Abstract: In this paper, we consider the Manpower System of an organization with two groups. Breakdown occurs in the two groups of the Manpower System due to attrition process. Group A consists of employees other than top management level executives; group B consists of top management level executives. Two models are studied in this paper. In Model-1, group A is exposed to Cumulative Shortage Process (CSP) due to attrition and group B has an exponential life distribution. In Model-2, both the groups A and B are exposed to CSP and have exponential thresholds. Inter occurrence time of shortages to them have exponential distribution. In each model, two cases are discussed. In one case, after the threshold, recruitment policy to compensate the shortages one by one is followed. Joint Laplace transforms of Time to Recruit and Recruitment time have been found. In the second case, recruitment policy of filling vacancies simultaneously is followed. Here, marginal Recruitment time distributions have been obtained

R Ramanarayanan

S Mythili

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International Journal of Pure and Applied Mathematics
Volume 77 No. 4 2012, 533-542
ISSN: 1311-8080 (printed version)
url: http://www.ijpam.eu
PA
ijpam.eu
PROBABILISTIC ANALYSIS OF TIME TO RECRUIT
AND RECRUITMENT TIME IN MANPOWER
SYSTEM WITH TWO GROUPS
S. Mythili1 §, R. Ramanarayanan2
1Department of Mathematics
Vel Tech Multi Tech Dr. Rangarajan
Dr. Sakuntala Engineering College
Avadi, Chennai, Tamil Nadu, INDIA
2Department of S&H
Vel Tech Dr. RR & Dr. SR Technical University
Avadi, Chennai, Tamil Nadu, INDIA
Abstract: In this paper, we consider the Manpower System of an organiza-
tion with two groups. Breakdown occurs in the two groups of the Manpower
System due to attrition process. Group A consists of employees other than top
management level executives; group B consists of top management level exec-
utives. Two models are studied in this paper. In Model-1, group A is exposed
to Cumulative Shortage Process (CSP) due to attrition and group B has an
exponential life distribution. In Model-2, both the groups A and B are exposed
to CSP and have exponential thresholds. Inter occurrence time of shortages
to them have exponential distribution. In each model, two cases are discussed.
In one case, after the threshold, recruitment policy to compensate the short-
ages one by one is followed. Joint Laplace transforms of Time to Recruit and
Recruitment time have been found. In the second case, recruitment policy of
filling vacancies simultaneously is followed. Here, marginal Recruitment time
distributions have been obtained.
AMS Subject Classification: 90B05
Received: February 6, 2012 c© 2012 Academic Publications, Ltd.
url: www.acadpubl.eu
§Correspondence author
534 S. Mythili, R. Ramanarayanan
Key Words: manpower system, attrition, shortage, cumulative shortage
process
1. Introduction
In an organization, the total flow out of the Manpower System (MPS) is termed
as shortage. The flow out of the MPS of an organization happens due to res-
ignation, dismissal and death. The shortages that have occurred due to the
outflow of manpower should be compensated by recruitment. But recruitment
cannot be made frequently since it involves cost. Therefore, the MPS is allowed
to undergo Cumulative Shortage Process (CSP). The basic idea is that accu-
mulating random amount of shortages due to successive attritions leads to the
breakdown of the system when the total shortage crosses a random threshold
level. The breakdown point or the threshold can also be interpreted as that
point at which immediate recruitment is necessitated.
The shortage of MPS depends on individual propensity to leave the orga-
nization which in turn depends on various factors as discussed before. Such
models have been discussed by Grinold and Marshall [7], Bartholomew and
Forbes [8]. For statistical approach one may refer to Bartholomew [1]. Lesson
[9] has given methods to compute shortages (Resignations, Dismissals, Deaths)
and promotion intensities which produce the proportions corresponding to some
descried planning proposals. Markovian models are designed for shortage and
promotion in MPS by Vassiliou [5]. Subramanian. V [10] has made an attempt
to provide optimal policy for recruitment, training, promotion and shortages in
manpower planning models with special provisions such as time bound promo-
tions, cost of training and voluntary retirement schemes.
Esary et al. [3] have discussed that any component or decice, when exposed
to shocks which cause damage to the device or system, is likely to fail when the
total accumulated damage exceeds a level called threshold. R. Ramanarayanan
and G. Sankaranarayanan [6], in their paper “On Correlated Life and Repair
times” have studied the same using Gaver D.P. [2]. Linton D.G. and Saw
J.G. [4] have done Reliability analysis of k-out-of-n : F system. In mapower
planning, threshold having SCBZ (setting the clock back to zero) property
has been discussed and the expected time to recruitment has been found by
Sathiyamoorthy R. and Parthasarathy. S. [11]. For MPS having threshold that
follows Exponentiated Exponential distribution, expected time to recruitment
and variance have been obtained by S. Parthasarathy and R. Vinoth [12].
In this paper, two groups of the MPS are exposed to shortage process.
PROBABILISTIC ANALYSIS OF TIME TO RECRUIT... 535
Group A consists of employees of MPS other than top level management exec-
utives. Group B consists of top level management executives.
We assume that after the threshold of MPS, recruitment takes places using
recruitment policies 1) recruitment for vacancies due to shortages one by one
(Additive Recruitment Policy) and 2) recruitment for vacancies due to shortages
simultaneously (Parallel Recruitment Policy).
Two models are discussed in this paper. In Model-1, group A is exposed to
CSP and group B has exponential life distribution. In Model-2, both the groups
A and B have exponential thresholds and are exposed to CSP. Inter-occurrence
time of shortages to them have exponential distribution.
In models 1 and 2 with additive recruitment policy, joint Laplace Transforms
of Time to recruit and Recruitment time are obtained. In both the models
with parallel recruitment policy, marginal Recruitment time distributions are
obtained.
2. Model-1
2.1. Model Description
In this model, MPS with two groups A and B is considered. Group A has em-
ployees of MPS other than top level management executives. Group B consists
of top level management executives. Group A is exposed to CSP and group B
has exponential life distribution.
The time to breakdown of MPS is T = min{T1, T2} where T1 and T2 are
the times to breakdown of groups A and B respectively. We assume that the
recruitment time R of a shortage is independent of the shortage magnitude.
2.2. Assumptions
1. Shortages occur in group A in accordance with inter occurrence time
distribution F (x) and
∫ ∞
0
xdF (x) < ∞.
2. N is the random number of shortages required for breakdown of group A.
The group survives k shortages with an arbitrary but known probability
Pk(= P (N > k)) for k = 0, 1, 2, . . . , {Pk} is assumed to be a decreasing
sequence of numbers in the unit interval subject only to
∞
∑
k=0
Pk < ∞,
P0 = 1.
536 S. Mythili, R. Ramanarayanan
3. Let pk(= Pk−1 − Pk), the probability that group A has a breakdown on
the kth shortage, be given by the generating function φ(S), 0 ≤ S ≤ 1.
4. T2 has an exponential distribution with parameter µ.
5. At time 0, the MPS comprising of the two groups A and B is free of
shortage due to manpower loss and put into operation. Groups A and B
function independently during time to recruit.
6. When the breakdown of MPS occurs, recruitment begins immediately.
Productive work does not happen in group B when the system breakdown
occurs due to group A. When it occurs due to group B, productive work
does not happen in group A and recruitment takes place for group A.
7. Recruitment time of the ith shortage is Ri. Ri’s are independent and
identically distributed r.v. with distribution function R(y), indepen-
dent of the shortage magnitudes. Recruitment time for group B is as-
sumed to be R independent of Ri’s but with distribution function R(y),
∫ ∞
0
ydR(y) < ∞.
2.3. Model-1.1
In this model, we assume that as per additive recruitment policy, recruitments
are done for all the shortages one at a time after time T . RT = R1 + R2 +
· · ·+Ri where i is the number of shortages occurred during T . When the MPS
breakdown occurs due to group A, we have to recruit for N shortages. Hence,
RT = R1 +R2 + · · ·+RN . In the other case, when the MPS breakdown occurs
due to group B, we have, RT = R1 +R2 + · · ·+Rn+1 as we have to rejuvenate
group B also with the n (< N) shortages that occurred in group A during T .
The joint distribution of T and RT is given as
P (T ≤ x,RT ≤ y) = P (T1 ≤ x,RT ≤ y, T2 > T1)
+ P (T2 ≤ x,RT ≤ y, T1 > T2)
=
∞
∑
n=1
∞
∑
n=1
pnRn(y)
∫ x
0
e−µzdFn(z)
+
∞
∑
n=0
PnRn+1(y)
∫ x
0
µe−µz [Fn(z)− Fn+1(z)]dz (2.1)
where Rn(y) and Fn(y) are the n fold convolutions of R(y) and F (y) respec-
tively. Fn(y) = 1 for y ≥ 0; Fn(y) = 0, otherwise.
PROBABILISTIC ANALYSIS OF TIME TO RECRUIT... 537
The term under the first summation symbol of the right side of (2.1) is the
joint probability that the MPS breakdown occurs due to group A on the nth
shortage during (0, x) and
n
∑
i=1
Ri ≤ y. The term under the second summation
symbol is the joint probability that MPS breakdown occurs due to group B
during (0, x), group A survives the n shortages occurred to it before threshold
level and
n+1
∑
i=1
Ri ≤ y.
From (2.1), we get,
E(e−ǫT e−ηRT ) =
∫ ∞
0
∫ ∞
0
e−ǫxe−ηy
∞
∑
n=1
pne
−µxdRn(y)dFn(x)
+
∫ ∞
0
∫ ∞
0
e−ǫxe−ηy
∞
∑
n=0
Pnµe
−µx[Fn(x)− Fn+1(x)]dRn+1(y)dx
=
∞
∑
n=1
pn[R
∗(η)F ∗(µ+ ǫ)]n
+
∞
∑
n=0
Pn[R
∗(η)]n+1[1− F ∗(µ+ ǫ)][F ∗(µ + ǫ)]n
µ
µ+ ǫ
where * denotes Laplace-Stieltjes transform.
Using the fact that Pi = 1−
i
∑
j=1
pj ,
E(e−ǫT e−ηRT ) = φ[R∗(η)F ∗(µ + ǫ)]
+
µR∗(η)[1 − F ∗(µ+ ǫ)]
(µ+ ǫ)[1−R∗(η)F ∗(µ+ ǫ)]
{1− φ[R∗(η)F ∗(µ+ ǫ)]} (2.2)
The dependence of Recruitment time on the corresponding Time to recruit
can be seen easily. From (2.2), for η = 0, ǫ = 0 we get respectively,
E(e−ǫT ) =
µ+ ǫφ[F ∗(µ + ǫ)]
(µ+ ǫ)
(2.3)
E(e−ηRT ) = φ[F ∗(µ)R∗(η)]
+
R∗(η)[1 − F ∗(µ)]
[1−R∗(η)F ∗(µ)]
{1− φ[R∗(η)F ∗(µ)]} (2.4)
538 S. Mythili, R. Ramanarayanan
From (2.3) and (2.4), by differentiation,
E(T ) =
1− φ(F ∗(µ))
µ
(2.5)
E(RT ) =
−R∗′(0){1 − φ(F ∗(µ))}
[1− F ∗(µ)]
(2.6)
where ′ denotes differentiation.
Similarly, from (2.2) we get,
E(TR) =
−R∗′(0)
[1− F ∗(µ)]
{
F ∗′(µ)φ′[F ∗(µ)] +
[1− φ(F ∗(µ))]
µ
}
+R∗′(0)
{
1
µ
F ∗(µ)φ′[F ∗(µ)] +
F ∗′(µ)[1 − φ(F ∗(µ))]
[1− F ∗(µ)]2
}
(2.7)
From (2.5), (2.6) and (2.7) covariance can be found.
2.4. Model-1.2
In this model, it is assumed that, after threshold, parallel recruitment policy is
adopted and all recruitments are done simultaneously. RT = max{R1, R2, . . . , Ri}
where i is the number of shortages occurred during T .
P (T ≤ x,RT ≤ y) =
∞
∑
n=1
[R(y)]npn
∫ x
0
e−µzdFn(z)
+
∞
∑
n=0
Pn[R(y)]
n+1
∫ x
0
µe−µz[Fn(z)− Fn+1(z)]dz (2.8)
From (2.8) marginal recruitment time distribution can be obtained as
P (RT ≤ y) = φ[R(y)F
∗(µ)] +
R(y)[1− F ∗(µ)]
1−R(y)F ∗(µ)]
{1− φ[R(y)F ∗(µ)]} (2.9)
3. Model-2
3.1. Model Description
In this model, we consider the MPS when both the groups A and B are exposed
to CSP. Recruitment begins immediately after MPS breakdown occurs. Joint
transform and marginal recruitment time distributions are obtained for models
2.1 and 2.2 respectively when the groups A and B of the MPS have exponential
thresholds and shortages occur to them in accordance with Poisson process.
PROBABILISTIC ANALYSIS OF TIME TO RECRUIT... 539
3.2. Assumptions
1. Shortages occur in group A and group B in accordance with two different
and independent Poisson processes with parameters λ and µ respectively.
2. The successive shortages X1,X2, . . . in group A and Y1, Y2, . . . in group
B are two sequences of independent and identically distributed positive
random variables. X and Y are independent with different distributions.
3. W1 and W2 are the thresholds of groups A and B respectively. They are
independent and exponentially distributed with different parameters c1
and c2 respectively.
4. At time 0, the groups A and B of the MPS are free of shortage and put
into operation. Groups A and B function independently during Time to
recruit.
5. When there is breakdown in one of the groups, the other group does
not do productive function and the two groups are taken for immediate
recruitment.
6. For all the shortages occurred in the two groups, recruitment is done.
The recruitment times Ri of the shortages are independent and identically
distributed r.v. with distribution R(y) independent of the magnitude of
the shortage. Let
∫ ∞
0
ydR(y) < ∞.
3.3. Probability of breakdown
P (group A survives k shortages)
= P (W1 > X1 +X2 + · · · +Xk)
=
∫ ∞
0
P (X1 +X2 + · · ·+Xk < y)c1e
−c1ydy
= ak (3.1)
where a = E(e−c1x) and 0 ≤ a < 1.
P (group A has breakdown on nth shortage) = (1−a)an−1, n = 1, 2, . . . (3.2)
540 S. Mythili, R. Ramanarayanan
Similarly
P (group B survives k shortages) = P (W2 > Y1 + Y2 + · · ·+ Yk) = b
k (3.3)
where b = E(e−c2y) and 0 ≤ b < 1.
P (group B has breakdown on the nth shortage) = (1− b)bn−1, n = 1, 2, . . .
(3.4)
3.4. Model-2.1
We proceed as in model-1.1. RT = R1 +R2 + · · ·+Ri where i is the number of
shortages that have occurred during T .
From (3.1) to (3.4),
P (T1 ≤ x,RT ≤ y, T2 > T1)
=
∞
∑
n=1
∞
∑
k=0
Rn+k(y)
∫ x
0
bke−µz
(µz)k
k!
(1− a)an−1λe−λz
(λz)n−1
(n− 1)!
dz (3.5)
where Rn(y) is the n fold convolution of R(y) with itself. The term under the
double summation is the joint probability that the MPS suffers breakdown dur-
ing the nth shortage to group A during (0, x), group B survives the k shortages
occurred during time to recruit and
∑n+k
i=1 Ri ≤ y.
Similarly we can find
P (T2 ≤ x,RT ≤ y, T1 > T2)
=
∞
∑
n=1
∞
∑
k=0
Rn+k(y)
∫ x
0
ake−λz
(λz)k
k!
(1− b)bn−1µe−µz
(µz)n−1
(n− 1)!
dz (3.6)
Using (3.5) and (3.6),
E(e−ǫT e−ηRT )
= λ(1 − a)R∗(η)
∫ ∞
0
e−ǫx
∞
∑
k=0
e−µx
[bµxR∗(η)]k
k!
∞
∑
n=1
e−λx
[aλxR∗(η)]n−1
(n− 1)!
dx
+ µ(1− b)R∗(η)
∫ ∞
0
e−ǫx
∞
∑
k=0
e−λx
[aλxR∗(η)]k
k!
∞
∑
n=1
e−µx
[bµxR∗(η)]n−1
(n− 1)!
dx
(3.7)
E(e−ǫT e−ηRT ) = R∗(η)
[λ(1− a) + µ(1− b)]
[(ǫ + λ+ µ)− (aλ+ bµ)R∗(η)]
(3.8)
PROBABILISTIC ANALYSIS OF TIME TO RECRUIT... 541
3.5. Model-2.2
We proceed as in model-1.2. RT = max{R1, R2, . . . , Ri} where i is the number
of shortages occurred during T . We get,
P (T ≤ x,RT ≤ y)
=
∞
∑
n=1
∞
∑
k=0
[R(y)]n+k
∫ x
0
bke−µz
(µz)k
k!
λ(1− a)an−1e−λz
(λz)n−1
(n− 1)!
dz
+
∞
∑
n=1
∞
∑
k=0
[R(y)]n+k
∫ x
0
ake−λz
(λz)k
k!
µ(1− b)bn−1e−µz
(µz)n−1
(n− 1)!
dz (3.9)
Marginal Recruitment time distribution is
P (RT ≤ y) = R(y)
[λ(1− a) + µ(1− b)]
[(λ+ µ)− (aλ+ bµ)R(y)]
(3.10)
where 0 ≤ a < 1 and 0 ≤ b < 1.
Acknowledgments
It is a pleasure to thank the management of Vel Tech Multi Tech Dr. Rangara-
jan Dr. Sakunthala Engineering College, Avadi, Chennai, Tamilnadu, India, for
providing necessary facilities to bring out this research paper in a short time.
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Probabilistic analysis of time to recruit and recruitment time in manpower system with two groups,

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