CHAPTER 1

INTRODUCTION

In any country the socio economic welfare of the people is a matter of prime

concern and interest. The people as well as those at the governance strive hard to

achieve the economic and social prosperity. However the economic welfare is a

causative factor of social prosperity. To a large extent depends upon the natural

resources available in that country. But the availability of the natural resources is

not alone enough. A well planned utilization of the same is very important.

Natural resources no doubt includes manpower as well that optimal utilization of

the same leads to prosperity and economic development of the country. So just as

planning is required for exploitation and utility of all natural resources, effective

planning is required for utilization of manpower. This fact being strongly

considered by the government of India that they have created a separate portfolio

in the name of HUMAN RESOURCES DEVELOPMENT. It is appropriate to

quote the observations of Grinold. R.C and K.T. Marshall [28] “Manpower

planning must be an ancient art, since manpower problems have existed for

centuries”.

Waker. J.W. [78] has rightly pointed out the need and importance of

manpower planning and utilization by stating that “Manpower planning refers to

the rather complex task of forecasting and planning for the right number and the

right kind of people at the right places at the right time to perform activities that

will benefit both the organization and the individual in it”.

According to Grinold and Marshall [28], “There is interaction between the

four ingredients of a manpower namely people, job, time and money. The growth

of an organization depends on better management of money, time, job and the

manpower. Bennison and Casson [13] stated that ‘organizations have become

more complex requiring a wider range of specialist’. A wider range of specialists

mean persons who have specialised in a particular field, persons who have

developed a special skill like operating special machineries, manufacturing skill,

marketing skill, etc. Any organization or a manufacturing company will have to

recruit such skilled persons for its growth, so, if they quit it will be very difficult to

replace or find substitute. More dangerous is that the clients may go behind them

or that good will of the company is likely to be lost. There are very many causes

for a business concern to get in to crises, may be because of non availability of a

particular raw material or power shortage or sudden competition coming up in the

market because of innovative marketing strategy or government policies or natural

calamities or war between countries. But certain causes are extraordinary, normal

and most common are shortage of money and manpower. Human Resource is vital

department in any company, sensitive and to be handled by experts only. The

failure to provide appropriate type of personnel and manpower result in business

getting in to crisis state. Therefore an overall need arises for improved and

sophisticated manpower management. Manpower shortage will lead to recruitment

process resulting in huge expenditure. The various costs associated with the policy

making in manpower management is indicated in Poornachandra Rao [54].

A greater emphasis was laid on manpower planning in twentieth century,

particularly after a boom in Information Technology throughout the world. Rapid

growth in information technology created a high demand for software and

hardware engineers. Industries to individuals all use computers for its multifarious

purposes that the demand for computer and its accessories have gone up, so

naturally manpower is required for such industries. There is growing competition

in business and quality products find a high demand with the public. So many

industries and manufacturing units have come up, resulting in high demand for

both skilled and non skilled labourers. More private sectors come up because of the

encouragement and assistance given by the government and for their participation

in nation’s development. This opened new and fast employment opportunities to

educated, skilled and unskilled people. In the field of medical service also the

need for highly skilled professional is very much on the increase nowadays. Hence

the need of proper planning of manpower resources and proper utilization of the

same has resulted in the emergence of manpower planning as a separate division of

management science, and Human Resources Management (HRM) is considered to

be a sensitive area of study and approach.

Problems that arise in manpower management are mainly due to the

disequilibrium that arises often. Disequilibrium in the sense that the amount of

demand and supply may not be equal. For example the demand may be more than

the supply. This causes manpower shortage. On the other hand the supply of

manpower more than the demand results in surplus manpower. In the former

production may suffer, in the latter laborers suffer because of retrenchment and

dislodgement. So manpower planning must be effectively handled that human

resources is fully utilized in every possible way

Attrition is one of the important factors that contribute to the disequilibrium

between the demand and supply of manpower. Though attrition is there in every

company or organization it is very much pronounced in the area of IT and software

companies. It is interesting to observe the news item in economic Times “High

attrition may hit size at mid term and IT firms”. The high attrition rate has become

pronounced in IT firms. This causes the disequilibrium between demand and

supply of specialists in manpower.

This loss of manpower affect adversely a company of its production and the

profit, its image with the public and its reputition as well. So Mathematical and

Stochastic models for the study of attrition in manpower is considered to be an

important area of investigation and lot of research models come up in this area for

effective handling of manpower.

1.1 Methods of Approach in Manpower Models

There are several areas of investigation and research in manpower planning.

In such studies the objectives are also different. One of the objectives of approach

is to obtain the likely time to recruitment which is necessary due to attrition. The

second approach is to determine the size of manpower supply to meet the demand

for the same which fluctuates and is of random character. Optimization techniques

are used for the same, in the sense that the techniques for determining the optimal

manpower at any point of time which neither affects the profit nor the production.

There are also Mathematical models which forecast the demand of manpower in

future.

A scientific and systematic approach to the study of manpower planning

was evolved during 1930’s. Effective models in manpower planning for time of

recruitment, optimal recruitment, attrition have been dealt using stochastic

processes, operations research and statistics. Pioneering studies in the areas of

manpower planning have been carried out by Lane and Andrew [40] and the

models studied by them were mostly descriptive with not much of computational

work. The leaving process which in other words known as wastage plays an

important role in policy decisions pertaining to manpower planning. The studies

by Rice and Bollinger [58] Silock [65] are mostly pertaining to the concept of

wastage. Young and Almond [81], Gani [24] have taken up studies relating to

stocks, flows and transitions in an organization. Application of the stochastic

process to manpower planning has been developed to a marked extent by

Barthlomew [6, 7], Charles et al., [17] have introduced the application of operation

research particularly linear programming and goal programming techniques in

manpower planning.

A systematic approach using modeling techniques can be traced to the work

of Seal [63], Vajda [72, 73]. The potential of this new approach could not be fully

exploited until the introduction of computers which can perform fast and accurate

calculations in 1960s and after then the Matlab making even difficult calculations

easy and computations fast and accurate. During 1970 Keenay and others

developed techniques for studying career prospects.

Situation plays an important role in developing manpower models. In such

models the behaviour of each of a system is simulated on the computer using

random numbers to determine the events which occur. Simulation models are

particularly useful in studying small systems where random fluctuations are likely

to be more serious. The simulation models are useful in the sense that realistic

operating rules in an organization with regard to manpower utilization could be

arrived at. But the computer infrastructure should be available for the application

of simulation methods. Computing expertise as well as full knowledge of the

manpower concepts is necessary in this respect. The application of simulation

models in manpower policies can be found in Bloomand Knight [14]

In manpower planning policies regarding promotions, recruitment, attrition

etc., the decisions should be based on the real life situations and ground realities

that exist from time to time. In such cases the application of stochastic processes

to manpower planning is more appropriate and in fact this type of an approach

involves a conceptualization of a real life problem in terms of an appropriate

stochastic model. The solutions arrived may not be as expected and hence a

reformulation of the model becomes necessary. This iterative process together

with other techniques of Operations Research will yield a desired result. A

detailed account of the application of stochastic processes in manpower planning

models can be seen from Barthlomew and Morris [10], Barthlomew and Smith [9],

Smith [66] Barthlomew [7]. Clough et al., [18] Grinold and Marshall [28].

Barthlomew and Forbes [11] and Bennison and Casson [13].

Manpower planning quite often arises in the context of evolving appropriate

policies which would reconcile any difference between the demand for manpower

on one hand and the supply on the other. It is often known as “Closing the

manpower gap”. Cost consideration are also taken into account while formulating

the optimal policies. It may be noted that the prediction of the future manpower

and the future supply for the same is an important aspect of manpower planning.

While making such predictions many factors such as technological changes,

productivity, market forces, trends and corporate strategies should all be taken in to

consideration. In the determination of future strength of manpower knowledge

about current manpower stocks, recruitments, wastage, promotions and labour

mobility should all be taken into consideration.

Manpower planning is an interdisciplinary activity. It requires the combined

technical skills of statisticians, psychologists, economists and behavioural

scientists together with the practical knowledge of managers and planners. In other

words a plan based on Operations Research has to be made for manpower to play

an effective role in the growth of the business. At the national level manpower

planning aims to make the best use of nation’s human resources. This involves the

attempts to forecast the demand and supply for people with various skills and

qualifications and bring them into balance.

One of the important aspects of manpower planning relates to the study of

social system that are subjected to continuous transformations. The changes may

come within the system itself as a need of the hour, or may well be dictated by the

circumstance prevailing outside the system. In order to have a better

understanding of the turmoil within the system, whatever may be the cause of it;

one has to observe the system through its constituents. Study of the manpower in

corresponding to system or an organization in particular, therefore necessarily boils

down to observing its members moving along various grades or set of classes.

Manpower planning may not have been scientifically thought of but

optimal use of manpower dates back to the times of the Roman census to the

accounting of slaves, and eventually to the population census towards the end of

the eighteenth century. Thomas Mathews and Adolphe Quetelet were early

contributors to the establishment of labor economics as an autonomous field in the

following century. Indeed, Quetelet’s emphasis on quantitative manpower analysis

made to reckon him as a founder of manpower planning.

The descriptive and prescriptive models used at the national level of

planning be categorized into three general groups (i) policy conditional forecasts

in which the value of the dependent variables are contingent on the attainments of

policy-oriented independent variables; (ii) onlookers forecasts which extrapolate

historical trends’ and (iii) optimizing forecasts which are driven by an objective

function.

In recent years the steady state Stochastic flow models have been

supplanted with dynamic formulations based on Mathematical programming

techniques with multi-attributed and multiple horizon objective functions to

regulate the trade-offs among promotion, recruitment and organizational expansion

and size. As opposed to static models, these models are characterised by a greater

appreciation of both the multidimensional aspects of planning that produce a viable

policy (recruitment, promotion, transfers, training, attrition career ladders) as well

as the data processing requirements to support it in practice.

The managers of any business concern are faced with, an extremely dynamic

systems of interrelationships. The stock of manpower, its flow either through

retrenchment or voluntary and recruitment have to be carefully monitored that the

production or any work in the business concern does not get affected because of

shortage of labour. Due to the complexity of such systems managers need

assistance to understand them first and then manage them.

It is not uncommon to find a business to face busy and lean periods. In any

business the two main characteristics to get affected to lead to above periods are

Money and Manpower. Reasons are aplenty for a workman to leave the

organization, may be because of better salary and scope for promotion or

otherwise. The leaving is likely to affect at times of busy periods. If more persons

are to leave then the business would be severely affected during busy periods. As

loss and shortcomings are inevitable and fund management have to be done during

busy period, it may require to borrow from private financial institutions or banks

or individuals, to get the same at the busy period is again probabilistic. Labor and

funds become dear during busy periods and may force to pay a heavy cost. The

occurrence of busy period and lean period are random and they occur alternately in

a business organization. It may not become necessary to have full strength of staff

and full availability of funds during busy periods. Their full usages are

intermittent. It is widely accepted that steady state probability is a satisfactory

measure for system which are operated continuously such as for manpower

planning and money management system. A systematic approach to manpower

system was made as early as 1947 by Vajda and others. For an effective stand on

manpower planning one can refer to Barthlomew and Forbes [12], Grinold and

Marshall [28] and Vajda [72]. Lesson [44] has give methods to compute wastages

(Resignation, dismissal and death) and promotion intensities which produce the

proportions corresponding to some desired planning proposals. Markovian models

are designed for wastage and promotion in manpower system by Vassiliou [77] V.

Subramanian [68] in his thesis has made an attempt to provide optimal policy for

recruitment training, promotion, and wastages in manpower planning models with

special provisions such as time bound promotions, cost of training and voluntary

retirement scheme. For other manpower models one may refer K. Setlhare [64]..

For n unit standby system one may refer Ramanarayanan and Usha [57]. Yadavalli

and Botha [80] have examined the same for a two unit system with introduction of

preparation time for the service facility and the confidence limits for stationary rate

of disappointment of an intermittently used system have been studied.

The concept of randomly varying environments in manpower planning

models which has not been studied at any depth before has been introduced. Now

in this technological world, machines are preferred more than human beings

because of complexities involved in production or packing or designing or any

other job. But machines develop problems over a period of time because of wear

and tear. They have to be repaired, and, may have to be replaced when they attain

the state of beyond repair. Business enters the worst crises states when crises

states occur during the time when the machines are under repair. There is

perfection in manufacture or packing or designing if done by machines hence if

machines get corrupted then the production is affected or quality of the product is

lost. The situation may warrant hiring of machine to continue with production. But

this would result in escalation of cost affecting the profitability of the company.

It is an undisputed truth that COMPUTER has made its way into homes just

like television. It does multifarious jobs. Nowadays all works are carried out in

computers, be it shops or banks or manufacturing concerns or educational

institutions. The records are well maintained and preserved for long years. The

main components of a computer are Software and Hardware. Because of electrical

fluctuations, viruses and improper use software or hardware may get affected. If

software gets affected, it can be replaced and work can be continued but this is not

the case if hardware gets corrupted, the work comes to a stand still and requires a

restart as a whole. The crises states become serious if software fails and become

worst if hardware gets affected. Such a state is called catastrophe. The cost of

expenditure will rise accordingly. These are new models developed in manpower

planning considering the modern trend in business.

