MARKOV ANALYSIS IN MANPOWER SYSTEMchain models. Markov chain models have been applied in examining...

MARKOV ANALYSIS IN MANPOWER SYSTEM

Thesis submitted in partial fulfilment for the award of

Degree of Doctor of Philosophy in Mathematics

By

V. AMIRTHALINGAM

Under the Guidance and Supervision of

Dr. P. MOHANKUMAR M.Sc., M.Phil., Ph.D.,

VINAYAKA MISSIONS UNIVERSITY

SALEM, TAMILNADU, INDIA

July - 2016


DECLARATION

I, V. AMIRTHALINGAM declare that the thesis entitled

“MARKOV ANALYSIS IN MANPOWER SYSTEM” submitted by

me for the Degree of Doctor of Philosophy in Mathematics is the record

of work carried out by me during the period from

January 2011 to July 2016 under the guidance of

Dr. P. MOHANKUMAR, M.Sc., M.Phil., Ph.D., Professor, Department

of Mathematics, Aarupadai Veedu Institute of Technology, Chennai and

that has not formed the basis for the award of any other degree, diploma,

associateship, fellowship, titles in this or any other university or other

similar institutions of higher learning.

Signature of the Candidate

Place: Salem

Date:


CERTIFICATE BY THE GUIDE

I, Dr. P. MOHANKUMAR M.Sc., M.Phil., Ph.D., certify that the

thesis entitled “MARKOV ANALYSIS IN MANPOWER SYSTEM”

submitted for the Degree of Doctor of Philosophy in Mathematics by

V. AMIRTHALINGAM is the record of research work carried out by

his during the period from January 2011 to July 2016 under my

guidance and supervision and that this work has not formed the basis for

the award of any degree, diploma, associateship, fellowship or other titles

in this University or any other University or Institutions of higher

learning.

Signature of the Supervisor with designation

Place: Salem

Date:

ACKNOWLEDGEMENT

At first, I would like to express my deepest gratitude to my

research supervisor and guide Dr. P. Mohankumar, without whose

guidance none of this would have been possible. His encouragement,

support, freedom and keen insight have been invaluable and I have indeed

been fortunate to have such an ideal advisor. He played a very important

role in leading me towards scientific maturity. I am indebted to his

support through the years on both scientific and personal matters.

I am grateful to the founders of Vinayaka Missions University

especially I thank respected Madam Founder Chancellor, Vinayaka

Missions University Mrs. Annapoorani Shanmugasundaram,

Dato Dr. S. Sharavanan, Vice Chairman, Vinayaka Missions and

respected Chancellor Dr. A. S. Ganesan, for their valuable support.

I thank the Vice-Chancellor Prof. Dr. V. R. Rajendran, and Registrar

Prof. Dr. Y. Abraham of Vinayaka Missions University and I express

my sincere thanks to Dr. K. Rajendran, Dean (Research) Vinayaka

Missions University, Salem, Tamilnadu, India.

I express my profound thanks to Dr. A. Nagappan, Principal,

VMKV Engineering College, Salem, Tamilnadu, India. Whose constant

encouragement to carry out this research activity and for providing me

with necessary help.

5

I am thankful to Dr. M. Nithya, Head of the Department of

Computer Sciences, VMKV Engineering College, Salem and my sincere

thanks are due to Dr. S. Kandasamy, Head of the Department of Science

and Humanities and my colleagues, Professors of VMKV Engineering

College, Salem Tamilnadu, India.

I am indebted to my wife, my son and friends for their timely and

valuable support and cooperation throughout my studies.

Above all, my sincere thanks are due to the Lord ‘The Supreme’

for providing opportunity of enlightment.

V. AMIRTHALINGAM

6

CONTENTS

CHAPTERS TITLES Page No

Chapter 1 INTRODUCTION 1

1.1 Overview … 1

1.1.1 Recruitment … 8

1.1.2 Promotion … 9

1.1.3 Wastages … 12

1.2 Techniques Used in Manpower Models … 13

1.2.1 Renewal theory … 13

1.2.2 Markov renewal theory … 16

1.2.3 Semi-Markov processes … 20

1.2.4 Stochastic point processes … 21

1.2.5 Product densities … 22

1.3 Heterogeneity … 23

Chapter 2 REVIEW OF LITERATURE 29

Chapter 3 MANPOWER PLANNING PROCESS WITH

TWO GROUPS USING STATISTICAL

TECHNIQUE

61

3.1 Introduction … 61

3.2 Description and Analysis of the Model

… 62

3.3 Main Result

… 63

3.4 Numerical Illustration … 65

3.5 Conclusion … 67

7

Chapter 4 OPTIMAL MANPOWER RECRUITMENT AND PROMOTION POLICIES FOR THE TWO

GRADE SYSTEM USING DYNAMIC

PROGRAMMING APPROACH

68


4.2 Manpower System Costs … 69

4.3 Mathematical Model … 72

4.4 Dynamic Programming Formulation … 74

4.5 The Manpower Planning Horizon Theorem … 77


4.7 Conclusions … 79

Chapter 5 APPLICATIONS OF DIFFERENCE EQUATIONS 80


5.2 Model 1: Description and Analysis of the Model … 81

5.2.1 Main Results … 84

5.2.2 Special Case … 93

5.2.3 Numerical Illustration … 94

5.2.4 Conclusions … 95




5.3.3 Numerical Illustrations … 100

5.4 Conclusions … 101

8

Chapter 6 MARKOV ANALYSIS OF BUSINESS WITH

TWO LEVELS AND MANPOWER WITH

THREE LEVELS

103


6.2 Assumptions … 104

6.3 System Analysis … 105



Chapter 7 EXPECTED TIME FOR RECRUITMENT IN A

TWO GRADEDMANPOWER SYSTEM

ASSOCIATED WITH CORRELATEDINTER-

DECISION TIMES WHEN

THRESHOLDDISTRIBUTION HAS SCBZ

PROPERTY

113





7.2.3 Numerical Illustration … 123

7.2.4 Conclusion … 124





References … 130

Publications … 142

1

CHAPTER 1

INTRODUCTION

1.1 OVERVIEW

The analyses of manpower systems have become very important

component of planned economic development of any organization or

nation. However, manpower planning depends on the highly

unpredictable human behavior and the uncertain social environment in

which the system functions. Hence the study of probabilistic or stochastic

models of manpower systems is very much essential. Several stochastic

models of manpower systems have been proposed and studied

extensively in the past (see Bartholomew (1967) and Vajda (1978)).

Various stochastic models of manpower systems can be classified broadly

into two types:

1) Markov Chain models

2) Renewal Models

In all these models, the manpower system is hierarchically graded

into mutually exclusive and exhaustive grades so that each member of the

system may be in one and only one grade at any given time. These grades

are defined in terms of any relevant state variables. Individuals move

between these grades due to promotions or demotions and to the outside

world due to dissatisfaction, retirement or medical reasons. If the size of

2

the grades is not fixed, then the state of the system at any time is

represented by a vector 2 ,( ,l nX t X t X t X t where the

component t represents the number in the thi grade at any time t .

Further the very nature of several manpower systems require to be

observed at, say, annual intervals. Accordingly, the system behaviour is

adequately described by a Markov chain, such models are called Markov

chain models.

Markov chain models have been applied in examining the structure

of manpower systems in terms of the proportion of staff in each grade or

age profile of staff under a variety of conditions and evaluating policies

for controlling manpower systems (see for example, Young and Almond

(1961), Young (1971), Forbes (1971a,b), Bartholomew (1973) and Gani

(1973)). In these works and in all of what followed the important question

was the control of the expected numbers in the various states by

recruitment control. The numbers of people in such categories change

over time through wastage, promotion flows and recruitment. Some of

these flows are subject to management control while others vary in a

random manner. Factors such as the need to offer adequate career

prospects or the requirement of the job will often dictate a desirable age

or grade structure and it is the manpower planner's task to determine

whether this can be achieved and , if so, how.

3

The limiting behavior of an expanding non-homogeneous Markov

system has practical importance as shown by the literature on manpower

systems (Vassiliou 1981a&b, 1982a). The limiting structure of the

expected class sizes was derived under certain conditions and the relative

limiting structure is shown to exist with a different set of conditions.

Mehlmann (1977) and Vassiliou (1982b) studied the limiting behavior of

the system with Poisson recruitment and observed that the number in the

various grades are asymptotically mutually independent Poisson.

Vassiliou (1984c) studied the asymptotic behavior of non-homogeneous

Markov systems under the cyclical behavior assumption and provided a

general theorem for the limiting structure of such systems. Vassiliou

(1986) later extended the results and provided a basic theorem for the

existence and determination of the limiting structure for the vector of

means, variances and covariances under more general possible

assumptions. He argued that the results are useful from the practical point

of view since they provide valuable information about the inherent

tendencies in the system.

The control of asymptotic variability of expectations, variances and

covariances in a Markov chain model is a major research area in

manpower systems. The earliest work on this subject was that of Pollard

(1966). The results were later extended by several authors (Vassiliou and

Gerontidis (1985), Vassiliou (1986), Vassiliou et al. (1990)). Attainable

4

and maintainable structures in Markov manpower systems under

recruitment control have been studied by Bartholomew (1977), Davies

(1975, 1982), Vassiliou and Tsantas (1984 a&b) and later Kalamatianou

(1987) analysed the same with pressure in grades. The concept of a non-

homogeneous Markov system in a stochastic environment (S-NHMS)

was introduced for the first time by Tsantas and Vassiliou (1993). The

problem of attaining the desired structure in an optimal way as well as

maintaining relative grade sizes applying recruitment control in a

stochastic environment as introduced in Bartholomew (1975, 1977) is

considered. More references in this and related topics an be found in

various papers by (Georgiou (1992), Tsantas (1995), Tsantas and

Georgiou (1994, 1998)). A Markov model responding to promotion

blockages has been proposed by Kalamatianou (1988). Raghavendra

(1991) has employed a Markov chain model in obtaining the transition

probabilities for promotion in a bivariate framework consisting of

seniority and performance rating. Georgiou and Vassiliou (1997) have

introduced phases in a Markov chain model and investigated the input

policies subject to cost objective functions. Yadavalli and Natarajan

(2001) studied a semi-Markov model in which a single grade system

allows for wastage and recruitment. The time dependent behaviour of

stochastic models of manpower system with the impact of pressure on

promotion was subsequently studied by Yadavalli et al. (2002).

5

Although a Markov model is simple and easy to implement, it does

not take into account existing knowledge of the distribution of length of

service until leaving. In such cases the mathematically intractable Semi-

Markov models approach is suggested (McClean 1991). The Semi-

Markov processes are a generalization of Markov processes in which the

probability of leaving a state at a given point in time may depend on the

length of time the state has been occupied (duration of stay) and on the

next state entered. However, there are several theoretical literatures on

Semi-Markov Models ( Pyke (1961 a & b), Ginsberg (1971), Mehlmann

(1979), McClean (1978, 1980, 1986)). A stochastic model of migration,

occupational and vertical mobility, based on the theory of Semi-Markov

process was derived by Ginsberg (1971). McClean (1978) extended the

assumption of simple Markov transitions between grades and the leaving

process to semi-Markov formulation which allows for inclusion of well-

authenticated leaving distributions such as the mixed exponential.

Moreover, the previous assumption of Poisson recruitment is generalized

to allow for a recruitment process which may vary with time, either as a

mixed exponential time dependent Poisson process or by assuming that

the number of recruits depends on the amount of capital owned by the

firm. The previous formulation is therefore extended to take into account

the fact that recruitment to a firm is a highly variable process and the

assumption of Poisson recruitment to each grade is therefore restrictive.

6

The concept of non-homogeneous semi-Markov systems found

important applications in manpower system particularly in the subjects of

variability, limiting distributions and maintainability of grade sizes

(Vasiliou and Papadopoulou (1992)).

On the other hand, there are several manpower systems where the

grade sizes are fixed by the budget or amount of work to be done.

Recruitment and promotion can occur only when vacancies arise through

leaving or expansion. There may be randomness in the method by which

vacancies are filled. The movements of individuals are characterised by

replacements (renewals) according to some probabilistic law, and such

models of manpower systems are called renewal models. The main

advantage of these models over the Markov chain models is that they are

closer to reality since the losses (wastages) occur continuously in time

and there is always the possibility that a new recruit may also leave

during the study period. White (1970) has used models of this kind to

study the flows of clergy of several large American denomination.

Stewmann (1975) has applied White's methods to the study of

recruitment and losses in a state police force. Bartholomew (1982) has

provided a detailed analysis of renewal models of manpower systems.

Sirvanci (1984) has applied renewal processes to forecast the manpower

losses of an organisation in order to determine whether the organisation

will be able to meet its demand for manpower under present conditions.

7

The distributions of completed length of service (CLS) in these

models have been fitted to actual data from industry by several

researchers (see Bartholomew, 1982). McClean (1976, 1978) has used a

mixed exponential distribution for CLS and estimated the parameters

using data for two companies. Agrafiotis (1983, 1984, and 1991) studied

the problem of labour turnover by using renewal process type models.

A satisfactory model of manpower system should provide answers to

the following questions:

1. How to provide estimates of manpower indicators of the system?

2. How to predict the future behaviour of the system under various

assumptions?

3. How to find optimum solutions to various policy problems subject

to various constraints given by the management?

4. How to avoid various problems by giving a warning before the

situation develops?

5. How to design manpower, which is related to various problems of

prediction in consultation with management?

In order to provide answers to questions raised above, the model

considered should incorporate the following main factors, which

predominantly determine the behaviour of a manpower system:

1. Recruitment

2. Promotion of employees

3. Wastages.

8

1.1.1 Recruitment

The sizes of various grades, which respond to the expansion,

promotions and wastages, are maintained at the desired level at any time

by a process called recruitment. The flow of recruitment can be

controlled by the management authorities. The recruitment can be made

in several ways. Vacancies can be filled as and when they arise or they

may be allowed to accumulate and then filled up at specified periods or

whenever the total number of vacancies attains a certain specific level, so

as to minimize the cost. The recruitment can be made by the organization

itself or by some external agencies to avoid delay and huge overhead

costs. Several organizations in South Africa do not recruit employees by

themselves (e.g. the preliminary process of senior level positions in

Statistics South Africa) but approach recognized recruiting agencies.

Usually, vacancies that arise are allowed to accumulate for a specified

period of time, or to attain a specified level and then these agencies are

requested to fill them up and to complete the process of recruitment in a

specified period of time. However, they may not be able to fill up all the

notified vacancies due to the non-availability of suitable candidates

with prescribed qualifications and experience. Further additional

vacancies may also arise during the period of recruitment process.

Therefore there may exist some vacancies even after the process of

recruitment is completed. In reality, many such manpower systems exist.

9

However, these types of models have not been considered in the

literature. Davies (1975) considered a fixed size Markov chain model that

suffered losses and admits recruits to various grades in such a manner that

the total grades in the system remain constant. In that paper, the

recruitments take place at integral points in time and at the time of

recruitment, no vacancy is left unfilled. Vassiliou et al. (1990) deal with a

non-homogeneous Markov manpower system, which allows recruitment

in each grade of the hierarchically graded manpower system. They have

obtained the limiting expected structure of the system by control over the

limit of the recruitment probabilities. Rao (1990) has considered a

manpower planning model with the objective of minimizing the

manpower cost with optimal recruitment policies. The recruitment size is

known and fixed in this model. Hence the study of a model where

vacancies are accumulated and then filled up deserves attention.

