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Hidden heterogeneity in manpower systems: A Markov-switching model approach
European Journal of Operational Research 210 (2011) 106–113
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Hidden heterogeneity in manpower systems: A Markov-switching model approach Marie-Anne Guerry ⇑ Department for Mathematics, Operational Research, Statistics and Information Systems for Management (MOSI), Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussel, Belgium
a r t i c l e
i n f o
Article history: Received 24 February 2009 Accepted 21 October 2010 Available online 5 November 2010 Keywords: Manpower planning Unobserved heterogeneity Mover-stayer Markov-switching model
a b s t r a c t In modeling manpower systems, it is of crucial importance to deal with heterogeneity. Until recently, manpower models are dealing with heterogeneity due to observable sources, neglecting heterogeneity due to latent sources. In this paper a two-step procedure is introduced. In the first step personnel groups homogeneous with respect to the transition probabilities are determined in a classical way by taking into account the observable sources of heterogeneity. In the second step heterogeneity caused by latent sources is handled. A multinomial Markov-switching manpower model is introduced that deals with heterogeneity due to latent sources for the internal flows as well as for the wastage flows. The model incorporates the mover-stayer principle. A re-estimation algorithm is presented to estimate the parameters of the Markov-switching manpower model. The switching approach offers a methodology to build a Markov model with personnel groups as states that are more homogeneous, and therefore can contribute to a better validity of the manpower model. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Manpower planning is an aspect of Human Resources Management, based on mathematical modeling. Modeling a manpower system concerns three types of flows, namely the recruitment flows, the internal personnel flows between the different personnel categories (among others promotion flows), and the wastage. The aggregated models used in manpower planning are based on the concept of homogeneous personnel groups, where the validity of such a model is determined in a large extend by the degree of homogeneity of the groups. It is therefore of crucial importance to deal with heterogeneity in this context. Ugwuowo and McClean are the authors of a good review article on this topic (Ugwuowo and McClean, 2000). In previous work attention is paid to algorithms to detect the personnel groups within a manpower system that are homogeneous with respect to the transition probabilities (as there are promotion probabilities). So far those algorithms only take into account the information that is available from the personnel database on the observable variables (as there are grade, seniority, gender and full time equivalent) to determine the homogeneous personnel groups (De Feyter, 2006; Guerry, 2008). Nevertheless in a manpower system, apart from the observable sources of heterogeneity, there is heterogeneity due to latent sources (Ugwuowo and McClean, 2000). In several concrete career studies there is pointed out the role of unobserved heterogeneity: for example in Verbakel and de Graaf (2008) unobserved heterogeneity is mentioned as possible reason for the male marriage ⇑ Tel.: +32 2 629 2049; fax: +32 2 629 2054. E-mail address: [email protected] 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.10.039
premium, next to higher productivity, employer favoritism and selection. These applied research studies give confirmation of the need of building a manpower model in which there is taken into account the observable as well as the hidden heterogeneity. In this paper, in tackling the problem of heterogeneity in a manpower system, the analysis is not restricted to observable variables but observable as well as latent sources of heterogeneity are taken into account. For discrete-time models that are time-homogeneous there is introduced a method in which there is dealt with both types of heterogeneity in determining homogeneous personnel subgroups. The method will be formulated as a two-step procedure (Section 3.1). In the first step of the procedure personnel groups, homogeneous with respect to transition probabilities, are determined by taking into account the available data on the observable sources (De Feyter, 2006; Guerry, 2008). In the second step heterogeneity caused by latent sources will be handled. A Markovswitching model will be introduced to model the mover-stayer principle for the promotion flows (and for any internal personnel transitions, in general) and the heterogeneity in the wastage flows due to unobservable sources. An algorithm is presented to get estimates for the parameters of the model based on the Baum–Welch EM-algorithm (Section 3.2). This way of taking into account heterogeneity due to latent sources will result in a supplementary subdivision of each of the personnel groups in movers and stayers. The approach therefore gives rise to personnel subgroups that have a higher degree of homogeneity without requiring additional or more detailed observations. A finer subdivision of the personnel of a manpower system in more homogeneous subgroups may improve the quality of the model. However, one has to be aware of the fact that a finer
M.-A. Guerry / European Journal of Operational Research 210 (2011) 106–113
subdivision in personnel subgroups results in a greater number of parameters to be estimated and therefore the gain in terms of the goodness of fit is not obvious (Bartholomew et al., 1991). A manpower model should have states that are personnel subgroups for which there is a good compromise between the level of the subdivision of the personnel system and the quality of the estimations for the parameters. Testing and comparing the goodness of fit of models built on personnel groups with different levels of subdivisions (finer and less fine) will give an answer to the question which subdivision of the personnel results in a model with a high validity. In Section 5 an illustration is presented to demonstrate the hidden Markov approach and its different aspects (among others parameter estimation and interpretation, and validation of the model) for a concrete manpower system. Facing the problem of heterogeneity within a personnel system, the two-step procedure presented in this paper offers the opportunity to end up with more homogeneous personnel subgroups and to improve the quality of the manpower model. The values of the parameters of the model provide in a description of characteristics of the push transitions, for the internal flows as well as for the wastage flows. Furthermore, based on the model extrapolations for the number of members in each of the personnel groups can be found under the hypothesis that the personnel strategy remains unchanged for the future. And what-if-analysis can be conducted in comparing several strategies.
2. Heterogeneity in manpower systems Modeling personnel systems based on Markov chains is a well-known approach (Bartholomew et al., 1991). The states are homogeneous groups of personnel for which the promotion (or in general the internal transition) and wastage probabilities are assumed to be equal for each of the individuals within a group. Nevertheless some flows depend on individual traits even within such a homogeneous group; and if there is a lack of observations on these sources of heterogeneity, parameter estimation is not possible for such subgroups in a classical Markov model. Earlier work (Ugwuowo and McClean, 2000) concerning manpower models points to the importance of making a distinction between two types of sources of heterogeneity, namely observable sources and latent sources. For the observable sources (among others gender, full time equivalent and seniority) information is available in the personnel dataset for each of the employees. The available data give the opportunity to distinguish homogeneous personnel groups based on the corresponding (observable) variables, resulting for example in a category of female employees who are working part-time and having a length of service between 5 and 10 years. The fact is that within each of these categories, in general, there is still heterogeneity without knowing its nature. These latent sources can be of individual or environmental type. This paper discusses the problem of how to deal with heterogeneity due to latent sources for which there are no observations available. This approach is of interest in manpower planning since for the wastage flows as well as for the promotion flows possibly there is heterogeneity due to latent sources (Bartholomew et al., 1991; Ugwuowo and McClean, 2000). Among others, Bartholomew et al. (1991) pointed out the complexity of modeling wastage. The terminology ‘wastage’ refers to the total loss of individuals from the manpower system for whatever reason. From the point of view of the employee, wastage refers to voluntary wastage (for example because of a more attractive job in another company) as well as involuntary wastage (for example because of forced retirement). From the point of view of management, wastage can be controlled (for example in the case where an employee is fired) as well as uncontrolled (for example in
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the case where an employee dies). In Agrafiotis (1984), besides controlled wastage, reasons for leaving are classified as being reasons depending on the internal structure of the company (e.g. lack of promotion prospects) and as being reasons not depending on the internal structure of the company (e.g. retirement and death). All these classifications make it clear that in modeling wastage, one has to deal with different underlying reasons for leaving. Furthermore, most of the time wastage is not the result of one but rather the result of a combination of reasons, and consequently a strict classification of the reason for leaving seems inappropriate. Often there is the supplementary difficulty that no observations are available for all these types of wastage. As pointed out for wastage flows, within a homogeneous personnel group determined by observable variables, for promotion rates heterogeneity due to latent sources is possible. Without knowing the real nature of the heterogeneity, it can be that one subgroup experiences fewer promotions in comparison with another subgroup that has more promotions. Those subgroups can be related to the mover-stayer principle. The concept of stayers and movers is introduced and studied in earlier work (Blumen et al., 1955; Goodman, 1961). Discrete-time mover-stayer models have been extensively used in studies of heterogeneous populations (among others in: Rossi and Schinaia, 1998; Major, 2008). And the intrinsic meaning of the concepts movers and stayers is not the same in all these models: the stayers can refer either to a subgroup of units that are moving infrequently or, in the simplest form of the model, to a subgroup of units that do not move. In Bartholomew et al. (1991) the idea of incorporating the mover-stayer principle in manpower planning models was introduced. In this paper, the problem of heterogeneity for the promotion flows in a manpower system due to latent sources will be formulated in terms of stayers and movers: movers are characterized by the fact that they experience faster career growth, reflected by higher promotion probabilities. Stayers change their grade less frequently or not at all. For both types of flows, wastage flows as well as promotion flows, latent sources of heterogeneity may be viewed as determining the rates. Since there are no observations available to quantify the impact of these sources on the flow rates, a (classical) Markov manpower model is not appropriate. In this paper, a Markov-switching model as a modeling tool is introduced in order to deal with heterogeneity due to latent sources in manpower systems.
