Transcript Document 7523609
Emanuele Borgonovo Structural Decision Market Return
Quantitative Methods for Management
First Edition Quantitative Methods for Management Emanuele Borgonovo 1
Quantitative Methods for Management
Chapter three: Models
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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 Quantitative Methods for Management Emanuele Borgonovo 3
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 Quantitative Methods for Management Emanuele Borgonovo 4
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: F=mg 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?
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Chapter II Introductory Elements of Probability theory
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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) Quantitative Methods for Management Emanuele Borgonovo 7
Kolmogorov Axioms
U B A
P ( U )
1 P ( A )
0 If A e B mutually esclusive P ( A
B )
P ( A )
P ( B ) events ,
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U
Areas and rectangles?
A B C D and U A B C D E • 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 Quantitative Methods for Management Emanuele Borgonovo 9
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(A|B)
U B
Quantitative Methods for Management AB
A
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Conditional Probability
• Suppose now that B has happened, i.e., you jumped into area B (and you cannot jump back!).
B
•You cannot but agree that: •P(A|B)=P(AB)/P(B) •Hence: P(AB)=P(A|B) *P(B) Quantitative Methods for Management AB
A
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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
Quantitative Methods for Management Thus, for independent events: P(AB)=P(A)*P(B) Emanuele Borgonovo 12
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?
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In the subjectivist approach, Probability changes with information
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Bayes’ theorem
• Hypothesis: A and B are two events. A has happened. • Thesis: P(B) changes as follows: P(B) before A Probability of A given B
P ( B A )
P ( B )
P ( A B ) P ( A )
New value of the Probability of B Probability of A Quantitative Methods for Management Emanuele Borgonovo 15
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 o ) P ( A ) P ( o A ) P ( o ) • P(A)=Probability of both rings being golden before the extraction =1/2 • P(o)=Probability of a golden ring=3/4 • P(o|A)=Probability that the extracted ring is golden given A=1 (since both rings are golden) • So:
P ( A o )
1 3 / / 2
1 4
2 / 3
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Bayes’ theorem Proof
Starting point P ( AB ) P ( AB ) Conditional Probability formula P ( A B ) P ( B ) P ( B A ) P ( A ) thesis P ( A B ) P ( B A ) P ( A ) P ( B ) Quantitative Methods for Management Emanuele Borgonovo 17
B
the Total Probability theorem
D U and
A
C 1 P ( A ) P ( B ) P ( C ) P ( D ) • the total Probability theorem states: given N mutually exclusive and exhaustive events A 1 , A 2 ,…,A N , the Probability of an event and in U can be decomposed in: P ( E ) P ( E A 1 ) P ( A 1 ) P ( E A 2 ) P ( A 2 ) ...
P ( E A N ) P ( A N ) • Bayes theorem in the presence of N events becomes : Quantitative Methods P ( A 1 E ) P ( E N A P ( E 1 A ) i ) P ( i 1 Emanuele Borgonovo A P ( 1 A ) i ) for Management 18
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. Quantitative Methods for Management Emanuele Borgonovo 19
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.
f ( x ) dx 1 • Note: f(x 0 )dx is the Probability that x lies in an interval dx around x 0 . Quantitative Methods for Management Emanuele Borgonovo 20
Cumulative Distribution Function
• Given a continuous random variable X, the Probability that X • Note: P ( X 1 x X 2 ) X 2 X 1 f ( x ) dx P ( x X 2 ) P ( x X 1 ) Quantitative Methods for Management Emanuele Borgonovo 21 • 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: P ( T t ) 1 e λ t • and by the density function: f ( t ) e t Quantitative Methods for Management Emanuele Borgonovo 22 • 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. Quantitative Methods for Management Emanuele Borgonovo 23 • Thus the change in the population is: • -N(t) dt=N(t+dt)-N(t)=dN(t) • Where the minus sign indicates that the number of working components has decreased. • Hence: dN ( N ( t t ) ) dt T 0 dN ( N ( t ) ) T 0 dt ln N ( T ) / N ( 0 ) T • Which solved leads: N ( T ) e T N ( 0 ) • 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. Quantitative Methods for Management Emanuele Borgonovo 24 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 f(t) 5 10 15 T/t Quantitative Methods for Management Emanuele Borgonovo 25 2 2 2 Percentile p: is the value x p of X such that the Probability of X being lower than x p is equal to p/100 Quantitative Methods for Management Emanuele Borgonovo 26 • Is a symmetric distribution around the mean • Pdf: f G ( x ) • Cdf: 1 2 e 1 ( 2 x ) 2 X G 1 ( 2 x ) 2 Quantitative Methods for Management Emanuele Borgonovo 27 P G f ( G x ( x ) X ) Distribuzione Normale Standard 3000 2500 2000 1500 1000 500 0 -4 -3 -2 -1 0 x 1 Cumulative Gaussian Distribution 2 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0 -5 -4 -3 -2 -1 x 0 1 2 3 3 4 4 Quantitative Methods for Management Emanuele Borgonovo 28 • Pdf f L ( x ) x 1 2 e 1 ( 2 ln x ) 2 0 x • Cdf P L ( x X ) 0 X x 1 2 e 1 ( 2 ln x ) 2 Quantitative Methods for Management Emanuele Borgonovo 29 .20 L 0.1 0 0 0 0.07 1 1 20 x P ( x X ) 0.5 Quantitative Methods for Management 0 0 0 0.07 40 50 20 x 40 50 Emanuele Borgonovo 30 • the failure rate of a car gear is 1/5 for year (exponential events). • What is the mean time to failure of the gear? t • What is the Probability of the gear being integer after 9 years? T ( 1 / 5 ) 9 Quantitative Methods for Management Emanuele Borgonovo 31 • 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) Quantitative Methods for Management Emanuele Borgonovo 32 • For the example of the two rings, determine: – P(B|o) – P(B|a) – 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 Quantitative Methods for Management Emanuele Borgonovo 33 • For the example of the two rings, determine: – P(B|o) • 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: P ( A 2 o ) P ( 2 o P ( 2 o A ) P 1 ( A ) A ) P 1 ( A ) P ( 2 o B ) P 1 ( B ) Quantitative Methods for Management Emanuele Borgonovo 34 • where, in the formula, subscript 1 indicates the probabilities after the information of the first extraction has been taken into account: – P 1 (B)=P(B or)=1/3 and P 1 (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: P ( A 2 o ) P ( 2 o P ( 2 o A ) P 1 ( A ) A ) P 1 ( A ) P ( 2 o B ) P 1 ( B ) 1 * 2 / 1 * 3 2 / 1 / 3 2 * 1 / 3 0 . 8 – 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 Quantitative Methods for Management Emanuele Borgonovo 35 Quantitative Methods for Management Emanuele Borgonovo 36 • 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 Quantitative Methods for Management Emanuele Borgonovo 37 • “ 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) Quantitative Methods for Management Emanuele Borgonovo 38 Problem identification Alternatives identification Model implementation Alternatives evaluation Yes Sensitivity Analysis Further Analysis? No Best Alternatives implementation Quantitative Methods for Management Emanuele Borgonovo 39 Quantitative Methods for Management Emanuele Borgonovo 40 • Objectives: – Maximize profit • Attributes: – Money • Alternatives: – Risky – Less Risky – Safe • Random events: – the Market • Consequences: – Profit or Loss Quantitative Methods for Management Emanuele Borgonovo 41 • Influence Diagrams Structural Decision Market Return • Decision Trees Less Risky How should the invest $1000? Risky Safe Market up prob_up Market down 1-prob_up Market up prob_up Market down 1-prob_up Quantitative Methods for Management Emanuele Borgonovo 42 • 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 ) Quantitative Methods for Management Emanuele Borgonovo 43 NODES ARCS • Informational Arcs = Decision • probabilistic Dependency Arcs = Random Event • Structural Arcs = utility Quantitative Methods for Management Emanuele Borgonovo 44 Decision Node Informational Arc Sequential Decisions Decision Node Chance Node Structural Conditional Arc Chance Node probabilistic Dependency Quantitative Methods for Management Value Node Emanuele Borgonovo 45 Quantitative Methods for Management Emanuele Borgonovo 46 • Influence Diagram for Case Study 2 shutdown_cost Leakage Rate Decisions I - Normal Makeup II - Shutdown III - Reduce Power IV - Isolate SG Leakage from primary to secondary, maximum rate of 20 l/hr Primary Cooling Chemical Volume Control System time_to_repair Value Secondary Cooling days_to_shutdown Deterministic Information core_damage_cost Quantitative Methods for Management Emanuele Borgonovo 47 Structural Quantitative Methods for Management Emanuele Borgonovo 48 • 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. Quantitative Methods for Management Emanuele Borgonovo 49 Less Risky How should the invest $1000? Risky Market up Market down 1-prob_up Market up Market down Safe Quantitative Methods for Management Emanuele Borgonovo 50 • Alternative Payoff or utility: E [ U i ] P i ( C j ) U C j • j=1…m i the • U j is the utility or the payoff of consequence j • P i (C j ) is the Probability that consequence C j one chose alternative the happens given that • In general, we will get: P(C j ) =P(E 1 E 2 … E N ), where E 1 E 2 … E N are the events that have to happen so that consequence C j is realized. Using conditional probabilities: • P(C