Stochastic Model to Determine the Expected Time to Recruitment with Three Sources of Depletion of Manpower under Correlated Interarrival Times

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Bonfring International Journal of Industrial Engineering and Management Science, Vol. 4, No. 1, February 2014 34

ISSN 2277-5056 | © 2014 Bonfring

Abstract--- It is a common phenomenon that some

personnel leave an organization after completing a certain

period of services to that organization voluntarily or

involuntarily due to death, retirement or termination. It

usually happen that whenever the policy decisions regarding

promotion and target of work or sales to be achieved are

revised, then, there will be exit of personnel, which in other

words called the attrition or wastage. In any organization

like marketing, industrial, software, the depletion of

manpower due to policy decisions is quite common. This

results in manpower attrition and then recruitment becomesnecessary. Frequent recruitment is not advisable due to the

cost of the same. Hence recruitment is postponed till a point

called the breakdown point of total depletion beyond which

the normal activities cannot be continued due to shortage of

manpower. This level of allowable manpower attrition is

called threshold. In this paper a Stochastic model to

determine expected time to recruitment with three sources of

depletion of manpower attrition under correlated interarrival

times has been derived. It provides the optimal solution

because; it takes into consideration the cost of frequent

recruitments, depletion of manpower and the cost of shortage

of manpower. The Stochastic model discussed in the paper is

not only applicable to industry as a whole but also in a widercontext of other applicable areas.

Keywords--- Attrition, Threshold, Depletion of Manpower,

Correlated, Interarrival Times, Optimal Solution

I. I NTRODUCTION

N any organization it usually happens that whenever the

policy decisions regarding pay, perquisites, promotion and

targets of work or sales to be achieved are revised then therewill be exit of personnel, which in other words is called the

wastage. The concept of wastage is dealt in details by

Barthlomew (1982). In the case of wastages has following i.i.drandom variables and the threshold level of wastage being arandom variable, the expected time to recruitment has already

been derived by Sathiyamoorthi and Elangovan (1998). It is a

common phenomenon that some personnel leave an

Dr. R. Elangovan, Professor, Department of Statistics, Annamalai

University, Annamalainagar. E-mail: [email protected]

R. Arulpavai, Assistant Professor, Department of Statistics, Government

Arts College, C.Mutlur, Chidambaram. E-mail: [email protected]

DOI: 10.9756/BIJIEMS.4802

organization after completing a certain period of services to

that organization voluntarily or involuntarily due to death,retirement or termination. It usually happen that whenever the

policy decisions regarding promotion and target of work or

sales to be achieved are revised, then, there will be exit of

personnel, which in other words called the attrition or wastage.

In any organization like marketing, industrial, software, thedepletion of manpower due to policy decisions is quite

common. This results in manpower attrition and then

recruitment becomes necessary. Frequent recruitment is not

advisable due to the cost of the same. Hence recruitment is postponed till a point called the breakdown point of total

depletion beyond which the normal activities cannot be

continued due to shortage of manpower. This level ofallowable manpower attrition is called threshold. In this

Chapter a Stochastic model to determine expected time to

recruitment with three sources of depletion of manpower

attrition under correlated interarrival times has been derivedusing the result of Gurland (1955) and Esary et al., (1973).

Some quite interesting results cited in the recent literature by

Arulpavai and Elangovan (2012, 2013). It provides the

optimal solution because; it takes into consideration the cost offrequent recruitments, depletion of manpower and the cost of

shortage of manpower. The Stochastic model discussed in theChapter is not only applicable to industry as a whole but alsoin a wider context of other applicable areas.

II. ASSUMPTION

• The depletion of manpower occurs whenever the

policy decisions are announced.

• The interarrival times between decision epochs are

random variables are constantly correlated and

exchangeable but not independent.

• The depletion of manpower also occurs due to transferof personnel and the interarrival times between

successive epochs of transfer are i.i.d. random

variables having exponential distribution with parameter λ.

• The total depletion due to the above three assumptions

crosses the threshold which itself is a random variable,

recruitment becomes necessary.

• The three types of depletion are linear and additive.

Stochastic Model to Determine the Expected Time

to Recruitment with Three Sources of Depletion of

Manpower under Correlated Interarrival Times

R. Elangovan and R. Arulpavai

I




ISSN 2277-5056 | © 2014 Bonfring

III. NOTATIONS

i. Xi - a continuous random variable representing the

amount of depletion of man hours due to the ith epoch

of policy decisions having exponential distribution

with parameter α.

• Y j – a continuous random variable representing the

amount of depletion of man hours due to the jth event

of transfer of personnel having exponential

distribution with parameter µ.• Zn – a continuous random variable representing the

amount of depletion of man hours due to the nth event

of transfer of personnel having with exponential

distribution parameter is denoted as η.

• h – Probability density function of Xi, k – Probabilitydensity function of Yi and m – Probability density

function of Zn

• Ui – Random variable denoting the interarrival times between the successive decision epochs. The ui’s are

constantly correlated with exponential distributionwith parameter a.

• V j – Random variable representing the interarrival

times between the successive epochs of transfer. vj’sare i.i.d. random variable with exponential distribution

with parameter λ.

• Wn –Random variable representing the interarrival

times between the successive epochs of transfer of nth

event with parameter β.

• m X X X X X ++++= ....~

321 ,

mY Y Y Y Y ++++= ....~

321 an

m Z Z Z Z Z ++++= ....~

321

• L(t) = P(T < t) =c.d.f of the time to recruitment of thesystem.

• S(t) = Survivor function P(T > t)

•

L*(S) = Laplace transform of L(t), h*(.) = Laplacetransform of h(.), k*(.)=

• Laplace transform of k(.) and p*(.) = Laplace

transform of p(.)

