MGS3100-Solved - Exercises Decision Analyisis
MGS3100-Solved - Exercises Decision Analyisis
MGS3100-Solved - Exercises Decision Analyisis
1. In the following payoff table (in $ million), 1) identify decision alternatives, states of nature,
and payoffs; 2) what does the number 29 mean? -12?
2. Your company is testing a site for drilling for oil. You may hit a dry well, a small oil well, or a
large oil well. Consider the following payoff table (in $ thousand) for your situation:
Small Large
Dry
Oil Oil Maximax Maximin Laplace
Well
Well Well
Drill -5000 1000 6000
Do not
0 0 0
Drill
1) What are the decision alternatives? What are the states of nature?
2) What would be the best decisions under each of the following criteria (Show your work by
filling in the blank cells in the table above)?
a) Maximax
b) Maximin
c) Laplace
3. Consider now that you have the following expectations based on your analysis of the site. You
believe that there is a 50% chance of hitting a small well and a 20% chance of hitting a large
well.
Small Large
Dry Well
Oil Well Oil Well
Drill -5000 1000 6000
Do not Drill 0 0 0
1
4. You own a pizza shop in a downtown mini-mall. It is Saturday morning, and you are trying
to decide how many pizzas to make to meet today's lunch hour demand. Based upon your
experience with Saturdays, you think that the probability of being able to sell 20 pizzas is
0.2, of being able to sell 40 pizzas is 0.3, and of being able to sell 50 pizzas is 0.5.
Suppose a pizza sells for $10 and has an incremental cost of $4.25. If you have leftover
pizzas, you can sell them to the homeless shelter for $1.25 each. If demand exceeds the
number of pizzas you have prepared, every disappointed customer costs you $0.25 worth of
lost customer goodwill.
1) Define the decisions, alternatives, states of nature, outcomes, revenues, costs, and payoffs
for this problem. Then construct the payoff table.
2) What is the decision making situation based on the knowledge about the state of nature?
3) Assume you did not have the probability knowledge. Identify the decision you would
make using each of the following criteria: Maximax, Maximin, and LaPlace.
4) Find the ER for each alternative. Based upon the ER criterion, what is the best decision?
Also find the ER with PI and the EVPI.
5) Draw the decision tree that this problem calls forth and solve it.
5. General Motors (GM) is planning their production strategy for their next model. Three
alternatives are being considered for their model Malibu: 30,000, 20,000, and 12,000. GM
decides to categorize the demand for Malibu for the next year as either High (H) or Low (L).
The payoffs measured in millions of dollars and probabilities of states of nature are presented
in the table below.
States of nature
Decision Alternatives High (H) Low (L)
Produce 30K 29 -12
Produce 20K 18 8
Produce 12K 3 11
Probabilities 0.62 0.38
6. For the above GM problem and the decision tree, it can hire a marketing research firm to help
estimate the demand more accurately. Consider the reliabilities of the marketing research firm
given below,
1) compute the posterior probabilities,
2) draw the revised decision tree (a blank tree is provided below).
3) compute EVPI and EVSI.
2
RELIABILITIES
High Low
Encouraging 0.80
Discouraging 0.70
PRIOR PROBABILITIES
High Low
0.62 0.38
POSTERIOR PROBABILITIES
High Low
Encouraging
Discouraging
=
=
Hig h 29
Low -1 2
Produ ce 30 K
Hig h 18
Produ ce 20 K
Low 8
Produ ce 12 K
En cou rag ing Hig h 3
Low 11
=
=
Low -1 2
Produ ce 30 K
Hig h 18
Produ ce 20 K
Low 8
Produ ce 12 K
Hig h 3
Low 11
3
7. Guido's EZ Loan Company pigeonholes credit applicants as Good and Bad risks. On the
average, 10% of applicants are Bad risks. Guido uses a sophisticated computerized credit
scoring model to attempt to discriminate between these 2 groups. A study of their experience
in using the model yielded interesting results. Guido's gives credit to good risks 90% of the
time. Bad risks get credit only 20% of the time.
Assume that a person selected at random from potential applicants to Guido's applies for
credit and is granted a loan. What is the probability that he/she is bad risk, P(Bad Risk|Credit
Granted)?
8. Your friend Tenfour Goodbuddy is an independent trucker whose truck is empty after his most
recent haul. He has found a deal to take a load to Birmingham, with a return load included,
for a total of $3000. Another deal would give him a load to Charlotte for $2500. Tenfour
thinks there's a 50-50 chance of finding a $2500 return load from Charlotte. Otherwise he
must return empty, with no revenue. Assume that the cost of a round trip to either city is the
same, and that he sees no other loads presently available.
Sometimes in the past he has called the Thunder Road Truck Stop (TRTS) in Charlotte to get
information. His friend there is very chatty, so it always costs him at least $10 for the call.
90% of the time when he had ended up with a return load, TRTS had told him "It's busy".
80% of the time when he had ended up with no return load (a "load of postholes"), TRTS had
said "It's slow".
4
Solutions
1. 1)
States of Nature
Decision Alternatives High Demand (H) Low Demand (L)
Produce 30,000 29 -12
Produce 20,000 18 8 Payoffs
Produce 12,000 3 11
2) 29 means that if you choose to produce 30,000 units and the demand is high, then you can
make $29 million. -12 means that if that if you choose to produce 30,000 units and the
demand is low, then you will lose $12 million.
2.
Small Large
Dry
Oil Oil Maximax Maximin Laplace
Well
Well Well
Drill -5000 1000 6000 6000 -5000 666.67
Do not 0 0 0
0 0 0
Drill
1) Decisions: Drill and Do not Drill; States of nature: Dry, Small, and Large Oil Well.