1.2 Background of Manpower Planning

In the early 1970s many companies were planning significant expansion.

During this period such companies were quick to realize that the key to success

was an adequate supply of appropriately skilled people. This led to the emergence

of human resource planning as a personal management tool. Manpower planning is

the process of ensuring that the optimal number of human resource is available at

the right time and at right place. Companies use analytic and rationale methods to

forecast their human resource requirements for which they need appropriate

analytical tools. Much effort was devoted to developing tools and techniques to

assist managers in their planning. Many of these were based on the theory of

Stochastic processes and more specifically on the concept of Markov chains,

Bowey (15)

In large organizations the flow of individuals between the various ranks is a

task which requires careful and detailed monitoring. A close and a careful study

has to be made in the individuals’ joining, staying and quitting the concern over a

period of time by the managers Such serious study would give a clear picture the

time to shortage, the time of recruitment. In a stable environment where the

features and characteristics of product and labor markets are expected to evolve in

a predictable and orderly fashion, a model of long term organization would

emerge. This would show the expected number of retirements, the expected

turnover of staff, within departments and for involuntary reasons. This can give a

broad and rather basic picture of staff turnover. Hence it can also be used to

provide valuable information on timings and rates for replenishing staff. To

maintain stable level of employees over time management requires data on where,

when and how many employees need to be recruited.

1.3 Approach to Manpower Planning

The simplest and oldest approach to manpower planning is perhaps the so

called replacement table method; a list of men or groups of men presently in the

systems organized by function and job level, provides a description of the current

inventory. The main problem, according to this approach, is to ensure that as men

quit, retire, or die, suitable candidates will be ready to move into their jobs.

Retirement statistics can be predicted with precision and coupled with historical

information, general actuarial data, an estimate about the future loss can be

estimated. Thus rough estimates can be made of where and when vacancies will

occur.

The replacement table method is laborious to carry through by hand

computations, and it usually reflects a static rather than a dynamic picture of an

organization’s structure and needs. The development of other approaches have

come since world war II in response to variety of particular problems.

1.4 Classification of Manpower Models

A computer model can basically be classified into two distinct categories.

In the first category, the flow rate of one grade of manpower professionals into the

same, higher or even lower grades over a fixed period of time is given. This

generates ultimately fixed proportions of manpower in different grades. These

types of models are known as Markovian models or models based on Markov

chains. In second category, given the distribution of the grade sizes, the flow rates

from one category of manpower to other is obtained on the basis of the given grade

structure or probability vector of different grades. These kinds of models are

known as Renewal models.

Concept of Manpower Planning

Process by which Management determines how the management should

move from its current manpower to its desired manpower utilization described as

formal planning is based on the following:

Establishment and recognition of future job requirement.

Scanning the organization through systematic manpower audit.

Assured supplies of qualified participants.

Development of available manpower.

Effective utilization of current and prospective workforce members.

1.5 Steps in Manpower Planning

Predict strategies for manpower plans.

Design job description and the job requirements.

Find adequate sources of recruitment.

Give boost to youngsters by appointment to higher posts.

Best motivation for internal promotion.

Closely observe the expected losses due to retirement, transfer and other

issues.

See for replacement due to accident, death, dismissals and promotion.

1.6 Need and Importance of Manpower Planning

The main factors which determine the behavior of a manpower system are:

(i) Recruitment

(ii) Promotion of employees

(iii) Wastage

(i) Recruitment

The sizes of various grades, which respond in the expansion, promotions and

wastages, are maintained at the desired level at any time by a process of

recruitment. 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. The recruitment can be made by the organization itself or

by some external agencies to avoid delay and huge overhead cost.

The concept of recruitment also plays an important role in planning.

Retirement is a part of flow. The number of recruits in each category is

represented as a recruitment vector. In dealing with the problems of maintaining a

given structure of an organization recruitment control plays an important role and

it is done by the proper choice of the retirement vector r and p, where r is the

number of recruits in each category and p is the transition probability matrix

describing the transitions of individuals between the grades of that organization.

ii) Promotion of Employees

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

recruitments whereas those in the higher are filled up by promotions. Some of the

promotion rules are :

a) The senior most in the grade is promoted.

b) Promotion is given at random.

c) Those that fill certain efficiency criterion along with some minimum

completed length of service are promoted.

As per the rule (a), the length of service is the sole criterion for promotion and

hence the management can control it. The rule (b), gives full freedom to the

management to promote any employee of their choice, which also is not desirable.