1.1.2 Promotion

Normally vacancies that arise in the lower grade are filled up by

recruitments where as those in the higher grades are filled up by

promotions. Further, promotions besides giving due recognition to

proficiency and credibility of the employees reduce the chance of an

efficient employee leaving the organization. Some of the promotion rules

are given below:

10

i The senior most in the grade is promoted.

ii Promotion is given at random.

iii Those who fill certain efficiency criterion along with some

minimum completed length of service are promoted.

As per the rule (i), the length of service is the sole criterion for

promotion and hence the management can control it. The rule (ii) gives

full freedom for the management to promote any employee of their

choice, which also is not desirable. Normally rule (iii) is preferred. Some

of the reasons, which influence the promotion policies, are (a) pressure

(b) efficiency and (c) length of service.

(a) Pressure

In a multi-graded hierarchical manpower system, a promotion

policy that is associated with constant promotion probabilities leaves a

proportion of employees qualified by completed length of service in a

lower grade un-promoted. This proportion increases and pressure starts

building up as time progresses. When pressure exceeds a certain level of

control, a high proportion of un-promoted employees could have serious

effect on the efficiency of the organization for several reasons such as

productive loss and wastage. The pressure can be quantified as a function

of the proportion of the people in a job grade according to Kalamatianou

(1987, 1988). She has quantified pressure in three stages and suggested

11

models to reduce the pressure by suitably changing the promotion

policies well in advance.

(b) Efficiency (training)

Training of manpower has long been recognized as an important

factor for improving the efficiency of the employees and for the

productive improvement. Further, when it is considered as a criterion for

promotion, it becomes very much effective. Mathematical models

incorporating training aspects have been studied by Guardabassi et al.

(1969), Grinold and Marshall (1977), Mehlmann(1980) and Vajda

(1978). Goh et al. (1987) have analysed the training problem within an

organisation using dynamic programming principles. These results were

recently generalised using Dynamic Programming by Yadavalli et al.

(2002).

(c) Length of service

Length of service in a grade should necessarily be a natural

criterion for promotion in order to create a healthy atmosphere among the

employees. However, for controlling the promotion, the management can

include other efficiency criterion along with it for promotion. This aspect

has been discussed by Bartholomew (1973, 1982), Glen (1977) and in the

thesis of Kamatianou (1983).

12

1.1.3 Wastages

When employees move from one grade to another, they are

exposed to different factors influencing them to leave the organization.

Various data indicate that the reasons for leaving can be classified into

the following cases:

(i) Discharge

(ii) Resignation

(iii) Redundancy

(iv) Retirement

(v) Medical retirement

Agrafotis (1984) has grouped the above cases into two main

reasons, normally, (a) unnatural and (b) natural. Unnatural reasons for

leaving depend on the internal structure of the company or organisation,

viz, lack of promotion prospects, job satisfaction, problem of adjustment,

etc., including the cases (i), (ii), and (iii) mentioned above. Natural

reasons for leaving the organisation do not depend on the internal

structure of the organisation, including the cases under (iv) and (v). In

analysing data on a number of companies, Agrafiotis (1984) has shown

that there is a significant difference in the wastage rates corresponding to

reasons (a) and (b) for leaving. However, the cases (iv) and (v) relating to

the natural leaving are entirely different and are to be discussed

separately, for an employee leaving by way of natural retirement after

having served the organisation completely cannot be grouped with an

13

employee who leaves the organisation by way of medical reasons. As

such, there are three different wastage rates:

(a) Due to internal structure

(b) Due to retirement

(c) Due to medical reasons

Unlike natural wastage the unnatural wastage can be controlled by

the management by resorting to better promotional prospects, improved

working conditions and training.

Some other manpower studies which investigated wastage

intensities are (Vassiliou (1976, 1982), Leeson (1981, 1982), McClean et

al. (1992)).

1.2 TECHNIQUES USED IN MANPOWER MODELS

Here, we present the various techniques used in the analysis of

models of manpower systems.

1.2.1 Renewal theory

Renewal theory forms an important constituent in the study of

stochastic processes and is extremely used in the analysis of manpower

models with recruitment. Feller (1941, 1968) made significant

contributions to renewal theory giving the proper lead. Smith (1958) gave

an extensive review and highlighted the applications of renewal theory to

a variety of problems. A lucid account of renewal theory is given by Cox

(1962).

14

Definition 1

Let : 1,2.,,,iX i be a collection of random variables, which are

non-negative, independent and identically distributed. Then the sequence

nX is called a renewal process. We assume that each of the random

variable iX has a finite mean . A renewal process is completely

determined by means of .f , the p.d.f of iX . Associated with the

renewal process is a random variable ,N t which represents the number

of renewals in the time interval (0, ].t N t is also known as the renewal

counting process (Parzen, 1962).

Definition 2

The expected value of N t is called the renewal function and is

denoted by H t . The derivative of H t if it exists, is denoted by h t

and is called the renewal density. The quantity h t dt has the

interpretation that it represents the probability that a renewal occurs in

, .t t dt We will have to identify this as what is known as the first order

product density for a more general process. The renewal density satisfies

the following integral equation:

0

t

h t f t f u h t u du

One of the important and useful theorems in application is the key-

renewal theorem (Smith, 1958).

15

Theorem

Let Q i satisfy the following conditions:

(i) 0Q t for all 0t

(ii) Q t is non-increasing

(iii) 0

Q t dt

Then,

0

limtQ Q t u dH u

0

1Q u du

Further details regarding renewal theory can be found in Smith

(1958), Feller (1968), Prabhu (1965) and Srinivasan (1974). We now

briefly indicate how renewal theory has been used in the study of

manpower models. The stochastic element in manpower systems occur

principally due to the loss mechanism arising out of staff moving out of

the system. The randomness may also be due to the method by which the

vacancies are filled. In the context of manpower planning, the renewal

process , 0N t t represents the number of recruitments required for

the given position for which the first person was employed at 0.t The

random time X between successive replacements is called the completed

length of service (CLS) and its distribution F x is termed as the CLS

distribution. Thus, during the operation period from 0t up to time ,t

while N t employees leave, an equal number need to be recruited in

order to keep a given position continuously staffed. To predict the value

16

of N t for any given time, its expected value, which is referred to as the

renewal function, may be used. The relationship between the CLS

distribution and the renewal density ,h t the derivative of ,H t is given

by the renewal equation

0

; 0.t

h t f t f u h t u du t

Where f t is the density of the CLS distribution .F t The

renewal density h t can be interpreted as the rate at which the losses

occur. On the other hand, .F t is the distribution of the time an

employee spends in the organisation before leaving. The renewal process

of personnel losses has been extensively studied by Bartholomew (1962,

1982) and Bartholomew and Forbes (1979).

1.2.2 Markov renewal theory

Let E be a finite set, N the set of non-negative integers and

[0, ).R Suppose we have, on a probability space , ,B P random

variables : , :n nX E T R defined for each n N so that

0 1 20 ....T T T

Definition 1

The stochastic process , , ;n nX T X T n N is said to be a

Markov renewal process with the state space E provided that

1 1 0 1 0 1, \ , ,... ; , ,...n n n n nP X j T T t X X X T T T

17

1 1, /n n n n nP X j T T t X

for all ,n N j E and .t R

We assume that ,X T is time-homogeneous, that is, for any

,i j E and t R

1 1, , , /n n n nQ i j t P X j T T t X i

independent of n. The family of probabilities

, , ; ,Q i j t i j E t R

is called a semi-Markov kernel over E. We assume that

, ,0 0Q i j for all , .i j E

For each pair ,i j the function , ,t Q i j t has all the properties

of a distribution function except that;

, , ,tP i j Q i j t

is not necessarily 1. It is easy to see that

, 0,P i j , 1j E

P i j

that is, ,P i j are the transition probabilities for some Markov chain

with state space E. It follows from the definition 1 and above that

1 0 1 0 1 \ , ; , .. ). (n n n nP X j X X X T T T P X j

for all ,n N j E

This implies that ;nX X n N is a Markov chain with state

space E and the transition matrix P.

18

1.2.2.1 Markov Renewal Functions

We write 1P A for the conditional probability 0[ ]/P A X i and

similarly iE for the conditional expectations given 0 .X i We also

assume that

0 1 2 0 0.P T T T

Let us define , ,nQ i j t as

, ,nQ i j t , ;n nP X j T t , ,i j E t R for all .n N

Then,

0 , ,Q i j t1

0ij

if i j

if i j

for all 0t and 0n

where ij is the Kronecker delta function.

We have the recursive relation

1 , ,nQ i j t 1

0, , , , |n

j E

Q i j ds Q j k t s

Where the integration is over [0,0) The expression , ,R i j t that

gives the expected number of renewals of the position j in the interval

[0,0) is given by

, ,R i j t 0

, ,n

n

n

Q i j t

This is finite for any ,i j E and .t The , ,R i j t are called

Markov renewal functions and the collection

19

, , ; , ,R R i j t i j E t R of these functions is called the

Markov renewal kernel corresponding to Q. We note that for fixed

,i j E the function , , ,i j R i j t is a renewal function. We can now

easily see from the various expressions above that 1

,aR I Q

where

1 is the unit matrix.

1.2.2.2 Markov Renewal Equations

The class of functions B which we will be working with is the set

of all functions

:f E R R

such that for every i E the function ,t f i t is Borel measurable and

EXR bounded over finite intervals and for every fixed j E the

functions , , ,ni j Q i j t and , , , , , ,i j R i j t i j R i j t is both

belong to B. For any function .f B the function Q f defined by

,Q f i t 1

0, , ,n

j E

Q i j ds f j t s

is well defined and Q f B again. Hence the operation can be repeated,

and the thn iterate is given by

,Q f i t 1

0, , ,n

j E

Q i j ds f j t s

We can replace Q by R, which is again a well-defined function,

which we will denote by ,R f that is for ,f B

20

R f 1

0, , , .

j E

R i j ds f j t s

A function f B is said to satisfy a Markov renewal equation if

for all i B and ,t R

,f i t 1

0, , , ,

j E

g i t Q i j ds f j t s

for some function g B

Limiting ourselves to functions ,f g B which are non-negative

and denoting this by ,B the Markov renewal equation now becomes

, ,f g Q f f g B

This Markov renewal equation has a solution .R g Every solution

f is of the form ,R f h where h satisfies , .h Q h h B For a more

detailed on Mark renewal equations see Cinclar (1975).

1.2.3 Semi-Markov processes

Let ,X T be a Markov renewal process with state space E and

semi-Markov kernel Q. Define .Sup

n nL T Then L is the lifetime of

, .X T If E is finite or if X is irreducible and recurrent, then L

almost surely. By weeding out those and t R for which

Sup

n nT we assume that Sup

n nT for all .

Then for any and t R there is some integer n N such

that 1 .n nT t T w We can therefore define a continuous time

21

parameter t t RY Y

with state space E by putting t nY X on

1.n nT t T The process t t RY Y

so defined is called a semi-Markov

process with state space E and a semi-Markov transition kernel

, , .Q Q i j t

1.2.4 Stochastic point processes

Stochastic point processes form a class of random process more

general than those considered in the previous sections. Since point

processes have been studied by many researchers with varying

backgrounds, there have been several definitions of them each appearing

quite natural from the view point of the particular problem under study

(see, for example, Bartlett (1966), Bhaba (1950), Harris (1963) and

Khinchine (1955)). A stochastic process is the mathematical abstraction,

which arises from considering such phenomena as a randomly located

population or a sequence of events in time. Typically, there is envisaged a

state space X and a set of points Xn from X representing the locations of

the different members of the population or the times at which the events

occur. Because a realization (or a sample path) of any of these

phenomena is just a set of points in time or space, a family of such

realizations has come to be called point processes (see Daley and Vere-

Jones, (1971)).

22

A comprehensive definition of a point process is due to Moyal

(1962) who deals with such process in a general space, which is not

necessarily Euclidean. Consider a set of objects each of whom is

described by a point x of a fixed set of points X. Such a collection of

objects, which we may call a population, may be stochastic if there exists

a well-defined probability distribution P on some field B of subsets of

the space of all states. We shall assume that the members of the

population are indistinguishable from one another. The state of the

population is defined as an unordered set 1 2, ,....,n

nX x x x representing

the situation where the population has n members with one of the states

1 2, ,...., nx x x . Thus the population state space is the collection of all

such nX with 0,1,2,....n where 0X denotes the empty population. A

point process is defined to be the triplet , ,B P . For a detailed treatment

of stochastic point processes with special reference to its applications the

reader is referred to Srinivasan (1974). A point process is called a regular

point process if the probability of occurrence of more than one event in

0, so .

1.2.5. Product densities

One of the ways of characterizing a general point process is

through product densities (Ramakrishnan (1950), Srinivasan (1974)).

These densities are analogous to the renewal density in the case of non-

23

renewal processes. Let ,t x denote the random variable representing

the number of events in the interval , ,t t x ,dxN t x the events in the

interval ,t x t x dx and , , , .P n t x P N t x n

The product density of order n is defined as

1 2, ,...,n nh x x x

1 2, ,... 01 2

, 1; 1,2,...lim

....n

i i

n

P N x i n

Where 1 2 .... ,nx x x or equivalently for a regular process

1 2, .....,h n x x x n

lim 1, 2,.... 0n

1 , 1; 1,2,...i n N x i i i n

where 1 2 .... .nx x x

These densities represent the probability of an event in each of the

intervals 1 1 1 2 2 2, , , ,... , .n n nx x x x x x x x x Even though the

functions 1 2, ,...n nh x x x are called densities it is important to note that

their integration will not give probabilities but will yield the factorial

moments. The ordinary moments can be obtained by relaxing the

condition that all the '

ix s are different.

1.3 HETEROGENEITY

The validity of the models described under section 1.2 depends

highly on the assumption that the manpower study be based on

homogeneous groups of individuals. This is a huge task, which can never

24

be attained in practice because human behaviour is highly unpredictable

and the environment on which the system operates is uncertain. However,

it is paramount that the researcher ensures that there is no major source of

heterogeneity. Individuals" differences depend on many factors such as

their motivation, performance and commitment to the employer.

The subject of homogeneity of individuals is fundamental in

virtually all fields of study. However, in the biomedical literature, it is a

well known fact that individuals differ substantially in their endowment

for longevity (see Manton (1981); Keyfitz (1978); Shepard and

Zeckhauser (1977). Hence it is important to try and understand the

impact of heterogeneity on the study results. In demography and public

policy analysis studies, it has been found that ignoring heterogeneity in

frailty results in biased results (Vaupel et al. (1979, 1985)).

According to Bartholomew et al. (1991) the analysis of individual

differences is of fundamental importance in the study of manpower

system, in particular, wastages (losses from the system). Any attempt to

describe wastage pattern must reckon with the fact that an individual's

propensity to leave a job depends on a great many factors, both personal

and environmental. Failure to recognise the effects of heterogeneity may

not only result in erroneous results when applying manpower models but

also complicate both the theoretical and empirical research due to the

composition of the population and the differential impact of economic,

25

environmental and social forces. The flow of people in manpower

systems, moving employees in various states can be subdivided into

recruitment stream, the transition between the state and the outflow from

the system. Considering a discreet time 0,1,...t we assume that the

individuals' transitions between the states take place either according to a

homogeneous Markov chain. Most of the work was based on

homogeneous Markov chain model introduced by Young and Almond

(1961), Gani (1963), Young (1971), and Sales (1971). Later on Young

and Vassiliou (1974), Vassiliou (1976, 1978) introduced the non-

homogeneous Markov chain model, which was reported by many

researchers to provide a good prediction in practice. Vassiliou (1982a)

introduced the more general framework of non-homogeneous Markov

model, which incorporates a great variety of applied probability models.

