Au4201315330

P. Saranya et al Int. Journal of Engineering Research and Applications www.ijera.com

ISSN : 2248-9622, Vol. 4, Issue 2( Version 1), February 2014, pp.315-330

www.ijera.com 315 | P a g e

Expected Time to Recruitment in A Two - Grade Manpower

System Using Order Statistics for Inter-Decision Times and

Attrition

J. Sridharan*, P. Saranya

**, A. Srinivasan***

*(Department of Mathematics, Government Arts College (Autonomous), Kumbakonam-1)

** (Department of Mathematics, TRP Engineering College (SRM Group), Trichirappalli-105)

*** (Department of Mathematics, Bishop Heber College (Autonomous), Trichirappalli-17)

ABSTRACT In this paper, a two-grade organization subjected to random exit of personal due to policy decisions taken by the

organization is considered. There is an associated loss of manpower if a person quits. As the exit of personnel is

unpredictable, a new recruitment policy involving two thresholds for each grade-one is optional and the other

mandatory is suggested to enable the organization to plan its decision on recruitment. Based on shock model

approach three mathematical models are constructed using an appropriate univariate policy of recruitment.

Performance measures namely mean and variance of the time to recruitment are obtained for all the models

when (i) the loss of man-hours and the inter decision time forms an order statistics (ii) the optional and

mandatory thresholds follow different distributions. The analytical results are substantiated by numerical

illustrations and the influence of nodal parameters on the performance measures is also analyzed.

Keywords-Manpower planning, Shock models, Univariate recruitment policy, Mean and variance of the time to

recruitment.

I. INTRODUCTION Consider an organization having two grades in which depletion of manpower occurs at every decision

epoch. In the univariate policy of recruitment based on shock model approach, recruitment is made as and when

the cumulative loss of manpower crosses a threshold. Employing this recruitment policy, expected time to

recruitment is obtained under different conditions for several models in [1], [2] and [3].Recently in [4],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 namely mean and variance of the time to recruitment 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. In [5-8] the authors have extended the results in [2] for a two-grade

system according as the thresholds are exponential random variables or geometric random variables or SCBZ

property possessing random variables or extended exponential random variables. In [9-17], the authors have

extended the results in [4] for a 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 the performance measures when (i) the loss of man-hours and the inter decision time forms an order

statistics (ii) the optional and mandatory thresholds follow different distributions.

II. 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 Xi be the loss of man-hours due to the ith

decision epoch , i=1,2,3…n Let XXXX n321 ,....,, be the order statistics selected from the sample

XXX n21 ,...., with respective density functions (.)(.),....(.), ggg )n(x)2(x)1(x . Let Ui be a continuous random

variable 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

1 (λ>0). Let U(1) (U(k)) be the smallest (largest) order

RESEARCH ARTICLE OPEN ACCESS




statistic with probability density function. (.)(.), ff )k(u)1(u.Let Y1, Y2 (Z1, Z2) denotes the optional (mandatory)

thresholds for the loss of man-hours in grades 1 and 2, with parameters 2121 ,,, ‎‎ respectively, where

𝜃1,𝜃2,𝛼1 ,𝛼2 are positive. It is assumed that Y1 < Z1 and 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 [18] 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

1ii

k

1ii

0kk

k

1ii

0kk

ZPYP)t(pYP)t()tT(P XXVXV (1)

For n=1, 2, 3…k the probability density function of X(n) is given by

k..3,2,1n,)]t(G1)[t(g)]t(G[kcn)t(g nk1nn)n(x

(2)

If )t(g)t(g )1(x

In this case it is known that

1k)1(x )t(G1)t(gk)t(g

(3)

By hypothesis ctce)t(g

(4)

Therefore from (3) and (4) we get,

kc

kc)(g

)1(x

(5)

If )t(g)t(g )k(x


)t(g)t(Gk)t(g1k

)k(x

(6)

Therefore from (4) and (6) we get

)kc)...(c2)(c(

c!k)(g

k*

)k(x

(7)

For r=1, 2, 3…k the probability density function of U(r) is given by

k..3,2,1r,)]t(F1)[t(f)]t(F[kcr)t(f rk1rr)r(u

(8)

If )t()t(f f )1(u


1k)1(u )t(F1)t(fk)t(f

(9) Therefore from (8) and (9) we get,

sk

k)s(f )1(u

(10)

If )t(f)t(f )k(u


)t(f)t(Fk)t(f1k

)k(u

(11)

)ks)...(2s)(s(

!k)s(f

k*

)k(u

(12)

It is known that

22

0s2

*22

0s

*

))T(E()T(E)T(Vandds

)s(l(d)T(E,

ds

))s(l(d)T(E

(13)

Case (i): The distribution of optional and mandatory thresholds follow exponential distribution

For this case the first two moments of time to recruitment are found to be




If )t()t(f),t()t(g fg )1(u)1(x

HHHHHHHHHCCCCCC 6,3I5,3I4,3I6,2I5,2I4,2I6,1I5,1I4,1I5I4I3I2I1I 6I(p)T(E

(14)

HHHHHHHHHCCCCCCT2

6,3I

2

5,3I

2

4,3I

2

6,2I

2

5,2I

2

4,2I

2

6,1I

2

5,1I

2

4,1I

2

6I

2

5I

2

4I

2

3I

2

2I

2

1I

2p2)(E

(15)

where for a = 1, 2…6. b=1, 2, 3 and d=4, 5, 6.

)1(k

1and

)1(k

1

DDH

DC

IdIbd,Ib

IaIa

(16)

)(),(),(),(),(),(21

*

)1(x6I2

*

)1(x5I1

*

)1(x4I21

*

)1(x3I2

*

)1(x2I1

*

)1(x1I gDgDgDgDgDgD

are given by (5) (17) If )t()t(f),t()t(g fg )1(u)k(x

MMMMMMMMMLLLLLL 6,3K5,3K4,3K6,2K5,2K4,2K6,1K5,1K4,1K5K4K3K2K1K 6K(p)T(E

(18)

MMMMMMMMMLLLLLLT2

6,3K

2

5,3K

2

4,3K

2

6,2K

2

5,2K

2

4,2K

2

6,1K

2

5,1K

2

4,1K

2

6K

2

5K

2

4K

2

3K

2

2K

2

1K

2p2)(E (19)

where for a = 1, 2…6. b=1, 2, 3 and d=4, 5, 6.

)1(k

1and

)1(k

1

DDM

DL

KdKbd,Kb

KaKa

(20)

)(),(),(),(),(),(21

*

)k(x6K2

*

)k(x5K1

*

)k(x4K21

*

)k(x3K2

*

)k(x2K1

*

)k(x1K gDgDgDgDgDgD

are given by (7) (21)

If

)t()t(f),t()t(g fg )k(u)k(x

QQQQQQQQQPPPPPP 6,3K5,3K4,3K6,2K5,2K4,2K6,1K5,1K4,1K5K4K3K2K1K 6K

(p)T(E (22)

DDDDDDDDDDDDDDDDDDDDDN

DDDNQQQQQQQQQPPPPPPT

3K1

1

3K1

1

3K1

1

2K1

1

2K1

1

2K1

1

1K1

1

1K1

1

1

1

1

1

1

1

1

1M

p

1

1

1

1

1

1M

1p2)(E

6K5K4K6K5K4K6K5K4K1K6K5K4K

2

2

3K2K1K

2

2

2

6,3K

2

5,3K

2

4,3K

2

6,2K

2

5,2K

2

4,2K

2

6,1K

2

5,1K

2

4,1K

2

6K

2

5K

2

4K

2

3K

2

2K

2

1K

2

(23)

where for a = 1, 2…6. b=1, 2, 3 and d=4, 5, 6.

k

1n2

k

1nKdKbd,Kb

KaKa

nDDQ

DP

1Mand

n

1N,

)1(

N,

)1(

N (24)

If )t()t(f),t()t(g fg )k(u)1(x

SSSSSSSSSRRRRRR 6,3I5,3I4,3I6,2I5,2I4,2I6,1I5,1I4,1I5I4I3I2I1I 6I(p)T(E

(25)

DDDDDDDDDDDDDDDDDDDDDN

DDDNSSSSSSSSSRRRRRRT

3I1

1

3I1

1

3I1

1

2I1

1

2I1

1

2I1

1

1I1

1

1I1

1

1

1

1

1

1

1

1

1M

p

1

1

1

1

1

1M

1p2)(E

6I5I4I6I5I4I6I5I4I1I6I5I4I

2

2

3I2I1I

2

2

2

6,3I

2

5,3I

2

4,3I

2

6,2I

2

5,2I

2

4,2I

2

6,1I

2

5,1I

2

4,1I

2

6I

2

5I

2

4I

2

3I

2

2I

2

1I

2

(26)

where for a = 1, 2…6, b=1, 2, 3 and d=4, 5

k

1n2

k

1nIdIbd,Ib

IaIa

nDDS

DR

1Mand

n

1N,

)1(

N,

)1(

N

(27)

Case (ii): The distributions of optional and mandatory thresholds follow extended exponential distribution with

shape parameter 2.


H16H4H4H4H8H8H8H8H4HHH

H2H2H2H2H4HHHH2H2H2H2H4HHH

H2H2H2H2H8H2H2H2H4H4H4H4H8H2H2H2

H4H4H4H4H8H2H2H2H4H4H4H4H8H2H2H2

H4H4H4H4C4CCCC2C2C2C2C4CCCC2C2C2C2

6,3I16,3I15,3I14,3I13,3I12,3I5,3I4,3I6,11I16,11I15,11I14,11I

13,11I12,11I5,11I4,11I6,10I16,10I15,10I14,0I113,10I12,10I5,10I4,10I6,9I16,9I15,9I14,9I



13,1I12,1I5,1I4,1I6I16I15I14I13I12I5I4I3I11I10I9I8I7I2I1Ip)T(E

(28)

H16H4H4H4H8H8H8H8H4HHHH2H2H2

H2H4HHHH2H2H2H2H4HHHH2H2H2

H2H8H2H2H2H4H4H4H4H8H2H2H2H4H4

H4H4H8H2H2H2H4H4H4H4H8H2H2H2H4

H4H4H4C4CCCC2C2C2C2C4CCCC2C2C2C2T

2

6,3I

2

16,3I

2

15,3I

2

14,3I

2

13,3I

2

12,3I

2

5,3I

2

4,3I

2

6,11I

2

16,11I

2

15,11I

2

14,11I

2

13,11I

2

12,11I

2

5,11I

2

4,11I

2

6,10I

2

16,10I

2

15,10I

2

14,10I

2

13,10I

2

12,10I

2

5,10I

2

4,10I

2

6,9I

2

16,9I

2

15,9I

2

14,9I

2

13,9I

2

12,9I

2

5,9I

2

4,9I

2

6,8I

2

16,8I

2

15,8I

2

14,8I

2

13,8I

2

12,8I

2

5,8I

2

4,8I

2

6,7I

2

16,7I

2

15,7I

2

14,7I

2

13,7I

2

12,7I

2

5,7I

2

4,7I

2

6,2I

2

16,2I

2

15,2I

2

14,2I

2

13,2I

2

12,2I

2

5,2I

2

4,2I

2

6,1I

2

16,1I

2

15,1I

2

14,1I

2

13,1I

2

12,1I

2

5,1I

2

4,1I

2

6I

2

16I

2

15I

2

14I

2

13I

2

12I

2

5I

2

4I

2

3I

2

11I

2

10I

2

9I

2

8I

2

7I

2

2I

2

1I

2p2)(E

(29)




where for a=1, 2, 3…16, b=1, 2, 3,7, 8, 9, 10,11 and d=4, 5, 6, 12, 13,14,15,16.

