Abstract
In the present competitive world, it is often said that “Time is Money” in almost every aspect of life. Time is a factor which affects the various real-life problems directly or indirectly. So, in order to incorporate the “time” as a factor in transportation problems (TPs), we have considered the probabilistic cost/profit function termed as “survival cost/profit” which is again a time-dependent function. In this study, we have assumed that the supply and demand quantities are varying between some specified intervals. Due to the variation in the supply and demand quantities, the value of the objective function is also obtained between interval which is bounded by lower and upper values. Based on the above-stated assumptions, we have developed a couple of mathematical optimization models for the TPs. The solution procedure has also been discussed to solve the proposed mathematical models. At last, a numerical illustration has been presented to show the validity of the model and solution procedure which is helpful in the decision-making process.
Similar content being viewed by others
Abbreviations
- \( x_{ij} \) :
-
Number of items transported from ith source to jth destination
- \( c_{ij} \) :
-
Transportation cost of unit item transported from ith source to jth destination
- \( p_{ij} \) :
-
Profit incurred over unit item transported from ith source to jth destination
- \( S_{ij} (t) \) :
-
Probability that items will remain in good condition when transported from ith source to jth destination which is function of time (t)
- \( S_{ij} (c_{ij} ) \) :
-
Survival cost function defined from ith source to jth destination
- \( S_{ij} (p_{ij} ) \) :
-
Survival profit function defined from ith source to jth destination
- \( \tilde{a}_{i} \) :
-
Varying supply quantity at ith source
- \( \tilde{b}_{j} \) :
-
Varying demand quantity at jth destination
- \( [\underline{{a_{i} }} ,\overline{{a_{i} }} ] \) :
-
Lower and upper bound on the supply quantity at ith source
- \( [\underline{{b_{j} }} ,\overline{{b_{j} }} ] \) :
-
Lower and upper bound on the demand quantity at jth destination
- \( u_{i} \) :
-
Dual variable associated with ith supply constraint
- \( v_{j} \) :
-
Dual variable associated with jth demand constraint
References
Hitchcock, F.: Optimum utilization of the transportation system. Econometrica 17, 136–146 (1941)
Liu, S.: The total cost bounds of the transportation problem with varying demand and supply. Omega 31(31), 247–251 (2003). https://doi.org/10.1016/S0305-0483(03)00054-9
Ananya, C., Chakraborty, M.: Cost-time minimization in a transportation problem with fuzzy parameters : a case study. J. Transp. Syst. Eng. Inf. Technol. 10(6), 53–63 (2010). https://doi.org/10.1016/S1570-6672(09)60071-4
Mahapatra, D.R., Roy, S.K., Biswal, M.P.: Multi-choice stochastic transportation problem involving extreme value distribution. Appl. Math. Model. 37(4), 2230–2240 (2013)
Maity, G., Roy, S.K.: Solving multi-choice multi-objective transportation problem: a utility function approach. J. Uncertain. Anal. Appl. 2(1), 11 (2014)
Ebrahimnejad, A.: A simplified new approach for solving fuzzy transportation problems with generalized trapezoidal fuzzy numbers. Appl. Soft Comput. J. 19, 171–176 (2014). https://doi.org/10.1016/j.asoc.2014.01.041
Roy, S.K.: Transportation problem with multi-choice cost and demand and stochastic supply. J. Oper. Res. Soc. China (2016). https://doi.org/10.1007/s40305-016-0125-3
Xie, F., Butt, M.M., Li, Z.: An upper bound on the minimal total cost of the supplies. Omega (2016). https://doi.org/10.1016/j.omega.2016.06.007
Liu, S.: Fractional transportation problem with fuzzy parameters. Soft. Comput. 20(9), 3629–3636 (2016). https://doi.org/10.1007/s00500-015-1722-5
