A Novel Approach Algorithm for Determining the Initial Basic Feasible Solution for Transportation Problems

In optimization, the transportation problem is of utmost significance. The fundamental idea for solving the transportation problem is to develop methods that lower the overall cost between the source and the destination. Numerous research techniques are given in the literature as solutions to transportation problems. The majority of strategies focus on locating the initial basic, feasible solution to the transportation problem, while some techniques concentrate on locating an optimal solution. To find an initial, basic, feasible solution, Vogel's approximation, the least-cost method, the northwest method, and other methods are used. The Modified Distribution Method and the Stepping Stone Method are gaining acceptance as the optimal solution to the transportation problem. In this research study, a model approach to figuring out the basic, feasible solution for transportation problems with balanced and unbalanced components is proposed based on the average unit cost value of columns and rows. A new table is created using the unit cost values and average unit cost values in the columns and rows and compared to the transportation problem to solve the problem by balancing demand and supply. The proposed methodology is effortless, easy to understand, and simple to use. Comparatively speaking, the algorithmic technique suggested by this work is less complex than the well-known meta-heuristic algorithms in the literature. Finally, provide a case study to demonstrate the proposed approach.


INTRODUCTION
Transportation problems can be expressed and resolved as a linear equation also with the expectation of minimizing the total cost of transportation problems. F. L. Hitchcock (1941) proposed the first mode of transportation. Afterward, the algorithm is implemented by T.C. Koopmans (1949) and G.B. Dantzig (1951). Dantzig became the first to propose a transportation problem-solving method. He invented the phrase "North West Coner Method (NWCM)" to describe his approach. This methodology depends on the specified location. Later, the Column Minimum Method (CAM) and Row Minimum Method (RAM) were also introduced. The primary goal of these was to find an initial basic feasible solution. These methods can provide a basic feasible solution to the transportation problem. Danzig later developed an algorithm called the least-cost method in 1963. Reinfeld and Vogel (1958) pioneered the VAM method. In addition, Goyal (1984) enhanced the VAM for the unbalanced transportation problem by including the dummy's maximum cost. The methods described above can be used to identify the most basic feasible solution to the transportation problem. After obtaining the initial basic feasible solution using these procedures, there are two main methods for determining the optimal solution. "Stepping Stone Method" (SSM) was the first optimal solution search method to be introduced. Charnes and Coope created this. The other optimal solution is the Modified Distribution Method (MODI), introduced in 1955.
Moreover, there are other models for evaluating the initial basic, feasible solution to the transportation problem. For example, Priyanka and Sushma devised the Average Transportation Cost Method (ATCM). The modified ATCM was developed by Priyanka and Sushma, which calculated a penalty equivalent two the average of two minimum costs per row and column. In 2002, Sabawi and Hayawi proposed a new solution to the transport model problem by using the difference within cost in the highest and lowest cost. Several new methods have recently been introduced, including Ekanayake (Parish, 1994), minimization of transport costs in an industrial company through linear programming (Prifti et al, 2020), optimal feasible solutions to a road freight transportation problem (Latunde et al,2020) and Analysis of Transportation Method in Optimization of Distribution Cost Using Stepping Stone Method and Modified Distribution (Febriani et al., 2021). Meanwhile, through the above-mentioned research methods, the research has been improved to an initial solution as well as to its optimal solution.
All of the research papers mentioned above have attempted to solve the problem in such a way that the cost of the transportation problem is minimized. Although this model is a method based on cost minimization, its various uses are carried out in operational research. It is used to carry out various tasks in daily life efficiently and effectively. Here, a mathematical approach is given to these problems, and the problem is guided so that the objective considered initially is a minimum value. Thus, obtaining an optimal solution is the primary objective of a transportation problem. It is best to get a basic solution to a transportation problem before getting the optimal one. It requires driving the solution so that it is close to the optimal solution or the optimal solution in the least number of steps. For that purpose, a new algorithm was introduced for this research, and it can be shown by the results obtained as a successful method that was guided by the abovementioned objectives. Several definitions are commonly used in transportation problems, and it is important to discuss them.

Preliminaries
1. Source: Where the commodities are located, a transportation problem is a supply. This is frequently where the factory started. The transport table shows that (S 1 , S 2 ..., S m ). 2. Destination: The demand for a transportation problem is the storage of goods supplied from sources. This is often referred to as "destination storage". The transport table represents this as (D 1 , D2... Dn'). 3. Supply limit: The supply limit is the number of goods available at a source to meet demand.
4. Demand requirement: The quantity of goods required to meet demand is referred to as the demand requirement. 5. Initial Basic Feasible Solution (IBFS): If the basic parameters are m + n-1, then the initial basic feasible solution applies to a situation with m sources and n destinations. 6. Optimal Solution: When the initial basic feasible solution is the optimum, it is called the optimum solution.

Mathematical formulation
When solving a transportation problem, it is mathematically derived. The equations can be presented as follows: A single product must be shipped from the warehouses to the outlets. Each warehouse has a specific supply m source denoted by S 1 , S 2 , ", S m with respective capacities a 1 , a 2 , ", a m and each outlet has a specific demand n denoted by D 1 , D 2 , ", D m with respective demands b 1 , b 2 , ", b n .
Additionally, consider the cost of transportation from i th source to j th -sink is the is C ij and the amount shipped is X ij , where i = 1, 2..., m and j = 1, 2..., n.

