Operation research and Management -VOGAL’S approximation method

1. CALCULATION OF OPTIMUM COST OF TRANSPORTATION
OF GOODS FROM GODOWNS TO DIFFERENT RETAILERS
OF A TOWN
A report submitted in partial fulfillment of
Requirement for the course
of
OPERATIONS RESEARCH
By
BOLISETTI SIVA PRADEEP
09BEM080
SCHOOL OF MECHANICAL AND BUILDING SCIENCES
Vellore – 632014, Tamil Nadu, India
APRIL 2012
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2. CONTENTS
Chapter No.
CHAPTER
Title
1
INTRODUCTION
Page No.
4
1.1 General introduction
1.2
Introduction to the case study
CHAPTER
2
LITERATURE REVIEW
5
CHAPTER
3
METHODOLOGY
6
3.1
Problem procedure
4
RESULTS AND DISCUSSION
4.1
Results and its significance
5
CONCLUSIONS
CHAPTER
CHAPTER
11
12
REFERENCES
13
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3. ABSTRACT
Transportation problem is used in many fields of business in the past and also these days,
as it is an efficient tool to optimize the transportation costs of the produced goods, which forms
one of the expenses to be considered the most. In this report, it is used to determine the optimum
cost of transportation of goods in a town to the different retailers of the town.
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4. I.INTRODUCTION
1.1
GENERAL INTRODUCTION
Transportation problem aims at minimizing the cost of transportation of similar goods from
different origins to different destinations.
In today’s highly competitive market, the pressure on organizations to find better ways to
create and deliver value to customers becomes stronger. How and when to send the products to
the customers in the quantities, that too in a cost-effective manner, has become more
challenging. Transportation models provide a powerful framework to meet this challenge. They
ensure the efficient movement and timely availability of raw materials and finished goods.
1.2 INTRODUCTION TO THE CASE STUDY
As a part of this case study, optimum cost of transportation of goods is found, for
transporting boxes of soaps by a distributor to different retailers of a town. Different datas are
collected such as the distance of individual retailer from different godown and the cost of
transportation for each kilometer is considered.
Wipro has various products under its brand name and among all the products, santoor
soaps are the most famous/most selling ones. The firm which supplies it in the city has 3
different godowns at different parts of the town and the boxes of soaps in different number are
stored at these paces according to the capacity of each one.
The town has nearly 30 retailers and considering all, will make the problem lengthy. For
this purpose, 4 main retailers (the orders given by them) are considered.
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5. II. LITERATURE REVIEW
In the literature, many papers of transportation problem come with new and modified
ideas to improve the cost efficiency of transportation.
Shiang Tai liu et al in the year 2005 came up withFuzzy total transportation cost
measures for fuzzy solid transportation problem.
A fuzzy number is an extension of a regular number in the sense that it does not refer to
one single value but rather to a connected set of possible values, where each possible value has
its own weight between 0 and 1. This weight is called the membership function. Fuzzy numbers
are extensions of real numbers. In this paper they developed a method that is able to derive the
fuzzy objective value of the fuzzy solid transportation problem when the cost coefficients, the
supply and demand quantities and conveyance capacities are fuzzy numbers.
Junb Bok Jo et al in the year 2007 published on non-linear fixed charge transportation
problem by spanning tree based genetic algorithm for non-linear fixed charge transportation
problem.
R.R.K Sharma and Saumya Prasad et al in a certain paper gave a heuristic that obtains
a very good starting solution for the primal transportation problem and this was expected to
enhance the performance of network simplex algorithm that obtains the optimal solution.
Yinzhen Li et al in the year 1998 ha published a study on improved genetic algorithm for
solving multi-objective solid transportation problem with fuzzy numbers. In this paper they have
presented improved genetic algorithm for solving fuzzy multi ojective solid transportation
problem in which the co-efficients of objective function are represented as fuzzy numbers.
Krzysztof Kowalski et al in the year 2007 published their study on Step Fixed Charge
Transportation problem (SFCTP) which is a variation of FCTP where the fixed cost is in the
form of a step function dependent on the load in a given route. They discussed the theory of
SFCTP and presented a computationally simple heuristic algorithm for solving small SFCTP’s.
In the year 2010, Amarpreethkaur et al of Patiala university have published a paper, in
which a new algorithm is proposed for solving a special type of fuzzy transportation problems by
assuming that a decision maker is uncertain about the precise values of transportation cost only
but there is no uncertainty about the supply and demand of the product. In the proposed
algorithmtransportation costs are represented by generalized trapezoidal fuzzy numbers.
Zhi-chunin the year 2010 have worked on the intermodal equilibrium, road toll pricing,
and bus system design issues in a congested highway corridor with two alternative modes – auto
and bus – which share the same roadway along this corridor.
Amarpreethkaur in the year 2010 proposed another paper on solving fuzzy
transportation problem using ranking function.
Preetwanisingh and P.K.Saxena in the year 1997 have published their work on multiple
objective transportation problem with additional restrictions.
