American Journal of Applied Mathematics

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Solving Triangular Fuzzy Transportation Problem Using Modified Fractional Knapsack Problem

Received: Aug. 01, 2023    Accepted: Aug. 21, 2023    Published: Oct. 28, 2023
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Abstract

A transportation problem (TP) is a specific part of a linear programming problem that arises in a collection of contexts and has received much attention in the literature. Minimizing transportation costs or time (one objective) is one of the primary goals of transportation problem-solving approaches. Supply, demand, and unit transportation costs may be uncertain in real-life applications due to many factors, such as multiple objectives. The goal of this paper is to look at the fuzzy transportation problem (FTP), which is crucial in TP with multiple objectives. In the literature, numerous techniques for dealing with FTPs are proposed. The cost, supply, and demand values of the FTPs are taken as symmetric triangular fuzzy numbers and then converted into crisp values using ranking techniques to solve the FTP. The initial solution is then obtained by Vogel’s approximation method (VAM), and the optimal solution is obtained by the modified distribution method (MODI). The proposed method is based on the Modified Fractional Knapsack Problem and introduces a new approach to solving the triangular fuzzy transportation problem. This paper analyses an alternative method using the fractional knapsack problem, which was modified using a minimum ratio test. To express the efficiency of the proposed method, it is compared with existing methods in the literature.

DOI 10.11648/j.ajam.20231105.11
Published in American Journal of Applied Mathematics ( Volume 11, Issue 5, October 2023 )
Page(s) 77-88
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Fractional Knapsack Problem, Fuzzy Transportation Problem, Harmonic Mean, Initial Feasible Solution, Minimum Ratio Test, Triangular Fuzzy Numbers

References
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[6] Chanas, S., & Kuchta, D. (1996). A concept of the optimal solution of the transportation problem with fuzzy cost coefficients. Fuzzy Sets and Systems. An International Journal in Information Science and Engineering, 82 (3), 299–305. https://doi.org/10.1016/0165-0114(95)00278-2
[7] Gani, A., & Razak, K. A. (2006). Two stage fuzzy transportation problem. https://www.semanticscholar.org/paper/b4f0502c84f55f8c14211ef8c893ef07055b9cca
[8] Kaur, A., & Kumar, A. (2011). A new method for solving fuzzy transportation problems using ranking function. Applied Mathematical Modelling, 35 (12), 5652–5661. https://doi.org/10.1016/j.apm.2011.05.012
[9] Bisht, D. C. S., & Srivastava, P. K. (2019). Fuzzy optimization and decision making. In Advanced Fuzzy Logic Approaches in Engineering Science (pp. 310–326). IGI Global.
[10] Srivastava, P. K., & Bisht, D. C. S. (2020). A segregated advancement in the solution of triangular fuzzy transportation problems. American Journal of Mathematical and Management Sciences, 1–11. https://doi.org/10.1080/01966324.2020.1854137
[11] Dinagar, D. S., Palanivel, K. The transportation problem in fuzzy environment. Int. J. Algorithms Comput. Math. 2009, 2, pp. 65–71.
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[14] Sinha P., Zoltners A. A. The Multiple-Choice Knapsack Problem. Operations Research 1979, 27, 3, pp. 431-627.
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[16] Lee, K. H. (2006). First course on fuzzy theory and applications (2005th ed.). Springer.
[17] Uthra, G., Thangavelu, K., & Amutha, B. (2017). An improved ranking for Fuzzy Transportation Problem using Symmetric Triangular Fuzzy Number. Ripublication.com. Retrieved July 12, 2023, from https://www.ripublication.com/afm17/afmv12n3_18.pdf
[18] Ramesh Kumar, M., & Subramanian, S. (2018). Solution of fuzzy transportation problems with triangular fuzzy numbers using. Acadpubl. Eu. Retrieved July 12, 2023, from https://acadpubl.eu/hub/2018-119-15/2/285.pdf
[19] Deepa, M., & Kannan, A. (2018). New approach to solve fully fuzzy transportation problem using ranking technique. Acadpubl. Eu. Retrieved July 12, 2023, from https://acadpubl.eu/hub/2018-119-15/3/413.pdf
[20] Afwat, A. E. M, Salama, A. A., Farouk, N. A New Efficient Approach to Solve Multi-Objective Transportation Problem in the Fuzzy Environment (Product approach). J. of Applied Eng. Research 2018, 13, 18, pp. 13660-13664.
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[22] Pandian, P., & Natarajan, G. (2011). Solving two stage transportation problems. In Communications in Computer and Information Science (pp. 159–165). Springer Berlin Heidelberg.
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  • APA Style

    Ekanayake Mudiyanselage Tharika Dewanmini Kumari Ekanayake, Ekanayake Mudiyanselage Uthpala Senerath Bandara Ekanayake. (2023). Solving Triangular Fuzzy Transportation Problem Using Modified Fractional Knapsack Problem . American Journal of Applied Mathematics, 11(5), 77-88. https://doi.org/10.11648/j.ajam.20231105.11