HC d,IbIa , are given by

(16)

)1

(),(),(),2

(),1

(

),()(),(),1

(),(

22gD2gD2gD2gD2gD

22gD2gD2gD2gD2gD

2

*

)1(x16I2

*

)1(x15I1

*

)1(x14I1

*

)1(x13I2

*

)1(x12I

21

*

)1(x11I2

*

)1(x10I1

*

)1(x9I2

*

)1(x8I21

*

)1(x7I

(30)

If )t()t(f),t()t(g fg )1(u)k(x

M16M4M4M4M8M8M8M8M4MMMM2M2M2M2M4MMMM2M2M2M2M4MMM

M2M2M2M2M8M2M2M2M4M4M4M4M8M2M2M2M4M4M4M4M8M2M2M2M4M4M4M4M8M2M2M2

M4M4M4M4L4LLLL2L2L2L2L4LLLL2L2L2L2

6,3K16,3K15,3K14,3K13,3K12,3K5,3K4,3K6,11K16,11K15,11K14,11K

13,11K12,11K5,11K4,11K6,10K16,10K15,10I14,0KI13,10K12,10K5,10K4,10K6,9K16,9K15,9K14,9K

13,9K12,9K5,9K4,9K6,8K16,8K15,8K14,8K13,8K12,8K5,8K4,8K6,7K16,7K15,7K14,7K


13,1K12,1K5,1K4,1K6K16K15K14K13K12K5K4K3K11K10K9KI8K7K2K1Kp)T(E

(31)

M16M4M4M4M8M8M8M8M4MMM

M2M2M2M2M4MMMM2M2M2M2M4

MMMM2M2M2M2M8M2M2M2M4M4

M4M4M8M2M2M2M4M4M4M4M8M2M2

M2M4M4M4M4M8M2M2M2M4M4M4M4

L4LLLL2L2L2L2L4LLLL2L2L2L2T

2

6,3K

2

16,3K

2

15,3K

2

14,3K

2

13,3K

2

12,3K

2

5,3K

2

4,3K

2

6,11K

2

16,11K

2

15,11K

2

14,11K

2

13,11K

2

12,11K

2

5,11K

2

4,11K

2

6,10K

2

16,10K

2

15,10K

2

14,10K

2

13,10K

2

12,10K

2

5,10K

2

4,10K

2

6,9K

2

16,9K

2

15,9K

2

14,9K

2

13,9K

2

12,9K

2

5,9K

2

4,9K

2

6,8K

2

16,8K

2

15,8K

2

14,8K

2

13,8K

2

12,8K

2

5,8K

2

4,8K

2

6,7K

2

16,7K

2

15,7K

2

14,7K

2

13,7K

2

12,7K

2

5,7K

2

4,7K

2

6,2K

2

16,2K

2

15,2K

2

14,2K

2

13,2K

2

12,2K

2

5,2K

2

4,2K

2

6,1K

2

16,1K

2

15,1K

2

14,1K

2

13,1K

2

12,1K

2

5,1K

2

4,1K

2

6K

2

16K

2

15K

2

14K

2

13K

2

12K

2

5K

2

4K

2

3K

2

11K

2

10K

2

9K

2

8K

2

7K

2

2K

2

1K

2p2)(E

(32)

where for a=1,2,3,4,5,….16,b=1,2,…8 and d=9,10…..16.

ML d,KbKa , are given by

(20) .


Q16Q4Q4Q4Q8Q8Q8Q8Q4QQQQ2Q2Q2

Q2Q4QQQQ2Q2Q2Q2Q4QQQQ2Q2Q2

Q2Q8Q2Q2Q2Q4Q4Q4Q4Q8Q2Q2Q2Q4Q4

Q4Q4Q8Q2Q2Q2Q4Q4Q4Q4Q8Q2Q2Q2Q4

Q4Q4Q4P4PPPP2P2P2P2P4PPPP2P2P2P2

6,3K16,3K15,3K14,3K13,3K12,3K5,3K4,3K6,11K16,11K15,11K14,11K13,11K12,11K5,11K




12,1K5,1K4,1K6K16K15K14K13K12K5K4K3K11K10K9K8K7K2K1Kp)T(E

(33)

DDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDDDD





DDDDDDDDDDN

DDDDD

DDDNQ16Q4Q4Q4Q8Q8Q8Q8Q4QQQ

Q2Q2Q2Q2Q4QQQQ2Q2Q2Q2Q4QQQ

Q2Q2Q2Q2Q8Q2Q2Q2Q4Q4Q4Q4Q8Q2Q2Q2

Q4Q4Q4Q4Q8Q2Q2Q2Q4Q4Q4Q4Q8Q2Q2Q2

Q4Q4Q4Q4P4PPPP2P2P2P2P4PPPP2P2P2P2T

6K3K16K3K15K3K14K3K13K3K12K3K5K3K4K3K

6K11K16K11K15K11K14K11K13K11K12K11K5K11K4K11K6K10K16K10K15K10K





4K1K6K16K15K14K13K12K5K4K

2

23K11K10K9K8K

7K2K1K

2

2

2

6,3K

2

16,3K

2

15,3K

2

14,3K

2

13,3K

2

12,3K

2

5,3K

2

4,3K

2

6,11K

2

16,11K

2

15,11K

2

14,11K

2

13,11K

2

12,11K

2

5,11K

2

4,11K

2

6,10K

2

16,10K

2

15,10K

2

14,10K

2

13,10K

2

12,10K

2

5,10K

2

4,10K

2

6,9K

2

16,9K

2

15,9K

2

14,9K

2

13,9K

2

12,9K

2

5,9K

2

4,9K

2

6,8K

2

16,8K

2

15,8K

2

14,8K

2

13,8K

2

12,8K

2

5,8K

2

4,8K

2

6,7K

2

16,7K

2

15,7K

2

14,7K

2

13,7K

2

12,7K

2

5,7K

2

4,7K

2

6,2K

2

16,2K

2

15,2K

2

14,2K

2

13,2K

2

12,2K

2

5,2K

2

4,2K

2

6,1K

2

16,1K

2

15,1K

2

14,1K

2

13,1K

2

12,1K

2

5,1K

2

4,1K

2

6K

2

16K

2

15K

2

14K

2

13K

2

12K

2

5K

2

4K

2

3K

2

11K

2

10K

2

9K

2

8K

2

7K

2

2K

2

1K

2

1

16

1

4

1

4

1

4

1

8

1

8

1

8

1

8

1

4

1

1

1

1

1

1

1

2

1

2

1

2

1

2

1

4

1

1

1

1

1

1

1

2

1

2

1

2

1

2

1

4

1

1

1

1

1

1

1

2

1

2

1

2

1

2

1

8

1

2

1

2

1

2

1

4

1

4

1

4

1

4

1

8

1

2

1

2

1

2

1

4

1

4

1

4

1

4

1

8

1

2

1

2

1

2

1

4

1

4

1

4

1

4

1

8

1

2

1

2

1

2

1

4

1

4

1

4

1

4

1

4

1

1

1

1

1

1

1

2

1

2

1

2

1

2)M(

p

1

4

1

1

1

1

1

1

1

2

1

2

1

2

1

2)M(

1

p2)(E

(34)

where for a=1,2,3,4,5,….16,b=1,2,…8 and d=9,10…..16.

QP d,KbKa , are given by

(24) .


S16S4S4S4S8S8S8S8S4SSSS2

S2S2S2S4SSSS2S2S2S2S4SSSS2

S2S2S2S8S2S2S2S4S4S4S4S8S2S2S2S4

S4S4S4S8S2S2S2S4S4S4S4S8S2S2S2S4

S4S4S4R4RRRR2R2R2R2R4RRRR2R2R2R2

6,3I16,3I15,3I14,3I13,3I12,3I5,3I4,3I6,11I16,11I15,11I14,11I13,11I




12,1I5,1I4,1I6I16I15I14I13I12I5I4I3I11I10I9I8I7I2I1Ip)T(E

(35)




DDDDDDDDDD






DDDDDDDDDDDDDDDDN

DDDDDDDDNS16S4S4S4S8S8

S8S8S4SSSS2S2S2S2S4SSSS2S2S2

S2S4SSSS2S2S2S2S8S2S2S2S4S4S4S4S8S2

S2S2S4S4S4S4S8S2S2S2S4S4S4S4S8S2S2S2

S4S4S4S4R4RRRR2R2R2R2R4RRRR2R2R2R2T

6I3I16I3I15I3I14I3I13I3I

12I3I5I3I4I3I6I11I16I11I15I11I14I11I13I11I12I11I5I11I4I11I





13I1I12I1I5I1I4I1I6I16I15I14I13I12I5I4I

2

2

3I11I10I9I8I7I2I1I

2

2

2

6,3I

2

16,3I

2

15,3I

2

14,3I

2

13,3I

2

12,3I

2

5,3I

2

4,3I

2

6,11I

2

16,11I

2

15,11I

2

14,11I

2

13,11I

2

12,11I

2

5,11I

2

4,11I

2

6,10I

2

16,10I

2

15,10I

2

14,10I

2

13,10I

2

12,10I

2

5,10I

2

4,10I

2

6,9I

2

16,9I

2

15,9I

2

14,9I

2

13,9I

2

12,9I

2

5,9I

2

4,9I

2

6,8I

2

16,8I

2

15,8I

2

14,8I

2

13,8I

2

12,8I

2

5,8I

2

4,8I

2

6,7I

2

16,7I

2

15,7I

2

14,7I

2

13,7I

2

12,7I

2

5,7I

2

4,7I

2

6,2I

2

16,2I

2

15,2I

2

14,2I

2

13,2I

2

12,2I

2

5,2I

2

4,2I

2

6,1I

2

16,1I

2

15,1I

2

14,1I

2

13,1I

2

12,1I

2

5,1I

2

4,1

2

6I

2

16I

2

15I

2

14I

2

13I

2

12I

2

5I

2

4I

2

3I

2

11I

2

10I

2

9I

2

8I

2

7I

2

2I

2

1I

2

1

16

1

4

1

4

1

4

1

8

1

8

1

8

1

8

1

4

1

1

1

1

1

1

1

2

1

2

1

2

1

2

1

4

1

1

1

1

1

1

1

2

1

2

1

2

1

2

1

4

1

1

1

1

1

1

1

2

1

2

1

2

1

2

1

8

1

2

1

2

1

2

1

4

1

4

1

4

1

4

1

8

1

2

1

2

1

2

1

4

1

4

1

4

1

4

1

8

1

2

1

2

1

2

1

4

1

4

1

4

1

4

1

8

1

2

1

2

1

2

1

4

1

4

1

4

1

4

1

4

1

1

1

1

1

1

1

2

1

2

1

2

1

2)M(

p

1

4

1

1

1

1

1

1

1

2

1

2

1

2

1

2)M(

1

p2)(E

(36)

where for a=1,2,3…16, b=1, 2, 3, 7, 8, 9, 10,11 and d=4, 5, 6, 12, 13,14,15,16.

