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Emanuele Borgonovo 1 Quantitative Methods for Management Emanuele Borgonovo Quantitative Methods for Management First Edition Decision Market Return Structural.

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1 Emanuele Borgonovo 1 Quantitative Methods for Management Emanuele Borgonovo Quantitative Methods for Management First Edition Decision Market Return Structural

2 Emanuele Borgonovo 2 Quantitative Methods for Management Chapter three: Models

3 Emanuele Borgonovo 3 Quantitative Methods for Management Models A Model is a mailmatical-logical instrument that the analyst, the manager, the scientist, the engineer develops to: –foretell the behaviour of a system –foresee the course of a market –evaluate an investment decision accounting for uncertainty factors Common Elements to the Models: –Uncertainty –Assumptions –Inputs Model Results

4 Emanuele Borgonovo 4 Quantitative Methods for Management Building a Model To build a reliable model requires deep acquaintance of: –the Problem –Important Events regarding the problem –Factors that influence the behavior of the quantities of interest –Data and Information Collection –Uncertainty Analysis –Verification of the coherence of the Model by means of empiric analysis and, if possible, analysis of Sensitivity Analysis

5 Emanuele Borgonovo 5 Quantitative Methods for Management 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?

6 Emanuele Borgonovo 6 Quantitative Methods for Management Chapter II Introductory Elements of Probability theory

7 Emanuele Borgonovo 7 Quantitative Methods for Management Probability Is it Possible to Define Probability? Yes, but there are two schools the first considers Probability as a property of events the second school asserts that Probability is a subjective measure of event likelihood (De Finetti)

8 Emanuele Borgonovo 8 Quantitative Methods for Management Kolmogorov Axioms U B A

9 Emanuele Borgonovo 9 Quantitative Methods for Management 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 Areas and rectangles? U C ABDand

10 Emanuele Borgonovo 10 Quantitative Methods for Management 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 A AB

11 Emanuele Borgonovo 11 Quantitative Methods for Management Conditional Probability Suppose now that B has happened, i.e., you jumped into area B (and you cannot jump back!). B A AB You cannot but agree that: P(A|B)=P(AB)/P(B) Hence: P(AB)=P(A|B) *P(B)

12 Emanuele Borgonovo 12 Quantitative Methods for Management 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 A AB B A Thus, for independent events: P(AB)=P(A)*P(B)

13 Emanuele Borgonovo 13 Quantitative Methods for Management Probability and Information Problem: you are given a box containing two rings. the box content is such that with the same Probability (1/2) the box contains two golden rings (event A) or a golden ring and a silver one (event B). To let you know the box content, you are allowed to pick one ring from the box. Suppose it is a golden one. –In your opinion, did you gain information from the draw? –the Probability that the oil one is golden is 50%? –Would you pay anything to have the possibility to draw from the box?

14 Emanuele Borgonovo 14 Quantitative Methods for Management In the subjectivist approach, Probability changes with information

15 Emanuele Borgonovo 15 Quantitative Methods for Management Bayes theorem Hypothesis: A and B are two events. A has happened. Thesis: P(B) changes as follows: P(B) before A New value of the Probability of B Probability of A Probability of A given B

16 Emanuele Borgonovo 16 Quantitative Methods for Management Let us come back to the ring problem Events: A: both rings are golden o: the picked up ring is golden the theorem states: P(A)=Probability of both rings being golden before the extraction =1/2 P(o)=Probability of a golden ring=3/4 P(o|A)=Probability that the extracted ring is golden given A=1 (since both rings are golden) So:

17 Emanuele Borgonovo 17 Quantitative Methods for Management Bayes theorem Proof Starting point Conditional Probability formula thesis

18 Emanuele Borgonovo 18 Quantitative Methods for Management U the Total Probability theorem 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: Bayes theorem in the presence of N events becomes : B A C D and

19 Emanuele Borgonovo 19 Quantitative Methods for Management Continuous Random Variables Till now we have discussed individual events. there are problems in which the event space is continuous. For example, think of the failure time of a component or the time interval between two earthquakes. the random variable time ranges from 0 to +. To characterize such events one resorts to Probability distributions.

20 Emanuele Borgonovo 20 Quantitative Methods for Management Probability Density Function f(x) is a Probability density function (pdf) if: –It is integrable –And –the integral of f(x) over - :+ is equal to 1. Note: f(x 0 )dx is the Probability that x lies in an interval dx around x 0.

21 Emanuele Borgonovo 21 Quantitative Methods for Management Cumulative Distribution Function Given a continuous random variable X, the Probability that X

22 Emanuele Borgonovo 22 Quantitative Methods for Management the exponential distribution Consider events that happen continuously in time, and with continuous time T. If the events are: –Independents –With constant failure rates the random variable T is characterized by an exponential distribution: and by the density function:

23 Emanuele Borgonovo 23 Quantitative Methods for Management Meaning of the Exponential Distribution We are dealing with a reliability problem, and we must characterize the failure time, T. T is a random variable: one does not know when a component is going to break. All one can say is that for sure the component will break between 0 and infinity. Thus, T is a continuous random variable. Let us consider that failures are independent. This is the case if the failure of one component does not influence the failure of the other components. Let us also consider constant failure rates. This is the case when repair brings the component as good as new and when the component does not age during its life. Under these Hypothesis, the failure times are independent and characterized by a constant failure rate at every dt. What is the Probability distribution of T? Let us consider a population of N(t) components at time t. If is the failure rate of a component, then N(t) dt is the number of failues in dt around time t.

24 Emanuele Borgonovo 24 Quantitative Methods for Management the Exponential Distribution 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: Which solved leads: N(T) is the number of components surviving till T. N(0) is the initial number of components. Set N(0)=1. then N(T)/N(0) is the Probability that a component survives till T.

25 Emanuele Borgonovo 25 Quantitative Methods for Management T/t Pdf and Cdf of the Exponential Distribution P(t

26 Emanuele Borgonovo 26 Quantitative Methods for Management Expected Value, Variance and Percentiles 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

27 Emanuele Borgonovo 27 Quantitative Methods for Management the Normal Distribution Is a symmetric distribution around the mean Pdf: Cdf:

28 Emanuele Borgonovo 28 Quantitative Methods for Management Graphs of the Normal Distribution Cumulative Gaussian Distribution x

29 Emanuele Borgonovo 29 Quantitative Methods for Management Lognormal Distribution Pdf Cdf

30 Emanuele Borgonovo 30 Quantitative Methods for Management Lognormal Distribution

31 Emanuele Borgonovo 31 Quantitative Methods for Management Problem II-1 and solution the failure rate of a car gear is 1/5 for year (exponential events). What is the mean time to failure of the gear? What is the Probability of the gear being integer after 9 years?

32 Emanuele Borgonovo 32 Quantitative Methods for Management Problem II-2 You are considering a University admission test for a particularly selective course. the admission test, as all tests test, is not perfect. Suppose that the true distribution of the class is such that 10% of the applicants are really qualified and 90% are not. then you perform the test. If a student is qualified, then the test will admit him/her with 90% Probability. If the student is not qualified he/her gets admitted at 10%. Now, let us consider a student that got admitted: –What is the Probability that the student is really qualified? –Is it a good test? How would you use it? –(Hint: use the theorem of Total Probability)

33 Emanuele Borgonovo 33 Quantitative Methods for Management Problem II-3 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

34 Emanuele Borgonovo 34 Quantitative Methods for Management Problem II-3 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:

35 Emanuele Borgonovo 35 Quantitative Methods for Management Problem II-3 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: –It is the same problem as in the example, but with adjourned probabilities. the Probability of being in B given that the picked ring is golden in two consecutive extractions, having put the ring back in the box after the first extraction –Solution: 1-P(A 2o)=0.2

36 Emanuele Borgonovo 36 Quantitative Methods for Management Chapter III: Introductory Decision theory

37 Emanuele Borgonovo 37 Quantitative Methods for Management An Investment Decision At time T, you have to decide whether, and how, to invest $1000. You face three mutually exclusive options: –(1) A risky investment that gives you $500 PV in one year if the market is up or a loss of $400 if the market is down –(2) A less risky investment that gives you $200 in one year or a loss of $160 –(3) the safe investment: a bond that gives you $20 in one year independently of the market

38 Emanuele Borgonovo 38 Quantitative Methods for Management Decision theory According to Laplace the theory leaves nothing arbitrary in choosing options or in making decisions and we can always select, with the help of the theory, the most advantageous choice on our own. It is a refreshing supplement to the ignorance and feebleness of the human mind. Pierre-Simon Laplace (March Beaumont-en-Auge - March Paris)

39 Emanuele Borgonovo 39 Quantitative Methods for Management Decision-Making Process Steps Problem identification Alternatives identification Model implementation Alternatives evaluation Sensitivity Analysis Further Analysis? Yes Best Alternatives implementation No

40 Emanuele Borgonovo 40 Quantitative Methods for Management Decision-Making Problem Elements Values and Objectives Attributes Decision Alternatives Uncertain Events Consequences

41 Emanuele Borgonovo 41 Quantitative Methods for Management Decision Problem Elements Objectives: –Maximize profit Attributes: – Money Alternatives: –Risky –Less Risky –Safe Random events: –the Market Consequences: –Profit or Loss

42 Emanuele Borgonovo 42 Quantitative Methods for Management Decision Analysis Tools Influence Diagrams Decision Trees Decision Market Return Structural Market up prob_up Market down 1-prob_up Less Risky Market up prob_up Market down Risky Safe How should the invest $1000? 1-prob_up

43 Emanuele Borgonovo 43 Quantitative Methods for Management Influence Diagrams Influence diagrams (IDs) are… a graphical representation of decisions and uncertain quantities that explicitly reveals probabilistic dependence and the flow of information ID formal definition: –ID = a network consisting of a directed graph G=(N,A) and associated node sets and functions ( Schachter, 1986 )

44 Emanuele Borgonovo 44 Quantitative Methods for Management ID Elements NODES = Decision = Random Event = utility ARCS Informational Arcs probabilistic Dependency Arcs Structural Arcs

45 Emanuele Borgonovo 45 Quantitative Methods for Management ID Elements Decision Node Chance NodeValue NodeChance Node Decision Node Conditional Arc probabilistic Dependency Informational Arc Sequential Decisions Structural

46 Emanuele Borgonovo 46 Quantitative Methods for Management Influence Diagram Levels 1. Physical Phenomena and Dependencies 2. Function level: node output states probabilistic relations (models) 3. Number level: tables of node probabilities

47 Emanuele Borgonovo 47 Quantitative Methods for Management Case Study 2 - Leaking SG tube Influence Diagram for Case Study 2

48 Emanuele Borgonovo 48 Quantitative Methods for Management Influence Diagram Decision Market Return Structural

49 Emanuele Borgonovo 49 Quantitative Methods for Management Decision Trees Decision Trees (DTs) are constituted by the same type of arcs of Influence Diagrams, but highlight all the possible event combinations. Instead of arks, one finds branches that emanate from the nodes as many as the Alternatives or Outcomes of each node. With respect to Influence Diagrams, Decision Trees have the advantage of showing all possible patterns, but their structure becomes quite complicated at the growing of the problem complexity.

50 Emanuele Borgonovo 50 Quantitative Methods for Management the Decision Tree (DT) Market up Market down 1-prob_up Less Risky Market up Market down Risky Safe How should the invest $1000?

51 Emanuele Borgonovo 51 Quantitative Methods for Management Decision Tree Solution Alternative Payoff or utility: j=1…m i spans all the Consequences associated to alternative the U j is the utility or the payoff of consequence j P i (C j ) is the Probability that consequence C j happens given that one chose alternative the 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 )

52 Emanuele Borgonovo 52 Quantitative Methods for Management example Market up P.up C1 Market down 1-P.up C2 Blue Chip Stock Market up P.up C3 Market down 1-P.up C4 Risky investment CD paying 5% C5 How should the invest $1000?

53 Emanuele Borgonovo 53 Quantitative Methods for Management Problem Solution Using the previous formula:

54 Emanuele Borgonovo 54 Quantitative Methods for Management the Best Investment for a Risk Neutral Decision - Maker Market up $200 Market down ($160) Blue Chip Stock $56 Market up $500; P = Market down ($600); P = Risky investment $60 CD paying 5% return = $50 How should the invest $1000?

55 Emanuele Borgonovo 55 Quantitative Methods for Management Run or Withdraw? You are the owner of a racing team. It is the last race of the season, and it has been a very good season for you. Your old sponsor will remain with you for the next season offering an amount of $50000, no matter what happens in the last race. However, the race is important and transmitted on television. If you win or end the race in the first five positions, you will gain a new sponsor who is offering you $100000, besides $10000 or $5000 praise. However there are unfavorable running conditions and an engine failure is likely, based on your previous data. It would be very bad for the image of you racing team to have an engine failure in such a public race. You estimate the damage to a total of -$ What to do? Run or withdraw? A) Elements of the problem: –What are your objectives –What are the decision alternatives –What are the attributes of the decision –What are the uncertain events –What are the alternatives

56 Emanuele Borgonovo 56 Quantitative Methods for Management Example of a simple ID Decision Engine failure Profit Final Classification

57 Emanuele Borgonovo 57 Quantitative Methods for Management From IDs to Decision Trees Out of first five $20,000; P = failure Engine failure $20,000 Win $110,000; P = In first five $105,000; P = Out of first five $50,000; P = No failure $94,500 Run Decision $57,250 Old sponsor $50,000 Withdraw Engine_failure=0 $50,000 Decision pfailure=0.5 pfive=0.30 pout=0.2 pwin=0.5 Run : $57,250

58 Emanuele Borgonovo 58 Quantitative Methods for Management Sequential Decisions Are decision making problems in which more than one decisions are evaluated one after the other. You are evaluating the purchase of a production machine. Three models are being judged, A B and C. the machine costs are 150, 175 and 200 respectively. If you buy model A, you can choose insurance A1, that covers all possible failues of A, and costs 5% of A cost, or you can choose insurance policy A2, that costs 3% of A cost, but covers only transportation risk. If you buy model B, insurance policy B1 costs 3% of B cost and covers all B failures. Insurance B2 costs 2% of B and covers only transportation. For model C, the most reliable, the insurance coverages cost 2% and 1.5% respectively. Based on this information and supposing that the machines production is the same, what will you choose? (failure Probability of A in the period of interest=5%) (failure Probability of B in the period of interest=3%) (failure Probability of C in the period of interest=2%

59 Emanuele Borgonovo 59 Quantitative Methods for Management Influence Diagram

60 Emanuele Borgonovo 60 Quantitative Methods for Management Decision Tree

61 Emanuele Borgonovo 61 Quantitative Methods for Management the Expected Value of Perfect Information Data and information collection is essential to make decisions. Sometimes firms hire consultants or experts to get such information. But, how much should one spend? One can value information, since it is capable of helping the decision-maker in selecting among alternatives the value of information is the added value of the information. the expected value of perfect information (EVPI) assumed that the source of information is perfect, and then: the definition is read as follows: how much is the decision worth with the new information and without N.B.: we will refer only to aleatory uncertainty

62 Emanuele Borgonovo 62 Quantitative Methods for Management Example: investing

63 Emanuele Borgonovo 63 Quantitative Methods for Management EVPI for the Example

64 Emanuele Borgonovo 64 Quantitative Methods for Management EVPI Result

65 Emanuele Borgonovo 65 Quantitative Methods for Management Problems

66 Emanuele Borgonovo 66 Quantitative Methods for Management How much to bid? Bob works for an energy production company. Your company is engaged in the decision of how much to bid to salvage the wreckage of the SS.Kuniang, a carbon transportation boat. If the firm wins, the boat could be repaired and could come back to its transportation activity again. Pending on the possible winning and on the decision is the result of a judgment by Coast Guard, which will be revealed only after the opening of the bids. That is, if the Coast Guard will assign a low value to the ship, this would mean that the ship is considered as recoverable. Otherwise, the boat will be deemed unusable. If you do not win, you will be forced to buy a new boat. Identify the decision elements Structure the corresponding ID and DT

67 Emanuele Borgonovo 67 Quantitative Methods for Management Influence Diagram with three events 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?