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    ACS Style

    Ekanayake Mudiyanselage Tharika Dewanmini Kumari Ekanayake; Ekanayake Mudiyanselage Uthpala Senerath Bandara Ekanayake. Solving Triangular Fuzzy Transportation Problem Using Modified Fractional Knapsack Problem . Am. J. Appl. Math. 2023, 11(5), 77-88. doi: 10.11648/j.ajam.20231105.11

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    AMA Style

    Ekanayake Mudiyanselage Tharika Dewanmini Kumari Ekanayake, Ekanayake Mudiyanselage Uthpala Senerath Bandara Ekanayake. Solving Triangular Fuzzy Transportation Problem Using Modified Fractional Knapsack Problem . Am J Appl Math. 2023;11(5):77-88. doi: 10.11648/j.ajam.20231105.11

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  • @article{10.11648/j.ajam.20231105.11,
      author = {Ekanayake Mudiyanselage Tharika Dewanmini Kumari Ekanayake and Ekanayake Mudiyanselage Uthpala Senerath Bandara Ekanayake},
      title = {Solving Triangular Fuzzy Transportation Problem Using Modified Fractional Knapsack Problem
    
    	
    },
      journal = {American Journal of Applied Mathematics},
      volume = {11},
      number = {5},
      pages = {77-88},
      doi = {10.11648/j.ajam.20231105.11},
      url = {https://doi.org/10.11648/j.ajam.20231105.11},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.ajam.20231105.11},
      abstract = {A transportation problem (TP) is a specific part of a linear programming problem that arises in a collection of contexts and has received much attention in the literature. Minimizing transportation costs or time (one objective) is one of the primary goals of transportation problem-solving approaches. Supply, demand, and unit transportation costs may be uncertain in real-life applications due to many factors, such as multiple objectives. The goal of this paper is to look at the fuzzy transportation problem (FTP), which is crucial in TP with multiple objectives. In the literature, numerous techniques for dealing with FTPs are proposed. The cost, supply, and demand values of the FTPs are taken as symmetric triangular fuzzy numbers and then converted into crisp values using ranking techniques to solve the FTP. The initial solution is then obtained by Vogel’s approximation method (VAM), and the optimal solution is obtained by the modified distribution method (MODI). The proposed method is based on the Modified Fractional Knapsack Problem and introduces a new approach to solving the triangular fuzzy transportation problem. This paper analyses an alternative method using the fractional knapsack problem, which was modified using a minimum ratio test. To express the efficiency of the proposed method, it is compared with existing methods in the literature.
    },
     year = {2023}
    }
    

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  • TY  - JOUR
    T1  - Solving Triangular Fuzzy Transportation Problem Using Modified Fractional Knapsack Problem
    
    	
    
    AU  - Ekanayake Mudiyanselage Tharika Dewanmini Kumari Ekanayake
    AU  - Ekanayake Mudiyanselage Uthpala Senerath Bandara Ekanayake
    Y1  - 2023/10/28
    PY  - 2023
    N1  - https://doi.org/10.11648/j.ajam.20231105.11
    DO  - 10.11648/j.ajam.20231105.11
    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
    SP  - 77
    EP  - 88
    PB  - Science Publishing Group
    SN  - 2330-006X
    UR  - https://doi.org/10.11648/j.ajam.20231105.11
    AB  - A transportation problem (TP) is a specific part of a linear programming problem that arises in a collection of contexts and has received much attention in the literature. Minimizing transportation costs or time (one objective) is one of the primary goals of transportation problem-solving approaches. Supply, demand, and unit transportation costs may be uncertain in real-life applications due to many factors, such as multiple objectives. The goal of this paper is to look at the fuzzy transportation problem (FTP), which is crucial in TP with multiple objectives. In the literature, numerous techniques for dealing with FTPs are proposed. The cost, supply, and demand values of the FTPs are taken as symmetric triangular fuzzy numbers and then converted into crisp values using ranking techniques to solve the FTP. The initial solution is then obtained by Vogel’s approximation method (VAM), and the optimal solution is obtained by the modified distribution method (MODI). The proposed method is based on the Modified Fractional Knapsack Problem and introduces a new approach to solving the triangular fuzzy transportation problem. This paper analyses an alternative method using the fractional knapsack problem, which was modified using a minimum ratio test. To express the efficiency of the proposed method, it is compared with existing methods in the literature.
    
    VL  - 11
    IS  - 5
    ER  - 

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Author Information
  • Department of Mathematics, Faculty of Natural Sciences, The Open University of Sri Lanka, Nugegoda, Sri Lanka

  • Department of Applied Sciences, Faculty of Physical Sciences, Rajarata University of Sri Lanka, Anuradhapura, Sri Lanka

  • Section