www.ijera.com 578 |P a g e
Expected Time To Recruitment In A Two Grade Manpower
System
J.SRIDHARAN, K.PARAMESWARI, A.SRINIVASAN
*Assistant Professor in Mathematics, Government Arts college (Autonomous), Kumbakonam- 612 020(T.N)
**Lecturer in Mathematics, St. Joseph’s college of Engineering & Technology, Thanjavur-613 005(T.N) ***Associate Professor in Mathematics, Bishop Heber College (Autonomous), Thiruchirappalli- 620 017 (T.N)
Abstract
In this paper a two graded organization is considered in which depletion of manpower occurs due to its policy decisions. Three mathematical models are constructed by assuming the loss of man-hours and the inter-decision times form an order statistics. Mean and variance of time to recruitment are obtained using an univariate recruitment policy based on shock model approach and the analytical results are numerically illustrated by assuming different distributions for the thresholds. The influence of the nodal parameters on the system characteristics is studied and relevant conclusions are presented.
Key words :
Man power planning, Univariate recruitment policy, Mean and variance of the time for recruitment, Order statistics, Shock model.I.
Introduction
Exits of personnel which is in other words known as wastage, is an important aspect in the study of manpower planning. Many models have been discussed using different kinds of wastages and also different types of distributions for the loss of man-hours, the threshold and the inter-decision times. Such models could be seen in [1] and [2]. Expected time to recruitment in a two graded system is obtained under different conditions for several models in [3],[4],[5],[6],[7],[8] and [9] according as the inter-decision times are independent and identically distributed exponential random variables or exchangeable and constantly correlated exponential random variables. Recently in [10] the author has obtained system characteristic for a single grade man-power system when the inter-decision times form an order statistics. The present paper extend the results of [10] for a two grade manpower system when the loss of man-hours and the inter decision times form an order statistics. The mean and variance of the time to recruitment of the system characteristic are obtained by taking the distribution of loss of man-hours as first order (minimum) and kth order (maximum) statistics respectively. This paper is organized as follows: In sections 2, 3 and 4 models I, II and III are described and analytical expressions for mean and variance of the time to recruitment are derived . Model I, II and III differ from each other in the following sense: While in model-I transfer of personnel between the two grades is permitted, in model-II this transfer is not permitted. In model-III the thresholds for the number of exits in the two grades are combined in order to provide a better allowable loss of manpower in the organization
compared to models I and II. In section 5, the analytical results are numerically illustrated and relevant conclusions are given.
II.
Model description and analysis for
Model-I
Consider an organization having two grades in which decisions are taken at random epochs in
)
,
[
0
and at every decision making epoch a random number of persons quit the organization. There is an associated loss of man-hour to the organization, if a person quits and it is linear and cumulative. Let Xi be the loss of man-hours due to the ith decision epoch, i=1,2,3…k. LetX
i,
i
1
,
2
,
3
...
k
are independent and identically distributed exponential random variables with density function g(.) and mean 1/c,(c>0). . LetX
(1),
X
(2),...
X
(k) be the orderstatistics selected from the sample
X
1,
X
2,...
X
k with respective density functions(.).
(.)....
(.),
( ) ( ))
( x xk
x
g
g
g
1 2 LetU
i,
i
1
,
2
,
3
...
k
are independent and identically distributed exponential random variables with density function f(.). LetU
(1),
U
(2),...
U
(k) be the order statistics selected from the samplek
U
U
U
1,
2,...
with respective density functions(.).
(.)....
(.),
( ) ( ))
( u uk
u
f
f
f
1 2 Let T be a continuouswww.ijera.com 579 |P a g e
(.)
(.)
(.),
(.),
* *() *( )*
k u
u
and
f
f
f
l
1 be the Laplacetransform of
l
(.),
f
(.),
f
u(1)(.)
and
f
u(k)(.)
respectively. Let YA and YB be independent random variables denoting the threshold levels for the loss of man-hours in grades A and B with parameters αA and αB respectively (αA,αB>0). In this model the threshold Y for the loss of man-hours in the organization is taken as max (YA,YB). The loss of manpower process and the inter-decision time process are statisticallyindependent. The univariate recruitment policy employed in this paper is as follows: Recruitment is done as and when the cumulative loss of man-hours in the organization exceeds Y. Let Vk(t) be the probability that there are exactly k-decision epochs in (0,t]. Since the number of decisions made in (0,t] form a renewal process we note that Vk(t)=Fk(t) -Fk+1(t), where F0(t)=1. Let E(T) and V(T) be the mean and variance of time for recruitment respectively.
III.