As the literature shows, the theory of non-homogeneous Markov systems

(NHMS) has flourished since then (Vassiliou, et al. (1990); Tsantas and

Vassiliou (1993); Georgiou (1992); Tsantas (1995))

A number of authors suggested tackling the problem of

heterogeneity by dividing the personnel system into more homogeneous

subsystems. The pioneering work on mover stayer models of labor

mobility by Blumen et al. (1955), Goodman (1961) and later

Bartholomew (1982) was one form of subdividing the population into

categories-the 'stayers' who hardly change their jobs and the 'movers'

26

who tend to change jobs frequently. Ugwuowo and McClean (2000)

proposed some techniques to deal with heterogeneity for modeling

wastage, though the problem exits in other flows within the personnel

system. To incorporate population heterogeneity into manpower

modeling, two strategies have been suggested: the use of observable

sources of heterogeneity as it affects wastage and the latent source of

heterogeneity that are impossible to observe but are known to affects the

key parameters of the model. Although the division of individuals in

homogeneous subcategories is a fundamental and important step in

application of the manpower planning techniques, there is still lack of

attention towards the way homogeneous groups can be attained in

practice. De Feyer (1996) presented a general framework to get more

homogeneous subgroups for using Markov Chain theory in manpower

planning. A general splitting-up approach is suggested as well as the use

of some statistical multivariate techniques is proposed to support the

splitting-up process.

Sathiyamoorthi and Elangovan (1998) have obtained the mean and

varianceof time to recruitment using shock model approach when (i)

loss of manpower is a non-negative integer valued random variable, (ii)

threshold for loss of manpower is a discrete random variable following

geometric distribution and (iii) the time between three consecutive

27

decision form a sequence of independent and identically distributed

random variables.

Mariappan and Srinivasan (2001,2002) have obtained the mean

and variance of the time for the recruitment using shock model approach

when (i) staff depletion are caused by the decision making epochs and the

inter - arrival time between consecutive decisions are exchangeable and

constantly correlated exponential random variables (ii) the sequences

associated with the cumulative loss of man hours due to the exodus of

personnel and the inter- decision times taken by the organization, form a correlated pair of renewal sequences. Sathyamoorthy and Parthasrathy (2002) have obtained the

expected time for recruitment when (i) loss of manpower is a

continuous random variable (ii) threshold for loss of manpower is a

continuous random variable having SCBZ property instead of

exponential distribution which has lack of memory property and the inter

decision times form a sequence of independent and identically distributed

random variables.

Saavithri and Srinivasan (2001) have obtained the expected

time for recruitment using some univariate policies of recruitment when

(i) the loss of man hours for each decision taken form a sequence

of independent and identically distributed random variables, with state

space (0,∞), (iv) survival time process and loss of man hours process are

independent with state space (0,∞).

28

Kasturri and Srinivasan (2007) have obtained the mean time to

recruitment for both the models when (i) the thresholds grades are

exponential random variables and (ii) the inter decision timeshare

exchangeable and constantly correlated exponential random

variables. Recently Kasturri and Srinivasan (2007) have

extended their results by assuming that the threshold distributions for the

two grades have SCBZ property.

29

CHAPTER 2

REVIEW OF LITERATURE

2 REVIEW OF LITERATURE

Manpower has to be wisely exploited for the steadfast growth of a

economy. This is the reason why there is Ministry Of Human Resources,

the aim is to implement plans to utilize the human resources available

through out the country for their growth and country's Development. This

is given as much an importance as any other discipline as economics,

psychology, law and public administration, industrial relations, computer

science and operations research. All the disciplines stated above are

themselves in a tremendous state of flux. Manpower planning requires a

keen study, this has necessitated the coming up of lot of literature. New

ways and means are suggested for optimum usage of manpower through

Economics, Operations Research and Mathematical Models. Research is

going on in every field for their growth and manpower planning does not

lag behind.

Manpower planning is historically rooted in the gathering of

manpower statistics dating from the times of the Roman census to the

accounting of slaves, and eventually to population census towards the

end of the eighteenth century Morton [56].

30

Historically, origin of the models of manpower systems could be

traced back to Seal [72]. However, simple models have been reported by

Edwards [20] to have been used by manpower planners long before then.

Mehlmann [54] has developed an optimal recruitment and transition

strategies for manpower systems using dynamic programming recursion

with the objective of minimizing a quadratic penalty function which

reflects the importance of correct manning of each grade under preferred

recruitment and transition patterns.

Lane and Andrew [46] has developed a lognormal model in which

the distribution of wastage was related to length of services and proposed

two methods of analysis. Cohort analysis, in which the wastage

characteristics of an initial homogeneous groups are observed over longer

periods of time; census analysis in which two sample points in time are

used to determine the wastage rates.

As alternative, approach to manpower planning is based on

optimization theory; the theoretical foundations of the optimization

approach have been developed in Holt et al. [35]. Holt developed a cost

model that includes both the costs of maintaining and changing the work

force. Holt uses a quadratic cost model that allows closed form solution

to be developed and finds that optimal staffing levels are based on the

weighted values of forecast demand.

31

A general description of the models and the methods for

developing mathematical models for manpower studies have been

discussed in a broad way by Dill et.al [19]. In this paper the authors have

explored some results in manpower and the development of some simple

stochastic models to take note of such issues in manpower planning.

Direct mathematical methods have been used for structuring such models

and the simulation methods have been used to a large extent in such

models.

Morton [56] presents a concise historical summary of forecasting

techniques starting from demographers' modified exponentials through to

renewal theory, stochastic processes, moving average ad exponential

smoothing. In recent years there has been a swing away from

demographic ally based forecasts towards econometric and input / output

models as well as Monte Carlo simulation. Moreover, recognition that

long lead time in manpower development makes planning particularly

vulnerable to changes in a policy variables. It has simulated research into

'Ideological" or target - related forecasting in which the study of

explicitly stated achievable future goals are undertaken through futurist

speculation or expert consensus in order to restrict the range of the

exogenous variables.

32

Walker [90] compresses forecasting, inventory, determining

qualitative and quantitative, recruitment, selection, training,

development, motivation and compensation into two constituents. The

determination of organizational needs and available manpower supply

within the organization at various times through forecasting; and

programming, the planning of activities which will result in the

recruitment of new employees for the organization, further development

activities for employees, designation of replacement for key managers,

and new expectations for effective top managements planning,

The author goes on to integrate the two basic elements in a time

frame short range (0-2) intermediate range (2-5 years), long range (5-10

years).

Considering recruitment and promotion as some of main activities

of the organization, Vajda [84] has discussed a very systematic account of

the applications of mathematical models to the problems of manpower. In

any organization the employees can be partitioned into different grades.

So, a population of employees in an organization has a well defined

structure in terms of the partition. The question is to find out the changes

in a given structure and how it gives rise to another structure after one or

more transitions. The second question is from a given structure in how

many steps a specified structure could be attained. The problem of

33

interest here is to find out in how many transitional steps the structure

which is identical with the starting could be attained. It implies the revisit

to the starting state in the terminology of stochastic processes.

An optimum manpower utilization model using mathematical

programming has been discussed by Schneider and Kilpatrick [71] for

health maintenance organization. The interaction between effective

manpower utilization, faculty requirements and available capital is

discussed in two basic models, one is an overall planning model and the

other is a subscriber maximization model. The objective used in these

models pertains to either minimum cost or minimum feasible use of

physicians through the substitution of physician extenders.

Girnold [30] has examined the problem of producing a commodity

with uncertain future demand with time lags in the production process

and with the commodity itself being a vital input in the production

process. Kurosu [45] research which is of relevance to job shops

situation described the influence of demand uncertainties on waiting

time, idle time and rate of losing customers. The study, modeled demand

as a queuing process and gone as far as prescribing timing and conditions

for temporary increasing or decreasing process capacity to absorb

fluctuations in demand but, however failed to consider the manpower

34

costs. Aderoba [2] has established a procedure for determining

appropriate levels of full time labour and over time engagement.

A mathematical model of a military manpower system with a view

to determine the optimal steady state, wage rate and force distribution by

length of service is by Jaquette and Nelson [39]. In this paper it is

assumed that the cost of hiring personnel is determined by military

manpower supply function which relates enlistment and re-enlistment

rates to military pay. The optimal force is defined as that force which

provides the greatest military capability for a given budget cost. Optimal

rates of pay are determined by maximizing the productivity index subject

to a budget constraint. Assuming the basic flow process as Markovian the

optimal rates of pay are determined. The steady state optimal policy for

the Cobb-Douglas type function is obtained using the Lagrangian

multiplier technique. Numerical results are also discussed.

The use of linear programming methods to derive optimal long

term policy has been discussed by Grinold and Marshall [31]. They have

introduced the long term planning horizon and uncertain conditions

pertaining to future manpower requirement. The input data as regarding

future requirements, budget constraints, cost, discount rate, utilization

factor and coefficient factor relating to How processes. The inflow is

taken as decision variable, the objective function being the discount cost.

35

The minimization of the same is discussed. A person leaves the rth

grade with probability pr. The length of time x to stay in grade r has a

probability density function rg x and a survival function rG x , in

case he leaves the grade .rS In case the person is promoted the

corresponding probability density function of the length of time to stay

which is y is f y with the corresponding survivor function .rF y

Under these assumptions the semi-Markov transitions between grades are

discussed. The mean grade size at time ' 't is obtained.

A particular aspect has received much attention in the examination

of movement structure of the state of these systems in terms of the

proportion of staff in each grade; and the evaluation of recruitment and

promotion policies for controlling them. Morgan [55], Vassilou [86] and

Lesson [48, 49] in their works have analyzed graded structures with

grades depending on promotion probabilities. The length of service is

considered as an important criterion in determining the staff flow. The

organizations such as civil service where large number of manpower is

required, the grades are sub divided in to several categories for

administrative purposes.

An interesting paper by Abounde and Mcclean [1] contains the

discussions about the model where a manpower system with a constant

level of recruitment is considered, It is related to the production planning

36

in the development of telephone services and linking the same to the

work force. The constant level of recruitment necessary to bring the

number of installations eventually up to their final is discussed. Also a

stochastic model is developed which evaluates the effect of implementing

the recruitment policies in term of changing distributions of staff

numbers, and the changing number of installations with time. Numerical

results are also provided.

Zanakis and Maret [94] have discussed a Markov chain model to

describe the manpower supply planning. In doing so, the two different

aspects of manpower planning namely the demand for manpower and

supply of manpower are considered. The authors have developed a

model, which indicates the How of personnel through an organization as

Markov chain. The authors described the stage interval like week, year,

ete, to define the time interval for transitions. Also the authors indicate

the need for the selection of exhaustive and non-overlapping stage to

which an employee can be classified. Using this information the method

of constructing the TPM is indicated. The authors have indicated the use

of the statistical procedures for testing the Markov chain assumptions.

The use of Chi-Square test for verifying the stationary and the first order

process in non-overlapping chains is indicated. The authors indicate the

method of obtaining the probability of an employee remaining in the

37

same grade of a specific length of time. The authors indicate that the

Markov chain model provides valuable insight into predicting future

organization manpower losses. Numerical examples are also worked out

to indicate the usefulness of the results.

Barthlomew [8] has discussed the form of subdividing the

population into categories, the 'stayers' who hardly change their jobs and

'movers' who tend to change jobs frequently, while Barthlomew and

Forbes [11] have developed a more specific application of the

principles, to the manpower planning problem. A basic model defines a

number of discrete manpower grades, with employees advancing or

leaving with fixed transition probabilities. The state of the system is

defined as the number of employees in each grade and the system is

analysed as a Markov chain.

Lesson [49] has considered the recruitment policies and their

effects on internal structures. Recruitment control refers to an effective

control of recruitment policies to obtain an optimal supply of manpower

for a system at any time. Generally recruitment levels are connected with

wastage and promotions in a system as well as the desired growth of the

system, hence controlling recruitment policies may help to attain a

desired structure, which could be maintained over a time.

38

The paper by Agrafotis [3] is on analysis of wastage and is worth

mentioning because of the deviation of this model from the conventional

models relating to the analysis of wastage in manpower system. The

author of this paper has presented a model which is designed to

investigate the effect on wastage of the internal structure of the company

and the promotion experience of its staff. Also a stochastic model has

been developed to depict the probability of the number of promotions to

an employee in the interval (0, t]. The estimation method for calculating

the probabilities is also discussed.

Mukherjee and Chattopadhayay [58] have discussed an optimal

recruitment policy. The authors have considered an organization in which

number of persons are recruited at time t . Every recruited person can be

in service for ' t ' years at the most. It is also assumed that the efficiency

of each recruited person is adversely affected by the longer duration of

service. The authors have derived a recruitment policy at interval of time

t . The optimal values of ' t ' which minimizes the total cost of unified

vacancies and forced recruitment have been worked out.

Gardner [28] has presented a research study on exponential

smoothing where its historical development was traced to the time of

second World War. The research critically commented on the merits of

various models and deferred with others based on his research as well as

39

research of other researchers. In the conclusion exponential smoothing

technique was also rated as one of the best forecasting methods.

An optimal planning of manpower training programmers has been

analyzed by Goh et. al [29]. Two different models are discussed in this

paper by assuming the finite planning period and an infinite planning

period. A finite slate Markov chain is used to model the manpower state

for the finite planning period and the optimum solution is computed

using the dynamic programming technique. A non-linear integer

programming problem is used to model the manpower state for a finite

planning period.

A model responding to promotion blockage is discussed by

Kalamatianou [42]. This model is proposed for manpower system in

which promotion probabilities are functions of seniority within grades.

Poornachandra Rao [63] has made attempts to identify various

costs associated with manpower planning system. Based on this a

manpower planning model with the objective of minimizing the

manpower system costs was formulated. The major limitation of the

model is the consideration of manpower costs in isolation of various

constraints and operating policies under which a manpower system

operates. The model proposed an integrated model which will minimize

40

the manpower costs in the presence of the system constraints and other

operating policies.

Raghavendra [64] has discussed a bivariate for manpower planning

system. The author indicates that in many developing countries there is

limited mobility of people from one organization to another. This results

in policies of promotion and recruitment, which will have long term

effects on the organization. It is also observed that in many organizations

especially in the public sector undertaking two types of policies on

promotions are followed (1 ) promotion by seniority ( length of service ),

(ii) promotion by performance rating. The author has taken the two

aspects in a bivariate frame work. So a Markov chain model is developed

to derive the estimates of the transition probabilities. It is also shown that

in presence of organizational objectives the promotional policies can be

translated in to respective levels in terms of either seniority or

performance rating or a combination of both. The author has obtained the

joint probability distributions of the two random variables namely x is the

seniority and y is the performance rating; the marginal distributions are

also obtained. The minimum level of seniority required for promotion

and the minimum level of performance rating required for promotion

have been estimated. Numerical examples are also furnished to explain

the usefulness of the model.

41

According to Barthlomew et al, [11] the analysis of individual

difference is of fundamental importance in the study of manpower

system, in particular, wastage (loss from the system). Any attempt to

describe wastage pattern must reckon with the fact an individual's

propensity to leave a job depends on many factors, both personal and

environmental.

McClean [52] has discussed a Mathematical model using which

provides a method to predict the growth of manpower needs of the

Northern Ireland Software Industry. Northern Ireland has been in an

enviable position regarding the quality and quantity of manpower in its

software industry. The growth of the software industry in turn results in

increased demand for manpower in IT industries. So the author has taken

two groups of staff (i) those who are working in IT firms (ii) those who

are working in firms which are IT users. Growth in the demand for

software personnel in IT firms depends upon the developments achieved

in software industry. The growth of manpower in IT firms will be

primarily dependent on factors specific to software sectors. The author

has used a transition model based on Markov analysis and using the

model the author has prepared a formula for predicting the demand for

manpower in the IT industry. This model will be very useful to take some

steps well in advance so that the training and recruitment could be

42

suitably modified and planned. The author has given the projection

figures for a period of 5 years in the future and it would give an idea of

the need for training sufficient number of personnel well in advance.

