Civil Engineering Reference
In-Depth Information
Table 6.8
Regret matrix
Evaluation criteria
Alternative
C
1
C
2
C
3
C
4
C
5
C
6
C
7
C
8
A
1
0.12
0.00
0.08
0.74
0.60
0.02
0.03
0.67
A
2
0.42
0.57
0.58
0.00
0.11
0.77
0.06
0.00
A
3
0.00
0.11
0.08
0.34
0.00
0.02
0.29
0.41
A
4
0.63
0.46
0.56
0.07
0.60
0.00
0.00
0.00
A
5
0.21
0.08
0.00
0.07
0.11
0.01
0.29
0.62
understanding of the costs and risks, the plant manager is criticized for
not choosing the alternative that would have yielded the best outcome.
The regret matrix is shown in Table 6.8. For example, for C
1
, the maximum
value is 0.89 (in Table 6.7), which is associated with A
3
. Thus, the regret
matrix is computed by subtracting the other values from 0.89. In a case
where a cost criterion is considered, where lower number is desirable, the
regret matrix will be negative, and thus a Maximin operator should be used.
Thus, for each alternative, the maximum values can be computed, and the
ranks can be shown to be:
(
)
>
(
)
>
(
)
>
(
)
>
(
)
A
041
.
A
062
.
A
063
.
A
074
.
A
077
.
3
5
4
1
2
Decision makers may well decide to take an action which maximizes the
minimum regret that they may later experience.
Hurwicz criterion
Often, the decision maker is neither a complete pessimist (maximin crite-
rion) nor optimist (maximax criterion). In this case, the Hurwicz criterion
(Hurwicz 1951a,b) furnishes the fl exibility to lie anywhere in between, by
computing a
w
-index for each alternative as:
(
)
w
-index of
a
=
wM
+
1
−
w m
[6.13]
i
i
i
where
M
i
and
m
i
are the maximum and minimum payoff or utility for action
a
i
over all possible states of nature, respectively. The
w
-index in Hurwicz
operators is used to incorporate the attitude of a decision maker in the
aggregation process. It characterizes the degree to which the aggregation is
like
OR
or
AND
operator. Thus, from Equation (6.13), when
w
0, the
Hurwicz operator becomes a
minimum
operator and conversely, when
w
=
1, the operator becomes a
maximum
operator. In general, the Hurwicz
=
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