University of Nigeria Research Publications

OGBU, Francis A.

Aut

hor

PG/M.Sc./83/1894

Title

A Model of Personnel Prediction in a Graded Organization

Facu

lty

Physical Sciences

Dep

artm

ent

Statistics

Dat

e

July, 1985

Sig

natu

re

OGBUj FRANC IS AKUJaRI ( P G / M . S C . / ~ ~ ; I ~ S ~ )

E.Sc, ( H o n s . ) , U . N . N .

DEPARTMENT OF STATISTICS UNIVERSITY OF N I G E R I A , MSURKA

NODEL OF PERSONNEL PRED1C;TION IN A GRADED ORGNIIZATIQ!!

OGBU, FRANCIS AKUJOBI

(Hans.), U.N.N?

BEING A PROJECT SUBMITTED TI? PARTIAL

FULFILMENT OF THE REQUIREMENTS FOB THE

AWARD OF M, ~ c ' . (STATISTICS) DEGREE

UNIVERSITY OF N I G E R I A , NSUKKA

DEPARTMENT OF S T A T I S T I C S

SUFERVXSORS:

1 . D R . P . I . U C H E 2 . M R . W . I . E . CHCXSXJ

This work is dedicated to my

dear mother, Mrs. Margaret OgbU who,

since the death of m y f a t h e r i n 196 1 ,

has shouldered the g r e a t t a s k o f

t r a i n i n g her six children up to

University level-

ACKNOWLEDGEMENT

It i s a p a r t i c v ! a r p l e a s u r e t o a c k n o w l e d g e t h e

s c h o l a r y a n d f a t h e r l y a d v i c e a n d g u i d a n c e o f my

S u p e r v i s o r s , D r . P . I . Uche and M r . W.I.E. Chukwu t h r o u g h

o u t t h e o r g a n i z a t i o n o f t h i s work.

My s i n c e r e g r a t i t u d e g o e s t o my c o l l e a g u e s a n d

f e l l o w s t u d e n t s f o r t h e i r immense e n c o u r a g e m e n t and a d v i c e

when t h e g o i n g seemed r o u g h . I an a l s o g r a t e f u l to

o f f i c e r s a n d s t a f f o f t h e P e r s o n n e l S e r v i c e s D e p a r t v e n t of

t h e U n i v e r s i t y o f N i g e r i a , Nsukka f a r t h e i r h e l p and

c o - o p e r a t i o n d u r i n g the c o l l e c t i o n o f d a t a w h i c h w a s u s e d

i n t h e a p p l i c a t i o n .

I a m h a p p y to t h a n k my b r o t h e r s a n d s i s t e r s and

o t h e r r e l e t i o n s f o r t h e i r mora l and f i n a n c i a l s u p p o r t . I

a l s o t h a n k M r . Bas i l I . Omeye f o r t h e t y p i n g o f t h i s work.

I am t h a n k f u l t o t h e a u t h o r i t i e s o f t h e U n i v e r s i t y o f

Maidugur;-, Y o l a Campus who made i t p o s s i b l e f o r m e t o

p u r s u e t h e M.Sc. ( S t a t i s t i c s ) d e g r e e c o u r s e .

F i n a l l y , many r e f e r e n c e s were c i t e d i n t h i s work;

t o their a u t h o r s and p u b l i s h e r s , I a m h i g h l y indepted.

OGBU, F R A N C I S AKUJOBL JULY, 1985.

ACKNOWLEDGEMENT .. . L a = a q . . TABLE OF CONTENTS . . @ a , . .

1 . INTRODUCTION AMi: LITERATURE REVIE!? . . 1 . . Introduction . . o o e o

1 . 2 Aims And Objec t i -ves I O . . 1 . 3 Literature Review . . . * . .

PAGE - .. ii

. . iii

2 . THE PROPOSED MODEL . * . O Y O .. ?

2.1 Assumptions .. . . . O * . . . 9

2 . 2 N o t a t i o n s .. e . . O . . .. $ 0

2.3 TheMo2e l .. . . e - . .. 1 1

2.4 Validation Of The Model ,. . . .. 14

APPLICATION .. . . 3 . 1 Collection Of Data . . . .. 17

3.2 Test For S t a t i o n a r i t y .. . . .. 18

3 . 3 C a l c u l a t i o n O f The Stochastic Matrix and Futrlre Grade Sizes . . . . 24

3 . 4 E x p e c t e d V i thdrawa l s . a . . . . 2P,

3 . 5 E x p e c t e d Lengths of S t a y . . . . 29

3 . 6 Variances and Standard Deviat ions of Lengths Of Stay . . . . . . .. 31

4 . "1:" PROBLEM OF CONTROL OF GRADE SIZES

4 . 1 C o n t r o l By Maintainability .

5 . CONCLUSION . . . . . . . . . .

REFERENCES

APPENDIX . a

CHAPTER 1

INTRODUCTION AND LITERATURE REVIEW

I . I 1 NTRODUCTION

I t i s a f a c t t h a t c a r e e r s o f p e o p l e i n a h i e r a r -

chical o r g a n i z a t i o n a r e n o t a l w a y s t h e same. E v e r y member

o f t h e o r g a n i z a t i o n a s p i r e s t o r i s e t o t h e t o p but n o t a l l

achieve this; some e v e n l e a v e t h e o r g a n i z a t i o n b e f o r e

r i s i n g t o any o f t h e t o p g r a d e s . F o r a l o n g e s t a b l i s h e d

o r g a n i z a t i o n , t h e v a r i o u s grades w i l l be composed o f

members who j o i n e d t h e o r g h n i z a t i o n a t d i f f e r e n t t ine and

on d i f f e r e n t grades. I n o r g a n i z a t i o n s l i k e t h r c i v i l

s e r v i c e and i n s t i t u t i o n s o f l e a r n i n g , i t i s n e c e s s a r y t o

manage and c o n t r o l s t a f f e f f i c i e n t l y s o t h a t o p t i m a l

s e r v i c e s w i l l b e rendernd.

S t u d i e s o f t h e nmvement of p e r s o n n e l t h r o u g h a

g r a d e d o r g a n i z a t i o n i s t h e n o f i n t e r e s t i n g i v i n g t h e

c a r e e r e x p e c t a t i o n s o f members of t h e o r g c - n i z a t i o n . Also,

t h e management of s u c h a g r a d e d o r g a n i z a t i o n would w i s h t o ,

among o t h e r t h i n g s , h a v e a n i d e a of t h e f u t u r e grade sizes

as this helps i n good b u d g e t i n g a n 2 p l a n n i n g . For instance,

i f t h e o r g a n i z a t i c n i s becornin, t o p l a d e n , t h e management

would want t o c5eck this, ma kin^ sure t h a t t h e method

employed d o e s not r e s u l t i n f r u s t r a t i o n among the s t a f f .

