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Appl. Stochastic Models Bus. Ind., 2006; 22:321–334
Published online 1 March 2006 in Wiley InterScience ( DOI: 10.1002/asmb.619
Modelling heterogeneity in manpower planning: dividing the
personnel system into more homogeneous subgroups
Tim De Feyter*,y
Center for Manpower Planning and studies, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium
Manpower planning is very useful for human resource management in large organizations. Most manpower models are concerned with the prediction of the future behaviour of the staff: they might leave the
organization, get promoted or acquire more and new skills. This behaviour can vary a lot among different
employees, what makes prediction difficult. It is common to tackle this problem by dividing the whole
heterogeneous personnel system in several more homogeneous subgroups. This approach is often used to
develop manpower planning models for prediction, control or optimization. Although the division in
homogeneous subcategories is a fundamental and important step in the application of the models, up till
now literature neglects to discuss a procedure to deal with this in practice. This paper suggests a general
framework to find the distinguished homogeneous subcategories by determining and considering observable sources of personnel heterogeneity. Copyright # 2006 John Wiley & Sons, Ltd.
Received 5 February 2005; Revised 2 December 2005; Accepted 20 January 2006
manpower planning; homogeneity; stochastic models; Markov models
The objective of manpower planning is to ensure that the right number of people with the right
qualifications is available in the organization in the future. Basically, manpower planning is
done in three steps [1, 2]: Firstly, the demand of personnel in the future is estimated by trying to
get an insight in the employees needed to fulfil its objectives. Secondly, the supply of personnel
in the future is forecasted: The organization needs to get an idea of what will happen with its
present and future personnel. Especially, leaving the organization and acquiring qualifications
are important for manpower planning. In the third step, the organization has to compare
demand and supply in the future and take actions to attune both. Recruitment and training are
*Correspondence to: T. De Feyter, Center for Manpower Planning and studies, Vrije Universiteit Brussel, Pleinlaan 2,
B-1050 Brussels, Belgium.
E-mail: [email protected]
Copyright # 2006 John Wiley & Sons, Ltd.
obvious actions that could be taken. Several operational research methods are developed to
support organizations in their manpower planning challenge. Because of several reasons [3–5],
almost all academic attention in the past is given to the development of methods for manpower
supply forecasting. Those models attempt to predict the future evolution of the current and
future employees. Several transitions between states are possible. The two most interesting
potential flows of an individual are ‘leaving the organization’ and ‘develop a broader range of
skills’. Those transitions influence, respectively, the quantitative and qualitative supply of manpower. This behaviour highly depends on the individual. Several factors cause differences
among employees in, for example, motivation, performance or commitment. This personnel
heterogeneity makes the manpower supply forecasting difficult. Ugwuowo and McClean [6]
made an overview of techniques to handle this heterogeneity. Several manpower planners suggest tackling this problem by dividing the whole heterogeneous personnel system in several more
homogeneous subgroups that form a partition of the total personnel system. This simplifies the
manpower supply prediction because it becomes acceptable to presume that everyone in the
same group evolves analogously. Consequently, the assumption can be made that every employee within a homogeneous group has the same probability to leave the organization or to
move towards another homogeneous group. The use of transition probabilities implies that the
techniques are only suitable for large organizations. The way that those transitions are estimated and modelled, depends on several different model assumptions. Manpower planners
commonly use time-homogeneous Markov models [1, 7], semi-Markov models [8, 9], non-time-homogeneous Markov models [10,11] or non-time-homogeneous semi-Markov models [12,13].
The validity of those models highly depends on the homogeneity of the subgroups. In one of
the most important reference works on manpower planning [1] the problem of heterogeneity in
the system is pointed out and the possibility is mentioned to use more general models,
which allow heterogeneity instead of using classical manpower analysis requiring homogeneous groups. Nevertheless, the recommend to use the Markov manpower models, because
the estimation and the number of parameters would make other models more difficult to use in
practice. In this paper, the discussion is restricted to time-discrete models where transitions
during well-defined successive periods are studied.
