Beat patrolling in urban areas A case study

Beat patrolling in urban areas A case study We define the following terms which are used throughout the paper: (i) Beat is a geographical unit/locality in our context. (ii) A beat point is a point (node) of some strategic importance and it must be covered at least once in one round. (iii) A beat route is a path which the patrol man may follow while performing a patrol. (iv) Beat numbers are the areas into which a given locality is divided for the purpose of patrolling/ supervision. (v) Beat patrolling is the procedure by which patrolling work is rendered in a given locality (for explanation see Figs. 1 and 2). The paper deals with the following: (i) To devise a plan for deciding beat numbers of given area and to check the usefulness or otherwise of the existing beats. (ii) To find out all possible routes in a given beat covering all the beat points and to rank them (by length) for the purpose of deployment. (iii) To devise a plan for deployment and choice of route number (hourly) covering all the beats on a 4 hour interval basis. (iv) To analyse crime with regard to type of crime and time of commission, and to use the analysis for deployment purposes in each beat. (v) To assess the relative vulnerability of different beats at various times.

J.P. S A K S E N A National Productivity Council, New Delhi-3, India Received July 1977 Revised May 1978

Urban police patrol work has been analysed in the present paper. Working on the crime-rate, as reported by the police department, a plan has been worked out to ensure an effective police patrol in a given locality. This locality is predominantly a "business/marketing area". Many problems connected with beat patrol like type of beat, route numbers, beatranking and others have been discussed. The analysis will help the police station to understand the beat patrol problem. Alternative beat routes have been suggested, together with deployment. Beat ranking may help in assessing the vulnerability of an area and as such proper steps may be taken in advance to prevent the occurrence of the crime. It is expected that patrolling will be more effective and useful after the implementation of this study.

1. Introduction In this paper, we propose to study the problem of beat patrolling in un urban areas. We are currently doing beat patrol work, but it is far from satisfactory. The beat-constables, who are supposed to patrol, are usually deployed to some work other than beat patrol in order to meet an emergent situation. Patrol duties, prior to this study, were assigned without any basis. In fact there was no patrolling work in many localities. With the increase in the population of big cities, the crime situation is deteriorating very fast. We wish to develop a new plan

2. Constraints of the beat problem These constraints were imposed by the problem formulators: (1) Length of a beat route must be about 3 km/2 miles. (2) Time of one round trip should be approximately 1 hour, effective working time being 40 minutes (3) Deployment should be based on reported crime. (4) Deployment should be single, double or multiple using one, two or three constables respectively. (5) Area of beat should be about 0.50 km 2 or less

The author wishes to thank the Director, Bureau of Police Research, for his kind permission to publish the report on beat patrol work. He is thankful to the referees for their valuable comments to improve the earlier drafts. which should substantially change the present situation and help in reducing crime thereby improving the social security of the public. What strategy should be adopted to meet this challenging situation? © North-Holland PublishingCompany European Jeurnal of Operational Research 3 (1979) 199-206. 199

200

J'.P. 8 a k m ~ / Beat I~trolltng In urban areas

•

DIVISIONBEAT 2 BEAT 3

Fig. 1. l ~ p showingjumdiction of PS.

depending on the density of the population in the beat. (6) Approximately ten beat points will be considered in each route. (7) IT~ese beat points should be covered once/n o n e hour. (Repetition is not allowed, as fas as possible, in this period.) (8) After 4 hours of beat patrolling covering around 12 km the constable will be put to light duty at the police station for the remaining period.

3. Methodology The analys ~ proceeds on the following lines: This police ..ration has ten ~ats, which have not been changed purposely. The population in beats 1 and 2 is very low; beats 8 through 10 are densely populated. Others are of average population size. No such relationship is commonly adopted between density of population and area of be~tt, but the m'~e is: The greater the population, the smalle~ the are t.