j ) =P(E 1 E 2 … E N )=P(E N | E 1 E 2 … )*…*P(E 2 | E 1 )*P(E 1 ) Quantitative Methods for Management Emanuele Borgonovo 51 Blue Chip Stock How should the invest $1000? Risky investment Market up P.up Market down C1 C2 1-P.up Market up C3 P.up Market down C4 1-P.up CD paying 5% C5 Quantitative Methods for Management Emanuele Borgonovo 52 • Using the previous formula: Risky j 2 1 j j 1 2 LessRisky j 2 1 j j 3 4 LessRisky j 2 1 j j 5 Quantitative Methods for Management Emanuele Borgonovo 53 the Best Investment for a Risk Neutral Decision - Maker Blue Chip Stock How should the invest $1000? Risky investment Market up 0.600 $56 Market down $200 ($160) 0.400 Market up $500; P = 0.600 $60 0.600 Market down ($600); P = 0.400 0.400 CD paying 5% return = $ 50 Quantitative Methods for Management Emanuele Borgonovo 54 • 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 Quantitative Methods for Management Emanuele Borgonovo 55 Decision Engine failure Final Classification Profit Quantitative Methods for Management Emanuele Borgonovo 56 Decision pfailure=0.5 pfive=0.30 pout=0.2 pwin=0.5 Engine failure Decision Run Run : $57,250 Withdraw Engine_failure=0 failure 0.500 $57,250 No failure 0.500 Out of first five $20,000 1.000 Win 0.500 In first five $94,500 0.300 Out of first five 0.200 Old sponsor $50,000 1.000 $20,000; P = 0.500 $110,000; P = 0.250 $105,000; P = 0.150 $50,000; P = 0.100 $50,000 Quantitative Methods for Management Emanuele Borgonovo 57 • 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 interest=5%) • (failure Probability of B in the period of interest=3%) • (failure Probability of C in the period of interest=2% Quantitative Methods for Management Emanuele Borgonovo 58 Decision Assicurazione Ruttura Costo Quantitative Methods for Management Emanuele Borgonovo 59 pA=0.05 pB=0.03 pC=0.02 A Decision B A : (£158) C 1 Assicurazione 1 : (£158) 2 1 1 : (£180) 2 1 1 : (£204) 2 -150-5%*(150) = (£158); P = 1.000 Sì (£161) No 0.050 0.950 -150-2%*150-150 = (£303) -150*(1+2%) = (£153) -(175+3%*(175)) = (£180) Sì (£184) No 0.030 0.970 -175-2%*175-175 = (£354) -175-2%*175 = (£179) -200-2%*200 = (£204) Sì (£207) No 0.020 0.980 -200-1.5%*200-200 = (£403) -200-200*1.5% = (£203) Quantitative Methods for Management Emanuele Borgonovo 60 • 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 Quantitative Methods for Management Emanuele Borgonovo 61 Decision Market Value P _UP =0.5 Quantitative Methods for Management Decision RISKY Market Up 0.500 £50 Down Up 0.500 0.500 £20 Down 0.500 SAFE Market=0 £500; P = 0.500 (£400); P = 0.500 £200 (£160) £20 Emanuele Borgonovo 62 EVPI for the Example Market P _UP =0.5 Quantitative Methods for Management Market Up 0.500 £260 Down 0.500 Value Decision Decision RISKY LESS RISKY RISKY : £500 SAFE RISKY LESS RISKY SAFE : £20 SAFE £500; P = 0.500 £200 £20 (£400) (£160) £20; P = 0.500 Emanuele Borgonovo 63 Quantitative Methods for Management Emanuele Borgonovo 64 Quantitative Methods for Management Emanuele Borgonovo 65 • 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 Quantitative Methods for Management Emanuele Borgonovo 66 • Given the following elements: – Alternatives 1 and 2 – Events: A=(up, down); (B=high, low);(C=good, bad); – Consequences C i (one distinct consequence for each event combination) – If A=Down happens, then C Adown 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? Quantitative Methods for Management Emanuele Borgonovo 67 • Influence Diagram the Skip Arc A Decision C B Consequences Quantitative Methods for Management Emanuele Borgonovo 68 • Corresponding Decision Tree B high A up 1 Decision low down B=0 C=0 C good bad good bad 2 up high low good bad good bad down B=0 C=0 Quantitative Methods for Management (No Payoff) (No Payoff) (No Payoff) (No Payoff) (No Payoff) (No Payoff) (No Payoff) (No Payoff) (No Payoff) (No Payoff) Emanuele Borgonovo 69 • Influence Diagram II A Decision C B Consequences Quantitative Methods for Management Emanuele Borgonovo 70 • Decision Tree II: B high A up 1 Decision low down B=0 high up low 2 down B=0 C good bad good bad good bad good bad good bad good bad Quantitative Methods for Management (No Payoff) (No Payoff) (No Payoff) (No Payoff) (No Payoff) (No Payoff) (No Payoff) (No Payoff) (No Payoff) (No Payoff) (No Payoff) (No Payoff) Emanuele Borgonovo 71 • • • Given the following Influence Diagram and Decision Tree, given P_High and P_High|High, P_high|low, find the value of the Alternatives as a function of the assigned probabilities. Supposing P_high=0.5 and P_high|high=P_high|low=0.3, find the preferred alternative. Sales Cost Decisione Vendite Costo high=0.5 P_Alte=0.3 P_high=0.5 Payoff Decision Invest high P_high Basso 1-P_high High 0 P_Alte|high Low -10 1- P_Alte|high High P_ P_Alte|high 20 Low 0 1- P_Alte|high Do not Invest 5 What would be the preferred decision if to a higher investment cost there would correspond a better sale result? Set: P_high|high=0.6 and P_high|low=0.2 Quantitative Methods for Management Emanuele Borgonovo 72 alto=0.5 P _Alte=0.3 P _alto=0.5 Decisione Investo Non-Investo : £5 Non-Investo Costo Alto 0.500 (£1) Basso 0.500 £5; P = 1.000 Vendite Alte 0.300 (£7) Basse Alte 0.700 0.300 £6 Basse 0.700 £0 (£10) £20 £0 Quantitative Methods for Management Emanuele Borgonovo 73 • 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? Quantitative Methods for Management Emanuele Borgonovo 74 • 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) Quantitative Methods for Management Emanuele Borgonovo 75 • • • 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? Quantitative Methods for Management Emanuele Borgonovo 76 Decisione Riparazione_10g_Perfetta Riparazione_2g_perfetta Perdite Quantitative Methods for Management Emanuele Borgonovo 77 Decisione Intervento_2g Riparazione_10g_Perfetta=0 P_10g=0.9 P_2g=0.3 years=3 Intervento_10g Riparazione_10g_Perfetta Riparazione_10g_Perfetta Riparazione_2g_perfetta=0 P_10g Riparazione_10g_non_Perfetta Riparazione_2g_perfetta=0 1-P_10g Riparazione_2g_perfetta Riparazione_2g_perfetta P_2g Riparazione_2g_non_perfetta 1-P_2g -50000-500000 -50000-500000-25000*0.15*365*years -250000-1000000 -1000000-250000-years*.05*25000*365 Quantitative Methods for Management Emanuele Borgonovo 78 • Three years Quantitative Methods for Management Emanuele Borgonovo 79 • 2 years • 4 years Quantitative Methods for Management Emanuele Borgonovo 80 Quantitative Methods for Management Emanuele Borgonovo 81 – One Way SA – Two Way SA – Tornado Diagrams – (Differential Importance Measure) – Monte Carlo – (Global SA) Quantitative Methods for Management Emanuele Borgonovo 82 • 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) Quantitative Methods for Management Emanuele Borgonovo 83 • Underline the critical dependencies of the outcome Tornado Diagram at Decision pfailure: 0.25 to 0.75 pwin: 0.3 to 0.7 pfive: 0.2 to 0.4 $49K $55K $61K $67K Expected Value $73K Sensitivity Analysis on pfailure $62K $59K $56K $53K $50K $47K $44K $41K $38K 0.450 0.525 0.600 pfailure 0.675 0.750 Run Withdraw Threshold Values: pfailure = 0.597 EV = $50K Quantitative Methods for Management Emanuele Borgonovo 84 – One way sensitivity – Two way sensitivity – Tornado Diagrams – Aleatory Uncertainty – Epistemic Uncertainty – Bayes‘ theorem for continuous distributions – Monte Carlo Method Quantitative Methods for Management Emanuele Borgonovo 85 • 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 Quantitative Methods for Management Emanuele Borgonovo 86 • 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 Sensitivity Analysis on pfailure $62K $59K $56K $53K $50K $47K $44K $41K $38K 0.450 Run Withdraw Threshold Values: pfailure = 0.597 EV = $50K 0.750 0.525 0.600 pfailure 0.675 Quantitative Methods for Management Emanuele Borgonovo 87 • 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 Quantitative Methods for Management Emanuele Borgonovo 88 • 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. Quantitative Methods for Management Emanuele Borgonovo 89 Tornado Diagram at Decision pfailure: 0.25 to 0.75 pwin: 0.3 to 0.7 pfive: 0.2 to 0.4 Quantitative Methods for Management $49K $55K $61K $67K Expected Value $73K Emanuele Borgonovo 90 • 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 Quantitative Methods for Management Emanuele Borgonovo 91 • 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 Quantitative Methods for Management Emanuele Borgonovo 92 • Identify how sensitivity analysis techniques work through analysis of several examples • Formulate a definition • Classify sensitivity analysis techniques accordingly Quantitative Methods for Management Emanuele Borgonovo 93 • Model Output: 1 2 n • Local Sensitivity Analysis: – Determines model parameter (x i ) relevance with all the x i fixed at nominal value • Global Sensitivity Analysis: – Determines x i distribution relevance of x i ’s epistemic/uncertainty Quantitative Methods for Management Emanuele Borgonovo 94 • Nominal Model output: – No uncertainty in the model parameters – and/or parameters fixed at nominal value • Local Decomposition: 1 1 2 2 n n • Local importance measured by fraction of the differential attributable to each parameter i x i o Quantitative Methods for Management Emanuele Borgonovo 95 • Uncertainty in U and parameters is considered • Sobol’’s decomposition theorem: 0 • Sobol’Indices i n 1 i i 1 i j n ij i j 12 ... n i 1 ...i s i i 1 ...i s x x i1 i1 i 1 ...i n i 1 n Ω 2 0 2 Quantitative Methods for Management Emanuele Borgonovo 96 Formal Definition of Sensitivity Analysis (SA) Techniques • SA technique are Operators on U: x 1 x 2 x n 1 2 n or I(x n ) I(x 1 ) or I(x 2 ) Quantitative Methods for Management Emanuele Borgonovo 97 • Importance relations: – X the set of