Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-27T16:18:28.640Z Has data issue: false hasContentIssue false
  • English
  • Français

Further counterexamples to the monotonicity property of t-step maintainable structures

Part of: Markov processes Operations research and management science

Published online by Cambridge University Press:  14 July 2016

John Haigh
John Haigh*
Affiliation:
University of Sussex
*
∗Postal address: Mathematics Division, University of Sussex, Falmer, Brighton BN1 9QH, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

Until Guerry's (1990) counterexample to a conjecture of Davies about three-state hierarchical organisations kept at constant size via annual promotion, wastage and recruitment, it was easy to believe that such structures maintainable in t steps would also be maintainable in t + 1 steps. Here we present further counterexamples, which show that t-step maintainability does not imply (t + 1)-step maintainability, for astonishingly large values of t.

Keywords

MANPOWER PLANNINGSUBSTOCHASTIC MATRICES

MSC classification

Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Secondary: 90B70: Theory of organizations, manpower planning
Type
Short Communications
Information
Journal of Applied Probability , Volume 29 , Issue 2 , June 1992 , pp. 441 - 447
Copyright
Copyright © Applied Probability Trust 1992 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bartholomew, D. J. (1975) A stochastic control problem in the social sciences. Bull. Int. Statist. Inst. 46, 670–680.Google Scholar
Bartholomew, D. J. (1977) Maintaining a grade or age structure in a stochastic environment. Adv. Appl. Prob. 9, 1–17.CrossRefGoogle Scholar
Bartholomew, D. J. (1979) The control of a grade structure in a stochastic environment using promotion control. Adv. Appl. Prob. 11, 603–615.CrossRefGoogle Scholar
Bartholomew, D. J. (1981) Mathematical Methods in Social Sciences (Handbook of Applicable Mathematics). Wiley, New York.Google Scholar
Bartholomew, D. J. and Forbes, A. F. (1979) Statistical Techniques for Manpower Planning. Wiley, New York.Google Scholar
Davies, G. S. (1973) Structural control in a graded manpower system. Management Sci. 20, 76–84.CrossRefGoogle Scholar
Davies, G. S. (1975) Maintainability of structures in Markov chain models under recruitment control. J. Appl. Prob. 12, 376–382.CrossRefGoogle Scholar
Davies, G. S. (1981) Maintainable regions in a Markov manpower model. J. Appl. Prob. 18, 738–742.CrossRefGoogle Scholar
Grinold, R. C. and Marshall, K. T. (1977) Manpower Planning Models. North-Holland, Amsterdam.Google Scholar
Grinold, R. C. and Stanford, R. E. (1974) Optimal control of a graded manpower system. Management Sci. 20, 1201–1215.CrossRefGoogle Scholar
Guerry, M.-A. (1991) Monotonic property of t-step maintainable structures in three-graded manpower systems: a counterexample. J. Appl. Prob. 28, 221–224.CrossRefGoogle Scholar
Haigh, J. (1983) Maintainability of manpower structures - counterexamples, results and conjectures. J. Appl. Prob. 20, 700–705. Correction (1986) J. Appl. Prob. 23, 849.CrossRefGoogle Scholar
Seneta, E. and Sheridan, S. (1981) Strong ergodicity of non-negative matrix products. Linear Algebra Appl. 37, 277–292.CrossRefGoogle Scholar
Stanford, R. E. (1982) The set of limiting distributions for a Markov chain with fuzzy transition probabilities. Fuzzy Sets and Systems 7, 71–78.CrossRefGoogle Scholar
Vajda, S. (1975) Mathematical aspects of manpower planning. Operat. Res. Quart. 26, 527–542.CrossRefGoogle Scholar
Vajda, S. (1978a) Mathematics of Manpower Planning. Wiley, New York.Google Scholar
Vajda, S. (1978b) Maintainability and preservation of a graded population structure. TIMS Studies in the Management Sciences 8, 219–230.Google Scholar
Vassiliou, P-C. G. (1984) Cyclic behaviour and asymptotic stability of non-homogeneous Markov systems. J. Appl. Prob. 21, 315–325.CrossRefGoogle Scholar
Vassiliou, P.-C. G. and Tsantas, N. (1984a) Stochastic control in non-homogenous Markov systems. Int. J. Com. Maths. 16, 129–155.Google Scholar
Vassiliou, P.-C. G. and Tsantas, N. (1984b) Maintainability of structures in nonhomogeneous Markov systems under cyclic behaviour and input control. SIAM J. Appl. Math. 44, 1014–1022.CrossRefGoogle Scholar
Vassiliou, P.-C. G. and Georgiou, A. C. (1990) Asymptotically attainable structures in nonhomogeneous Markov systems. Operat. Res. 38, 537–545.CrossRefGoogle Scholar
Young, A. and Almond, G. (1961) Predicting distributions of staff. Computer J. 3, 246–250.CrossRefGoogle Scholar