Dynamic modelling for age distribution and age- based policies in manpower planning

Dynamic modelling for age distribution and agebased policies in manpower planning Pratap K. J. Mohapatra, Department Kharagpur,

of Industrial India

Purnendu Mandal, and Biswajit Mahanty Engineering

and Management,

Indian

Institute

of Technology,

Model&g for age and retirement becomes essential in many situations where promotion is age based and retirement takes place on attaining a particular age. This paper presents a step-by-step approach to model&g retirement and average age of population through system dynamics. The equations for a deterministic situation, with fixed residence time, are presented first, and then the equations are modified to accommodate age distribution of individuals in the manpower pool. The proposed model has been tested for plausibility under various test conditions. The paper also discusses a few agebased policies. The model behavior with these policies is presented and discussed. Keywords: manpower

planning,

age-based policies,

Introduction Age and retirement are important considerations in manpower planning. Both are related in the sense that retirement takes place on the attainment of a particular age. Age, however, can be important in many other situations. Age can be an indicator of the skill level of the employees. Age can indicate the level of activity in an organization. Age also can be a basis for promotion. Thus modelling for age is very useful in the overall framework of manpower planning. System dynamics models depicting such phenomena are rare. Meadows et al. have considered a disaggregated age structure while modelling population.’ They have considered four distinct age groups, and the outflow of each group has been considered as a firstorder delay process with a delay time constant equal to the residence time in each age group. This process only suggests the number of persons in any particular age group but does not give any idea as to the average age of persons in that age group. An alternative modelling procedure is suggested in this paper. This procedure helps in explicitly computing average age of individuals in a particular category. This calculated age is used to determine age-based retirement of employees from an organization. The model has been tested to render logically sound and consistent results. Later in the paper a few illustrative applications of age-based policies are reported. Address reprint requests to Dr. Mandal at the Department of Industrial Engineering and Management, Indian Institute of Technology, Kharagpur, India. Received 12 October October 1991

192

Appl.

Math.

1989; revised

6 September

1991; accepted

age-distribution,

1992, Vol. 16, April

dynamics

Modelling age of individuals order delay process

undergoing a first-

Consider a situation where persons join and leave a manpower pool. A system dynamics model will represent this situation as a first-order exponential delay process. In such a process the flow rate of persons (POUT) from the manpower pool will depend upon the total number of persons (P) present in the system (the level variable) and the average residence time (RT). If persons enter the pool with an average incoming age of AGEIN years and the inflow rate of persons is PIN, the problem is to compute the average age of a person residing in the pool. The approach followed in this paper for computing the average age of a person in a pool is similar to that used by Coyle for determining age of products in a company.2 Figure 1 depicts a flow diagram of the modelling procedure. The second level variable, personage (PA) in Figure I, indicates the sum of the ages of all the persons residing in the pool at any time. Total number of persons in the pool is an accumulation of persons joining and leaving the pool: f

P(t)

= PIN(t)

- POUT(t)

(1)

where P = persons in the pool (men) PIN = persons incoming (men/wk) POUT = persons outgoing (men/wk) No effort has been made in this paper to define the inflow into the pool, PIN. The outflow from the pool is modelled here as the outflow from a first-order delay:

10

POUT(t)

Modelling,

system

= P(t)IRT

(2)

0 1992 Butterworth-Heinemann

Age distribution

and age-based PAZNP(t)

policies: P. K. J. Mohapatra

et al.

= P(t)ICC (5)

cc=52

where PAZNP

incoming due to persons residing in the pool (manyrlwk) P = persons in the pool (men) CC = conversion coefficient (wk/yr)

-. -s Figure 1.

where POUT

The conversion coefficient (CC) takes a value of 52 (wk/yr), since the solution time interval is expressed in weeks. If, however, the solution time interval is in months, then the conversion coefficient will assume a value of 12 (months/yr), and so on. The outgoing person-age is a product of the outgoing persons from the pool and their average age. The average outgoing age equals the sum of the average incoming age and the average residence time:

-__--

Flow diagram to determine

age

= persons outgoing from the pool

(men/wk) P = persons in the pool (men) RT = average residence time (wk)

