DECISION THEORY

PROF. KAUSHIK PAUL

ASSOCIATE PROFESSOR

OPERATIONS AREA

IILM GRADUATE SCHOOL OF MANAGEMENT

GREATER NOIDA

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TOPICS OF DISCUSSION

INTRODUCTION

PAYOFF MATRIX

REGRET TABLE

DECISION MAKING UNDER UNCERTAINTY

DIFFERENT METHODS OF DECISION MAKING UNDER UNCERTAINTY

DECISION MAKING UNDER RISK

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TOPICS OF DISCUSSION

DIFFERENT METHODS OF DECISION MAKING UNDER RISK MAXIMUM LIKELIHOOD PRINCIPLE

MAXIMUM EXPECTATION PRINCIPLE

CONCEPT OF DECISION TREE

DECISION MAKING WITH PERFECT INFORMATION AND ECONOMIC VALUE OF PERFECT INFORMATION (EVPI)

DECISION MAKING WITH REVISED PROBABILITIES ( USING BAYE’S THEOREM)

EXPECTED OPPORTUNITY LOSS/REGRET PRINCIPLE

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INTRODUCTION EXTERNAL ENVIRONMENT IMPOSES CERTAIN CONSTRAINTS ON US.

BASED ON OUR RESPONSE TO SUCH CONSTRAINTS, WE GET DIFFERENT PAYOFFS.

WE CAN NOT CHANGE THE PAYOFF MATRIX.

AT BEST, WE CAN USE THE AVAILABLE INFORMATION JUDICIOUSLY TO ARRIVE AT THE OPTIMAL DECISION AND MAXIMIZE OUR PAYOFFS IN THE LONG RUN.

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EXAMPLE A WHOLE SALER OF FRUITS BUYS STRAWBERRIES AT $ 20 A CASE

AND SELLS THEM AT $ 50 A CASE. THE PRODUCT IS PERISHABLE BY NATURE AND CAN NOT BE STORED OVERNIGHT. IT HAS BE SOLD ON THE DAY OF THE PURCHASE ITSELF.

FROM EXPERIENCE, THE WHOLE SALER KNOWS THAT THE DAILY DEMAND WILL RANGE BETWEEN 10 TO 13 CASES.

EVERY CASE OF STRAWBERRIES BOUGHT AND NOT SOLD WOULD LEAD TO A MARGINAL LOSS OF $ 20, WHILE EVERY CASE THAT COULD NOT BE SOLD BECAUSE OF STOCK OUT WOULD LEAD TO AN OPPORTUNITY LOSS OF $ 30.

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PAYOFF MATRIX IN US $

POSSIBLE DEMAND IN

CASES

POSSIBLE STOCK ACTION IN CASES

10 11 12 13

10 300 280 260 240

11 300 330 310 290

12 300 330 360 340

13 300 330 360 390

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REGRET TABLE IN US $

POSSIBLE DEMAND IN

CASES

POSSIBLE STOCK ACTION IN CASES

10 11 12 13

10 0 20 40 60

11 30 0 20 40

12 60 30 0 20

13 90 60 30 0

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DECISION MAKING UNDER UNCERTAINTY HERE, WE HAVE NO INFORMATION ABOUT THE

LIKELIHOOD( PROBABILTY) OF ANY PARTICULAR STATE OF DEMAND.

IN SUCH A CONDITION, WE ARE MAKING A DECISION UNDER UNCERTAINTY.

FOLLOWING PRINCIPLES ARE USED TO TAKE DECISIONS UNDER UNCERTAINTY:

LAPLACE PRINCIPLE MAXIMIN OR MINIMAX CRITERION MAXIMAX. OR MINIMIN. CRITERION HURWICZ CRITERION SALVAGE PRINCIPLE

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LAPLACE PRINCIPLE ASSUMES ALL

EXTERNAL CONSTRAINTS ( HERE DEMAND OF STRAWBERRIES IN CASES TO BE EQUIPROBABLE).

MAXIMUM PAYOFF IS ACHIEVED BY PURSUING THE STRATEGY OF STOCKING 12 UNITS.

STOCK ACTION ( CASES)

PAYOFF (US $)

10 300

11 317.5

12 322.5

13 315

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MAXIMIN. OR MINIMAX PRINCIPLE

HERE, MINIMUM PAYOFFS FROM EACH STRATEGY IS CONSIDERED. THE STRATEGY WITH MAX. OF THESE MINIMUM PROFITS IS SELECTED. THIS IS KNOWN AS MAXIMIN. PRINCIPLE. FOR COSTS, THE STRATEGY WITH MIN. OF MAXIMUM COSTS IS CHOSEN. THAT IS KNOWN AS MINIMAX. PRINCIPLE. THIS IS A PESSIMISTIC DECISION MAKING.

