a separable programming approach to allocating manpower
TRANSCRIPT
A SEPARABLE PROGRAMMING APPROACH TOALLOCATING MANPOWER FOR THECALIFORNIA HIGHWAY PATROL
Daniel Carev Schneible
tlittl'u'li t> Ls, U %i' \J> b .
IIATT Q: '
UnnffWHf f! r :
' r:J r '
(iUiilut oy, uuiiiut H
i^
[: i
A SEPARABLE PROGRAMMING APPROACH TO
ALLOCAT I NG MANPOWE R FOR THJ
CALIFORNIA HIGHWAY PATROL
by
Daniel Carey Schm;ible
Thesis Advisor: R W. Botterworth
March 197 3
Ap.pn.avzd (fiK puhtic sizZzaie.; dii^AAbtUA.on untanLttd.
T153559
A Separable Programming Approach
to
Allocating Manpower for the California Highway Patrol
by
Daniel Carey SchneibleLieutenant, United States Navy
B.A. , Saint Bernard College, 1965
Submitted in partial fulfillment of therequirements for the degree of
MASTER OF SCIENCE IN OPERATIONS RESEARCH
from the
NAVAL POSTGRADUATE SCHOOLMarch 19 7 5
Library
Naval Postgraduate SchoolMonterey, California 93940
ABSTRACT
This paper presents a mathematical approach to
formulating a problem in the allocation of manpower for
the Monterey County Area of the California Highway Patrol
(CHP) . The technique employed is to formulate the man-
power allocation problem of the CHP as a nonlinear pro-
gramming problem. The problem turns out to be separable,
and the Mathematical Programming System/360 for the IBM 360
computer is subsequently used. This project was undertaken
with the cooperation of the CHP.
TABLE OF CONTENTS
I. INTRODUCTION ------------------ 4
II. PROBLEM DEFINITION --------------- 7
A. CONSTRAINTS CONSIDERED ----------- 7
B. DATA ANALYSIS- --------------- 15
III. FORMULATION- ------------------ 24
A. MODEL SELECTION- -------------- 24
B. COMPUTER UTILIZATION ------------ 31
C. DATA ANALYSIS- --------------- 31
IV. SUMMARY- -------------------- 33
A. RESULTS OF MANPOWER ALLOCATION ------- 33
B. RESULTS OF DATA ANALYSIS - - - - - - - - - - 38
V. CONCLUSIONS- ------------------ 39
APPENDIX A - PRE -LIEUTENANT'S CLASSPERSONNEL DEPLOYMENT- ---------- 41
APPENDIX B - APPLICATION OF SEPARABLEPROGRAMMING MODEL ------------ 44
APPENDIX C - MANPOWER DISTRIBUTION VS. ACCIDENTRATE DISTRIBUTION FOR 2000 DOUBLE-UP REQUIREMENT- ------------- 46
APPENDIX D - MANPOWER DISTRIBUTION VS. ACCIDENTRATE DISTRIBUTION FOR 2200 DOUBLE-UP REQUIREMENT- ------------- 63
LIST OF REFERENCES ------------------ 80
INITIAL DISTRIBUTION LIST- -------------- 81
FORM DD1475---------------------82
I. INTRODUCTION
The California Highway Patrol (CUP) has the primary
duty of reducing automobile accidents which result in
property damage, bodily injury, or death. The manner in
which they approach this task is through the employment of
patrol officers throughout various areas of responsibility.
The patrol officers issue warnings and citations to viola-
tors of traffic laws, inspect the mechanical conditions of
automobiles, and assist those involved in traffic accidents.
The criterion used for allocation of manpower by the CHP
is based on the idea that manpower strength should be
tailored to the accident rate over the day by hour of the
day [Ref.s 1 and 2]
.
Reports of all accidents and traffic violations are
filed with the CHP Headquarters in Sacramento, California.
There the reports are tabulated and published quarterly 1
the "Accident Enforcement and Patrol Summary" (ALPS) , wh ch
is forwarded to the local areas. The distribution of acci-
dents are obtained from the AEPS and are used as the basis
for determining the required assignment of officers. To
meet the required assignment, the Areas distribute the
officers among shifts during the day with each officer
working an eight- and-one-half -hour shift under normal
circumstances
.
To determine the optimal number of officers to assign to
each shift, the CUP utilizes computations described on page 2
of Appendix A. A close examination of these computations
reveals that they are only valid when considering a three-
shift, day with no overlap of shifts. When more than three
shifts are considered, and therefore the shifts overlap,
the methodology breaks down due to the inability to account
for the effect of interaction between shifts. However,
if the procedure in Appendix A is used, it does provide a
starting point from which refinements can then be made.
Presently refinements are made through trial and error.
To illustrate, an assignment is made and the local area
operates for a period of time. When the AEPS is received
from Sacramento, a close examination is made to determine
the weak areas and more adjustments are made. The feedback
cycle for these adjustments . is on the order of six or more
months
.
To assist the CHP in obtaining a method to solve the
problem of allocation, it was decided to model the con-
straints utilizing nonlinear programming techniques. The
nonlinear problem evolved into one which was piecewise
linear and therefore became suited to separable programming
techniques. In separable form it was then formatted to fit
the Mathematical Programming System (MPS)/560 for the IBM
360 computer which was used to solve the problem.
In examining data available at the Monterey County
Area, a secondary benefit evolved which resulted in a
closer examination of accident rates and distributions.
No analytical techniques were involved in this effort, but
closer examination of the accident trends in various areas
of the county evolved into more accurate descriptions of
accident distributions.
II. PROBLEM DEFINITION
This portion of the paper is divided into two sections.
The first examines the constraints that were considered in
selecting a model for the manpower allocation problem and
the second considers the data available for studying trends
in accident statistics.
A. CONSTRAINTS CONSIDERED
One of the major problems which face the CMP in solving
the problem of manpower allocation is the large number of
constraints which must be considered. Through various
interviews with the CHP,- it was determined that many of
the factors considered by scheduling officers while allo-
cating their men are not suitable for an analytical
approach. It therefore became necessary to isolate, as
much as possible, those factors which needed to be consic red
utilizing management techniques from the constraints that
would be applicable to a mathematical model. The applic; jle
constraints are discussed in the following paragraphs.
The Monterey Area has 56 officers available for assignment,
This number takes into account the officers who are on limited
vacations. When the officers are on duty, they work 20 to 21
days per month on a five-day work week with two consecutive
days off.
The Monterey Area requires that each hour of the day be
covered by at least one shift. This requires that there be
a minimum of three shifts per day. For administrative
reasons, the maximum number of shifts allowed for one day is
five
.
In order to meet the scheduling criterion stated in the
introduction, the scheduling of officers must consider two
factors: the occurrence of accidents varies with the day
of the week (Figure 1) , and the distribution of accidents
over a day varies by hour of the day (Figure 2) . The
officers are thereby scheduled accordingly. The scheduling
is accomplished on a monthly basis and, barring unforeseen
circumstances, remains fixed for that period of time.
The number of patrol cars available is dependent upon
the day of the week and the hour of day. On normal working
days, Monday through Friday, there are approximately IS
patrol cars available between 0800 and 1700 hours. On
weekends, holidays, and after 1700 hours on normal working
days there are approximately 20 patrol cars available.
Other considerations taken into account are that an
officer working alone after dark makes fewer enforcement
stops than a two-man car, and drinking drivers pose special
accident prevention problems from 2200 to 0230 hours.
For these reasons and also for safety, the CHP has a state-
wide policy which requires that cars will carry two men
during hours of darkness. An exception to this policy may
be granted by the zone commander which allows single-man
units to be operated until 2200 hours.
MONTEREY COUNTY ACCIDENT RATEBY DAY OE WEEK FOR 19 71
FIGURE 1
l <
1 9•j^C'*CC^*ix-,.-,,,.
1 8J
:
1 7 -:
-
xx-x-x-:
1 6vlv!v!v.v!v!v!
C-yy.'yyy.
yyyyyyy..•.\V.\VJ
1 5x :
::x :x :>
XvX\v>x :
::x :x : i
1 4*X*X*X*X*X'X*X
.\V.V.VJX\"X\v3•XvX'Xl
1 34
ps^><^r^-vx*xx\ x*X\;X;!;!;!;!•!;!;!;)
£::#:?]Wl 2 L*r-.T"^."i* J vXv/Xio *xx*x :•;•:•:•: ::•:>:•:•:•:•:•:•: ;.;.;.;.;.;.-.]
