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Methodology and Computing in Applied Probability

, Volume 20, Issue 3, pp 935–955 | Cite as

Arrangement Increasing Resource Allocation

  • Qi Feng
  • J. George Shanthikumar
Article
  • 42 Downloads

Abstract

Consider a system composed of several units. The performance of each unit can be affected by providing a portion of a limited amount of costly resources available. An allocation of resources to a unit results in a unit’s response that depends on the level of resources allocated to it and some of its random parameters. In this paper we consider cases where each unit has one or two random parameters. The overall performance of the system is mapped by a function on the vector of responses generated by all the units in the system. Our interest is in identifying the conditions on the response function of the units, the system performance function and the random parameters under which the random system performance as a function of the resource allocation has stochastic arrangement increasing property. This allows one to substantially reduce the number of allocation that needs to be searched to identify an optimal allocation that maximizes the expected utility derived from the system response as a result of the resource allocation.

Keywords

Allocation Arrangement increasing Joint stochastic orders Joint stochastic increasing vector Stochastic functions 

Mathematics Subject Classification (2010)

60E15 62E10 90B05 90B25 

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References

  1. Belzunce F, Martínez-Puertas H, Ruiz JM (2013) On allocation of redundant components for systems with dependent components. Eur J Oper Res 230:573–580MathSciNetCrossRefzbMATHGoogle Scholar
  2. Cai J, Wei W (2014) Some new notions of dependence with applications in optimal allocation problems. Insur Math Econ 55:200–209MathSciNetCrossRefzbMATHGoogle Scholar
  3. Chen Z, Hu T (2008) Asset proportions in optimal portfolios with dependent default risks. Insur Math Econ 43(2):223–226MathSciNetCrossRefzbMATHGoogle Scholar
  4. Cheung KC (2006) Optimal portfolio problem with unknown dependency structure. Insur Math Econ 38:167–175MathSciNetCrossRefzbMATHGoogle Scholar
  5. Cheung KC (2007) Optimal allocation of policy limits and deductibles. Insur Math Econ 41:291–382MathSciNetCrossRefzbMATHGoogle Scholar
  6. Cheung KC, Yang H (2004) Ordering optimal proportions in the asset allocation problem with dependent default risks. Insur Math Econ 35:595–609MathSciNetCrossRefzbMATHGoogle Scholar
  7. Cheung KC, Yang H (2008) Ordering of optimal portfolio allocations in a model with a mixture of fundamental risks. J Appl Probab 45:55–66MathSciNetCrossRefzbMATHGoogle Scholar
  8. Fang R, Li X (2016) On allocating one active redundancy to coherent systems with dependent and heterogeneous components’ lifetimes. Nav Res Logist 63:335–345MathSciNetCrossRefGoogle Scholar
  9. Hennessy DA, Lapan HE (2002) The use of archimedean copulas to model portfolio allocations. Math Financ 12:143–154MathSciNetCrossRefzbMATHGoogle Scholar
  10. Hu F, Wang Y (2010) Optimal allocation of policy limits and deductibles in a model with mixture risk and discount factors. J Comput Appl Math 234:2953–2961MathSciNetCrossRefzbMATHGoogle Scholar
  11. Hu S, Wang Y (2014) Stochastic comparison and optimal allocation for policy limits and deductibles. Communications in Statistics–Theory and Methods 43:151–164MathSciNetCrossRefzbMATHGoogle Scholar
  12. Hu T, Wang Y (2009) Optimal allocation of active redundancies in r-out-of-n systems. J Stat Plann Inference 139(10):3733–3737MathSciNetCrossRefzbMATHGoogle Scholar
  13. Kijima M, Ohnishi M (1996) Portfolio selection problems via the bivariate characterization of stochastic dominance relations. Math Financ 6:237–277MathSciNetCrossRefzbMATHGoogle Scholar
