Lecture 3 S11 (1)

Lecture 3 S11 (1) - Lecture 3 Source: /1 / D.R.Anderson,...

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Lecture 3 Source: /1 / D.R.Anderson, D.J.Sweeney, T.A.Williams. Quantitative Methods for Business, South- Western College Publishing, 11-th edition. Chapters 4, 5. Contents: 3.1 Decision making with probabilities 3.2 Utility and decision making 3.3 Utility: other considerations 3.1 Decision making with probabilities In many decision-making situations, we can obtain probability assessments for the states of nature. When such probabilities are available, we can use the expected value approach to identify the best decision alternative. Let us first define the expected value of a decision al- ternative and then apply it to the PDC problem. Let N = the number of states of nature , P ( s j ) = the probability of state of nature s j . Because one and only one of the N states of nature can occur, the probabilities must satisfy two conditions: P ( s j ) > 0 for all states of nature (3.1) 1 ) ( ) ( ) ( ) ( 2 1 1 = + + + = = N N j j s P s P s P s P (3.2) The expected value (EV) of decision alternative d i is defined as follows: ) ( ) ( ) ( 1 , j N j i j i s d V s P d EV = = (3.3) Let us return to the PDC problem to see how the expected value approach can be applied. Example (3.1) of the expected value approach in PDC problem (Lecture 2) Let the a priori probability for s 1 (strong demand) be 0.8 and for s 2 (week demand) – 0.2, i.e., 8 . 0 ) ( 1 = s P and 2 . 0 ) ( 2 = s P . Using the payoff values in Table 2.1 and equation (3.3) , we compute the expected value for each of the three decision alternatives as follows: 8 . 7 ) 7 ( 2 . 0 ) 8 ( 8 . 0 ) ( 1 = + = d EV 2 . 12 ) 5 ( 2 . 0 ) 14 ( 8 . 0 ) ( 2 = + = d EV 2 . 14 ) 9 ( 2 . 0 ) 20 ( 8 . 0 ) ( 3 = - + = d EV Thus, using the expected value approach, we find that the large condominium complex, with an expected value of $14.2 million, is the recommended decision. The calculations required to identify the decision alternative with the best expected value can be conveniently carried out on a decision tree. Figure 3.1 shows the decision tree for the PDC problem with state-of-nature branch probabilities. Working backward through the decision tree, we first compute the expected value at each chance node. That is, at each chance node, we weight each possible payoff by its probability of occurrence. By doing so, we obtain the expected values for nodes 2, 3, and 4, Because the decision maker controls the branch leaving decision node 1 .and because we are trying to maximize the expected profit, the best decision alternative at node 1 is d 3 . Thus, the decision tree analysis leads to a recommendation of d 3 with an expected value of $14.2
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million. Note that this recommendation is also obtained with the expected value approach in conjunction with the payoff table. Other decision problems may be substantially more complex than the PDC problem, but if a
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This note was uploaded on 06/07/2011 for the course ECON 3 taught by Professor Dd during the Spring '11 term at Alvin CC.

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Lecture 3 S11 (1) - Lecture 3 Source: /1 / D.R.Anderson,...

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