Although Markov manpower models are by nature stochastic, most research on those models
is done from a deterministic point of view: the actual number of employees in every homogenous
group is a random variable, but for analysis, it is replaced by its expected value [14,15]. Relatively few researchers consider other stochastic aspects, like dividing the personnel system into
homogeneous subgroups. This might explain why almost all publications in the domain of
manpower planning start with a given personnel classification [16,17]. In most examples, the
different grades in the hierarchy of the enterprise are used to split-up the personnel system in
subcategories. In practice, it is seldom the case that a division only based on the grades results in
acceptable forecasting results. Although it might be one of the most difficult steps in an
application of manpower models and it is important to get reliable results, there is a lack of
attention in literature towards the way that homogeneous groups can be attained.
We applied the manpower planning methodology to the personnel system of a Belgian Federal Office. It is not the main objective of this paper to discuss the results of this analysis. In this
paper, we will, above all, report on the followed procedure for dividing the personnel system
into more homogeneous personnel categories. A general framework is suggested to partition
manpower data in order to handle heterogeneity and to improve predictions of future behaviour. A similar idea has been developed previously in survival literature [18,19]. Tree-based
Copyright # 2006 John Wiley & Sons, Ltd.
Appl. Stochastic Models Bus. Ind., 2006; 22:321–334
DOI: 10.1002/asmb
methods are suggested to investigate the effects of several variables on the survival rate and to
partition the population (e.g. cancer patients) into several homogeneous subsets in order to
improve the estimation of the survival distribution. This method, however, is not applicable in
Markov manpower analysis: while in survival analysis the objective is finding groups homogeneous with respect to the survival probability, the objective in manpower analysis is to reach
homogeneity with respect to probabilities of transitions which are not defined in advance. Since
the groups are not known at the start of the manpower analysis, it is not yet possible to identify
the transitions between them. Moreover, in survival analysis, only binary outcomes (survive or
not survive) are studied. In Markov manpower models, however, several outcomes (i.e. transitions) need to be investigated. In the next section, all difficulties with creating homogeneous
subgroups in Markov manpower analysis are explained in more detail and a general splitting-up
approach is suggested to overcome those difficulties. In Sections 3 and 4, the use of some
statistical multivariate techniques is proposed to support the splitting-up process. In Section 5,
the general splitting-up procedure is illustrated by an example. Finally, the last section contains
some topics for further research on handling heterogeneity in manpower planning.
To be able to use Markov models for manpower supply prediction, the decision maker should
go through some preparing phases. On the one hand, there should be determined whether timehomogeneous Markov models, semi-Markov models or non-time-homogeneous Markov models are appropriate. This fully depends on the characteristics of the considered personnel system.
On the other hand, the homogeneous subgroups need to be identified and the parameters of the
model have to be estimated. In this section, a general approach is suggested, covering those
preparing phases.
2.1. Historical data set
For this approach, a historical data set of all transitions is required. Explicitly, a complete
overview of the characteristics of all former and current employees is needed. The characteristics
involved are all observable factors (e.g. sex, number of children, full-time equivalent) of which
the organization has information at its disposal and which are believed to possibly have a direct
or indirect influence on the employee’s transition behaviour. Since manpower planning is
interested in the employee’s transitions in time, the data set should include all changes in the
considered states or in the influential factors. Since manpower planning is a longitudinal study,
it is desirable to have data on an as large as possible period, as long it is believed to be relevant
for the time interval for which the predictions will be made. In this way, a complete overview of
every employee’s profile at every point of time is available. This allows drawing conclusions
about the homogeneity in the personnel system. In order to decide on the appropriate Markov
model, the general splitting-up procedure starts with an attempt to fit the easiest possible model,
namely the time-homogeneous Markov chain, to the personnel system. For that, if indeed
several observations in time are used, the assumption should be checked whether the effects of
the characteristics on heterogeneity did not change in time.
The characteristics in the data set are used as criteria to divide the personnel system in several
homogeneous groups, in such way that it can be assumed that everyone in the same
Copyright # 2006 John Wiley & Sons, Ltd.
