APPLIED STOCHASTIC MODELS IN BUSINESS AND INDUSTRY Appl. Stochastic Models Bus. Ind., 2006; 22:321–334 Published online 1 March 2006 in Wiley InterScience (www.interscience.wiley.com). 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 SUMMARY 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 KEY WORDS: manpower planning; homogeneity; stochastic models; Markov models 1. INTRODUCTION 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. y E-mail: [email protected] Copyright # 2006 John Wiley & Sons, Ltd. 322 T. De FEYTER 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 HETEROGENEITY IN PERSONNEL 323 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. 2. MODEL PREPARATION FOR MARKOV MANPOWER PLANNING SYSTEMS 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 324 T. De FEYTER 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 system. 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 Dataset Split-up w.r.t. study objective All characteristics used No Groups too small for further division No Stepwise split-up w.r.t. transition probabilities between temporal groups Yes Yes 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 325 HETEROGENEITY IN PERSONNEL 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 Nij Ni with Nij ¼ Copyright # 2006 John Wiley & Sons, Ltd. X t nij ðtÞ and Ni ¼ X ni ðt 1Þ t Appl. Stochastic Models Bus. Ind., 2006; 22:321–334 DOI: 10.1002/asmb 326 T. De FEYTER Start stepwise split-up No All transitions between temporal groups investigated No Split-up w.r.t. characteristic that contributes most to homogeneity All characteristics used No Groups too small for further division Yes Yes Yes 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 procedure: * 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 HETEROGENEITY IN PERSONNEL * 327 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]. 3. STATISTICAL SUPPORT FOR THE GENERAL SPLITTING-UP PROCEDURE 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 328 T. De FEYTER 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 HETEROGENEITY IN PERSONNEL 329 4. CLASSIFICATION RULES FOR NEW RECRUITS 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. 5. ILLUSTRATION 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 330 T. De FEYTER 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): log * p# 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 HETEROGENEITY IN PERSONNEL 331 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 332 T. De FEYTER Average Linkage (Within Group) Node 0 n Category % Cluster 1 Cluster 2 Cluster 1 Cluster 2 Total 24.8 75.2 27 82 100.0 109 Children Adj. P-value=0.000, Chisquare=109.000, df=1 <= 0 >0 Node 1 Category % n Node 2 Category % Cluster 1 Cluster 2 100.0 0.0 7 0 Cluster 1 Cluster 2 24.8 27 Total Total n 0.0 100.0 0 82 75.2 82 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 Group Group Group Group Group 1 2 3 4 5 6 Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Wastage 0.769 0.662 0.024 0 0 0 0 0.148 0 0 0 0 0 0.005 0.768 0 0 0 0.063 0.037 0.049 0.788 0 0 0.05 0 0 0 0.728 0 0 0 0 0.073 0.253 0.90 0.118 0.148 0.159 0.139 0.019 0.10 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. 6. CONCLUSION AND FURTHER RESEARCH 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 HETEROGENEITY IN PERSONNEL 333 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]. 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