the model parameters; – Binary relation x x j iff I(x i )I( x j ) x i ~ x j iff I(x i )I( x j ) x x j iff I(x i )I( x j ) x x j iff I(x i )I( x j ) • Importance relations induced by importance measures are complete preorder Quantitative Methods for Management Emanuele Borgonovo 98 • In many situation decision-maker interested in joint importance: I ( x , x ) I ( x x ) i j i j • An Importance measure is additive if: I ( x i , x j ) I ( x i ) I ( x j ) • DIM is additive always • S i are additive iff f(x) additive and x j ’s are uncorrelated Quantitative Methods for Management Emanuele Borgonovo 99 Techniques that fall under the definition of Local SA techniques IMPORTANCE MEASURE DIM L Tornado Diagrams One Way Sensitivity Fussell-Vesely Risk Achievement Worth EQUATION TYPE ADDITIVE dU x i dU U x i x i U x 0 U ( x 0 ) x i x 0 U ( x 0 ) U x i U 0 Local Local Local Local Local Local Yes No No No No No Quantitative Methods for Management Emanuele Borgonovo 100 IMPORTANCE MEASURE Sobol’ Indices Extended Fast Morris Pearson Smirnov Standardized regression coefficients EQUATION TYPE D i 1 ...i s D S i 2 p 1 A 2 p w i 2 j 1 A 2 j d ( ) i B 2 p w i B j 2 f ( x 1 ,..., x i ,..., x n ) i cov( U , i U x i ) sup Y 1 ( X i ) Y 2 ( X i ) Global Global Global Global Global b k k Global ADDITIVE No No No No No No Quantitative Methods for Management Emanuele Borgonovo 101 Sensitivity Analysis in Risk-Informed Decision-Making and Regulation • Risk Metric: • x i 1 2 is undesired event Probability n • Fussell-Vesely fractional Importance: i i • Tells us on which events regulator has to focus attention Quantitative Methods for Management Emanuele Borgonovo 102 • 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 Quantitative Methods for Management Emanuele Borgonovo 103 Quantitative Methods for Management Emanuele Borgonovo 104 Quantitative Methods for Management Monte Carlo Simulation at Decision 1.000 0.900 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.100 0.000 $10K $40K $70K Value $100K $130K Emanuele Borgonovo 105 • Distinction between Aleatory Uncertainty ed Epistemic Uncertainty • Epistemic Uncertainty and Bayes‘ theorem • Monte Carlo Method for uncertainty propagation Quantitative Methods for Management Emanuele Borgonovo 106 • 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 “ Eit ”, 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. Quantitative Methods for Management Emanuele Borgonovo 107 • the Probability of Earthquakes is usually modeled through a Poisson model: t n • 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). Quantitative Methods for Management Emanuele Borgonovo 108 • the Poisson Probability that n events happen on 0-t is: P ( n , t ) e t ( t ) n n ! • the sum on n=0... of P(n,t) is, obviously, equal to 1. n 0 e t ( t ) n n ! e t n 0 ( t ) n n ! e t e t 1 • the Probability of k>N is given by: n N 1 e t ( t ) n n ! 1 n N 0 e t ( t ) n n ! • E[n]= t n 0 ne t ( t ) n n ! e t n 0 ( t ) n n 1 ! e t t n 0 ( t ) n 1 n 1 ! e t t e t Quantitative Methods for Management Emanuele Borgonovo 109 • 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( ): Epistemic distribution for the frequency of earthquakes u ( u ( ) ) 0 1 / 5 1 / 50 1 / 50 1 / 1 / 5 50 1 / 5 7 6 5 8 Quantitative Methods for Management 2 1 4 3 0 0 0.02 0.04 0.06 0.08 0.1 0.12 lambda 0.14 0.16 0.18 0.2 Emanuele Borgonovo 110 • 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: t n Quantitative Methods for Management Emanuele Borgonovo 111 • This expression tells us that not necessarily all Poisson distributions weight the same. Thus: P ( n , t ) p ( n , t , ) d • In our case: u( )=c; e t ( t ) n n ! u ( ) d E [ P ( n , t )] p ( n , t , λ ) d λ c e λ t ( λ t ) n n ! d λ • Hence, there is an expected Probability! Quantitative Methods for Management Emanuele Borgonovo 112 • the MOW will depend on m parameters , ,…: • the event Probability (P(t)) will be: Quantitative Methods for Management Emanuele Borgonovo 113 • the failure time of a series of components is characterized by the exponential Probability function : df e t dt • From the available data, it emerges that: 1 / 8 1 1 / / 5 10 p 0 . 5 p 0 . 3 p 0 . 2 • What is the mean time to failure? Quantitative Methods for Management Emanuele Borgonovo 114 • E[t]= ( 0 t 1 e 1 t dt ) 0 . 5 ( 0 t 2 e 2 t dt ) 0 . 5 ( 0 t 3 e 3 t dt ) 0 . 1 1 / 1 0 . 5 1 / 2 0 . 3 1 / 3 0 . 1 Quantitative Methods for Management Emanuele Borgonovo 115 • 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…. Quantitative Methods for Management Emanuele Borgonovo 116 • the Probability density of a parameter, after having obtained evidence and, changes as follows: π ( λ ) L ( E λ ) π 0 ( λ ) L ( E λ ) π 0 ( λ ) d λ • • • L(E ) = MOW likelihood 0 ( ) is the pdf of Distribution before the evidence, called Prior ( ) is the pdf of after the evidence, called Posterior Distribution Quantitative Methods for Management Emanuele Borgonovo 117 • Let us take Bayes‘ theorem for discrete events: P ( A j E ) n P ( E A P ( E j A ) i ) P ( A P ( j A ) i ) Probability that a parameter of the MOW distribution assumes a certain value, given a certain evidence • Thus, event A j is: takes on value * • Hence: P(A j ) 0 ( )d 0 ( )=prior density • therefore: P(E A j ) 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!!! Quantitative Methods for Management Emanuele Borgonovo 118 • 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: i n 1 i i 0 • Substituting the various terms, one finds Bayes‘ theorem for continuous random variables we have shown before Quantitative Methods for Management Emanuele Borgonovo 119 • What is the MOW? • It is a binomial distribution with parameter p: P ( k , n k ) n k p k ( 1 p ) n k • 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: π 0 ( p ) 1 if 0 p 1 π 0 ( p ) 0 otherwise • Let us get some evidence. • At the first tossing it is head • At the second tail • At the third head Quantitative Methods for Management Emanuele Borgonovo 120 • • • 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)=p 2 (1-p) – Prior: 0 Quantitative Methods for Management π 1 ( p ) L ( h p ) π 0 ( p ) L ( h p ) π 0 ( p ) dp p 1 1 0 pdp 2 p π 2 ( p ) L ( t L ( t p ) π 1 ( p ) p ) π 1 ( p ) dp 6 ( p p 2 ) p ( 1 p ) 1 0 ( p p 2 ) dp π 3 ( p ) L ( h p ) π 2 ( p ) L ( h p ) π 2 ( p ) dp 12 ( p 2 p 3 ) p 2 ( 1 p ) 1 0 p 2 ( 1 p ) dp π 3 ( p ) L ( hth L ( hth p ) π 0 ( p ) p ) π 2 ( p ) dp 12 ( p 2 p 3 ) p 2 ( 1 p ) 1 1 0 p 2 ( 1 p ) 1 dp Emanuele Borgonovo 121 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 3 2 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Quantitative Methods for Management Emanuele Borgonovo 122 • Likelihood – Poisson t n • Prior distribution – Gamma 0 α α 1 βλ • Posterior: gamma ' ' 1 ' • with: ' r ' t Quantitative Methods for Management Emanuele Borgonovo 123 • Likelihood – Normal f X ( x ) σ x 1 2 π e 1 2 ( x μ x σ x ) 2 • Prior distribution: – Normal π 0 ( m ) σ μ 1 2 π e 1 2 ( m μ x σ μ ) 2 • Posterior: Normal f G ( x ) σ ' x 1 2 π e 1 ( 2 x μ ' ) 2 σ ' x • with: μ ' μ ( σ x ) 2 ( σ x ) 2 n x ( σ 0 μ ) 2 n ( σ 0 μ ) 2 σ ' x ( σ x ( σ μ ) 2 / n ) 2 ( σ μ ) 2 ( σ x ) 2 / n Quantitative Methods for Management Emanuele Borgonovo 124 • Likelihood – Binomial n k p k ( 1 p ) n k • Prior: – Beta π 0 ( p ) p ( q 1 ) ( 1 p ) r 1 • Posterior, Beta: π 1 ( p ) p ( q ' 1 ) ( 1 p ) r ' 1 • with: q ' q k r ' r n k Quantitative Methods for Management Emanuele Borgonovo 125 Summary of Conjugate Distributions MOW - Likelihood Binomiale Poisson Normal Normal Negative binominal Prior Distribution Beta Gamma Normal Posterior Distribution Beta Gamma Normal Gamma Beta Gamma Beta Quantitative Methods for Management Emanuele Borgonovo 126 • Investment: E [ U Risky ] E [ U LessRisky E [ U LessRisky ] ] j 2 1 P ( C j ) C j 2 1 j 2 1 P ( C P ( C j ) j j ) C j P.up C j P.up C 1 1 C 5 C 3 ( 1 P.up) ( 1 P.up) C 2 C 4 • Suppose that P.up is characterized by a uniform pdf between 0.3 and 0.7 E [ U Risky P . up ] 2 P ( C j ) C j j 1 • How does the decision changes? P.up U 1 ( 1 P.up) U 2 • It is necessary to propagate the uncertainty in the model Quantitative Methods for Management Emanuele Borgonovo 127 • It is the same problem of the MOW … E U Risky ( P.up U 1 E [ U Risky P . up ] f ( P . up ) dP . up ( 1 P.up) U 2 ) f ( P . up ) dP . up • Repeating for the other decisions and comparing the resulting mean values, one gets the optimal decision. • Recall that: E [ g ( x 1 , x 2 ,..., x n )] g ( x ) f ( x ) d x Quantitative Methods for Management Emanuele Borgonovo 128 • 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 Quantitative Methods for Management Emanuele Borgonovo 129 • 1) Random Number Generator “u” between 0 and 1 0 1 u • 2) Numbers u are generated with a uniform Distribution • 3) Suppose that parameter is uncertain and characterized by the cumulative distribution reported below: 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 Distribuzione cumulativa esponenziale 0.1 0.2 0.3 0.4 0.5 x 0.6 0.7 0.8 0.9 1 Quantitative Methods for Management Emanuele Borgonovo 130 1 0 • Inversion theorem: Distribuzione cumulativa esponenziale 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 x F 1 ( u ) 0.6 0.7 0.8 0.9 1 • the values of sampled in this way have the Probability distribution from which we have inverted Quantitative Methods for Management Emanuele Borgonovo 131 • Let us evaluate the volume of the yellow solid through the Monte Carlo method. V 0 V Quantitative Methods for Management V lim n n in N V 0 Emanuele Borgonovo 132 • 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. Quantitative Methods for Management Emanuele Borgonovo 133 • Strategy Selection Frequency • Decision Value Distribution Quantitative Methods for Management Monte Carlo Simulation at Decision 1.000 0.900 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.100 0.000 $10K $40K $70K Value $100K $130K Emanuele Borgonovo 134 • 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: t=15, 22, 25. – Update the epistemic distribution based on the new data – What is the new mean time to failure? Quantitative Methods for Management Emanuele Borgonovo 135 • • • • • • • 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. Quantitative Methods for Management Emanuele Borgonovo 136 E [ U LessRisky P _ up ] C 5 E [ U Risky P_up ] j 2 1 P ( C j ) C j P_up C 1 ( 1 P_up) C 2 E [ U LessRisky P _ up ] j 2 1 P ( C j ) C j P_up C 3 ( 1 P_up) C 4 E [ U LessRisky P _ up ] j 2 1 P ( C j ) C j 1 C 5 • Substituting: E[U Risky ]=50, E[U Safe ]= 20, E[U Less Risky ]= 20 E [ U Less Risky ] 1 0 E [ U Less Risky P_up ] f ( P _ up ) dP _ up E P_up ( C 3 C 4 ) 0 . 