• f(.) = p.d.f. of ui, g(.) = p.d.f. of vi and s(.) = p.d.f. of

wn.The probability that the total depletion of manpower on

”m” occasions of decision making and ”n” and “q” occasions

of transfer of personnel does not exceed the threshold level is

given as,

[ ] daecaQ A Z Y X Pcz

z y x−

∞

++∫=<++0

~~~ )(~~~

daecaQcz

z y x−

∞

++∫=0

~~~ )(

)(*~~~ cCQ z y x ++=

But we know that

)(1

)( ** s f s

sF mm =

)(1

)( ** sgs

sG nn =

)(1

)( ** sss

sS qq =

But we know that

[ ]c

cqc A Z Y X P

z y x)(~~~

*~~~ ++=<++

)()()(***~~~ cqcqcq z y x

=

[ ] [ ] [ ]qnmcqcqcq )()()( ***=

[ ] [ ] [ ]qnmc pck ch )()()(

***= (1)

Since x, y and z are independent and xi, y j and zn

IV. R ESULTS

s(t)=P(T<t) Probability that there are exactly “m” occasions of

policy making and “n”, “p” occasions of transfer and the totaldepletion does not cross the threshold A in (0, t).

)(1][)( t st T pt L −=<=

( )( ) ( )

( )( ) ( )

( )( ) ( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −−

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −−

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −−−=

∑

∑

∑

∞

=

−

∞

=

−

∞

=

−

1

1**

1

1**

1

1**

)(11

)(11

)(111

q

qq

n

nn

m

mm

c pt sc p

ck t Gck

cht F ch

(2)

( )( ) ( ) ( )( ) ( )

( )( ) ( )

( )( ) ( )( ) ( ) ( )⎪⎩

⎪⎨⎧

⎜⎜

⎝

⎛ −−−

−+

−+−=

∑∑

∑

∑∑

∞

=

−−∞

=

−∞

=

−∞

=

−∞

=

1

1*1*

1

**

1*

1

*

1*

1

*1*

1

*

)()(11

)(1

)(1)(1

n

nn

m

m

m

q

q

q

n

n

nm

m

m

ck t Gcht f ck ch

c pt sc p

ck t gck cht f ch

( ) ( ) ⎭⎬⎫+ −

∞

=

∞

=− ∑∑

1*

11

1* )()( n

n

n

m

mm ck t gcht F

( )( ) ( )( ) ( ) ( )⎪⎩

⎪⎨⎧

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −−− ∑∑

∞

=

−∞

=

−

1

1*

1

1*** )()(11

q

qq

m

mm c pt scht f c pch

( ) ( )⎭⎬⎫

+ −∞

=

∞

=

− ∑∑ 1*

11

1*)()(

q

q

q

m

m

m c pt S cht F

( )( ) ( )( ) ( ) ( )⎪⎩

⎪⎨⎧

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ ⎟

⎠

⎞⎜⎝

⎛ −−− ∑∑

∞

=

−∞

=

−

1

1*

1

1*** )()(11q

q

q

n

n

nc pt sck t gc pck

( ) ( ) ⎟⎟

⎠

⎞⎜⎜

⎝

⎛ ⎟⎟

⎠

⎞⎜⎜

⎝

⎛ ∑∑

∞

=

−∞

=

−

1

1*

1

1*)()(

q

qq

n

nn c pt sck t G

( ) ( )⎭⎬⎫

+ −∞

=

∞

=

− ∑∑ 1*

11

1* )()(q

q

q

n

n

nc pt sck t G

( )( ) ( )( ) ( )( ) ( )⎪⎩

⎪⎨⎧

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −−−+ ∑

∞

=

−

1

1****)(111

m

mm cht f c pck ch




ISSN 2277-5056 | © 2014 Bonfring

( ) ( ) ( )∑ ∑∑∞

=

∞

=

−−∞

=

−+1 1

1*1*

1

1* )()()(m q

q

q

n

n

n

m

m c pt sck t gcht F

( ) ( ) ( )∑ ∑∑∞

=

−∞

=

−∞

=

−

⎪⎭

⎪⎬⎫

++1

1*

1

1*

1

1* )()()(

m

q

q

qn

n

nm

m c pt sck t Gcht F

( )( ) ( ) ( )( ) ( )

( )( ) ( )

( )( ) ( )( ) ( ) ( )

( ) ( )ck t t

m

mm

ck t

t

t m

m

m

c pt t

ck t t m

m

m

eecht F

dt eecht f ck ch

eec p

eeck cht f ch

*

*

*

*

1

1*

0

1*

1

**

*

*1*

1

*

)(

)(11

1

1)(1

λ λ

λ λ

µ µ

λ λ

λ

λ

µ

λ

−∞

=

−

−−∞

=

−

−−∞

=

∑

∫∑

∑

+

−−−

−+

−+−=

( )( ) ( )( ) ( ) ( )

( ) ( )∑

∫∑∞

=

−−

−∞

=

−

+

−−−

1

1*

01

1***

*

*

)(

)(11

m

c pt t mm

c pt

t

t

m

mm

eecht F

dt eecht f c pch

µ µ

µ µ

µ

µ

( )( ) ( )( ) ( ) ( )

( ) ( )c pt t ck t

t

t

t

c pt t ck t t

eedt ee

dt eeeec pck

**

**

0

0

** 11

µ µ λ λ

µ µ λ λ

µ λ

µ λ

−−

−−

∫

∫

+

−−−

( )( ) ( )( ) ( )( ) ( )