Mathematical Model
The total transportation cost is Minimize

=1 =1
Subject to the constraints b j , j=1,2,",n and 3. X ij ≥0 for all i=1, 2, ", m and j=1,2,",n Note that in this case, the sum of the supplies and demands equal the overall.
i.e., = =1 =1 . Such problems are called balanced transportation problems and otherwise, i.e., Add a dummy origin to the transportation table, and set the cost for such an origin to be zero. The availability of this source.

Transportation tableau
The transportation problem can be described using a mathematical model based on linear programming, and it is typically displayed in a transportation tableau.
After the transportation problem is formulated as mentioned above, researchers have developed various algorithms to find solutions to it, and some of the popular methods used to find basic solutions are shown below.

Northwest Corner Method (NWCM)
The following steps are used to determine the transportation cost by the northwest corner method. 1.
Step 1-Begin with each upper-left cell. the transportation table's northwest corner. 2.
Step 2-For the first row and the first column, Allocate an item equal to the needed value. If the allocation is the same as the supply from the first source, move it down vertically, that is, repeat the above steps for the second row and the first column. 3.
Step 3-Allocate the same sources as the supply on the second row as well. Finish the problem by the demanded number of items in a row from left to right horizontally and column downwards in this way. 4.
Step 4-Then calculate the relevant transportation cost.

Least cost method (LCM)
The least cost method is another useful strategy for identifying the initial basic feasible solution to the transportation problem. The next step should be taken for that method. 1. Find the cell's position in the transportation problem table that has the lowest unit cost. Remove the row or column after completing as much of the supply or demand for this cell as you can. Cross that row and that column if both the column and the row are satisfied simultaneously. 2. For all uncrossed rows and columns, it will choose the cell with the lowest unit cost to equalize supply and demand. Complete the supply or demand, and remove that row or column. 3. To solve the problem, pick the cell with the lowest unit cost, delete every row and column, so that all the supply and demand are met, and finish the problem. 4. Then calculate the relevant transportation cost.

Vogel's approximation method (VAM)
The efficient answer can be found using the VAM, which is close to it. It follows the steps below to find the basic feasible solution values. Step 8. Calculate the transportation problem's initial basic feasible solution.
The solutions obtained using the algorithm presented in this research paper can be shown by the following transportation problems. It has been shown in detail how the final answer was reached according to the relevant steps.

Comparison of the Numerical Example with the New Method
In this section, the effectiveness of several well-known methods including the least cost method, northwest corner method, Vogel's approximation method, row minimum method, column minimum method, and some other wellliked methods are compared using results from various problems. Comparative evaluation is carried out and displayed. The detailed representation of the numerical data of tables 25 and 26.  [6, 3, 5, 4; 5, 9, 2, 7; 7, 12, 17, 9] S i = [22,15,8] D j = [7,12,17,9] C ij = [20, 22, 17, 4; 24, 37, 9, 7; [4, 6, 9, 5; 2, 6, 4, 1; 5, 7, 2, 9] S i = [16,12,15] D j = [12,14,9,8] C ij = 19, 8, 3, 4; 12, 14, 20, 2; 3, 9, 23, 25] CAM 377  2712  606  1010  80  1695  840  RuAM  374  2792  606  965  88  1680  840  Proposed  Method   374  2424  606  975  80  1720  840   Optimum  Solution   374  2424  606  960  80  1650  840 The analysis of six different methods for getting an initial basic feasible solution, including the proposed approach method, is the focus of this part. It will also be compared with the optimum solution. This analysis is expected to confirm the justification of the new methodology. The bar graph prepared according to the data values represented in tables 25 and 26 is shown above.    It is demonstrated from such graphs that there exist solutions in the new methodology that are competitively close to the initial basic feasible solution found through other methods. Along with the optimal answer, the proposed method's solutions were evaluated. The proposed methodology produces similar results for thirteen of the fifteen problems as the numerically optimal method. As a result, these studies back up the effectiveness of the proposed approach.

CONCLUSION
The main goal of this article was to present a novel method for estimating the cost (basic feasible solution) of a transportation problem. A unique approach based on the mean value of rows and columns was presented here. This method offered an algorithm for finding an abasic feasible solution to transportation problems that are both balanced and unbalanced. The primary goals of solving a transportation problem are to reduce the time it takes to solve the problem, improve the proposed algorithm to make it easier to understand, and show that the initial basic feasible solution is closer to or near to the optimum solution. The article's proposed model looked at eight problems of balance and seven problems of unbalanced transportation problems. The solutions to those problems were able to reach the initial basic feasible solution and also competitive approximated answers derived from other main transport problem-solving methods. Furthermore, thirteen out of the fifteen problems examined obtained the optimum solution. This comparative analysis also leads to the reasonable conclusion that this method is effective when compared to other methods. It has also been shown that effective solutions can be obtained by solving problems using this method. This can be demonstrated as a model that can be successfully applied to real-world problems. That solves the problem in such a way as to minimize a specific objective such as cost, time, and distance.