5 ( C 1 C 2 ) E [ U Risky ] 1 0 E [ U Risky P_up ] f ( P _ up ) dP _ up E P_up ( C 1 1 0 P_up C 2 ) C 1 0 . 5 ( C 1 ( 1 C 2 P_up) ) C 2 f ( P _ up ) dP _ up Quantitative Methods for Management Emanuele Borgonovo 137 • c) Let us use Bayes’s theorem to update the prior uniform distribution – – – evidence: up,down, down,down,down,up,down,up,down,up,down,up,up,up L(E|P_up): L(E | P_up) 15 ! 7 ! 8 ! ( P _ up ) 7 ( 1 P _ up ) 8 Prior: 0 uniform bewteen 0 and 1 • Bayes’theorem: L(E | P_up) 15 ! 7 ! 8 ! ( P _ up ) 7 ( 1 P _ up ) 8 0 1 15 ! 7 ! 8 ! ( P _ up ) 7 ( 1 P _ up ) 8 1 1 dP _ up ( P _ up ) 7 ( 1 P _ up ) 8 0 1 ( P _ up ) 7 ( 1 P _ up ) 8 dP _ up • • • Posterior Distribution 3.5 3 2.5 2 1.5 1 0.5 0 p0 p1 E[p_up]=0.47 d) Posterior Decision: E[U Risky ]=23, E[U Safe ]= 20, E[U Less Risky ]= 9.2 Quantitative Methods for Management Emanuele Borgonovo 138 Quantitative Methods for Management Emanuele Borgonovo 139 • • • • • 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 decide? 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 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? Quantitative Methods for Management Emanuele Borgonovo 140 Decisione Decision Invest P=0.1 Pmigl=0.5 P_500=0.5 P_500_down=0.5 P_500_up=0.7 Quantitative Methods for Management Not Invest Clienti=0 Servizio=0 Servizio Clienti Guadagno Service Improves Pmigl Does not improve 1-Pmigl Clienti More than 50 P_50_up Less than 50 1-P_50_up 10000 -5000 -5000 0 Emanuele Borgonovo 141 Quantitative Methods for Management Emanuele Borgonovo 142 • 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 Quantitative Methods for Management Emanuele Borgonovo 143 • 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. Quantitative Methods for Management Emanuele Borgonovo 144 Opzione Valore • Here is a diagram for the Choice 1 Opction X1 2 X2 3 X3 X1=0.0 X2=0.0 X3=0.0 X4=0.0 X5=0.0 4 X4 5 X5 Quantitative Methods for Management Emanuele Borgonovo 145 • Indifference Curves: x 2 x 1 • Points on the same curve leave you indifference Quantitative Methods for Management Emanuele Borgonovo 146 • You can associate a numerical value representing you preferences to each indifference curve: x 2 x 1 1 2 n • V(x) is the function that says how much of x i one is willing to exchange for an increase or decrease in x k Quantitative Methods for Management Emanuele Borgonovo 147 • V(x) is a value function if it satisfies the following properties: • a) x ' x ' ' v ( x ' ) v ( x ' ' ) • b) x ' x ' ' v ( x ' ) v ( x ' ' ) Quantitative Methods for Management Emanuele Borgonovo 148 • For the “first job choice”, suppose that you value function is as follows: v ( x ) 3 / x 1 2 4 x 2 x 3 2 • 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? Quantitative Methods for Management Emanuele Borgonovo 149 Opzione Utilità Evento Casuale 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)) 1 P11 P12 P13 P14 U1 U2 U3 U4 Decision 2 3 4 P41 P42 P43 P44 U1 U2 U3 U4 Quantitative Methods for Management Emanuele Borgonovo 150 • 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: E U ( x 1 ) U ( x 2 ) Quantitative Methods for Management Emanuele Borgonovo 151 – One attribute Problem. Suppose that alternative 1 produces x 1 and the 2 x 2 , then 1 2 if x 1 >x 2 – Let us take two Alternatives 1 and 2, with x 1 >x 2 , given with certainty. – the value function will give us: v(x 1 )>v(x 2 ) – Let us now consider the following problem: 1 P1 2 1-P1 XI – To choose one need u(x1) and u(2) X1 X2 Quantitative Methods for Management Emanuele Borgonovo 152 0.9 0.8 1 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 1 Quantitative Methods for Management Distributions over attribute x 2 1 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 3 4 x 5 6 7 8 9 10 Emanuele Borgonovo 153 Quantitative Methods for Management Emanuele Borgonovo 154 • Given the lottery: 1 P1 X1 XN 2 1-P1 X3 • the value of x such that you are indifferent between x* for certain and playing the lottery. 1 P1 X1 X2 • In equations: 1-P1 2 u ( x *) E X* • N.B.: if you are risk neutral, then x*=E[x] Quantitative Methods for Management Emanuele Borgonovo 155 • a decision-maker is risk averse if preferisce sempre the expected value of a lottery alla lottery 1 2 : £10 2 £40 £10 0.500 0.500 £10; P = 1.000 • 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 Quantitative Methods for Management Emanuele Borgonovo 156 • the Risk Premium (“RP”) of a lottry is the difference between the expected value of the lottery and your Certainty Equivalent for the lottery: RP E x * • 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 E[x]=0. the insurance premium is how much one would pay to avoid a lottery: Quantitative Methods for Management Emanuele Borgonovo 157 • the Risk Aversion function is defined as: • Or, equivalently: r ( x ) : u " ( x ) u ' ( x ) r ( x ) d dx ln( u ' ( x )) • Supposing a constant risk aversion one gets an exponential utility function: d dx ln( u ' ( x )) ln( u ' ( x )) x u ' ( x ) e x ) u ( u ( x x 0 u ' ) ( t ) dt 0 x x e t dt u ( x ) u ( x 0 ) u ( x ) a e x b 1 ( e x e x 0 ) Quantitative Methods for Management Emanuele Borgonovo 158 • Constant Risk Aversion U 1 e • Compute constant Equivalent (CE): through Certainty 0 . 5 e x 1 0 . 5 e x 1 / 2 e CE Quantitative Methods for Management Emanuele Borgonovo 159 TwoStock prob_up=0.6 Market Decision Market up Blue Chip Stock -3 0.600 Market Down 1-exp(-200/70) = 1 1-exp(-(-160)/70) = -9 0.400 Market up Risky Investment 0.600 -2,110 Market Down 1-exp(-500/70) = 1 1-exp(-(-600/70)) = -5,278 0.400 Bond=1 1-exp(-50/70) = 1; P = 1.000 Quantitative Methods for Management Emanuele Borgonovo 160 Sensitivity Analysis on prob_up 400.0 0.0 -400.0 -800.0 -1,200.0 -1,600.0 -2,000.0 -2,400.0 -2,800.0 -3,200.0 0.40 0.52 0.64 0.76 prob_up Quantitative Methods for Management 0.88 1.00 Blue Chip Stock Risky Investment Bond Threshold Values: prob_up = 0.96 EV = 0.5 prob_up = 1.00 EV = 0.9 Emanuele Borgonovo 161 • Linear: u=ax – Risk Properties: • Risk Neutral • Exponential: u ( x ) a e cx b – Risk Properties: • - sign: Constant Risk Aversion, + sign: Constant Risk Proneness • Logarithmic: – Risk Properties: u ( x ) • Decreasing Risk Aversion a ln( x ) b Quantitative Methods for Management Emanuele Borgonovo 162 Quantitative Methods for Management Emanuele Borgonovo 163 • For the following three utility functions, u ( x ) a e x b u ( x ) a ln( x ) b • compute: – the risk aversion function r(x) – the risk premium for 50/50 lotteries – the insurance premium Quantitative Methods for Management Emanuele Borgonovo 164 • 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? Quantitative Methods for Management Emanuele Borgonovo 165 • 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: ( x x s ( x ) 1 e 1 a x 1 2 ) 2 3 b c • where x and 10. 1 is the vacation cost in kEUR, x 2 is distance in km and x 3 is a merit coefficient regarding relax/amusement to be assigned between 1 • What do you choose? Quantitative Methods for Management Emanuele Borgonovo 166 Quantitative Methods for Management Emanuele Borgonovo 167 Quantitative Methods for Management Emanuele Borgonovo 168 • 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. Quantitative Methods for Management Emanuele Borgonovo 169 • 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 Quantitative Methods for Management Emanuele Borgonovo 170 1 2 n • 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 Quantitative Methods for Management Emanuele Borgonovo 171 1 In Out 2 n • 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 Quantitative Methods for Management Emanuele Borgonovo 172 Quantitative Methods for Management Emanuele Borgonovo 173 • An event (and) can be True or False • State Variable or Indicator: X j • Properties: 1 0 if E has happened if E has not happened – (X J ) n =Xj – X j X j 0 X j J • This simple definition enables one to use algebraic operations to describe the logical behavior of systems. Quantitative Methods for Management Emanuele Borgonovo 174 • Let E i denote the event “the i-th component failed.” • Let X T • X T denote the event: “the System failed”. 