< 1 »
!•!•!';•;•!•!•;'*•!•!•!'!•;•!•> /;*!*>!*!*x-; ;•;•;•;*! ;';i;-!*;*!-;*!*; ;
H l 1i-I't'i'i'^l'l^-.'yilXyX^N^v^^C-X'Xviv^X^viylX^!;!;^.
IvXvX-j2 •y.'yy.'yAw , ~O 1 yyyyyy.'APJ yyyy'-:- '.'
W o•'.'.'.•yyy.'j
WV.WVJ;.;.;. ;.;.;.;.;J
8
7
6
5> :x :x:x-
4 >x . x-x-;X;X;X;X
iX*X*X\- X;X;X;X;
i«tAr_Jl-!. V* I
c£^ w o D i—
i
H 2o £3 w JTj Pi < Ds H te; H Uh CO CO
I?<N
COCM
CN
CS1
w
oPL,
O
O
CO
w<Pi rH
H en
wQ Pit-i O
< >H
OHXo
>> OPiw >-<
H PQ
O
I"<
CO
CN
E
oooo
O
o
owo
o
oCOO
O
o o—. o O o
oCIo o o
10
To model this manpower allocation problem as described
above, it was necessary to make various assumptions. The
relevant constraints which were quantified and used to
formulate the problem precisely are discussed in the follow-
ing paragraphs.
Because the distribution of accidents that governs the
allocation of manpower is given in percentages, it was
decided to model the problem in a form that would give
required percentages of resources as a solution. This would
also allow more general results that are applicable to a
variety of combinations of manpower and cars that are
available
.
Due to the nature of the manpower constraints, it can
be seen that there will always be total utilization of the
number of men scheduled for any given day. The same state-
ment cannot be made about the number of cars available.
If, in fact, the number of men scheduled for each hour of
the day does not equal the total number of cars available,
the remaining cars will sit idle. With the above in mind,
it was then decided to formulate the requirements in terms
of percentages of manpower. It should be noted that once
the manpower solution is obtained, it can then be translated
into the number of men required which in turn has a direct
linear relationship with the number of cars required. This
is true even when considering the two-man cars required
after dark.
11
With one-man cars patrolling during daylight and two-
man cars patrolling after dark, it is assumed a two-man
car has the same effectiveness as a one-man car. With this
assumption, each man that is placed in a two -man car is
half as effective in matching the accident distribution as
each man in a one-man car. Because the model uses manpower
requirements, the following transformation was made to main-
tain consistency between the distribution of accidents over
the day and the required manpower allocation. The percent-
ages of accidents during the hours of the day that required
two-man cars on duty were doubled. In essence this means
that for each increment in accident rate after dark, the
manpower allocation is twice that required during daylight.
The resulting curve was then renormalized to cause the
percentages of accidents over the day to sum to unity.
Figure 3 shows what the resulting accident distribution
becomes when the hour of darkness double-up period is set
at 2000 hours. Figure 4 shows the same curve when the
2200 hours mandatory double-up policy is enforced.
To simplify the model, the 8-1/2 hour shifts were con-
sidered to be eight hours long since one-half hour is
spent at the station and not patrolling. Also, each day of
the week was assumed to have the same distribution of acci-
dents. A more detailed discussion of this assumption is
made in the next section.
12
to
Wcrj
:=>Oi—
1
flH oo
E LOH ot—i
^ QPS
to <H2: oW oQ o1—1 CMUU Z
' < ww
p. =£
O Hpq
^ PQOHH HH E^3 OPQ K-
1
h-1 PJC* &Hco wl-H JO PQ
C3oQ
CM
'c»
c<
-~?C I
K
o
ooC3
o
Ooo
o
o
o
"I 'I
o o>— o o o o o
no
C* —o o
13
cn
(NP4
w&Dot—
1
Ph oo
n: lo
H on^ xi
Pi
C/D 05
H2s
: oW oQ csj
(—1 (SI
Ucj ;s< ww
ix, ~=
O HW
^ pqohH HH K^3 OPQ HHl-H W(^ £=Hco wl-H JQ CQDOO
oIN
O
CO
N.
O
«n
"*
co
cs
oOCO
O
OOotn
o
o
o
o
T—~1
—
—T— 1 1 1
O oo
00O o o
o O o
14
B. DATA ANALYSIS
The data analysis, as stated earlier, was a secondary
benefit which evolved from examining the assumption that
each day of the week had the same distribution of accidents
by hour of the day. Basically, in order to model the problem
over a one-day period, it was necessary to determine if there
was a correlation between the total number of accidents by
day and the distribution of accidents by time of day.
The Monterey Area maintains a three-by-five index card
on each accident. The data on the cards were transferred
to computer cards in the format shown in Figure 5. The
distribution for each day of the week was then plotted
against the overall accident rate. Although no statistical
tests were run, the trends of each day approximated the
overall accident rate curve. These comparisons are shown
in Figures 6 through 12.
15
Pi
y-1
P-,
<<Q
etc
o
H
o
<
S313IH3A jo >i3awn\T
SISA1VNV N0IID3HIQ
3Q03 3SnVD
A1IH3A3S
3N 1
1
HV3A
3JLVQ
HINOW
l^ola ~~<fT N0w7"VGf
H3HWflN IV39
16
1
I
s_
CM
COCM
CMCM
pq
i—
l
_LL
C4
ocs
CV-
CS
wQ •
I—i 00o >u<
o
W rH
H CDCJ rHU< oio
-1
f.
>o
TI
I I
l-
r
i
1
1
I—
1
>. rHaj rci
T3 rM
C <D
o >s o
ro
C>i
o
oCOorvooowr»
O
oCOo
o
o o—. o -c
o ono
fS —
17
"3-
cs
CM
CS
r-l
rT
o
o
co
w WH HSi SH HZ s:W wQ Q rHr—
1
i—
i
t^-
U • u CTl
U CO u rH<><o>-" _3
< -J PJh
Q <£00 KW wD >H o
r'i
rI
CN
r
"
i
U
1 1
1 1
1 1
X i—
1
ctf r-i
nd cd
to r-l
<D d)
3 >H o
ooCO
o
o«oo
o«roCOo
O
o o-—. o o a
on <m —o o o
18
r
L_
cm
CM
CMCM
CO
m
i—
i
P-h
CM
oCM
CO
wH<£ wK HH SP̂J HQ ^i—
:
P-J rHu Q t-~
u . 1—1 CTi
< WUH>u>H < c£< oQ .-} PhCO JPJP2 gjQ PJw >^ o
7*I
U"rr
1
XrS
h3 i—
i
t/i i—i
<D cti
d Mtt CD
<d >S O
iit.
CO
CM
oCO
O
OoomO
otooCMo
o o—
.o o
1/1
o omo
19
en
W
uI—
I
TI
oCM
o
wH pq
S H
HP-,
^ HW ^Q w fHi—
i
P r-~
U . i—
i
cr>
u 72 u rH< > < OS>~> o< -3 U-.
Q -JC/3 <tj
C*I c2D wDC >H o
r
i
i
I
I
L.-
m
ex
ooCOo
X05 .—
I
X) i—i
V) nj
U Jm
3 <L)
^ >H O
ro•oomO<<•
COo<No
o o-—. o o o o
20
cm
'.•>
ex
cmCM
w
CD
Ph
CM
oCM
o
03
Wpq HH
HH "Z2: m rHw Q r-Q • (—
i
cr»
—
i
75 CJ i-H
CD > CDCJ < cd< O
J P>H kJ< si—
i
WPi >P-, o
X rHnj .eti
TJ ?H
•H (1)
J-. >P-. o
I
—
1
ri
O O CO-~ o o
4--L,
fc» lAy *
o
tfl
"J-
O
CM
O
ooCO
o
o<oo
—
1
rmo
OCOOCMo
f-* o ""> ^f n cm >-O o O O O O o
21
n cm
COCM
CMCM
w
oCL<
L _.
rr
c<«
oCM
o
CO
wH W<£ Hcd
2H^ HW ZQ W ,-H
l-H Q r-u • \—\ cr>
u CO U t-i
< > u< pi
>H o< J PhQ _}
£3 sH w< >CO o
...J
n>srt rHT3 I—
1
^ oS
3 5h
4-> <D
03 >co o
K
vO
ro
CM
OOCO
OKOOO
o
oCOoCMo
—li
; ;
—
O O en r^— o o o •O "1 "*
o o o o o o
22
III. FORMULATION
A. MODEL SELECTION
Prior to formulating a model, a close examination of the
constraints was made. When taken singly, the constraints
were observed to be linear and this initially led to the
choosing of some form of linear programming model.
