A Simple Method to Find Optimum Efficient Basic Solutions to Bi-Objective Transportation Problems

B. S. Surya Prabhavati*, V. Ravindranath**
* Department of Mathematics, CMR Institute of Technology, Hyderabad, India.
** Department of Mathematics, Jawaharlal Nehru Technological University, Kakinada, Andhra Pradesh, India.
Periodicity:January - June'2022
DOI : https://doi.org/10.26634/jmat.11.1.18668


This paper proposes a simple method to find an optimum and efficient basic solution to bi-objective transportation problem (BOTP) using weighted goal programming approach. Preferential weights for the goals (objectives) are derived from analytic hierarchy process (AHP). Solution obtained by the proposed approach has been found to be the nearest basic feasible solution to the ideal solution in terms of Euclidean distance measure. The method is illustrated with numerical examples taken from the literature and solutions compared in terms of the number of iterations, computational complexity and proximity to the ideal solution. The modified minimum supply and demand method is used to obtain initial basic feasible solutions that are found to be optimal, and hence this method is computationally simpler compared to other linear programming methods. It has been observed that weighted objective function optimization yields optimal efficient BOTP solutions using appropriate weights for the objective functions.


Bi-Objective Transportation Problem, Efficient Basic Solution, Modified Minimum Supply Demand Method, Multi-Objective Transportation Problem.

How to Cite this Article?

Prabhavati, B. S. S., and Ravindranath, V. (2022). A Simple Method to Find Optimum Efficient Basic Solutions to Bi-Objective Transportation Problems. i-manager’s Journal on Mathematics, 11(1), 1-11. https://doi.org/10.26634/jmat.11.1.18668