Table 1 Beat No. 1

Beat No. 2

Beat No. 3

Beat No. 4

Beat No. 5

Cell

Distance

Cell

Distance

Cell

Distance

CeU

Dismn~

CeU

Dismn~

(iS, 1) (IS, 9) (1,2) (1,9) (2,3) (2,5) (3,4) (3,5) (4,5) (4,6) (5,6) (6,7) (6,8) (7.8) (7,9) (7,10)

15.2 80 11 7.2 1.7 2.6 1.6 1.7 2.8 2.1 2.7 3.1 1.2 3.3 1.2 1.8

(IS, 1) (1,2) (1,3) (1,4) (2,3) (2,4) (3,4) (3,5) (3,6) (4,5) (4,8) ($,6) (5,9) (6,9) (7,9) (7,10)

7.3 1.3 3.4 3.8 2.5 3.4 3.1 1.2 1.7 3.1 1.0 0.5 3.4 2.9 4.0 0.9

(IS, 1) (1,2) (1,10) (2,3) (3,4) (4,5) (5,6) (6,7) (7,8) (8,9) (9,10)

4.2 3.6 1.0 2.7 2.0 1.8 1.7 2.9 3.3 0.8 1.0

(IS, 1) (IS, 5) (1,2) (1,10) (2,3) (3,4) (4,5) (5,6) (6,7) (7,8) (8,9) (9,10)

5.7 2.5 0.8 1.2 3.2 1.3 2.5 2.0 2.2 0.7 1.0 3.2

(PS, 1) (1,2) (1,7) (1,8) (2.3) (3,4) (4,5) (5,6) (6,7) (6,8) (7,8) (7,10) (8,9) (9,10)

0.4 0.8 1.8 0.8 1.7 1.0 1.8 t.:~ 12 T.9 2.3 1.0 1.3 1.1

(8,10)

1,8

(8,10)

2.0

(9,10)

3.C

(9,10)

2.5

Note: PS stands for Police Station.

Distance are measured in units of 125 metres.

201

ZP. Saksena / Beat patrolling in urban areas

vals and by type of crime for 6 months (Table 3 through 6) and has been analysed. Beat schedules have been drawn for all the ten beats (Table 7) and a ranking has been given to different beats, according to crime-behaviour (Table 8). Here, two estimates have been used for the purpose of analysis. The first relates to deployment of constables. Three types of deployment - single/double/multiple have been proposed (Table 7). Regarding type of crimes, weights have been given for different categories. For the purpose of this analysis, categories I, II, 111, IV (defined in Table 5) have been given weights 4, 3, 2 and 1 respectively. The last section contains some concluding remarks and answers the problem posed in the beginrdng of the paper.

Fig, 2. Beat patrolling.

The police station has supplied ten beat-points (Fig. 1) in each beat, which have been incorporated. The distance matrix (Table 1) gives the distance from a given beat-point to another beat point in a given beat. Based on the above matrix, alternative routes have been worked out (Table 2). The problem is essentially of the travelling salesman type [6]. The path is to be changed every hour in order to pay a surprise visit at a given point and to incorporate the concept of randomness in a given route. (The criminal should not be able to judge the arrival of a patrol man at a given point during a specified time.) Crime data have been recorded at 4-hourly inter-

4. A network approach Design o f beats and beat routes:

(1) Here the problem is: "How should the given area be divided into the least number of beats, such that each beat has a specified area whose upper limit is fixed, and encloses wholly or partly a given colony/ locality?" It is like the districting problem [3,41.

Summary of distance-matrix

Beat No. 6

Bet No. 7

Beat No. 8

Beat No. 9

Beat No. 10

Cell

Distance

Cell

Distance

Cell

Distance

Cell

Distance

Cell

Distance

~S,I) (1,2) (2,3) (3,4) (4,5) (5,6)

0.5 2.8 2.1 1.6 2.9 2.6

(PS,l) (1,2) (2,3) (3,4) (4,5A) (5,5A)

4.8 2.4 3.2 3.1 2.1 2.9

(PS,I) (PS,8) (1,2) (1,9) (2,3) (3,4)

12.0 9.4 0.5 0.8 0.5 1.7

(PS, I) (1,2) (I,I0)

7.8 0.7 1.4

(PS,7) (1,2) (I,I0)

I 1.2 0.9 I.I

3.4

(5,6)

4.5

(4,5)

0.9

3.6 2.6 1.8

(5,8) (6,7) (6,8) (7,8) (8,9) (9,10)

4.4 1.0 2.1 3.0 1.3 2.5

(5,6) (6,7) (6,8) (7,9) (8,10) (9,10)

1.7 1.3 2.0 1.0 0.8 1.0

(2,3) (2,4) (3,4) (3,6) (3,8) (4,5) (4,6) (5,6)

0.7 4.1 4.3 3.0 1.6 0.3 4.1 3.4

(2,3) (3,4) ¢'4,5) (5,6) (5,9) (6,7) (7,9) (7,10)

1.0 1.1 0.3 2.6 0.8 0.3 1.0 0.9

(6,7) (7,8) (8,9) (9,10)

(5,7) (6,7) (7,8) (8,9) (9,10)

3.5 0.7 1.7 1.5 1.9

(8,9) (8,10) (9,10)

1.4 1.8 2.8

202

Y.I'. ~ ~

/ Beat patroUlng in urban areas

Table 2 Beat routt~ Beat No.

