Transcript
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TRANSPORTATION PROBLEM
UNIT II
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Transportation Model/Problem
TP is essentially an LPP.
But because of certain special characteristics
associated with the problem, we have aspecial algorithm to solve a TP.
It is not solved by applying the simplex
method.
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Transportation Model/Problem
Although in principle it can be solved by the
simplex method but its application would
make the problem very unmanageable.
The TP, along with the assignment problem is
a special type of LPP, therefore we have
special purpose algorithms for them.
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TP: Important Features
TP normally used for minimising
transportation cost, although the algorithm
can be applied on some other types of
problems also.
A TP deals with problems related to
minimisation of cost when goods have to be
transported from different points of supply todifferent points of demand.
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TP: Important Features
The amounts available at different points of
supply and amounts required at different
points of demand are known to us.
The unit cost of transportation is also known
to us.
In a general TP we may assume m points of
supply and n points of demand.
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TP: Important Features
We have to design a transportation plan which
would minimise the total cost of
transportation.
Total number of variables = m x n.
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TP: Formulation
General notation of variable: xij (amount of goodstransported from ith point of supply to jth point of demand).
Cij = cost coefficient attached with xij. Total number of constraints = m+n.
There would be m supply and n demandconstraints.
Availability at supply points: b1, b2, b3, , bm.
Requirement at demand points: d1, d2, d3, , dn.
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TP: Formulation
This is a balanced TP because total supply is equalto total demand.
Because total supply = total demand, thereforeone constraint becomes redundant.
T
otal number of effective constraints = m+n-1. Number of basic variables = number of constraints = m+n-1
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There are 3 coalieries which can supply coal in
the following quantities:
a: 14 tonnes, b: 12 tonnes, c: 5 tonnes.
There are 3 consumers who require coal in the
following quantities:
A: 6 tonnes, B: 10 tonnes, C: 15 tonnes.
The cost of transporting 1 tonne of coal from
any one point a, b or c to any other point A, B orC are as follows:
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The problem is to minimise total cost of
transportation.
6 4 1 a8 9 2 b
4 3 6 c
A B C
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North West Corner Rule
1. Select the north-west (upper left) corner cell.
2. Allocate as many units as possible.
3. Adjust supply and demand figures in therespective row and column.
4. If row supply is exhausted, move down one row.
5. If column demand is exhausted, move right onecolumn.
6. Repeat steps 2-5 till total available quantity isfully allocated.
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Least Cost Method
1.Select the cell with the lowest transportation
cost.
2.Allocate as many units as possible.3.Adjust supply and demand figures in the
respective row and column.
4.Repeat steps 1-3 till total available quantity is
fully allocated.
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Vogels Approximation Method
Best method of obtaining initial solution.
Consists of determining (row and column)penalties.
Row penalty is calculated by subtracting theminimum cost element from the second leastcost element in the row.
Column penalty is calculated by subtractingthe minimum cost element from the secondleast cost element in the column.
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Stepping Stone Method
Determine an initial basic feasible solution
using any of the above three methods.
Make sure that the number of occupied cellsis m+n-1.
Select an unoccupied cell.
Beginning at this cell, trace a closed path.
Move only horizontally or vertically.
Turn only at occupied cells.
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Stepping Stone Method
The cells at the turning points are called the
stepping stones.
Assign plus and minus signs alternatively toeach stepping stone, starting with a plus sign
at the unoccupied cell to be evaluated.
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Stepping Stone Method
Compute the net change in cost along theclosed path by finding the algebraic sum of the unit costs of the stepping stones,
considering the plus/minus signs assigned inthe previous step.
A negative sum indicates a decrease intransportation cost.
Repeat these steps for all the unoccupiedcells.
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Modified Distribution Method
Determine an initial basic feasible solution.
Compute the values of the row and column
variables (ui and v j). Compute the improvement index for each
unoccupied cell,
If all improvement indices are positive or zero,
the optimum solution has been reached.
