Mathematical programming

人の管理技術 Operation research

Finding an initial basic feasible solution is the prime requirement to obtain an optimal solution for the transportation problems. In this article, a new approach is proposed to find an initial basic feasible solution for the transportation problems. The method is also illustrated with numerical examples.

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A New Approach to Solve Transportation Problems

In this article, the crisp, fuzzy and intuitionistic fuzzy optimization problem is formulated. The basic definitions and notations related to optimization problems are given in the preliminaries section. Algorithms for solving the optimization problems using fuzzy and intuitionistic fuzzy set is presented in this article. Then, with the help of the proposed algorithm the optimal solution of the crisp, fuzzy and intuitionistic fuzzy optimization problems are determined. A new theorem related to type-2 fuzzy/type-2 intuitionistic fuzzy optimization problems is proposed and proved. Some new and concrete results related to type-2 fuzzy/type-2 intuitionistic fuzzy optimization problems are presented. To illustrate the proposed method, some real-life numerical examples are presented. The proposed article provides seven fully worked examples with screenshots of output summaries from the software used in the computations for better understanding. The advantages of the proposed approach as compared to other existing work are also specified. Detail analyses of the comparative study as well the discussion are given. To show the advantages of the proposed approach, superiority analysis is discussed. Comparison analysis and the advantages of the proposed operators are also discussed. Some managerial applications and the advantages of the proposed approach are given. Finally, conclusion and future research directions are also given.

https://link.springer.com/content/pdf/10.1007/s13198-019-00941-3.pdf

Algorithms for solving the optimization problems using fuzzy and intuitionistic fuzzy set

In order to proceed with a minimal total cost solution technique, it is necessary to start with an initial feasible solution (IFS). Thus IFS acts as a foundation to a minimal total cost solution technique to the transportation problem. Generally, better is the initial feasible solution lesser is the number of iterations of obtaining the minimal total cost solution. Here, first we demonstrate a deficiency of a recently developed method in obtaining the minimal total cost solution to this problem. Then we develop a better polynomial time (O(N3) (N, higher of the numbers of source and destination nodes)) heuristic solution technique to obtain a better initial feasible solution to the transportation problem. The developed heuristic is coded using C++ programming language. Comparative studies of this heuristic with the best available ones in the literature on results of some numerical problems are carried out to show better performance of the current one. Our heuristic is found to lead to the minimal total cost solution in most cases (88.89%) of the studied numerical problems. We develop a better heuristic to obtain a better IFS to the transportation problem.We find that 88.9% of the solved problems by JHM led to the optimal solution.We demonstrated that the ZSM does not provide the optimal solution all the time.The developed JHM is coded using C++ programming language. Transportation of products from sources to destinations with minimal total cost plays an important role in logistics and supply chain management. All algorithms start with an initial feasible solution in obtaining the minimal total cost solution to this problem. Generally, better is the initial feasible solution lesser is the number of iterations of obtaining the minimal total cost solution. Here, first we demonstrate a deficiency of a recently developed method in obtaining the minimal total cost solution to this problem. Then we develop a better polynomial time (O(N3) (N, higher of the numbers of source and destination nodes)) heuristic solution technique to obtain a better initial feasible solution to the transportation problem. Because of the intractability of carrying out enormous calculations in this heuristic technique without a soft computing program, this technique is coded using C++ programming language. Comparative studies of this heuristic with the best available ones in the literature on results of some numerical problems are carried out to show better performance of the current one. Our heuristic is found to lead to the minimal total cost solution in most cases (88.89%) of the studied numerical problems.

An efficient heuristic to obtain a better initial feasible solution to the transportation problem

The study in this paper is to discuss the limitation of Vogel's Approximation Method (VAM) and developed an improved algorithm after resolving this limitation for solving transportation problem. Vogel's Approximation Method (VAM) is the more efficient algorithm to solve the transportation problem but it has some limitations when highest penalty cost appear in two or more row or column. For that case VAM does not give any logical solution. In this paper we stand a logical approach for this problem and developed an algorithm named by ―Logical Development of Vogel's Approximation Method (LD-VAM)‖ where feasible solution from this method are very close to optimal solution more than VAM.

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Logical Development Of Vogel's Approximation Method (LD-VAM): An Approach To Find Basic Feasible Solution Of Transportation Problem

In Operation Research, obtaining significant result for Transportation Problems is very important now-a-days. Vogel’s Approximation Method (VAM) is the very efficient algorithm to solve the transportation problem for feasible solution which is nearer to optimal solution. In this paper we identified a computational error in VAM and approach a logical development of VAM algorithm. The main concept of VAM is to determine penalty cost which obtains from the difference of smallest and next to smallest cost in each row or column and make maximum allocation in lowest cost cell of that row or column which have largest penalty. The difficulty arises when smallest cost and next to smallest cost have same magnitude. In that case we find a very logical concept to resolve this and developed a new algorithm “Advanced Vogel’s Approximation Method (AVAM)” to find a feasible solution of transportation problem which is very close to optimal solution more than VAM. Keyword: AVAM, VAM, Penalty Cost, Feasible Solution, Error Estimation, Transportation Problem (TP).

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Advanced Vogel’s Approximation Method (AVAM): A New Approach to Determine Penalty Cost for Better Feasible Solution of Transportation Problem

Industries require planning in transporting their products from production centres to the users end with minimal transporting cost to maximize profit. This process is known as Transportation Problem which is used to analyze and minimize transportation cost. This problem is well discussed in operation research for its wide application in various fields, such as scheduling, personnel assignment, product mix problems and many others, so that this problem is really not confined to transportation or distribution only. In the solution procedure of a transportation problem, finding an initial basic feasible solution is the prerequisite to obtain the optimal solution. Again, development is a continuous and endless process to find the best among the bests. The growing complexity of management calls for development of sound methods and techniques for solution of the problems. Considering these factors, this research aims to propose an algorithm “Incessant Allocation Method” to obtain an initial basic feasible solution for the transportation problems. Several numbers of numerical problems are also solved to justify the method. Obtained results show that the proposed algorithm is effective in solving transportation problems.

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Incessant Allocation Method for Solving Transportation Problems

Maximum Flow Problem (MFP) discusses the maximum amount of flow that can be sent from the source to sink. Edmonds-Karp algorithm is the modified version of Ford-Fulkerson algorithm to solve the MFP. This paper presents some modifications of Edmonds-Karp algorithm for solving MFP. Solution of MFP has also been illustrated by using the proposed algorithm to justify the usefulness of proposed method.

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Aminur Rahman Khan

Papers

A New Approach to Solve Transportation Problems

Logical Development Of Vogel's Approximation Method (LD-VAM): An Approach To Find Basic Feasible Solution Of Transportation Problem

Advanced Vogel’s Approximation Method (AVAM): A New Approach to Determine Penalty Cost for Better Feasible Solution of Transportation Problem

Incessant Allocation Method for Solving Transportation Problems

Modified EDMONDS-KARP Algorithm to Solve Maximum Flow Problems