MAXIMUM PAYOFF IS ACHIEVED BY PURSUING THE STRATEGY OF STOCKING 10 UNITS.

STOCK ACTION ( CASES)

MIN. PAYOFF (US $)

10 300

11 280

12 260

13 240

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MAXIMAX. OR MINIMIN. PRINCIPLE

HERE, MAXIMUM PAYOFFS FROM EACH STRATEGY IS CONSIDERED. THE STRATEGY WITH MAX. OF THESE MAXIMUM PROFITS IS SELECTED. THIS IS KNOWN AS MAXIMAX. PRINCIPLE. FOR COSTS, THE STRATEGY WITH MIN. OF THE MINIMUM COSTS IS CHOSEN. THAT IS KNOWN AS MINIMIN. PRINCIPLE. THIS IS A HIGHLY OPTIMISTIC DECISION MAKING.

MAXIMUM PAYOFF IS ACHIEVED BY PURSUING THE STRATEGY OF STOCKING 13 UNITS.

STOCK ACTION ( CASES)

MAX. PAYOFF (US $)

10 300

11 330

12 360

13 390

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HURWICZ PRINCIPLE INDEX OF OPTIMISM= α

CRITERION VALUE = α( MAX. PROFIT) + (1- α)(MIN. PROFIT)

FOR COSTS, CRITERION VALUE = α( MIN.COST) + (1- α)(MAX.COST)

α = 0 STANDS FOR MAXIMIN. OR MINIMAX. PRINCIPLE

α = 1 STANDS FOR MAXIMAX. OR MINIMIN. PRINCIPLE

FOR OUR EXAMPLE, LET US ASSUME AN INDEX OF OPTIMISM OF 60% (α = 0.6)

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HURWICZ PRINCIPLE CONTINUED

THE STRATEGY WITH THE MAXIMUM HURWICZ CRITERION VALUE IS CHOSEN.

HERE, THE MAXIMUM HURWICZ CRITERION VALUE IS ACHIEVED BY PURSUING THE STRATEGY OF STOCKING 13 UNITS.

STOCK ACTION ( CASES)

HURWICZ CRIETRION VALUE

(US $)

10 300

11 310

12 320

13 330

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SALVAGE PRINCIPLE

HERE, WE SELECT THE STRATEGY THAT MINIMIZES THE MAXIMUM REGRET. THIS IS ALSO A PESSIMISTIC DECISION MAKING. THIS IS CLOSE TO THE MINIMAX. PRINCIPLE BUT RESULTS MAY VARY.

HERE, THE MINIMUM OF MAX. REGRET (SALVAGE) VALUE IS ACHIEVED BY PURSUING THE STRATEGY OF STOCKING 12 UNITS.

STOCK ACTION ( CASES)

MAX. REGRET VALUE

(US $)

10 90

11 60

12 40

13 60

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DECISION MAKING UNDER RISK

NOW, LET US ASSUME THAT BASED ON THE SALES OF PAST 100 DAYS, THE WHOLESALER HAS FOLLOWING INFORMATION ABOUT THE MARKET DEMAND.

DAILY SALES ( CASES)

NO. OF DAYS SOLD

PROB. OF DEMAND

10 15 0.15

11 20 0.20

12 40 0.40

13 25 0.25

TOTAL 100 1.00

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DECISION MAKING UNDER RISK

HERE, WE HAVE ADDITIONAL INFORMATION ON THE PROBABILITY OF EACH DEMAND STATE.

WHEN WE HAVE PROBABILITIES ASSOCIATED WITH EACH DEMAND STATE AVAILABLE TO US, THE DECISION MAKING TECHNIQUE IS CALLED DECISION MAKING UNDER RISK.

FOLLOWING PRINCIPLES ARE USED TO ARRIVE AT THE OPTIMAL DECISION.

MAXIMUM LIKELIHOOD PRINCIPLE MAXIMUM EXPECTATION PRINCIPLE MINIMUM EXPEXCTED OPPORTUNITY LOSS/REGRET

PRINCIPLE

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MAXIMUM LIKELIHOOD PRINCIPLE

HERE, WE CHOOSE TO STOCK ACCORDING TO THAT DEMAND STATE WHICH HAS MAXIMUM PROBABILITY OF OCCURRENCE. IN THIS CASE, THE WHOLE SALER SHOULD STOCK 12 CASES, IF HE ADOPTS THIS PRINCIPLE.

DAILY SALES ( CASES)

PROB. OF DEMAND

10 0.15

11 0.20

12 0.40

13 0.25

TOTAL 1.00

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MAXIMUM EXPECTATION PRINCIPLE

HERE, WE CHOOSE THAT STRATEGY WHICH HAS MAXIMUM EXPECTED PAYOFF. THIS IS THE MOST ACCEPTABLE PRINCIPLE SINCE, THE EXPECTED PAYOFF WILL ALWAYS COME TRUE IN THE LONG RUN.