1.

2.

3.

Distance f~om I'S to

A v e r a p distance Travelled per trip inkm

Total time taken (Ave.) in nfin

Anea in km 2

Alternative routes

30

0.5

Route No. 1. 9 - 7 - 1 0 - 8 - 6 ~ - 3 - 5 - 2 - 1 - 9 . Route No. 2. 9 - 1 - 2 - 5 - 3 ~ - 6 - 8 - 1 0 - 7 - 9 . Route No. 3. 1 - 2 - 5 - 3 - 4 - 6 - 7 - 8 - 1 0 - 7 - 9 - 1 .

2.8 3.5

Lenth in km

2.8

1.0 BP(9)

2.8

0.89 BP(I)

2.6

35

0.42

Route Route Route Route Route

No. No. No. No. No.

1. 1 - 4 - 8 - 1 0 - 7 - 9 - 6 - 5 - 3 - 2 - 1 . 2. 1 - 2 - 3 - 5 - 6 - 9 - 7 - 1 0 - 8 - 4 - 1 . 3. 1 - 2 - 3 - 5 - 6 - 9 - 1 0 - 7 - 8 - 4 - 1 . 4. 1 - 4 - 8 - 1 0 - 7 - 1 0 - 9 - 6 - 5 - 3 - 2 - 1 . 5. 1 0 - 7 - 9 - 5 - 6 - 3 - 2 ~ - 8 - 4 - 1 .

2.6 2.6 2.4 2.4 2.8

0.33

2.6

25

0.55

Route Route Route Route

No. No. No. No.

1. 2. 3. 4.

1-2-3-4-5-6-7-8-9-10-1. 1-10-9-8-7-6-5-4-3-2-1. 10-9-8-7-6-5-4-3-2-1-10. 9-10-1-2-3-4-5-6-7-8-9.

2.6 2.6 2.6 2.6

BP(I)

4.

0.31 BP(5)

2.4

40

0.50

Route Route Route Route

No. No. No. No.

1. 2. 3. 4.

5-6-7-8-9-10-1-2-3-4-5. 5-4-3-2-1-10-9-8-7-6-5. 1-2-3-4-5-6-7-8-9-10-1. 1-10-9-8-7-6-5-4-3-2-!.

2.3 2.3 2.4 2.4

5.

0.05

1.8

30

0.16

Route Route Route Route

No. No. No. No.

1. 2. 3. 4.

1-2-3-4-5-6-7-~9-10-1. 1-10-9-8-7-6-5-4-3-2-1. 7-I0-9-8-7-6-5~-5-2-1--7. 7-1-2-3-4-5-6-7-8-9-10-7.

1.7 1.7 1.8 1.8

2.9

40

0.30

Route Route Route Route

No. No. No. No.

1. 2. 3. 4.

1-2-3-4-5-6-7-8-9-10-1. 1-10-9-8-7-6-5-4-3-2-1. 1-2-3-7-6-5-4-8-9-10-1. 1-10-9-8-4-5-6-7-3-2-1.

2.9 2.9 3.3 3.3

3.5

45

0.35

Route Route Route Route

No. No. No. No.

1. 2. 3. 4.

1-2-3-4-5A-5-6-7-8-9-10-1. 1-10-9-8-7-6-5-5A-4-3-2-1. 10-9-8-7-6-5A-5-4-3-2-1-10. 10-1-2-3-4-5A-5-7-6-8-9-10.

3.5 3.5 3.3 3.3

1.7

45

0.19

Route Route Route Route Route Route

No. No. No. No. No. No.

1. 2. 3. 4. 5. 6.

8-10-7-6-5~-3-2-1-9-10-8. 8-10-9-1-2-3-4-5-6-7-9-10-8. 1-2-3~-5-6-8-10-9-7-9-1. 1-9-7-9-10-8-6-5-4-3-2-1. 8-10-9-7-9-1-2-3-4-5-6-8. 8-6-5-4-3-2-1-9-7-9-10-8.

