Decision Analysis
Chapter Topics
The payoff table and decision trees
Criteria for decision making
Opportunity loss 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. 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) 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) 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 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) Sell Hot Dogs (A2)
Event (Ei) Cool Weather (E1)
x11 =$50
x12 = $100
Warm Weather (E2)
x21 = $200
x22 = $125
xij = payoff (profit) for event i and action j
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. 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 Low Moderate High
Course of Action (Aj) Production Process A B C 70 80 100 120 120 125 200 180 160
Action C “dominates” Action D Action D is “inadmissible”
D 100 120 150
Decision Tree: Example Food Vendor Profit Tree Diagram x11 = $50 x21 = $200 x12 = $100
x22 =$125
Opportunity Loss: Example Highest possible profit for an event Ei
- Actual profit obtained for an action Aj Opportunity Loss (lij ) 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
Opportunity Loss: Table Alternative Course of Action Event Dogs
Optimal Action
Profit of Optimal Action
Sell Soft Drinks Sell Hot
Cool Weather
Hot Dogs
100
100 - 50 = 50
100 - 100 = 0
Warm Weather
Soft Drinks
200
200 - 200 = 0
200 - 125 = 75
Decision Criteria
Expected Monetary Value (EMV)
Expected Opportunity Loss (EOL)
The expected profit for taking an action Aj The expected loss for taking action Aj
Expected Value of Perfect Information (EVPI)
The expected opportunity loss from the best decision
Decision Criteria -- EMV Expected Monetary Value (EMV) =
Sum (monetary payoffs of events) ×
(probabilities of the events)
Number of events N
ΕΜV ΕΜ j =
∑ Xij Pi i=1
EMVj = expected monetary value of action j Xi,j = payoff for action j and event i Pi = probability of event i occurring
Decision Criteria -- EMV Table Example: Food Vendor Pi Event
.50 Cool .50 Warm
MV xijPi Soft Drinks $50 $50 ×.5 = $25 $200
$200 ×.5 = 100
EMV Soft Drink = $125
MV Hot Dogs $100
xijPi
$100×.50 = $50
$125
$125×.50 = 62.50
EMV Hot Dog = $112.50
Highest EMV = Better alternative
Decision Criteria -- EOL Expected Opportunity Loss (EOL)
Sum
(opportunity losses of events) × (probabilities of events)
ΕΟL ΕΟ j =
N
∑ lij Pi i =1
EOLj = expected opportunity loss of action j li,j = opportunity loss for action j and event i Pi = probability of event i occurring
Decision Criteria -- EOL Table Example: Food Vendor Pi
Event Op Loss Soft Drinks
.50 Cool .50 Warm
lijPi
Op Loss Hot Dogs
$50
$50×.50 = $25
$0
0
$0 ×.50 = $0
$75
EOL Soft Drinks = $25
lijPi
$0×.50 = $0 $75 ×.50 = $37.50
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 Computation Expected Profit Under Certainty = .50($100) + .50($200) = $150 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 σ2 for Soft Drink = (50 -125)2 ×.5 + (200 -125)2 ×.5 = 5625
σ for Soft Drink = 75 CVfor Soft Drinks = (75/125) × 100% = 60%
σ2 for Hot Dogs = 156.25 σ for Hot dogs = 12.5 CVfor 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 =
EMV j
σj
Return to Risk Ratio Example: Food Vendor RRR Soft Drinks = 1/CVSoft Drinks = 1.67
RRR Hot Dogs = 1/CVHot Dogs = 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
Excel spreadsheet for the food vendor example
Microsoft Excel Worksheet
Decision Making with Sample Information Prior Probability
Permits revising old 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 70% of the time. 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
Prior Probability
