Chapter Topics. The payoff table and decision trees. Criteria for decision making

Chapter Topics

The payoff table and decision trees

Opportunity loss

Criteria for decision making

Expected monetary value
Expected monetary value
Expected opportunity loss
Return to risk ratio

Expected profit under certainty

Decision making with sample information
Decision under uncertainty

Definition

Analisis keputusan (decision analysis)

melibatkan penggunaan sebuah proses

rasional untuk memilih beberapa alternatif terbaik.

terbaik.

Pemilihan alternatif “terbaik” bergantung pada

kualitas data yang digunakan dalam mendeskripsikan situasi keputusan.

Ada tiga kategori proses pengambilan

keputusan:

Pengambilan keputusan dibawah kondisi pasti (data diketahui deterministik)

(data diketahui deterministik)

Pengambilan keputusan dibawah beresiko

(data dideskripsikan dengan distribusi probabilitas)
Pengambilan keputusan dibawah kondisi

ketidakpastian

(data tidak diketahui bobotnya, yang

merepresentasikan tingkat relevansi dalam proses keputusan)

Pengambilan keputusan dibawah

kondisi pasti

Linear programming (Programa linier)
Analytic Hierarchy Process (AHP)

Pengambilan keputusan dibawah

beresiko

Data dideskripsikan dengan distribusi

probabilitas

Didasarkan pada

kriteria nilai harapan

(

expected value criteria

) (

expected value criteria

)

Alternatif keputusan dibandingkan

berdasarkan pada maksimasi profit yang diharapkan atau minimasi biaya yang

Langkah-langkah pengambilan

keputusan

Daftar semua alternatif (

courses of action

)

yang mungkin

Daftar semua

events

or

outcomes

or

states of

nature

yang mungkin

nature

yang mungkin

Tentukan “payoffs”

(Kaitkan sebuah payoff dengan setiap pasangan

alternatif dan event)

Gunakan kriteria keputusan (

decision criteria

)

List Possible Actions or Events

Two Methods of Listing

Payoff Table (Step 1)

Consider a food vendor determining whether to sell soft drinks or hot dogs.

Course of Action (A_j) Sell Soft Drinks (A₁)

x_ij = payoff (profit) for event i and action j Event (E_i)

Cool Weather (E₁) x₁₁ =$50 x₁₂ = $100 Warm Weather (E₂) x₂₁ = $200 x₂₂ = $125

Payoff Table (Step 2)

Do Some Actions Dominate?

Action A “dominates” action B if the payoff of

action A is at least as high as that of action B under any event and is higher under at least one event.

one event.

Action A is “inadmissible” if it is dominated by

any other action(s).

Inadmissible actions do not need to be

considered.

Non-dominated actions are called

Payoff Table (Step 2)

Do Some Actions Dominate?

(continued) Event (E_i) Level of Demand Course of Action (A_j) Production Process A B C D Low 70 80 100 100 Moderate High 120 120 125 120 200 180 160 150

Action C “dominates” Action D
Action D is “inadmissible”

Decision Tree:

Example

Food Vendor Profit Tree Diagram

x₁₁ = $50

x₂₁ = $200

x₂₂ =$125 x₁₂ = $100

Opportunity Loss:

Example

Highest possible profit for an event E_i

-

Actual profit obtained for an action A_j Opportunity Loss (l_ij )

Opportunity Loss (l_ij )

Event: Cool Weather

Action: Soft Drinks Profit x₁₁ : $50

Alternative Action: Hot Dogs Profit x₁₂ : $100

Opportunity Loss l₁₁ = $100 - $50 = $50 Opportunity Loss l₁₂ = $100 - $100 = $0

Event Optimal Profit of Sell Soft Drinks Sell Hot Dogs Action Optimal

Opportunity Loss: Table

Alternative Course of Action

Dogs Action Optimal Action

Cool Hot 100 100 - 50 = 50 100 - 100 = 0 Weather Dogs

Warm Soft 200 200 - 200 = 0 200 - 125 = 75 Weather Drinks

Decision Criteria

Expected Monetary Value (EMV)

The expected profit for taking an action Aj

Expected Opportunity Loss (EOL)