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Factory Warehouse Capacity
(Supply)W1 W2 W3 W4
F
1 21 16 25 13 11F2 17 18 14 23 13
F3 32 27 18 41 19
Requirement
(Demand)
6 10 12 15 43
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W1 W2 W3 W4
F121 16 25 13
11
F2 17 18 14 23 13
F3
32 27 18 41
19
6 10 12 15 43
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W1 W2 W3 W4
F121 16 25 13
11
F2 17 18 14 23 13
F3
32 27 18 41
19
6 10 12 15
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W1 W2 W3 W4
F121 16 25 13
116
F2 17 18 14 23 13
F3
32 27 18 41
19
6 10 12 15
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W1 W2 W3 W4
F121 16 25 13
56
F2 17 18 14 23 13X
F3
32 27 18 41
19X
0 10 12 15
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W1 W2 W3 W4
F121 16 25 13
56 5
F2 17 18 14 23 13X
F3
32 27 18 41
19X
0 10 12 15
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W1 W2 W3 W4
F121 16 25 13
06 5 X X
F2 17 18 14 23 13X
F3
32 27 18 41
19X
0 5 12 15
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W1 W2 W3 W4
F121 16 25 13
06 5 X X
F2 17 18 14 23 13X 5
F3
32 27 18 41
19X
0 5 12 15
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W1 W2 W3 W4
F121 16 25 13
06 5 X X
F2 17 18 14 23 8X 5
F3
32 27 18 41
19X X
0 0 12 15
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W1 W2 W3 W4
F121 16 25 13
06 5 X X
F2 17 18 14 23 8X 5 8
F3
32 27 18 41
19X X
0 0 12 15
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W1 W2 W3 W4
F121 16 25 13
06 5 X X
F2 17 18 14 23 0X 5 8 X
F3
32 27 18 41
19X X
0 0 4 15
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W1 W2 W3 W4
F121 16 25 13
06 5 X X
F2 17 18 14 23 0X 5 8 X
F3
32 27 18 41
19X X 4
0 0 4 15
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W1 W2 W3 W4
F121 16 25 13
06 5 X X
F2 17 18 14 23 0X 5 8 X
F3
32 27 18 41
15X X 4
0 0 0 15
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W1 W2 W3 W4
F121 16 25 13
06 5 X X
F2 17 18 14 23 0X 5 8 X
F3
32 27 18 41
15X X 4 15
0 0 0 15
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W1 W2 W3 W4
F121 16 25 13
06 5 X X
F2 17 18 14 23 0X 5 8 X
F3
32 27 18 41
0X X 4 15
0 0 0 0
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W1 W2 W3 W4
F121 16 25 13
11
F2 17 18 14 23 13
F3
32 27 18 41
19
6 10 12 15
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W1 W2 W3 W4
F121 16 25 13
1111
F2 17 18 14 23 13
F3
32 27 18 41
19
6 10 12 15
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W1 W2 W3 W4
F121 16 25 13
0X X X 11
F2 17 18 14 23 13
F3
32 27 18 41
19
6 10 12 4
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W1 W2 W3 W4
F121 16 25 13
0X X X 11
F2 17 18 14 23 1312
F3
32 27 18 41
19
6 10 12 4
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W1 W2 W3 W4
F121 16 25 13
0X X X 11
F2 17 18 14 23 112
F3
32 27 18 41
19X
6 10 0 4
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W1 W2 W3 W4
F121 16 25 13
0X X X 11
F2 17 18 14 23 11 12
F3
32 27 18 41
19X
6 10 0 4
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W1 W2 W3 W4
F121 16 25 13
0X X X 11
F2 17 18 14 23 01 X 12 X
F3
32 27 18 41
19X
5 10 0 4
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W1 W2 W3 W4
F121 16 25 13
0X X X 11
F2 17 18 14 23 01 X 12 X
F3
32 27 18 41
1910 X
5 10 0 4
