New Approach to Obtain the Maximum Flow in a Network and Optimal Solution for the Transportation Problems

The maximum flow problem is also one of the highly regarded problems in the field of optimization theory in which the objective is to find a feasible flow through a flow network that obtains the maximum possible flow rate from source to sink. The literature demonstrates that different techniques have been developed in the past to handle the maximum amount of flow that the network can handle. The Ford-Fulkerson algorithm and Dinic's Algorithm are the two major algorithms for solving these types of problems. Also, the Max-Flow Min-Cut Theorem, the Scaling Algorithm, and the Push–relabel maximum flow algorithm are the most acceptable methods for finding the maximum flows in a flow network. In this novel approach, the paper develops an alternative method of finding the maximum flow between the source and target nodes of a network based on the "max-flow." Also, a new algorithmic approach to solving the transportation problem (minimizing the transportation cost) is based upon the new maximum flow algorithm. It is also to be noticed that this method requires a minimum number of iterations to achieve optimality. This study's algorithmic approach is less complicated than the well-known meta-heuristic algorithms in the literature.

movement of resources. This problem leads to the concept of a flow network. Aviation-based armed forces scientists T. E. Harris and F. S. Ross (1956) published a classified report considering the rail network that linked the Soviet Union's satellites. Harris and Ross, studying the rail network, modeled it as a graph with 44 vertices and 105 edges, representing joins between those locales in the rail network. The system unraveling technique is based on the Ford-Fulkerson max-stream calculation. The network unraveling method is based on the Ford-Fulkerson max-flow algorithm. Ford and Fulkerson (1962) defined flows as an area of operations research. This concept was implemented by many scientists, statisticians, mathematicians, and engineers.
In the previous few years, different types of algorithms for solving this problem have been proposed, for example, the network unraveling technique based on the Ford-Fulkerson max-flow algorithm (1956), A Parallel Ford-Fulkerson Algorithm For Maximum Flow Problem (Harris and Ross, 1956), Maximum Flow Problem (Ahuja, 1989;Karzanova, 1974). The Maximum Flow Network Interdiction Problem (Douglas, 2010), etc. In like manner, transportation problems have been generally considered in Operation Research. It is one of the fundamental problems of the network flow problem (Dinice, 1970;Elias et al., 1956;Fulkerson & Dantzig, 1955), which is normally used to minimize the transportation cost for ventures with a couple of sources to a couple of objectives.
The fundamental TP was first proposed by F. L. Hitchcock in 1941, and then independently by T. C. Koopmans in 1947, after which the logical solution procedures from the simplex algorithm were further developed (Kulkarni & Datar, 2010;Mallick et al., 2016), primarily by Dantzig and then by Charnes et al. (1953). The Simplex Method isn't reasonable for the transportation problem, especially for large scale transportation problems, in view of the interesting structure of the model by charges and Cooper (1954) who made the Stepping Stone method. The literature witnesses that different techniques (Ahmedat et al., 2016(Ahmedat et al., -2017Babu at el., 2013;Deshmukh, 2012;Girmay, 2013;Hitchcock, 1941;Khan, 2015;Kulkarni, 2010;Pannerselvam, 2010) have been developed in the past to solve the transportation problem. In this study, we examine a novel approach for solving the maximum-flow problems and TP problems (TPMFP), which is based on the use of the Predecessor cost matrix. The proposed algorithm is basic, straightforward, and simple to actualize. This method is also to be noticed that, requires a minimum number of steps to reach optimality as compare the obtained results with the regular methods. Finally, this procedure is outlined with some numerical examples. We present a few new effective algorithms for the summarized maximum flow problem.

The Key Definitions
Definition 1 (Edmonds, 1972;Fulkerson, 1955). A flow network is a directed graph G = (V, E), with a source s ∈ V, a sink t ∈ V, and each edge (i, j) ∈ E has a non-negative capacity c(i, j).
Definition 2 (Ford, 1962). Given a flow network = ( , ), a flow in G is described by a function : × → ℝ Satisfying i.
iii. Flow conservation: For each , ∈ , then ∑ ( , ) = ∑ ( , ) ∈ ∈ Flow into j Flow out of j (Dimitri, 1998). Predecessor cost matrix The algorithm represents an n-node network as a square matrix with n-rows and n-columns. Entry (i, j) of the matrix gives the weight w from node i to node j.
Definition 4 (Dimitri, 1998). Flow Capacity Update. (Augmenting path) We are given a weighted, directed graph = ( , ) .There is a weight of path (P) from s to t for all edges( , ) ∈ , sub graph of = ( , ); ⊆ and ⊆ s.t. ii. is the path from s to t.
iii. For all ∈ ; the unique direct path from s to t.
iv. Max flow-carrying path P has been found from s to t ( * 0 for every arc ( , ) on the path), and ( ) = * |: ( , ) }, where m is number of iteration.