Subramaniam [77] has studied the optimal time for the withdrawal

of the voluntary retirement scheme in manpower planning. A period of

length T years is considered during which the employees are permitted to

go on voluntary retirement on k selected epochs. As and when the staff

strength reaches a level, which is called threshold level the voluntary

retirement scheme is withdrawn because the staff strength reaches the

required level. The optimal strength of T years which is numerically

illustrated with graphs is obtained based on certain assumptions.

An analytical model which deals with finding the size of each grade

in a hierarchical organization has been developed by Kenway et. al [43].

The size of each grade and proportion of employees in each grade are

derived using a number of assumptions regarding the demand in that

organization, the wastage, rate of retirement, etc. The concept of

promotion is also taken into account. It is assumed that the demand for

manpower increases exponentially at a rate 'p' The wastage rate ,w x t

for employees of age x at a time ' t ' is defined as the proportion ,w x t

t of staff at age x at time t that leaves the organization tt t N N t

43

is the total size of the system at time ,t p t is taken to be the proportion

of the employees in the top grade. R t 5t denotes the total number of

recruits in the interval tt t . A constraint taken up is

t

op t xt g xt dt . Where ,f x t is defined as the population of all

employees age x at time ' 't who have been promoted to the top grade

and ,g x t is the age distribution of all employees at time t . The

constraint that demotion is not permitted is represented as , f x a t a

for all ,x t and a . Using all the above the authors have obtained a model

for the sizes of the different grades at future point of time ' '.t

Thio [80] has discussed the need for retention strategies especially

in turbulent times. According to the author it is commonly assumed that

the retention of staff in an organization is an indicator of the health of the

organization, as well as its unhealthy state. But it is difficult to establish

the links between attrition and unhealthy state. The author has made an

explanatory attempt of synthesizing some ideas about attrition and

retention. To some extent the staff turnover is inevitable and cannot be

beneficial. The process of attrition makes way for the recruitment of new

blood and also facilitates the career progression of those who remain in

the organization. However high and unexpected turnover can be a

reflection of negative job attitudes and low staff morale. It may warrant

44

counter measures. Remedial measures are necessary to manage attrition

in a way that causes least dislocation to the work of the organization.

The author has also indicated that the reduced staff strength due to

attrition will result in a loss of customers or clients to the organization.

The author has taken up the issue of attrition with particular reference to

certain welfare organizations in Singapore. The author has reiterated

certain retention strategies such as recognition of work turnover by the

staff as a key measure which helps in the management of attrition. The

management must provide participative opportunities available for the

staff. It must develop a congenial environment to work for the staff.

There must be informal communication among the staff, job rotation and

recognition for work turnover, etc. Retention of key staff requires

continuing leadership influence and management affiliation. Even though

attrition is not always harmful, it should not be dealt passively.

Estimation for semi-Markov manpower models in a stochastic

environment has been discussed by McClean and Montgomery' [53].

Manpower planning, often needs to estimate and predict distributions of

duration in various grades in the hierarchical set up of a company. A

methodology for fitting a stochastic environment to manpower data for

both the non-homogeneous semi-Markov system in a stochastic

environment (S-NHMS) and the non-homogeneous semi-Markov system

45

in a stochastic environment (S-NHSMS) is discussed in this paper. These

models provide means of describing changes in the environment between

contiguous periods, in particular homogeneous time period and the

process governing movements between such periods. Thus providing a

description of time heterogeneity. The predictions of future movements

of staff through the system are made.

Koley [44] has brought out the importance of human resource

investments in order to place any organization in a comfortable position

and on the appreciating track. This paper suggests that to build up the

human resource, investments on employee recruitments, training and

development besides cost arising due to wastages and salaries may be

used for decision making purpose. It suggests the use of available tools

and techniques like, works study, learning curve, activity based costing,

decision tree and risk analysis, life cycle cost approach to assess the cost

of making managers as investments. A company where there is no dearth

of qualified manpower can be one among the richest in the world to build

up the manpower.

In many organizations, the number of sanctioned positions may be

vacant year after year. Huge amount is spent by many organizations for

the requirement of specialists as well as training and development of such

persons. The recruitments process involves locating the right type of

46

candidates from inside and outside the organization through interval

circulars, external advertisements etc.

The author suggests that the expenditure on the projects on HR

development and related activities should be carefully decided so that the

control of cost over HR can be decided by using PERT and CPM

methods. It is also necessary to measure the HR productivity.

It becomes vital to decide the effectiveness of various strategic

moves of human resource managers from time to time. The build up of

HR costs and investment figures is not to put the man on the balance

sheet but to use those for decision making purposes.

Gupta and Kundu [32] have studied some properties of a new

family of distributions, namely Exponentiated Exponential distribution.

The Exponentiated Exponential family has two parameters (sales and

shape) similar to a Weibull or a Gamma family. It is observed that many

properties of this family are quite similar to those of a Weibull or a

Gamma family; therefore this distribution can be used as a possible

alternative to a Weibull or a Gamma distribution. Some numerical

experiments are performed to see how the maximum likelihood

estimators and their asymptotic results work for finite sample sizes.

47

Sathyamoorthy et al [70] have discussed a manpower model for

estimating the propensity to leave the primary job. In this paper they have

discussed the method of deriving the propensity of individuals to leave

the primary job in an organization which leads to attrition. Cox's

regression approach has been used to derive the level of propensity of an

individual in a primary job in an organization. The authors have taken up

the exponential distribution as a special case to estimate the propensity to

leave the organization. The specialists holding the prmary job have some

covariates of personnel character. These covariates also contribute to the

intensity or degree of propensity to leave the job. In addition to the

degree of propensity generated by the Completed Length of Service

(CLS) in the organization, the covariates also contribute and hence the

combined influence of both namely CLS and the covariates decide the

degree of propensity to leave the job.

Sathiyamoorthy and Parthsarathy [68] have considered a two grade

organization in which the mobility of personnel from one grade to the

other is permitted as to compensate the loss of manpower which is larger

among the two grades. They have considered the case in which the Max

2|Y Y is taken to be the threshold level of the organization where iY

andY2 are the individual thresholds of the grades respectively. They have

48

obtained an expression for the expected time for recruitment in a two

grade organization.

Sathiyamoorthy and Parthasarathy [69] have used the idea of

change of prarmeter for the threshold distribution after the truncation

point. This idea is similar to SCBZ property where the parameter

undergoes a change. Assuming the truncation point is itself a random

variable following exponential distribution, which is taken for the

threshold level. The expected time for recruitment is also obtained using

the shock model approach and the results are compared when there is no

change of parameters for the threshold distribution.

Charles et al., [16] have examined the interaction effects of

maintenance policies on batch plant scheduling in a semiconductor water

fabrication facility. The purpose of the work is the improvement of the

quality of maintenance department activities by the implementation of

optimized preventive maintenance strategies and comes within the scope

of total productivity maintenance strategy.

The production of semiconductor devices is carried out in a water

lab. In this production environment equipment breakdown or procedure

drifting usually induces unscheduled production interruptions.

Jeeva et al., [41] have discussed frequent wastage or exit of

personnel common in many administrative and production oriented

49

organizations. Once the accumulated number of exits from the

organization reaches a certain threshold level, it could be viewed as a

"breakdown point". The time to attain breakdown point is an important

characteristic for the management of the organization. A shock model

approach is proposed to obtain the expectation and variance of the time to

attain the threshold level.

Elangovan et al., [21] have discussed a model using which the

optimal level of hiring expertise service in manpower planning has been

discussed. It has been assumed that the cost of hiring experts in some

chosen areas of human activity is fairly high and hence it would be

advantageous to use the Mathematical methods to find the optimal

duration for which the hiring of experts in terms of man hours. If the

number of hours of contract is more than the requirements it would be a

financial loss. Again if the number of hours of contract is below the

requirements or demand it would prove to be financial loss due to

shortages. Hence taking the demand into consideration the exact or

optimal size of number of hours of purchase is determined. In doing so

the demand for man hours of expert service is assumed to be a random

variable and is following the so called exponential distribution.

Appropriate costs of excess manpower as well shortage costs are

assumed in obtaining the optimal solution. Another interesting variation

50

in this model is also discussed in this paper. It is further assumed that the

demand for expert manpower undergoes change from time to time. Also

if the demand for expert manpower is beyond a particular level then the

cost of hiring also undergoes a change. All these modifications and

additional assumptions make this model very much in agreement with

real life situations. For each kind of the situation that arises in practice

the optimal policy for hiring expertise has been obtained by the authors.

Numerical examples of different types are taken up and the situation of

optimal types are derived and the graphs are provided.

Sureshkumar [78] has developed a stochastic model in which the

prediction of the likely time to recruitment due to the depletion of

manpower in a two grade organization. The manpower planning studies

about depletion of manpower due to leaving of personnel, known as

attrition. This is also called as 'wastage' The attrition takes place on

successive occasions of policy decisions regarding pay revision,

perquisites and when targets are fixed. The recruitment is not taken up on

every occasion of attrition, but the deficiency in manpower is managed

by transfer of persons from one grade to the other where the attrition is

more pronounced. The authors have also introduced the concept of

threshold level for cumulative attrition. If the attrition or wastage crosses

the threshold level then the recruitment has to be done. This is with a

51

view to reduce the cost of frequent recruitments. The various costs

involved in recruitment are discussed in detail by Poornachandra Rao

[63]. The expected time to recruitment is predicted assuming the wastage

as random variable on successive occasions of attrition. The authors have

discussed two such models. In the first model the transfer of personnel

from one grade to another is permitted. In the second model the transfer

is not permitted. These mathematical models serve as projection

techniques so that the management can adopt suitable strategies to

contain the level of attrition and also decide suitable policies to deal with

the consequences. Numerical illustrations are also provided to support the

findings.

Anantharaj et. al [4] have discussed on the method of arriving at

the optimal time intervals between recruitments. When attrition takes

place on successive occasions over a period of time and cumulative

attrition when reaches a particular level called the threshold the

recruitment becomes necessary to make up the loss of manpower.

Recruitments very often are not advisable since it involves costs of

several nature. Hence the determination of the optimal time interval

between recruitments, become necessary. In deriving the optimal time

periods between the recruitments, the authors have used the shock model

and cumulative damage process approach. The cost arising due to the

settlement of gratuity and other compensation packages, the cost arising

52

due to breakdown of regular work schedule are all taken care of in

obtaining the optimal solution for the problem. Numerical illustrations

have been provided on the basis of simulated data to prove the validity of

application of this model and also the behaviour of the optimal solution

obtained when changes take place in the influencing variables.

Laura Roe [47] has indicated that a twenty five percent IT industry

average turnover rate persists all over. This requires the recruitment and

hiring of same number of employees to make up the gap. Some

suggestions as how to improve the retention rates are suggested by the

author. Many of them may look impossible, but are critically important

from the view of retention strategies. Some suitable strategies suggested

for promoting retention are:

The hiring process should be as good as possible, since retention

starts with good hiring process.

The technical staff should be motivated and told how project is, to

their team and company.

Training is important factor that contributes to the retention.

Technicians especially those at the risk of leaving should be

assigned technically challenging tasks.

53

If necessary internal assignments should be assigned to use leading

edge technologies.

Managers and IT staffer relationship have a profound effect on

retention. Provision of high quality work place and corporate atmosphere.

In addition to the above the author has suggested a number of other

retention strategies.

Srinivasan and Sudha [76] have considered four grades

organizations with policy of recruitment. The mean and variance of time

to recruitment are derived by assuming random threshold following non

identical exponential distribution for each grade and the threshold for the

organization as the minimum of four thresholds.

Mallikarjunan [51] has given an overview of the causes and

remedies for employee attrition. According to the author employee

attrition is caused not only by natural inevitabilities like disability, death,

retirement and resignation, but also by the burgeoning mobility of human

resources or the human capital. One of the toughest problems that

confront HR managers is employee attrition. Due to the vertical growth of

the business. Process, services and products, skilled and even semi skilled

workers find a matrix of possible avenues for self development.

54

The author has indicated that the nature of the business, the nature

of responsibility shouldered by them is the reasons for the attrition of

employees, This is very much relevant to the software industry. He also

indicated that the employee attrition can be classified into two categories

namely (i) drive attrition, which is caused by the policy practice and

treatment of the employer in the industry,(ii) Drag attrition as a result of a

number of uncertainties faced by the employee in the working

environment such as absence of opportunity for advancement in career,

absence of opportunity to achieve mental and functional growth.

A few industries have been found to be in the constant threat of this

syndromes of attrition and they are Information Technology and

Hardware Industry. Information Technology and Software Development

Industry Call Centres. Business process outsourcing industry. Other

industries like pharmaceuticals, manufacturing, etc.

Arivazhagan et al [5] have developed a mathematical model which

can be used to estimate the likely time at which the enrolment for

recruitment should be stopped. According to the authors these are many

organizations which are providing service in the HRM sector. The supply

of skilled laborers and specialists is one of the main areas of activities in

such organizations. They keep a reserve or inventory of skilled personnel

and whenever there is demand or request from organizations or industries

55

the supply is the main activity. The enrolment has to be stopped at a

particular level called optimal enrolment. Assuming that the allowable

level of enrolment as the threshold level the expected time to stop

enrolment has been found out. Numerical illustrations are also given.

Suvro Ray Chaudhuri [79] has given a detailed account of

employee attrition and the methods of predicting the attrition rate and

also the strategy for mitigation of the rate of attrition. According to the

author the manpower attrition is similar to the customer switching

problems in the case of products. The author has used the Markov

analysis to predict attrition. Human resource professionals are under

increased pressure from a different kind of a corporate problem which

causes no less harm to human capital assets. The American Productivity

and Quality Centre (APQC), has made three different categories of

knowledge that suffer due to attrition. They are as follows:

Cultural knowledge. This includes management practices, values,

respect for hierarchy, and decision flows.

Historical knowledge - This includes the organization's journey

from the day it was founded till the present.

Functional knowledge - This includes technical, operational.

Process and client information.

56

From the organization point of view the counter strategy is to

predict attrition zones which depend on the critically or type of

knowledge, that is important to organization and there by evolve plans to

counter loss of human assets from those positions. The attrition is one of

the main areas in the field of knowledge management because it is easier

to fill up any position by recruitment but filling the knowledge gap is not

easy. The author says that the organizations have to spend huge sums of

money on recruitment. This is due to the fact that the functional

knowledge of the new persons may not be equal to that of the person who

has left the organization. The author has used the Markov Analysis by

taking the employees as internal customers. The purpose of Markov

Analysis is to predict the rate of attrition based on the present data. The

transition probability matrix which is a basic tool in Markov Analysis is

also discussed and the solution is derived to predict the future rate of

attrition from the organization.

It is quite reasonable to think that recruitment cannot be done as

and when manpower leaves, a threshold can be kept upto which loss of

manpower can be allowed, after then the recruitment can be done. This

idea of recruitment to start after the depletion of manpower reaching a

threshold is brought in the mathematical model of R. Elangovan, R.

Sathyamoorthy and E. Susiganeshkumar[22]. The expected time to

57

recruitment is derived in this model As the exit of personnels is

unpredictable, J. B. Esther Clara and A. S. Srinivasan [24] have

constructed a mathematical model of new bivariate recruitment policy

involving optional and mandatory thresholds for the loss of manpower in

the organization, based on shock model approach and cumulative damage

process to enable the organization to plan its decision on recruitment.

Assuming different distributions for optional and mandatory thresholds,

expected time to recruitment is obtained.

Based on shock model approach, a mathematical model is

constructed by J.B.Esther Clara and A.Srinivasan [25] using an

appropriate univariate max policy of recruitment and an analytical

expression for the mean time to recruitment is obtained under suitable

conditions on the loss of manhours, inter-decision times and thresholds.

Ishwarya. G., Mariappanan. P and Srinivasan .A. [37] in their

paper have discussed the problem of time to recruit when the thresholds

follow extented exponential distibution with shape parameter 2 which is

more general than exponential distribution.

M. Jeeva and Fernandes Jayashree Felix [41] have developed a

manpower model where the recruitment process follows a pre-emptive

repeat priority service discipline. The transient behaviour of the

applicants waiting for the recruitment process is discussed in this model

58

and mean and variance of the high priority and low priority applicants are

determined.

When the Manpower System of an organization is exposed to

Cumulative Shortage Process due to attritions that cause manpower loss,

breakdown occurs at threshold level. S.Mythili and R.Ramanarayanan

[60], in this paper have considered the Manpower System with two

groups A and B. Group A consists of man-power other than top

management level executives. Group B consists of top management level

executives. Group A is exposed to cumulative shortage process and

shortage process of group B has varying shortage rates. Recruitment is

done to all the shortages of the two groups. The expected lime to recruit

and recruitment time are determined.

S.Mythili and R.Ramanarayanan [59], have considered manpower

system of an organization for which Attrition Reduction Strategy (ARS)

is applied prior to recruitment. In this paper recruitment policy of filling

vacancies one by one and parallel recruitment policy of filling vacancies

simultaneously are considered.

Nabendu Sen and Manish Nandi [61] have formulated a strategic

planning using the Goal Programming approach to Rubber Plantation

Planning in Tripura

59

A project of a company requires processing in several stages for

completion of the same. K.Usha, A.C.Tamil Selvi and R.Ramanarayanan

[81] have considered n intermittent stages and the project visits and

revisits these n stages before it moves to the completion stage 1n . The

probability generating function of the number of paths of specific type

namely x to y the project executes before completion is determined. Its

expected value and the variance are found. Here every path x to y is

treated as a recruitment of a batch of employees. K. Usha,

P.Leelathilagam and R.Ramanarayanan [82] have Considered a project of

a company with n intermittent stages. The project visits and revisits these

stages before it enters the completion stage 1n . Five stages have been

considered where one stage indicates changes in company policy for

manpower, one stage indicates project modification for manpower, one

stage indicates shortage of manpower, one stage indicates recruitment of

manpower and one stage indicates training of manpower. In this paper, is

obtained the probability generating function of the number of pentagonal

loops the project forms in the respective stages before completion. Its

expected value and the variance are determined.

Vinod Kumar Mishra and Lai Sahab Singh [88], in this paper, a

deterministic inventory model is developed for deteriorating items in

which shortages are allowed and partially backlogged. Deterioration rate

60

is constant. Demand rate is linear function of time, backlogging rate is

variable and is dependent on the length of the next replenishment. The

model is solved analytically by minimizing the total inventory cost.

61

CHAPTER 3

MANPOWER PLANNING PROCESS WITH TWO GROUPS

USING STATISTICAL TECHNIQUE

3.1 INTRODUCTION

Employees are the most important asset for a business. They serve

to create or promote an organization's culture, and they significantly

affect the success of a business. In challenging economic times, the cost

of hiring inefficient personnel may prove to be detrimental to the

profitability of an organization. An effective and thorough manpower-

recruiting process requires an employer to carefully choose the most

talented employees who will positively benefit the organization or

business. A needs analysis initiates the manpower recruiting process. This

phase entails identifying a vacant position or creating one to meet new

needs that have arisen in the organization. this may be an entry mid- or

upper-level management position. The employer then develops a job

description describing the duties involved with this position. Criteria such

as skills and competencies, experience, age, and education that best serve

the position are also identified. Using this information, the employer

prepares a standard application form to collate information provided by

the applicants, in addition to their own resumes. The vacancy is then

advertised.

62

3.2 DESCRIPTION AND ANALYSIS OF THE MODEL

In this sectiona two graded manpower system is considered and the

description of the model is given below.

Assumptions

1. Group A is given at the most k observation times each with

exponential distribution with parameter ' ' before recruitment. On

completion of the first exponential observation time, recruitment is

done with probability , or, the second observation starts with

probability , where 1. The process is repeated upto i

observations for 1 1.i k On completion of the thk observation,

recruitment is done with probability 1 . If 1T is the time to recruit due

to group A and 1 2; , ..X X are the shortages caused by manpower

loss in group A, then, 1

1

k

j

j

T X

with probability 1k for 1 1i k

or, 1

1

k

j

j

T X

with probability 1k Group B has shortage process with

varying shortage rates. At time 0, the shortage rate of the group is .

Let 2 be the time at which breakdown of group B occurs

necessitating immediate recruitment.

2. Recruitment for MPP starts if either of the groups A or B has a

breakdown. All the shortages due to manpower loss are compensated

by recruitment.

63

3. When recruitment is done due to breakdown of group A, recruitment

time corresponding to the thi observation is ,iR 1 i k When the

breakdown occurs due to group B, recruitment is done for shortages in

group B and also for shortages in group A for the number of

observations completed. All the recruitment times are independent and

identically and distributed random variables with distribution function

R y such that

( )

y

o

ydR y .

3.3 MAIN RESULT

Based on the assumptions, recruitment starts at time

1 2min ,T T T . Identifying the exponential phase time of the

modified Erlangian, the pdf of time 1T is given by

12

1

0

( )! 1 !

i kk

x k x

i

i

x xf x e e

i k

(3.1)

2

,P T x Rt yx y

2( )1 ( ) ( ) ( )

1!

x xxH x e r y e r y

22 3( )

( ) ....2!

xxe r y

2 12 ( 1) 1( ) ( )

( ) ( )( 2)! ( 1)!

k kx k k x k kx x

e r y e r yk k

(3.2)

64

The first term corresponds to breakdown due to group A and the

second terms corresponds to breakdown due to group B Using (3.1) and

(3.2), we get,

2

,P T x Rt yx y

1 2

21*

0 !

ik

ix x x i

i

xe e e r y

i

1 2

1

1 *

1 !

k

x x x k kxe e e r y

k

1 2

1* 1

1 2

0 !

ik

ix x x i

i

xe e e r y

i

(3.3)

(3.3) can be simplified by taking Double Laplace transform.

T RtE e e

1

1 1

1 1 1

( ) ( ) ( )

( )

k k

r r r

r

1

2 2

2 2 2

( ) ( ) ( )

( )

k k

r r r

r

1 1

1 2

( ) ( ).

k k

k kr r

(3.4)

for 0 and 0 , we obtain from (3.6)

( )tE e

1 1

1 2

1 2

1 1

k k

65

1 1

1 2

k k

k k

1 2

1 2

1 2

1 1

.

k k

(3.5)

RtE e

1

1 1

1 1 1

( ) ( ) ( )

( )

k k

r r r

r

1

2 2

2 2 2

( ) ( ) ( )

( )

k k

r r r

r

1 1

1 2

( ) ( )k k

k kr r

(3.6)

From (5) and (6), by differentiating,

1 1 2 2

1 1( ) 1 1 .

k k

E T

(3.7)

1 1

1 1 2 2

( ) ( ) 1 1 1

k k

E Rt E R

(3.8)

3.4 NUMERICAL ILLUSTRATION

By giving different values to the parameters in E T and E Rt

by varying λ from 1 to 10, we present the graphs of E T and E Rt .

66

Table 3.1

1 0.07, 2 0.05, 0.4, 0.6, 2,k ( ) 3E R

1 2 3 4 5 6 7 8 9 10

E T 3.479 1.923 1.320 1.003 0.809 0.678 0.583 0.512 0.456 0.411

E Rt 2.71 3.32 3.45 3.35 3.29 3.25 3.22 3.19 3.17 3.16

67

Table 3.2

1 0.07, 2 0.05, 0.6, 0.4, 2,k ( ) 3E R

1 2 3 4 5 6 7 8 9 10

E T 2.876 1.566 1.069 0.810 0.652 0.546 0.469 0.411 0.366 0.330

E Rt 2.640 2.500 2.431 2.385 2.350 2.321 2.298 2.278 2.260 2.245

3.5 CONCLUSION

From tables 3.1 and 3.2, we observe the behavior of E T and E Rt

i.e., mean time to recruit and mean Recruitment time for fixed values of

1 2, , , ,k and E R . When the parameter λ increases, the value of

E T increases and E Rt decreases. When α increases, both E T and

E Rt decrease.

68

CHAPTER 4

OPTIMAL MANPOWER RECRUITMENT AND PROMOTION

POLICIES FOR THE TWO GRADE SYSTEM USING

DYNAMIC PROGRAMMING APPROACH

4.1 INTRODUCTION

Bartholomew and Forbes (1979) have described the state of the art

I n various facts of manpower planning. Edwards (1983) has surveyed

various models on their assumption and application and concluded that

good presentations of results are more important than theoretical

sophistications. Price and Piskor (1972) have developed Goal

programming model of manpower planning system for financial,

manning, promotion and manpower accounting.

Zanakis and Maret (1981) have formulated a Markovian goal

programming model with pre-emptive priorities and provided a more

flexible and realistic tool for manpower planning problems. Mehlmann

(1980) has developed optimal recruitment and transition strategies for

manpower systems using dynamic programming. He has formulated a

dynamic programming recursion with the objective of minimizing a

quadratic penalty function which reflects the importance of correct

manning of each grade under preferred recruitment and transition

patterns.

69

While the models developed in the manpower planning literature

have considered financial and labor costs and the system and resulted in

the form of recursive optimization. A dynamic programming model has

been found to be analogous to the Wagner–Whitin (1958) model, based

on the cost data, it generates the optimal recruitment and promotion

schedules for future periods.

4.2 MANPOWER SYSTEM COSTS

Manpower system costs depend upon the various factors. The

various costs associated with manpower system consist of the following:

i) Recruitment and Promotion costs

ii) Overstaffing costs Understaffing costs Retention costs Wastage

costs.

iii) Recruitment and promotions costs

iv) Cost of advertising

v) Cost of conducting written test

vi) Cost of information processing

vii) Cost of manpower working on the processing of application

viii) Cost of administrative authority which determines recruitment

and promotion policies

ix) Costs incurred in the form of payment to the interview

committee members or the wages of the people on the interview

committee.

70

x) T.A paid to the candidates which is optional.

xi) Cost of medical examination done by the organization

xii) Cost of training people

xiii) Miscellaneous expenditure, including postage, telephone calls etc.

The actual components of recruitment and promotion cost depends

upon the procedure followed by the organization for recruitment while

the above components are indicative only. Even though the charges are

paid by applicants for processing, it is not proportional to the

actual recruitment and promotion costs met by many organizations.

The cost of advertising and cost of administrative authority from a

fixed component is independent of people recruited or promoted based

on the suitability of the candidates.

The costs like traveling expenses are paid to interviewing people

and also depend on the policy of each organization in determining the

number of candidates to be interviewed. According to management’s

policy, if the people to be called are a predetermined ratio, which is

proportional to number of candidates selected or interviewed and

remains constant.

A fixed and a variable component per recruiter or promoter is

applicable for all the costs like conducting written test, manpower

working on the processing of applications, Information processing,

71

medical examination and training the people. The fixed costs are higher

if the selection process is in groups like military recruitment process.

(a) Overstaffing Costs

Overstaffing costs are those incurred owing to an unutilized

workforce. These costs are analogous to the inventory costs in a

production / inventory situation.

(b) Understaffing Costs

Understaffing costs are those resulting from decreased

productivity and loss of goodwill (in a profit-motive organization) as a

result of the non-availability of the workforce.

(c) Wastage Costs

The costs result from the retrenchment or retirement of the

employee.

(d) Retention Costs

There are certain costs which are involved in retaining an

employee in an organization. These costs consist of (i) probation costs,

(ii) training and development costs, and (iii) internal mobility costs.

Probation costs are those incurred owing to the learning effect of

an employee during a probationary period. The training and

development costs are different from the recruitment costs and are

72

incurred owing to the development programmes which an employee

undergoes during the course of his service to the organization. Internal

mobility costs are the costs involved in demotion or transfer of an

employee within the organization.

4.3 MATHEMATICAL MODEL

The following assumptions are made while formulating the

manpower planning problem to determine optimal recruitment and

promotion policies:

The recruitment and promotion size are known and fixed.

Recruitment and promotion at a particular grade is considered.

Recruitment, promotion and overstaffing costs are known and fixed.

Understaffing is not allowed in both the grades.

Notations

1. :R t Recruitment in any period .t

:S t Fixed recruitment cost in period .t

:P t Promotion at any period .t

:Q t Cost of promotion / period.

:i t Cost of overstaffing per recruiter or promoter per period.

:l t Number of people recruited / promoted in an earlier

period for the requirements of period .t

1 :x t Number of people recruited in period t at Grade 1.

2 :x t Number of people recruited in period t at Grade.

73

2. :y t Number of people promoted in period t from Grade 1 to

Grade 2.

1 :v Variable cost of recruitment at Grade 1 / employee

recruited.

2 :v Recruitment at Grade 2 / employee recruited.

:u Variable cost of promotion at Grade 1 to Grade 2.

Overstaffing cost not allowed for Grade 2 since it was for higher

level, not necessary. Since, we need to satisfy all requirements on time,

so that understaffing is prohibited.

The requirement cost in period t is given by the conncave function:

1 , 1,2.i iS t x t v x t i (4.1)

The promotional cost in period t is given by:

Q t y t uy t (4.2)

The overstaffing cost is i t I t . The total cost of recruitment

for the T — period planning interval is:

1

T

i i i

i

S t x t v x t i t I t

(4.3)

The total cost of promotion for the T — period planning interval is:

1

T

t

Q t y t uy t i t I t

(4.4)

Thus the total cost of recruitment and promotion for the T — period

planning interval is:

74

1

, 1,2.T

i i i

i

S t x t Q t y t v x t uy t i

(4.5)

Here we take 0 0 0i l without loss of generality.

The problem is to minimize this sum, subject to the constraint that

all recruitments and promotions must be met on time, and since the

variable cost of recruitment and promotions are constant. we have that

are constant in (3)

1

T

i

t

Vv R t

and 1

T

t

u P t

are constants

Thus the problem may therefore be stated as Minimize

1

, 1,2.T

i

i

S t x t Q t y t i t I t i

(4.6)

Subject to

1 1

, 1,2,... : 1,2t t

ik k

k k

x R t T i

and

1 1

, 1,2,...t t

k k

k k

y P t T

4.4 DYNAMIC PROGRAMMING FORMULATION

Theorem

The well-known Wagner—Whit in model is characterized to

determine economic lot size with this model. The fixed recruitment

and promotion cost is analogous to the set-up cost and the overstaffing

75

cost is analogous to the cost of carrying inventory in an inventory

system.

The propositions of Wagner — Whitin model, which facilitate

formulation of Dynamic programming recursion are thus given.

