Decision Analysis

• Steps in Decision making

• Decision analysis with decision tables

• Types of decision (decision modelling

environments)

• Decision trees

Steps in Decision making

1. Clearly define the problem2. List all possible alternatives3. Identify all possible outcomes for each

alternative4. Identify the payoff for each alternative &

outcome combination5. Use a decision modeling technique to choose

an alternative

Decision analysis with decision table

• The quantitative data of many decision situations can be arranged in a standardized tabular form known as decision table.

• Also known as payoff table• Decision table typically contains four elements. They

are:• The alternative courses of action• The state of nature• The probabilities of state of nature• The payoffs

The alternative course of action

• DM involves two or more options• One, and only one of these alternatives must

be selected• Alternative courses of actions are designated

by a1,a2,a3…… an• Number of alternative may be either finite or

infinite. As for example…….

The States of Nature

• Also known as Events or possible futures that a decion maker can not control

• A state of nature can be – A state of economy (e.g. inflation)– A weather condition– A political development– Or other situations

• Investment example

• One goal: maximize the yield after one year

• Yield depends on the status of the economy

(the state of nature)– Solid growth

– Stagnation

– Inflation

Possible Situations

1. If solid growth in the economy, bonds yield 12%; stocks 15%; time deposits 6.5%

2. If stagnation, bonds yield 6%; stocks 3%; time deposits 6.5%

3. If inflation, bonds yield 3%; stocks lose 2%; time deposits yield 6.5%

Decision Table

Alternatives

.5 .3 .2

Solid G Stagnation Inflation

A1 Bonds 12 6 3

A2 Stocks 15 3 -2

A3 Time deposits

6.5 6.5 6.5

State of nature

The probabilities of States of Nature & Payoffs

• Likelihood of these states of nature

• Payoff also known as outcome• Payoff can be measured in terms of money, market share or

other measures.

Types of decision modelling environment

Type 1: Decision making under certainty

Type 2: Decision making under uncertainty

Type 3: Decision making under risk

Decision making under certainty

• The consequence of every alternative is known

• Usually there is only one outcome for each alternative

• This seldom occurs in reality

Decision making under uncertainty• Probabilities of the possible outcomes are not

known. Choice here is made by organizational policy the attitude of the decision maker toward risk or both.

• Decision making methods:1. Laplace2. Maximin3. Maximax4. Coefficient of Optimism(Hurwicz Criterion)5. The criterion of Regret(Savage’s Criterion

Example

• The palm tree Hotel is considering the construction of an additional wing.

Mgt is evaluating the possibility of adding 30,40,50 rooms. The success of

the addition depends on a combination of local government legislation

and competition in the field, four states of nature are being considered.

They are shown together with the anticipated payoff. Mgt cannot agree

on the probabilities of the state of nature. The problem is : how many

rooms to build in order to maximize the return on investment.

Laplace (The criterion of equal probabilities

Alternative/State of Nature

s1 s2 s3 s4

A1=30 10 5 4 -2

A2=40 17 10 1 -10

A3=50 24 15 -3 -20

E(a1) =.25*10+.25*5+.25*4+.25*-2 =4.25

E(a2) =.25*17+.25*10+..25*1+25*-10 =4.5 Largest Expected Yield

E(a3) .25*24+.25*15+.25*-3+.25*-20 =4

Criterion of Pessimism (Maximin)

Alternative/State of Nature

s1 s2 s3 s4 Worst Best of Worst

A1=30 10 5 4 -2 -2 -2*

A2=40 17 10 1 -10 -10

A3=50 24 15 -3 -20 -20

Criterion of Optimism(Maximax)Alternative/State of Nature

s1 s2 s3 s4 Best Best of Bests

A1=30 10 5 4 -2 10

A2=40 17 10 1 -10 17

A3=50 24 15 -3 -20 24 24*

Coefficient of Optimism(Hurwicz)

Hurwicz suggested that the best alternative is the one with the highest (in maximization) weighted value (WV).