SR d,IbIa , are given by

(27).

Case (iii): The distributions of optional and mandatory thresholds possess SCBZ property.


HqqqqHqpqqHqqqHqpqqHppqqHpqq

HqqqHpqqHqqpqHqppqHqpqHqppqHpppq

HppqHqpqHppqHqqqHqpqHqqHqpqHppq

HpqHqqHpqHqqqpHqpqpHqqpHqpqpHppqpHpqp

HqqpHpqpHqqppHqpppHqppHqpppHppppHppp

HqppHpppHqqpHqppHqpHqppHpppHppHqp

HppHqqqHqpqHqqHqpqHppqHpqHqqHpq

HqqpHqppHqpHqppHpppHppHqpHppCqqCqp

CqCqpCppCpCqCpCqqCqpCqCqpCppCpCqCp

16,8I42115,8I32114,8I32113,8I42112,8I411,8I31

10,8I419,8I16,7I42115,7I32114,7I32113,7I42112,7I4

11,7I3110,7I419,7I16,6I4115,6I3114,6I3113,6I4112,6I4

11,6I310,6I49,6I16,5I42115,5I32114,5I32113,5I42112,5I411,5I31

10,5I419,5I16,4I42115,4I32114,4I32113,4I42112,4I411,4I31

10,4I419,4I16,3I4115,3I3114,3I3113,3I4112,3I411,3I310,3I4

9,3I16,2I4215,2I3214,2I3213,2I4212,2I411,2I310,2I49,2I

16,1I4215,1I3214,1I3213,1I4212,1I411,1I310,1I49,1I16I415I3

14I313I412I411I310I49I48I27I16I15I24I23I12I21I2

3431 2 32

21 2 43431 2 3

221 2 43431 3

111 43431 2 32

21 2 43431 2 32

21 2 43431 311

1 43432 3222 4

3432 3222 434

33(p

1211)T(E

(37)

HqqqqHqpqq

HqqqHqpqqHppqqHpqqHqqqHpqqHqqpqHqppqHqpq

HqppqHpppqHppqHqpqHppqHqqqHqpqHqqHqpq

HppqHpqHqqHpqHqqqpHqpqpHqqpHqpqpHppqp

HpqpHqqpHpqpHqqppHqpppHqppHqpppHpppp

HpppHqppHpppHqqpHqppHqpHqppHpppHpp

HqpHppHqqqHqpqHqqHqpqHppqHpqHqqHpq

HqqpHqppHqpHqppHpppHppHqpHppCqqCqp

CqCqpCppCpCqCpCqqCqpCqCqpCppCpCqCpT

2

16,8I421

2

15,8I321

2

14,8I321

2

13,8I421

2

12,8I4

2

11,8I31

2

10,8I41

2

9,8I4

2

16,7I421

2

15,7I321

2

14,7I321

2

13,7I421

2

12,7I4

2

11,7I31

2

10,7I41

2

9,7I4

2

16,6I41

2

15,6I31

2

14,6I31

2

13,6I41

2

12,6I4

2

11,6I3

2

10,6I4

2

9,6I4

2

16,5I421

2

15,5I321

2

14,5I321

2

13,5I421

2

12,5I4

2

11,5I31

2

10,5I41

2

9,5I41

2

16,4I421

2

15,4I321

2

14,4I321

2

13,4I421

2

12,4I4

2

11,4I31

2

10,4I41

2

9,4I41

2

16,3I41

2

15,3I31

2

14,3I31

2

13,3I41

2

12,3I4

2

11,3I3

2

10,3I4

2

9,3I4

2

16,2I42

2

15,2I32

2

14,2I32

2

13,2I42

2

12,2I4

2

11,2I3

2

10,2I4

2

9,2I4

2

16,1I42

2

15,1I32

2

14,1I32

2

13,1I42

2

12,1I4

2

11,1I3

2

10,1I4

2

9,1I4

2

16I4

2

15I3

2

14I3

2

13I4

2

12I4

2

11I3

2

10I4

2

9I4

2

8I2

2

7I1

2

6I1

2

5I2

2

4I2

2

3I1

2

2I2

2

1I2

2

34

31 2 3221 234

31 2 3221 2343

1 31113431 2 3

2223431 2 3

2223431 31

113432 3222

3432 322234

33p

12112)(E

(38)

where for a=1,2….16, b=1,2,3,4,5,6,7,8 and d=9,10,11,12,13,14,15,16.

)1(k

1and

)1(k

1

BBH

BC

IdIbd,Ib

IaIa

(39)

444

44

4

333

33

3

222

22

2

111

11

1pppp ,,,

pqpqpqpq 44332211 1,1,1,1 (40)

)3

(),3

(),(),3

(

),33(),(),(),(

),(),(),(),(

),11(),(),(),(

4

*

)1(x16I44

*

)1(x15I3

*

)1(x14I34

*

)1(x13I

44

*

)1(x12I33

*

)1(x11I4

*

)1(x10I44

*

)1(x9I

21

*

)1(x8I221

*

)1(x7I1

*

)1(x6I221

*

)1(x5I

22

*

)1(x1411

*

)1(x3I2

*

)1(x2I22

*

)1(x1I

gBgBgBgB

gBgBgBgB

gBgBgBgB

gBgBgBgB

(41)





MqqqqMqpqqMqqqMqpqqMppqqMpqqMqqqMpqq

MqqpqMqppqMqpqMqppqMpppqMppqMqpqMppqMqqq

MqpqMqqMqpqMppqMpqMqqMpqMqqqpMqpqpMqqp

MqpqpMppqpMpqpMqqpMpqpMqqppMqpppMqppMqppp

MppppMpppMqppMpppMqqpMqppMqpMqppMppp

MppMqpMppMqqqMqpqMqqMqpqMppqMpqMqq

MpqMqqpMqppMqpMqppMpppMppMqpMppLqq

LqpLqLqpLppLpLqLpLqqLqpLqLqpLppLpLqLp

16,8K42115,8K32114,8K32113,8K42112,8K411,8K3110,8K419,8K

16,7K42115,7K32114,7K32113,7K42112,7K411,7K3110,7K419,7K16,6K41

15,6K3114,6K3113,6K4112,6K411,6K310,6K49,6K16,5K42115,5K32114,5K321

13,5K42112,5K411,5K3110,5K419,5K16,4I42115,4I32114,4I32113,4I421

12,4K411,4K3110,4K419,4I16,3I4115,3I3114,3I3113,3K4112,3K4

11,3K310,3K49,3K16,2K4215,2K3214,2K3213,2K4212,2K411,2K310,2K4

9,2K16,1K4215,1K3214,1K3213,1K4212,1K411,1KI310,1K49,1K16K4

15K314K313K412K411K310K49K48K27K16K15K24K23K12K21K2

3431 2 3221 2 4

3431 2 3221 2 43

431 3111 434

31 2 3221 2 4343

1 2 3221 2 43431 3

111 43432 322

2 43432 3222 43

433(p

1211)T(E

(42)

MqqqqMqpqqMqqqMqpqqMppqqMpqqMqqqMpqqMqqpq

MqppqMqpqMqppqMpppqMppqMqpqMppqMqqqMqpq

MqqMqpqMppqMpqMqqMpqMqqqpMqpqpMqqpMqpqp

MppqpMpqpMqqpMpqpMqqppMqpppMqppMqpppMpppp

MpppMqppMpppMqqpMqppMqpMqppMpppMpp

MqpMppMqqqMqpqMqqMqpqMppqMpqMqqMpq

MqqpMqppMqpMqppMpppMppMqpMppLqqLqp

LqLqpLppLpLqLpLqqLqpLqLqpLppLpLqLpT

2

16,8K421

2

15,8K321

2

14,8K321

2

13,8K421

2

12,8K4

2

11,8K31

2

10,8K41

2

9,8K4

2

16,7K421

2

15,7K321

2

14,7K321

2

13,7K421

2

12,7K4

2

11,7K31

2

10,7K41

2

9,7K4

2

16,6K41

2

15,6K31

2

14,6K31

2

13,6K41

2

12,6K4

2

11,6K3

2

10,6K4

2

9,6K4

2

16,5K421

2

15,5K321

2

14,5K321

2

13,5K421

2

12,5K4

2

11,5K31

2

10,5K41

2

9,5K41

2

16,4K421

2

15,4K321

2

14,4K321

2

13,4K421

2

12,4K4

2

11,4K31

2

10,4K41

2

9,4K41

2

16,3K41

2

15,3K31

2

14,3K31

2

13,3K41

2

12,3K4

2

11,3K3

2

10,3K4

2

9,3K4

2

16,2K42

2

15,2K32

2

14,2K32

2

13,2K42

2

12,2K4

2

11,2K3

2

10,2K4

2

9,2K4

2

16,1K42

2

15,1K32

2

14,1K32

2

13,1K42

2

12,1K4

2

11,1K3

2

10,1K4

2

9,1K4

2

16K4

2

15K3

2

14K3

2

13K4

2

12K4

2

11K3

2

10K4

2

9K4

2

8K2

2

7K1

2

6K1

2

5K2

2

4K2

2

3K1

2

2K2

2

1K2

2

3431 2 3221 23

431 2 3221 234

31 3111343

1 2 32223431 2 3

2223431 31

113432 3222

3432 322234

33p

12112)(E

(43)

where for a=1,2….16, b=1,2,3,4,5,6,7,8 and d=9,10,11,12,13,14,15,16.

