forecasting the numbers and types of enlisted … · 2. thebasicmodel...
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LIBRARYTECHNICAl REPORT SECTIONNAVAL POSTGRADUATE SCHOOIMONTEREY* CALIFORNIA 93940
NPS55-77-24
NAVAL POSTGRADUATE SCHOOL
Monterey, California
FORECASTING THE NUMBERS AND TYPES OF
ENLISTED PERSONNEL IN THE UNITED STATES
MARINE CORPS: AN INTERACTIVE COHORT MODEL
by
Kneale T. Marshall
May 1977
Approved for public release; distribution unlimited
spared for
:
dquarters Marine CorpsFEDDOCS e Mpi20D 208.14/2:NPS-55-77-24 hington, DC 20380
NAVAL POSTGRADUATE SCHOOLMonterey, California
Rear Admiral Isham Linder Jack R. BorstingSuperintendent Provost
This work was supported by the Manpower Planning Division(MPI20) of the Marine Corps through the Navy Personnel Researchand Development Lab, San Diego.
Reproduction of all or part of this report is authorized.
Prepared by:
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1. REPORT NUMBER
NPS55-77-24
2. GOVT ACCESSION NO 3. RECIPIENT'S CATALOG NUMBER
4. TITLE (and Subtitle)
Forecasting the Numbers and Types ofEnlisted Personnel in the United StatesMarine Corps: An Interactive CohortModel
5. TYPE OF REPORT ft PERIOD COVERED
Technical Report6. PERFORMING ORG. REPORT NUMBER
7. AUTHORfs;
Kneale T. Marshall
8. CONTRACT OR GRANT NUMBERfsJ
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Naval Postgraduate SchoolMonterey, CA 9394
10. PROGRAM ELEMENT, PROJECT, TASKAREA ft WORK UNIT NUMBERS
N6822177WR70052
11. CONTROLLING OFFICE NAME AND ADDRESS
Headquarters Marine Corps, Code MPI20Washington, DC 20380
12. REPORT DATE
May 197713. NUMBER OF PAGES31
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Approved for public release; distribution unlimited.
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18. SUPPLEMENTARY NOTES
19. KEY WORDS (Continue on reverse aide It neceaaary and Identity by block number)
ManpowerForecasting
PersonnelCohort model
20. ABSTRACT (Continue on reverae aide If neceaaary and Identity by block number)
In this report we develop a model to forecast the total enlistedMarine Corps strength at the end of each quarter for one or twoyears into the future. The method involves a simple cohortmodel. The model is programmed in APL with a number of featureswhich allow the user to interact in the forecasting procedure.He can introduce further projected retention policy changes orchanges in the recruit population which might affect future
DD ,;FORM ,^7 ,AN 73 1473 EDITION OF 1 NOV 65 IS OBSOLETE
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TABLE OF CONTENTS
PAGE
1. Introduction 1
2. The Basic Model 2
3. Continuation Rate Estimation 8
4. Steady State Results , . . 12
5. Interactive Program Illustration 17a. Data Input and Storage 17b. Endstrength Forecasts 20
References 26
Appendix 27
1. Introduction
In this report we develop a model to forecast the total
enlisted Marine Corps strength at the end of each quarter for
one or two years into the future. The method involves a simple
cohort model similar to that described in Zacks and Haber [1975]
The model is programmed in APL with a number of features which
allow the user to interact in the forecasting procedure. He
can introduce further projected retention policy changes or
changes in the recruit population which might affect future
predictions.
In Section 2 we describe the cohort model and present
the equations for forecasting. In Section 3 we discuss a number
of ways of determining the parameters in the model from past
data. In Section 4 we describe how steady state results can
be obtained. In Section 5 we describe the use of the APL
programs which comprise the model, and present illustrative
examples. Detailed and documented APL functions are given
in the appendix.
2 . The Basic Model
It has been shown previously (see, for example, Zacks and
Haber [19 75] that retention behavior patterns of enlisted marines
are reasonably consistent within certain subgroups of the total
population. It has been found that the important characteristics
to be used in forming the subgroups appear to be race, educa-
tional level, and length of first term enlistment (FTE) of new
recruits. For race we take the groups Caucasian (C) and non-
Caucasian (NC) ; for education we take the groups high school
graduate (HS) and non-high school graduate (LHS) ; for FTE we
have 2, 3 and 4 year enlistments. Thus all recruits and current
marines can be uniquely placed into one of the 12 cohorts (groups)
in Table 1.