⎪⎩

⎪⎨⎧

−−−+ ∑∞

=

−

1

1**** )(111

m

mm cht f c pck ch

( ) ( )dt eedt ee

t

c pt t ck t

t

t ∫∫ −−

00

** µ µ λ λ µ λ

( ) ( ) ( )∑ ∫∞

=

−−−+1 0

1* **

)(

m

t

c pt t ck t t mm dt eeeecht F

µ µ λ λ µ λ

( ) ( ) ( )∑ ∫∞

=

−−−+1 0

1* **

)(

m

c pt t ck t

t

t mm eedt eecht F

µ µ λ λ µ λ

( )( ) ( ) ( )( ) ( )( )

( )( ) ))(1(*

1*1*

1

*

*

*

1

1)(1

c pt

ck t m

m

m

ec p

eck cht f ch

−−

−−−∞

=

−+

⎟⎟ ⎠ ⎞⎜⎜

⎝ ⎛ −+−= ∑

µ

λ

µ

λ

( )( ) ( ) ⎟ ⎠

⎞⎜⎝

⎛ −−− −−−

∞

=∑ ))(1(1*

1

* *

1)(1 ck t m

m

mecht f ch λ

( )( ) ( )( ) ( ) ⎟ ⎠

⎞⎜⎝

⎛ −−− −−−

∞

=∑ ))(1(1*

1

***

)(11 ck t m

m

m echt F ck ch λ λ

( )( ) ( ) ( )

( )( ) ( )( ) ( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −−−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −−−

∑

∑∞

=

−

−−∞

=

−

−−

1

1***

))(1(

1

1**

))(*1(

*

)(11

1)(1

m

mm

c pt

m

mm

c pt

echt F c pch

echt f ch

µ

µ

µ

( )( ) ( ) ( )( )( )( )( ) ⎟

⎠

⎞⎜⎝

⎛ −−−

−−−

−−−−

−−+−

))(*

1(*

**

))(1(*

))(1(*

11

11

c pt

eec p

eeck

ck t

c pt ck t t

µ

µ

λ

λ

µ λ λ

( )( ) ( ) ( )

( ) ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−

−−+

−−

−−∞

=

−∑))(1(

))(1(

1

1**

*

*

1

1)(1

ck t

c pt

m

mm

e

echt f ch

λ

µ

( )( ) ( )( ) ( ) ( )

( ) ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−

−−+

−−

∞

=

+−−∑))(1(

1

1***

*

*

1

)(11

c pt

m

ck t t mm

e

echt F ck ch

µ

λ λ λ

( )( ) ( )( ) ( ) ( )

( ) ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−

−−+

−−

∞

=

+−−∑))(1(

1

1***

*

*

1

)(11

ck t

m

c pt t mm

e

echt F c pch

λ

µ µ µ

(3) Taking Laplace Transform on both side

L[f(t)]=F(s)

( )( ) ( ) ( ) ( )( ) ( )( )( )

( )( ) ( )))(1(*

1*

1

1**

*

*

1

1)(1

c pt

ck t

m

mm

e Lc p

e Lck t f Lchch

−−

−−∞

=

−

−+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −+−= ∑

µ

λ

µ

λ

( )( ) ( ) ( )( )⎟ ⎠

⎞⎜⎝

⎛ −−− ∑

∞

=

−−−

1

))(1(1***

1)(1m

ck t

m

met f Lchch λ

( )( ) ( )( ) ( ) ( )⎟ ⎠

⎞⎜⎝

⎛ −−− ∑

∞

=

−−−

1

))(1(1****

)(11m

ck t

m

met F Lchck ch λ λ

( )( ) ( ) ( )( )⎟ ⎠

⎞⎜⎝

⎛ −−− ∑

∞

=

−−−

1

))(1(1** *

1)(1m

c pt

m

met f Lchch

µ

( )( ) ( )( ) ( ) ( )( )( )⎟

⎠

⎞⎜

⎝

⎛ −−− ∑

∞

=

−−−

1

11*** *

)(11m

c pt

m

met F Lchc pch µ µ

( )( ) ( )( )( )( ) ( )( )( )( ) ( )( )( )⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛ −+−−

−−−−

−−

c pt ck t

ck t

e L

ck e Lck

**

*

11

*

1*1

1µ λ

λ λ

λ

( )( ) ( )( )( )( ) ( )( )( )( ) ( )( )( )⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛ −+−−

−−−−

−−

c pt ck t

c pt

e L

c pe Lc p

**

*

11

*

1*1

1µ λ

µ µ

µ




ISSN 2277-5056 | © 2014 Bonfring

( )( ) ( ) ( ) ( )⎜⎝

⎛ −−+ ∑ ∑

∞

=

∞

=

−−−

1 1

))(1(1***

)()(1m m

ck t

mm

met f Lt f Lchch λ

( )( )( ) ( )( ) ( )( )( )⎟ ⎠

⎞+− ∑∑

∞

=

−−−−∞

=

−−

1

11

1

1***

)()(m

c pt ck t

m

m

c pt

m et f Let f L µ λ µ

( )) ( )) ( )

( )( )( ) ( )( ) ( )( )( )⎟⎟ ⎠ ⎞

⎜⎜⎝

⎛ −

−−+

∑ ∑∞

=

∞

=

−−−−−−

−

1 1

111

1***

***

)()(

11

m m

c pt ck t mck t m

m

et F Let F L

chck ch

µ λ λ λ

( )( ) ( )( ) ( ) 1*** 11 −−−+ m

chc pch

( )( )( ) ( )( ) ( )( )( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −∑ ∑

∞

=

∞

=

−−−−−−

1 1

111***

)()(

m m

c pt ck t m

c pt m et F Let F L

µ λ µ µ

…( 7)