3. Markov-switching manpower models Hidden Markov chains are as modeling technique widespread in the engineer sciences, among others in speech and form recognition (Aas et al., 1999; Levinson et al., 1983). More recently hidden Markov chains are increasingly used in econometrics (Gregoir and Lenglart, 2000) and in other domains. In manpower planning, Markov models are established and this in contrast with hidden Markov models. In this paper a hidden Markov model is introduced that takes into account the specificity of a manpower system and the fact that there are observable as well as latent sources of heterogeneity in such a system. A hidden Markov chain is generated by two probability mechanisms. Namely firstly, an unobserved Markov chain {Xt} that is hidden in the observations. And secondly, a set of probability distributions, one for each state of the Markov chain that produces the observations {Yt}. Markov-switching models are generalizations of hidden Markov chains in which the distribution of the observed variable Yt is not only depending on Xt but also on Yt1, Yt2, Yt3, . . . (Cappé et al., 2005). An application based on Markov-switching models is for
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example the model of Hamilton for an economic analysis of the business cycle (Hamilton, 1989). A Markov-switching model offers great flexibility in approximating complex probability distributions and provides the possibility to consider different probability distributions for each of the states of the underlying Markov chain. Furthermore the Markov-switching model approach allows one to estimate the parameters of the unobserved Markov chain for which there are no observations required. For these reasons Markov-switching models are useful to model a manpower system for which no observable variables are available or adequate to bring insights in the mover-stayer principle. For such a manpower system a classical Markov model is not able to clarify the fact that for a personnel category some employees are movers and do have higher promotion probabilities than others. In what follows a Markov-switching model is introduced for manpower systems with heterogeneity caused by latent sources and with lack of data. The characteristics and modeling strengths of Markov-switching models are a powerful means to model phenomena as wastage and promotion flows. This approach can improve the validity of the manpower model since it is based on personnel groups that are more homogeneous. 3.1. Two-step procedure In order to deal with heterogeneity caused by observable sources as well as with heterogeneity caused by latent sources a two-step procedure is introduced. In the first step of the procedure personnel groups H1, . . . , Hk homogeneous, with respect to transition probabilities in between the groups, will be determined by taking into account the available data on the observable variables. Those personnel groups H1, . . . , Hk can be found as the result of a multinomial logistic regression analysis (De Feyter, 2006) or by a profile based approach (Guerry, 2008). A person of the manpower system at time t is a member of one of these personnel groups H1, . . . , Hk. This phenomenon is described by the process {Yt} with states H1, . . . , Hk and transition probabilities pij = Pr(Yt+1 = HjjYt = Hi) (1 6 i, j 6 k). For a time-homogeneous Markov chain, the maximum likeli^ij can be computed based on the observed flows hood estimator p ntij , being the number of persons that are at time t in state Hi and P t at time t + 1 in state Hj, and the stocks sti ¼ kþ1 j¼1 nij , i.e. the number of persons in state Hi at time t (Bartholomew et al., 1991, p. 113) as
P t t nij ^ pij ¼ P t : t si
ð3:1Þ
The additional state Hk+1 corresponds with the leaving state and the flow nti;kþ1 refers to the number of members of personnel group Hi at time t that has left the manpower system at time t + 1. In a similar way to the internal transition probabilities, for each group Hi (1 6 i 6 k) the wastage probability pi,k+1 = Pr(Yt+1 = Hk+1jYt = Hi) can be estimated by
P ^i;kþ1 ¼ p
t t ni;kþ1 P : t t si
The leaving state can be considered as an absorbing state: pk+1,k+1 = 1. In the second step of the procedure, a Markov-switching model is introduced to deal with the heterogeneity due to latent sources within each of the personnel groups H1, . . . , Hk. This stage results in a supplementary subdivision of each of the personnel groups Hi (i = 1, . . . , k) into the subgroup of movers C i1 and the subgroup of stayers C i2 , being the states of the unobserved n o Markov chain X it as part of the Markov-switching model. The
subgroup of movers C i1 is characterized by a greater value for the transition probabilities from Hi to Hj (j – i) in comparison with the subgroup of stayers C i2 . Concerning the personnel group Hi, the unobserved Markov chain is characterized by the with Cilm ¼ Pr X itþ1 ¼ (2 2) probability matrix Ci ¼ Cilm C im X it ¼ C il Þ for l, m = 1, 2. The matrix Ci is the matrix of the transition probabilities for the states of movers and stayers within the personnel group Hi. The transition probability from the state C il Hi ðl ¼ 1; 2Þ into Hj will be denoted by pilj ¼ Pr Y tþ1 ¼ Hj X it ¼ C il ¼ PrðY tþ1 ¼ Hj Y t ¼ Hi ; X it ¼ C il Þ. For the personnel group Hi, the tth observation (t = 1, . . . , T) concerns the observed flows for the time period [t, t + 1], i.e. data on the number ntij of people of Hi promoted to each of the personnel groups Hj (j = 1, . . . , k) or moved to the wastage status Hk+1(j = k + 1) in time period [t, t + 1]. These observed flows will be gathered into ti ¼ nti1 ; . . . ; ntik ; nti;kþ1 . the flow vector n Let us denote by Z it the random vector with jth component (j = 1, . . . , k + 1) referring to the number of persons moved in time period [t, t + 1] from group Hi to group Hj. Since the outcome of each trial is into one of these k + 1 categories H1, . . . , Hk+1, the model results in a multinomial Markov-switching model: Conditional on X it ¼ C il , the flow vector Z it is multinomial distributed with paramPkþ1 t eters sti ¼ j¼1 nij (being the stock of the personnel group Hi at time Pkþ1 i t), and probabilities pil1 ; . . . ; pil;kþ1 satisfying j¼1 pl;j ¼ 1. For the time period [t, t + 1], the probability that the outcome for n o ti ¼ nti1 ; . . . ; Z it equals the flow vector n
the observed process
nt ; nt Þ, conditional to the fact that the unobserved Markov chain nik oi;kþ1 X it is in state C il , can be expressed by the state-to-observation ti X it ¼ C il : probability Pil;nt ¼ Pr Z it ¼ n i
Pil;nt ¼ i
sti t ni1 ; . . . ; nti;kþ1
pil1
nti1
nt i;kþ1 . . . pil;kþ1
¼ 1; 2 and t ¼ 1; . . . ; T:
for l ð3:2Þ
In what follows the notation Pi ¼ Pilt refers to the (2 T)-matrix with elements Pilt ¼ Pil;nt (for l = 1, 2 and t = 1, . . . , T). The i (2 (k + 1))-probability matrix with elements pilj (for l = 1, 2 and j = 1, . . . , k + 1) will be denoted by Pi. The flows that are considered in the Markov-switching model to deal with the heterogeneity within the personnel group Hi are schematically represented in Fig. 3.1. Concerning the outgoing flows from personnel group Hi, the graphical representation is restricted to the flows into the group Hj (0 6 j 6 k + 1) (dotted arcs). Since for each personnel group Hi (i = 1, . . . , k) a Markov chain with transition matrix Ci is considered the obtained model is, according to the terminology used in Hagenaars and McCutcheon (2002), a latent mixed Markov model for several groups. 3.2. Estimating the parameters of the markov-switching manpower model In what follows a re-estimation algorithm is presented resulting in estimations for the parameters Ci, Pi and Pi of the Markovswitching manpower model. This algorithm is inspired by the Baum–Welch EM-algorithm (see e.g. MacDonald and Zucchini, 1997) and is taking into account the specificity of the manpower system. In this context the EM-algorithm will be formulated in terms of the characteristics of the multinomial manpower model.
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Fig. 3.1. Flows in the Markov-switching model.
The parameters of a Markov-switching model can be estimated given aggregated data for each type of flow. For the personnel group Hi it concerns observations, for some time periods, on the number of members being promoted to group Hj (j = 1, . . .k) and the number of members having left the system (transition to Hk+1). These observed flows are gathered into the vectors ti ¼ nti1 ; . . . ; ntik ; nti;kþ1 for t = 1, . . . , T. There is no information ren quired separately for the subgroups of and stayers of Hi. n movers o For the unobserved Markov chain X it , the probability that the process is in state C il at time t, conditional on the observations 1i ; . . . ; n Ti , is denoted by n
1i ; . . . ; Z iT ¼ n Ti : uil ðtÞ ¼ Pr X it ¼ C il Z i1 ¼ n
The initial distribution over the states is then
C i1
(movers) and
(stayers)
Further, the conditional probability
1i ; . . . ; Z iT ¼ n Ti X it1 ¼ C il ; X it ¼ C im Z i1 ¼ n
v il;m ðtÞ ¼ Pr
is introduced. The re-estimation formulas for the parameters Ci, Pi and Pi of the Markov-switching model are expressed in terms of the forward 1i ; . . . ; Z it ¼ n ti ; X it ¼ C il satisfying probabilities ail;t ¼ Pr Z i1 ¼ n
ail;t ¼ dil :Pil;n1 ¼ ul ð1Þ:Pil;n1 for l ¼ 1; 2 and t ¼ 1; ¼
i
i
i m;t1 :
a
i ml :
i t l;n i
C P
m
for l ¼ 1; 2 and t from 2 to T
ð3:3Þ
i i tþ1 and ; . . . ; Z iT ¼ the backward probabilities bl;t ¼ Pr Z tþ1 ¼ n i i i Ti X t ¼ C l Þ for which the following holds n
bil;t ¼ 1 for l ¼ 1; 2 and t ¼ T; X i Pm;ntþ1 :bim;tþ1 :Cilm for l ¼ 1; 2 and for t from T 1 to 1: ¼ m
1i ; . . . ; Z iT ¼ n Ti X it1 ¼ C il ; X it ¼ C im Z i1 ¼ n
ail;t1 :Pim;nti :bim;t :Cil;m
¼
P
l
i
ð3:4Þ By definition of the conditional probabilities uil ðtÞ and lows for l = 1, 2 and for t from 1 to T that
ai :bi 1i ; . . . ; Z iT ¼ n Ti ¼ P l;t l;t i uil ðtÞ ¼ Pr X it ¼ C il Z i1 ¼ n i l al;t :bl;t
v il;m ðtÞ, it foland
ð3:5Þ
ail;t :bil;t
ð3:6Þ
:
Concerning the flows from Hi (1 6 i6 k) and the T observations 1i ; n 2i ; . . . ; n Ti , the likelihood is LiT ¼ Pr Z i1 ¼ n 1i ; . . . ; Z iT ¼ n Ti . n Since there are no observations concerning the Markov chain with states C i1 and C i2 , in the Expectation-step for the conditional probabilities uil ðtÞ and v il;m ðtÞ expectations are considered, based on the current parameter estimates for Ci and Pi and the formulas (3.5) and (3.6), to express the log-likelihood as T X 2 X 2 X
C i2
di ¼ ui1 ð1Þ; ui2 ð1Þ :
X
v il;m ðtÞ ¼ Pr
t¼2
log Cil;m :v il;m ðtÞ þ
l¼1 m¼1
T X 2 X t¼1
log Pil;nt :uil ðtÞ:
l¼1
i
In the Maximization-step, the first part of this expression is maximized with respect to the parameters Cil;m ðl; m ¼ 1; 2Þ under the constraints Cil;1 þ Cil;2 ¼ 1 ðl ¼ 1; 2Þ; and the second part is maximized with respect to the parameters Pil;nt ðl ¼ 1; 2Þ under the coni ditions pil;1 þ þ pil;kþ1 ¼ 1 ðl ¼ 1; 2Þ. It can be verified that the optimization method of the Lagrange multipliers results in the following expressions
PT
vi
l;m Cil;m ¼ P2 t¼2 PT j¼1
ðtÞ
i t¼2 v l;j ðtÞ
for l; m ¼ 1; 2;
ð3:7Þ
PT
i t t¼1 ul ðtÞ:nij
pil;j ¼ PT
i t t¼1 ul ðtÞ:si
for l ¼ 1; 2 and j ¼ 1; . . . ; k þ 1:
ð3:8Þ
All these properties give rise to the re-estimation algorithm presented in Fig. 3.2 that results in estimations for the parameters of the multinomial Markov-switching manpower model, given the 1i ; n 2i ; . . . ; n Ti and for chosen starting values for Cil;m ; dil observations n i and pl;j (resulting in starting values for Pil;nt for t = 1, . . . , T by Fori mula (3.2) for l, m = 1, 2 and j = 1, . . . , k+1. At each iteration, the forward and backward probabilities are computed based on the observations and the estimations for Ci, Pi and diso far. This outcome for the forward and backward probabilities is used to compute a re-estimation for Ci, Pi and di. After each iteration, the likelihood is improved except at a critical point of the likelihood (MacDonald and Zucchini, 1997, p. 63). The likelihood will be (locally or globally) maximized by the presented iterative procedure.
