The probability values of various states of nature, discussed so far, were prior probabilities. Such probabilities are either computed from the past data or assigned subjectively. It is possible to revise these probabilities in the light of current information available by using the Bayes' Theorem. The revised probabilities are known as posterior probabilities.
Example : A manufacturer of detergent soap must determine whether or not to expand his productive capacity. His profit per month, however, depend upon the potential demand for his product which may turn out to be high or low. His payoff matrix is given below:
Do not Expand Expand
High Demand Rs 5,000 Rs 7,500
Low Demand Rs 5,000 Rs 2,100
On the basis of past experience, he has estimated the probability that demand for his product being high in future is only 0.4 Before taking a decision, he also conducts a market survey. From the past experience he knows that when the demand has been high, such a survey had predicted it correctly only 60% of the times and when the demand has been low, the survey predicted it correctly only 80% of the times.
If the current survey predicts that the demand of his product is going to be high in future, determine whether the manufacturer should increase his production capacity or not? What would have been his decision in the absence of survey?
Let H be the event that the demand will be high. Therefore,P(H) = 0.4 and P(H) = 0.6
Note that H andH
are the only two states of nature.Let D be the event that the survey predicts high demand. Therefore,P(D/ H) = 0.60 and P(D/H) = 0.80
We have to findP(H/D) and P(H /D).
For this, we make the following table:
From the above table, we can writeP(H/D)= 0.24/0.36= 2/3 and P(H/D)= 0.12/0.36= 1/3.
The EMV of the act 'don't expand' = 5000*2/3+5000*1/3= RS 5,000and the EMV of the act 'expand'
=7500*2/3+2100*1/3= RS 5,700
Since the EMV of the act 'expand' > the EMV of the act 'don't expand', the manufacturer should expand his production capacity.
It can be shown that, in the absence of survey the EMV of the act 'don't expand' is Rs 5,000 and the EMV of the act expand is Rs 4,260. Hence, the optimal act is 'don't expand'.
Decision Tree Approach
The decision tree diagrams are often used to understand and solve a decision problem. Using such diagrams, it is possible to describe the sequence of actions and chance events. A decision node is represented by a square and various action branches stem from it. Similarly, a chance node is represented by a circle and various event branches stem from it. Various steps in the construction of a decision tree can be summarised as follows:
(i) Show the appropriate action-event sequence beginning from left to right of the page.
(ii) Write the probabilities of various events along their respective branches stemming from each chance node.
(iii) Write the payoffs at the end of each of the right-most branch.
(iv) Moving backward, from right to left, compute EMV of each chance node, wherever encountered. Enter this EMV in the chance node. When a decision node is encountered, choose the action branch having the highest EMV. Enter this EMV in the decision node and cutoff the other action branches.
Following this approach, we can describe the decision problem of the above example as given below:
Case I: When the survey predicts that the demand is going to be high
Thus, the optimal act to expand capacity.
Case II: In the absence of survey
Thus, the optimal act is not to expand capacity.
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