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Industries require planning in transporting their products from production centres to the users end with minimal transporting cost to maximize profit. This process is known as Transportation Problem which is used to analyze and minimize transportation cost. This problem is well discussed in operation research for its wide application in various fields, such as scheduling, personnel assignment, product mix problems and many others, so that this problem is really not confined to transportation or distribution only. In the solution procedure of a transportation problem, finding an initial basic feasible solution is the prerequisite to obtain the optimal solution. Again, development is a continuous and endless process to find the best among the bests. The growing complexity of management calls for development of sound methods and techniques for solution of the problems. Considering these factors, this research aims to propose an algorithm “Incessant Allocation Method” to obtain an initial basic feasible solution for the transportation problems. Several numbers of numerical problems are also solved to justify the method. Obtained results show that the proposed algorithm is effective in solving transportation problems.

Transportation Problem (TP) is one of the subclasses of Linear Programming Problems in which the objective is to transport various quantities of a single homogeneous commodity that are initially stored at various origins to different destinations in such a way that the total transportation cost is minimum. To achieve this objective we must know the amount and location of available supplies and the quantities demanded. In addition, we also know the unit transportation cost of the commodity to be transported from various origins to various destinations. Few examples of TP are summarized in

Source | Destination | Commodity | Objective |
---|---|---|---|

Plants | Markets | Finished goods | Minimizing total cost of shipping |

Plants | Finished goods warehouses | Finished goods | Minimizing total cost of shipping |

Finished goods warehouses | Markets | Finished goods | Minimizing total cost of shipping |

Suppliers | Plants | Raw materials | Minimizing total cost of shipping |

Suppliers | Raw materials warehouses | Raw materials | Minimizing total cost of shipping |

Raw materials warehouses | Plants | Raw materials | Minimizing total cost of shipping |

Balanced transportation problems and unbalanced transportation problems are the types of TPs. If the sum of the supplies of all the sources is equal to the sum of the demands of all the destinations, the problem is termed as a balanced TP. On the other hand, the problem is termed as unbalanced TP. The basic steps for obtaining an optimum solution to a TP are:

・ Step 1: Mathematical formulation of the TP.

・ Step 2: Verify the TP: either it is balanced or unbalanced. If the problem is unbalanced, first balance it.

・ Step 3: Determine the Initial Basic Feasible Solution (IBFS).

・ Step 4: Verify the optimality condition of the IBFS. If the solution is not optimal, improve it for obtaining optimal solution.

The basic TP was first developed by Hitchcock [

The tabular form of a TP is a matrix within a matrix shown in _{ij}, indicating the cost of shipping one unit from the i-th origin to the j-th destination. Super impose of this matrix is the matrix of transportation variable x_{ij}, indicating the amount shipped from i-th source to j-th destination. Right and bottom sides of the transportation table point out the amounts of supplies a_{i} available at source i and the amount demanded b_{j} at destination j.

The TP can be stated as an allocation problem in which there are m sources (suppliers) and n destinations (customers). Each of the m sources can allocate to any of the n destinations at a per unit carrying cost c_{ij} (unit transportation cost from source i to destination j). Each sources has a supply of a_{i} units, _{j} units,

Minimize:

Subject to: