On the Non-Homogeneity of Markovian Manpower Model with Application

The effect of the heterogeneity in sex factor is considered in the Markovian manpower modeling model. The estimation of the transition probability matrix of the staff flow was done using the maximum likelihood method. The expected future durations of the individuals in the manpower system were obtained under the aggregated and disaggregated (male and female) staff flow as well as their variances. Test for the heterogeneity of the staff based on sex factor was performed and the result of the heterogeneity test shows that there is difference between the disaggregated staff flows. The computed entrant probabilities also indicate that the male group has higher probabilities of attaining higher grades than the female group.

Keywords:

Subject: Computer Science and Mathematics - Probability and Statistics

1. Introduction

Manpower modeling is a very important methodology in human resource management for businesses and industries [1,2,3]. Its general aim is to best match future manpower needs and resources to satisfy the organizational objectives[4]. However, manpower modeling involves three critical flows which includes; flow into the system through personnel recruitment, internal personnel flows across categories (via promotion), and the wastage flows by mean of retirement or resignation, death etc[5]. There are factors which are inherent in any manpower system [6] and must be put into consideration while modeling a manpower system. Apart from the factors associated to human behavior which exhibits high degree of variability in manpower system [1,6], there are other factors which are crucial in modeling manpower systems which includes grade (sub-class of individuals in form of salary bands or work function), duration variables (including age and length of stay in the system), gender, motivation, performance or commitment[2,6,7]. All these factors constitute heterogeneity in manpower system, and can be a subject of classifying members of manpower system into homogeneous groups [5]. However, [6,8] categorized these factors into observable and non-observable(latent) factors.

There are many approaches and consideration while modeling the heterogeneity in manpower system; see [1,3,8] and references therein. Guerry[9] considered the non-observable factors; the mover-stayer, to determine the personnel subgroups that have a higher propensity for homogeneity with respect to transition probabilities using markov-switching model. In the same vain, [8]extended the non-observable factors ‘mover-stayer’ to mover-mediocre-stayer and used the hidden markov model. A multinomial markov-switching model approach was used to investigate the non-observable factor (mover-mediocre-stayer) in intra-departmental and interdepartmental transition in departmentalized manpower system [6].

The assumption of homogeneity is usually central in the modeling manpower system irrespective of the approach used[4,9]. In the markovian model, the challenge of heterogeneity of units in a given state was examined by [10] and suggested the disaggregation of the state into homogeneous sub-groups or states. De Feyter [2] proposed a general framework of getting more homogeneous subgroups using Markov chain theory in manpower planning. However, [11] noted that the heterogeneity or otherwise the homogeneity of the individuals in a manpower system is evident on the time-heterogeneity of the transition probability of the Markov chain. Sales [12] defines the test statistic for testing the time-homogeneity of the transition probability matrix of a Markovian manpower model. There is no doubt that the disaggregation of units into homogeneous sub-groups in situations where heterogeneity exist within units is a way forward in having a reliable statistical inference in manpower modeling. In relation to manpower modeling, effort should be made to examine if there exist heterogeneity in the assumed classified (homogeneous) group. And as pointed out by [10], if a group is non-homogeneous and interest is to disaggregate the group into sub-homogeneous groups, to what point can the subgroups be reasonably homogeneous. Furthermore, [9] also noted that the increase in sub-homogeneous groups result to a greater number of parameters to be estimated thereby affecting the goodness-of-fit of the model.

In this paper, we examined the effect of heterogeneity in the sex group of the academic personnel system of the Ebonyi State University under the assumption of time-homogeneous Markov chain model. The remainder of the paper is organized as follows: Section 2 considered the Markovian manpower model including the maximum likelihood estimation of the transition probability matrix, and the expected future duration in the manpower system is considered in Section 3. The Predication of the future cohort size is considered in Section 4, while Section 5 considers the modeling of heterogeneity in sex factor. Finally, Section 5 and Section 6 considered the real-life application and conclusion, respectively.