Theorem 1

There exist an optimal program such that:

0iI t x t for all t and 1,2.i

Theorem 2

The minimum cost policy has the property that the recruitment

cost x takes the values 0, , 1 ,...,( ) ( ) ( ) ( ) ( ) ( ) 1 ... R t R t R t R t R t R T

and the promotion cost y takes the values ( ) ( )0, , 1 ,...,( )P t P t P t

( ) ( )1 ... ( ).P t P t P T

Table 4.1

Hypothetical Data for a 5 Year Planning Period of a Manpower System

Year R P S in 000’s Q in 000’s I in 000’s

1 79 41 728 540 15

2 34 10 705 220 12

3 52 14 698 385 16

4 61 38 714 412 14

5 25 8 708 398 16

76

Theorem 3

There exist an optimal program such that if R is satisfied by

some **x t and P is satisfied by some ** , ** *,y t t t then R t

and , ** 1,..( ) / ., * 1P t t t t are also satisfied by **x t and **y t .

Theorem 4

Given that 0I t for period t , it is optimal to consider periods

1 to 1t by themselves. Let F t denote the minimal cost program

for periods 1 to t, then:

F t 1

11

min 1 .,t t

j tn j k h

S j Q j i h R k P k F j

1S t Q t F t (4.7)

The above recursion, stated in words, means that the minimum

cost for the first ‘t’ periods comprised a fixed recruitment and promotion

cost in period ,j plus the charges for satisfying requirements R k and

promotion , 1,...,( )P k k j t by recruiting and promoting manpower in

period ,j which results in overstaffing cost, plus the cost of adopting an

optimal policy in periods 1 to 1j taken by themselves. We state

below the manpower planning horizon theorem analogous to the

Wagner — Whitin planning horizon theorem which further amplifies

determination of optimal policies.

77

t

4.5 THE MANPOWER PLANNING HORIZON THEOREM

If the minimum in (1) occurs for ** *j t t at any period t, then

in periods *t t it is sufficient to consider only ** .t j t If * **t t

then it is sufficient to consider programmes such that ( )* 0x t

and ( )* 0y t . Wagner—Whitin algorithm can be made use of to

determine the optimal recruitment and promotion policies. The

algorithm at period *, * 1,2,...t t N may be stated as:

Consider the policies of recruiting and promoting at (period

**, ** 1,2,.. . *.t t t

The total cost of these t* different policies by adding the

fixed recruitment cost, promotion cost and overstaffing costs

associated with the recruitment and promotion at period t** and the

cost of acting optimally for periods 1 to ** 1 t considered by

themselves. The latter cost has been determined previously in the

computations for periods 1,2, * 1.t t

(3) From the t* alternatives, select the minimum cost policy for

periods 1 to t* considered independently.

(4) Proceed the process to period * 1t or stop if * .t N

78

Table 4.2

The Calculations of the Manpower Planning Problem Presented

Year 1 2 3 4 5

S 728 705 698 714 708

Q 540 220 385 412 398

i 15 12 16 14 16

R 79 34 52 61 25

P 41 10 14 38 8

1268 1928* 3011 3846* 4952

2193 2720* 5690 4308*

4595

Minimum cost 1268 1928 2720 3846 4308

Optimum policy 1 2 2.3 4 5


Table 4.1 shows the hypothetical data for a 5 year planning period

of a manpower system. Table 4.2 summarizes the calculations of the

manpower planning presented in Table 4.1. Thus the optimal policy may

be stated as follows: Recruit and promote in period 4,

4 4 86 46 132x y and use the optimal policy for periods 1 to 4,

implying (2) Recruit and promote in period 2, 2 2 86 24 100x y and

use the optimal policy for periods 1 to 2, implying (3) Recruit and

promote in period 1, 1 1 79 41 120.x y The total cost of this policy is

4308.

79

4.7 CONCLUSIONS

In this paper an attempt has been made to obtain the optimal

number of recruits and promotions made so that the total cost incurred is

minimum in the manpower planning system along with the various costs

like recruitment costs, promotion costs, overstaffing costs, wastage costs

and retention costs. There are two types of cost have been taken into

account namely fixed and variable costs. The model has been found to

be analogous to the Wagner-Whitin model in a production or inventory

situation. The major limitation of the model is the fact that it is

considered in isolation from the various constraints and operating

policies under which a manpower system operates. As another constraint

of the model is that, it is assumed that there is no overstaffing in the

higher grade. This model can also be discussed without this constraint as

further work.

80

CHAPTER 5

EXPECTED TIME FOR RECRUITMENT IN A TWO GRADED

MANPOWER SYSTEM ASSOCIATED WITH CORRELATED

INTER-DECISION TIMES – A SHOCK MODEL APPROACH

5.1 INTRODUCTION

For a single graded manpower system, Sathiyamoorthy and

Elangovan (1998) have obtained the mean and variance of the time to

recruitment assuming that the random variables

denoting staff depletion are independent and identically

distributed random variables, using cumulative damage model

concept without using any cost structure. Mariappan and

Srinivasan (2001(a)) have extended the result. when the

interdecision times are exchangeable and constantly correlated

exponential random variables for a single graded system. For a

two grade system, Sathiyamoorthy and Parthasarathy (2002)

have obtained the mean time to recruitment under suitable

conditions.

In this chapter, an organization with two grades each

having its own threshold level is considered. Two mathematical models

are considered employing two different univariate policies of recruitment.

In model 1, recruitment is made whenever the threshold crossing

takes place in both the grades and in model 2, recruitment is made

81

whenever the threshold crossing takes place in any one of the grades. The

objective of this chapter is to find the mean and variance of time to

recruitment in the organization for both the models assuming that (i) the

inter-decision times are exchangeable constantly correlated and (ii) the

threshold distribution is continuous.

The rest of this chapter is organised as follows: In section

5.2, description of Model 1 is given and based upon the aforesaid

univariate policy of recruitment, analytical expressions for mean and

variance of the time to recruitment are obtained. A closed form of these

analytical expressions are obtained by assuming specific distributions.

For a better understanding, the results are numerically illustrated and

relevant conclusions are presented. In section 5.3 Model 2 is described

and a similar computation is carried out using a different univariate

policy of recruitment. Model 2 of this chapter extends the results of

Sathiyamoorthy and Elangovan (1998) for a two graded manpower

system when the inter-decision times are correlated.

5.2 MODEL 1: DESCRIPTION AND ANALYSIS OF THE MODEL

In this section a two graded manpower system is considered and

the description of the model is given below:

82

Assumptions:

1. An organization having two grades (grades A and B) takes

policy decisions at random epochs in [0, ) and at every

decision making epoch, a random number of persons quit

the organization.

2. There is an associated loss of manpower to the organization if a person

quits and it is linear and cumulative.

3. Each grade has its individual independent threshold. If the

total number of persons who leave the organization crosses

the maximum of the two thresholds, recruitment becomes

necessary. In other words, recruitment is made whenever

the cumulative number of exits in the three grades crosses

both the thresholds.

4. Mobility of manpower from one grade to the other grade is

allowed.

Notations:

iU : time between the 1th

i and thi decision epoch. 'iU s are

exchangeable and constantly correlated exponential random variables.

iX :discrete random variable denoting the total number of

persons who leave the organization from the two

83

grades at the thi decision epoch. 1,2,... 'ii X s independent and

identically distributed random variables.

,A BY Y :continuous random variables denoting the threshold levels

for grades A and B respectively and the distribution of AY

and BY is exponential.

Y : max ,A BY Y Y

kV t :probability that there are exactly k decision making epochs

in (0, ]t .

T :a continuous random variable denoting the time for

recruitment in the organization.

L t :cumulative distribution function of T.

.AH :distribution function of AY follows exponential

distribution with the parameter 1

.BH :distribution function of BY follows exponential

distribution with the parameter 2

.H : distribution function of Y.

1a 1

0

: r

i

r

e P X r

84

2a 2

0

: r

i

r

e P X r

3a 1 2

0

:r

i

r

e P X r

and 1 , 1,2,3i ia a i

.kF : cumulative distribution function of 1

k

i

i

U

*: Laplace – Stieltje’s transform.

R : correlation between any iU and jU i j

,n x 1

0

:

x

r ne d

a : mean of , 1,2...iU i

b : 1a R

m

1:

1m m s

bs

E T : Mean time for recruitment

V T : Variance of time to recruitment

5.2.1 Main Results

In this subsection, expressions for , * ,L t L s E T and V T

are obtained.

First we shall obtain the distribution .H x

The distribution functions of AY and BY are given by

11 x

AH x e

85

and

21 x

BH x e

respectively.

Since max , ,A BY Y Y the distribution function H x is

given by

BH x H x H x

1 21 21x xx xH x e e e

(5.2.1)

Next we shall obtain .L t

0k

P T t P

{exactly k decisions in (0, ]t and the threshold

levels are not crossed}

0 1

k

k i

k i

V t P X Y

P T t 0 1

k

k i

k i

V t P X Y

(5.2.2)

Using the law of total probability and (5.2.1)

1

k

i

i

P X Y

1 1 1 1

/k k k k

i i i

i i i i

P Y X X r P X r

0 1

k

i

r i

P Y r P X r

86

1 2

0 1 0 1

k kr r

i i

r i r i

e P X r e P X r

1 2

0 1

kr

i

r i

e P X r

(5.2.3)

Consider

1

0 1

kr

i

r i

e P X r

1

1 1

0 1

k

i

i

k Xr

i

r i

e P X r E e

1 1

1

kX

i

E e

i.e., 1

1

0 1

kr k

i

r i

e P X r a

(5.2.4)

Similarly

2

0 1

kr

i

r i

e P X r

(5.2.5)

1 2

3

0 1

kr k

i

r i

e P X r a

(5.2.6)

Using (5.2.4), (5.2.5) and (5.2.6) in (5.2.3) we get

1

k

i

i

P X Y

1 2 3

k k ka a a (5.2.7)

87

From (5.2.1) and (5.2.7), we get

P T t 1 2 3

0

k k k

k

k

V t a a a

P T t 1 1 2 3

0

k k k

k k

k

F t F t a a a

(5.2.8)

Consider

1

0

k

k k i

k

F t F t a

,

1

0

k

k k i

k

F t F t a

1

0

1 , 1,2,3,..k

k i

k

F t a i

(5.2.9)

Since

1 ,L t P T t using (5.2.8) and (5.2.9) we get

L t 1 1 1

1 1 2 2 3 3

0

k k k

k

k

F t a a a a a a

(5.2.10)

(5.2.10) gives the distribution of T.

Next we find the Laplace-Stieltje’s transform of L t . (i.e.,) * .L s

From Gurland (1955) the cumulative distribution function of

kF x is given by

kF x

10

,

1 ,11

i

ii

xk i

RK bR

k iR kR

(5.2.11)

88

Where 1 2 .... k is the characteristics function of the joint

distribution of any k random variables from 1.k k

X

The Laplace-Stieltje’s transform of kF t is given by

*

kF s 0

st

ks e F t dt

1

1 1 / 1 1k

bs kRbs R bs

(5.2.12)

Taking Laplace-Stieltje’s transform on both sides of (5.2.10)

*L s 1 1 1

1 1 2 2 3 3

1

k k k

k

k

F t a a a a a a

(5.2.13)

Using (5.2.12) in (5.2.13) we get

*L s

1 1 1

1 1 2 2 3 3

1

1 11 1

k k k

k

k k

F t a a a a a a

kRbsbs

R bs

*L s

1 1 1

1 1 2 2 3 3

1 11

1

kk k k

k

ma a a a a a

kR m

R

(5.2.14)

(5.2.14) gives the Laplace-Stieltje’s transform of L t

Where

1

1m

bs

We now obtain E T

89

Since E T 0

* ,s

dL s

ds

From (5.2.14) we get

E T

1 1 1

1 1 2 2 3 3

1

0

11

1

kk k k

k

s

d ma a a a a a

kR mds

R

(5.2.15)

Now for 1,2,3i

11

0

11

1

kk

i i

k

s

d ma a

kR mds

R

1 4

1

21

0

11

1 1

11

1

k

k

i l

k

s

dmkR

kR m dm dskm m

R ds Ra a

kR m

R

(5.2.16)

1

1

11

k

i i

k

Ra a kb

R

11 1

11

kk

i i

k

d ma a

kR mds

R

1

1

1,

1

k

i i

k

a a kbR

Since

,1

ba

R

for 1,2,3i

90

1

1

1

1

k

i i

k

a a kbR

1

1

k

i i

k

a a ka

21 2 3 ...i i ia a a a

i

a

a (5.2.17)

From (5.2.16) and (5.2.17), we get

11 1

11

kk

i i

k

d ma a

kR mds

R

,i

a

a 1,2,3i (5.2.18)

Using (5.2.18) in (5.2.15) we get

E T 1 2 3

1 1 1a

a a a

i.e., E T 1 2 3

1 1 1

1

b

R a a a

(5.2.19)

(5.2.19) gives the mean time for recruitment.

Now, we obtain V T . It is known that

22V T E T E T (5.2.20)

and

2

2

2 0*

s

dE T L s

ds (5.2.21)

Now for 1,2,3i

91

2

1

21 1

11

kk

i i

k

d ma a

kR mds

R

21

21 1 1

kk

i i

k

d ma a

ds kC m

(5.2.22)

Suppose

.1

RC

R

From (5.2.22) one can show that

2

21 1

11

k

k

d m

kR mds

R

1

3 21

21

1 1 1 1

k k

i i

k

dmkC

dm dsa a kC mds kC m kC m

2 21 1

221 1

k k

dmkC

dm d m dm dskm k mds ds ds kC m

2

2

21

2 2 2

11

1 1 1

k

k

dmk m

d mdsm

dskC m kC m

(5.2.23)

Since 2

2

20, 1; ; 2

dm d ms m b b

ds ds from (5.2.23)

2

1

21 1

11

kk

i i

k

d ma a

kR mds

R

92

= 1 2 2 2 2 2 2 2 2 2

1

2 2 1k

i i

k

a a k C b k Cb kCb k Cb k k b

1,2,3i (5.2.24)

Since

1

1

k

i

k

ka

21 2 3 ....a a

and

2 1

1

k

i

k

k a

21 4 9 ....a a

From (5.2.24),

2

1

21 1

11

kk

i i

k

d ma a

kR mds

R

2

2

2

2 1

1i

a

R a

(5.2.25)

Using (5.2.25) in (5.2.21) we get

2E T

22 22

2

1 2 3

2 1 1 1

1

a

R a a a

(5.2.26)

Using (5.2.19) and (5.2.26) in (5.2.20), we get

V T

22 22

2

1 2 3

2 1 1 1

1

a

R a a a

2

2

1 2 3

1 1 1a

a a a

(5.2.27)

Equation (5.2.27) gives the variance of time to recruitment.

93

5.2.2 Special Case

In this section explicit expressions for the mean and variance of

time to recruitment are obtained by assuming specific distributions.

Suppose , 1,2...iX i follows Poisson distribution with

parameter . In this case

1a 1

0

r

i

r

e P X r

1

0 !

rr

r

e er

i.e., 1a 11 e

e

Doing similar computations for 2a and 3a and using in (5.2.19)

and in (5.2.27)we get the following results.

E T 31 2 11 1

1 1 1

1 11 1ee e

b

R ee e

(5.2.28)

and

V T 31 2

2

2 2 2211 1

2 1 1 1

111 1

ee e

a

Ree e

1 2 3

2

21 1 1

1 1 1

11 1e e e

a

ee e

(5.2.29)

Equations (5.2.28) and (5.2.29) gives the explicit expression for

mean and variance of the time to recruitment for the special case.

94

5.2.3 Numerical Illustration

In this section, the analytical expression obtained in (5.2.29) and

(5.2.30) are numerically illustrated and relevant conclusions are made.

