Geoscience Reference
In-Depth Information
and substituting the given parameters into the above equations, we obtain the
following system of equations:
8
<
1048
x
p
1
þ
1040
x
p
2
ˉ
p
1
þ
6
ˉ
p
2
¼
10085
þ
15
:
5
:
5
1040
x
p
1
þ
1054
x
p
2
ˉ
p
1
ˉ
p
2
¼
10083
þ
6
:
5
þ
8
:
5
7
z
p
1
ˉ
p
1
¼
0
16
z
p
2
ˉ
p
2
¼
0
:
5
x
p
1
6
5
x
p
2
þz
p
1
15
:
:
¼
60
5
x
p
1
5
x
p
2
þz
p
2
6
:
8
:
¼
64
:
Solution of the above system yields:
x
p
1
¼
1
04and
x
p
2
¼
7
:
:
50
:
Hence, the optimal values of link flows are:
f
a
¼ f
b
¼ f
c
¼ f
f
¼ f
g
¼ x
p
1
þx
p
2
¼
8
f
d
¼ x
p
1
¼
1
04, and
f
e
¼ x
p
2
¼
7
:
:
:
:
54,
50
As seen above, the optimal flow of the disaster relief item on link
e
(air
transportation to the affected region) was considerably higher than that on link
d
(ground transportation). This is because the humanitarian organization, in this
example, chooses the quicker mode of transportation, but at a higher cost. The
optimal time deviations on paths
p
1
and
p
2
with respect to the target of 72 h are:
z
p
1
¼
4
85 and
z
p
2
¼
6
:
:
47
:
Neither of the two transportation modes to the affected area would be able to
satisfy the target time requirement. Interestingly, the time deviation is higher on the
path that contains the air route, which is due to the majority of the load being
allocated to this mode.
The value of the projected demand at point
R
1
was:
v
R
1
¼ x
p
1
þ x
p
2
¼
8
:
54,
which is the amount that needs to be pre-positioned at the storage facility. The
projected demand was closer to the upper bound of the uniform distribution range
of the demand for the relief item at point
R
1
. If the organization seeks to reduce the
consequences of the shortage of relief items in the affected region, the decision-
maker should assign a higher unit shortage penalty so as to be able to better meet the
uncertain demand within the given time limit. This obviously will result in a higher
operational cost and, yet, in a lower social cost.
The optimal values of the Lagrange multipliers corresponding to the time goal
constraints were:
p
1
¼
33
p
2
¼
103
ˉ
:
97 and
ˉ
:
55
:
