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Chapter 1 - Elements of Decision Theory

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Regardless of the decision problem, there are some elements that are common to all the problems. i) An objective to be achieved The objective depends on the type of problem on which a decision has to be made, e.g. the ideal inventory level, reducing the downtime of a piece of machinery or maximizing profits. ii) Course of action This is the available alternative from which the decision must be taken. Uncertainty arises due to uncontrollable factors associated with the states of nature. v) Pay-off Also known as conditional profit or conditional economic consequence, a Pay-off is a calculable measure of the benefit or value of an action and it represents the net gain from different combinations of alternatives and events. Then the ( ,i j)th element aij of the table is the conditional profit associated with the ith event and the jth alternative.

It is the difference between the maximum benefit and the benefit of the chosen action. Depending on the information available, the decision environment may be one of the following types: He knows that at the end of one year, the share prices of the three companies would be Rs.

Example 1: A person wants to invest in one of three investment plans: stocks, bonds or a savings account. A moderate change in the composition of the existing product with a new packaging at a moderately increased price P2. A very small change in the composition of the existing product with a new packaging at a slightly higher price P3.

The following table gives the payouts in terms of annual profit from each of the strategies:

Table 1.1: Conditional pay-off matrix  Courses of action States of Nature
Table 1.1: Conditional pay-off matrix Courses of action States of Nature

Telecommunications 5.5

Decision making under risk

A decision should therefore be made which will maximize the benefits in the long run subject to the neutral occurrence of the events. Choosing a decision with the largest expected value or pay-off is a strategy that will maximize benefits in the long run. Each payoff is assigned a probability, which can be chosen subjectively depending on the decision maker or can be calculated based on past data or experience.

The expected value of an action is then the weighted sum of the conditional payoffs, where the weights are the corresponding probabilities. Due to the perishable nature of the product, the unsold cakes ultimately yield him nothing. Sol: If D denotes the demand for pies and S denotes the supply, then the conditional payout function for the baker is given by .

Now we will get the expected payoff of each possible decision, which is the sum of the products of each conditional outcome and its probability. If he bakes 22 cakes a day, this would earn him an expected daily payment of Rs. It should be noted that no other number of cakes would pay him more in the long run than 22 cakes a day.

If his supply falls short of the demand, the result is a cash loss of Rs. Note: It can be noted that when adding up the respective elements of the contingent payouts and contingent losses tables, we get the maximum payout associated with that event i.e. the contingent loss is the difference between the best payout and the with that decision related payout on which contingent loss is calculated. This loss arises because he does not have the demand in advance.

Now suppose that the unsold cakes are not only thrown away at the end of the day, but can be sold the next day, albeit at a reduced price, i.e. the cakes have a reserve value. This is the reality of most products and most products have a scrap value. For a given cake salvage value, the decision to bake 23 units per day is the optimal decision.

The optimal strategy changed due to the fact that the conditional profits increased by the salvage value of the pie and the expected losses decreased. Thus, the optimal strategy depends on the extent to which expected losses can be covered by the residual value of the product.

Table 1.27: Conditional pay-off
Table 1.27: Conditional pay-off

When the product has more than one salvage value

According to the decision rule, an additional unit should be reserved as long as the probability of selling it is greater than p. Thus, to justify another unit of cake, the cumulative probability of its sale must be at least 0.57. So the baker must bake 22 units of cake to realize the maximum profit.

If, for each event, we calculate p MP and (1-p) ML, then we get the following table. Now, consider the case where the baker may realize a salvage value in the unsold cake.

Sequential decision-making

If he has chosen (a) first and he is running the business successfully, then he can expand his business and choose (b) also;. iii). If he has chosen (b) initially and he is running the business successfully, then he can expand his business and choose (a) as well. The initial investment in both options is Rs, which can be financed at an EMI of Rs.

The businessman must choose one of these options according to the following four states of nature (i) Both (a) and (b) are successful ( )a b.

Continuous random variable – Use of normal distribution

Construct a contingent loss table from the above data. a) What is the cost of uncertainty and the expected value of perfect information. Since these investments must be made in the future, the company anticipates different market conditions, expressed in terms of states of nature. Further, there is an additional fine of Rs. 500 per report for not meeting demand.

An investor is given the following investment options and percentage rates of return Table 1.45. Due to the nature of vaccines, all unused vials must be discarded at the end of the week. Using marginal analysis, determine the number of bottles to be purchased per week if the physician spends Rs.

The company has a policy of inspecting every piece before it is shipped to retailers. The company has defined five inspection categories according to the percentage of defects contained in each lot. The first alternative will cost Rs. 6,00,000 while the second alternative will cost the company Rs. 10 for each defective item returned.

In case the machine fails and there are no spare parts, the company's cost of repairing the plant would be Rs. The optimal number of part units based on (i) the Minimax principle;. iv) Criterion of expected monetary value. She is considering two investment alternatives. i) Plant expansion at an estimated cost of Rs.

The company believes that demand will be high or moderate during the payback period. Based on the analysis, ALPHA consultants predict whether demand will be high or moderate. The cost of hiring ALPHA consultants is Rs. i) Based on the above information, determine the optimal decision for the company to make; ii) Find out if it is advisable to use ALPHA consultants and if so, will the optimal decision made in part (i) change?

Figure

Table 1.1: Conditional pay-off matrix  Courses of action States of Nature
Table 1.2: Conditional opportunity loss matrix
Table 1.5: Conditional pay-off matrix
Table 1.8: Pay-off matrix Strategies States of
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References

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