It has 4 phases 1Action or acts 2State of nature or events or outcome 3Pay off and pay off table or pay off matrix Decision A decision problem may be represented by tree diagram Decision making problems deals with the selection of single act from a set of acts ID: 795898
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Slide1
DECISION THEORY
Slide2It’s deals with a very scientific and quantitative way of coming to decision.
It has 4 phases.
1.Action or acts.
2.State of nature or events or outcome.
3.Pay off and pay off table or pay off matrix.
Decision
A decision problem may be represented by tree diagram
Slide3Decision making problems deals with the selection of single act from a set of acts.
There can be 2 or more acts denoted by A1,A2,A3….An.
action space A = {A1 A2 A3 ……….An}Decision tree of acts Tabular form of reprehensive acts
Action or acts
action actsA1 A2 ………AnA1 A2 . An
Slide4Each act is associated with one or more events or state of natures.
There events are the outcome of consequence of an act.
Events are denoted by E1 E2 ….EnE = {E1 E2 ………En} is a set of events.Tree diagram of events. Tabular form of events.
Events or state of natures
E EE1 E2 ………EnE1 E2 . En
Slide5In decision problems it is required to measure the degree to which the decision maker’s objectives is achieved.
Monetary value is used
or a measure to represent achievement or lack of achievement.This monetary gain or loss is called a pay off.Pay off is expressed as profit, loss cost satisfaction etc.
Pay off & pay off table
E AE1E2
……EnA1
P11
P12
P1n
A2
P21
P22
P2n
.
....AmAm1Am2.Amn
PAY OFF TABLE
TREE DIAGRAM OF PAY OFF
Slide6Once a pay off table is read no its turn to some decision.
There are 3 decisions making situations.
Decision under uncertainty.(without problem) Decision under risk.(with problem)Decision under certainty.
Decision making situations
Slide7The probabilities of the states of nature is not known.
Decision is taken on the basis of 4 criteria.
Maxi min or mini maxMaxi max or mini minMini max reg.Laplace.
Decision under uncertainty.(without problem
)
Slide8Maxi min => maximize the minimum
Minimax => minimize the maximum
Maximin : find the pay off using maximinMinimum profit/pay off for Miximum pay off of minimum profit.
A2 act is chosen.
Pacimistic approachA18A2
40A3-25
E
A
E1
E2
E3A187050A2504540
A3-25-10
0
A2
40
Slide9Minimax :- find the pay off using minimax
Maximum cost
minimax of maximum
cost = 100 A3 act is chosen.A1
700A2900
A3
100
E
A
E1E2E3A150700500A210
500900A3100
6080
Slide10Maximax
=> maximum of maximum profit (optimistic approach)
Maximum pay off =
Maximum of maximum pay off =
A1 = act is chosen according to maximaxMinimum criteria.Minimine pay off minimum of minimum cost =
A3 act is chosen according to minimum
A1
7
A2
4
A3
6
E AE1
E2E3E4
A1-5
070
A2
-4
-3
3
4
A3
-6
-7
6
2
A1 =
7
A3 =
-7
A1
-5
A2
-4
A3
-7
Slide112)
1)Cal the maximum of E (regret pay off) 3)take max of each row
max reg. minimum of
4)take minimum of this (max of reg. pay off) A3 act is chosen
Minimax regret or minimax opportunity loss
E
A
E1
E2
E3E4A11812149A2151411
11A313
161716
E
A
E1
E2
E3
E4
A1
18
12
14
9
A2
15
14
11
11
A313
161716E118E2
16
E3
17
E4
16
A1
7A26A35
Slide12 200
Find the average pay off for each act.Find the maximum av from step(1)
A1 is chosen.
Laplase(equally likely criteria)
E AE1E2
E3
AV
A1
200
200
200
200
A2
175205195195.6A3150180210180
Slide13In such problems uncertainty is there but probability is given may be from past experience.
In such problems 2 methods are used:
Using EMV(expected monetary value)Using EOL(expected opportunity loss)Decision making under risk(probability given)
Slide14A baker buys veg cutlet at rs.2 & sell it for rs.5. at the end of the day unsold veg cutlets are given to the poor for free of cost.