Definition 5.
Network flow algorithms depend on the idea of augmenting paths (predecessor cost matrix) and more than once find a path of positive capacity from s to t and add it to the flow. It tends to be demonstrated that the flow through a network is optimal if and only if it contains no augmenting paths.
The value of a Max flow is defined as | | = ∑ ( ) ( , )∈ which is the total amount out of the source (also equal the total amount into the sink).

Max Flow
The problem can be formulated as follows; For , = A, B, C…... L, M, N. (Rayand & Hosain, 2007) The total transportation cost is

Illustration of the Proposed Algorithm
A wide range of decision problems that can be modeled as network optimization problems and solved efficiently and effectively can be solved using network models. Some of these decision problems, such as transportation or commodity flow, are physical in nature. On the other hand, many network problems, such as the critical path activity network in project management, are abstract representations of processes or activities. Researchers have improved a number of algorithms for solving maximal-flow problems over the last fifty years. In this algorithm, we presented a new algorithm for calculating the maximum flow in a network. The new maximum flow algorithm also serves as the foundation for a new algorithmic approach to solving the transportation problem (minimizing transportation costs).

New Algorithm
Step 1. Construct the Predecessor cost matrix Step 2. Start with 1 st row (source node ) and select any weighted node k with * 0.
Step 3. Next Select k th row and select any another weighted node i ( * ), after that select i th row and select any weighted node j ( * ), next select j th row and select any weighted node j ( * ), and label every node ℎ with * 0 until either sink t has been labeled ( * .. ). (There is a path (P) from s to t ) Step 4. Find the max flow ( ) = * |: ( , ) } Step 5. Flow is updated (Definition 4) Step 6. If the * = 0( = 2,3,4, , , , ) or node k with * .. = 0, yes go to Step 7, no go to Step 3.

Proposed Algorithm
The proposed algorithm is given below: Step 1: first initialize the flow f to 0, for each edge (u, v) ∈ E [G], Step2: f (u, v) = f (v, u) = 0 Step 3: Calculate maximum capacity C in the flow network and then calculate = 11 Step 4: while D ≥ 1 Step 5: If there exists an augmenting path p from s to t in the residual network with capacity at least D the select it and go to step 6; otherwise go to step 9.
Step 9: D = Step10: The flow is maximum

New Algorithm for TP
Step-1. Construct a Predecessor Transportation Cost Table (PTCT) from the given transportation problem.
Step-2. Ensure whether the PTCT is balanced or not, if not, make it balanced and complete PTCT of order m x n.
Step 5. ∅ cell placed at the starting nodes with the minimum value of the PTCT to make the first allocation Step 6. Next utilizing the above Algorithm and goes to head.
Step 7. Then terminate and calculate the total transportation cost

Comparative Assessment with Max Flow
A numerical example has been unraveled for finding the maximum value of a Maximum flow problem by utilizing proposed algorithm which is given underneath (Dash et al., 2019). Consider a pipeline system in an Azimpur colony to supply gas in different zones of an Azimpur Colony of Dhaka in the Bangladesh. The pipeline has a communicated capacity in per unit per hour between any two zones which given as a maximum flow at which gas can flow by the pipe between those two zones. Presently, Consider I need to supply gas from the source zone to the sink zone, assume the source zone is say A and the sink zone is F and gas passes into 4 others zones before getting from source to sink. Suppose B, C, D, E, are of these 4 zones and pipeline between any two zones has indicated capacity. Demonstrates the input information's which have given to the problem talked above in the following Table 1  Calculate the maximum amount of gas which can flow from A to F.
Step 3. Table 6. Predecessor cost matrix Now we construct the following table to compare different algorithms with our proposed algorithm Table 7. Assessment of the residual obtained by different methods By implies new algorithm yields better outcomes that will be resulted in maximum flow in a network flow problem involving a fewer number of iterations just as fewer augmentations of this we needed to propose.