Appl. Stochastic Models Bus. Ind., 2006; 22:321–334
DOI: 10.1002/asmb
homogeneous subgroup has the same transition probabilities. Therefore, the impact of every
characteristic on the transition probabilities needs to be investigated. For this purpose, all
profiles in the data set are grouped with respect to the values of the considered characteristic and
the transition probabilities are compared between those different groups. In case that the difference is significant, the characteristic is selected as criterion for splitting-up the personnel
2.2. Splitting-up process
Since the groups are not known at the start of the splitting-up process, it is not yet possible to
compute the transition probabilities between them. Therefore, we suggest dividing the personnel
system in several stages. Every stage results in a temporal classification, which needs to be
reconsidered in the next stage. The splitting-up process is summarized in Figure 1. In the first
stage of the splitting-up process, the objective of the manpower study needs to be considered
and groups are created according to the characteristics which are necessary to study the problem
at hand. If the interest is in the future number of employees with a certain characteristic
(e.g. grade, salary, qualifications or experience level), this characteristic has to be taken into
account in the division in distinctive groups, although it might not contribute to any homogeneity. In the easiest case, one is only interested in the future available personnel of the
organization in its entirety. That way, it is unnecessary to introduce any division yet. In the
second stage of the splitting-up process, the homogeneity of the temporal subgroups created in
the previous stage is investigated. Next to the transitions between those subgroups, the wastage
is considered. In case that the problem definition did not put on any subdivision, the wastage is
temporally the only flow considered. The objective is to investigate for each subgroup which
Split-up w.r.t.
study objective
Groups too
small for
further division
Stepwise split-up
w.r.t. transition
probabilities between
temporal groups
Combine groups with equal
transition probabilities
Final classification
Figure 1. Summary of the splitting-up process.
Copyright # 2006 John Wiley & Sons, Ltd.
Appl. Stochastic Models Bus. Ind., 2006; 22:321–334
DOI: 10.1002/asmb
division is necessary to get homogeneity with respect to the considered flows. After this is
realized, it is important to take into account that every introduction of a new group brings along
new needs for further investigation of the heterogeneity. Indeed, new transitions were introduced and everyone in every group needs to have comparable transition probabilities for every
possible flow. As a consequence, in the third and following stages of the splitting-up process, the
homogeneity of the temporal subgroups created in the previous stage is investigated, but only
with respect to flows introduced in this previous stage. The other flows do not have to be
considered, since their corresponding transition probabilities were already investigated in the
stage(s) before. In principle, the division goes on until homogeneity is reached for all possible
transitions in all created subgroups. Nevertheless, this division should be done very carefully.
This relates to the trade-off problem between homogeneity and the number of people in every
group: the final homogeneous groups need to be large enough because this will enable reliable
estimations of the transition probabilities as well as reliable manpower supply predictions.
Besides, the conclusion that a specific criterion is significant for the explanation of differences in
transition probabilities might not be reliable if it is based on a small number of observations.
Consequently, it might be that it is not desirable anymore to introduce more characteristics for
further division, although sources of heterogeneity still exist. It is the challenge to find an
acceptable equilibrium between homogeneity and group size. With this in mind, we suggest
introducing in the second and following stages of the splitting-up process, only one criterion for
division at the time. The criterion that contributes most to homogeneity is used for division first.
This stepwise approach has two important advantages: Firstly, it becomes possible to keep an
eye on the size of the subgroups such that the manpower planner can weigh the group sizes and
homogeneity against each other. In this evaluation, it is important to take into account that
small groups will lead to less accurate estimations of the transition probabilities. Consequently,
large differences in estimated transition probabilities between subgroups might not be caused by
differences in the real probabilities, but rather by bad estimations. Secondly, it might be that
there is a strong correlation between several characteristics, such that the introduction of only
one of them as division criterion is enough to get reasonable homogeneity. This also has a
positive effect on the equilibrium between homogeneity and group size. The stepwise approach
is illustrated in Figure 2.
After all sources of heterogeneity are identified, the division in subcategories still needs
reconsideration in the final stage of the splitting-up process. Some criteria might have created
subgroups with equal transition probabilities. For example, the splitting-up procedure might
identify the individual’s sex as a separating variable regarding to the wastage probabilities while
men with a high length of service could have the same probability to leave as women. In the final
stage of the splitting-up process, such subgroups should be combined. Otherwise those interaction effects would cause non-relevant divisions and avoidable smaller groups. Eventually, the
resulting subgroups are large and homogeneous enough and a definitive choice of the homogeneous groups is made.
