Paradox Algorithm in Application of a Linear Transportation Problem

Osuji George A.¹, Opara Jude², Nwobi Anderson C.³, Onyeze Vitus² and Iheagwara Andrew I.⁴

¹Department of Statistics, Nnamdi Azikiwe University, Awka Anambra State Nigeria

²Department of Statistics, Imo State University, Owerri Nigeria

³Department of Statistics, Abia State Polytechnic, Aba Nigeria

⁴Department of Planning, Research and Statistics, Ministry of Petroleum and Environment Owerri Imo State Nigeria

Cite this paper

Osuji George A., Opara Jude, Nwobi Anderson C., Onyeze Vitus and Iheagwara Andrew I.. Paradox Algorithm in Application of a Linear Transportation Problem. American Journal of Applied Mathematics and Statistics. 2014; 2(1):10-15. doi: 10.12691/AJAMS-2-1-3

Abstract

Paradox seldom occurs in a linear transportation problem, but it is related to the classical transportation problem. For specific reasons of this problem, an increase in the quantity of goods or number of passengers (as used in this paper) to be transported may lead to a decrease in the optimal total transportation cost. Two numerical examples were used for the study. In this paper, an efficient algorithm for solving a linear programming problem was explicitly discussed, and it was concluded that paradox does not exist in the first set of data, while paradox exists in the second set of data. The Vogel’s Approximation Method (VAM) was used to obtain the initial basic feasible solution via the Statistical Software Package known as TORA. The first set of data revealed that paradox does not exist, while the second set of data showed that paradox exists. The method however gives a step by step development of the solution procedure for finding all the paradoxical pair in the second set of data.

Keywords

transportation paradox, paradoxical range of flow, transportation problem, linear programming, paradoxical pair, VAM

Copyright

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References

[1]	Adlakha V. and Kowalski, K. (1998): A quick sufficient solution to the more-for-less paradox in a transportation problem, Omega 26(4):541-547.
[2]	Appa G.M. (1973): The Transportation problem and its variants, Oper. Res. Q. 24:79-99.
[3]	Arora S.R. and Ahuja A. (2000): A paradox in a fixed charge transportation problem. Indian J. pure appl. Math., 31(7): 809-822, July 2000 printed in India.
[4]	Charnes A.; Cooper W.W. and Henderson (1953): An Introduction to Linear programming (Wiley, New Work).
[5]	Charnes A. and Klingman D. (1971): The more-for-less paradox in the distribution model, Cachiers du Centre Etudes de Recherche Operaionelle 13; 11-22.
[6]	Deineko, V.G; Klinz, B and Woeginger, G.J. (2003): Which Matrices are Immune against the Transportation Paradox? Discrete Applied Mathematics, 130:495-501.
[7]	Dantzig, G.D. (1963): Linear Programming and Extensive (Princeton University Press, NJ).
[8]	Dantzig G.B. (1951): Application of the simplex method to a transportation problem, in Activity Analysis of Production and Allocation (T.C. Koopmans, ed.) Wiley, New York, pp.359-373.
[9]	Ekezie, D.D.: Ogbonna, J.C. and Opara, J. (2013): The Determination of Paradoxical Pairs in a Linear Transportation Problem. International Journal of Mathematics and Statistics Studies. Vol. 1, No. 3, p.p.9-19, September 2013.
[10]	Finke, G.(1978): A unified approach to reshipment, overshipment and postoptimization problems, lecture notes in control and information science, Vol.7, Springer, Berlin, 1978, p.p.201-208.
[11]	Gupta, A. Khanna S and Puri, M.C. (1993): A paradox in linear fractional transportation problems with mixed constraints, Optimization 27:375-387.
[12]	Hadley G. (1987): Linear Programming (Narosa Publishing House, New Delhi).
[13]	Hitchcock, F.L. (1941): The distribution of a product from several resources to numerous localities, J. Math. Phys. 20:224-230.
[14]	Joshi, V. D. and Gupta, N. (2010): On a paradox in linear plus fractional transportation problem, Mathematika 26(2):167-178.
[15]	Opara, J. (2009): Manual on Introduction to Operation Research, Unpublished.
[16]	Klingman, D. and Russel, R. (1974): The transportation problem with mixed constraints, operational research quarterly, Vol. 25, No. 3: p.p. 447-455.
[17]	Klingman, D. and Russel, R. (1975): Solving constrained transportation problems, Oper. Res. 23(1):91-105.
[18]	Storoy, S. (2007): The transportation paradox revisited, N-5020 Bergen, Norway.
[19]	Szwarc W. (1971): The transportation paradox, Nav. Res. Logist. Q.18:185-202.