IN OUR EXAMPLE, THE WHOLE SALER SHOULD CHOOSE THAT STRATEGY WHICH WILL MAXIMIZE HIS EXPECTED PROFIT.

NOW, WE HAVE TO EXAMINE THE EXPECTED PROFIT FOR EACH STRATEGY ( STOCK ACTION).

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EXPECTED PROFIT FROM PURSUING A STRATEGY OF STOCKING 10 CASES

DEMAND IN CASES

CONDITIONAL PROFIT

PROB. OF DEMAND

EXPECTED PROFIT

(US $)

10 300 0.15 45.0

11 300 0.2 60.0

12 300 0.4 120.0

13 300 0.25 75.0

TOTAL 1.00 300.0

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EXPECTED PROFIT FROM PURSUING A STRATEGY OF STOCKING 11 CASES

DEMAND IN CASES

CONDITIONAL PROFIT

PROB. OF DEMAND

EXPECTED PROFIT

(US $)

10 280 0.15 42.0

11 330 0.2 66.0

12 330 0.4 132.0

13 330 0.25 82.5

TOTAL 1.00 322.5

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EXPECTED PROFIT FROM PURSUING A STRATEGY OF STOCKING 12 CASES

DEMAND IN CASES

CONDITIONAL PROFIT

PROB. OF DEMAND

EXPECTED PROFIT

(US $)

10 260 0.15 39.0

11 310 0.2 62.0

12 360 0.4 144.0

13 360 0.25 90.0

TOTAL 1.00 335.0

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EXPECTED PROFIT FROM PURSUING A STRATEGY OF STOCKING 13 CASES

DEMAND IN CASES

CONDITIONAL PROFIT

PROB. OF DEMAND

EXPECTED PROFIT

(US $)

10 240 0.15 36.0

11 290 0.2 58.0

12 340 0.4 136.0

13 390 0.25 97.5

TOTAL 1.00 327.5

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DECISION TREE SO, THE MAXIMUM PROFIT COMES FROM PURSUING A STRATEGY

OF STOCKING 12 CASES.

NOW, HOW DO WE REPRESENT THIS LOGIC DIAGRAMATICALLY? WELL, WE USE A DIAGRAM CALLED A DECISION TREE.

IN A DECISION TREE, WE REPRESENT A DECISION ( STATEGY) WITH A TECTANGLE WHILE AN OUTCOME ( DEMAND IN THIS CASE) IS REPRESENTED BY A CIRCLE.

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$ 335.0

$ 300.0

DECISION TREE FOR THE WHOLESALER PROBABILITIES

$ 335.0

$ 327.5

$ 322.5

10,45.0, 0.15

13,75,0.25

11,60,0.2

12,120.0,0.4

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DECISION MAKING WITH PERFECT INFORMATION (EVPI)

IF WE GO BACK TO THE PAYOFF MATRIX, WE HAVE MARKED OUR BEST DECISIONS FOR EACH DEMAND SCENARIO. IF WE COULD HAVE PERFECT INFORMATION ABOUT THE MARKET DEMAND, OUR EXPECTED PAYOFF TABLE WOULD HAVE LOOKED LIKE THIS.

PAYOFF MATRIX IN US $

MARKET DEMAND

( CASES)

CONDITIONAL PROFIT

PROB. OF DEMAND

EXPECTED PROFIT

( US $)

10 300 0.15 45.0

11 330 0.2 66.0

12 360 0.4 144.0

13 390 0.25 97.5

TOTAL 1.00 352.5

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DECISION TREE FOR THE WHOLESALER WITH PERFECT INFORMATION 10, 300.0

13, 240.0

11, 280.0

12, 260.0

$ 352.5

$ 300.0

$ 330.0

$ 360.0

$ 390.0

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ECONOMIC VALUE OF PERFECT

INFORMATION ( EVPI)

INCREASE IN EXPECTED PROFIT WITH PERFECT INFORMATION = ( 352.5- 335) $

= 17.5 $

THIS IS KNOWN AS THE ECONOMIC VALUE OF PERFECT INFORMATION ( EVPI).

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DECISION MAKING WITH REVISED

PROBABILITIES LET, THERE BE A MARKET RESEARCH FIRM, THAT PROVIDES ADDITIONAL

INFORMATION ( FORECASTING) ABOUT THE POSSIBLE STATE OF DEMAND, AND CHARGES A FEE FOR THE SAME.

WHAT IS THE ADDITIONAL VALUE OF THIS FORECAST AND HOW MUCH CAN THE WHOLESALER PAY FOR IT?

