Expected Value for Option S 2
EV C1 ¼ (0.25)(23) + (0.5)(31) + (0.25)(38) ¼ 30.75
EV C2 ¼ (0.2)(26) + (0.7)(34) + (0.1)(43) ¼ 33.3
EV C3 ¼ (0)(0) + (1)(25) + (0)(0) ¼ 25
EV AGG ¼ (0.16)(30.75) + (0.36)(33.33) + (0.48)(25) ¼ 28.919
Expected Value for Option S 3
EV C1 ¼ (0.1)(32) + (0.8)(37) + (0.1)(44) ¼ 37.2
EV C2 ¼ (0.08)(19) + (0.84)(22) + (0.08)(24) ¼ 21.93
EV C3 ¼ (0)(0) + (1)(44) + (0)(0) ¼ 44
EV AGG ¼ (0.16)(37.2) + (0.36)(21.93) + (0.48)(44) ¼ 34.967
Variance of Option S 1
Var Cl ¼ (0.2)(30
2 ) + (0.6)(43
2 ) + (0.2)(58
2 )À(43.4)
2
¼ 78.64
Var C2 ¼ (0.1)(32
2 ) + (0.75)(44
2 ) + (0.15)(60
2 )À(45.2)
2
¼ 51.36
Var C3 ¼ (1)(30
2 ) À (30)
2
¼ 0
Var AGG ¼ (0.16
2 )(78.64) + (0.36
2 )(51.36) + (0.48
2 )(0) ¼ 8.669
Variance of Option S 2
Var Cl ¼ (0.25)(23
2 ) + (0.5)(31
2 ) + (0.25)(38
2 )À(30.75)
2
¼ 28.19
Var C2 ¼ (0.2)(26
2 ) + (0.7)(34
2 ) + (0.1)(43
2 )À(33.3)
2
¼ 20.41
Var C3 ¼ (1)(25
2 ) À (25)
2
¼ 0
Var AGG ¼ (0.16
2 )(28.19) + (0.36
2 )(20.41) + (0.48
2 )(0) ¼ 3.367
Variance of Option S 3
Var Cl ¼ (0.1)(32
2 ) + (0.8)(37
2 ) + (0.1)(44
2 )À(37.2)
2
¼ 7.36
Var C2 ¼ (0.08)(19
2 ) + (0.84)(22
2 ) + (0.08)(24
2 )À(21.93)
2 ¼ 0.969
Var C3 ¼ (1)(44
2 ) À (44)
2
¼ 0
Var AGG ¼ (0.16
2 )(7.36) + (0.36
2 )(0.969) + (0.48
2 )(0) ¼ 0.314
The expected values for the three options give a ranking consistent with the basic
raw scores listed in Table 15.8. The expected values represent the scores one can
assign to each option with a confidence level of 50%, in other words an expected
mean, average score. If one wishes to increase the confidence level associated with
these scores, the variances must be utilised.
Firstly, the square root of the variance yields the standard deviation of each mean
score. For confidence levels greater than 50%, the option score is obtained by
decreasing the expected value by z times the standard deviation, where z is a value
obtained from a standard normal distribution curve. Values of z for confidence levels
between 50% and 99% are given in Table 15.10. The z values associated with 50%,
85% and 99% are given in Table 15.11.
15.5 Probabilistic Additive Weighting
425
EV C1 ¼ (0.25)(23) + (0.5)(31) + (0.25)(38) ¼ 30.75
EV C2 ¼ (0.2)(26) + (0.7)(34) + (0.1)(43) ¼ 33.3
EV C3 ¼ (0)(0) + (1)(25) + (0)(0) ¼ 25
EV AGG ¼ (0.16)(30.75) + (0.36)(33.33) + (0.48)(25) ¼ 28.919
Expected Value for Option S 3
EV C1 ¼ (0.1)(32) + (0.8)(37) + (0.1)(44) ¼ 37.2
EV C2 ¼ (0.08)(19) + (0.84)(22) + (0.08)(24) ¼ 21.93
EV C3 ¼ (0)(0) + (1)(44) + (0)(0) ¼ 44
EV AGG ¼ (0.16)(37.2) + (0.36)(21.93) + (0.48)(44) ¼ 34.967
Variance of Option S 1
Var Cl ¼ (0.2)(30
2 ) + (0.6)(43
2 ) + (0.2)(58
2 )À(43.4)
2
¼ 78.64
Var C2 ¼ (0.1)(32
2 ) + (0.75)(44
2 ) + (0.15)(60
2 )À(45.2)
2
¼ 51.36
Var C3 ¼ (1)(30
2 ) À (30)
2
¼ 0
Var AGG ¼ (0.16
2 )(78.64) + (0.36
2 )(51.36) + (0.48
2 )(0) ¼ 8.669
Variance of Option S 2
Var Cl ¼ (0.25)(23
2 ) + (0.5)(31
2 ) + (0.25)(38
2 )À(30.75)
2
¼ 28.19
Var C2 ¼ (0.2)(26
2 ) + (0.7)(34
2 ) + (0.1)(43
2 )À(33.3)
2
¼ 20.41
Var C3 ¼ (1)(25
2 ) À (25)
2
¼ 0
Var AGG ¼ (0.16
2 )(28.19) + (0.36
2 )(20.41) + (0.48
2 )(0) ¼ 3.367
Variance of Option S 3
Var Cl ¼ (0.1)(32
2 ) + (0.8)(37
2 ) + (0.1)(44
2 )À(37.2)
2
¼ 7.36
Var C2 ¼ (0.08)(19
2 ) + (0.84)(22
2 ) + (0.08)(24
2 )À(21.93)
2 ¼ 0.969
Var C3 ¼ (1)(44
2 ) À (44)
2
¼ 0
Var AGG ¼ (0.16
2 )(7.36) + (0.36
2 )(0.969) + (0.48
2 )(0) ¼ 0.314
The expected values for the three options give a ranking consistent with the basic
raw scores listed in Table 15.8. The expected values represent the scores one can
assign to each option with a confidence level of 50%, in other words an expected
mean, average score. If one wishes to increase the confidence level associated with
these scores, the variances must be utilised.
Firstly, the square root of the variance yields the standard deviation of each mean
score. For confidence levels greater than 50%, the option score is obtained by
decreasing the expected value by z times the standard deviation, where z is a value
obtained from a standard normal distribution curve. Values of z for confidence levels
between 50% and 99% are given in Table 15.10. The z values associated with 50%,
85% and 99% are given in Table 15.11.
15.5 Probabilistic Additive Weighting
425