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Finding an initial basic feasible solution is the prime requirement to obtain an optimal solution for the transportation problems. In this article, a new approach is proposed to find an initial basic feasible solution for the transportation problems. The method is also illustrated with numerical examples.

Transportation problem is famous in operation research for its wide application in real life. This is a special kind of the network optimization problems in which goods are transported from a set of sources to a set of destinations subject to the supply and demand of the source and destination, respectively, such that the total cost of transportation is minimized. The basic transportation problem was originally developed by Hitchcock in 1941 [

・ Phase 1: Mathematical formulation of the transportation problem.

・ Phase 2: Finding an initial basic feasible solution.

・ Phase 3: Optimize the initial basic feasible solution which is obtained in Phase 2.

In this paper, Phase 2 has been focused in order to obtain a better initial basic feasible solution for the transportation problems. This problem has been studied since long and is well known by Abdur Rashid et al. [

Again, some of the well reputed methods for finding an initial basic feasible solution of transportation problems developed and discussed by them are North West Corner Method (NWCM) [

In this paper, a new algorithm is proposed to find an initial basic feasible solution for the transportation problems. A comparative study is also carried out by solving a good number of transportation problems which shows that the proposed method gives better result in comparison to the other existing heuristics available in the literature.

Generally the transportation model is represented by the network in _{ij} and the amount shipped, x_{ij}. The amount of supply at source i is S_{i}, and the amount of demand at destination j is d_{j}. The objective of the model is to determine the unknowns’ x_{ij} that will minimize the total transportation cost while satisfying the supply and demand restrictions.

Considering the above notations, the transportation problem can be stated mathematically as a linear programming problem as:

Minimize:

Subject to:

The objective function minimizes the total cost of transportation (Z) between various sources and destinations. The constraint i in the first set of constraints ensures that the total units transported from the source i is less than or equal to its supply. The constraint j in the second set of constraints ensures that the total units transported to the destination j is greater than or equal to its demand.

In the proposed approach, an allocation table is formed to find the solution for the transportation problem. That’s why this method is named as Allocation

・ Step-1: Construct a Transportation

・ Step-2: Ensure whether the TP is balanced or not, if not, make it balanced.

・ Step-3: Select minimum odd cost (MOC) from all the cost cells of TT. If there is no odd cost in the cost cells of the TT, keep on dividing all the cost cells by 2 (two) till obtaining at least an odd value in the cost cells.

・ Step-4: Form a new table which is to be known as allocation table (AT) by keeping the MOC in the respective cost cell/cells as it was/were, and subtract selected MOC only from each of the odd cost valued cells of the TT. Now all the cell values are to be called as allocation cell value (ACV) in AT.

・ Step-5: At first, start the allocation from minimum of supply/demand. Allocate this minimum of supply/ demand in the place of odd valued ACVs at first in the AT formed in Step-4. If demand is satisfied, delete the column. If it is supply, delete the row.

・ Step-6: Now identify the minimum ACV and allocate minimum of supply/demand at the place of selected ACV in the AT. In case of same ACVs, select the ACV where minimum allocation can be made. Again in case of same allocation in the ACVs, choose the minimum cost cell which is corresponding to the cost cells of TT formed in Step-1 (i.e. this minimum cost cell is to be found out from the TT which is constructed in Step-1). Again if the cost cells and the allocations are equal, in such case choose the nearer cell to the minimum of demand/supply which is to be allocated. Now if demand is satisfied delete the column and if it is supply delete the row.

・ Step-7: Repeat Step-6 until the demand and supply are exhausted.

・ Step-8: Now transfer this allocation to the original TT.

・ Step-9: Finally calculate the total transportation cost of the TT. This calculation is the sum of the product of cost and corresponding allocated value of the TT.

A company manufactures motor cars and it has three factories F_{1}, F_{2} and F_{3} whose weekly production capacities are 300, 400 and 500 pieces of cars respectively. The company supplies motor cars to its four showrooms located at D_{1}, D_{2}, D_{3} and D_{4} whose weekly demands are 250, 350, 400 and 200 pieces of cars respectively. The transportation costs per piece of motor cars are given in the transportation

Factories | Showrooms | Production capacity | |||
---|---|---|---|---|---|

D_{1} | D_{2} | D_{3} | D_{4} | ||

F_{1} | 3 | 1 | 7 | 4 | 300 |

F_{2} | 2 | 6 | 5 | 9 | 400 |

F_{3} | 8 | 3 | 3 | 2 | 500 |

Demand | 250 | 350 | 400 | 200 |

Allocation of various cells in the allocation table for Example-1 is shown in Allocation