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M.-A. Guerry / European Journal of Operational Research 210 (2011) 106–113
Fig. 3.2. Re-estimation algorithm.
By using multiple sets of starting values, the problem of local extreme points may be avoided (Hagenaars and McCutcheon, 2002, p. 337). 4. Forecasting and control Based on the Markov-switching manpower model presented and the estimations for the parameters of the model, a wastage analysis and an evaluation of the impact of alternative promotion and recruitment strategies can be made. In manpower planning a wastage analysis is crucial. Wastage creates vacancies and so provides opportunities for promotion and recruitment, giving rise to corresponding recruitment costs. It is therefore of great importance to detect bottlenecks with respect to wastage and to obtain knowledge of the characterization of those groups of personnel having for example a rather high wastage probability. The answer to these questions will be given by the evaluation of the values for the parameters pi1;kþ1 and pi2;kþ1 ði ¼ 1; . . . ; kÞ. Besides, in manpower planning modeling wastage gives the opportunity to have, based on extrapolations, information on the outflow of the system in the future. Taking into account this information, control actions can be planned. The interpretation and labeling of the latent classes is straight forward in the context of a mover-stayer analysis: For each personnel category Hi, the group of stayers will be characterized by a greater value for the parameter pi2;i in comparison with the value for pi1;i (characterizing the group of movers). A pair wise comparison of the values of the parameters pi1;j and pi2;j ðj ¼ 1; . . . ; kÞ, gives information onto what extend there is in the company heterogeneity due to latent sources concerning the promotion from the personnel group Hi. The difference in value of pi1;j and pi2;j gives an indication for the degree of latent class separation, whereby a high degree of latent class separation implies a high degree of homogeneity for the latent classes (Collins and Lanza, 2010). The manpower model presented is based on a two-step procedure in which in the first stage the approach deals with heterogeneity due to observable sources (and results in personnel groups H1, . . . , Hk). In the second stage the Markov-switching manpower model in addition deals with heterogeneity due to latent sources (resulting in a supplementary subdivision of Hi in C i1 and C i2 ). The goal is now to get a forecast for the stock hi(t), i.e. the number of members of the personnel group Hi at time t, by taking into account the supplementary insight according to movers and stayers from the Markov-switching approach. By the parameter estimations for Ci and di (i = 1, . . . , k) in the Markov-switching model, the evolution of the distribution ui ðtÞ ¼ ui1 ðtÞ; ui2 ðtÞ of the members of the personnel group Hi over the subgroups of movers and stayers can be forecasted by
ui ðtÞ ¼ ui ðt 1Þ:Ci
where ui ð1Þ ¼ di :
Let us denote c(t) the stockvector over the 2k subgroups, as a result of considering for each personnel group Hi (i = 1, . . . , k) a subdivision of the hi(t) members in movers C i1 and stayers C i2 ; the components of c(t) can be computed as:
c2i2þl ðtÞ ¼ uil ðtÞ:hi ðtÞ for i ¼ 1; . . . ; k and l ¼ 1; 2:
ð4:1Þ
The flow from the personnel category Hi (i = 1, . . . , k) to the personnel category Hj (j = 1, . . . , k) can be calculated based on the characterisation of the hidden states as follows:
c2i1 ðtÞ:pi1;j þ c2i ðtÞ:pi2;j :
ð4:2Þ
The evolution of the stockvector h(t) over the personnel categories Hi (i = 1, . . . , k) can be expressed in terms of the (2k k)-matrix P with elements p2i2þl;j ¼ pil;j (i, j = 1, . . . , k and l = 1, 2):
hðt þ 1Þ ¼ cðtÞ:P þ RðtÞ:
ð4:3Þ
The vector R(t) stands for the recruitment vector in which Ri (t) refers to the number of new recruits in category Hi in time period t. As in the classical manpower models, the vectors R(t) depend on and reflect the recruitment policy. In an iterative way, based on the forecast h(t + 1), the components of the stockvector c(t + 1) can be found by formula (4.1), giving result to a forecast for the stockvector h(t + 2) over the personnel categories Hi (i = 1, . . . , k) by formula (4.3). By what-if analyses, the impact of alternative promotion and recruitment strategies on the evolution of the stockvector h(t) can be examined. 5. Illustration The goal of this illustration is to concretize the way in which a particular manpower system can be analyzed on the one hand only based on information of observable variables available from the personnel dataset and on the other hand by taking into account observable as well as latent sources of heterogeneity. These two approaches, of determining homogeneous personnel subgroups for the manpower system under study, result on the one hand in a (classical) Markov model and in the other hand in a Markov switching model. Moreover based on the illustration there is pointed out in concrete terms in what way the estimated parameters for both types of model can be interpreted and in what way the goodness of fit can be compared for the two obtained manpower models. For the organization under study a personnel dataset is available with information for each employee over the past nine years on his/her grade, monthly number of working hours, age, seniority, number of children, etc. Internal flows and wastage flows (as in Table 5.1) as well as incoming flows (as in Table 5.2) can be quantified based on this personnel dataset.
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M.-A. Guerry / European Journal of Operational Research 210 (2011) 106–113 Table 5.1 Observed flows for the subsystem of female employees. H1
H2
H3
H4
H9
t = 1: t = 2: t = 3: t = 4: t = 5: t = 6: t = 7: t = 8: t = 9:
H1
15 17 23 29 36 43 50 56 64
1 1 1 1 1 1 1 1 1
15 12 4 6 3 3 2 2 1
1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1
t = 1: t = 2: t = 3: t = 4: t = 5: t = 6: t = 7: t = 8: t = 9:
H2
1 1 1 1 1 1 1 1 1
21 31 41 50 58 67 74 82 89
1 1 1 1 1 1 1 1 1
1 1 1 2 2 2 2 3 3
1 1 1 1 1 1 1 1 1
t = 1: t = 2: t = 3: t = 4: t = 5: t = 6: t = 7: t = 8: t = 9:
H3
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
21 22 13 11 8 7 6 6 6
21 39 37 43 44 50 53 57 61
1 1 1 1 1 1 1 1 1
t = 1: t = 2: t = 3: t = 4: t = 5: t = 6: t = 7: t = 8: t = 9:
H4
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
20 16 31 32 40 43 48 52 56
24 34 48 59 68 75 83 90 97
1 1 2 2 2 2 3 3 3
Table 5.2 Observed incoming flows for the subsystem of female employees.
