DOI : 10.17577/IJERTCONV4IS24002
- Open Access
- Total Downloads : 14
- Authors : P. Saranya, A. Srinivasan
- Paper ID : IJERTCONV4IS24002
- Volume & Issue : AMASE – 2016 (Volume 4 – Issue 24)
- Published (First Online): 24-04-2018
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
Stochastic Models on Time to Recruitment in a Two Grade Manpower System using Univariate Recruitment Policy
P. Saranya*
Department of Mathematics,
TRP Engineering College (SRM Group), Trichy-621105, Tamil Nadu, India.
A. Srinivasan
PG & Research Department of Mathematics, Bishop Heber College,
Trichy -620017, Tamil Nadu, India.
Abstract In this paper for a two grade manpower system involving optional and mandatory exponential thresholds, the analytical results for some performance measures related to time to recruitment are obtained using a univariate policy of recruitment by considering different forms of inter-decision times and loss of manpower in the system.
Keywords Two grade manpower system; loss of manpower; Univariate policy of recruitment, geometric process; Correlated random variables and Mean time to recruitment.
-
INTRODUCTION
In any organization, depletion of manpower is quite common whenever policy decisions are announced .Frequent recruitment to compensate this depletion is costlier and hence suitable policy decision on recruitment has to be designed. In this context, several authors have obtained performance
two-grade system involving two thresholds by assuming different distributions for thresholds under different condition of inter-decision time and wastage. The objective of the present paper is to obtain performance measures for a two grade manpower system with exponential thresholds by considering different forms of loss of manpower and inter decision times.
-
MODEL DESCRIPTION AND ANALYSIS OF MODEL I Consider an organization taking decisions at random
epoch in (0, ) and at every decision epoch a random number
of persons quit the organization. There is an associated loss of man-hours if a person quits. It is assumed that the loss of man-hours are linear and cumulative. Let Y1, Y2 (Z1, Z2) denotes the optional (mandatory) thresholds for the loss of man-hours in grades 1 and 2, with parameters
measures namely mean and variance of time to recruitment using different policies of recruitment based on shock mode
,
1 2
, ,
1 2
respectively, where
,
1 2
, ,
1 2
are
approach. Employing the univariate recruitment policy, the expected time to recruitment is obtained under different conditions for several models in [1], [2] and [3].In [5],for a single grade man-power system with a mandatory exponential threshold for the loss of manpower ,the authors have obtained the system performance measures when the inter-decision times form an order statistics .In [2] ,for a single grade manpower system, the author has considered a new recruitment policy involving two thresholds for the loss of man-power in the organization in which one is optional and the other is mandatory and obtained the mean time to recruitment under different conditions on the nature of thresholds ,inter -decision times and loss of man-hours . In [6-9] the authors have extended the results in [2] for a two- grade system according as the thresholds are exponential random variables or extended exponential random variables or SCBZ property possessing random variables or geometric random variables. In [10],the authors have obtained performance measures by assuming that the inter-decision times for the two grades form same geometric process .In [11-19], the authors have extended the results in [5] for a
positive. It is assumed that Y1 < Z1and Y2 < Z2. Write Y = Max (Y1, Y2) and Z=Max (Z1, Z2), where Y (Z) is the optional (mandatory) threshold for the loss of man-hours in the organization. The loss of man- hours, optional and mandatory thresholds are assumed as statistically independent. Let T be the time to recruitment in the organization with cumulative distribution function L (.), probability density function l (.), mean E (T) and variance V (T). Let F k(.) be the k fold convolution of F(.). Let l*(.) and f*(.), be the Laplace transform of l (.) and f (.), respectively. Let Vk(t) be the probability that there are exactly k decision epochs in (0, t]. It is known from Renewal theory that Vk(t) = Fk(t) – Fk+1(t) with F0(t) = 1.Let p be the probability that the organization is not going for recruitment whenever the total loss of man- hours crosses optional threshold Y. The Univariate recruitment policy employed in this paper is as follows: If the total loss of man-hours exceeds the optional threshold Y, the organization may or may not go for recruitment. But if the total loss of man-hours exceeds the mandatory threshold Z, the recruitment is necessary.
-
MAIN RESULTS
k
k
k
(1)
P(T t)
Vk(t)P Xi Y p Vk(t)P Xi Y P Xi Z
k 0
i 1
k 0
i 1
i 1
We now obtain some performance measures related to time to recruitment for different forms of wastage and inter-decision times.