In the splitting-up procedure, the transition probabilities need to be estimated to enable an
evaluation of the differences between them. For convenience, we suggested initially trying to
model the personnel system in the easiest possible way, namely as a time-homogeneous Markov
chain. An estimator for the transition probabilities is given by Anderson and Goodman [20]:
p# ij ¼
with Nij ¼
Copyright # 2006 John Wiley & Sons, Ltd.
nij ðtÞ
and Ni ¼
ni ðt 1Þ
Appl. Stochastic Models Bus. Ind., 2006; 22:321–334
DOI: 10.1002/asmb
Start stepwise
All transitions
temporal groups
Split-up w.r.t.
characteristic that
contributes most to
Groups too
small for
further division
Temporal classification
Figure 2. Summary of the stepwise splitting-up approach.
with p# ij equal to the estimated transition probability between group i and group j, nij ðtÞ equal to
the observed transitions between group i and group j in time period t and ni ðt 1Þ equal to the
observed number of employees in group i at the start of period t. Statistical tests are available to
compare transition probabilities between subsamples of the data set [21].
2.3. Semi-Markov models and non-time-homogeneous Markov models
As already mentioned, next to the classification of employees in homogeneous groups, the
preparing phase of the use of Markov theory in manpower planning also includes determining
the appropriate Markov model. Therefore, the general splitting-up procedure initially tries to fit
the time-homogeneous Markov model to the personnel system. In case the system should be
modelled by another Markov model, the decision maker will find out that the model assumptions of the time-homogeneous Markov model do not hold or get stuck during the splitting-up
The time-homogeneous Markov model assumes time-homogeneous transition probabilities. A statistical procedure to check the assumption of time-homogeneity is available in
References [1,20]. It should be mentioned that exceptional transitions in one or more time
periods would cause a violation of the assumption of time-homogeneity. It goes without
saying that such outliers should be deleted from the data set [1]. In case that the personnel
system still does not fit with the assumption of time homogeneity and that it is allowed by
the objective of the manpower planning study, the time intervals could be adapted. In this
way, periodic, cyclic or seasonal fluctuations can be neutralized. If, for example, higher and
lower wastage probabilities are observed, respectively, at the start and the end of a calendar year, time intervals of one year instead of intervals of six months would solve the
non-time-homogeneity. If the time intervals cannot be adapted or the adaptation does not
Copyright # 2006 John Wiley & Sons, Ltd.
Appl. Stochastic Models Bus. Ind., 2006; 22:321–334
DOI: 10.1002/asmb
solve the problem of time-homogeneity, non-time-homogeneous Markov models are
appropriate [22].
Secondly, the decision maker could get stuck during the splitting-up process. The problem
could be that a deterministic variable influences homogeneity. The most important examples are age and length of service. It is not inconceivable that those variables influence the
individual’s wastage and promotion behaviour. Because in that case it is most likely
impossible to create homogeneous groups with individuals of the same age or length of
service (for reason of group size), the time-homogeneous Markov model is not suitable and
semi-Markov models are appropriate [1].
In the general splitting-up procedure suggested in the previous section, the only statistical
techniques used are those to compute and compare transition probabilities and those to check
the model assumption of time-homogeneity. In this section, some multivariate statistical techniques are suggested to support and improve the general approach.
3.1. Multinomial logistic regression analysis
First of all, instead of the stepwise splitting-up approach, multinomial logistic regression analysis can be used to evaluate the relationship between the observable factors and the transition
probabilities. The relationship is investigated between on the one hand the transition outcomes
(and implicitly the transition probabilities) and on the other hand the observable factors and
their interaction effects. In case the objective of the manpower planning study does not put on
any initial subdivision in the first stage of the splitting-up process, the analysis in the second step
of the process is restricted to an binary logistic regression: the regression model gets one with
two possible outcomes (namely, the states ‘stayed in the organization’ or ‘left the organization’).
In the other case and in the following stages of the splitting-up process, the data set needs to
contain a dependent outcome variable for every individual at every time point that refers to the
transition made by the individual in that time interval (namely, the flow ‘left the organization’ or
‘stayed in or moved to’ one of the temporal subgroups between which the transitions are
investigated at that stage of the splitting-up process). The classical procedures for selecting the
best subset of predictor variables in regression analysis generally apply to logistic regression and
are used here to identify the variables for dividing the personnel system [23]. In a logistic
regression analysis, all possible influencing sources and all transitions from out of the temporal
groups are investigated at once, while in the stepwise splitting-up approach every variable and
transition is investigated separately. Besides, the final reconsideration stage becomes redundant
because the significance of the interaction effects is immediately investigated by the regression
analysis. So, logistic regression analysis has the advantage that the investigation procedure is
faster, all the more because statistical software is readily available for use. SAS (with PROC
logistic and PROC catmod) as well as SPSS (with the logistic regression, NOMREG and
GENLOG procedure) provide support for this analysis.