Main results
The survival function of T is given by
0 1
k
k
i i
k
t
P
X
Y
V
t
T
P
(
)
(
)
(
)
0 0
dx
x
g
x
y
p
t
V
kk
k
(
)
(
)
(
)
(1)
Case 1:YA and YB follow exponential distribution with parameters αA and αB respectively. In this case it is shown
that
e
e
e
g
x
dx
t
V
x
Y
p
x x x kk k
B A B
A
(
)
)
(
)
(
( )
0 0
(2)
From (1) and (2) we get
(
)
(
)
(
)
(
)
(
)
(
)
)
(
* * *3
0
1
k
B A k B k A k k
k
t
F
t
g
g
g
F
t
T
P
Since
(
)
1
(
)
(
)
l
(
t
)
dt
d
t
l
and
t
T
P
t
L
(4)
from (3) and (4) it is found that
)
(
))
(
)(
(
)]
(
[
))
(
)(
(
)]
(
[
))
(
)(
(
)]
(
[
)
(
* *
* *
* *
5
1
1
1
1
1 1
1
1
1
k
k B A k
B A
k
k B k
B k
k A k
A
g
t
f
g
g
t
f
g
g
t
f
g
t
l
Taking Laplace transform on both sides of (5) it is found that
(
)
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
* ** *
* *
* *
* *
* *
*
6
1
1
1
1
1
1
B A B A
B B
A A
g
s
f
s
f
g
g
s
f
s
f
g
g
s
f
s
f
g
s
l
The probability density function of rth order statistics is given by
)
(
..
,
,
,
)]
(
)[
(
)]
(
[
)
(
)
(
1
1
2
3
7
1
k
r
t
F
t
f
t
F
kc
r
t
f
ur
r r
kr
If f(t)=fu(1)(t)
then
f
*(
s
)
f
u*(1)(
s
)
(8)
From (7) it is found that
(
)
(
)
)
(
)
(
)
(
1
9
1 1
ku
t
k
f
t
f
t
f
www.ijera.com 580 |P a g e
s
k
k
s
f
u
)
(
* )
(1
(11)
It is known that
)
(
))
(
(
)
(
)
(
)
(
(
)
(
,
))
(
(
)
(
* *
12
2 2
0 2 2 2
0
T
E
T
E
T
V
and
ds
s
l
d
T
E
ds
s
l
d
T
E
s s
Therefore from (6), (11) and (12) we get
(
)
)
(
T
1
V
1V
2V
313
E
(
)
)
(
2
3214
2 2 2 1 2 2
V
V
V
T
E
Where
(
)
)
(
)
(
,
)
(
* **
15
1
1
1
1
1
1
3 2
1
B A B
A
g
V
and
g
V
g
V
If f(t)=fu(k)(t)
In this case
f
*(
s
)
f
u*(k)(
s
)
From (7) it is found that
(
)
(
)
(
)
)
(
)
(
16
1
t
f
t
F
t
f
uk
kFrom(10) , (16) and on simplification we get
)
(
)
)...(
)(
(
!
)
(
* )
(
17
2
k
s
s
s
k
s
f
k
k
u
Therefore from (6),(17) and (12) we get
(
)
)
(
18
1
3 2 1
1
n
V
V
V
T
E
k
n
(
)
)
(
19
1
1
2
3 2 1 2 1
2
3 2 3 2 2 2 1 2 1 2
2
1 2
V
V
V
n
V
V
V
V
V
V
n
T
E
k
n k
n
In (18) & (19) V1,V2 and V3 are given by (15).
The probability density function of nth order statistics is given by
)
(
..
,
,
,
)]
(
)[
(
)]
(
[
)
(
)
(
1
1
2
3
20
1
k
n
x
G
x
g
x
G
kc
n
x
g
xn
n n
kn
If g(x)=gx(1)(x)
then in(13),(14),(18) and (19)
g
*(
)
g
*x(1)(
)
for
A,
Band
A
BFrom (20) it is found that
(
)
(
)
)
(
)
(
)
(
1
21
1 1
kx
x
k
g
x
g
x
g
Since by hypothesis
g
(
x
)
ce
cx(22)
from (21) and (22) we getB A B
A
x
and
kc
kc
g
,
,
)
(
* )
(1
(23)
In (13),(14),(18) and (19)
g
*(
A),
g
*(
B)
&
g
*(
A
B)
are given by (23) when s=1.and
V
(
T
)
E
(
T
2)
(
E
(
T
))
2If g(x)=gx(k)(x)
then
g
*(
)
g
*x(k)(
)
for
A,
Band
A
Bwww.ijera.com 581 |P a g e
From (20) it is found that
(
)
(
)
(
)
)
(
)
(
24
1
x
g
x
G
x
g
xk
kFrom(22),(24) and on simplification we get
)
(
,
)
)...(
)(
)(
(
!