Revising Probabilities Example:Food Vendor
Revised Probability (Bayes’s Theorem)
P ( F1 | E1 ) = 0.80 P ( F1 | E2 ) = 0.30 P ( E1 ) = 0.50 P ( E2 ) = 0.50 P ( E1 ) P ( F1 | E1 )
.50)(.80) ( P ( E1 | F1 ) = = = .73 P ( F1 ) (.50)(.80) + (.50)(.30) P ( E2 ) P ( F1 | E2 ) P ( E2 | F1 ) = = .27 P ( F1 )
Revised EMV Table Example: Food Vendor Pi Event .73 Cool
Soft Drinks $50
.27 Warm
$200
xijPi $36.50 54
EMV Soft Drink = $90.50
Hot Dogs $100 125
xijPi $73 33.73
EMV Hot Dog = $106.75
Revised probabilities Highest EMV = Better alternative
Revised EOL Table Example: Food Vendor Pi
Event Op Loss Soft Drink
.73 Cool .27 Warm
$50 0
lijPi
$36.50 $0
EOL Soft Drinks = 36.50
OP Loss Hot Dogs
lijPi
$0
0
75
20.25
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 σ2 for Soft Drinks = (50 -90.5)2 ×.73 + (200 -90.5)2 ×.27 = 4434.75
σ for Soft Drinks = 66.59 CVfor Soft Drinks = (66.59/90.5) × 100% = 73.6%
σ2 for Hot Dogs = 123.1875 σ for Hot dogs = 11.10 CVfor Hot dogs = (11.10/106.75) × 100% = 10.4%
Revised Return to Risk Ratio RRR Soft Drinks = 1/CVSoft Drinks = 90.50/66.59
RRR Hot Dogs = 1/CVHot Dogs = 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 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 averse person, once reaching a goal, assigns less value to each incremental $1. A risk seeker assigns more value to each incremental $1. A risk neutral person assigns the same value to each incremental $1.
Three Types of Utility Curves
$
$
$
Risk Averter:
Risk Seeker:
Risk-Neutral:
Utility rises slower than payoff
Utility rises faster than payoff
Maximizes Expected payoff and ignores risk
Decision under Uncertainty
Melibatkan alternatif-alternatif kegiatan ai yang mana payoff nya bergantung pada state of nature secara (acak random) sj. 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 Kriteria Kriteria Kriteria
Laplace Minimax/Maximin Savage Hurwicz
Kriteria Laplace
Didasarkan pada prinsip alasan ketidakcukupan. Jika payoff v(ai, sj) mewakili gain (untung), alternatif terbaik adalah:
1 max ai n
v (a i , s j ) ∑ j =1 n
Jika payoff v(ai, sj) mewakili loss (rugi), alternatif terbaik diperoleh dengan mengubah maksimasi menjadi minimasi.
Kriteria Minimax/Maximin
Didasarkan pada prinsip the best out of the worst possible conditions. Jika payoff v(ai, sj) mewakili loss (rugi), alternatif terbaik: minmax v (a i , s j ) ai sj
Jika payoff v(ai, sj) mewakili gain (untung), alternatif terbaik: max min v (a i , s j ) ai sj
Kriteria Savage regret
Mengubah matriks payoff v(ai, sj) dengan matriks regret r(ai, sj) dimana:
v(ai , s j ) − min {v(ak , s j )} , ak r (ai , s j ) = v(ak , s j )} − v( ai , s j ), { max ak
jika
v adalah
loss
jika
v adalah
gain
Kriteria Hurwicz
0≤α≤1 Jika payoff v(ai, sj) mewakili gain (untung), alternatif terbaik: max α max v (a i , s j ) + (1 − α ) min v (a i , s j ) sj ai sj Jika payoff v(ai, sj) mewakili loss (rugi), alternatif terbaik: minα min v (a i , s j ) + (1 − α ) max v (a i , s j ) ai sj sj
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
Kriteria Laplace
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(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
a4
25
15
7
0
25
Kriteria Hurwicz Alternatif Row min Row max α(Row min)+(1-α)(Row max) a1 a2 a3 a4
5 7 12 15
25 23 21 30
25 - 20 α 23 - 16 α 21 - 9 α 30 - 15 α
Menggunakan α yg tersedia, dapat ditentukan alternatif optimum. Sebagai contoh, α=0.5, a1 atau a2 adalah alternatih optimum.
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
Provided criteria for decision making
Opportunity loss Expected 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