Fixing 1 20.5; 0.2; 2a and varying R and the values of

E T and V T are computed and tabulated in Table 5.2.1.

Table 5.2.1

R 1 2 3 4 5

0.5

E T 0.5049 0.3746 0.3441 0.3353 0.3329

V T 0.3944 0.2281 0.1957 0.1869 0.1846

0.4

E T 0.6311 0.4682 0.4301 0.4191 0.4161

V T 0.3943 0.2305 0.1985 0.1897 0.1874

0.2

E T 0.9467 0.7023 0.6452 0.6286 0.6242

V T 0.8872 0.5787 0.4466 0.4268 0.4170

0.5

E T 1.5147 1.1237 1.0323 1.0058 0.9988

V T 3.5497 2.0532 1.7617 1.6818 1.6612

0.8

E T 3.7866 2.8092 2.5807 2.5146 2.4969

V T 61.7530 35.2841 30.1531 28.7490 28.3840

95

5.2.4 Conclusions

From the above table, we make the following observations.

(i) The mean and variance of time to recruitment increases as

decreases, keeping other parameters fixed. In other words when the

average loss of manhours decreases, the mean time for recruitment

increases.

(ii) The mean and variance of time to recruitment

increases as R increases, keeping other parameters fixed.

(iii) The mean and variance of recruitment decreases for negative

correlation and increases for positive correlation when both and R

varies.


Assumptions:

1. An organization having two grades (grade A and grade B) takes policy

decisions at random epochs in [0, ) and at every decision making

epoch, a random number of persons quit the organization.

2. There is an associated loss of manpower to the organization if a

person quits and it is linear and cumulative.

3. Each grade has its individual independent threshold. If the total

number of persons who leave the organization crosses the minimum of

96

the two thresholds, recruitment becomes necessary. In other words,

recruitment is made whenever the cumulative number of exits in the

two grades crosses any one of the thresholds.

4. Mobility of manpower from one grade to the other grade is not

allowed.

Notations:

iU : Time between the 1th

i and thi decision epoch. 'iU s are

exchangeable and constantly correlated exponential random variables.

iX : Discrete random variable denoting the total number of

persons who leave the organization from the two

grades at the thi decision epoch. 1,2,... 'ii X s independent and

identically distributed random variables.

,A BY Y : continuous random variables denoting the threshold levels

for grades A and B respectively and the distribution of AY and BY is

exponential.

Y : max ,A BY Y Y

kV t : probability that there are exactly k decision making epochs

in (0, ]t .

T :a continuous random variable denoting the time for

recruitment in the organization.

97

L t :cumulative distribution function of T.

.AH :distribution function of AY follows exponential

distribution with the parameter 1 .

.BH :distribution function of BY follows exponential

distribution with the parameter 2 .


1a 2

0

: r

i

r

e P X r

3a 3 3: 1a a

.kF : cumulative distribution function of 1

k

i

i

U

*: Laplace – Stieltje’s transform.

R : correlation between any iU and jU i j

,n x 1

0

:

x

r ne d

.

a : mean of , 1,2...iU i

b : 1a R

m

1:

1m m s

bs

98

E T : mean time for recruitment

V T : variance of time to recruitment

5.3.1 Main Results

In this subsection, expressions for , * ,L t L s E T and V T

are obtained.

The threshold distributions for grade A and grade B are given by

11 x

AH x e

and

21 x

BH x e

Respectively.

Since max , ,A BY Y Y the distribution function H x is

given by

1 2x xH x e e (5.3.1)

Next we shall obtain .L t

0k

P T t P

{k instants of exits in (0, t]} and the cumulative

loss of manpower in these k decisions does not reach the threshold

levels}.

0 1

k

k i

k i

V t P X Y

(5.3.2)

99

Proceeding as in Model l, it is found that

1

k

i

i

P X Y

1 2

3

1

i

kU k

i

E e a

(5.3.3)

1k k kV t F t F t 0,1,2k

From (5.3.2) and (5.3.3) we get

P T t 1 3

0

k

k k

k

F t F t a

(5.3.4)

As in model 1 , * ,L t L s E T and V T can be obtained as

follows:

L t 1

3 3

1

k

k

k

a F t a

(5.3.5)

*L s

13 3

1 11

1

kk

k

ma a

kR m

R

(5.3.6)

E T 3 31

b a

a R a

(5.3.7)

V T

2

2

3

1 21

bR

a R

(5.3.8)

Equations (5.3.7) and (5.3.8) gives the mean and variance of time

to recruitment for Model 2 respectively.

100

5.3.2 Special Case

Assume that , 1,2...iX i follows Poisson distribution with

parameter

In this case, we get

1 2

3

ra E e

1 2

0

r

i

r

e P X r

1 2

0 !

rr

r

e er

i.e.,

1 2

3

e

a e

(5.3.9)

E T

1 2

1e

b

e R

(5.3.10)

and

V T

1 2

2

21 2

1e

bR

e R

(5.3.11)

Equation (5.3.10) and (5.3.11) gives mean and variance of time

to recruitment for this model.

5.3.3 Numerical Illustrations

In this section, the analytical expression obtained in (5.3.10) and

(5.3.11) are numerically illustrated and relevant conclusions are made

101

Fixing 1 20.5; 0.2; 2a and varying R and the values of

E T and V T are computed and tabulated in Table 5.3.1.

Table 5.3.1

R 1 2 3 4 5

0.5

E T 0.7094 0.4740 0.4040 0.3803 0.6383

V T 0.6950 0.3103 0.2287 0.1997 0.1873

0.4

E T 0.8277 0.5530 0.4748 0.4436 0.4296

V T 0.7421 0.3313 0.2442 0.2132 0.2000

0.2

E T 1.2415 0.8296 0.7122 0.6655 0.6445

V T 1.6698 0.7455 0.5495 0.4798 0.4499

0.5

E T 1.6554 1.1061 0.9496 0.8893 0.8593

V T 3.7841 1.6895 1.2453 1.0872 1.0196

0.8

E T 4.9661 3.3183 2.8489 2.6619 2.2778

V T 112.3491 50.1611 36.9729 32.2786 30.2729

5.4 CONCLUSIONS


(i) The meantime for recruitment increases as decreases, keeping

other parameters fixed. In other words, when the average loss of

manhours decreases, the mean and variance for recruitment increases.

102

(ii) The mean and variance of time to recruitment increases as R

increases, keeping other parameters fixed.

(iii) The mean and variance of recruitment decreases for negative

correlation and increases for positive correlation when both and R

varies.

103

CHAPTER 6

MARKOV ANALYSIS OF BUSINESS WITH TWO LEVELS

AND MANPOWER WITH THREE LEVELS

6.1 INTRODUCTION

Nowadays labour has become a buyers market as well as seller's

market. Any company normally runs on commercial basis wishes to keep

only the optimum level of any resources needed to meet company's

requirement at any time during the course of the business and manpower

is not an exception. This is spelt in the sense that a company does not

want to keep manpower more than what is required. Hence, retrenchment

and recruitment are common and frequent in most of the companies now.

Recruitment is done when the business is busy and shed manpower when

the business is lean. Equally true with the labour, has the option to switch

over to other jobs because of better working condition, better emolument,

proximity to their living place or other reasons. Under such situations the

company may face crisis because business may be there but manpower

may not be available. If skilled labourers and technically qualified

persons leave the business the seriousness is worst felt and the company

has to hire paying heavy price or pay overtime to employees.

In this chapter considered are two characteristics namely

manpower and business. A formula is derived for the steady state rate of

crisis and the steady state probabilities. The situations may be that the

104

manpower may be fully available, insufficiently available or hardly

available, but business may fluctuate between full availability to nil

availability. The steady state probabilities of the continuous Markov

chain describing the transitions in various states are derived and critical

states are identified for presenting the cost analysis. Numerical

illustrations are provided.

6.2 ASSUMPTIONS

1. There are three levels of Manpower namely Manpower is full, is

moderate and Manpower is nil.

2. There are two levels of business namely (I) business is fully

available (2) business is lean or nil.

3. The time T during which the manpower remains continuously

moderate and becomes nil has exponential distribution with

parameter 10. The time R required to complete recruitment for

filling up of vacancies from level nil to moderate level is

exponentially distributed with parameter 01.

4. The time T' during which the Manpower remains continuously full

and becomes nil has exponential distribution with parameter 20 and

the time R' required to complete full recruitment from nil level is

exponentially distributed with parameter 02 .

105

5. The period of time T" during which the Manpower is continuously

full becomes moderate has exponential distribution with parameter

21 and the period of time R" required for recruitment from

insufficient to full is exponentially distributed with parameter 12 .

Random variables T and R; T’ and R; T" and R" are all independent.

6. The busy and lean periods of the business are exponentially

distributed with parameters ‘a’ and ‘b’ respectively.

6.3 SYSTEM ANALYSIS

The Stochastic Process X t describing the state of the system is a

continuous time Markov chain with 4 points state space as given below in

the order of Manpower and Business

0 0 , 0 1 , 1 0 , 1 1 , 2 0 , 2 1S (6.3.1)

where

2 -Refers to full availability in the case of manpower

1 -Refers to semi availability or insufficiently available manpower

and it refers to busy period in the case of business.

0 -Refers to shortage/lean/non availability manpower or business.

The infinitesimal generator Q of the continuous time Markov

chain of the state space is given below which is a matrix of order 4.

106

Q =

MP/B (0 0) (0 1) (1 0) (1 1) (2 0) (2 1)

(6.3.2)

(0 0) 1 b 01 0 02 0

(0 1) a 2 0 01 0 02

(1 0) 10 0 3 b 12 0

(11) 0 10 a 4 0 12

(2 0) 20 0 21 0 5 b

(2 1) 0 20 0 21 a 4

1 02 02 2 02 02 2 10 12= ( ) ( ) ( ) , , ,b a b

4 10 12 5 20 21 4 21 21, ,( ) .( ) ( )a b a (6.3.3)

The occurrences of transitions in both manpower and business are

independent, the individual infinitesimal generator of them are given by:

1. The infinitesimal generator of business is given below by a matrix of

order

B =

B 0 1

0 b b

1 a a

and the steady state probabilities are

0B

a

a b

and 1 .B

b

a b

2. The infinitesimal generator of manpower is given below by the matrix

of order 3,

107

M =

M 0 1 2

0 01 02 01 02

1 10 10 12 12

2 20 21 21 20

The steady state probabilities of manpower are:

00

0 1 2

,M

d

d d d

1

1

0 1 2

,M

d

d d d

2

2

0 1 2

M

d

d d d

Where

0 20 12 20 10 21 10d

1 20 12 20 10 21 10d

2 10 02 12 02 12 01d

The steady state probability vector of the matrix Q can be derived

easily by using

0Q and 1e

000 2

0

,

i

ad

Z d

0

01 2

0

,

i

bd

Z d

1

10 2

0

,

i

ad

Z d

111 2

0

,

i

bd

Z d

0

20 2

0

,

i

ad

Z d

1

21 2

0

,

i

ad

Z d

(6.3.4)

Where 0 1

2

0

2 i d d d Zd a b

108

When the business is available, either full manpower or moderate

manpower must be available. When it is not so this will create heavy loss,

we shall call this situation as crisis.

The crisis state is {(0 1)} and the crises occur when there is full

business but manpower is NIL.

Now the rate of crisis in steady state (C) is obtained as follows.

P(crisis in[ ]tt t )

0 1 / 0 0 0 0P X t t X t P X t

0 1 / 2 1 2 1P X t t X t P X t

0 1 / 11 11 .P X t t X t P X t O t

Taking limit as 0,t

00 20 21 10 11tC bP t P t P t

00 20 21 10 11[ ] 1 tC t bP t P t P t

that is

00 10 21 10 11[ ]C b

Using the steady state probabilities, obtained

0 10 0 10 1

bC ad d d

ZY (6.3.5)

Where Z a b and 0 1 2 .Y d d d

109


Now taking the values of the parameters in the model as below, can

find the steady probabilities and the rate of crises using the formulas

(6.3.4) and (6.3.5) respectively.

10 = 4, 21 = 5, 20 = 2, 12 = 8, 01 = 4, 02 = 7, a = 8 and b = 9

Steady state

probability Value

00 0.0818

01 0.0922

10 0.1432

11 0.1411

20 0.2455

21 0.2742

Total 1.0000

Now assigning the values 9, 12, 15, 17b and 19 we calculate the

corresponding rate of crisis and are given below in the table:

b Y

9 1.9355

12 2.1913

15 2.3819

17 2.4865

19 2.5701

110

The graph of the steady state crisis is given below taking the values

of b on the x-axis and the value of C∞ on y axis.

Y Values

0

0.5

1

1.5

2

2.5

3

1 2 3 4 5

Y

We find that as the value of parameter b increases the crisis

rate also increases. Also we observe that the cost of doing business

is very heavy if the manpower is full but there is no business. Under

circumstances we should fetch business at premium rate or offer

heavy discount to get business. The cost of business is comparatively

low when the business is full and the manpower is also full. The

same holds in the case of manpower is moderate whereas the

business may be dull or busy.

6.5 CONCLUSION

Though there may be many factors (characteristics) affecting a

business, the most vital among all are manpower and money. If these are

kept strong all other factors can be managed. Finance must very

meticulously handled by experienced person with specialization. Charted

111

accountants, persons with company secretary ship qualification, cost

accountants, Income tax specialist, sales tax specialist, etc must be

employed and salary should not stand as hurdles to employ such

specialist. Skilled labourers must be carefully handled. They are assets to

the company, their genuine demands must be met but at the same time

optimal manpower should only be maintained, this must be done in the

interest of the company and keeping in mind the welfare of the

employees. There can be retrenchments, but must be carried out in a wise

way as may not affect the morale of the employees. Proper training must

be given to the employees.

Every chapter is special in its own way. A business may be

governed by environments but by comparing their strength, we can

always reduce to two characteristics which may bring in a crisis state. A

crisis state is one where business gets affected because of shortage

occurring in the characteristics governing the business. Application of

game theory can reduce the number of environments to two. The

application of the formulas in the models will give the rate of crisis. Now

a business concern can take precautionary measures to avoid coming of

such situations. Cost at different situations can be worked out using the

formulas in the models. Necessary steps may be taken to avoid incurring

heavy expenditure. A manufacturing concern depends on machines for its

production.

112

The failure of machineries will result in heavy expenditure for

production. Such situations are dealt deriving formulas to determine cost

and the rate of crisis. This model is sure to have its application for a

manufacturing concern. Just like machines the computers have

importance in any business concern and have become totally

indispensable. A model dealing with its application gives formulas and is

a very useful to avoid a crisis state occurring due to software or hardware

failure. Instead of two levels (0 for nil availability and 1 for full

availability) three levels of manpower is dealt. Models with this new

concept are useful because we find many business concerns run the

business with inadequate or insufficient staff. Formulas will help to over

come crisis states and avoid incurring heavy costs because of manpower.

113

CHAPTER - 7

EXPECTED TIME FOR RECRUITMENT IN A TWO

GRADEDMANPOWER SYSTEM ASSOCIATED WITH

CORRELATEDINTER-DECISION TIMES WHEN THRESHOLD

DISTRIBUTION HAS SCBZ PROPERTY

7.1 INTRODUCTION

For a single graded system, Sathiyamoorthy and Elangovan

(1998(a)) have obtained the mean and variance of the time to recruitment.

(i) When the number of exits forms a sequence of independent and

identically distributed random variables,

(ii) The random threshold is geometric and (iii) the inter-decision

times are independent and identically distributed random variables. Later,

for the same manpower system, Sathiyamoorthy and Parthasarathy (2003)

have obtained the mean and variance of the time to recruitment when

(i) The loss of manhours process isa sequence of independent and

identically distributed random variables and (ii) the random threshold has

Setting the Clock Back to Zero (SCBZ) property.