R(2) R(3) R(4) R(5) R(6) R(7) R(8) R(9)
H3, H4 (concerning female employees) on the one hand and the groups H5, H6, H7, H8 (concerning male employees) on the other hand form two subsystems for which in between there are no transitions. In what follows the Markov-switching approach is discussed for the subsystem of the females (the discussion is similar for the subsystem of the male employees). Table 4.1 provides information, for each time period [t, t + 1], on the observed number ntij of women of group Hi (i = 1, 2, 3, 4) that has transferred to group Hj (j = 1, 2, 3, 4, 9) within the considered time period. For the female employees, as a result of Formula (3.1), the Markov model with states H1, H2, H3, H4, H9 is characterized by the transition matrix PM with elements (PM)i,j = pi,j, the transition probability from state Hi to state Hj:
0
0:80059 0:02381 0:12798 0:02381 0:02381
C B B 0:01732 0:91775 0:01732 0:03030 0:01732 C C B B PM ¼ B 0 0 0:21076 0:77130 0:01794 C C: C B 0 0 0:36200 0:61746 0:02054 A @ 0 0 0 0 1 Based on the re-estimation algorithm presented in Fig. 3.2, the parameters of the multinomial Markov-switching manpower model are estimated, resulting in the matrix PMS with elements ðP MS Þ2i2þl;j ¼ pil;j , the transition probability from the subgroup of movers/stayers of Hi to state Hj:
0
PMS
0:49363 B 0:88802 B B B 0:03221 B B 0:01439 B B B 0 ¼B B 0 B B B 0 B B 0 B B @ 0 0
H1
H2
H3
H4
16 12 14 12 12 11 10 11
13 13 13 12 13 11 13 12
5 0 6 3 6 6 7 7
4 6 6 5 5 6 6 5
In the considered manpower system, two grades (gr 1 and gr 2) are distinguished and no demotions occur. A study as in De Feyter (2006) or Guerry (2008) of the personnel dataset let us conclude that significant differences in transition probabilities are imposed by the observed variables gender and full time equivalent (FTE). More specifically, the outcomes female (f), male (m), FTE less than or equal to 0.5 (FTE 6 0.5) and FTE greater than 0.5 (FTE > 0.5) determine 8 personnel groups Hi:
H1 : gr1; f ; FTE 6 :5 H2 : gr1; f ; FTE > :5 H3 : gr2; f ; FTE 6 :5 H4 : gr2; f ; FTE > :5; H5 : gr1; m; FTE 6 :5 H6 : gr1; m; FTE > :5 H7 : gr2; m; FTE 6 :5 H8 : gr2; m; FTE > :5; that are homogeneous with respect to the transition probabilities. The group H9 refers to the leaving state. Concerning the flows between these groups Hi, there are 9 observations (t = 1, . . . , t = 9) available in time. The groups H1, H2,
1
1
0:03078
0:41402 0:03078
0:03078
0:02064
0:05007
0:02064 C C C 0:03221 C C 0:01439 C C C 0:01683 C C: 0:01919 C C C 0:02222 C C 0:02022 C C C A 1
0:02064
0:87024 0:03221 0:03312 0:92660 0:01439 0:03023 0
0:12600
0
0:36323 0:61759
0
0:44444 0:53333
0
0:35730 0:62247
0 0
0 0
0:85717
0 0
1
The element (PM)1,3 = p1,3 of the transition matrix of the Markov model, for example, stands for the probability that a female employee from grade 1 and working part-time (FTE 6 0.5) is after one period of time still working part-time (FTE 6 0.5) but being promoted into grade 2. For that same internal transition, the corresponding elements of the transition matrix PMS of the Markov switching model, ðPMS Þ1;3 ¼ p11;3 and ðPMS Þ2;3 ¼ p12;3 , give the transition probability, respectively for the female movers and the female stayers. The transition probability (PM)1,3 = p1,3 is estimated based on available data for the observable variables grade and FTE, without taking into account the fact there is heterogeneity due to latent factors. The transition probabilities ðPMS Þ1;3 ¼ p11;3 and ðPMS Þ2;3 ¼ p12;3 give more detailed information, namely for the movers and stayers respectively, and result from the parameter estimation procedure for the switching manpower model (Section 3.2). For a woman of the state H1, the Markov model results in an estimation for the transition probability to the state H3 equal to 0.12798. Since for the latent classes within H1 hold that p11;3 > p12;3 , the classes C 11 and C 12 are labeled as respectively the movers and stayers of H1. In the Markov-switching model, for the movers within the group H1 the transition probability to H3 is estimated to be equal to 0.41402. And for the stayers within the group H1 this transition probability is estimated to be equal to 0.05007. The two-step procedure therefore gives rise to a supplementary subdivision of the group H1 of the female part-time working with FTE 6 0.5 of grade 1, resulting in a subgroup of movers (with a promotion probability to grade 2 equal to 0.41402) and a subgroup of stayers (with a promotion probability to grade 2 equal to 0.05007).
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1
for members of H 1 for movers of H 1 for stayers of H 1
probability
0.8
significantly better than the Markov model in fitting the personnel data. The fact that the Markov-switching model is based on more homogeneous personnel groups, by taking in addition latent variables into account, results in a model with a better validity in comparison with the Markov model that is only based on observed variables.
0.6
6. Conclusions 0.4
0.2
0
staying in H 1
promoting from H 1 to H 3
Fig. 5.1. Transition probabilities conditional on latent class membership.
Since, for example, H1 can be partitioned in two subgroups for which the transition probability to H3 equals respectively 0.41402 and 0.05007, the states H1, H2, H3, H4, H9 of the classical Markov model are not really homogenous with respect to the transition probabilities. Remark that the Markov model gives an estimation for the transition probability for the members of group H1 in global (without making the distinction between movers and stayers) to H3 equal to 0.12798. In this way and as could be expected, concerning the promotions from H1 to H3, the transition probability conditional on latent class membership is greater for the movers than for the stayers; and the (unconditional) transition probability in the classical Markov model is a value in between the conditional probability for the stayers and the conditional probability for the movers. (see Fig. 5.1) For the personnel categories H1, H2, H3, H4 the number of recruitments can be computed from the information on the internal and wastage flows provided in Table 5.1: For 2 6 t 6 9 and for P each personnel group Hg g 2 {1, 2, 3, 4} holds that 4j¼1 ntgj þ ntg9 ¼ stg , P4 t1 t the stock of Hg at time t, and i¼1 nig ¼ sg Rg ðtÞ, the number of members of Hg at time t who were already in the manpower system at time t 1. Therefore the number of members recruited in Hg at time t can be computed from the information on the flows P P in Table 5.1 as Rg ðtÞ ¼ 4j¼1 ntgj þ ntg9 4i¼1 nt1 ig . These results for the incoming flows, characterized by the recruitment vectors R(t) are gathered in Table 5.2. In both the Markov and the Markovswitching manpower model these recruitment flows are considered as deterministic and known. In order to measure the fit of the Markov-switching manpower model and to compare the goodness of fit for the Markov-switching model and the Markov model, a 3-fold cross validation is considered. The model was trained on the observations at t = 1, 2, 3, 4, 5, 6, parameters were estimated (according to the procedure of Section 3.2) and predictions on the flows were made for t = 7, 8, 9 (according to the approach in Section 3). This way of testing the models results for the Markov model in v2 = 83.18246 and for the Markov-switching model in v2 = 24.2473. In validating the models, the number of degrees of freedom equals 3k2 = 64. Since v2:05 ð64Þ ¼ 83:67526 the Markov model as well as the Markov-switching model are acceptable models. Let us denote the likelihood for the Markov-switching model by Q Q 1i ; . . . ; Z iT ¼ n Ti and in an analogous LðMSÞ ¼ ki¼1 LiT ¼ ki¼1 Pr Z i1 ¼ n LðMÞ way the likelihood for the Markov model by L(M). Since 2 log LðMSÞ
is chi-square distributed with k2 + 3k = 28 degrees of freedom and LðMÞ ¼ 41; 4 > v2:05 ð28Þ, the Markov-switching model is since 2 log LðMSÞ
A Markov-switching manpower model is introduced to face with the problem of hidden heterogeneity in the wastage flows as well as in the internal transitions (as there are promotions) within a manpower system. In comparison with the (classical) Markov manpower models, the Markov-switching approach results in personnel categories that are more homogeneous by taking into account heterogeneity due to latent sources. There is presented an algorithm that results in estimations for the transition probabilities for the movers as well as for the stayers of each personnel group without requiring additional data on the flows for those subgroups. In practice, based on a comparison of the goodness of fit for the Markov model and the Markov-switching model, one can decide which type of model is the most appropriate for the manpower system under study. In this way the procedure results in a manpower model that characterizes the internal and wastage flows of the personnel in a good way. The manpower model is useful in predicting for the future the number of members in each of the personnel categories under the assumption that the transition probabilities are constant in time. More in general based on the model, what-if-analyses can be conducted in comparing and evaluating alternative personnel strategies; and a decision can be made on what personnel strategy results in a preferable evolution for the number of members in each of the personnel categories. Acknowledgements The author thanks the reviewers for their remarks and valuable suggestions. This research is partially funded by the research project FWOAL453 of the Research Foundation Flanders. References Aas, K., Eikvil, L., Huseby, R.B., 1999. Applications of hidden Markov chains in image analysis. Pattern Recognition 32 (4), 703. Agrafiotis, G.K., 1984. A grade-specific stochastic model for the analysis of wastage in manpower systems. Journal of the Operational Research Society 35, 549–554. Bartholomew, D.J., Forbes, A.F., McClean, S.I., 1991. Statistical Techniques for Manpower Planning, second ed. John Wiley & sons, Chichester. Blumen, I., Kogan, M., McCarthy, P.J., 1955. The Industrial Mobility of Labor as a Probability Process. Cornell University, W.F. Humphrey Press Inc., Geneva, New York. Cappé, O., Moulines, E., Rydén, T., 2005. Inference in Hidden Markov Models. Springer, New York. Collins, L.M., Lanza, S.T., 2010. Latent Class and Latent Transition Analysis. With Applications in the Social, Behavioral, and Health Sciences. John Wiley & Sons, Inc., Hoboken, New Jersey. De Feyter, T., 2006. Modelling heterogeneity in manpower planning: dividing the personnel system into more homogeneous subgroups. Applied Stochastic Models in Business and Industry 22 (4), 321–334. Goodman, L.A., 1961. Statistical methods for the mover-stayer model. Journal of American Statistical Association 56, 841–868. Gregoir, S., Lenglart, F., 2000. Measuring the probability of a business cycle turning point by using a multivariate hidden Markov chain. Journal of Forecasting 19 (2), 81–102. Guerry, M.A., 2008. Profile based push models in manpower planning. Applied Stochastic Models in Business and Industry 24 (1), 13–20. Hagenaars, J.A., McCutcheon, A.L. (Eds.), 2002. Applied Latent Class Analysis. Cambridge University Press, Cambridge. Hamilton, J.D., 1989. A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, 357–384. Levinson, S.E., Rabiner, L.R., Sondhi, M.M., 1983. An introduction to the application of the theory of probabilistic functions of a Markov process to automatic speech recognition. Bell System Technical Journal 62, 1035–1074.
M.-A. Guerry / European Journal of Operational Research 210 (2011) 106–113 MacDonald, I.L., Zucchini, W., 1997. Hidden Markov and Other Models for DiscreteValued Time Series. Chapman & Hall, London. Major, K., 2008. Income disparities among Hungarian micro-regions: the moverstayer model. Acta Oeconomica 58 (2), 127–156. Rossi, C., Schinaia, G., 1998. The mover-stayer model for the HIV/AIDS epidemic in action. Interfaces 28 (3), 127–143.
