Transcript TRANSPORTATION MODEL

TRANSPORTATION MODEL
presented BY,
MANEET KUMAR
MANI SHANKAR
MANINDER PAL SINGH
MANOJ KUMAR
MANISH KUMAR GARG
MADHU MAYA
INTRODUCTION
• 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.
Assumptions
• 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 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).
Let xij be the amount of commodity to be shipped from the source i to
the destination j.
Thus the problem becomes the LPP
m
minimize
z
i 1
n
c
j 1
ij
xij
We make an important assumption that the problem is a
balanced one. That is, total availability equals total demand
m
n
 a  b
i 1
i
j 1
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)
Assignment vs transportation
ASSIGNMENT
TRANSPORTATION
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. ai = 1
Supply & demand is not unity
i.e. ai ≠ 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.
Important Terms
 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+n1,then this condition is called Degeneracy in transportation problem
METHODS
 NWCM(North West Corner Method)
 CM(Cost Minima)
 RM(Row Minima)
 LCM(Least Cost Method)
 VAM(Vogel’s Approximation Method
Problem
D1
D2
D3
Supply
S1
8
5
6
120
S2
15
10
12
80
S3
3
9
10
80
Demand
150
80
50
280
Finding the basic feasible solution by VAM
D1
D2
D3
Supply
S1
8(70)
5
6(50)
120/50/0
[1]
[1]
[1]
S2
15
10(80)
12
80/0
[2]
[2]
[2]
S3
3(80)
9
10
80/0
[6]
Demand
150/70/0
80/0
50/0
280
[5]
[4]
[4]
[7]
[5]
[6]
[5]
[6]
OPTIMALITY
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
MODI method
VARIANTS IN TRANSPORTATION
 Unbalanced Transportation Problems.
 Maximization Problem.
 Different Production Costs.
 No allocation in a particular cell/cells.
 Over Time Production.