The solution to the Profit maximization transportation problem using new transportation algorithm

. Any engineering system requires engineers to make numerous managerial and technological decisions during the design, building, and maintenance phases. All such decisions aim to either maximize the desired benefit or minimize the amount of effort needed. There is no solitary approach that can effectively address all optimization problems. As a result, number of optimization techniques have been created to address various optimization problems. Transportation problem (TP) formulation is the most significant programing applications in optimization which is attached to everyday life and used in industries under logistics. This study proposed a new algorithm to obtain a basic feasible solution (BFS) for the maximization TP. The approach outlined in this study yields an initial solution that is close to or optimal in most scenarios. A variety of numerical examples are used to demonstrate the new technique. The proposed technique is effective for analyzing balanced or unbalanced transportation problems with maximization objective function.


Introduction
Optimization is essential in machine design as mechanical components need to create in the most effective way.When designing machine parts, optimization assists in several ways to lower material costs, provide better component service, boost production rate, and many other similar factors [1].The TP is a special case of LPP in the area of Optimization and has wide applications.Many physical models are converted into transportation challenges, which often feature traffic, assignment, and inventory concerns.Supply chains and logistics management employ transportation modules as a significant tool to get the best price and ultimately improve services [2].In 1781, French Mathematician Gaspard Monge formalized the transportation problem to transport soil at the lowest cost [3].Tolstoi [4].Frank Hitchcock formulated an algorithm to solve these problems in 1941 [5].Dantzig [6,7] proposes the "North West Corner Method (NWCM)" and "least-cost method (LCM)".Reinfeld and Vogel [8] pioneered the VAM method and Goyal [9] enhanced the VAM for the unbalanced TP.The strategies mentioned above can be utilised to find BFS of TP.
The optimality of the solution obtained by the above methods is checked using "Stepping Stone Method" (SSM) introduced by Charnes and Cooper [10] and "Modified Distribution Method (MODI)" which was introduced in 1955 [11].Several researchers have since improved the procedures and developed new approaches."Advanced Vogel's Approximation Method (AVAM)" was introduced by Das et.al. [12].Different variations for calculating penalties for VAM method was proposed by [13,14,15].Some of the well-known techniques created and explored for locating an initial basic workable answer to transportation challenges include SECM [16], ATCM [17], TOCM-MEDM [18], LCMM [19].
Further, several innovative methods for finding IBFS to unbalanced problems have just recently been developed.Snehee et al. [20] presented a case study of oil transport in Nigerian cities to minimize the total cost where the solution is provided using Python code.A new algorithm is proposed by Ekanayake [21] to solve balanced and unbalanced cost minimization problems.A study on applications of transportation problems in the Steel Industry was given by Rehile [22].An Improved Algorithm for Optimal Solution of Unbalanced Transportation Problems is provided by Gill et al [23].Fuzzy approach to solve TP was used by many researchers [24,25,26].
The research has been developed to find IBFS and optimal solution with the aforementioned techniques.The objective of all the mentioned research is to minimize the transportation cost.The profit maximization models are very effective in optimization.It is useful to complete a variety of daily jobs effectively and efficiently.These problems are addressed mathematically, considering objective function as profit maximization.A novel strategy was introduced to find solution of TP which is close to or optimal in most scenarios.
Comparative study shows the novel method is superior than the previous methods proving its efficiency.

The mathematical model for the Maximization transportation problem
Let there be a sources having (

Proposed Algorithm
For the method recommended in this paper, use the stages listed below: Step-1: To maximize the profit, construct a transportation matrix using the provided data.
Step-2: Verify that the TP is balanced (supply = demand); if not, balance it by adding dummy source (supply < demand) or destination (supply > demand) with zero profit component.
Step-3: Subtract each matrix element from its highest element to convert it into an equivalent minimization problem.
Step-4: Choose the first column's lowest value, subtract it by other values in the same column.Repeat the steps to the remaining columns to form column reduce matrix.
Step-5: Similar process is applied to the rows as well.Select the minimum value in the first row of equivalet minimization problem, subtract it by other values in the same row and repeat the steps to the remaining rows to get row reduce matrix.
Step-6: Add the column and row reduce matrices to get modified minimization martix.
Step-7: Subtract the highest value from the lowest in each respective row and column to determine the penalty.
Step-8: Check for the highest penalty in row /column and assign greatest supply/demand to the lowest value.If there is a tie for penalties then check for minimum value, if there is a tie for minimum value then break the tie by selecting the greatest allocation of demand/supply to the minimum value and still if allocations are the same then choose any cell for the allocation.
Step-9: Step-7 should be repeated until the demand and supply are consumed.
Step-10: Calculate the Total profit from the original Transportation matrix.