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Ugwuowo, F.I., McClean, S.I., 2000. Modelling heterogeneity in a manpower system: a review. Applied Stochastic Models in Business and Industry 16, 99–110. Verbakel, E., de Graaf, P.M., 2008. Resources of the partner: support or restriction in the occupational career? Developments in The Netherlands between 1940 and 2003. European Sociological Review 24 (1), 81–95.
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Hidden heterogeneity in manpower systems: A Markov-switching model approach Marie-Anne Guerry ⇑ Department for Mathematics, Operational Research, Statistics and Information Systems for Management (MOSI), Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussel, Belgium
a r t i c l e
i n f o
Article history: Received 24 February 2009 Accepted 21 October 2010 Available online 5 November 2010 Keywords: Manpower planning Unobserved heterogeneity Mover-stayer Markov-switching model
a b s t r a c t In modeling manpower systems, it is of crucial importance to deal with heterogeneity. Until recently, manpower models are dealing with heterogeneity due to observable sources, neglecting heterogeneity due to latent sources. In this paper a two-step procedure is introduced. In the first step personnel groups homogeneous with respect to the transition probabilities are determined in a classical way by taking into account the observable sources of heterogeneity. In the second step heterogeneity caused by latent sources is handled. A multinomial Markov-switching manpower model is introduced that deals with heterogeneity due to latent sources for the internal flows as well as for the wastage flows. The model incorporates the mover-stayer principle. A re-estimation algorithm is presented to estimate the parameters of the Markov-switching manpower model. The switching approach offers a methodology to build a Markov model with personnel groups as states that are more homogeneous, and therefore can contribute to a better validity of the manpower model. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Manpower planning is an aspect of Human Resources Management, based on mathematical modeling. Modeling a manpower system concerns three types of flows, namely the recruitment flows, the internal personnel flows between the different personnel categories (among others promotion flows), and the wastage. The aggregated models used in manpower planning are based on the concept of homogeneous personnel groups, where the validity of such a model is determined in a large extend by the degree of homogeneity of the groups. It is therefore of crucial importance to deal with heterogeneity in this context. Ugwuowo and McClean are the authors of a good review article on this topic (Ugwuowo and McClean, 2000). In previous work attention is paid to algorithms to detect the personnel groups within a manpower system that are homogeneous with respect to the transition probabilities (as there are promotion probabilities). So far those algorithms only take into account the information that is available from the personnel database on the observable variables (as there are grade, seniority, gender and full time equivalent) to determine the homogeneous personnel groups (De Feyter, 2006; Guerry, 2008). Nevertheless in a manpower system, apart from the observable sources of heterogeneity, there is heterogeneity due to latent sources (Ugwuowo and McClean, 2000). In several concrete career studies there is pointed out the role of unobserved heterogeneity: for example in Verbakel and de Graaf (2008) unobserved heterogeneity is mentioned as possible reason for the male marriage ⇑ Tel.: +32 2 629 2049; fax: +32 2 629 2054. E-mail address: [email protected] 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.10.039
premium, next to higher productivity, employer favoritism and selection. These applied research studies give confirmation of the need of building a manpower model in which there is taken into account the observable as well as the hidden heterogeneity. In this paper, in tackling the problem of heterogeneity in a manpower system, the analysis is not restricted to observable variables but observable as well as latent sources of heterogeneity are taken into account. For discrete-time models that are time-homogeneous there is introduced a method in which there is dealt with both types of heterogeneity in determining homogeneous personnel subgroups. The method will be formulated as a two-step procedure (Section 3.1). In the first step of the procedure personnel groups, homogeneous with respect to transition probabilities, are determined by taking into account the available data on the observable sources (De Feyter, 2006; Guerry, 2008). In the second step heterogeneity caused by latent sources will be handled. A Markovswitching model will be introduced to model the mover-stayer principle for the promotion flows (and for any internal personnel transitions, in general) and the heterogeneity in the wastage flows due to unobservable sources. An algorithm is presented to get estimates for the parameters of the model based on the Baum–Welch EM-algorithm (Section 3.2). This way of taking into account heterogeneity due to latent sources will result in a supplementary subdivision of each of the personnel groups in movers and stayers. The approach therefore gives rise to personnel subgroups that have a higher degree of homogeneity without requiring additional or more detailed observations. A finer subdivision of the personnel of a manpower system in more homogeneous subgroups may improve the quality of the model. However, one has to be aware of the fact that a finer
M.-A. Guerry / European Journal of Operational Research 210 (2011) 106–113
subdivision in personnel subgroups results in a greater number of parameters to be estimated and therefore the gain in terms of the goodness of fit is not obvious (Bartholomew et al., 1991). A manpower model should have states that are personnel subgroups for which there is a good compromise between the level of the subdivision of the personnel system and the quality of the estimations for the parameters. Testing and comparing the goodness of fit of models built on personnel groups with different levels of subdivisions (finer and less fine) will give an answer to the question which subdivision of the personnel results in a model with a high validity. In Section 5 an illustration is presented to demonstrate the hidden Markov approach and its different aspects (among others parameter estimation and interpretation, and validation of the model) for a concrete manpower system. Facing the problem of heterogeneity within a personnel system, the two-step procedure presented in this paper offers the opportunity to end up with more homogeneous personnel subgroups and to improve the quality of the manpower model. The values of the parameters of the model provide in a description of characteristics of the push transitions, for the internal flows as well as for the wastage flows. Furthermore, based on the model extrapolations for the number of members in each of the personnel groups can be found under the hypothesis that the personnel strategy remains unchanged for the future. And what-if-analysis can be conducted in comparing several strategies.
2. Heterogeneity in manpower systems Modeling personnel systems based on Markov chains is a well-known approach (Bartholomew et al., 1991). The states are homogeneous groups of personnel for which the promotion (or in general the internal transition) and wastage probabilities are assumed to be equal for each of the individuals within a group. Nevertheless some flows depend on individual traits even within such a homogeneous group; and if there is a lack of observations on these sources of heterogeneity, parameter estimation is not possible for such subgroups in a classical Markov model. Earlier work (Ugwuowo and McClean, 2000) concerning manpower models points to the importance of making a distinction between two types of sources of heterogeneity, namely observable sources and latent sources. For the observable sources (among others gender, full time equivalent and seniority) information is available in the personnel dataset for each of the employees. The available data give the opportunity to distinguish homogeneous personnel groups based on the corresponding (observable) variables, resulting for example in a category of female employees who are working part-time and having a length of service between 5 and 10 years. The fact is that within each of these categories, in general, there is still heterogeneity without knowing its nature. These latent sources can be of individual or environmental type. This paper discusses the problem of how to deal with heterogeneity due to latent sources for which there are no observations available. This approach is of interest in manpower planning since for the wastage flows as well as for the promotion flows possibly there is heterogeneity due to latent sources (Bartholomew et al., 1991; Ugwuowo and McClean, 2000). Among others, Bartholomew et al. (1991) pointed out the complexity of modeling wastage. The terminology ‘wastage’ refers to the total loss of individuals from the manpower system for whatever reason. From the point of view of the employee, wastage refers to voluntary wastage (for example because of a more attractive job in another company) as well as involuntary wastage (for example because of forced retirement). From the point of view of management, wastage can be controlled (for example in the case where an employee is fired) as well as uncontrolled (for example in
107
the case where an employee dies). In Agrafiotis (1984), besides controlled wastage, reasons for leaving are classified as being reasons depending on the internal structure of the company (e.g. lack of promotion prospects) and as being reasons not depending on the internal structure of the company (e.g. retirement and death). All these classifications make it clear that in modeling wastage, one has to deal with different underlying reasons for leaving. Furthermore, most of the time wastage is not the result of one but rather the result of a combination of reasons, and consequently a strict classification of the reason for leaving seems inappropriate. Often there is the supplementary difficulty that no observations are available for all these types of wastage. As pointed out for wastage flows, within a homogeneous personnel group determined by observable variables, for promotion rates heterogeneity due to latent sources is possible. Without knowing the real nature of the heterogeneity, it can be that one subgroup experiences fewer promotions in comparison with another subgroup that has more promotions. Those subgroups can be related to the mover-stayer principle. The concept of stayers and movers is introduced and studied in earlier work (Blumen et al., 1955; Goodman, 1961). Discrete-time mover-stayer models have been extensively used in studies of heterogeneous populations (among others in: Rossi and Schinaia, 1998; Major, 2008). And the intrinsic meaning of the concepts movers and stayers is not the same in all these models: the stayers can refer either to a subgroup of units that are moving infrequently or, in the simplest form of the model, to a subgroup of units that do not move. In Bartholomew et al. (1991) the idea of incorporating the mover-stayer principle in manpower planning models was introduced. In this paper, the problem of heterogeneity for the promotion flows in a manpower system due to latent sources will be formulated in terms of stayers and movers: movers are characterized by the fact that they experience faster career growth, reflected by higher promotion probabilities. Stayers change their grade less frequently or not at all. For both types of flows, wastage flows as well as promotion flows, latent sources of heterogeneity may be viewed as determining the rates. Since there are no observations available to quantify the impact of these sources on the flow rates, a (classical) Markov manpower model is not appropriate. In this paper, a Markov-switching model as a modeling tool is introduced in order to deal with heterogeneity due to latent sources in manpower systems.