Normally rule (c) is preferred. Some of the reasons, which influence the

promotion policies are (i) pressure, (ii) efficiency and (iii) length of service.

Promotion denotes that an individual has the competencies, i.e. the skills,

knowledge and attitude required to perform effectively at the next higher rank.

The competencies reflect the knowledge and skills exhibited in observable

behavior in the relevant areas of work. Promotion provides motivation with

additional salary, responsibility and authority to perform well and is an important

part of performance management.

iii) Wastage

Wastage in manpower planning terminology refers to the leaving process of

persons from an organization. It is the most fundamental concept that plays a key

role in manpower planning. In fact wastage has impact upon the manpower

system. Wastage arises due to individual decisions to leave the organization and is

hence outside the direct control of the management. In organizations where the

number of jobs is controlled, it is wastage which creates vacancy and so provides

opportunity for promotion and recruitments. Hence the measurement of wastage is

very important for the successful formulation of manpower policies, Statistical

analysis of data on wastage is found to be very useful for making policy decisions.

The term wastage is also used to refer to the total loss of individuals from a system

for whatever reasons. Wastage can be either voluntary or involuntary. Involuntary

wastage arises for reasons beyond the control of individuals such as death, illness,

redundancy and retirement and to a large extent it is predictable and it presents few

difficulties in manpower planning. On the other hand voluntary or natural wastage

refers to the leaving of an individual of his own choice such as taking to another

job etc. It is not predictable. The recruitment policy, promotions are all based on

the extent of wastage that occur in an organization. In the study of wastage many

factors are introduced and it may be noted that the Complete Length Of Service

(CLS) plays an important role in the study of manpower models. Forecasting the

future wastage is an important aspect of manpower planning and the cohort

analysis. Markov chain models are used in the forecast of wastage. The survivor

function of reliability function is also used in the study of wastage and its

measurement.

When employees move from one grade to another, they are exposed to

different factors influencing them to leave organization. Various data indicate that

the reasons for leaving can be classified in to the following cases:

i) Discharge

ii) Resignation

iii) Redundancy

iv) Retirement

v) Medical treatment

vi) Death

vii) Transfer.

1.7 Completed Length of Service (CLS)

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. The leaving pattern or wastage

process in any organization is also related to completed length of service. This

aspect has been discussed by Barthlomew [7, 8].

1.8 Cost in Manpower System

A manpower system may be compared to an inventory system in which

there is demand for products and at the same time replenishment is made by the

organization. In many inventory system the holding cost, reorder cost, and the

shortage cost are taken into consideration while formulating an optimum inventory

decision or policy. In the case of a manpower system the depletion of manpower

occurs due to wastage and the same should be compensated by replenishment of

manpower. Hence recruitments are made. The optimal policy or decisions are

taken by considering the associated cost. The various costs associated with policy

decisions in a manpower system are given in Poornachandra Rao [54]. The

following are the costs associated with manpower system.

i) Recruitment cost

ii) Cost of excess or overstaffing cost

iii) Shortage cost otherwise called understaffing cost

iv) Cost of retrenchment/ retirement

v) Retention cost

(i) Recruitment cost

This is the cost incurred in the process of recruitment. Recruitment costs

can be broadly classified in to two categories, viz. fixed and variable, which are

proportional to the number of people recruited. The following are the different

components of recruitment cost:

a) Cost of advertising

b) Cost of administrative authority, which determines recruitment policy.

c) Cost of manpower working on the processing of applications

d) Cost of information processing

e) Cost of conducting written tests

f) Cost incurred in the form of payment to the interview committee members.

g) Travelling expenses paid to candidates.

h) Cost of medical examination done by the organization.

i) Cost of training people

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

ii) Overstaffing costs

Overstaffing costs are those incurred owing to an unutilized work force.

These costs are analogous to the inventory costs in a production / inventory

situation.

iii) Understaffing costs

Understaffing costs are those resulting from decreased productivity and loss

of good will on account of failing to meet the demand of customers in time as a

result of the non-availability of the workforce.

(iv) Firing / Retirement costs

These costs result from the retrenchment or retirement of the employees.

(v) In addition to the various costs identified above, there are certain costs which

are involved in retaining an employee in an organization. These are termed as

retention costs, and they consist of (a) probation costs, (b) training and

development costs, (c) internal mobility costs. Probation costs are those incurred

owing to the learning of an employee during a probationary period. The training

and development are recruitment costs, and are 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 the promotion,

demotion or transfer of an employee within the organization.