The p o i n t t o n o t e h e r e i s that n o t t h a t a n increase i n t h e

number o f s t n f f a t t h e t o p grades r e l a t i v e t o t h o s e o n the

l ower g r a d e s i s u n e e s i r a b l e , b u t t h a t the f o r m e r c o s t s

more i n s a l a r y . Thcre i s o f c o u r s e a need t o m a i n t a i n

b a l a n c e be tween v a r i o t 3 c a d r e of s t a f f .

I n r e c e n t y e a r s , a number of models f o r manpower

p l a n n i n g have been suggested and a p p l i e d i n r ,overnmen'tal

and p u b l i c a g e n c i e s , i n d u s t r i e s , e d u c a t i o n , h e a l t h as w e l l

as in the m i l i t a r y . The f u n c t i o n a l forms of the models

i n e l u d e : - r e g r e s s i o n models , s i m u l a t i o n models ( b o t h

s t o c h a s t i c and d e t e r m i n i s t i c ) , and Markov c h a i n mode l s .

Many o f t h e s e models d e a l w i t h different forms o r s t r d c t u r e s

of t h e o r g a n i z a t i o n a s a whole w h i l e o t h e r s d e a l w i t h

s p e c i f i c a s p e c t s o f t h e o r g a n i z a t i o n s u c h as w i t h d r a w a l s

(waet s e e s ) , Length o f s e r v i c e , a t t a i n a b i l i t y of d e s i r e d

s t r u c t u r e s and m a i n t a i n a b i l i t y of e x i s t i n g s t r u c t u r e s . F o r

i n s t a n c e , Zanakie P - 980I], u s i n g a s i m p l e l i n e a r r e g r e s s i o n

model, mode l l ed t h e w i t h d r a w a l and promot ion p r o c e s s e s a s

l i n e a r f u n c t i o n s o f growth and t h e p o p u l a t i o n e rowth pe r

perLod was expressed a s

(i) t o t a l gains o r l o s s e s

(ii) a c c e l e r a t e d g a i n o r l o s s , and

( i i i ) c u m n u l a t i v e a c c e l e r a t i o n .

A stochastfc programming model was d e v e l o p e d by

Abernnthy e t a1 u 9 7 2 7 for t h e p lanninp , and a r l e d u l i n g of

manpower ( ~ u t s c s t a f f i n g ) . I n t h i s model t h e p r o c e s s f o r

e t a f f i n g s e r v i c e s was d i v i d e d into three d e c i s i o n l e v e l s ,

name l y : -

3 .

i n g t h e o p e r a t i n g p r o c e d u r e

for s e r v l c e c e n t r e s a n d f o r t h e s t a f f c o n t r o l

p r o c e s s i t s e l f .

( b ) S t a f f p l a n n i n g ( h i r i n g , d i s c h a r g e a n d t r a i n i n g ) , and

( c ) S h o r t - t e r m s c h e d u l i n ~ o f a v a i l a b l e s t a f f w i t h i n

the c o n s t r a i n t s d e t e r m i n e d by (a) a n d ( b ) a b o v e . I

Then t h e p l a n n i n g and s c h e d u l i n g s t a g e s were f o r m u l a t e d as

s t o c h a s t i c p rogramming p r o b l e m s and b o t h iterative s o l u t i o n

~ r o c e d u r e ( u s i n g random l o s s f u n c t i o n ) a n d n o n - i t e r a t i v e

s o l u t i o n p r o c e d u r e f o r a c h a n c e - c o n s t r a i n e d f o r m u l a w e r e used.

P e r s ~ n n e l s u p p l y i n a h i e r a r c h i c a l o r g a n i 7 a t i . c n c a n

a l s o be y r e d i c t e d u s i n g Markov C h e i n t o model t h e f l o w o f

p e o p l e t h r o u g h t h e v a r i o u s g r a d e s ( s k i l l s o r * p o s i t i o n levels).

I n cne of s u c h s t u d i e s Younf: and Almond p 9 6 1 3 c o n s i d e r e d

a n o r g a n i z a t i o n w h i c h i s s t i l l e x p a n d i n g and f o u n d t h a t t h z

d i f f e r e n c e s between a c t u a l and p r e d i c t e d r e s u l t s were s m a l l .

G e n e r a l l y , t h e p u r p o s e o f a l l t h e s e manpower p l a n n i n g

m o d e l s i s t o e x a m i n e t h e l i k e l y e f f e c t s o f d i f f e r e n t

r e . L u i t m e - t , p r o m o t i o n , and w i t h d r a w a l policies o n f u t u r e

manpower needs and r e s o u r c e s a t t h e various l e v e l s o r

g r a d e s i n the o r g a n i z a t i o n . I n t h i s work, we u s e d t h e

Markov C h a i n t o d e s c r i b e t h e method of y r e d i c t i n g t h e

number and d i s t r i 3 u t i o n o f s t ~ f f among t h e v a r i o u s g r a d e s Ln

f u t u r e y e a r s f o r a graded o r ~ a n i z a t i o n s t h a t h s a f i x e 6

s i z e b u t where w i t h d r a w a l s a r e random. A m a t h e m a t i c r l n o d e l

was obtained snd applied to j u n i o r staff of the

University of Nigeria, Nsukka ( U . H . R . ) . In other words,

the work will be conaerned with describing a stochastic

model for predicting the movement of personnel through

various career levels in a graded organization. Ve shall

also look at the problem of control associated with t h e

movement.

1 . 2 AIMS AND O B J E C T I V E S

In this work our aims are as follows:

(a) To describe a stochastic model which can be u b e d to

p r e d i c t the composition of a graded organization which

has a fixed size, or to p r e d i c t the distribution of

staff among the various grades of such a hierarchical

organization.

( b ) To investigate the options available for such an

organization for the control of its sizes - control strateqies.

(c) To apply the model to the data on the Junior Staff at

U.N.P. so as

(i) to estimate the transition probabilities

(ii) t o estimate the expected length of stay in 3.

grade and in the entire system; and their

etandard errors.

(iii) t o estimate the unconditional probability cS en

an entrant to a grade to dttain higher 2rades .

(iv) to estimate e x p e c t e ? future grade sizes and

expec ted wit~~r?rawzl:!.

1 .3 LITERATURE REVIEW

Literature on manpower planning using Markov ~ h a 5 n s

abound. Bartholomew 2 9 8 2 3 discussed extensively Markov

Chain rnoder,3 arid his worb provides a major basis for this

work. His work covers manpower planning models both in t h e

deterministic and stochastic environments.

In the educetional s e c t o r , manpower plannix3 usin?

Harkav chain modzl kas been stuaied by many marrpower planners

and academics. Townstad E 9 6 q discussed and gtlve many

practical examples of manpower planning in this sector.

Gani p96g w a s one of the proponents of the Markovian model

in education. He conseructed a simplified model for students

progress through a university for some few future years and

then used it to p r e d i c z the enrolment and the number of

bachelor's degree awarded in an Australian University.