68 Emanuele Borgonovo 68 Quantitative Methods for Management Solution Influence Diagram the

69 Emanuele Borgonovo 69 Quantitative Methods for Management Solution Corresponding Decision Tree

70 Emanuele Borgonovo 70 Quantitative Methods for Management Solution Influence Diagram II

71 Emanuele Borgonovo 71 Quantitative Methods for Management Solution Decision Tree II:

72 Emanuele Borgonovo 72 Quantitative Methods for Management 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. 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 Sales_Costs High Sales P_Alte|high 0 Low 1- P_Alte|high -10 high Cost P_high High P_ P_Alte|high 20 Low 1- P_Alte|high 0 Basso 1-P_high Invest Decision Do not Invest 5 high=0.5 P_Alte=0.3 P_high=0.5

73 Emanuele Borgonovo 73 Quantitative Methods for Management Solution Sales_Costs

74 Emanuele Borgonovo 74 Quantitative Methods for Management Breakdown in Production An industrial system composed from two lines has experience a breakdown in one line. Production, therefore, is reduced by 50%. the management asks you collaboration on the following decision. It is explained to you that there are two ways to proceed: 1) an intermediate repair, of the duration of two days, with a repair cost of EUR 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 EUR With Probability P_10g the line will be as before the breakdown. –According to you, the residual life of the system is important for the decision? –Suppose that there are still three years of life for the system. –Which strategy should you carry out? –Determine the decision problem elements. Draw the Influence Diagram and the corresponding Decision Tree. Find the value or values of the probabilities for which a complete repair is more convenient than a partial one. –What would you would advise to the director of the system to do based on the engineer estimates?

75 Emanuele Borgonovo 75 Quantitative Methods for Management EVPI Problems Determine the EVPI for the random event nodes in the previous IDs and DTs of the following problems: Sales_Costs (lez. 2) Production break-down (lez.2)

76 Emanuele Borgonovo 76 Quantitative Methods for Management Troubles in Production One of the two production lines of the plant you manage has broke down. the plant production capacity is therefore halved. the management faces the following decision and asks you a collaboration. Technically one can a: 1) perform an temporary repair, lasting two days, and costing For every lost production day one has a revenue loss of 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 With Probability P_10g the line will be as good as new. In your opinion, the residual plant life is relevant to this decision? Suppose that there are still three years of life for the plant. What should one decide? –Identify the decision making elements –Draw the Influence Diagram for the problem –Find the values of the probabilities for which one or the other intervention is more convenient –What would your suggestion to the plant director be? –What would happen if the plant life were 2 and 4 years instead of 3?

77 Emanuele Borgonovo 77 Quantitative Methods for Management Influence Diagram

78 Emanuele Borgonovo 78 Quantitative Methods for Management Decision Tree

79 Emanuele Borgonovo 79 Quantitative Methods for Management Probability Values Three years

80 Emanuele Borgonovo 80 Quantitative Methods for Management 2 and 4 years 2 years 4 years

81 Emanuele Borgonovo 81 Quantitative Methods for Management Chapter IV Elements of Sensitivity Analysis

82 Emanuele Borgonovo 82 Quantitative Methods for Management Sensitivity Analysis Various Types of SA –One Way SA –Two Way SA –Tornado Diagrams –(Differential Importance Measure) Uncertainty Analysis –Monte Carlo –(Global SA)

83 Emanuele Borgonovo 83 Quantitative Methods for Management How do we use SA? a) To check model correctness and robustness b) To Further interrogate the model –Questions: What is the most influential parameter with respect to changes? What is the most influential parameter on the uncertainty (data collection)

84 Emanuele Borgonovo 84 Quantitative Methods for Management Underline the critical dependencies of the outcome Sensitivity Analysis (Run or withdraw)

85 Emanuele Borgonovo 85 Quantitative Methods for Management Summary Sensitivity Analysis –One way sensitivity –Two way sensitivity –Tornado Diagrams Uncertainty Analysis –Aleatory Uncertainty –Epistemic Uncertainty –Bayes theorem for continuous distributions –Monte Carlo Method

86 Emanuele Borgonovo 86 Quantitative Methods for Management Sensitivity Analysis By sensitivity analysis one means the study of the change in results (output) due to a change in one of the model parameters (input) the simplest Sensitivity Analysis types are: –One way sensitivity –Two way sensitivity –Tornado diagrams

87 Emanuele Borgonovo 87 Quantitative Methods for Management One-way Sensitivity Analysis A one way sensitivity is obtained changing the Model input variables one at a time, and registering the change in the decision value. It enables the analyst to study the change in value of each of the alternatives with respect to the change in the input parameter under consideration

88 Emanuele Borgonovo 88 Quantitative Methods for Management Two-way Sensitivity Analysis In a Two-way Sensitivity Analysis two parameters are varied at the same time. Instead of a line, one obtains a plane, in which each region identifies the preferred alternative that correspond to the combination of the two parameter values

89 Emanuele Borgonovo 89 Quantitative Methods for Management Tornado Diagrams the analysis is focused on the preferred decision An interval of variation for each input parameter is chosen the parameters are changed one at a time, while keeping the oilrs at their reference value the change in output is registered the output change is shown by means of a horizontal bar the most important variable is the one that corresponds to the longest bar.

90 Emanuele Borgonovo 90 Quantitative Methods for Management Example of a Tornado Diagram

91 Emanuele Borgonovo 91 Quantitative Methods for Management Upsides and Downsides Upsides –Easy numerical calculations –Results immediately understandable Downsides –Input range of variation not considered together with the output range: should not be used to infer parameter importance –One or two parameters can be varied at the same time

92 Emanuele Borgonovo 92 Quantitative Methods for Management Sensitivity Analysis and Parameter Importance Parameter importance: –Relevance of parameter in a model with respect to a certain criterion Sensitivity Analysis used to Determine Parameter Importance Concept of importance not formalized, but extensively used –Risk-Informed Decision Making –Resource allocation Need for a formal definition

93 Emanuele Borgonovo 93 Quantitative Methods for Management Process Identify how sensitivity analysis techniques work through analysis of several examples Formulate a definition Classify sensitivity analysis techniques accordingly

94 Emanuele Borgonovo 94 Quantitative Methods for Management Sensitivity Analysis Types Model Output: Local Sensitivity Analysis: – Determines model parameter (x i ) relevance with all the x i fixed at nominal value Global Sensitivity Analysis: –Determines x i relevance of x i s epistemic/uncertainty distribution

95 Emanuele Borgonovo 95 Quantitative Methods for Management the Differential Importance Measure Nominal Model output: –No uncertainty in the model parameters –and/or parameters fixed at nominal value Local Decomposition: Local importance measured by fraction of the differential attributable to each parameter

96 Emanuele Borgonovo 96 Quantitative Methods for Management Global Sensitivity Indices Uncertainty in U and parameters is considered Sobols decomposition theorem: SobolIndices

97 Emanuele Borgonovo 97 Quantitative Methods for Management Formal Definition of Sensitivity Analysis (SA) Techniques SA technique are Operators on U: x1x1 x2x2 xnxn I(x 1 )I(x n ) I(x 2 ) or

98 Emanuele Borgonovo 98 Quantitative Methods for Management Importance Relations Importance relations: –X the set of the model parameters; – Binary relation x i x j iff I(x i x j ) x i ~ x j iff I(x i x j ) x i x j iff I(x i x j ) Importance relations induced by importance measures are complete preorder