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 E 1 or E 2 or …. or E n is true. • From a set point of view: E 1 E 2 ... E n E1 E3 • From a logical p.o.v., we get the following expression: X T 1 ( 1 X 1 ) ( 1 X 2 ) ... ( 1 X n ) 1 i n 1 ( 1 X i ) Quantitative Methods for Management Emanuele Borgonovo 175 • Let E i • Let X T denote the event “the i-th component has failed.” denote the event: “the system has failed”. • the condition for failure of the system is that all component fail. This happens if E 1 and E 2 and … E n are true at the same time. • From a Set point of view: E 1 E 2 ... E n E1 E3 E2 • the logical expression is: T 1 2 n i n 1 i Quantitative Methods for Management Emanuele Borgonovo 176 • 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 (X T ) as function of the individual failure events. Quantitative Methods for Management Emanuele Borgonovo 177 • Let A i denote the event “the i-th component is working (=Not failed).” • Let Y T denote the event: “the system is working”. • For a series system: all the components must be working for the system to work. Thus: A 1 , A 2 , … and A n must be true at the same time Y T Series n Y i i 1 • In parallel: for the system to work it is sufficient that just one component is working. Thus: A 1 or A 2 or A n must be true. Y T Parallel 1 i n 1 ( 1 Y i ) Quantitative Methods for Management Emanuele Borgonovo 178 • 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 1 In 2 Out 3 Quantitative Methods for Management Emanuele Borgonovo 179 • Let us find X T for a 2/3 system • Events: E 1 , E 2 , E 3 , Indicators: X 1 , X 2 , X 3 • Events that provoke a failure: E 1 E 2 E 3 , E 1 E 2 , E 1 E 3 , E 3 E 2 • Let us denote E 1 E 2 E 3 =Z 1 , E 1 E 2 =M 1 , E 1 E 3 = M 2 , E 3 E 2 • For X T to happen: Z 1 (M 1 M 2 M 3 ). . = M 3 . • the structure function expression is: X T 1 ( 1 Z 1 ) • Let us go to a level below: ( 1 M 1 ) ( 1 M 2 ) ( 1 M 3 ) X T 1 ( 1 X 1 X 2 X 3 ) ( 1 X 1 X 2 ) ( 1 X 1 X 3 ) ( 1 X 2 X 3 ) • Let us solve the calculations, noting that (X i ) n =X i : T 1 2 1 3 2 3 1 2 3 Quantitative Methods for Management Emanuele Borgonovo 180 • We recall that, for generic Events: P ( n i 1 E i ) j P ( E i ) ij i, j P ( E i E j ) ijk i, j k P ( E i E j E i ) ijkh i, j P ( E i E h k j E k E h ) ... ( 1 ) n P ( E 1 E 2 ... E n ) • We recall that, if the events are independent: P ( E i E j ,..., E n ) s n 1 P ( E s ) • Rare Event Approssimation: neglect all terms corresponding to multiple events Quantitative Methods for Management Emanuele Borgonovo 181 • In practice: the System Failure Probability is computed from the solved Structure Function, substituting to indicator X Event Probability. i the corresponding T 1 2 1 3 2 3 1 2 3 T 1 2 1 3 2 3 1 2 3 Quantitative Methods for Management Emanuele Borgonovo 182 • • • • the System Failure Probability is: P(X T )=P[(Z 1 (M 1 M 2 M 3 )]= P[(Z 1 Z 2 ]=P(Z 1 )+P(Z 2 )-P(Z 1 Z 2 ) where: – P(Z 1 )=P( E 1 E 2 E 3 ) – P(Z 2 )=P(M 1 M 2 M 3 )= P(M 1 )+ P(M 2 )+P(M 3 )-P(M 1 M 2 )- P(M 1 M 3 ) P(M 3 M 2 )+P(M 1 M 2 M 3 ). Ma: M 1 = E 1 E 2 , M 2 = E 3 E 1 , M 3 = E 3 E 2 . Noting, that: M 1 M 2 = M 1 M 3 = M 2 M 3 = M 1 M 2 M 3 = E 1 E 2 E 3 . Substituting: P(Z 2 )=P(E 1 E 2 )+ P(E 3 E 1 )+P(E 3 E 2 )-P(E 1 E 2 E 3 )- P(E 1 E 2 E 3 ) P(E 1 E 2 E 3 )+P(E 1 E 2 E 3 ). Thus: • P(Z 2 )=P(E 1 E 2 )+ P(E 3 E 1 )+P(E 3 E 2 )-2P(E 1 E 2 E 3 ) – P(Z 1 Z 2 )=P( E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 3 E 2 )=P(E 1 E 2 E 3 ) Thus: P(X T )= P(E 1 E 2 E 3 )+P(E 1 E 2 )+ P(E 3 E 1 )+P(E 3 E 2 )-2P(E 1 E 2 E 3 )-P(E 1 E 2 E 3 )= P(E 1 E 2 )+ P(E 2 E 1 )+P(E 3 E 2 )-2P(E 1 E 2 E 3 ) Quantitative Methods for Management Emanuele Borgonovo 183 Quantitative Methods for Management Emanuele Borgonovo 184 • For the following systems compute: – the Structure Function for System Failure – the Structure Function for System Operation – the Failure Probability – the Operation Probability 1 2 3 1 2 3 2/3 4 1 3 2 4 Quantitative Methods for Management Emanuele Borgonovo 185 • for the following system: 1 2 3 4 • Compute the Failure Probability supposing independent events and denoting the component failure probability by p. • Repeat the computation starting with the system success function, Y T . Verify that the two results coincide. Quantitative Methods for Management Emanuele Borgonovo 186 Quantitative Methods for Management Emanuele Borgonovo 187 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 Quantitative Methods for Management Emanuele Borgonovo 188 • Event Trees : represent the sequence of events that lead to the event top. Initiating Event Event 1 Event 2 Top Event Sì No Sì No Quantitative Methods for Management Emanuele Borgonovo 189 • 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. Quantitative Methods for Management Emanuele Borgonovo 190 High Pressure Pipe Room Building Top Event Yes No No No Quantitative Methods for Management Emanuele Borgonovo 191 • 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 And Or event Base • Basic Event of detail. : 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 Quantitative Methods for Management Emanuele Borgonovo 192 • 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 Quantitative Methods for Management Engine Static Filter Emanuele Borgonovo 193 • 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: Quantitative Methods for Management Emanuele Borgonovo 194 Engine Aeration 1 Static filter Elettr. Quantitative Methods for Management Mech. Fault Install. Emanuele Borgonovo 195 • Engine=A • Static filter =B • Electr.=1, Mech=2, Fault=3, Install.=4 X 1 T 1 1 1 1 [ 1 ( ( 1 ( 1 X 1 A ) X 2 ( 1 X 1 ) B ) ( 1 X 2 X 1 X 2 )] )] 1 [ 1 [ 1 ( X 3 ( 1 X 4 X 3 ) X 3 ( 1 X 4 ) X 4 )] X 1 X 2 X 1 X 3 X 4 X 3 X 4 X 2 X 3 X 4 X 1 X 3 X 1 X 2 X 3 X 1 X 4 X 1 X 2 X 1 X 4 X 2 X 3 X 4 X 2 X 3 X 1 X 2 X 3 X 4 X 2 X 4 • What are the minimal cut sets? Quantitative Methods for Management Emanuele Borgonovo 196 • If the event probabilities are low (rare events), then lower the event intersection probabilities will be. • One neglects the probabilities of intersections. • the Failure Probability is computed as sum of the minimal cut sets Probabilities: P ( X T ) P ( X 1 ) P ( X 2 ) P ( X 3 ) P ( X 4 ) Quantitative Methods for Management Emanuele Borgonovo 197 High pressure Pipe Aeration 1 Aeration 2 Top Event yes No No No Aeraz. 1 engine filter Quantitative Methods for Management Electr. Mech. Fault Installaz. Emanuele Borgonovo 198 • From the Event Tree: • Expanding: P ( Top ) P ( Aeraz . 1 Aeraz . 2 Condutt . ) * P ( Condutt . AP ) P ( Aeraz . 1 Aeraz . 2 , Condutt ., AP ) P ( Aeraz . 2 Cond . , AP ) P ( Cond AP ) • the conditional probabilities are found solving the corresponding Fault Trees Quantitative Methods for Management Emanuele Borgonovo 199 Quantitative Methods for Management Emanuele Borgonovo 200 • Given a system, tet us denote with s the Probability that the system fails between t and t+dt • It must hold that: 0 s Quantitative Methods for Management Emanuele Borgonovo 201 • The Reliability of a system between 0 and t is the Probability that the system fulfills its function between 0 and t • The Unreliability of a system between 0 and t is the Probability that the system breaks within time T: t F ( t ) Pr( T t ) f ( t ) d t 0 • Thus the Reliability [R(t)] is related to the failure time pdf as follows: t R ( t ) 1 Pr( • Note that, if f(t) is continuous: T t ) 1 0 f ( t ) d t Quantitative Methods for Management Emanuele Borgonovo 202 (t) Infant Mortality Aging Useful Life Quantitative Methods for Management t Emanuele Borgonovo 203 • Failure rate, (t), is the Probability that the system si rompa between t and t+dt, given that is sopravvissuto fino a t. • Dalla definition segue immediatamente the relaction with the Reliability and the function densità: • Thus: ( t ) f ( t ) R ( t ) R ' ( t ) R ( t ) Quantitative Methods for Management Emanuele Borgonovo 204 • From the above definition, there follows: ( t ) f ( t ) R ( t ) R ' ( t ) R ( t ) • Relationship R(t) (t): R ( t ) e 0 t ( t ) d t • Relationship f(t) (t): f ( t ) ( t ) R ( t ) ( t ) e 0 t ( t ) d t Quantitative Methods for Management Emanuele Borgonovo 205 • The mean time to failure is defined as: 0 Quantitative Methods for Management Emanuele Borgonovo 206 Quantitative Methods for Management Emanuele Borgonovo 207 • Series: P ( T S t ) R S ( t ) 1 F S ( t ) P ( T 1 T 2 ... T N • if independence is assumed: t ) P ( T S t ) P ( T 1 t ) P ( T 2 t ) P ( T N t ) • Thus: R s ( t ) i n 1 R i ( t ) • Faulure rate: S ( t ) i n 1 i ( t ) Quantitative Methods for Management Emanuele Borgonovo 208 • Failure Probability of the system: P ( T S t ) F S ( t ) 1 R S ( t ) P ( T 1 T 2 ... T N • if independent: t ) P ( T S t ) P ( T 1 t ) P ( T 2 t ) P ( T N t ) • Thus: F s ( t ) i n 1 F i ( t ) R S ( t ) 1 F s ( t ) 1 i N 1 1 R i ( t ) • Failure Rate: 1 s ( t ) 1 1 ( t ) 1 2 ( t ) ... 1 n ( t ) 1 1 ( t ) 2 ( t ) 1 1 ( t ) 3 ( t ) 1 ( t ) 1 2 ( t ) 3 ( t ) 1 ( t ) 1 2 ( t ) 3 ( t ) .... ( 1 ) n 1 1 ( t ) 2 ( t ) 3 ( t ) ... n ( t ) Quantitative Methods for Management Emanuele Borgonovo 209 • A standby system is a system where a subsystem is operational and the other subsystems become operational only after the failure of the system which is operating at the time of failure. • An example is the fifth wheel of a car. • In this case the System Reliability is given by: – 1) Two components: – Thus: R S 2 ( t ) R 1 ( t ) t P ( T S f 1 ( t 1 2 ) t ) R 2 ( t P ( T 1 t 1 ) dt 1 t ) P ( T 1 t T 2 t T 1 ) 0 – where 2 indicates that there are two components in standby, while the subscript denotes the second component – 2) Three components: – Thus: R S 3 ( t ) R S 2 ( t ) P ( T S 3 t ) P ( T 1 t ) P ( T 1 t T 2 t 0 P ( T 1 t f 1 ( t 1 ) 0 t 1 f 2 t ( t 2 ) T 2 t R 3 ( t T 1 t 1 T 3 t 2 ) t dt 1 T 1 dt 2 t T 2 ) T 1 ) Quantitative Methods for Management Emanuele Borgonovo 210 • For a standby system, it holds that: T T 1 T 2 ... T N • then P(t • If these distributions are exponential and the failure rates identical: R n s n 1 i 0 e t ( t ) i !i Quantitative Methods for Management Emanuele Borgonovo 211 • If a system is called in function and does not respond (i.e. does not begin to work), one talks about a “failure on demand”. • For a standby system, one denotes with q the failure on demand probability : • and R S 2 ' ( t ) R 1 ( t ) ( 1 q ) 0 t f 1 ( t 1 ) R 2 ( t t 1 ) dt 1 R S 3 ( t ) R S 2 ' ( t ) ( 1 q ) 2 t 0 f 1 ( t 1 ) t 0 t 1 f 2 ( t 2 ) R 3 ( t t 1 t 2 ) dt 1 dt 2 Quantitative Methods for Management Emanuele Borgonovo 212 Quantitative Methods for Management Emanuele Borgonovo 213 • Write the Fault Trees for the following systems and derive the structure function: 1 2 3 1 2 3 2/3 4 1 3 2 4 Quantitative Methods for Management Emanuele Borgonovo 214 • Una delle sequenze incidentali di un piccolo reattore di ricerca prevede la spaccatura della conduttura principale del circuito idraulico primario. Se la conduttura si rompe, si ha perdita immediata di raffreddamento del nocciolo - la zona del reattore dove avviene la reazione nucleare. L’incidente si può evitare se il sistema di raffreddamento ausiliario interviene per tempo e se il sistema di spegnimento del reattore interviene con successo. L’insuccesso dello spegnimento può avvenire se uno dei seguenti avvenimenti si realizza: mancata lettura del segnale per un guasto al software [P(Sof|alta press.)