~ a r s ! 1 a 1 1 & 3 7 ~ described both the Markovian model which

u s e s cross-sectional data of an organization in a given

time period to precict the com~osition of the organization

in t h e following t i n e period(s1, and the cohort model which

follows each groa? a f newly entering people (. ~hoxt) over

t h e f r life times in t h z organization. He then compared the

two types with v z a p e c t to the prediction of s t u c e n t enrol-

ment a c the sophomore, junior, and sdnior levels st

Berkeley U n i v e r s i t y . R e coo- luded t h e t when c o n s t a n t

c o h o r t sizes are u s e d , tk2 Mzrkov Chatn node1 qi.ves

e s s e n t i a l l y t h e s a n e p r e d i c t i o n as the c o h o r t model , the

c o m p l e x i t y of t h e l a t e r n o t w i t h s t a n d i n g . O t h e r stcdies

o n manpower p l a n n i n b i n e d u c a t i o n u s i n g Markov c h a i n m o d e l

include that of S t o n e 597g. I n t h i s study, Stone likened

t h e f l o w of p e q p l e i n t h e e d u c a t i o n s y s t e m t o a demographic

process w i t h w i t h d r a w a l s a s d e a t h s and r e c r u i t m e n t s a s

b i r t h s . Thonstad 9697 , i n f o r e c a s t i n g s c h o o l a t t e n d a n c e

i n a l l p a r t s of zhe s c h o o l systen ae w e l l a s E i n a l

g r a d u a t i o n f rom a l l t h e d i f f e r e n t t y p e s of s c h o o l s , mode l l ed

t h e problem u s i n g Markov Chain a p p r o a c h . Uche E 9 7 8 7 , i n

s t r - 3 y i n g t h e e d u c a t i o n a l s y s t e n of N i g e r i a , u s e d t h e Narkov

Chain r m d e l and showed t h a t t h e s y s t e n in e x i s t e n c e was

ill p r e p a r e d for t h e p r o d u c t i o n o f t h e much-needed nanponer

i n t h e c o u n t r y . A q a p p l i c a t i o n of t h e Markov Cha in model

io t h e s e c r e t a r i a l u n l t o f t h e i ? n i v e r s i t y o f Nigeria was

done by 3gbua3u i n 1950. Also a t t h e U n i v e r s i t y of N i ~ e r i a ,

Anyanwu D98g s t u d i e d t h e s t a f f i n g p rob lem of n u r s e s i n

t h e U n i v e - s i t y MeGical C e n t r e u s i n g t h e Markov a p p r o a c h .

H e l ooked a t t h e p r o b l e m o f c a l c u l a t i n g t h e e x p e c t e d

number of n u r s e s i n e a c h g r a d e a t a given t i l e f c r

p a v t i c u l a t r e c r u i t m ~ n t p o l i c i e s . H i s c a l c u l a t i o n s w a s

t h r o u g h t h e s p e c t r a l r e p r e s e n t a t i o n of t h e T r a n s i t i o n

P r o b a b i l i t y Matrix.

In tht i n d l : s t r i a l s a c t o r , s t u d i e s and a p p l i c a t i o n s

of Markov Chain manpower models ? .nclude t h e work by

Zanaki G98g. He l s e d t h e M-lrkov Chain to madel t h e man-

power s u p p l y of e n g i n e e r s i n a l a r g e compa.~y. I n t h a t work,

i n s t e a d of t h e u s u a l p r a c t i c e of t r e a t i n g g a i n s ( r e c r u i t ~ e n t s !

as a separate i n p u t d e c t o r , t h e y w e r e made p a r t o f the

transition probability m a t r i x to p r o v i d e a s i m p l e and

u n i f i e d p i c t u r e of a l l t r a n s i t i o n s . H e remarked t n a t l o n g e r

p e r i o d s of 0 5 s e r v a t i o n are n o t always a d v a n t a g e o u s b e c a u s ~

even t h o u ~ h t h e y y i e l d b e t t e r e s t i m a t e s of the t r a n s i t i o n

probabilities due t o p o l i c y o r o r g a n i z a t i o n cnanses , i n c r e a s e

t h e risk of n o n - s t a t i o n a r i t y which Leeds t o p o o r prediction.

H e also recommended t h a t r e g r e s s i o n models c o u l d be used t o

o b - a i n b e t t e r e s t i m a t e s o f p romot ion and wastage p r o b a b i : i t i e s

i n o r g a a i z a t i o n s experiencing growth o r d e c l i n i n g

p o p u l a t i o n t r e n d s . G r i n o l d p97g i n v e s t i g a t e d t h e p rob lem

of producing a comno+i ty with u n c e r t a i n f u t u r e demand with

time lags in t h e p r o d u c t i o n p r o c e s s and t h e commodity i t s e l f

b e i n g a v i t a l i n p u t i n t h e p r o d u c t i o n p r o c e s s . R e m o d e l l e d

t h e p rob lem u s i n g t h e Markov Chain a p p r o a c h and a p p l i e d it

t o t h e U n i t e d S t a t e s Naval A ~ i a t i o n System.

There has a l s o been several s t u d i e s c n s p e c i f i c

a s p e c t s of t h e Markovian model i n manpower -1ann i r rg . F c r

e - a m p l e , a stochastic node1 f o r the d e s c r i ? t i o n , p r e d i c t i o n

and c o n t r o l of w a s t a g e in h i e r a c h i c a l o r g a n i z a t i o n s w a s

studied 537 V a e s i l i c r d P 9 7 6 - j . - - Davies P ? S ~ , examined

the problen of maintainir.2 thz (e-x~ected) grade sizes c f

a hierarchical orgsnization ae. zach a t e l (time unit)

with 3 fixed promotion policy and control on recruits.

Davies p98221, also discussed the control of g r a d e sizes

in a partially stochastic Mnrkov manpower planling noJal.

Here, the discussion was mainly on the probabi'ity of

attaining the d e s i r e d structures and structural paths

using recruitment control. The limiting behaviours of

manpower s y s t e m where the non-homogenous Mark~v Chain

has independent ~ o i s s o n input has been studied by

Vaseiliou 98g . Other works include those of Lesson p 9 ~ g on th.-

determination of wastage and pronotion intensities

requ ired to bring about any desired s e ~ of future grade

distributions; and Glen p 9 7 7 3 which outlined an

appropriate method f a r the determination of the 1ere;th

of bervi?e distribution in Markov manpower planning

modclq. It is pertinent to say that the above-mentioned

references do not e x h a u s t the literature on Markov Chain

approaches to manpower planning modelling.

CHAPTER 2

THE PROPOSEP EODEL

2.1 ASSUMPTIONS

In this work we consider a graded system which has

constant size s o that the basic set of quantities which

we shall be dealing b;th w i l l be the stock - the number of people in each grade at n given time. Changes In c h i s

s t o c k occur as a result of flows in and out of the s y s t z n

(recruitments, promotions end withdrawals).