)1(k

1and

)1(k

1

BBM

BL

KdKbd,Kb

KaKa

(44)

)3

(),3

(),(),3

(

),33(),(),(),(

),(),(),(),(

),11(),(),(),(

4

*

)k(x16K44

*

)k(x15K3

*

)k(x14K34

*

)k(x13K

44

*

)k(x12K33

*

)k(x11K4

*

)k(x10K44

*

)k(x9K

21

*

)k(x8K221

*

)k(x7K1

*

)k(x6K221

*

)k(x5K

22

*

)k(x4K11

*

)k(x3K2

*

)k(x2K22

*

)k(x1K

gBgBgBgB

gBgBgBgB

gBgBgBgB

gBgBgBgB

(45)

If )t()t(f),t()t(g fg )k(u)k(x

QqqqqQqpqqQqqqQqpqqQppqqQpqqQqqq

QpqqQqqpqQqppqQqpqQqppqQpppqQppq

QqpqQppqQqqqQqpqQqqQqpqQppqQpq

QqqQpqQqqqpQqpqpQqqpQqpqpQppqp

QpqpQqqpQpqpQqqppQqpppQqppQqppp

QppppQpppQqppQpppQqqpQqppQqp

QqppQpppQppQqpQppQqqqQqpqQqq

QqpqQppqQpqQqqQpqQqqpQqppQqp

QqppQpppQppQqpQppPqqPqpPqPqp

PppPpPqPpPqqPqpPqPqpPppPpPqPp

16,8K42115,8K32114,8K32113,8K42112,8K411,8K3110,8K41

9,8K16,7K42115,7K32114,7K32113,7K42112,7K411,7K31

10,7K419,7K16,6K4115,6K3114,6K3113,6K4112,6K411,6K3

10,6K49,6K16,5K42115,5K32114,5K32113,5K42112,5K4

11,5K3110,5K419,5K16,4K42115,4K32114,4K32113,4K421

12,4K411,4K3110,4K419,4K16,3K4115,3K3114,3K31

13,3K4112,3K411,3K310,3K49,3K16,2K4215,2K3214,2K32

13,2K4212,2K411,2K310,2K49,2K16,1K4215,1K3214,1K32

13,1K4212,1K411,1K310,1K49,1K16K415K314K313K4

12K411K310K49K48K27K16K15K24K23K12K21K2

3431 2 322

1 2 43431 2 32

21 2 43431 31

11 43431 2 3

221 2 4343

1 2 3221 2 434

31 3111 434

32 3222 434

32 3222 4343

3(p

1211)T(E

(46)




BB

qqqq

BB

pqqq

BB

qqq

BB

qpqq

BB

ppqq

BB

pqq

BB

qqq

BB

qpq

BB

qqpq

BB

pqpq

BB

qpq

BB

qppq

BB

pppq

BB

ppq

BB

qpq

BB

ppq

BB

qqq

BB

pqq

BB

qq

BB

qpq

BB

ppq

BB

pq

BB

qq

BB

qp

BB

qqqp

BB

pqqp

BB

qqp

BB

qpqp

BB

ppqp

BB

pqp

BB

qqp

BB

qpp

BB

qqpp

BB

pqpp

BB

qpp

BB

qppp

BB

pppp

BB

ppp

BB

qpp

BB

ppp

BB

qqp

BB

pqp

BB

qp

BB

qpp

BB

ppp

BB

pp

BB

qp

BB

pp

BB

qqq

BB

pqq

BB

qq

BB

qpq

BB

ppq

BB

pq

BB

qq

BB

pq

BB

qqp

BB

pqp

BB

qp

BB

qpp

BB

ppp

BB

pp

BB

qp

BB

pp

B

qq

B

pq

B

q

B

qp

B

pp

B

p

B

q

B

pN

B

qq

B

pq

B

q

B

qp

B

pp

B

p

B

q

B

pN

QqqqqQqpqqQqqqQqpqqQppqqQpqqQqqq

QpqqQqqpqQqppqQqpqQqppqQpppq

QppqQqpqQppqQqqqQqpqQqqQqpq

QppqQpqQqqQpqQqqqpQqpqpQqqp

QqpqpQppqpQpqpQqqpQpqpQqqpp

QqpppQqppQqpppQppppQpppQqpp

QpppQqqpQqppQqpQqppQpppQpp

QqpQppQqqqQqpqQqqQqpqQppq

QpqQqqQpqQqqpQqppQqpQqpp

QpppQppQqpQppPqqPqpPqPqp

PppPpPqPpPqqPqpPqPqpPppPpPqPpT

16K8K

431

15K8K

431

14K8K

321

13K8K

431

12K8K

431

11K8K

31

10K8K

421

9K8K

241

16K7K

431

15K7K

431

14K7K

321

13K7K

431

12K7K

431

11K7K

31

10K7K

421

9K7K

241

16K6K

43

15K6K

43

14K6K

31

13K6K

43

12K6K

43

11K6K

3

10K6K

41

9K6K

14

16K5K

43

15K5K

431

14K5K

321

13K5K

431

12K5K

431

11K5K

31

10K5K

421

9K5K

241

16K4K

43

15K4K

431

14K4K

321

13K4K

431

12K4K

431

11K4K

31

10K4K

421

9K4K

241

16K3K

43

15K3K

43

14K3K

31

13K3K

43

12K3K

43

11K3K

3

10K3K

41

9K3K

14

16K2K

43

15K2K

43

14K2K

32

13K2K

43

12K2K

43

11K2K

3

10K2K

42

9K2K

42

16K1K

43

15K1K

43

14K1K

32

13K1K

43

12K1K

43

11K1K

3

10K1K

42

9K1K

24

16K

43

15K

43

14K

3

13K

43

12K

43

11K

3

10K

4

9K

42

28K

21

7K

21

6K

1

5K

21

4K

21

3K

1

2K

2

1K

22

2

2

16,8K421

2

15,8K321

2

14,8K321

2

13,8K421

2

12,8K4

2

11,8K31

2

10,8K41

2

9,8K4

2

16,7K421

2

15,7K321

2

14,7K321

2

13,7K421

2

12,7K4

2

11,7K31

2

10,7K41

2

9,7K4

2

16,6K41

2

15,6K31

2

14,6K31

2

13,6K41

2

12,6K4

2

11,6K3

2

10,6K4

2

9,6K4

2

16,5K421

2

15,5K321

2

14,5K321

2

13,5K421

2

12,5K4

2

11,5K31

2

10,5K41

2

9,5K41

2

16,4K421

2

15,4K321

2

14,4K321

2

13,4K421

2

12,4K4

2

11,4K31

2

10,4K41

2

9,4K41

2

16,3K41

2

15,3K31

2

14,3K31

2

13,3K41

2

12,3K4

2

11,3K3

2

10,3K4

2

9,3K4

2

16,2K42

2

15,2K32

2

14,2K32

2

13,2K42

2

12,2K4

2

11,2K3

2

10,2K4

2

9,2K4

2

16,1K42

2

15,1K32

2

14,1K32

2

13,1K42

2

12,1K4

2

11,1K3

2

10,1K4

2

9,1K4

2

16K4

2

15K3

2

14K3

2

13K4

2

12K4

2

11K3

2

10K4

2

9K4

2

8K2

2

7K1

2

6K1

2

5K2

2

4K2

2

3K1

2

2K2

2

1K2

2

1

2

1

2

11

2

1

2

1

2

111

2

1

2

11

2

1

2

1

2

111

1

1

1

11

1

1

1

1

1

111

21

1

2

11

2

1

2

1

2

111

21

1

2

11

2

1

2

1

2

111

1

1

1

11

1

1

1

1

1

111

2

1

2

11

2

1

2

1

2

111

2

1

2

11

2

1

2

1

2

11111111

11)M(

p

11111111)M(

1

3431 2 322

1 23431 2 3

221 2343

1 311134

31 2 32223

431 2 322

23431 31

113432 3

222343

2 3222343

3p

12112)(E

(47)

where for a=1,2….16, b=1,2,3,4,5,6,7,8 and d=9,10,11,12,13,14,15,16.

)1(

Nand

)1(

N

BBQ

BP

KdKbd,Kb

KaKa

(48)


SqqqqSqpqqSqqqSqpqqSppqqSpqqSqqq

SpqqSqqpqSqppqSqpqSqppqSpppqSppq

SqpqSppqSqqqSqpqSqqSqpqSppqSpqSqq

SpqSqqqpSqpqpSqqpSqpqpSppqpSpqpSqqp

SpqpSqqppSqpppSqppSqpppSppppSpppSqpp

SpppSqqpSqppSqpSqppSpppSppSqpSpp

SqqqSqpqSqqSqpqSppqSpqSqqSpqSqqp

SqppSqpSqppSpppSppSqpSppRqqRqpRq

RqpRppRpRqRpRqqRqpRqRqpRppRpRqRp

16,8I42115,8I32114,8I32113,8I42112,8I411,8I3110,8I41

9,8I16,7I42115,7I32114,7I32113,7I42112,7I411,7I31

10,7I419,7I16,6I4115,6I3114,6I3113,6I4112,6I411,6I310,6I4

9,6I16,5I42115,5I32114,5I32113,5I42112,5I411,5I3110,5I41

9,5I16,4I42115,4I32114,4I32113,4I42112,4I411,4I3110,4I41

9,4I16,3I4115,3I3114,3I3113,3I4112,3I411,3I310,3I49,3I

16,2I4215,2I3214,2I3213,2I4212,2I411,2I310,2I49,2I16,1I42

15,1I3214,1I3213,1I4212,1I411,1I310,1I49,1I16I415I314I3

13I412I411I310I49I48I27I16I15I24I23I12I21I2

3431 2 322

1 2 43431 2 32

21 2 43431 311

1 43431 2 322

1 2 43431 2 322

1 2 43431 3111 4

3432 3222 43

432 3222 434

33(p

1211)T(E

(49)




