Emanuele Borgonovo

1

• Emanuele Borgonovo
• Quantitative Methods for Management
• First Edition

2
Chapter threeModels
3
Models

• A Model is a mailmatical-logical instrument that
  the analyst, the manager, the scientist, the
  engineer develops to
• foretell the behaviour of a system
• foresee the course of a market
• evaluate an investment decision accounting for
  uncertainty factors
• Common Elements to the Models
• Uncertainty
• Assumptions
• Inputs
• Model Results

4
Building a Model

• To build a reliable model requires deep
  acquaintance of
• the Problem
• Important Events regarding the problem
• Factors that influence the behavior of the
  quantities of interest
• Data and Information Collection
• Uncertainty Analysis
• Verification of the coherence of the Model by
  means of empiric analysis and , if possible,
  analysis of Sensitivity Analysis

5
Example the law of gravity

• We want to describe the vertical fall of a body
  on the surface of the earth. We adopt the Model
• Fmg
• for the fall of the bodies
• Hypothesis (?)
• Punctiform Body (no spins)
• No frictions
• No atmospheric currents
• Does the model work for the fall of a body placed
  to great distance from the land surface?

6
Chapter IIIntroductory Elements of Probability
theory
7
Probability

• Is it Possible to Define Probability?
• Yes, but there are two schools
• the first considers Probability as a property of
  events
• the second school asserts that Probability is a
  subjective measure of event likelihood (De
  Finetti)

8
Kolmogorov Axioms
U
B
A
9
Areas and rectangles?
U

• Suppose one jumps into the area U randomly. Let
  P(A) be the Probability to jump into A. What is
  its value?
• It will be the area of A divided by the area of
  U P(A)A/U
• Note that in this case P(U)P(A) P(B) P(C)
  P(D) P(E), since there are no overlaps

10
Conditional Probability

• Consider events A and B. the conditional
  Probability of A given B, is the Probability of A
  given the B has happened. One writes P(AB)

U
B
AB
A
11
Conditional Probability

• Suppose now that B has happened, i.e., you jumped
  into area B (and you cannot jump back!).

B
AB
A

• You cannot but agree that
• P(AB)P(AB)/P(B)
• Hence P(AB)P(AB) P(B)

12
Independence

• Two events, A and B, are independent if given
  that A happens does not influence the fact that B
  happens and vice versa.

B
B
A
AB
A
Thus, for independent events P(AB)P(A)P(B)
13
Probability and Information

• Problem you are given a box containing two
  rings. the box content is such that with the
  same Probability (1/2) the box contains two
  golden rings (event A) or a golden ring and a
  silver one (event B). To let you know the box
  content, you are allowed to pick one ring from
  the box. Suppose it is a golden one.
• In your opinion, did you gain information from
  the draw?
• the Probability that the oil one is golden is
  50?
• Would you pay anything to have the possibility to
  draw from the box?

14
In the subjectivist approach, Probability changes
with information
15
Bayes theorem

• Hypothesis A and B are two events. A has
  happened.
• Thesis P(B) changes as follows

16
Let us come back to the ring problem

• Events
• A both rings are golden
• o the picked up ring is golden
• the theorem states
• P(A)Probability of both rings being golden
  before the extraction 1/2
• P(o)Probability of a golden ring3/4
• P(oA)Probability that the extracted ring is
  golden given A1 (since both rings are golden)
• So

17
Bayes theorem Proof
Starting point
Conditional Probability formula
thesis
18
the Total Probability theorem
U
and

• the total Probability theorem states given N
  mutually exclusive and exhaustive events A1,
  A2,,AN, the Probability of an event and in U can
  be decomposed in
• Bayes theorem in the presence of N events becomes
  

19
Continuous Random Variables

• Till now we have discussed individual events.
  there are problems in which the event space is
  continuous. For example, think of the failure
  time of a component or the time interval between
  two earthquakes. the random variable time ranges
  from 0 to ?.
• To characterize such events one resorts to
  Probability distributions.

20
Probability Density Function

• f(x) is a Probability density function (pdf) if
• It is integrable
• And
• the integral of f(x) over -? ? is equal to 1.
• Note f(x0)dx is the Probability that x lies in
  an interval dx around x0.

21
Cumulative Distribution Function

• Given a continuous random variable X, the
  Probability that Xltx is given by
• If f(x) is continuous, then
• Note

22
the exponential distribution

• Consider events that happen continuously in time,
  and with continuous time T.
• If the events are
• Independents
• With constant failure rates
• the random variable T is characterized by an
  exponential distribution
• and by the density function

23
Meaning of the Exponential Distribution

• We are dealing with a reliability problem, and we
  must characterize the failure time, T. T is a
  random variable one does not know when a
  component is going to break. All one can say is
  that for sure the component will break between 0
  and infinity. Thus, T is a continuous random
  variable.
• Let us consider that failures are independent.
  This is the case if the failure of one component
  does not influence the failure of the other
  components.
• Let us also consider constant failure rates.
  This is the case when repair brings the component
  as good as new and when the component does not
  age during its life.
• Under these Hypothesis, the failure times are
  independent and characterized by a constant
  failure rate ? at every dt. What is the
  Probability distribution of T?
• Let us consider a population of N(t) components
  at time t. If ? is the failure rate of a
  component, then N(t)?dt is the number of failues
  in dt around time t.

24
the Exponential Distribution

• Thus the change in the population is
• -N(t)?dtN(tdt)-N(t)dN(t)
• Where the minus sign indicates that the number of
  working components has decreased.
• Hence
• Which solved leads
• N(T) is the number of components surviving till
  T. N(0) is the initial number of components. Set
  N(0)1. then N(T)/N(0) is the Probability that a
  component survives till T.