In d e v e l o p i n g a model o f t h e flow of personnel

through the system, we have to take i n t o consideration the

recruitment, promotion axd withdrawal processes of t h a t

sys tem. We s h a l l take as states or grades the salary grade

lev,le and e h a l l denote t h e number o f such states by k, with

the ( k + O t h state denotiap the a b s o r b t i o n state. We shall

assume t h a t a l l promotions occur at the end o f the year

(annually) that no nenber of the organization is promoted

twice a y e a r , Thus the stage interval will be taken as

one year,

New appointments and transfers of s e r v i c e i n t o t h e

organizati~n will constitute the recruitment process. We

shall also assume t h a t recruitments can be made into any of

the k grades and that the nnmb5r o f recruits at any tine

p e r t o d i s randox. ' I 3 e recruit men,^ vector will be denoted by

r .

10.

Furthor, we assume that withdrawals occur dae to

resignations, retirements, deaths and dismissals. The

withdrawal vector b P . shall denote by V.

For promotions, ac assume that these are directly

related to number 0 2 vacancies occuring higher up, the

level of competence and ;he length of service. T h e transi-

tion pr b a b i l t t y matrix (TPM) which signifies the

probabilities of movement from grade to grade will be denoted

by P. We shall also assume that promotions are nsde only

to the next higher grade, i . e . , there axe no demotions and

no jumps.

Finally, we shall assume that individual movements

are independent and that !-ransitions are constant over tine,

2.2 NOTATIONS -- We now define the following additional notations we

wish to use in the texc:

xi(o) = the initial grade size of grsde i.

N = the ( f i z q d ) size of the system

x~(L) = the No. of person3 in grade i at time t,

i= 1 , 2 , ...., k; t = lp29....0T.

x It) = t h e No. of persons 5-50 move to grade j from i j

grade E at t i m e t ,

x :k+l) = eh, recruitment to grade j at time ( + + I ) o,j

x ,k+l(t) = the S o . of persons vho wit3draw fro,m g r e d e i i

at tine t.

x i ( t 4 - 1 1 = x i t + = t o t a l r e c r u i t m e n t i n t o t h e i

SyStaT a t t i m e ( t + l )

' i j ( t ) = Prob. of a person i n g r a d e i ~ o v i n g , t o p,rade j at:

rime t . ( I f t r a n s i t i o n i s s t a t i o n a r y t h e n P . . 1 J

'ij (t) = P . ., f o r a l l t ) . 1J

P . = Prob. of a n i n d i v i d u a l wi thdrawing from g r a d e i l,k+l

P = Prob. of r e c r u i t m n n t i n t o g r a d e 2 . o , i

F o r r e a s o n s of convenience , we s h a l l u s e t h e n o t a t i o n s w . and 1

r . i n s t e a d of P . 1 1 9 k + l and Po ; t o d e n o t e p r o b a b i l i t i e s o f

9 - withdrawing from and r e c r u i t m e n t i n t o g rade i , r e s p e c t i v e l y .

2 . 3 THE FIODEL Thus the p r o p o s e d mode l w i l l be specified by t h e

(a) A m a t r i x , P, a £ k a c s i t i o ? p r o b a b i l i t i e s g o v e r n i n s t h e

movement within t h e system. P i s denote2 by

( b ) A v e c t o r o f w ~ i h d r a w a l p r o b a b i l i t i e s d e n o t e d by

( c ) A v e c t o r of r e c r u i t a l e n t p r o b a b i l i t i e s which we d e n o t e

We shall n o t e t h a t k

and

A l s o

Having s p e c i f i e d all t h e s e we c a n now g ive t h e number o f

persons (stock) in grade j at time t + l . Given t h e s t o c k a t

t i m e t, the number i n grade j a t t i m e ( t * l ) c o n s i s t s o f :

t h e new e n t r a n t s i n t o g r ~ d e j, t h o s e who remained i n grade j

during (t, t + l ) and t h o s e who noved from i t c j durrng (t,t+l).

When the above r e l a t i o n s h i p i s w r i t t e n m a t h e m a t i c a l l y , w e

have

If w e d e n o t e those who r-ained i n grade j ( s u r v i v o r s i n qrade

t h r - equat : on ( 4 ) becomes

Since f o r t - o, the expected ~ r a d e s i z e s are random

v a r i a b l e s and can not be predicted with certainity, we shall

concern o u r s e l v e s with the exneVred grade sizes. Given the

s t o c k a t rime t as x;(t), the f l o w from i t o j , x.. h a s a - 1.1 '

binomial d i s t r i b u t i o n , i.e. x i j 1 R(x; (t) , ?, . : i i-d bence J-J

the expected flow f r o m grade i to grade j in time (r+l)

will be xi(t) P i j . Taking ex?ectarions tern by term and

using the 'barv sign to denote expected values, we obta irL:

Furthermore, we have t h a t i n an organization with a fixed

size, the total number of recruitments muat be equal to the

number of withdrawals. That is, we m u s t have

R(t+l) 1 C t ) i

Hence it follows t h a t

and t h e r e f o r e

A c c o ~ ! i n g l y , taking expectations in (6) and u s i n g ( 7 ) , (8)?

( 9 ) and ( l o ) , we derive the b a s i c p r e d i c t i o n equation a s

Equat ion ( 1 1 ) can be e x p r e s s e d u s i n g matrix n o t a t i o n as

G ( t + l ) = x(t) CP + w 1 r )

= x ( t ) Q o .. (12) where Q is a stochastic matrix with the (ij)th e lernant g i v e n

We c a n o b s e r v e that t h e m a t r i x Q i s n o t t r i a n g u l a r i n

patter^ and s h a l l t h e r e f m e n o t v s e t h e s p e c t r a l r e p r e s e n t -

a t i o n of Q as g i v e n by Barthol .~mew P 9 8 3 , b e c a u s e of t h e

c3mplex i ty i n v o l v e d w h e n the d g e n v a l u e s of t h e m a t r i x d o n o t

c o r r e s p o n d w i t h t h e d i a g o n a l e l e m e n t s ; r a t h e r we s h a l l a p p l y

t h e theory of Matkov Chains and u s e the b - h a v i o u r o f Q t o

discuss and answer q u e s t i o n s abou t t h e model d e s c r i b e d .

Using t h e p r e d i c t e d v a l u e a t t ime ( : + I ) we o b t a i n t h a t

f o r ( t + 2 ) and s o on. T*lat is ve have

G ( t + 2 ) = x ( t ~ l ) ~

- G ( t * 3 ) = xtt+2:q

e t c .

2 . 4 V A L I D A T I C N G'r' THE IIiODEL (TEST FOR STATIONARI~Y) -- - The p-redic.:!.~:, e q o a t i 3 n , G ( t + l ) = x ( t ) ~ i s t r u e whethe r

the p r o b a b i l i t i e s a r e c o n s t a n t o r got. But i f t h e a s s u m p t i o n s

of s t a t i o n ~ r i t y i s nct v a l i d a t e d , w e would have t o u p d a t e t h e

m a t r i x It b e f o r e ucing i t to 1 i s a i c t f o r each new t ime p e r i o d .

