Decision Theory

transcript

DECISION THEORY

Presenter:Apolonio Cabangal Jr.

TOPICS TO DISCUSS

IntroductionPayoff MatrixRegret TableDecision Making Under UncertaintyDifferent Methods of Decision Making Under UncertaintyDecision Making Under Risk

TOPICS TO DISCUSS(CONTINUED)

Different methods of decision making under risk

Maximum likelihood principlesMaximum expectation principlesConcept of decision treeDecision 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

INTRODUCTION

External environment imposes certain constrain on us. Based on our response to such constrains, we get different payoffs.We cannot 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

EXAMPLE

A wholesaler of fruits buy strawberries at P 20 a case and sells them at P 50 a case. The product is perishable by nature and cannot be stored overnight. It has be sold on the day of purchase itself.From experience, the wholesaler knows that the daily demand will range between 10 to 13 cases.Every case of strawberries bought and would not sold will lead to a marginal loss of P 20, while every case that could not be sold because of stock out would lead to an opportunity loss of P 30.

PAYOFF MATRIX IN PHP

POSSIBLE DEMAND IN

POSSIBLE STOCK ACTION IN CASES

10 11 12 13

10 300 280 260 24011 300 330 310 29012 300 330 360 34013 300 330 360 390

REGRET TABLE IN PHP

Possible Demands 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

DECISION UNDER UNCERTAINTYHere, we are no information about the likelihood(probability) of any particulars state of demand.In such a condition, we are making decision under uncertainty.The following principles used to take decisions under uncertainty: Laplace principles Maximin or Minimax criterion Maximax or Minimin criterion Hurwicz criterion Salvage principle

LAPLACE PRINCIPLE

Stock Action(Case

Payoff(PHP)

10 30011 317.512 322.513 315

Assumes all external constrains(here demand of strawberries in cases to be equiprobable)

Maximum Payoff is achieved by pursuing the strategy of stocking 12 units.

MAXIMIN. OR MINIMAX. PRINCIPLE

Stock Action(Cases)

Min. Payoff(PHP)

10 300

11 280

12 260

13 240

Here, minimum payoffs from each strategy with max. of this profits is selected. This is known as maximin principle. For cost, the strategy with min of maximum cost is chosen. That is known as minimax. Principle. This is pessimistic decision making.Maximum payoffs is achieved by pursuing the strategy of stocking 10 units.

MAXIMAX. OR MINIMIN PRINCIPLE

Stock Action(cas

MAX. Payoff(PH

10 300

11 330

12 360

13 390

Here are the maximum payoffs from each strategy with max. of these maximum profit is selected. This is known as maximax. principle for cost, the strategy with min. of the minimum cost is chosen. That is known as minimin. principle. This is highly optimistic decision making.Maximum payoff is achieved by pursuing the strategy of stocking 13 units.

HURWICZ PRINCIPLE

Index of optimism = αCriterion Value= α(Max. Profit) + (1- α)(Min. Profit)For costs,Criterion Value = α(Min. Cost) + (1- α)(Max. Cost)Criterion Value = α(Max. Payoff) + (1- α)(Min. payoff)

α=0 stand for maximin and minimax principleα=1 stand for maximax or minimin principleFor our example, let us assume for an index of optimism of 60%(α=0.6)

HURWICZ PRINCIPLE CONTINUED

Stock Action(Case

Hurwicz Criterion

Value (Php)

10 300

11 310

12 320

13 330

The strategy with maximum Hurwicz criterion value is chosen.Here, the maximum hurwicz criterion value is achieved pursuing the strategy of stocking 13 units.

SALVAGE PRINCIPLE

Stock Action(Case

Max. Regret Value(PHP)

Here, we select the strategy that minimizes the maximum regret. This is also pessimistic decision making.Here, the minimum of max. regret(salvage) value is achieved by pursuing the strategy of stocking 12 units.

DECISION MAKING UNDER RISK

Now, let us assume that based on the sales of past 100 days, the wholesaler has the 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

DECISION MAKING UNDER RISK(CONTINUATION)

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 techniques is called decision making under risk.The following principles are used to arrived at the optimal decision.

Maximum likelihood principleMaximum expectation principleMinimum expected opportunity loss/regret principle

MAXIMUM LIKELIHOOD PRINCIPLE

Here, we choose to stock according to that demand state which has maximum probability of occurrence. In this case, the wholesaler should stock 12 cases, if he adopts this principles.

Daily Sales(Cases) Prob. Of Demand

10 0.15

11 0.20

12 0.40

13 0.25

Total 1.00

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 wholesaler should choose that strategy which will maximize his expected profit. Now, we have to examine the expected profit for each strategy(stock action).

