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

Transkripsi

1 Decision Analysis

2 Chapter Topics The payoff table and decision trees Opportunity loss Criteria for decision making Expected monetary value Expected opportunity loss Return to risk ratio Expected profit under certainty Decision making with sample information Decision under uncertainty Utility

3 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.

4 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)

5 Pengambilan keputusan dibawah kondisi pasti Linear programming (Programa linier) Analytic Hierarchy Process (AHP)

6 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

7 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 )

8 List Possible Actions or Events Two Methods of Listing Payoff Table Decision Tree

9 Payoff Table (Step 1) Consider a food vendor determining whether to sell soft drinks or hot dogs. Event (E i ) Course of Action (A j ) Sell Soft Drinks (A 1 ) Sell Hot Dogs (A 2 ) Cool Weather (E 1 ) x 11 =$50 x 12 = $100 Warm Weather (E 2 ) x 21 = $200 x 22 = $125 x ij = payoff (profit) for event i and action j

10 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.

11 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 Moderate High Action C dominates Action D Action D is inadmissible

12 Decision Tree: Example Food Vendor Profit Tree Diagram x 11 = $50 x 21 = $200 x 12 = $100 x 22 =$125

13 Opportunity Loss: Example Highest possible profit for an event E i - Actual profit obtained for an action A j Opportunity Loss (l ij ) Event: Cool Weather Action: Soft Drinks Profit x 11 : $50 Alternative Action: Hot Dogs Profit x 12 : $100 Opportunity Loss l 11 = $100 - $50 = $50 Opportunity Loss l 12 = $100 - $100 = $0

14 Opportunity Loss: Table Event Dogs Alternative Course of Action Optimal Profit of Sell Soft Drinks Sell Hot Action Optimal Action Cool Hot = = 0 Weather Dogs Warm Soft = = 75 Weather Drinks

15 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 Expected Value of Perfect Information (EVPI) The expected opportunity loss from the best decision

16 Decision Criteria -- EMV Expected Monetary Value (EMV) = Sum (monetary payoffs of events) (probabilities of the events) Number of events ΕΜV j = N i = 1 X ij P i 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

17 Decision Criteria -- EMV Table Example: Food Vendor P i Event MV x ij P i MV x ij P i Soft Hot Drinks Dogs.50 Cool $50 $50.5 = $25 $100 $ = $50.50 Warm $200 $200.5 = 100 $125 $ = EMV Soft Drink = $125 EMV Hot Dog = $ Highest EMV = Better alternative

18 Decision Criteria -- EOL Expected Opportunity Loss (EOL) Sum (opportunity losses of events) (probabilities of events) N ΕΟL = j l ij P i i =1 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

19 Decision Criteria -- EOL Table Example: Food Vendor P i Event Op Loss l ij P i Op Loss l ij Pi 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 Lowest EOL = Better Choice

20 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

21 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

22 Taking Account of Variability Example: Food Vendor σ 2 for Soft Drink = (50-125) ( ) 2.5 = 5625 σ for Soft Drink = 75 CV for Soft Drinks = (75/125) 100% = 60% σ 2 for Hot Dogs = σ for Hot dogs = 12.5 CV for Hot dogs = (12.5/112.5) 100% = 11.11%

23 Return to Risk Ratio Expresses the relationship between the return (expected payoff) and the risk (standard deviation) RRR = Return to Risk Ratio = EMV j σ j

24 Return to Risk Ratio Example: Food Vendor RRR = 1/CV = 1.67 Soft Drinks Soft Drinks RRR = 1/CV = 9 Hot Dogs Hot Dogs 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.

25 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

26 Decision Making with Sample Information Permits revising old probabilities based on new information Prior Probability New Information Revised Probability

27 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. F 1 = Cool forecast F 2 = Warm forecast E 1 = Cool Weather = 0.50 E 2 = Warm Weather = 0.50 P(F 1 E 1 ) = 0.80 P(F 1 E 2 ) = 0.30 Prior Probability

28 Revising Probabilities Example:Food Vendor Revised Probability (Bayes s Theorem) P F E = 0.80 P F E = 0.30 ( ) ( ) P E = 0.50 P E = 0.50 ( ) ( ) P E F 1 2 ( ) P E ( ) P E P F E ( ) ( ) ( ) ( ) ( )( ) = = =.73 P( F ) ( )( ) ( )( ) P E P F E F1 = =.27 P( F ) 1

29 Revised EMV Table Example: Food Vendor P i Event Soft x ij P i Hot x ij P i Drinks Dogs.73 Cool $50 $36.50 $100 $73.27 Warm $ EMV Soft Drink = $90.50 EMV Hot Dog = $ Revised probabilities Highest EMV = Better alternative