25
Pdf and Cdf of the Exponential Distribution
P(tltT)
f(t)
T/t
26
Expected Value, Variance and Percentiles
Percentile p is the value xp of X such that the
Probability of X being lower than xp is equal to
p/100
27
the Normal Distribution

• Is a symmetric distribution around the mean
• Pdf
• Cdf

28
Graphs of the Normal Distribution
Cumulative Gaussian Distribution
10000
9000
8000
7000
6000
5000
4000
3000
2000
1000
0
-5
-4
-3
-2
-1
0
1
2
3
4
x
29
Lognormal Distribution

• Pdf
• Cdf

30
Lognormal Distribution
31
Problem II-1 and solution

• the failure rate of a car gear is 1/5 for year
  (exponential events).
• What is the mean time to failure of the gear?
• What is the Probability of the gear being integer
  after 9 years?

32
Problem II-2

• You are considering a University admission test
  for a particularly selective course. the
  admission test, as all tests test, is not
  perfect. Suppose that the true distribution of
  the class is such that 10 of the applicants are
  really qualified and 90 are not. then you
  perform the test. If a student is qualified,
  then the test will admit him/her with 90
  Probability. If the student is not qualified
  he/her gets admitted at 10. Now, let us consider
  a student that got admitted
• What is the Probability that the student is
  really qualified?
• Is it a good test? How would you use it?
• (Hint use the theorem of Total Probability)

33
Problem II-3

• For the example of the two rings, determine
• P(Bo)
• P(Ba)
• the Probability of being in A given that the
  picked ring is golden in two consecutive
  extractions, having put the ring back in the box
  after the first extraction
• the Probability of being in B given that the
  picked ring is golden in two consecutive
  extractions, having put the ring back in the box
  after the first extraction

34
Problem II-3

• For the example of the two rings, determine
• P(Bo)
• Solution there are only two possible events, A
  or B. Thus, P(B?or)1-P(A?or)1/3
• P(B?a)
• P(B?a)1, since B is the only event that has a
  silver ring. One can also show it using Bayes
  theorem
• P(B?a)P(a?B)P(B)/P(a?B) P(B)P(a?A)P(A).
  Since P(a?A)0, one gets 1 at once.
• the Probability of being in A given that the
  picked ring is golden in two consecutive
  extractions, having put the ring back in the box
  after the first extraction
• Using Bayes theorem

35
Problem II-3

• where, in the formula, subscript 1 indicates the
  probabilities after the information of the first
  extraction has been taken into account
• P1(B)P(B?or)1/3 and P1(A)P(A?or)2/3.
• One can note that P(2o?A)1, and P(2o?B)1/2.
  P(2o?B) is the Probability to pick a golden ring
  at the second run, given that one is in state B.
• Thus, we have all the numbers to be substituted
  back in the theorem
• It is the same problem as in the example, but
  with adjourned probabilities.
• the Probability of being in B given that the
  picked ring is golden in two consecutive
  extractions, having put the ring back in the box
  after the first extraction
• Solution 1-P(A?2o)0.2

36
Chapter IIIIntroductory Decision theory
37
An Investment Decision

• At time T, you have to decide whether, and how,
  to invest 1000. You face three mutually
  exclusive options
• (1) A risky investment that gives you 500 PV in
  one year if the market is up or a loss of 400 if
  the market is down
• (2) A less risky investment that gives you 200
  in one year or a loss of 160
• (3) the safe investment a bond that gives you
  20 in one year independently of the market

38
Decision theory According to Laplace

• the theory leaves nothing arbitrary in choosing
  options or in making decisions and we can always
  select, with the help of the theory , the most
  advantageous choice on our own. It is a
  refreshing supplement to the ignorance and
  feebleness of the human mind.
• Pierre-Simon Laplace
• (March 28 1749 Beaumont-en-Auge - March 5 1827
  Paris)

39
Decision-Making Process Steps
40
Decision-Making Problem Elements

• Values and Objectives
• Attributes
• Decision Alternatives
• Uncertain Events
• Consequences

41
Decision Problem Elements

• Objectives
• Maximize profit
• Attributes
• Money
• Alternatives
• Risky
• Less Risky
• Safe
• Random events
• the Market
• Consequences
• Profit or Loss

42
Decision Analysis Tools

• Influence Diagrams

• Decision Trees

43
Influence Diagrams

• Influence diagrams (IDs) are
• a graphical representation of decisions and
• uncertain quantities that explicitly reveals
• probabilistic dependence and the flow of
• information
• ID formal definition
• ID a network consisting of a directed graph
  G(N,A) and associated node sets and functions
  (Schachter, 1986)

44
ID Elements

• NODES

• ARCS
• Informational Arcs
• probabilistic Dependency Arcs
• Structural Arcs

45
ID Elements
Structural
46
Influence Diagram Levels

• 1. Physical Phenomena and Dependencies
• 2. Function level node output states
  probabilistic relations (models)
• 3. Number level tables of node probabilities

47
Case Study 2 - Leaking SG tube

• Influence Diagram for Case Study 2

48
Influence Diagram
49
Decision Trees

• Decision Trees (DTs) are constituted by the same
  type of arcs of Influence Diagrams, but highlight
  all the possible event combinations.
• Instead of arks, one finds branches that emanate
  from the nodes as many as the Alternatives or
  Outcomes of each node.
• With respect to Influence Diagrams, Decision
  Trees have the advantage of showing all possible
  patterns, but their structure becomes quite
  complicated at the growing of the problem
  complexity.

50
the Decision Tree (DT)
51
Decision Tree Solution

• Alternative Payoff or utility
• j1mi spans all the Consequences associated to
  alternative the
• Uj is the utility or the payoff of consequence j
• Pi(Cj) is the Probability that consequence Cj
  happens given that one chose alternative the
• In general, we will get P(Cj) P(E1E2 EN),
  where E1E2 EN are the events that have to happen
  so that consequence Cj is realized. Using
  conditional probabilities
• P(Cj) P(E1E2 EN)P(EN E1E2 )P(E2
  E1)P(E1)

52
example
53
Problem Solution

• Using the previous formula

54
the Best Investment for a Risk Neutral Decision -
Maker
55
Run or Withdraw?

