A STUDY OF MATHEMATICAL METHOD OF TRANSPORTATION PROBLEM

1. A STUDY OF MATHEMATICAL METHOD OF TRANSPORTATION
PROBLEM BASED ON RAW MATERIAL
Shivang Ranjan Guide Name
M.Sc. (Mathematics) Ms. Anshika Agarwal
Roll No. 1620808017 Assistant Professor
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2. Content
 Introduction.
 Objective.
 Methodology.
 Calculation of Transportation Cost .
 North-West corner method (NWCM)
 Least Cost Method.
 Vogel’s Approximation Method.
 Conclusions.
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3. Introduction
 A transportation problem basically deals with the problem which
aims to find the best way to fulfill the demand and supply. While
trying to find the best way, generally a variable cost of shipping the
product from one supply point to a demand point or a similar
constraint should be taken into consideration. Here under this study
main problem to the company is that the company is not able to
fulfill 100% demand of the orders that is a common problem in
different sectors, and we will try to know all those factors which are
involved to lateness of the orders, like cost, shortage of raw
material, production and final product orders.
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4.  The task is to evaluate production schedule, for this we need to
evaluate orders with respect of raw material and final delivery.
According to initial information the mill has 55 to 60 tons
production per day while the orders per day are 72 tons. It is may be
difficult to equalize the demand and production but we can try to
minimize this gap between production and delivery orders.
Coordination between production and transportation of raw material
and distribution planning has received increased attention in global
companies. To minimize the total cost, manufacturing firms should
integrate their production and logistics decisions, especially when
considering the rising costs of transportation and distribution of
finished products.
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5.  There are many problems in the literature have dealt with a single
commodities and products. Here in the study we use linear
programming model for a multi-type of waste paper used for paper
production and raw material problem to minimize the total
transportation cost. In their formulation, which include different raw
material categories, and potential warehouses that receive the raw
material from several places. While studying, the problem is
formulated with several decision variables and their values and used
different transportation methods to solve the problem.
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6. Objective
 The objective of this part of study is to minimize the cost of raw
material for the company, and make sure the availability of raw
material at optimized cost. The optimum sequence of work orders is
first of all sought and reliability is also checked with different
alternatives of sources. Finding best sequence of orders is a difficult
task and has a strong relationship with reliability of the production
system as optimum sequence or orders with better work content
make a better flow of the production system. Here in the study we
will also check the alternative costs and opportunity costs while
finding the optimize solution of this problem, using the method of
multipliers and its sensitivity analysis.
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7. What is transportation ?
 Transportation is the movement of people and goods from one place
to another. The term is derived from the Latin trans ("across") and
portare ("to carry"). Industries which have the business of providing
equipment, actual transport, or goods and services used in transport
of goods or people make up a large broad and important sector of
most national economies, and are collectively refered to as transport
industries.
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8. History of Transportation:-
 Two-wheel chariot - world’s first form of wheeled transportation -
invented in Sumeria, around 3500 BC. This eventually led to
invention of four-wheel chariot in due course.
 Cornelis Drebbel invented the first submarine in 1620 AD
 Leonardo da Vinci - first to seriously theorize about flying machines
- with over 100 drawings that illustrated his theories on flight –
1492 AD
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9. I have taking the Ramaa Shyama Pvt Limited company problem
Which is situated in Bareilly District and facing the transportation
problem.
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10. Research Methodology
Mathematical method to solved transportation problem by three
method .
 1. North West Corner Method (NWCM)
 2. Lowest Cost Entry Method
 3. Vogel’s Approximation Method (VAM)
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11. North West Corner Method (NWCM)
 The North West corner rule is a method for computing a basic
feasible solution of a transportation problem where the basic
variables are selected from the North–West corner .
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12. Algorithm to solve NWCM
 Step 1: start with the cell at the upper left (North-West) corner of the
transportation matrix and allocate commodity equal to the minimum
of the rim values first row and first column, i.e. min(a1,b1).
 Step 2: if allocation made in step 1 is equal to the supply available
at first source (a1, in first row), then move vertically down to the
cell (2, 1) in the second row and first column. Apply step 1 again
for, for next step.
 Step 3: Continue the procedure step by step till an allocation is
made in the south-east corner of the transportation table
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13. 13