( )[ ] ( ) ( )( ) ( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −== ∑

∞

=

−

1

*1*** )(1

m

mm

s f chchslt L L

( )( ) ( )( ) ⎟⎟

⎠ ⎞

⎜⎜⎝ ⎛ ⎟⎟

⎠ ⎞⎜⎜

⎝ ⎛

−+−+

ck sck

*

*

1

11λ

λ

( )( )( )( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛ ⎟⎟

⎠

⎞⎜⎜⎝

⎛

−+−+

c psc p

*

*

1

11

µ µ

( )( ) ( ) ( )

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −− ∑

∞

=

−

1

1** )(1

m

mm

t f Lchch

( )( ) ( ) ( ) ( )( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −+ ∑

∞

=

−−−

1

11** *

)(1

m

ck t m

met f Lchch

λ

( )( ) ( )( ) ( )

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−−

−∑ ∫

∞

=

−−

−

1

))(1(

0

1***

*

)(

11

m

ck t

t

m

m

dt et f L

chck ch

λ λ

( )( ) ( ) ( )⎟ ⎠

⎞⎜⎝

⎛ −− ∑

∞

=

−

1

1**)(1

m

m

mt f Lchch

( )( ) ( ) ( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −+ ∑

∞

=

−−−

1

))(1(1** *

)(1

m

c pt m

met f Lchch

µ

( )( ) ( )( ) ( )

( )( )⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−−

−∑ ∫

∞

=

−−

−

1 0

1

1***

*

)(

11

m

t

c pt m

m

dt et f L

chc pch

µ µ

( )( ) ( )( )

( )( ) ( )( ) ( )( )( )( )( )t c pck

ck t

e Lck

e Lck

**

*

11*

1*

1

1

−+−−

−−

−+

−−µ λ

λ

λ

λ

( )( ) ( )( )

( )( ) ( )( ) ( )( )( )( )t c pck

c pt

e Lc p

e Lc p**

*

11*

1*

1

1

−+−−

−−

−+

−−µ λ

µ

µ

µ

( )( ) ( ) ( ) ( )( ) ( )

( )∑

∑∞

=

−−

−∞

=

− −−−+

1

))(1(

1**

1

1**

*

)(

1)(1

m

ck t m

m

m

mm

et f L

chcht f Lchch

λ

( )( ) ( ) ( )( )( )∑∞

=

−−−−−1

11** *

)(1

m

c pt m

met f Lchch

µ

( )( ) ( ) ( )( ) ( )( )( )( )∑∞

=

−+−−−−−1

111** **

)(1

m

t c pck m

met f Lchch

µ λ

( )( ) ( )( ) ( ) ( )( )∑ ∫∞

=

−−−

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ −−+

1 0

11*** *

)(11m

t

ck t

m

mdt et f Lchck ch

λ λ

( )( ) ( )( ) ( )

( )( ) ( )( )( )∑ ∫∞

=

−+−−

−

⎟⎟

⎠

⎞⎜⎜

⎝

⎛

−−−

1 0

11

1***

**

)(

11

m

t

t c pck m

m

dt et f L

chck ch

µ λ λ

( )( ) ( )( ) ( ) ( )( )∑ ∫∞

=

−−−

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ −−+

1 0

11*** *

)(11m

t

c pt

m

mdt et f Lchc pch µ µ

( )( ) ( )( ) ( )

( )( ) ( )( )( )∑ ∫∞

=

−+−−

−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−−−

1 0

11

1***

**

)(

11

m

t

t c pck m

m

dt et f L

chc pch

µ λ µ (4)

[ ] ( )( )( )( ) ( )( )[ ]( )( )

( )( ) ( )( )[ ] ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−+−+

−−+

⎟⎟ ⎠

⎞

⎜⎜⎝

⎛

−+−+−−=

2**

*

2**

*

*

11

1

111)(

c pck s

c p

c pck sck sl

dsd

µ λ

µ

µ λ λ

( )( ) ( ) ( )( ) ( )( )( )∑∞

=

− −+−+−+1

***1** 111m

m

mc pck s f

ds

d chch µ λ

( )( ) ( )( ) ( )

( )( ) ( )( )( )(

( )( ) ( )( )( )( )

( )( ) ( )( )( ))( )( ) ( )( )( )∑∞

=

−

−+−+−+−+−

−+−+

−+−+

−−−1

2**

***

***

**

1***

1111

11

11

11

m

m

m

m

c pck sc pck s f

c pck s f ds

d

c pck s

chck chµ λ

µ λ

µ λ

µ λ

λ

( )( ) ( )( ) ( )

( )) ( )))

( )( ) ( )( )( )( )

( )( ) ( )( )( ))( )( ) ( )( )( )∑

∞

=

−

−+−+

−+−+−

−+−+

−+−+

−−−1

2**

***

***

**

1***

11

11

11

11

11

m

m

m

m

c pck s

c pck s f

c pck s f ds

d

c pck s

chc pch

µ λ

µ λ

µ λ

µ λ

µ

( 5)




ISSN 2277-5056 | © 2014 Bonfring

[ ] ( )( )( )( ) ( )( )[ ]

( )( )( )( ) ( )( )[ ] 2**

*

2**

*

0*

11

1

11

1)(

c pck

c p

c pck

ck sl

ds

d s

−+−

−−

−+−

−−==

µ λ

µ

µ λ

λ

( )( ) ( ) ( )( )

( )( )∑∞

=

−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−+

−−+

1*

**1**

1

11

m

mm

c p

ck f

ds

d chch

µ

λ

( )( ) ( )( ) ( )

( )( ) ( )( )( ) ( )( ) ( )( )( )( )

( )( ) ( )( )( )∑∞

=

−

−+−

−+−−+−

−−−

12**

*****

1***

11

1111

11

m

m

m

c pck

c pck f ds

d c pck

chck ch

µ λ

µ λ µ λ

λ

( )( ) ( )( ) ( ) ( )( ) ( )( )( )