Case (i): Let Xi be the loss of man hours due to the ith decision epoch , i=1,2,3 forming a sequence of independent and
identically distributed exponential random variables with mean
1 (c>0), probability density function g(.),Ui are exchangeable
c
and constantly correlated exponential random variables denoting inter-decision time between (i-1)th and ith decision, i=1,2, 3.k. with cumulative distribution function F(.), probability density function f(.) and mean u .
Using law of total probability
k
P X
i
Y 1 PY xg
k
xdx 1 H(x)gk
xdx
(2)
i1 0 0
Since the distribution of thresholds follow exponential distribution it can be shown that
k x
k
k
x
x
PX Y e
1 e
2 e
1 2 g
(x) dx
(3)
Similarly,
i
i1
k
0
x
x
k
k
x
PX
Z e
1 e
2 e
1 2 g
(x) dx
(4)
i
i1
0
Using (3) and (4) in (1) and on further simplification we get,
* k * k * k
P(T t) [ F (t) F (t) ] g ( ) g ( )
g ( )
k0 k
k1
1
1
1
*
2
k *
1 2
k *
k *
k *
k *
(5)
k
p [F (t) F
(t) ] 1 g ( )
g ( )
g ( )
g ( )
g ( )
g ( )
k0 k
k1
2
1 2
1
2
1 2
Since L(t) 1 P(T t) from (5)
*
1 g ( ) F
* k1 * *
1 k1
k
1
2 k1
k
2
*
1 g ( ) F
* k1 * *
1 k1
k
1
2 k1
k
2
k1
1 g* ( ) F
* k1
g ( )
p
1
g* ( ) F
* k1
( )
1 2 k1 k
1 2
1 k1
k
1
k1 k1
g* ( )g* ( ) F (t) g* ( )g* ( ) 1 g* ( )g* ( ) F (t) g* ( )g* ( )
1 1 k 1 1 2 1 k 2 1
k1 k1
1 g* ( ) F
* k1
g ( )
p
1
g* ( ) F
* k1
( )
1 2 k1 k
1 2
1 k1
k
1
k1 k1
g* ( )g* ( ) F (t) g* ( )g* ( ) 1 g* ( )g* ( ) F (t) g* ( )g* ( )
1 1 k 1 1 2 1 k 2 1
k1 k1
L(t)
1
(t) g ( )
(t)
1 g (
) F (t) g (
)
(t) g
* * * *
k1 * * k1
1 g ( ) g ( ) F (t) g
g ( ) 1 g ( ) F (t) g ( )
1 2
1 k1 k 1 2
1
2 k1 k 2
* *
* *
k1 * *
* *
k1
1 g ( )g ( ) F (t) g ( )g ( )
1 g (
)g ( ) F (t) g ( )g ( )
1 2 k1 k 1
2
2 2 k1 k 2 2
* * *
* k1 *
*
k1
1 g ( )g ( ) F (t) g (
)g 1 g (
) F (t) g (
)
1 2
*
2 k1 k
*
1 2
*
2
* k1
1 2 k1 k
* *
1 2
*
* k1
1 g (1 2 )g ( ) F (t) g (1 2 )g ( )
1 g (1 2 )g (2 ) F (t) g (1 2 )g (2 )
(6)
* *
1 k1 k
*
1
*
k1
k1 k
1 g (1 2 )g (
) F (t) g ( )g (
2 1 2
)
2
1 k1 k
1
By assumption {Ui} is a sequence of exchangeable and constantly correlated random variable each following the exponential
k
distribution .As in [4] , we get the cumulative distribution function of Ui as
i1
1
k i
k j1 y
b
b
e
ki j1
(y / b)
F (t) 1
(7)
k 1 k i01 k
j0
(k i j 1)!
