MARKOV CHAIN MODELS IN DISCRETE TIME SPACE AND APPLICATION TO PERSONNEL MANAGEMENT

.C. E. Okorie

ABSTRACT

A Markov chain probability model is found to fit personnel data of recruitment and promotion pattern in El-Amin International School, Minna. Manpower planning is a useful tool for human resource management in large organizations. Classical Manpower Planning models are analytical time –discrete push and pull models. A mixed push-pull model is developed for in this study. This model allows taking into account push and pull transitions of employees through an organization at the same time. In fact, in any organization, the present number of staff in each level is known and at any particular time each member of staff is in a particular grade either by promotion or recruitment into that grade, We consider a Markov model formulated to assist in making promotion, recruitment policies for the next time period given that existing staff structure is known.  Data from El-Amin International School, Minna is used on the formulated model. The data is collected for a period of ten years,from 2000-2010 The result shows that the probability of those on promotion is  of the entire personnel and that of the teachers retained but no promotion is  while new recruitment is .

Keywords: Stochastic, Transition, Markov chain, recruitment, promotion, pull, push.

INTRODUCTION

Manpower systems are hierarchical in nature and consists of a finite number ordered grades for which internal movement or promotion of staff is possible from one grade to another though there is no promotion beyond the highest grade. Members of staff in the same grade have certain common characteristics and attributes (such as rank, trade, age, or experience) and the grades are mutually exclusive and exhaustive so that any staff must belong to one but only one grade at any time (Georgiou and Tsantas, 2002)

Markov chain theory is one of the Mathematical tools used to investigate dynamic behaviours of a system (e.g. workforce system, financial system, health service system) in a special type of discrete-time stochastic process in which the time evolution of the system is described by a set of random variables. It is worth mentioning that variables are called random if their values cannot be predicted with certain and discrete-time means that the state of the system can be viewed only at discrete instant rather than at any time (Howard, 1971).

Stochastic model are influential and have been used widely in health care management. Markov chain models have been applied to many areas of health related problems (Parker and Caine, 1996). Some mathematical models of diseases in populations (Epidemometric models) have also been employed to study leprosy disease. McClean (1991) has studied, the incidences of new cases as a result of prevalence of overt cases, using different equation and McClean and Montgomery (2000) have adopted the kinship coefficient to determine correlation between leprosy rates in village of different distances apart. The mixed push –pull model is capable of incorporating this additional constraint. This model is based on the assumption of the classical pull models, in which vacancies arise in case that the number of employees in a specific group is less than the desired one. It allows the organization to choose a policy to fulfil those vacancies. According to the pull strategy, the vacancies are filled by promotions or by external recruitment. Besides, in the mixed push-pull model, push promotions are possible in case not enough people had the opportunity to promote after all vacancies at higher levels were filled.

Homogeneous Markov chain models having time independent (or stationary) transition probability have been applied to manpower planning in Winston (1994), Bartholomew et al (1991) and Ekoko (2006).  Alem (1985) and krishnamurty (1988) asserted manpower mobility from one organization  to another results in policies of promotion and recruitment that are within a systematic and qualitative framework in some sectors of the economy. The bivariate model in Raghavendra (1991) is a non –homogeneous Markov model by which promotion and recruitment policies are derived given the required future structure.  The model in Raghavendra (1991) uses two fundamental equations: one is the probability equation and the other is for determining the number of staff in each grade in the next time period.

Aim and Objectives

The ultimate aim of this study is to apply Markov model for recruitment and promotion systems and the objectives include the following:

  • Develop analytical time-discrete push and pull models.
  • Consider constant promotion probabilities over time.
  • Estimate transition and the future number of employees in an organization using push and pull models.

MATERIALS AND METHODS

Mixed push and pull mode is developed for this study. The model uses the assumptions that push as well as pull promotions are possible to occur in the same system at the same time. An example of a personnel system requiring a model in which both push and pull transitions occur, is an organization in which vacancies are filled by promotions from groups of employees that succeeded in an examination. A transition between the group of people that not yet passed an examination and the group of people that succeeded in the examination happens with a certain probability. This is a typical push movement. Meanwhile, the actual promotion (only if there is a vacancy at another level) has to be considered as a pull transition. A mixed push-pull model has an advantage from the practitioner’s point of view. Often, organizations promote employees because of several reasons: Obviously, vacancies at higher levels can be filled by promotions from lower levels. The mixed push-pull model allows considering several reasons for promotion at the same time. Under Markovian assumptions, the equation for determining the number of staff in each grade in the next time period is

                                     (1.1)

Where  is the current time period and (  is the next time period.