2. Materials and Methods

Let

〈X_{t}, t \geq 0〉

denotes a stochastic process. Then, if

X_{t}

assumes a positive integer valued number and

P (X_{t} = j| X_{0}, X_{1}, X_{2}, \dots, X_{t} = i) = P (X_{t + 1} = j| X_{t - 1} = i)

X_{t}

is referred to as a time-homogeneous Markov chain. Say, for

i, j = 1,2, 3, \dots, H

P = {(P}_{i, j})

is the transition probability matrix (TPM) of the time-homogeneous Markov chain and

H

represents the number of states in the Markov chain.

Let

H

denotes the number of grades in a manpower system and the grades are mutually exclusive and collectively exhaustive;

t

denotes the time horizon of the manpower system (

t

is in year), and

n_{i} (t)

and

n_{i j} (t)

are the number of individuals in the

i^{t h}

grade at time

t,

and at time interval

(t - 1, t),

respectively. Individual in grade, say, 1 may move to grade 2 by promotion and may leave the system at the end of the year. The number of individuals leaving the system at time t is denoted by

W_{i} (t) .

The probability that an individual in grade

i

moves to grade

j

at time interval

(t - 1, t)

is represented by

p_{i j}

and

p_{i 0} (t)

represents the probability of leaving the system. Let

N

denotes the matrix of flow indicating

n_{i j} (t)

, then

N

is defined as

N = [\begin{matrix} n_{11} (t) & \dots & n_{1 H} (t) \\ ⋮ & ⋱ & ⋮ \\ n_{H 1} (t) & \dots & n_{H H} (t) \end{matrix}];

(1)

and the TPM,

P,

is given by

P = [\begin{matrix} p_{11} & \dots & p_{1 H} \\ ⋮ & ⋱ & ⋮ \\ p_{H 1} & \dots & p_{H H} \end{matrix}] = (P_{i j});

(2)

where

p_{i j} \geq 0

. Define

n_{i} (t) = \sum_{H} n_{i j} (t)

, the maximum likelihood estimate,

\hat{P},

P

is given by

\hat{P} = \begin{matrix} 1 \\ ⋮ \\ h \end{matrix} [\begin{matrix} {\hat{p}}_{11} & \dots & {\hat{p}}_{1 H} \\ ⋮ & ⋱ & ⋮ \\ {\hat{p}}_{H 1} & \dots & {\hat{p}}_{H H} \end{matrix}] = ({\hat{P}}_{i j});

(3)

where

{\hat{p}}_{i j} = \frac{n_{i j} (t)}{n_{i} (t)}

. However, for over the time period

= 1,2, \dots, T,

{\hat{p}}_{i j} = \frac{\sum_{t = 1}^{T} n_{i j} (t)}{\sum_{t = 1}^{T} n_{i} (t)}

Let

R_{j} (t)

denote the number of new individuals in grade

j

at time

t,

respectively. Then

p_{0 j}

is the associated probability of entering the

j^{t h}

grades. Then according to [13], the total number of individuals in the manpower system can be obtained using

N_{j} (t + 1) = \sum_{i = 1}^{H} P_{i j} (t) N_{i} (t) + R_{i} (t); j = 1,2, \dots, H

(2)

2.1. Expected Future Duration in a Manpower System

It is very critical in manpower planning to determine in advance how long an employee is expected to last while in the system. According to[5], it serves as a measure of the career prospect of an employee. The expected future duration,

μ

, can be obtained as

μ = {(1 - P)}^{- 1};

(3)

where

μ = μ_{i j}

is a matrix of the same dimension with the TPM,

P .

The corresponding variance of the expected future duration, according to [14], is given by

σ_{i j}^{2} = \{\begin{matrix} 2 μ_{i j} μ_{j j} - μ_{i j} - μ_{i j}^{2} i < j \\ μ_{i i}^{2} - μ_{i i} i = j \end{matrix}

(4)

Furthermore, the probability of an entrant in grade

i

attaining higher grade, denoted by

λ_{i j}

, is given by

λ_{i j} = \{\begin{matrix} \frac{μ_{i j}}{μ_{j j}}; i < j \\ \frac{μ_{i i}}{μ_{i i}}; i = j \end{matrix}

(5)

2.2. Prediction of Future Cohort Sizes

Planning for future recruitment and the determination the size of the recruit are important aspects of manpower system[2,15,16]. Let

n (t)

be a vector comprising elements

n_{i} (t),

then the predicted future cohort sizes denoted by

n (t + 1)

is defined as

n (t + 1) = Q Υ (t);

(6)

where

Q = p^{0} P

, and

Υ (t) = \sum_{i = 1}^{H} n_{i} (t)

. The parameter

P

is the TPM and

p^{0} = n_{i} (t) / Υ (t)

is the probability vector of the based year.