I n o t h e r w o r d s , we would be d e a l i n g with e q u a t i o n s of t h e

I n this s e c t i o n d e s h a l l g i v e a test f o r t e s t i n 8 t h e

a s s u n p t i o n of stationary t r a n s i t i o n p r o b a b i l i t i e s . A s s u m p t i o n

of constant r r a n a i t i o r i p r o b a b i l i t i e s i m p l i e s t h a t

P. .(t) = Pi: fcr all t and consequently 1.1 - J

q i j (t) = 4:. Gar all c . 1. j

We state OUT hypotheses as follows:

Ho: T r a n s i t i o ~ probabilities are constant over time

i . e . P i j ( t l = 'ij

for all t

H~ : Transition probabilities are not constant over time.

Then, under the null hypothesis of stationarity, the maximum

likelihood e s t i ~ a t e s of P . .(t) and P are g i v e n by 1.l i j

For a given i, we can form an (k+l)xT contigency table

representing the joint estimates P (t) and for j = 1,2p...,k+l, ij

and t = 1?2, . . . .? as shown below

2, (Tj a - s * i! 'i,k+l (TI

This is the l a y o u t f o r the standard test o f homogeneity i n

concigency t a b l e s discussed by Anderson an3 Gaodmar. E95g.

Thus, testing the above hypothesis is equivalent to

testing that the random variablearepresented by the T rows

have tne same d l a t ~ i h u t i o n or that t h v r e are k+l constants k+ 1

Pi], Pi29.... 'i.k+l 7ith 1 P i j = 1 such that the j - 1

probability associ=ted with-the j t h column is equal to P; - j in all T rGWSr

At the lsvel of significance a, we have that:

(a ) Transitions from a r o w state i a r a stationary if

where m is the No. of these P i j v s for which P. > o :j

(b ) The entire transition probability matrix (TPM) is

constant over time if

k kol T 2

i = l j ( i ) t = i lJ J lpij < x a k ( m - l ) ( ~ - ~ )

The above statistics are also sunmarized by Zanaki et z l

F 9 8 g . We shall use the above to test our dsta far

stationarity kefore prediction will be made.

CHAPTER 3

APP?aI CATION

The stochastic model described in this work w a s

applied to the junior ztaff cadre of the University of

Nigeria. The states are the salary grade levels USS 01 - 05. Thus there are five transient states and one absorbtion state

(which comprises those who either move out of the junior

staff cadre or leave the services of the University entirely,

3 . ! COLLECTION OF DATA - The data u s e d were collected from the Personnel

Services Department of the University of Nigeria. It covered

staff in both the Nsukka and Enugu C a q u s e s of the University,

Every staff has a personal file kept by Personnel Services

Department where records of caree- attainments of the

individual arc documented. It was not possible to go through

all the personal files of the 5,765 junior staff in the

omoloyment of the Univerity. Therefore a random sample

500 files were selected. Information reearding year

recruitment, promotions and withdrawals were extracted

from this, we estimated the transition probabilities,

recruitment rates and withdrawal probabilities. The po

covered is 1975 to 1983 .

We were able to get the number of withdrawals bec

their personal files are still kept in the same filing

I 8.

together with those of staff still in service. With the

register maintained by the Department, w e were able to

pull o y t files that fell into OLT saap le .

There were some problems encountered in the process

of collecting the 4ata 3uch as file locations; some files

that fell into our sample were not in their cabinets.

However, with the file movement reg i s ter , maintained by t h e

Department, we were able to trace such files. T % e r e was

also the problem of doubting if information could not be

let out to unwanted people, but with the promise of

couiidentiality, 1 was allowed to go through. the file o .

Information was ~ ' z u , vhere necessary, call-ected

from the Planning Division of the Vice-Chancellor's Office.

Summary of the data collected is given in the appendix.

3 . 2 TES'ZIb?C FOR STAT1ONA';ITY

We applied the test described in section 2.4 to the

d a t a . For the applications we hzve that the states are

i = 1 , 2 , .... 6 and the times of observation are

t = 1,2,....6 ( 1 9 7 5 - 1 9 8 3 ) . Our hypotheses are:

H : Transitions from a r o w state i are stationary (a>

0

PA: Trensitions from a row s t a t e i are not stationary.

Ho: The entire transition ?robability matrix (TPM) is (b 1

conszant over time

: The entire Transition Probability Matrix (TPL:) is

t on time.

Ic+l T

TI m 1 / P i j and w e r e j e c t j (i) t-::!

the n u l l hypothesis a: u i f T I > X 2 ( ~ - l ) ( r n - ~ ) a

For ( b ) the tes t statistic i s

r e j ec t ed i f T2 > X2(k)(m-~)(~-l). a

Po?: easy follon through, we shall g i v e the transitions

calculated for each t i m e o f observetion and illustrate the

procedure by using i = 1 . The rcsults for the o ~ h c l - s t a t e s ,

i = 2 , 3 , 4 , 5 , 6 shall only be stated.

TRANSITIOR RATIOS FOR 1 9 7 8

TRANSIT ION R A T I O S ?'OR 1979

TRANSITION R A T I O S FOR 1980

T R A N S I T I O N RATIOS FOR 1981

TRANSITION RATIOS FOB 1 9 8 2

3 4 4 I?

.oooo

TRANSITION RATIOS FOR 1983

Now, to test for example, that transitions from zrade or

state i = 1 t o the other grades are constant over time V s

t h a t they are dependent on time, we arrange the transition

ratios for stnee I in the contingency table belo,w. All t h e

t e s t s are a t t h e 5% level of significance.

The T e s t Statistic is

A t 5 % l e v e l of significance, the cut o f f p o i n t is the

v a l u e of ~ ~ ~ ~ ~ ( 1 ) = 11.31. S i n c e our t e s t statistie

X: = 1 0 . 2 2 2 2 < X 2 , 0 5 ( 1 0 ) * 13 .31 , we have no b a s i s t o reject

the n u l l hypothesis of r h z t t r a n s i t i o n s from state 1 t o t h e

other states are s ~ a t i o n a r y over t i m e . The r e s u l t s f o r

similar t e s t s f o r grades 2 , 3 , 4 , 5 and that for the entire

TPM are s e t down i n t h e t a b l e b e l o w .

Table 1 : RESULTS OF TEST OF STATLOMARITY OF TRANSITIOF PRCIBABILITIES

1 1 1 0 . 2 2 2 2 1 18.31 / 10 I ACCEPT I

I STATE VALUE OF TEST S T A T I S T I C ( X ?

