NATIONAL BUREAU OF STANDARDS REPORT

1834

HOW MANY GUARDS?

By

Ho Knight

I* Hiehard gar^g®

<NBS>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Charles Sawyer, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Ast Director

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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

3737-60-0002 I August 1952 1834

HOW MANY GUARDS?

By

W® R® Knight

I® Richard Savage

Statistical Engineering Laboratory

To

Administrative Services

Division 42

National Bureau of Standards

The publication, rf

unless permission li

25. D. C. Such p prepared If that

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

In part, Is prohibited

landards, Washington

has been specifically

lort for Its own use,
I
FOREWORD

This report was prepared in the Statistical Engineering

Laboratory of the National Bureau of Standards for use by

NBS Division l\2 9 Administrative Services* The Statistical

Engineering Laboratory is Section 11.3 of the National Applied

Mathematics Laboratories (Division 11, National Bureau of

Standards }|> and is concerned with the development and application

of modern statistical methods in the physical sciences and

engineering.

J. H. 'Curtiss

Chief, National Applied

Mathematics Laboratories

A. ¥ o As tin

Director

National Bureau of Standards

HOW MANY GUARDS ?

By

Wo R & Knight

lo Richard Savage

Summary s In preparing work programs for the guard

fore© at th© National Bureau of Standards one should take into

leave

account the fact that guards take sick/and some of their

annual leave unexpected! y, which gives rise to th© questions

of whether and by what amount th© number of guards should

exceed the number of posts to be filled, and thus the theory

of probability is called into play® This report gives an

outline of a method for preparing a work program for a guard

fore®, th© outline taking into account th© element of probab^

ility that is present® It is further shown that several

different work programs can be devised that cost essentially

the same amount, and consequently th© director of these programs has some flexibility in choosing th© method for running

the fore©

a

It should be emphasized that this ire port only gives an

outline of a method for devising an economic method for running

the guard force since not all of the fiscal complexities have

been taken into account® In practice the detailed analysis

and design of work programs can only be done with the intimate

cooperation between the analyst and the accounting sectiona

of th© guard system®
'
— 2 »

Introduc tion g Mr® D&lzaXX,, th© chief of the Adminis traitive

Services Division*, National Bureau of Standards , has noticed

that probability comes into th© planning of the guard force

since guards take some of their leave £ annual and sick)

unexpectedly and consequently he cannot always plan on having

ms many gusrc $ tc tually coming to work as were asked to com©

to work*. When guards are unexpectedly absent it is often

necessary to hire extra men at overtime rates in order to

make sure that all the guards posts are filled® It is the

purpose ©f this report to show how one can take the above into

account in order to devise work programs that are economical

The Problem s Mr® Dalzell must hire by the year enough

guards to take care of all his posts 0 The lookkeeping year5

is 364 days of exactly £2 weeks | this gives 260 weekdays and

104 Satui6 days and Sundays , some of which are th© 8 holidays „ On weekdays 13 posts must be filled | on Saturdays, Sundays 9 and holidays it is only necessary to fill 12® However, if

a holiday falls on Sunday, only 12 posts need fee filled the

following Mondsy© #

This Is because the Bureau will be closed the following

Monday as the other employees get that day off to make

up for the holiday®

If w© assume that of th© 8 holidays, on© will fall on

Saturday, one on Sunday, and th© neat on weekdays, then there

are 253 days upon which 13 posts must be filled, and 103 days

(in addition to the 8 holidays which are treated separately

for reasons which will shortly become clear „) on which IS

posts must foe fill© do The days upon which 13 posts must be

filled will be hereafter referred to as. weekdays, and the days

upon which 12 posts must be filled other than holidays will*

be referred to as having fallen on a weekend, even though this

may not b® strictly the case,

Hie absence rate among the guards is not exactly known * but it is somewhere between four and six percent. In numerical

Illustrations it will, for purposes of this paper, be taken

§ss five percent.

There are 361). days in th© year. After d educ ting for

weekends, holidays, annual and sick leave, etc,, there are

212 days on which a man must w©rk„ Of this, 15 days are sick

leaves w© suppose that the guards are sick unexpectedly, hence

they will b© asked to work during these 15 days, but are sick

and therefore do not com© 0 Also there will be a few days on

which a guard is asked to work, Hut does not come because he

has taken unexpected annual leave* Thus while a guard works

212 days plus whatever sick leave he does not use, he is asked

to work on more than 212 days by the amount of unused sick

leave on which he must work, plus the amount of used sick leave
,

-
©n which he was asked to work but didn»t, plus whatever unexpected annual leave he has taken* In shorts Each guard

is asked to work 227 days plus unexpected annual leave* For

purposes of numerical Illustration 5 days of unexpected annual

leaver per man will be assumed although no figures are available

on this*

The guards are paid at an annual rate on the basis of 260

days per year (all weekdays)* In addition to this basic

annuel pay, they get time and a half for any overtime except

on holidays* When they are called upon to work overtime en

a holiday, they receive double the regular daily pay* When a

Holiday happens to fall on a scheduled work day, guards receive

their regular daily pay plus a bonus consisting of a full day»s

pay* It will be seen that regular and overtime pa$f is thus

the same on holidays, the only difference, being that on regular

time, half of the holidays pay is included in the regular pay

check, while on overtime all of it is separate*

There are two choices that must be made in this situation®

First, how many men should be asked to work on a given day?