1.7 1.7 1.7 1.7 1.5 1.5

2.0

45

0.10

Route Route Route Route

No. No. No. No.

I. 2. 3. 4.

1-2-3-4-5-6-7-8-9-10-1. 1-10-9-8-7-6-5-4-3-2-1. 1-10-9-8-3-6-7-5-4-2-1. 1-2~-5-7-6-3-8-9-10-1.

2.0 2.0 2.3 2.3

1.6

30

0.10

Route Route Route Route

No. No. No. No.

I. 2. 3. 4.

7-6-5-4-3-2-1-10-9-8-7. 7-8-9-10-1-2-3-4-5-6-7. 7-6-5-9-8-10-1-2-3-4-5-6-7. 7-6-5-4-3-2-1-10-8-9-5-6-7.

1.6 1.6 1.8

BP(1)

6.

0.05

BP(I)

7.

0.5

BP(1)

8.

1.2

BP(8)

9

1.0

BP(I)

10

1.4 BP(7)

23.90 km

Note: BP stands for beat point.

3.65 hours 3.45 km 2

1.8

Table 3 Crime analysis (details for beat 3 only) (&hours intervals) Beat no. 3

Month Hours *

Oct.

Nov.

Dec.

Jan.

Febr.

March

Total

Max.

Min.

0-4 4-8 8-12 12-16 16-20 20-24 Timing unknown:

0 0 3 2 1 0

2 0 0 0 3 1

0 0 0 2 1 1

0 0 0 4 3 1

0 0 0 1 1 1

1 0 1 1 2 0

3 0 4 10 11 4

2 0 3 4 3 1

0 0 0 0 0 0

3

-

2

-

1

2

8

3

0

Total

9

6

6

8

4

7

40

16

0

Total

Max.

Min.

Time of the day at which the crime was committed.

Table 4 Summary of crime analysis (4-hour intervals) Month Hour~ *

Oct.

Nov.

Dec.

Jan.

Febr.

March

0-4 4-8 8-12 12-16 16-20 20-24

1 1 6 10 9 9

5 1 8 9 8 8

2 1 2 5 5 4

1 2 1 10 9 9

1 0 1 14 6 10

1 1 5 5 10 4

11 6 23 53 47 44

8 5 14 23 21 17

0 0 0 0 1 1

Timings unknown:

14

5

11

11

7

17

65

24

1

Total

50

44

43

249

112

3

.

.

30 .

.

.

.

.

.

.

.

.

.

.

.

43 .

.

.

.

.

.

.

.

.

.

.

39 .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

* Time of the day at which the crime was committed.

Table 5 Crime analysis (details for beat 4 only) (by type 0_f_cri___me). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . weight index Category I

- 4 points

Heinous crime: dacoity, murder, attempt to murder, robbery, riots. Category ii - Non-heinous crimes: buzglary by day/night, thefts of all kinds, cheating, kidnapping, abduction, molestation of women and miscellaneous Indian penal codes cases. Category Iil - Vigilance of police: arms act, opium ~ct, gambling act, excise act and other acts. Category IV - Preventive action: (i) resolving conflict between two people/groups (case not registered), (ii) a drunkard kept in detention for overnight (case not registered), (iii) driving in an irregular manner, violating traffic rules, etc.

- 3

points

- 2 points - 1 point

Beat No. 4

Category Oct.

Nov.

Dec.

Jan.

Febr.

March

Total

Weighted total

Max.

Min.

1 11 III IV

1 6 0 2

0 8 1 1

1 4 1 0

0 10 2 0

0 8 2 0

0 11 1 0

2 47 7 3

S 141 14 3

I 11 2 2

0 4 1 0

Total:

9

10

6

12

10

12

59

166

16

5

204

J.P. $ a ~

/ Beat p a t m ~ / n

urban a r e ~

Table 6 S u m n m y of a i m e a ~ t y ~ (by type of crime) Month Category

Oct.

Nov.

Dec.

Jan.

Febr.

March

Total

Weighted total

Max.

Min.

