Abstract
We formulate and solve probabilistic constrained stochastic programming problems, where we prescribe lower and upper bounds for \(k\)-out-of-\(n\) and consecutive-\(k\)-out-of-\(n\) reliabilities in the form of probabilistic constraints. Problems of this kind arise in many practical situations where stockout in a number of consecutive or a certain fraction of all periods causes irreparable damage. Examples are determination of reservoir capacities used for irrigation or capital reserves of banks. The optimal solution is typically obtained by the solution of a“reliability equation” involving discrete or continuous random variables. The reliability here is approximated by the use of binomial Boolean probability bounds. For \(k\)-out-of-\(n\) reliability, the properties of Gamma distribution are used to approximate the reliability equation. For the consecutive \(k\)-out-of-\(n\) case, Binomial probability bounds (\(S_1, S_2, S_3\) sharp lower bounds, Hunter’s upper bound and Cherry tree upper bound) are used to create lower and upper bounds for the reliability constraint and therefore to solve the reliability equation. A bi-section algorithm is finally applied to determine the optimal capacity level.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Boole, G. (1854). Laws of thought. American reprint of 1854 edition. New York: Dover.
Bonferroni, C. E. (1937). Teoria statistics delle classi e calcolo delle probability. In Volume in onore di Riccardo Dalla Volta, Universita di Firenze, pp. 1–62.
Bukszár, J., & Prékopa, A. (2001). Probability bounds with cherry trees. Mathematics of Operations Research, 26, 174–192.
Bukszár, J., & Szántai, T. (2002). Probability bounds given by hypercherry trees. Optimization Methods and Software, 17, 409–422.
Chao, M. T., Fu, J. C., & Koutras, M. V. (1995). Survey of the reliability studies of consecutive-k-out-of-n: F and related systems. IEEE Transactions on Reliability, 44, 120–127.
Chernobai, A. S., Svetlozar, T. R., & Fabozzi, F. J. (2008). Operational risk: A guide to basel II capital requirements, models, and analysis. London: Wiley.
Tomescu, I. (1986). Hypertrees and Bonferroni inequalities. Journal of Combinatorial Theory, Series B, 41, 209–217.
Dawson, D., & Sankoff, A. (1967). An inequality for probabilities. Proceedings of the American Mathematical Society, 18, 504–507.
Habib, A., & Szántai, T. (2000). New bounds on the reliability of the consecutive k-out-of-r-from-n: F system. Reliability Engineering and System Safety, 68(2), 97–104.
Hunter, D. (1976). An upper bound for the probability of a union. Applied Probability Trust, 13, 597–603.
Kruskal (1956). On the shortest spanning subtree of a graph and the traveling salesman problem, Proceedings of the American Mathematical Society, 7, 48–50.
Kwerel, S. M. (1975a). Most stringent bounds on aggregated probabilities of partially specified dependent probability systems. Journal of American Statistics Association, 70(350), 471–479.
Kwerel, S. M. (1975b). Bounds on the probability of a union and intersection of m events. Advanced Applied Probability, 7, 431–448.
Prékopa, A. (1971). Logarithmic concave measures with application to stochastic programming. Acta Scientiarum Mathematicarum, 32, 301–316.
Prékopa, A. (1972). A class of stochastic programming decision problems. Mathematische Operations forschung und Statistik, 3, 349–354.
Prékopa, A. (1973). Stochastic programming models for inventory control and water storage problems, Inventory Control and Water Storage. In: Colloquia Mathematica Societatis Janos, Bolyai, 7, 229–245. North Holland Publishing Company.
Prékopa, A. (1987). Boole–Bonferroni inequalities and linear programming. Operations Research, 36, 145–162.
Prékopa, A. (1988) Boole-Bonferoni inequalities and linear programming, Operations Research, 36(1), 145–162.
Prékopa, A. (1990). Sharp bounds on probabilities using linear programming, Operations Research, 38, 227–239.
Prèkopa, A. (1995). Stochastic programming. Budapest, Hungary: Kluwer.
Prékopa, A., Vizvári B., & Regös, G. (1998). A method of disaggregation for bounding probabilities of Boolean functions of events. Rutcor Research Report, pp. 21–97.
Prékopa, A., Szántai, T., & Zsuffa, I. (2010). Természetes Vizfolyások Ontozoviz Készle]’enek Meghatározását Célzo Matematikai Modellek. Alkalmazott Matematikai Lapok, 27, 175–188.
Szántai, T. (1998). Bounds for the reliability of k-out-of-connected-(r, s)-from-(m, n): F lattice systems. Lecture Notes in Economics and Mathematical Systems, 458, 223–237.
Veneziani, P. (2008). Graph-based upper bounds for the probability of the union of events. Mathematics Faculty Publications, Department of Mathematics, The College at Brockport: State University of New York, Brockport.
Vizvári, B. (2007). New upper bounds on the probability of events based on graph structures. Mathematical Inequalities and Applications, 10, 217–228.
Wood, G. R. (1992). The bisection method in higher dimensions. Mathematical Programming, 55, 319–337.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Unuvar, M., Ozguven, E.E. & Prékopa, A. Optimal capacity design under \(k\)-out-of-\(n\) and consecutive \(k\)-out-of-\(n\) type probabilistic constraints. Ann Oper Res 226, 643–657 (2015). https://doi.org/10.1007/s10479-014-1712-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-014-1712-5