  14. Landsberger M, Meilijson I (1990) Demand for risky financial assets: a portfolio analysis. J Econ Theory 20:204–213MathSciNetCrossRefzbMATHGoogle Scholar
  15. Laniado H, Lillo RE, Pellerey F, Romo J (2012) Portfolio selection through an extremality stochastic order. Insur Math Econ 51:1–9MathSciNetCrossRefzbMATHGoogle Scholar
  16. Lapan HE, Hennessy DA (2002) The use of archimedean copulas to model portfolio allocations. Econ Theory 19:747–772CrossRefzbMATHGoogle Scholar
  17. Li C, Li X (2016a) Some new results on allocation of coverage limits and deductibles to mutually independent risks. To appear in Communications in Statistics - Theory and MethodsGoogle Scholar
  18. Li C, Li X (2016b) Sufficient conditions for ordering aggregate heterogeneous random claim amounts. Insur Math Econ 70:406–413Google Scholar
  19. Li X, You Y (2015) Permutation monotone functions of random vector with applications in financial and actuarial risk management. Adv Appl Probab 47:270–291MathSciNetCrossRefzbMATHGoogle Scholar
  20. Li X, You Y, Fang R (2016) On weighted k-out-of-n systems with statistically dependent component lifetimes. Probab Eng Inf Sci 30(2):533–546MathSciNetCrossRefzbMATHGoogle Scholar
  21. Liyanage L, Shanthikumar JG (1993) Allocation through stochastic schur convexity and stochastic transposition increasingness. In: Shaked M, Tong YL (eds) Stochastic inequalities, vol 22. Institute of Mathematical Statistics, pp 253–273Google Scholar
  22. Lu Z, Meng L (2011) Stochastic comparisons for allocations of policy limits and deductibles with applications. Insur Math Econ 48(3):338–343MathSciNetCrossRefzbMATHGoogle Scholar
  23. Manesh SF, Khaledi BE (2015) Allocation of policy limits and ordering relations for aggregate remaining claims. Insur Math Econ 65:9–14MathSciNetCrossRefzbMATHGoogle Scholar
  24. Marshall AW, Olkin I, Arnold BC (2011) Inequalities: theory of majorization and its applications. Springer-Verlag, New YorkCrossRefzbMATHGoogle Scholar
  25. Misra N, Misra K, Dhariyal ID (2011) Active redundancy allocations in series systems. Probab Eng Inf Sci 25:219–235MathSciNetCrossRefzbMATHGoogle Scholar
  26. Muller A, Stoyan D (2002) Comparison methods for stochastic models and risks. Wiley, West SussexzbMATHGoogle Scholar
  27. Pellerey F, Semeraro P (2005) A note on the portfolio selection problem. Theor Decis 59:295–306MathSciNetCrossRefzbMATHGoogle Scholar
  28. Righter R, Shanthikumar JG (1992) Extension of the bivariate characterization for stochastic orders. Adv Appl Probab 24:506–508MathSciNetCrossRefzbMATHGoogle Scholar
  29. Shaked M, Shanthikumar JG (2006) Stochastic orders, 1st edn. Springer, New YorkzbMATHGoogle Scholar
  30. Shanthikumar JG (1987) Stochastic majorization of random variables with proportional equilibrium rates. Adv Appl Probab 19:854–872MathSciNetCrossRefzbMATHGoogle Scholar
  31. Shanthikumar JG, Yao DD (1991) Bivariate characterization of some stochastic order relations. Adv Appl Probab 23:642–659MathSciNetCrossRefzbMATHGoogle Scholar
  32. You Y, Li X (2014) Optimal capital allocations to interdependent actuarial risks. Insur Math Econ 57:104–113MathSciNetCrossRefzbMATHGoogle Scholar
  33. Zhao P, Chan PS, Li L, Ng HKT (2013) On allocation of redundancies in two-component series systems. Nav Res Logist 60:588–598MathSciNetCrossRefzbMATHGoogle Scholar
  34. Zhao P, Zhang Y, Chen J (2017) Optimal allocation policy of one redundancy in a n-component series system. Eur J Oper Res 257(2):656–668MathSciNetCrossRefzbMATHGoogle Scholar
  35. Zhuang W, Chen Z, Hu T (2009) Optimal allocation of policy limits and deductibles under distortion risk measures. Insur Math Econ 44:409–414MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Krannert School of ManagementPurdue UniversityWest LafayetteUSA

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