Before proceeding with the formulation, it is necessary
to define the terms which will be used in developing the
model. The objective function in this problem is the
benefit from the match of manpower to the accident rate.
The benefit is in terms of percentage of manpower allocation
and therefore its maximum value is unity. Let the term
"time period" represent those hours of the day when the
shifts on duty remain constant. Figure 13 shows the shifts
that were used by the Monterey Area during 1971 and their
associated time periods. A time period may contain from
one to eight hours. Let Y. be the percentage of manpower
assigned to time period i and X. be the percentage of
manpower assigned to shift j
.
The following is a discussion of the objective function.
The benefit in a time period increases as the manpower
assigned to it increases. The rate of the increase in
benefit is dependent upon the number of hours in the time
period. A useful analogy to benefit is the area under the
accident rate curve. If a time period extends over four
hours of the day, a unit increase in manpower benefits all
24
«tf
k>
w
l-H
PL, H^::=3
ou
PSwHo
psoPL,
CO«oi—
i
PSwP-,
H
pl,
rnCO
E-p-
CO
eg
PL,
CO
O
O
PC,
CO CO
E-t
PL,
CO
QCO oPS w i—
i
PO § PSO i-h cyX E-* Ph
25
four hours and therefore the area is increasing at a rate
of four to one. It is at this point that the linearity
breaks down.
To better describe the nonlinearity of the objective
function, it is advantageous to utilize a graphical descrip-
tion. Figure 14 depicts a sample time period consisting of
four hours. The percentage of accidents is plotted on the
vertical axis and the hours are plotted on the horizontal
axis. As the percentage of manpower assigned to the time
period increases along the vertical axis, the benefit
increases with a rate of four to one. This rate of
increase is depicted as a slope in Figure 15 which has
benefit, b, plotted as a function of Y.
The slope of the benefit function remains constant
until Y reaches the .07 manpower level. At this point any
further increase in Y causes excess manpower in the first
hour and is considered to have no additional benefit. How-
ever, the remaining three hours still accumulate benefit.
It can be seen from Figure 15 that the slope of the benefit
function decreases each time the percentage of manpower
exceeds the value of the accident percentage for the next
lowest hour. When Y reaches .15, any additional increase
in manpower will give zero additional benefit to the time
period.
The benefit derived from each time period is a piece-
wise linear function of Y, denoted by B(Y) = b. The model
can now be written as:
26
maximize: Benefit (Y)
subject to: The constraints on the X's
Where Y = f (X)
.
Let Y. be the manpower allotted time period i, (i=l...k),
and X. be the manpower allotted to shift j, (j=l...m). Y is
now a linear function of X which is described by the matrix
multiplication Y = AX where Y is the k x 1 vector of manpower
assigned to the time periods and X is the m x 1 vector of
manpower assigned to the shifts. The matrix A is a k x m
matrix whose elements a., satisfy
1 if shift j is on duty in time period i
a . .=
1 3 if shift j is not on duty in time period i.
The model can be rewritten as
:
kmaximize : £ B (Y,
)
i=lK
ksubject to: I Y. - [A]X. = 0.
i = l
The next step was to write the nonlinear model in
separable form. To accomplish this, the "Lambda Method"
was used as presented in Ref. 3. Using this method the
additional variable X is used to indicate the intervals of
the separable function as shown on Figure 15. The resulting
problem is written as follows
:
28
k ymaximize: Z Z b..X..
i=l j=l 13 1J
Nk
subiect to: Z Y.X.. - £(X) =0 V i=l,...k.ject to: Z Y.X.. - £(X) =
j=l 3 3
Nk
Z A. .
j=l 13= 1
MZ X =
1=1.125
V i=l,...k
X. . > VX.
.
X, > vx£
.
K equals the number of time periods per day.
N, equals the number of X's per time period.
M equals the number of shifts to be considered.
The value of the sum of the X's in the above model
needs to be explained further. Intuitively it can be
reasoned that since each shift will contain a percentage
of the total manpower, the total sum of the X's, which
relates to the percentage assigned, should equal unity.
The difficulty occurs when considering the units of the
Y's and the X's. Again the point can best be explained
graphically. Figure 16 depicts a hypothetical five-hour
day with two three-hour shifts that overlap during the third
hour. As can be seen from the computations, the total allo-
cation of manpower over each hour of the day must sum to
unity. From this the sum of X-, + X?must equal 1/3 or
29
.30
.25
.20
,15
.10
ssxx\Xl
+ X2
MEN ON SHIFT 1
X2
= % MEN ON SHIFT 2
THEREFORE:
HOUR 1 2
X-, + X.
FIGURE 16
X.
ax
- [X1
x2 ]
+ x2
+ x2
= 1
3X1
+ 3X2
= 1
x:
+ X 1/3
30
unity divided by the number of hours per shift. This
concept can be enlarged to encompass a 24-hour day with five
eight-hour shifts. It now becomes:
8 Y+ 8 Y
+ 8 +8 +8 =1Xl
X2
X3 *4 A
5
or Xx
+ X2
+ X3
+ X4
+ X5
= 1/8 = .125.
For illustrative purposes the model is written out in
complete form (See Appendix B) for the 1971 starting times
used by the Monterey Area of the CHP. The accident rate
distribution used is the one in Figure 3.
B. COMPUTER UTILIZATION
Because of the extensive manipulation of data required,
the computer was utilized to formulate the separable pro-
gramming model for each set of starting times examined. A
program was written (not included in this thesis) in
Fortran IV which, when given the starting times and the
constraining distribution of accidents as data, punched th
data on cards in a format adaptable to the MPS/560. The
allocation of manpower per shift was then computed. These
values were then read into a third program which plotted
the manpower allocation curves superimposed on the accident
rate curves.
C. DATA ANALYSIS
The technique employed in the analysis of the data was
to examine graphically the accident trends over the various
51
areas of the county. Plots of the various areas were made
in an attempt to identify specific accident trends related
to day of the week and time of the day. The results of
these plots are discussed in the summary.
32
IV. SUMMARY
A. RESULTS OF MANPOWER ALLOCATION
Upon completion of the model, the next step was to
obtain useful results. The approach taken was not to
arrive at a single solution, but rather a set of alterna-
tive solutions from which comparisons could be made.
Seventeen sets of starting times and two accident rate
curves were chosen for comparison. (See Table I and
Table II.)
One measure of effectiveness which could be used for
comparison is the value of the objective function for each
solution. The maximum value possible is unity, but it
is not attained because of the nature of the constraints.
The value of the objective function gives relative compari-
sons. If more information is needed, a second method may
be used. This method plots the manpower distribution super-
imposed on the accident rate versus time of day, thereby
allowing a visual comparison of the manpower and accident
rate (Appendix C and Appendix D) . An interesting comparison
to make at this point is between the allocation of manpower
used in 1971 by the Monterey Area and the solution obtained
from this model. (See Figures 17 and 18.) This gives an
indication of the effectiveness of the model in assigning
manpower for a problem of this nature.
There is a third measure which must also be considered.