PAOUT

The second-level variable, person-age (PA), accumulates the incoming and the outgoing flows of age of persons associated with the inflows and outflows of persons into and out of the pool. When there is no inflow or outflow of persons to and from the pool, the inflow and the outflow associated with the level person-age (PA) also assume zero values. However, persons residing in the pool continue to age with time. To take care of such aging with time, an additional inflow rate has to be associated. This level, person-age (PA), is therefore formulated as the following:

2PA(t) = PAIN(t)

+ PAZNP(t)

- PAOUT

= person-age

= POUT(t)

x AGEOUT

AGEOUT

where PAOUT

= AGEZN

= persons outgoing from the pool

(men/wk) AGEOUT = average age outgoing (yr) AGEZN = average age incoming (yr) RT = average residence time (wk)

CC = conversion

coefficient

(3)

(7)

It is to be noted that (7) is undefined person in the pool).

The inflow rate PAIN is defined as the product of persons incoming into the pool and their average age:

Initialization

where PAIN PIN AGEZN

= PIN(t)

x AGEZN

(wk/yr)

The average age of a person residing in the pool at any time now equals the person-age in the pool divided by the persons residing in the pool:

where PA = person-age in the pool (man-yr) PAIN = person-age incoming into the pool (man-yr/wk) PAZNP = person-age incoming due to persons residing in the pool (man-yr/wk) PAOUT = person-age outgoing from the pool (man-yr/wk)

PAIN(t)

(6)

+ RTICC

where AGE

= average age of a person residing in

the pool (yr) PA = person-age in the pool (man-yr) P = persons in the pool (men)

(4)

= person-age

incoming into the pool (man-yr/wk) = persons incoming into the pool (man/wk) = average age incoming (yr)

Naturally, a knowledge of the average age of the incoming population is required to compute PAIN. The value of AGEZN is therefore assumed given. As indicated earlier, persons residing in the pool age with time and cause the value of the level, person-age (PA), to rise. The relevant inflow rate (PAZNP) is therefore directly proportional to the level of persons residing in the pool and equals P (man-wk/wk) in magnitude. But since age is expressed in years, the inflow rate will be preferred to be expressed as P/52 (manyrlwk).

if P = 0 (no

of level variables

Unless otherwise known, the level variables will be initialized assuming steady-state conditions. Thus the initial level of persons in the pool equals the product of the inflow rate of persons and their average residence time: P(0) = PIN(O)

x RT

(8)

The initial value of the person-age is the product of the initial value of persons in the pool and their average age. Assuming a uniform distribution of age of persons in the pool, the average age comes out to be the sum of the incoming age and half the average residence time: PA(O) = P(0) x (AGEZN

+ f x RTICC)

(9)

To run the model, certain other variables and parameters must be specified/given. These are the time-varying incoming rate of the population PIN(t), the average

Appl.

Math. Modelling,

1992, Vol. 16, April

193

Age distribution

and age-based policies:

incoming age AGEIN,

P. K. J. Mohapatra

et al.

and the average residence time

RT.

Retirement

at a fixed age

When persons retire on attaining a fixed age, a lirstorder delay structure is not applicable, since such a structure allows some persons to reside in the delay for a time, which can make their age much higher than the retirement age. The outflow from the pool should be dependent on the age, and not necessarily only on the level of population in the pool. The approach followed in this paper to tackle this situation assumes that the age of the population is normally distributed with a mean equal to the average age computed in a manner discussed in the previous section. The model for the case of retirement at a fixed age Figure 2 is a causal loop diagram showing relations

among population, retirement, incoming rate, retirement age, and age of population. This diagram also shows how age can be used to decide desired population, retirement age, or incoming age; these are represented by double lines. It is assumed that the age of the persons residing in the manpower pool will follow a normal distribution. The number of persons retiring will therefore depend on the average age of the persons. At a particular instant, all the persons falling to the left of the retirement age line in the normal curve should retire. When the average age (A) equals the retirement age (RA), it is expected that about half the population should retire. When the average age is less than the retirement age, it is expected that less than half the population should retire. And when the average age is more than the retirement age, it is expected that more than half the population should retire. Therefore the area under the normal curve to the left of the ordinate RA gives the probability of retirement. This probability can be stored as a table function whose x-axis is the random variate (RA - A)ISZGMA. The probability of a person retiring is then given as the following: PR(t) = 0.5 - f(Rs&;jt))

RA ?A

PR(t) = 0.5 + f(“:;;;E)

RA
(lo)

where PR = probability of retirement (dimensionless) RA = retirement age (yr) A = age of population (yr) SIGMA = standard deviation of the distribution of the age of the population (yr) f(x) = nonlinear function indicating the area under the normal distribution bounded by the normal variates 0 and x; 0.5 is the area bounded by normal variates 0 and +m The maximum retirement rate will depend on the size of the population P. It will also depend on the solution

194

Appl.

Math. Modelling,

1992, Vol. 16, April

Figure 2.

Causal loop diagram showing

influences of average

age

time interval DT, so that the total number of persons retiring in DT time will not exceed the size of the population P. The actual retirement rates, however, will, depend on the retirement probabilities as given by the probability function PR(t): R(t) = g

x PR(t)

(11)

where DT is time step used in the Euler integration of equations (1) and (3). The outflow from the level of population in the pool is the retirement rate. Thus (2) is not valid in this case. The variable POUT(t) should be replaced by R(t) in (1) and (6). Further, the average outgoing age AGEOUT now equals the retirement age RA. Therefore the equivalence of (1) and (6) is given by $P(t)

= PZN(t) - R(t)

PAOUT

= R(t) x RA

(12) (13)

Thus (3), (4), (7), (lo), (ll), (12), and (13) define the model for the case of retirement at a fixed age. The model, however, cannot be run because the standard deviation SIGMA is still to be defined. In the next section we give a procedure by which SIGMA can be computed as a function of its past value. This means that an initial value of SIGMA must be specified. The following section suggests a simple method of initializing SIGMA. the standard deviation SIGMA In what follows, a procedure is laid out to update the value of the standard deviation of age with changing incoming and retirement rates as time progresses. Let at a particular instant there be P persons in the pool with ages {a,, a2, . . . , ap} which are arranged in an increasing sequence. Let the average age of a person in the pool be given by AGE; then the variance of the age is given by

Modelling

SIGMA2

1

= (p _ 1) x [(aI - AGE)*

+ (a2 - AGE)* + . . . + (ap - AGE)*1

(14)

Age

distribution

After DT time, every person in the pool ages by DT weeks or DTl52 years. Some of them attain the retirement age and retire. Let this number be RP. During this elapsed DT time, some new persons will also enter the pool. Let the number of such incoming persons be M, and let the age of each such person be constant at dollar (a0 = AGEIN). The following relations will RP=RxDT M = PIN

{

policies:

P. K. J. Mohapatra

ao, ao, . . . Mtimes, a, + g,

et al.

a2 DT

DT

+z,...,

aP-RP

+

52 l

Here, ao, a,, u2, . . . , are assumed to be in years, whereas the time step DT is assumed to be in weeks. We adopt the notation that the new average of the age sequence is AGE and the previous average age is PAGE. Similarly, the new standard deviation is denoted by SIGMA, and the old standard deviation by

(15) x DT

and age-based

(16)

PSIGMA.

where R = rate of retirement (men/wk) PIN = persons incoming into the pool (men/wk)

The new variance is given as

The age sequence of the population in the pool, arranged in increasing order, is then given by 2

1 ‘IGMA2

=

(p

+

+

M

_

RP

_

1)

x

M x (AGEIN

(LZ~+~-AGE)~+.

