Decision Analysis
Version 3.1
c
2010,
2009, 2008, 2002, 1998
Maria Antónia Carravilla
José Fernando Oliveira
FEUP
Nothing is more difficult, and therefore more precious, than to be able to
decide
Napoleon Bonaparte (Maxims, 1804)
To be, or not to be: that is the question
To be, or not to be: that is the question:
Whether ’tis nobler in the mind to suffer
The slings and arrows of outrageous fortune,
Or to take arms against a sea of troubles,
And by opposing end them? To die: to sleep;
No more; and by a sleep to say we end
The heart-ache and the thousand natural shocks
That flesh is heir to, ’tis a consummation
Devoutly to be wish’d. To die, to sleep;
To sleep: perchance to dream: ay, there’s the rub;
For in that sleep of death what dreams may come
When we have shuffled off this mortal coil,
Must give us pause: there’s the respect
That makes calamity of so long life;
For who would bear the whips and scorns of time,
The oppressor’s wrong, the proud man’s contumely,
The pangs of despised love, the law’s delay,
The insolence of office and the spurns
That patient merit of the unworthy takes,
When he himself might his quietus make
With a bare bodkin? who would fardels bear,
To grunt and sweat under a weary life,
But that the dread of something after death,
The undiscover’d country from whose bourn
No traveller returns, puzzles the will
And makes us rather bear those ills we have
Than fly to others that we know not of?
William Shakespeare
To be, or not to be (Hamlet act 3, scene 1)
Characteristics of a decision problem
http://decision-analysis.society.informs.org/Field/FieldLexicon.html
Decision A decision is an allocation of resources. It is irrevocable, except that a new
decision may reverse it.
Decision Maker The decision maker is one who has authority over the resources
being allocated. Presumably, he makes the decision in order to further some objective,
which is what he hopes to achieve by allocating the resources.
Alternatives At the time of the decision, the decision maker has available to him at
least two alternatives, which are the courses of action that he might take. When he
chooses an alternative and commits to it, he has made the decision and then events come
into play.
Events These are those uncontrollable elements that we sometimes call luck. Different
alternatives that the decision maker might choose might subject him to different events.
Outcome In every case the alternatives combine with the events to produce the
outcome. The outcome is the result of the decision situation and is measured on the scale
of the decision maker’s values. Since the outcome is the result not only of the chosen
alternative but also of the events, it is itself an uncertainty.
Decision problems – Some Key Distinctions
http://decision-analysis.society.informs.org/Field/FieldLexicon.html
decision vs. objective
Example: To accelerate an R&D program is an objective, not a decision.
To allocate the funds in an effort to accelerate the program is a decision.
Why it’s important: The decision might not succeed in achieving the objective. One might spend the funds and
yet, for any number of reasons, achieve no acceleration at all.
good decision vs. good outcome
Example: Someone who buys a lottery ticket and wins
the lottery obtains a good outcome. Yet, the decision to buy the lottery ticket may or may not have been a
good decision.
Why it’s important: A bad decision may lead to a good outcome and conversely a good decision may lead to a
bad outcome. The quality of a decision must be evaluated on the basis of the decision maker’s alternatives,
information, values, and logic at the time the decision was made.
strategy vs. goal
Example: Launching two new products a year is a goal. Investing in additional
personnel, while at the same time stopping the funding of some stalled projects, is a strategy intended to lead
to that goal.
Why it’s important: Strategy describes a collection of actions that the decision maker takes. The outcome of
the actions is uncertain, but one of the possible outcomes is the attainment of the goal.
decision vs. prioritization
Example: To assert that one would rather fund development
project A than development project B, and project B than project C, is a prioritization. Actually funding
project A is a decision.
Why it’s important: A prioritization might be an intermediate step en route to a decision, and one might even
use a prioritization as a tool to aid in a decision.