I H m IV

1 39 6 4

0 37 2 7

2 25 3 0

0 32 9 2

0 28 6 2

1 32 6 3

4 193 32 18

16 579 64 18

3 59 18 11

0 11 1 0

Total:

50

46

30

43

36

42

247

677

91

12

Table 7 Beat patzol schedule S - single beat - upto 30% of the maximum of total crime in a ~ven beat in a given period - (No. of constables I) D - double beat - 30-60% of the maximum of the total crime in a given beat in a given period- (No. of constables 2) M - multiple - over 60~: - (No. of constables 3) Time interval

0-4 4-8 8-12 12-16 16-20 20-24

Beat No. 1

Beat No. 2

Beat No. 3

Type of beat

Type of beat

Type of beat

S S S S S S

Route No. (hour-wise) I

!I

111

IV

1 2 3 1 2 3

2 3 2 2 3 2

3 1 1 3 1 1

1 2 2 1 2 2

S S S D D D

Route No. (hour-wise) I

II

III

IV

1 2 3 4 5 1

2 3 4 5 1 2

3 4 5 1 2 3

4 5 1 2 3 4

S S S D D S

Route No. (hour-wise) I

II

III

IV

3 3 3 3 3 3

1 2 4 1 2 4

2 4 1 4 1 2

4 1 2 2 4 1

Beat No. 4

Beat No. 5

Beat No. 6

Type of beat

Type of beat

Type of beat

Route No. (hour-wise) 1

II

ili

IV

4 4 4 4 4 4

1 2 3 1 2 3

2 3 1 3 1 2

3 1 2 2 3 1

Route No. (hour-wise)

Route No. (hour-wise)

1

I1

111

IV

I

I1

III

IV

2 2 2 2 2 2

3 4 1 4 1 3

1 3 4 1 3 4

4 1 3 3 4 1

3 3 3 3 3 3

2 4 1 4 1 2

1 2 4 1 2 4

4 1 2 2 4 1

0-4 4-8 8-12 12-16 16-20 20-24

S S S D M S

Time interval

Beat No. 7

Beat N,, 8

Beat No. 9

Type of beat

Type of beat

Type of beat

0-4 4-8 8-12 12-16 16-20 20-24

S S S S S S

Route No. (houx-wise) I

11

111

IV

2 3 4 4 1 2

3 4 3 1 2 4

4 2 1 2 3 3

1 1 2 3 4 1

S S S S S S

S S S S S S

Rou'~." No. (hour-wise) 1

II

!II

IV

6 5 4 3 2 1

1 ~ 5 4 3 2

2 1 6 5 4 3

3 2 1 6 5 4

S S S S S S

Route No. (hour-wise) I

II

III

IV

3 3 3 3 3 3

2 4 I 4 ! 2

I 2 4 I 2 4

4 I 2 2 4 1

J.P. Saksena / Beat patrolling in urban areas Table 7 (continued)

Time Interval

5. Beat-mutes Beat No. 10

Type of beat 0-4 4-8 8-12 12-16 16-20 20-24

205

Route No. (hour-wise)

S S S S S S

I

H

III

IV

1 2 3 4 3 4

2 3 4 3 4 1

3 4 2 2 1 2

4 1 I 1 2 3

(2) for the construction of beat routes, we can apply the algorithm of the travelling salesmanproblem [2,6]. In our problem, this algorithm could not be applied as very few nodes, with restrictions on movements, are available. Direct enumeration gave satisfactory results. (3) The next application is: The problem is a "routing problem" which can be def'mes as: "Given a set of nodes, how to f'md the shortest path from a given origin to a given destination?" This will become the travelling salesman problem if origin and destination coincides. Many algorithms are available for finding the shortest path, for example see [ 1,2]. The routing problem with alternative routes, such that the Kth route is no longer than a fixed predetermined value. (The constraint is that the beat route length be around 3 km).

We eoUected data on distances between pairs of beat points in every beat (Table 1). There are only a few links between nodes of this matrix, because we are moving along straight roads most of the time. Beat patrol on linear beats is the most prevalent type of patrol in the country. The control room of the police station is intended to have complete information regarding all beats at any time. This will be looked after by a senior police officer, having overall charge of beat work. He will change the sequence of beat routes every day/hour (Table 7) at random by throwing dices having 3/4/6 faces and noting the numbers. He will generate a set of 240 (10 X 24) random numbers daily which will be the sequence of route numbers for the ten beats, each beat requiring 24 numbers in the next 24 hours. As such, Table 7, which is of strategic importance in this work, will go on changing and no set pattern of behaviour will be avagable for any beat at any time. No previous information can be had anywhere for the culprit to evade the constables on patrol duty. "Fais will also ensure that the thne between visits at a point (node) is completely random. The possible beat routes have been shown in Table 2. In each beat alternative routes exist and they will be used for deployment purposes. Since the beat constable has to change his route every hour in order to pay surprise visits, the control room must guide him as the appropriate route every hour.

