1

Statistical Analysis Decision Analysis

updated 9.11.01

NTUSY-521-N

SMUEMIS 5300/7300

Systems Analysis MethodsDr. 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, commonstocks, or certificates of deposit (time deposits). The Trust’s objective is to maximize the yield on the investment over a one-year period. Theproblem is that the economic situation is uncertainand no one is able to predict the exact movementsof the stock or even the bond markets. Misty’sresearchers informed her that they expected theeconomy 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 ofstagnation, and a 20% chance for inflation overthe next year. Past experience indicated the following trends:

1. If there is growth, bonds will yield 12%, stockswill 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%, stockswill drop 2% and time deposits will yield 6.5%.

How should Misty maximize the investment yieldover 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 (decisionvariables)

• 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 ofnature’ (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 politicaldevelopment (election of a certain candidate) orother situation which the decision maker cannotcontrol.

11

Use of Decision Tables

Probabilities of the states of nature:

• A question may be asked: ‘What is the likelihoodof these states of nature occurring?

• Whenever it is possible to answer this question interms of explicit chances (or probabilities), theinformation is recorded at the top of the table.

• The probabilities are given either in percent or inpercentage fractions

12

Use of Decision Tables

• Since it is assumed that one and only one of thegiven states of nature will occur in the future, thenthe 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 acertain alternative and a specific state that is given in that cell within the body of the table located at the intersection of the alternative inquestion (given by a row) and the specific stateof nature (given by a column).

• The payoff is designated by oij where i indicatesthe row and j the column

• The payoffs can be thought of as conditionalsince a specific payoff results from a specific stateof nature occurring but after a certain alternativecourse of action has been taken.

14

General Structure of a Decision Table

p1

s1

p2

s2

. . .

. . .

pm

sm

a1 o11 o11 o11 o1m

a2 o21 o21 o21 o2m

.

.

.

.

.

.

.

.

.

. . . . . . . . .

.

.

.an on1 on1 . . . onw

Alternative courses of

action

States of nature

15

Decision Making Under Certainty

• In decision making under certainty, it is assumedthat complete information is available so that thedecision maker knows exactly what the outcome ofeach 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) isone in which the decision maker must considerseveral possible states of nature, each with a given probability of occurrence.

• Thus, in risk situations, it is assumed that thelong-run probabilities of occurrence of the givenstates of nature (and their conditional outcomes)are known or can be estimated.

17

Decision Making Under Risk

• Less information is available than in decisionmaking under certainty since it is not definitelyknown which outcome will occur. The actualoutcome 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 whichseveral outcomes are possible for each courseof action.

• In contrast to the risk situation, the decisionmaker does not know, or cannot estimate, theprobability of occurrence of the possible states ofnature.

19

Relationship Between Decision Situations and Techniques

Analysis Decision Situations Certainty Risk Uncertainty

Decision tables X X XDecision trees X X XLinear programming XBranch and bound XInteger programmingXGoal programming XDistributionMaximum flow XCMP XShortest route XPERT XDynamic programming X XMarkov chains XInventory X XQueuing XSimulation X XForecasting X

20

Decision Making Process

The classical definition of risk and uncertainty canbe viewed from a different perspective by lookingat the process of making a decision. This processconsists of the following steps:

• Consider the alternatives (opportunities) and thepossible uncertainties concerning the anticipatedconsequences.

• Draw a decision table (or a decision tree)

• Probabilities of each of the states of nature areassigned, either objectively or else through the decision maker’s subjective judgement.

21

Decision Making Process

• Evaluate the results in light of the criterion ofchoice

• Make the decision

22

Decision Under Certainty

23

Complete Enumeration

• Complete enumeration means examining everypayoff, one at a time, comparing the payoffs toeach other (e.g. in pairs), and discarding inferiorsolutions.

• The process continues until all payoffs are examined

24

Summary

Decision making under certainty involves thefollowing steps:

• Determine the alternative courses of action

• Calculate (or assess) the payoffs, one for eachcourse of action.

• Select the one with the best payoff (e.g., largestprofit or smallest cost), either by complete enumeration, by using an algorithm, or by theuse 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 naturedi = decision alternativePij = payoff for decision i under state j

27

Decision Making Under Certainty

• The most elementary of the decisions undercertainty, the states of nature are known.

• Examine the payoffs under different decisionalternatives

• Select the alternative with the largest payoff

28

Example

The following table lists the payoffs for eachpossible investment decision under each possiblestate 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 Growth

Stocks -$500 $700 $2200

Investment Bonds -$100 $600 $900 DecisionAlternative CD’s $300 $500 $750

Mixture -$200 $650 $1300

(0.25) (0.45) (0.30)