Decision Analysis

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

Utility

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 diperkirakan

Langkah-langkah pengambilan

keputusan

Daftar semua alternatif (courses of action) yang mungkin

Daftar semua events or outcomes or states of nature yang mungkinnature yang mungkin

Tentukan payoffs

(Kaitkan sebuah payoff dengan setiap pasangan alternatif dan event)

Gunakan kriteria keputusan (decision criteria) (Evaluasi kriteria untuk memlilih alternatif terbaik )

List Possible Actions or Events

Two Methods of Listing

Payoff Table Decision Tree

Payoff Table (Step 1)

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

Course of Action (Aj)

Sell Soft Drinks (A1)

xij = payoff (profit) for event i and action j

Event (Ei)

Cool Weather (E1) x11 =$50 x12 = $100

Warm Weather (E2) x21 = $200 x22 = $125

Sell Hot Dogs (A2)

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 admissible.

Payoff Table (Step 2)Do Some Actions Dominate?

(continued)

Event (Ei)Level of Demand

Course of Action (Aj)Production ProcessA B C D

Low 70 80 100 100ModerateHigh

120 120 125 120200 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 Ei - Actual profit obtained for an action Aj

Opportunity Loss (lij )Opportunity Loss (lij )

Event: Cool Weather Action: Soft Drinks Profit x11 : $50Alternative Action: Hot Dogs Profit x12 : $100

Opportunity Loss l11 = $100 - $50 = $50Opportunity 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 decision

Expected Monetary Value (EMV) =Sum (monetary payoffs of events) (probabilities of the events)

Decision Criteria -- EMV

N

Number of events

Xij PiVj ==== N

EMVj = expected monetary value of action jXi,j = payoff for action j and event iPi = probability of event i occurring

i = 1

Decision Criteria -- EMV Table Example: Food Vendor

Pi Event MV xijPi MV xijPiSoft HotDrinks 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

EMV Hot Dog = $112.50

Decision Criteria -- EOL

Expected Opportunity Loss (EOL)Sum (opportunity losses of events) (probabilities of events)

Lj ==== lijPi

EOLj = expected opportunity loss of action jli,j = opportunity loss for action j and event iPi = probability of event i occurring

i =1

N

Decision Criteria -- EOL Table Example: Food Vendor

Pi Event Op Loss lijPi Op Loss lijPiSoft 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

Lowest EOL = Better Choice

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 willing to pay to obtain perfect information

EVPI ComputationExpected Profit Under Certainty

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

= $150

Expected Monetary Value of the Best AlternativeExpected 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 VariabilityExample: Food Vendor

2 for Soft Drink = (50 -125)2 .5 + (200 -125)2 .5 = 5625

for Soft Drink = 75 for Soft Drink = 75CVfor Soft Drinks = (75/125) 100% = 60%

2 for Hot Dogs = 156.25 for Hot dogs = 12.5CVfor Hot dogs = (12.5/112.5) 100% = 11.11%

Return to Risk Ratio

Expresses the relationship between the return (expected payoff) and the risk (standard deviation)

RRR = Return to Risk Ratio = jEMV

RRR = Return to Risk Ratio = j

j

EMV

Return to Risk RatioExample: Food Vendor

Soft Drinks Soft DrinksRRR = 1/CV = 1.67

Hot Dogs Hot DogsRRR = 1/CV = 9 Hot Dogs Hot DogsRRR = 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 boxesmeasures 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

PriorProbability

probabilities based on new information

NewInformation

RevisedProbability

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 1 2| 0.80 | 0.30

0.50 0.50

P F E P F E

P E P E

= =

= =

Revised Probability (Bayess Theorem)

( ) ( )

( ) ( ) ( )( )( )( )

( ) ( ) ( )( )

( ) ( ) ( )( )

1 2

1 1 11 1

1

2 1 22 1

1

0.50 0.50

| .50 .80| .73

.50 .80 .50 .30

|| .27

P E P E

P E P F EP E F

P F

P E P F EP E F

P F

= =

= = =+

= =

Revised EMV Table Example: Food Vendor

Pi Event Soft xijPi Hot xijPiDrinks 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 alternativeRevised probabilities