99 Emanuele Borgonovo 99 Quantitative Methods for Management Additivity Property In many situation decision-maker interested in joint importance: An Importance measure is additive if: DIM is additive always S i are additive iff f(x) additive and x j s are uncorrelated

100 Emanuele Borgonovo 100 Quantitative Methods for Management Techniques that fall under the definition of Local SA techniques

101 Emanuele Borgonovo 101 Quantitative Methods for Management Global Importance Measures

102 Emanuele Borgonovo 102 Quantitative Methods for Management Sensitivity Analysis in Risk-Informed Decision-Making and Regulation Risk Metric: x i is undesired event Probability Fussell-Vesely fractional Importance: Tells us on which events regulator has to focus attention

103 Emanuele Borgonovo 103 Quantitative Methods for Management Summary of the previous concepts Formal Definition of Sensitivity Analysis Techniques Definition of Importance Relations Definition enables to: –Formalize use of Sensitivity Analysis –Understand role of Sensitivity Analysis in Risk- informed Decision-making and in the use of model information

104 Emanuele Borgonovo 104 Quantitative Methods for Management Chapter V Uncertainty Analysis

105 Emanuele Borgonovo 105 Quantitative Methods for Management Uncertainty Analysis

106 Emanuele Borgonovo 106 Quantitative Methods for Management Summary Distinction between Aleatory Uncertainty ed Epistemic Uncertainty Epistemic Uncertainty and Bayes theorem Monte Carlo Method for uncertainty propagation

107 Emanuele Borgonovo 107 Quantitative Methods for Management Uncertainty Aleatory Uncertainty: –From Alea, die: Alea jacta est It refers to the realization of an event. –Example: the happening of an earthquake Epistemic Uncertainty: –From GreeK, knowledge it reflects our lack of knowledge in the value of the Aleatory Model input parameters. the aleatory model or model of the world is the model chosen to represent the random event.

108 Emanuele Borgonovo 108 Quantitative Methods for Management Example: Model of the World the Probability of Earthquakes is usually modeled through a Poisson model: that rappresents the Probability that the number of earthquakes between 0 and t is equal to n. the Poisson Distribution holds for independent events, in which next events (arrivals) are not influenced by previous events and the Probability of an event in a given interval of time is the same independently of the time where the interval is located the Model chosen to describe the arrivals of earthquakes is given the non-humble name of "model of the world" (MOW).

109 Emanuele Borgonovo 109 Quantitative Methods for Management Some useful information on Poisson Distributions the Poisson Probability that n events happen on 0-t is: the sum on n=0... of P(n,t) is, obviously, equal to 1. the Probability of k>N is given by: E[n]= t

110 Emanuele Borgonovo 110 Quantitative Methods for Management the Corresponding Epistemic Model Now,in spite of all the efforts and studies, it is unlikely that a scientist would tell you: the rate ( ) of arrivals of earthquakes is exactly xxx. More likely, he will indicate you a range where the true value of lies. For example cuold be between 1/5 and 1/50 (1/years). Suppose that the scientist state of knowledge on can be expressed by a uniform distribution u( ):

111 Emanuele Borgonovo 111 Quantitative Methods for Management Combining the Epistemic Model and MOW We have been dealing with two Models: MOW: the events happen according to a Poisson Distribution Epistemic Model: Uniform Uncertainty Distribution then, what is the Probability of having 1 earthquake in the next year? Answer: there is no unique Probability, but a p(n,t, ) for all values of. Thus, we have to write:

112 Emanuele Borgonovo 112 Quantitative Methods for Management …. This expression tells us that not necessarily all Poisson distributions weight the same. Thus: In our case: u( )=c; Hence, there is an expected Probability!

113 Emanuele Borgonovo 113 Quantitative Methods for Management In General the MOW will depend on m parameters,,…: the event Probability (P(t)) will be:

114 Emanuele Borgonovo 114 Quantitative Methods for Management An problem the failure time of a series of components is characterized by the exponential Probability function : From the available data, it emerges that: What is the mean time to failure?

115 Emanuele Borgonovo 115 Quantitative Methods for Management Solution E[t]=

116 Emanuele Borgonovo 116 Quantitative Methods for Management Continuous form of Bayes Theorem Epistemic Uncertainty and Bayes theorem are connected, in that we know that we can use evidence to update probabilities. For example, suppose to have a coin in your hands. will it be a fair with, i.e., will the Probability of tossing the coin lead to 50% head and tails?. How can we determine whether it is a fair coin? ….let us toss it….

117 Emanuele Borgonovo 117 Quantitative Methods for Management Formula the Probability density of a parameter, after having obtained evidence and, changes as follows: L(E ) = MOW likelihood 0 ( ) is the pdf of before the evidence, called Prior Distribution ( ) is the pdf of after the evidence, called Posterior Distribution

118 Emanuele Borgonovo 118 Quantitative Methods for Management From discrete to continuous Let us take Bayes theorem for discrete events: Let us go to continuous events: our purpose is to know the Probability that a parameter of the MOW distribution assumes a certain value, given a certain evidence Thus, event 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!!!

119 Emanuele Borgonovo 119 Quantitative Methods for Management From discrete to continuous the denominator in Bayes theorem expresses the sum of the probabilities of the evidence given all the possible states (the total Probability theorem). In the case of epistemic uncertainty these events are all possible values of. Thus: Substituting the various terms, one finds Bayes theorem for continuous random variables we have shown before

120 Emanuele Borgonovo 120 Quantitative Methods for Management Is it a fair coin? What is the MOW? It is a binomial distribution with parameter p: What is the value of p? Suppse we do not know anything about p. Let us assume a uniform prior distribution between 0 and 1: Let us get some evidence. At the first tossing it is head At the second tail At the third head

121 Emanuele Borgonovo 121 Quantitative Methods for Management Result First tossing –Evidence: h. –MOW: L(h p)=p –Prior: 0 Second tossing: –Evidence: t –MOW: L(t p)=(1-p) –Prior: 1 Third tossing: –Evidence: h –MOW: L(h p)=p –Prior: 2 Equivalently: –Evidence: h, t, h –L(hth p)=p 2 (1-p) –Prior: 0

122 Emanuele Borgonovo 122 Quantitative Methods for Management Graph

123 Emanuele Borgonovo 123 Quantitative Methods for Management Conjugate Distributions Likelihood –Poisson Posterior: gamma Prior distribution –Gamma with:

124 Emanuele Borgonovo 124 Quantitative Methods for Management Conjugate Distributions Likelihood – Normal Posterior: Normal Prior distribution: –Normal with:

125 Emanuele Borgonovo 125 Quantitative Methods for Management Conjugate Distributions Likelihood – Binomial Posterior, Beta: Prior: –Beta with:

126 Emanuele Borgonovo 126 Quantitative Methods for Management Summary of Conjugate Distributions MOW - LikelihoodPrior Distribution Posterior Distribution BinomialeBeta PoissonGamma Normal Gamma Negative binominalBeta

127 Emanuele Borgonovo 127 Quantitative Methods for Management Epistemic Uncertainty in Decision-Making Problems Investment: Suppose that P.up is characterized by a uniform pdf between 0.3 and 0.7 How does the decision changes? It is necessary to propagate the uncertainty in the model

128 Emanuele Borgonovo 128 Quantitative Methods for Management Analytical Propagation of Uncertainty It is the same problem of the MOW … Repeating for the other decisions and comparing the resulting mean values, one gets the optimal decision. Recall that:

129 Emanuele Borgonovo 129 Quantitative Methods for Management the Monte Carlo Method Sampling a value of P.up For all sampled P.up the Model is re-evaluated. Information: –Frequency of the preferred alternative –Distribution of each individual Alternative

130 Emanuele Borgonovo 130 Quantitative Methods for Management the core of Monte Carlo 1) Random Number Generator u between 0 and 1 2) Numbers u are generated with a uniform Distribution 3) Suppose that parameter is uncertain and characterized by the cumulative distribution reported below: 0 1 u

131 Emanuele Borgonovo 131 Quantitative Methods for Management Inversion theorem Inversion theorem: the values of sampled in this way have the Probability distribution from which we have inverted 1 0

132 Emanuele Borgonovo 132 Quantitative Methods for Management Example Let us evaluate the volume of the yellow solid through the Monte Carlo method. V V0V0

133 Emanuele Borgonovo 133 Quantitative Methods for Management Application to ID and DT For every Model parameter one creates the corresponding epistemic distribution Run nr. 1: One generates n random numbers between 0 and 1, as many as the uncertain variables are One samples the value of each of the parameters inverting from the corresponding distribution Using these values one evaluates the model One keeps record of the preferred alternative and of the value of the decision the procedure is repeated N times.