[1]. Abd El-Wahed, W. F. (2001). A multi-objective transportation problem under fuzziness. Fuzzy Sets and Systems, 117(1), 27-33. https://doi.org/10.1016/S0165-0114(98)00155-9
[2]. Akilbasha, A., Pandian, P., & Natarajan, G. (2018). An innovative exact method for solving fully interval integer transportation problems. Informatics in Medicine Unlocked, 11, 95-99. https://doi.org/10.1016/j.imu.2018.04.007
[3]. Ammar, E. E., & Youness, E. A. (2005). Study on multiobjective transportation problem with fuzzy numbers. Applied Mathematics and Computation, 166(2), 241-253. https://doi.org/10.1016/j.amc.2004.04.103
[4]. Aneja, Y. P., & Nair, K. P. (1979). Bicriteria transportation problem. Management Science, 25(1), 73-78. https://doi.org/10.1287/mnsc.25.1.73
[5]. Bit, A. K., Biswal, M. P., & Alam, S. (1992). Fuzzy programming approach to multicriteria decision making transportation problem. Fuzzy Sets and Systems, 50(2), 135-141. https://doi.org/10.1016/0165-0114(92)90212-M
[6]. Chanas, S., & Kuchta, D. (1996). A concept of the optimal solution of the transportation problem with fuzzy cost coefficients. Fuzzy Sets and Systems, 82(3), 299-305.
[7]. Das, S. K., Goswami, A., & Alam, S. S. (1999). Multiobjective transportation problem with interval cost, source and destination parameters. European Journal of Operational Research, 117(1), 100-112. https://doi.org/10.1016/S0377-2217(98)00044-7
[8]. Gallagher, R. J., & Saleh, O. A. (1994). Constructing the set of efficient objective values in linear multiple objective transportation problems. European Journal of Operational Research, 73(1), 150-163. https://doi.org/10.1016/0377-2217(94)90154-6
[9]. Gupta, A., & Kumar, A. (2012). A new method for solving linear multi-objective transportation problems with fuzzy parameters. Applied Mathematical Modelling, 36(4), 1421-1430. https://doi.org/10.1016/j.apm.2011.08.044
[10]. Gupta, A., Kumar, A., & Kaur, A. (2012). Mehar's method to find exact fuzzy optimal solution of unbalanced fully fuzzy multi-objective transportation problems. Optimization Letters, 6(8), 1737-1751. https://doi.org/10.1007/s11590-011-0367-2
[11]. Hitchcock, F. L. (1941). The distribution of a product from several sources to numerous localities. Journal of Mathematics and Physics, 20(1-4), 224-230. https://doi.org/10.1002/sapm1941201224
[12]. Ignizio, J. P. (1976). Goal Programming and Extensions. Washington, D.C: Lexington Books.
[13]. Isermann, H. (1979). The enumeration of all efficient solutions for a linear multiple‐objective transportation problem. Naval Research Logistics Quarterly, 26(1), 123-139. https://doi.org/10.1002/nav.3800260112
[14]. Keshavarz, E., & Khorram, E. (2011). A fuzzy bi-criteria transportation problem. Computers & Industrial Engineering, 61(4), 947-957. https://doi.org/10.1016/j.cie.2011.06.007
[15]. Kirca, Ö., & Şatir, A. (1990). A heuristic for obtaining and initial solution for the transportation problem. Journal of the Operational Research Society, 41(9), 865-871. https://doi.org/10.1057/jors.1990.124
[16]. Lee, S. M. (1972). Goal Programming for Decision Analysis. PA: Philadelphia, Auer Bach Publishers.
[17]. Li, L., & Lai, K. K. (2000). A fuzzy approach to the multiobjective transportation problem. Computers & Operations Research, 27(1), 43-57. https://doi.org/10.1016/S0305-0548(99)00007-6
[18]. Maity, G., Roy, S. K., & Verdegay, J. L. (2019). Time variant multi-objective interval-valued transportation problem in sustainable development. Sustainability, 11(21), 6161. https://doi.org/10.3390/su11216161
[19]. Natarajan, P. P. G. (2010). A new method for finding an optimal solution of fully interval integer transportation problems. Applied Mathematical Sciences, 4(37), 1819-1830.
[20]. Nomani, M. A., Ali, I., & Ahmed, A. (2017). A new approach for solving multi-objective transportation problems. International Journal of Management Science and Engineering Management, 12(3), 165-173. https://doi.org/10.1080/17509653.2016.1172994
[21]. Ojha, A., Mondal, S. K., & Maiti, M. (2011). Transportation policies for single and multi-objective transportation problem using fuzzy logic. Mathematical and Computer Modelling, 53(9-10), 1637-1646. https://doi.org/10.1016/j.mcm.2010.12.029
[22]. Pandian, P., & Natarajan, G. (2010). A new algorithm for finding a fuzzy optimal solution for fuzzy transportation problems. Applied Mathematical Sciences, 4(2), 79-90.
[23]. Pandian, P., & Anuradha, D. (2011). A new method for solving bi-objective transportation problems. Australian Journal of Basic and Applied Sciences, 5(10), 67-74.
[24]. Prabhavati, B. S. S., & Ravindranath, V. (2020a). A simple and efficient method to solve fully interval and fuzzy transportation problems, International Journal Mathematics in Operational Research, (In Press).
[25]. Prabhavati, S., & Ravindranath, V. (2020b). A New Approach for Finding a Better Initial Feasible Solution to Balanced or Unbalanced Transportation Problems. In Numerical Optimization in Engineering and Sciences (pp. 359-369). Springer, Singapore. https://doi.org/10.1016/j.matpr.2021.01.175
[26]. Ringuest, J. L., & Rinks, D. B. (1987). Interactive solutions for the linear multiobjective transportation problem. European Journal of Operational Research, 32(1), 96-106. https://doi.org/10.1016/0377-2217(87)90274-8
[27]. Saaty, T. L. (1994). How to make a decision: The analytic hierarchy process. Interfaces, 24(6), 19-43. https://doi.org/10. 1287/inte.24.6.19
[28]. Sharma, J. K. (2006). Operations Research Theory and Application (3rd ed). Macmillan India Ltd.
[29]. Sudha, G., & Ganesan, K. (2021). A solution approach to time variant multi-objective interval transportation problems in material aspects. Materials Today: Proceedings. Advance Online Publication. https://doi.org/10.1016/j.matpr.2021.01.175
[30]. Taha, H. A. (2006). Operation Research- An Introduction, (8th ed). Prentice-Hall of India.
[31]. Venkatasubbaiah, K., Acharyulu, S G., & Mouli, K. V. V. C. (2011). Fuzzy goal programming method for solving multiobjective transportation problems. Global Journal of Research in Engineering, 11(3), 4-10.
[32]. Zaki, S. A., Abd Allah, A. M., Geneedi, H. M., & Elmekawy, A. Y. (2012). Efficient multiobjective genetic algorithm for solving transportation, assignment, and transshipment problems. Applied Mathematics, 3(1), 92-99. https://doi.org/10.4236/am.2012.31015
If you have access to this article please login to view the article or kindly login to purchase the article

Purchase Instant Access

Single Article

North Americas,UK,
Middle East,Europe
India Rest of world
Pdf 35 35 200 20
Online 35 35 200 15
Pdf & Online 35 35 400 25

Options for accessing this content:
  • If you would like institutional access to this content, please recommend the title to your librarian.
    Library Recommendation Form
  • If you already have i-manager's user account: Login above and proceed to purchase the article.
  • New Users: Please register, then proceed to purchase the article.