Decision Analysis and Markov Chains: Mathematical Models

Regret and Expected Value in Decision Making

Regret: This represents lost opportunities, defined as the difference between the best and the worst alternative of a decision. It involves maximizing the highest and the lowest minimum (taking the greatest of odds within the columns).

Expected Value: This is the amount of benefits for each alternative weighted decision, where the weight is the probability (often represented in a tree criteria).

Decision Matrix: Optimistic, Pessimistic, and Hurwicz

Example:

• B: -2, 5, 8 | Optimistic: 8 | Pessimistic: -2 | Hurwicz: (a)(8) + (1-a)(-2)
• M: -5, 10, 12 | Optimistic: 12 | Pessimistic: -5 | Hurwicz: (a)(12) + (1-a)(-5)
• A: -8, 6, 15 | Optimistic: 15 | Pessimistic: -8 | Hurwicz: (a)(15) + (1-a)(-8) (more)

Minimax Regret Penalty and Probabilistic Criterion

Minimax Regret Penalty: Calculated as the maximum of each column minus the specific column value.

• FSG Minimum Penalty (Max): 0, 5, 7 (Max: 7)
• Probabilistic (more): 3, 0, 3 (Max: 3) Minimum
• Calculation: 6, 4, 0 (Max: 6)

Probabilistic Criterion: This allows for the incorporation of knowledge regarding the relative probability of each outcome.

Expected Value of Perfect Information (EVPI)

Expected Value of Information: This is calculated as the Expected Profit with Perfect Information minus the Gain Expected without Perfect Information.

Maximum value of each decision (row): -2, 10, 15. Gain with perfect information = (%)(-2) + (%)(10) + (%)(15).

Expected Value: It determines if it is worth obtaining perfect information to increase profit. It provides an estimate of how expensive it is to "know the future." This process involves:

1. Design and execution of market research.
2. Review of the probabilities of different outcomes based on the results of market investigation.
3. Identification of the optimal decision using the revised probabilities.

Markov Chains and Transition Probabilities

Markov Chains: These are useful for studying the evolution of a system through different tests. They analyze successive time periods where the status or outcome of the system at a particular period cannot be determined accurately.

Transition Probabilities: These describe the manner in which the system changes from one period to another.

Chains with Stationary Transition Probabilities

• Finite number of states.
• Probabilities remain constant over time.
• The likelihood of a period depends on the previous period.

Hypothesis: The probability of each state depends only on the current state and not on how that state was reached (known as the Memoryless Property).

Formulas:
P1 = P21 / (P12 + P21)
P2 = P12 / (P12 + P21)

Bayes Theorem and Conditional Probability

Bayes Theorem:
P(B | A) = [P(B) * P(A | B)] / P(A)
Where P(A) = Σ [P(Bi) * P(A | Bi)]