134 Emanuele Borgonovo 134 Quantitative Methods for Management Results Strategy Selection Frequency Decision Value Distribution

135 Emanuele Borgonovo 135 Quantitative Methods for Management Problem V-1 the mean time to failure of a set of components is characterized by an exponential distribution with parameter. Suppose that is described by a uniform epistemic distribution between 1/100 and 1/10. –Which is the MOW? Which is the epistemic model? –What is the mean time to failure? Suppose you registered the following failure times: t=15, 22, 25. –Update the epistemic distribution based on the new data –What is the new mean time to failure?

136 Emanuele Borgonovo 136 Quantitative Methods for Management Problem V-2: Investing We are again thinking of how to invest. Actually, we were not aware of the bayesian approach before. Thus we start using data about P_up in Bayesian way. After 15 working days we get the evidence: up,down, down,down,down,up,down,up,down,up,down,up,up,up. Assuming that each day is independent of the previous one: a) Which are the MOW and the epistemic model? b) What is the best decision without incorporating the evidence? c) What is the distribution of P_up after the evidence? d) What do you decide when the new information is incorporated in the model? Solution: –the MOW is the model of the events that accompany the decision. It is our ID or DT. More in specific, there is a second mode which is the one utilized for modeling the fact that the market can be up or down. This is a binomial distribution with parameter P_up. –the epistemic model is the set of the uncertainty distributions used to characterize the lack of knowledge in the model parameters. In this case, it is the distribution of P_up. We need to choose a prior distribution for P_up. We choose a uniform distribution between 0 and 1. b) We write the alternative payoffs as a function of P_up.

137 Emanuele Borgonovo 137 Quantitative Methods for Management Prob. 5-2 Substituting: E[U Risky ]=50, E[U Safe ]= 20, E[U Less Risky ]= 20

138 Emanuele Borgonovo 138 Quantitative Methods for Management Investment c) Let us use Bayess theorem to update the prior uniform distribution –evidence: up,down, down,down,down,up,down,up,down,up,down,up,up,up –L(E|P_up): –Prior: 0 uniform bewteen 0 and 1 Bayestheorem: Posterior Distribution E[p_up]=0.47 d) Posterior Decision: E[U Risky ]=23, E[U Safe ]= 20, E[U Less Risky ]= 9.2

139 Emanuele Borgonovo 139 Quantitative Methods for Management Problems Apply the one way, two way and Tornado Diagrams SA to the IDs and DTs of the previous chapters: Discuss your results

140 Emanuele Borgonovo 140 Quantitative Methods for Management Bayesian Decision You are the director of a library shop. To improve the sales, you are thinking of hiring additional sale personnel. This should, in your opinion, improve the service level in the shop. If this happens, you expect an increase in costumer number, and correspondingly, an increase in revenue sales. Suppose that the number of people entering the shop is, any day, distributed according to a Poisson distribution with uncertain. the prior distribution of is a gamma with mean equal to 55 and standard deviation equal to 15. the cost increase due to the hiring is 5000EUR for month. If the service quality improves and the library receives more than 50 customers per day, revenues increase would amount at 15000EUR (on the average). If less than 50 customers visit the shop, then revenues would not increase (and you loose the 5000EUR). What to you decide? You decide to monitor the number of customers on the next 6 working days: 75,45,30,80,72,41. You update the Probability. What do you decide now? How much do you expect to gain now? Perform a sensitivity analysis on the probabilities. What information do you get?

141 Emanuele Borgonovo 141 Quantitative Methods for Management Influence Diagram More than 50 Clienti P_50_up Less than 50 1-P_50_up Improves Service Pmigl Does not improve 1-Pmigl Invest Decision Not Invest Clienti=0 Servizio=0 0 P=0.1 Pmigl=0.5 P_500=0.5 P_500_down=0.5 P_500_up=0.7

142 Emanuele Borgonovo 142 Quantitative Methods for Management Chapter VI Introduction to Decision theory

143 Emanuele Borgonovo 143 Quantitative Methods for Management Summary Preferences under Certainty –Indifference Curves –the Value Function [V(x)]: properties –Preferential independence Preferences under Uncertainty –Axioms of rational choice –utility Function [U(x)] in one dimension –Risk Aversion Preferences with Multiple Objectives –Multi-attribute utility Function

144 Emanuele Borgonovo 144 Quantitative Methods for Management Preferences Under Certainty Example: you are choosing your first job. You select your attributes as: location (measured in distance from home), starting salary and career perspectives. You denote the attributes as x1, x2, x3. you have to select among five offers a1, a2,…,a5. Every offer gives you certain values of x1, x2, x3 for certain. How do you decide? It is a multi-attribute decision problem in the presence of certainty, since once you decide you will receive x1,x2,x3 for certain. In this case you have to establish how much of one attribute to forego to receive more of anoilr attribute.

145 Emanuele Borgonovo 145 Quantitative Methods for Management 1 Opction X1 2 X2 3 X3 4 X4 5 X5 X1=0.0 X2=0.0 X3=0.0 X4=0.0 X5=0.0 Preferences under Certainty Here is a diagram for the Choice

146 Emanuele Borgonovo 146 Quantitative Methods for Management Structuring Preferences Indifference Curves: Points on the same curve leave you indifference x1x1 x2x2

147 Emanuele Borgonovo 147 Quantitative Methods for Management the Value Function You can associate a numerical value representing you preferences to each indifference curve: V(x) is the function that says how much of x i one is willing to exchange for an increase or decrease in x k x1x1 x2x2

148 Emanuele Borgonovo 148 Quantitative Methods for Management V(x) V(x) is a value function if it satisfies the following properties: a) b)

149 Emanuele Borgonovo 149 Quantitative Methods for Management Example For the first job choice, suppose that you value function is as follows: where x1 measures the distance from home in 100km, x2 is the career perspective measured on a scale from 0 a 10 and x3 the starting salary in kEUR. Suppose to have received the following offers: –(1, 5, 20), (5, 4, 10), (8,3,60), (10, 5, 20), (10,2,40) Which one would you pick?

