Decision Theory

Expected Value of a Decision Alternative The expected value of a decision alternative is the sum of weighted payoffs for the decision alternative. The expected value (E) of decision alternative di is defined as: where: N = the number of states of nature P(sj) = the probability of state of nature sj Vij = the payoff corresponding to decision alternative di and state of nature sj

Example ( 1 ) : Marketing Strategy If data are available for the quantity sold and the corresponding probability in one of the companies shown in the following table: Find the expected value of the units sold ? 14 12 10 Quantity 0.2 0.5 0.3 Probability

E ( q ) = ∑ p* q = 10 * 0.3 + 12 * 0.5 + 14* 0.2 = 3 + 6 + 2.8 = 11 . 8

Example ( 2) If data are available for the quantity sold and the corresponding probability in one of the companies shown in the following table: The purchase price 20 pounds and the selling price 30 pounds Find the quantity to be maintained ? 12 11 10 Quantity 0.3 0.5 0.2 Probability

Profit = selling price - purchase price 20 = 30 - 20 = 10 0.3 0.5 0.2 12 11 10 100 110 80 120 90 60

The quantity to be maintained is 11 E ( q ) = ∑ p * q E ( 10 ) = 100 * 0.2 + 100 * 0.5 + 100 * 0.3 = 100 E ( 11) = 80 * 0.2 + 110 * 0.5 + 110 * 0.3 = 16 + 55 + 33 = 104 E ( 12) = 60 * 0.2 + 90 * 0.5 + 120 * 0.3 = 12 + 45 + 36 = 93 The quantity to be maintained is 11

Example ( 3) If data are available for the quantity sold and the corresponding probability in one of the companies shown in the following table: The purchase price 20 pounds and the selling price 30 pounds Find the quantity to be maintained ? 14 12 10 Quantity 0.3 0.5 0.2 Probability

Profit = selling price - purchase price 20 = 30 - 20 = 10 0.3 0.5 0.2 24 12 10 100 120 60 140 80 20

The quantity to be maintained is 12 E ( q ) = ∑ p * q E ( 10 ) = 100 * 0.2 + 100 * 0.5 + 100 * 0.3 = 100 E ( 12) = 60* 0.2 + 120 * 0.5 + 120 * 0.3 = 12+ 60 + 36 = 108 E ( 14) = 20 * 0.2 + 80* 0.5 + 140 * 0.3 = 4 + 40 + 42 = 86 The quantity to be maintained is 12