In the general splitting-up approach, the stepwise introduction of criteria was recommended
because it brings along some interesting advantages. Those advantages still hold when the
suggested multivariate analysis techniques are used: The procedure for selecting the best subset
Copyright # 2006 John Wiley & Sons, Ltd.
Appl. Stochastic Models Bus. Ind., 2006; 22:321–334
DOI: 10.1002/asmb
of predictor variables reduces the number of criteria used to split-up the data set. In case of
strong correlation between two or more influencing variables, the regression procedure for
variables reduction will lead to the retention of only one of those correlating variables. The
adequate number of criteria will be used for system division what leads to an acceptable equilibrium between homogeneity and group size.
In the previous section, we discussed the suggestion to try to fit an as easy as possible model,
namely a time-homogeneous Markov model, to the personnel system. This brings along the
need for checking the model assumption of time-homogeneity. If regression analysis is used, this
investigation can be done at the same time as the investigation of the significance of the
characteristics. Time can be brought in as a predictor variable, for which the significance is
tested in the regression procedure. An analogue approach is valid for the assumption of timehomogeneous effects of the characteristics on the transition probabilities. To check this assumption, interaction effects between every variable and time can be brought into the regression
model as predictor variables. In case such an interaction effect seems significant, the effect is not
time-homogeneous. This approach of time-homogeneity allows choosing that set of predictor
variables (and in that way also the division in subgroups) that does not contain the time variable
or time dependent characteristics. The manpower planner should measure out the loss of homogeneity against the advantage of using the easiest manpower planning model.
3.2. Cluster analysis
The regression analysis results in the identification of the significant variables and allows
implementing continuous variables that potentially are sources of heterogeneity in the system,
such as ‘the average number of overtime hours a week’. Without using this statistical technique,
the analysis of continuous variables is difficult: to investigate the effect of a specific variable on
heterogeneity, the transition probabilities are compared between groups divided using this specific variable as splitting-up criterion. This method works well with variables measured on a
nominal or ordinal scale. Individuals with different values for those variables are separated and
transition probabilities between the separated groups can be compared. The use of variables
measured on an interval or ratio scale on the contrary is more difficult: The manpower planner
tries to find the boundaries on the scale that separate individuals with most significant difference
in transition probabilities. Since the large number of potential boundaries, this might be a hard
or impossible task. Although regression analysis discovers whether the variable is significant or
not, those boundaries are not identified either.
Therefore, cluster analysis can be used. Profiles with comparable transition probabilities are
clustered together. The criteria for clustering are the estimated transition probabilities, which
are computed using the fitted regression function (with the selected predictor variables). Each
cluster will be selected as a more homogeneous subgroup. If, for example, there exists a significant relationship between the number of overtime hours and the promotion probability,
cluster analysis identifies the boundaries for separating the individuals in groups. In case people
with on average less than 5 h overtime a week have comparable promotion probabilities that are
different from those of people working more than 5 h overtime a week, they will be separated
into different clusters. It goes without saying that in the cluster analysis there should be taken
account of the fact that small groups need to be avoided. Cluster analysis is supported by SAS
as well as by SPSS.
Copyright # 2006 John Wiley & Sons, Ltd.
Appl. Stochastic Models Bus. Ind., 2006; 22:321–334
DOI: 10.1002/asmb
Once the proper Markov model is identified, homogeneous groups are selected and the transition probabilities are estimated, the manpower planner is ready to apply the model for prediction, control or optimization. To predict the evolution of manpower supply, every current
and new recruited employee must be assigned to one of the homogeneous groups. In this way,
the future flows can be forecasted by using the estimated transition probabilities for every
group. In the general splitting-up procedure suggested in the second section, every group is well
defined based on the division criteria selected and every individual is easily classified in the
adequate homogeneous group according to his profile. Meanwhile, if cluster analysis is used for
dividing all individuals in several groups, it might be impossible to find out how the groups are
defined. Especially when several variables are taken into account in the personnel division, it will
be difficult to gain a clear view of the clustering rules quickly. This is not a problem for the
current employees, because the cluster analysis results will assign them to the proper cluster. On
the other hand, the unclear composition of the homogeneous groups is a problem for the future
employees, since they should be assigned to one of the homogeneous clusters at the moment that
they are recruited. Moreover, also for using the manpower model for control or optimization,
the planner should know how the homogeneous groups are defined. Therefore, a classification
tree can be build based on a classification algorithm, e.g. the C4.5 algorithm [24]. Since the
individuals are clustered based on transition probabilities that are estimated by taking linear
combinations of the significant variables, classification algorithms will reveal the way individuals were clustered. The classification tree can be used to assign future employees to one of the
designed clusters, such that a manpower supply forecast can be performed. There are several
software programs available which are able to build classification trees, like, e.g. SPSS, Ctree
[25] or CART.