)
(
* )
(
25
3
2
A B A Bk
k
x
for
and
kc
c
c
c
c
k
g
In (13),(14),(18) and (19)
g
*(
A),
g
*(
B)
&
g
*(
A
B)
are given by (25) when s=k and 22
))
(
(
)
(
)
(
T
E
T
E
T
V
Case 2:
YA and YB follow extended exponential distribution with scale parameters αA and αB respectively and
shape parameter 2. In this case it can be shown that
If f(t)=fu(1)(t)
(
)
)
(
T
1
2
V
12
V
24
V
32
V
42
V
5V
6V
7V
826
E
(
)
)
(
T
22
22
V
122
V
224
V
322
V
422
V
52V
62V
72V
8227
E
)
(
)
(
)
(
,
)
(
,
)
(
,
)
(
* *
* *
*
28
2
1
1
2
1
1
2
2
1
1
2
1
2
2
1
1
8 7
6 5
4
B A
B A B
A B
A
g
V
and
g
V
g
V
g
V
g
V
where
when n=1,in (26)&(27) V1,V2,V3,V4,V5,V6,V7 and V8 are given by (15),(28) and (23).
when n=k,in (26)&(27) V1,V2,V3,V4,V5,V6,V7 and V8 are given by (15),(28) and(25). If f(t)=fu(k)(t)
Proceeding as in case(i) it can be found that
(
)
)
(
2
2
4
2
2
29
1
8 7 6 5 4 3 2 1
1
n
V
V
V
V
V
V
V
V
T
E
k
n
(
)
)
(
30
1
1
2
2
4
2
2
1
1
2
2
4
2
2
2
1 2 2
1 8 7 6 5 4 3 2 1
2 2
1 2 8 2 7 2 6 2 5 2 4 2 3 2 2 2 1 2 2
k
n k
n
k
n
n
n
V
V
V
V
V
V
V
V
n
V
V
V
V
V
V
V
V
T
E
when n=1,in (26)&(27) V1,V2,V3,V4,V5,V6,V7 and V8 are given by (15),(28) and (23).
when n=kin (26) (27) V1,V2,V3,V4,V5,V6,V7 and V8 are given by (15),(28) and (25).
Case 3:
YA follows extended exponential distribution with scale parameters αA and shape parameter 2 and YB
www.ijera.com 582 |P a g e
Proceeding as in case 1 it can be shown that
(
)
)
(
T
1
2
V
1V
2V
42
V
3V
731
E
(
)
)
(
2
2
2
232
7 2 3 2 4 2 2 2 1 2
2
V
V
V
V
V
T
E
when n=1, in (31) & (32) V1,V2,V3,V4 and V7 are given by (15),(28) and (23).
when n=k, in (31) & (32) V1,V2,V3,V4 and V7 are given by (15),(28) and (25). If f(t)=fu(k)(t)
Proceeding as in case (i) it can be shown that
(
)
)
(
2
2
33
1
7 4 3 2 1 1
V
V
V
V
V
n
T
E
k
n
2
1 2 7 2 4 2 3 2 2 2 1 2
2
1
2
2
2
kn
n
V
V
V
V
V
T
E
)
(
k
n k
n
n
n
V
V
V
V
V
1 2 2
1 7 4 3 2 1
2
2
2
1
1
1
(34)
when n=1, in (33) & (34) V1,V2,V3,V4 and V7 are given by (15),(28) and (23).
when n=k, in (33) & (34) V1,V2,V3,V4 and V7 are given by (15),(28) and (25). Case 4:
The distributions of YA has SCBZ property with parameters αA,µ1 & µ2, and the distribution of YB has SCBZ property with parameters αB,µ3 & µ4. In this case it can be shown that
If f(t)=fu(1)(t)
(
)
)
(
T
1
p
1V
9p
2V
10p
1p
2V
13p
1q
2V
14p
2q
1V
15q
1q
2V
16q
1V
11q
2V
1235
E
(
)
)
(
T
22
2p
1V
92p
2V
102p
1p
2V
132p
1q
2V
142p
2q
1V
152q
1q
2V
162q
1V
112q
2V
12236
E
where)
(
)
(
)
(
,
)
(
,
)
(
)
(
,
)
(
,
)
(
,
)
(
*
* *
*
* *
* *
37
1
1
1
1
1
1
1
1
1
1
1
1
1
2
1
1
4 2 16
3 1 15
4 1 14
3 1 13
4 12
2 11
3 10
1 9
g
V
and
g
V
g
V
g
V
g
V
g
V
g
V
g
V
B A
B A
B A
when n=1,in(35)&(36)V9,V10,V11,V12 ,V13,V14,V15 and V16 are given by (37) and (23).
when n=k,in(35)&(36)V9,V10,V11,V12 ,V13,V14,V15 and V16 are given by (37) and (25).