Secondly, how many men should be hired to cover the posts for

the year? There will be any number of programs that can be

used, depending on the number of men asked to work each day

©f the year and the total number of men hired* The cost of

any program will be the sum of the following items t

(a) The basic annual salaries of the men hired*

(b) isdditional holiday pay*

(c) Overtime pay at the end of the year to fill vacant posts

in the event that not enough men were hired to fill all

of the posts that had been planned upon*
• • ' • I

.. t '

.A

A.';. • • . v *:.i t
• 5

09

This item comes into the bookkeeping since we cannot

hire fractions of men*

(d) The amount of overtime that must be paid to hav® the

posts filled when the men are absent® This will be

equal to the amount of overtime that would be paid

in one day (average) summed over all the days of the

year. Thus in a simple case, it would be 253 times

the amount of overtime that would, on the average,

be paid on a day when T men were asked to come to

work to fill 13 posts, *plua 103 times the overtime

that would be paid on a day when T' men were asked

to come to fill 12 posts® 0

The formula determining this quantity is derived in

Appendix 1, and a table for a few selected values is

included Appendix II® The average amount of overtime that will be paid during one day is given by the

expression?

T-l T

overtime = c» [TA2B(A,T~1„ Y) - (T-P)2B(A,T,Y) 3 Y=(T-1)-P Y=T-P

Where? P the number of posts that must be filled

T •“ the number of men asked to come to work

Y * the number of men asked who don't come

A^ * the average absence rate

c'=s one man's daily pay at overtime rates

(continued on bottom of next page)
/

.

'

-

. "-V

’ • V' ' i

.

. v,. ^ s
«*> <=»

Thus , given a prospective program, the sum of these

four items will give the cost of such a program® The problem

is to find the program with th® least possible cost® Unfortunately there is no direct method for doing this® However,

common sense will eliminate all but a few of all the possible

programs o For example, it is clear that at least as many men

must be asked to come to work as there are posts to be filled,

and that, given any reasonable absence rate, it should not

be necessary to ask very many more men to come to work than

there are posts to be filled® Also if B^ equals the number

of days per year on which T men are asked to come to work, and

W equals the number of days per year that one man can be asked

to come to work, then the total number of men to hire to

cover the yeai^s posts would be i

%TBfp/W - number ©f men to hire®

Unfortunately this will not usually be an integer, but it

will b© seen that the number of men to hire must be an integer

in th© neighbor of this quantity® Thus in practice it will

suffice to simply calculate the cost of a few programs which

common sense will point out® The one of these resulting programs

TFoolnofe continuing from page 5)

b(a,t„y) = r

*

Y U-i.)*-Y

O 9 This function and its partial summations over Y are

tabulated in the National Bureau of Standards Applied

Mathematics Series No® 6®

” 7

"

which proves cheapest is* of course * the on© to select®

The following programs will be figured in units of

on© man 0 s pay per day® That is* the number w one^ refers t©

an amount of money equal to the amount one man would receive

in on© day* the number ?,lon© and on© half” refers to an amount

of money equal to that which one man would receive in on© day

at ovoi* time rates* etc® Thus one man 9 s basic yearly pay would

be represented as ^260® 51

Program I®

Let us examine the following programs 13 men will b©

asked to work on weekdays* 12 on weekends® It has been pointed

out that the overtime rate for holidays is the same as the

regular rat© for holidays* therefor© let us always have men

work holidays on an overtime basis! this will enable fewer

men to be hired for the year® Sine© the holidays are now

all overtime* they will not be considered in determining

i how many men we need to fill our regular posts this number

beings

number of men 2£>3xX3 + 103x12 *. -to o

according to the formula given on page 6® As was remarked

at th© time* the result is not an integer® It will be

necessary to examine both a program hiring 20 men, and on®

hiring 19 I taking 20 firsts
'

.
- 8 - Since we '^really9* need 19o m©n s let us consider the

hiring of 20 which will give us a few Man days ^leftover98

at the end of the years however* w© can always absorb some

of this surplus by asking some men to work holidays on

regular time instead of on overtime was planned® The

exact amount of the surplus will b© 115 man®dayss

13x253 + 12x103 = 4525 ~ total number of man days needed®

232x20 s I4.6 itO * total number of man days available

if 20 men ar© hired®

“TX5 - difference between man days needed

and man days available TM surplus

(or shortage if too few men ar©

hired)

it now only remains to determine the cost of program Is

(a) 5200

(b) 96

(c) 0

(d) 253 times the overtime 339®

that would be paid on

on© day if 13 men were

asked to come to fill 13

posts* plus 103 times the

overtime that would be paid

Basic annual pay for 260 men©

Mditional holiday pay*

By asking some of the surplus

of 115 man days to work holi®

days* on regular time* half

of this has been brought under

th© regular basic annual pay!

if w© had had to Mr® all of

them at overtime* "it would have

been 2x12x8 s 192®

If less man days ar© available

than ar® needed* men must be

hired at overtime rates at the

end of th© year to make up the

shortage! however, in this case

there is a surplus rather

than a shortage®

Overtime due to absence,.

