MGS 3100 Business Analysis. Chapter 8 Decision Analysis II. Construct tdecision i Tree. Example: Newsboy. Decision Tree

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1 MGS 3100 Business Analysis Chapter 8 Decision Analysis II Decision Tree An Alternative e (Graphical) Way to Represent and Solve Decision Problems Under Risk Particularly l Useful lfor Sequential Decisions Two Basic Symbols (square) = decision or choice node (circle) = chance or event node Decision Analysis 2 Construct tdecision i Tree General Procedure Start with a decision node followed by several branches representing decision alternatives Each alternative branch leads to a decision node or a chance node which is followed by several branches representing possible states of nature Repeat above steps as necessary until all scenarios have been considered Put all probabilities and payoffs on the tree Decision Analysis 3 Example: Newsboy Decision Tree Decision Analysis 4

2 Solve Decision Tree Backward Approach Folding Back At each chance node (circle), calculate the ER (sum of values times probabilities) and write it above (or below) the node At each decision node (square), find the maximum ER and write it above the node, and then cut off all decision branches except the optimal one (the one with the highest ER) Repeat above steps until the left most node is reached Decision Analysis 5 Example: Newsboy Solution 22.5 =MAX(-85,-12.5,22.5,7.5) =(0) + (-50) + 04(-100) (-100) + (-150) Decision Analysis 6 Ui Using Decision i Tree for Sequential Decisions Sequential Decision Problems A sequence of decisions s with later ae decision(s) depending on the results of previous decision(s) Solution Process Construct the tree following time and/or logical sequence, from left to right Analyze the tree by working backward, from right to left Incorporating New Information How Much Worth Is New Information? Expected Value of Sample (Imperfect) Information EVSI measures the maximum worth or value of sample information that we would like to pay for in order to improve our decisions EVSI = max ER with sample information max ER without sample information Decision Analysis 7 Decision Analysis 8

3 Incorporating New Information Bayes Theorem A systematic way of revising probabilities as new information becomes available Basic Terminology Prior Probability: P(A) initial knowledge about an outcome A prior to obtaining i relevant information i Likelihood: P(B A) information about B given A, which is usually obtained from historical data Posterior Probability: P(A B) our modified knowledge about A after taking observed information B into account Example: Prior and Posterior Probability Roll a die Possible outcomes: Let event A =, what is the probability of A? P(A) = 1/6 Prior Probability Now if we have observed the event B = an odd number, what is the probability of A? P(A B) = 1/3 Posterior Probability Decision Analysis 9 Decision Analysis 10 Example: Prior and Posterior Probability Medical Diagnosis i A = a medical problem P(A) = Prior probability B = some symptoms or test results P(A B) = Posterior probability bbili What is P(B A)? Likelihood or Reliability B New information Decision Analysis 11 Bayes Theorem A well known result from the probability theory that can help us find the posterior probabilities If we consider two events A1 = having heart problem A2 = don t have heart problem and also some new information B = EKG test result positive Then according to the Bayes Theorem, PA ( 1 and B) PA ( 1) PB ( A1) PA ( 1 B) = = PB ( ) PA ( ) PB ( A) + PA ( ) PB ( A) P(A1 and B) P(A1 B) joint probability P(B) marginal probability Decision Analysis 12

4 How to Apply Bayes Theorem Computing Posterior Probability bilit Organize (list) input data: Prior probability: P(A 1 ), P(A 2 ),, P(A m ) Likelihood: P(B j A i ), i = 1, 2,, m; j = 1, 2,, n Calculate joint and marginal probability Joint probability = Prior probability Likelihood Marginal probability = Sum of joint probabilities Cl Calculate l posterior probability bbili Joint probability divided by marginal probability Joint & Marginal Probability Table A 1 A 2 A m Marginal B 1 P(A 1 B 1 ) = P(B 1 A 1 ) P(A 1 ) P(A 2 B 1 ) = P(B 1 A 2 ) P(A 2 ) P(A m B 1 ) P(B 1 ) = sum of the 1st row B 2 P(A 1 B 2 )= P(A 2 B 2 )= P(A m B 2 ) P(B 2 ) = sum of P(B 2 A 1 ) P(A 1 ) P(B 2 A 2 ) P(A 2 ) the 2nd row B n P(A 1 B n ) P(A 2 B n ) P(A m B n ) P(B n ) = sum of the nth row Decision Analysis 13 Decision Analysis 14 A B C D E F G 1 Computing Posterior Probability (Genreal Layout) 2 3 Prior probability 4 Event A1 A2 Am Checksum 5 Probability P(A1) P(A2) P(Am) Likelihood 8 9 A1 A2 Am 10 B1 P(B1 A1) P(B1 A2) P(B1 Am) 11 B2 P(B2 A1) P(B2 A2) P(B2 Am) Bn P(Bn A1) P(Bn A2) P(Bn Am) 14 Checksum Joint probability A1 A2 Am Marginal 19 B1 =P(B1 A1)*P(A1) ( ) =P(B1 A2)*P(A2) ( ) =P(B1 Am)*P(Am) ( ) =SUM(this row) 20 B2 =P(B2 A1)*P(A2) =P(B2 A2)*P(A2) =P(B2 Am)*P(Am) =SUM(this row) Bn =P(Bn A1)*P(A1) =P(Bn A2)*P(A2) =P(Bn Am)*P(Am) =SUM(this row) Posterior probability A1 A2 Am Checksum 28 B1 P(A1 B1) = P(A1B1)/P(B1) P(A2 B1) = P(A2B1)/P(B1) P(Am B1) = P(AmB1)/P(B1) 1 29 B2 P(A1 B2) = P(A1B2)/P(B2) P(A2 B2) = P(A2B2)/P(B2) P(Am B2) = P(AmB2)/P(B2) Bn 32 P(A1 Bn) = P(A1Bn)/P(Bn) P(A2 Bn) = P(A2Bn)/P(Bn) P(Am Bn) = P(AmBn)/P(Bn) 1 15 Example: South Mountain A B C D E F G 1 Posterior Probability - South Mountain Power Company 2 3 Prior probability 4 Dmd High Dmd Low 5 P(DH) P(DL) 6 7 Likelihood P(Pos DH) 8 Dmd High Dmd Low 9 Positive Negative 0.8 P(Neg DL) Joint probability 13 Dmd High Dmd Low Marginal 14 Positive 5 5 P(Pos) 15 Negative P(DH Pos) P Posterior probability 19 Dmd High Dmd Low Chksum 20 Positive Negative P(DL Neg) P(DH Pos) Decision Analysis 16

5 Some Important End Notes For a decision i analysis problem, one important tstep is to identify which decision making situation it fits in: Under Uncertainty or Under Risk If Under Ignorance, choose one of the criteria: Maximax, Maximin, i LaPlace You will not use all of them in a REAL situation Which one should you use? Depends on If Under Risk, then max ER criterion is typically used Some Important End Notes In DecisionMaking Under Risk, identify how many decisions in the situation and what they are. If only one decision, this is a single stage problem Solve the problem with a Payoff Table or a Decision Tree If multiple decisions, this is a multi stage problem Solve the problem with a Decision i Tree Solve Problems with Decision Tree Build the Tree (with Time & Logical sequence) Solve the Tree (with Backward approach) Decision Analysis 17 Decision Analysis 18