Example 1:
A firm owns facilities at 4 places (W1, W2, W3, W4) and it has manufacturing plants at 3 places (P1, P2, P3).Table 1.shows the net profit for each unit, along with manufacturing plants daily production(supply) and facilities daily requirements (demand).Find the optimal schedule to maximize the profit.7 provides the optimal answer for the given Profit Maximization problem using the new approach.

Example 2:
A company that produces industrial chemicals has three plants A, B, and C, each of which has the capacity to produce 300 kg, 200 kg, and 500 kg, respectively, of a certain chemical every day.In plants A, B, and C, the cost of production per kg is 0.7, 0.6, and 0.66 rupees respectively.Based on the following criteria, four bulk customers have ordered the products:  13 shows the allocations according to the new method.

Example 3:
Consider the Mumbai-based transporter's problem with the shipment of fruits and veggies.First, the three product destinations (Andheri, Vile Parle, and Santacruz) and their demand requested in the actual period are identified.These locations emphasize grocery sales in various areas of city.Afterwards, sources are also defined, although, in this case, they are replaced by products offered (Broccoli, Pineapple, and Pepper).There is demand of 460kg, 220kg, and 350kg selected products at Andheri, Vile Parle, and Santacruz respectively.
The cost and selling prices are presented in Table 15.

Example 4:
A company needs to decide on an investment plan for each of the next three years.The maximum investment amounts for each of the four investment kinds have been defined, and investment capital has been allotted for the next three years.It is assumed that investments made in any given year would last to the end of the three-year planning horizon.The net investment return for a rupee until the end of the planned horizon is shown in Table 17.Find the optimal investment plan to maximize the net return.What will be the net investment return for three-year planning horizon?The net investment return for three-year planning is 50.9 lacs.This is not the optimal return (51.3 lacs) but it's better that VAM (47.9 lacs) or any other known method of finding basic feasible solution.

Illustrations Result Analysis
The new algorithm is employed to solve balanced and unbalanced maximization TP.From the given data Profit Maximization problem is formulated and then solved using new technique.The results were compare with different existing methods to explain the efficiency of the new method.The comparison has been made with VAM, LCM, SECM, and NWCM and is presented in Table 19.It is observed that proposed method gives relatively better BFS than those obtained by NWCM, SECM, LCM and VAM.In few cases, it equals to the results of VAM, but its efficiency is always more than the other methods.

Conclusion
The new algorithm employed in this paper can be used with a great deal of success in solving maximization transportation problems.A comparison of the new method is done with VAM, LCM, SECM, and NWCM by considering four real-life numerical examples.It is observed that the new algorithm gives better answer.From the comparison table, we can observe that the optimum solution gained by the new algorithm is better than that of other methods.The proposed method is simple, easy to understand, and saves a lot of time and effort while finding the optimum solution.
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Table 1 :
Data matrix for Example 1 The above problem is balanced as total supply = total demand =1000 Applying the algorithm of the new technique, equivalent minimization matrix i.e.Table 2 is obtained.

Table 2 :
Equivalent Minimization MatrixNow applying the steps of the proposed algorithm we obtained column reduce and row reduce matrix given by Table3, 4 and modified minimization matrix in Table5.

Table 4 :
Row reduce Matrix

Table 6
illustrate the assignments according to the new method.

Table 6 :
The TP matrix after applying new method for example 1.

Table 7 :
Optimal solution for Example 1.

Table 8 :
Data for the example 2.

Customer Price offered Rs. / kg per day Kg required I
From plants to customers shipping costs (paise per kg) are in the table below:

Table 9 :
From plants to customers shipping costs (paise per kg) for example 2. Find the optimal schedule to maximize the profit of the firm.Solution: From the given data of example 2, Profit maximization matrix is formulated in

Table 11 ,
where dummy plant is added with total production capacity 150 kg and unit profit coefficient equal to zero.

Table 11 :
Balanced Profit maximization matrix for example 2. of the new technique, equivalent minimization matrix i.e.Table12is obtained.For ease of calculations matrix is multiplied by 100.

Table 12 :
Equivalent Minimization Matrix

Table 13 :
The TP matrix after applying new method for example 2.Optimal schedule for maximization of profit for the given problem is shown in Table14.

Table 14 :
Optimal solution for Example 2.

Table 15 .
Costs and Sale prices in Indian Rupees From the given data of example 3, Optimal solution applying new algorithm is presented in Table16.

Table 16 :
Optimal schedule to maximize the Profit for example 3

Table 17 :
Data matrix for Example 4

Table 18 :
Investment plan to maximize net returns for example 4

Table 19 .
Comparative study between New Algorithm and classical methods.