In this section, the general splitting-up procedure is illustrated by an example. Consider an
organization that is interested in the future evolution of the distribution of its employees over
three different grades. It is only possible to be promoted from grade 1 to grade 2 and from grade
2 to grade 3. A historical data set ðn ¼ 2090Þ is available containing information about every
employee at 1st January of the past seven years. The following variables are at our disposal:
grade (1,2 or 3), sex, full-time equivalent (part-time or full-time), marital status (married, not
married) and number of children. Employees who are promoted or recruited in grade 2 are
immediately evaluated with respect to their leadership abilities (high or low). The results of this
evaluation are also available in the data set. We will try to fit a time-homogeneous Markov
manpower planning model to the data. Instead of using only three groups (grades 1, 2 and 3), we
will try to improve the validity of the model by dividing the personnel system into more
homogeneous subgroups.
In the first stage of the splitting-up process, the personnel system is divided considering the
study objective. This results in three temporal subgroups (grades 1, 2 and 3) in order to be able
to study the future size of each of those subgroups. In the following stages, the heterogeneity of
temporal subgroups with respect to the transition probabilities is investigated by a stepwise
Copyright # 2006 John Wiley & Sons, Ltd.
Appl. Stochastic Models Bus. Ind., 2006; 22:321–334
DOI: 10.1002/asmb
multinomial logistic regression analysis (with backward elimination). All possible interaction
effects are included in the model, even as the year in which the transitions were observed.
The second stage in the splitting-up process results in five temporal groups that are considered
to be homogeneous with respect to promotion and wastage probabilities:
The analysis of grade 3 is restricted to a binary logistic regression analysis. None of the
variables seems to have a significant effect on the wastage probability. Consequently, grade
3 is selected as a final homogeneous subgroup (final subgroup 6).
Only the variable leadership abilities (Wald w2 ¼ 28:404; p50.001) seem to have an effect
on the probability to promote from grade 2 to grade 3. This results in a division of grade 2
into two temporal subgroups, namely employees in grade 2 with, respectively, low and high
leadership abilities.
The promotion probabilities of employees in grade 1 are only significantly influenced
by the interaction effect between sex and fulltime equivalent (FTE) (Wald w2 ¼ 19:155;
p50.001). It seems that part-time women have a lower promotion and a higher
wastage probability than men and full-time women. Therefore, part-time women in
grade 1 are separated from all other employees in this grade. Without using statistical
support, the general splitting-up procedure would create four temporal subgroups (parttime men, part-time women, full-time men and full-time women). Three of those
groups would be recombined in the final stage of the procedure. Since the interaction
effects are immediately investigated in the regression analysis, this final stage becomes
redundant in this application. Remark that the investigated transition probabilities in the
analysis are the promotion probabilities from grade 1 to each of the temporal subgroups of
grade 2.
An overview of the temporal classification after the first two stages is given in Figure 3.
In the third stage, more partitioning is possible since the groups are still large enough and not
all variables in the data set are used. The homogeneity in the two subgroups in grade 1 and two
subgroups in grade 2 is further examined. In this stage only the transitions between those
subgroups are considered:
The organization does not allow full-timers to start working part-time. This means that the
temporal subgroup containing all men and full-time women in grade 1 is accepted as a final
subgroup (final subgroup 1). Only the transitions from a part-time to a full-time status
need to be investigated. The stepwise binary logistic regression analysis only identifies the
number of children (Wald w2 ¼ 31:957; p50.001) as a variable with a significant influence
on the decision of part-time women to become a full-time employee. Since number of
children is a continuous variable, a cluster analysis is necessary to divide the part-time
women into more homogeneous groups. The cluster variable is the predicted probability p#
to become a full-time employee, which is estimated by the fitted logistic regression model
(with only the number of children as independent variable):
¼ 1:491 þ 3:996 number of children
1 p#
A hierarchical cluster analysis (with cluster method ‘Within-groups linkage’) results in two
new temporal subgroups, in which all employees have similar transition probabilities. A
classification tree identifies the classification rules used to assign part-time women to one of
Copyright # 2006 John Wiley & Sons, Ltd.
