.333
Hi
50,500
48,400
45,000
Select j: maxj{Hj=maxk[V(jk)]+(1-)mink(V(jk)
Hurwicz Principle
=
Alternative
Crane 1
Crane 2
Crane 3
Prob
0.3
Low Gr
43,000
37,000
30,000
0.333
Payoff
Med. Gr.
60,000
52,000
57,000
0.333
High Gr.
68,000
75,000
80,000
0.333
Hi
50,500
48,400
45,000
Select j: maxj{Hj=maxk[V(jk)]+(1-)mink(V(jk)
max{50500, 48400, 45000} = 50,500
Hurwicz Principle
=
Alternative
Crane 1
Crane 2
Crane 3
Prob
0.3
Low Gr
43,000
37,000
30,000
0.333
Payoff
Med. Gr.
60,000
52,000
57,000
0.333
High Gr.
68,000
75,000
80,000
0.333
Hi
50,500
48,400
45,000
Select j: maxj{Hj=maxk[V(jk)]+(1-)mink(V(jk)
max{50500, 48400, 45000} = 50,500
choose A1
Savage Principle
(Minimax Regret)
Savage Principle
Probability
Alternatives S1 = 10%
A1
15,163
A2
16,536
A3
18,397
S2= 15%
13,409
13,465
14,240
S3= 20%
11,962
10,934
10,840
Build table of regrets: Rjk = maxj[V(jk)] - V(jk)
(max in each column less cell value)
Savage Principle
(Minimax Regret)
Savage Principle
Probability
Alternatives S1 = 10%
A1
15,163
A2
16,536
A3
18,397
S2= 15%
13,409
13,465
14,240
S3= 20%
11,962
10,934
10,840
S2= 15%
831
775
0
S3= 20%
0
1,028
1,122
Table of Regrets
Probability
Alternatives S1 = 10%
A1
3,234
A2
1,861
A3
0
Savage Principle
(Minimax Regret)
Table of Regrets
Probability
Alternatives S1 = 10%
A1
3,234
A2
1,861
A3
0
S2= 15%
831
775
0
S3= 20%
0
1,028
1,122
Minimize the maximum regret
Min {3,234 1,861 1,122} = 1,122
choose A3
Class Problem
Crane 1
Crane 2
Crane 3
Low Gr
43,000
37,000
30,000
Med. Gr.
60,000
52,000
57,000
High Gr.
68,000
75,000
80,000
Being somewhat of a pessimist, you constantly worry
about lost opportunities. Compute a regret matrix and
determine an alternative which minimizes the
maximum regret.