• You are the owner of a racing team. It is
  the last race of the season, and it has been a
  very good season for you. Your old sponsor will
  remain with you for the next season offering an
  amount of 50000, no matter what happens in the
  last race. However, the race is important and
  transmitted on television. If you win or end the
  race in the first five positions, you will gain a
  new sponsor who is offering you 100000, besides
  10000 or 5000 praise. However there are
  unfavorable running conditions and an engine
  failure is likely, based on your previous data.
• It would be very bad for the image of you
  racing team to have an engine failure in such a
  public race. You estimate the damage to a total
  of -30000.
• What to do? Run or withdraw?
• A) Elements of the problem
• What are your objectives
• What are the decision alternatives
• What are the attributes of the decision
• What are the uncertain events
• What are the alternatives

56
Example of a simple ID
57
From IDs to Decision Trees
58
Sequential Decisions

• Are decision making problems in which more than
  one decisions are evaluated one after the other.
• You are evaluating the purchase of a production
  machine. Three models are being judged, A B and
  C. the machine costs are 150, 175 and 200
  respectively. If you buy model A, you can choose
  insurance A1, that covers all possible failues of
  A, and costs 5 of A cost, or you can choose
  insurance policy A2, that costs 3 of A cost, but
  covers only transportation risk. If you buy
  model B, insurance policy B1 costs 3 of B cost
  and covers all B failures. Insurance B2 costs 2
  of B and covers only transportation. For model
  C, the most reliable, the insurance coverages
  cost 2 and 1.5 respectively. Based on this
  information and supposing that the machines
  production is the same, what will you choose?
• (failure Probability of A in the period of
  interest5)
• (failure Probability of B in the period of
  interest3)
• (failure Probability of C in the period of
  interest2

59
Influence Diagram
60
Decision Tree
61
the Expected Value of Perfect Information

• Data and information collection is essential to
  make decisions. Sometimes firms hire consultants
  or experts to get such information. But, how
  much should one spend?
• One can value information, since it is capable of
  helping the decision-maker in selecting among
  alternatives
• the value of information is the added value of
  the information.
• the expected value of perfect information (EVPI)
  assumed that the source of information is
  perfect, and then
• the definition is read as follows how much is
  the decision worth with the new information and
  without
• N.B. we will refer only to aleatory uncertainty

62
Example investing
63
EVPI for the Example
64
EVPI Result
65
Problems
66
How much to bid?

• Bob works for an energy production company. Your
  company is engaged in the decision of how much to
  bid to salvage the wreckage of the SS.Kuniang, a
  carbon transportation boat. If the firm wins, the
  boat could be repaired and could come back to its
  transportation activity again. Pending on the
  possible winning and on the decision is the
  result of a judgment by Coast Guard, which will
  be revealed only after the opening of the bids.
  That is, if the Coast Guard will assign a low
  value to the ship, this would mean that the ship
  is considered as recoverable. Otherwise, the
  boat will be deemed unusable. If you do not win,
  you will be forced to buy a new boat.
• Identify the decision elements
• Structure the corresponding ID and DT

67
Influence Diagram with three events

• Given the following elements
• Alternatives 1 and 2
• Events A(up, down) (Bhigh, low)(Cgood,
  bad)
• Consequences Ci (one distinct consequence for
  each event combination)
• If ADown happens, then CAdown is directly
  realized
• Draw the ID corresponding to the problem
• Draw the corresponding Decision Tree
• If C now depends on both A and B outcomes, how
  does the ID become?
• How does the DT change?

68
Solution

• Influence Diagram the

69
Solution

• Corresponding Decision Tree

70
Solution

• Influence Diagram II

71
Solution

• Decision Tree II

72
Sales_Costs

• Given the following Influence Diagram and
  Decision Tree, given P_High and P_HighHigh,
  P_highlow, find the value of the Alternatives
  as a function of the assigned probabilities.
  Supposing P_high0.5 and P_highhighP_highlow0.
  3, find the preferred alternative.
• What would be the preferred decision if to a
  higher investment cost there would correspond a
  better sale result? Set
• P_highhigh0.6 and P_highlow0.2

73
Solution Sales_Costs
74
Breakdown in Production

• An industrial system composed from two lines has
  experience a breakdown in one line. Production,
  therefore, is reduced by 50. the management asks
  you collaboration on the following decision. It
  is explained to you that there are two ways to
  proceed 1) an intermediate repair, of the
  duration of two days, with a repair cost of
  EUR500000. For every day of production loss of
  EUR25000 for day is sustained (Full production
  amounts at EUR50000). From the engineer
  estimates, the Probability of perfect repair in
  two days is equal to P_2g. In the case in which
  the repair it is not perfect (partial repair),
  the line will come back with a loss of 15 of the
  productive ability 2) a more incisive
  intervention, of the duration of 10 days, with a
  cost of repair of EUR1000000. With Probability
  P_10g the line will be as before the breakdown.
• According to you, the residual life of the system
  is important for the decision?
• Suppose that there are still three years of life
  for the system.
• Which strategy should you carry out?
• Determine the decision problem elements. Draw the
  Influence Diagram and the corresponding Decision
  Tree. Find the value or values of the
  probabilities for which a complete repair is more
  convenient than a partial one.
• What would you would advise to the director of
  the system to do based on the engineer estimates?