After choosing a set of starting times, the number of patrol
33
CO<
CDn-qOPh
Ph:=>
wJPQ
OQoooCXI
CO>COwS
CDzI—
I
H<HCO
Hi—
i
CO
pq> S5n oH r-t
CD HW CD•-5 55pq :=>
O P-
LO
H
00
p< topqOh
HpqCD
pq
COpq
CD ^t-
55
Pi to
<Hco •>
(XJ
H
a:CO
CO
IOo
to
COvO
ovO
en
vo CD t—
1
«* 1—
1
rH f» «* vO CT> CO <* en rHt- CT) vo co LO CD vO [-- r-~ f- vO CO o enCO CO CO CO CO CO OD CO oo CO CO CO en CO
vO oo "5* CO »* «tf VO <3- CO ** CO CD <3" CO CO CXI «=t
to CX] CM CO <3- «* to <3- CD «* CD 'St «* o o to "3-
<—
1
cxi Cv) cxi t—
I
iH I—
1
to (XI rH <* O t—
1
CXI cxi CXI rH
O CO <* CO CO CD o CD O(XI vD CXJ vO CO vO CXI CD CDto rH CXI t—
1
CXI IO to CD CX] to
CDCDCD
vO CD CD CD vO «*vO CD •=t t-» vOIO CXJ CXJ rH CXI
oo vQ CO «tf CXI
rj- LO "«* o toCxi CXI CXI to CXI to
o CD CD "3- CD c CD CO «* CDVO vO VO CD vO vO vO CO CD voto to to to to to to (X) to to
•vi- vO CO vo O (XI «* CO CD O •vl- O CD o vO <Z) CDvo to CX] to CO LO vO (XJ CD CD "3- O CD CD cn (XJ oCD rH rH rH CD rH CD rH CD CD |H o CD O o ^ o
CO CDvO
*3-
COCO CO (XJ *3-
o vO to ccrH I—i CXJ rH
COvO (XJ CO OO <XJ
to to o o torH (XI r—
1
H cxi
K + -K *•K -K 4J
+ -r. r. -k
rr cxi
cd m <-hNNN(MNNNNN(MHNMNNM(NJ £ —! -HCxicxicxjcxicxicxicxjcxirvicxicxicxicxicxicxicxicxi.r-lj^^
E— CO CO
cncocncoocoocxjcnenrHOcncncnoo 4-> c/> c/i
rHrHrHrHCXJrHCXJfNJrH^HfNlCXJrHrHrHCXJCXlLMCDO•H 4~> -M
- X Ctj aN'sJ-^fKltONNin^tN'tN to to ^r- «3- co C £rH rH rH rH <-{<-*
rH E £* * ., ~ r- >h -h
CDCDCDOCDOCDCD'^-CXlCDCXJrHtOCDCD'^-CnrHrHrHrHrHrHrHrHrHi—IrHrHrHrHrHrHrHi—IrHrHPqpq
vOvOvOvOvOvOvOLOvOvOLOvOvO ,OvOOvO-S:+-XCDCDCDCDCDCDOCDCDCDOOCDCDOOO *
34
> ^1—
H
O LO VO CN] rH tO rH rH "C-J- to vO ri (XI o o rH ^t- tO
H i—
I
CO CO CO CO CO CO C> LO LO r-- LO CO CO >o CO r-~- lo
u H CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO co CO
m CJ*-> :zPQ DO Ph
>-Ui—
i
Jo CO CD O <=* o vO CO O O \D CN) vO CD "Bt «=t «=t OPh LO o VO v£5 CD vo cn vO CD CD rH to r-- M3 CN CN) CN) CD
eg fO to to rH CN) rH ** •vj- CN) "=t rH rH CN) CN) ^f ^>
ex,
J3
m_iCQ "<* CD <3" vO VO "=* v£> CD CD "=fr CD VO CD CO CO CD CD
E=> H "5* cn) CM vO C-^ LO CN] LO CD CD CN] CD LO CN) o CD CD CD
O Ph cni rH CD rH CN] Cn) CN) CD CD CN) CD CN] to CN) Cn] CD CDQ rH
ECo COo
i—
i
Csl Pin ro w o CO ** CO ^J- CD CO vO CN] <* VO <=* «=t CN] CO ^O CN)
a, to cni CD vO O CD CD CN] LO LO CO LO co co LO CO O) LO
W • to CNl CnI CNl to CN) IO CN] to to (NJ to to to Cn) Cn] to
^ co HPQ > Z< WH CO U
2 Pi CN) vo vO (NJ •=* ^f (NJ CO o o CO o CD o -=t ^t- on w (NJ t~- r^ t-O CD CD CO f-. o o CD o CD O o O o CD
H ex, o rH rH rH rH rH CD CN) CD O rH O O CD rH rH CD
U2;1—
1
Hctf \D VO o CD vO 'O o MD CO o «=t tT \D O o o CO< rH r- rO f- CNl h- CT> r-^ t-O *t r^ ^r co fO rH r-^ r^ «*
H i—
i
r-
1
rH rH H o rH rH CN) rH H r~\ <H CN) rH rH CN)
CO
HPhi—i
ECCO
X X X -X * •X
«=t CM
CO * + }C X + 3; K .£ X tn +-> +J
W CD m <^
S LO CnI CNl CNl Cn] (NJ CNl CN] CN) Cn) (N) rH CNl CN) CN) CN] CN) (NJ e •H •Hr-
H
cm CNl CNl CNl CNl CN) CNl CN) CN] (N) CNl CNl CN) CNl CNl CN] CN] •H x [-H
H t-1 CO CO
CD ^f CTi CO cn CO CD CO CD CN) CTi ZT> rH o CTi CTi cn o o -P (/) (^
Z I—
1
rH rH rH CN] rH CNl CN) rH -H CNl CNl rH rH rH (NJ (NJ CH o CD
i—
i
•H M +J
H X cj aS
cx: to (X) «* «=t to to CN) CN) LO "«* CN] •<* CN) rH to to »* «=t CO r^ »—
< t—
1
rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH •H .--i
H rH 6 £CO r>- H •H
(X) O O O o CD o O CD <* CN] CD CN) rH to O CD '^f CD rH rHH T-t rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH cq rU
i—
i
EC 1—
1
O >o O o o o O LO vO \D LO VO vo VD O ^a vO jj + •K
co o o CD O CD o CD O CD O CD O O O O CD CD 4<
35
t-^
wei:=>
IDt—i
Ph
wHsH^mQ
Pi nO UHh U<^
o •
t-H COH > rHO r^J Z CT>
O O rHCO t-H
H•J < •
pq uQ OO -1X ^<aw3=OOhzz
I
1.1
n
L__
o
o
I CO
|u")
<D <u+-> +-> .
Cti cti
or; bS
?H +j
o c2 CD
o "dOh •HCJ URJ . uS <
O f>— o
ooCDo
o
O
o
oCOo<No
o•* no o ex —
O O
36
oo
w«aj
DUh-
1
P-i
wH
H^Wni—
i
e£ oo uU-, <^ •
o to1—1 > rHH r-
^ z en•JOHO •-(
to H<
(X uK oCJ -JJ<c^w3^oCu•?:
<:s
1
tot
PI
L.
:i
+->
cci
oPh
+->
•Hou<
1 h
cm
o
o
»oCO
o
O
o
o"$
o
O
O
o 0>—. o o O
mO
vO O o o
cars required to meet the implied manpower allotments must
be computed. This effort is within the capability of the
local areas and includes consideration of the numbers of
men and cars available.
B. RESULTS OF THE DATA ANALYSIS
By examining the accident distribution plots for
various areas of Monterey County, a better description of the
specific trends throughout the county was obtained. Because
of the large number, the plots were not included in this
paper but were given to the Monterey Area CUP for their
use. A note of caution should be given as to the use of
this data. Because of the relatively small data base of
one year, any trends must be used only as rough guidelines
for decision making. This is due to the unknown parent
distribution from which the data was obtained.
38
V . CONCLUSIONS
The method developed in this thesis is considered to be
a valid starting point in solving the overall problem of
manpower allocation. The results obtained thus far have
immediate application, giving the CUP a reasonable alter-
native to their present method. However, there are many
other questions that remain unanswered and should be
addressed further.
As mentioned in the introduction, allocation of manpower
is made from considering the distribution of accidents from
the previous six to 12 months. No attempt was made to
fully describe the distribution of accidents, thereby
establishing a fixed distribution and the pertinent parameters
If this were accomplished, the model could be re-employed to
obtain results which would be applicable for a long period
of time and not be dependent upon the previous year's distri-
bution. This approach could then be further implemented to
describe differences in the accident distributions throughout
various beats in any area. The next logical step would then
be to construct a model for deployment of officers over the
various beats.
As for the model itself, it was employed in a simple
manner in solving the problem of manpower allocation. It
is recognized that, to solve the model, access to a computer
facility is necessary. However, it should also be recognized
39
that the model can be applied to the CMP problem or any
other similar type problem in a myriad of ways. As an
example, one could enlarge the time period from a day to a
week in order to determine a manning schedule. This concept
could be extended to cover a period of a month or even a
year to assist in determining optimal times for vacations.
Also, another variation would be to establish upper and
lower bounds on the number of men required.
40
APPENDIX A
PRE -LIEUTENANT'S CLASS
PERSONNEL DEPLOYMENT
1. There are approximately 1784 annual man-hours available
to you for scheduling per man. This considers vacation,
sick leave, etc.
2. Divide 1784 by the 52 weeks in the year and we find that
each man can provide us with approximately 34 man-hours per
week.
3. To determine the number of weekly man-hours available to
us for scheduling, we multiply 34 man-hours times the number
of men available to us for traffic enforcement. In other
words, if we have a 44-man squad we will assume that after
special detail and the like are considered, there will be
40 men available to us for traffic supervision purposes
during the month. Thirty- four multiplied by 40 provides
the total of man-hours available to us for each week of the
month, or 1360 man-hours.
4. In reviewing our Deployment Control Reports, we find
that in the last six months, or during a similar season
last year, our particular area experienced:
12.8% of its total accidents on Mondays12.3-0
11.9112.4%16.6%18.0%16.0 ?
5
TuesdaysWednesdaysThursdaysFridaysSaturdaysSundays
41
5. To determine the manpower assignments in proportion to
the accident experience by day of week, we must multiply
1360 (weekly hours available) by the per cent of accidents
each day of the week. Here is how it should look using
the foregoing accident experience.
Mondays - 12.8% of 1360 = 174 man-hoursTuesdays - 12.3% of 1360 = 167 man-hoursWednesdays - 11.91 of 1360 = 162 man-hoursThursdays - 12.4% of 1360 = 169 man-hoursFridays - 16.6% of 1360 = 226 man-hoursSaturdays - 18.0% of 1360 = 245 man-hoursSundays - 16.0% of 1360 = 217 man-hours
TOTALS 100% 1360 man-hours
6. To convert these daily requirements into men we divide
the man-hours required each day by eight hours (one man's
shift) and come up with the number of men we should have on
duty each day of the week to correspond with the accident
experience
.
Mondays - 174 divided by 8 = 2 2 men (nearest whole number)Tuesdays - 167Wednesdays - 162Thursdays - 169Fridays - 226Saturdays - 245
'
Sundays - 217
8 = 21 men (
8 = 20 men (
8 = 21 men (
8 = 2 8 men (
8 = 51 men (
8 = 2 6 men (
7. The same principles can be applied in determining the
number of men to assign to each respective shift. A time
of day study over a six-month or longer period should provid*
a reasonably accurate guide to the accident pattern. For
example, let us assume that such a study revealed that our
shifts should be established, generally, as follows:
42
Shift #1 - 0001 - 0800 hours - 10% of the accidentsShift #2 - 0700 - 1S00 hours - 25% of the accidentsShift #3 - 1400 - 2200 hours - 50% of the accidentsShift y/4 - 1600 - 2400 hours - 15% of the accidents
In applying the 40 available men to this pattern, we would
assign 10% or four men to Shift #1, 25% or 10 men to Shift
#2, 50% or 20 men to Shift #3, and 15% or six men to Shift
#4.
Many other factors would normally enter the picture,
such as vacations, fixed post assignments, injury time off,
and the like, which would reduce the available manpower.
However, the foregoing is provided as a guide and to offer
a basis from which to begin.
43
APPENDIX B
APPLICATION OF. SEPARABLE PROGRAMMING MODEL
FOR SHIFT STARTING TIMES 06, 10, 14, 19, 22
AND FOR A 2 000 TWO -MAN CAR POLICY
OBJECTIVE FUNCTION
maximize
:
i-\p?o3 +.kAi3 -t-jrt^ +-^~?>3S +
-^ 1¥3 + XX3^3 J
r
i-U>Je, 7^-- 0SJ 3I7 +-OS-3 J^7 ~t- 0i- 1 JJ7+- 0C'l J* 7J
44
CONSTRAINTS
subject toSUDJt'LL LU . y _ /\
Dirt*.***?,** 1- *** *~%3
OS
lDL,. of/2 S.os7l,+. o*?„+M+?s&<>hr*sfe
°
Vln.2
gi***/
£*3iw
45
H COPiDo
H s en^ rHw CDQ O •»
i—
i
LO ^tU CD rHU< I
CD
^ tH CD rHo r-. CD
u HH CTi CD •»
H i—
1
CNI vO
X E> CD\—
i -J • QQ o ' CO W CO"Z 00 > J mw PQ ^cu 1-1 ^ D t—
i
Ph w o O H< Q —
1
OO H oSS < pq 2
u H i—
i
o <i H—1 Ph Pil-J << c—
«
HCO
Pi ww Q H:s i—
i
PL,
o U i—
i
P< u Xz < CO
46
COex
cn
CM
oCM
H CO CNJ
So
CXI
H ac oo^ rHw oQ o \
rH LO <*CJ CD rHCJ< I
CD^ rH o rH
o t~-~ CDt—
I
G\ O #\
H rH CO OC3 CDkJ • Oo CO w COCO > hJ w
eo <i,
-1 :z D (—1
w o o HQ r—
(
QO H uS < W z
CJ H r—
1
o-J £
H-J << H H
^: COc£ ww Q H^ r—
1
Pho u r—
1
PL, o K•z < CO (II
CD
OP.
CD+->
cU
oi
CD
Xt•HOo<
I
oI-00
CO
-
oocoo
ooo
o«<?
oCOoCNJ
o
t" ll
\
o o— oCOO o
c* —o o
47
H
WQi—
i
CJCJ<
O r-
i—
I
E-
JO 00CO >
owQ -<
O HS <
C_5
O-J
<
woPL,
2
00
Ore
ooo
I
E->
2
CM
oO i—
I
oCD.(X) vO
CDoPJ CO
O E
W
2;
H<H E-
X COpqQ Ht—I P-,
U l-H
CJ< CO CD CD
4-> 4->
c3 nJ
Pi c^
J^ •P
0) ££ 0)
o TJp. •HC UOS us <
O-
i -
t>
1
cop— o o
L
I
L _
Jo*
Id
CM
OCM
CO
<0
O
C>
oo«n
o<*o
oOSo
o ono o o
48
^-rr
^3'
COMCMCM
c<
wH CO (XI
2C*
oH ffi coZ r-
1
W oQ o #\
r-H lo tou o rHu< i o
z rH o rHo t^~ ot—
1
CX> CD •\
H r-
1
(XI ^OD CD.-J • nO CO W COco > —J pq
cq 2-i 'Z D \—i
m o O HQ 1—
1
Qo H CD2 < W ^:
cj H h~i
o <£ HhJ c2 Pi1-5 << H H
^: COPi pqpq Q H^ i—
i
P-,
O CJ i—
i
P* u •"r-"
2: < CO
„J.ri
i
1
L_
CD CD
-M 4-> 1
rt 03
OS oi
u +->
CD FJ
£ CD
O n3P< •H« UBj oS <
o—
'l
co•" o o
._!
,r.
iO
c*i
cs
o
ooCOoN.o>©o
o
OCOOCMO
o oXT
omo o o
49
wH CO eg
3 Poeg
H X oss Cs)
w oP o r\
i—
i
LO toCJ o rHu< i
o^ iH o rHo r- ot—1 cn o *s
H rH CNJ ^D
O Oh4 • PO CO W COco > -J w
CQ s-3 ^ D h-
1
w o O Hp K—
1
Po H CJ
S < w 2;CJ H i—
i
o <^ HJ K C*—) << H HZ COex; Ww P H^ t—
i
CL,
o u 1—
1
P4 CJ X^ < co<s
50
M-l
H CO (XI
<£* Pi (XI
K DO #*
H a o^ OJW CDQ CD tf\
i—i LO (Nl
U CD rHu< 1
CD^ rH CD rH
o t"» OHH Cn CD *\
H rH cnj vD
D CDvA . Qo CO W COto ^> -J w
pq*r-»
J ^ D 11
W o o HQ n QO H t3*>* < w 2:
U H —
i
O 5yA << H H
:z COPi wm Q E-h
^ i—
i
Pho U 1—
1
P-. u K^ < CO (D 0)
+-> 4-)
tf 0}
Pi Pi
rH Md) c£ a>
o 'OP. •H.c u03 L>
s <
O o> towm o o
52
I
L_
wHK 00 (XI
c* (X)
H ^)^: o «\
w m (X)
Q (Xl
t—
i
oU o •V
u LO LO
< o rH
^ rH 1#\
o [-- O1—
1
CTl o rH
H rH oD o •>
-5 • (XI LO
O' 00 O00 > Q
W 00-5 z -4 ww o 03 S« HH O h-
1
o H O Hs < Q
U UO W z:-4 H i—
i
-J <tj H< £
<Pi P-H p-.
w z 00^ wo Q Ha, i—
i
IXz u 1—
1
u K< 00cm
o
o
<D+->
M
•HUo<
HI
I
Li
o
TI
ooCOo
ooomO«*•
oOo
o
I" 1
o e>—. o
a> rxO O o
—r~
ono
CM —o o
53
wH CO CM<^ os CN)
K J=>o r\
H K en^ r-\
w oQ o •N
i—
i
LO «*U CD rHCJ< I
»>
^^ I—
1
o l—
1
o 1-- oK-
1
en o #1
H i—
i
CN) MD
O CD-J • Qo CO W COCO > hJ W
PQ r^q -7 D t—
1
w o O HQ 1—
1
Qo H CDs < W Z
CJ H l-H
o <£ HkJ K eSJ << H H.
2 COc* Ww Q H:s i—i CIh
o U t—
1
a, CJ X^ < CO CD CD
•(-> M 1
aJ cd
c* C^
5h +->
CD C£ CD
O ndPL. •Hc ua3 us <
T~o ©•*- o
54
wH co CX)
3Pi
o(XI
H ac cr>
2: rHw oo O r\
1—
1
LO (X)
u o t—
I
u< 1
CX]
^ rH o r—
1
o r-- CDhH c^ CD *»
H i—i (XI VD^> O—] • nO CO w COCO > j W
PQ sJ ^ E> 1—
1
w o O HQ i—
i
QO H 'O2 < W z
O H 1—
1
o < H—1 Pd PihJ << H HZ COPi Ww P H^ t—
<
P-,
o u i—
i
Oh u a:2; < CO CD CD
M POj rt
Pi Pi
U +->
<D a^ CD
O T3a. •Hp! ocd uS <
o o-— o
I
L_-
u
o
O
oCOo
OOo
o
orto
o
o of*> p« —o o o
55
H CO (XI
<j" pS Csl
PS :=>
o •\
H E en:s r-H
w oQ o 9\
h-
1
LO =3-
O o rH
u< 1
O^ I—
1
O rHo t^ o\—\ c*> o r.
H r-i (NI vO!=> O-q • nO CO w COto > ^ w
PQ <i-3 ^ CD 1—
1
W o O HO n «o H •'O
s < W ^U H 1—
(
o g PihJ << H Er1
Z COcsJ Ww o H£s 1—
1
Pho U i—
i
a, o X^ < co
L__
a> CD
+-> •p
rt ai
a! Pi
u 4->
o C£ CD
o tJa •Hc L>
crj us <
o O- 00— o o
56
wH CO c-o
so
H EC o^ CN]
W oQ o >
n LO cxi
U o iHU< 1
CN1
^ r-
1
o i—
1
o r-. o1—
I
CTl CD >
H rH Csl vOED OJ oo CO cq COco > kJ w
CQ SkJ ^ D i—
i
w o O HQ h—
1
Qo H '(J
s < W ZU H i—
i
o <cj H-J K OS_q << H H
2; COoi ww o H^ 1—
1
P_o CJ i—
i
PL, CJ E2: < CO
P«S
woPL,
2:
E-
WQi—
i
Uu<
o
I
J.
___.!
r
I
I©*
;ea
v9
Vfl
to
OS
—4
oCOo
o
oto
o
o
o
o
o o OIN —o o
57
wH CO CN]
<3* & CNJ
o2 Do •N
H ffi <3\
z rHw oQ CD »M LO i—
1
u CD rHu< 1
rH
^ r—
1
CD rHo l>- CDr—l CTi CD •*
H rH 04 vOD O-J • «O CO W COCO > -J w
CQ SJ s D i—
i
w o O HQ 1—
1
QO H •CJ2 < W ^:
CJ H r—
1
o 2l-q << H e
2: COc*: ww o H^ i—
i
CHo CJ l—
I
Pm CJ rc2 < CO © <U
4-> +->
HS 03
c^; cr:
rH M<u P3
£ CD
o T3a, •H£ ocrj uS <
O
|CS>
is
•o
If}
<
ooe>O
ooo
o
o
o<No
o o-— o o •0
o o o o
HwH-l
uu<
S5Ot—
i
H
O coco >-J 2w oO Hs <u
o-3
<Pi!
W&OPL,
2:
co
o
Ooo
I
:=>
OQWH
(VI
CM
i—
I
t-O
tOO i-H
Ooo
Qpq co
r L3z;i—
i
Hp*<CO
< CO CD <D+-> +->
03 cj
c* OS
u +->
<L> P!
£ a)
O t3P* •Hc oRj oS <
———T—
-
o o CO•>- o o
59
wH CO CM
3 Pi
oCM
H K CTl
^ i—
1
W oQ o #\
t—
i
LO toU CD I—
1
u< 1
CD^ pH CD rHo r- CD1—
1
CTl O •
H rH CM vO
^ CDhJ • QD CO W COCO > -J w
pq 2-3 ^ D h-
1
w o O Ho t—
1
Qo H Us: < W ^:U H i—
i
o <£ HkJ p2 PiJ << H E^
S5 COat ww Q t—
'
;s i—
i
[Xo CJ 1—
1
Ph C_J DC^ < co
60
wH CO CXI
3 O(X)
H 33 o55 CsJ
W CDQ O *N
i—i LO •=^"
U o i—
1
u< 1
O55 i-H o i—
I
O r-- ol-H en o •N
H i—
1
CM vO30 oi-4 . QO CO W 00co > J w
PQ S-J 55 33 I-H
w O O HQ i—
i
QO H IDs < W "Z
U H t—i
o <£ H-J K CtJ
J <K H H
55 COc£ ww o H^ l-H Pho u I-H
PL, u 3355 < CO <D 0)
+-> +->
a aj
& Pi
u +->
<u Pi
£ <D
o nd(X •Hc U05 us <
o O- 00•- o o
LJ
ot
CMCM
CM
o
ea
N.
O
'•r,
<
en
CM
o
ooCOoIS.
o
o
o
OcnOCMO
•oo o o o o
61
wH CO CN)
SO
H 5C o2 CMW OQ O •\
I—
i
LT) ^t-
U O i—i
u< '
55 rH o rHo t^. on en CD •S
H t—
I
(X) vD
D Ok4 • QO 00 m COCO > _; w
PQ SK^! 55 53 H-l
W O O HQ i—
i
«O H •X3
S < pq SU H k—
i
oJ 3 C5J << H H
Z CO»; Ww Q H^ i—i P-,
o U t—
1
CX, u ac55 < CO
r A
I
i
CM
O
o <D
+-> +->
aj cd
PS c£
rH +->
0) C£ CD
o TJCu •Hfi urt u^ <
o o-— o
-f-~
CO
M
L_
1
I
i r
ooc*J
o
O
o
o**oCOoOSo
o o onO
ex —o o
62
wH CO CX)
3OS
OH X en^ i—
I
W oQ o #s
(-H LO ^fu o r—
1
u< 1
oP2; I—
1
CD i—
1
o r^ OQ t—
I
Ol CM #,
H i—
i
ro vOX D oi—
i
hJ • QQ o CO W CO^ CO > ^ cyW CQ ^.Oh -] 7Z D nO-, w o O H< Q 1—
1
QO H Os < W ^
CJ H i—
i
o1-5 S H
Ohj << HCO
oi pqW Q H^ i—
i
PhO CJ i—
i
a, u J"H
7Z < CO CD 0)
M Maj CCS
OS oi
fn »->
0) C£ <D
o -dp. •HC uaJ uS <
O 0- <*» f> J in_ o O O o
63
wH CO cm
3PSDO
cm
H K oo2 rHw oQ o #.
