Transportation Models MBA PPT

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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

Documents

Transportation Models MBA PPT