TRANSPORTATION PROBLEMS

63.1 INTRODUCTION

A scooter production company produces scooters at the units

situated at various places (called origins) and supplies them to the places where the depot (called destination) are situated. Here the availability as well as requirements of the various depots are finite and constitute the limited resources.

This type of problem is known as distribution or transportation problem in which the key idea is to minimize the cost or the time of transportation.

In previous lessons we have considered a number of specific

linear programming problems. Transportation problems are also linear programming problems and can be solved by simplex

method but because of practical significance the transportation problems are of special interest and it is tedious to solve them through simplex method. Is there any alternative method to solve such problems?

63.2 OBJECTIVES

After completion of this lesson you will be able to:

s olve the transportation problems by

(i) North-West corner rule;

(ii) Lowest cost entry method; and

(iii)Vogel’s approximation method.

t est the optimality of the solution.

112 :: Mathematics

63.3 MATHEMATICAL

FORMULATION

TRANSPORTATION PROBLEM

OF

Let there be three units, producing scooter, say, A 1 , A 2 a nd A 3 f rom where the scooters are to be supplied to four depots say B 1, B 2, B3 a nd B 4.

Let the number of scooters produced at A1 , A 2 a nd A 3 b e a1, a2 a nd a3 r espectively and the demands at the depots be b 2 , b 1, b 3 a nd b 4 r espectively.

We assume the condition

a1+ a2 + a3 = b 1 + b 2 + b 3 + b 4

i.e., all scooters produced are supplied to the different depots. Let the cost of transportation of one scooter from A1 t o B 1 b e c 11. Similarly, the cost of transportations in other casus are also shown in the figure and 63.1 Table 1.

Let out of a1 s cooters available at A 1, x 11 b e taken at B 1 depot, x12 b e taken at B 2 d epot and to other depots as well, as shown in the following figure and table 1.

a1

b1

x 11 ( c 11 )

x1 2 ( c )

A1

c 1)

x 21( 2

12

x

13

(c

13

a2

x

14

A2

x 31

a3

A3

(c

14

)

( c 31

x3

b2

x 22( c 22)

)

)

)

( c 32

2

24

)

x 33( c 33

B2

b3

x23 ( c )

23

x

(c

24

B1

B3

)

b4

B4

x 34 ( c 34 )

Fig 63.1

Total number of scooters to be transported form A 1 t o all

destination, i.e., B1 , B2 , B 3, and B 4 m ust be equal to a1 . ∴

x11 + x 12+ x 13+ x 14 = a1

(1)

T ransportation Problems :: 113

Similarly, from A 2 a nd A 3 t he scooters transported be equal to a2 a nd a3 r espectively.

∴

x21+ x22+ x 23+ x 24 = a2

(2)

and

x31+ x32+ x 33+ x 34 = a3

(3)

On the other hand it should be kept in mind that the total

number of scooters delivered to B 1 f rom all units must be equal to b 1 , i.e.,

x11+ x21+ x 31 = b 1

x12+ x22+ x 32 = b 2

(5)

x13+ x23+ x 33 = b 3

(6)

x14+ x24+ x 34 = b 4

Similarly,

(4)

(7)

With the help of the above information we can construct the

following table :

Table 1

Depot

Unit

To B 1

To B 2

To B 3

To B 4

S tock

From A 1

x 11 ( c 11 )

x 12 ( c 12 )

x 13 ( c 13 )

x 14 ( c 14 )

a1

From A 2

x 21 ( c 21 )

x 22 ( c 22 )

x 23 ( c 23 )

x 24 ( c 24 )

a2

From A 3

x 31 ( c 31 )

x 32 ( c 32 )

x 33 ( c 33 )

x 34 ( c 34 )

a3

∑

i,j

Requirement

b1

b2

b3

b4

The cost of transportation from A i (i=1,2,3) to B j ( j=1,2,3,4) will be equal to

S=

where the symbol

quantities c ij x ij

j = 1 ,2,3,4.

∑

i,j

c ij x ij ,

(8)

put before c ij x ij s ignifies that the

m ust be summed over all i = 1 ,2,3 and all

114 :: Mathematics

Thus we come across a linear programming problem given by

equations (1) to (7) and a linear function (8).

We have to find the non-negative solutions of the system

such that it minimizes the function (8).

Note :...