I—
i
LO <3"
U O rHu< 1
O2 rH o rHo r-- or—
I
CTi CM ^
H i—
1
(XI vOD CD-J • Qo CO m COto > -j w
pq S>-} 55 D t-H
w o O HQ 1—
1
QO H cj
SS < pq 2;U H i—
i
o < HhJ cd OS-J << H En
Z COoi Ww Q F-H
^ n P-i
o U i—
i
a, O »uz < CO (1) <u
+-> +->
cd oj
OS C^
f-t +->
0) C> <1>
o ^a&, •Hc uaJ uS <
O Cr-
- o<*> s vi ^ > IA N -Q OOOO OOO
64
J"
COCM
CM
CM
wH 00 CO<£ Pi (N)
c2 £3o n
H ffi CX>
^ r—
1
W oQ o •<
t—
i
LO M-U o rHu< 1
o^ rH o rHo r-^. CDr—
1
CT) CNJ •*
H rH Cx) vOD O-1 • QO 00 W 0000 > -J w
pq <i,
—
!
^; D r—
1
w o O Ho r-H Qo H CJ2 < W £:U H i—
i
OS
-J << H H^ 00
t* WW Q F-^ i—
i
PhO U i—
i
cx u 3^z < 00 CD CD
4-> 4->
nJ crj
Ctf Pi
rH P0) CJ
£ CD
o T3Ph •Hc oCC5 uS <
oCM
Cr-
CO
N.
<0
lO
CO
CM
OoCOo
oooLO
o
«roCOOCM
O
&- ©a r« sS In -*• m r4 —o Q> o o o o O li
65
wH CO CXI
<* c< CN)
K :=>
o •>
H DC oo2: rHw oQ o •N
K-H LO rou o rHu< 1
O^ rH o rHo r~- o1—
1
CT> rj *>
H rH Cn] <o:=> okJ • Qo CO W COCO > ,-5 w
m SkJ ^ D r—
1
w o o HQ r-H QO H •oS < P4 ^U 1—
|
i—
i
O •ei HH K (X,-J << H. H
2: COC* wcu p E-h
«£ 1—
1
PhO u i—
i
CL, CJ DC^: < co <D to
+-> 4->
rt oj
P< c£
?H MCD c£ aO ^dPh •Hfi ucrj os <
r---J
L__J-
L_L
_ „ t
CN
CO
CN
CNCN
CN
oCN
00
•O
"J-
CO
CN
o
Oooo
o
O•Oo
mO<*
oCO
OCN
O
t—»—»—«—
r
6- CO N O Ifl tj*ff» •* >OOOOOOO9O
66
-4~-
co
CM
<N
wH 00 CxJ
<£ C* rsi
p. I=>o »
H K oiz; CNJ
w oQ o Vl
l-H LO toU O i—l
u< 1
O•^ I—
1
CD i—
1
o r^. OK—
1
CT) r-j *\
H i—l CM vDD O-3 • OO CO W COCO > -J w
rq S-5 ^ E3 mW o O HQ H-H RO H uS < W 2U H l-H
o <f H-q k o*J << H E-
2: COPi ww R H^ i—
i
Pho U i—
i
Pn u *T")
2: < CO+->
cd
in
O
cci
+->
Pi
+J
o
•HUu<
U
1 r-
o o~i
—
o
CN
oCN
CO
m
CO
i
CN
O
oooo
O
ooomO
COOcsO
o o o O
67
CM
roCM
CM
C4
wH 00 eg<3j e* cnI
ex Do «
H au CO^ rHw oQ CD *>
t—
i
LO Ox]
U o rHu< 1
o^ rH o rHo r^ CDn cr> rq *\
H rH rg ^D:=> OkJ • Qo 00 W 00CO > H-3 W
pq s-4 ^ :=> r—
1
w o O H« 1—
1
Qo H CDs < W 2:
U H r—
1
o5
-} <<< H H
Z 00C^ Ww Q H^ i—
i
tin
o U t—i
Ph u K2: < 00
r
i—
i
03 0)
P +-> ,#
crj 05
CX" (X
u •M<D c£ (D
O M3Ph •Hc u03 us <
«
o s1
Mo
r*i
i
I
I
i
—
CM
oCM
O
co
is.
MO
in
co
CM
oooooKOOO
om-
oCOoCMo
O o o o 3 o q
68
pq
H2:wo1—
1
uu<r-l
i—
1
CO>
ol-H
H<CJo-J
<
wOex.
CO
Do
ooo
1
oCDCNl
•
cm
QW-JpqDOQpq
E-^:cqQi—
i
CJu<
CM
OCNl
CNl
rH
#^
OT—
1
CD
COcq
i—
i
HCD
l-H
H<HCO
l-H
CO
Manpower
Rate
'-' » — ' r
1
1
1
1
1
1
1
1
I
1
r*~*
f
1
1
o1—
1
H!
i
OCO 1
L-5pq
1
O
CD+->
c*;
Mfi<U
•HUu<r!
1
1
1
1
1
1
I
1
1
f • T
1
1
1 7 1 1 1 '
CN
COCN
CN
CN
C-l
CDC-J
o
00
m
0")
CN
o
0"
aCOo
ooomO
orooCN
o
O ocj— O o O a Sa-o O
iN -O O
69
I
ll-
CM
©CM
wH CO CN]
<t| C* Osl
c2 :=)
o -
H E CsJ
iz; CNI
w oQ O •«<
i—i LO LOU o 1—
1
u< 1
o2 r—
1
o r-H
o r-- o1—1 en CN)
H rH Osl LO
D o-J • QO 00 W COCO > —1 w
PQ s>-} ^ D 1—
1
W O O HO t—
i
QO H •us < W ^
CJ H 1—
1
o2
H.-J &,-} << E- H
^: 00pej wW Q H& i—
i
LL,
o U h—
1
(X CJ a:^ < 00 <D CD
+-> M0j cd
(K (X
5-i +->
CD r-1
£ <D
O T3P. •Hc Ua us <
I
<T>
00
CO
IS)
PO
<M
o
©oCO©
oto©in
©
o
ofv!
©
©
9. % O o O o F O 5$ -O O
70
wH CO (XI
«rf & cm(X C3o »
H ac en^ rHW oQ o **
n ' LO -=*
U o i—
I
u< 1
«3-
^ I—
1
O T-i
o f^ o1—
I
CD CM #1
H .—
1
CM vOD CDhJ • nO CO w COto > k4 w
pq s-J p?: £3 h-l
W o O HQ i—i QO H US < W 2:U H i—
i
os
,-3 << H. H
2; CO(X wW Q H^ i—
i
Pho U K—
1
a, U X^: < CO <D 0)
+-> 4-> t
cd nj
as CC
Jh +->
<D C£ <D
O t3CL, •Ha ua u*i <
1 t
o «-o
71
wH CO co<f & COc2 Oo m
H ffi CT)
2: iHw oQ o *
t—
i
LO (XI
U o rHu< 1
CMPS t—
1
o i—
1
o r- o1—1 CT) CS1 *s
H t—
1
rv] vOD O-q • QO CO W COCO > -J w
PQ s-q z D 1—
1
w o O HQ 1—
1
QO E- OS < pq S
CJ H t—
i
o <£ H-J K oih-q << H E-
^ coc^ Wpq Q t—
i
^ i—
i
Pho CJ 1—
1
&4 u 1~"'
;z; < CO
1|
1
1 •
<D <D4-> Pcd 05
Cr^ rt
Jm p0) pi
£ 0)
o t3D, •HPi oOS uS <
cn:
CM
II
1
l_„ __I
r1
-i r"
CM -.
O«
C-J
©
o»
co
to
l/>
TO
»"4
©
©
CO©
o
o
o
©
o
o o o D o*4-
o
72
o
o
H
QUu
O ooCO >-J 2;w oQ M
<UO
<
woOh
<
o
w
CnJ
CNI
00
Oa:
ooO
I
O rHO(XI .
*
OQW oo
w
< E-
<E- E-Z 00WQ Hi—
i
PuU l-H
U K< 00
+->
CD
op.