The overtime that would

be* on the average, paid

in one day if T men are

asked to come to fill P

posts can be figured by the

• 9

“

formula given on page 5>

or can be looked up in

appendix II for these

particular cases®

Total cost of program I

for one year in terms of

one man«s daily pay®

Program I* « Now let us see what will be the cost if 33 men are asked

’ 12 asked to work weekends 9 to work weelkd ays,/and 19 men are hired instead of 20® This

will give a shortage of men at the end of the year, the exact

nuutoer beings

13x2^3 + 12x103 " 4525 “ Total man days needed

19x232 - I4

.I4.08 s 81 » » available

117 - Han days short! these will

have to be made up at over*

time rates®

The cost of the programs

(a) 19x260 49M> Basic annual pay for 19 men

(b) 2x12x8 192 Mditional holiday pay'

(all at overtime rates®)

(c) (3/2),*117 175.5 Overtime to make up for

the shortage of 11? man days*

(d) 330.9 Overtime du© to absence®

This is figured in the same

way as in the previous example®

However, the last 117 man

days of the year will be

hired at overtime rate®, hence

absence does not cost anything

extra on those days© This means

that instead of 253 days

during which overtime must be

paid for absence there are

' only 253^(117/13) such days®

in one day if 12 men were

asked to come to fill 12

posts

* 10 —

(We arrange to take up the

shortages on weekdays only

as this proves a trifle

less expensive «

)

S&y&Tli Total cost of program js.

Program I XT , Let us now see what will happen if more men are asked

to come to work than there are posts to b© filled ® Let 14 men

be asked to come to work on weekdays, and 13 on weekends

(12 on holidays, of course®) This requires hirings

s 21 men

In this case we will simply round to 21 men® This will

result in a small shortage®

14x2^3 + 13x103 35 48@1 Man days needed

21x232 - 48?g Mm days available

Man days short.

The cost of program -XX/ :

(a) 21x260 5460

(b) 2 xl2x8 192

(c) (3/2)x8 12

Basic annual pay

Additional holiday pay

Overtime due to shortage of

men® (reason that 8 rather

than 9 is used is that on

the last day 13 rather than

14 men are asked to come to

work.

)

(d) 252x(over time on one 98.6 Overtime due to absence.

day if 14 men are asked On the last day only 4 out

to fill 13 posts) + 103x of 13 men are on regular

(overtime, if 13 men are time! the rest get overasked to fill 12) + time anyway.
.
<® 11 «=

(i|/l3) times (overtime if

13 men are asked to come to

fill 13 posts*)

06 Total cost of program II

In this particular case it seems that it is better not

to ask more men to come to work than there are posts to fill*

It also appears that it doesn't make much difference whether

20 or 19 men are hired, thus giving some freedom of choice to

the administrator of the guard force® It should be r emembered

that these examples are merely a rough treatment to illustrate

the general method; not only have fiscal details been omitted

for the sake of simplicity, but also two possible programs

[(1) asking 1^ men to work weekdays and 12 on weekends, and

(2) asking 13 men to work weekdays and 13 on weekends] have

not been treated, because it seems probable that they would have

proved to be more expensive than programs I and I s ©

Appendix I

Derivation of Overtime Pay Due to Absence

The average overtime that will be paid on a day when

T men are asked to come to work to fill P posts hereafter

denoted F(A,T,P) where & is the absence rate, will be

-

- T / T

x F(*,T»P) = Z f(X,P)lxl A

T-X (1-A) X

X=0

Where; X = The number of men who actually appear for work

on that day when asked (~ T-Y)

F (X,P) ” The amount of overtime that must be paid if X men

appear to fill P posts*

an absence rate of A,

asked to come to work#

is the probability that, given

X men will appear when T men were

P /T | X

F(A,T,P) =o' 2 (P-X)(xJ A (1-A)

X=0

Making the substitution Y — T-X®

= c> 2 [(P-T)(y) A

Y

(1-A)

T"T

"T" P

*

= ='

p-T)

y=T_|(y) a

y

(i-a)

t'x

+ c'TA Z (

T

Y^js'

f

(1-A)

Y“(T“>1)^P

Now, for simplicity of notation, defines

Thus givings

r t-i t

p 8 c«JTA2 B (A,T®1 ,Y) - (T-P) 2 B(A,T,Y)

7 y~(T“1)®p y^t-p

Appendix II

Average overtime that will b e paid in one day to make

up for absences. In terms of one man* s daily payj overtime

at one and a half time.

A = absence rate

5 % 6f0

P-13 Tt-11; •187 .282 .391

L

13 • 780 .975 1.170

P-12 CT"13

L 1£

• 162 .245 •341

• 720 .900 I .080

0 r~t-i T

- - P = f

I TA2 B(A,T-X,Y) - (T-P) 2 B (A,T,Y)

> Y^(T»i}=P Y=T“P

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