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W1 W2 W3 W4
F121 16 25 13
0X X X 11
F2 17 18 14 23 01 X 12 X
F3
32 27 18 41
910 X
5 0 0 4
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W1 W2 W3 W4
F121 16 25 13
0X X X 11
F2 17 18 14 23 01 X 12 X
F3
32 27 18 41
95 10 X
5 0 0 4
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W1 W2 W3 W4
F121 16 25 13
0X X X 11
F2 17 18 14 23 01 X 12 X
F3
32 27 18 41
45 10 X
0 0 0 4
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W1 W2 W3 W4
F121 16 25 13
0X X X 11
F2 17 18 14 23 01 X 12 X
F3
32 27 18 41
45 10 X 4
0 0 0 4
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W1 W2 W3 W4
F121 16 25 13
0X X X 11
F2 17 18 14 23 01 X 12 X
F3
32 27 18 41
05 10 X 4
0 0 0 0
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Vogels Approximation Method
VAM
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W1 W2 W3 W4
F121 16 25 13
11
F2 17 18 14 23 13
F3
32 27 18 41
19
6 10 12 15
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W1 W2 W3 W4 Row
Penalty
F121 16 25 13
11 3
F2
17 18 14 23
13
F332 27 18 41
19
6 10 12 15 43
Column
Penalty
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W1 W2 W3 W4 Row
Penalty
F121 16 25 13
11 3
F2
17 18 14 23
13 3
F332 27 18 41
19
6 10 12 15 43
Column
Penalty
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W1 W2 W3 W4 Row
Penalty
F121 16 25 13
11 3
F2
17 18 14 23
13 3
F332 27 18 41
19 9
6 10 12 15 43
Column
Penalty
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W1 W2 W3 W4 Row
Penalty
F121 16 25 13
11 3
F2
17 18 14 23
13 3
F332 27 18 41
19 9
6 10 12 15 43
Column
Penalty 4
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W1 W2 W3 W4 Row
Penalty
F121 16 25 13
11 3
F2
17 18 14 23
13 3
F332 27 18 41
19 9
6 10 12 15 43
Column
Penalty 4 2
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W1 W2 W3 W4 Row
Penalty
F121 16 25 13
11 3
F2
17 18 14 23
13 3
F332 27 18 41
19 9
6 10 12 15 43
Column
Penalty 4 2 4
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W1 W2 W3 W4 Row
Penalty
F121 16 25 13
11 3
F2
17 18 14 23
13 3
F332 27 18 41
19 9
6 10 12 15 43
Column
Penalty 4 2 4 10
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W1 W2 W3 W4 Row
Penalty
F121 16 25 13
11 3
F2
17 18 14 23
13 3
F332 27 18 41
19 9
6 10 12 15 43
Column
Penalty 4 2 4 10
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W1 W2 W3 W4 Row
Penalty
F121 16 25 13
11 311
F2
17 18 14 23
13 3
F332 27 18 41
19 9
6 10 12 15 43
Column
Penalty 4 2 4 10
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W1 W2 W3 W4 Row
Penalty
F121 16 25 13
0 311
F2
17 18 14 23
13 3
F332 27 18 41
19 9
6 10 12 15 43
Column
Penalty 4 2 4 10
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W1 W2 W3 W4 Row
Penalty
F121 16 25 13
0 311
F2
17 18 14 23
13 3
F332 27 18 41
19 9
6 10 12 4 43
Column
Penalty 4 2 4 10
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W1 W2 W3 W4 Row
Penalty
F121 16 25 13
0 3X X X 11
F2
17 18 14 23
13 3
F332 27 18 41
19 9
6 10 12 4 43
Column
Penalty 4 2 4 10
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W1 W2 W3 W4 Row
Penalty
F121 16 25 13
0 -X X X 11
F2
17 18 14 23
13 3
F332 27 18 41
19 9
6 10 12 4 43
Column
Penalty 15 9 4 18
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W1 W2 W3 W4 Row
Penalty
F121 16 25 13
