1)
How is an optimal solution to a Transportation problem changed if a constant
is added
to each cost in
(i) one row say rth row
(ii) one column say kth column?
By how much is the optimal value of the objective function changed?
Solution:
Consider the standard form of the objective function;
11
mn
ij ij
ij
zcx
Now if we add
to each cost of rth row, the total cost(objective function) will become;
1( ) 1 1
1( ) 1 1 1
11
mn n
new ij ij rj rj
iirj j
mn n n
ij ij rj rj rj
iirj j j
mn
ij ij r
ij
new old r
zcxcx
cx cx x
cx a
zz a
Since
ar is constant, so if some feasible solution of the original TP minimizes zold then
the feasible solution will also minimizes the znew.
On adding
to every cost of rth row of a TP, the optimal solution will not change,
while the total transportation cost will be increased by (
ar).
Similarly, we can show that on adding constant
to every cost of kth column of a TP,
the optimal solution ‘ll remain unaltered, while the total transportation cost will be
increased by
bk.
2)
A company has three plants A,B and C, and three ware houses X, Y and Z. The number
of units available at the plants is 60, 70 and 80 and the demand at X, Y and Z are 50,80
and 80 respectively. The unit cost of the transportation is given as follow:
Find the allocation so that the total transportation cost is minimum.
Solution:
We solve this by Least Cost method.
X Y Z
A 8 7 3
B 3 8 9
C 11 3 5
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