Proceedings International Conference On Advances In Engineering And Technology
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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 = ISBN NO: 978 - 1503304048
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Proceedings International Conference On Advances In Engineering And Technology
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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 Demand
10 55
8 45
10 30
40 130 (Total)
D3
Supply
By using Ones Method allocations are obtained as follows D1 S1
11
S2
12
S3
10
Demand
ISBN NO: 978 - 1503304048
D2 9 50 5
55
10
14 8
35
45
6
30
40
11
50
10
40 30
130
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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 S1 13 18 30 S2 55 20 25 S3 30 6 50 Demand 4 7 6
D4 8 40 10 12
Supply 8 10 11 29 (Total)
D4
Supply 8
By using Ones Method allocations are obtained as follows D1
D2
S1
13
S2
55
20
S3
30
6
4
Demand
D3
18
30 25
4
7
4
40
6
50
3
4
8
10
10
11
8
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 S1
4
S2
2
S3
3
Demand
ISBN NO: 978 - 1503304048
D2 1
17 5
22
D3
D4
2
4
3
2
2
5
2
45
45
20
20
18
4
D5 4
15
3
35
4 18
Supply 60
15
40
30
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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
S1
19
S2
70
30
S3
40
8
Demand
5
5
D3
30
D4
50 2 6
10
40
60
7
70
8
2
9
20 7
Supply 7
12
18
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
Problem No.
Problem Size
Ones Method
NWCM
LCM
VAM
MODI
3.1
3X3
1200
1200
1200
1200
1200
3.2
3X4
412
484
516
476
412
3.3
3X5
290
363
305
290
290
3.4
3X4
743
1015
814
779
743
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.
ISBN NO: 978 - 1503304048
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Proceedings International Conference On Advances In Engineering And Technology
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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.
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