- AGE)2 +

u, + g

- AGE

. . + (,,,,,.+g-AGEj2]

As per the steps outlined in the appendix,

can be expressed

SIGMA

1 ‘IGMA2 = (P + DT x (PIN - R) - 1)

x

(17)

(P - 1) x PSIGMA2

as the following: + DT x PIN

x (AGEIN

2

-DTxRx

RA-$-PAGE C

2

+(P-(DTxR))x ( PAGE+RA

We note from equation (18) that the lrew standard deviation is a function of its old value, previous and new averages, the incoming and retirement ages, the inflow and the outflow rates, the population size, and the step size DT. The new average and the new standard deviation values will be stored as PAGE and PSZGMA, respectively, for use in the next solution time interval. initial value of the standard

PAGE-AGE+%

>

+2x

SpeciJLing

- AGE)2

deviation

SIGMA If real-life data are available, one can calculate the initial value of standard deviation of the distribution of age directly. Otherwise a feasible value of the standard deviation has to be estimated. A procedure is suggested for this purpose. In theory, a normal distribution is valid for any value of the random variable lying between - ~0and + ~0.But a practical range of age A is obtained from the considerations that it should be more than the age of the incoming population AGEZN and should not exceed the retirement age RA. Therefore AGEINsAsRA

One way to specify the initial value of SIGMA is to consider that initially no person in the pool has an age that violates the above constraint. Since we know that about 99.98% of the area under the normal distribution

(18)

>I

lies within +3.5 x SIGMA limits, one can specify the following upper and lower bounds for SIGMA: AGE

- AGEIN 3.5

% SIGMA

I

RA - AGE 3.5

(19)

where AGE is the mean age of the population. Normal distribution, which is assumed to hold for the age, is symmetric, so that the inequality (19) can be expressed by the following equality to express the initial value of SIGMA: SIGMA

= (RA - AGEIN)/

(20)

Testing the proposed model The model of the previous section does not represent an exponential delay process; it nevertheless depicts a delay situation where the outflow rate is governed by the age distribution and the retirement age and, of course, by the population size of the manpower pool. Three cases are considered for the purpose of testing this model. The following parameter values are assumed for all the cases: AGEZN

= 30 (v-1 RA = 50 (v-J P= 2600 (men) (initial value) PA = 80,600 (man-yr) (initial value)

Appl. Math. Modelling,

1992, Vol. 16, April

195

Age distribution

and age-based policies:

P. K. J. Mohapatra

These initial values are so chosen that the initial average age of the population is 3 1 years (= 80,600/2600) and the initial standard deviation is 2.86 (= [50 301/7) years. It can be shown that with these values of average and standard deviation the probability of retirement, and therefore the actual retirement rate, are zero. The model was simulated using the FORTRAN-based DYMOSIM software package3 developed at the Indian Institute of Technology, Kharagpur, in a PC environment. Two sets of graphical outputs were obtained for each run, one for age-related variables and the other for number-related variables. However, these two sets of graphs were juxtaposed for easy reading of the values. For all cases the solution time interval DT was taken as 1 week. The first case is for a constant incoming population, PIN = 5 (men/wk). Figure 3 shows the behavior of the system for the case. This case represents a simple delay process (not exponential delay) with a constant inflow rate. Therefore we expect a steady-state situation to persist. In this case the inflow rate PIN is 5 (men/wk). The average residence time is the difference between the retirement age and the incoming age. It becomes equal to 20 years (= 50 - 30), or 1040 weeks. Thus the steady-state value of the population P is expected to be equal to the product of the constant inflow rate and the average residence time. In this case it is 5200 (men) (= 5 x 1040). Since the initial value of P is 2600, we expect P to rise to 5200 almost asymptotically. We also expect the steady-state average age of the population to be equal to the sum of the incoming age and half of the average residence time. It comes to 40 years (= 30 + (50 - 30)/2). Since the initial value of the average age equals 31 years (= 80,600/2600), we expect the value of average age to rise to 40 years systematically. After an initial fluctuation the standard deviation of age, SZGMA, stabilizes to a value of 3.15 years after 22 years. Figure 3 depicts such a behavior. The second case assumes a step rise of incoming population PIN from its initial value of 5 men/wk to 10 men/wk occurring at TIME = 104 weeks. The behavior of the system is presented in Figure 4. In this case we expect the final steady-state value of population to be 10,400 (= 10 x 1040) men and that of average age to be 40 years. We also expect these values to be achieved asymptotically. Figure 4 conforms to such expectations. It is seen from Figure 4 that retirement rate is more than the inflow rate at the end of the simulation run. It means that the system has not stabilized even after 40 years! The standard deviation of age also does not stabilize; its value becomes around 3.15 years at the end of the simulation run. In the third case the incoming population is made dependent on a desired level of population (PD): PIN = (PD - P)IUFD

Appl.