The expected loss for taking action Aj

The expected loss for taking action Aj

Expected Value of Perfect Information (EVPI)

The expected opportunity loss from the best

Expected Monetary Value (EMV) =

Sum

(monetary payoffs of events) × (probabilities of the events)

Decision Criteria -- EMV

∑

∑

∑

∑

N Number of events

X

_ij

P

_i

ΕΜ

ΕΜ

ΕΜ

ΕΜ

V

_j

====

∑

∑

∑

∑

N

EMV_j = expected monetary value of action j

X_i,j = payoff for action j and event i P_i = probability of event i occurring

Decision Criteria -- EMV Table

Example: Food Vendor

P_i Event MV x_ijP_i MV x_ijP_i Soft Hot Drinks Dogs .50 Cool $50 $50 ×.5 = $25 $100 $100×.50 = $50 .50 Cool $50 $50 ×.5 = $25 $100 $100×.50 = $50 .50 Warm $200 $200 ×.5 = 100 $125 $125×.50 = 62.50

EMV _{Soft Drink} = $125

Highest EMV = Better alternative

Decision Criteria -- EOL

Expected Opportunity Loss (EOL)

Sum

(opportunity losses of events) × (probabilities of events)

ΕΟ

ΕΟ

ΕΟ

ΕΟ

L

_j

====

∑

∑

∑

∑

_l

_ij

_P

_i

EOL_j = expected opportunity loss of action j

l_i,j = opportunity loss for action j and event i

P_i = probability of event i occurring

i =1

Decision Criteria -- EOL Table

Example: Food Vendor

P_i Event Op Loss l_ijP_i Op Loss l_ijPi

Soft Drinks Hot Dogs

.50 Cool $50 $50×.50 = $25 $0 $0×.50 = $0

.50 Warm 0 $0 ×.50 = $0 $75 $75 ×.50 = $37.50

EOL _{Soft Drinks} = $25 EOL _{Hot Dogs} = $37.50

EVPI

Expected Value of Perfect Information (EVPI)

The expected opportunity loss from the best

decision

Expected Profit Under Certainty

-

Expected Monetary Value of the Best Alternative

EVPI

(should be a positive number)

Represents the maximum amount you are

EVPI Computation

Expected Profit Under Certainty

= .50($100) + .50($200) = $150

Expected Monetary Value of the Best Alternative Expected Monetary Value of the Best Alternative

= $125

EVPI = $150 - $125 = $25 = Lowest EOL

= The maximum you would be willing to spend to obtain perfect information

Taking Account of Variability

Example: Food Vendor

σσσσ

2

_{for Soft Drink}

= (50 -125)

2

××××

.5 + (200 -125)

2

××××

.5 = 5625

σσσσ

for Soft Drink = 75

σσσσ

for Soft Drink = 75

CV

_{for Soft Drinks}

= (75/125)

××××

100% = 60%

σσσσ

2

_{for Hot Dogs = 156.25}

σσσσ

_{for Hot dogs = 12.5}

Return to Risk Ratio

Expresses the relationship between the return

(expected payoff) and the risk (standard deviation)

RRR = Return to Risk Ratio = EMVj
RRR = Return to Risk Ratio = j

j

EMV

Return to Risk Ratio

Example: Food Vendor

Soft Drinks Soft Drinks

RRR

= 1/CV

= 1.67

Hot Dogs Hot Dogs

RRR

_{Hot Dogs}

= 1/CV

_{Hot Dogs}

= 9

RRR

= 1/CV

= 9

You might want to sell hot dogs. Although soft drinks have the higher Expected Monetary

Value, hot dogs have a much larger return to risk ratio and a much smaller CV.

Decision Making in PHStat

PHStat | decision-making | expected monetary

value

Check the “expected opportunity loss” and

“measures of valuation” boxes “measures of valuation” boxes

Excel spreadsheet for the food vendor

example

Microsoft Excel Worksheet

Decision Making

with Sample Information

Permits

revising

old

probabilities based on New Prior Probability probabilities based on new information New Information Revised Probability

Revised Probabilities

Example: Food Vendor

Additional Information: Weather forecast is COOL.

When the weather was cool, the forecaster was correct 80% of the time.

