Statistical Analysis Decision Analysis updated 9.11.01

1. SMU EMIS 5300/7300 NTU SY-521-N Systems Analysis Methods Dr. Jerrell T. Stracener, SAE Fellow Statistical Analysis Decision Analysis updated 9.11.01

2. Example Misty is in charge of a Trust’s investment department. She had just been authorized to invest a large sum of money in one (and only one) of three alternatives: corporate bonds, common stocks, or certificates of deposit (time deposits). The Trust’s objective is to maximize the yield on the investment over a one-year period. The problem is that the economic situation is uncertain and no one is able to predict the exact movements of the stock or even the bond markets. Misty’s researchers informed her that they expected the economy to be in one of three states: growth, stagnation or inflation. The researchers estimated

3. Example continued a 50% chance for growth, a 30% chance of stagnation, and a 20% chance for inflation over the next year. Past experience indicated the following trends: 1. If there is growth, bonds will yield 12%, stocks will yield 15% and time deposits 6.5%. 2. If stagnation prevails, bonds will yield 6%, stocks 3% and time deposits 6.5%. 3. If there is inflation, bonds will yield 3%, stocks will drop 2% and time deposits will yield 6.5%. How should Misty maximize the investment yield over the next year?

4. Important Characteristics of a Decision • A decision is made at an instant in time • A decision is made based of the information available at the time that it is made.

5. Decision Analysis • Decision analysis is the analysis of choices in a particular situation. • Decision analysis can be defined in many ways • Decision analysis is the systematic evaluation of specific alternative choices • Decision analysis is a term used to describe body of knowledge and professional practice for the logical illumination of decision-making solutions. • Decision analysis is a normative, rather descriptive, approach. It does not attempt to describe a decision process. It shows how a decision maker subscribing to a particular set of logical rules would make decisions in order to maximize attainment of his or her objectives. • Decision analysis provides philosophical foundations and logical quantitative procedures for decision making. • Decision analysis provides a basis for documentation of the state of information at the time a decision is made and a basis for communication with others.

6. Decision Making Process • Establish Need or Requirement • Define Problem • Postulate Alternatives • Evaluate Alternatives • Select “Best” Alternative • Implement • Monitor • Revise

7. Decision Analysis with Decision Tables

8. Use of Decision Tables • The quantitative data of many decision situationscan be arranged in a standardized tabular form • known as a decision table (or a payoff table). • The object is to enable a systematic analysis of • the problem. • Many concepts used in decision tables are • common to all decision situations.

9. Use of Decision Tables • Decision tables typically contain 4 elements: • The alternative courses of action (decision • variables) • The states of nature (uncontrollable variables) • The probabilities of the states of nature • (uncontrollable variables) • The payoffs (result or outcome variables)

10. Use of Decision Tables • States of nature: • At the top of the table, the possible ‘states of • nature’ (also called or possible futures) are listed. • They are generally labeled s1, s2, …, sm. • A state of nature can be a state of economy (inflation), a weather condition (rain), a political • development (election of a certain candidate) or • other situation which the decision maker cannot • control.

11. Use of Decision Tables • Probabilities of the states of nature: • A question may be asked: ‘What is the likelihood • of these states of nature occurring? • Whenever it is possible to answer this question in • terms of explicit chances (or probabilities), the • information is recorded at the top of the table. • The probabilities are given either in percent or in • percentage fractions

12. Use of Decision Tables • Since it is assumed that one and only one of the • given states of nature will occur in the future, then • the sum of the probabilities must always be one, • i.e., • p1 + p2 + … + pm = 1 • where • p1 = probability of s1 occurring, • p2 = probability of s2 occurring, and so on.

13. The Payoffs • The payoff (or the outcome) associated with a • certain alternative and a specific state that is • given in that cell within the body of the table • located at the intersection of the alternative in • question (given by a row) and the specific state • of nature (given by a column). • The payoff is designated by oij where i indicates • the row and j the column • The payoffs can be thought of as conditional • since a specific payoff results from a specific state • of nature occurring but after a certain alternative • course of action has been taken.

