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 Table 1 [1] .

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 [2] and then the systematic solution procedures from the simplex algorithm were further developed, primarily by Dantzig [3] and then by Charnes et al. [4] . In the solution procedure of TP, IBFS is known as the fundamental stage for finding an optimal solution. The well recognized classical methods, for finding an IBFS for the transportation problems are North West Corner Rule (NWCR) [5] [6] , Least Cost Method (LCM) [5] [6] and Vogel’s Approximation Method (VAM) [5] - [7] . Again researchers worked and are working on transportation problem to develop new algorithm to find a better IBFS for TPs [8] - [34] , and these methods may be used to solve maximization transportation problems [35] - [38] and also time minimization transportation problems [29] [39] - [41] . Like other researchers, in this paper an effective procedure for finding an IBFS for the cost minimizing TPs is proposed, and the proposed method is also used to solve profit maximization TP. Exceptionality and applicability of the method are also studied and explained in this study.

The tabular form of a TP is a matrix within a matrix shown in Table 2. Cost matrix is one of them that representing unit transportation cost c_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, and each destination has a demand of b_j units,. The objective is to determine which routes are to be selected and the size of the shipment on those routes, so that the total transportation cost of meeting demand, given the supply constraints, is minimized. Hence the mathematical formulation of cost minimization TP is,

Minimize:

Subject to:

, for all i and j.