When the weather was warm, the forecaster was correct When the weather was warm, the forecaster was correct 70% of the time. Prior Probability F₁ = Cool forecast F₂ = Warm forecast E₁ = Cool Weather = 0.50 E₂ = Warm Weather = 0.50 P(F₁ | E₁) = 0.80 P(F₁ | E₂) = 0.30

Revising Probabilities

Example:Food Vendor

(

)

(

)

( )

( )

1 | 1 0.80 1 | 2 0.30 0.50 0.50 P F E P F E P E P E = = = =

Revised Probability (Bayes’s Theorem)

( )

( )

(

) ( ) (

_{( )}

)

_{( )( ) ( )( )}

( )( )

(

) ( ) (

_{( )}

)

1 2 1 1 1 1 1 1 2 1 2 2 1 1 0.50 0.50 | .50 .80 | .73 .50 .80 .50 .30 | | .27 P E P E P E P F E P E F P F P E P F E P E F P F = = = = = + = =

Revised EMV Table

Example: Food Vendor

P_i Event Soft x_ijP_i Hot x_ijP_i

Drinks Dogs

.73 Cool $50 $36.50 $100 $73

.27 Warm $200 54 125 33.73

EMV _{Soft Drink} = $90.50 EMV _{Hot Dog} = $106.75

Highest EMV = Better alternative

Revised EOL Table

Example: Food Vendor

P_i Event Op Loss l_ijP_i OP Loss l_ijPi

Soft Drink Hot Dogs

.73 Cool $50 $36.50 $0 0

.73 Cool $50 $36.50 $0 0

.27 Warm 0 $0 75 20.25

EOL _{Soft Drinks} = 36.50 EOL _{Hot Dogs} = $20.25

Revised EVPI Computation

Expected Profit Under Certainty

= .73($100) + .27($200) = $127

Expected Monetary Value of the Best Alternative = $106.75

EPVI = $127 - $106.75 = $20.25

= The maximum you would be willing to spend to obtain perfect information

Taking Account of Variability:

Revised Computation

σσσσ

2

_{for Soft Drinks}

= (50 -90.5)

2

××××

.73 + (200 -90.5)

2

××××

.27 = 4434.75

σσσσ

for Soft Drinks = 66.59

σσσσ

for Soft Drinks = 66.59

CV

_{for Soft Drinks}

= (66.59/90.5)

××××

100% = 73.6%

σσσσ

2

_{for Hot Dogs = 123.1875}

σσσσ

for Hot dogs = 11.10

Revised Return to Risk Ratio

Soft Drinks Soft Drinks

RRR

= 1/CV

= 90.50/66.59

Hot Dogs Hot Dogs

RRR

_{Hot Dogs}

= 1/CV

_{Hot Dogs}

= 9.62

RRR

= 1/CV

= 9.62

You might want to sell Hot Dogs. Hot Dogs have a much larger return to risk ratio.

Revised Decision

Making in PHStat

PHStat | decision-making | expected monetary

value

Check the “expected opportunity loss” and

“measures of valuation” boxes “measures of valuation” boxes

Use the revised probabilities

Excel spreadsheet for the food vendor

example

Microsoft Excel Worksheet

Utility

Utility is the idea that each incremental $1 of

profit does not have the same value to every individual

A risk averserisk averse person, once reaching a goal, assigns

A risk averserisk averse person, once reaching a goal, assigns

less value to each incremental $1.

A risk seekerrisk seeker assigns more value to each

incremental $1.

A risk neutralrisk neutral person assigns the same value to

Three Types of Utility Curves

$ $ $

Risk Averter:

Utility rises slower than payoff

Risk Seeker:

Utility rises faster than payoff

Risk-Neutral:

Maximizes

Expected payoff and ignores risk

Decision under Uncertainty

Melibatkan alternatif-alternatif kegiatan a_i

yang mana payoff nya bergantung pada

state

of nature

secara (acak random) s_j.

Payoff atau outcome yang terkait dengan

Payoff atau outcome yang terkait dengan

kegiatan a_i dan state s_j ditulis dengan v(a_i, s_j).

Distribusi probabilitas setiap s_j tidak diketahui

Payoff Matrix

S1 S2 … Sn

a1 V(a₁, s₁) V(a₁, s₂) … V(a₁, s_n)

a2 V(a₂, s₁) V(a₂, s₂) … V(a₂, s_n)

… … … … …

Pengambilan keputusan

Kriteria Laplace

Kriteria Minimax/Maximin
Kriteria Savage

Kriteria Laplace

Didasarkan pada prinsip alasan ketidakcukupan.

Jika payoff v(a_i, s_j) mewakili gain (untung), alternatif terbaik

adalah:    

∑

n v(a ,s ) 1 max

Jika payoff v(a_i, s_j) mewakili loss (rugi), alternatif terbaik

diperoleh dengan mengubah maksimasi menjadi minimasi.

   

∑

= j j i a_i n v a s 1 ) , ( 1 max

Kriteria Minimax/Maximin

Didasarkan pada prinsip the best out of the

worst possible conditions.

Jika payoff v(a_i, s_j) mewakili loss (rugi),

alternatif terbaik: alternatif terbaik:

Jika payoff v(a_i, s_j) mewakili gain (untung),

alternatif terbaik:       ) , ( max min _{i j} s a v a s j i       ) , ( min max _{i j} s a_{i j} v a s

Kriteria Savage regret

Mengubah matriks payoff v(a_i, s_j) dengan

matriks regret r(a_i, s_j) dimana:

{

}



{

}

{

}

( , ) min ( , ) , ( , ) max ( , ) ( , ), k k i j k j a i j k j i j a v a s v a s r a s v a s v a s  ₋  =  −  

jika v adalah loss

Kriteria Hurwicz

0 ≤ α ≤ 1

Jika payoff v(a_i, s_j) mewakili gain (untung),

alternatif terbaik:

 

Jika payoff v(a_i, s_j) mewakili loss (rugi),

alternatif terbaik:       α − + α max ( , ) (1 )min ( , ) max _{i j} s j i s a_{i j} v a s _j v a s       α − + α min ( , ) (1 )max ( , ) min _{i j} s j i s a_{i j} v a s _j v a s

Contoh Pengambilan Keputusan

dalam lingkungan tidak pasti

Cost matriks (loss): dalam ribuan

s1 s2 s3 s4

a1 5 10 18 25

a2 8 7 12 23

a3 21 18 12 21

Nilai ekspektasi untuk setiap alternatif kegiatan: E(a₁) = ¼ (5+10+18+25) = 14,500 E(a₂) = ¼ (8+7+12+23) = 12,500 (optimum) E(a ) = ¼ (21+18+12+21) =18,000

Kriteria Laplace

E(a₃) = ¼ (21+18+12+21) =18,000 E(a₄) = ¼ (30+22+19+15) = 21,500 Jadi alternatif 2 (yaitu a₂) yang terpilih.

Kriteria Minimax

s1 s2 s3 s4 Row max a1 5 10 18 25 25 a2 8 7 12 23 23 a3 21 18 12 21 21 (minimax) a4 30 22 19 15 30

Kriteria Savage

Matriks regret ditentukan dengan mengurangkan 5,

7, 12 dan 12 dari kolom-kolom 1, 2, 3 dan 4. Jadi

s1 s2 s3 s4 Row max

a1 0 3 6 10 10

a2 3 0 0 8 8 (minimax)

a3 16 11 0 6 16

Kriteria Hurwicz

Alternatif Row min Row max α(Row min)+(1-α)(Row max) a1 a2 a3 5 7 12 25 23 21 25 - 20 α 23 - 16 α 21 - 9 α

Menggunakan α yg tersedia, dapat ditentukan

alternatif optimum. Sebagai contoh, α=0.5, a₁ atau a₂ adalah alternatih optimum.

a3 a4 12 15 21 30 21 - 9 α 30 - 15 α

EXERCISES: OPERATIONS RESERCH

7

TH

_{EDITION (HAMDY A. THAHA)}

PROBLEM SET 14.2B
PROBLEM SET 14.3A

Chapter Summary

Described the payoff table and decision trees

Opportunity loss

Provided criteria for decision making

Expected monetary valueExpected monetary value
Expected opportunity loss
Return to risk ratio

Introduced expected profit under certainty
Discussed decision making with sample

information