----Decision Analysis

1 2528162216

12528 1

2 2

2 3

(Alternative, State of Nature, Payoff)Aii=1mA=A1A2AmSij=1nS=S1S2Sn

SjAirij=RAiSj1234

1(Pay off Matrix)R=rijmn i=12mj=12n2(payoff table)

(Decision Making without probability)

(,conservative approach)

r*

A3

S1S2S3

A13614-8-8A2201600A3141033

(optimistic approach)

r*

A1

S1S2S3

A13614-836A22016020A31410314

(Hurwicz decision criterion)

r*01=1=0dii

=0.6

= 18.4 d2=0.620+0.40= 12 d3=0.614+0.43= 9.6A1

S1S2S3

A13614-8-836A220160020A314103314

(minimum regret approach)

h*hijSjAih*

(i=1mj=1n) 6

=11A1

S1S2S3A1021111A2160316A3226022

Laplace decision criterionn1/n

r*ERAiAi

1/3

A1

(Decision Making with Probability)

120 7

25(S1)26(S2)27(S3)28(S4)PS1=0.1PS2=0.3PS3=0.5PS4=0.125A115015015015026A213415615615627A311814016216228A4102124146168

r*

4

A3

25(S1)26(S2)27(S3)28(S4)PS1=0.1PS2=0.3PS3=0.5PS4=0.125A115015015015026A213415615615627A311814016216228A4102124146168

(expected value approach)

Ak

ERA1=0.1150+0.3150+0.5150+0.1150=150.0ERA2=0.1134+0.3156+0.5156+0.1156=153.8ERA3=0.1118+0.3140+0.5162+0.1162=151.0ERA4=0.1102+0.3124+0.5146+0.1168=137.2

A2

25(S1)26(S2)27(S3)28(S4)PS1=0.1PS2=0.3PS3=0.5PS4=0.125A115015015015026A213415615615627A311814016216228A4102124146168

AkhijSjAi

ELA1=0.10+0.36+0.512+0.118=9.6ELA2=0.116+0.30+0.56+0.112=5.8ELA3=0.132+0.316+0.50+0.16=8.6ELA4=0.148+0.332+0.516+0.10=22.4

A2

25(S1)26(S2)27(S3)28(S4)PS1=0.1PS2=0.3PS3=0.5PS4=0.125A106121826A216061227A332160628A44832160

ERAiELAi

(decision tree)2

S1S2S3ER(Ai)A13614-816.2A22016014A3141039.8

1

7 0.80.250030010011

11

2

(expected value of perfect information, EVPI)S1

S1S2S3ER(Ai)A13614-816.2A22016014A3141039.8

rj*SjEPPIER* EVPI=EPPIER*

r1* =36 r2* =16 r3*=3 EPPI=0.336+0.516+0.23=19.4A1 ER*=max{16.2149.8}=16.2=ERA1 EVPI=EPPIER*=19.416.2=3.23.23.2

(Bayes Decision)(Sample Information)(Expected Value of sample information)(prior probabilities)(posterior probabilities)

12

PSjSj PBk|SjSjBkPSj|BkBkSj

3

ABABAPA|B10010%ABPA=10/100 PB|A=90/99

90%30%75%S1S2 B1B2 B1 PB1|S1=90%PS1=75% PS2=25%PB1|S2=30% PS1|B1=0.9PS1=75%

S1S2B1B2B2PB2|S1=10%PS1=75% PS2=25%PB2|S2=70%

9 101.50.80.20.50.30.71211.5 2

PB1 = P(S1)P(B1|S1)+P(S2)P(B1|S2) +PS3PB1|S3 = 0.30.8+0.50.5+0.20.3 = 0.55

PB2 = P(S1)P(B2|S1)+P(S2)P(B2|S2) +PS3PB2|S3 = 0.30.2+0.50.5+0.20.7 = 0.45

ERA1=P(S1|B1)r11 + P(S2|B1)r12 + P(S3|B1)r13 =0.436436 + 0.454514 + 0.1091(8) =21.2() ER(A2) =0.436420 + 0.454516 + 0.10910 =16() ER(A3) =0.436414 + 0.454510 + 0.10913 =10.98() ER*(B1) =max{21.21610.98}=21.2()=ERA1A1

ERA1=P(S1|B2)r11 + P(S2|B2)r12 + P(S3|B2)r13 =0.133336 + 0.555614 + 0.3111(8) =10.09 ERA2=0.133320+ 0.555616 + 0.31110 =11.56 ERA3=0.133314 + 0.555610 + 0.31113 =8.36 ER*B2=max{10.0911.568.36} =11.56=ERA2A2

ERI=P(B1)ER*(B1)+P(B2)ER*(B2) =0.5521.2 + 0.4511.56=16.862 EVSI=ERIER* =16.86216.2=0.6621.5

PDCPDC (1)30 (2)60 (3)90

PDC : (s1)(s2)(d1)(d2)(d3)8142075-9

. ()1. : max max rij=r31=20 , d3 : . 2. max min rij=r12=7 , d1: .