( )( ) ( )( )( )∑∞

=

−

−+−

−+−−−+

12**

***1***

11

1111

m

mm

c pck

c pck f chck ch

µ λ

µ λ λ

( )( ) ( )( ) ( )

( )( ) ( )( )( ) ( )( ) ( )( )( )( )( )( ) ( )( )( )∑

∞

=

−

−+−

−+−−+−

−−−

12**

*****

1***

11

1111

11

m

m

m

c pck

c pck f dsd c pck

chc pch

µ λ

µ λ µ λ µ

( )( ) ( )( ) ( ) ( )( ) ( )( )( )

( )( ) ( )( )( )∑∞

=

−

−+−

−+−−−+

12**

***1***

11

1111

m

mm

c pck

c pck f chc pch

µ λ

µ λ µ

(6)

))(1())(1(

1)(

**c pck

T E −+−

=µ λ

( )( ) ( ) ( )( ) ( )( )( )

( )( ) ( )( )∑∞

=

−

−+−

−+−−−

1**

***1**

11

111

m

mm

c pck

c pck f chch

µ λ

µ λ

The expression for the c.d.f of the partial sum

Sm=u1+u2+…+um. when the random variables ui, i = 1, 2, 3, m

are exchangeable, exponentially distributed and are withconstruct with correlation has derived that the c.d.f of

( )( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛

+−++

=

)1()1(11

1*

bs R

mRbsbs

s f m

m (7)

Where b=a(1-R), a being the parameter of the exponential

distribution. Noting that the interarrival times between

decision making epochs are constantly correlated exponential

variables with parameter a.

( )( ) ( )[ ]⎟⎟

⎠ ⎞⎜⎜

⎝ ⎛

++−++−=

MRbsbs Rbsbs Rs f

mm11)1(

)1()1(*

( )) ( )))=−+− c pck f m

*** 11 µ λ

( ) ( )( ) ( )( )( )( )( )( ) ( )( )( ){ }

( ) ( )( ) ( )( )( )( ){( )( ) ( )( )( )} ⎟⎟

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

−+−+

−+−+−

−+−+

−+−+−

c pck MRb

c pck b R

c pck b

c pck b Rm

**

**

**

**

11

1111

111

1111

µ λ

µ λ

µ λ

µ λ

(8)

))(1())(1(

1)(

**c pck

T E −+−

=µ λ

( )( ) ( )

( ) ( )( ) ( )( )( )( )( )( ) ( )( )( ){ }

( ) ( )( ) ( )( )( )( ){( )( ) ( )( )( )} ⎟

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

−+−+

−+−+−

−+−+

−+−+−

−− −

c pck MRb

c pck b R

c pck b

c pck b R

chch

m

m

**

**

**

**

1**

11

1111

111

1111

1

µ λ

µ λ

µ λ

µ λ

(9)h(.),k(.) and p(.) are exponentially distributed with

parameters α, µ, and η.

( )c

ch+

=α

α * , ( )c

ck +

= β

β * , ( )c

c p+

=η

η * ,

( )α +

=−c

cch*1 , ( )

β +=−

c

cck

*1 , ( )η +

=−c

cc p

*1

η

µ

β

λ

++

+

=

c

c

c

cT E

1)(

( )

( ) ( )

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟ ⎠

⎞⎜⎜⎝

⎛

++

++

⎟⎟ ⎠

⎞⎜⎜⎝

⎛

++

+−−

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ ⎟⎟ ⎠

⎞⎜⎜⎝

⎛

++

++−

++

+

⎟ ⎠

⎞⎜⎝

⎛ +

⎟ ⎠

⎞⎜⎝

⎛ +

−∑

∞

=

−−

η

µ

β

λ

η

µ

β

λ

η

µ

β

λ

η

µ

β

λ

α

α

α

c

c

c

cmRb

c

c

c

cb R R

c

c

c

cb R

c

c

c

c

cc

c

m

mm

11

11

1

11

In the case of three sources of manpower deletion assume thatm=3, the resulting expression becomes,

η µ

β λ

+++

=

cc

cc

T E 1

)(

( )

( ) ( )( )( )

( ) ( )⎪⎩

⎪⎨⎧

⎭⎬⎫

⎟⎟

⎠

⎞

⎜⎜⎝

⎛

++

++

⎟⎟

⎠

⎞

⎜⎜⎝

⎛

++

+−−

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ +

++

⎟⎟

⎠

⎞⎜⎜⎝

⎛ ⎟⎟

⎠

⎞⎜⎜⎝

⎛

+++++

+

−⎟ ⎠

⎞⎜⎝

⎛ +

⎟ ⎠

⎞⎜⎝

⎛ +

−

−

η

µ

β

λ

η

µ

β

λ

η

µ

β

λ

η β

β µ λ η

α

α

α

c

c

c

c Rb

c

c

c

cb R R

c

c

c

c

cc

ccccb

Rcc

c

m

311

1

1

1

2

(10)




ISSN 2277-5056 | © 2014 Bonfring

( )

( )( ) ( )( ) ( )( )( )

( ) ( )

⎟⎟ ⎠ ⎞⎜⎜

⎝ ⎛

++

+⎭⎬⎫⎟⎟

⎠ ⎞⎜⎜

⎝ ⎛

++

++

⎟⎟ ⎠

⎞⎜⎜⎝

⎛

++

+−−

⎟⎟ ⎠

⎞⎜⎜⎝

⎛

++++++++

−⎟ ⎠

⎞⎜⎝

⎛ +

⎟ ⎠

⎞⎜⎝

⎛ +

−

++

+

=

−

η µ

β λ

η µ

β λ

η

µ

β

λ

η β

β µ η λ η β

α

α

α

η

µ

β

λ

cc

cc

cc

cc Rb

c

c

c

cb R R

cc

ccccbcc

Rcc

c

c

c

c

c

m

3

11

1

1

1

2

(11)

