EVPI helps to determine the worth of an insider who possesses perfect information. The expected value with perfect information is the amount of profit foregone due to uncertain conditions affecting the selection of a course of action. Given the perfect information, a decision-maker is supposed to know which particular state of nature will be in effect. Thus, the procedure for the selection of an optimal course of action, for the decision problem given in example 18, will be as follows:
If the decision-maker is certain that the state of nature S1 will be in effect, he would select the course of action A3, having maximum payoff equal to Rs 200.
Similarly, if the decision-maker is certain that the state of nature S2 will be in effect, his course of action would be A1 and if he is certain that the state of nature S3 will be in effect, his course of action would be A2. The maximum payoffs associated with the actions are Rs 200 and Rs 600 respectively.
The weighted average of these payoffs with weights equal to the probabilities of respective states of nature is termed as Expected Payoff under Certainty (EPC).
Thus, EPC = 200 * 0.3 + 200 * 0.4 + 600 *0.3 = 320
The difference between EPC and EMV of optimal action is the amount of profit foregone due to uncertainty and is equal to EVPI.
Thus, EVPI = EPC - EMV of optimal action = 320 - 194 = 126
It is interesting to note that EVPI is also equal to EOL of the optimal action.
Cost of Uncertainty
This concept is similar to the concept of EVPI. Cost of uncertainty is the difference between the EOL of optimal action and the EOL under perfect information.
Given the perfect information, the decision-maker would select an action with minimum opportunity loss under each state of nature. Since minimum opportunity loss under each state of nature is zero, therefore,
EOL under certainty = 0 *0.3 + 0 *0.4 + 0 * 0.3 = 0
Thus, the cost of uncertainty = EOL of optimal action = EVPI
The expected values of perfect information solved examples are given below
Example : A group of students raise money each year by selling souvenirs outside the stadium of a cricket match between teams A and B. They can buy any of three different types of souvenirs from a supplier. Their sales are mostly dependent on which team wins the match. A conditional payoff (in Rs.) table is as under:
Type of Souvenir I II III
Team A wins 1200 800 300
Team B Wins 250 700 1100
(i) Construct the opportunity loss table.
(ii) Which type of souvenir should the students buy if the probability of team A's winning is 0.6?
(iii) Compute the cost of uncertainty.
(i) The Opportunity Loss Table
(ii) EOL of buying type I Souvenir = 0 * 0.6 + 850 * 0.4 = 340
EOL of buying type II Souvenir = 400 *0.6 + 400 * 0.4 = 400.
EOL of buying type III Souvenir = 900 *0.6 + 0* 0.4 = 540.
Since the EOL of buying Type I Souvenir is minimum, the optimal decision is to buy Type I Souvenir.
(iii) Cost of uncertainty = EOL of optimal action = Rs. 340
Example : The following is the information concerning a product X :
(i) Per unit profit is Rs 3.
(ii) Salvage loss per unit is Rs 2.
(iii) Demand recorded over 300 days is as under:
Units demanded : 5 6 7 8 9
No. of days : 30 60 90 75 45
Find : (i) EMV of optimal order.
(ii) Expected profit presuming certainty of demand.
(i) The given data can be rewritten in terms of relative frequencies, as shown below:
Units demanded: 5 6 7 8 9
No of days : 0.1 0.2 0.3 0.25 0.15
From the above probability distribution, it is obvious that the optimum order would lie between and including 5 to 9.
Let A denote the number of units ordered and D denote the number of units demanded per day.
If D ≥A, profit per day = 3A, and if D < A, profit per day = 3D – 2(A – D) = 5D – 2A.
Thus, the profit matrix can be written as
From the above table, we note that the maximum EMV = 19.00, which corresponds to the order of 7 or 8 units. Since the order of the 8th unit adds nothing to the EMV, i.e., marginal EMV is zero, therefore, order of 8 units per day is optimal.
(ii) Expected profit under certainty
= (5* 0.10+ 6*0.20+ 7 * 0.30+8* 0.25+ 9 * 0.15)* 3 = Rs 21.45
Alternative Method: The work of computations of EMV's, in the above example, can be reduced considerably by the use of the concept of expected marginal profit. Let p be the marginal profit and l be the marginal loss of ordering an additional unit of the product. Then, the expected marginal profit of ordering the Ath unit, is givenby
=Π .P(D ≥ A) -λ.P(D < A) =Π .P(D ≥ A) -λ. [1 - Π(D ≥ A)]
= (Π + λ ).P (D ≥ A) - λ .... (1)
The computations of EMV, for alternative possible values of A, are shown in the following table :
In our example, Π = 3 and λ = 2
Thus, the expression for the expected marginal profit of the Ath unit
= (3 + 2)P(D ≥ A)- 2 = 5P(D ≥ A)- 2.
Since the expected marginal profit (EMP) of the 8th unit is zero, therefore, optimal order is 8 units.
Marginal analysis is used when the number of states of nature is considerably large. Using this analysis, it is possible to locate the optimal course of action without the computation of EMV's of various actions. An order of A units is said to be optimal if the expected marginal profit of the Ath unit is non-negative and the expected marginal profit of the (A + 1)th unit is negative. Using equation (1), we can write
(Π+λ)P(D ≥A)-λ≥ 0 and .... (2)
(Π+λ)P(D ≥A +λ)-λ < 0 .... (3)
From equation (2), we get
Writing the probability distribution, given in example 20, in the form of less than type cumulative probabilities which is also known as the distribution function F(D), we get
Units demanded(D) : 5 6 7 8 9
F(D) : 0.1 0.3 0.6 0.85 1.00
We are given Π= 3 and λ = 2
∴ Π/ Π+λ= 3/5 = 0.6
Since the next cumulative probability, i.e., 0.85, corresponds to 8 units, hence, the optimal order is 8 units.
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Probability Distribution Of A Random Variable
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