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

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 situations can 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 s 1, s 2, …, s m. 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., p 1 + p 2 + … + p m = 1 where p 1 = probability of s 1 occurring, p 2 = probability of s 2 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 o ij 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 tablesXXX Decision treesXXX Linear programmingX Branch and boundX Integer programmingX Goal programmingX Distribution Maximum flowX CMPX Shortest routeX PERTX Dynamic programmingXX Markov chainsX InventoryXX QueuingX SimulationXX ForecastingX

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 s 1 s 2 s 3...s n d 1 P 1,1 P 1,2 P 1,3... P 1,n d 2 P 2,1 P 2,2 P 2,3... P 2,n Decisiond 3 P 3,1 P 3,2 P 3,3... P 3,n Alternative d m P m,1 P m,2 P m,3... P m,n

26 Decision Table where: s j = state of nature d i = decision alternative P ij = 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 RapidGrowth Stocks -$500$700$2200 InvestmentBonds -$100$600$900 Decision AlternativeCD’s $300$500$750 Mixture -$200$650$1300 (0.25)(0.45)(0.30)