Mathematical Methods

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MATHEMATICAL METHODS

1. Finding An Initial Basic Feasible Solution:
An initial basic feasible solution to a transportation problem can be found by any one of the three following methods: I. North West Corner Rule
II. The Least Cost Method
III. Vogel’s Approximation Method

1. North West Corner Rule
The North West corner rule is a method for computing a basic feasible solution of a transportation problem, where the basic variables are selected from the North-West Corner (i.e. top left corner). The standard North West corner rule instructions are paraphrased below: Steps:

Step 1.Select the north west (upper left hand) corner cell of the transportation table and allocate as many units as possible equal to the minimum between available supply and demand. Step 2.Adjust the supply and demand numbers in the respective rows and columns. Step 3.If the demand for the first cell is satisfied, then move horizontally to the next cell in the second row. Step 4.If the supply for the first row is exhausted, then move down to the first cell in the second row. Step 5.If for any cell, supply equals demand, then the next allocation can be made in either in the next row or column. Step 6.Continue the process until all supply and demand values are exhausted.

2. The Least-Cost Method
The least-Cost Method is a method for computing a basic feasible solution of a transportation problem, where the basic variables are chosen according to the unit cost of transportation. This method is very useful because it reduces the computation and the time required to determine the optimal solution. The following steps summarize the approach. Steps:

Step 1.Indentify the box having minimum unit transportation cost. Step 2.If the minimum cost is not unique, then you are liberty o choose any cell. Step 3.Choose the value of the corresponding cell as much as possible subject to the capacity and requirement constraints. Step 4.Repeat steps 1-3 until all restrictions are satisfied.

3. Vogel’s Approximation Method
The Vogel Approximation (Unit penalty) method is an iterative procedure for computing a basic feasible solution of a transportation problem. This method is preferred over the two methods i.e. North West Corner Rule and Least cost Method, because the initial basic feasible solution obtained by this method is either optimal or very close to the optimal solution. The standard instructions are paraphrased below:

Steps:
Step 1.Indentify the boxes having minimum and next to minimum transportation cost in each row and write the difference (Penalty) along the side of the table against the corresponding row. Step 2.Indentify the boxes having minimum and next to minimum transportation cost in each column and write the difference (Penalty) along the side of the table against the corresponding column. Step 3.Indentify the row and column with the largest penalty, breaking ties arbitrarily. Allocate as much as possible to the variable with the least cost in the selected row or column. Adjust the supply and demand and cross out the satisfied row or column. If a row and column are satisfies simultaneously, only one of them is crossed out and remaining row or column is assigned a zero supply or demand. Any row or column with zero supply or demand should not be used in computing future penalties. Step 4.

a. If exactly one row or one column remains uncrossed out, stop. b. If only one row or column with positive supply or demand remains uncrossed out, determine the basic variables in the row or column by the Least-Cost Method. c. If all uncrossed out rows and column have (assigned) zero supply and demand, determined the zero basic variables by the Least-Cost Method. Stop. d. Otherwise, recomputed the penalties for the uncrossed out rows and columns, then go to step 3. (Notice that the rows and columns with assigned zero supple and demand...
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