Math.

Modelling,

Figure 3.

The constant incoming

rate

Figure 4.

A step change in the incoming

(21)

The values of the new parameters PD (persons desired) and HFD (hiring-firing delay) were taken as the following:

196

et al.

1992, Vol. 16, April

rate

Age distribution and age-based policies: P. K. J. Mohapatra

et al.

F=P I=mN hs: 4.Pu1G7.3321 2.6E*izZ.I44 ‘:zz,

PD = 4500(men) HFD = 26(weeks) Figure 5 shows the dynamic behavior of the system for this case. Here the inflow rate is high in the beginning but falls gradually to zero as the value of population in the pool increases to its desired value. A high inflow rate initially keeps the average age of the population low, but it rises after the fifty-second week or so as the inflow rate reduces considerably. The low average age is responsible for low retirement for about 12 years (624 weeks), during which time population achieves the desired value of 4500 men. However, with rising age, retirement becomes considerable, population level comes down from its desired value, and inflow rate spurts up during the thirteenth to sixteenth years. This high value of inflow rate increases population level back to its desired value and brings down average age for a while. From the twentieth year (1040th week) onward the system achieves a steady-state condition, a well-known property of a proportional controller. The standard deviation of age stabilizes around the value of 3.3 years. Under all three test conditions the model displays plausible behavior. Thus the modelling of age and retirement can be considered an acceptable representation of reality. The next section presents a few applications of age-based policies.

I

.xsum

S

4. S:Eu:xj

:

**

.A .AS .A5 .As. .

s

.

5

. . .

B

.

%a. . SfA. . SA. 5X~.ilt--S----Qt 5 . 5 5

5

cl.5

Figure 5.

:: .

RI .HI

..H ..R

5

H

L+----*-i-r ..R

8.,XlEc:i: 3.lXE+O,

.P .P .P .F’ .F .F’ .F .F .F’

iR

.

.

.F .F’ .F’ F .F .P .F’

H R

s s

c\.

1w.OI”*^;

R . R ..I

4.mT3ix P 4.ME+ol

*--I--*

-;-*+----I-,

--I A .A .A 67 5 A. Ia. A. A.5

ifE+w * F-*-*.R I . RI H . R . R . R . H H . R

.

. .

.

.P . P .P

The incoming rate depends on desired population

Designing age-based policies For the purpose of illustration, three types of age-based policies are considered here. It is assumed that an organization is interested in maintaining its pool of employees with a comparatively low average age. Three ways are considered here for lowering the average age: (a) increasing the number of new recruits (whose ages are invariably less than the average age of the employees of the organization), (b) reducing the retirement age (i.e., tempting the older employees to leave the organization by offering higher preretirement benefits), and (c) recruiting younger employees (i.e., reducing the age of the incoming population).

i:-*A .A .AS .pIs .As. .

5

s

5

.

*R-*-l.R RI . H . R . R . H

I--I--1 I

.

.P .I= .P

: F

.P .F’

Recruitment rate as a fhction of average age Equation (21) defines the incoming population as a function of desired and actual persons in the organization. When the organization perceives that the average age of its employees is quite high, it may follow a policy of going for fresh recruitments, in anticipation of older employees leaving the organization in future, instead of waiting for them to retire. To model this situation, PD in (21) is defined as a variable and as a nonlinear table function of average age AGE in the following fashion: AGE = 36 PD = 4500

37 4500

38 6750

39 6750

40 6750

41 6750

42 6750

Figure 6 shows the behavior of the system for this case. With this policy the behavior of the system during the

Figure 6.

Appl.