Comparative Assessment with TP
This section gives execution relationships over the distinctive eminent methods -NWCR, LCM, VAM, New method (TPMFP) and to find the optimal result by the solutions obtained from disparate problems. Comparable assessments are performed and illustrated in the immediately following sections. The detailed representation of the numerical information of Table is given in Appendix A. The TPMFP method's efficiency was also tested by solving five benchmark problems, and performance comparisons between the various well-known methods (NWCM, LCM, VAM, and IAM) are provided in Table 8. It is clear that the TPMFP method produces a comparatively more appealing result. It can also be demonstrated that the TPMFP provides a motivating initial feasible solution (IFS) by requiring a minimum number of iterations. Table 8 is depicted as a bar graph in Figure 1 and as a line graph in Figure 2 to demonstrate the significance of the proposed method, TPMFP, over other considered existing methods in terms of transportation cost, IFS, and percentage deviation (PD), respectively:

Balance Transportation Problem
According to Table 8 and Figures 1 and 2, TPMFP is the best IFS, almost identical to the optimal solution (OS) in every case, obtained by the MODI method for TPs, and better than the solution obtained by the other existing methods such as NWCM, LCM, VAM, and IAM. The proposed TPMFP achieves more efficient IFS for a diverse set of TPs in fewer iterations.
The next section gives execution relationships for the distinctive eminent methods-NWCR, LCM, VAM, and the new method-and how to find the best result from the solutions obtained from disparate problems. Comparable assessments are performed and illustrated in the immediately following sections. The detailed representation of the numerical information in the table is given in Appendix B. As shown in Table 9, the proposed TPMFP method achieves a more promising IFS than the traditional algorithms considered in this study, as shown in Figure 3 and 4. Table 6 is depicted in Figure 3 as a bar graph and in Figure 4 as a line graph to demonstrate the significance of the proposed method, NEWA, in terms of transportation cost and IFS and PD, respectively: This section also compares the performance of various well-known methods (NWC, LCM, VAM, IAM, and TPMFP) developed using maximization TPs. The sections that follow perform and illustrate comparative assessments. Table 10 contains a detailed representation of the numerical data. Appendix C includes detailed data representations of the following five problems: The TPMFP method's efficiency has also been tested by solving five benchmark problems and comparing performance across various well-known methods (NWCM, LCM, VAM, and IAM) in Table 10, where it is discovered that the TPMFP method yields a comparatively better result in all but one, which is not a better result. It can also be argued that the TPMFP may provide an outstanding IFS by requiring a minimum number of iterations. Table 10 also compares the bar graphs in Figure 5 and the line graphs in Figure 6 required to achieve the IFS for the proposed TPMFP algorithm. The comparative results shown in Table 10 are also represented using bar graphs and line graphs, as shown in Figures 5 and 6 Table 10 are presented in Figure 6:  Comparative results acquired by NWCM, LCM, VAM, EDM and Ah. et al. and the proposed method for the ten benchmark instances are shown in the following Table 11. Detailed data information portrayal of these ten problems is provided in Appendix D:  To compare the effectiveness of the TPMFP to existing methods, 10 different TPs were benchmarked and solved using NWCM, LCM, and VAM, EDM, Ah.et.al, and TPMFP with the results shown in Table 11. TPMFP clearly achieved the best OS in the fewest number of iterations for all problems. The comparative data from Table 11 is also depicted using bar graphs and line graphs, as shown in Figures 7 and 8.  Table 11.
This section also compares the performance of various well-known methods (NWC, LCM, VAM, EDM, Ah. et al., and PROPOSED METHOD) as well as the solutions obtained from disparate problems. In the sections that follow, comparative assessments are performed and illustrated. Appendix E contains a detailed representation of the numerical data.  . Let us consider the following TP to find out the minimum transportation cost.

Numerical Illustration (TP)
A numerical example has been developed for resolving the transportation problem using the proposed algorithm, which is provided below.

Conclusion
Any mathematical algorithm's modification is a never-ending process to achieve the best possible result. This paper presents some new strategies for taking care of the maximal network flow problem with applications to solving the transportation problem. By means of this, we expect to propose an algorithm that will realize maximum flow in a network flow problem, including a lower number of augmentations. For this, we considered different kinds of algorithms, for example, Ford-Fulkerson, Edmonds-Karp, Md. Al-Amin Khan, Faruque Ahmed, Chintan, and Deepak Garg, modified Edmonds-Karp algorithms. This proposed algorithm requires fewer iterations and augmentations to obtain maximum flow in a network flow problem more precisely in comparison with the other prominent algorithms. In addition, Transportation of product dissemination from several original points to several destinations minimizes the cost of transportation. Numerous researchers have focused on solving this problem by utilizing various approaches. The Northwest, Least Cost, and Vogel's Approximation methods are the most noticeable and renowned for finding an initial feasible solution for a TP out of all the existing methods and techniques in the writing. The Modified Distribution (MODI) Method and the Stepping Stone Method are the most acceptable methods for finding the minimal total cost solution to the transportation problem. These well-known techniques for minimizing total cost begin with an Initial Feasible Solution (IFS). In this way, an IFS acts as the foundation of an optimal cost solution technique for any TP. The better the IFS, the fewer iterations it takes to reach the optimal solution. However, in this research paper, we discuss a new alternative method, a modified max flow algorithm, which often gives an optimal solution to the transportation problem.
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