Post on 05-Apr-2018
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Transportation Models
Module 3
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The Transportation Model
The transportation model is a special class of LPPs that
deals with transporting(=shipping) a commodity from
sources (e.g. factories) to destinations (e.g. warehouses).
The objective is to determine the shipping schedule that
minimizes the total shipping cost while satisfying supplyand demand limits.
We assume that the shipping cost is proportional to the
number of units shipped on a given route.
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A Transportation Model Requires
The origin points, and the capacity or supplyper period at each
The destination points and the demand per
period at each The cost of shipping one unit from each
origin to each destination
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We assume that there are m sources 1,2, , m and n destinations 1,
2, , n. The cost of shipping one unit from Source i to Destination
j is cij.We assume that the availability at source i is ai(i=1, 2, , m) and
the demand at the destination j is bj(j=1, 2, , n). We make an
important assumption: the problem is a balanced one. That is
n
j
j
m
i
i ba11
That is, total availability equals total demand.
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We can always meet this condition by introducing
a dummy source (if the total demand is more than
the total supply) or a dummy destination (if thetotal supply is more than the total demand).
Let xij be the amount of commodity to be shipped
from the source i to the destination j.
Special Case
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definitions
Feasible solution-any set of non negativeallocations which satisfies row and columnrequirement
Basic feasible solution-a feasible solution is calledbasic feasible solution if the number of nonnegative allocations is equal to m+n-1 where m isthe no of rows and n is the number of columns
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Steps involved in solution of
transportation problem To find an initial basic feasible solution
(IBFS)
To check the above solution for optimality To revise the solution
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Methods to determine IBFS
North West corner rule
Row minima method
Column minima method
Matrix minima method
Vogels approximation method
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North West corner rule
19 30 50 10
70 30 40 60
40 8 70 20
F1
F2
F3
W1 W2 W3 W4
Fa
ctory
Warehouses
Requirement
Capacity
5 8 7 14
7
9
18
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5
19
2
30 50 10
706
303
40 60
40 8
4
70
14
20
W1 W2 W3 W4
F1
F2
F3
Factory
Warehouses
5 8 7 14
7
9
18
Requirement
Capacity
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Steps in solving a transportation
problem1. Check whether given transportation problem isbalanced
2. Find IBFS using VAM and TTC
3. To check for optimality and find out the value ofDij= Cij ( ui+vj)
4. To revise the solution if obtained solution is notoptimal (i.e. if all the values of D are not positive)
5. Recheck for optimality
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How to find out the value of Dij= Cij ( ui+vj)?
1. To find the values of ui and vj using theformula u + v = c
2. To find ui+vj for empty cells
3. To find Dij= Cij ( ui+vj )
4. Where c is the original cost given in theproblem
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How to revise the solution?
Mark + in the place where there is a negative value
Proceed with the loop
Direction of loop can be changed at only placeswhere there is a allotment
mark + and where the loop changes itsdirection
Observe cells and take the least allocation
Add the value of where + is there and subtract
the value of where is there
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Model of a loop
- 25
5
+ 352
- 11
3 +
70
20 +
10
7
15
9 -
LOOP
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Different cases of transportation problem
Unbalanced transportation problem
Degeneracy case (when total no ofallocations m+n-1)
Maximisation transportation problem
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Converting unbalanced to
balanced transportation problem
15 8 11
14 9 10
w1 w2 w3
F1
F2
capacity
requirement
9
8
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10 5 6
15 8 11
14 9 10
F1
F2
w1 w2 w3 capacity
requirement
9
8
4
Soln. - add a dummy Raw
D 0 0 0Dummy Raw
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How to resolve Degeneracy?
In order to resolve degeneracy a very smallvalue is allocated in the least costindependent cell
Independent cell-a cell from which a loopcan not be formed
Identify the independent cell in the matrixfirst and then allocate
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Resolving degeneracy
(60)
3
(50)3
(20)9
(80)
3
()
5
LeastCostIndependent
cell
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Maximisation transportation problem
Always maximisation problems need to be
converted into minimisation problem It can be done by subtracting all other
elements in the matrix from the highestelement in matrix
Note: if a given transportation problem is notbalanced and is of maximisation type firstbalancing to be done and then need to beconverted into minimisation type
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How to identify a maximisation problem?
Maximisation generally done for profit..hence any questions that appear with profithas to be converted into minimisation type
While writing final answer it is to be takencare that profit is written and not the cost
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Maximisation to minimisation
80 90 100
70 50 60
20 10 0
30 50 40