BB

qqqq

BB

pqqq

BB

qqq

BB

qpqq

BB

ppqq

BB

pqq

BB

qqq

BB

qpq

BB

qqpq

BB

pqpq

BB

qpq

BB

qppq

BB

pppq

BB

ppq

BB

qpq

BB

ppq

BB

qqq

BB

pqq

BB

qq

BB

qpq

BB

ppq

BB

pq

BB

qq

BB

qp

BB

qqqp

BB

pqqp

BB

qqp

BB

qpqp

BB

ppqp

BB

pqp

BB

qqp

BB

qpp

BB

qqpp

BB

pqpp

BB

qpp

BB

qppp

BB

pppp

BB

ppp

BB

qpp

BB

ppp

BB

qqp

BB

pqp

BB

qp

BB

qpp

BB

ppp

BB

pp

BB

qp

BB

pp

BB

qqq

BB

pqq

BB

qq

BB

qpq

BB

ppq

BB

pq

BB

qq

BB

pq

BB

qqp

BB

pqp

BB

qp

BB

qpp

BB

ppp

BB

pp

BB

qp

BB

pp

B

qq

B

pq

B

q

B

qp

B

pp

B

p

B

q

B

pN

B

qq

B

pq

B

q

B

qp

B

pp

B

p

B

q

B

pNSqqqqSqpqqSqqqSqpqqSppqqSpqq

SqqqSpqqSqqpqSqppqSqpqSqppqSpppqSppq

SqpqSppqSqqqSqpqSqqSqpqSppqSpqSqq

SpqSqqqpSqpqpSqqpSqpqpSppqpSpqpSqqp

SpqpSqqppSqpppSqppSqpppSppppSpppSqpp

SpppSqqpSqppSqpSqppSpppSppSqpSpp

SqqqSqpqSqqSqpqSppqSpqSqqSpqSqqp

SqppSqpSqppSpppSppSqpSppRqqRqpRq

RqpRppRpRqRpRqqRqpRqRqpRppRpRqRpT

16I8I

431

15I8I

431

14I8I

321

13I8I

431

12I8I

431

11I8I

31

10I8I

421

9I8I

241

16I7I

431

15I7I

431

14I7I

321

13I7I

431

12I7I

431

11I7I

31

10I7I

421

9I7I

241

16I6I

43

15I6I

43

14I6I

31

13I6I

43

12I6I

43

11I6I

3

10I6I

41

9I6I

14

16I5I

43

15I5I

431

14I5I

321

13I5I

431

12I5I

431

11I5I

31

10I5I

421

9I5I

241

16I4I

43

15I4I

431

14I4I

321

13I4I

431

12I4I

431

11I4I

31

10I4I

421

9I4I

241

16I3I

43

15I3I

43

14I3I

31

13I3I

43

12I3I

43

11I3I

3

10I3I

41

9I3I

14

16I2I

43

15I2I

43

14I2I

32

13I2I

43

12I2I

43

11I2I

3

10I2I

42

9I2I

42

16I1I

43

15I1I

43

14I1I

32

13I1I

43

12I1I

43

11I1I

3

10I1I

42

9I1I

24

16I

43

15I

43

14I

3

13I

43

12I

43

11I

3

10I

4

9I

42

28I

21

7I

21

6I

1

5I

21

4I

21

3I

1

2I

2

1I

22

2

2

16,8I421

2

15,8I321

2

14,8I321

2

13,8I421

2

12,8I4

2

11,8I31

2

10,8I41

2

9,8I4

2

16,7I421

2

15,7I321

2

14,7I321

2

13,7I421

2

12,7I4

2

11,7I31

2

10,7I41

2

9,7I4

2

16,6I41

2

15,6I31

2

14,6I31

2

13,6I41

2

12,6I4

2

11,6I3

2

10,6I4

2

9,6I4

2

16,5I421

2

15,5I321

2

14,5I321

2

13,5I421

2

12,5I4

2

11,5I31

2

10,5I41

2

9,5I41

2

16,4I421

2

15,4I321

2

14,4I321

2

13,4I421

2

12,4I4

2

11,4I31

2

10,4I41

2

9,4I41

2

16,3I41

2

15,3I31

2

14,3I31

2

13,3I41

2

12,3I4

2

11,3I3

2

10,3I4

2

9,3I4

2

16,2I42

2

15,2I32

2

14,2I32

2

13,2I42

2

12,2I4

2

11,2I3

2

10,2I4

2

9,2I4

2

16,1I42

2

15,1I32

2

14,1I32

2

13,1I42

2

12,1I4

2

11,1I3

2

10,1I4

2

9,1I4

2

16I4

2

15I3

2

14I3

2

13I4

2

12I4

2

11I3

2

10I4

2

9I4

2

8I2

2

7I1

2

6I1

2

5I2

2

4I2

2

3I1

2

2I2

2

1I2

2

1

2

1

2

11

2

1

2

1

2

111

2

1

2

11

2

1

2

1

2

111

1

1

1

11

1

1

1

1

1

111

21

1

2

11

2

1

2

1

2

11

1

21

1

2

11

2

1

2

1

2

111

1

1

1

1

1

1

1

1

1

1

111

2

1

2

11

2

1

2

1

2

111

2

1

2

11

2

1

2

1

2

111

1111111)M(

p

111111

11)M(

1

3431 2 32

21 23431 2 32

21 23431 311

13431 2 322

23431 2 322

23431 3111

3432 32223

432 322234

33p

12112)(E

(50)

where for a=1,2….16, b=1,2,3,4,5,6,7,8 and d=9,10,11,12,13,14,15,16.

)1(

Nand

)1(

N

BBS

BR

IdIbd,Ib

IaIa

(51)

III. MODEL DESCRIPTION AND ANALYSIS OF MODEL-II For this model, the optional and mandatory thresholds for the loss of man-hours in the organization are

taken as Y=min (Y1, Y2) and Z=min (Z1, Z2). All the other assumptions and notations are as in model-I.

Case (i): The distribution of optional and mandatory thresholds follow exponential distribution

For this case the first two moments of time to recruitment are found to be

Proceeding as in model-I, it can be shown for the present model that


)(p)T(E HCC 6,3I6I3I (52)

HCC IIIpTE

2

6,3

2

6

2

3

2 (2)( (53)

where for a = 1, 2…6. b=1, 2, 3 and d=4, 5, 6 HC d,IbIa , are given by

(16)


)(p)T(E MLL 6,3K6K3K

(54)

MLLT2

6,3K

2

6K

2

3K

2(p(2)(E

(55)

where for a = 1, 2…6. b=1, 2, 3 and d=4, 5, 6 ML d,KbKa , are given by

(20).


)(p)T(E QPP 6,3K6K3K

(56)

DDDN

DNQPPT

3K1

1

1

1M

p

1

1M

1p2)(E

6K6K

2

23K

2

2

2

6,3K

2

6K

2

3K

2

(57)

where for a = 1, 2…6. b=1, 2, 3 and d=4, 5, 6 QP d,KbKa , are given by

(24)





)(p)T(E SRR 6,3I6I3I (58)

DDDN

DNSRR

I

Mp

MpTE

III

III

31

1

1

1

1

112)(

66

2

2

3

2

2

2

6,3

2

6

2

3

2

(59)

where for a = 1, 2…6. b=1, 2, 3 and d=4, 5, 6 SR d,IbIa , are given by

(26).

Case (ii): The distribution of optional and mandatory thresholds follow extended exponential distribution


HH2H2H4H2H4H4H8H2H4H4

H8H4H8H8H16CC2C2C4CC2C2C4

16,11I13,11I12,11I6,11I16,8I13,8I12,8I6,8I6,7I13,7I12,7I

6,7I16,3I13,3I12,3I6,3I16I13I12I6I11I8I7I3Ip)T(E

(60)

HH2H2H4H2H4H4H8H2H4

H4H8H4H8H8H16CC2C2C4CC2C2C4T2

16,11I

2

13,11I

2

12,11I

2

6,11I

2

16,8I

2

13,8I

2

12,8I

2

6,8I

2

16,7I

2

13,7I

2

12,7I

2

6,7I

2

16,3I

2

13,3I

2

12,3I

2

6,3I

2

6I1

2

13I

2

12I

2

6I

2

11I

2

8I

2

7I

2

3I

2p2)(E

(61)

where for a=3, 6,7,8,11,12, 13,16 ,b=3,7,8,11,and d=6,12,13,16 HC d,IbIa, are given by

(16)


MM2M2M4M2M4M4M8M2M4M4

M8M4M8M8M16LL2L2L4LL2L2L4

16,11K13,11K12,11K6,11K16,8K13,8K12,8K6,8K6,7K13,7K12,7K

6,7K16,3K13,3K12,3K6,3K16K13K12K6K11K8K7K3Kp)T(E

(62)

LL2L2L4L2L4L4L8L2L4L4

L8L4L8L8L16LL2L2L4LL2L2L4T2

16,11K

2

13,11K

2

12,11K

2

6,11K

2

16,8K

2

13,8K

2

12,8K

2

6,8K

2

16,7K

2

13,7K

2

12,7K

2

6,7K

2

16,3K

2

13,3K

2

12,3K

2

6,3K

2

16K

2

13K

2

12K

2

6K

2

11K

2

8K

2

7K

2

3K

2p2)(E

(63)

where for a=3, 6,7,8,11,12, 13,16 ,b=3,7,8,11,and d=6,12,13,16 ML d,KbKa , are given by

(20).


QQ2Q2Q4Q2Q4Q4Q8Q2Q4Q4

Q8Q4Q8Q8Q16PP2P2P4PP2P2P4

16,11K13,11K12,11K6,11K16,8K13,8K12,8K6,8K6,7K13,7K12,7K

6,7K16,3K13,3K12,3K6,3K16K13K12K6K11K8K7K3Kp)T(E

(64)

DDDDDDDDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDN

DDDDNQQ2Q2Q4Q2Q4Q4Q8

Q2Q4Q4Q8Q4Q8Q8Q16PP2P2P4PP2P2P4T

16K11K13K11K12K11K6K11K16K8K13K8K12K8K6K8K6K7K13K7K

12K7K6K7K16K3K13K3K12K3K6K3K16K13K12K6K

2

2

11K8K7K3K

2

2

2

16,11K

2

13,11K

2

12,11K

2

6,11K

2

16,8K

2

13,8K

2

12,8K

2

6,8K

2

16,7K

2

13,7K

2

12,7K

2

6,7K

2

16,3K

2

13,3K

2

12,3K

2

6,3K

2

16K

2

13K

2

12K

2

6K

2

11K

2

8K

2

7K

2

3K

2

1

1

1

2

1

2

1

4

1

2

1

4

1

4

1

8

1

2

1

4

1

4

1

8

1

4

1

8

1

8

1

16

1

1

1

2

1

2

1

4)M(

p

1

1

1

2

1

2

1

4)M(

1

p2)(E

(65)

where for a=3, 6,7,8,11,12, 13,16 ,b=3,7,8,11,and d=6,12,13,16 QP d,KbKa , are given by

(24) .


SS2S2S4S2S4S4S8S2S4S4

S8S4S8S8S16RR2R2R4RR2R2R4

16,11I13,11I12,11I6,11I16,8I13,8I12,8I6,8I6,7I13,7I12,7I

6,7I16,3I13,3I12,3I6,3I16I13I12I6I11I8I7I3Ip)T(E

(66)

DDDDDDDDDDDDDD

DDDDDDDDDDDDDDDDDDDD

DDN

DDDDNSS2S2S4S2S4S4

S8S2S4S4S8S4S8S8S16RR2R2R4RR2R2R4T

16I11I13I11I12I11I6I11I16I8I13I8I12I8I

6I8I6IK7I13I7I12I7I6I7I16I3I13I3I12I3I6I3I16I13I

12I6I

2

211I8I7I3I

2

2

2

16,11I

2

13,11I

2

12,11I

2

6,11I

2

16,8I

2

13,8I

2

12,8I

2

6,8I

2

16,7I

2

13,7I

2

12,7I

2

6,7I

2

16,3I

2

13,3I

2

12,3I

2

6,3I

2

6I1

2

13I

2

12I

2

6I

2

11I

2

8I

2

7I

2

3I

2

1

1

1

2

1

2

1

4

1

2

1

4

1

4

1

8

1

2

1

4

1

4

1

8

1

4

1

8

1

8

1

16

1

1

1

2

1

2

1

4)M(

p

1

1

1

2

1

2

1

4)M(

1

p2)(E

(67)

where for a=3, 6,7,8,11,12, 13,16 ,b=3,7,8,11,and d=6,12,13,16 SR d,IbIa , are given by

(26).

Case (iii): The distributions of optional and mandatory thresholds possess SCBZ property.


