Computation of three-stage stochastic transportation planning under an uncertain environment

Document Type : Original Article


1 iit kharagpur

2 IIT Kharagpur

3 Department of Mathematics IIT Kharagpur


Stochastic programming is often used to solve optimization problems where parameters are uncertain. In this article, we have proposed a mathematical model for a three-stage transportation problem, where the parameters, namely transport costs, demand, unload capacity and external purchasing costs are uncertain. In order to remove the uncertainty, we have proposed a new transformation technique to reformulate the uncertain model deterministically with the help of Essen inequality. The obtained equivalent deterministic model is nonlinear. Furthermore, we have provided a theorem to ensure that the deterministic model gives a feasible solution. Finally, a numerical example, following uniform random variables, is presented to illustrate the model and methodology.


Atalay, K. D., & Apaydin, A. (2011). Gamma distribution approach in chance-constrained stochastic programming model. Journal of Inequalities and Applications, 2011(1), 1-13.‏
Birge, J. R., & Louveaux, F. (2011). Introduction to stochastic programming. Springer Science & Business Media.‏
Charnes A, Cooper W.W. (1959). Chance-constrained programming. Managementscience. 6(1), pp. 73-79.
Hulsurkar, S., Biswal, M. P., & Sinha, S. B. (1997). Fuzzy programming approach to multi-objective stochastic linear programming problems. Fuzzy Sets and Systems, 88(2), 173-181.‏
Jiménez, F., & Verdegay, J. L. (1998). Uncertain solid transportation problems. Fuzzy sets and systems, 100(1-3), 45-57.‏
Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions, volume 2 (Vol. 289). John wiley & sons.‏
Liu, B., Iwamura, K. (1998). Chance constrained programming with fuzzy parameters. Fuzzy sets and systems, 94(2), pp. 227-237.
Olson, D. L., Swenseth, S. R. (1987). A linear approximation for chance-constrained programming. Journal of the Operational Research Society, 38(3), pp. 261-267.
Petrov, V. V. (1995). Limit theorems of probability theory: sequences of independent random variables.‏
Prékopa, A. (2013). Stochastic programming (Vol. 324). Springer Science & Business Media.‏
Roy, S. K. (2014). Multi-choice stochastic transportation problem involving Weibull distribution. International Journal of Operational Research, 21(1), 38-58.‏
Roy, S. K., & Midya, S. (2019). Multi-objective fixed-charge solid transportation problem with product blending under intuitionistic fuzzy environment. Applied Intelligence, 49(10), 3524-3538.‏
Roy, S. K., Mahapatra, D. R., & Biswal, M. P. (2012). Multi-choice stochastic transportation problem with exponential distribution. Journal of Uncertain Systems, 6(3), 200-213.‏
Ruszczyński, A., & Shapiro, A. (2003). Stochastic programming models. Handbooks in operations research and management science, 10, 1-64.‏
Sagratella, S., Schmidt, M., & Sudermann-Merx, N. (2020). The noncooperative fixed charge transportation problem. European Journal of Operational Research, 284(1), 373-382.‏
Shen, J., & Zhu, K. (2020). An uncertain two-echelon fixed charge transportation problem. Soft Computing, 24(5), 3529-3541.‏
Singh, S., Pradhan, A., & Biswal, M. P. (2019). Multi-objective solid transportation problem under stochastic environment. Sādhanā, 44(5), 105.‏
Symonds, G. H. (1967). Deterministic solutions for a class of chance-constrained programming problems. Operations Research, 15(3), 495-512.‏
van Beek, P. (1972). An application of Fourier methods to the problem of sharpening the Berry-Esseen inequality. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 23(3), 187-196.‏
Wang, X., & Ning, Y. (2017). Uncertain chance-constrained programming model for project scheduling problem. Journal of the operational research society, 1-9.‏
Williams, A. C. (1963). A stochastic transportation problem. Operations Research, 11(5), 759-770.‏
Yang, L., & Liu, L. (2007). Fuzzy fixed charge solid transportation problem and algorithm. Applied soft computing, 7(3), 879-889.‏