Cohort No. FTE Education Race
1 2 LHS C
2 2 LHS NC
3 2 HS C
4 2 HS NC
5 3 LHS C
6 3 LHS NC
7 3 HS C
8 3 HS NC
9 4 LHS C
10 4 LHS NC
11 4 HS C
12 4 HS NC
Table 1: The twelve cohorts of the enlisted Marine Corps
personnel.
Let S. (t;u) be the "stock" of enlisted Marine Corps
personnel in cohort type i at time t with length of service
(LOS) equal to u (for a detailed descripton of cohort models
the reader should see Grinold and Marshall [1977]). The time
periods are taken to be quarters, and the phrase "at time t"
means the last day of quarter t. For consistency the LOS is
also measured in quarters. If a person enters the Marine Corps
in quarter t and is counted as being present at time t,
then we say he has LOS equal to 1. Thus, if he enters in t
and is counted at time t+k , k >_ , then he has LOS equal to
k+1 . Table 2 gives the stocks at the end of March 1976 of
marines who were listed as being still in their first term
enlistment.
Let q. (u) be the fraction of those in cohort type i
with LOS equal to u at some time t who will continue in
service to time t+1 with LOS u+1. The q. (u) are commonly
called "continuation rates." By using this notation we are
assuming that they are independent of the actual time t. This
assumption is modified later. Let g. (t) be the number of
new recruits who enter in cohort type i in period t, and
let m be the maximum number of periods a person can spend
in the Marine Corps. The total stock S(t+1) of marines at
(t+1) is given by
12 m(1) S(t+1) =
I £ S. (t+l;u) ,
i=l u=l
where
(1.1) Si(t+l;l) =
g;.(t+l) q..(0)
(1.2) Si(t+l;u) = S
i(t;u-1) q^u-1), u = 2,3,..., m
In order to use equation (1) to forecast end strength
it is necessary to know (i) the cohort stocks {S. (t;u)}
for all i and all u, (ii) the continuation rates {q. (u)
}
for all i and all u, and (iii) the future recruit inputs
{g. (t+1) } for all i. It is the determination of these
three sets of data to which we now turn.
The new recruit input in future years will be given
to the model by the user and thus we can dispose of (iii)
.
Table 2 shows the stocks of marines who are still in their
first term enlistment at some given t for each cohort i,
but only for u = 1,2,..., 20. Marines can reenlist and
remain in service 30 years; thus m is 4 x 30 = 120. But
as the LOS of a marine increases beyond his FTE period there
is little distinction to be found among the 12 cohorts. These
marines essentially form what is called the "career" force,
and continuation in service of these people is governed by a
different set of factors than those affecting first term
marines. Thus, we modify equation (1) in the following way
to reduce both the size of the model and the number of parameters
to be estimated.
Cohort Number
LOS 1 2 3 4 5 6 7 8 9 10 11 12
1 3 3 2 1706 301 1424 422 2419 410 2982 5882 5 4 14 2 931 134 1256 299 2635 497 3418 7973 48 12 221 33 907 132 1683 387 2441 462 5563 1166J* 165 29 879 117 495 81 1041 236 1749 336 4276 9075 527 92 806 137 928 211 645 147 3143 897 2563 5826 543 154 804 202 672 210 495 187 2372 793 2013 5037 845 235 1926 541 736 247 936 299 2180 697 3841 8168 380 128 899 387 301 108 560 196 1107 356 2685 7009 197 80 248 85 794 200 545 143 1981 583 1433 340
10 83 39 140 51 547 124 436 108 1487 435 1179 33411 43 27 174 34 508 122 876 185 1688 387 2846 62412 23 31 50 21 414 96 460 124 1721 400 1892 40713 26 13 52 22 126 39 123 50 1502 394 1147 35014 20 12 25 11 61 18 70 28 1187 360 1176 40415 30 19 74 24 52 24 123 41 1282 354 2409 53016 31' 20 24 18 40 11 62 28 932 206 1562 28117 22 7 17 9 26 6 25 11 325 70 285 8618 14 4 7 3 19 8 14 4 150 17 186 4419 6 5 14 5 11 33 10 119 17 279 612 4 1 6 4 10 2 19 3 101 32 157 28
Table 2 Stocks of enlisted Marines on 31 March 1976.
