Weitere Kapitel dieses Buchs durch Wischen aufrufen
This paper introduces three methods for approximating distribution of weighted sum of exponential variates. These methods are useful for transforming chance constraints into their equivalent deterministic constraints when the technologic coefficients are exponential random variables. Hence, the equivalent deterministic constraint is obtained by three methods which are normal approximation and first- and second-term Edgeworth series expansions, respectively. These methods are based on normal approximation related to the central limit theorem (CLT). Furthermore, the exact distribution of weighted sum of exponential variates is presented by using convolution technique. The fourth method is proposed for deriving deterministic equivalent of chance constraint by using this exact distribution. The fifth method is transforming the exponential variates into the chi-squared variates. Illustrative examples are given for the purpose of comparing the solutions of these five methods. Additionally, the optimal solution for Example 1 of Biswal et al. (1998. European Journal of Operational Research 111:589–597) is extended to a global solution by using three methods.
Bitte loggen Sie sich ein, um Zugang zu diesem Inhalt zu erhalten
Sie möchten Zugang zu diesem Inhalt erhalten? Dann informieren Sie sich jetzt über unsere Produkte:
Aringhieri, R. (2005). A Tabu search algorithm for solving chance-constrained programs. Journal of the ACM, V(N), 1–14.
Birge, J. R., & Louveaux, F. (1997). Introduction to stochastic programming. Springer series in operations research and financial engineering. New York: Springer.
Biswal, M. P., Biswal, N. P., & Li, D. (1998). Probabilistic linear programming problems with exponential random variables: A technical note. European Journal of Operational Research, 111, 589–597. CrossRef
Bitran, G. R., & Thin-Y, L. (1990). Distribution-free, uniformly-tighter linear approximations for chance-constrained programming. MIT Sloan School Working Paper #3111–90-MSA.
Charnes, A., & Cooper, W. W. (1959). Chance-constrained programming. Management Science, 5, 73–79. CrossRef
Chiralaksanakul, A., & Mahadevan, S. (2005). First-order approximation methods in reliability-based design optimization. Journal of Mechanical Design, 127(5), 851–857. CrossRef
DePaolo, C. A., & Rader, D. J., Jr. (2007). A heuristic algorithm for a chance constrained stochastic program. European Journal of Operational Research, 176, 27–45. CrossRef
Feller, W. (1966). An introduction to probability theory and its applications (Vol. II). New York: Wiley.
Gurgur, C. Z., & Luxhoj, J. T. (2003). Application of chance-constrained programming to capital rationing problems with asymmetrically distributed cash flows and available budget. The Engineering Economist, 48(3), 241–258. CrossRef
Hansotia, B. J. (1980). Stochastic linear programs with simple recourse: The equivalent deterministic convex program for the normal, exponential and Erlang cases. Naval Research Logistics Quarterly, 27, 257–272. CrossRef
Hillier, F. S., & Lieberman, G. J. (1990). Introduction to mathematical programming. New York: Hill Publishing Company.
Jeeva, M., Rajagopal, R., Charles, V., & Yadavalli, V. S. S. (2004). An Application of stochastic programming with Weibull Distribution cluster based optimum allocation of recruitment in manpower planning. Stochastic Analysis and Applications, 22(3), 801–812. CrossRef
Kall, P., & Wallace, S. W. (2003). Stochastic programming. 2nd Edition. Wiley.
Kampas, A., & White, B. (2003). Probabilistic programming for nitrate pollution control: Comparing different probabilistic constraint approximations. European Journal of Operational Research, 147, 217–228. CrossRef
Kendall, M. G. (1945). The advanced theory of statistics. Volume I. Charles Griffin Company Limited.
Kibzun, A. I. (1991). Probabilistic optimization problems. Technical Report. pp 91–34.
Kolbin, V. V. (1977). Stochastic programming. Boston: D Reidel Publishing Company. CrossRef
Lehmann, E. L. (1999). Elements of large sample theory. New York: Springer Verlag. CrossRef
Liu, B. (2009). Theory and practice of uncertain programming. 3rd Edition. UTLAB.
Petrov, V. V. (1975). Sums of independent random variables. New York: Springer-Verlag. CrossRef
Poojari, C. A., & Varghese, B. (2008). Genetic algorithm based technique for solving chance constrained problems. European Journal of Operational Research, 185, 1128–1154. CrossRef
Sahoo, N. P., & Biswal, M. P. (2009). Computation of a multi-objective production planning model with probabilistic constraints. International Journal of Computer Mathematics, 86(1), 185–198. CrossRef
Sengupta, J. K. A. (1970). Generalization of some distribution aspects of chance constrained linear programming. International Economic Review, 11, 287–304. CrossRef
Shapiro, A., Dentcheva, D., & Ruszczyński, A. (2009). Lectures on stochastic programming: Modeling and theory. MPS-SIAM series on optimization (Vol. 9). Philadelphia: SIAM and MPS. CrossRef
Taha, H. A. (1997). Operations research on introduction. Upper Saddle River: Prentice Hall.
Varghese, B., & Poojari, C. (2004). Genetic algorithm based technique for solving chance constrained problems arising in risk management. Technical Report CARISMA 54 pages.
Wallace, D. L. (1958). Asymptotic approximations to distributions. The Annals of Mathematical Statistics, 29(3), 635–654. CrossRef
Yılmaz, M. (2007). Edgeworth series approximation for Chi-Square type chance constraints. Communications Faculty of Sciences University of Ankara Series A1, 56(2), 27–37.
Yılmaz, M. (2009). Edgeworth series approximation for gamma type chance constraints. Selcuk Journal of Applied Mathematics, 10(1), 75–89.
Yılmaz, M., & Topçu, B. (2008). Some comments on solving probabilistic constrained stochastic programming problems. Selcuk Journal of Applied Mathematics, 9(2), 29–44.
- Solutions to Chance-Constrained Programming Problems with Exponential Random Variables by Edgeworth Approximation
- Chapter 1
Neuer Inhalt/© Stellmach, Neuer Inhalt/© Maturus, Pluta Logo/© Pluta