Roy, S.K., Maity, G., Weber, G.W., Alparslan Gok, S.Z.: Conic scalarization approach to solve multi-choice multi-objective transportation problem with interval goal. Ann. Oper. Res. 253(1), 599–620 (2017). https://doi.org/10.1007/s10479-016-2283-4
Roy, S.K., Maity, G., Weber, G.W.: Solving multi-objective both-way grey transportation problem using utility function for selection of goals. CEJOR 25(2), 417–439 (2017). https://doi.org/10.1007/s10100-016-0464-5
Maity, G., Roy, S.K., Verdegay, J.L.: Multi-objective transportation problem with cost reliability under uncertain environment. Int. J. Comput. Intell. Syst. 9(5), 839–849 (2016). https://doi.org/10.1080/18756891.2016.1237184
Gulati, D.R.T.R.: Time optimization in totally uncertain transportation problems. Int. J. Fuzzy Syst. (2016). https://doi.org/10.1007/s40815-016-0176-y
Maity, G., Roy, S.K.: Solving a multi-objective transportation problem with nonlinear cost and multi-choice demand. Int. J. Manag. Sci. Eng. Manag. 11(1), 62–70 (2016). https://doi.org/10.1080/17509653.2014.988768
Roy, S.K., Maity, G.: Minimizing cost and time through single objective function in multi-choice interval valued transportation problem. J. Intell. Fuzzy Syst. 32(3), 1697–1709 (2017). https://doi.org/10.3233/JIFS-151656
Biswas, A., Modak, N.: On solving multiobjective transportation problems with fuzzy random supply and demand using fuzzy goal programming. Int. J. Oper. Res. Inf. Syst. 8(3), 54–81 (2017). https://doi.org/10.4018/IJORIS.2017070104
Shiripour, S., Mahdavi-amiri, N., Mahdavi, I.: A nonlinear model for a capacitated random transportation network. J. Ind. Prod. Eng. 32(8), 500–515 (2017). https://doi.org/10.1080/21681015.2015.1078419
Narayanamoorthy, S., Ranjitha, S.: An approach to solve unbalanced intuitionisitic fuzzy transportation problem using intuitionistic fuzzy numbers. Int. J. Pure Appl. Math. 117(13), 411–419 (2017)
Ebrahimnejad, A., Luis, J.: A new approach for solving fully intuitionistic fuzzy transportation problems. Fuzzy Optim. Decis. Mak. (2017). https://doi.org/10.1007/s10700-017-9280-1
Niroomand, S.: A multi-objective based direct solution approach for linear programming with intuitionistic fuzzy parameters. J. Intell. Fuzzy Syst. (2018). https://doi.org/10.3233/JIFS-171504
Roy, S.K., Ebrahimnejad, A., Verdegay, J.L., Das, S.: New approach for solving intuitionistic fuzzy multi-objective transportation problem. Sādhanā 43(1), 3 (2018). https://doi.org/10.1007/s12046-017-0777-7
Ahmad, F., Adhami, A.Y.: Neutrosophic programming approach to multiobjective nonlinear transportation problem with fuzzy parameters. Int. J. Manag. Sci. Eng. Manag. (2018). https://doi.org/10.1080/17509653.2018.1545608
Ahmad, F., et al.: Single valued neutrosophic hesitant fuzzy computational algorithm for multiobjective nonlinear optimization problem. Neutrosophic Sets Syst. (2018). https://doi.org/10.5281/zenodo.2160357
Sys, L. Inc.: LINGO 13.0-optimization modeling software for linear, nonlinear, and integer programming (2011)
Acknowledgements
Authors are very thankful and overwhelmed to all the anonymous reviewers and editors for the insightful comments to enhance the readability of the manuscript. The first author is also very thankful to Mr. Abdul Nasir Khan for his valuable suggestions and eternal support.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ahmad, F., Adhami, A.Y. Total cost measures with probabilistic cost function under varying supply and demand in transportation problem. OPSEARCH 56, 583–602 (2019). https://doi.org/10.1007/s12597-019-00364-5
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12597-019-00364-5