Decision Analysis

• Decision Analysis provides a formal analytic framework for decision making under uncertainty. It is used to determine optimal strategies where a decision maker is faced with several decision alternatives and an uncertain, or risky, pattern of future events. The decision making process involves the following steps:

a) Identification of the various possible outcomes, called states of nature or events, Ei ‘s, for the decision problem. The events are beyond the control of the decision maker.

b) Identification of all the courses of action Aj’s or the strategies that are available to the decision maker. He has control over choice of these.

c) Determination of pay-off values which describes the consequences resulting from the different combinations of the acts and events. Pay-off’s are denoted by Vij

d) Choosing best alternative among the others using some criteria.

• Pay-off Table: It depicts the economics of the problem. A pay-off is a conditional value-a conditional profit/loss or cost. It is conditional in the sense that associated with each course of action, is certain profit/loss, given that certain event has occurred. Thus the profit/loss resulting by the adoption of a certain strategy is dependent upon the particular event that may occur.

• Regret (Opportunity Loss) table: It is defined as the amount of pay-off foregone by not adopting the optimal course of action which would give the highest pay-off for each possible event .

• The greater the departure from the optimal strategy the lesser the profit earned and consequently the greater the opportunity loss (regret).

Different decision rules under uncertainty

1) Laplace Principle: Assigning equal probabilities to the possible payoffs for each action and then selecting that alternative which corresponds to the maximum expected payoff.

2) Maximin Principle: It is based upon the conservative approach (extreme pessimism) to assume that the worst possible is going to happen. The decision maker considers each strategy and locate the minimum payoff for each and then selects that alternative which maximizes the minimum payoff.

3) Minimax Principle: Basically used when costs involved. Here first determine the maximum possible cost for each alternative and then select that alternative corresponds to the minimum of the above costs.

4) Maximax principle: It is based on extreme optimism. Here one may selects that particular strategy which corresponds to the maximum of the maximum payoff for each strategy.

5) Minmin Principle: Usually used when costs are involved. Here find the minimum for each alternatives and then select that alternative which minimizes the above minimum costs.

6) Hurwicz principle: This principle lies between maximin and maximax criterion. The criterion provides a mechanism by which balance between extreme pessimism and extreme optimism is made by weighting (between 0 and 1) them with certain degrees of optimism and pessimism. Let α be the degree of optimism 0 <= α <= 1. Find maximum and minimum payoff for each alternative and obtain the quantity:

h = α (maximum) + (1- α) minimum for each alternative. Then choose the alternative which has maximum h.

7) Savage principle: This criterion is based on the regret or opportunity loss. Here find the regret for the ith alternative when event j occurs. Determine maximum regret for each alternative and then choose the alternative which has minimum regret among the all other regrets.

ith regret = ( max. payoff – ith payoff) if payoffs are profits = (ith payoff – max. payoff) if payoffs are costs

Shivam Bookstore (SB) Problem

• SB sells a particular book of tax laws for Rs.100. It purchases the book for rs. 80 per copy. Since some of the tax laws change every year, the copies unsold at the end of a year become outdated and can be disposed off Rs.30 each. According to past experience, the annual demand for this book is between 18 an d 23 copies. Assuming the order can be placed only once during a year, decide how many copies of the book should be purchased for the next year.

a) Construct Payoff tableb) Construct regret tablec) Evaluate each principle stated above and state your decision.

Take α = 0.6 for Hurwicz principle.

1) Payoff table.

Given S =100, P = 80 and scrap = 30. Let D denote the demand and Q the quantity to be purchased.

When D >= Q, Profit = 100Q – 80Q = 20Q.

When D < Q, Profit = 100D + 30(Q-D) – 80Q = 70D-50Q.

Hence P = 20Q when D>=Q

= 70D-50Q when D<Q

Payoff Table is given below.

