IS466: Decision Support Systems
Tutorial 2
Chapther-2: Decision Analysis
Exercise-1
The KSU’s bookstore must decide how many economic textbooks to order for the next
semester’s class. The bookstore believes that seven, eight, nine, or ten sections of the
course will be offered next semester; each section contains 40 students. The publisher
is offering bookstores a discount if they place their orders early. If the bookstore orders
too few text and runs out, the publisher will air express additional books at the
bookstore’s expense. If it orders too many texts, the store can return unsold texts to the
publisher for a partial credit. The bookstore is considering ordering either 280, 320, 360,
or 400 texts in order to get the discount. Taking into account the discounts, air express
expenses, and credits for returned textbooks, the bookstore manager estimates the
following resulting profits:
Number of
Number of introductory economic classes offered
textbooks
7
8
9
10
ordered
280
S.R. 2800
S.R. 2720
S.R. 2640
S.R. 2480
320
S.R. 2600
S.R. 3200
S.R. 3040
S.R. 2880
360
S.R. 2400
S.R. 3000
S.R. 3600
S.R. 3440
400
S.R. 2200
S.R. 2800
S.R. 3400
S.R. 4000
1) What is the optimal decision if the bookstore manager uses the maximax criterion?
2) What is the optimal decision if the bookstore manager uses the maximin criterion?
3) What is optimal decision if the bookstore manager uses the principle of insufficient
reason criterion?
Based on conversations held with the chair of the economics department, suppose that
the bookstore manager believes that the following probabilities hold:
a) P(7 classes offered) = 0.1
b) P(8 classes offered) = 0.3
c) P(9 classes offered) = 0.4
d) P(10 classes offered) = 0.2
4) Using the expected value criterion, determine how many economics books the
bookstore manager should purchases in order to maximize the store’s expected
profit. Do you think the expected value is appropriate for this problem?
5)Based on the above probabilities, determine the expected value of perfect
information. Interpret its meaning
Solution
Number of introductory Economics
Maximum Minimum
classes offered
Payoff
Payoff
Number of Textbooks to
order
7
8
9
10
280
2800
2720
2640
2480
2800
2480
320
2600
3200
3040
2880
3200
2600
360
2400
3000
3600
3440
3600
2400
400
2200
2800
3400
4000
4000
2200
1) The Maximax criterion belongs to the decision making under uncertainty. Decision
under uncertainty can be used when there is no knowledge about the probability of the
states of nature occurring and is based on the decisionj maker’s attitude toward life.
Maximax criterion is based on the best possible scenario and it fits both an optimistic
and an aggressive decision maker.
Maximax criterion= select the decision alternative with the maximum of maximum
payoff. The maximum of the maximum payoff is equal to 4000 and so the optimal
decision alternative for our manager is “number of textbooks to sold= 400”
2) Maximin is another criterion belonging to the decision making under uncertainty
model. The Maximin criterion is based on the worst-case scenario and it fits both a
pessimistic and conservative decision maker.
Maximin criterion= select the decision with the maximum of minimum payoff. The
maximum of the minimum payoff is 2600 and so the optimal decision alternative for our
manager is “number of textbooks to order=320”.
3) The principle of insufficient reason is to assume that each state of nature is equally
likely to occur. It fits a decision maker who is neither pessimistic nor optimistic. The
optimal decision alternative can be found by adding up the payoffs for each decision
alternative and selecting the alternative with the highest sum.
Principle of insufficient reason criterion= The alternative with the maximum of the sum
on the payoff is 12440, so the optimal decision alternative for our manager is “number of
textbooks to sold= 360”
4) In the decision making under risk, the states of nature have probabilities of
occurrences. For each decision calculate its expected payoff:
Expected Payoff =  (probability*Payoff).
Then we select the decision with the best expected payoff.
