Iaetsd ones method for finding an optimal

1. ONES METHOD FOR FINDING AN OPTIMAL
SOLUTION FOR TRANSPORTATION
PROBLEM
Pushpa Latha Mamidi
Assistant Professor of Mathematics, Vishnu Institute of Technology, Bhimavaram, A.P., India
Email: pushpamamidi@gmail.com
Abstract: In this paper, a new method named Ones Method is proposed for finding an optimum solution
for a wide range of transportation problems, directly. The new method is based on allocating units to the
cells in the transportation matrix initiating with maximum number of ones starting with minimum
demand/supply to the cell and then try to find an optimum solution to the given transportation problem.
The proposed method is a systematic procedure, easy to apply and can be utilized for all types of
transportation problem. A numerical illustration is established and the optimality of the result yielded by
this method is also checked.
Keywords: Transportation Problem, Assignment Problem, Optimal solution, i.b.f.s, VAM
I. INTRODUCTION
Transportation problem is used to transport various amounts of single homogeneous commodity that
are initially stored at various origins, to different destinations in such a way that the total transportation
cost is a minimum. It is a special class of Linear Programming Problem.
In 1941 Hitchcock[1] developed the basic transportation problem along with the constructive method
of solution and after that in 1949 Koopams [4] discussed the problem in detail. Again in 1951 Dantzig[9]
formulated the Transportation Problem as L.P.P.
The simplex method is not suitable for the Transportation Problem especially for large scale
transportation problem due to its special structure of the model in 1954 Charnes and Cooper[10] was
developed Stepping Stone Method for the efficiency reason. For obtaining an optimum solution for
Transportation Problem it was required to solve the problem in two stages. In the first stage the initial
basic feasible solution (i.b.f.s) was obtained by using any one of the methods such as North West Corner
Rule, Row Minima, Column Minima, Least Cost, Vogle’s Approximation methods. Then finally MODI
method was used to get an optimum solution.
In last few years P.Pandian et.al[13], Sudhakar et.al[6], N.M.Deshmukh[2], G.Reena Patel et.al[5],
Aramerthakannan et.al[7], Abdul Quddoos[3], ezhil rannan[11] and many others proposed different
methods for finding an optimum solution directly. This paper presents a new approach for finding an
optimum solution directly with a systematic procedure.
Mathematical Representation
Let there are ‘m’ origins, Oi having ai (i=1,2…….m) units of source which are to be transported to ‘n’
destinations Dj’s with bj (j=1,2….n) units of demand respectively. Let Cij be the cost of shipping one unit
product from ith
origin to jth
destination and xij be the amount to be shipped form ith
origin to jth
destination.
It is also assumed that total availabilities satisfy the total requirements i.e.,
Mathematically the problem can be stated as
Min Z =
Proceedings International Conference On Advances In Engineering And Technology
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2. Subject to and
And for all i and j
II PROPOSED METHOD [ONES METHOD]
Step 1: Construct the transportation table
Step 2: Select minimum element from each row and divide with each element in the corresponding row so
that each row contains at least one ones.
Step 3: Select column minimum from each column and divide with each element in the corresponding
column.
Step 4: In the reduced cost matrix there will be at least one one in each row and column, then find the row
or column having maximum number of ones. If tie occurs then select the minimum demand/ supply value.
Allocate the minimum of supply/demand at the place of one and delete the row or column where supply or
demand depleted.
Step 5: After performing step 4, delete the row or column for further calculation where supply from a
given source is depleted or the demand for a given destination is satisfied.
Step 6: Check whether the resultant matrix possesses at least one one in each row/column then go to step 7
otherwise repeat step 2 and step 3.
Step 7: Repeat step 4 to step 6 until and unless all the demands are satisfied and all the supplies are
exhausted.