If f(t)=fu(k)(t)
Proceeding as in case (i) it can be shown that
(
)
)
(
38
1
12 2 11 1 16 2 1 15 1 2 14 2 1 13 2 1 10 2 9 1 1
V
q
V
q
V
q
q
V
q
p
V
q
p
V
p
p
V
p
V
p
n
T
E
k
n
www.ijera.com 583 |P a g e
and
21 2 2
1 2 15 1 2 2 14 2 1 2 13 2 1 2 10 2 2 9 1 2
2
2
1
2
1
k
n k
n
n
n
V
q
p
V
q
p
V
p
p
V
p
V
p
T
E
)
(
2
16 2 1 2 12 2 2 111
V
q
V
q
q
V
q
k
n k
n
n
n
V
q
q
V
q
V
q
1 2 2
1 16 2 1 12 2 11 1
2
1
1
1
-
k
n k
n
n
n
M
q
p
M
q
p
M
p
p
M
p
M
p
1 2 2
1 15 1 2 14 2 1 13 2 1 10 2 9 1
2
1
1
1
(39)
when n=1,in(35)&(36)V9,V10,V11,V12 ,V13,V14,V15 and V16 are given by (37) and (23).
when n=k,in(35)&(36)V9,V10,V11,V12 ,V13,V14,V15 and V16 are given by (37) and (25).
IV.
Model description and analysis for Model-II
For this model
Y
min(
Y
A,
Y
B)
. All the other assumptions and notations areas in model-I. Then the values of
E
(
T
)
&
E
(
T
2)
when
r
1
and
r
k
are
given
by
case 1: If f(t)=fu(1)(t)
Proceeding as in case 1 it can be shown that
(
)
)
(
T
1
V
340
E
(
)
)
(
T
22
2V
3241
E
when n=1,in (40) & (41) V3 is given by (15) and (23).
when n=k,in (40) & (41) V3 is given by (15) and (25). If f(t)=fu(k)(t)
Proceeding as in case 1 it can be shown that
(
)
)
(
42
1
3
1
n
V
T
E
k
n
(
)
)
(
43
1
1
1
2
3 2
1 2 2
1 2 3 2
2
1 2
V
n
n
V
n
T
E
k
n k
n k
n
when n=1,in (42) & (43) V3 is given by (15) and (23).
when n=k,in (42) & (43) V3 is given by (15) and (25). 2
2
)
(
(
))
(
)
(
T
E
T
E
T
V
and
Case 2: If f(t)=fu(1)(t)
Proceeding as in case 1 it can be shown that
(
)
)
(
1
4
V
3V
62
V
42
V
544
k
T
E
(
)
)
(
2
4
2
2
5245
2 4 2 6 2 3 2 2 2
V
V
V
V
k
T
E
www.ijera.com 584 |P a g e when n=k,in (44) & (45) V3,V4,V5 and V6 are given by (15),(28) and (25).
If f(t)=fu(k)(t)
Proceeding as in case 1 it can be shown that
(
)
)
(
4
2
2
46
1
5 4 6 3 1
V
V
V
V
n
T
E
k
n
2
1 2 5 2 4 2 6 2 3 2
2
1
2
2
4
2
kn
n
V
V
V
V
T
E
)
(
4
2
2
1
1
(
47
)
1
1 2 2
1 5 4 6 3
2
k
n k
n
n
n
V
V
V
V
when n=1,in (46) & (47) V3,V4,V5 and V6 are given by (15),(28) and (23).
when n=k,in (46) & (47) V3,V4,V5 and V6 are given by (15),(28) and (25). Case 3:
If f(t)=fu(1)(t)
Proceeding as in case 1 it can be shown that
(
)
)
(
1
2
V
3V
448
k
T
E
(
)
)
(
2
2
4249
2 3 2 2 2
V
V
k
T
E
when n=1,in (48) & (49) V3 and V4 are given by (15),(28) and (23).
when n=k,in (48) & (49) V3 and V4 are given by (15),(28) and (25). If f(t)=fu(k)(t)
Proceeding as in case 1 it can be shown that
(
)
)
(
2
50
1
4 3 1
V
V
n
T
E
k
n
(
)
)
(
2
2
1
1
2
1
1
51
1 2 2
1 4 3 2 2
1 2 4 2 3 2 2
k
n k
n k
n
n
n
V
V
n
V
V
T
E
when n=1,in (50) & (51) V3 and V4 are given by (15),(28) and (23).
