Comparison of Methods of Solving Transportation Problems (TP) and Resolving the Associated Variations

Awogbemi Clement Adeyeye; Alagbe Samson Adekola; Osamo Caleb Kehinde

doi:doi:10.11648/j.ajtas.20221105.11

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Comparison of Methods of Solving Transportation Problems (TP) and Resolving the Associated Variations

Awogbemi Clement Adeyeye, Alagbe Samson Adekola, Osamo Caleb Kehinde

Published in American Journal of Theoretical and Applied Statistics (Volume 11, Issue 5)

Received: 10 March 2022 Accepted: 14 May 2022 Published: 28 September 2022

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Abstract

Transportation Problem is a Linear Programming application to physical distribution of goods and services from various origins to several destinations such that the cost of transportation is minimal. In this study, five different methods were employed to solve transportation problems arising from unequal demand and supply of goods and variations. The methods considered in terms of North West Corner Rule, Least Cost Method, Vogel’s Approximation Method, Row Minima Method and Column Minima Method were compared. Necessary and sufficient condition for the existence of a feasible solution to the transportation problem was initiated and established. Unbalanced transportation problems were resolved using Vogel’s Approximation Method (VAM) and Modified Distribution (MODI) methods. The five methods compared produced different results with VAM generating the least transportation cost and better solution. The least value of the transportation costs obtained by the five methods is VAM with the most economical initial feasible solution. It was also established that, out of m + n constraint equations, only m + n-1 equations are linearly independent. With the MODI method, economic values were generated for the dual variables, u_is and v_js associated with the source and demand points respectively.

Published in	American Journal of Theoretical and Applied Statistics (Volume 11, Issue 5)
DOI	10.11648/j.ajtas.20221105.11
Page(s)	140-149
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

Transportation Problems, Origins, Destinations, Unbalanced Transportation Problem, Optimal Solution, Optimality Test

References

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[3]	Farzana, S. R. and Safiqul, I. (2020). A Comparative Study of Solving Methods of Transportation Problem in Linear Programming Problem. British Journal of Mathematics and Computer Science, 35 (5), pp 45- 57.
[4]	Funso, A. (1995). Practical Operational Research for Developing Countries (A Process Frame Work Approach. Panaf Press, Lagos, Nigeria.
[5]	Hamdy, A. T. (2017). Operations Research, An Introduction. Pearson Education Limited, 10^th ed., England.
[6]	Joshi, R. V. (2015). Optimization Techniques for Transportation Problems of Three Variables. IOSR Journal of Mathematics, 9 (1), pp 46-50.
[7]	Kalavathy, S. (2013). Operations Research. VIKAS Publishing House, PVT LTD.
[8]	Mohammad, H. and Farzana, S. F. (2018). A New Method for Optimal Solutions of Transportation Problems in LPP. Journal of Mathematics, 10 (5), pp 60-75.
[9]	Natarajan, A. M., Balasubramani, P. and Tamilarasi, A. (2005). Operations Research. Pearson Education, Singapore.
[10]	Ocotlan, D. P., Jorge, A. R., Beatriz, B. L., Alejandro, F. P. and Richardo, A. B. C. (2013). A Survey of Transportation Problems. Journal of Applied Mathematics, 2014, pp 1-17.
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Awogbemi Clement Adeyeye, Alagbe Samson Adekola, Osamo Caleb Kehinde. (2022). Comparison of Methods of Solving Transportation Problems (TP) and Resolving the Associated Variations. American Journal of Theoretical and Applied Statistics, 11(5), 140-149. https://doi.org/10.11648/j.ajtas.20221105.11

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ACS Style

Awogbemi Clement Adeyeye; Alagbe Samson Adekola; Osamo Caleb Kehinde. Comparison of Methods of Solving Transportation Problems (TP) and Resolving the Associated Variations. Am. J. Theor. Appl. Stat. 2022, 11(5), 140-149. doi: 10.11648/j.ajtas.20221105.11

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AMA Style

Awogbemi Clement Adeyeye, Alagbe Samson Adekola, Osamo Caleb Kehinde. Comparison of Methods of Solving Transportation Problems (TP) and Resolving the Associated Variations. Am J Theor Appl Stat. 2022;11(5):140-149. doi: 10.11648/j.ajtas.20221105.11

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@article{10.11648/j.ajtas.20221105.11,
  author = {Awogbemi Clement Adeyeye and Alagbe Samson Adekola and Osamo Caleb Kehinde},
  title = {Comparison of Methods of Solving Transportation Problems (TP) and Resolving the Associated Variations},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {11},
  number = {5},
  pages = {140-149},
  doi = {10.11648/j.ajtas.20221105.11},
  url = {https://doi.org/10.11648/j.ajtas.20221105.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20221105.11},
  abstract = {Transportation Problem is a Linear Programming application to physical distribution of goods and services from various origins to several destinations such that the cost of transportation is minimal. In this study, five different methods were employed to solve transportation problems arising from unequal demand and supply of goods and variations. The methods considered in terms of North West Corner Rule, Least Cost Method, Vogel’s Approximation Method, Row Minima Method and Column Minima Method were compared. Necessary and sufficient condition for the existence of a feasible solution to the transportation problem was initiated and established. Unbalanced transportation problems were resolved using Vogel’s Approximation Method (VAM) and Modified Distribution (MODI) methods. The five methods compared produced different results with VAM generating the least transportation cost and better solution. The least value of the transportation costs obtained by the five methods is VAM with the most economical initial feasible solution. It was also established that, out of m + n constraint equations, only m + n-1 equations are linearly independent. With the MODI method, economic values were generated for the dual variables, uis and vjs associated with the source and demand points respectively.},
 year = {2022}
}