THE AGENCY FORECASTS THE DEMAND BY RATING IT AS ABOVE NORMAL (AN) OR BELOW NORMAL (BN).

IT HAS BEEN OBSERVED IN THE PAST THAT IN 80% OF THE INSTANCES, WHEN THE DEMAND WAS 10 CASES, THE AGENCY HAD FORECASTED BELOW NORMAL (BN). IN 60% OF THE INSTANCES, WHEN THE DEMAND WAS 11 CASES, THE AGENCY HAD FORECASTED BELOW NORMAL (BN). IN 30% OF THE INSTANCES, WHEN THE DEMAND WAS 12 CASES, THE AGENCY HAD FORECASTED BELOW NORMAL (BN). IN 20% OF THE INSTANCES, WHEN THE DEMAND WAS 13 CASES, THE AGENCY HAD FORECASTED BELOW NORMAL (BN).

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REVISED PROBABILITIES (USING BAYE’S THEOREM)

FORECAST EVENT P( EVENT) P(FORECAST/EVENT)

P(FORECAST & EVENT)

REVISED PROB. (EVENT/FORECAST)

ABOVE NORMAL (AN)

10 0.15 0.2 0.03 0.05

11 0.2 0.4 0.08 0.14

12 0.4 0.7 0.28 0.47

13 0.25 0.8 0.2 0.34

TOTAL 1.00 P(AN) = 0.59 1.00

BELOW NORMAL

(BN)

10 0.15 0.8 0.12 0.29

11 0.2 0.6 0.12 0.29

12 0.4 0.3 0.12 0.29

13 0.25 0.2 0.05 0.13

TOTAL 1.00 P(BN) = 0.41 1.00

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$ 335.165

DECISION TREE FOR THE REVISED PROBABILITIES

$ 335.0

$335.165

$ 348.14

$ 316.5

A

D

C

B

A’

B’

C’

D’

DO NOT BUY FORECAST

BUY FORECAST

EVPI= $ 0.165

10

13

11

12

AN, 0.59

BN, 0.41

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NODE ‘A’

$ 300.0

10,300.0,0.05

11,300,0.14

12,300.0, 0.47

13, 300.0, 0.34

STOCKING 10 CASES

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NODE ‘B’

$ 327.46

10,280.0,0.05

11,330,0.14

12,330.0, 0.47

13, 330.0, 0.34

STOCKING 11 CASES

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NODE ‘C’

$ 348.14

10,260.0,0.05

11, 310.0,0.14

12,360.0, 0.47

13, 360.0, 0.34

STOCKING 11 CASES

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NODE ‘D’

$ 345.08

10,240.0,0.05

11, 290.0,0.14

12,340.0, 0.47

13, 390.0, 0.34

STOCKING 11 CASES

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NODE ‘ A’ ’

$ 300.0

10,300.0,0.29

11,300,0.29

12,300.0, 0..29

13, 300.0, 0.13

STOCKING 10 CASES

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NODE ‘ B’ ’

$ 315.50

10,280.0,0.29

11,330,0.29

12,330.0, 0..29

13, 330.0, 0.13

STOCKING 11 CASES

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NODE ‘ C’ ’

$ 316.50

10,260.0,0.29

11,310,0.29

12,360.0, 0..29

13, 360.0, 0.13

STOCKING 12 CASES

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NODE ‘ D’ ’

$ 303.00

10,240.0,0.29

11,290,0.29

12,340.0, 0..29

13, 390.0, 0.13

STOCKING 13 CASES

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MINIMUM EXPECTED

REGRET/OPPORTUNITY LOSS PRINCIPLE THIS PRINCIPLE WOULD GIVE THE SAME ANSWER AS THE

PREVIOUS PRINCIPLE.

THIS IS BECAUSE, ER(j)+ EP(j)= EPPI

( EXPECTED PAYOFF UNDER PERFECTINFORMATION)

HENCE, THE STRATEGY THAT WOULD MINIMIZE ER(j) WOULD AUTOMATICALLY MAXIMIZE EP(j).

THE MIN. ER(j) IS ALSO THE EVPI IN THIS CASE.

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EXPECTED OPPORTUNITY LOSS/REGRET FOR DIFFERENT STOCK ACTIONS

STOCK ACTION

( CASES)

EXPECTED REGRET

( US $)

10 52.5

11 30.0

12 17.5

13 25.0

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EVSI AND EFFICIENCY OF EVSI

IN OUR EXAMPLE, ECONOMIC VALUE OF SAMPLE INFORMATION ( EVSI) = (335.165- 335) $

= 0.165 $

EFFICIENCY OF EVSI = 0.165/335 = 0.05 %

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QUESTIONS PLEASE

THANK YOU