Let c(t) be the total number of enlisted marines in
service time t with 21 or more quarters of service.
Thus
12 mc(t) =
I I S. (t;u)
.
i=l u=21
Now assume that q.(u) = q for all u >_ 20 and i = 1,2,..., 12
Using (1.2) it is easy to show that
12(2) c(t+l) = [c(t) + I S. (t;20)]q .
i=lX
Equations (1) and (2) are now combined to give the forecasting
equation
12 20(3) S(t+1) =11 S. (t+l
;u) + c(t+l)
i=l u=l
where
(3.1) Si(t+l;l) = g i
(t+l) qi (0)
(3.2) S. (t+l;u) = S.(t;u-1) q.(u-l), u = 2,3,.. .,20
12(3.3) c(t+l) = [c(t) + I S. (t;20)]q.
i=l
Equation (3) requires only 241 continuation rates com-
pared to 1452 in (1). This gives a considerable saving in data,
computation, and storage requirements. Figure 1 illustrates
the flows assumed in (3).
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3. Continuation Rate Estimation
Before (3) can be used for forecasting we must obtain
values of q^Cu), u = 0,1,. ..,19, i = 1,2,. ..,12, and q.
These values are determined from historical data on past
stocks.
Assume that t = is the most recent time for which
stocks are available. By using these and the stocks at
t = -1, let
S. (0;u+l)(4) V U) = sV^u) '
u = 0,1,..., 19; i = 1,2,..., 12 ,
and
(5) q = c(0)12
c(-l) +I S. (-1;20)
i=lX
Thus all the parameters can be estimated from the data of two
consecutive periods.
If data from more than two periods is available
we can use it to obtain smoother estimates, ones less susceptible
to random fluctuations in the stocks. Assume that data is
available for periods 0,-1,-2, ... ,-k. Then we modify (4) and
(5) to be
(6) q, (u) =
-(k-1)
I Si(j;u+1)
-k
I Si(j;u)
u = 0,1,... ,19; i = 1,2,... ,12
and
(7) q =
-(k-1)
I c.(j)
JJlP_-k 12
I (c(j) + J Si(j;20)}
j=-l i=l
The values obtained using (6) and (7) for any given k
are said to be determined using rate method k.
Recruit attrition in the first six months of service
is measured by (l-q.(0)) and (l-q.(l)), and is considered
controllable by the Marine Corps. Estimates from past data
have little meaning. In the interactive APL model described
in detail in Section 4, recruit attrition is entered directly
by the user at the computer terminal. The six month attrition
of LHS and HS recruits is normally about 20% and 10% respec-
tively. The conversion of these into q. (0) and q. (1)
is illustrated for some cohort i if a six month rate r is
entered. They are calculated by
(8) q..(0) =qi
(l) = /l - (r/100) ,
which spreads attrition evenly over the six month period.
Table 3 gives the continuation rates q.(u) obtained
using rate method 2 with t = equal to 3-31-1976, and re-
cruit attrition for LHS and HS equal to 15% and 12%, respec-
tively. The value obtained for q was 0.980.
The reader will notice that some of the q. (u) in
Table 3 exceed 1.0. Although this could be in part due to
errors in the data base, it is also in part due to the return
from unauthorized absences. Note that this phenomenon occurs
most frequently when the LOS is 2 and 3. This is a period
immediately following the end of basic training, when many
Marines return after being AWOL. Thus they are counted in
the numerator of (4) or (6) but are not present to be counted
in the denominator.
10
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4 . Steady State Results
The model summarized mathematically in equation (3)
is used for short-term forecasting (1 or 2 years) under a
given recruitment policy. The basic model can also be used to
show the long-run, or steady-state, effects of a fixed recruit-
ment policy. Although it is unlikely that the system parameters
will remain constant over many periods or that recruitment
policies will remain unchanged, the long run behavior of the
system under a given policy is often useful in showing trends.
These trends can act as warning signals of future problems.