HqqqqHqpqqHqpqqHppqqHqqpqHqppq

HqppqHpppqHqqqpHqpqpHqpqp

HppqpHqqppHpqppHqpppHpppp

CqqCqpCqpCppCqqCqpCqpCpp

16,8I42115,8I32113,8I42112,8I416,7I42115,7I321

13,7I42112,7I416,5I42115,5I32113,5I421

12,5I416,4I42115,4I42113,4I42112,4I4

16I415I313I412I48I27I15I24I2

3431 2 334

31 2 3343

1 2 33331 2 3

3433(p

1211)T(E (68)




HqqqqHqpqqHqqqHqpqqHppqqHqqpq

HqppqHqppqHpppqHqqqpHqpqpHqpqp

HppqpHqqppHqpppHqpppHpppp

CqqCqpCqpCppCqqCqpCqpCppT

2

16,8I421

2

15,8I321

2

14,8I321

2

13,8I421

2

12,8I4

2

16,7I421

2

15,7I321

2

13,7I421

2

12,7I4

2

16,5I421

2

15,5I321

2

13,5I421

2

12,5I4

2

16,4I421

2

15,4I321

2

13,4I421

2

12,4I4

2

16I4

2

15I3

2

13I4

2

12I4

2

8I2

2

7I1

2

5I2

2

4I2

2

3431 2 33

431 2 3343

1 2 33431 2 3

3433p

12112)(E

(69)

where for a=4,5,7,8, 12, 13,15,16 ,b=4,5,7,8 and d=12, 13,15,16. HC d,IbIa, are given by

(39) .


MqqqqMqpqqMqpqqMppqqMqqpqMqppq

MqppqMpppqMqqqpMqpqpMqpqp

MppqpMqqppMpqppMqpppMpppp

LqqLqpLqpLppLqqLqpLqpLpp

16,8K42115,8K32113,8K42112,8K416,7K42115,7K321

13,7K42112,7K416,5K42115,5K32113,5K421

12,5K416,4K42115,4K42113,4K42112,4K4

16K415K313K412K48K27K15K24K2

3431 2 334

31 2 3343

1 2 33331 2 3

3433(p

1211)T(E

(70)

MqqqqMqpqqMqqqMqpqq

MppqqMqqpqMqppqMqppqMpppqMqqqp

MqpqpMqpqpMppqpMqqppMqpppMqppp

MppppLqqLqpLqpLppLqqLqpLqpLppT

2

16,8K421

2

15,8K321

2

14,8K321

2

13,8K421

2

12,8K4

2

16,7K421

2

15,7K321

2

13,7K421

2

12,7K4

2

16,5K421

2

15,5K321

2

13,5K421

2

12,5K4

2

16,4K421

2

15,4K321

2

13,4K421

2

12,4K4

2

16K4

2

15K3

2

13K4

2

12K4

2

8K2

2

7K1

2

5K2

2

4K2

2

343

1 2 33431 2 33

431 2 3343

1 2 33433p

12112)(E

(71)

where for a=4,5,7,8, 12, 13,15,16 ,b=4,5,7,8 and d=12, 13,15,16. ML d,KbKa , are given by

(44).


QqqqqQqpqqQqpqqQppqqQqqpqQqppq

QqppqQpppqQqqqpQqpqpQqpqp

QppqpQqqppQpqppQqpppQpppp

PqqPqpPqpPppPqqPqpPqpPpp

16,8K42115,8K32113,8K42112,8K416,7K42115,7K321

13,7K42112,7K416,5K42115,5K32113,5K421

12,5K416,4K42115,4K42113,4K42112,4K4

16K415K313K412K48K27K15K24K2

3431 2 334

31 2 3343

1 2 33331 2 3

3433(p

1211)T(E

(72)

DD

qqqq

DD

pqqq

DD

qpqq

DD

ppqq

DD

qqpq

DD

pqpq

DD

qppq

DD

pppq

DD

qqqp

DD

pqqp

DD

qpqp

DD

ppqp

DD

qqpp

DD

pqpp

DD

qppp

DD

pppp

D

qq

D

pq

D

qp

D

ppN

D

qq

D

pq

D

qp

D

ppNQqqqqQqpqqQqqqQqpqq

QppqqQqqpqQqppqQqppqQpppqQqqqp

QqpqpQqpqpQppqpQqqppQqpppQqppp

QppppPqqPqpPqpPppPqqPqpPqpPppT

16K8K

431

15K8K

431

13K8K

431

12K8K

431

16K7K

431

15K7K

431

13K7K

431

12K7K

431

16K5K

43

15K5K

431

13K5K

431

12K5K

431

16K4K

43

15K4K

431

13K4K

431

12K4K

431

16K

43

15K

43

13K

43

12K

432

28K

21

7K

21

5K

21

4K

212

2

2

16,8K421

2

15,8K321

2

14,8K321

2

13,8K421

2

12,8K4

2

16,7K421

2

15,7K321

2

13,7K421

2

12,7K4

2

16,5K421

2

15,5K321

2

13,5K421

2

12,5K4

2

16,4K421

2

15,4K321

2

13,4K421

2

12,4K4

2

16K4

2

15K3

2

13K4

2

12K4

2

8K2

2

7K1

2

5K2

2

4K2

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

21

1

2

1

2

1

2

1

21

1

2

1

2

1

2

1111)M(

p

1

111)M(

1

343

1 2 33431 2 33

431 2 3343

1 2 33433p

12112)(E

(73)

where for a=4,5,7,8, 12, 13,15,16 ,b=4,5,7,8 and d=12, 13,15,16.

QP d,KbKa , are given by

(48).


SqqqqSqpqqSqpqqSppqqSqqpqSqppq

SqppqSpppqSqqqpSqpqpSqpqp

SppqpSqqppSpqppSqpppSpppp

RqqRqpRqpRppRqqRqpRqpRpp

16,8I42115,8I32113,8I42112,8I416,7I42115,7I321

13,7I42112,7I416,5I42115,5I32113,5I421

12,5I416,4I42115,4I42113,4I42112,4I4

16I415I313I412I48I27I15I24I2

3431 2 334

31 2 3343

1 2 33331 2 3

3433(p

1211)T(E

(74)




DD

qqqq

DD

pqqq

DD

qpqq

DD

ppqq

DD

qqpq

DD

pqpq

DD

qppq

DD

pppq

DD

qqqp

DD

pqqp

DD

qpqp

DD

ppqp

DD

qqpp

DD

pqpp

DD

qppp

DD

pppp

D

qq

D

pq

D

qp

D

ppN

D

qq

D

pq

D

qp

D

ppNSqqqq

SqpqqSqqqSqpqqSppqqSqqpqSqppqSqppq

SpppqSqqqpSqpqpSqpqpSppqpSqqppSqppp

SqpppSppppRqqRqpRqpRppRqqRqpRqpRppT

16I8I

431

15I8I

431

13I8I

431

12I8I

431

16I7I

431

15I7I

431

13I7I

431

12I7I

431

16I5I

43

15I5I

431

13I5I

431

12I5I

431

16I4I

43

154I

431

13I4I

431

12I4I

431

16I

43

15I

43

13I

43

12I

432

28I

21

7I

21

5I

21

4I

212

2

2

16,8I421

2

15,8I321

2

14,8I321

2

13,8I421

2

12,8I4

2

16,7I421

2

15,7I321

2

13,7I421

2

12,7I4

2

16,5I421

2

15,5I321

2

13,5I421

2

12,5I4

2

16,4I421

2

15,4I321

2

13,4I421

2

12,4I4

2

16I4

2

15I3

2

13I4

2

12I4

2

8I2

2

7I1

2

5I2

2

4I2

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

21

1

2

1

2

1

2

1

21

1

2

1

2

1

2

1111)M(

p

1111)M(

1

3

431 2 3343

1 2 33431 2 334

31 2 33433p

12112)(E

(75)

where for a=4,5,7,8, 12, 13,15,16 ,b=4,5,7,8 and d=12, 13,15,16. SR d,IbIa , are given by

(51).

IV. 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=Y1+Y2 and Z=Z1+ Z2 .All the other assumptions and notations are as in model-I. Proceeding as in

model-I, it can be shown for the present model that

Case (i): The distributions of optional and mandatory thresholds follow exponential distribution.


HAAHAAHAAHAACACACACA 5,2I525,1I514,1I424,1I414I45I51I12I2p)T(E

(76)

HAAHAAHAAHAACACACACAT 5,2I5,1I4,1I4,1I4I

p1I2I

2)(E2

52

2

51

2

42

2

41

2

4

2

5I5

2

1

2

2

2

(77)

where

21

15

21

24

21

12

21

21 AAAA ,,,

for a=1, 2, 4, 5, b=1,2and d=4, 5 HC d,IbIa , are given by equation (16).


MAAMAAMAAMAALALALALA 5,2K525,1K514,1K424,1K414K45K51K12K2

p)T(E (78)

MAAMAAMAAMAALALALALAT 5,2K5,1K4,1K4,1K4K

p1K2K

2)(E2

52

2

51

2

42

2

41

2

4

2

5K5

2

1

2

2

2 (79)

where for a=1, 2, 4, 5, b=1,2and d=4, 5 ML d,KbKa , are given by

(20).

If


QAAQAAQAAQAAPAPAPAPA 5,2K525,1K514,1K424,1K414K45K51K12K2

p)T(E

(80)

DD

AA

DD

AA

DD

AA

DD

AA

DA

DA

N

DA

DA

NQAAQAAQAAQAAPAPAPAPAT

5K2K

5

5K1K

5

4K1K

4

4K1K

4

4K

4

5K

52

2

1K

1

2K

22

2

2

52

2

51

2

42

2

41

2

4

2

5K5

2

1

2

2

2

1

2

1

1

1

2

1

1

11)M(

p

11)M(

15,2K5,1K4,1K4,1K4K

p1K2K

2)(E

(81)

where for a=1, 2, 4, 5, b=1,2and d=4, 5 QP d,KbKa , are given by

(24) .


SAASAASAASAARARARARA 5,2I525,1I514,1I424,1I414I45I51I12I2p)T(E

(82)

DD

AA

DD

AA

DD

AA

DD

AA

DA

DA

ND

AD

AN

SAASAASAASAARARARARAT

5I2I

5

5I1I

5

4I1I

4

4I1I

4

4I

4

5I

52

21I

1

2I

22

2

2

52

2

51

2

42

2

41

2

4

2

5I5

2

1

2

2

2

1

2

1

1

1

2

1

1

11)M(

p

11)M(

1

5,2I5,1I4,1I4,1I4Ip

1I2I2)(E

(83)

where for a=1, 2, 4, 5, b=1,2and d=4, 5 SR d,IbIa , are given by

(27).

Case (ii): If the distributions of optional and mandatory thresholds follow extended exponential distribution

with shape parameter 2.