Appl. Stochastic Models Bus. Ind., 2006; 22:321–334
DOI: 10.1002/asmb
Figure 3. Stage 1 (w.r.t. the study objective) and stage 2 (homogeneity w.r.t. promotion and wastage
probabilities) in the splitting-up process.
those two clusters. This classification tree is given in Figure 4. From this classification tree
and the fitted regression model, we can conclude that part-time women with children have
a significant lower probability to become a full-time employee than part-time women
without children.
Since the evaluation of the leadership abilities of employees in grade 2 is only made once
(i.e. when the employee arrives in grade 2), no transitions are possible between the subgroups in grade 2. The two temporal subgroups in grade 2, created in the second stage of
the splitting-up process, are accepted as final subgroups (final subgroups 4 and 5).
Since the number of employees in the subgroups resulting from the third stage becomes rather
small (n ¼ 27 and 82), no further partitioning is considered. The two clusters are accepted as
final subgroups (final subgroups 2 and 3). Since none of the final selected regression models
Copyright # 2006 John Wiley & Sons, Ltd.
Appl. Stochastic Models Bus. Ind., 2006; 22:321–334
DOI: 10.1002/asmb
Average Linkage (Within Group)
Node 0
Cluster 1
Cluster 2
Cluster 1
Cluster 2
100.0 109
Adj. P-value=0.000, Chisquare=109.000, df=1
<= 0
Node 1
Node 2
Cluster 1
Cluster 2
Cluster 1
Cluster 2
Figure 4. Stage 3 in the splitting-up process: classification of part-time women in grade 1.
Table I. Estimated transition probabilities between the final groups.
Group 1
Group 2
Group 3
Group 4
Group 5
Group 6
contains time as an explanatory variable, it is not unreasonable to accept the assumption of
time-homogeneous transition probabilities. The next step the manpower planning analysis is
estimating those transition probabilities. Therefore, we use the estimator given by Anderson and
Goodman [20]. The transition probabilities are given in Table I. The estimated probabilities
clearly show the importance of the division of the different grades into more subgroups. Employees in grade 2 (final subgroups 4 and 5), for example, have very different promotion
probabilities, depending on their leadership abilities.
The main contribution of this paper is the presentation of a general framework to get more
homogeneous subgroups for using Markov chain theory in manpower planning. Although the
division in homogeneous subcategories is a crucial step in an application of those manpower
planning methods, up till now, literature has neglected to suggest a procedure to deal with this in
practice. The general framework results in groups that are homogeneous considering the
Copyright # 2006 John Wiley & Sons, Ltd.
Appl. Stochastic Models Bus. Ind., 2006; 22:321–334
DOI: 10.1002/asmb
observed sources of heterogeneity in the data set. Although, it might be that there exist some
latent sources of heterogeneity that are impossible to observe or for which data were not
collected by the Human Resource Information System. In case that the manpower planners use
regression analysis to support the general splitting-up approach, whether or not there are latent
sources of heterogeneity will be discovered by the results of this analysis. Regression analysis
provides indeed a measure to quantify the variation in the transition probabilities that is not
explained by the variables in the model. Ugwuowo and McClean [6] already proposed some
techniques to handle latent heterogeneity for modelling wastage, but the problem also exists for
the other flows within the personnel system. Using hidden Markov chain theory might settle this
difficulty. Our research center is presently investigating the application of this theory to manpower planning to be able to handle the problem of latent sources of heterogeneity in the
division of the personnel system into more homogeneous subgroups [26].
I am very grateful to the referees for their valuable comments, which resulted in a substantial improvement
of the paper. I would like to thank Prof. Dr M.-A. Guerry for her suggestions during the realization of the
1. Bartholomew DJ, Forbes AF, McClean SI. Statistical Techniques for Manpower Planning (2nd edn). Wiley:
Chichester, 1991.