Desired population

Math. Modelling,

depends on average age

1992, Vol. 16, April

197

Age distribution

and age-based

policies:

P. K. J. Mohapatra

first six years is similar to that obtained for the third case depicted, in Figure 5. The system responds to the policy of increasing recruitment rate as average age reaches a value greater than 37 years. In the rr;nge 37-38 years of average age the desired population PD increases from 4500 to 6750 (a rise of 50%). The difference (DZFFP) between the desired population and the actual population (DZFFP = DP - P) becomes positive after the average age rises beyond 37 years, and therefore the recruitment rate PIN assumes a positive value. With fresh recruitment (with younger age) the average age of the employees remains low. The average age increases slowly primarily due to aging of people with time. As age rises, retirement rises. The system ultimately achieves steady-state values after 10 years or so. The standard deviation of age stabilizes to a value of 3.3 years. Retirement

age as a function

of average

et al.

3.1cE-Ilt I.zcE~

5. *t**M 4.xew

Fc

s

.KE4W .oE*m * R-*-r-*

.fIV,----i A .A .AS

s

.

s

l_.R

I

i. .

.AS .Fs. . s . SA . . sa. m.o*--S--4t---t

.

sa: L

.R

l4s AS

*I . . L..HI L ..RI L..R L..R L..R L..R L ..R L ..R

AS AS AS

I_ ..R L..R L ..R

93. AS AS AS AS &S

L

es.

*c4o.t,;.^---I

.

RI R R R R R

L. L. L. L. L. L.H

.

4. oE*:G 4. Il:E+m -t--I--l .P .F’ .F’ .P .P .P .F’ .P F

-L----t+----t-* IR

P *Fi

: . .

. .

.

.P .I-

. . . . . .

.F’ .F .P .P .F’ .P .F

. . . .

.P .P .P

. .

t-L-----t+---I-” . . .

.

age

The organization may even decide to motivate older employees to leave the organization in the form of a “golden handshake.” If this policy succeeds, this can be assumed to be equivalent to a reduction of the retirement age. This case has been modelled by considering the retirement age as a nonlinear table function of the average age: = 36 Average age Retirement age = 50

37 50

38 49

39 40 48 47

41 46

42 46

7 shows the result of such a policy. Up to the end of the sixth year the behavior with this policy is exactly similar to that for the previous policy. Thereafter, with the average age exceeding 37 years, the retirement age falls, and there is a spurt of retirement during the tenth through twelfth years. There is a fall in the population; the inflow rate therefore rises, and average age falls. Ultimately the system settles down at or about the sixteenth year. The average age of the employees settles at 39 years (= 30 + (48 - 30)/2), which is a lower figure compared to that for the previous policy (40 years). The standard deviation of age settles to a value of 3.0 years.

Figure 7.

Retirement rate depends

on average age

Figure

Incoming

age as a function

of average

37

38

39

40

41

42

= 30

30

30

28

26

24

24

of the system for this case. In this policy the steady-state value of the recruitment age is reduced from 30 to 28 years. The average age is reduced only marginally. The reduced average age reduces the retirement rate. Therefore the incoming rate of population is also reduced. This explains why the policy of reduced incoming age is not

198

8 depicts the behavior

Appl.

Math. Modelling,

. . . .

.cs. sxl.mi--S-I-*

s s

.t

Average age = 36 Figure

sm. SC& SCA SC.A sc.cI SC.62 SC. SC.

.c .c

age

In the earlier two cases the age of a new recruit was assumed to be 30 years. The organization may decide to go for younger recruits in response to growing average age of the employees. This policy is modelled by considering incoming age (AGEZN) as a nonlinear table function of the average age of the employees: AGEZN

. . .

1992, Vol. 16, April

c. c t.

c

.

. c . c Icr(o.ol--;-* .

.H . HI . R R

.c

:R

. H . R .H 4 . R R--l . A.R 6. R . s A.. * S..R . .RI . AS s ..R . 54 ..H . s LI ..H s a ..IR

.C.SPI..R .C.SA..R .C.SCI..R . c . c 2cg).o*--c-*--I

Figure 8.

I

.

..H

SQ SR

..R

‘--w-i

I. .

:

R

;-R---k----t+--r-~ . . .

.SA..R

.

.

se

Incoming

. . . .

..R R

. . . . . .

. .

: . . . .

-*+--* . . . . . . . .

.SO..R

.SA

P *+--I--* .P .P H .P .P .P .P .P .P .P

.

