Manpower Planning with Uncertain Requirements

This paper examines a longitudinal model of a manpower system in which the demands for effective manpower is determined by the state of a ! aiite Markov chain, and there are delays in training an adequate supply of effective manpower. We present an operational method of calculating optimal appointment policies. This calculation can in turn be used to find the equilibrium operating rules for the system. The model is a useful device for measuring the impact of alternate assumptions about continuation rates, manpower utilization policies, the demands, and the transition probabilities in the demand process. -Jijn(iijfiliüiiifc«li>j;ir.,ii.i";.i4-^ i,i-..J.^i-..i .i--(..;'i,-;"i-.JVi L'..r:;if.&\-jüi.'dilrtra*tiAh& ■^..l■-^^.■. :.;.:,.:;.! .-.:.,■ ■.."ilJ.;>,.ü;:'>.-':-V---.--'i t&&*mm B^SSBSpiWTSBSWB^^ 0. INTRODUCTION This paper considers the problem of providing an adequate supply of effective manpower when demand for manpower is uncertain and there are time lags in the training of effective personnel. The model described in this paper is useful in three specific ways: (i) it can calculate optimal manpower input policy for alternate control objectives, (ii) it can be used to measure the effects of alternate training, retention, aiJ utilization policies on system performance, and finally (iii) it can measure the sensitivity of optimal policies and system performance to various assumptions on the nature of the stochastic nature of the demand process. In Section '\ we present a model for the flow of personnel. This model attempts to capture the essential features of the manpower system in a simple framework by utilizing the longitudinal stability of various manpower cohorts. The stochastic nature of the demand process is described in Section 2, At time t , the demand for trained manpower is determined by the state of a finite Markov chain. In Section 3 we present two alternate control objectives. The first objective is to minimize the expected, weighted, discounted square of the error between the effective supply and demand for manpower. The second objective is to minimize the expected discounted square error between the actual manpower stocks and ideal stocks for any demand state. In Section 4 we describe the form of the optimal policy and optimal value function. Numerical solutions are presented for the two control objectives. In Section 5 we outline several uses for the model. The first is the determination of the expected manpower stocks when the optimal appointment policy is used. Calculatior of the equilibrium stocks can then be used to calculate the expected cost of the appointment policy along with some expected measure fsystem performance. These calculations allow us to measure the tradeoff between cost and performance for alternate manpower policies. In Section 5 we alsr . ... , „■ ■ .-■..*■..• ,v/.i.i'>■!—(.—^ ,^,:-,..!^,j1>^.-...-:iit --■..[,,-^I:u:;/'J-i'.;»^^v,^;ii.>>V.;^^ «.I^.f^.^v^-./.^^-i^iv. ^■' Wirtjit«: »A^H^wJ i.u-« jyAw* MwapjBPBBaBPpawiwgwwwswwp^gpinswwg!^ consider th" effect of errors in the specification of the stochastic law of motion. We find it is relatively easy to discover the effects of these errors. Finally in Section 5 we consider the effect of possible alterations in the stock of manpower, either by releasing people from the system or allowing personnel inflow that has a past accumulation of experience. The appendix is devoted to the theoretical aspect of the model we find that the optimization procedure Is a hybrid of the linear-qu idratic optimal control problem ([1],[4],[7]) and the Markov decision problem. We are able to show that the optimal decision rules are linear and that the optimal policy functions are quadratic. In addition, under special assumptions on "he problem data we can show that the optimal decision rules converge to an equilibrium optimal appointment policy. The model was motivated by the problem of regulating the supply of naval aviators. We do ,iot pretend to present an operational solution to that real problem, however, a numerical example, using the naval aviator problem as a setting, is used throughout the paper. The author is the sole source of the data for the numerical example. The paper is based on a longitudinal manpower flow model; Oliver and Hookins [5],[6], that has been used in a similar context by Grinold, Marshall, and Oliver, [2], and Grinold and Oliver, [3]. <*p!m!***^mm'r^iGmm BHBBBBWiWiBWPWBWWWBWBgpaTOllW^ 1. THE FLOW OF MANPOWER This section desci'ibes the basic manpower flow process. For a more detailed description of a related deterministic model see [2] and [3]. The state of the manpower system is observed at discrete equally spaced time points t = 0,1,2,3, ..., T ; where T is a planning horizon. The time interval (t l,t] after time t 1 and up to and including time t will be called period t . At each time t individuals in the system are classified according to their length of service (LOS), The length of service is simply the number of occasions the individual has been classified. Thus individuals entering in period t will have length of service equal to one at time t > since they are in their first period of service at time t . Individuals entering in period t 1 that are still present at timi t will be in their second period of service. We assume that m is the maximum number of periods an 4individual can remain in the system. It follows that individuals in the f-Vi system at time t in their m period of service must leave the system in period t + 1 . The stock of manpower at time t is described by the length of service distribution s(t) ; s(t) is ; vector with m components, where s. (t) is the number of individuals at time t with LOS equal to k . (1) s(t) = [s.co.s.ct), ..., s (or i Z m In each period t a number of new acaessions f(t) enter the manpower system. The change in the manpower system from time t to time t + 1 depends on three factors: the stock at time t , the accessions in period t + 1 , and the continuation rates of the stock, of manpower at time t . The continuation rate q, for k = 1,2, ..., m 1 is defined to be the fraction Caution: In [2] and [2] the length of service is defined to be the number of completed periods of service, and is then one less than the measure used here.