The Company
The increasing environmental concerns among consumers, led recently to the creation of a company dedicated
only to the development and production of environmentally friendly toys. These toys are made of non-toxic
and biodegradable materials. The material used is an agglomerate of wood fibers known commercially as MDF
(Medium Density Fibers) and the connections between the pieces are made with threads of natural latex. All
parts are painted with paints based on natural, non-toxic products. The packages are built from cardboard,
made with recycled paper.
Initially the company will try to establish itself in the national market, but it is expected to expand its sales to
the international market in the near future. Given that future development, the product line developed is
named
.
The launch of the new product line will be based on an articulated doll represented in the following picture.
The constituent parts of the doll can be painted in any color palette of the company, salmon, green, yellow and
natural color
. The articulated doll pictured on the left has all the pieces painted salmon and
the colored one on the right has the additional accessory Bouler hat.
The Company
(contd.)
The product line is completed by two optional accessories, painted in green in the picture, that can be
produced in any color of the palette. The optional accessories are mounted after the assembly of the doll, as
shown in the picture.
Another accessory, the “skirt”, must be mounted during the assembly of the doll, as shown in the next picture.
– The problem
During its last meeting of the
board the issue was strategy.
One international company made a very interesting offer to buy the
brand. On the other hand the board did ask, a couple of
months before, for an analysis of a possible internationalization of the brand.
The results of that analysis are also at hand. The third alternative would be
to leave everything as it is: keep the brand and not internationalize.
The future is however very uncertain in what concerns the ecological
awareness of the buyers. Their ecological awareness could rise, implying an
high increase in the sales, or the ecological awareness could reduce, due to
the problems in the world economy, implying a reduction of the sales.
– Definitions
For the decision problem of
the decision maker –
the alternatives –
the events –
the outcomes –
define:
– Definitions
the decision maker – will be the board of directors of the company
the alternatives – will be:
• Keep the Brand
• Keep the Brand and Internationalize
• Sell the Brand
the events – will be
• The rise of the ecological awareness
• The fall of the ecological awareness
the outcomes – there will be an outcome for each pair
(alternative,uncertainty).
Decision Matrix
The payoff is a quantitative measure of the value to the decision maker of
the consequences of the outcome. For each combination of an alternative
and an event, the decision maker knows what the resulting payoff would be.
The payoff could be, for example, the profit or net monetary gain or can be
the expected value of the measure of the consequences. The Decision
Matrix is commonly used to provide the payoff for each combination of an
action and an event.
Uij = U (ai ; θj )
Events
Actions
θ1
θ2
θ3
...
θn
a1
U11
U12
U13
...
U1n
a2
U21
U22
U23
...
U2n
a3
..
.
U31
..
.
U32
..
.
U33
..
.
...
..
.
U3n
..
.
am
Um1
Um2
Um3
...
Umn
– Decision Matrix
The Decision Matrix for
is represented in the following table:
Events
Ecological
Ecological
awareness
awareness
rises
falls
Keep the Brand
1500
0
Keep the Brand and Internationalize
2000
-400
Sell the Brand
500
500
Actions
Deciding with perfect information
If the decision maker knows what event will happen, he will decide in order
to maximize the payoff.
Considering that event θ0 will happen, the action a0 that should be taken
would be:
a0 : U (a0 , θ0 ) = maxai U (ai , θ0 )
– Deciding with perfect information
Events
Ecological
Ecological
awareness
awareness
rises
falls
Keep the Brand
1500
0
Keep the Brand and Internationalize
2000
-400
Sell the Brand
500
500
Actions
If the decision maker knows that the ecological awareness will rise
then he/she will “Keep the Brand and Internationalize”.
If the decision maker knows that the ecological awareness will fall
then he/she will “Sell the Brand”.