The following table shows the sales of veg cutlets during the past 100 days.
total = 100days
Decision under risk by(EMV) method consider
Daily sale10111213
No. of days1520
40
25
Slide15Now the question is how many veg cutlets the baker has to stock every day in order to maximise his profit?
The 4 events are:
E1 = demand for 10 cutlets E2 = .. .. 11 .. E3 = .. .. 12 ..
E4 = .. .. 13 ..
The 4 acts are: A1 = stock of 10 cutlets profit on 1 cutlet = rs.3
A2 = .. .. 11 .. A3 = .. .. 12 .. A4 = .. .. 13 ..
Slide16net profit is called conditional pay off
Conditional pay off for each act event combination
Pay off for A1.E1 = 10×3 = 30Pay off for A1.E2 = 10×3 = 30 as 50 onPay off for A2.E1 = 10×3 -2 = 28
Pay off for A2.E2 =
11×3
=
33 as soon.
P(selling 10 cutlets) = 15/100 = 0.15
P(selling
11
cutlets) =
20/100 = 0.20
P(selling 12 cutlets) = 40/100 = 0.40P(selling 13 cutlets) = 25/100 = 0.25
E AE1 10
E2 11E3 12
E4 1310 A1
30
30
30
30
11 A2
28
33
33
33
12 A3
26
31
36
36
13 A4
24
29
34
39
Slide17Expected conditional pay off is given by the multiplying each conditional pay off by the corresponding probability,
expected conditional pay off for A1.E1 = 30(.15) = 4.5 expected conditional pay off for A1.E2 = 30(.20) = 6 expected conditional pay off for
A1.E3 = 30(.40)
= 12 expected conditional pay off for A1.E4 = 30(.25) = 7.5And so on…Table for expected conditional pay off
E
A
E1
E2
E3
E4A14.56127.5A24.26.613.28.25
A33.96.2
14.49
A43.65.8
13.6
9.75
Slide18EMV (expected monetary value) for A1 = 4.5+ 6 + 12 + 7.5 = 30
EMV
(expected monetary value) for A2 = 32.5EMV (expected monetary value) for A3
= 33.5EMV
(expected monetary value) for A4 = 32.5 Since, EMV is maximum for act 3 i.e. A3 = 33.5 act A3 is chosen.i.e. 12 veg cutlets are to be stocked every day for maximum profit
Slide19It is same as (EMV) method only the difference is;
After finding the conditional pay off regret pay off has to found. This new table is called conditional opportunity loss table(COL).
The product of col and the corresponding probability given expected COL.The sum of all expected COL is act wise given EOL.The minimum of EOL is selected as or act.Decision under risk by(EOL) method
Slide20A newspaper boy purchases magazines at rs.3 each & sales them at rs.5 each. He cannot return the unsold magazines. The probability distribution of the demand for the magazine is given below.
Determine how many copies of magazines should he purchases daily by EOL method
Demand
16
17181920probability0.1
0.150.20.250.3
Slide21demand
stock conditional pay off table
For conditional pay off;
cp = 3 & sp = 5Profit is rs.2 on each magazine.
conditional pay off for A1E1 = 16 × 2 = 32
A2E1 = 32 – 3 = 29
A3E1 = 32 – 6 = 26
E
AE1 16E2 17E3 18E4 19E5 20A1 1632
323232
32A2 1729
3434
34
34
A3 18
26
31
36
36
36
A4 19
23
28
33
38
38
A5 20
20
25
30
35
40
0.1 0.15 0.2 0.25 0.3
Slide22CONDITIONAL OPPORTUNITY LOSS TABLE
For E1 = (32~x) , x
Є
E1For
E2 = (34~x) , x Є E2 and so on…
E A
E1
E2
E3
E4
E5
A1
0
2
468A230246A3
6302
4A4
963
0
2
A5
12
9
6
3
0
Slide23Expected COL = p(E) × COL
minimum EOL A3 is chosen
18 magazine should be purchasedBoth COL & EMV are same result
E A
E1
E2
E3
E4
E5
EOL
A1
0
0.30.81.52.45A20.300.4
11.83.5
A30.6
0.450
0.5
1.2
2.75
A4
0.9
0.9
0.6
0
0.6
3
A5
1.2
1.35
1.2
0.75
0
4.5
Slide24THANK YOU