<D
•MOj
a+j
0)
•Huu<
73
"3-
CM
COCM
<MCM
CM
CVS
wH C^O CM
$o
CM
H ac O^ CMW oQ o »\
i—
i
LO CMU o rHCJ< i
CM^ l—f o i—
1
o r- ot—
1
CT) CM rv
H rH CM voP o,-} • GO 00 W 00CO > -3 W
PQ S-1 ^ P hHw o O H« i—
i
Qo H COs < pq 2O H 1—
I
o1-3 2 E-"
J << H H
S; 00Pn* Ww Q H^ >-H Hho U 1—
1
Oh u E:z; < oo <D CD
+-> +->
ctf erf
«! rt
u 4->
<D a•>
0)
O H3D, •HC UC3 us <
L
cv>
oo
CO
if}
c<-»
CM
o
oCOo
©ino
©
COoCMO
O O O a S o o o o
74
wH CO CM<£ Pi Os)
Oh DO e
H K CTl
Jz; rHw CDQ O *>
i—
i
LO <*U CD rHCJ< 1
CD£: rH O rHo r>- ot—i CT) oo #\
H rH cn \oS CDH^ • nO CO w COCO > •J w
PQ S—I 53 D r—
1
w O O Ho i—
i
Qo E- C3s < W 2:U H r—
1
o <£ H,-3 K OSk-J << H H
^: COC^ wW Q H^ i—
i
UhO U rHa, u K;? < CO
75
cs
COCS
wH CO CS<t" Pi cnj
K t=>o •«
H ac cr>
2: rHw oQ o •*«
t—
i
LO toCJ o rHC_J
< 1
to!? rH o r—
1
O r~- ot—
1
CT> CNl r.
H r-
1
cxi vOS O-J • QO 00 W COCO > i-l W
CQ *><
—1 ^ £> i—
i
w o O H« r-
1
Qo H C3s < W •z
CJ H *—i
O <£ H-1 K oi-4 << H S-
•z copcj wW Q E-h
j» t—l CX,
o U 1—
I
Cl, u J-l
^ < CO 0) CD+-> +->
ctf 03
cd »;
Jh •MCD a^ CD
o TJPh •H£ ort uS <
r - - ir j ^ «,•»"
i •f—r
1
I
L_.
CMCS
CS
o
o
CO
m
CO
cs
ooCO
oKooa
o
oCOocsO
G- oc>
O O o is"9-
o ON —
O
76
- ft-'
.f-
CM
cocm
cmCN
wH CO CM<C Pi CMK po f>
H X Ol^ I—
1
W o« o »
i—
i
LO tou o 1—f
u< 1
#1
orH o rH
o r^. or—
1
CT> On] #>
H rH CN] vOZ3 O-3 • QO CO W 00CO > —J w
rq S-J ^ D •—
i
w o o HQ l—l nO H US < w p^
CJ H i—
i
o <f H-j &-, PS-i << H H
Z CO& Ww Q H^ n Pho U i—
i
P-. U X:z; < CO
ri
i
i
L_.
I ,_
I
CD 0)4-> +->
CCj rt
pS pS
U +->
CD Pi
£ oO xsP, •Ha ua CJ
S <
cm
oCM
00
—*i o
Irt
CO
CM
ooCOo
o
o
o
oCO
oCMo
X O ov0
o o o o o t>
77
wH CO CM<* Pi CMp2 DO ^
H as oZ (XI
W oQ CD '•<
i—
i
LO "3"
U o i—
1
U< 1
CD^ rH o rHo t-. o1—
1
en CM .*
H T-i CM MDas OhJ • Qo CO W COCO > -1 w
PQ *i•-q Z a> t—
1
w o o HQ r-
1
QO H oS < W ^U H r—
1
o <rf f-H
h-J ex psJ << E-h Eh
2: COCC ww Q E-i
<£ n Uho CJ t-H
0-, u n-1
^ < CO a> <r>
+-> +->
nj nJ
Pi PS
J-t •M
© PS
£ 0)
o TJCu •HCS oRj CJ
s <
O <y ^ r^ -£ trt
O D O O Ci O O
78
pqH CO ro<f c*: Csl
c2 do r
H K ois (Nl
w oQ o m
i—
i
LO •st
U o i—
i
u< 1
^ rH o rHo t-^ o1—1 en CMH i—
i
CNJ v£>
O O-J • QO 00 W 00CO > -3 PJ
PQ S-} ^ D i—
i
w o O Ho 1—
1
Qo H Os < pq ^U H i—
i
o <£ H-J K o:-3 << H H
2: COfii ww Q H:s i—
i
Uho U r—
1
ex, u »-th
^ < CO 0) <D
M +->
Ctj aj
Pi Di
rM 4->
CO c£ CD
o X!a, •H<-> urt oS <
r
cm
CMCM
-4
CM
©CM
CO»-!
r-t
ID
in»-»
«=»
»-«
COv-4
CM»-)
*-i
«-«
O
ooCOo
©
c
in
©COO
CM©
2 O0%
oeno o
I
K\ CMO O
79
LIST OF REFERENCES
1. Operation 101, State of California, Department ofCalifornia Highway Patrol, Operational Planning andAnalysis Division, Operational Analysis Section,November 1969.
2. Operation 500, State of California, Department ofCalifornia Highway Patrol, April 197 2.
3. Hadely, G., Nonlinear and Dynamic Programming, AddisonWesley Publishing Co.
80
INITIAL DISTRIBUTION LIST
No. Copies
1. Defense Documentation Center 2
Cameron StationAlexandria, Virginia 22314
2. Library, Code 0212 2
Naval Postgraduate SchoolMonterey, California 93940
3. Professor R. W. Butterworth, Code 55 Bd 1
Department of Operations Researchand Administrative Sciences
Naval Postgraduate SchoolMonterey, California 93940
4. California Highway Patrol 1
ATTN: Operations Planning SectionPost Office Box 898Sacramento, California 92504
5. Mathematics Department 1
Saint Bernard CollegeSaint Bernard, Alabama 35138
6. Chief of Naval Personnel 1
Pers libDepartment of the NavyWashington, D. C. 20570
7. Naval Postgraduate School 1
Department of Operations Researchand Administrative Sciences
Monterey, California 95940
8. LT Daniel Carey Schneible, USM 1
Naval Postgraduate SchoolBox 1993Monterey, California 93940
9. Professor G. H. Howard, Code 55Hk 1
Department of Operations Researchand Administrative Sciences
Monterey, California 93940
UNCLASSIFIED
DOCUMENT CONTROL DATA -R&D.Security CssiticUon ot titie, M.oi^W^j^^^'>f_'» r "> c,a
.
s.i.'JLl -.u . , .—
»
~——»"—
"
"' • * ~~—-~—«• ''— b«, HtPORT 5ECUF1ITY CLASSIFICATIONJToTiTT-UNG activity (Corporal* nulhorj
Naval Postgraduate SchoolMonterey, California 93940
Unclassified!b. GROUP
EPOflT 1ITLF.
A Separable Programming Approach to Allocating Manpower for
the California Highway Patrol
>ESCr<iPTivE NOTES (Type ol report and. inclusive dates)
Ma s t e rj_s_ Tli e sis; March 1975JjTHORlSI (First name, middle initial, latt name)
Daniel Carey Schneible
lEPOd T DATE
March 19 7 3
7*. TOTAL NO. OF PAGES
83
CONTRACT OR GRANT NO.
PROJEC T NO.
DISTRIBUTION STATEMENT
Approved for public release; distribution unlimited
. SUPPLEMENTARY NOTES 12. SPONSOMNG MILITARY ACTIVITY
Naval Postgraduate SchoolMonterey, California 93940
A es TR A C T
This paper presents a mathematical approach to formulating
problem in the allocation of manpower for the Monterey County A 3a
of the California Highway Patrol (CHP) . The technique employed is
to formulate the manpower allocation problem of the CHP as a no -
linear programming problem. The problem turns out to be separable,
and the Mathematical Programming System/360 for the IBM 560 computer
is subsequently used. This project was undertaken with the corpora-
tion of the CHP.
FORW1 NOV 65
>/N 0101 -807-681 1
D i A "7 *
)
(PAGE 1) UNCLASSIFIED82 "Security Cltssificetion A-3M08
UNOA^TJiim™-Sccurity Clfiasificstion _™_»r
KEY WORDS
Nonlinear Programming ModelSeparable Programming ModelManpower AllocationManpower AssignmentCalifornia Highway PatrolAccident Rate
ft TJ
R O l_ E
J
FORMt NOV e i.
S/N 0)01-807-6821
(BACK)3 UNCLASSIFIED
83Security Classification A- 31 409
(8 NOV 7*2 36l6
i4#S40Schneible
A separable program-
ming approach to allo-
cating manpower for the
California Highway Pa-
trol .
I e NOV 7K2 3 6 16
Schna ib)e^M
A separable program-ming approach to allo-cating manpower for theCalifornia Highway Pa-trol .