We will use the given probabilities to calculate the expected value (EV) of each decision
alternative:
EV(number of maximum textbook to order=280)=2800*0.1 + 2720*0.3 + 2640*0.4 +
2480*0.2=2648
Sum Of
Payoff
10640
11720
12440
12400
EV(number of maximum textbook to order=320)=2600*0.1 + 3200*0.3 + 3040*0.4 +
2880*0.2=3012
EV(number of maximum textbook to order=360)=2400*0.1 + 3000*0.3 + 3600*0.4 +
3440*0.2=3268
EV(number of maximum textbook to order=400)=2200*0.1 + 2800*0.3 + 3400*0.4 +
4000*0.2=3200
Since order 360 textbooks is the highest expected value, it is the optimal solution.
This criterion seems appropriate from the Bookstore's point of view if their objective is to
maximize profit. If their objective is to serve student needs, this is probably not the
appropriate criterion.
5) Expected value of perfect information =EVIP
EVPI = ERPI – EREV
EREV=Expected return without additional information as to which state of nature will
occur prior to making decision
ERPI = Expected return with perfect information as to which state of nature will occur
prior to making decision.
ERPI=(Probability of 1st state of nature)*(Best outcome of 1st state of nature) + ….
ERPI=0.1*2800 + 0.3*3200 + 0.4*3600 + 0.2*4000=3480
EREV=3268
EVPI = 3480 - 3268 = 212 S.R.
Interpretation of EVPI: EVPI represents the expected gain in profit from knowing with
certainty how many economics course sections will be ordered (i.e. knowing with
certainty the future state of nature that was going to occur).
Exercise-2
(Bayesian Analysis)
Ahmed has inherited an important amount of money. He has selected three potential
investments he believes appropriate for him: watches, mobiles and cars.
Helped by his family, experienced in business, Ahmed has constructed the following
payoff table giving its estimate of the expected profit resulting from investing in watches,
mobiles, or cars. The state of nature, corresponding to requests on the market, are
“Little number of requests”, “average number of requests”, “big number of requests”,
and “no requests”.
Investements
Watches
Mobiles
Cars
No
200000
500000
900000
Perceived Market requests
Little
Average
300000
700000
600000
1200000
500000
1300000
Big
1300000
1800000
2200000
Based on past market’s requests, suppose that Ahmed believes that the following
probabilities hold for the state of nature: P(No)=0.20, P(Little)=0.40, P(Average)=0.30,
P(Big)=0.10.
Moreover, Ahmed can hire the noted economic expert Ali to give his opinion on as to
whether the forecast predicts “negative” or “positive” econometric growth which
indirectly influence the market’s requests. Suppose the following probabilities hold for
Ali’s prediction:
P(Ali predicts economics’ growth will be positive|No requests)=0.15
P(Ali predicts economics’ growth will be positive|Little requests)=0.25
P(Ali predicts economics’ growth will be positive|Average requests)=0.50
P(Ali predicts economics’ growth will be positive|Big requests)=0.80
P(Ali predicts economics’ growth will be negative| No requests)=0.85
P(Ali predicts economics’ growth will be negative| Little requests)=0.75
P(Ali predicts economics’ growth will be negative| Average requests)=0.50
P(Ali predicts economics’ growth will be negative| Big requests)=0.20
1) If Ali predicts the economics’ growth will be positive, what is the probability the there
will be no requests on the market?
2) What is Ahmed’s optimal strategy if Ali predicts the economics’ growth will be i)
positive or ii) negative?
3) What is the expected value of Ali’s information?
Solution
Bayesian
statistics play a vital role in assessing the value of additional sample
information obtained from other sources as marketing surveys or experiments, which
can assist in decision-making process. The decision maker can use this input to revise
or fine tune the original probability estimates and possibly improve decision making.
Hence a Bayesian approach enables the decision maker to revise initial probability
estimates in light of additional information. The original probability estimates are known
as the set of prior or priori probabilities. A set of revised or posterior probabilities is
obtained based on knowing the results of sample or indicator information.
