module 3-transportation models

8/6/2019 Module 3-Transportation Models

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Module 3



3



• How much should be shipped from severalsources to several destinations

– Sources: Factories, warehouses, etc.

–Destinations: Warehouses, stores, etc.

• Transportation models

– Find lowest cost shipping arrangement

– Used primarily for existing distribution systems



• 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



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 b j (j=1, 2, …, n). We make an

important assumption: the problem is abalanced

one. That is

∑∑==

=n

j

j

m

i

i ba11

That is, total availability equals total demand.



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.



c11 c12 c1n a1

c21 c22 c2n a2

cm1 cm2 cmn am

b1

b2

bn

So

u

r

c

e

1

2

.

.

m

Destination

1 2 . . n Supply

Demand



Definitions

• Feasible solution -any set of non negative

allocations which satisfies row and column

requirement

• Basic feasible solution-a feasible solution iscalled basic feasible solution if the number of non

negative allocations is equal to m+n-1 where m is

the no of rows and n is the number of columns



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



Methods to determine IBFS

• North West corner rule

• Row minima method

• Column minima method

• Matrix minima method

• Vogel’s approximation method



19

30 50 10

70 30 40 60

40 8

70 20

F1

F2

F3

W1 W2 W3 W4

F a

c t o

r y

Warehouses

Requirement

Capacity

5 8 7 14

7

9

18



5

19

2

30 50 10

70

6

30

3

40 60

4 14

W1 W2 W3 W4

F1

F2

F3

F a c t o r y

Warehouses

5 8 7 14

7

9

18

Requirement

Capacity



1. Check whether given transportation problem isbalanced

2. Find IBFS using VAM and TTC

3. To check for optimality and find out the value of Dij= Cij – ( ui+v j)

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



• Total no of allocations=m+n-1

• Where m is the total no of rows

• n is the total no of columns



How to find out the value of Dij= Cij – ( ui+v j)?



Model of a loop

-θ 25

5

+θ 352

-θ 113 +θ

70

20 +θ

10

7

15

9 -θ

LOOP



• Unbalanced transportation problem

• Degeneracy case (when total no of allocations ≠ m+n-1)

• Maximisation transportation problem



Converting unbalanced to

balanced transportation problem

15

8 11

14 9 10

w1 w2 w3

F1

F2

capacity

requirement

9

8

C a p a c i t y d o e s n o t t a l l y w i t h r e q u i r e m e n t



10 5 6

15

8 11

14 9 10

F1

F2

w1 w2 w3 capacity

requirement

9

8

Soln. - add a dummy Raw

D0 0 0 Dummy Raw



• In order to resolve degeneracy a very smallvalue Δ 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 Δ



Resolving degeneracy

(60)

3

(50)3

(20)9

(80)

3

(Δ)

5

Least

Cost



Maximisation transportation 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



Maximisation to minimisation

80 90 100

70 50 60

20 10 0

30 50 40

module 3-transportation models

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