Decision Analysis - UPpaginas.fe.up.pt/~mac/ensino/docs/OR/DecisionAnalysis.pdf · Decision Analysis Version 3.1 c 2010, 2009, 2008, 2002, 1998 Maria Ant onia Carravilla Jos e Fernando

Decision Analysis

Version 3.1

c©2010, 2009, 2008, 2002, 1998

Maria Antonia Carravilla

Jose Fernando Oliveira

FEUP

Nothing is more difficult, and therefore more precious, than to be able todecide 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


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 andyet, for any number of reasons, achieve no acceleration at all.

good decision vs. good outcome Example: Someone who buys a lottery ticket and winsthe lottery obtains a good outcome. Yet, the decision to buy the lottery ticket may or may not have been agood decision.

Why it’s important: A bad decision may lead to a good outcome and conversely a good decision may lead to abad 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 additionalpersonnel, while at the same time stopping the funding of some stalled projects, is a strategy intended to leadto that goal.

Why it’s important: Strategy describes a collection of actions that the decision maker takes. The outcome ofthe 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 developmentproject A than development project B, and project B than project C, is a prioritization. Actually fundingproject A is a decision.

Why it’s important: A prioritization might be an intermediate step en route to a decision, and one might evenuse 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 dedicatedonly to the development and production of environmentally friendly toys. These toys are made of non-toxicand 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. Allparts 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 tothe 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 andthe 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 beproduced in any color of the palette. The optional accessories are mounted after the assembly of the doll, asshown 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 ofmonths before, for an analysis of a possible internationalization of the brand.The results of that analysis are also at hand. The third alternative would beto leave everything as it is: keep the brand and not internationalize.

The future is however very uncertain in what concerns the ecologicalawareness of the buyers. Their ecological awareness could rise, implying anhigh increase in the sales, or the ecological awareness could reduce, due tothe problems in the world economy, implying a reduction of the sales.

– Definitions

For the decision problem of define:

the decision maker –

the alternatives –

the events –

the outcomes –

– 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 ofthe consequences of the outcome. For each combination of an alternativeand 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 bethe expected value of the measure of the consequences. The DecisionMatrix is commonly used to provide the payoff for each combination of anaction 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

Actions awareness awareness

rises falls

Keep the Brand 1500 0

Keep the Brand and Internationalize 2000 -400

Sell the Brand 500 500

Deciding with perfect information

If the decision maker knows what event will happen, he will decide in orderto maximize the payoff.

Considering that event θ0 will happen, the action a0 that should be takenwould be:

a0 : U(a0, θ0) = maxaiU(ai, θ0)

– Deciding with perfect information

Events



rises falls

Keep the Brand 1500 0

Keep the Brand and Internationalize 2000 -400

Sell the Brand 500 500

If the decision maker knows that the ecological awareness will risethen he/she will “Keep the Brand and Internationalize”.

If the decision maker knows that the ecological awareness will fallthen he/she will “Sell the Brand”.

Decision criterion – Laplace

This method considers that the state of nature probabilities are all thesame. If there are n events, then the probability of each one will be 1

n . Theaction to choose is the one for which:

maxai

{1n

∑nj=1 U(ai; θj)

}

– Decision criterion – Laplace

Using the Decision Matrix of , and using the Laplace decisioncriterion:

Events



rises falls 1n

∑nj=1 U(ai; θj)

Keep the Brand 1500 0 15002

Keep the Brand and Internationalize 2000 -400 16002

Sell the Brand 500 500 10002

Considering this decision criterion the action “Keep the Brand andInternationalize” should be chosen. This action corresponds to:

16002 = maxi 1

n

∑nj=1 U(ai; θj).

Decision criterion – Maximin (pessimist)

In this criterion the decision maker’s problem is viewed as a “game” againstnature. The decision maker considers that the worst event will occur. Theaction to be chosen should then be the one that:

maxai

{minθj

U(ai; θj)}

– Decision criterion – Maximin (pessimist)

Using the Decision Matrix of , and using the Maximin(pessimist) decision criterion:

Events



rises falls minθjU(ai; θj)


Keep the Brand and Internationalize 2000 -400 -400


Considering this decision criterion the action “Sell the Brand” should bechosen. This action corresponds to:

500 = maxi{minθjU(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 makersmay express regret because they now know what event has taken place andmay wish they had selected a different action. The Savage criterion intendsto minimize this regret.

To apply the Savage criterion, the Decision Matrix must be transformedinto 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 (moderatepessimist)

Transform the Decision Matrix of into a Regret Matrix, andapply the Minimax decision criterion:

Events



rises falls maxθjP (ai; θj)


Keep the Brand and Internationalize 0 900 900


Considering this decision criterion the action “Keep the Brand” should bechosen. This action corresponds to:

500 = minai

{maxθjP (ai; θj)

}.

Decision criterion – Maximum Expected Value (MEV)

Using the best available estimates of the probabilities h(θj) of the events θj(the pior probabilities), such that

∑j h(θj) = 1.

Calculate the expected value of the payoff for each decision alternative.

