An Integrated Disaster Relief Supply Chain Network Model with Time Targets and Demand Uncertainty - Regional Science Matters - page 287

Therefore, the expected penalty assigned to the humanitarian organization due
to the shortage and surplus of the relief item at each demand point is equal to:

¼ λ

k þ λ k , k k þ λ k k k E k E

E λ Δ Δ Δ Δ k ¼ 1, ... , n R , ð 15 : 5 Þ

λ k is the unit penalty corresponding to the shortage of the relief item at
demand point k which can represent the social cost of a death due to the inefficiency
of the relief distribution system. In order to avoid such shortages, a large penalty
should be assigned by the authorities, which can vary across the demand points.
Furthermore,

where

λ + denotes the unit penalty associated with the surplus of the
relief item. This penalty is taken into account so as to minimize the over-shipping
of goods, which results in congestion and additional efforts. Depending on the
criticality of the situation, the unit surplus penalty can be expected to be lower than
that of shortage, for a given demand point, and can be equal to zero. Similar ideas
have been applied in the case of critical needs (Nagurney et al. 2011 ) and human
blood (Nagurney et al. 2012a ; Nagurney and Masoumi 2012 ).
x p represents the flow of the disaster relief goods on path p joining node 1 with
a demand node which must be nonnegative, since the goods are procured, stored,
and shipped in nonnegative quantities, that is,

x p 0, 8p ∈P: ð 15 : 6 Þ

The projected demand at destination node k , v k , is equal to the sum of flows
on all paths belonging to P k , that is:
v k X
p∈P k

x p , k ¼ 1, ... , n R : ð 15 : 7 Þ

The relationship between the flow on link a , f a , and the path flows is as follows:
f a ¼ X
p∈P

x p ʴ ap , 8a ∈ L : ð 15 : 8 Þ

ʴ ap is an indicator of link a 's relation with path p , and is equal to 1 if link
a is contained in path p and is 0, otherwise.
Next, we present the expressions that capture the time aspect of our integrated
disaster relief supply chain model.
Let

Here,

˄ a denote the completion time of the activity on link a , which is assumed
to be a linear function of the flow of the product on that link. We have:

˄ a fðÞ¼g a f a þ h a , 8a ∈ L , ð 15 : 9 Þ

where h a 0, and g a 0. We allow (some of) these terms to take on zero
values for modeling flexibility purposes, as we shall show in a forthcoming
numerical example.