14. General Structure of a Decision Table

15. Decision Making Under Certainty • In decision making under certainty, it is assumed • that complete information is available so that the • decision maker knows exactly what the outcome of • each course of action will be. • The decision maker thus becomes a perfect • predictor of the future.

16. Decision Making Under Risk • A decision under risk (also known as a • probabilistic or stochastic decision situation) is • one in which the decision maker must consider • several possible states of nature, each with a • given probability of occurrence. • Thus, in risk situations, it is assumed that the • long-run probabilities of occurrence of the given • states of nature (and their conditional outcomes) • are known or can be estimated.

17. Decision Making Under Risk • Less information is available than in decision • making under certainty since it is not definitely • known which outcome will occur. The actual • outcome depends on which state of nature occurs. • For example, the number of umbrellas a store • sells in a month depends on how much rain falls • during the month.

18. Decision Making Under Uncertainty • In decision making under uncertainty, the • decision maker considers situations in which • several outcomes are possible for each course • of action. • In contrast to the risk situation, the decision • maker does not know, or cannot estimate, the • probability of occurrence of the possible states of • nature.

19. Relationship Between Decision Situations and Techniques AnalysisDecision Situations Certainty Risk Uncertainty Decision tables X X X Decision trees X X X Linear programming X Branch and bound X Integer programming X Goal programming X Distribution Maximum flow X CMP X Shortest route X PERT X Dynamic programming X X Markov chains X Inventory X X Queuing X Simulation X X Forecasting X

20. Decision Making Process • The classical definition of risk and uncertainty can • be viewed from a different perspective by looking • at the process of making a decision. This process • consists of the following steps: • Consider the alternatives (opportunities) and the • possible uncertainties concerning the anticipated • consequences. • Draw a decision table (or a decision tree) • Probabilities of each of the states of nature are • assigned, either objectively or else through the • decision maker’s subjective judgement.

21. Decision Making Process • Evaluate the results in light of the criterion of • choice • Make the decision

22. Decision Under Certainty

23. Complete Enumeration • Complete enumeration means examining every • payoff, one at a time, comparing the payoffs to • each other (e.g. in pairs), and discarding inferior • solutions. • The process continues until all payoffs are • examined

24. Summary • Decision making under certainty involves the • following steps: • Determine the alternative courses of action • Calculate (or assess) the payoffs, one for each • course of action. • Select the one with the best payoff (e.g., largest • profit or smallest cost), either by complete • enumeration, by using an algorithm, or by the • use of an analytical model.

25. Decision Table States of Nature s1 s2 s3 . . . sn d1 P1,1 P1,2 P1,3 . . . P1,n d2 P2,1 P2,2 P2,3 . . . P2,n Decision d3 P3,1 P3,2 P3,3 . . . P3,n Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dm Pm,1 Pm,2 Pm,3 . . . Pm,n

26. Decision Table where: sj = state of nature di = decision alternative Pij = payoff for decision i under state j

27. Decision Making Under Certainty • The most elementary of the decisions under • certainty, the states of nature are known. • Examine the payoffs under different decision • alternatives • Select the alternative with the largest payoff

28. Example • The following table lists the payoffs for each • possible investment decision under each possible • state of the economy. • The largest payoff comes with a stock • investment under a rapid-growth economic • scenario, with a payoff of $2200 per year on an • investment of $10,000 • The lowest payoff occurs under a stock • investment during stagnant economic times, • with an annual loss of $500 on the $10,000 • investment

29. Decision Table with State of Nature Probabilities State of the Economy Stagnant Slow Rapid Growth Growth Stocks -$500 $700 $2200 Investment Bonds -$100 $600 $900 Decision Alternative CD’s $300 $500 $750 Mixture-$200 $650 $1300 (0.25) (0.45) (0.30)