Members of staff could stay in the same grade, move to another grade or leave the system. There is therefore a probability equation that governs the way promotion is carried out in each level. The probability equation is given as;

                                                                         (1.2)

For all

 The promotion and new recruitment to any grade in an organization follows a prescribed policy as expressed in Krishnamurthy, (1988). These proportion and recruitment are specified in the policy to be translated into estimates of the probability   of moving from state  to state  in a time period . Let  represent the proportion of staff to be promoted from grade level  .The  represent the proportion of newly recruited staff to grade level .

As stated earlier  are the probabilities of double promotion respectively.

From period  and starting from the highest grade level

                   (1.3)

But , from

                                                                                                  (1.4)

Where in the highest grade  is the probability of double promotion into grade  in period  i.e. .

And substituting for  in (1.3) and simplifying we obtain:

(1.5)

 can be easily determined  since  is assumed known and given.

Since the number of promotions and recruitments to grade  should follow the ratio

 respectively, it follows that

                                     (1.6)

And                                   (1.7)

Equations (1.6) and (1.7) would give the number of promotions from grade (k-1) to k and the number of new recruitments to grade k respectively. From (1.6),

                                                 (1.8)

Equation (1.6) and (1.7) would give the number of promotions from grade  to  and the number of new recruitments to grade  respectively.

                                                 (1.9)

For example, at  we have;

                                   (1.10)

Using equations (1.7) and (1.8) and (1.9), the number of promotions, the number of recruitments and the transition probabilities can be estimated successfully for all other state of the organization.

Development of the model and model assumptions-The Mixed Push-pull Model

We consider an organization in which the total population of employees is divided into  homogeneous groups. The homogeneous groups form a partition of the total population. The number of people in group  at time  is denoted by  We use a discrete time scale.  The length of one time interval is chosen in such a way that it can be assumed that one employee can make at most one transition during the time interval. This implicates the assumption that vacancies are not filled instantaneous. This is a realistic hypothesis since it takes time to perform a promotion or recruitment decision. This assumption also implicates that a vacancy does not disappear in the company when it is filled by a promotion. When an employee is promoted from group  to group , the initial vacancy in group  created a new vacancy in group .

In fact, the initial vacancy moves in the opposite direction of the employee. This means that  is possibly smaller than the desired number of employees  at time .

Classical pull models most often assume that vacancies which need to be filled in the next time interval are determined at the end of the period in which they turned up.

This way, the model is given by;

(1.11)

Where

 being a row vector formed by the vacancies in every group to be filled in time interval

 matrix with elements

 diagonal matrix formed by the voluntary wastage probabilities

 is the probability that an employee in group  will leave the organization in time period

 denoting the row vector  and  denoting the  row stock vector .

         (1.12)

                                              (1.13)

 becomes the identity matrix because there will be no push promotions. The push recruitments  also becomes zero. The model (1.12) becomes:

                                                                (1.14) For computational reasons, we put

 and

(1.12) becomes:

                                                                        (1.15)

We use the Jordan normal form theorem (Gantmacher,1964), which allows us to rewrite  as;

                                                                              (1.16)

With  non-singular matrix and  a block diagonal matrix with m the number of eigenvalues of  and

                                   (1.17)

With  a  matrix with eigenvalue

 To incorporate the vacancies arising out of all pull promotions in the coming time period, (1.11) in the mixed push-pull model needs to be replaced by:

                     (1.18)

Since the vacancies within one time period evolve as a chain satisfying the Markov properties with  acting as a transition matrix, the initial vacancies as estimated by (1.11) need to be multiplied by the fundamental matrix. Indeed, it is very well known that the fundamental matrix gives the expected number of visits to each state before absorption occurs (Bartholomew et al, 1991).

Results and  Discussion

The implication of models in this study on a common data for the purpose of comparison is the concern in this chapter. A teacher is seen to be recruited and also promoted when the conditions specified are met. The states considered are promotion, retained and recruited status of the teacher available over the period.

