Engineering Risk Benefit Analysis
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1 Engineering Risk Benefit Analysis 1.155, 2.943, 3.577, 6.938, , , , 22.82, ES.72, ES.721 A 1. The Multistage ecision Model George E. Apostolakis Massachusetts Institute of Technology Spring 2007 A 1. The Multistage ecision Model 1
2 Why decision analysis? A structured way for ranking decision options by: 1. Enumerating the immediate and later choices available to the M 2. haracterizing the relevant uncertainties 3. Quantifying the relative desirability of outcomes 4. Providing rules for ranking the decision options, thus helping the M to select the best one. A 1. The Multistage ecision Model 2
3 Value of formal analysis Provides a systematic way to process large amounts of information ecision making process is explicit and enhances communication Provides formal rules for quantifying preferences A 1. The Multistage ecision Model 3
4 Limitations of A The theory is for an individual decision maker. This reduces considerably its applicability in practice. (But, great normative tool.) In most cases there is no satisfactory way to combine the utility function of several people As with all formal analysis, the results are no better than the quality of the model and its supporting assessments The required inputs may not be easily obtainable A 1. The Multistage ecision Model 4
5 Manufacturing Example ecision: To continue producing old product (O) or convert to a new product (N). The payoffs depend on the market conditions: s: strong market for the new product m: mild market for the new product w: weak market for the new product A 1. The Multistage ecision Model 5
6 Manufacturing Example Payoffs Earnings (payoffs): L 1 : $150,000, old product, P[L 1 /O] = 1.0 L 2 : $300,000, new product and the market is strong, P[s] = P[L 2 /N] = 0.3 L 3 : $100,000, new product and the market is mild, P[m] = P[L 3 /N] = 0.5 L 4 : -$100,000, new product and the market is weak, P[w] = P[L 4 /N] = 0.2 A 1. The Multistage ecision Model 6
7 Building the decision tree ecision Options N Payoff depends on market Payoffs L 2 $300K L 3 $100K L 4 -$100K O L 1 $150K A 1. The Multistage ecision Model 7
8 ecision Tree ecision Options N States of Nature P[s]= 0.3 P[m]= 0.5 P[w]=0.2 Payoffs L 2 $300K L 3 $100K L 4 -$100K O P [s m w] = 1 L 1 $150K A 1. The Multistage ecision Model 8
9 Non-Probabilistic ecision Rules Maximin Rule: hoose option with the largest smallest payoff (risk averse M). N: -$100 O: $150 hoose O Maximax Rule: hoose option with the largest payoff (risk taker). N: $300 O: $150 hoose N A 1. The Multistage ecision Model 9
10 Probabilistic ecision Rule The maximin and maximax rules are incomplete because they ignore uncertainties. We include probabilities by taking expected values of the payoffs (slide 31, RPRA 2). ecision Rule: Maximize the expected monetary value (EMV) of the earnings (payoffs). In the decision tree, work from right to left and compute expectations. A 1. The Multistage ecision Model 10
11 alculation of the EMV EMV[N] = 0.3x x x(-100) = $120K EMV[O] = 1.0x150 = $150K Option O has the largest EMV, therefore it should be chosen. A 1. The Multistage ecision Model 11
12 alculation of the EMV (cont d) EMV[N]=$120 s, $90 w, -$20 m, $50 L 2 $300 L 3 $100 L 4 -$100 EMV[O]=$150 $150 L 1 $150 Best ecision: O A 1. The Multistage ecision Model 12
13 A New ecision The M considers the possibility of commissioning a survey to be able to better judge the future market. The survey costs $20,000. There are now two decisions (multistage model): The initial decision of whether to buy the survey The terminal decision of whether to market the new product. A 1. The Multistage ecision Model 13
14 Strong The survey results can be: P(s/L 2 ) = 0.8 P(s/L 3 ) = 0.2 P(s/L 4 ) = 0.0 Mild P(m/L 2 ) = 0.2 P(m/L 3 ) = 0.6 P(m/L 4 ) = 0.3 Weak P(w/L 2 ) = 0.0 P(w/L 3 ) = 0.2 P(w/L 4 ) = A 1. The Multistage ecision Model 14