HSSHSSHSSHSSHSSHSSHSSHSSHSSHSS

HSSHSSHSSHSSHSSHSSCSCSCSCSCSCSCSCS

15,10I845,10I414,10I410,4I5415,2I835,2I314,2I34,2I5315,9I825,9I2

14,9I24,9I5215,1I815,1I114,1I14,1I515I85I714I64I510I42I39I21I1

76767

676p)T(E

(84)




HSSHSSHSSHSSHSSHSSHSSHSSHSSHSSHSS

HSSHSSHSSHSSHSSCSCSCSCSCSCSCSCST

8 15,10I7 5,10I6 14,10I10,4I8 15,2I7 5,2I6 14,2I4,2I8 15,9I7 5,9I6 14,9I

4,9I8 15,1I7 5,1I6 14,1I4,1I5I5I14I4Ip

10I2I9I1I2)(E

2

4

2

4

2

4

2

54

2

3

2

3

2

3

2

53

2

2

2

2

2

2

2

52

2

1

2

1

2

1

2

51

2

8

2

7

2

6

2

5

2

4

2

3

2

2

2

1

2

(85)

Where for a=1, 2, 4, 5, 9,10,14,15 b=1, 2,9,10 and d=4, 5, 14, 15. HC d,IbIa, are given by

(15)

2S

2S

2S

2S

2S

2S

2S

2S

2121

2

18

2121

2

17

2121

2

26

2121

2

25

2121

2

14

2121

2

13

2121

2

22

2121

2

21

,4

,4

4,,

4

(86)


MSSMSSMSSMSSMSSMSS

MSSMSSMSSMSSMSSMSSMSSMSS

MSSMSSLSLSLSLSLSLSLSLS

15,10K845,10K414,10K410,4K5415,2K835,2K3

14,2K34,2K5315,9K825,9K214,9K24,9K5215,1K815,1K1

14,1K14,1K515K85K714K64K510K42K39K21K1

767

6767

6p)T(E

(87)

MSSMSSMSSMSSMSSMSS

MSSMSSMSSMSSMSSMSSMSSMSS

MSSMSSLSLSLSLSLSLSLSLST

8 15,10K7 5,10K6 14,10K10,4K8 15,2K7 5,2K

6 14,2K4,2K8 15,9K7 5,9K6 14,9K4,9K8 15,1K7 5,1K

6 14,1K4,1K15K5K14K4Kp

10K2K9K1K2)(E

2

4

2

4

2

4

2

54

2

3

2

3

2

3

2

53

2

2

2

2

2

2

2

52

2

1

2

1

2

1

2

51

2

8

2

7

2

6

2

5

2

4

2

3

2

2

2

1

2

(88)

where for a=1, 2,4,5 ,9,10,14,15 b=1,2,9,10 and d=4,5,14,15 ML d,KbKa , are given by

(20).

If


QSSQSSQSSQSSQSSQSS

QSSQSSQSSQSSQSSQSSQSSQSS

QSSQSSPSPSPSPSPSPSPSPS

15,10K845,10K414,10K410,4K5415,2K835,2K3

14,2K34,2K5315,9K825,9K214,9K24,9K5215,1K815,1K1

14,1K14,1K515K85K714K64K510K42K39K21K1

767

6767

6p)T(E

(89)

DD

SSDD

SSDD

SSDD

SSDD

SSDD

SS

DD

SSDD

SSDD

SSDD

SSDD

SSDD

SSDD

SSDD

SS

DD

SSDD

SS

DS

DS

DS

DS

NDS

DS

DS

DS

NQSSQSSQSSQSSQSSQSS

QSSQSSQSSQSSQSSQSSQSSQSS

QSSQSSPSPSPSPSPSPSPSPST

10K1

4

10K1

4

10K1

4

10K1

4

2K1

3

2K1

3

2K1

3

2K1

3

9K1

2

9K1

2

9K1

2

9K1

2

1K1

1

1K1

1

1K1

1

1K1

1

1111)M(

p

111

1)M(

18 15,10K7 5,10K6 14,10K10,4K8 15,2K7 5,2K

6 14,2K4,2K8 15,9K7 5,9K6 14,9K4,9K8 15,1K7 5,1K

6 14,1K4,1K15K5K14K4Kp

10K2K9K1K2)(E

15K

8

5K

7

14K

6

4K

5

15K

8

5K

7

14K

6

4K

5

15K

8

5K

7

14K

6

4K

5

15K

8

5K

7

14K

6

4K

5

15K

8

5K

7

14K

6

4K

52

210K

4

2K

3

9K

2

1K

12

2

2

4

2

4

2

4

2

54

2

3

2

3

2

3

2

53

2

2

2

2

2

2

2

52

2

1

2

1

2

1

2

51

2

8

2

7

2

6

2

5

2

4

2

3

2

2

2

1

2

(90)

where for a=1, 2,4,5 ,9,10,14,15 b=1,2,9,10 and d=4,5,14,15 QP d,KbKa , are given by

(24) .


SSSSSSSSSSSSSSSSSS

SSSSSSSSSSSSSSSSSSSSSSSS

SSSSSSRSRSRSRSRSRSRSRS

15,10I845,10I414,10I410,4I5415,2I835,2I3

14,2I34,2I5315,9I825,9I214,9I24,9I5215,1I815,1I1

14,1I14,1I515I85I714I64I510I42I39I21I1

767

6767

6p)T(E

(91)

DD

SSDD

SSDD

SSDD

SSDD

SSDD

SS

DD

SSDD

SSDD

SSDD

SSDD

SSDD

SSDD

SSDD

SS

DD

SSDD

SS

DS

DS

DS

DS

NDS

DS

DS

DS

NSSSSSSSSSSSSSSSSSS

SSSSSSSSSSSSSSSSSSSSSSSS

SSSSSSRSRSRSRSRSRSRSRST

10I1

4

10I1

4

10I1

4

10I1

4

2I1

3

2I1

3

2I1

3

2I1

3

9I1

2

9I1

2

9I1

2

9I1

2

1I1

1

1I1

1

1I1

1

1I1

1

1111)M(

p

111

1)M(

18 15,10I7 5,10I6 14,10I10,4I8 15,2I7 5,2I

6 14,2I4,2I8 15,9I7 5,9I6 14,9I4,9I8 15,1I7 5,1I

6 14,1I4,1I5I5I14I4Ip

10I2I9I1I2)(E

15I

8

5I

7

14I

6

4I

5

15I

8

5I

7

14I

6

4I

5

15I

8

5I

7

14I

6

4I

5

15I

8

5I

7

14I

6

4I

5

15I

8

5I

7

14I

6

4I

52

210I

4

2I

3

9I

2

1I

12

2

2

4

2

4

2

4

2

54

2

3

2

3

2

3

2

53

2

2

2

2

2

2

2

52

2

1

2

1

2

1

2

51

2

8

2

7

2

6

2

5

2

4

2

3

2

2

2

1

2

(92)

where for a=1, 2,4,5 ,9,10,14,15 b=1,2,9,10 and d=4,5,14,15 SR d,IbIa , are given by

(27).

Case (iii):The distributions of optional and the mandatory thresholds possess SCBZ property.





HRRHRRHRR

HRRRRHRRHRRHRRHRRRRHRRHRRHRRHRRRRHRR

HRRHRRHRRRRCRRCRRCRRCRRCRRCRRCRRCRR

14,6I161511,6I121110,6I1413

9,6I1096514,3I161511,3I121110,3I14139,3I10943

14,2I161511,2I121110,2I14139,2I1098714,1I1615

11,1I121110,1I14139,1I1092114I161511I1211

10I14139I1092I876I653I431I21p)T(E

(93)

HRRHRRHRR

HRRRRHRRHRRHRR

HRRRRHRRHRRHRR

HRRRRHRRHRRHRR

HRRRRCRRCRRCRR

CRRCRRCRRCRRCRRT

14,6I11,6I10,6I

9,6I14,3I11,3I10,3I

9,3I14,2I11,2I10,2I

9,2I14,1I11,1I10,1I

9,1I14I11I10I

9Ip

2I6I3I1I2)(E

2

1615

2

1211

2

1413

2

10965

2

1615

2

1211

2

1413

2

10943

2

1615

2

1211

2

1413

2

10987

2

1615

2

1211

2

1413

2

10921

2

1615

2

1211

2

1413

2

109

2

87

2

65

2

43

2

21

2

(94)

where for a=1, 2, 3,6,9,10,11,14 b=1, 2,3,6and d=9,10,11,14. HC d,IbIa, are given by

(39).

43

43416

443

3444

15

43

43314

343

3433

13

43

43412

4343

4344

11

443

43310

4343

4333

9

21

2118

211

2111

7

21

2126

221

2122

5

121

2124

2121

2122

3

221

2112

2121

2111

1

qqR

qpR

qqR

pqR

qpR

ppR

pqR

ppR

qqR

qpR

qqR

pqR

qpR

ppR

pqR

ppR

,,

3

,,,,,

,,,,

(95)


MRRMRRMRRMRRRR

MRRMRRMRRMRRRRMRRMRRMRRMRRRRMRR

MRRMRRMRRRRLRRLRRLRRLRRLRRLRRLRRLRR

14,6K161511,6K121110,6K14139,6K10965

14,3K161511,3K121110,3K14139,3K10943

14,2K161511,2K121110,2K14139,2K1098714,1K1615

11,1K121110,1K14139,1K1092114K161511K1211

10K14139K1092K876K653K431K21p)T(E

(96)

MRRMRRMRRMRRRR

MRRMRRMRRMRRRR

MRRMRRMRRMRRRR

MRRMRRMRRMRRRRLRR

LRRLRRLRRLRRLRRLRRLRRT

14,6K11,6K10,6K9,6K

14,3K11,3K10,3K9,3K

14,2K11,2K10,2K9,2K

14,1K11,1K10,1K9,1K14K

11K10K9Kp

2K6K3K1K2)(E

2

1615

2

1211

2

1413

2

10965

2

1615

2

1211

2

1413

2

10943

2

1615

2

1211

2

1413

2

10987

2

1615

2

1211

2

1413

2

10921

2

1615

2

1211

2

1413

2

109

2

87

2

65

2

43

2

21

2

(97)

where for a=1, 2, 3,6,9,10,11,14 b=1, 2,3,6and d=9,10,11,14 ML d,KbKa , are given by

(44).


QRRQRRQRRQRRRR

QRRQRRQRRQRRRR

QRRQRRQRRQRRRRQRR

QRRQRRQRRRRPRRPRRPRRPRRPRRPRRPRRPRR

14,6K161511,6K121110,6K14139,6K10965

14,3K161511,3K121110,3K14139,3K10943

14,2K161511,2K121110,2K14139,2K1098714,1K1615

11,1K121110,1K14139,1K1092114K161511K1211

10K14139K1092K876K653K431K21p)T(E

(98)

BBRR

BBRR

BBRR

BBRR

RR

BBRR

BBRR

BBRR

BBRR

RRBBRR

BBRR

BBRR

BBRR

RR

BBRR

BBRR

BBRR

BBRR

RRB

RRB

RRB

RRBRR

N

BRR

BRR

BRR

BRR

NQRRQRRQRR

QRRRRQRRQRRQRRQRRRR

QRRQRRQRRQRRRRQRR

QRRQRRQRRRRPRRPRR

PRRPRRPRRPRRPRRPRRT

14I6I

1615

11I6I

1211

10I6I

1413

9I6I

10965

14I3I

1615

11I3I

1211

10I3I

1413

9I3I

10943

14I2I

1615

11I2I

1211

10I2I

1413

9I2I

10987

14I1I

1615

11I1I

1211

10I1I

1413

9I1I

10921

14I

1615

11I

1211

10I

1413

9I

1092

2

2I

87

6I

65

3I

43

1I

212

2

2

1615

2

1211

2

1413

2

10965

2

1615

2

1211

2

1413

2

10943

2

1615

2

1211

2

1413

2

10987

2

1615

2

1211

2

1413

2

10921

2

1615

2

1211

2

1413

2

109

2

87

2

65

2

43

2

21

2

1111

11111111

11111111M

p

1111M

114,6K11,6K10,6K

9,6K14,3K11,3K10,3K9,3K

14,2K11,2K10,2K9,2K14,1K

11,1K10,1K9,1K14K11K

10K9Kp

2K6K3K1K2)(E

(99)

where for a=1, 2, 3,6,9,10,11,14 b=1, 2,3,6and d=9,10,11,14 QP d,KbKa , are given by

(48).