75
EVPI Problems

• Determine the EVPI for the random event nodes in
  the previous IDs and DTs of the following
  problems
• Sales_Costs (lez. 2)
• Production break-down (lez.2)

76
Troubles in Production

• One of the two production lines of the plant you
  manage has broke down. the plant production
  capacity is therefore halved. the management
  faces the following decision and asks you a
  collaboration. Technically one can a 1) perform
  an temporary repair, lasting two days, and
  costing 500000. For every lost production day
  one has a revenue loss of 25000 for day (the
  total daily production value is 50000). Based on
  the Engineer estimates, the Probability of
  perfect repair in two days is P_2g . In the case
  of an imperfect repair, the production capacity
  will be lowered by 15. 2) perform a more
  incisive repair, lasting 10 days, and costing
  1000000. With Probability P_10g the line will
  be as good as new.
• In your opinion, the residual plant life is
  relevant to this decision?
• Suppose that there are still three years of life
  for the plant. What should one decide?
• Identify the decision making elements
• Draw the Influence Diagram for the problem
• Find the values of the probabilities for which
  one or the other intervention is more convenient
• What would your suggestion to the plant director
  be?
• What would happen if the plant life were 2 and 4
  years instead of 3?

77
Influence Diagram
78
Decision Tree
79
Probability Values

• Three years

80
2 and 4 years

• 2 years
• 4 years

81
Chapter IVElements of Sensitivity Analysis
82
Sensitivity Analysis

• Various Types of SA
• One Way SA
• Two Way SA
• Tornado Diagrams
• (Differential Importance Measure)
• Uncertainty Analysis
• Monte Carlo
• (Global SA)

83
How do we use SA?

• a) To check model correctness and robustness
• b) To Further interrogate the model
• Questions
• What is the most influential parameter with
  respect to changes?
• What is the most influential parameter on the
  uncertainty (data collection)

84
Sensitivity Analysis (Run or withdraw)

• Underline the critical dependencies of the
  outcome

85
Summary

• Sensitivity Analysis
• One way sensitivity
• Two way sensitivity
• Tornado Diagrams
• Uncertainty Analysis
• Aleatory Uncertainty
• Epistemic Uncertainty
• Bayes theorem for continuous distributions
• Monte Carlo Method

86
Sensitivity Analysis

• By sensitivity analysis one means the study of
  the change in results (output) due to a change in
  one of the model parameters (input)
• the simplest Sensitivity Analysis types are
• One way sensitivity
• Two way sensitivity
• Tornado diagrams

87
One-way Sensitivity Analysis

• A one way sensitivity is obtained changing the
  Model input variables one at a time, and
  registering the change in the decision value.
• It enables the analyst to study the change in
  value of each of the alternatives with respect to
  the change in the input parameter under
  consideration

88
Two-way Sensitivity Analysis

• In a Two-way Sensitivity Analysis two parameters
  are varied at the same time.
• Instead of a line, one obtains a plane, in which
  each region identifies the preferred alternative
  that correspond to the combination of the two
  parameter values

89
Tornado Diagrams

• the analysis is focused on the preferred decision
• An interval of variation for each input parameter
  is chosen
• the parameters are changed one at a time, while
  keeping the oilrs at their reference value
• the change in output is registered
• the output change is shown by means of a
  horizontal bar
• the most important variable is the one that
  corresponds to the longest bar.

90
Example of a Tornado Diagram
91
Upsides and Downsides

• Upsides
• Easy numerical calculations
• Results immediately understandable

• Downsides
• Input range of variation not considered together
  with the output range should not be used to
  infer parameter importance
• One or two parameters can be varied at the same
  time

92
Sensitivity Analysis and Parameter Importance

• Parameter importance
• Relevance of parameter in a model with respect to
  a certain criterion
• Sensitivity Analysis used to Determine Parameter
  Importance
• Concept of importance not formalized, but
  extensively used
• Risk-Informed Decision Making
• Resource allocation
• Need for a formal definition

93
Process

• Identify how sensitivity analysis techniques work
  through analysis of several examples
• Formulate a definition
• Classify sensitivity analysis techniques
  accordingly

94
Sensitivity Analysis Types

• Model Output
• Local Sensitivity Analysis
• Determines model parameter (xi) relevance with
  all the xi fixed at nominal value
• Global Sensitivity Analysis
• Determines xi relevance of xis
  epistemic/uncertainty distribution

95
the Differential Importance Measure

• Nominal Model output
• No uncertainty in the model parameters
• and/or parameters fixed at nominal value
• Local Decomposition
• Local importance measured by fraction of the
  differential attributable to each parameter

96
Global Sensitivity Indices

• Uncertainty in U and parameters is considered
• Sobols decomposition theorem
• SobolIndices

97
Formal Definition of Sensitivity Analysis (SA)
Techniques

• SA technique are Operators on U

98
Importance Relations

• Importance relations
• X the set of the model parameters
• Binary relation
• xi xj iff I(xi)gtI(xj)
• xixj iff I(xi)I(xj)
• xi xj iff I(xi)ltI(xj)
• xi xj iff I(xi)ltI(xj)
• Importance relations induced by importance
  measures are complete preorder

99
Additivity Property

• In many situation decision-maker interested in
  joint importance
• An Importance measure is additive if
• DIM is additive always
• Si are additive iff f(x) additive and xjs are
  uncorrelated

100
Techniques that fall under the definition of
Local SA techniques
101
Global Importance Measures
102
Sensitivity Analysis in Risk-Informed
Decision-Making and Regulation

• Risk Metric
• xi is undesired event Probability
• Fussell-Vesely fractional Importance
• Tells us on which events regulator has to focus
  attention

103
Summary of the previous concepts

• Formal Definition of Sensitivity Analysis
  Techniques
• Definition of Importance Relations
• Definition enables to
• Formalize use of Sensitivity Analysis
• Understand role of Sensitivity Analysis in
  Risk-informed Decision-making and in the use of
  model information

104
Chapter VUncertainty Analysis
105
Uncertainty Analysis
106
Summary

• Distinction between Aleatory Uncertainty ed
  Epistemic Uncertainty
• Epistemic Uncertainty and Bayes theorem
• Monte Carlo Method for uncertainty propagation

107
Uncertainty

• Aleatory Uncertainty
• From Alea, die Alea jacta est
• It refers to the realization of an event.
• Example the happening of an earthquake
• Epistemic Uncertainty
• From GreeK Epist???, knowledge
• it reflects our lack of knowledge in the value
  of the Aleatory Model input parameters. the
  aleatory model or model of the world is the model
  chosen to represent the random event.

