ENGRI 1101 Engineering Applications of OR
Fall 2008 Handout 7
The transportation and assignment problems
Of all the models that we shall study in this course, the
transportation problem
is one of the most often
used in practice.
Imagine that there are
m
warehouses that combine to serve
n
clients (perhaps retail
stores). Imagine that there is only one commodity that is stored in these warehouses and is subsequently
shipped to the clients; for example, suppose that there are oranges stored at the warehouses that must be
shipped to grocery stores.
Each warehouse
i
= 1
, . . . , m
has a given supply of oranges that it can ship;
call this supply
s
i
(say, the number of bushels of oranges on hand). Each grocery store
j
= 1
, . . . , n
has a
given demand for this commodity (the number of bushels of oranges that is expected to be sold); call this
demand
d
j
,
j
= 1
, . . . , n
. It costs
c
ij
dollars to ship each bushel of oranges from warehouse
i
to grocery
store
j
; that is, to ship 10 bushels costs 10
c
ij
, and to ship 10,000,000 bushels costs 10
,
000
,
000
c
ij
.
In
the transportation problem, we wish to decide how to ship oranges from warehouses to stores so that the
demand is met in a feasible way, so as to minimize the total shipping charges.
The input for the transportation problem consists of
m
and
n
, which specify the number of warehouses
and stores, respectively;
s
i
,
i
= 1
, . . . , m
, which specify the supply at each warehouse
i
;
d
j
,
j
= 1
, . . . , n
,
which specify the demand at each store
j
; and
c
ij
,
i
= 1
, . . . , m,
j
= 1
, . . . , n
, which specify the perunit
shipping cost from each warehouse
i
to each store
j
.
A solution to the transportation problem specifies the amount shipped from each warehouse to each store.
We might denote such a solution
x
ij
,
i
= 1
, . . . , m
,
j
= 1
, . . . , n
. What conditions must be satisfied for
x
to specify a feasible solution? First, if
x
ij
,
i
= 1
, . . . , m
,
j
= 1
, . . . , n
, specifies the amounts shipped,
then clearly each
x
ij
≥
0 . Furthermore, the total amount shipped from warehouse
i
is exactly equal to
∑
n
j
=1
x
ij
. For this to be feasible, there must exist sufficient supply at this warehouse; that is,
n
summationdisplay
j
=1
x
ij
≤
s
i
,
for each
i
= 1
, . . . , m.
(1)
On the other hand, the total amount shipped to the
j
th store is exactly equal to
∑
m
i
=1
x
ij
. For this to be
feasible, this shipment must meet the required demand; that is,
m
summationdisplay
i
=1
x
ij
≥
d
j
,
for each
j
= 1
, . . . , n.
(2)
These are the only requirement for
x
to specify a feasible solution.
What is the cost associated with a feasible solution
x
ij
,
i
= 1
, . . . , m
,
j
= 1
, . . . , n
? The cost associated
with shipping from warehouse
i
to store
j
is equal to
c
ij
x
ij
, and so the total cost is
m
summationdisplay
i
=1
n
summationdisplay
j
=1
c
ij
x
ij
.
(3)
More formally, the transportation problem is find
mn
values
x
ij
,
i
= 1
, . . . , m
,
j
= 1
, . . . , n
, such that
(1) and (2) are satisfied, so as to minimize the function (3).
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 Spring '05
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 Optimization, Equals sign, tij xij

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