a separable programming approach to allocating manpower

A SEPARABLE PROGRAMMING APPROACH TOALLOCATING MANPOWER FOR THECALIFORNIA HIGHWAY PATROL

Daniel Carev Schneible

DSrSgNaval Postgraduate School

Monterey, California 93940

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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

S337£

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

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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.

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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.

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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

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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

FIGURE 14

b\ A;l A3 * 4.

1,0

y

27

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

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X-, + X.

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X.

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+ x2

+ x2

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+ 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

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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

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44

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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


3. Professor R. W. Butterworth, Code 55 Bd 1

Department of Operations Researchand Administrative Sciences


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



8. LT Daniel Carey Schneible, USM 1

Naval Postgraduate SchoolBox 1993Monterey, California 93940

9. Professor G. H. Howard, Code 55Hk 1



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


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


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 .

3 2768 001 00445DUDLEY KNOX LIBRARY

a separable programming approach to allocating manpower

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