150 Emanuele Borgonovo 150 Quantitative Methods for Management Preferences under Uncertainty P11 U1 P12 U2 P13 U3 P14 U P41 U1 P42 U2 P43 U3 P44 U4 4 Decision Suppose one has to choose between lotteries that offer a mix the previous job offers: to choose one does not use the value function, but must resort to the utility function (U(x))

151 Emanuele Borgonovo 151 Quantitative Methods for Management utility Function the utility function is the appropriate one to express preferences over the distributions of the Attributes. Given two distributions 1 and 2 on the Consequences, Distribution 1 is more or as much desirable than Distribution 2 if and only if:

152 Emanuele Borgonovo 152 Quantitative Methods for Management Utility vs. Value –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: –To choose one need u(x1) and u(2) P1 X1 1-P1 X2 XI 1 2

153 Emanuele Borgonovo 153 Quantitative Methods for Management Stochastic Dominance x Probability distributions over x Distributions over attribute x 1 2 Distribution 1 is dominated by distribution 2, if obtaining more of x is preferable. Vice versa, if less of x is preferable, then Distribution 2 is dominated by distribution 1

154 Emanuele Borgonovo 154 Quantitative Methods for Management One Attribute Utility Functions

155 Emanuele Borgonovo 155 Quantitative Methods for Management Certainty Equivalent Given the lottery: the value of x such that you are indifferent between x* for certain and playing the lottery. In equations: N.B.: if you are risk neutral, then x*=E[x] P1 X1 1-P1 XN X3 1 2 P1 X1 1-P1 X2 X* 1 2

156 Emanuele Borgonovo 156 Quantitative Methods for Management definition of Risk Aversion a decision-maker is risk averse if preferisce sempre the expected value of a lottery alla lottery Hp: increasing utility function. Th: You are risk averse if the Certainty Equivalent of a lottery is always lower than the expected value of the lottery You are risk averse if and only if your utility function utility is concave (£20) £ £ 20 1 £10 2 £10; P = : £10

157 Emanuele Borgonovo 157 Quantitative Methods for Management Risk Premium and Insurance Premium the Risk Premium (RP) of a lottry is the difference between the expected value of the lottery and your Certainty Equivalent for the lottery: Intuitively, the Risk Premium is the quantity of attribute you are willing to forego to avoid the risks connected with the lottery. Suppose now that E[x]=0. the insurance premium is how much one would pay to avoid a lottery:

158 Emanuele Borgonovo 158 Quantitative Methods for Management Mailmatical Definition the Risk Aversion function is defined as: Or, equivalently: Supposing a constant risk aversion one gets an exponential utility function:

159 Emanuele Borgonovo 159 Quantitative Methods for Management Risk Preferences Constant Risk Aversion Compute constant through Certainty Equivalent (CE):

160 Emanuele Borgonovo 160 Quantitative Methods for Management Investment Results with Risk Aversion Market up Market exp(-200/70) = 1 Market Down exp(-(-160)/70) = -9 Blue Chip Stock Decision -3 Market up exp(-500/70) = 1 Market Down exp(-(-600/70)) = -5,278 -2,110 Bond=1 1-exp(-50/70) = 1; P = TwoStock prob_up=0.6 Risky Investment

161 Emanuele Borgonovo 161 Quantitative Methods for Management A quale value accetterei linvestimento rischioso

162 Emanuele Borgonovo 162 Quantitative Methods for Management Esempi of funzioni utility Linear: u=ax –Risk Properties: Risk Neutral Exponential: –Risk Properties: - sign: Constant Risk Aversion, + sign: Constant Risk Proneness Logarithmic: –Risk Properties: Decreasing Risk Aversion

163 Emanuele Borgonovo 163 Quantitative Methods for Management Problems

164 Emanuele Borgonovo 164 Quantitative Methods for Management problem VI-1 For the following three utility functions, compute: –the risk aversion function r(x) –the risk premium for 50/50 lotteries –the insurance premium

165 Emanuele Borgonovo 165 Quantitative Methods for Management Problem VI-2 Consider a 50/50 lottery. Determine your Risk Aversion constant, assuming an exponential utility function. Reexamine some of the problems discussed till now utilizing instead of the monetary payoff the corresponding exponential utility function with the constant determined above. How do the decisions change?

166 Emanuele Borgonovo 166 Quantitative Methods for Management Problem VI-3 You are analyzing some alternatives for your next vacations: –A guided tour through Italian cultural cities (Rome, Florence, Venice, Siena …an infinite list..), duration 10 days, cost 500EUR, for a total of 1500km by bus. –A journey to the Caribbean, lasting 1 week, cost 2000EUR, by plane. –15 days in a wonderful mountain in Trentino, for a cost of 2000EUR, with 500km of promenades. Do you need a utility or a value function to decide? Suppose that, after some thinking, you discover to have the following three attribute utility function: where x 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 and 10. What do you choose?

167 Emanuele Borgonovo 167 Quantitative Methods for Management Chapter VII the Logic of Failures

168 Emanuele Borgonovo 168 Quantitative Methods for Management Elements of Reliability theory

169 Emanuele Borgonovo 169 Quantitative Methods for Management Safety and Reliability Safety and Reliability study the performance of systems. Reliability and safety study cover two wide areas: –System Failures and Failure Modes Structure Function –Failure Probability Failure Data Analysis the approach can be static or dynamic. Static approach is analytically simpler and is more diffuse.

170 Emanuele Borgonovo 170 Quantitative Methods for Management Systems A system is a set of components connected through some logical relations with respect to operation and failure of the system More simple structures are: –Series –Parallel

171 Emanuele Borgonovo 171 Quantitative Methods for Management Series Every component is critical w.r.t. the system being able to perform its mission. the fault of just one component is sufficient to provoke system failure Redundancy: 0 12n

172 Emanuele Borgonovo 172 Quantitative Methods for Management Parallel Systems Each of the components is capable of assuring that the system accomplish its tasks. Thus, to provoke the failure of the system, all the components must be contemporarily failed Redundancy: n n In Out

173 Emanuele Borgonovo 173 Quantitative Methods for Management Elements of System Logics

174 Emanuele Borgonovo 174 Quantitative Methods for Management Boolean Logic An event (and) can be True or False State Variable or Indicator: Properties: –(X J ) n =Xj – where is the complementary of X J This simple definition enables one to use algebraic operations to describe the logical behavior of systems.

175 Emanuele Borgonovo 175 Quantitative Methods for Management Series Systems Let E i denote the event the i-th component failed. Let X T denote the event: the System failed. X T 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 From a logical p.o.v., we get the following expression: E1E2E3 E1E2E3

176 Emanuele Borgonovo 176 Quantitative Methods for Management Parallel Systems Let E i denote the event the i-th component has failed. Let X T 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 the logical expression is: E1E2E3

177 Emanuele Borgonovo 177 Quantitative Methods for Management the Structure Function In general, a system will be formed by a combination of series or parallel elements, or other logics (as we will see next). One defines the Structure Function of a system the logical expression that expresses the top event (X T ) as function of the individual failure events.

178 Emanuele Borgonovo 178 Quantitative Methods for Management the Logic of Performance 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 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.

179 Emanuele Borgonovo 179 Quantitative Methods for Management n/N Logics n/N logics are intermediate logics between series and parallel. N represents the total number of components in the system and n the number of components that must contemporarily fail to break the system. As an example, a system has a 2/3 logic if it has 3 components and when two components have failed the system failes In Out 2/3

180 Emanuele Borgonovo 180 Quantitative Methods for Management Example: 2/3 System Logics 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 = M 3. For X T to happen: Z 1 (M 1 M 2 M 3 ). the structure function expression is: Let us go to a level below: Let us solve the calculations, noting that (X i ) n =X i :

181 Emanuele Borgonovo 181 Quantitative Methods for Management Probability Sum Rules We recall that, for generic Events: We recall that, if the events are independent: Rare Event Approssimation: neglect all terms corresponding to multiple events

182 Emanuele Borgonovo 182 Quantitative Methods for Management Golden Rule In practice: the System Failure Probability is computed from the solved Structure Function, substituting to indicator X i the corresponding Event Probability.