III. NUMERICAL EXAMPLES
3.1 Consider the following cost minimizing transportation problem with 3 origins and 3 destinations.
D1 D2 D3 Supply
S1 11 9 6 40
S2 12 14 11 50
S3 10 8 10 40
Demand 55 45 30 130 (Total)
By using Ones Method allocations are obtained as follows
D1 D2 D3 Supply
S1 11 9 6 40
S2 12 14 11 50
S3 10 8 10 40
Demand 55 45 30 130
10 30
50
5 35
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3. The total cost associated with these allocations is
9(10) + 6(30) + 12(50) + 10(5) + 8(35) = 1200
3.2 Consider the following cost minimizing transportation problem
D1 D2 D3 D4 Supply
S1 13 18 30 8 8
S2 55 20 25 40 10
S3 30 6 50 10 11
Demand 4 7 6 12 29 (Total)
By using Ones Method allocations are obtained as follows
D1 D2 D3 D4 Supply
S1 13 18 30 8 8
S2 55 20 25 40 10
S3 30 6 50 10 11
Demand 4 7 6 12
The total cost associated with these allocations is
13(4) + 8(4) + 20(4) + 25(6) + 6(3) + 10(8) = 412
3.3 Find the most economical shipment to minimize the transportation cost for the following
transportation problem
D1 D2 D3 D4 D5 Supply
S1 4 1 2 4 4 60
S2 2 3 2 2 3 35
S3 3 5 2 4 4 40
Demand 22 45 20 18 30 135 (Total)
By using Ones Method allocations are obtained as follows
D1 D2 D3 D4 D5 Supply
S1 4 1 2 4 4 60
S2 2 3 2 2 3 35
S3 3 5 2 4 4 40
Demand 22 45 20 18 30
4 4
4 6
3 8
45 15
17 18
5 20 15
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4. The total cost associated with these allocations is
1(45) + 4(15) + 2(17) +2(18) + 3(5) + 2(20) + 4(15) = 290
3.4 Consider the following cost minimizing transportation problem with three origins and four
destinations
D1 D2 D3 D4 Supply
S1 19 30 50 10 7
S2 70 30 40 60 9
S3 40 8 70 20 18
Demand 5 8 7 14 34 (Total)
By using Ones Method allocations are obtained as follows
D1 D2 D3 D4 Supply
S1 19 30 50 10 7
S2 70 30 40 60 9
S3 40 8 70 20 18
Demand 5 8 7 14
The total cost associated with these allocations is
19(5) + 10(2) + 30(2) + 40(7) + 8(6) + 20(12) = 743
IV. COMPARISION OF TOTAL COST OF TRANSPORTATION PROBLEM FROM VARIOUS
METHODS
V. CONCLUSION
In this paper a new and simple method was introduced for solving transportation problem. Thus it can
be concluded that the optimum solution obtained by the Ones Method is same as that of MODI method.
As this method requires less time and is very easy to understand and apply, so it will be very useful for
decision makers who are dealing with logistic and supply chain problems.
Problem
No.
Problem
Size
Ones
Method
NWCM LCM VAM MODI
3.1 3 X 3 1200 1200 1200 1200 1200
3.2 3 X 4 412 484 516 476 412
3.3 3 X 5 290 363 305 290 290
3.4 3 X 4 743 1015 814 779 743
5 2
2 7
6 12
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5. REFERENCES
[1] F.L. Hitckcock, “The distribution of a product from several sources to numerous localities”, Journal of
Mathematical Physics, vol 20, pp.224-230, 2006.
[2] N.M. Deshmukh, “An innovative Method for solving Transportation Problem”, International Journal of
Physics and Mathematical Sciences, vol 2(3), pp.86-91,2012.
[3] Abdul Quddoos,Shakeel Javaid, M.M. Khalid, “A New Method for Finding an Optimal Solution for
Transportation Problem”, vol 4 No. 7, pp. 1271-1274, 2012.
[4] Koopams TC, “Optimum Utilization of the Transportation system” in Proc. International statistical
conference, Washington D.C.
[5] Reena G. Patel, P.H. Bhathawala, “The New Global Approach to a Transportation Problem”, Vol 2(3),
pp.109-113, 2014.
[6] Sudhakar VJ, Arunnsankar N, Karpagam T, “A new approach for find an optimal solution for
Transportation Problems”, European Journal of Scientific Research 68, pp.254-257.
[7] S. Aramuthakannan, P.R. Kandasamy, “Revised Distribution Method of finding optimal solution for
Transportation Problems”, vol 4(5), pp.39-42, 2013.
[8] Abdallah A.Hlayel, Mohammad A. Alia, “Solving Transportation Problems using the Best Candidates
Method”, vol. 2(5), pp.23-30,2012.
[9] Dantzig GB, Linear Programming and Extensions, New Jersey, Princeton University press
[10] Charnes, Cooper, “The Stepping-Stone method for explaining linear programming calculation in
transportation problems”, Management Science vol 1(1), pp.49-69
[11] S. Ezhil Vannan, S,Rekha, “A New Method for obtaining an optimal solution for Transportation
Problems”,vol 2(5), pp.369-371,2013.
[12] Taha H.A., Operations Research, Prentice Hall of India, New Delhi, 2004
[13] P.Pandian and G.Natarajan, “A New method for finding an optimal solution for Transportation
problems”, International Journal of Math.sci & Eng., Appls., vol 4, pp.59-65,2010.
Proceedings International Conference On Advances In Engineering And Technology
ISBN NO: 978 - 1503304048
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