1.9 Stocks and Flows

In an organization with hierarchical system we have a number of classes or

categories of employees. At any epoch time T the number of persons in a particular

class is denoted as ni (t) ; i = 1, 2, ……,k. The number of classes and ni (t) are

called the stocks. Hence we have stock vector denoted as n(T) = [n1(t), n2 (t) … nk

(t)]. With the passage of time the movement of persons from one category to the

other takes place and it is called flow. nij (T - 1) to (T). The flows can be

represented in the form of square matrix as follows:

n11(t – 1) n12(t – 1)…...........n1k(t – 1)

n21(t – 1) n22(t – 1) …………n2k(t – 1)

…… …… ……. ……. ………

…………. ……………. …………….

nk1(t – 1) nk2(t – 1) …………nkk(t – 1)

The flows represented in the form of a matrix refer to the transfers,

promotions or demotions in an organization. It may be observed that the flows in

any manpower system may be due to push factors. The persons are promoted from

a lower grade to higher grade because a vacancy arises at the higher grade in an

organization and it is due to acquiring a new qualification it is called push flow.

The push and pull factors play a key role in occurrence of change in the manpower

structure of the organization.

1.10 Push and Pull Model

The push and pull model are based on the principles of Markov process; the

pull models use the renewal theory. Pull models are developed for organizations

where a promotion only takes place when there is a vacancy in another group. Pull

model employees are pulled up once there is a vacancy at another level.

Employees in push models have certain probability to leave or to be promoted

independently from vacancies at other levels, Push models treat people as passive

consumers whose needs can be anticipated and shaped by centralized decision

makers. Pull model treat people as networked creators who are uniquely

positioned to transform uncertainty from a problem into an opportunity.

1.11 Some Basic Preliminary Concepts used for the Development of Models

1.11.1 Shock model and cumulative damage model

The concept of shock model and cumulative damage process is an attractive

one, which helps in the interpretation of the behaviors of complex mechanisms. A

component or device exposed to shocks which cause damage to the device or

system is likely to fail when the total cumulated damage exceed a level called

threshold. Considered here a device exposed to stocks. Suppose that shocks

cause damage and the damages accumulate additively. Let the device fail when

total damage exceeds a threshold level. Assume that the damages X1, X2 … Xn

caused by successive shocks are mutually independent identically distributed

random variables with distribution function F(.), independent of the threshold

whose distribution function G(.). Then the probability that the device survives k

damages is denoted as

Pk = k = 1,2,…….

Where Fk(x) is the k fold convolution of F(x) with itself and F0(x) = 1.

The reliability R(t) of the device is given by

R(t) =

Where vk(t) is the probability that k damages are caused during (0 t]. The above

model has been considered by Esary, Marshall and Prochan [22]. Hameed and

Prochan 130] with the underlying process generating shocks as poisson process,

non homogeneous poisson process and birth process respectively. Ramanarayanan

[56] has considered a cumulative damage process introducing the concept of

alertness of the worker.

1.11.2 Markov process

A random process X(t) is said to be a Markov process if for any time instant

t1< t2< t3< ……< tn+1, the random process satisfies

FX [ X (tn+1) ≤ xn+1 / X (tn) = xn, X (tn-1) = xn-1, …… X (t1) = x1 ] = FX [ X (tn+1) ≤ xn+1 /

X (tn) = xn, ]

Where

tn = present time

tn+1 = some point in future

t1, t2 ,…….. .. .. tn-1 = various points in past.

1.11.3 Renewal process (Recurrent process)

A renewal process is a counting process such that the time until the first

event occurs has some distribution F, the time between the first and second event is

independent of the time of the first event, the same distribution F, and so on.

When an event occurs, we say that a renewal has taken place.

Generally, the Markov renewal process is a counting process which keeps

track of the total number of visits to each site, while the semi Markov process only

records at each point of time the single state at which the process finds itself.

Chapman Kolomogrov Equations

The transition probability Pij ( t + T ) is the probability that given that the

state was i at epoch 0, it is at state j at epoch t + T; but in passing from state i to

the state j in time t + T the process passes through some state k in time t. Thus

Pij ( t + T ) = { ( ) /kP X t k X ( 0 ) = i } P{ X ( 0 ) = i , X( t ) =

k }

Since {X(t)} is a Markov process,

P { X ( 0 ) = i , X( t ) = k } = P{ X( t ) = k }

= Pkj ( T )

Thus we have, Pij ( t + T ) = Pik ( t ) Pkj ( T ) for all i and j and t ≥ 0,T ≥ 0. In

matrix notation it is P(t+T) = P(t)P(T). These hold for discrete space Markov Chain

Denote the right hand derivative at zero by

.aij = when t = o; i j (1)

.aij = when t = o (2)

From the above we get,

( t ) = ai j t + O t) i

( t ) = 1 + ai i t + O t)

It can be seen from the above that aij ≥ 0 i and aii < o

(t) = 1.