EXPECTED PROFIT FROM PURSUING A STRATEGY OF STOCKING 10 CASES

Demand in Cases

Conditional Profit

Prob. of Demand

Expected Profit(PHP)

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

Demand in

Conditional

Profit

Prob. of

Demand

Expected

Profit

10 280 0.15 42.0

11 330 0.20 66.0

12 330 0.40 132.0

13 330 0.25 82.5

Total 1.00 322.5

Demand in

Conditional

Profit

Prob. of

Demand

Expected

Profit

10 260 0.15 39.0

11 310 0.20 62.0

12 360 0.40 144.0

13 360 0.25 90.0

Total 1.00 335.0

Demand in

Conditional

Profit

Prob. of

Demand

Expected

Profit

10 240 0.15 36.0

11 290 0.20 58.0

12 340 0.40 136.0

13 390 0.25 97.5

Total 1.00 327.0

DECISION TREE

So, the maximum profit come from pursuing a strategy of stocking 12 cases.Now, how do we represent this logic diagrammatically? Well, we use a diagram called decision tree.In a decision tree, we represent a decision(strategy) with a rectangle while an outcome(demand in this case) is represented by a circle.

DECISION TREE FOR THE WHOLESALER PROBABILITIES

P 335.0

P 300.0

P 335.0

P 327.5

P 322.5

10,45.0, 0.15

13,75,0.25

11,60,0.2

12,120.0,0.4

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 PHP.

Market Demand(Cas

Conditional Profit

Prob. of Demand

Expected Profit in

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

DECISION TREE FOR THE WHOLESALER WITH PERFECT INFORMATION

10, 300.0

13, 240.0

11, 280.0

12, 260.0

P 300.0

P 330.0

P 360.0

P 390.0

$ 352.5

ECONOMIC VALUE OF PERFECT INFORMATION ( EVPI)

Increase in expected profit with perfect information.= ( 352.5- 335) P

= 17.5 P This is known as economic value of perfect information(EVPI)

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% 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).

REVISED PROBABILITIES(USING BAYE’S THEOREM)

Foreca

stEvent P(Event)

P(Forecast

/Event)

P(Foreca

st&Event

Revised

(Event/Forec

Normal

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

Normal

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

DECISION TREE FOR THE REVISED PROBABILITIES

P 335.0

P 335.165

P 348.14

P 316.5

DO NOT BUY FORECAST

BUY FORECAST

EVPI= P 0.165

AN, 0.59

BN, 0.41

P 335.165

NODE ‘A’

P 300.0

10,300.0,0.05

11,300,0.14

12,300.0, 0.47

13, 300.0, 0.34

STOCKING 10 CASES

NODE ‘B’

P 327.46

10,280.0,0.05

11,330,0.14

12,330.0, 0.47

13, 330.0, 0.34

STOCKING 11 CASES

NODE ‘C’

P 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

NODE ‘D’

P 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

NODE ‘ A’ ’

P 300.0

10,300.0,0.29

11,300,0.29

12,300.0, 0..29

13, 300.0, 0.13

STOCKING 10 CASES

NODE ‘ B’ ’

P 315.50

10,280.0,0.29

11,330,0.29

12,330.0, 0..29

13, 330.0, 0.13

STOCKING 11 CASES

NODE ‘ C’ ’

P 316.50

10,260.0,0.29

11,310,0.29

12,360.0, 0..29

13, 360.0, 0.13

STOCKING 12 CASES

NODE ‘ D’ ’

P 303.00

10,240.0,0.29

11,290,0.29

12,340.0, 0..29

13, 390.0, 0.13

STOCKING 13 CASES

MINIMUM EXPECTED REGRET/OPPORTUNITY LOSS PRINCIPLE

This principle would give same answer as the previous principle.This is because, ER(j)+Ep(j)=EPPI (Expected Payoff under Perfect Information)Hence, the strategy that would minimize ER(j) would automatically maximize EP(j).The Min. ER(j) is also the EVPI in this case.

EXPECTED OPPORTUNITY LOSS/REGRET FOR DIFFERENT STOCK ACTIONS

Stock Action(Cases) Expected Regret (PHP)

10 52.5

11 30.0

12 17.5

13 25.0

EVSI AND EFFICIENCY OF EVSI

In our example, economic of sample information(EVSI)=(335.165- 335) P

= 0.165 P Efficiency of EVSI = 0.165/335 = 0.05 %

QUESTIONS PLEASE

THANK YOU

Decision Theory

Documents