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### Non-probability decision rules

Dr. Yan Liu

Department of Biomedical, Industrial & Human Factors Engineering

Wright State University

Types of Decision Making Environment

Decision maker knows with certainty the consequences of every alternative or decision choice (Non-probability decision making)

Decision maker can assign the probabilities of the various outcomes (decision making under risk)

Decision maker can neither predict nor describe the probabilities of the various outcomes (decision making under uncertainty)

2

Types of Non-Probabilistic Decision Rules

Lexicographic Ordering

Satisficing

Maxmax Payoff

Maximin Payoff

Minimax Regret

Laplace

Hurwitz Principle

3

Desirable Properties of Decision Rules

Transitivity

If alternative A is preferred to alternative B and alternative B is preferred to alternative C, then alternative A is preferred to alternative C

Column Linearity

The preference relation between two alternatives is unchanged if a constant is added to all entries of a column (value) of the decision table

Addition/Deletion of Alternatives

The preference relation between two alternatives is unchanged if another alternative is added/deleted from the decision table

Addition/Deletion of Identical Columns

The preference relation between two alternatives is unchanged if a column with the same value in all alternatives is added/deleted to the decision table

4

Lexicographic Ordering

V1≥V2≥ ∙∙∙≥Vn, n values are ordered in order of importance

Compare different decision alternatives on the most important value, and continue until one alternative is the best

C > A > B

Non-exhaustive comparisons and can be efficient when there are many values

5

Satisficing/Minimum Aspiration Level

Select any alternative which satisfies the minimum aspiration levels (the minimum acceptable criteria) of all values

May not be optimal because not all alternatives will be considered as long as one satisfactory alternative is found

6

Maximax Payoff

Select the alternative which results in the maximum of maximum payoffs; an optimistic criterion

Payoff Table

Maximum Payoff

$1,000

$10,000

$5,000

$8,000

B > D > C > A

7

Maximum Payoff

$10,000

$10,000

$9,000

$8,000

A =B > C > D

Maximax payoff violatescolumn linearity

8

Maximum Payoff

$8,000

$10,000

$8,000

$8,000

B > A = C = D

Maximax payoff violatesaddition/deletion of identical columns

9

Maximin Payoff

Select the alternative which results in the maximum of minimum payoffs; a pessimistic criterion

Payoff Table

Minimum Payoff

$1,000

-$7,000

$0

-$2,000

A > C > D > B

Maximin payoff violatescolumn linearity and addition/deletion of identical columns

10

Minmax Regret

Select the alternative which results in the minimum of maximum regret.

Regret is the difference between the maximum payoff possible for a specific outcome and the payoff actually obtained when a specific alternative is chosen and that outcome is encountered

Regret Table

Payoff Table

Maximum Regret

$9,000

$8,000

$5,000

$3,000

D > C > B > A

11

Payoff Table

Maximum Regret

$9,000

$11,000

$5,000

$6,000

$11,000

C > D > A > B

Minmax regret violates addition/deletion of alternatives

12

Laplace

Calculate the average of each alternative by assuming that the outcomes are equally likely to occur, and select the alternative with the largest average

Payoff Table

Average

$1,000

$1,166.7

$1,933.3

$2,233.3

13

Hurwicz Principle

Select the alternative that has the largest weighted average of its maximum and minimum payoffs; the weight of the maximum payoff is , referred to as the coefficient of optimism, and the weight of the minimum payoff is 1-

- if =1, then Hurwicz criterion is the same as Maxmax payoff
- if =0, then Hurwicz criterion is the same as Maxmin payoff

Payoff Table

=0.4

Hurwicz Score

$1,000

10,000*0.4+(-7,000)*0.6 = - $200

5,000*0.4+0*0.6 = $2,000

8,000*0.4+(-2,000)*0.6 = $2,000

14

Hurwicz Scores of Alternatives with Respect to α

A: Hurwicz score = 1000

B: Hurwicz score = 10000∙α + (-7000)∙(1-α) = 17000α-7000

C: Hurwicz score = 5000∙α + 0∙(1-α) = 5000α

D: Hurwicz score = 8000∙α + (-2000) ∙(1-α) = 10000α-2000

Hurwicz score = Max. payoff ∙α + Min. payoff ∙(1-α)

15

α=0.2

α=0.4

When 0≤α<0.2, A is the best alternative

When 0.2≤α≤0.4, C is the best alternative

When 0.4≤α≤5/7, D is the best alternative

When α>5/7, B is the best alternative

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