Abstract
In this paper, we propose a two-stage stochastic linear programming problem considering some of the left hand side and right hand side of linear constraints parameters as interval discrete random variables with known probability distribution and rest of the parameters are precisely known. Both the randomness and discrete intervals are simultaneously considered for the model parameters. Further, the concepts of best optimum and worst optimum solution are studied in two-stage stochastic programming. To solve the stated problem, first we remove the randomness from the problem and formulate an equivalent deterministic linear programming model with interval coefficients. Then the deterministic model is solved using the solution procedure of linear programming with interval coefficients. We obtain the upper bound and lower bound of the objective function as the best and the worst value respectively. It highlights the possible risk involved in the decision making process. A numerical example is presented to demonstrate the usefulness of the proposed methodology.
Similar content being viewed by others
References
Allahdadi, M., Nehi, H.M.: Fuzzy linear programming with interval linear programming approach. Adv. Model. Optim. 13(1), 1–12 (2011)
Beale, E.M.L.: On minimizing a convex function subject to linear inequalities. J. R. Stat. Soc. 17B, 173–184 (1955)
Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer, New York (1997)
Chinneck, J.W., Ramadan, K.: Linear programming with interval coefficients. J. Oper. Res. Soc. 51, 209–220 (2000)
Dantzig, G.B.: Linear programming under uncertainty. Manage. Sci. 1, 197–206 (1955)
Dantzig, G.B., Madansky, A.: On the solution of two-stage linear programs under uncertainty. In: 4th Berkeley Symposium on Statistics and Probability, vol. 1, pp. 165–176. University California Press, Berkeley (1961)
Han, Y.C., Huang, G.H., Li, C.H.: An interval-parameter multi-stage stochastic chance-constrained mixed integer programming model for inter-basin water resources management systems under uncertainty. In: Fifth International Conference on Fuzzy Systems and Knowledge Discovery, IEEE 2008, pp. 146–153
Kambo, N.S.: Mathematical Programming Techniques. Affiliated East-West Press, New Delhi (1984)
Li, Y.P., Huang, G.H.: An inexact two-stage mixed integer linear programming method for solid waste management in the City of Regina. J. Environ. Manag. 81, 188–209 (2006)
Li, Y.P., Huang, G.H.: Interval-parameter two-stage stochastic nonlinear programming for water resources management under uncertainty. Water Resour. Manag. 22, 681–698 (2009)
Molai, A.A., Khorram, E.: Linear programming problem with interval coefficients and an interpretation for its constraints. Iran. J. Sci. Technol., Trans. A Sci. 31, 369–390 (2007)
Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs (1966)
Rosenthal, R.E.: GAMS—A User’s Guide. GAMS Development Corporation, Washington, DC (2010)
Sahinidis, N.V.: Optimization under uncertainty: state-of-the-art and opportunities. Comput. Chem. Eng. 28, 971–983 (2004)
Schrage, L.: Optimization Modeling with LINGO, 6th edn. LINDO Systems Inc., Chicago (2006)
Su, J., Huang, G.H., Xi, B.D., Qin, X.S., Huo, S.L., Jiang, Y.H., Chen, X.R.: Long-term panning of waste diversion under interval and probabilistic uncertainties. Resour. Conserv. Recycl. 54, 449–461 (2010)
Suprajitno, H., Mohd, I.B.: Linear programming with interval arithmetic. Int. J. Contemp. Math. Sci. 5, 323–332 (2010)
Talla, N.F., Guo, R.: Foundation and formulation of stochastic interval programming. PGD thesis, African Institute for Mathematical Sciences, Cape Town, South Africa (2006)
Tong, S.: Interval number and fuzzy number linear programming. Fuzzy Sets Syst. 66, 301–306 (1994)
Walkup, D.W., Wets, R.J.B.: Stochatsic programs with recourse. SIAM J. Appl. Math. 15, 1299–1314 (1967)
Xu, Y., Huang, G.H., Qin, X.: Inexact two-stage stochastic robust optimization model for water resources management under uncertainty. Environ. Eng. Sci. 26, 1765–1776 (2009)
Zhou, F., Guo, H.C., Huang, K., Huang, G.H.: The interval linear programming: a revisit. Environ. Inform. Arch. 5, 101–110 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Barik, S.K., Biswal, M.P. & Chakravarty, D. Two-stage stochastic programming problems involving interval discrete random variables. OPSEARCH 49, 280–298 (2012). https://doi.org/10.1007/s12597-012-0078-1
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12597-012-0078-1