2.3. Modeling the Heterogeneity

We consider the number of individuals in the manpower system consists of male and female individuals which constitute heterogeneity. Accordingly, the number of individuals in the manpower system is disaggregated into two homogeneous groups with the corresponding matrix of flows, respectively, denoted as

N_{1}

and

N_{2}

. Similarly, the estimated transition probability matrix, the expected future duration, the variance of the expected future duration, and the probability of an entrant in grade

i

attaining higher grade for the two groups, respectively, denoted as

{\hat{P}}_{1}

and

{\hat{P}}_{2},

μ_{1}

and

μ_{2},

σ_{1}^{2}

and

σ_{2}^{2},

and

λ_{1 i j}

and

λ_{2 i j}

Considering the existence of heterogeneity in the sex category of the manpower population under the time-homogeneous Markov model, a test to validate the assumption is

H_{0} : σ_{1}^{* 2} = σ_{2}^{* 2} v s H_{1} : σ_{1}^{* 2} \neq σ_{2}^{* 2}

. The corresponding test statistic is

F = σ_{2}^{* 2} / σ_{1}^{* 2};

where

σ_{i}^{* 2}, i = 1,2

is the mean variance of the expected future duration.

3. Results

In order to illustrate the effect of heterogeneity in the sex category under time-homogeneous Markovian manpower model, data from the manpower system of the Ebonyi state University, Nigeria were used. The data set covers a period between 2013/2014 to 2016/2017 academic session. Table 1, Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, Table 9, Table 10, Table 11 and Table 12 show the aggregated and disaggregated staff matrix of flow of the period considered.

Table 1. Aggregated(

N (t)

) staff matrix of flow for 2013/2014 session.

Table 1. Aggregated(

N (t)

) staff matrix of flow for 2013/2014 session.

	1	2	3	4	5	6	7	$W_{i}$	$n_{i} (t)$
1	51	0	0	0	0	0	0	0	51
2	0	95	3	0	0	0	0	1	99
3	0	0	138	3	0	0	0	2	143
4	0	0	0	135	9	0	0	2	146
5	0	0	0	0	140	4	0	1	145
6	0	0	0	0	0	51	3	2	56
7	0	0	0	0	0	0	38	2	40
$R_{j}$	11	11	29	15	8	0	1		75

Table 2. Disaggregated (

N_{1} (t)

) staff matrix of flow for 2013/2014 session.

Table 2. Disaggregated (

N_{1} (t)

) staff matrix of flow for 2013/2014 session.

	1	2	3	4	5	6	7	$W_{i}$	$n_{i} (t)$
1	31	0	0	0	0	0	0	0	31
2	0	67	2	0	0	0	0	1	70
3	0	0	110	2	0	0	0	2	114
4	0	0	0	81	5	0	0	1	87
5	0	0	0	0	98	3	0	1	102
6	0	0	0	0	0	41	2	2	45
7	0	0	0	0	0	0	28	1	29
$R_{1 j}$	9	9	23	12	6	0	1		60

Table 3. Disaggregated (

N_{2} (t)

) staff matrix of flow for 2013/2014 session.

Table 3. Disaggregated (

N_{2} (t)

) staff matrix of flow for 2013/2014 session.

	1	2	3	4	5	6	7	$W_{i}$	$n_{i} (t)$
1	20	0	0	0	0	0	0	0	20
2	0	28	1	0	0	0	0	0	29
3	0	0	28	1	0	0	0	0	29
4	0	0	0	54	4	0	0	1	59
5	0	0	0	0	42	1	0	0	43
6	0	0	0	0	0	10	1	0	11
7	0	0	0	0	0	0	10	1	11
$R_{2 j}$	2	2	6	3	2	0	0		15

Table 4. Aggregated (

N (t)