183 Emanuele Borgonovo 183 Quantitative Methods for Management Proof 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 )

184 Emanuele Borgonovo 184 Quantitative Methods for Management Problems

185 Emanuele Borgonovo 185 Quantitative Methods for Management Problem VII-1 For the following systems compute: –the Structure Function for System Failure –the Structure Function for System Operation –the Failure Probability –the Operation Probability /

186 Emanuele Borgonovo 186 Quantitative Methods for Management Problem VII-2 for the following system: Compute the Failure Probability supposing independent events and denoting the component failure probability by p. Repeat the computation starting with the system success function, Y T. Verify that the two results coincide

187 Emanuele Borgonovo 187 Quantitative Methods for Management Chapter VIII Elements of Reliability

188 Emanuele Borgonovo 188 Quantitative Methods for Management Cut and Path sets Failure Logic By cut set one means an event/set of events whose happening causes system failure By minimal cut set one means a cut set that does not have other cut sets as subsets Success Logic By path set one means an event/set of events whose happening causes system to work By minimal path set one means a path set that does not have other path sets as subsets

189 Emanuele Borgonovo 189 Quantitative Methods for Management Even Trees Event Trees: represent the sequence of events that lead to the event top. Initiating Event Event 1Event 2Top Event Sì No

190 Emanuele Borgonovo 190 Quantitative Methods for Management Example One has to establish the sequence of events that lead to leakage of toxic chemicals from a production plant. High pressure in one of the pipes can cause a breach in the pipe itself, with leakage of toxic material in the room where the machine works. the filtering of the air conditioning could prevent the passage of the toxic gas to the outside of the room. A fault on the air circulation system due to air filter fault or maintenance error, would lead to the diffusion of the gas to the entire firm building. At this point, public safety would be protected only by the building air circulation system, last barrier for the gas going to the outsides. Draft the event tree for this sequence.

191 Emanuele Borgonovo 191 Quantitative Methods for Management Gas Leakage Fault - Tree No High Pressure PipeRoom Yes No BuildingTop Event

192 Emanuele Borgonovo 192 Quantitative Methods for Management Fault Trees Fault Trees: represent the logical connection among failures that lead to the failure of a barrier they are characterized by a set of logic symbols that connect a series of events Basic Event: is the event that represents the base of the fault-tree. From a physical point of view, it represents the failure of a component or of part of it. From a modeling point of view, it represents the lowest level of detail. AndOrevent Base

193 Emanuele Borgonovo 193 Quantitative Methods for Management Example Let us consider the failure of the aeration system. Suppose that the system is composed by two main parts: an suction engine and a static filter. the failure of the aeration system, thus, happens either due to engine failure or for filter fault. Aeration 1 Engine Static Filter

194 Emanuele Borgonovo 194 Quantitative Methods for Management Example We could however realize that the level of detail could be Further increased. In fact, we discover that the engine can brake for a failure of its mechanical components and, in particular, of the fan or for a fault of the electric feeder. the filter can break because of wrong installation after maintenance or for an intrinsic fault. the fault tree becomes as follows:

195 Emanuele Borgonovo 195 Quantitative Methods for Management Level II Aeration 1 EngineStatic filter FaultInstall. Elettr.Mech.

196 Emanuele Borgonovo 196 Quantitative Methods for Management Engine=A Static filter =B Electr.=1, Mech=2, Fault=3, Install.=4 What are the minimal cut sets? From Fault Trees to Structure Functions

197 Emanuele Borgonovo 197 Quantitative Methods for Management Rare Events Approximation 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:

198 Emanuele Borgonovo 198 Quantitative Methods for Management Event Trees & Fault Trees No High pressure PipeAeration 1 yes No Aeration 2Top Event Aeraz. 1 engine filter Fault Installaz.Electr.Mech.

199 Emanuele Borgonovo 199 Quantitative Methods for Management Probability of the Top Event From the Event Tree: Expanding: the conditional probabilities are found solving the corresponding Fault Trees

200 Emanuele Borgonovo 200 Quantitative Methods for Management Definitions

201 Emanuele Borgonovo 201 Quantitative Methods for Management Failure Density Given a system, tet us denote with the Probability that the system fails between t and t+dt It must hold that:

202 Emanuele Borgonovo 202 Quantitative Methods for Management Reliability 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: Thus the Reliability [R(t)] is related to the failure time pdf as follows: Note that, if f(t) is continuous:

203 Emanuele Borgonovo 203 Quantitative Methods for Management General Fault rate t (t) Infant Mortality Useful Life Aging

204 Emanuele Borgonovo 204 Quantitative Methods for Management Hazard/Failure Rate 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:

205 Emanuele Borgonovo 205 Quantitative Methods for Management Legami between R(t), f(t) and (t) From the above definition, there follows: Relationship R(t)- (t): Relationship f(t)- (t):

206 Emanuele Borgonovo 206 Quantitative Methods for Management time medio of failure (MTTF) The mean time to failure is defined as:

207 Emanuele Borgonovo 207 Quantitative Methods for Management System Reliability

208 Emanuele Borgonovo 208 Quantitative Methods for Management Reliability of systems in Series Series: if independence is assumed: Thus: Faulure rate:

209 Emanuele Borgonovo 209 Quantitative Methods for Management Reliability of systems in Parallelo Failure Probability of the system: if independent: Thus: Failure Rate:

210 Emanuele Borgonovo 210 Quantitative Methods for Management Reliability of Standby Systems 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: –where 2 indicates that there are two components in standby, while the subscript denotes the second component –2) Three components: –Thus:

211 Emanuele Borgonovo 211 Quantitative Methods for Management Standby Systems with const. failure rates For a standby system, it holds that: then P(t

212 Emanuele Borgonovo 212 Quantitative Methods for Management Failure on Demand 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

213 Emanuele Borgonovo 213 Quantitative Methods for Management Problems

214 Emanuele Borgonovo 214 Quantitative Methods for Management problem VIII-1 Write the Fault Trees for the following systems and derive the structure function: /

215 Emanuele Borgonovo 215 Quantitative Methods for Management Problema VIII-2 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. Lincidente si può evitare se il sistema di raffreddamento ausiliario interviene per tempo e se il sistema di spegnimento del reattore interviene con successo. Linsuccesso 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 -4 ], mancato arrivo del segnale per un guasto del sistema elettrico [P(E|alta press.)= ], mancato sganciamento delle barre per un guasto meccanico [P(Bar|alta press.)= ]. Il sistema di raffreddamento ausiliario è costituito da due pompe in parallelo, con rateo di guasto 1/10000 [1/h] e probabilita di guasto on demand di Le pompe devono funzionare per 100 ore affinche limpianto sia fuori pericolo. Determinare: –Lalbero degli eventi –Gli alberi dei guasti –La probabilità di fondere il reattore dato che si è verificato lincidente in un anno dato che la frequenza di eventi di alta pressione e per anno.

216 Emanuele Borgonovo 216 Quantitative Methods for Management Problema VIII-3 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?

217 Emanuele Borgonovo 217 Quantitative Methods for Management Problema VIII-4 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. Cè una probabiltà di 0.1 che il tempo sia molto caldo (Temperatura>30 gradi) durante lestate (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 ed E 2 3) Scrivere le probabilità: P(S), P(U), P(M) Suggerimenti: Utilizzate alberi degli eventi e dei guasti per determinare la funzione di struttura e poi passate alle probabilità

218 Emanuele Borgonovo 218 Quantitative Methods for Management Problema VIII-5: Distribuzione Weibull Dato un componente con rateo di guasto: 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.

219 Emanuele Borgonovo 219 Quantitative Methods for Management Problema VIII-6 Dato un componente con il tasso di guasto (t) seguente: calcolare: R(t), f(t), e il MTTF del componente t l(t)

220 Emanuele Borgonovo 220 Quantitative Methods for Management Problema VIII-7 Calcolare lespressione dellaffidabilità [R(t)] di un sistema k su n con rateo di guasto generico. Calcolare la stessa espressione con distribuzioni esponenziali

221 Emanuele Borgonovo 221 Quantitative Methods for Management Problema VIII-8 Calcolare laffidabilità 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

222 Emanuele Borgonovo 222 Quantitative Methods for Management Problema VIII-9 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?

223 Emanuele Borgonovo 223 Quantitative Methods for Management Problema VIII-10 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?