30 Revised EOL Table Example: Food Vendor P i Event Op Loss l ij P i OP Loss l ij Pi Soft Drink Hot Dogs.73 Cool $50 $36.50 $ Warm 0 $ EOL Soft Drinks = EOL Hot Dogs = $20.25 Lowest EOL = Better Choice

31 Revised EVPI Computation Expected Profit Under Certainty =.73($100) +.27($200) = $127 Expected Monetary Value of the Best Alternative = $ EPVI = $127 - $ = $20.25 = The maximum you would be willing to spend to obtain perfect information

32 Taking Account of Variability: Revised Computation σ 2 for Soft Drinks = ( ) ( ) 2.27 = σ for Soft Drinks = CV for Soft Drinks = (66.59/90.5) 100% = 73.6% σ 2 for Hot Dogs = σ for Hot dogs = CV for Hot dogs = (11.10/106.75) 100% = 10.4%

33 Revised Return to Risk Ratio RRR = 1/CV = 90.50/66.59 Soft Drinks Soft Drinks RRR = 1/CV = 9.62 Hot Dogs Hot Dogs You might want to sell Hot Dogs. Hot Dogs have a much larger return to risk ratio.

34 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

35 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 risk neutral person assigns the same value to each incremental $1.

36 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

37 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 kegiatan a i dan state s j ditulis dengan v(a i, s j ). Distribusi probabilitas setiap s j tidak diketahui atau tidak dapat ditentukan.

38 Payoff Matrix S1 S2 Sn a1 V(a 1, s 1 ) V(a 1, s 2 ) V(a 1, s n ) a2 V(a 2, s 1 ) V(a 2, s 2 ) V(a 2, s n ) am V(a m, s 1 ) V(a m, s 2 ) V(a m, s n )

39 Pengambilan keputusan Kriteria Laplace Kriteria Minimax/Maximin Kriteria Savage Kriteria Hurwicz

40 Kriteria Laplace Didasarkan pada prinsip alasan ketidakcukupan. Jika payoff v(a i, s j ) mewakili gain (untung), alternatif terbaik adalah: 1 n max v ( ai, sj ) ai n j = 1 Jika payoff v(a i, s j ) mewakili loss (rugi), alternatif terbaik diperoleh dengan mengubah maksimasi menjadi minimasi.

41 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: min max ai sj v( a i, s Jika payoff v(a i, s j ) mewakili gain (untung), alternatif terbaik: max min ai sj v( a i, s j j ) )

42 Kriteria Savage regret Mengubah matriks payoff v(a i, s j ) dengan matriks regret r(a i, s j ) dimana: r( a, s ) i j { } v( ai, s j ) min v( ak, s j ), ak = max { v( a, )} k s j v( ai, s j ), ak jika v adalah loss jika v adalah gain

43 Kriteria Hurwicz 0 α 1 Jika payoff v(a i, s j ) mewakili gain (untung), alternatif terbaik: max αmax ai sj v( a i, s Jika payoff v(a i, s j ) mewakili loss (rugi), alternatif terbaik: j ) + (1 α)min s j v( a i, s j ) min α ai min s j v( a i, s j ) + (1 α)max s j v( a i, s j )

44 Contoh Pengambilan Keputusan dalam lingkungan tidak pasti Cost matriks (loss): dalam ribuan s1 s2 s3 s4 a a a a

45 Kriteria Laplace Nilai ekspektasi untuk setiap alternatif kegiatan: E(a 1 ) = ¼ ( ) = 14,500 E(a 2 ) = ¼ ( ) = 12,500 (optimum) E(a 3 ) = ¼ ( ) =18,000 E(a 4 ) = ¼ ( ) = 21,500 Jadi alternatif 2 (yaitu a 2 ) yang terpilih.

46 Kriteria Minimax s1 s2 s3 s4 Row max a a a (minimax) a

47 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 a a (minimax) a a

48 Kriteria Hurwicz Alternatif Row min Row max α(row min)+(1-α)(row max) a α a α a α a α Menggunakan α yg tersedia, dapat ditentukan alternatif optimum. Sebagai contoh, α=0.5, a 1 atau a 2 adalah alternatih optimum.

49 EXERCISES: OPERATIONS RESERCH 7 TH EDITION (HAMDY A. THAHA) PROBLEM SET 14.2B PROBLEM SET 14.3A

50 Chapter Summary Described the payoff table and decision trees Opportunity loss Provided criteria for decision making 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