I

These show that the t r a n s i t i o n s from each g r a d e of t h e

s y s t e m to the other gradeds and in t h e e n t i r e T P Y are constant,

thus validating the assumption w e earlier made i n the

development o f t3.a nodel.

CUT OFF POINT AT a = 0.05

2

3

4

5

TPM

DEGREE OF I FREEDOM(df) I

i

1 3 . 4 7 3 2

1 3 . 9 3 4 0

7 . 3 4 1 3

1 . 9 3 4 9

47.4054

18.31

18.31

18.31

1 8 . 3 1

9 6 . 2 2

10 ACCE35'

10 ACCEPT

10 ACCEPT

10

75 ACCEPT

3 . 3 CALCULATIONS OF THE STOCHASTIC MATRIX Q AND FUTURE GRADE SIZES

We define the stochastic matrix Q in the matrix focm

as Q = {P + ) 3 ' ~ 3 where P is the transition probability matrix

(TPM), W is the vec<ox aoE withdrawal probability, and r is

the recruitment probability vector. The elements of Q are

d e f i n e d by a i j - P i j + r.w J i'

From the data (197f l -1983) we obtained the transition

table below.

The last column g i v e s the estimation of the withdrawal pro-

from 1 2 3 4 5 ra

babilities while the last row gives that of recruitment.

1

The remaining 5x5 matrix is the TPM. From these and using

.fl5?!39 .Os61 , , 0 1 4 9

the relation connecting Q, P, W , and r , we o b t a i n Q as

25 .

An e x p l a n a t i o n abou t t h e e l e m e n t s of t h s m a t r i x 12 i s

necessary h e r e as i t seem: tha t an e n t r y such as q5, would

mean a d e n o t i a n f r o m g r a d e l e v e l 5 to g r a d e l e v e l 1 , which

i s c o n t r a r y t o o u r assumption of n o d e n o t i o n i n the rnodei.

One of t h e i m p l i c a t i o n s of the assumpt ions o f fixed size i n

t h e model i s t h a t t o t a l number of w i t h d r a w a l s must e q u z l

t o t a l number of r e c r u i t m e n t s . Hence each p e r s o n who l e a v e s the

organization can be paired w i t h a new e n t r a n t a n d t h e two

changes t r e a t e d as one. A s a r e s u l t , a t r a n s i t i c n frore 3 r a d e

i to g r a d e j can e i t h e r t a k e p l a c e w i t h i n t h e s y s t e m o r by

loss f r o m grade i and rep lacemen t t o g r a d e j with t o t a l

p r o b a b i l i t y qi ' i j

+ r a w J i '

FUTURE GRADE S I Z E S

Using 1983 a s t h e " ~ t i 2 year (tao) and w i t h t h e stock

a t t h i s t i m given by

x ( o ) = (i264 - I346 1666 870 61g

and t h e f i x e d size, N = 5785, we p r o j e c t t h e s t r u c t u r e f o r t h e

nex t f i v e years ahead u s i i ~ g o u r p r e d i c t i o n e q u a t i o n which

i s g i v e n by

x ( t - : - ~ ) = % ( t i 9

Thus we o b t a i n

From the results above, we find t h a t for the system under

examination the top grades are increasing while the Lower

ones are decreasing in s i z e . Ve hnve seen the structural

p a t t e r n f o r t h e five y e 3 r c ahead, we also have to investigate

what the st: :uceam will he in the long run. In other words,

we get the limiting b ~ h a v i o u r o f the structure o f x ( t ) as

P r o n o u r prsdi::tion equstion, thd predicted stock z f t e r

From T time periods is ;(T) = ;(T-I)Q or ;(T) = x(o)Q . :s ;yy , YE know t h a t u n d e r very g e n e r a l

conditioq, .,- J p e r i o d i c i t y an2 irreducibility) which will

be satisfied

where n = (IT]. T ~ ~ . . . ~ " ) o < w < I s Ir ja l 5 j ... (13 )

Hence x(=) = X ( O ) T

a TJn . . = ( 1 9 )

where N is t h e t o t a l f i x e d s i z e of the s y s t e n . We can obtain

n by s o l v i n g the aysten of linear equat ions

subject t o

Forming the e b o v e linear e q u a t i o n s we have

n, 4 ?r2 9 r3 + T 4

+ I T 5 3 1

Solving, w e o b t a i n t h e solution as

When these results are used in equation (19) we have the

limiting behaviour of x(t) as E + - as x(a) =

E051, 880, 1531, 1102, 120g. This is in line with tho

trend shown by the projection for the f i v e years ahead. Pe

observe t h a t i n the long run, t h e f i r s t three grades

(grades 1-3) will decrease from 1264, 1346 and 1666 t o

1051, 880, and 1531 respectively. Grades 4 and 5 will in-

crease to 1 1 0 2 and 1201 iron 870 and 619 respec t ive ly . Thus

we have found a limiting s t r u c t u r e which does not depenc! on

the starting structure.

3 . 4 EXPECTED WITHDRAWALS

Our model was developed on t h e assumption of f i x e d

total sizes; withdrawals and recruitments are random. We

are then j u r t i z l e d to talk of expected withdrawals instead of

t o t a l withdrawals. "

Given x . ( t ) as t h e expec ted grade s i z e or the structure J

a t the rime t, the expected withdrawal at the end of t h e

time t is g i v e n by

where w . i s the probability of an i n d i v i d u a l withdrawing from 3

grade j. With w given as j

w = [2.0149, . 0 3 4 9 , . 6 3 5 , . 1 2 3 6 ' , . 17357 j

for j = I,2,.. ..5 and employing equation (21) we zijtein t h c

f a l l o w i n g results f o r t h e f i v e 1

a

3

IC

P

=

m

3

the v e c t o r s of the expec ted v~ i thdrawa1 . s i n f u t u r c

y e a r s , we observe that w h i l e the exTceted withdrawals from

t h e sys tem for grade l e v e l s 1 , 2 and 3 are d e c r e a s i n g , those

f o r grade l e v e l ( 4 ) and ( 5 ) a r e i n c r e a s i n g . Thi-s i s t h e

same trend w e observed e a r l i e r when we p r o j e c t e d t h e erade

structure f o r fu ture years.

3.5 THE EXPECTED LENGTH OF STAY IN A GRADE

I t i s o f i n t e r e s t t o b o t h the employer (management)

and employee ( s t a f f ) i n an o r g a n i z a t i o n t o have an i d e a 0 5

the l e n g t h of t ine an employee i s l i k e l y t o spend on a g i v z n

grade , The mean t o t a l t i m e s p e n t is the system Is also

u s e f u l .