( ) ( )( )( )( ) ( )( )( )

22

2 20

2

1 1s

d E T l s

ds k c p cλ µ

∗

= ∗ ∗= =

− + −

( )( ) ( ) ( )( ) ( )( )( )2

1 * * *

21

1 1 1m

m

m

d h c h c f k c p c

dsλ µ

∞−∗ ∗

=

+ − − + −∑

( )( ) ( ) ( )( ) ( )( )( ) ( )( ) ( )( )( )

( )( ) ( )( )( )

21 * * * * *

2

* *

1 1 1 1 1

1 1

m

m

d h c h c f k c p c k c p c

dsk c p c

λ µ λ µ

λ µ

−∗ ∗⎛ ⎞− − + − − + −⎜ ⎟

⎜ ⎟−⎜ ⎟− + −⎜ ⎟⎝ ⎠

( )( ) ( ) ( )( ) ( )( )( ) ( )( ) ( )( )( )

( )( ) ( )( )( ) ( )( ) ( )( )( )

( )( ) ( )( )( )

1 * * * * *

1

* * * * *

3* *

2 1 1 1 1 1

1 1 1 1

1 1

m

m

m

m

d h c h c k c p c f k c p c

ds

f k c p c k c p c

k c p c

λ µ λ µ

λ µ λ µ

λ µ

∞−∗ ∗

=

⎛ ⎞− − + − − + −⎜ ⎟

⎜ ⎟⎜ ⎟− − + − − + −⎜ ⎟+⎜ ⎟− + −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

∑

( )( )( ) ( )( )( )

2

2

2

1 1

E T

k c p cλ µ ∗ ∗=

− + −

( )( ) ( ) ( )( ) ( )( )( )

( )( ) ( )( )( )

1 * * *

1

2* *

2 1 1 1

1 1

m

m

m

h c h c f k c p c

k c p c

λ µ

λ µ

∞−∗ ∗

=

⎛ ⎞− − + −⎜ ⎟⎜ ⎟−⎜ ⎟− + −⎜ ⎟⎝ ⎠

∑

( )2 2 E T

c c

c c

λ µ

β η

=⎛ ⎞

+⎜ ⎟+ +⎝ ⎠

( )

( )

11

1

21

1 1

1 1

2

mm

m

m

c c R b

c c

c c c c R b mRbc c c cc

c c c c

c c

λ µ

β η

λ µ λ µ β η β η α

α α λ µ

β η

−−

− ∞

=

⎧ ⎫⎛ ⎞⎧ ⎫⎛ ⎞⎪ ⎪⎜ ⎟− + +⎨ ⎬⎜ ⎟⎪ ⎪+ +⎜ ⎟⎝ ⎠⎩ ⎭⎪ ⎪⎜ ⎟⎧ ⎫⎛ ⎞ ⎛ ⎞⎪ ⎪⎜ ⎟− + + + +⎨ ⎬⎜ ⎟ ⎜ ⎟⎪ ⎪⎜ ⎟+ + + +⎝ ⎠ ⎝ ⎠⎩ ⎭⎪ ⎪⎛ ⎞ ⎛ ⎞ ⎝ ⎠−⎨ ⎬⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠ ⎛ ⎞⎪ ⎪+⎜ ⎟⎪ ⎪+ +⎝ ⎠⎪ ⎪

⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭

∑

2

c c

c c

λ µ

β η

=⎛ ⎞

+⎜ ⎟+ +⎝ ⎠

( )

( )

1

1

1

1 11

1

1 1

m

m

m

c c R b

c cc

c c c cc c c c R b mRb

c cc c c c

λ µ

β η α

α α λ µ λ µ λ µ

β η β η β η

−

− ∞

=

⎧ ⎫⎛ ⎞⎧ ⎫⎛ ⎞ ⎛ ⎞⎪ ⎪⎜ ⎟− + +⎨ ⎬⎜ ⎟ ⎜ ⎟+ +⎪ ⎪⎜ ⎟⎝ ⎠⎛ ⎞⎛ ⎞ ⎩ ⎭ ⎜ ⎟−⎨ ⎬⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟+ + ⎧ ⎫ ⎛ ⎞⎝ ⎠⎝ ⎠ ⎛ ⎞ ⎛ ⎞⎪ ⎪⎜ ⎟ +− + + + + ⎜ ⎟⎨ ⎬ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎪ ⎪⎜ ⎟ + ++ + + + ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎩ ⎭⎝ ⎠⎩ ⎭

∑

put m=3

2 2( ) E T

c c

c c

λ µ

β η

=⎛ ⎞

+⎜ ⎟+ +⎝ ⎠

( ) ( ) ( )

( ) ( ) ( ) ( )( )

( )

22

1 1

1 1 3

c cc R

c c c c b c c c c

c c c c c c R b Rb

c c c c c c

β η α

α α β η λ η µ β

λ µ λ µ λ µ

β η β η β η

⎛ ⎞⎛ ⎞+ +⎛ ⎞⎛ ⎞⎜ ⎟− − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟+ + + + + + + +⎝ ⎠⎝ ⎠ ⎝ ⎠⎜ ⎟⎧ ⎫⎜ ⎟⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎪ ⎪− + + + + +⎨ ⎬⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟+ + + + + +⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭⎝ ⎠

(12)

[ ]22)()()( T E T E T V −=

2

c c

c c

λ µ

β η

= ⎛ ⎞+⎜ ⎟+ +⎝ ⎠

( ) ( ) ( )