Bayesian Analysis
Prior Probability
Additional
Posterior
Information
Probability
The Bayesian approach utilizes Bayes’ Theorem to revise the prior probabilities. This
theorem states:
Given events B and A1, A2, A3,……,An, where A1, A2, A3,……,An are mutually exclusive
and collectively exhaustive, posterior probability is given by:
P(Ai|B)=(P(B|Ai)P(Ai))/(P(B|A1)P(A1)+P(B|A2)P(A2)+…+P(B|An)P(An)).
A convenient way of calculating the posterior probabilities is to use a tabular approach
which utilizes 5 columns:
- Column 1 - State of nature: A listing of the state of the nature for the pb.
-
-
Column 2 - Prior probabilities: the prior probabilities estimates (before obtaining
sample information) for the state of nature, P(Ai).
Column 3 - Conditional probabilities: the known conditional probabilities of obtaining
sample information given a particular state of nature, P(Ai).
Column 4 – Joint probabilities: the joint probabilities of a particular state of nature
and sample information occurring simultaneously, P(B∩Ai)=P(Ai)P(B|Ai). These
numbers are calculated for each row by multiplying the number in the second
column by the number in the third column. The sum of this column is the marginal
probability, P(B).
Column 5 - Posterior probabilities: the posterior probabilities found by Bayes’
theorem, P(Ai|B). Since
P(Ai |B) = P(B∩Ai)/P(B), these number are calculated for
each row by dividing the number in the fourth column by the sum of the number in
the fourth column.
The following table gives the tabular approach for calculating posterior probabilities
assuming that the opinion of Ali regarding the market Growth is “positive”:
States
Prior
Conditional
Joint
Posterior
of Nature Probabilities Probabilities Probabilities Probabilites
Si
P(Si)
P(Positive|Si) P(Positive∩Si) P(Si|Positive)
No
0.2
0.15
0.03
0.083
Little
0.4
0.25
0.1
0.278
Average
0.3
0.5
0.15
0.417
Big
0.1
0.8
0.08
0.222
P(Interesting)
0.36
The following table gives the tabular approach for calculating posterior probabilities
assuming that the opinion of Ali regarding the market growth is “Negative”:
Conditional
Joint
Posterior
States
Prior
Probabilities
Probabilities
Probabilites
of Nature Probabilities
P(not
P(not
P(Si|not
Si
P(Si)
interesting|Si)
interesting∩Si)
interesting)
No
0.2
0.85
0.17
0.266
Little
0.4
0.75
0.3
0.469
Average
0.3
0.5
0.15
0.234
Big
0.1
0.2
0.02
0.031
P(Non
Interesting)
0.64
1) So P(game will be dull|Jim predicts game interesting)= 0.083
2) Ali predicts generates two sets of probabilities for the set of nature, one based on a
Positive growth predict and the other on Negative growth predict. We can now
determine the optimal strategy on commercials purchases using the expected value
criterion. If Ali’s predict is an positive growth, the revised expected values for the
decision alternatives are:
EV(Watches Positive growth)= 0.083*(-200000) + 0.278*300000 +
0.417*700000 + 0.222*1300000= 647222.22
EV(Mobiles Positive Growth)= 0.083*(-500000) + 0.278*600000 +
0.417*1200000 + 0.222*1800000=1025000.0
EV(Watches Negative Growth)=0.266*(-200000) + 0.469*300000 +
0.234*700000 + 0.031*1300000=292187.50
……, etc
The following table listing the expected values that we have generated for each decision
alternative using the posterior probabilities assuming an positive growth predict from Ali
and the posterior probabilities assuming an Negative growth predict from Ali.
Decision
Alternative
Watches
Mobiles
Cars
EV
prior
420000
680000
630000
EV
EV
Growth Positive Growth Negative
647222.22
292187.50
1025000.00
485937.50
1094444.44
368750.00
So, (i) If Ali prediction is interesting, buy cars, (ii) If Ali prediction is non-interesting, buy
Mobiles.