108
Example Model of the World

• the Probability of Earthquakes is usually modeled
  through a Poisson model
• that rappresents the Probability that the number
  of earthquakes between 0 and t is equal to n.
• the Poisson Distribution holds for independent
  events, in which next events (arrivals) are not
  influenced by previous events and the Probability
  of an event in a given interval of time is the
  same independently of the time where the interval
  is located
• the Model chosen to describe the arrivals of
  earthquakes is given the non-humble name of
  "model of the world" (MOW).

109
Some useful information on Poisson Distributions

• the Poisson Probability that n events happen on
  0-t is
• the sum on n0...? of P(n,t) is, obviously, equal
  to 1.
• the Probability of kgtN is given by
• En?t

110
the Corresponding Epistemic Model

• Now,in spite of all the efforts and studies, it
  is unlikely that a scientist would tell you the
  rate (? ) of arrivals of earthquakes is exactly
  xxx. More likely, he will indicate you a range
  where the true value of ? lies. For example ?
  cuold be between 1/5 and 1/50 (1/years). Suppose
  that the scientist state of knowledge on ? can be
  expressed by a uniform distribution u(? )

111
Combining the Epistemic Model and MOW

• We have been dealing with two Models
• MOW the events happen according to a Poisson
  Distribution
• Epistemic Model Uniform Uncertainty Distribution
• then, what is the Probability of having 1
  earthquake in the next year?
• Answer there is no unique Probability, but a
  p(n,t, ?) for all values of ?.
• Thus, we have to write

112
.

• This expression tells us that not necessarily all
  Poisson distributions weight the same. Thus
• In our case u(?)c
• Hence, there is an expected Probability!

113
In General

• the MOW will depend on m parameters ?, ?,
• the event Probability (P(t)) will be

114
An problem

• the failure time of a series of components is
  characterized by the exponential Probability
  function
• From the available data, it emerges that
• What is the mean time to failure?

115
Solution

• Et

116
Continuous form of Bayes Theorem

• Epistemic Uncertainty and Bayes theorem are
  connected, in that we know that we can use
  evidence to update probabilities.
• For example, suppose to have a coin in your
  hands. will it be a fair with, i.e., will the
  Probability of tossing the coin lead to 50 head
  and tails?.
• How can we determine whether it is a fair coin?
• .let us toss it.

117
Formula

• the Probability density of a parameter, after
  having obtained evidence and, changes as follows
• L(E??) MOW likelihood
• ?0(?) is the pdf of ? before the evidence, called
  Prior Distribution
• ?(?) is the pdf of ? after the evidence, called
  Posterior Distribution

118
From discrete to continuous

• Let us take Bayes theorem for discrete events
• Let us go to continuous events our purpose is to
  know the Probability that a parameter of the MOW
  distribution assumes a certain value, given a
  certain evidence
• Thus, event Aj is ? takes on value ?
• Hence P(Aj)??0(?)d? ?0(?)prior density
• therefore P(E?Aj) has the meaning of Probability
  that the evidence and is realized given that ?
  equals ? . One writes L(E ? ? ) and it is the
  likelihood function
• Note it is the MOW!!!

119
From discrete to continuous

• the denominator in Bayes theorem expresses the
  sum of the probabilities of the evidence given
  all the possible states (the total Probability
  theorem). In the case of epistemic uncertainty
  these events are all possible values of ?. Thus
• Substituting the various terms, one finds Bayes
  theorem for continuous random variables we have
  shown before

120
Is it a fair coin?

• What is the MOW?
• It is a binomial distribution with parameter p
• What is the value of p?
• Suppse we do not know anything about p. Let us
  assume a uniform prior distribution between 0 and
  1
• Let us get some evidence.
• At the first tossing it is head
• At the second tail
• At the third head

121
Result

• First tossing
• Evidence h.
• MOW L(h?p)p
• Prior ?0
• Second tossing
• Evidence t
• MOW L(t?p)(1-p)
• Prior ?1
• Third tossing
• Evidence h
• MOW L(h?p)p
• Prior ?2
• Equivalently
• Evidence h, t, h
• L(hth?p)p2(1-p)
• Prior ?0

122
Graph
123
Conjugate Distributions

• Likelihood
• Poisson
• Posterior gamma

• Prior distribution
• Gamma
• with

124
Conjugate Distributions

• Likelihood
• Normal
• Posterior Normal

• Prior distribution
• Normal
• with

125
Conjugate Distributions

• Likelihood
• Binomial
• Posterior, Beta

• Prior
• Beta
• with

126
Summary of Conjugate Distributions
MOW - Likelihood Prior Distribution Posterior Distribution
Binomiale Beta Beta
Poisson Gamma Gamma
Normal Normal Normal
Normal Gamma Gamma
Negative binominal Beta Beta
127
Epistemic Uncertainty in Decision-Making Problems

• Investment
• Suppose that P.up is characterized by a uniform
  pdf between 0.3 and 0.7
• How does the decision changes?
• It is necessary to propagate the uncertainty in
  the model

128
Analytical Propagation of Uncertainty

• It is the same problem of the MOW
• Repeating for the other decisions and comparing
  the resulting mean values, one gets the optimal
  decision.
• Recall that

129
the Monte Carlo Method

• Sampling a value of P.up
• For all sampled P.up the Model is re-evaluated.
• Information
• Frequency of the preferred alternative
• Distribution of each individual Alternative

130
the core of Monte Carlo

• 1) Random Number Generator u between 0 and 1
• 2) Numbers u are generated with a uniform
  Distribution
• 3) Suppose that parameter ? is uncertain and
  characterized by the cumulative distribution
  reported below

131
Inversion theorem
1
0

• Inversion theorem
• the values of ? sampled in this way have the
  Probability distribution from which we have
  inverted

132
Example

• Let us evaluate the volume of the yellow solid
  through the Monte Carlo method.

