1
Dr. Maddah
ENMG 500 Engineering Management I
12/20/05
The Transportation Problem (2)
•
Balance Condition and Number of Basic Variables
¾
Consider the LP formulation for the TP
1
1
1
1
(TP)
min
subject to
,
1,
,
,
1,
,
0
m
n
ij
ij
i
j
n
ij
i
i
j
m
ij
j
j
i
ij
Z
c x
x
s
i
m
u
x
d
j
n
v
x
=
=
=
=
=
=
=
←
=
=
←
≥
∑∑
∑
∑
…
…
¾
A necessary condition for (TP) to have feasible condition is
“balance.”
That is,
1
1
m
n
i
j
i
j
s
d
=
=
=
∑
∑
.
¾
If balance is violated, then an artificial balance could be
established by adding “dummy” nodes.
¾
For example, if
1
1
,
m
n
i
j
i
j
s
d
=
=
>
∑
∑
then adding a dummy destination,
n
+1, with demand
1
1
1
m
n
n
i
j
i
j
d
s
d
+
=
=
=
−
∑
∑
and
c
i
,
n
+1
= 0,
i
= 1, …,
m
,
restores balance.
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2
¾
The balance condition implies that any of the
m
+
n
constraints
can be removed without affecting the optimal solution.
1
¾
This implies that the number of basic variables in
m
+
n
−
1
.
•
Duality Results
¾
Let
u
i
,
i
= 1, …,
m
, and
v
j
,
j
= 1, …,
m
, be the dual variables
associated with supply and demand constraints respectively.
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 Fall '09
 DrBacelMaddah
 Linear Programming, Optimization, Supply And Demand, TP, Vogel

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