3) The expected value of Ali’s information is called expected value of sample
information (EVSI):
EVSI(Expected Value of Sample Information) = ERSI (Expected Return with Sample
Information) – EREV(Expected Return Without Additional Information)
ERSI = (largest expected return for an positive growth )*(probability of an positive
growth) + (largest expected return for an negative growth )*(probability of an Negative
Growth)= 1094444.44*0.36+ 485937.50*0.64=563159.99
ERSI = 705000
EREV = expected return without Ali’s prediction = 680000
EVSI = ERSI-EREV=SAR 25,000
Exercise-3
(Decision trees)
In order to increase tax revenue, residents of a town are considering legalizing
motorcycle shows. An election will be held in November on this issue. At present, the
town’s principal hotel is scheduled to be sold at a sealed bid auction. Ali, a local real
estate speculator, is interested in biding on the property and is considering bidding
either 1.2 Million, 1.4 Million, or 1.8 Million.
Ali estimates that if he bids 1.2 Million, he will have a 25% chance of buying the
property; if he bids 1.4 Million, he will have a 40% chance of buying the property; and if
he bids 1.8 Million, he will have an 80% chance of buying the property. If he is
successful in buying the property, and the town approves legalized the game
(motorcycle shows), he estimates he will be able to sell the hotel for 1.2 Million. If the
town defeats legalized the game, Ali estimates that the hotel could be sold for 1.1
Million. Based on current polling data published in the town newspaper, he believes
there is 55% chance the town will vote to legalize the game.
Ali is considering conducting his own survey to gain some insights into voter attitudes
toward the game, prior to placing his bid for the property. The poll will cost 20000 to
conduct and will indicate whether or not the majority of people polled favor the game. Ali
believes that the following conditional probabilities hold for the survey:
P(a majority favor the game|legalized game approved)=0.9
P(a majority do not favor the game|legalized the game defeated)=0.70
That is, if the voters will approve legalized the game, Ali believes there is 0.90
probability that the majority of people surveyed will favor the game; and if the voters will
defeat legalized the game, there is a 0.70 probability that the majority of people
surveyed will not favor the game,
Draw a decision tree for this problem and determines Ali’s optimal decision strategy.
Solution
The following problem consists of a sequence of dependent decisions. Unfortunately,
the payoff table is limited to problems where the decision maker needs to make a single
decision. Decision tree is useful in analyzing decision problem consisting of a sequence
of dependent decisions.
Decision tree is a chronological representation of the decision process by a network that
utilizes two types of node: decision nodes (the squares) and state of nature (chance)
nodes (the circles). The root of the tree corresponds to the present time, and the tree is
constructed outward into the future.
Each branch emanating from a decision node corresponds to a decision alternative and
includes the cost or benefit value associated with that decision. Each branch emanating
from a state of nature node corresponds to a particular state of nature labeled with the
probabilities of that state of nature will occur, given all previous occurrences (decisions
and states of nature) prior to the state of nature node.
Initially, Ali faces two decisions alternatives: (1) do poll and (2) do not poll. So the initial
decision tree corresponds to this decision: the root of the tree is a decision node with
two branches lead out from that node. The value of “do not do poll” branch is 0 (there is
no cost to this decision), while the –20 (to read as –20000SR) on the “do poll” branch is
the cash flow with making a poll.
Let us consider the decision branch corresponding to “do not do poll”. Ali faces four
decisions: (1) do nothing, (2) bid 1.2 Million SR, (3) bid 1.4 Million, or (4) bid 1.8 Million.
Thus, we find a decision node at the end of this branch and four branches
(corresponding to the four decision alternatives) leaving the node. The values on the
four decision branches are 0.
If Ali does not do poll and decides to do nothing, its total return is 0. If, on the other
hand, he decides to bid 1.2 Million SR, he will know if will win or loss the bid. This is a
chance or state of nature, represented by a circle, and will leads to two branches: “win
the bid” or “loss the bid”, on which the values, 0.25 and 0.75, respectively, are the
corresponding state of nature probabilities. Also, the values –1200 and 0, respectively,
are the corresponding cash flows associated with each state of nature.