2) Maximax – Drill
Maximin – Do Not Drill
Laplace – Drill
3.
Small Large
Dry Well ER
Oil Well Oil Well
Drill -5000 1000 6000 200
Do not Drill 0 0 0 0
Probability 0.3 0.5 0.2
1) Expected Returns for both decisions are in the last column of the above table.
Drilling is better with ER = 200.
5
4. 1). Decisions: since we assume that the decision of opening the pizza shop for this particular
Saturday has been made, the only decision here is to determine how many pizzas to make today.
Alternatives: there are three alternatives the pizza shop should consider (based on the possible demand
levels): make 20 pizzas, make 40 pizzas, and make 50 pizzas.
States of nature: the uncertainty involved with this decision is the demand level. Hence the three states
of nature are: demand of 20, 40, and 50.
Outcomes: the outcomes can be understood as the states of nature or as the consequence of the
combination of different alternatives and states of nature.
Revenues: the total income of the pizza shop for any combination of a particular alternative (make
quantity) and a particular state of nature (demand). It is calculated as:
Revenue = 10 min(demand, make quantity) + 1.25 max(0, make quantity - demand).
Costs: the total cost for any combination of a particular alternative (make quantity) and a particular
state of nature (demand). It is defined as:
Cost = 4.25(make quantity) + 0.25 max(0, demand - make quantity).
Payoffs: the net income (profit) which is calculated as: Payoff = Revenue - Cost.
Payoff Table
Demand
Make Quantity 20 40 50
20 $115.00 $110.00 $107.50
40 $55.00 $230.00 $227.50
50 $25.00 $200.00 $287.50
4). Expected returns for the three alternatives are $109.75, $193.75, and $208.75 respectively. The best
alternative is to make 50 pizzas. With perfect information, the ER is $235.75. EVPI = $27.
6
Dem and 20
115
0.2
Dem and 50
107.5
0.5
Dem and 20
55
0.2
Dem and 50
227.5
0.5
Dem and 20
25
0.2
Dem and 50
287.5
0.5
7
0.62
High
29
Produce 30K 29 29
0 13.42 0.38
Low
-12
-12 -12
0.62
High
18
Produce 20K 18 18
2
14.2 0 14.2 0.38
Low
8
8 8
0.62
High
3
Produce 12K 3 3
0 6.04 0.38
Low
11
11 11
6) Solve the decision tree and find the best production strategy.
Produce 20K (EV=14.2)
6. 1)
Encouraging = E
Discouraging =D
High = H
Low = L
RELIABILITIES
High Low
Encouraging P(E|H)= 0.80 P(E|L)= 0.30
Discouraging P(D|H)= 0.20 P(D|L)= 0.70
PRIOR PROBABILITIES
High Low
0.62 0.38
POSTERIOR PROBABILITIES
High Low
Encouraging P(H|E)= 0.813 P(L|E)= 0.187
Discouraging P(H|D)= 0.318 P(L|D)= 0.682
8
2)
MGS 3100 Bayes' Theorem
P(H|E) = 0.813
P(L|E) = 0.187
Produce 12K
Encouraging 4.495 High 0.813 3
P(H|D) = 0.318
17.376 P(L|D) = 0.682
Produce 12K
8.456 High 0.318 3
Low 0.682 11
3) EVSI = Max ER with sample info – Max ER w/o sample info = 17.376 – 14.2 = 3.176
EVPI = EV w/ PI – EV w/o PI = 29(0.62) + 11(0.38) – 14.2 = 7.96
P (CG | BR ) P ( BR )
P ( BR | CG ) =
P (CG )
P (CG | BR ) P ( BR )
=
P (CG | BR ) P ( BR ) + P (CG | GR ) P (GR )
0.20 × 0.10
= ≈ 0.0241
0.20 × 0.10 + 0.90 × 0.90
9
That is, the probability that the person is a bad risk given that he/she has been granted a loan is about
0.0241 or 2.41%.
Note: if you are not comfortable with the above formula based calculation, you may use the following
tabular form as:
6. 1). The decision tree is given below. Notice that the marginal probabilities (P(B), P(S)) and the
posterior probabilities (P(RL|B) etc.) have been included in the tree. These probabilities are obtained
based on the calculation of Part (b).
10
$3000
Birmingham
RL $5000
0.8182*
Busy Charlotte
0.55 ** 0.1818*
NL $2500
0.45 ** $3000
Call
Slow Birmingham
RL $5000
0.1111*
Charlotte
0.8889*
No NL $2500
call $3000
Birmingham
RL $5000
0.5
Charlotte
0.5
NL $2500
Marginal
RL NL probabilities
B 0.45 0.1 0.55
S 0.05 0.4 0.45
11
$3000
Birmingham
$4545
$4545 RL $5000
0.8182*
Busy Charlotte
**
$3850 0.55 0.1818*
NL $2500
0.45** $3000
Call
Slow Birmingham
$3850
RL $5000
$2777.5 0.1111*
$3000 Charlotte
0.8889*
No NL $2500
call $3000
Birmingham
RL $5000
$3750 0.5
$3750 Charlotte
0.5
NL $2500
The decision rule is: Call Thunder Road. If things are slow in Charlotte, take the Birmingham road; If
things are busy in Charlotte, take the Charlotte load.
Without the call, you would just go to Charlotte, with an expected return of $3750. With the call, your
ER becomes $3850. So, EVSI = 3850-3750=$100. That is, the call can, on the average, improve profit
by $100. Since the call costs $10, the net worth of the call is $90.
12