WV(a1) .7*10+(1-.7)*(-2)=6.4

WV(a2) .7*17+(1-.7)*(-10)=8.9

WV(a3) .7*24+(1-.7)*(-20)=10.8 (Maximum)*

Alternative/State of Nature

s1 s2 s3 s4

A1=30 10 5 4 -2

A2=40 17 10 1 -10

A3=50 24 15 -3 -20

The criterion of regret (savage’s Criterion)Alternative/State of Nature

s1 s2 s3 s4

A1=30 10 5 4 -2

A2=40 17 10 1 -10

A3=50 24 15 -3 -20

Alternative/State of Nature

s1 s2 s3 s4 Largest regret(Worst)

A1=30 24-10=14 15-5=10 4-4=0 -2- (-2)=0 14

A2=40 24-17=7 15-10=5 4-1=3 -2 (-10)=8 8* Minimum

A3=50 24-24=0 15-15=0 4-(-3)=7 -2-(-20)=18 18

Example2• The manager of an advertising agency has to make decisions between

three available programs (a1,a2,a3)There are three possible futures that can be expected ie market rises, market falls, no change in the market. The manager can estimate the yields in each case but can not estimate the possibilities of the various future occurring.

Programs s1 s2 s3

a1 3 6 -1

a2 8 5 4

a3 -4 7 12

Which program will the manager select if he uses the following decision approachesi. Laplaceii. Pessimisticiii. Otimisticiv. Hurwicz criterionv. Minimax regret (Savage)

Decision making under Risk

• Where probabilities of outcomes are available

• Expected payoff criterion: Here the Decision maker selects the alternative with the best expected(average) payoff

EMV (for alternative i) = ∑(probability of outcome) x (payoff of outcome)

• EOL criterion:

Investment Problem Decision Table Model

States of Nature

Solid Stagnation Inflation

Alternatives Growth

Bonds 12% 6% 3%

Stocks 15% 3% -2%

CDs 6.5% 6.5% 6.5%

Table 5.3: Decision Under Risk and Its Solution

Solid Stagnation InflationExpected

Growth Value

Alternatives .5 .3 .2

Bonds 12% 6% 3% 8.4% *

Stocks 15% 3% -2% 8.0%

CDs 6.5% 6.5% 6.5% 6.5%

How to find the Expected payoff

• E(a1)=12*.5+6*.3+3*.2=8.4

• E(a2)=15*.5+3*.3+-2*.2=8.0

• E(a3)=6.5*.5+6.5*.3+6.5*2=6.5

Expected Opportunity Loss

• Also called regret criterion

• Opportunity loss is defined as the relative loss

resulting from selecting an alternative.

• Minimizing expected regret. Regret is always

bad and is to be avoided or minimized

EOL/Regret Criterion

Solid Stagnation InflationExpected

Growth Value

Alternatives .5 .3 .2

Bonds 12% 6% 3% 2.35* Smallest

Stocks 15% 3% -2%

CDs 6.5% 6.5% 6.5%

EOL/Regret Criterion

Solid Stagnation Inflation Expected Value

Growth

Alternatives .5 .3 .2

Bonds 15-12=3 6.5-6=.5 6.5- 3=3 =3*.5+.5*.5*.3+3*.2=2.35*

Stocks 15-15=0 6.5- 3=3. 6.5-( -2) =2.75

CDs 15-6.5=8.5 6.5 -6.5=0 6.5-6.5 =4.25

Example2

• Find the best alternative in the following decisions table: Use both an expected value and an EOL

State of natureAlternative

.6

s1

.1

s2

.2

s3

.1

s4

a1 3 5 8 -1

a2 6 5 2 0

a3 0 5 6 4

Example3

• A survey conducted over last 20 yrs in Hamburg Germany indicated that in 8 of them the winter was mild, in 7 of them it was cold and in remaining 5 it was very cold. A company sells 1000 heavy coats in mild year,1300 in cold year and 2000 in very cold year. Find the yearly expected profit of the company if a coat cost 85 euro & sold to 123 euro.

Decision Trees