3. Markov-switching manpower models Hidden Markov chains are as modeling technique widespread in the engineer sciences, among others in speech and form recognition (Aas et al., 1999; Levinson et al., 1983). More recently hidden Markov chains are increasingly used in econometrics (Gregoir and Lenglart, 2000) and in other domains. In manpower planning, Markov models are established and this in contrast with hidden Markov models. In this paper a hidden Markov model is introduced that takes into account the specificity of a manpower system and the fact that there are observable as well as latent sources of heterogeneity in such a system. A hidden Markov chain is generated by two probability mechanisms. Namely firstly, an unobserved Markov chain {Xt} that is hidden in the observations. And secondly, a set of probability distributions, one for each state of the Markov chain that produces the observations {Yt}. Markov-switching models are generalizations of hidden Markov chains in which the distribution of the observed variable Yt is not only depending on Xt but also on Yt1, Yt2, Yt3, . . . (Cappé et al., 2005). An application based on Markov-switching models is for
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example the model of Hamilton for an economic analysis of the business cycle (Hamilton, 1989). A Markov-switching model offers great flexibility in approximating complex probability distributions and provides the possibility to consider different probability distributions for each of the states of the underlying Markov chain. Furthermore the Markov-switching model approach allows one to estimate the parameters of the unobserved Markov chain for which there are no observations required. For these reasons Markov-switching models are useful to model a manpower system for which no observable variables are available or adequate to bring insights in the mover-stayer principle. For such a manpower system a classical Markov model is not able to clarify the fact that for a personnel category some employees are movers and do have higher promotion probabilities than others. In what follows a Markov-switching model is introduced for manpower systems with heterogeneity caused by latent sources and with lack of data. The characteristics and modeling strengths of Markov-switching models are a powerful means to model phenomena as wastage and promotion flows. This approach can improve the validity of the manpower model since it is based on personnel groups that are more homogeneous. 3.1. Two-step procedure In order to deal with heterogeneity caused by observable sources as well as with heterogeneity caused by latent sources a two-step procedure is introduced. In the first step of the procedure personnel groups H1, . . . , Hk homogeneous, with respect to transition probabilities in between the groups, will be determined by taking into account the available data on the observable variables. Those personnel groups H1, . . . , Hk can be found as the result of a multinomial logistic regression analysis (De Feyter, 2006) or by a profile based approach (Guerry, 2008). A person of the manpower system at time t is a member of one of these personnel groups H1, . . . , Hk. This phenomenon is described by the process {Yt} with states H1, . . . , Hk and transition probabilities pij = Pr(Yt+1 = HjjYt = Hi) (1 6 i, j 6 k). For a time-homogeneous Markov chain, the maximum likeli^ij can be computed based on the observed flows hood estimator p ntij , being the number of persons that are at time t in state Hi and P t at time t + 1 in state Hj, and the stocks sti ¼ kþ1 j¼1 nij , i.e. the number of persons in state Hi at time t (Bartholomew et al., 1991, p. 113) as
P t t nij ^ pij ¼ P t : t si
ð3:1Þ
The additional state Hk+1 corresponds with the leaving state and the flow nti;kþ1 refers to the number of members of personnel group Hi at time t that has left the manpower system at time t + 1. In a similar way to the internal transition probabilities, for each group Hi (1 6 i 6 k) the wastage probability pi,k+1 = Pr(Yt+1 = Hk+1jYt = Hi) can be estimated by
P ^i;kþ1 ¼ p
t t ni;kþ1 P : t t si
The leaving state can be considered as an absorbing state: pk+1,k+1 = 1. In the second step of the procedure, a Markov-switching model is introduced to deal with the heterogeneity due to latent sources within each of the personnel groups H1, . . . , Hk. This stage results in a supplementary subdivision of each of the personnel groups Hi (i = 1, . . . , k) into the subgroup of movers C i1 and the subgroup of stayers C i2 , being the states of the unobserved n o Markov chain X it as part of the Markov-switching model. The
subgroup of movers C i1 is characterized by a greater value for the transition probabilities from Hi to Hj (j – i) in comparison with the subgroup of stayers C i2 . Concerning the personnel group Hi, the unobserved Markov chain is characterized by the with Cilm ¼ Pr X itþ1 ¼ (2 2) probability matrix Ci ¼ Cilm C im X it ¼ C il Þ for l, m = 1, 2. The matrix Ci is the matrix of the transition probabilities for the states of movers and stayers within the personnel group Hi. The transition probability from the state C il Hi ðl ¼ 1; 2Þ into Hj will be denoted by pilj ¼ Pr Y tþ1 ¼ Hj X it ¼ C il ¼ PrðY tþ1 ¼ Hj Y t ¼ Hi ; X it ¼ C il Þ. For the personnel group Hi, the tth observation (t = 1, . . . , T) concerns the observed flows for the time period [t, t + 1], i.e. data on the number ntij of people of Hi promoted to each of the personnel groups Hj (j = 1, . . . , k) or moved to the wastage status Hk+1(j = k + 1) in time period [t, t + 1]. These observed flows will be gathered into ti ¼ nti1 ; . . . ; ntik ; nti;kþ1 . the flow vector n Let us denote by Z it the random vector with jth component (j = 1, . . . , k + 1) referring to the number of persons moved in time period [t, t + 1] from group Hi to group Hj. Since the outcome of each trial is into one of these k + 1 categories H1, . . . , Hk+1, the model results in a multinomial Markov-switching model: Conditional on X it ¼ C il , the flow vector Z it is multinomial distributed with paramPkþ1 t eters sti ¼ j¼1 nij (being the stock of the personnel group Hi at time Pkþ1 i t), and probabilities pil1 ; . . . ; pil;kþ1 satisfying j¼1 pl;j ¼ 1. For the time period [t, t + 1], the probability that the outcome for n o ti ¼ nti1 ; . . . ; Z it equals the flow vector n
the observed process
nt ; nt Þ, conditional to the fact that the unobserved Markov chain nik oi;kþ1 X it is in state C il , can be expressed by the state-to-observation ti X it ¼ C il : probability Pil;nt ¼ Pr Z it ¼ n i
Pil;nt ¼ i
sti t ni1 ; . . . ; nti;kþ1
pil1
nti1
nt i;kþ1 . . . pil;kþ1
¼ 1; 2 and t ¼ 1; . . . ; T:
for l ð3:2Þ
In what follows the notation Pi ¼ Pilt refers to the (2 T)-matrix with elements Pilt ¼ Pil;nt (for l = 1, 2 and t = 1, . . . , T). The i (2 (k + 1))-probability matrix with elements pilj (for l = 1, 2 and j = 1, . . . , k + 1) will be denoted by Pi. The flows that are considered in the Markov-switching model to deal with the heterogeneity within the personnel group Hi are schematically represented in Fig. 3.1. Concerning the outgoing flows from personnel group Hi, the graphical representation is restricted to the flows into the group Hj (0 6 j 6 k + 1) (dotted arcs). Since for each personnel group Hi (i = 1, . . . , k) a Markov chain with transition matrix Ci is considered the obtained model is, according to the terminology used in Hagenaars and McCutcheon (2002), a latent mixed Markov model for several groups. 3.2. Estimating the parameters of the markov-switching manpower model In what follows a re-estimation algorithm is presented resulting in estimations for the parameters Ci, Pi and Pi of the Markovswitching manpower model. This algorithm is inspired by the Baum–Welch EM-algorithm (see e.g. MacDonald and Zucchini, 1997) and is taking into account the specificity of the manpower system. In this context the EM-algorithm will be formulated in terms of the characteristics of the multinomial manpower model.
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Fig. 3.1. Flows in the Markov-switching model.
The parameters of a Markov-switching model can be estimated given aggregated data for each type of flow. For the personnel group Hi it concerns observations, for some time periods, on the number of members being promoted to group Hj (j = 1, . . .k) and the number of members having left the system (transition to Hk+1). These observed flows are gathered into the vectors ti ¼ nti1 ; . . . ; ntik ; nti;kþ1 for t = 1, . . . , T. There is no information ren quired separately for the subgroups of and stayers of Hi. n movers o For the unobserved Markov chain X it , the probability that the process is in state C il at time t, conditional on the observations 1i ; . . . ; n Ti , is denoted by n
1i ; . . . ; Z iT ¼ n Ti : uil ðtÞ ¼ Pr X it ¼ C il Z i1 ¼ n
The initial distribution over the states is then
C i1
(movers) and
(stayers)
Further, the conditional probability
1i ; . . . ; Z iT ¼ n Ti X it1 ¼ C il ; X it ¼ C im Z i1 ¼ n
v il;m ðtÞ ¼ Pr
is introduced. The re-estimation formulas for the parameters Ci, Pi and Pi of the Markov-switching model are expressed in terms of the forward 1i ; . . . ; Z it ¼ n ti ; X it ¼ C il satisfying probabilities ail;t ¼ Pr Z i1 ¼ n
ail;t ¼ dil :Pil;n1 ¼ ul ð1Þ:Pil;n1 for l ¼ 1; 2 and t ¼ 1; ¼
i
i
i m;t1 :
a
i ml :
i t l;n i
C P
m
for l ¼ 1; 2 and t from 2 to T
ð3:3Þ
i i tþ1 and ; . . . ; Z iT ¼ the backward probabilities bl;t ¼ Pr Z tþ1 ¼ n i i i Ti X t ¼ C l Þ for which the following holds n
bil;t ¼ 1 for l ¼ 1; 2 and t ¼ T; X i Pm;ntþ1 :bim;tþ1 :Cilm for l ¼ 1; 2 and for t from T 1 to 1: ¼ m
1i ; . . . ; Z iT ¼ n Ti X it1 ¼ C il ; X it ¼ C im Z i1 ¼ n
ail;t1 :Pim;nti :bim;t :Cil;m
¼
P
l
i
ð3:4Þ By definition of the conditional probabilities uil ðtÞ and lows for l = 1, 2 and for t from 1 to T that
ai :bi 1i ; . . . ; Z iT ¼ n Ti ¼ P l;t l;t i uil ðtÞ ¼ Pr X it ¼ C il Z i1 ¼ n i l al;t :bl;t
v il;m ðtÞ, it foland
ð3:5Þ
ail;t :bil;t
ð3:6Þ
:
Concerning the flows from Hi (1 6 i6 k) and the T observations 1i ; n 2i ; . . . ; n Ti , the likelihood is LiT ¼ Pr Z i1 ¼ n 1i ; . . . ; Z iT ¼ n Ti . n Since there are no observations concerning the Markov chain with states C i1 and C i2 , in the Expectation-step for the conditional probabilities uil ðtÞ and v il;m ðtÞ expectations are considered, based on the current parameter estimates for Ci and Pi and the formulas (3.5) and (3.6), to express the log-likelihood as T X 2 X 2 X
C i2
di ¼ ui1 ð1Þ; ui2 ð1Þ :
X
v il;m ðtÞ ¼ Pr
t¼2
log Cil;m :v il;m ðtÞ þ
l¼1 m¼1
T X 2 X t¼1
log Pil;nt :uil ðtÞ:
l¼1
i
In the Maximization-step, the first part of this expression is maximized with respect to the parameters Cil;m ðl; m ¼ 1; 2Þ under the constraints Cil;1 þ Cil;2 ¼ 1 ðl ¼ 1; 2Þ; and the second part is maximized with respect to the parameters Pil;nt ðl ¼ 1; 2Þ under the coni ditions pil;1 þ þ pil;kþ1 ¼ 1 ðl ¼ 1; 2Þ. It can be verified that the optimization method of the Lagrange multipliers results in the following expressions
PT
vi
l;m Cil;m ¼ P2 t¼2 PT j¼1
ðtÞ
i t¼2 v l;j ðtÞ
for l; m ¼ 1; 2;
ð3:7Þ
PT
i t t¼1 ul ðtÞ:nij
pil;j ¼ PT
i t t¼1 ul ðtÞ:si
for l ¼ 1; 2 and j ¼ 1; . . . ; k þ 1:
ð3:8Þ
All these properties give rise to the re-estimation algorithm presented in Fig. 3.2 that results in estimations for the parameters of the multinomial Markov-switching manpower model, given the 1i ; n 2i ; . . . ; n Ti and for chosen starting values for Cil;m ; dil observations n i and pl;j (resulting in starting values for Pil;nt for t = 1, . . . , T by Fori mula (3.2) for l, m = 1, 2 and j = 1, . . . , k+1. At each iteration, the forward and backward probabilities are computed based on the observations and the estimations for Ci, Pi and diso far. This outcome for the forward and backward probabilities is used to compute a re-estimation for Ci, Pi and di. After each iteration, the likelihood is improved except at a critical point of the likelihood (MacDonald and Zucchini, 1997, p. 63). The likelihood will be (locally or globally) maximized by the presented iterative procedure.