Now if Ali wins the bid, he will know if the game has been approved or denied. This is
chance or state of nature and from this node we will have two branches “win bid” and
“loss bid” with 0.55 (from the newspaper polling data) and 0.45, respectively, the
corresponding state of nature probabilities; and with 2.1 Millions and 1.1 Million,
respectively the corresponding cash flow associated with each state of nature.
At the end of each path trough the tree is the total profit or loss connected with that
particular set of decisions and outcome, calculated by adding the cash flow on the
decision branches, making up the path. For example, if Ali does not do poll, bids 1.2
Million, win the bid, and the finally the game is approved, the profit will be
0
+ 0 – 1200 + 2100 = 900.
The same reasoning is applied for the case Ali bids 1.4Million or 1.8Million.
Let us now consider the decision process if Ali does do poll. Consequently, he will
predict that the majority will either be favor or do not favor. This chance event is
represented by round state of nature node. We have first to calculate the probability that
Ali will predict majority favor or majority do not favor using Bayes’ theorem.
The conditional probabilities are:
P(a majority favor to the game|legalized game approved)=0.90
So, P(a majority not in favor to the game|legalized game approved)1-0.9=0.1
P(a majority not in favor to the game|legalized game defeated)=0.70
So, P(a majority favor to the game|legalized game defeated)=1-0.70=0.3
The prior probabilities are:
P(legalized game approved)=0.55
So, P(legalized gamed defeated)=1-0.55=0.45
The tabular approach, used in the previous exercise, will allow deducing the posteriori
probabilities:
Majority Favor
States Prior
Approved 0.55
Conditional
P(Majority favor|state)
0.9
Joint
0.495
Posterior
P(State nature|Majority favor)
0.785714
Defeated
0.45
.
0.3
P(Majority favor)
0.135
0.63
0.214286
Majority Not in Favor
Conditional
States Prior P(Majority not in favor|state) Joint
Approved 0.55
0.1
0.055
Defeated 0.45
0.7
0.315
P(Majority not in favor)
0.37
Posterior
P(State nature|Majority not in
favor)
0.148649
0.851351
Consequently, if Ali does do poll, he will lead it to two branches –“majority favor” or
“majority do not favor”, on which the values, 0.63 and 0.37, respectively, are the
corresponding state of nature probabilities.
Once Ali’s prediction is known, Ali is now faced with the same decision choices as when
he doesn’t do poll:(1) do nothing, (2) bid 1.2 Million SR, (3) bid 1.4 Million, or (4) bid 1.8
Million. The differences here are the probabilities for the state of nature which
correspond to our posterior probabilities and not prior probabilities.
The end of each path through the tree is obtained as previously: summing the cash
flows on the decision branches making up the path.
To determine the optimal strategy we work backward from the ends of each branch until
we come to either a state of nature node or a decision node.
At a state of nature node, we calculate the expected value of the node using the ending
node values for each branch leading out of the node and the probability associated with
that branch. The expected value is the sum of the products of the branch probabilities
and corresponding ending node values. This sum becomes the value for the sate of
nature node. Example, consider the possible paths reached if Ali do not pool, bid 1.2
Million, two outcomes (branches) are possible: the game approved and the game is
denied. The expected return at this state of nature node is equal: (probability game
approved)*(expected return if game approved) +(probability game denied)*(expected
return if game denied); (0.55*900) + (0.45*(-100))=450.
At a decision node, the branch that has the highest ending node value is the optimal
solution. The highest ending node value, in turn, becomes the value for the decision
node.
Thus, if Ali decides to not do poll, the corresponding expected values that result if it
does nothing, bids 1.2Million, bids 1.4Million, or bids 1.8Million, are 0, 112.5, 100, and –
120, respectively; then bidding 1.2Million is the optimal solution. The expected return
corresponding to bidding 1.2Million (112.5) then becomes the expected return
corresponding to the decision node do not do poll.
If Ali decides to do poll, and if he predicts majority favor, he should bid 1.4Million. on the
other hand, if he predicts majority do not favor, he should do nothing.