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Problems (Decision Making Under Uncertainty)(1) The given matrix shows the payoff (Profit in Rs.) of differ-

ent strategies (alternatives). A 1, A2 & A3 against states ofnature (conditions) E 1, E 2, E 3 and E 4.

Indicate the decision taken under the following approach:

(i) Laplace Criterion (ii) Maximin Criterion

(iii) Maximax Criterion (iv) Regret Criterion and

(v) Hurvicz Criterion, the degree of optimism being 0.7.

Solution-

(i) Laplace Criterion: Since the probabilities of the states of nature are not known, we assume that they areequal. In this case we assume that each of the four states of nature has probability 1/4 of occurrence. Thusthe expected pay off (profits) corresponding to each of the three strategies is as follows

Strategy A1 : D(4000 - 100 + 6000 + 18000) = 6975

Strategy A2 : D(20000 + 5000 + 400 + 0) = 6350

Strategy A3 : D(20000 + 15000 - 2000 + 1000) = 8500

(ii) Maximin Criterion: For the given pay off matrix the minimum assured payoff for each alternative are

A1: -100, A 2: 400, A 3: -2000

Since the maximum of these minimum payoffs is 400, thealternative A 2 is selected according to the Maximin Criterion.

(ii) Maximax Criterion: For the given pay off matrix the Maximumassured payoff for each alternative are

A1: 18000, A 2: 20000, A 3: 20000

Since the maximum of these maximum payoffs is 20000, anyone of the alternatives A 2 & A3 can be selected according to the Maximax Criterion.

(ii) Regret Criterion: When event E 1 Occurs, the regretpayoffs for A 1, A2 & A3 are obtained by subtractingthe each payoff from the maximum payoff i.e. 20000;When event E 2 Occurs, the regret payoffs for A 1,A2 & A3 are obtained by subtracting the each payofffrom the maximum payoff i.e. 15000; When eventE3 Occurs, the regret payoffs for A 1, A2 & A3 areobtained by subtracting the each payoff from themaximum payoff i.e. 6000 & When event E 4 Occurs, the regret payoffs for A 1, A2 & A3 are obtained bysubtracting the each payoff from the maximum payoff i.e. 18000.

Since the alternative A 1 corresponds to the minimum of the maximum regrets, the decision maker wouldchoose A

1.

(ii) Hurvicz Criterion: As shown in the table given, we calculate the payoff values corresponding to each of thealternatives.

Since the payoff corresponding to the alternative A 2 is maximum, the decision maker would choose A 2.

(Strategies) Alternative

States of Nature A1 A2 A3

E1 4000 20000 20000

E2 -100 5000 15000E3 6000 400 -2000En 18000 0 1000

(Strateg ies) AlternativesStates of Nature

A1 A2 A3

E1 4000 20000 20000E2 -100 5000 15000E3 6000 400 -2000En 18000 0 1000

Column Minimum -100 400 -2000Column M aximum 18000 20000 20000

(Strategies) Alternatives Regret PayoffsStates ofNature

A1 A2 A3 A1 A2 A3

E1 4000 20000 20000* 16000 0 0E2 -100 5000 15000* 15100 10000 0E3 6000* 400 -2000 0 5600 8000En 18000* 0 1000 0 18000 17000

Maximum Regret 16000 18000 17000

(Strategies) AlternativesStates of Nature

A1 A 2 A 3 E 1 4000 20000 20000E 2 -100 5000 15000E 3 6000 400 -2000E n 18000 0 1000

(i) M axim um Payoff 18000 20000 20000(i i) M inimum Payoff -100 400 -2000

H = 0.7 X (i) + 0.3 X (ii) 12570 14000 13400

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