=10 press.)= 10 -3 -4 ], mancato arrivo del segnale per un guasto del sistema elettrico [P(E|alta press.)= 10 guasto on demand di 10 -3 affinche’ l’impianto sia fuori pericolo. Determinare: -5 ], mancato sganciamento delle barre per un guasto meccanico [P(Bar|alta ]. Il sistema di raffreddamento ausiliario è costituito da due pompe in parallelo, con rateo di guasto 1/10000 [1/h] e probabilita’ di . Le pompe devono funzionare per 100 ore – L’albero degli eventi – Gli alberi dei guasti – La probabilità di fondere il reattore dato che si è verificato l’incidente in un anno dato che la frequenza di eventi di alta pressione e’ 0.0001 per anno. Quantitative Methods for Management Emanuele Borgonovo 215 • • • • Un test di polizia per la determinazione del grado di alcool nei guidatori, ha probabilità 0.8 di essere corretto, cioè di dare risposta positiva quando il contenuto di alcool nel sangue è elevato o negativa quando il contenuto è basso. Coloro che risultano positivi al test, vengono sottoposti ad un esame da parte di un dottore. Il test del dottore non fa mai errori con un guidatore sobrio, ma ha un 10% di errore con guidatori ebbri. I due test si possono supporre indipendenti. 1) Determinare la frazione di guidatori che, fermati dalla polizia subiranno un secondo test che non rivela alto contenuto di alcool 2) Qual è la probabilità a posteriori che tale persona abbia un alto contenuto di alcool nel sangue? 3) Quale frazione di guidatori non avrà un secondo test? Quantitative Methods for Management Emanuele Borgonovo 216 • • • • • Un impianto elettrico ha due generatori (1 e 2). A causa di manutenzioni e occasionali guasti, le probabilità che in una settimana le unità 1 e 2 siano fuori serivizio (eventi che chiamiamo E 1 ed E 2 rispettivamente) sono 0.2 e 0.3 rispettivamente. C’è una probabiltà di 0.1 che il tempo sia molto caldo (Temperatura>30 gradi) durante l’estate (chiamiamo H questo evento). In tal caso, la domanda di elettricità potrebbe aumentare a causa del funzionamento dei condizionatori. La prestazione del sistema e la potenzialità di soddisfare la domanda può essere classificata come: – Soddisfacente (S): se tutte e due le unità sono funzionanti e la temperatura è inferiore a 30 gradi – Marginale (M) : se una delle due unità è funzionante e la temperatura è maggiore di 30 gradi – Insoddisfacente (U): se tutte e due le unità sono non funzionanti 1) Qual è la probabilità che esattamente una unità sia fuori servizio in una settimana? 2) Definire gli eventi S, M e U in termini di H, E 1 3) Scrivere le probabilità: P(S), P(U), P(M) ed E 2 Suggerimenti: Utilizzate alberi degli eventi e dei guasti per determinare la funzione di struttura e poi passate alle probabilità Quantitative Methods for Management Emanuele Borgonovo 217 • Dato un componente con rateo di guasto: ( t ) t 1 • con e 0 t calcolare: • R(t), f(t), il MTTF e la varianza del tempo medio di guasto • R(t) è detta distribuzione di Weibull • Disegnare (t),f(t) ed R(t) per =-1,1, 2. Dedurne che la Weibull può essere utilizzata per descrivere il tasso di guasto di componenti in tutta la vita del componente. Quantitative Methods for Management Emanuele Borgonovo 218 • Dato un componente con il tasso di guasto (t) seguente: 10 9 λ ( t ) t 1 1 1 1 100 t 2 se se se 0 t 1 1 t 10 t 10 8 7 6 5 4 3 2 1 0 0 • calcolare: 5 t • R(t), f(t), e il MTTF del componente 10 15 Quantitative Methods for Management Emanuele Borgonovo 219 • Calcolare l’espressione dell’affidabilità [R(t)] di un sistema k su n con rateo di guasto generico. • Calcolare la stessa espressione con distribuzioni esponenziali Quantitative Methods for Management Emanuele Borgonovo 220 • Calcolare l’affidabilità annuale di un sistema con 4 componenti in serie con ratei di guasto [1/h]: (1/6000, 1/8000, 1/10000, 1/5000). • Confrontatela con quella di un sistema in cui i componenti sono messi in: – Parallelo – In logica 3/4 – In logica 2/4 Quantitative Methods for Management Emanuele Borgonovo 221 • Due componenti identici, con tasso di guasto =3x10 -7 [1/h] devono essere messi in parallelo o standby. Determinate la configurazione migliore e il guadagno in affidabilità (in percentuale). • Supponete ora che il sistema di switch sia difettoso, con probabilità q=0.01. Quale delle due configurazioni è più conveniente? Quantitative Methods for Management Emanuele Borgonovo 222 • Considerate un sistema in standby di due componenti diversi, con densità di guasto esponenziali. Il MTTF del primo componente è 2 anni, quello del secondo è 3 anni. Calcolate: • La densità di guasto del sistema • Il MTTF • Cosa succede se i due componenti sono identici con MTTF di 2.5 anni? Quantitative Methods for Management Emanuele Borgonovo 223 Sof. Rottura Primario Spegnimento Raffreddamento Top Event No Si’ Si’ Raffreddamento Spegnimento On Demand El. Bar. Pompa 1. Pompa 2 Quantitative Methods for Management Emanuele Borgonovo 224 • Assumiamo eventi rari. • La frequenza si calcola dalla combinazione degli eventi: F f rottura P ( Raff Rottura Pr im ) P ( Spegn Rottura Pr im . ) • dove: f rottura =.000001 per anno • P(Spegn|rottura prim.)=P(Sof|rottura prim.)+P(E|rottura prim.)+P(Bar|rottura prim.)=0.00111 • Quindi: F 1 / q 1 e ( 1 10000 100 ) 2 1000000 . 00109 . 0011 2 . 2 10 9 Quantitative Methods for Management Emanuele Borgonovo 225 • Calcolare l’affidabilità annuale di un sistema con 4 componenti in serie con ratei di guasto [1/h]: (1/6000, 1/8000, 1/10000, 1/5000). – R ( t ) e 1 t e 2 t e 3 t e – Ore in un anno: 8760. – Sostituendo i numeri: 4 t R ( t ) e 8760 6000 e 8760 8000 8760 e 10000 e 8760 5000 • Confrontatela con quella di un sistema in cui i componenti sono messi in: 0 . 006 R ( 1 ( 1 8760 ( 1 8760 – Parallelo: t ) • Risultato: 0.11 e 6000 ) e 8000 ) ( 1 – ¾ supponendo I ratei di guasto =1/8000. 8760 e 10000 ) ( 1 e 8760 5000 ) – 2/4 supponendo I ratei di guasto =1/8000 • Risultato: 0.41 0 . 75 Quantitative Methods for Management Emanuele Borgonovo 226 • Due componenti identici, con tasso di guasto =3x10 -7 [1/h] devono essere messi in parallelo o standby. Determinate la configurazione migliore e il guadagno in affidabilità (in percentuale) per t=7 anni (61320hs). R p ( t ) 1 ( 1 61320 e 30000000 ) ( 1 61320 e 30000000 ) 0 . 99967 R S 2 ( t ) R 1 ( t ) t f 1 ( t 1 ) R 2 ( t t 1 ) dt 1 e t ( 1 t ) 0 . 99983 0 – Il guadagno di affidabilita’ e’ dell’ordine del 10^-2% (0.0002), quindi trascurabile • Supponete ora che il sistema di switch sia difettoso, con probabilità q=0.01. Quale delle due configurazioni è più conveniente? R p ( t ) 1 ( 1 61320 e 30000000 ) ( 1 61320 e 30000000 ) 0 . 99967 R S 2 ( t ) R 1 ( t ) ( 1 q ) t 0 f 1 ( t 1 ) R 2 ( t t 1 ) dt 1 e t ( 1 ( 1 q ) t ) 0 . 99965 Quantitative Methods for Management Emanuele Borgonovo 227 Quantitative Methods for Management Emanuele Borgonovo 228 Quantitative Methods for Management Emanuele Borgonovo 229 • Sistemi riparabili o manutenibili: il sistema puo’ ritornare a funzionare dopo la rottura • Indisponibilta’ istantanea: – q(t):= P(sistema indisponibile per T=t) • Indisponibilita’ limite: t lim q ( t ) • Indisponiblita’ media in T: • Indisponibilta’ media limite: T 0 q ( t ) dt q q lim T T lim T 0 q ( t ) dt T Quantitative Methods for Management Emanuele Borgonovo 230 • La disponibilita’ istantanea e’ il complementare della indisponibilita’. Le altre definizioni seguono immediatamente • Note: la disponibilita’/indisponibilita’ non e’ una densita’ di probabilita’ e l’indisponibilita’ media non e’ una probabilita’. • Interpretazione: la disponibilita’/indisponibilita’ media e’ la frazione media di tempo in cui il sistema e’ disponibile in [0 T]. • Le riparazioni/manutenzioni introducono periodicita’ nel problema Quantitative Methods for Management Emanuele Borgonovo 231 Quantitative Methods for Management Emanuele Borgonovo 232 Calcolo della Indisponibilita’: un unico componente, una sola modalita’ di guasto • Evoluzione temporale: t t r t t r t • A t=0 il sistema entra in funzione dopo la manutenzione. Dopo un tempo t= t torna di nuovo in manutenzione. La manutenzione dura t r . t e’ il tempo in cui il componente e’ soggetto a rotture casuali con (t). • Si nota che il problema e’ periodico, di periodo T= t r + t • Durante t il sistema ha una indisponibilita’ istantanea pari alla sua probabilita’ di rottura, se, come da ipotesi, non ci sono riparazioni: q ( t ) F S ( t ) 1 R ( t ) 1 e t 0 ( t ' ) dt ' Quantitative Methods for Management Emanuele Borgonovo 233 Calcolo della Indisponibilita’: un unico componente, una sola modalita’ di guasto • L’indisponibilta’ istantanea risulta quindi: • Da cui l’ind. Media: q q ( t ) 1 1 e 0 t ( t ' ) dt ' 0 t 1 e t 0 ( t ' ) dt ' dt t t r t t r t r 0 t t t t t t r • Supponiamo cost e 1. Quindi utilizziamo approssimaz. Taylor: 