Page 1 of 8

Journal for Studies in Management and Planning

Available at http://internationaljournalofresearch.org/index.php/JSMaP

e-ISSN: 2395-0463

Volume 01 Issue 09

October 2015

Available online: http://internationaljournalofresearch.org/ P a g e | 351

Markov Chain Models in Discrete Time Space and

Application to Personnel Management

1.C. E. Okorie

Department of Mathematics and Statistics, Federal University Wukari, Taraba State, Nigeria.

E-mail of corresponding author: Chyokanmelu@yahoo.com

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 0.21 of the entire personnel

and that of the teachers retained but no

promotion is 0.52 while new recruitment is

0.27.

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

Page 2 of 8

Journal for Studies in Management and Planning

Available at http://internationaljournalofresearch.org/index.php/JSMaP

e-ISSN: 2395-0463

Volume 01 Issue 09

October 2015

Available online: http://internationaljournalofresearch.org/ P a g e | 352

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:

(1) Develop analytical time-discrete

push and pull models.

(2) Consider constant promotion

probabilities over time.

(3) 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

Nt  p tN t P t R t

i d d

k

j ij

    

1

1

(1.1)

Page 3 of 8

Journal for Studies in Management and Planning

Available at http://internationaljournalofresearch.org/index.php/JSMaP

e-ISSN: 2395-0463

Volume 01 Issue 09

October 2015

Available online: http://internationaljournalofresearch.org/ P a g e | 353

for j = 1, 2, ... . . k

Where t is the current time period and

(t + 1) 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

1   

 P t P t

d

k

j ij

(1.2)

For all i = 1, 2, ... . . k

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

probabilityPij(t) of moving from state

i to state j in a time period t. Let

ej(< 1) represent the proportion of

staff to be promoted from grade level

j − 1 to j .The (1 − ej) represent the

proportion of newly recruited staff to

grade level j.

As stated earlier Pd

(t) and Pd

(t)Nd

(t)

are the probabilities of double promotion

respectively.

From period t to period t + 1 and

starting from the highest grade level k.

 

  N t P tN t P tN t P tN t P t

k k k k kk k d d k

     1 1

1

(1.3)

But , from (1.2)and (1.3).

P t P t

kk d

1

(1.4)

Where in the highest grade k, Pd

(t) is the

probability of double promotion into grade

k in period t i.e. Pd

(t) = P(k−2)k

(t).

 

  P t P t

d k 2 k



And substituting for Pkk in (1.3) and

simplifying we obtain:

         1 1          1

'

1 1           P t N t R t N t P t N t P t N t N t

k k k k k d k d d k

(1.5)

N′k

(t + 1) can be easily determined since

Pd

(t) is assumed known and given.

Since the number of promotions and

recruitments to grade k should follow the

ratio

ek: (1 − ek

) respectively, it follows that

 

     1

'

1 1

    P t N t e N t

k k k k k

(1.6)

And

  1   1

'

R t   e N t  k k k

(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),

 

 

  













 







t

t

t

N

e N

P

k

k k

k k

1

'

( 1)

1

(1.8)

Equation (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.

P t P i t p t

d

ij j

1 1 

(1.9)

j = 1, 2, ... . . k − 1

For example, at j = k − 1, we have;