Optimality of Sequential Quality Control via Stochastic Orders

  • David D. Yao
  • Shaohui Zheng
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 19)


In recent years, stochastic orders in general and stochastic convexity in particular have been demonstrated as playing a central role in the optimal design and control of stochastic systems (refer to the wide-ranging applications presented in a recent monograph by Shaked and Shanthikumar [16]; also, refer to Shaked and Shanthikumar [15] and Shanthikumar and Yao [17], among many others). A somewhat less known but equally useful property, stochastic submodularity, and its many applications have been illustrated in Chang and Yao [4] and in Chang, Shanthikumar, and Yao [5].


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  1. [1]
    Albin, S. L., and Friedman, D. J. The impact of clustered defect distributions in 1C fabrication. Management Sci. 35, 1066–1078, 1989.CrossRefGoogle Scholar
  2. [2]
    Blischke, W. R. Mathematical models for analysis of warranty policies. Math. Comput. Modeling 13, 1–16, 1990.zbMATHCrossRefGoogle Scholar
  3. [3]
    Beutler, F. X, and Ross, K. W. Optimal policies for controlled Markov chains with a constraint. J. Math. Ana. Appl. 112, 236–252, 1985.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Chang, C. S., and Yao, D. D. Rearrangement, majorization, and stochastic scheduling. Math. Oper. Res. 18, 658–684, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Chang, C. S., Shanthikumar, J. G., and Yao, D. D. Stochastic convexity and stochastic majorization. In: Yao, D. D. (ed.), Stochastic Modeling and Analysis of Manufacturing Systems, Chapter 5. Springer-Verlag, New York, 1994.Google Scholar
  6. [6]
    Chen, J., Yao, D. D., and Zheng, S. Quality control for products supplied with warranty. Oper. Res, to appear.Google Scholar
  7. [7]
    Chen, J., Yao, D. D., and Zheng, S. Sequential Inspection of Components in an Assembly System. Working Paper, IEOR Department, Columbia University, New York, NY 10027, 1997.Google Scholar
  8. [8]
    Derman, C. Finite State Markovian Decision Processes. Academic Press, New York, 1970.zbMATHGoogle Scholar
  9. [9]
    Feinberg, E. A. Constrained semi-Markov decision processes with average rewards. ZOR Math. Meth. Oper. Res. 39, 257–288, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Kallenberg, L. Linear Programming and Finite Markovian Control Problems. Math Centre Tracts 148, Mathematisch Centrum, Amsterdam, 1983.Google Scholar
  11. [11]
    Keilson, J., and Sumita, U. Uniform stochastic ordering and related inequalities. Can. J. Stat. 10, 181–198, 1982.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Ross, S. M. Stochastic Processes. Wiley, New York, 1983.zbMATHGoogle Scholar
  13. [13]
    Ross, S. M. Introduction to Stochastic Dynamic Programming. Academic Press, New York, 1983.zbMATHGoogle Scholar
  14. [14]
    Scarf, H. The optimality of (S, s) policies in the dynamic inventory problem. In: Arrow, K. J., Karlin, S., and Suppes, P. (eds.), Mathematical Methods in the Social Sciences. Stanford University Press, Stanford, CA, 1960, pp. 196–202.Google Scholar
  15. [15]
    Shaked, M., and Shanthikumar, J. G. Stochastic convexity and its applications. Adv. Appl. Prob. 20, 427–446, 1988.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    Shaked, M., and Shanthikumar, J. G. Stochastic Orders and Their Applications. Academic Press, New York, 1994.zbMATHGoogle Scholar
  17. [17]
    Shanthikumar, J. G, and Yao, D. D. Strong stochastic convexity: closure properties and applications. J. Appl. Prob. 28, 131–145, 1991.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    Shanthikumar, J. G., and Yao, D. D. Bivariate characterization of some stochastic order relations. Adv. Appl. Prob. 23, 642–659, 1991.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    Topkis, D. M. Minimizing a submodular function on a lattice. Oper. Res. 26, 305–321, 1978.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    Yao, D. D., and Zheng, S. Sequential inspection under capacity constraints. Oper. Res., to appear.Google Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • David D. Yao
  • Shaohui Zheng

There are no affiliations available

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