Taking Laplace-Stieltjes transform on both sides, we have
1k
F
F
*(s)
k
(1 )(1 bs)
(1 )(1 bs) kbs
(8)
*
*
d (F (s)
uk
and
ds k
d2 *
s 0
2 2 2 2 b
F (s) u {(1 )k (1 )k} , where u
(9)
It is known that
ds2 k
d(l*(s))
s0
2 d2 (l*
(s)
1
2 2
E(T)
ds
s0
, E(T )
ds2
s0
and
V(T) E(T
) (E(T))
(10)
Using (6) and (9) in (10) we get the first two moments
1 1 1
E(T) u * * *
1 1 1
p * * *
1 g
( )
1
1 g
( )
2
1 g
( )
1 2
1 g
( )
1
1 g
( )
2
1 g
( )
1 2
1
1 g*( )g*( )
1 g*(
1
)g*( )
1 g*(
1
)g*( )
1
1 g*( )g*(
) 1 g*(
1
)g*( )
1 1 2 1
1 2 1 1 2 2 2
1 1 1 1
1 g*(
1
)g* 2
( )
1
1 g*
( )g*(
1 1
)
2
1 g*(
2
)g*(
1
)
2
1 g*(
1
)g*(
2 1
)
2
(11)
1 2g*( ) 1 2g*( ) 1 2g*( ) 1 2g*( ) 1 2g*( ) 1 2g*( )
E(T 2 ) 2 u 2 1 2 1 2 p 1 2 1 2
* 2
* 2 *
2 * 2
* 2 * 2
(1 g ( )) (1 g ( )) (1 g (
)) (1 g ( ))
(1 g ( )) (1 g ( ))
1 2
1 2 1
2 1 2
1 2g*( )g*( ) 1 2g*( )g*( ) 1 2g*( )g*( ) 1 2g*( )g*( ) 1 2g*( )g*( )
1 1 2 1 1 2 1 1 2 2 2
(1 g*( )g*( ))2 (1 g*( )g*( ))2 (1 g*( )g*( ))2 (1 g*( )g*( ))2 (1 g*( )g*( ))2
1 1 2 1 1 2 1 1 2 2 2
1 2g*( )g*( ) 1 2g*( )g*( ) 1 2g*( )g*( ) 1 2g*( )g*( )
1 2 1 1 1 2 2 1 2 1 2 1 2
* * 2 * *
2 * *
2 * *
2
(1 g ( )g ( )) (1 g ( )g (
))
(1 g ( )g (
))
(1 g ( )g (
))
1 2 1
1 1 2
2 1 2
1 2 1
2
(12)
where
g*()
c c
for , ,
1 2 1
, , ,
2 1 2 1 2 .
(11) gives the expected time to recruitment, (11)and (12) together with (10) gives the variance of time to recruitment.
Case (ii):
i
i
Let X be the loss of man-hours due to the ith decision epoch , i=1,2,3k Let X1, X2, X3,….Xn be the order
statistics selected from the sample X1, X2,….Xn with respective density functions gx(1) (.), gx(2) (.),….g x(n) (.)
If Ui be exchangeable and constantly correlated exponential random variables denoting inter-decision times between (i-1) and i th decision epoch ,i=1,2,3.
Suppose
g(t) g (t)
x(1) .
The first two moments are given by (11) and (12) where
g*
x(1)
()
kc kc
. (13)
Suppose
g(t) g (t) The first two moments are given by (11) and (12) where
x(k) .
k
g*
x(k)
()
k!c
( c)( 2c)…( kc)
(14)
(14)
If Ui form a geometric process with parameter a , a>0 then we find that
k * s
f *(s) f
k r1
*
, k 1,2,3….
a
a
r1
d
(15)
Suppose recruitment
g(t) g
x(1)
(t)
. Since l
-
(l(t)) and using (6),(15) in (10) we get the first two moments of time to
ds
E(T) aE(U )C C
-
C pC
-
C C H
-
H H H
-
H H
-
H H H
(16)
1 1 2
3 4 5 6
1,4
1,5
1,6
2,4
2,5
2,6
3,4
3,5
3,6
2 a 2
(1 g*( ))
(1 g*( ))
(1 g*( ))
E(U )a 2
2(1 g*( ))
1 g*( )
2(1 g*( ))
E(T 2 ) U1 1 1 2 1 2 1 1 1 1 2
2 2 * 2 * 2 * 2 * 2 * *
(a 1) a ( ) a g ( ) a g ( ) (a 1) a g ( ) a g ( ) a g ( )
1 2 1 2 1 1 2
1 g*( )
2(1 g*( ))
1 g*( )
2 a 2
(1 g*( ))
(1 g*( ))
(1 g*( ))
2 1 2 1 2 p U1 1 2 1 2
2 * *
2 *
2 2 * 2 * 2 *
2 1
2 1
a g ( )
2
a g ( )
1 2
a g ( )
(a 1)
a g ( )
1
a g ( )
2
a g ( )
1 2
(1 g*( )g*( )) (1 g*( )g*( )) (1 g*( )g*( )) (1 g*( )g*( )) (1 g*( )g*( ))
1 1 2 1 1 2 1 1 2 2 2
a 2 g*( )g*( ) a 2 g*( )g*( ) a 2 g*( )g*( ) a 2 g*( )g*( ) a 2 g*( )g*( )
1 1 2 1 1 2 1 1 2 2 2
(1 g*( )g*( )) (1 g*( )g*( )) (1 g*( )g*( )) (1 g*( )g*( ))
1 2 2 1 1 2 2 1 2 1 2 1 2
2 * *
2 * *
2 * *
2 * *
a g ( )g ( )
a g ( )g ( )
a g ( )g ( )
a g ( )g ( )
1 2 2
1 1 2
2 1 2
1 2 1 2
pE(U )a 2 2(1 g*( )) 1 g*( ) 2(1 g*( )) 1 g*( ) 2(1 g*( )) 1 g*( )
1 1 1 2 2 1 2 1 2
(a 1)2 a g*( ) a 2 g*( ) a g*( ) a 2 g*( ) a g*( ) a 2 g*( )
1 1 2
2 1 2 1 2
2(1 g*( )g*( )) (1 g*( )g*( )) 2(1 g*( )g*( )) (1 g*( )g*( )) 2(1 g*( )g*( ))
1 1 1 1 2 1 2 1 1 2 1
a g*( )g*( ) a 2 g*( )g*( ) a g*( )g*( ) a 2 g*( )g*( ) a g*( )g*( )
(17)
1 1 1 1 2 1 2 1 1 2 1
(1 g*( )g*( )) 2(1 g*( )g*( )) (1 g*( )g*( )) 2(1 g*( )g*( ))
1 2 1 2 2 2 2 1 2 2
a 2 g*( )g*( ) a g*( )g*( ) a 2 g*( )g*( ) a g*( )g*( )
1 2 1 2 2 2 2 1 2 2
(1 g*( )g*( )) 2(1 g*( )g*( )) (1 g*( )g*( )) 2(1 g*( )g*( ))
1 2 2 1 1 2 1 1 2 2 1 2
a 2 g*( )g*( ) a g*( )g*( ) a 2 g*( )g*( ) a g*( )g*( )
1 2 2
1 1 2
1 1 2
2 1 2
(1 g*( )g*( )) 2(1 g*( )g*( )) (1 g*( )g*( ))
2 1 2 1 2 1 2 1 2 1 2
2 * *
* * 2 *
*
2 2
2 2
a g ( )g ( )
2 2
a g ( )g ( )
2 2
a g ( )g ( )
In (16) and (17)
E(U
1
1
) f
1
*' (0) , 2
U1
1
f *''
(0) (f
1 1
*' (0))2 ,
C 1
c a D
c
and
H
b,d
1
a D
a
, for c=1,2,36 , b=1,2,3 and d=4,5,6. (18)
g
g
D *( ),D
*(
),D *( ),D *( ),D
*(
),D *( ) .are given by (13)
g
g
1 1 2
2 3 1 1 4 1 5
2 6 1 2
g
g
g
g
g
g
g
g
(16) gives the expected time to recruitment, (16) and (17) together with (10) gives the variance of time to recruitment.
Suppose
g(t) g (t) . The first two moments are given by (16), (17) and (14).
x(k)
Case (iii):
Let Xi be a exponential random variable denoting the loss of man-hours due to ith decision epoch, i=1,2,3.with cumulative distribution function G (.) and probability density function g(.)
If Ui are exchangeable and constantly correlated exponential random variables
Considering the first term of (1) and conditioning upon y we get
k k
P X Y P
X Y h(y) dy G
(y) h(y) dy
i
i1
i k
0 i1 0
Since Xis are assumed to be identical constantly correlated and exchangeable exponential random variables with parameter ,
c.d.f of the partial sum S
k
X X
1 2
…… X
k
is given by Gurland(1955) as
ki
k i, y
G (y) (1 ) b
(19)
k i0
1 ki1 k i 1!
where is the constant correlation between Xi and Xj , ij.
y
y b z
k i1
k i,
b
e z
0
dz and b 1
1 2
1 2
Since the thresholds follow exponential distribution
y
h(y) e 1
1
y
e 2
2 1
e y 2
(20)
Using (19),(20) and on further simplification we get
k
(21)
F (t) F
(t)
PXi Y
1 F
-
-
-
-
F
(t) W W W
k0 k
Similarly,
k1
i1
k 0 k
k1
1k 2k 3k
F (t) F (t) PXi Y
F (t) F
(t)
1 1 W W W
k
k k1
k k1
1k 2k 3k
k0
i1
k0
(22)
PXi Z 1 W
-
W W
(23)
k
4k 5k 6k
1
1
W k k 1
1
1
i1
, W
2k
(23)
1
k 1
(b
1
1)
1 k(b
1
1) k
(b
2
1)
1 k(b
2
1) k
1
1
W k k 1
3
3
, W
4k
1
k 1
(b(
1
) 1)
2
1 k(b(
1
) 1) k
2
(b
1
1)
1 k(b
1
1) k
1
1
W k k 1
5
5
, W
6k
1
k 1
(b
2
1)
1 k(b
2
1) k
(24)
(b(
1
) 1)
2
1 k(b(
1
) 1) k
2
Using (21) ,(22) and (23) in (1) we get
P(T t) 1
k0
F (t) F
k k1
(t)
W W W
1k 2k 3k
-
p
k0
F (t) F
k k1
(t)
(25)
W W
4k 5k
W 1
6k k0
F (t) F
k k1
(t) W
1k
-
W W W W
2k 3k 4k 5k
-
W
6k
Since L(t) 1 P(T t) from (25).
L(t) 1 1
k0
F (t) F
k k1
(t)
W W W
1k 2k 3k
-
p
k0
F (t) F
k k1
(t)
W W
4k 5k
W p1
6k k0
F (t) F
k k1
(t) W
1k
-
-
W W W W
2k 3k 4k 5k
-
W
6k
(26)
Proceeding in the same way we get the first two moments
E(T) 1 u (W W
-
W ) p(W
-
W W )
k 0
p(1 ) (W
1k
1k 2k
-
-
-
W W
2k 3k
3k 4k
) (W W
4k 5k
5k 6k
-
W )
6k
(27)
2
1
1
E(T2 ) 2b
k(1 2 ) 1)(W W
-
W ) p(W
-
W W )
k 0
p(1 )(W
1k
-
-
-
W W
2k 3k
1k 2k
)(W W
4k 5k
3k 4k
-
W )
6k
5k 6k
(28)
(27) gives the expected time to recruitment,(27) and (28) together with (10) gives the variance of time to recruitment.
If Ui form a geometric process with parameter a, then the first two moments are given by
1
E(T) E(U
)1
(W W
k
-
W ) p(W
-
W W )
1 k 0 a 1k
2k 3k
4k 5k 6k
p(1 ) (W
1k
-
-
-
W W
2k 3k
) (W W
4k 5k
-
W )
6k
(29)
k
2 k
2
E(T 2 ) (1 ) V(U 1 (E(U
))2 1 1
(W W W )
)
1 2k
1 i1
j1 1k 2k 3k
k0
a
j1 a
j1 a
-
p(W W
4k 5k
W ) p(1 )(W
6k 1k
-
-
W W
2k 3k
)(W
4k
-
W W )
5k 6k
(30)
(29) gives the expected time to recruitment, (29) and (30) together with (10) gives the variance of time to recruitment.
-
-
MODEL DESCRIPTION AND ANALYSIS FOR MODEL-II
For this model, the optional and mandatory thresholds for the loss of man-hours in the organization are taken
as Y min(Y
A
, Y ) and Z min(Z
B A
, Z ) . All the other assumptions and notations are as in model-I.
B
Case (i): Let Xi be the loss of man hours due to the ith decision epoch , i=1,2,3 forming a sequence of independent and
identically distributed exponential random variables with mean
1 (c>0), probability density function g(.),Ui are exchangeable
c
and constantly correlated exponential random variables denoting inter-decision time between (i-1)th and ith decision, i=1,2, 3.k with cumulative distribution function F(.), probability density function f(.) and mean u .
Proceeding as in previous model we get the first two moments
1
E(T) u
1 g*(
1
)
2
1
-
p
1 g*(
1
)
2
1 g*(
1
1
)g*(
2 1
)
2
(31)
1 2g*( )
1 2g*( )
1 2g*( )g*( )
E(T2 ) 2u 2 1 2
1 2
1 2 1 2
* 2 p *
2 * *
2
(1 g ( )) (1 g (
))
(1 g ( )g (
))
1 2 1 2
1 2 1
2
(32)
(31) gives the expected time to recruitment, (31) and (32) together with (10) gives the variance of time to recruitment.
Case (ii):
Let Xi be the loss of man-hours due to the ith decision epoch , i=1,2,3k Let X1, X2, X3,….Xn
be the order statistics
selected from the sample X1, X2,….Xn with respective density functions gx(1) (.), gx(2) (.),….g x(n) (.) ,
If Ui are exchangeable and constantly correlated exponential random variables.
Suppose
The first two moments are given by (31),(32) and (13).
