Abstract
Convexity and submodularity are useful second-order properties in optimization. Majorization and the related arrangement ordering are key properties that support pairwise-interchange arguments. Together these properties play an important role in resource allocation, production planning, and scheduling. Here we present the essentials of the recently developed theory of stochastic convexity and stochastic majorization, emphasizing their interplay. We illustrate the many applications of the theory through examples in production systems, including a random yield model, a joint setup problem, a manufacturing process with trial runs, a production network with constant work-in-process (WIP) and WIP-dependent production rate, and scheduling in tandem production lines and parallel assembly systems.
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References
Brown, M. and Solomon, H. 1973. Optimal Issuing Policies under Stochastic Field Lives. Journal of Applied Probability, 10, 761–768.
Chang, C.S. 1990. A New Ordering for Stochastic Majorization: Theory and Applications. Advances in Applied Probability, 24, 604–634.
Chang, C.S., Chao, X.L. and Pinedo, M. 1991. Monotonicity Results for Queues with Doubly Stochastic Poisson Arrivals: Ross’s Conjecture. Advances in Applied Probability, 23, 210–228.
Chang, C.S., Chao, X.L., Pinedo, M. and Shanthikumar, J.G. 1991. Stochastic Convexity for Multidimensional Process and Its Applications. IEEE Transactions on Automatic Control, 36, 1347–1355.
Chang, C.S. and Yao, D.D. 1990. Rearrangement, Majorization, and Stochastic Scheduling. Mathematics of Operations Research, 18, 658–684.
Day, P.W. 1972. Rearrangement Inequalities. Canadian Journal of Mathematics, 24, 930–943.
Fox, B. 1966. Discrete Optimization via Marginal Analysis. Management Science, 13, 210–216.
Gallego, G., Yao, D.D. and Moon, I. 1993. Optimal Control of a Manufacturing Process with Trial Runs. Management Science, 39, 1499–1505.
Hardy, G.H., Littlewood, J.E. and Polya, G. 1952. Inequalities. Cambridge U. Press, London.
Liyanage, L. and Shanthikumar, J.G. 1992. Allocation through Stochastic Schur Convexity and Stochastic Arrangement Increasing-ness. In: Stochastic Inequalities, M. Shaked and Y.L. Tong (eds.), IMS Lecture Notes/Monographs Series, 22, 253–273.
Marshall, A.W. and Olkin, I. 1979. Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.
Pinedo, M. and Weber, R.R. 1984. Inequalities and Bounds in Stochastic Shop Scheduling. SIAM Journal of Applied Mathematics, 44, 869–879.
Righter, R. 1992. Scheduling. In: Stochastic Orders, M. Shaked and J.G. Shanthikumar (eds.), to appear.
Rockafellar, R.T. 1970. Convex Analysis. Princeton U. Press, Princeton, N.J.
Ross, S.M. 1983. Stochastic Processes. Wiley, New York.
Shaked, M. and Shanthikumar, J.G. 1988. Stochastic Convexity and Its Applications. Advances in Applied Probability, 20, 427–446.
Shaked, M. and Shanthikumar, J.G. 1990. Convexity of a Set of Stochastically Ordered Random Variables. Advances in Applied Probability, 22, 160–167.
Shaked, M. and Shanthikumar, J.G. 1990. Parametric Stochastic Convexity and Concavity of Stochastic Processes. Ann. Inst. Statist. Math., 42, 509–531.
Shaked, M. and Shanthikumar, J.G. 1992. Optimal Allocation of Resources to Nodes of Parallel and Series Systems. Advances in Applied Probability, 24, 894–914.
Shanthikumar, J.G. 1987. Stochastic Majorization of Random Variables with Propotional Equilibrium Rates. Advances in Applied Prob-ability, 19, 854–872.
Shanthikumar, J.G. and Yao, D.D. 1986. The Preservation of Likelihood Ratio Ordering under Convolution. Stochastic Processes and Their Applications, 23, 259–267.
Shanthikumar, J.G. and Yao, D.D. 1986. The Effect of Increasing Service Rates in Closed Queueing Networks. Journal of Applied Probability, 23, 474–483.
Shanthikumar, J.G. and Yao, D.D. 1988. On Server Allocation in Multiple-Center Manufacturing Systems. Operations Research, 36, 333–342.
Shanthikumar, J.G. and Yao, D.D. 1988. Second-Order Properties of the Throughput in a Closed Queueing Network. Mathematics of Operations Research, 13, 524–534.
Shanthikumar, J.G. and Yao, D.D. 1989. Second Order Stochastic Properties in Queueing Systems. Proceedings of IEEE, 77 (1), 162–170.
Shanthikumar, J.G. and Yao, D.D. 1991. Bivariate Characterization of Some Stochastic Order Relations. Advances in Applied Probability, 23, 642–659.
Shanthikumar, J.G. and Yao, D.D. 1991. Strong Stochastic Convexity: Closure Properties and Applications. Journal of Applied Probability, 28 (1991), 131–145.
Shanthikumar, J.G. and Yao, D.D. 1992. Spatiotemporal Convexity of Stochastic Processes and Applications. Probability in the Engineering and Information Sciences, 6, 1–16.
Stoyan, D. 1983. Comparison Methods for Queues and Other Stochastic Models. Wiley, New York.
Topkis, D.M. 1978. Minimizing a Submodular Function on a Lattice. Operations Research, 26, 305–321.
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Chang, CS., Shanthikumar, J.G., Yao, D.D. (1994). Stochastic Convexity and Stochastic Majorization. In: Yao, D.D. (eds) Stochastic Modeling and Analysis of Manufacturing Systems. Springer Series in Operations Research. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2670-3_5
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DOI: https://doi.org/10.1007/978-1-4612-2670-3_5
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