True capacity of the reservoir
Indicators : 1 : Less than 100 : 2 : Larger than 100
I 1 : Capacity >105 0.1 0.8
I 2 : Capacity < 95 0.7 0.1
I 3 : 95Capacity105 0.2 0.1
Table 3.1: Likelihood of the true capacity of the reservoir given the trial pump test results.
Given that a test well is established and a trial pump test conducted with the result that a
capacity is indicated smaller than 95 water volume units a posterior decision analysis can be
performed to identify whether the optimal decision is to establish a well locally or if it is more
optimal to construct a pipeline to the existing well.
Therefore, the posterior probabilities given the new information Pz@@: can be given as:
21 1
12
21 1 2 2 2
[ | ] [ ] 0.7 0.6 0.42
[ | ] 0.913
[ | ] [ ] [ | ] [ ] 0.7 0.6 0.1 0.4 0.46
PI P
PI
PI P PI P
::
:
:: ::
@
@@
@@
22 2
22
21 1 2 2 2
[|][] 0.1 0.4 0.04
[ | ] 0.087
[ | ] [ ] [ | ] [ ] 0.7 0.6 0.1 0.4 0.46
PI P
PI
PI P PI P
::
:
:: ::
@@ @
@@
which are also shown in Figure 3.5. Having determined the updated probabilities the posterior
expected values ECI'' 2 of the utility corresponding to the optimal action alternative is
readily obtained as:
>?
>?
| 212 min [ | ] (100 10) [ 22 | ] 10; 100
min 101.3;100 100
ECI P I P I
MU
@@ @@::@@
and indicated in boxes in Figure 3.5.
Considering the additional information the optimal decision has been switched to a 2.
Action / Choice Event Concequence
a 1
a 2
2
10 MU
(100 + 10) MU
100 MU
101
100
PI( | 12 ) 0.913
PI( 22 | ) 0.087
1
Figure 3.5: Illustration of decision/event tree for water supply decision problem.