I t has been e s t a b l i s h e d , Bartholomew p9fl27, - t h s t t h e

mean l e n g t h o f t i m e spent i n a grede i n the system i s s i v ~ n

by

E ~ T I = ( I -PI - ' . . . ( 1 8 )

where f is t h e identity matrix and P, t h e t r a n s i t i o n

p r o b e b i l i t y matrix. Thus an entrant t o grade j i n t h e BY-

stem i a e x T e c t e d t o have a l e n g t h o f Btay k

u b . R j

. . * ( 1 9 ) R= 1 3

where u.E i~ the j l th element of (1-P)-I. We s h a l l now J

ca l cu la te these values. Using the matrix P ( as a l r e a d y

d e f i n e d ) we f i n d

I .I01 1 . 0 8 6 1 0 0 0

0 , 2 8 7 6 .2533 0 0

I = 0 f .2667 .2032 0

0 0 0 .3090 . I 7 9 4

0 C 0 0 , 1 7 3 5

TOTAL

The above shorn the t o t a l expected l e n g t h of s t a y in t h e

system as w e l l as the t'ne an individual stays i n a given

g r a d e . For e x a m p l e , on enterin3 g r a d e I , an Lndi.vidue1

expects t o spend 9 . 8 9 1 7 years i n t h i s g r a d e , 2 . ? f 1 3 years in

the second grade, 2 . 3 1 2 1 years Cn the t h i r d , 1 . 8 4 9 6 i n +.he

fourth and 1 . 9 1 3 6 i n t h e fifth grade. On the w h o l e , -

e n t r a n t into t h i s s y s t e m i s e x p e c t e d t o s p e n d 19 ypars.

The above result s h o u l d be expected c o n s i d e r i n g t h s f a c t

t h a t employees on g r a d e I are mainly those with t h e l e a s t

q u a l i f i c a t i o n a n d their r i s e t h r o u g h the s u b s e q u e n t gra6es

i s d u e t o l e n g t h o f s e r v i c e a n d e x p e r i e n c e r a t h e r , than

q u a l i f i c a t i o n ,

S i m i l a r l y , an entrant into grede 2 is e x p e c t e d t o spend

a total cf 11 .1976 y e a r s i n t h e system w h i c h i s d i v i d e d into

3 . 4 7 7 3 y e a r s in t h e s e c o n d grade , 3.3029 i n t h e t h i r d ,

2 , 1 7 I8 in t h e f o u r t h and 2.2456 in the fifth grade. We

observe, g e n e r a l l y , that t52 t o t a l e x p e c t e d l e n g t h of

s e r v i c e decreases as one ascends t h e h e i r a r c h y . T h i s result

r e f l e c t s t h e i n c r e a s e of wastage w i t h i n c r e a s i n g s e n i o r i t y .

3 . 6 VARf ANCE AND STANDARD DEVIATION OF LEMGTH OF STAY IN A GRADE --.

oE stay The v a r i a n c e o f icngth i s a n e a s u r e of the v a r i a b i l i t y /

of l e n g t h o f stay i n a g r a d e . This i s g i v e n by

The corresponding standard E r r o r s are

We observe from the above natrices of variance and standard

error t h a t there i s high v a r i a b i l i t y i n the e x p e c t e d l e n g t h s

o f s t a y i n a g iven grade .

THE P R O B A B I L I T Y OF AN ENTRANT TO GRADE i TO ATTAIN HIGHEX S R A D E S

We c a l c u l a t e the probebilities that an e n t r a n t t o a n y

grade i a t t a i n s h i g h e r g r a d e s . We note that t h i s i s an

u n c o n d i t i o n a l p r o b a b i l i t y , a ~ d i s g i v e n by X i j - 'ij - -- r;hCr."

9 j .- 1

'ij i s the i - j t h element of t h e fundamental maryip: (1-2)

and X . . denotes the p r o b a b i l i t y t h a t an entrant t.o grs . : ? i 1 J

will attain grade j.

Thus we o b t a i n the f o l l o w i n g result:

From these results we observe t h a t sn entrant t o grade 1 has

a chance of about 86X of ever being in grade 2, 744 of being

promoted to grade 3, 574 of being promoted to grade 4 and

33% of b c i n g promoted t o grade 5 . S i m i l a r l y an entrant t o

grade 3 has a 75Z chance o f b e i n g promoted t o grade 4 snd a

44Z chance of being pronoted t o grede 5 .

When control is effected using promotion, it is

called pronotion control. The immediate consequences of

changes in pronotion rates are predictable, however, the

long-term effects are much less predictable since changes in

promotion policy affect the career pros;.ects and expectations

of staff. It cen also have adverse effects on staff

> morale. Hence t7e shall consider only control through

recruitment.

As the name suggests, the maintainability asbect of

control involves sustaining ,?:e status quo. Thus if

x = x i i = 1 . 1 is the structure which we wish to

maintain, then it must satisfy

X = xQ ... ( 2 1 )

That is, the problem of maintainability is then to find a

matrix Q which satisfies (21).

Attainability on the ~ t h e r hand is concerned vith

whether or not a desired srructrt.:o (i.o. a goal) can be

reached and if so by what means.

CONTSOL BY H A I N T A L P A B I L I T Y :

If x* is the structure which we want to maintain,

then the control probleu f-bre is finding a matrix Q so that

x* = x*Q is satisfied. Earlier in our node1 we obtainec

Q as a function of P , TJ and 'PC, i.e. r; = P+ W V l f . T?le czn

then state the prablem as findine the suitable vector

component of P, IJ and r s o that the equation

i s s a t i s f i e d . We have n o t e d that p r o n o t i o n and wastage

f l o w s are n o t u s u a l l y c o n v e n i e n t neana of c o n t r o l b e c a u s e

of o b v i o u s consequences . Be s h a l l t h e r e f o r e c o n s i d e r o n l y

c o n t r o l u s i n g r e c r u i t m e n t f l o w s . Thus we l o o k f o r a v e c t o r

s u c h that e q u a t i o n ( 2 2 ) w i l l be s a t i s f i e d . S o l v i n g f o r

.a* i n ( 2 2 ) we a b t a i n

I f a t a l l the e l n e n t s of r* s o o b t a i n e d ere a l l n o n n e g a t i v e

a n d sum t o 1 , we know t h a t t h e s t r u c t u r e can be m a i n t a i n e d

t h r o u g h r e c r u i t m e n t , o t 5 e t w i s e n o t . E q u i v a l e n t l y , a s t r u c t u r e

x* c a n be m a i n t a i n e d if

x* > x*? - o r ~"(1-P) - > 0.

We s h a l l now a p p l y t h e above t o s e e i f o u r structure in o u r

example i s m a i n t a i n a b l e o r n o t .

S u p p o s e we want t o m a i n t a i n the p r e s e n t structure

which i s ~ i v e n as

*(a) = c 2 6 4 1 3 4 6 1666 0 7 0 61921

With the w a s t a g e v e c t o r !J a s

a = [.0149 . a 3 4 9 .0635 .I296 .I7351 -

we have

Then

X(I-P) F 2 7 , 7 ; ' C 4 27E ,2 ;92 1 0 3 . 3 8 0 4 - 6 9 . 7 0 1 2 - 4 p . 6 8 1 5 7

= 3 2 6 2 ' 7 1rJ3 , 2 4 3 9 -. 1779 .-. 12:a The above result shows t h a t n l l ths efenents o f the vector

T c a ~ c u l a t e d are n o t a l l positive (r and r b e i n g 4 5

n e g a t i v e ) . The implication b e i n ? t h 2 t the ?resent s t r u ~ ~ v r e

can n o t be msi r - t a ined lly recruitment control.