Mariappan and Srinivasan (2001(a)) have also obtained the

meantime for recruitment in a single graded system using shock model

approach when the inter-decision times are correlated exchangeable and

exponential random variables also they have obtained the mean time for

recruitment in a single graded system using shock model approach when

114

the bi-variate process formed by loss of manpower and inter-decision

times forms a correlated renewal sequence.

In this chapter, an organization with two grades subjected to loss of

manpower due to staff depletions caused by policy decisions taken by the

organization is considered. Assuming that each grade has its own random

threshold whose distribution has SCBZ property and the inter-decision

times are exchangeable and constantly correlated exponential random

variables, two mathematical models are constructed based upon an

appropriate univariate policy of recruitment. The objective of this chapter

is to find the mean time for recruitment in the organization for both the

models.

The rest of this chapter is organised as follows: In Model 1 is

given, analytical expression for mean time to recruitment is obtained and

the special cases are discussed.

In Model 2 is described and a similar computation is carried out

using a different univariate policy of recruitment. In section 4.4, both the

models are numerically illustrated and relevant conclusions are made.


Assumptions

1. An organization having two grades (grades A and B) takes policy

decisions at random epochs in 0, and at every decision making

epoch, a random number of persons quit the organization.

115

2. There is an associated loss of man hours to the organization if a


3. Each grade has its own threshold level and the

thresholddistribution has SCBZ property. Recruitment is

madewhenever the total number of exits exceeds the threshold

level in both the grades.

4. The inter–decision times are exchangeable constantly correlated

exponential random variables.

5. The process which generates the number of exits and the threshold

are mutually independent.

6. Mobility of man power from one grade to the other is permitted.

Notations


i and thi decision epoch. ’iU s are

exchangeable and constantly correlated exponential random variables .

iX : discrete random variable denoting the total number of persons

who leave the organization from the two grades at the thi decision

epoch. 11,2,..... 'i X S independent and identically distributed random

variable

,A BY Y : continuous random variables denoting the threshold of

levels for the grades A and B respectively and the distributions AY and

BY follows SCBZ property.

116

,(: )max ,A BY Y Y Y

kV t : probability that there are k decisions in (0,t].

.g : probability density function of , 1,2, ..iX i

)*(.g : Laplace transform of .g

.kg : k -fold convolution of .g

.f : probability density function of inter-decision times

.kf : k-fold convolution of .f

.kF : k-fold convolution of .F

.AH : distribution function of AY with parameters 1 2, and 1

.BH : distribution function of YB with parameters 3 4, and

2


T : time for recruitment in the organization

L t : distribution function of T

*L s : Laplace-Stieltje’s transform of L t .

R: correlation between any iU and ,jU i j

1

0

, :

x

nn x e d

a : mean of , 1,2......iU i

b : 1a R

117

m :

1

1m m s

bs

E T : mean time for recruitment .

7.2.1 Main Results

In this subsection an analytical expression for the mean time to

recruitment is obtained.

Since Y= max ,( ),A BY Y , the distribution of Y is given by,

A BH x H x H x

1 1 A BH x H x H x

The probability distribution of the thresholds YA and YB for the two

grades respectively are given by,

1 1 2

1 11x x

AH x p e q e

And

3 2 4

2 21x x

BH x p e q e

where

1 2 11 1

1 1 2 1 1 2

; ;p q

3 4 22 2

2 3 4 2 3 4

;p q

Where 1 2and are the parameters of the exponentially distributed

truncated random variables.

Since max , ,A BY Y Y

118

1 1 3 3 1 2

1 21x x

H x p e p e

2 4 2 4 1 3 1 2

1 2 1 2 1 2

x x x rq e q e q q e p p e

3 4 1 2 3 2 2 4

1 2 1 2 1 2

x x xp q e q p e q q e

(7.2.1)

The probability that the threshold level is not reached till ‘t’

is, 0k

P T t P

{ k instants of exits in (0,t] }and the cumulative loss of

manpower in these k decisions does not reach the threshold level}

0 1

k

k

k i

V t P X Y

(7.2.2)

We now calculate 1

.k

i

P X Y

Using the law of total probability and (7.2.1),

1 0 0 0 0

|k

i i

i r k k k

P X Y P Y X X r P X Y

0 0

i

r k

P Y r P X r

1 1 3 2

1 2

0 0 0 0

r r

i i

r k r k

p e p X r p e X r

2 4

1 2

0 0 0 0

r r

i i

r k r k

q e p X r q e p X r

1 3 1 2 1 4 1

1 2 1 2

0 0 0 0

r r

i i

r k r k

p p e p X r p q e p X r

119

2 3 2 2 4

2 1 1 2

0 0 0 0

r r

i i

r k r k

p q e p X r q q e p X r

(7.2.3)

write

1 1 3 2

1 2

0 0

; ;r r

i i

r r

a e p X r a e p X r

1 4

3 4

0 0

; ;r r

i i

r r

a e p X r a e p X r

1 3 1 2 1 3 1

5 6

0 0

; ;r r

i i

r r

a e p X r a e p X r

1 3 1 2 1 3 1

5 6

0 0

; ;r r

i i

r r

a e p X r a e p X r

Now

0

, 1,2,.......ixr

i

r

e p X r E e i

and

0 1 1

i

kXr

i

r i i

e p X r E e

1 1 1 1

1 1

0 0 1 0

kr

i i

r k i k

p e p X r p E e X

1 1

1 1 0

1 1

0 0 1

i

k

k Xr

i

r k i

p e p X r p E e

= 1 1

kp a (7.2.5)

Similarly for other terms of (7.2.3) we can show that

120

3 2

2 2 2

0 0

r k

i

r k

p e p X r p a

(7.2.6)

2

1 1 3

0 0

r k

i

r k

q e p X r q a

(7.2.7)

4

2 2 4

0 0

r k

i

r k

q e p X r q a

(7.2.8)

1 3 1 2

1 2 1 2 5

0 0

r k

i

r k

p p e p X r p p a

(7.2.9)

1 4 1

1 2 1 2 6

0 0

r k

i

r k

p q e p X r p q a

(7.2.10)

2 3 2

2 1 2 1 7

0 0

r k

i

r k

p q e p X r p q a

(7.2.11)

2 4

2 41 2 1 2 1 2 8

0 0

k

r k

i

r k

q q e p X r q q d q q a

(7.2.12)

Using the results (7.2.5) to (7.2.12) in (7.2.3)

1 1 2 2 1 3 2 4

0 1 2 5 1 2 6 2 1 7 1 2 8

k k k k

k k k k kk

p a p a q a q ap T t V t

p p a p q a p q a q q a

Now L*(s) can be obtained as follows

1 1 2 2 1 3 2 4

1

0 1 2 5 1 2 6 2 1 7 1 2 8

k k k k

k k k k k kk

p a p a q a q ap T t F t F t


(7.2.13)

Since 1L t p T t p T t

121

1 1 1 2 2 2 1 3 3 2 4 4

0 1 2 5 5 1 2 6 6 2 1 7 7 1 2 8 8

k k k k

k k k k kk

p a a p a a q a a q a aL t F t

p p a a p q a a p q a a q q a a

(7.2.14)

From Gurland, J (1955), Laplace - Stieltje’s transform of kF t is

given by

1*

1 11 1

k

k

F skRbs

bsR bs

Taking Laplace-Stieltjes transform on both sides of (7.2.14), we get

1

1*

11 1

k

k

bsL s X

kRbs

R bs

1 1 1 2 2 2 1 3 3 2 4 4

1 2 5 5 1 2 6 6 2 1 7 7 1 2 8 8

k k k k

k k k k

p a a p a a q a a q a a


*L s = 1 1

11

k

k

m

kR m

R

1 1 1 2 2 2 1 3 3 2 4 4

1 2 5 5 1 2 6 6 2 1 7 7 1 2 8 8

k k k k

k k k k

p a a p a a q a a q a a


(7.2.15)

Since 0

*s

dE T L s

ds (7.2.16)

Using (7.2.15) in (7.2.16) the mean time for recruitment is found to be

122

1 2 1 2 1 2 1 2 2 1 1 2

1 2 3 4 5 6 7 81

b p p q q p p p q p q q qE T

R a a a a a a a a

(i.e) 1 2 1 2 1 2 1 2 2 1 1 2

1 2 3 4 5 6 7 8

p p q q p p p q p q q qE T a

a a a a a a a a

(7.2.17)

Where , 1,2......ia i and 1i ia a are given by (7.2.4),(7.2.17)

gives the mean time for recruitment.

7.2.2 Special Case

Suppose , 1,2...iX i follows Poisson distribution with parameter

1 1

1

0

r

i

r

a e p X r

1 1

0

.

!

rr

r

ee

r

1 1

0 !

r

r

ee

r

1 1.ee e

1 11

1

e

a e

(7.2.18)

Similarly,

3 1 1 3 1 22 41 11 1

2 3 4 5; ; ;e ee e

a e a e a e a e

3 2 21 4 1 2 411 1

6 7 8; ;ee e

a e a e a e

(7.2.19)

Using (7.2.18) and (7.2.19) in (7.2.17) we get

123

1 1 23 2

4 1 4 11 3 1 2

2 43 2 2

1 2 1

1 11

2 1 2 1 2

1 11

2 1 1 2

11

1 11

1 11

11

e ee

e ee

ee

p P q

e ee

q p q p qE T a

e ee

p q q q

ee

(7.2.20)

Equation (7.2.20) gives the meantime for recruitment for the

special case.

7.2.3 Numerical Illustration

In this section model 1 is numerically illustrated and relevant

conclusions are made.

Fixing 1 2 1 2 3 42; 0.8; 0.5; 0.4; 0.3; 0.6; 0.2B and

varying R and , the values of E T are computed and tabulated in table

7.2.1.

Table 7.2.1

R

1 2 3 4 5

0.4 0.9962 0.5307 0.3748 0.2967 0.2503

0.2 1.1622 0.6192 0.4372 0.3462 0.2920

0.2 1.7434 0.9288 0.6558 0.5192 0.4380

0.4 2.3245 1.2384 0.8744 0.6923 0.5840

0.8 6.9734 3.7151 2.6233 2.0769 1.7521

124

7.2.4 Conclusion


(i) The mean time for recruitment increases as decreases,

keeping other parameters fixed. In other words, when the number

of exits decreases on the average, the mean time for recruitment

increases.

(ii) The mean time for recruitment increases as R increases,

keeping other parameters fixed.

(iii) The meantime to the recruitment decreases for negative

correlation and increases for positive correlation when R and

varies simultaneously.

7.3 Model 2: Description and Analysis of the Model

Assumptions

1. An organization having two grades A and B takes policy decisions

at random epochs in 0, and at every decision making epoch, a

random number of persons quit the organization.

2. There is an associated loss of man hours to the organization if a


3. Each grade has its own threshold level and the threshold

distribution has SCBZ property. Recruitment is made whenever the

total number of exits exceeds in any one of the two threshold.

125

4. The inter – decision times are exchangeable and constantly

correlated.

5. The process which generates the number of exits and the threshold

are mutually independent.

6. Mobility of man power from one grade to the other is permitted.

Notations


i and thi decision epoch. ’iU s are

exchangeable and constantly correlated exponential random variables .

iX : discrete random variable denoting the total number of persons

who leave the organization from the two grades at the thi decision

epoch.11,2,..... 'i X s independent and identically distributed random

variable

,A BY Y : continuous random variables denoting the threshold of

levels for the grades A and B respectively and the distributions AY and

BY follows SCBZ property.

,(: )min ,A BY Y Y Y

kV t : probability that there are k decisions in (0,t].

.g : probability density function of X

)*(.g : Laplace transform of .g

.kg : k -fold convolution of .g

126

.f : probability density function of inter-decision times

.kf : k-fold convolution of .f

.kF : k-fold convolution of .F

.AH : distribution function of AY with parameters 1 2, and

1 .BH : distribution function of BY with parameters 3 4,

and 2 .H : distribution function of Y.

T: time for recruitment in the organization

L t : distribution function of T

*L s : Laplace-Stieltje’s transform of L t .

R: correlation between any iU and ,jU i j

1

0

, :

x

nn x e d

a : mean of , 1,2......iU i

b : 1a R

m :

1

1m m s

bs

E T : mean time for recruitment .

7.3.1 Main Results

In this subsection an analytical expression for the mean time to

recruitment is obtained.

As in chapter 2, the distribution of Y is given by,

127

1 H x 1 3 1 2 3 4 1

1 2 1 2

x xp p e p q e

2 3 2 2 4

1 2 1 2

x xq p e q q e

(7.3.1)

As in model 1, it can be show that

L t 1 2 5 1 2 6 2 1 7 1 2 8

1

k k k k

k

k

F t p p a p q a p q a q q a

(7.3.2)

and

*L s

11 2 5 1 2 6 2 1 7 1 2 8

1 /1 / 1 1k

k k k kk

bs kRbs R bs


1 2 5 5 1 2 6 6

1 2 1 7 7 1 2 8 8

1

11 1

k k k

k kk

p p a a p q a abs

kRbs p q a a q q a a

R bs

*L s

1 2 5 5 1 2 6 6

1 2 1 7 7 1 2 8 81

11

k kk

k kk

p p a a p q a am

kR m p q a a q q a a

R

(7.3.3)

Where 1

1m

bs

Since E T 0

*s

dL s

ds from (7.3.4) one can found that

E T 1 2 1 2 2 1 1 2

5 6 7 81

b p p p q p q q q

R a a a a

i.e., E T 1 2 1 2 2 1 1 2

5 6 7 8

p p p q p q q qa

a a a a

128

Where , 5,6,7,8ia i and 1i ia a are given by (7.2.4)

Equation (7.3.4) gives the mean time for requirement.

7.3.2 Special Case

Suppose , 1,2,..iX i follows Poisson distribution with parameter .

5a 1 3 1 2

0

r

i

r

e P X r

1 3 1 2

0 !

rr

r

ee

r

1 3 1 2e e

i.e., 5a 1 3 1 21 e

e

Similarly,

1 4 11

6 ;e

a e

3 2 21

7 ;e

a e

2 41

8

e

a e

Now,

E T 2 41 3 1 2 1 4 3 21 2

1 2 1 2 2 1 1 2

11 1 1 ee e e

p p p q p q q qa

ee e e

Equation (4.3.7) is an analytical expression to the mean time for

recruitment. In this section model 2 is numerically illustrated and relevant

conclusions are made.

Fixing 1 2 1 2 3 42; 0.8; 0.5; 0.4; 0.3; 0.6; 0.2b

and varying R and , the values of E T are computed and tabulated in

Table 7.3.1.

129

T

R

1 2 3 4 5

0.4 0.3246 0.2079 0.1729 0.1581 0.1508

0.2 0.3787 0.2425 0.2018 0.184 0.1760

0.2 0.5681 0.3638 0.3026 0.2766 0.2640

0.4 0.7574 0.4850 0.4035 0.3688 0.3520

0.8 2.2723 1.4551 1.2106 1.1065 1.0559

7.4 CONCLUSION


(i) The mean time for recruitment increases as decreases, keeping

other parameters fixed. In otherwords, when the number of exits

decreases on the average, the mean time for recruitment increases.

(ii) The mean time to recruitment increases as R increases, keeping

other parameters fixed.

(iii) The meantime to the recruitment decreases for negative correlation

and increases for positive correlation when R and varies

simultaneously.

112

130

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