Suppose
g(t) g (t)
x(1)
g(t) g (t)
x(k)
The first two moments are given by (31) ,(32) and (14) If Ui form a geometric process with parameter a
Suppose
g(t) g (t)
x(1)
E(T) aE(U
1
)C
3
p(C
6
H )
3,6
(33)
2 a 2
(1 g*(
))
E(U )a 2
2(1 g*(
))
1 g*(
)
E(T2 ) U1 1 1 2 1 1 1 2 1 2
(a 2
1)
a 2 g*(
1
2 )
(a 1)2
a g*(
1
)
2
a 2 g*(
1
2
2
)
2 a 2 (1 g*( )) (1 g*( )g*( ))
p U1 1 2 1 2 1 2
(a 2
2
1) a
g*(
1
)
2
a 2 g*(
1
)g*(
2 1
2 )
E(U )a 2 2(1 g*( )) 1 g*( ) 2(1 g*( )g*( )) (1 g*( )g*( ))
1 1 2 1 2 1 2 1 2 1 2 1 2
(a 1)2
a g*(
1
)
2
a 2 g*(
1
)
2
a g*(
1
)g*(
2 1
)
2
a 2 g*(
1
)g*(
2 1
2 )
Where g*
x(1)
()
1
,
2 1
-
is given by (13)
2
(34)
-
gives the expected time to recruitment, (33) and (34) together with (10) gives the variance of time to recruitment.
Suppose
g(t) g (t)
x(k)
The first two moments are given by (33),(34) and (14).
Case (iii):
Let Xi be a exponential random variable denoting the loss of man-hours due to ith decision epoch, i=1,2,3.with cumulative distribution function G (.) and probability density function g(.)
If Ui are exchangeable and constantly correlated exponential random variables.
The first two moments are given by
E(T) 1 u
k0
W pW
3k 6k
p(1 )W W
3k 6k
(35)
E(T2 ) 2b k(1 2 ) 1)
2
W
-
pW
p(1 )W W
1
1
k0
3k 6k
3k 6k
(36)
where
W ,(a=3,6) are given by (24).
ak
If Ui forms geometric process then the first two moments are given by
1
E(T) E(U ) 1
1
W pW p(1 )W W
k k k k k
k0 a
k
3 6
2 k
3 6
2
(37)
E(T2 ) (1 )
V(U
1 (E(U) )2
)
)
1
1 W
-
pW
p(1 )W W
k0
1 2k
a
1 j1 a
i1
j1 a
j1 3k 6k
3k 6k
(38)
where W ,(a=3,6) are given by (24).
ak
-
-
-
-
MODEL DESCRIPTION AND ANALYSIS OF MODEL-III
For this model, the optional and mandatory thresholds for the loss of man-hours in the organization are taken as
Y Y Y
1 2
and Z Z Z
1 2
.All the other assumptions and notations are as in model-I.
k
e
x
2
x
e 1
PX
Y 1 2 g
(x) dx
0
0
i
i1
1 2
k 1 2
k
* k * k
(i.e) PX
Y 1 g ( )
2 g
( )
i
i1
1 2
2 1
1 2
Case (i): Let Xi be the loss of man hours due to the ith decision epoch , i=1,2,3 forming a sequence of independent and identically distributed exponential random variables with mean 1 (c>0), probability density function g(.),Ui are exchangeable
c
and constantly correlated exponential random variables denoting inter-decision time between (i-1)th and ith decision, i=1,2, 3.k with cumulative distribution function F(.), probability density functin f(.) and mean u .
Proceeding as in model I we get the first two moments
A A A A A A
A A A A
A A
E(T) u 2 1 p 5 4 1 4 2 4 1 5 2 5
* * *
* * *
* * * *
* *
1 g ( ) 1 g ( ) 1 g ( )
1 g ( ) 1 g ( )g ( )
1 g ( )g ( ) 1 g ( )g ( ) 1 g ( )g ( )
2 1 2
1 1 1
2 1 1 2 2
(39)
2
1 2g*( )
1 2g*( )
1 2g*( )
1 2g*( )
E(T2 ) 2u 2 A
2 A
1 A
2 A
1
2
2
(1 g*( ))2 2
1 (1 g*( ))2
1
p
5 (1 g*(
2
))2
4 (1 g*( ))2
1
1 2g*( )g*( )
1 2g*( )g*( )
1 2g*( )g*( )
1 2g*( )g*( )
-
A A
1 1 A A
2 1 A A
1 2 A A
2 2
1 4 *
* 2 2 4
* * 2 1 5
* * 2 2 5
* * 2
(1 g ( )g ( ))
(1 g ( )g ( ))
(1 g ( )g ( ))
(1 g ( )g ( ))
1 1 2 1 1 2
2 2
(40)
In (39) and (40) A
2 , A 1 , A 2 , A 1
1
1 2
2
1 2
4
1 2
5
1 2
(39) gives the mean time to recruitment, (39) and (40) together with (10) gives the variance of the time to recruitment .