V0
V
133
Application to ID and DT

• For every Model parameter one creates the
  corresponding epistemic distribution
• Run nr. 1
• One generates n random numbers between 0 and 1,
  as many as the uncertain variables are
• One samples the value of each of the parameters
  inverting from the corresponding distribution
• Using these values one evaluates the model
• One keeps record of the preferred alternative and
  of the value of the decision
• the procedure is repeated N times.

134
Results

• Strategy Selection Frequency

• Decision Value Distribution

135
Problem V-1

• the mean time to failure of a set of components
  is characterized by an exponential distribution
  with parameter ?. Suppose that ? is described by
  a uniform epistemic distribution between 1/100
  and 1/10.
• Which is the MOW? Which is the epistemic model?
• What is the mean time to failure?
• Suppose you registered the following failure
  times t15, 22, 25.
• Update the epistemic distribution based on the
  new data
• What is the new mean time to failure?

136
Problem V-2 Investing

• We are again thinking of how to invest.
  Actually, we were not aware of the bayesian
  approach before. Thus we start using data about
  P_up in Bayesian way. After 15 working days we
  get the evidence up,down, down,down,down,up,down,
  up,down,up,down,up,up,up. Assuming that each day
  is independent of the previous one
• a) Which are the MOW and the epistemic model?
• b) What is the best decision without
  incorporating the evidence?
• c) What is the distribution of P_up after the
  evidence?
• d) What do you decide when the new information is
  incorporated in the model?
• Solution
• the MOW is the model of the events that accompany
  the decision. It is our ID or DT. More in
  specific, there is a second mode which is the one
  utilized for modeling the fact that the market
  can be up or down. This is a binomial
  distribution with parameter P_up.
• the epistemic model is the set of the uncertainty
  distributions used to characterize the lack of
  knowledge in the model parameters. In this case,
  it is the distribution of P_up. We need to choose
  a prior distribution for P_up. We choose a
  uniform distribution between 0 and 1.
• b) We write the alternative payoffs as a function
  of P_up.

137
Prob. 5-2

• Substituting EURisky50, EUSafe 20,
  EULess Risky 20

138
Investment

• c) Let us use Bayess theorem to update the prior
  uniform distribution
• evidence up,down, down,down,down,up,down,up,down,
  up,down,up,up,up
• L(EP_up)
• Prior ?0 uniform bewteen 0 and 1
• Bayestheorem
• Posterior Distribution
• Ep_up0.47
• d) Posterior Decision EURisky23, EUSafe
  20, EULess Risky 9.2

139
Problems

• Apply the one way, two way and Tornado Diagrams
  SA to the IDs and DTs of the previous chapters
• Discuss your results

140
Bayesian Decision

• You are the director of a library shop. To
  improve the sales, you are thinking of hiring
  additional sale personnel. This should, in your
  opinion, improve the service level in the shop.
  If this happens, you expect an increase in
  costumer number, and correspondingly, an increase
  in revenue sales. Suppose that the number of
  people entering the shop is, any day, distributed
  according to a Poisson distribution with ?
  uncertain. the prior distribution of ? is a gamma
  with mean equal to 55 and standard deviation
  equal to 15. the cost increase due to the hiring
  is 5000EUR for month. If the service quality
  improves and the library receives more than 50
  customers per day, revenues increase would amount
  at 15000EUR (on the average). If less than 50
  customers visit the shop, then revenues would not
  increase (and you loose the 5000EUR). What to you
  decide?
• You decide to monitor the number of customers on
  the next 6 working days 75,45,30,80,72,41.
• You update the Probability. What do you decide
  now?
• How much do you expect to gain now?
• Perform a sensitivity analysis on the
  probabilities. What information do you get?

141
Influence Diagram
142
Chapter VIIntroduction to Decision theory
143
Summary

• Preferences under Certainty
• Indifference Curves
• the Value Function V(x) properties
• Preferential independence
• Preferences under Uncertainty
• Axioms of rational choice
• utility Function U(x) in one dimension
• Risk Aversion
• Preferences with Multiple Objectives
• Multi-attribute utility Function

144
Preferences Under Certainty

• Example you are choosing your first job. You
  select your attributes as location (measured in
  distance from home), starting salary and career
  perspectives. You denote the attributes as x1,
  x2, x3. you have to select among five offers a1,
  a2,,a5. Every offer gives you certain values of
  x1, x2, x3 for certain. How do you decide?
• It is a multi-attribute decision problem in the
  presence of certainty, since once you decide you
  will receive x1,x2,x3 for certain.
• In this case you have to establish how much of
  one attribute to forego to receive more of anoilr
  attribute.

145
Preferences under Certainty

• Here is a diagram for the Choice

146
Structuring Preferences

• Indifference Curves
• Points on the same curve leave you indifference

147
the Value Function

• You can associate a numerical value representing
  you preferences to each indifference curve
• V(x) is the function that says how much of xi one
  is willing to exchange for an increase or
  decrease in xk

148
V(x)

• V(x) is a value function if it satisfies the
  following properties
• a)
• b)

149
Example

• For the first job choice, suppose that you
  value function is as follows
• where x1 measures the distance from home in
  100km, x2 is the career perspective measured on a
  scale from 0 a 10 and x3 the starting salary in
  kEUR.
• Suppose to have received the following offers
• (1, 5, 20), (5, 4, 10), (8,3,60), (10, 5, 20),
  (10,2,40)
• Which one would you pick?