( ) ( ) ( ) ( )( )

( )

22

1 1

1 1 3

c cc R

c c c c b c c c c

c c c c c c R b Rb

c c c c c c

β η α

α α β η λ η µ β

λ µ λ µ λ µ

β η β η β η

⎛ ⎞⎛ ⎞+ +⎛ ⎞⎛ ⎞⎜ ⎟− − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟+ + + + + + + +⎝ ⎠⎝ ⎠ ⎝ ⎠⎜ ⎟⎧ ⎫⎜ ⎟⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎪ ⎪− + + + + +⎨ ⎬⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟+ + + + + +⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭⎝ ⎠

( ) ( )( )

( )( ) ( ) ( )( )

( )

222

2

311

1

11

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

⎪⎩

⎪⎨⎧

⎭⎬⎫

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ +

++

+⎟⎟ ⎠

⎞⎜⎜⎝

⎛ ⎟⎟ ⎠

⎞⎜⎜⎝

⎛ +

++

+−

⎟⎟ ⎠

⎞⎜⎜⎝

⎛

++++++++

−⎟ ⎠

⎞⎜⎝

⎛ +

⎟ ⎠

⎞⎜⎝

⎛ +

−

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ +

++

−

η

µ

β

λ

η

µ

β

λ

β µ η λ η β

η β

α

α

α

η

µ

β

λ

c

c

c

c Rb

c

c

c

cb R

ccccbcc

cc R

cc

c

c

c

c

c

(13)

V. NUMERICAL EXAMPLES

The parameters such as R, α, µ, η, β and λ are taken indifferent inputted data values and keeping c=0.2 and b=0.5 are

fixed. The expected time to recruitment and its variance are

shown in table 1 to table 6 and figure 1 to figure 6respectively.

Table. 1: Variations in E(T) and V(T) for the Variations in R

keeping α=1.0, λ=1.5, µ=2.0, η=3.6 and β=4.0 are fixed

R E(T) V(T)

0.20.4

0.60.81.0

5.18065.2224

5.28745.40215.6595

37.094252.1782

75.6130116.9880209.6770




ISSN 2277-5056 | © 2014 Bonfring

Fig. 1: Variations in E(T) and V(T) for the variations in R

From table 1 it is observed that, if the value of R, which is

the parameter of the correlation between the inter-decision

time increases and other inputted data values α=1.0, λ=1.5,

µ=2.0, η=3.6 and β=4.0 are fixed, the expected time to

recruitment and its variance increases, as shown in table 1 and

figure 1 respectively.

Table. 2: Variations in E(T) and V(T) for the variations in α

and µ=2.0, λ=1.5, R=0.6, η=3.6 and β=4.0

α E(T) V(T)

1.01.5

2.02.5

3.0

5.28745.36505

5.417985.45536

5.48294

75.61375.6647

75.693175.7098

75.7203

Fig. 2: Variations in E(T) and V(T) for the Variations in α

From table 2 it is observed that, if the value of α, which isthe parameter of the random variable representing the amount

of depletion at every epoch of policy decision increases and

other inputted data values µ=2.0, λ=1.5, R=0.6, η=3.6 and

β=4.0 are fixed, the expected time to recruitment and its

variance increases, as shown in table 2 and figure 2

respectively.

Table. 3: Variations in E(T) and V(T) for the Variations in µ

and α=1.0, λ=1.5, R=0.6, η=3.6 and β=4.0

µ E(T) V(T)

1.21.4

1.6

1.82.0

6.87916.3956

5.9768

5.61055.2874

145.9590121.9820

103.0620

87.908275.6130

Fig. 3: Variations in E(T) and V(T) for the Variations in µ

In table 3, if the value of µ, which is the parameter of therandom variable representing the amount of depletion at every

epoch of transfer of personnel increases and other inputted

data values α=1.0, λ=1.5, R=0.6, η=3.6 and β=4.0 are fixed,the results shows that the expected time to recruitment and its

variance decreases, which is also depicted in table 3 and figure3 respectively.

Table. 4: Variations in E(T) and V(T) for the Variations in η

and µ=2.0, α=1.0, R=0.6, λ=1.5 and β=4.0

η E(T) V(T)

0.20.40.60.8

1.0

0.92171.32721.70232.0512

2.3769

0.23241.19292.89005.2923

3302




ISSN 2277-5056 | © 2014 Bonfring

Fig. 4: Variations in E(T) and V(T) for the Variations in η

In table 4, if the value of η, which is the parameter of the

interarrival times between nth transfer of personnel increases

and other inputted data values µ=2.0, α=1.0, R=0.6, λ=1.5

and β=4.0 are fixed, the results shows that the expected timeto recruitment and its variance increases, which is also

depicted in table 4 and figure 4 respectively.

Table. 5: Variations in E(T) and V(T) for the Variations in λ

and µ=2.0, α=1.0, R=0.6, η=3.6 and β=4.0

λ E(T) V(T)

0.5

1.01.5

2.02.5

7.1635

6.08105.2874

4.68014.2000

161.1360

107.608075.6130

55.201541.5196

Fig. 5: Variations in E(T) and V(T) for the Variations in λ

In table 5, if the values of λ, which is the parameter of the

successive epochs of first transfer increases and other values

µ=2.0, α=1.0, R=0.6, η=3.6 and β=4.0 are fixed, the results

shows that the expected time to recruitment and its variance

increases, which is also depicted in table 5 and figure 5respectively.