Using (1), find

= 0 (3)

= - aii

A = ( aij ) is called transition density matrix. Its off diagonal elements are

non-negetive and sum of elements of any row is zero.

1.12 Author’s Work

The first chapter deals with the introduction to the thesis with a review of

relevant research articles.

Chapter 2 deals with literature survey of manpower planning of the past and

present.

Chapter 3 deals with a business organization under varying conditions

which are restricted to depend on only three characteristics viz, manpower, money

and business under fluctuating conditions of availability and shortage in all the

three characteristics. The different states have been discussed under the

assumptions that change from full availability to shortage and shortage to full

availability occurs in exponential times with different parameters. An expression

for “Rate of crisis under steady state (C∞)” is arrived and the steady state costs

have also been worked by assigning different costs for the characteristics under

different conditions. Derived are expression for rate of crisis under steady state

conditions and also for the steady state probabilities considering availability and

shortage in money and manpower in a business which is assumed to have busy

periods and lean periods alternately, whereas in the second model, besides these

three characteristics money, manpower and business, some environments are

considered that affect the eight states. The different environments may be

government policies, famine or drought, demand for the article of manufacture,

scarcity of some raw materials that may be essential for the manufacture or war

time, etc.

A portion of this chapter with the title “Analysis of Critical States and

Total Cost under Varying Availabilities of Manpower, Money and Business”

has been published in the proceedings of National Seminar on ‘Applications of

Hidden Markov Models in Engineering Fields’, 23.4.2010, at Sri Sakthi Institute

of Engineering and Technology, Coimbatore.

The full content of this chapter in the present form has been published as a

paper with the title “An Analysis of Manpower, Money and Business with

Random Environments” in Int. Journal of Applied Mathematics, Volume 23,

No. 5, 2010, 927940.

Chapter 4 deals with a manufacturing concern. An additional characteristic

has also been brought in to play namely the machinery. A manufacturing concern

apart from being dependent mostly on money, manpower and business is critically

dependent on machines that produce. In the event of machines getting repaired, the

alternatives are to stop production till the machine gets ready for production or to

engage another equivalent one in its place and continue with production but with

higher cost and replace it with the original one after the repair. In this machine

dependent model for production we consider a business organization under varying

conditions of manpower, money and business and which fluctuate between

availability and shortage. The different states have been discussed under the

assumptions that change from availability to shortage and vice versa occurs in

exponential times with different parameters. Expressions for “Rate of crisis under

steady state (C∞)” and steady state probabilities are derived and numerical

examples are worked out to explain the effect of change of parameters for

determining the rate of crisis graphically and expected cost is also calculated.

This chapter has been published as a paper with the title “A Crisis State

Analysis of Manufacturing Concern subject to Varying Manpower, Money

and Business” in Int. Journal of Applied Mathematics, Volume 24, No.4, 2011,

539–548.

Chapter 5 deals with a business concern that depends not only on money,

manpower, but also on computer systems, which has become indispensable and a

dominating part for survival and prosperity of business. System failures can be

because failure of software and software induced hardware failure. In the former,

a substitute can be used to run the show without interruption, where as hardware

failure leads to catastrophe, necessitating a restart as a whole. An analysis is made

on a model where crises occur not only on account of money and manpower

becoming lean but also under situations of software failure and software induced

hardware failure. Expressions for steady state probabilities and rate of crises are

derived and numerical examples are discussed and cost analysis is carried out.

The content of this chapter has been published as a paper with the title

“System with Stochastic Manpower Hardware and Software Interaction” in

Int. Journal of Contemporary Mathematical Sciences, Volume, 6, 4144, 2011.

Chapter 6 deals with a business organization under varying conditions

which are restricted to depend only on manpower under fluctuating conditions of

full availability, moderate availability and nil availability in the case of manpower

and full availability and nil availability in the case business. The different states

have been discussed under the assumption that the transitions from one state to

another in both manpower and business occur in exponential times with different

parameters. An expression for “Rate of Crisis under steady state (C∞)” is arrived

and steady state costs have also been worked by assuming different costs for the

parameters under different conditions.