150
Preferences under Uncertainty
Suppose one has to choose between lotteries that
offer a mix the previous job offers to choose
one does not use the value function, but must
resort to the utility function (U(x))
151
utility Function

• the utility function is the appropriate one to
  express preferences over the distributions of the
  Attributes.
• Given two distributions 1 and 2 on the
  Consequences , Distribution 1 is more or as much
  desirable than Distribution 2 if and only if

152
Utility vs. Value

• One attribute Problem. Suppose that alternative
  1 produces x1 and the 2 x2, then 1?2 if x1gtx2
• Let us take two Alternatives 1 and 2, with x1gtx2,
  given with certainty.
• the value function will give us v(x1)gtv(x2)
• Let us now consider the following problem
• To choose one need u(x1) and u(2)

153
Stochastic Dominance
Distribution 1 is dominated by distribution 2,
if obtaining more of x is preferable. Vice
versa, if less of x is preferable, then
Distribution 2 is dominated by distribution 1
154
One Attribute Utility Functions
155
Certainty Equivalent

• Given the lottery
• the value of x such that you are indifferent
  between x for certain and playing the lottery.
• In equations
• N.B. if you are risk neutral, then xEx

156
definition of Risk Aversion

• a decision-maker is risk averse if preferisce
  sempre the expected value of a lottery alla
  lottery
• Hp increasing utility function. Th You are
  risk averse if the Certainty Equivalent of a
  lottery is always lower than the expected value
  of the lottery
• You are risk averse if and only if your utility
  function utility is concave

157
Risk Premium and Insurance Premium

• the Risk Premium (RP) of a lottry is the
  difference between the expected value of the
  lottery and your Certainty Equivalent for the
  lottery
• Intuitively, the Risk Premium is the quantity of
  attribute you are willing to forego to avoid the
  risks connected with the lottery.
• Suppose now that Ex0. the insurance premium
  is how much one would pay to avoid a lottery

158
Mailmatical Definition

• the Risk Aversion function is defined as
• Or, equivalently
• Supposing a constant risk aversion one gets an
  exponential utility function

159
Risk Preferences

• Constant Risk Aversion
• Compute constant ? through Certainty Equivalent
  (CE)

160
Investment Results with Risk Aversion
Risky Investment
161
A quale value accetterei linvestimento rischioso
162
Esempi of funzioni utility

• Linear uax
• Risk Properties
• Risk Neutral
• Exponential
• Risk Properties
• - sign Constant Risk Aversion, sign Constant
  Risk Proneness
• Logarithmic
• Risk Properties
• Decreasing Risk Aversion

163
Problems
164
problem VI-1

• For the following three utility functions,
• compute
• the risk aversion function r(x)
• the risk premium for 50/50 lotteries
• the insurance premium

165
Problem VI-2

• Consider a 50/50 lottery. Determine your Risk
  Aversion constant, assuming an exponential
  utility function.
• Reexamine some of the problems discussed till now
  utilizing instead of the monetary payoff the
  corresponding exponential utility function with
  the constant determined above. How do the
  decisions change?

166
Problem VI-3

• You are analyzing some alternatives for your next
  vacations
• A guided tour through Italian cultural cities
  (Rome, Florence, Venice, Siena an infinite
  list..), duration 10 days, cost 500EUR, for a
  total of 1500km by bus.
• A journey to the Caribbean, lasting 1 week, cost
  2000EUR, by plane.
• 15 days in a wonderful mountain in Trentino, for
  a cost of 2000EUR, with 500km of promenades.
• Do you need a utility or a value function to
  decide?
• Suppose that, after some thinking, you discover
  to have the following three attribute utility
  function
• where x1 is the vacation cost in kEUR, x2 is
  distance in km and x3 is a merit coefficient
  regarding relax/amusement to be assigned between
  1 and 10.
• What do you choose?

167
Chapter VIIthe Logic of Failures
168
Elements of Reliability theory
169
Safety and Reliability

• Safety and Reliability study the performance of
  systems.
• Reliability and safety study cover two wide
  areas
• System Failures and Failure Modes
• Structure Function
• Failure Probability
• Failure Data Analysis
• the approach can be static or dynamic. Static
  approach is analytically simpler and is more
  diffuse.

170
Systems

• A system is a set of components connected through
  some logical relations with respect to operation
  and failure of the system
• More simple structures are
• Series
• Parallel

171
Series

• Every component is critical w.r.t. the system
  being able to perform its mission.
• the fault of just one component is sufficient to
  provoke system failure
• Redundancy 0

172
Parallel Systems

• Each of the components is capable of assuring
  that the system accomplish its tasks.
• Thus, to provoke the failure of the system, all
  the components must be contemporarily failed
• Redundancy n-1

173
Elements of System Logics
174
Boolean Logic

• An event (and) can be True or False
• State Variable or Indicator
• Properties
• (XJ)nXj
• where is the
  complementary of XJ
• This simple definition enables one to use
  algebraic operations to describe the logical
  behavior of systems.

175
Series Systems

• Let Ei denote the event the i-th component
  failed.
• Let XT denote the event the System failed.
• XT takes the name of Top Event.
• For the system failure, by definition of series,
  it is enough that one single component failes.
  Thus it is enough that E1 or E2 or . or En is
  true.
• From a set point of view E1?E2 ? ... ? En
• From a logical p.o.v., we get the following
  expression

176
Parallel Systems

• Let Ei denote the event the i-th component has
  failed.
• Let XT denote the event the system has failed.
• the condition for failure of the system is that
  all component fail. This happens if E1 and E2
  and En are true at the same time.
• From a Set point of view E1?E2 ? ... ? En
• the logical expression is

177
the Structure Function

• In general, a system will be formed by a
  combination of series or parallel elements, or
  other logics (as we will see next).
• One defines the Structure Function of a system
  the logical expression that expresses the top
  event (XT) as function of the individual failure
  events.