when n=k,in (50) & (51) V3 and V4 are given by (15),(28) and (25). 2
2
))
(
(
)
(
)
(
T
E
T
E
T
V
and
Case 4: If f(t)=fu(1)(t)
Proceeding as in case 1 it can be shown that
(
)
)
(
1
p
1p
2V
13p
1q
2V
14p
2q
1V
15q
1q
2V
1652
k
T
E
(
)
)
(
2 22
2p
1p
2M
132p
1q
2M
142p
2q
1M
152q
1q
2M
16253
k
T
E
when n=1,in (52) & (53) V13,V14,V15 and V16 are given by (37) and (23).
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Proceeding as in case 1 it can be shown that
(
)
)
(
54
1
16 2 1 15 1 2 14 2 1 13 2 1 1
V
q
q
V
q
p
V
q
p
V
p
p
n
T
E
k
n
k
n k
n k
n
n
n
n
V
q
q
V
q
p
V
q
p
V
p
p
T
E
1 2 2
1 2
1 16 2 1 15 1 2 14 2 1 13 2 1 2
2
2
1
1
1
)
(
1
2
p
1p
2V
132
p
1q
2V
142
p
2q
1V
152
q
1q
2V
162
(
55
)
when n=1,in (54) & (55) V13,V14,V15 and V16 are given by (37) and (23).
when n=k,in (54) & (55) V13,V14,V15 and V16 are given by (37) and (25).
V.
Model description and analysis for Model-III
For this model
Y
Y
A
Y
B. All the other assumptions and notations are as in model-I. Then the values ofE
(
T
)
&
E
(
T
2)
when
n
1
and
n
k
are
given
by
case 1: If f(t)=fu(k)(t)
Proceeding as in case 1 it can be shown that
)
(
)
(
1
2 1
56
V
V
T
E
B A
B
B A
A
)
(
)
(
22
2 22 12
57
V
V
T
E
B A
B
B A
A
when n=1,in (56) & (57) V1 and V2 are given by (15) and (23).
when n=k,in (56) & (57) V1 and V2 are given by (15) and (25). If f(t)=fu(k)(t)
Proceeding as in case 1 it can be shown that
)
(
)
(
58
1
1 2
1
V
V
n
T
E
B A
B
B A
A k
n
)
(
)
(
59
1
1
1
2
1 2
2 1
2 2
1 2
1 2
2 2
2
1 2
V
V
n
n
V
V
n
T
E
B A
B
B A
A
k
n k
n
B A
B
B A
A k
n
when n=1,in (58) & (59) V1 and V2 are given by (15) and (23).
when n=k,in (58) & (59) V1 and V2 are given by (15) and (25). Case 2:
If f(t)=fu(1)(t)
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)
(
)
(
60
2
2
2
2
1
2
4
4
4
2
4
1
8 7 2 1
V
V
k
V
V
k
T
E
B A A B A A B A B B A B B A A B A A B A B B A B
)
(
)
(
61
2
2
2
2
2
2
4
4
4
2
4
2
2 8 2 7 2 2 2 2 2 1 2 2 2
V
V
k
V
V
k
T
E
B A A B A A B A B B A B B A A B A A B A B B A B
when n=1,in (60) & (61) V1,V2,V7 and V8 are given by (15),(28) and (23).
when n=k,in (60) & (61) V1,V2,V7 and V8 are given by (15),(28) and (25).
If f(t)=fu(k)(t)
Proceeding as in case 1 it can be shown that
)
(
)
(
62
2
2
2
2
1
2
4
4
4
2
4
1
8 7 1 2 1 1
V
V
n
V
V
n
T
E
B A A B A A B A B B A B k n B A A B A A B A B B A B k n
2 8 2 7 2 2 1 2 2 2 1 2 2 1 22
2
2
2
1
2
2
4
4
4
2
4
1
2
V
V
n
V
V
n
T
E
B A A B A A B A B B A B k n B A A B A A B A B B A B k n
)
(
)
(
63
2
2
2
2
1
1
2
4
4
4
2
4
1
1
8 7 2 1 2 2 1 2 1 2 1 2 2 1
V
V
n
n
V
V
n
n
B A A B A A B A B B A B k n k n B A A B A A B A B B A B k n k n
when n=1,in (62) & (63) V1,V2,V7 and V8 are given by (15),(28) and (23).
when n=k,in (62) & (63) V1,V2,V7 and V8 are given by (15),(28) and (25). Case 3:
If f(t)=fu(k)(t)