Let L. be the random lifetime of an individual of
cohort type i. Let Q.(£) = P[L. > £] ; then
9) Q. U) = n q. (u) , I = 0,1,2,.. . ,m .
1u=0
Now let X . be the average time spent in the Marine Corps
of an individual in cohort type i, and let g. be the
fixed quarterly recruit input into this cohort. The stocks
S(t) converge to a constant stock level S, where
12(10) S = I X g
i=l
and
m(11) X. = I Q. U)
1 1=0
12
Recall that we assumed that for u _> 20 and all
i = 1,2,..., 12, q. (u) = q, a constant. Recall also that m
m-19 99is 120 so that q = q is negligible for values of q
of interest. From (9) we have
£
Q. (£) = n q. (u) , £ = 0,1,. .. ,19u=0
£-19= Q
i(19) q* ., £ > 19
Using these with (11) and (10) gives for the steady state
stock level
12 i-18 Q. (19) -,
(12) S =I I Q (£) + -i—— g
i=i L £=0 J- -l q j i
To illustrate the use of (12) , Table 4 gives the life
distributions Q.
(u ) for the continuation rates in Table 3i
for £ <_ 19. Using the constant recruitment policy shown in
Table 5 and q = 0.980, the steady state stocks will become
163,149.
In addition to calculating S, many more steady state
calculations of interest can be made. For example, the steady
state fraction with less than high school education is given
by
(A 1^1 + A2 g 2
+ A5 g 5
+ A6 g 6
+ A9 g 9
+ A10g 10
)/S*
13
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Cohort No. Number of Recruits
1
2
3
4
5 1428
6 252
7 2142
8 378
9 2652
10 468
11 3978
12 702
TOTAL 12,000
Table 5 : Quarterly input of recruits into each cohort
15
Similarly, the steady state fraction of non-Caucasions in the
force is given by
(A2 g 2
+ A4g 4
+ A6 g 6
+ A8 g 8
+ A10g10
+ A12 g 12
)/S*
16
5 . Interactive Program Illustration
In this section we describe the use of a set of inter-
active APL functions for both data input and enlisted end-strength
forecasting. These functions are available in Scientific Time
Sharing* s APL + system in a workspace called ENLISTED . Listings
and documentation of the functions are given in the Appendix.
a) Data Input and Storage
There are 3 functions used to input, display and correct
the stocks {S. (t;u)} for any given time period t. These are
(i) INPUTSTK
(ii) DISPLAYSTK
(iii) CORRECTSTK
INPUTSTK is a function that requires no arguments. It asks the
user for the time period t, and for each cohort 1=1,2,... 12 it
asks for the 20 numbers {S. (t?u) , u=l , 2 , . . . , 20 . } . After these
are entered it asks for the remainder of the force, c(t) . A
sample use of INPUTSTK is shown in Figure 2. The data is stored
in a file called '4894733 STOCKS' ; details of the file format
can be found in the Appendix.
DISPLAYSTK is a monadic function which takes as its right
hand argument a 2 element vector of month and year, and displays
the stocks for that time period with suitable headings. Table 6
demonstrates the use of DISPLAYSTK for data on 31 March 1976.
CORRECTSTK is a monadic function which takes as its right
hand argument a 2 element vector of month and year. After using
DISPLAYSTK to observe the stocks on file, CORRECTSTK can be used
to make any necessary corrections. A sample use of CORRECTSTK
17
cr> <r> u"> ct>
t-« to ^-4 «-(
CO ^-H 00
r^ cm c^ r- r-*-H CM *-l ,~l
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jt o .zr cm jr-i CM i-< tH ^H
co co co co co,-1 cm ,_| ,_| ^
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cm lJ cm kjQ r\i >-3
18
Oj«-<^3-cf\'cr ro t-< cm r- cc*-• *—« o ro i/) (\
CT> LO U0 -3" Lf> «-• co o^h i-> r- a>
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Cocoocccnp^*-tcoo<x>r^tDLOOO coCNC^coootDCTOmcoo>0 coT-tcorr)OOT-iLO*-''»-tco(£>r^
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r~CMtor^cD«-<Lnr~cor^cocMWCflOCKNrlJJJffiaO)
(DCMt-- ^ncT'CO^-co.rrcor- o,-i x h j o cm r~ ro cm cn
CJHcd
En
cooocMTHfnr^CMfnr^^-tCMCoir)
CM tTHiDrOJ^lft^r-» CM lT> *-t
CMLO^-jCMt-ts-tocrimc^cor^«-* corou")cncocn*-<cn
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% 5£ * 3; =S *
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0*-<cNcr>;ru">t£>r-~cocno*-iCN
19
is shown in Figure 3.
