presented BY ,
MANEET KUMAR
MANI SHANKAR
MANINDER PAL SINGH
MANOJ KUMAR
MANISH KUMAR GARG
MADHU MAYA

• Introduced by “T.C.KOOPMANS” in 1947, who presented a study called optimum utilization of “Transportation System”.
• 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 supply and demand limits.

• A product is transported from a number of sources to a number of destinations at the minimum possible cost.
• Each source is able to supply a fixed number of units of the product, and each destination has a fixed demand for the product.
• The linear programming model has constraints for supply at each source and demand at each destination.
• The shipping cost is proportional to the number of units shipped on a given route .

 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 c ij
.
 We assume that the availability at source i is a i
(i=1, 2, …, m) and the demand at the destination j is b j
(j=1, 2, …, n).
Let x ij be the amount of commodity to be shipped from the source i to the destination j.
Thus the problem becomes the LPP minimize z
 i m 

1 j n 

1 c ij x ij

We make an important assumption that the problem is a
balanced one. That is, total availability equals total demand i m 

1 a i
 j n 

1 b j
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 the total supply is more than the total demand)

Number of jobs is equal to the number of facility.
It is not necessary that number of jobs is equal to the number of
Facility.
Supply & demand is unity i.e. a i
= 1 Supply & demand is not unity i.e. a i
≠ 1
Number of unit allocated to a cell
Can be either one or zero.
Number of unit allocated to a cell
Can be more than zero.

 Feasible solution -
A set of non-negative values xij i=1,2,3……m, j=1,2,3……n that satisfies the rim condition is called a feasible solution to the transportation problem.
 Basic Feasible solution –
A feasible solution to a m x n transportation problem that contains no more than m + n – 1 non-negative allocations is called a basic feasible solution to the transportation problem

 Optimal solution -
A feasible solution (not necessarily the basic) that minimizes the transportation cost ot maximizes the profit is called an optimal solution
 Non degeneracy –
If a basic feasible solution to a (m x n ) transportation problem has total number of non negative allocation equals to m+n-1, then this condition is called Degeneracy in transportation problem.
 Degeneracy –
If a basic feasible solution to a (m x n ) transportation problem has total number of non negative allocation is less then m+n-
1,then this condition is called Degeneracy in transportation problem

 NWCM(North West Corner Method)
 CM(Cost Minima)
 RM(Row Minima)
 LCM(Least Cost Method)
 VAM(Vogel’s Approximation Method

S
1
S
2
S
3
Demand
D
1
8
15
3
150
D
2
5
10
9
80
D
3
6
12
10
50
Supply
120
80
80
280

Finding the basic feasible solution by VAM
S
1
S
2
D
1
8
(70)
15
D
5
2
D
3
6
(50)
12
S
3
3
(80)
10
(80)
9 10
Demand 150/70/0 80/0 50/0
[5] [4] [4]
[7] [5]
[5]
[6]
[6]
Supply
120/50/0 [1] [1] [1]
80/0
80/0
280
[2]
[6]
[2] [2]

Optimality test is done to find out ,whether the obtained feasible solution is optimal or not.
Optimality test is performed only on the feasible solution in which ,
(a) Number of allocation is m+n-1, where m = number of rows and n = number of columns
(b) These allocation should be in independent position

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