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M.-A. Guerry / European Journal of Operational Research 210 (2011) 106–113
Fig. 3.2. Re-estimation algorithm.
By using multiple sets of starting values, the problem of local extreme points may be avoided (Hagenaars and McCutcheon, 2002, p. 337). 4. Forecasting and control Based on the Markov-switching manpower model presented and the estimations for the parameters of the model, a wastage analysis and an evaluation of the impact of alternative promotion and recruitment strategies can be made. In manpower planning a wastage analysis is crucial. Wastage creates vacancies and so provides opportunities for promotion and recruitment, giving rise to corresponding recruitment costs. It is therefore of great importance to detect bottlenecks with respect to wastage and to obtain knowledge of the characterization of those groups of personnel having for example a rather high wastage probability. The answer to these questions will be given by the evaluation of the values for the parameters pi1;kþ1 and pi2;kþ1 ði ¼ 1; . . . ; kÞ. Besides, in manpower planning modeling wastage gives the opportunity to have, based on extrapolations, information on the outflow of the system in the future. Taking into account this information, control actions can be planned. The interpretation and labeling of the latent classes is straight forward in the context of a mover-stayer analysis: For each personnel category Hi, the group of stayers will be characterized by a greater value for the parameter pi2;i in comparison with the value for pi1;i (characterizing the group of movers). A pair wise comparison of the values of the parameters pi1;j and pi2;j ðj ¼ 1; . . . ; kÞ, gives information onto what extend there is in the company heterogeneity due to latent sources concerning the promotion from the personnel group Hi. The difference in value of pi1;j and pi2;j gives an indication for the degree of latent class separation, whereby a high degree of latent class separation implies a high degree of homogeneity for the latent classes (Collins and Lanza, 2010). The manpower model presented is based on a two-step procedure in which in the first stage the approach deals with heterogeneity due to observable sources (and results in personnel groups H1, . . . , Hk). In the second stage the Markov-switching manpower model in addition deals with heterogeneity due to latent sources (resulting in a supplementary subdivision of Hi in C i1 and C i2 ). The goal is now to get a forecast for the stock hi(t), i.e. the number of members of the personnel group Hi at time t, by taking into account the supplementary insight according to movers and stayers from the Markov-switching approach. By the parameter estimations for Ci and di (i = 1, . . . , k) in the Markov-switching model, the evolution of the distribution ui ðtÞ ¼ ui1 ðtÞ; ui2 ðtÞ of the members of the personnel group Hi over the subgroups of movers and stayers can be forecasted by
ui ðtÞ ¼ ui ðt 1Þ:Ci
where ui ð1Þ ¼ di :
Let us denote c(t) the stockvector over the 2k subgroups, as a result of considering for each personnel group Hi (i = 1, . . . , k) a subdivision of the hi(t) members in movers C i1 and stayers C i2 ; the components of c(t) can be computed as:
c2i2þl ðtÞ ¼ uil ðtÞ:hi ðtÞ for i ¼ 1; . . . ; k and l ¼ 1; 2:
ð4:1Þ
The flow from the personnel category Hi (i = 1, . . . , k) to the personnel category Hj (j = 1, . . . , k) can be calculated based on the characterisation of the hidden states as follows:
c2i1 ðtÞ:pi1;j þ c2i ðtÞ:pi2;j :
ð4:2Þ
The evolution of the stockvector h(t) over the personnel categories Hi (i = 1, . . . , k) can be expressed in terms of the (2k k)-matrix P with elements p2i2þl;j ¼ pil;j (i, j = 1, . . . , k and l = 1, 2):
hðt þ 1Þ ¼ cðtÞ:P þ RðtÞ:
ð4:3Þ
The vector R(t) stands for the recruitment vector in which Ri (t) refers to the number of new recruits in category Hi in time period t. As in the classical manpower models, the vectors R(t) depend on and reflect the recruitment policy. In an iterative way, based on the forecast h(t + 1), the components of the stockvector c(t + 1) can be found by formula (4.1), giving result to a forecast for the stockvector h(t + 2) over the personnel categories Hi (i = 1, . . . , k) by formula (4.3). By what-if analyses, the impact of alternative promotion and recruitment strategies on the evolution of the stockvector h(t) can be examined. 5. Illustration The goal of this illustration is to concretize the way in which a particular manpower system can be analyzed on the one hand only based on information of observable variables available from the personnel dataset and on the other hand by taking into account observable as well as latent sources of heterogeneity. These two approaches, of determining homogeneous personnel subgroups for the manpower system under study, result on the one hand in a (classical) Markov model and in the other hand in a Markov switching model. Moreover based on the illustration there is pointed out in concrete terms in what way the estimated parameters for both types of model can be interpreted and in what way the goodness of fit can be compared for the two obtained manpower models. For the organization under study a personnel dataset is available with information for each employee over the past nine years on his/her grade, monthly number of working hours, age, seniority, number of children, etc. Internal flows and wastage flows (as in Table 5.1) as well as incoming flows (as in Table 5.2) can be quantified based on this personnel dataset.
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M.-A. Guerry / European Journal of Operational Research 210 (2011) 106–113 Table 5.1 Observed flows for the subsystem of female employees. H1
H2
H3
H4
H9
t = 1: t = 2: t = 3: t = 4: t = 5: t = 6: t = 7: t = 8: t = 9:
H1
15 17 23 29 36 43 50 56 64
1 1 1 1 1 1 1 1 1
15 12 4 6 3 3 2 2 1
1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1
t = 1: t = 2: t = 3: t = 4: t = 5: t = 6: t = 7: t = 8: t = 9:
H2
1 1 1 1 1 1 1 1 1
21 31 41 50 58 67 74 82 89
1 1 1 1 1 1 1 1 1
1 1 1 2 2 2 2 3 3
1 1 1 1 1 1 1 1 1
t = 1: t = 2: t = 3: t = 4: t = 5: t = 6: t = 7: t = 8: t = 9:
H3
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
21 22 13 11 8 7 6 6 6
21 39 37 43 44 50 53 57 61
1 1 1 1 1 1 1 1 1
t = 1: t = 2: t = 3: t = 4: t = 5: t = 6: t = 7: t = 8: t = 9:
H4
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
20 16 31 32 40 43 48 52 56
24 34 48 59 68 75 83 90 97
1 1 2 2 2 2 3 3 3
Table 5.2 Observed incoming flows for the subsystem of female employees.