0 -X X X 11
F2
17 18 14 23
13 34
F332 27 18 41
19 9
6 10 12 4 43
Column
Penalty 15 9 4 18
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W1 W2 W3 W4 Row
Penalty
F121 16 25 13
0 -X X X 11
F2
17 18 14 23
9 34
F332 27 18 41
19 9X
6 10 12 0 43
Column
Penalty 15 9 4 -
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W1 W2 W3 W4 Row
Penalty
F121 16 25 13
0 -X X X 11
F2
17 18 14 23
9 36 4
F332 27 18 41
19 9X
6 10 12 0 43
Column
Penalty 15 9 4 -
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W1 W2 W3 W4 Row
Penalty
F121 16 25 13
0 -X X X 11
F2
17 18 14 23
3 46 4
F332 27 18 41
19 9X X
0 10 12 0 43
Column
Penalty - 9 4 -
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W1 W2 W3 W4 Row
Penalty
F121 16 25 13
0 -X X X 11
F2
17 18 14 23
3 46 4
F332 27 18 41
19 9X X
0 10 12 0 43
Column
Penalty - 9 4 -
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W1 W2 W3 W4 Row
Penalty
F121 16 25 13
0 -X X X 11
F2
17 18 14 23
3 46 3 4
F332 27 18 41
19 9X X
0 10 12 0 43
Column
Penalty - 9 4 -
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W1 W2 W3 W4 Row
Penalty
F121 16 25 13
0 -X X X 11
F2
17 18 14 23
0 -6 3 X 4
F332 27 18 41
19 9X X
0 7 12 0 43
Column
Penalty - - - -
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W1 W2 W3 W4 Row
Penalty
F121 16 25 13
0 -X X X 11
F2
17 18 14 23
0 -6 3 X 4
F332 27 18 41
19 -X 7 X
0 7 12 0 43
Column
Penalty - - - -
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W1 W2 W3 W4 Row
Penalty
F121 16 25 13
0 -X X X 11
F2
17 18 14 23
0 -6 3 X 4
F332 27 18 41
12 -X 7 12 X
0 0 12 0 43
Column
Penalty - - - -
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W1 W2 W3 W4 Row
Penalty
F121 16 25 13
0 -X X X 11
F2
17 18 14 23
0 -6 3 X 4
F332 27 18 41
0 -X 7 12 X
0 0 0 0 43
Column
Penalty - - - -
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W1 W2 W3 W4
F121 16 25 13
1111
F2 17 18 14 23 136 3 4
F3
32 27 18 41
197 12
6 10 12 15
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W1 W2 W3 W4
F121 16 25 13
11
F217 18 14 23
1 12
F332 27 18 41
5 10 4
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W1 W2 W3 W4
F121 16 25 13
11
F217 18 14 23
1 12
F332 27 18 41
5 10 4
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W1 W2 W3 W4
F121 16 25 13
11
F217 18 14 23
1 12
F332 27 18 41
5 10 4
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W1 W2 W3 W4
F121 16 25 13
11
F217 18 14 23
1 12
F332 27 18 41
5 10 4
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F1W1F1W4F3W4F3W1 21-13+41-32=17
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W1 W2 W3 W4
F121 16 25 13
11
F217 18 14 23
1 12
F332 27 18 41
5 10 4
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W1 W2 W3 W4
F121 16 25 13
11
F217 18 14 23
1 12
F332 27 18 41
5 10 4
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W1 W2 W3 W4
F121 16 25 13
11
F217 18 14 23
1 12
F332 27 18 41
5 10 4
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F1W1F1W4F3W4F3W1 21-13+41-32=17
F1W2F1W4F3W4F3W2 16-13+41-27=17
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W1 W2 W3 W4
F121 16 25 13
11
F217 18 14 23
1 12
F332 27 18 41
5 10 4
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W1 W2 W3 W4
F121 16 25 13
11
F217 18 14 23
1 12
F332 27 18 41
5 10 4
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W1 W2 W3 W4