*v-t SC3

.C.SL..R .C.SI..R

.c

.F .P .F’ .P .I= .P .F’ .P

A A

c

.C.SR..R .C.SR..R .C.SCI..R .t.sa..R .C.Sc?..H .C.SPI..R . c . c 1EtO.Ol--c-*

I

“-;-t+-----;-~ L

--* .P .P .F’ .P . P .P .F .P .P

. . . .

.P .P .P .P .P .P .P .P . P

. . . . . . . .

: . .

rate depends on average age

8. ~:030; 8. tXE+c

1

Age distribution

and age-based

policies:

P. K. J. Mohapatra

et al.

(b) finite (beta) distribution instead of the assumed normal distribution of the age of the resident population, (c) both retirement and promotion occurring as outflows from the population, and (d) hierarchical manpower structures. In any case, application of the proposed model has to consider the contingencies local to the specific situation of the real systems.

very effective in drastically bringing down the average age of the employees. The standard deviation of age settles to a value of 3.6 years. Discussion Age of employees in an organization is often an indicator of productivity of the organization, particularly in the area of scientific and technological innovations. This paper has attempted to make an aggregate model of the age structure of employees in an organization. This model has been tested for plausibility under various conditions. Hypothetical age-based policies on recruitment and retirement have also been formulated and tested. The work can be extended to include more realistic cases of (a) age distribution of incoming population,

References 1

Meadows, D. L. Dynamics of Growth in a Finite World. Productivity Press, Cambridge, MA, 1984 Coyle, R. G. Technical note: The calculation of average age of a group of items. Dynamica, 1976, 3, Part I, 35-39 Bora, M. C. and Mohapatra, P. K. J. DYMOSIM-A Fortranbased software package for simulating system dynamics models.

2 3

Proceedings of the XV Annual Convention search Society oflndia. Kharagpur, 1982

of Operational

Re-

Appendix Equation (17) can be expressed

as the following: P-RP

1 ‘IGMA2

=

(p

+

M

_

RP

_

MX(AGEZN-AGE)‘+

1)

~

)I 2

ai-AGEf~

i=l

(AlI

Now, 2

a;-AGE+$)2=p~p(

ai

-

PAGE

-t PAGE

- AGE + $ ) 2

=

(a; - PAGE)’ i= pr[

+

PAGE

+

- AGE + g

(

1

2 x (a; - PAGE)

)

x (PAGE

- AGE + Dg

642)

Also, PAGE

= 5

aj

(A3)

j=I

We have P-RP ,Fl

P-RP (ai

-

PAGE)* =

2

(a; - PAGEJ2 + RP

i=l

x (RA-pAGE-g)2-R~~ = 1$1 (a; - PAGE)’

(,A-PAGE-~)*

- RP (RA

- PAGE

-

(A41 2

RA-PAGE-g

=(p-1)xPSIGMA2-RPx

)

and 2

PAGE

(P - RP) x

- AGE + DzT )

(A3

Also, P-RP 2

i=l

X(ai-PAGE)X

PAGE-AGE+% )

Appl.

Math. Modelling,

1992, Vol. 16, April

199

Age

distribution

and age-based policies: P. K. J. Mohapatra

= 2 x

PAGE

- AGE + g

c

x ‘kRP(a; )

et al.

- PAGE)

i=l PpRP

=2x

PAGE-AGE+% c

x )

,z,

ai

-

(P - RP) X PAGE

F

I

P-RP

=2x

PAGE-AGE+DFT c

-RPx

x )

(RA-gj

2

a; + RP x (RA - g)

F i= I

-(P-RP,xPAGE] gai-RPx

-

(P - RP) x PAGE

i=l

=2x

I

PxPAGE-RPxRA+RPx$

-PxPAGE+RPxPAGE =2x

PAGE+RA

Putting equations (A4), (AS), and (A6) into equation (A2), we get an expression pg(

ai-AGE+%

DT

for

’ )

without the ao, a,, . . . , up terms. Inserting this expression rearranging, we obtain equation (18).

200

(A6)

Appl. Math. Modelling,

1992, Vol. 16, April

into equation (Al), using equations (15) and (16), and