b) Endstrength Forecasts
The function ENDSTR is used to calculate end-strengths for
given recruitment policies using equation (3). It requires no
arguments, but interactively asks the user for the following
data
:
(i) Number of periods to project (call this n)
(ii) Base Period
After these it asks for the following data for recruits for each
of the next n periods:
(iii) Percent Caucasian
(iv) Percent 3-Year Enlistments (it is currently assumed
that there are no 2-Year Enlistments)
(v) Percent of high school graduates in Caucasian recruits
(vi) Percent of high school graduates in non-caucasian
recruits
(vii) Total recruits in each period
(viii) First 6-month attrition percent for high school
graduates
(ix) First 6-month attrition percent for those without
high school education
The answers to (iii) through (ix) are used to spread the recruits
in each period over the 12 cohorts. The next input required is
(x) Continuation rate method (see section 3)
(xi) Does the user wish to change the continuation rates.
If the answer to (xi) is yes, the user is asked for
(xi-a) High school graduate factor
(xi-b) Non high school graduate factor
20
CORRECTSTK 3 76TO END, ENTER COHORT NO. 0. TO CHANGE >20, USE COHORT NO. 1, LOS 2 1
COHORT NO.?Q:
12LOS?D:
6
CURRENT: -5 5 3
D:503
COHORT NO.?Q:
1
LOS?D:
21CURRENT: -4815 2
A/£W?
D:48049
COHORT NO.?D:
Figure 3. Demonstration of CORRECTSTK
21
These factors are used to multiply all the continuation rates
for the given cohort type. This approach is used rather than
asking the user for changes in each of the 241 rates.
After answers have been given to questions (i) through
(xi) the end-strengths are calculated using equation (3) and
stored on a temporary file. When the calculations are completed
a report is printed showing the recruit policies used and the
end-strengths obtained. Following this printed report the user
is asked if he wishes to continue. If the answer is yes, the
user can then pick a subset or all of the questions (i) through
(xi) to enter different data and rerun the model. An example
run follows in Figure 4.
22
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25
References
Zacks, S. and Haber, S. E. , "A Procedure for Forecasting theSize of a Force Subject to Random Withdrawal," Report No.T-312, Institute of Man. Sci., George Washington University,Washington, D.C. 20037 (1975)
.
Grinold, R. C. and Marshall, K. T., "Manpower Planning Models ,"
North-Holland Press, New York (1977) .
26
Appendix
1. Files
The forecast model requires 2 files for execution. The
file '4894733 STOCKS 1 contains historical data on enlisted Marines
(for details of file creation, manipulation and security, see
the booklet on the APL+ file subsystem from Scientific Time Share
Corp.) component 1 of this file contains data for 30 June 1975,
component 2 the data for 30 September 1975, etc. The format of
the data is shown in Table 6 of the main report.
The file '4894733 FORECAST' is used to store the forecasted
force and attrition whenever the function ENDSTR is used. It
is used to facilitate the printing of various statistics on
the forecasted force and attrition. Its use will greatly
simplify satisfying any future request and allow the user to
display many kinds of data without re-running ENDSTR.
The format of each component of this file is shown in
Table 7. Each component is a 24 x 21 matrix. Rows 1-12 of
component 1 give the base period stocks as in Table 6 with
column 21 having the stock with LOS > 20 in the top row,
followed by 11 zeros. Rows 13-24 of component 1 contain
zeros. Component i (i >_ 2) contains the forecasted stocks
and attrition for (i-1) periods after the base period. Rows
1-12 contain the forecasted stocks in the same format as com-
ponent 1. Rows 13-24 contain the forecasted attrition from
each cohort-LOS cell in the given period.
2
.
Functions
The use of the main functions is discussed in section
5 of the main report. This appendix contains detailed listings
of the functions.
27
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