The content of this chapter has been published as a paper with the title

“Stochastic Analysis of Business with Two Levels and Manpower with Three

Levels” in Int. Journal of Contemp. Math. Sciences, Volume 7, No. 29, 2012,

14211428.

Chapter 7 deals with a business organization under fluctuating conditions of

availability of manpower and business with a special emphasis given to a new and

prevailing idea of business to go off with the manpower leading to crisis state. The

different states have been discussed under the assumption that changes from

availability to shortage and shortage to availability occur in exponential times with

different parameters. An expression for “Rate of Crisis under steady state (C∞)” is

derived and steady state costs have also been worked by assuming different costs

for the parameters under different conditions.

The content of this chapter has been published as a paper with the title

“Stochastic Analysis of Manpower Levels Affecting Business” in International

Mathematical Forum, Volume 7, No. 44, 2012, 21572166.

Chapter 8 deals with two models of a business organization under varying

conditions which are restricted to depend on manpower, money and business under

fluctuating conditions of availability of manpower and business with a special

emphasis given to a new and prevailing idea of frequent changes taking place in

manpower. In the first model are considered three phases of manpower viz.

manpower is fully available, insufficiently available and manpower not at all

available and the transitions take place from full availability to nil availability and

from nil availability to full availability in case of money and business but in the

case of manpower it is assumed that the transitions can be from any state to any

state among the three possibilities. The different states have been discussed under

the assumption that changes from availability to shortage and shortage to

availability in the case business and money and the changes from nil to moderately

available or full availability and moderately available to full availability and

reverse cases occur in exponential times with different parameters. An expression

for “Rate of Crisis under Steady State (C∞)” is arrived and steady state cost have

also been worked by assuming different costs for the parameters under different

conditions.

The results obtained for the model has been published as apaper with the

title “Stochastic Analysis of Three Levels of Manpower Affecting Business

Continuous Time Markov Chain Approach”– in the proceedings of the Fourth

National Conference on ‘Mathematical Modeling in Global Perspective

(NCMMGP)’, 27.04.2012, Velammal Engineering College, Chennai.

There may be fluctuations in manpower in any organization, but care will

always be taken to see that shortages do not become acute. Full manpower is rarely

maintained because of various reasons. So in the second model an intermediate

level is introduced. At this level of manpower the business may not suffer because

of manpower. Three phases of manpower viz manpower is fully available,

moderately or insufficiently available and manpower not at all available or scarcely

available are considered and the transitions take place from full availability to nil

availability and from nil availability to full availability in case of money and

business but in the case of manpower it is assumed that the transitions can be from

nil availability to moderate availability and vice versa and from moderate

availability to full availability and vice versa. No transitions take place between the

lowest and the highest. The different states have been discussed under the

assumption that changes from one state to another in any three characteristic occur

in exponential times with different parameters. An expression for “Rate of Crisis

under steady state (C∞)” is arrived and steady state cost has also been worked by

assuming different costs for the parameters under different conditions.

The results obtained for the second model has been published as a paper

with the title “A Probabilistic Analysis of Three Levels of Manpower Affecting

Business - Continuous Time Markov Chain Model”, in Applied Mathematical

Sciences, Vol. 7, 2013, no. 3, 127 - 134,

Chapter 9 deals with a business organization under fluctuating conditions of

availability of manpower and business with a special emphasis given to a new and

prevailing idea of business to go off with the manpower leading to crisis state. The

different states have been discussed under the assumption that changes from

availability to shortage and shortage to availability occur in random times with

general distributions. Time dependent and time independent probabilities of the

states of the system are presented. Two models are treated. In model 1, manpower

and business fluctuate independently and in model 2, business on hand goes off

with manpower. The probabilities describing the transitions in various states are

derived for general distributions using integral equations applying renewal

theoretic arguments. Numerical illustrations are given.

The content of this chapter has been submitted for publication with the title

“Stochastic Analysis of Manpower and Business”.

Chapter 10 Conclusion

Keeping the three main characteristics, the crisis state, rate of crisis and

steady state probabilities are studied with application of continuous time Markov

chain under different environments. The fluctuations in all the characteristics are

assumed to follow exponential distribution but for further Research other

distributions may also be used and the various operating characteristics may be

studied. The concept of randomly varying environments in manpower planning

models is introduced which has not been studied at any depth earlier. The idea of

business to go off with manpower is a new concept which has not been touched

earlier. These ideas have been brought into the models which have a wide scope

for further research. The idea of deriving the probabilities describing the transitions

in various states with general distributions using integral equations and renewal

theoretic arguments presented in the last chapter is new and may be useful for

studies in similar manpower models.