Table. 6: Variations in E(T) and V(T) for the Variations in β

and µ=2.0, α=1.0, R=0.6, λ=1.5 and η=3.0

β E(T) V(T)

1.0

2.03.04.05.0

2.6959

3.90954.71385.28745.7173

12.1014

34.271556.244675.613092.1903

Fig. 6: Variations in E(T) and V(T) for the Variations in β

In table 6, if the values of β, which is the parameter of the

successive epochs of nth transfer increases and other valuesµ=2.0, α=1.0, R=0.6, λ=1.5 and η=3.0 are fixed, the resultsshows that the E(T) and V(T) increases, which is also depicted

in table 6 and figure 6 respectively.

VI. CONCLUSION

The models discussed in this paper, indicate how the real

life situations which may arise in the management of

manpower planning. The conceptualizations of mathematical

and stochastic model help the process of finding the optimalsolutions and also the implementation of the same. However it

is very important to identify the appropriate probability

distribution that would portray the realities. The identification

appropriate distribution is an important step. Hence the surveymethod can help the identification of such distribution. The

tests for goodness of fit of such distribution are real life data is

an important procedure. Once this is taken care of then themodels can be used in solving real life problems. Also many

of inventory models of practical use can be suitably

incorporated as model for future research.

ACKNOWLEDGMENT

The authors very much thankful to Mr. P. Rajesh,

Programmer, Engineering Mathematics Section, Faculty of

Engineering and Technology, Annamalai University for his

valuable help in the computer simulation work.

R EFERENCES

[1] R. Arulpavai and R. Elangovan, “Determination of Expected Time to

Recruitment - A Stochastic Approach”, International Journal of

Operations Research and Optimization, Vol. 3, No. 2, pp. 271-282,2012.

[2] R. Arulpavai and R. Elangovan, “A stochastic model to determine the

optimal time interval between recruitment using shock model approach”,

Proceeding of the International Conference on Frontiers of Statistics and

its Applications, Bonfring Publication, India. pp.1-14, 2013.[3] Arulpavai, R and R. Elangovan, “Determination of Expected Time to

Recruitment when the Breakdown Threshold has Four Components”,




ISSN 2277-5056 | © 2014 Bonfring

The International Innovative Research Journal for AdvancedMathematics, Vol.1, No. 1, pp.1-21, 2013.

[4] R. Arulpavai and R. Elangovan, “Stochastic Model to Determine the

Optimal Manpower Reserve at Four Nodes in Series”, International

Journal of Ultra Scientist of Physical Sciences, Vol. 25(1) A, pp.139-

155, 2013.[5] D. J. Bartholomew, “The Stochastic Model for Social Processes”, 3nd

ed., John Wiley and Sons, New York, 1982.

[6] D. J. Bartholomew, A. F. Forbes, and S. I. McClean, “Statistical

Techniques for Manpower Planning”, Wiley, Chichester, 1991.[7] D. J. Bartholomew and A. F. Forbes, “Statistical Techniques for

Manpower Planning”, John Wiley and Sons, Chichester, 1979.

[8] J. D. Esary, W. Marshall and F. Prochan, “Shock Models and wear

process”, Annals of probability, Vol.1, No.4, pp.627-650, 1973.[9] J. Gurland, “Distribution of the Maximum of the Arithmetic Mean of

Correlated Random Variables”, Annals of mathematical statistics, Vol.26, pp. 294-300, 1955.

[10] R. Sathiyamoorthi and R. Elangovan, “A Stochastic Model for Optimum

Training Duration”, Indian Association for Productivity, Quality and

Reliability (IAPQR) Transactions, Vol. 23, No.2, pp. 127-131, 1998a.[11] R. Sathiyamoorthi and R. Elangovan, “Shock Model Approach to

Determine the Expected time for Recruitment”, Journal of Decision and

Mathematical Science, Vol.3, No.1-3, pp. 67-78, 1998b.

Dr. R. Elangovan, is working as Professor, Departmentof Statistics, Annamalai University, Annamalainagar,

Tamilnadu. He is having more than 22 years of P.G.

teaching experience. He obtained his B.Sc., M.Sc.,Degrees in Statistics from Annamalai University in1982-1984, and M.Phil., Ph.D., degree from the same

institution in 1988 and 2000. He obtained the B.Ed.,and M.Ed., degrees also in Education from Annamalai

University in 1987 and 1994.He obtained the M.B.A. (H.R.M.) from the same

institution during 2013. He has an excellent academic record of guiding 42

students for M.Phil. degree also and he has published 54 research papers inleading National and International journals, currently guiding 5 students for

their Ph.D. programme in statistics. Five Students are already awarded Ph.D.

degree under his guidance. He has attended nearly 46 International and 26

National level seminars and presenting papers. He is the editor of variousreputed journals in India. He has written 3 books and written 5 lessons and

reports for the Directorate of Distance Education of Annamalai University. He

visited France, Hang Hong, Paris and USA and delivered invited talk and also

chaired the technical session in the conference. He worked as a Statistician in

Aravind Eye Hospital, Madurai from (1987-1988), as Assistant Technical

Officer in Christian Medical College, Vellore (1988-1989) before joiningAnnamalai University (1990) as Lecturer. He has gained the reputation of

being an excellent teacher. (E-mail: [email protected])

Mrs. R. Arulpavai, M.Sc., M.Phil., B.Ed., working as

Guest Lecturer in the Department of Statistics,

Government Arts College, C. Mutlur, Chidambaram

since 2004 to till date (10 Year). She is currently

pursuing Ph.D. degree in Statistics at Manonmaniyam

Sundaranar University, Thirunelveli. She has published7 papers in International and National level journals and

published 3 books in statistics. She has participated and

presented a papers in 7 National, International level

seminars and workshops. She has gained reputation of being a good teacher

for teaching statistics to the graduate students.(E-mail: [email protected])

Documents

Stochastic Model to Determine the Expected Time to Recruitment with Three Sources of Depletion of Manpower under Correlated Interarrival Times