Case (ii):
Let Xi be the loss of man-hours due to the ith decision epoch , i=1,2,3k Let X1, X2, X3,….Xn
be the order
statistics selected from the sample X1, X2,….Xn with respective density functions gx(1) (.), gx(2) (.),….g x(n) (.) ,
If Ui are exchangeable and constantly correlated exponential random variables.
If
The first two moments are given by (39),(40) and (13) .
If
g(t) g (t)
x(1)
g(t) g (t)
x(k)
The first two moments are given by (39) ,(40)and (14)
If Ui form a geometric process with parameter a, then the first two moments are given by
Suppose
g(t) g (t)
x(1)
E(T) aE(U )A C
-
A C
-
pA C
-
A C
-
A A H
-
-
A A H
-
A A H
-
A A H
1 2 2 1 1
5 5 4 4
1 4 1,4
2 4 2,4
1 5 1,5
2 5 2,5
(41)
2 a 2
(1 g*( ))
(1 g*( ))
E(U )a 2
2(1 g*( ))
1 g*( )
a
a
E(T 2 ) U1 A 1 2 A 1 1 1 A
1 2 2
2 2 2 *
1
2 *
2 2 *
2 *
(a 1)
a g ( )
2
g ( )
1
(a 1)
a g ( )
2
a g (2 )
2(1 g*( ))
1 g*( )
2 a 2
(1 g*(
))
(1 g*( ))
(1 g*( )g*( ))
A 1 1 1 p U1 A 1 2 A 1 1 A A 1 1
1 *
2 *
2 5 2 *
4 2 *
1 4 2 * *
a g ( )
a g ( )
(a 1)
a g ( )
a g ( )
a g ( )g ( )
1 1
2
1
1 1
(1 g*( )g*( )) (1 g*( )g*( )) (1 g*( )g*( ))
A A 2 1 A A 1 2 A A 2 2
2 4 2 * * 1 5 2 * * 2 5 2 * *
a g ( )g ( )
a g ( )g ( )
a g ( )g ( )
2 1
1 2
2 2
pE(U )a 2 2(1 g*( )) 1 g*( ) 2(1 g*( )) 1 g*( )
)
)
1 A 1 2 2 A 1 1 1
2 5 *
2 * 4 *
2 *
(a 1)
a g ( )
2
a g (
2
a g ( )
1
a g ( )
1
2(1 g*( )g*( ))
1 g*( )g*( )
2(1 g*( )g*( ))
1 g*( )g*( )
A A 1 1 1 1 1 A A 1 2 1 2 1
1 4 * *
2 * * 2 4 * *
2 * *
a g ( )g ( )
a g ( )g ( )
a g ( )g ( )
a g ( )g ( )
1 1
1 1 2 1
2 1
2(1 g*( )g*( )) 1 g*( )g*( ) 2(1 g*( )g*( )) 1 g*( )g*( )
A A 1 1 2 1 2 A A 1 2 2 2 2
1 5 * *
2 * * 2 5 * *
2 * *
a g ( )g ( )
a g ( )g ( )
a g ( )g ( )
a g ( )g ( )
1 2
1 2 2 2
2 2
(42)
(41) gives the mean time to recruitment , (41) and (42) together with (10) gives the variance of the time to recruitment .
Suppose
g(t) g (t)
x(k) .
where
g*
x(1)
() , ,
1
,
2 1 2
is given by (13).
The first two moments are given by (41),(42) and (14)
Case (iii):
Let Xi be a exponential random variable denoting the loss of man-hours due to ith decision epoch, i=1,2,3.with cumulative distribution function G (.) and probability density function g(.)
If Ui form a geometric process with parameter a.
The first two moments of time to recruitment is given by
1
E(T) (1 )u (W
-
W ) p(W
-
W )
k 0
p(1 )(W
k
-
W
33k
)(W
34k
-
W
35k
)
36k
(43)
33k
2
34k
35k
36k
1
1
E(T2 ) 2b
k(1 2 ) 1)(W
-
W ) p(W
-
W )
k 0
p(1 )(W
-
W )(W
33k
-
W
34k
)
35k
36k
33k
34k
35k
36k
(44)
W 1
33 k
(
)(b
1)k 1
1 k)(b
1) k] [(
1 2 2 2
W 2
34k
(
)(b
1)k 1
1 k)(b
1) k] [(
1 2 1 1
W 1
(45)
35k
(
)(b
1)k 1
1 k)(b
1) k] [(
1 2 2 2
W 2
36k
(
)(b
1)k 1
1 k)(b
1) k] [(
1 2 1 1
(43) gives the expected time to recruitment, (43) and (44) together with (10) gives the variance of time to recruitment.