1 e t 0 ( t ' ) dt ' 1 e t 1 ( 1 t ...) t • Sostituiamo nella ind. Media, e assumiamo t r << t : q 1 T 2 t r T Quantitative Methods for Management Emanuele Borgonovo 234 • Guasto in funzionamento: f (t) [1/T] • Guasto in hot standby: s (t) [1/T] • Guasto a seguito di manutenzione errata: 0 , 1 , 2 …. Dove: – 0=incondizionale, – 1=dato che 1 manutenzione errata, – 2= dato che 2 manutenzioni errate • Guasto on demand: Q 0 ,Q 1 etc. Quantitative Methods for Management Emanuele Borgonovo 235 • Consideriamo per un componente i modi di guasto indicati in precedenza. • A t=0 il componente puo’ essere gia’ guasto se disabilitato dall’erronea manutenzione. Questo evento ha probabilita’ 0 . Con probabilita’ (1 0 ) il componente invece potra’ invece aver superato con successo la manutenzione. In questo caso il componente potra’ rompersi “on demand” (E1) o con tasso di guasto (t) (E2). Si ha: P(E1 E2)=P(E1)+P(E2)-P(E1E2)=Q 0 +F(t)-Q 0 F(t). • Riassumendo, tra 0 e t si ha: q(t)= 0 +(1 0 )*[Q 0 +F(t)-Q 0 F(t)]. • Introduciamo ora una probabilita’ esponenziale per le rotture. Utilizziamo la approssimazione di Taylor. Abbiamo: q(t)= 0 +(1 0 )*[Q 0 +(1-Q 0 ) t]. • Quindi l’indisponibilita’ istantanea e’: q ( t ) 1 0 ( 1 0 t ) Q 0 t t t t r Q 0 t 0 t t Quantitative Methods for Management Emanuele Borgonovo 236 • L’indisponibilita’ media sull’intervallo 0 t + t r e’: q 0 t ( 1 0 ) Q 0 t t t r t 2 2 Q 0 t 2 2 t r • Due assunzioni: 1) Eventi rari 2) t + t r t 0 0 t t t r Quantitative Methods for Management Emanuele Borgonovo 237 Componente t Q 0 0 • La funzione struttura e’: • X C =1- (1-X t ) (1-X Q0 )(1-X 0 )(1-X t )= • = X t +X Q0 + X 0 + X t termini di ordine superiore…. • Approssimazione eventi rari: • X C = X t +X Q0 + X 0 + X t t Quantitative Methods for Management Emanuele Borgonovo 238 • Sostituzioni successive • Sostituzioni distanziate t t r1 t r2 t t r1 t r2 t r r t t +2 t r t r t r • Periodo: t +2 t r Indisponibilita’ media e’ la somma di piu’ termini: q q R q C q D q M “R”: random, “C” common cause, “D” demand e “M” maintenance r t t +2 t r Quantitative Methods for Management Emanuele Borgonovo 239 • Causa comune: sono quei guasti che colpiscono il sistema come uno e rendono inutili le ridondanze e/o annullano indipendenza condizionale dei guasti. • Es.: difetto di fabbrica in parallelo di componenti identici • Errori in manutenzione: human errors • Human Reliability • CC e HR sono due importanti rami dello studio dell’affidabilita’ dei sistemi Quantitative Methods for Management Emanuele Borgonovo 240 • Come stabilire una politica di replacement ottimale? • Costruzione della funzione obiettivo – i) Individuazione del Criterio – ii)Costruzione della funzione obiettivo o utilita’ – iii) Ottimizzazione Quantitative Methods for Management Emanuele Borgonovo 241 • 1 componente soggetto replacement periodico e manutenzione periodica • Criterio = disponibilita’ media • Funzione obiettivo: q( t ) • t – ottimale: t 2 t r t r =24 h, =1/10000 (1/h) t ott =700hr • Con =1/100000 (1/h) t ott =2200hr Quantitative Methods for Management Emanuele Borgonovo 242 • Ottimizzazione in considerazione del costo di sostituzione e della disponibilita’ • Funzione obiettivo: t q t t • t ottimale: dB ( t ) d d 2 B ( t ) d 2 t t 0 0 • Occorre introdurre vita del’impianto L. • Si ha: dove c 0 e’ il costo unitario di riparazione Quantitative Methods for Management Emanuele Borgonovo 243 • Introduciamo poi il costo della indisponibilita’: Cq • definito come multiplo del costo singola riparazione. • Funzione energia: E ( ) ( t 0 Q0 t r ) d d t E ( ) a c0 1 2 t r t 2 1 L t 2 c0 • Intervallo ottimale: t 2 t r L 1 a 2 Quantitative Methods for Management Emanuele Borgonovo 244 c0 5 L 70000 a 10 t r 24 1 10000 0 0.001 Q0 0.001 3 4000 2000 59.363 0 0 0 Quantitative Methods for Management 5 10 4 t 1 10 5 100000 t Emanuele Borgonovo 245 • Il modello si applica al meglio a componenti in standby o sistemi di sicurezza passivi. • Infatti si ipotizza che il componente sia rimpiazzato secondo un intervallo di tempo prestabilito t . • Si valuta percio’ la convenienza rispetto alla minimizzazione del costo di replacement e/o alla massimizzazione della disponibilita’ • Per sistemi in funzionamento occorre considerare invece la possibilita’ di riparare il sistema Quantitative Methods for Management Emanuele Borgonovo 246 Quantitative Methods for Management Emanuele Borgonovo 247 • Analogamente alla rottura, anche il processo di riparazione di un componente ha delle caratteristiche di casualita’. Per esempio, non si sa il tempo necessario alla individuazione del guasto, cosi’ come puo’ essere non noto a priori il tempo necessario all’arrivo delle parti di ricambio o il tempo richiesto dall’esecuzione della riparazione. Tutto cio’ viene condensato in una quantita’ analoga al rateo di guasto, e, precisamente, il tasso di riparazione (t) . E’ uso comune assumere un tasso di riparazione costante e spesso questa assunzione non e’ peggiore di quella di assumere (t) costante.- Ne seguono: rip ( Rip t ( t ) ) MTTR 1 0 e t e t t e t 1 • Dove rip(t) e’ la densita’ di riparazione, ovvero la probabilita’ che la riparazione avvenga tra t e t+dt e Rip(t) e’ la probabilita’ che la riparazione avvenga entro t. Notiamo che (t) e’ la probabilita’ che il componente sia riparato tra t e t+dt dato che non e’ stato ancora riparato a t. Quantitative Methods for Management Emanuele Borgonovo 248 • Consideriamo un sistema composto da due componenti, di cui uno in standby. • Per modellizzare questo problema occorre un approccio diverso sia dai due casi precedenti . • Occorre introdurre gli stati del sistema • Nell’esempio. Il sistema puo’ essere: in funzione con il componente 1 funzionante (stato 1), in funzione ma con il componente 2 funzionante e il componente 1 in riparazione (stato 2), (stato 3) con entrambe i componenti rotti. Da 3 puo’ tornare a 2 e da 2 ad 1. Puo’ passare da 1 a 3 se c’e’ failure on demand. 1 2 3 Quantitative Methods for Management Emanuele Borgonovo 249 • Stato del sistema al tempo t e’ indipendente dalla storia del sistema. • Questa assunzione e’ alla base dei processi stocastici di Markov. • In particolare, supponiamo che il sistema possa avere M stati e denotiamo con X t lo stato del sistema al tempo t. Allora X t potra’ assumere valori 1,2,….M. • Cosa accade in dt? • Il sistema puo’ transitare in un altro stato (eventualmente con dei vincoli): i j Quantitative Methods for Management Emanuele Borgonovo 250 • Indichiamo con P ij la probabilita’ che il sistema passi dallo stato i allo stato j M ij P P 11 21 P M 1 P 12 P 1 M P MM • Proprieta’: • 1) i M 1 P ij 1 Quantitative Methods for Management Emanuele Borgonovo 251 • Applichiamo uno schema a stati per il sistema in standby. Otteniamo: P 13 1 P 12 2 P 23 3 P 21 P 32 Quantitative Methods for Management Emanuele Borgonovo 252 dP A P dt • Dove A e’ la matrice di transizione del sistema, P e’ il vettore delle probabilita’ degli stati del sistema. Quantitative Methods for Management Emanuele Borgonovo 253 P 12 1 2 1 2 P 21 • Esempio: componente soggetto a rottura e riparazione. 2 stati: in funzione o in riparazione, con tassi di guasto e riparazione. • Chi sono P 12 e P 21 ? Sono le probabilita’ di transizione in dt. Quindi: P 12 = e P 21 = • La matrice di transizione e’ costruita con le seguenti regole: • (+) se il salto e’ in entrata allo stato, (-) se il salto e’ in uscita • Prendiamo lo stato 1: si entra in 1 da due con tasso tasso (-). • Quindi: dP 1 dt P 1 P 2 (+), si esce con Quantitative Methods for Management Emanuele Borgonovo 254 • • Analogamente: Quindi: 1 2 1 2 dP 2 dt P 1 P 2 • La matrice di transizione e’: A • dP Il sistema di equazioni differenziali diventa: dt dP 1 dt P 1 P 2 dP 2 dt P 1 P 2 Quantitative Methods for Management Emanuele Borgonovo 255 • E’ la probabilita’ che a t il componente sia nello stato 1. Occorre risolvere il sistema di equazioni differenziali lineari precedente. Modo piu’ usato in affidabilita’ e’ mediante trasformata di Laplace. • Con trasf. Laplace, le equazioni da differenziali diventano algebriche. Dopo aver lavorato con equazioni algebriche, occorre poi antitrasformare. • Si ottiene dunque la disponibilita’ come funzione del tempo. A questo punto due disponibilita’ interessano: quella asintotica e quella media. Il risultato per un componente singolo soggetto a riparazioni e rotture e’ il seguente: Quantitative Methods for Management Emanuele Borgonovo 256 • Disponibilita’ istantanea: P 1 ( t ) = μ μ + λ + λ μ + λ e ( μ + λ ) t • Disponibilita’ asintotica: t lim → ∞ P 1 ( t ) = μ μ + λ MTTF = MTTF + MTTR • Interpretazione: tempo che occorre in media alla riparazione diviso il tempo totale • Disponibilita’ media su T: ( ) 2 exp T ( ) ( ) 2 Quantitative Methods for Management Emanuele Borgonovo 257 • Calcolare l’ indisponibilita’ media di un componente in standby soggetto a sostituzione periodica con le seguenti probabilita’ di guasto per t =5000: 0 0.002 t r 24 Q0 0.002 1 15000 • (Soluzione: q=.175) • Calcolare l’intervallo di sostituzione ottimale e l’indisponibilita’corrispondente, con L=70000, a=10 e a= . • (Soluzione: t =14500, q=0.5; t =849, q=0.06) Quantitative Methods for Management Emanuele Borgonovo 258f ( x )
dP dx
the exponential distribution
Meaning of the Exponential Distribution
the Exponential Distribution
Pdf and Cdf of the Exponential Distribution
Expected Value, Variance and Percentiles
Expected Variance Value : V :
E
( E
x
E
) xf
( x ) dx
( x
E
)
f ( x ) dx
E
E
S tan dard Deviation : V
the Normal Distribution
P
( x
X )
1 2
e
dx
Graphs of the Normal Distribution
Lognormal Distribution
Lognormal Distribution
f
( x )
L
Problem II-1 and solution
t
e
dt
1 /
5
P ( t
9 )
1
P ( t
9 )
1
( 1
e
)
e
16 .
5 %
Problem II-2
Problem II-3
Problem II-3
Problem II-3
Chapter III: Introductory Decision theory
An Investment Decision
Decision theory According to Laplace
Decision-Making Process Steps
Decision-Making Problem Elements
• Values and Objectives • Attributes • Decision Alternatives • Uncertain Events • Consequences
Decision Problem Elements
Decision Analysis Tools
Influence Diagrams
ID Elements
ID Elements
Influence Diagram Levels
1. Physical Phenomena and Dependencies 2. “Function level”: node output states probabilistic relations (models) 3. “Number level”: tables of node probabilities
Case Study 2 - Leaking SG tube
Influence Diagram
Decision Market Return
Decision Trees
the Decision Tree (DT)
Decision Tree Solution
example
Problem Solution
E [ U
]
P ( C
)
C
P.up
C
( 1
P.up)
C
E [ U
]
P ( C
)
C
P.up
C
( 1
P.up)
C
E [ U
]
P ( C
)
C
1
C
Run or Withdraw?
Example of a simple ID
From IDs to Decision Trees
Sequential Decisions
Influence Diagram
Decision Tree
the Expected Value of Perfect Information
EVPI
E [ Knowing ]
E [ BeforeKnow ing ]
Example: investing
EVPI Result
EVPI
E [ Knowing ]
E [ r ]
260
50
210
Problems
How much to bid?
Influence Diagram with three events
Solution
Solution
Solution
Solution
Sales_Costs
Solution Sales_Costs
Breakdown in Production
EVPI Problems
Troubles in Production
Influence Diagram
Decision Tree
Probability Values
2 and 4 years
Chapter IV Elements of Sensitivity Analysis
Sensitivity Analysis
• Various Types of SA
• Uncertainty Analysis
How do we use SA?
Sensitivity Analysis (Run or withdraw)
Summary
• Sensitivity Analysis
• Uncertainty Analysis
Sensitivity Analysis
One-way Sensitivity Analysis
Two-way Sensitivity Analysis
Tornado Diagrams
Example of a Tornado Diagram
Upsides and Downsides
Sensitivity Analysis and Parameter Importance
Process
Sensitivity Analysis Types
U
f(x
, x
,..., x
)
the Differential Importance Measure
dU
x f
dx
f x
dx
...
f
x
dx
DIM(x
)
dU
dU x
Global Sensitivity Indices
U
f( x )
f
f
( x
)
f
( x
, x
)
...
f
( x ) S
(x
)
D
D
f
dx
...dx
f
( x ) d x
f
I(x)^ [U
f(x
, x
,..., x
)]
Importance Relations
Additivity Property
U U
Global Importance Measures
R
f(x
, x
,...,
x
)
FV(x
)
(R, x
) R
Summary of the previous concepts
Chapter V Uncertainty Analysis
Uncertainty Analysis
Summary
Uncertainty
Example: Model of the World
P ( n , t )
e
(
t )
n !
Some useful information on Poisson Distributions
the Corresponding Epistemic Model
Combining the Epistemic Model and MOW
p ( n , t ,
) d
e
(
t )
u (
) d
n !
….
In General
MOW ( t
,
,....
)
P ( t )
MOW ( t
,
,....
) f (
,
,....) d
d
.....
An problem
Solution
6 .
9
Continuous form of Bayes Theorem
Formula
From discrete to continuous
From discrete to continuous
P ( E A
)
P ( A
)
L ( E
)
(
) d
Is it a fair coin?
Result
Graph
Conjugate Distributions
P ( n , t )
e
(
t )
n !
π
( λ , α , β )
β
λ
e
Γ ( α )
(
,
' ,
' )
'
(
' ) e
Conjugate Distributions
Conjugate Distributions
Epistemic Uncertainty in Decision-Making Problems
Analytical Propagation of Uncertainty
the Monte Carlo Method
the core of Monte Carlo
Inversion theorem
Example
Application to ID and DT
Results
Problem V-1
Problem V-2: Investing
Prob. 5-2
Investment
Problems
• Apply the one way, two way and Tornado Diagrams SA to the IDs and DTs of the previous chapters: • Discuss your results
Bayesian Decision
Influence Diagram
Chapter VI Introduction to Decision theory
Summary
Preferences Under Certainty
Preferences under Certainty
Structuring Preferences
the Value Function
V ( x )
v ( x , x ,...,
x )
V(x)
Example
Preferences under Uncertainty
utility Function
Utility vs. Value
Stochastic Dominance
One Attribute Utility Functions
Certainty Equivalent
definition of Risk Aversion
Risk Premium and Insurance Premium
IP :
x *
Mailmatical Definition
Risk Preferences
r
Investment Results with Risk Aversion
A quale value accetterei l’investimento rischioso
Esempi of funzioni utility
Problems
problem VI-1
u ( x )
a
x
Problem VI-2
Problem VI-3
Chapter VII the Logic of Failures
Elements of Reliability theory
Safety and Reliability
Systems
Series
Parallel Systems
Elements of System Logics
Boolean Logic
Series Systems
Parallel Systems
X
X
X
...
X
X
the Structure Function
the Logic of Performance
n/N Logics
Example: 2/3 System Logics
X
X
X
X
X
X
X
2 X
X
X
Probability Sum Rules
Golden Rule
X
X
X
X
X
X
X
2 X
X
X
P ( X
)
P ( E
E
)
P ( E
E
)
P ( E
E
)
2 P ( E
E
E
)
Proof
Problems
Problem VII-1
Problem VII-2
Chapter VIII Elements of Reliability
Cut and Path sets
Even Trees
Example
Gas Leakage Fault - Tree
Fault Trees
Example
Example
Level II
From Fault Trees to Structure Functions
Rare Events Approximation
Event Trees & Fault Trees
Probability of the Top Event
P ( Top )
P ( Condutt .
Aeraz .
1
Aeraz .
2 AP )
Definitions
Failure Density
f
( t ) dt
f
( t ) dt
1
Reliability
R ' ( t )
f ( t )
General Fault rate
Hazard/Failure Rate
( t )
R ( t ) dt
f ( t ) dt
Legami between R(t), f(t) and
(t)
time medio of failure (MTTF)
MTTF
t
f ( t ) dt
System Reliability
Reliability of systems in Series
Reliability of systems in Parallelo
Reliability of Standby Systems
Standby Systems with const. failure rates
Failure on Demand
Problems
problem VIII-1
Problema VIII-2
Problema VIII-3
Problema VIII-4
Problema VIII-5: Distribuzione Weibull
Problema VIII-6
Problema VIII-7
Problema VIII-8
Problema VIII-9
Problema VIII-10
Prob. VIII-2 Soluzione
Prob. VIII-2 Soluzione
Problema VIII-8 Soluzione
Problema VIII-9 Soluzione
Capitolo IX Decisioni Operative: Ottimizzazione delle Manutenzioni
Decisioni Operative
• Decisioni di Affidabilita’ o Reliability Design • Decisioni di Optimal Replacement • Decisioni di ispezione ottimale • Decisioni di riparazione ottimale
Indisponibilita’
Disponibilita’
Effetto delle manutenzioni
Modi di Guasto
Indisponibilita’ istantanea con piu’ modi di guasto
Indisponibilita’ media con piu’ modi di guasto
q
Q
2
Rappresentazione equivalente
Il caso di due componenti
Modi di guasto
Modelli decisionali corrispondenti
Esempio 1
Esempio 2
B (
)
C
q (
)
C (
)
Esempio 2
Esempio 2
1.185
10 4
Applicazione del modello
Riparazioni
Il tasso di riparazione
(t)
Esempio
Assunzioni
Matrice di transizione
Esempio
Equazioni di Markov/Kolmogorov
Costruzione della matrice di transizione
La matrice di transizione
Disponibilita’ asintotica e media
Risultati per un componente
Problema IX-1