R(2) R(3) R(4) R(5) R(6) R(7) R(8) R(9)
H3, H4 (concerning female employees) on the one hand and the groups H5, H6, H7, H8 (concerning male employees) on the other hand form two subsystems for which in between there are no transitions. In what follows the Markov-switching approach is discussed for the subsystem of the females (the discussion is similar for the subsystem of the male employees). Table 4.1 provides information, for each time period [t, t + 1], on the observed number ntij of women of group Hi (i = 1, 2, 3, 4) that has transferred to group Hj (j = 1, 2, 3, 4, 9) within the considered time period. For the female employees, as a result of Formula (3.1), the Markov model with states H1, H2, H3, H4, H9 is characterized by the transition matrix PM with elements (PM)i,j = pi,j, the transition probability from state Hi to state Hj:
0
0:80059 0:02381 0:12798 0:02381 0:02381
C B B 0:01732 0:91775 0:01732 0:03030 0:01732 C C B B PM ¼ B 0 0 0:21076 0:77130 0:01794 C C: C B 0 0 0:36200 0:61746 0:02054 A @ 0 0 0 0 1 Based on the re-estimation algorithm presented in Fig. 3.2, the parameters of the multinomial Markov-switching manpower model are estimated, resulting in the matrix PMS with elements ðP MS Þ2i2þl;j ¼ pil;j , the transition probability from the subgroup of movers/stayers of Hi to state Hj:
0
PMS
0:49363 B 0:88802 B B B 0:03221 B B 0:01439 B B B 0 ¼B B 0 B B B 0 B B 0 B B @ 0 0
H1
H2
H3
H4
16 12 14 12 12 11 10 11
13 13 13 12 13 11 13 12
5 0 6 3 6 6 7 7
4 6 6 5 5 6 6 5
In the considered manpower system, two grades (gr 1 and gr 2) are distinguished and no demotions occur. A study as in De Feyter (2006) or Guerry (2008) of the personnel dataset let us conclude that significant differences in transition probabilities are imposed by the observed variables gender and full time equivalent (FTE). More specifically, the outcomes female (f), male (m), FTE less than or equal to 0.5 (FTE 6 0.5) and FTE greater than 0.5 (FTE > 0.5) determine 8 personnel groups Hi:
H1 : gr1; f ; FTE 6 :5 H2 : gr1; f ; FTE > :5 H3 : gr2; f ; FTE 6 :5 H4 : gr2; f ; FTE > :5; H5 : gr1; m; FTE 6 :5 H6 : gr1; m; FTE > :5 H7 : gr2; m; FTE 6 :5 H8 : gr2; m; FTE > :5; that are homogeneous with respect to the transition probabilities. The group H9 refers to the leaving state. Concerning the flows between these groups Hi, there are 9 observations (t = 1, . . . , t = 9) available in time. The groups H1, H2,
1
1
0:03078
0:41402 0:03078
0:03078
0:02064
0:05007
0:02064 C C C 0:03221 C C 0:01439 C C C 0:01683 C C: 0:01919 C C C 0:02222 C C 0:02022 C C C A 1
0:02064
0:87024 0:03221 0:03312 0:92660 0:01439 0:03023 0
0:12600
0
0:36323 0:61759
0
0:44444 0:53333
0
0:35730 0:62247
0 0
0 0
0:85717
0 0
1
The element (PM)1,3 = p1,3 of the transition matrix of the Markov model, for example, stands for the probability that a female employee from grade 1 and working part-time (FTE 6 0.5) is after one period of time still working part-time (FTE 6 0.5) but being promoted into grade 2. For that same internal transition, the corresponding elements of the transition matrix PMS of the Markov switching model, ðPMS Þ1;3 ¼ p11;3 and ðPMS Þ2;3 ¼ p12;3 , give the transition probability, respectively for the female movers and the female stayers. The transition probability (PM)1,3 = p1,3 is estimated based on available data for the observable variables grade and FTE, without taking into account the fact there is heterogeneity due to latent factors. The transition probabilities ðPMS Þ1;3 ¼ p11;3 and ðPMS Þ2;3 ¼ p12;3 give more detailed information, namely for the movers and stayers respectively, and result from the parameter estimation procedure for the switching manpower model (Section 3.2). For a woman of the state H1, the Markov model results in an estimation for the transition probability to the state H3 equal to 0.12798. Since for the latent classes within H1 hold that p11;3 > p12;3 , the classes C 11 and C 12 are labeled as respectively the movers and stayers of H1. In the Markov-switching model, for the movers within the group H1 the transition probability to H3 is estimated to be equal to 0.41402. And for the stayers within the group H1 this transition probability is estimated to be equal to 0.05007. The two-step procedure therefore gives rise to a supplementary subdivision of the group H1 of the female part-time working with FTE 6 0.5 of grade 1, resulting in a subgroup of movers (with a promotion probability to grade 2 equal to 0.41402) and a subgroup of stayers (with a promotion probability to grade 2 equal to 0.05007).
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1
for members of H 1 for movers of H 1 for stayers of H 1
probability
0.8
significantly better than the Markov model in fitting the personnel data. The fact that the Markov-switching model is based on more homogeneous personnel groups, by taking in addition latent variables into account, results in a model with a better validity in comparison with the Markov model that is only based on observed variables.
0.6
6. Conclusions 0.4
0.2
0
staying in H 1
promoting from H 1 to H 3
Fig. 5.1. Transition probabilities conditional on latent class membership.
Since, for example, H1 can be partitioned in two subgroups for which the transition probability to H3 equals respectively 0.41402 and 0.05007, the states H1, H2, H3, H4, H9 of the classical Markov model are not really homogenous with respect to the transition probabilities. Remark that the Markov model gives an estimation for the transition probability for the members of group H1 in global (without making the distinction between movers and stayers) to H3 equal to 0.12798. In this way and as could be expected, concerning the promotions from H1 to H3, the transition probability conditional on latent class membership is greater for the movers than for the stayers; and the (unconditional) transition probability in the classical Markov model is a value in between the conditional probability for the stayers and the conditional probability for the movers. (see Fig. 5.1) For the personnel categories H1, H2, H3, H4 the number of recruitments can be computed from the information on the internal and wastage flows provided in Table 5.1: For 2 6 t 6 9 and for P each personnel group Hg g 2 {1, 2, 3, 4} holds that 4j¼1 ntgj þ ntg9 ¼ stg , P4 t1 t the stock of Hg at time t, and i¼1 nig ¼ sg Rg ðtÞ, the number of members of Hg at time t who were already in the manpower system at time t 1. Therefore the number of members recruited in Hg at time t can be computed from the information on the flows P P in Table 5.1 as Rg ðtÞ ¼ 4j¼1 ntgj þ ntg9 4i¼1 nt1 ig . These results for the incoming flows, characterized by the recruitment vectors R(t) are gathered in Table 5.2. In both the Markov and the Markovswitching manpower model these recruitment flows are considered as deterministic and known. In order to measure the fit of the Markov-switching manpower model and to compare the goodness of fit for the Markov-switching model and the Markov model, a 3-fold cross validation is considered. The model was trained on the observations at t = 1, 2, 3, 4, 5, 6, parameters were estimated (according to the procedure of Section 3.2) and predictions on the flows were made for t = 7, 8, 9 (according to the approach in Section 3). This way of testing the models results for the Markov model in v2 = 83.18246 and for the Markov-switching model in v2 = 24.2473. In validating the models, the number of degrees of freedom equals 3k2 = 64. Since v2:05 ð64Þ ¼ 83:67526 the Markov model as well as the Markov-switching model are acceptable models. Let us denote the likelihood for the Markov-switching model by Q Q 1i ; . . . ; Z iT ¼ n Ti and in an analogous LðMSÞ ¼ ki¼1 LiT ¼ ki¼1 Pr Z i1 ¼ n LðMÞ way the likelihood for the Markov model by L(M). Since 2 log LðMSÞ
is chi-square distributed with k2 + 3k = 28 degrees of freedom and LðMÞ ¼ 41; 4 > v2:05 ð28Þ, the Markov-switching model is since 2 log LðMSÞ
A Markov-switching manpower model is introduced to face with the problem of hidden heterogeneity in the wastage flows as well as in the internal transitions (as there are promotions) within a manpower system. In comparison with the (classical) Markov manpower models, the Markov-switching approach results in personnel categories that are more homogeneous by taking into account heterogeneity due to latent sources. There is presented an algorithm that results in estimations for the transition probabilities for the movers as well as for the stayers of each personnel group without requiring additional data on the flows for those subgroups. In practice, based on a comparison of the goodness of fit for the Markov model and the Markov-switching model, one can decide which type of model is the most appropriate for the manpower system under study. In this way the procedure results in a manpower model that characterizes the internal and wastage flows of the personnel in a good way. The manpower model is useful in predicting for the future the number of members in each of the personnel categories under the assumption that the transition probabilities are constant in time. More in general based on the model, what-if-analyses can be conducted in comparing and evaluating alternative personnel strategies; and a decision can be made on what personnel strategy results in a preferable evolution for the number of members in each of the personnel categories. Acknowledgements The author thanks the reviewers for their remarks and valuable suggestions. This research is partially funded by the research project FWOAL453 of the Research Foundation Flanders. References Aas, K., Eikvil, L., Huseby, R.B., 1999. Applications of hidden Markov chains in image analysis. Pattern Recognition 32 (4), 703. Agrafiotis, G.K., 1984. A grade-specific stochastic model for the analysis of wastage in manpower systems. Journal of the Operational Research Society 35, 549–554. Bartholomew, D.J., Forbes, A.F., McClean, S.I., 1991. Statistical Techniques for Manpower Planning, second ed. John Wiley & sons, Chichester. Blumen, I., Kogan, M., McCarthy, P.J., 1955. The Industrial Mobility of Labor as a Probability Process. Cornell University, W.F. Humphrey Press Inc., Geneva, New York. Cappé, O., Moulines, E., Rydén, T., 2005. Inference in Hidden Markov Models. Springer, New York. Collins, L.M., Lanza, S.T., 2010. Latent Class and Latent Transition Analysis. With Applications in the Social, Behavioral, and Health Sciences. John Wiley & Sons, Inc., Hoboken, New Jersey. De Feyter, T., 2006. Modelling heterogeneity in manpower planning: dividing the personnel system into more homogeneous subgroups. Applied Stochastic Models in Business and Industry 22 (4), 321–334. Goodman, L.A., 1961. Statistical methods for the mover-stayer model. Journal of American Statistical Association 56, 841–868. Gregoir, S., Lenglart, F., 2000. Measuring the probability of a business cycle turning point by using a multivariate hidden Markov chain. Journal of Forecasting 19 (2), 81–102. Guerry, M.A., 2008. Profile based push models in manpower planning. Applied Stochastic Models in Business and Industry 24 (1), 13–20. Hagenaars, J.A., McCutcheon, A.L. (Eds.), 2002. Applied Latent Class Analysis. Cambridge University Press, Cambridge. Hamilton, J.D., 1989. A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, 357–384. Levinson, S.E., Rabiner, L.R., Sondhi, M.M., 1983. An introduction to the application of the theory of probabilistic functions of a Markov process to automatic speech recognition. Bell System Technical Journal 62, 1035–1074.
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