If Ui form a geometric process with parameter a
Then the first two moments are given by
1
E(T) (1 )E(U )
(W
k
W ) p(W
W )
a
a
1 k0
33k
34k
35k
36k
p(1 )(W
33k
-
-
W
34k
)(W
35k
-
W )
36k
k
2 k
2
(46)
E(T 2 ) (1 )
V(U
) 1 (E(U
)) 2
1
1
(W
-
W )
j1 a
j1 a
k0
1 2k
a
1
i1
j1 a
j1
33k
34k
-
p(W
-
-
35k
-
W
36k
) p(1 )(W
33k
-
W
34k
)(W
35k
W )
36k
(47)
Where
W ,a=33,34,35,36 are given by (45) .
ak
(46) gives the expcted time to recruitment, (46) and (47) together with (10) gives the variance of time to recruitment.
-
-
NUMERICAL ILLUSTRATIONS
The mean and variance of the time to recruitment for all the models are given in the following tables for the cases (i) and (ii),
Case (i):Table I(a) (Effect of on performance measures)
0.4,
1 2
0.6,
1
0.5,
2
0.8, p 0.8, c 2.5
|
MODEL I |
MODEL II |
MODEL III |
||||
|
E(T) |
V(T) |
E(T) |
V(T) |
E(T) |
V(T) |
|
|
0.6 |
26.9969 |
699.2067 |
10.8231 |
139.4303 |
34.1896 |
1.3954e+003 |
|
0.7 |
35.9959 |
1.4628e+003 |
14.4308 |
282.2801 |
45.5861 |
2.8775e+003 |
|
0.8 |
53.9938 |
3.8617e+003 |
21.6462 |
724.4484 |
68.3791 |
7.5047e+003 |
Case (ii):
Table II (a) (Effect of and k on performance measures)
0.4,
1 2
0.6,
1
0.5,
2
0.8, p 0.8, c 1.5
|
g(t) g (t) x(1) |
|||||||
|
k |
MODEL I |
MODEL II |
MODEL III |
||||
|
E(T) |
V(T) |
E(T) |
V(T) |
E(T) |
V(T) |
||
|
0.6 |
2 |
31.7785 |
964.2689 |
12.4203 |
184.3391 |
40.4002 |
1.9427e+003 |
|
0.7 |
2 |
42.3713 |
2.0227e+003 |
16.5604 |
374.701 |
53.8669 |
4.015e+003 |
|
0.8 |
2 |
63.5569 |
5.352e+003 |
24.8405 |
965.0629 |
80.8003 |
1.0491e+004 |
|
0.6 |
1 |
17.4197 |
296.0506 |
7.6103 |
68.1898 |
21.7516 |
570.7102 |
|
0.6 |
2 |
31.7785 |
964.2689 |
12.4203 |
184.3391 |
40.4002 |
1.9427e+003 |
|
0.6 |
3 |
46.1127 |
2.1034e+003 |
17.1967 |
356.3135 |
59.0232 |
4.1246e+003 |
Table II (b) (Effect of and k on performance measures)
0.4,
1 2
0.6,
1
0.5,
2
0.8, p 0.8, c 1.5
|
g(t) g (t) x(k) |
|||||||
|
K |
MODEL I |
MODEL II |
MODEL III |
||||
|
E(T) |
V(T) |
E(T) |
V(T) |
E(T) |
V(T) |
||
|
0.6 |
2 |
11.824 |
136.893 |
5.3347 |
33.1588 |
14.7110 |
260.1546 |
|
0.7 |
2 |
15.7667 |
280.3428 |
7.1129 |
64.8872 |
19.6147 |
527.0417 |
|
0.8 |
2 |
23.65 |
726.7705 |
10.6694 |
161.4127 |
29.4220 |
1.3534e+003 |
|
0.6 |
1 |
17.4197 |
296.0506 |
7.6103 |
68.1898 |
21.7516 |
570.7102 |
|
0.6 |
2 |
11.825 |
136.893 |
5.3347 |
33.1588 |
14.7110 |
260.1546 |
|
0.6 |
3 |
9.8102 |
94.6229 |
4.5346 |
23.7928 |
12.1695 |
177.7643 |
VI. FINDINGS From the tables we observe the following which agree with reality
case (i) :
case (ii)
-
As increases , the mean and variance of time to recruitment increase for all the models .
-
As increases , the mean and variance of time to recruitment increase for all models when the probability density function of loss of manpower is the probability density function of first order statistics and k-th order statistics.
-
As k increases , the mean and variance of time to recruitment increase for all models when the probability density function of loss of manpower is the probability density function first order statistics and decrease when the probability density function of loss of manpower is the probability density function of k-th order statistics.