~; . , ~," I-'-- , it " ELSEVIER European Journal of Operational Research 100 (1997) 81-96 EUROPEAN JOURNAL OF OPERATIONAL RESEARCH Theory and Methodology Cost models in nonhomogeneous Markov systems A.C. Georgiou a,., P.-C.G. Vassiliou b a Department of Business Administration, University of Macedonia, 156 Egnatia, P.O. Box 1591, 54 006 Thessaloniki, Greece b Statistics and Operations Research Section, Department of Mathematics, Aristotle University of Thessaloniki, 54 006 Thessaloniki, Greece Received September 1993; revised December 1995 Abstract We study the problem of introducing the phases through which a nonhomogeneous Markov system (NHMS) passes, as time goes to infinity. For each phase an appropriate objective function is introduced and a cost problem is formulated. Appropriate input policies subject to the cost objective functions which are established on the Markov manpower system are investigated. Finally, a procedure is provided that summarises the phases, the objective functions and the control policies commencing from the initial structure resulting to its stationary form. © 1997 Published by Elsevier Science B.V. Keywords: Modelling; Manpower planning; Markov processes 1. Introduction The control of manpower systems and NHMS has been of considerable concern in recent times. In a series of articles and books beginning back in the early 70's, the problem of finding appropriate re- cruitment policies was considered, and various math- ematical models were developed according to several criteria and practical considerations. For examples see Bartholomew (1973, 1982), Vajda (1975, 1978), Grinold and Stanford (1974), Mehlmann (1980), Davies (1973, 1975) and Vassiliou and Tsantas (1984a,b). Another important aspect of the subject concerns the asymptotic behaviour of homogeneous or nonhomogeneous Markov systems (Bartholomew, * Corresponding author. 0377-2217/97/$17.00 © 1997 Published by Elsevier Science B.V. All SSDI 0377-22 17(96)00379-7 1982; Feichtinger, 1976; Feichtinger and Mehlmann, 1976; Vassiliou, 1981a,b, 1982). Vassiliou and Georgiou (1990) introduced a new aspect of the classical problem of providing condi- tions under which a limiting relative structure of a NHMS exists. The determination of these limits, as the time parameter tends to infinity, was the second main objective in that work. The expected structure of a hierarchical manpower system with k grades at any time point t was defined by a vector of k entries, denoting expected numbers of members. Then, the relative expected structure was employed in the analysis, through the division of the initial expected population vector by the total number of members in the system. This relative structure consti- tutes a stochastic vector. The article focused on the investigation of which relative structures are possible as limiting ones, provided that control was exercised fights reserved