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DETERMINISTIC EQUIVALENTS OF CHANCE CONSTRAINED STOCHASTIC PROGRAMMING PROBLEMS

Year 2010, Volume: 1 Issue: 1, 1 - 18, 29.07.2011

Abstract

Many real life problems which are modeled as linear programming problems where coefficients appear as random variables. In this case, such problems are called as stochastic programming problem. The basic approach in the stochastic programming is solving the problem with known methods by converting the problem from a probability structure to a deterministic structure. The chance constraints in this programming approach can be forced from being the random coefficients to deterministic one according to their specific levels.

In this study when the coefficients which are random variables have normal and Chi-square distributions, obtaining of deterministic models that are equivalent to chance constrained stochastic programming models are examined. The importance of choosing the correct distributions of coefficients is explained by a numeric example.

References

  • Charnes, A. ve Cooper, W.W. (1959). Chance Constrained Programming. Management Science 6, 73- 79.
  • Déak, I. (1998). Linear Regression Estimators for Multinormal Distributions in Optimization of Stochastic Programming Problems. European Journal of Operational Research 111, 555-568.
  • Hillier, F.S. ve Lieberman, G.J. (1990). Introduction to Mathematical Programming. Hill Publishing Company, New York.
  • Hulsurkar, S., Biswal, M.P. ve Sinha, S.B. (1997). Fuzzy Programming Approach to Multi-Objective Stochastic Linear Programming Problems. Fuzzy Sets and Systems 88,173-181.
  • Jana, R.K. ve Biswal, M.P. (2004). Stochastic Simulation-Based Genetic Algorıthm for Chance Constraint Programming Problems with Continuous Random Variables. International Journal of Computer Mathematics 81(9), 1069-1076.
  • Kampas, A. ve White, B. (2003). Probabilistic Programming for Nitrate Pollution Control: Comparing Different Probabilistic Constraint Approximations. European Journal of Operational Research 147, 217-228.
  • Kolbin, V.V. (1977). Stochastic Programming. D. Reidel Publishing Company, Boston.
  • Patnaik, P.B. (1949). The Non-Central 2 χ and F- Distributions and Their Applications. Biometrika 36,202-232.
  • Resh, M. (1970). Chance Constrained Programming of the Machine Loading Problem with Stochastic Proccessing Times. Management Science 17,48-65.
  • Sengupta, J.K. (1970). A Generalization of Some Distribution Aspects of Chance Constrained Linear Programming. International Economic Review 11, 287-304.
  • Sengupta, J.K. (1972). Stochastic Programming: Methods and Applications. North-Holland Publishing Company, Amsterdam.
  • Symonds, G.H. (1967). Deterministic Solutions for A Class of Chance Constrained Programming Problems. Operations Research 15, 495-512.
Year 2010, Volume: 1 Issue: 1, 1 - 18, 29.07.2011

Abstract

References

  • Charnes, A. ve Cooper, W.W. (1959). Chance Constrained Programming. Management Science 6, 73- 79.
  • Déak, I. (1998). Linear Regression Estimators for Multinormal Distributions in Optimization of Stochastic Programming Problems. European Journal of Operational Research 111, 555-568.
  • Hillier, F.S. ve Lieberman, G.J. (1990). Introduction to Mathematical Programming. Hill Publishing Company, New York.
  • Hulsurkar, S., Biswal, M.P. ve Sinha, S.B. (1997). Fuzzy Programming Approach to Multi-Objective Stochastic Linear Programming Problems. Fuzzy Sets and Systems 88,173-181.
  • Jana, R.K. ve Biswal, M.P. (2004). Stochastic Simulation-Based Genetic Algorıthm for Chance Constraint Programming Problems with Continuous Random Variables. International Journal of Computer Mathematics 81(9), 1069-1076.
  • Kampas, A. ve White, B. (2003). Probabilistic Programming for Nitrate Pollution Control: Comparing Different Probabilistic Constraint Approximations. European Journal of Operational Research 147, 217-228.
  • Kolbin, V.V. (1977). Stochastic Programming. D. Reidel Publishing Company, Boston.
  • Patnaik, P.B. (1949). The Non-Central 2 χ and F- Distributions and Their Applications. Biometrika 36,202-232.
  • Resh, M. (1970). Chance Constrained Programming of the Machine Loading Problem with Stochastic Proccessing Times. Management Science 17,48-65.
  • Sengupta, J.K. (1970). A Generalization of Some Distribution Aspects of Chance Constrained Linear Programming. International Economic Review 11, 287-304.
  • Sengupta, J.K. (1972). Stochastic Programming: Methods and Applications. North-Holland Publishing Company, Amsterdam.
  • Symonds, G.H. (1967). Deterministic Solutions for A Class of Chance Constrained Programming Problems. Operations Research 15, 495-512.