2. Becket TP. Developing staffing strategies that work: implementing pragmatic, nontraditional approaches. Public
Personnel Management 2000; 29(4):465–476.
3. Bell D. New demands on manpower planning. Personnel management plus February 1994; 7.
4. Meehan RH, Ahmed SB. Forecasting human resources requirements: a demand model. Human Resource Planning
1990; 13(4):297–307.
5. Smith AR, Bartholomew DJ. Manpower planning in the United Kingdom: an historical review. Journal of the
Operational Research Society 1988; 39(3):235–248.
6. Ugwuowo FI, McClean SI. Modelling heterogeneity in a manpower system: a review. Applied Stochastic Models in
Business and Industry 2000; 16:99–110.
7. Parker B, Caine D. Holonic modeling: human resource planning and the two faces of Janus. International Journal of
Manpower 1996; 17(8):30–45.
8. McClean SI, Montgomery EJ. Estimation for semi-Markov manpower models in a stochastic environment. SemiMarkov Models and Applications, Janssen J, Limnios N (eds). Kluwer Academic Publishers: Dordrecht, 2000;
9. Yadavalli VSS, Natarajan R. A semi-Markov model of a manpower system. Stochastic Analysis and Applications.
2001; 19(6):1077–1086.
10. Georgiou AC, Vassiliou P-CG. Cost models in nonhomogeneous Markov systems. European Journal of Operations
Research 1997; 100:81–96.
11. Yadavalli VSS, Natarajan R, Udayabhaskaran S. Time dependent behaviour of stochastic models of manpower
system}impact of pressure on promotion. Stochastic Analysis and Applications 2002; 20(4):863–882.
12. Janssen J, Manca R. Salary cost evaluation by means of nonhomogeneous semi-Markov processes. Stochastic
Models 2002; 18(1):7–23.
13. McClean S, Montgomery E, Ugwouwo F. Non-homogeneous continuous time Markov and semi-Markov manpower models. Applied Stochastic Models and Data Analysis 1998; 13:191–198.
14. Bartholomew DJ. Maintaining a grade or age structure in a stochastic environment. Advances in Applied Probability
1977; 9:1–17.
15. Tsantas N. Ergodic behaviour of a Markov chain model in a stochastic environment. Mathematical Methods of
Operations Research 2001; 54:101–117.
Copyright # 2006 John Wiley & Sons, Ltd.
Appl. Stochastic Models Bus. Ind., 2006; 22:321–334
DOI: 10.1002/asmb
16. Georgiou AC, Tsantas N. Modelling recruitment training in mathematical human resource planning. Applied Stochastic Models in Business and Industry 2002; 18:53–74.
17. Guerry MA. Using fuzzy sets in manpower planning. Journal of Applied Probability 1999; 36(1):155–162.
18. Huang X, Soong S, McCarthy WH, Urist MM, Balch CM. Classification of localized melanoma by the exponential
survival trees method. Cancer 1997; 79(6):1122–1128.
19. Intrator O, Kooperberg C. Trees and splines in survival analysis. Statistical Methods in Medical Research 1995;
20. Anderson TW, Goodman LA. Statistical inferences about Markov chains. Annals of Mathematical Statistics 1957;
21. Bickenbach F, Bode E. Markov or not Markov}this should be the question. Paper presented at the 42nd Congress of
the European Regional Science Association, Dortmund, 27–31 August, 2002 (Kiel Institute of World Economics,
Germany Paper).
22. Vassiliou P. The evolution of the theory of non-homogeneous Markov systems. Applied Stochastic Models and Data
Analysis 1998; 13:159–176.
23. Neter J. Applied Linear Statistical Models (3rd edn). McGraw-Hill: Boston, U.S.A., 1996.
24. Quinlan JR. Improved use of continuous attributes in C4.5. Journal of Artificial Intelligence Research 1996; 4:77–90.
25. Saha Angshuman. Classification tree in Excel.
26. Guerry MA. Hidden Markov chains as a modelisation tool in manpower planning. Report MOSI/18. VUB-MOSI
Working Papers: Brussels, 2005.
Copyright # 2006 John Wiley & Sons, Ltd.
Appl. Stochastic Models Bus. Ind., 2006; 22:321–334
DOI: 10.1002/asmb
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