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TY  - JOUR
T1  - Comparison of Methods of Solving Transportation Problems (TP) and Resolving the Associated Variations
AU  - Awogbemi Clement Adeyeye
AU  - Alagbe Samson Adekola
AU  - Osamo Caleb Kehinde
Y1  - 2022/09/28
PY  - 2022
N1  - https://doi.org/10.11648/j.ajtas.20221105.11
DO  - 10.11648/j.ajtas.20221105.11
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 140
EP  - 149
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20221105.11
AB  - Transportation Problem is a Linear Programming application to physical distribution of goods and services from various origins to several destinations such that the cost of transportation is minimal. In this study, five different methods were employed to solve transportation problems arising from unequal demand and supply of goods and variations. The methods considered in terms of North West Corner Rule, Least Cost Method, Vogel’s Approximation Method, Row Minima Method and Column Minima Method were compared. Necessary and sufficient condition for the existence of a feasible solution to the transportation problem was initiated and established. Unbalanced transportation problems were resolved using Vogel’s Approximation Method (VAM) and Modified Distribution (MODI) methods. The five methods compared produced different results with VAM generating the least transportation cost and better solution. The least value of the transportation costs obtained by the five methods is VAM with the most economical initial feasible solution. It was also established that, out of m + n constraint equations, only m + n-1 equations are linearly independent. With the MODI method, economic values were generated for the dual variables, uis and vjs associated with the source and demand points respectively.
VL  - 11
IS  - 5
ER  -

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Author Information

Awogbemi Clement Adeyeye

Department of Statistics, National Mathematical Centre, Abuja, Nigeria
Alagbe Samson Adekola

Department of Computer Science, Isaac Jasper Boro College of Education, Sagbama, Nigeria
Osamo Caleb Kehinde

Department of Banking and Finance, Veritas University, Abuja, Nigeria

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APA Style

Awogbemi Clement Adeyeye, Alagbe Samson Adekola, Osamo Caleb Kehinde. (2022). Comparison of Methods of Solving Transportation Problems (TP) and Resolving the Associated Variations. American Journal of Theoretical and Applied Statistics, 11(5), 140-149. https://doi.org/10.11648/j.ajtas.20221105.11

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ACS Style

Awogbemi Clement Adeyeye; Alagbe Samson Adekola; Osamo Caleb Kehinde. Comparison of Methods of Solving Transportation Problems (TP) and Resolving the Associated Variations. Am. J. Theor. Appl. Stat. 2022, 11(5), 140-149. doi: 10.11648/j.ajtas.20221105.11

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AMA Style

Awogbemi Clement Adeyeye, Alagbe Samson Adekola, Osamo Caleb Kehinde. Comparison of Methods of Solving Transportation Problems (TP) and Resolving the Associated Variations. Am J Theor Appl Stat. 2022;11(5):140-149. doi: 10.11648/j.ajtas.20221105.11

Copy | Download

@article{10.11648/j.ajtas.20221105.11,
  author = {Awogbemi Clement Adeyeye and Alagbe Samson Adekola and Osamo Caleb Kehinde},
  title = {Comparison of Methods of Solving Transportation Problems (TP) and Resolving the Associated Variations},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {11},
  number = {5},
  pages = {140-149},
  doi = {10.11648/j.ajtas.20221105.11},
  url = {https://doi.org/10.11648/j.ajtas.20221105.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20221105.11},
  abstract = {Transportation Problem is a Linear Programming application to physical distribution of goods and services from various origins to several destinations such that the cost of transportation is minimal. In this study, five different methods were employed to solve transportation problems arising from unequal demand and supply of goods and variations. The methods considered in terms of North West Corner Rule, Least Cost Method, Vogel’s Approximation Method, Row Minima Method and Column Minima Method were compared. Necessary and sufficient condition for the existence of a feasible solution to the transportation problem was initiated and established. Unbalanced transportation problems were resolved using Vogel’s Approximation Method (VAM) and Modified Distribution (MODI) methods. The five methods compared produced different results with VAM generating the least transportation cost and better solution. The least value of the transportation costs obtained by the five methods is VAM with the most economical initial feasible solution. It was also established that, out of m + n constraint equations, only m + n-1 equations are linearly independent. With the MODI method, economic values were generated for the dual variables, uis and vjs associated with the source and demand points respectively.},
 year = {2022}
}

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TY  - JOUR
T1  - Comparison of Methods of Solving Transportation Problems (TP) and Resolving the Associated Variations
AU  - Awogbemi Clement Adeyeye
AU  - Alagbe Samson Adekola
AU  - Osamo Caleb Kehinde
Y1  - 2022/09/28
PY  - 2022
N1  - https://doi.org/10.11648/j.ajtas.20221105.11
DO  - 10.11648/j.ajtas.20221105.11
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 140
EP  - 149
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20221105.11
AB  - Transportation Problem is a Linear Programming application to physical distribution of goods and services from various origins to several destinations such that the cost of transportation is minimal. In this study, five different methods were employed to solve transportation problems arising from unequal demand and supply of goods and variations. The methods considered in terms of North West Corner Rule, Least Cost Method, Vogel’s Approximation Method, Row Minima Method and Column Minima Method were compared. Necessary and sufficient condition for the existence of a feasible solution to the transportation problem was initiated and established. Unbalanced transportation problems were resolved using Vogel’s Approximation Method (VAM) and Modified Distribution (MODI) methods. The five methods compared produced different results with VAM generating the least transportation cost and better solution. The least value of the transportation costs obtained by the five methods is VAM with the most economical initial feasible solution. It was also established that, out of m + n constraint equations, only m + n-1 equations are linearly independent. With the MODI method, economic values were generated for the dual variables, uis and vjs associated with the source and demand points respectively.
VL  - 11
IS  - 5
ER  -

Copy | Download