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ENGRI 1101 Engineering Applications of OR
Fall 2008 Homework 6
Homework 6 Solutions
1. (a) We’re looking at a transportation problem with the additional twist that the demand of the stores
does not necessarily have to be satisﬁed if one wants to account for shortfall cost (
k
j
per unit
shortfall for store
j
) instead. We wish to determine the amount to ship from each warehouse
i
to each store
j
; let
x
ij
be a variable that will specify the amount shipped for each
i
= 1
,...,m
,
j
= 1
,...,n
. Let’s start oﬀ with the constraints that we can use from the general transportation
problem: each warehouse can only ship as much as it has in stock
n
X
j
=1
x
ij
≤
s
i
for all
i
= 1
,...,m
(1)
and all shipping quantities have to be nonnegative (this is important and should not be forgotten!):
x
ij
≥
0 for all
i,j
(2)
The most elegant (and the simplest) way of formulating this problem as a mathematical program
is to introduce a new variable
y
j
, the shortfall for store
j
. Note that each
y
j
is a
variable
, so it’s
not known beforehand (like the input parameters
c
ij
and
k
j
for example) but instead it will be
treated in a similar manner as the
x
ij
’s, as part of the solution. First of all we set each
y
j
to be
nonnegative since shortfall is either there or not, it doesn’t make sense for it to be negative:
y
j
≥
0 for all
j
= 1
,...,n
(3)
The constraint involving the demand in each store now has the following shape:
m
X
i
=1
x
ij
+
y
j
≥
d
j
for all
j
= 1
,...,n
(4)
This says that the sum of all the shipments plus the shortfall
y
j
has to be at least the demand.
Note that it does not make any explicit statements about the size of the shipments: if nothing is
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