Decision criterion – Laplace
This method considers that the state of nature probabilities are all the
same. If there are n events, then the probability of each one will be n1 . The
action to choose is the one for which:
o
n P
n
1
maxai n j=1 U (ai ; θj )
– Decision criterion – Laplace
Using the Decision Matrix of
criterion:
, and using the Laplace decision
Events
Ecological
Ecological
awareness
awareness
rises
falls
Keep the Brand
1500
0
Keep the Brand and Internationalize
2000
-400
Sell the Brand
500
500
Actions
1
n
Pn
j=1
Considering this decision criterion the action “Keep the Brand and
Internationalize” should be chosen. This action corresponds to:
Pn
1600
1
j=1 U (ai ; θj ).
2 = maxi n
U (ai ; θj )
1500
2
1600
2
1000
2
Decision criterion – Maximin (pessimist)
In this criterion the decision maker’s problem is viewed as a “game” against
nature. The decision maker considers that the worst event will occur. The
action to be chosen should then be the one that:
maxai minθj U (ai ; θj )
– Decision criterion – Maximin (pessimist)
Using the Decision Matrix of
(pessimist) decision criterion:
, and using the Maximin
Events
Ecological
Ecological
awareness
awareness
rises
falls
minθj U (ai ; θj )
Keep the Brand
1500
0
0
Keep the Brand and Internationalize
2000
-400
-400
Sell the Brand
500
500
500
Actions
Considering this decision criterion the action “Sell the Brand” should be
chosen. This action corresponds to:
500 = maxi minθj U (ai ; θj ) .
Decision criterion – Savage (moderate pessimist)
The Savage criterion is also called the minimization of opportunity loss
(regret) criterion.
After decisions have been made and the events occurred, decision makers
may express regret because they now know what event has taken place and
may wish they had selected a different action. The Savage criterion intends
to minimize this regret.
To apply the Savage criterion, the Decision Matrix must be transformed
into a Regret Matrix, by using the following transformation:
P (ai ; θj ) = maxak {U (ak ; θj )} − U (ai ; θj )
and the Minimax criterion must be applied to the Regret Matrix.
minai maxθj P (ai ; θj )
– Decision criterion – Savage (moderate
pessimist)
Transform the Decision Matrix of
apply the Minimax decision criterion:
into a Regret Matrix, and
Events
Ecological
Ecological
awareness
awareness
rises
falls
maxθj P (ai ; θj )
Keep the Brand
500
500
500
Keep the Brand and Internationalize
0
900
900
Sell the Brand
1500
0
1500
Actions
Considering this decision criterion the action “Keep the Brand” should be
chosen. This action corresponds to:
500 = minai maxθj P (ai ; θj ) .
Decision criterion – Maximum Expected Value (MEV)
Using the best available estimates of the probabilities h(θj ) of the events θj
P
(the pior probabilities), such that j h(θj ) = 1.
Calculate the expected value of the payoff for each decision alternative.
P
∀ai V Eai = j {h(θj ) × U (ai ; θj )}
Choose the decision alternative a0 with the maximum expected payoff:
a0 : maxai {V Eai }
– Decision criterion – Maximum Expected
Value (MEV)
Using the Decision Matrix of
, and the MEV criterion:
Events
Ecological
Ecological
awareness
awareness
rises
falls
h(θj )
0.70
0.30
Keep the Brand
1500
0
1050
Keep the Brand and Internationalize
2000
-400
1280
Sell the Brand
500
500
500
Actions
minθj U (ai ; θj )
Considering this decision criterion the action “Keep the Brand and
Internationalize” should be chosen. This action corresponds to the
maximum expected value (MEV) 1280.
– Decision criterion – Expected Opportunity
Loss (EOL)
Using the Regret Matrix of
, and the EOL criterion:
Events
Ecological
Ecological
awareness
awareness
rises
falls
h(θj )
0.70
0.30
Keep the Brand
500
500
500
Keep the Brand and Internationalize
0
900
270
Sell the Brand
1500
0
1050
Actions
minθj U (ai ; θj )
Considering this decision criterion the action “Keep the Brand and
Internationalize” should be chosen. This action corresponds to the minimum
expected opportunity loss (EOL) 270.
Decision Trees
An alternative way to structure a decision problem pictorially is with a
decision tree. A decision tree depicts chronologically se sequence of actions
and events as they unfold.
Decision trees are very useful to represent complex decision problems, with
sequences of actions and events that occur over time.
Nodes (or Forks) in a decision tree
• Decision nodes, represented by a square, indicate that a decision needs
to be made at that point in the process.
• Event nodes, represented by a circle, indicate that a random event
occurs at that point.
– Decision Tree
Draw the decision tree for
, including the actions, the events
with their a priori probabilities and the outcomes:
Additional information can be of any use?
Until now we considered situations in which decision makers chose between
alternative actions based only on a priori information about the problem
and did not attempt to obtain any additional information.
Some questions we could ask at this moment:
• Is it worth to get additional information?
• What type of additional information should we get?
• What should we do with the additional information?
• How much would we pay for the additional information?
Expected Value of Perfect Information (EVPI)
In the absence of data on the credibility of the information provider, you
can not assign value to the information. You can however determine the
expected increase in the expected value if the information is perfect, which
is really an upper limit to this value.
This upper limit is known as the Expected Value of Perfect Information
(EVPI) and can be obtained by three different methods:
1. by subtracting the Maximum Expected Value (with uncertainty), of the
Maximum Expected Value (with perfect information);
2. by doing an “incremental” analysis;
3. by calculating the minimum value for the expected Minimum
Opportunity Loss.
– EVPI
Maximum Expected Value (with perfect information) - Maximum Expected Value (with
uncertainty)
P
j
h(θj ) × maxai U (ai , θj ) − maxai
nP
o
j h(θj ) × U (ai , θj )
Events
Ecological
Ecological
awareness
awareness
rises
falls
h(θj )
0.70
0.30
Keep the Brand
1500
0
1050
Keep the Brand and Internationalize
2000
-400
1280
Sell the Brand
500
500
500
maxai U (ai , θj )
2000
500
1550
Actions
EVPI = M V Epi - MVE = 1550 - 1280 = 270
EOL = 270
minθj U (ai ; θj )
“Credibility” Matrix
The “Credibility” Matrix is a way to “measure” the credibility of the
experience or of the consulting firm.
Considering P (rk |θj ) as the probability of the experience having result rk ,
given that the event is θj :
Experience
P
Events
Results
θ1
θ2
θ3
...
θJ
r1
P (r1 |θ1 )
P (r1 |θ2 )
P (r1 |θ3 )
...
P (r1 |θJ )
r2
P (r2 |θ1 )
P (r2 |θ2 )
P (r2 |θ3 )
...
P (r2 |θJ )
r3
..
.
P (r3 |θ1 )
..
.
P (r3 |θ2 )
..
.
P (r3 |θ3 )
..
.
...
..
.
P (r3 |θJ )
..
.
rK
P (rK |θ1 )
P (rK |θ2 )
P (rK |θ3 )
...
P (rK |θJ )
1
1
1
...
1
k
P (rk |θj )
“Credibility” Matrix in case of Perfect Information
If we have perfect information (maximum credibility), considering that the
result ri predicts that the evnt θi will occur, the “Credibility” Matrix will be:
Experience
Events
Results
θ1
θ2
θ3
...
θJ
r1
1
0
0
...
0
r2
0
1
0
...
0
r3
..
.
0
..
.
0
..
.
1
..
.
...
..
.
0
..
.
rJ
0
0
0
...
1
– Decision Tree in case of perfect information
Lets make a point:
We know:
P (θj ) – a priori probability of an event θj
we also know:
P (rk |θj ) – probability of the experience having result rk , given that the
event is θj (“credibility” of the experience)
but what we need to know are the probabilities of the events after the
information of the experiences (a posteriori probabilities).
