An Innovative Algorithmic Approach for Solving Profit Maximization Problems

: The new algorithmic technique developed in this article to solve the profit maximization problems using transportation algorithm of Transportation Problem (TP) has three basic parts; first converting the maximization problem into the minimization problem, second formatting the Total Opportunity Table (TOT) from the converted Transportation Table (TT), and last allocations of profits using the Row Average Total Opportunity Value (RATOV) and Column Average Total Opportunity Value (CATOV). The current algorithm considers the average of the cell values of the TOT along each row identified as RATOV and the average of the cell values of the TOT along each column identified as CATOV. Allocations of profits are started in the cell along the row or column which has the highest RATOVs or CATOVs. The Initial Basic Feasible Solution (IBFS) obtained by the current method is better than some other familiar methods which is discussed in this paper with the three different sized examples.


Introduction
Transportation problems in linear programming show the mathematical optimal ways to get the solutions of transportation problems. Transportation algorithm is an effective tool to get optimal profit. Some of the usual algorithms to solve the transportation problems are North West Corner (NWC) Method, Matrix Minima Method, and Vogel's Approximation Method (VAM). Afterwards many researchers provide many helpful algorithms to get IBFS of transportation problems. Some of the methods and algorithms that the current work has gone through are: P. Pandian and G. Natarajan's 'A New Approach for Solving Transportation Problems with Mixed Constraints' [1]; 'An Innovative Method for Solving Transportation Problem' [2] by N. M. Deshmukh; 'Modified Vogel's Approximation Method for Unbalance Transportation Problem' [3] [9]. Md. Amirul Islam et al. [10] calculate the Difference Indicators by taking the difference of the largest and the next largest cell value of each row and each column of the TOT for the allocation of units of the TT to determine maximum profit. The above mentioned algorithms cited in this article are beneficial to find the IBFS to solve profit maximization objective. The current research also ads a useful algorithm which gives a better IBFS in the profit maximization problems.

Algorithm
The developed algorithm in the current work involves three parts: 1. Converting the maximization problem into the minimization problem 2. Algorithm for Total Opportunity Table (TOT) 3. Algorithm for allocation Conversion of the problem Subtract the highest profit from the other profits in the TT and place the subtracted values in a new a TT. In this way the maximization problem becomes the minimization problem.

Algorithm for TOT
Step 1 Subtract the smallest entry from each of the elements of every row of the new TT and place them on the right-top of corresponding elements.
Step 2 Apply the same operation on each of the columns and place them on the right-bottom of the corresponding elements.
Step 3 Form the TOT whose entries are the summation of right-top and right-bottom elements of Steps 1 and 2.

Algorithm for Allocation
Step 1 Place the average of total opportunity values of cells of the TOT in the right side along each row in the first bracket identified as Row Average Total Opportunity Value (RATOV) and the average of total opportunity values of cells in the below along each column in the first bracket identified as Column Average Total Opportunity Value (CATOV).
Step 2 Identify the highest element among the RATOVs and CATOVs, if there are two or more highest elements; choose the highest element along which the smallest valued element is present. If there are two or more smallest elements, choose any one of them arbitrarily.
Step 3 Allocate ij x = min ( Step 5 Repeat Steps 1 to 4 until the rim requirement satisfied.
Step 6 Calculate ij ij x p p 1 1 , P being the maximum profit where ij p is the profit unit of i-th row and j-th column of the TT corresponding to the basic cells of TOT.

Numerical Examples
Example 01 Three products P 1 , P 2 , and P 3 are produced in three machines M 1 , M 2 , and M 3 and their profit margins are given in the following table. We want to maximize the profit by the current algorithm. Solution We subtract the maximum profit 380 from other profits of the TT. The row differences and column differences from the lowest row and the lowest column are:  The TOT is: The allocations with the help of RATOVs and CATOVs are: The allocations in the original TT are: The maximum profit is  We want to maximize the profit by the current algorithm. Solution We subtract the maximum profit 25 from other profits of the TT. The row differences and column differences from the lowest row and the lowest column are: The TOT is: The allocations with the help of RATOVs and CATOVs are: The allocations in the original TT are: The maximum profit is  We want to maximize the profit by the current algorithm. Solution We subtract the maximum profit 8 from other profits of the TT. The row differences and column differences from the lowest row and the lowest column are:  The TOT is: The allocations with the help of RATOVs and CATOVs are: The allocations in the original TT are:

Comparison of Results
The current algorithm mentioned in the article gives optimal or near optimal profit. However, a comparison of the developed work with the three existing conventional methods is presented in case of the three above examples.

Conclusion
The developed method considers all the opportunities of the cell values of the TOT by taking averages of the cell values. On the other hand, some other methods take some of the cell values only (ie. the lowest and the next lowest, the highest and the lowest etc.). The results or outcomes of the present algorithm are optimal or near optimal solutions while several examples were tested.