224 Emanuele Borgonovo 224 Quantitative Methods for Management Prob. VIII-2 Soluzione Rottura Primario Spegnimento No Si Raffreddamento Top Event Spegnimento Bar. Sof. El. Si Raffreddamento Pompa 1.Pompa 2 On Demand

225 Emanuele Borgonovo 225 Quantitative Methods for Management Prob. VIII-2 Soluzione Assumiamo eventi rari. La frequenza si calcola dalla combinazione degli eventi: dove: f rottura = per anno P(Spegn|rottura prim.)=P(Sof|rottura prim.)+P(E|rottura prim.)+P(Bar|rottura prim.)= P(Raff| rottura prim.)= = Quindi:

226 Emanuele Borgonovo 226 Quantitative Methods for Management Problema VIII-8 Soluzione Calcolare laffidabilità annuale di un sistema con 4 componenti in serie con ratei di guasto [1/h]: (1/6000, 1/8000, 1/10000, 1/5000). – –Ore in un anno: –Sostituendo i numeri: Confrontatela con quella di un sistema in cui i componenti sono messi in: –Parallelo: –¾ supponendo I ratei di guasto =1/8000. Risultato: 0.11 –2/4 supponendo I ratei di guasto =1/8000 Risultato: 0.41

227 Emanuele Borgonovo 227 Quantitative Methods for Management Problema VIII-9 Soluzione 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). –Il guadagno di affidabilita e dellordine 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?

228 Emanuele Borgonovo 228 Quantitative Methods for Management Capitolo IX Decisioni Operative: Ottimizzazione delle Manutenzioni

229 Emanuele Borgonovo 229 Quantitative Methods for Management Decisioni Operative Decisioni di Affidabilita o Reliability Design Decisioni di Optimal Replacement Decisioni di ispezione ottimale Decisioni di riparazione ottimale

230 Emanuele Borgonovo 230 Quantitative Methods for Management Indisponibilita 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: Indisponiblita media in T: Indisponibilta media limite:

231 Emanuele Borgonovo 231 Quantitative Methods for Management Disponibilita La disponibilita istantanea e il complementare della indisponibilita. Le altre definizioni seguono immediatamente Note: la disponibilita/indisponibilita non e una densita di probabilita e lindisponibilita 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

232 Emanuele Borgonovo 232 Quantitative Methods for Management Effetto delle manutenzioni

233 Emanuele Borgonovo 233 Quantitative Methods for Management Calcolo della Indisponibilita: un unico componente, una sola modalita di guasto Evoluzione temporale: A t=0 il sistema entra in funzione dopo la manutenzione. Dopo un tempo t= torna di nuovo in manutenzione. La manutenzione dura r. e il tempo in cui il componente e soggetto a rotture casuali con (t). Si nota che il problema e periodico, di periodo T= r + Durante il sistema ha una indisponibilita istantanea pari alla sua probabilita di rottura, se, come da ipotesi, non ci sono riparazioni: r t r

234 Emanuele Borgonovo 234 Quantitative Methods for Management Calcolo della Indisponibilita: un unico componente, una sola modalita di guasto Lindisponibilta istantanea risulta quindi: Da cui lind. Media: Supponiamo cost e 1. Quindi utilizziamo approssimaz. Taylor: Sostituiamo nella ind. Media, e assumiamo r << :

235 Emanuele Borgonovo 235 Quantitative Methods for Management Modi di Guasto 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.

236 Emanuele Borgonovo 236 Quantitative Methods for Management Indisponibilita istantanea con piu modi di guasto Consideriamo per un componente i modi di guasto indicati in precedenza. A t=0 il componente puo essere gia guasto se disabilitato dallerronea 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 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 lindisponibilita istantanea e:

237 Emanuele Borgonovo 237 Quantitative Methods for Management Indisponibilita media con piu modi di guasto Lindisponibilita media sullintervallo 0 + r e: Due assunzioni: 1) Eventi rari 2) + r

238 Emanuele Borgonovo 238 Quantitative Methods for Management Rappresentazione equivalente La funzione struttura e: X C =1- (1-X t ) (1-X Q0 )(1-X 0 )(1-X )= = X t +X Q0 + X 0 + X -termini di ordine superiore…. Approssimazione eventi rari: X C = X t +X Q0 + X 0 + X Componente 0 t Q0Q0

239 Emanuele Borgonovo 239 Quantitative Methods for Management Il caso di due componenti Sostituzioni successive Periodo: +2 r Sostituzioni distanziate r1 r2 r1 r r r r r +2 r Indisponibilita media e la somma di piu termini: R: random, C common cause, D demand e M maintenance

240 Emanuele Borgonovo 240 Quantitative Methods for Management Modi di guasto 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 dellaffidabilita dei sistemi

241 Emanuele Borgonovo 241 Quantitative Methods for Management Modelli decisionali corrispondenti Come stabilire una politica di replacement ottimale? Costruzione della funzione obiettivo –i) Individuazione del Criterio –ii)Costruzione della funzione obiettivo o utilita –iii) Ottimizzazione

242 Emanuele Borgonovo 242 Quantitative Methods for Management Esempio 1 1 componente soggetto replacement periodico e manutenzione periodica Criterio = disponibilita media Funzione obiettivo: q( ) ottimale: – r =24 h, =1/10000 (1/h) ott =700hr Con =1/ (1/h) ott =2200hr

243 Emanuele Borgonovo 243 Quantitative Methods for Management Esempio 2 Ottimizzazione in considerazione del costo di sostituzione e della disponibilita Funzione obiettivo: ottimale: Occorre introdurre vita delimpianto L. Si ha: dove c 0 e il costo unitario di riparazione

244 Emanuele Borgonovo 244 Quantitative Methods for Management Esempio 2 Introduciamo poi il costo della indisponibilita: definito come multiplo del costo singola riparazione. Funzione energia: Intervallo ottimale:

245 Emanuele Borgonovo 245 Quantitative Methods for Management Esempio 2

246 Emanuele Borgonovo 246 Quantitative Methods for Management Applicazione del modello 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. 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

247 Emanuele Borgonovo 247 Quantitative Methods for Management Riparazioni

248 Emanuele Borgonovo 248 Quantitative Methods for Management Il tasso di riparazione (t) 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 allarrivo delle parti di ricambio o il tempo richiesto dallesecuzione 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: 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.

249 Emanuele Borgonovo 249 Quantitative Methods for Management Esempio 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 Nellesempio. 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 ce failure on demand

250 Emanuele Borgonovo 250 Quantitative Methods for Management Assunzioni 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

251 Emanuele Borgonovo 251 Quantitative Methods for Management Matrice di transizione Indichiamo con P ij la probabilita che il sistema passi dallo stato i allo stato j Proprieta: 1) 2)Se allora stato i e detto assorbente

252 Emanuele Borgonovo 252 Quantitative Methods for Management Esempio Applichiamo uno schema a stati per il sistema in standby. Otteniamo: P 23 P 32 P 21 P 12 P 13

253 Emanuele Borgonovo 253 Quantitative Methods for Management Equazioni di Markov/Kolmogorov Dove A e la matrice di transizione del sistema, P e il vettore delle probabilita degli stati del sistema.

254 Emanuele Borgonovo 254 Quantitative Methods for Management Costruzione della matrice di transizione 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 (+), si esce con tasso (-). Quindi: 12 P 21 P 12 12

255 Emanuele Borgonovo 255 Quantitative Methods for Management La matrice di transizione Analogamente: Quindi: La matrice di transizione e: Il sistema di equazioni differenziali diventa:

256 Emanuele Borgonovo 256 Quantitative Methods for Management Disponibilita asintotica e media 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:

257 Emanuele Borgonovo 257 Quantitative Methods for Management Risultati per un componente Disponibilita istantanea: Disponibilita asintotica: Interpretazione: tempo che occorre in media alla riparazione diviso il tempo totale Disponibilita media su T:

258 Emanuele Borgonovo 258 Quantitative Methods for Management Problema IX-1 Calcolare l indisponibilita media di un componente in standby soggetto a sostituzione periodica con le seguenti probabilita di guasto per =5000: (Soluzione: q=.175) Calcolare lintervallo di sostituzione ottimale e lindisponibilitacorrispondente, con L=70000, a=10 e a=. (Soluzione: =14500, q=0.5; =849, q=0.06)


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