PDC : (s1)(s2)max rijmin rij(d1)(d2)(d3)8142075-98142075-9

3. : min max hij=h22=6 , d2 : .

PDC : rijhijmax hij(s1)(s2)(s1)(s2)(d1)(d2)(d3)8142075-91260021612616

. () 1. PDC0.80.2 ER(d1)=0.8 8+0.2 7=7.8 ER(d2)=0.8 14+0.2 5=12.2 ER(d3)=0.8 20+0.2 (-9)=14.2 , d3: .

PDC : (s1)(s2)ER(di)P(s1)=0.8P(s2)=0.2(d1)(d2)(d3)8142075-97.812.214.2

PDC14.214.212.27.8

. () 2. () , , $ 20 million . , $ 7 million . , EPPI=0.8 20+0.2 7=17.4 d3: , $ 14.2 million . , : EVPI=EPPI ER*=17.4 14.2 = 3.2

1. (indicators) , (Sample Information, or Indicator) . PDC, (two indicators): I1=(favorable market research report) () I2=(unfavorable market research report) () . ()

2. Bayes,. .

(I1)(I2)(s1)(s2)P(I1/s1)=0.9P(I1/s2)=0.25P(I1/s2)=0.10P(I2/s2)=0.75

(I1)

Sj

P(sj)P(I1/sj)

P(I1 sj)

P(sj/I1)s1s20.80.20.90.250.720.05P(I1)=0.770.93510.0649

(I2)

Sj

P(sj)P(I2/sj)

P(I2 sj)

P(sj/I2)s1s20.80.20.10.750.080.15P(I1)=0.230.34780.6522

3. ()8.1315.8218.111.098.137.3518.1113.427.94

4. : PDC$ 15.82 million . d3: , $ 14.2 million . , PDC: , : , ; , ; 5. EVSI=15.8214.2=1.60 million $ 6. E=EVSI/EVPI 100%=1.60/3.2 100%=50%

7. (risk profile) . , 8. PDC

-9 514200.77 0.06=0.050.23 0.65=0.150.23 0.35=0.080.77 0.94=0.72

(utility and decision making)

10 ABA0.70.3B0.90.113 13

ABAB

101000

A1A2A1x2A2px11px3x1>x2>x3,ux1x1A1A2 pux1+1pux3=ux2x2x1x3434

1x1x2x3ppA1A22px1x3x2x2A1A23px2x3x1px1x3A1A2Von NeumannMorgensternVMp=0.5x1x3 0.5ux1+0.5ux3=ux2x2

11 20050VMu200=1u50=0

1200.5600.251700.75500200153

153

12 AB0.60.420050120203

ERA=0.6200 + 0.450=100 ERB=0.6120 + 0.420=64A EuA= 0.61 + 0.40 = 0.6 EuB= 0.60.5 + 0.40.1 = 0.34A EuA= 0.61 + 0.40 = 0.6 EuB= 0.60.8 + 0.40.35 = 0.62B

, ,:

: P(s1)=0.3P(s2)=0.5P(s2)=0.2A (d1)B (d2)(d3)3000050000 0 20000-20000 0-50000-30000 0

1.: ER(d1)=0.3 30000+0.5 20000+0.2 (-50000) =9000 ER(d2)=0.3 50000+0.5 (-20000)+0.2 (-30000) =-1000 ER(d3)=0.3 0+0.5 0+0.2 0 =0, A.::30000, , d1d2, d3;

2. : A1: p50000(1-p)-50000; A2: 1X;: p, A1A2?: U(X)=pU(50000)+(1p)U(-50000) , X. X=30000, :p=0.95, U(30000)=0.95U(50000)+(10.95)U(-50000) =0.95 10+0.05 0 = 9.5

2. ,

(X)5000030000200000-20000-30000-500000.950.900.750.550.4010.09.59.07.55.54.00.0

3. : ER(d1)=0.3 9.5+0.5 9.0+0.2 0.0=7.35 ER(d2)=0.3 10+0.5 5.5+0.2 4.0=6.55 ER(d3)=0.3 7.5+0.5 7.5+0.2 7.5=7.5, d3, ..

P(s1)=0.3P(s2)=0.5P(s2)=0.2A (d1)B (d2)(d3)9.5107.59.05.57.50.04.07.5

;;;;, ;

THE END