Engineering Risk Benefit Analysis 1.155, 2.943, 3.577, 6.938, 10.816, 13.621, 16.862, 22.82, ESD.72, ESD.721 DA 1. The Multistage Decision Model George E. Apostolakis Massachusetts Institute of Technology Spring 2007 DA 1. The Multistage Decision Model 1 Why decision analysis? A structured way for ranking decision options by: 1. Enumerating the immediate and later choices available to the DM 2. Characterizing the relevant uncertainties 3. Quantifying the relative desirability of outcomes 4. Providing rules for ranking the decision options, thus helping the DM to select the “best” one. DA 1. The Multistage Decision Model 2 Value of formal analysis • Provides a systematic way to process large amounts of information • Decision making process is explicit and enhances communication • Provides formal rules for quantifying preferences DA 1. The Multistage Decision Model 3 Limitations of DA • 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 DA 1. The Multistage Decision Model 4 Manufacturing Example • Decision: 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 DA 1. The Multistage Decision Model 5 Manufacturing Example Payoffs • Earnings (payoffs): L1: $150,000, old product, P[L1/O] = 1.0 L2: $300,000, new product and the market is strong, P[s] = P[L2/N] = 0.3 L3: $100,000, new product and the market is mild, P[m] = P[L3/N] = 0.5 L4: -$100,000, new product and the market is weak, P[w] = P[L4/N] = 0.2 DA 1. The Multistage Decision Model 6 Building the decision tree Decision Options Payoffs L2 $300K N Payoff depends on market L3 $100K L4 -$100K O L1 $150K DA 1. The Multistage Decision Model 7 Decision Tree Decision Options States of Nature Payoffs L2 $300K P[s]= 0.3 P[m]= 0.5 L3 $100K N P[w]=0.2 O L4 -$100K P[s ∪ m ∪ w ] = 1 L1 $150K DA 1. The Multistage Decision Model 8 Non-Probabilistic Decision Rules • Maximin Rule: N: Choose option with the largest smallest payoff (risk averse DM). -$100 O: $150 Choose O • Maximax Rule: N: Choose option with the largest payoff (risk taker). $300 O: $150 Choose N DA 1. The Multistage Decision Model 9 Probabilistic Decision 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). • Decision Rule: Maximize the expected monetary value (EMV) of the earnings (payoffs). • In the decision tree, work from right to left and compute expectations. DA 1. The Multistage Decision Model 10 Calculation of the EMV EMV[N] = 0.3x300 + 0.5x100 + 0.2x(-100) = $120K EMV[O] = 1.0x150 = $150K Option O has the largest EMV, therefore it should be chosen. DA 1. The Multistage Decision Model 11 Calculation of the EMV (cont’d) L2 $300 s, $90 m, $50 L3 $100 EMV[N]=$120 w, -$20 L4 -$100 EMV[O]=$150 $150 L1 $150 Best Decision: O DA 1. The Multistage Decision Model 12 A New Decision • The DM 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): ¾ The initial decision of whether to buy the survey ¾ The terminal decision of whether to market the new product. DA 1. The Multistage Decision Model 13 The survey results can be: Strong P(s/L2) = 0.8 P(s/L3) = 0.2 P(s/L4) = 0.0 Mild P(m/L2) = 0.2 P(m/L3) = 0.6 P(m/L4) = 0.3 Weak P(w/L2) = 0.0 1.0 P(w/L3) = 0.2 1.0 P(w/L4) = 0.7 1.0 DA 1. The Multistage Decision Model 14 The new decision tree *Entries in earnings column do not yet take account of the cost of the survey. Purchase C survey Survey response "strong" Survey response "mild" Make old product Product earns* C 300,000 D Make new C product Make old product 150,000 C D 100,000 -100,000 150,000 300,000 Make new C product 100,000 -100,000 D Survey response "weak" Do not purchase survey C Make old product C D No new information D is obtained 150,000 300,000 Make new C product Make old product 100,000 -100,000 C 150,000 300,000 Make new C product 100,000 -100,000 DA 1. The Multistage Decision Model Figure by MIT OCW. 15 New inputs • The earnings must be reduced by the survey cost of $20K: L1 = $130K, L2 = $280K, L3 = $80K, L4 = -$120K • The probabilities of the states of nature (the probabilities of earnings) must also be updated to reflect the survey findings. DA 1. The Multistage Decision Model 16 Bayes’ Theorem (Slide 16, RPRA 2) • The mutually exclusive and exhaustive states are: L2, L3, and L4 • Evidence: “Survey result is strong” P( L j / s ) = P (s / L j )P ( L j ) 4 ∑ P(s / L j )P( L j ) 2 j = 2, 3,4 DA 1. The Multistage Decision Model 17 Calculations for “survey result is s” Payoff Prior Likelihood Prob. L2 L3 L4 0.3 0.5 0.2 1.0 P(s/ L2)=0.8 P(s/ L3)=0.2 P(s/ L4)=0.0 Product 0.24 0.10 0.00 0.34 DA 1. The Multistage Decision Model Posterior Probability P(L2/s)=0.706 P(L3/s)=0.294 P(L4/s)=0.000 1.000 18 Results for m and w P(L2/m) = 0.143 P(L2/w) = 0.000 P(L3/m) = 0.714 P(L3/w) = 0.417 P(L4/m) = 0.143 1.000 P(L4/w) = 0.583 1.000 DA 1. The Multistage Decision Model 19 The updated decision tree s O C 1 L1 N L2 C .706 .294 0 O C 1 L1 N C .143 .714 .143 L2 L3 O C 1 L1 N C 0 .417 .583 L2 L3 O C 1 L1 150,000 L2 C .3 .5 .2 300,000 100,000 -100,000 D .34 S C .42 m D .24 D w S C 1 D D N 130,000 280,000 L3 80,000 L4 -120,000 130,000 280,000 80,000 L4 -120,000 130,000 280,000 80,000 L4 -120,000 L3 L4 Figure by MIT OCW. DA 1. The Multistage Decision Model 20 Optimal terminal decisions 1. Solve “backwards in time.” 2. Determine the best solution at every terminal node, conditional on the DM’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. DA 1. The Multistage Decision Model 21 Decision tree solution 130,000 221,300 s D O C 221,300 N C .34 130,000 C . 42 m D O . 24 C 80,000 N C 130,000 130,000 D w D O C -36,600 N C 150,000 150,000 S L1 130,000 .706 .294 0 L2 L3 L4 280,000 80,000 -120,000 1 L1 130,000 .143 .714 .143 L2 L3 1 L1 0 .417 .583 L2 L3 1 L1 .3 .5 .2 L2 130,000 161,008 S 1 C 1 O 150,000 D C 120,000 N C 280,000 80,000 L4 -120,000 130,000 280,000 80,000 L4 -120,000 150,000 300,000 L3 100,000 L4 -100,000 Figure by MIT OCW. DA 1. The Multistage Decision Model 22 Best decision 1. Assume that the DM 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 Do not buy survey: EMV = $150,000 4. Best initial decision: Buy survey DA 1. The Multistage Decision Model 23 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) DA 1. The Multistage Decision Model 24 General form of a decision tree G1 P(H1 |C1 ) H1 C1 P(C 1 ) D1 D2 P(C 2 ) P(C n ) C2 G2 Gk Cn D3 Dm GENERAL FORM OF A DECISION TREE DA 1. The Multistage Decision Model 25 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 DM must select one of the initial acts Aj. 3. The Aj 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 a1, a2, …, of the initial decision, the DM must select the next decision B1, B2, … DA 1. The Multistage Decision Model 26 The multistage decision model (cont’d) 5. Finally, a terminal chance node, b1, b2,…, occurs. 6. As a result of the initial decision Aw, its chance outcome ax, the terminal decision By, and its chance outcome bz, the total consequence C(Aw, ax, By, bz) 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. DA 1. The Multistage Decision Model 27