P (θj |rk ) – probability of a certain event θj , given that the experience had
result rk
– External Consulting
The board of
considered the possibility of using an external
consultancy firm with some credibility in assessing future ecological
awareness.
Analyzing the portfolio of the company in that area, the board concluded
that the company’s “Credibility” Matrix P (rk |θj ) would be:
Events
P (rk |θj )
Ecological
Ecological
awareness
awareness
Experience results
rises
falls
Prediction of rise of the ecological awareness (r1 )
0.7
0.5
Prediction of reduction of the ecological awareness (r2 )
0.3
0.5
Bayes Theorem is a tool that allows us to calculate P (θj |rk ) and P (rk )
given P (rk |θj ) and P (θj )
Reverend Thomas Bayes (1702–1761)
Bayes set down his findings on probability in “Essay Towards
Solving a Problem in the Doctrine of Chances” (1763), published posthumously in the Philosophical Transactions of the
Royal Society. That work became the basis of a statistical technique, now called Bayesian estimation, for calculating the
probability of the validity of a proposition on the basis of a
prior estimate of its probability and new relevant evidence.
The article as sent to the “Royal Society” by his friend Richard
Price, who wrote:
I now send you an essay which I have found among the papers of our
a
deceased friend Mr Bayes, and which, in my opinion, has great merit...
In an introduction which he has written to this Essay, he says, that his
design at first in thinking on the subject of it was, to find out a method
by which we might judge concerning the probability that an event has to
happen, in given circumstances, upon supposition that we know nothing
concerning it but that, under the same circumstances, it has happened a
certain number of times, and failed a certain other number of times.
a http://www.britannica.com/EBchecked/topic/56807/Thomas-Bayes
Bayes’ Theorem
With Bayes’ Theorem we can calculate P (θj |rk ) (probability of event θj
given that the result of the experience was rk ), just by knowing P (rk |θj )
and P (θj ).
P (θj |rk ) =
P (θj , rk )
P (rk |θj ) × P (θj )
=P
P (rk )
j P (rk |θj ) × P (θj )
Experience
(1)
Events
P
P (θj |rk )
results
P (rk )
θ1
θ2
θ3
...
θJ
r1
P (r1 )
P (θ1 |r1 )
P (θ2 |r1 )
P (θ3 |r1 )
...
P (θJ |r1 )
1
r2
P (r2 )
P (θ1 |r2 )
P (θ2 |r2 )
P (θ3 |r2 )
...
P (θJ |r2 )
1
r3
..
.
P (r3 )
..
.
P (θ1 |r3 )
..
.
P (θ2 |r3 )
..
.
P (θ3 |r3 )
..
.
...
..
.
P (θJ |r3 )
..
.
1
..
.
rK
P (rK )
P (θ1 |rK )
P (θ2 |rK )
P (θ3 |rK )
...
P (θJ |rK )
1
j
– External Consulting
Applying Bayes’ Theorem, we obtain the a posteriori probabilities of
occurrence of each event, given the various possible outcomes of the
experiment, as represented in the following table:
Events
P (θj |rk )
Ecological
Ecological
awareness
awareness
Experience results
Prk
rises
falls
Prediction of rise of the ecological awareness (r1 )
0.64
0.77
0.23
Prediction of reduction of the ecological awareness (r2 )
0.36
0.58
0.42
– External Consulting – Decision tree
References
• Frederick S Hillier, Gerald J Lieberman (2005). Introduction to
Operations Research – eighth edition, Mc Graw-Hill.
• Informs – Decision Analysis Society
http://decision-analysis.society.informs.org/ Consulted on November
2009
• Murteira, Bento (1981). Introdução à Teoria da Decisão.
• Ravindram, Philips e Solberg (1987). Operations Research, Principles
and Practice. John Wiley & Sons.
• Taha, Hamdy A. (1997). Operations Research, an Introduction. Prentice
Hall.
• Winston, Wayne L. (1994). Operations Research, Applications and
Algorithms Duxbury Press.