Maria.J.Alves et al in the year 2003 published a paper on Interactive decision support. In
this paper they presented a linear programming solution method called TRIMAP, that is
dedicated in solving three-objective transportation problems.
Mitsuo gen in the year 1999 proposed a paper on spanning tree based genetic algorithm
for Bacteria transportation problem. This transportation problem has a special data structure
solution.
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6. Veena adhlaka in the year 2004 worked on ‘more-for-less algorithm on fixed
transportation problem. The more-for-less (MFL) phenomenon in distribution problems occurs
when it is possible to ship more total goods for less (or equal) total cost, while shipping the same
quantity or more from each origin and to each destination.In this paper, they developed a simple
heuristic algorithm to identify the demand destinations and the supply points to ship MFL in
FCTPs. The proposed method builds upon any existing basic feasible solution. It is easy to
implement and can serve as an effective tool for managers for solving the more-for-less paradox
for large distribution problems.
III. METHODOLOGY
For a simple transportation problem, the availability must be equal to the requirement.
Therefore an assumption is made stating that the total order given by 4 retailers will be equal to
the total stock present. The actual stock that is noted to be available while collecting the data is
scaled down to be equal to the order.
For example, if the total order given by the four retailers under consideration is 125 boxes
of soaps, and the actual stock present at the godowns is noted to be 100 in the first, 100 in the
second and 50 in the third godown, it is assumed to be 50, 50 and 25 respectively.
The distance of each retailer from each individual godown is estimated, and the cost is
calculated based on the distance.
3.1 DATA COLLECTED:
(i)
DISTANCES
G1 to R1  4 Kms
G1 to R2  0.75 Kms
G1 to R3  2.5 Kms
G1 to R4  4.5 Kms
G2 to R1  2 Kms
G2 to R2  0.5 Kms
G2 to R3  4 Kms
G2 to R43.25 Kms
Gi number of the godown
Ri number of the retailer
6
G3 to R1 1 Kms
G3 to R22.75 Kms
G3 to R3  2 Kms
G3 to R4  5 Kms

7. (ii)
COSTS
G1 to R1  16Rs
G1 to R2  3 Rs
G1 to R3  10 Rs
G1 to R4  18 Rs
G2 to R1  8 Rs
G2 to R2  2 Rs
G2 to R3  16 Rs
G2 to R413 Rs
G3 to R1 4 Rs
G3 to R2 11 Rs
G3 to R3  8 Rs
G3 to R4  20 Rs
3.2PROBLEM PROCEDURE:
STEP I:
Table 3.1
16
3
10
18
25(7)
8
2
16
13
35(6)
4
11
8
20
20(4)
15(4)
20(1) 15(2) 30(5)
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8. STEP II :
The initial solution is iterated for an IBFS using VOGAL’S approximation method. It is
followed by following sequence of steps:
Table3.2
16
3(20)
10
18
25(7)
8
2
16
13
35(6)
4
11
8
20
20(4)
15(2)
30(5)
15(4) 20(1)
Table 3.3
16
10
(5) 18
5 (6)
8
16
13
35 (5)
4
8
20
20 (4)
15 (4)
15 (2)
30 (5)
60/60
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9. Table 3.4
8
16
13
35 (5)
4
8
(10)
20
20 (4)
15 (4) 10 (8) 30 (7) 55/55
Table 3.5
8
13
35 (5)
4
(10)
20
10 (16)
15 (4)
30 (7)
45/45
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10. 8
(5)
13
(30)
35
5
30
35/35
Table 3.6
The initial basic feasible solution is found to be:
Table 3.7
16
3
(20)
10
8
(5)
2
16
4
(10)
11
8
20
(5)
15
15
18
25
13 (30) 35
(10)
20
20
30
STEP III :
This IBFS is solved using MODI method, and the solution is calculated.
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11. IV. RESULTS
The total transportation cost is estimated as:
(3*15) + (10*10) + (5*2) + (30*13) + (4*5) + (5*8) = Rs.645.
DISCUSSION:
The cost estimated is well behind the cost which is being incurred during the regular
process followed by the distributor. If this way of transportation is followed, greater profits can
be made.
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12. V. CONCLUSION
Thus, in this case study optimal cost of transportation is found. Though some assumptions are
made while solving the problem, it does not affect the final value much. The transportation problem can
thus be used in minimizing the cost of transportation in both small scale workplaces, as discussed through
the case study and also large work places like transport of finished manufactured goods from the factories
to different places, and transport of goods which are unloaded in huge quantities in ships, which come
from other countries. It is made evident that an operations research technique when used, can save money,
man-power, and time.
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13. VI. REFERENCES
WWW.SCIENCEDIRECT.COM
WWW.WIKIPEDIA.COM
OPERATIONS RESEARCH BY PREM KUMAR GUPTHA AND D.S.HYRA
OPERATIONS RESEARCH BY MANMOHAN AND KANTHI SWAROOP
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