178
the Logic of Performance

• Let Ai denote the event the i-th component is
  working (Not failed).
• Let YT denote the event the system is
  working.
• For a series system all the components must be
  working for the system to work. Thus A1 , A2 ,
  and An must be true at the same time
• In parallel for the system to work it is
  sufficient that just one component is working.
  Thus A1 or A2 or An must be true.

179
n/N Logics

• n/N logics are intermediate logics between series
  and parallel.
• N represents the total number of components in
  the system and n the number of components that
  must contemporarily fail to break the system.
• As an example, a system has a 2/3 logic if it has
  3 components and when two components have failed
  the system failes.

2/3
180
Example 2/3 System Logics

• Let us find XT for a 2/3 system
• Events E1, E2, E3, Indicators X1, X2, X3
• Events that provoke a failure E1E2 E3, E1 E2 ,
  E1E3, E3 E2 .
• Let us denote E1E2 E3Z1, E1 E2 M1, E1 E3 M2,
  E3 E2 M3.
• For XT to happen Z1?(M1 ? M2 ? M3).
• the structure function expression is
• Let us go to a level below
• Let us solve the calculations, noting that
  (Xi)nXi

181
Probability Sum Rules

• We recall that, for generic Events
• We recall that, if the events are independent
• Rare Event Approssimation neglect all terms
  corresponding to multiple events

182
Golden Rule

• In practice the System Failure Probability is
  computed from the solved Structure Function,
  substituting to indicator Xi the corresponding
  Event Probability.

183
Proof

• the System Failure Probability is
• P(XT)P(Z1?(M1 ? M2 ? M3) P(Z1
  ?Z2P(Z1)P(Z2)-P(Z1Z2)
• where
• P(Z1)P(E1E2 E3)
• P(Z2)P(M1 ? M2 ? M3) P(M1) P(M2)P(M3)-P(M1M2)-
  P(M1M3)-P(M3 M2)P(M1 M2 M3). Ma M1 E1E2, M2
  E3E1, M3 E3E2. Noting, that M1 M2 M1 M3 M2
  M3 M1 M2 M3 E1E2E3. Substituting
  P(Z2)P(E1E2) P(E3E1)P(E3E2)-P(E1E2E3)-
  P(E1E2E3)-P(E1E2E3)P(E1E2E3). Thus
• P(Z2)P(E1E2) P(E3E1)P(E3E2)-2P(E1E2E3)
• P(Z1Z2)P(E1E2E3? E1E2 ?E3E1 ? E3E2)P(E1E2E3 )
• Thus P(XT) P(E1 E2 E3)P(E1E2)
  P(E3E1)P(E3E2)-2P(E1E2E3)-P(E1E2E3) P(E1E2)
  P(E2E1)P(E3E2)-2P(E1E2E3)

184
Problems
185
Problem VII-1

• For the following systems compute
• the Structure Function for System Failure
• the Structure Function for System Operation
• the Failure Probability
• the Operation Probability

186
Problem VII-2

• for the following system
• Compute the Failure Probability supposing
  independent events and denoting the component
  failure probability by p.
• Repeat the computation starting with the system
  success function, YT. Verify that the two results
  coincide.

187
Chapter VIIIElements of Reliability
188
Cut and Path sets

• Failure Logic
• By cut set one means an event/set of events whose
  happening causes system failure
• By minimal cut set one means a cut set that does
  not have other cut sets as subsets
• Success Logic
• By path set one means an event/set of events
  whose happening causes system to work
• By minimal path set one means a path set that
  does not have other path sets as subsets

189
Even Trees

• Event Trees represent the sequence of events
  that lead to the event top.

190
Example

• One has to establish the sequence of events that
  lead to leakage of toxic chemicals from a
  production plant. High pressure in one of the
  pipes can cause a breach in the pipe itself, with
  leakage of toxic material in the room where the
  machine works. the filtering of the air
  conditioning could prevent the passage of the
  toxic gas to the outside of the room. A fault on
  the air circulation system due to air filter
  fault or maintenance error, would lead to the
  diffusion of the gas to the entire firm building.
  At this point, public safety would be protected
  only by the building air circulation system, last
  barrier for the gas going to the outsides.
• Draft the event tree for this sequence.

191
Gas Leakage Fault - Tree
192
Fault Trees

• Fault Trees represent the logical connection
  among failures that lead to the failure of a
  barrier
• they are characterized by a set of logic symbols
  that connect a series of events
• Basic Event is the event that represents the
  base of the fault-tree. From a physical point
  of view, it represents the failure of a component
  or of part of it. From a modeling point of
  view, it represents the lowest level of detail.

And
Or
event Base
193
Example

• Let us consider the failure of the aeration
  system. Suppose that the system is composed by
  two main parts an suction engine and a static
  filter. the failure of the aeration system,
  thus, happens either due to engine failure or for
  filter fault.

Aeration 1
Static Filter
Engine
194
Example

• We could however realize that the level of detail
  could be Further increased. In fact, we discover
  that the engine can brake for a failure of its
  mechanical components and, in particular, of the
  fan or for a fault of the electric feeder. the
  filter can break because of wrong installation
  after maintenance or for an intrinsic fault.
• the fault tree becomes as follows

195
Level II
196
From Fault Trees to Structure Functions

• EngineA
• Static filter B
• Electr.1, Mech2, Fault3, Install.4
• What are the minimal cut sets?