15 The new decision tree *Entries in earnings column do not yet take account of the cost of the survey. Survey response "strong" Make old product Make new product Product earns* 150, , , ,000 Purchase survey Survey response "mild" Make old product Make new product 150, , , ,000 Survey response "weak" Make old product Make new product 150, , ,000 o not purchase survey No new information is obtained Make old product Make new product -100, , , , ,000 A 1. The Multistage ecision Model Figure by MIT OW. 15
16 New inputs The earnings must be reduced by the survey cost of $20K: L 1 = $130K, L 2 = $280K, L 3 = $80K, L 4 = -$120K The probabilities of the states of nature (the probabilities of earnings) must also be updated to reflect the survey findings. A 1. The Multistage ecision Model 16
17 Bayes Theorem (Slide 16, RPRA 2) The mutually exclusive and exhaustive states are: L 2, L 3, and L 4 Evidence: Survey result is strong P(L j / s) = P(s / L 4 2 j P(s / L )P(L j j )P(L ) j ) j = 2,3,4 A 1. The Multistage ecision Model 17
18 alculations for survey result is s Payoff Prior Likelihood Product Posterior Prob. Probability L P(s/ L 2 )= P(L 2 /s)=0.706 L P(s/ L 3 )= P(L 3 /s)=0.294 L P(s/ L 4 )= P(L 4 /s)= A 1. The Multistage ecision Model 18
19 Results for m and w P(L 2 /m) = P(L 2 /w) = P(L 3 /m) = P(L 3 /w) = P(L 4 /m) = P(L 4 /w) = A 1. The Multistage ecision Model 19
20 The updated decision tree.34 s O N L 1 L 2 L 3 L 4 130, ,000 80, ,000 S m O N L 1 L 2 L 3 L 4 130, ,000 80, ,000 w O N L 1 L 2 L 3 L 4 130, ,000 80, ,000 S 1 O N L 1 L 2 L 3 L 4 150, , , ,000 A 1. The Multistage ecision Model Figure by MIT OW. 20
21 Optimal terminal decisions 1. Solve backwards in time. 2. etermine the best solution at every terminal node, conditional on the M s being there. 3. Find the EMV for each terminal node and the decision option that maximizes the EMV. 4. An arrow indicates the best decision for each terminal node. A 1. The Multistage ecision Model 21
22 ecision tree solution ,300 s 130,000 O 221,300 N L 1 L 2 L 3 L 4 130, ,000 80, , ,008 S ,000 S 1 130,000 m 130,000 w 150, ,000 O 80,000 N 130,000 O -36,600 N 150,000 O 120,000 N L 1 L 2 L 3 L 4 L 1 L 2 L 3 L 4 L 1 L 2 L 3 L 4 130, ,000 80, , , ,000 80, , , , , ,000 Figure by MIT OW. A 1. The Multistage ecision Model 22
23 Best decision 1. Assume that the M makes the best decision at each terminal node, if it is reached. 2. Find the EMV for the initial node. 3. Buy survey: EMV = $161,008 o not buy survey: EMV = $150, Best initial decision: Buy survey A 1. The Multistage ecision Model 23
24 Best terminal decision If survey result is strong Market new product (EMV = $221,200) If survey result is mild Market old product (EMV = $130,000) If survey result is weak Market old product (EMV = $130,000) A 1. The Multistage ecision Model 24
25 General form of a decision tree G 1 P(H 1 1 ) H P( 1 ) P( 2 ) P( n ) 2 n G 2 G k 3 m GENERAL FORM OF A EISION TREE A 1. The Multistage ecision Model 25
26 The multistage decision model 1. Each stage consists of a decision node followed by a chance node for each of the decision options available in this stage. 2. The M must select one of the initial acts A j. 3. The A j may be viewed as learning experiments providing, at specified costs, opportunities for obtaining partial or complete information about present uncertainties. 4. Following the probabilistic results a 1, a 2,, of the initial decision, the M must select the next decision B 1, B 2, A 1. The Multistage ecision Model 26
27 The multistage decision model (cont d) 5. Finally, a terminal chance node, b 1, b 2,, occurs. 6. As a result of the initial decision A w, its chance outcome a x, the terminal decision B y, and its chance outcome b z, the total consequence (A w, a x, B y, b z ) is obtained. 7. The solution proceeds sequentially backwards in time, i.e., we first identify the best decision at each terminal node, then the best decision one stage earlier assuming that, whatever chance event results from this stage, it will be followed by the best of the available terminal decisions. A 1. The Multistage ecision Model 27
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