Discrete State time Markov model. A follow-up summary statistics for 2009/2010 academic session on 130 teachers in El-Amin International school, Minna provides the following transition matrices for the second term. The transition count matrix for the number of teachers in first term

Is given as:

The transition count matrix for those who returned back to the school in second tern and those who were recruited to make up the required  teachers needed is ;

The following estimates of transition probabilities are obtained

In this study of Attitudinal  changes of teachers moving from this school and coming into this school based on the grade/ rank, it is now pertinent to ask whether these three sets of transition probabilities reflect the same behaviour on the part of teachers from one term to another. If so, the data can be pooled to give a single transition count matrix and hence a single set of estimates.

The pooled transition count matrix obtained is;

The pooled estimates of elements of the transition probability matrix obtained are;

The model can be represented by a single transition count matrix.

Thus, the maximum likelihood estimate of the transition probability matrix is given by;

Calculating we have,

Corrected to 2 decimal places and for , we find that gets closer to exactly  that is as  increases,

The limit state probability vector is given by

This shows that  of the teachers get promoted,  are retained in the school but without promotion and  of the teachers are recruited in the school at the beginning of the term.

Based on the plan of this school the desired personnel distribution  is fixed over time and it is given by

The current personnel size in every group is smaller than the desired ones. Consequently, both aspects of the mixed push and pull model have an influence on the changes or transition.

Under this recruitment policy

This school never reaches the desired personnel structure  The decision maker might consider changing its policy. So the optimal recruitment policy is

It is clear that there exist a structural problems in this school. The promotion system is not compatible with the desired personnel structure. So the school should consider trying to influence and change its promotion system.

Conclusion

According to the optimal recruitment policy, it is clear that there exists a structural problem in the organization studies (El-Amin International school, Minna). The promotion system is not compatible with the desired personnel structure. The school should reconsider its (push) promotion and or recruitment policy to increase its personnel size to the desired personnel size.Also, the organization should consider trying to influence and change its promotion system.

Aknowledgement and References

[1] Abubakar,U.Y.(2004) ’’A Three-State-Semi Markov Model in Continuous Time to study the Relapse Cases of Leprosy Disease After the Treatment Using Dapsone and Multi-Drug Therapy (M D T)’’; Proceeding of the 41st Annual National Conference of Mathematical Association  Nigeria. Page 15-23

[2] Alam, M.A.(1985), Steady State Career Structure Model

For Indian Army Officers, Defence Mgmt. Vol.12, 34-42

[3] Bartholomew, D. J., Forbes A. F.,&McClean, S. I. (1991). Statistical techniques for manpower planning, Chichester; Wiley.

[4] De Feyter, T. (2006),Modelling heterogeneity in Manpoer Planning; dividing the

personnel system in more homogeneous subgroups. Applied Stochastic Models in Business and Industry, 22(4), 321-334.

[5] Georgious, A. C. &Tsantas, N. (2002). Modelling recruitment training in

mathematical human resource planning. Applied Stocastic Models in Business and Industry,18,53-74.

[6] Ekoro,P.O., (2006).’’Analyzing Academic Staff Ptomotion Criteria in Nigeria Universities Using a Nonergordic Markov chain Model, The Nigerian Academic Forum, Vol. 11,No. 3 130-137.

[7] Krishnamurthy,G.S.(1988),’’Out of Turn Promotions: Government in a Fix,’’Deccan Herald

[8 ]McClean, S. (1991). Manpower planning models and their estimation. European Journal of Operation Research, 51, 179-187.

[9] McClean, S.I. &Montgomery, E. J. (2000). Estimation for semi-Markov manpower models in a stocastic environment.In.J.Janssen & N.Limnios (Eds.), Semi-Markov models and applicationa (pp. 219-227). Dordrecht: Kluwer Academic.

[10] Parker, B.,& Caine, D. (1996). Holonic modeling: human resource planning and the two faces of Janus, International Journal of Manpower, 17(8), 30-45.

[11] Raghavendra, B. G.(1991), “ A Bivariate Model for Markov manpower planning system, Journal Opl.Res.Soc.Vol. 42, No. 7, 565-570.

[12] Staff Condition of Service of El-Amin International School, Minna. Revised Edition (2008), page 3and 7.

 

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