A1

18

A219

A320

A421

A522

A623

D118

360 310 260 210 160 110

D219

360 380 330 280 230 180

D320

360 380 400 350 300 250

D421

360 380 400 420 370 320

D522

360 380 400 420 440 390

D623

360 380 400 420 440 460

2) Regret table:

A118

A219

A320

A421

A522

A623

D118

0 50 100 150 200 250

D219

20 0 50 100 150 200

D320

40 20 0 50 100 150

D421

60 40 20 0 50 100

D522

80 60 40 20 0 50

D623

100 80 60 40 20 0

1) Laplace principle: Here we assume equal probability of 1/6 to all alternatives. Since the maximum expected payoff attains for the action A2, the book store manager would by 19 copies of the book.

A1

18

A2

19

A3

20

A4

21

A5

22

A6

23

D1:18 360 310 260 210 160 110

D2:19 360 380 330 280 230 180

D3:20 360 380 400 350 300 250

D4:21 360 380 400 420 370 320

D5:22 360 380 400 420 440 390

D6:23 360 380 400 420 440 460

Expetd

360 363.3 365 350 323.3 285

2) Maximin criterion: Find minimum payoff for each strategies. Hence strategy A1 is selected because it has max. payoff out the minimums.

3) Maximax criterion: Find maximum payoff for each strategies and select max. strategy. Here the manager will choose strategy A6.

A1 A2 A3 A4 A5 A6

360 310 260 210 160 110

A1 A2 A3 A4 A5 A6

360 380 400 420 440 460

4) Hurwicz criterion: Here α = 0.6 so that (1- α) = 0.4. Calculate h = α(max.value) + (1- α) (min.value). Choose the maximum h. Hence bookstore go for strategy A1.

5) Savage criterion: Find the maximum regret for each strategy. Then choose the minimum regret. Hence bookstore must go for strategy A2.

Max Min h

A1 360 360 360

A2 380 310 352

A3 400 260 344

A4 420 210 336

A5 440 160 328

A6 460 110 320

A1 A2 A3 A4 A5 A6

100 80 100 150 200 250

• When a decision maker chooses from among several options whose probabilities of occurrence can be stated, he is said to take decisions under risk. The commonly used criterion is the expected monetary value (EMV) criterion.

A1

18

(0.05)

A2

19

(0.1)

A3

20

(0.3)

A4

21

(0.4)

A5

22

(0.1)

A6

23

(0.05)

D1

18

360 310 260 210 160 110

D2

19

360 380 330 280 230 180

D3

20

360 380 400 350 300 250

D4

21

360 380 400 420 370 320

D5

22

360 380 400 420 440 390

D6

23

360 380 400 420 440 460

EMV 360 376.5 386 374.5 335 288.5

Since the max.EMV is with strategy

A3. hence buy 20 copies

• An alternative approach to maximizing the expected monetary value (EMV) is to minimize expected opportunity loss (EOL) or expected regret. In this case, we have Max (EMV) = Min (EOL) .

A1

18

(0.05)

A2

19

(0.1)

A3

20

(0.3)

A4

21

(0.4)

A5

22

(0.1)

A6

23

(0.05)

D118

0 50 100 150 200 250

D219

20 0 50 100 150 200

D320

40 20 0 50 100 150

D421

60 40 20 0 50 100

D522

80 60 40 20 0 50

D623

100 80 60 40 20 0

EOL 51 34.5 25 36.5 76 122.5

Since the min.EOL is with strategy

A3. hence buy 20 copies

• Expected value of perfect information is defined as the difference between expected payoff of perfect information and max EMV,

• i.e. EVPI = EVWPI-EVWOPI.Where EVWOPI=max (EMV) EVWPI = (0.05)*360 + …

+(0.05)*460 = 411This is the highest profit that

the store can make when perfect information is available.

Hence EVPI = 411-386 = 25. Thus the store should pay no more than Rs.25/year for obtaining such information.

Acts A1

18

A2 19

A3

20

A4 21

A5 22

A6 23

Payoff

360 380 400 420 440 460

Probs

0.05 0.1 0.3 0.4 0.1 0.05

Suppose that an agency undertakes to supply this

information. When no information is

available, profit =386 andWhen information is available

Profit = 411