SRRSRRSRRSRRRR

SRRSRRSRRSRRRRSRRSRR

SRRSRRRRSRRSRRSRRSRRRR

RRRRRRRRRRRRRRRRRRRRRRRR

14,6I161511,6I121110,6I14139,6I10965

14,3I161511,3I121110,3I14139,3I1094314,2I161511,2I1211

10,2I14139,2I1098714,1I161511,1I121110,1I14139,1I10921

14I161511I121110I14139I1092I876I653I431I21p)T(E

(100)

BBRR

BBRR

BBRR

BBRR

RRBBRR

BBRR

BBRR

BBRR

RRBBRR

BBRR

BBRR

BBRR

RRBBRR

BBRR

BBRR

BBRR

RRB

RRB

RRB

RRBRR

NBRR

BRR

BRR

BRR

NSRRSRRSRRSRRRRSRR

SRRSRRSRRRRSRRSRRSRR

SRRRRSRRSRRSRRSRRRRRRR

RRRRRRRRRRRRRRRRRRRRRT

14I6I

1615

11I6I

1211

10I6I

1413

9I6I

10965

14I3I

1615

11I3I

1211

10I3I

1413

9I3I

10943

14I2I

1615

11I2I

1211

10I2I

1413

9I2I

10987

14I1I

1615

11I1I

1211

10I1I

1413

9I1I

10921

14I

1615

11I

1211

10I

1413

9I

1092

22I

87

6I

65

3I

43

1I

212

2

2

1615

2

1211

2

1413

2

10965

2

1615

2

1211

2

1413

2

10943

2

1615

2

1211

2

1413

2

10987

2

1615

2

1211

2

1413

2

10921

2

1615

2

1211

2

1413

2

109

2

87

2

65

2

43

2

21

2

11111

111111111

111111M

p

111

1M

114,6I11,6I10,6I9,6I14,3I

11,3I10,3I9,3I14,2I11,2I10,2I

9,2I14,1I11,1I10,1I9,1I14I

11I10I9Ip

2I6I3I1I2)(E

(101)

where for a=1, 2, 3,6,9,10,11,14 b=1, 2,3,6and d=9,10,11,14 SR d,IbIa , are given by

(51).

V. NUMERICAL ILLUSTRATIONS The mean and variance of the time to recruitment for the above models are given in the following

tables for the cases(i),(ii),(iii) respectively by keeping .8.0p,8.0,5.0,6.0,4.0 2121

1,2.0,5.1,1,5.0,9.0,8.0,4.0,7.0,4.0,7.0,3.0,6.0444333222111

fixed and varying c, k one at a time and the results are tabulated below .

Table – I (Effect of c, k, λ on performance measures)

MODEL-I

c 1.5 1.5 1.5 0.5 1 1.5

k 1 2 3 2 2 2

λ 1 1 1 1 1 1

Case (i)

r=1

n=1 E 6.9679 6.3557 6.1484 2.5225 4.4423 6.3557

V 25.6665 19.3505 17.3979 3.6283 9.9659 19.3505

n=k E 6.9679 2.3650 1.3080 1.0798 1.7248 2.3650

V 25.6665 3.1717 1.0148 0.8753 1.8559 3.1717

r=k

n=1 E 6.9679 19.0671 33.8154 7.5676 13.3270 19.0671

V 25.6665 161.4276 488.6549 28.1272 81.0386 161.4276

n=k E 6.9679 7.0959 7.1940 3.2394 5.1745 7.0959

V 25.6665 24.4003 24.2223 6.6552 13.9834 24.4003

Case(ii)

r=1

n=1 E 7.7324 7.0834 6.8728 2.7970 4.9371 7.0834

V 43.6104 36.0639 33.5832 5.7270 17.6908 36.0639

n=k E 7.7324 3.5027 1.4977 1.2753 2.1966 3.5027

V 43.6104 3.1907 1.4998 1.0853 2.3381 3.1907

r=k

n=1 E 7.7324 21.2503 37.7996 8.3909 14.8114 21.2503

V 43.6104 319.2949 998.6626 50.7338 156.1554 319.2949

n=k E 7.7324 8.0163 8.2373 3.7368 5.8959 8.0163

V 43.6104 38.4822 35.3702 7.2110 19.4178 38.4822

Case(iii)

r=1

n=1 E 6.7761 6.1834 5.9831 2.4527 4.3208 6.1834

V 29.2323 23.1644 21.2936 3.9963 11.6062 23.1644

n=k E 6.7761 2.2433 1.2782 1.0315 1.6384 2.2433

V 29.2323 3.6124 1.1285 0.8973 2.0285 3.6124

r=k

n=1 E 6.7761 18.5503 32.9067 7.3582 12.9625 18.5503

V 29.2323 196.1130 608.2081 31.0609 87.1721 196.1130

n=k E 6.7761 6.7298 7.0299 3.0944 4.9151 6.7298

V 29.2323 28.0247 26.4670 6.0124 14.9796 28.0247




MODEL-II

c 1.5 1.5 1.5 0.5 1 1.5

k 1 2 3 2 2 2

λ 1 1 1 1 1 1

Case (i)

r=1

n=1 E 3.0491 2.4841 2.2929 1.1966 1.8442 2.4841

V 7.1810 4.4459 3.6707 1.1672 2.5502 4.4459

n=k E 3.0491 1.0669 0.6046 0.6467 0.8546 1.0669

V 7.1810 0.9564 0.3209 0.4026 0.6514 0.9564

r=k

n=1 E 3.0441 7.4522 12.6107 3.5897 5.5326 7.4522

V 7.1810 40.0132 111.0337 10.5046 22.9520 40.0132

n=k E 3.0441 3.2008 3.3253 1.9402 2.5638 3.2008

V 7.1810 6.4733 6.0806 2.3297 4.1530 6.4733

Case(ii)

r=1

n=1 E 4.4497 3.8440 3.6369 1.6790 2.7663 3.8440

V 10.6673 6.6718 5.4893 1.7016 3.8162 6.6718

n=k E 4.4497 1.5259 0.8502 0.7959 1.1631 1.5259

V 10.6673 1.4048 0.4678 0.5362 0.9324 1.4048

r=k

n=1 E 4.4497 11.5320 20.0024 5.0370 8.2990 11.5320

V 10.6673 52.3579 144.2232 11.9566 28.8134 52.3579

n=k E 4.4497 4.5776 4.6762 2.3877 3.4894 4.5776

V 10.6673 9.5917 9.0489 3.2339 6.0657 9.5917

Case(iii)

r=1

n=1 E 2.9380 2.3862 2.1980 1.1588 1.7761 2.3862

V 7.1321 4.5063 3.7663 1.1488 2.5532 4.5063

n=k E 2.9380 1.0271 0.5761 0.6355 0.8291 1.0271

V 7.1321 0.9467 0.3126 0.3945 0.6393 0.9467

r=k

n=1 E 2.9380 7.1587 12.0889 3.4765 5.3284 7.1587

V 7.1321 33.4547 94.9477 6.4008 17.4633 33.4547

n=k E 2.9380 1.0271 0.5761 0.6355 0.8291 1.0271

V 7.1321 0.9467 0.3126 0.3945 0.6393 0.9467

MODEL-III

c 1.5 1.5 1.5 0.5 1 1.5

k 1 2 3 2 2 2

λ 1 1 1 1 1 1

Case (i)

r=1

n=1 E 8.7025 8.0800 7.8698 3.1046 5.5952 8.0800

V 34.8211 26.8975 24.4103 4.8250 13.6770 26.8975

n=k E 8.7025 2.9422 1.6226 1.2767 2.118 2.9422

V 34.8211 4.2573 1.3523 1.0932 2.4337 4.2573

r=k

n=1 E 8.7025 24.2401 43.2829 9.3139 16.7855 24.2401

V 34.8211 225.9177 691.1652 37.2158 111.9030 225.9177

n=k E 8.7025 8.8266 8.9241 3.83 6.3355 8.8266

V 34.8211 32.4314 31.1711 7.2859 17.6793 32.4314

Case(ii)

r=1

n=1 E 11.9437 11.2767 11.0507 4.1989 7.7414 11.2767

V 44.2200 33.4379 30.0115 6.1929 17.2159 33.4379

n=k E 11.9437 4.0179 2.2079 1.6500 2.8369 4.0179

V 44.2200 5.4334 1.7299 1.4021 3.1300 5.4334

r=k

n=1 E 11.9437 33.8301 60.7777 12.5968 23.2243 33.8301

V 44.2200 278.3881 841.5110 47.3386 139.4601 278.3881

n=k E 11.9437 12.0536 12.1431 4.95 8.5106 12.0536

V 44.2200 40.8646 39.0814 9.3189 22.4964 40.8646

Case(iii)

r=1

n=1 E 8.4755 7.8757 7.6735 3.0222 5.4512 7.8757

V 40.0820 32.4776 30.0962 5.3736 16.0863 32.4776

n=k E 8.4755 2.8689 1.5833 1.2449 2.0589 2.8689

V 40.0820 4.8388 1.5246 1.1397 2.6751 4.8388

r=k

n=1 E 8.4755 23.6271 42.2037 9.0665 16.3536 23.6271

V 40.0820 276.5466 864.3347 42.3183 133.8743 276.5466

n=k E 8.4755 8.6067 8.7081 3.7341 6.1768 8.6067




V 40.0820 37.8112 36.6164 7.7674 19.9579 37.8112

VI. FINDINGS The influence of nodal parameters on the performance measures namely mean and variance of the time

to recruitment for all the models are reported below.

i. It is observed that the mean time to recruitment decreases, with increase in k for the cases r=1,n=1 and

r=1 ,n=k but increases for the cases r=k,n=1 and r=k,n=k.

ii. If c increases, the average number of exits increases, which, in turn, implies that mean and variance of

the time to recruitment increase for all the models.

VII. CONCLUSION Note that while the time to recruitment is postponed in model-III, the time to recruitment is advanced

in model-I and II .Therefore from the organization point of view, model III is more preferable.

REFERENCES [1] D.J. Bartholomew, & A.F. Forbes, “Statistical Techniques for Manpower Planning”( John Wiley and

Sons, 1979).

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estimation of expected time to recruitment, Acta Ciencia Indica, 5(2), 2009, 501-508.

[5] A. Srinivasan, V. Vasudevan, Variance of the time to recruitment in an organization with two grades,

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