∀aiV Eai

=∑j {h(θj)× U(ai; θj)}

Choose the decision alternative a0 with the maximum expected payoff:

a0 : maxai{V Eai

}

– Decision criterion – Maximum ExpectedValue (MEV)

Using the Decision Matrix of , and the MEV criterion:

Events




h(θj) 0.70 0.30




Considering this decision criterion the action “Keep the Brand andInternationalize” should be chosen. This action corresponds to themaximum expected value (MEV) 1280.

– Decision criterion – Expected OpportunityLoss (EOL)

Using the Regret Matrix of , and the EOL criterion:

Events




h(θj) 0.70 0.30


Keep the Brand and Internationalize 0 900 270


Considering this decision criterion the action “Keep the Brand andInternationalize” should be chosen. This action corresponds to the minimumexpected opportunity loss (EOL) 270.

Decision Trees

An alternative way to structure a decision problem pictorially is with adecision tree. A decision tree depicts chronologically se sequence of actionsand events as they unfold.

Decision trees are very useful to represent complex decision problems, withsequences 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 needsto be made at that point in the process.

• Event nodes, represented by a circle, indicate that a random eventoccurs at that point.

– Decision Tree

Draw the decision tree for , including the actions, the eventswith their a priori probabilities and the outcomes:

Additional information can be of any use?

Until now we considered situations in which decision makers chose betweenalternative actions based only on a priori information about the problemand 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, youcan not assign value to the information. You can however determine theexpected increase in the expected value if the information is perfect, whichis 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 theMaximum Expected Value (with perfect information);

2. by doing an “incremental” analysis;

3. by calculating the minimum value for the expected MinimumOpportunity Loss.

– EVPI

Maximum Expected Value (with perfect information) - Maximum Expected Value (with

uncertainty)∑j h(θj)×maxaiU(ai, θj)−maxai

{∑j h(θj)× U(ai, θj)

}Events




h(θj) 0.70 0.30




maxaiU(ai, θj) 2000 500 1550

EVPI = MVEpi - MVE = 1550 - 1280 = 270

EOL = 270

“Credibility” Matrix

The “Credibility” Matrix is a way to “measure” the credibility of theexperience 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 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)∑k P (rk|θj) 1 1 1 . . . 1

“Credibility” Matrix in case of Perfect Information

If we have perfect information (maximum credibility), considering that theresult 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 theevent is θj (“credibility” of the experience)

but what we need to know are the probabilities of the events after theinformation of the experiences (a posteriori probabilities).

P (θj |rk) – probability of a certain event θj , given that the experience hadresult rk

– External Consulting

The board of considered the possibility of using an externalconsultancy firm with some credibility in assessing future ecologicalawareness.

Analyzing the portfolio of the company in that area, the board concludedthat 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)

a

Bayes set down his findings on probability in “Essay TowardsSolving a Problem in the Doctrine of Chances” (1763), pu-blished posthumously in the Philosophical Transactions of theRoyal Society. That work became the basis of a statistical te-chnique, now called Bayesian estimation, for calculating theprobability of the validity of a proposition on the basis of aprior estimate of its probability and new relevant evidence.The article as sent to the “Royal Society” by his friend RichardPrice, who wrote:

I now send you an essay which I have found among the papers of our

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.

ahttp://www.britannica.com/EBchecked/topic/56807/Thomas-Bayes

http://www.britannica.com/EBchecked/topic/56807/Thomas-Bayes

Bayes’ Theorem

With Bayes’ Theorem we can calculate P (θj |rk) (probability of event θjgiven 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)

=P (rk|θj)× P (θj)∑j P (rk|θj)× P (θj)

(1)

Experience Events

results P (rk) θ1 θ2 θ3 . . . θJ∑j P (θj |rk)

r1 P (r1) P (θ1|r1) P (θ2|r1) P (θ3|r1) . . . P (θJ |r1) 1



......

......

.... . .

......

rK P (rK) P (θ1|rK) P (θ2|rK) P (θ3|rK) . . . P (θJ |rK) 1

– External Consulting

Applying Bayes’ Theorem, we obtain the a posteriori probabilities ofoccurrence of each event, given the various possible outcomes of theexperiment, 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 toOperations Research – eighth edition, Mc Graw-Hill.

• Informs – Decision Analysis Societyhttp://decision-analysis.society.informs.org/ Consulted on November2009

• Murteira, Bento (1981). Introducao a Teoria da Decisao.

• Ravindram, Philips e Solberg (1987). Operations Research, Principlesand Practice. John Wiley & Sons.

• Taha, Hamdy A. (1997). Operations Research, an Introduction. PrenticeHall.

• Winston, Wayne L. (1994). Operations Research, Applications andAlgorithms Duxbury Press.

http://decision-analysis.society.informs.org/

Documents

Decision Analysis - UPpaginas.fe.up.pt/~mac/ensino/docs/OR/DecisionAnalysis.pdf · Decision Analysis Version 3.1 c 2010, 2009, 2008, 2002, 1998 Maria Ant onia Carravilla Jos e Fernando