Revised EOL Table Example: Food Vendor

Pi Event Op Loss lijPi OP Loss lijPiSoft 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

Lowest EOL = Better Choice

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

2 for Soft Drinks

= (50 -90.5)2 .73 + (200 -90.5)2 .27 = 4434.75

for Soft Drinks = 66.59 for Soft Drinks = 66.59

CVfor Soft Drinks = (66.59/90.5) 100% = 73.6%

2 for Hot Dogs = 123.1875 for Hot dogs = 11.10CVfor Hot dogs = (11.10/106.75) 100% = 10.4%

Revised Return to Risk Ratio

Soft Drinks Soft DrinksRRR = 1/CV = 90.50/66.59

Hot Dogs Hot DogsRRR = 1/CV = 9.62Hot Dogs Hot DogsRRR = 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 boxesmeasures 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 each incremental $1.

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 aiyang mana payoff nya bergantung pada state of nature secara (acak random) sj.

Payoff atau outcome yang terkait dengan Payoff atau outcome yang terkait dengan kegiatan ai dan state sj ditulis dengan v(ai, sj).

Distribusi probabilitas setiap sj tidak diketahui atau tidak dapat ditentukan.

Payoff Matrix

S1 S2 Sn

a1 V(a1, s1) V(a1, s2) V(a1, sn)

a2 V(a2, s1) V(a2, s2) V(a2, sn)

am V(am, s1) V(am, s2) V(am, sn)

Pengambilan keputusan

Kriteria Laplace

Kriteria Minimax/Maximin

Kriteria Savage

Kriteria Hurwicz

Kriteria Laplace

Didasarkan pada prinsip alasan ketidakcukupan.

Jika payoff v(ai, sj) mewakili gain (untung), alternatif terbaik adalah:

n

sav ),(1

max

Jika payoff v(ai, sj) mewakili loss (rugi), alternatif terbaik diperoleh dengan mengubah maksimasi menjadi minimasi.

=jji

asav

ni 1),(

1max

Kriteria Minimax/Maximin

Didasarkan pada prinsip the best out of the worst possible conditions.

Jika payoff v(ai, sj) mewakili loss (rugi), alternatif terbaik:alternatif terbaik:

Jika payoff v(ai, sj) mewakili gain (untung), alternatif terbaik:

),(maxmin jisa

savj

i

),(minmax jisa

savji

Kriteria Savage regret

Mengubah matriks payoff v(ai, sj) dengan matriks regret r(ai, sj) dimana:

{ } { }{ }

( , ) min ( , ) ,( , )

max ( , ) ( , ),

k

k

i j k ja

i j

k j i ja

v a s v a sr a s

v a s v a s

=

jika v adalah loss

jika v adalah gain

Kriteria Hurwicz

0 1 Jika payoff v(ai, sj) mewakili gain (untung), alternatif terbaik:

Jika payoff v(ai, sj) mewakili loss (rugi), alternatif terbaik:

+ ),(min)1(),(maxmax ji

sji

sasavsav

jji

+ ),(max)1(),(minmin ji

sji

sasavsav

jji

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

a4 30 22 19 15

Nilai ekspektasi untuk setiap alternatif kegiatan:

E(a1) = (5+10+18+25) = 14,500

E(a2) = (8+7+12+23) = 12,500 (optimum)

E(a ) = (21+18+12+21) =18,000

Kriteria Laplace

E(a3) = (21+18+12+21) =18,000

E(a4) = (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

a4 25 15 7 0 25

Kriteria Hurwicz

Alternatif Row min Row max (Row min)+(1-)(Row max)

a1

a2

a3

5

7

12

25

23

21

25 - 20 23 - 16 21 - 9

Menggunakan yg tersedia, dapat ditentukan alternatif optimum. Sebagai contoh, =0.5, a1atau a2 adalah alternatih optimum.

a3

a4

12

15

21

30

21 - 9 30 - 15

EXERCISES: OPERATIONS RESERCH 7TH 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

Addressed the concept of utility