14.  Total = 10× 200 + 16 × 450 + 16 × 130 + 14 × 90 + 15 × 270 +
16 × 128 + 18 × 122 + 18 × 200 + 22 × 68 + 51 × 292
= Rs. 40822
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15. Least Cost Method
 Matrix minimum method is a method for computing a basic feasible
solution of a transportation problem where the basic variables are
chosen according to the unit cost of transportation.
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16. Steps to solve the Least Cost Method
 Identify the box having minimum unit transportation cost (cij).
 If there are two or more minimum costs, select the row and the
column corresponding to the lower numbered row
 If they appear in the same row, select the lower numbered column.
 Choose the value of the corresponding xij as much as possible
subject to the capacity and requirement constraints.
 If demand is satisfied, delete the column .
 If supply is exhausted, delete the row.
 Repeat steps 1-6 until all restrictions are satisfied.
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17. 17

18.  Total Cost = 10 × 200 + 14 × 450 + 16 × 182 + 14 × 38 + 17 × 198 + 18 × 72 +
18 × 250 + 17 × 200 + 22 × 68 + 51 × 292
= Rs. 40694
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19. By Vogel’s Approximation Method
 Vogel’s approximation (penalty or regret) method is a heuristic
method and is preferred more than the other two method describe
above. In this method, each allocation is made on the basis of the
opportunity cost that would have been incurred if the allocation in
certain cell with minimum unit transportation cost were missed. In
this method allocation are made so that they are penalty cost
minimize. The advantage of this method is that it is give an initial
solution which is nearer to an optimal solution or is the optimal
solution itself.
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20. Step to solve the Vogel’s Approximation Method
 Identify the boxes having minimum and next to minimum
transportation cost in each row and write the difference (penalty)
along the side of the table against the Corresponding row.
 Identify the boxes having minimum and next to minimum
transportation cost in each column and write the difference (penalty)
against the corresponding column.
 Identify the maximum penalty. If it is along the side of the table,
make maximum allotment to the box having minimum cost of
transportation in that row.
 If the penalties corresponding to two or more rows or columns are
equal, select the top most row and the extreme left column.
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21. 21

22. Total Cost- 10 × 200 + 14 × 450 + 16 × 140 + 14 × 38 + 14 × 42 +
17 × 80 + 18 × 190 + 30 × 250 + 18 × 200 + 19 × 360
= Rs. 35598
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23. Comparison of both method
 North West Corner Method (NWCM) - Rs.40882
 Lowest Cost Entry Method - Rs.40692
 Vogel’s Approximation Method (VAM)- Rs.35598
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24. Conclusion
The transportation methods found the most optimum cost of the
transportation of raw material. In this procedure the study shows that
Vogel’s Approximation Method is most efficient, its gives Rs.35598.
So study can decide that cost allocation plan of Vogel’s Approximation
Method is the reasonable plan for acquiring raw material for the plant.
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25. References
 http://www.sciencedirect.com
 http://college.cengage.com/mathematics/larson/elementary_linear/4e
/shared/downloads/c09s4.pdf
 M.S. Sodhi, "What about the 'O' in O.R.?" OR/MS Today,
December, 2007, p. 12, http://www.lionhrtpub.com/orms/orms-12-
07/frqed.html
 Introduction to operation research, 8th Edition, Hamdy A. Taha,
University of Arkansas, Fayetteville.
 International Journal of Management Science and Engineering vol.1
(2006) No.1, pp.47-52
 Erlander S.B (2010) Cost-Minimizing Choice Behavior in
Transportation Planning: A Theoretical. Page 8-10
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26.  http://www.scribd.com/doc/7079581
 http://www.scribd.com/doc/32121115/Transportation-Model-
Management-Science
 Wikipedia (2010): Transportation Theory.
http://en.wikipedia.org/wiki/Transportation_theory
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27. Thank You
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