Let us go further by comparing the present structure

with the structure that could have been considering the

National Univsraitieo Comm5ssion (NUC) directive to

universities o n staff strength in Nigerian Universities. The

N.U.C. directed or tecommended that the ratio of junior

staff to student population in N i g e r i a n universities shou ld

be 1:2** . s u b j e c t tc a maximum of 5000 j u n i o r staff. T h i s

actually w i l l give us t h total number of junior s t a f f and

not t h e number in the respective grades. Using the

recruitment vector and t h e maximum number of junior staff

(5000) as prescribed by the N . * J . C . we can approximate what

the desired structure would be. Thus we find that the desired

structure is P-pproximately

The question now is, tag this d e s i r e d structure be

a t t a i n e d from the present structure of x(o) 3

F264 - 1346 1666 870 b i g ? When x ( o ) is compared

with x(D), we observe that each grade size in x(o) is

greater than the corresponding erade size in x ( D ) . This

shows that the desired ptructure can not be attained from the

**Obtained frc--. the P l a n ~ i n g Div:-sion of the t??-ce-Chancn- ?-ore, P f f i c e , U,??.??. .

presenT acructure u s i n g recruitment control. Therefore,

if the desired stracture is to be a t t a i n e d , a freeze on

promotion and retrenchment of staff are t h e only means

by which t h c s could be attained. How this is to be

achieved will c l e a r l y i n v ~ l v e a number of management

factors and we do not ~ i s h to go into this here.

CHAPTER 5

CONCLUSION :

In t h i ~ work ve have d e s c r i b e d and mathematically

obtsi~ied a model cf personnel prediction in a graded

organization. By assuming constant s i z e p we mainly con-

centrated on the structure or the composition of staff in the

various grade8 of the organization. For structures that do

not show any undes:rable features, an alternative analysis

would be the length of service distribution of staff in eech

grade in the system. This could have been achieved by

further classifyi.ne rhe employees in each grade according

to c o m p l e t e d length of serqice and fitting a length of

service distributioi:,

In collecting the dzta for the application, we used

the p e r i o d 1978 - i 9 E 3 (a total of eix years). A snaller

time period c o u l d heve been used to yield more accurate

esttmates of the TPM but stationatity would have been leas

likely. When the t e s t s for stationarity was applied, it

gave r o s i t i v e results, thus validating our Assumption of

constant transition probabilities.

We found that, given the present structure as

x ( o ) = p 2 5 4 1 3 4 6 1666 870 6191 , the structure

inthelonarunisx(m) p 5 3 1 880 1531 1102 120g

assuming that thd conditions u n d e r which the m o d e l was

bnilt r e m a i n s the same.

The e x p e c t e d t o t a l l e n g t h of s t a y of an i n d i v i d u a l who

e n t e r s t h e u n i v e r s i t y ' s j u n i o r s t a f f c a d r e o n grade 1 i s

19 y e a r s which Is d i s t r i b u t e d a s f o l l o w s : - 9 . 8 9 years

i n g r a d e 1 , 2.96 y e a r s i n g r a d e 2 , 2 . 8 1 years i n g r a d e 3 ,

1.85 y e a r s i n g r ~ d e 4 ~ n d 7.00 y e a r s i n g r a d e 5 . This i o

t o be e x p e c t e d . The e x p l a n a t i o n b e i n g t h a t t h o s e r e c r u i t e O

on g r a d e ! are usually t h o s e w i t h t h e l o w e s t o r no

q u a l i f i c a t i o n a t all; hence t h e i r p r o m o t i o n t o t h e n e x t

h i g h e r g r a d e i s s t r i c t l y b a s e d on year o f s e r v i c e . Once

p romoted t o g r a d e 2 , subr.-que-:t p r o m o t i o n s f o l l o w t h e n o r m a l

p o l i c y of a t l e a s t 2-3 y e s r s . T r z n s i t i o n o u t o f grade 5 cen

take p l a c e e i t h e r when t h e s t a f f i s p r o m o t e d t o t h e o f f i c e r ' s

g r a d e , retires, i s d i s m i s s e d , o r d i e s .

We a l s o c a l c u l a t e d and e x p l a i n e d t h e f u n d a m e n t a l

m a t r i x (I-P)"' . The s t a n d a r d e r r o r s c a l c u l a t e d shav c h a t

t h e r e i s h i g h v a r i a b i l i t y i n t h e e x p e c t e d l e n g t h s o f s t a y .

When we i n v e s t i g a t e d t h e p o s a i h i l i t y of m a i n t a i n i n g

t h e p r e s e n t s t r u c t u r e , we d i s c o v e r e d t h a t w i t h t h e g i v e n

t r a n s f e r ( p r o m o t i o n ) a n d wastage rates, it i s not p o s s i b l e

t o f i n d r e c r u i t m e n t r a t e ( s ) w h i c h w i l l m a i n t a i n t h e

s t r u c t u r e e i t h e r i n one s t e p , o r i n a s t e p by s t e p p r o c c s s .

A l s o we d i d show that t i le d e q i r e d (NUCPs i n s t r u c t i o n ) s t a f f b e

s t r u c t u r e c o n l d :lot a t t a i n e d f r o m t h e p r e s e n t s t r u c t u r e / -

u s i n g the r e c r u 5 r m e n t a s p e c t o f c o n t r o l o n l y , e x c e p t y e ~ h ~ ~ s

by influencing wastage and promotion rates (retrenchment

and freezing of promotion) with their undesirable consequences,

of course.

We combined the various aspects of wastage

(resignation, retirement, d e a t h , etc) to form one absorbtion

s t a t e in order to satisfy better the assumptions of

stationarity of the TPM. Perhaps it wouid be worthy a p F l y i n g

the model where each is taken as a n absorbtion state on its

own. This, in addition co sp6:cifying the estimates of

transition into each state, c o u l d also give insight as to

what extent retrenchment per-se, can be carried out to

achieve the d e a i r e d structure.

O n the model, we comment as f ollovrs :-

The prediction was nade on the assumption of stationary

transitions. If this waa not validated, the stochastic

matrix would have to be updated for each new time period

before predictions c o u l d be made. Finally the assumption of

constant size could be modified to constant relative sizes

of each grade or we could a l l o w each grade to grow in sons

s p e c i f i c manner,

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45. A P P E N D I X I

SUMMARY OF TBE DATA COLLECTED FROM 1 9 7 5 - 1 9 8 3

TOTAL ( 1 9 7 6 - 1 9 8 3 )