ŞANS KISITLI STOKASTİK PROGRAMLAMA PROBLEMLERİNİN DETERMİNİSTİK EŞİTLİKLERİ

Year 2010, Volume: 1 Issue: 1, 1 - 18, 29.07.2011

Abstract

Doğrusal programlama problemi olarak modellenen birçok gerçek hayat probleminde katsayılar rasgele değişken olarak ortaya çıkar. Bu durumda kurulan probleme stokastik programlama problemi adı verilmektedir. Stokastik programlamanın çözümünde temel yaklaşım, problemin olasılıksal bir yapıdan deterministik bir yapıya dönüştürülerek bilinen yöntemlerle çözülmesidir. Stokastik programlama tekniklerinden biri olan şans kısıtlı programlama yaklaşımı, rasgele kısıtları belirli seviyelerine göre deterministik duruma getirmeyi amaçlar. Bu çalışmada rasgele değişken olan katsayıların normal dağılıma ve Ki-kare dağılımına sahip olması durumunda ortaya çıkan şans kısıtlı stokastik programlama modelleri kullanılarak deterministik modellerin oluşturulması süreci incelenmiştir. Katsayıların dağılımının doğru olarak seçilmesinin önemliliği sayısal bir örnekle açıklanmış ve irdelenmiştir.

References

  • Charnes, A. ve Cooper, W.W. (1959). Chance Constrained Programming. Management Science 6, 73- 79.
  • Déak, I. (1998). Linear Regression Estimators for Multinormal Distributions in Optimization of Stochastic Programming Problems. European Journal of Operational Research 111, 555-568.
  • Hillier, F.S. ve Lieberman, G.J. (1990). Introduction to Mathematical Programming. Hill Publishing Company, New York.
  • Hulsurkar, S., Biswal, M.P. ve Sinha, S.B. (1997). Fuzzy Programming Approach to Multi-Objective Stochastic Linear Programming Problems. Fuzzy Sets and Systems 88,173-181.
  • Jana, R.K. ve Biswal, M.P. (2004). Stochastic Simulation-Based Genetic Algorıthm for Chance Constraint Programming Problems with Continuous Random Variables. International Journal of Computer Mathematics 81(9), 1069-1076.
  • Kampas, A. ve White, B. (2003). Probabilistic Programming for Nitrate Pollution Control: Comparing Different Probabilistic Constraint Approximations. European Journal of Operational Research 147, 217-228.
  • Kolbin, V.V. (1977). Stochastic Programming. D. Reidel Publishing Company, Boston.
  • Patnaik, P.B. (1949). The Non-Central 2 χ and F- Distributions and Their Applications. Biometrika 36,202-232.
  • Resh, M. (1970). Chance Constrained Programming of the Machine Loading Problem with Stochastic Proccessing Times. Management Science 17,48-65.
  • Sengupta, J.K. (1970). A Generalization of Some Distribution Aspects of Chance Constrained Linear Programming. International Economic Review 11, 287-304.
  • Sengupta, J.K. (1972). Stochastic Programming: Methods and Applications. North-Holland Publishing Company, Amsterdam.
  • Symonds, G.H. (1967). Deterministic Solutions for A Class of Chance Constrained Programming Problems. Operations Research 15, 495-512.
There are 12 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Kumru Atalay

Ayşen Apaydın

Publication Date July 29, 2011
Published in Issue Year 2010 Volume: 1 Issue: 1

Cite

APA Atalay, K., & Apaydın, A. (2011). DETERMINISTIC EQUIVALENTS OF CHANCE CONSTRAINED STOCHASTIC PROGRAMMING PROBLEMS. Anadolu University Journal of Science and Technology B - Theoretical Sciences, 1(1), 1-18.
AMA Atalay K, Apaydın A. DETERMINISTIC EQUIVALENTS OF CHANCE CONSTRAINED STOCHASTIC PROGRAMMING PROBLEMS. AUBTD-B. July 2011;1(1):1-18.
Chicago Atalay, Kumru, and Ayşen Apaydın. “DETERMINISTIC EQUIVALENTS OF CHANCE CONSTRAINED STOCHASTIC PROGRAMMING PROBLEMS”. Anadolu University Journal of Science and Technology B - Theoretical Sciences 1, no. 1 (July 2011): 1-18.
EndNote Atalay K, Apaydın A (July 1, 2011) DETERMINISTIC EQUIVALENTS OF CHANCE CONSTRAINED STOCHASTIC PROGRAMMING PROBLEMS. Anadolu University Journal of Science and Technology B - Theoretical Sciences 1 1 1–18.
IEEE K. Atalay and A. Apaydın, “DETERMINISTIC EQUIVALENTS OF CHANCE CONSTRAINED STOCHASTIC PROGRAMMING PROBLEMS”, AUBTD-B, vol. 1, no. 1, pp. 1–18, 2011.
ISNAD Atalay, Kumru - Apaydın, Ayşen. “DETERMINISTIC EQUIVALENTS OF CHANCE CONSTRAINED STOCHASTIC PROGRAMMING PROBLEMS”. Anadolu University Journal of Science and Technology B - Theoretical Sciences 1/1 (July 2011), 1-18.
JAMA Atalay K, Apaydın A. DETERMINISTIC EQUIVALENTS OF CHANCE CONSTRAINED STOCHASTIC PROGRAMMING PROBLEMS. AUBTD-B. 2011;1:1–18.
MLA Atalay, Kumru and Ayşen Apaydın. “DETERMINISTIC EQUIVALENTS OF CHANCE CONSTRAINED STOCHASTIC PROGRAMMING PROBLEMS”. Anadolu University Journal of Science and Technology B - Theoretical Sciences, vol. 1, no. 1, 2011, pp. 1-18.
Vancouver Atalay K, Apaydın A. DETERMINISTIC EQUIVALENTS OF CHANCE CONSTRAINED STOCHASTIC PROGRAMMING PROBLEMS. AUBTD-B. 2011;1(1):1-18.