F121 16 25 13
11
F217 18 14 23
1 12
F332 27 18 41
5 10 4
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W1 W2 W3 W4
F121 16 25 13
11
F217 18 14 23
1 12
F332 27 18 41
5 10 4
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W1 W2 W3 W4
F121 16 25 13
11
F217 18 14 23
1 12
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F1W1F1W4F3W4F3W1 21-13+41-32=17
F1W2F1W4F3W4F3W2 16-13+41-27=17
F
1W3F
1W4F
3W4F
3W1F
2W1F
2W3 25-13+41-32+17-14=24
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F1W1F1W4F3W4F3W1 21-13+41-32=17
F1W2F1W4F3W4F3W2 16-13+41-27=17
F1W3F1W4F3W4F3W1F2W1F2W3 25-13+41-32+17-14=24
F2W2F3W2F3W1F2W1 18-27+32-17=6
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F1W1F1W4F3W4F3W1 21-13+41-32=17
F1W2F1W4F3W4F3W2 16-13+41-27=17
F1W3F1W4F3W4F3W1F2W1F2W3 25-13+41-32+17-14=24
F2W2F3W2F3W1F2W1 18-27+32-17=6
F2W4F3W4F3W1F2W1 23-41+32-17=-3
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F1W1F1W4F3W4F3W1 21-13+41-32=17
F1W2F1W4F3W4F3W2 16-13+41-27=17
F1W3F1W4F3W4F3W1F2W1F2W3 25-13+41-32+17-14=24
F2W2F3W2F3W1F2W1 18-27+32-17=6
F2W4F3W4F3W1F2W1 23-41+32-17=-3
F3W2F3W1F2W1F2W3 18-32+17-14=-11
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F121 16 25 13
11
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F1W1F1W4F3W4F3W3F2W3F2W1 21-13+41-18+14-17=28
F1W2F1W4F3W4F3W2 16-13+41-27=17
F1W3F1W4F3W4F3W3 25-13+41-18=35
F2W2F2W3F3W3F3W2 18-14+18-27=-5
F2W4F3W4F3W3F2W3 23-41+18-14=-14
F3W1F2W1F2W3F3W3 32-17+14-18=11
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W1 W2 W3 W4
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F121 16 25 13
11
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F3 32 27 18 41
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F1W1F1W4F2W4F2W1 21-13+41-32=17
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F1W2F1W4F2W4F2W3F3W3F3W2 16-13+23-14+18-27=3
F1W3F1W4F2W4F2W3 25-13+23-14=21
F2W2F2W3F3W3F3W2 18-14+18-27=-5
F3W1F2W1F2W3F3W3 32-17+14-18=11
F3W4F3W3F2W3F2W4 41-18+14-23=14
W1 W2 W3 W4
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6 3- 4
F3 32 27 18 41
10- 9+
W1 W2 W3 W4
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11
F217 18 14 23
6 3 4
F3 32 27 18 41
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F1W1F1W4F2W4F2W1 21-13+41-32=17
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F1W2F1W4F2W4F2W2 16-13+23-18=8
F1W3F1W4F2W4F2W2F3W2F3W3 25-13+23-18+27-18=26
F2W3F3W3F3W2F2W2 14-18+27-18=5
F3W1F2W1F2W2F3W2 32-17+18-27=6
F3W4F3W2F2W2F2W4 41-27+18-23=9
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u111
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u21 12
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u3
5 10 4
v1 v2 v3 v4
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Degeneracy
A manufacturing company has three plants X Y
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A manufacturing company has three plants X, Y
and Z, which supply to the distributors locatedat A, B, C, D and E. Monthly plant capacities are
80, 50 and 90 units respectively. Monthly
requirements of distributors are 40, 40, 50, 40
and 80 units respectively. Unit transportationcosts are given below in rupees:
F T
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