6. Beat-schedules Table 8 Beat ranking according to crime-behaviour

Beat No.

Weighted total crime (Table 6)

Beat ranking accordingto crimebehaviour

Beat patrol

13 70 112 166

10 5 2 1

s S/D S/D S/D/M

5

73

4

S

6 7 8 9 10

53 36 82 43 29

6 8 3 7 9

S S S S S

1 2 3 4

schedule

(Table 7)

We analyse the crime behaviour in different beats at different times of day. Two types of data for six months (October through March) were recorded. The firs;, as shown in Table 3 and 4 gives the number of crimes committed during successive 4-hour intervals for each beat. Table 5 and 6 gives the same data, when recorded by type of crime. Four categories of crime severity (recorded in terms of points) are used. The total, weighted total, maximum, minimum of these values are calculated. Based on the information given, beat patrol schedules (Table 7) are prepared for each beat t,n a four hourly basis. Deployment was decided on the basis of the average number of crimes recorded during the period of observation. Here we specify the route number to be followed at a given time in a g~ven beat.

206

J.P. Saktena / Beat patrolling in urban areas

Table 9 Manpov~g requi~ed for the beat work Beat Total distance Numberof No. travelled/day constables in km required as per schedule 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

16.8 15.2 15.6 14.0 10.5 18.3 20.4 9.8 12.9 10.0

6 9 8 9 6 6

Total 143.5

68

Approx. cost of deployment at 1~.300/- per constable per month

6 6 6 Rs.20, 400/per month or Rs.713/- per day

These are chosen with the aim of changing the beat route every hour in a beat. These sequences can be generated by throwing a dice having 3, 4, 5 and 6 faces. Table 8 compares the beats in respect of crime and suggested deployment. Table 9 gives the manpower required to complete the above work with co~ts. We can ascertain from this the relative vulnerability of different beats, weighted total crime, beat ranking and the measures taken to combat the given situation. It is clear that the deployment should be hea, T at those places and at those times where and when there is a greater likelihood of occurrence of a crirae. This will help in minimizing loss, which is the objective of this study.

7. Summary and conclusions If t~,¢ suggestions given in the paper are followed, patrolling will be more effective and useful. The beat ranking may help in assessing the vulnerability of the area and as such proper steps may be taken in advance

to prevent the occurrence of the crime. In particular, the following answers are provided to problems posed in the Section 1. (i) Beat numbers 5, 8, 9, 10 do not satisfy the constraint on area of a beat. it is suggested "altar beat No. 5 be grouped with beat No. 6 while 8, 9 and 10 be grouped into one beat. Thus we shall have seven beats ~stead of ten, thereby reducing operating costs. In the new situation, emerging from this regrouping, all the 30 points of beat numbers 8, 9 and 10 cannot be incorporated. We have to select only ten strategic points out of these and then work out new beat routes/schedules etc. Similar work can be done for the other group (5,6). The details of this work are not discussed here and no plan developed by the above approach has been iml~lemented so far. (ii) Table 2 gives the required alternative beat routes to be used for deployment purposes. Off) Table 7 gives the beat schedules on an hourly basis to be used for deployment purposes. (iv) Table 8 gives the ranking of beats on the basis on crimes committed and from Table 7 the hours when they are vulnerable. From these two tables, we infer that the data given, is in consonance.

References

[1 ] R. Bellman,On a routing problem, Quart. Appl. Math. 16 (1958). [2] R. Bellman, Dynamic programmiz,gtreatment of the travelling salesman problem, Assoc. Comput. Mach. (l) (1962). [3, R.S. Garfinkel and G.L. Nembauser, Optimal political redistricting by implicit enumeration techniques, Management Sci. 16 (8) (1970) B495-B508. [4] S.W. Hess, J.B. Weaver,.LH. Siegfeld, J.N. Whelan and P.A. Zitlan, Non-partisa,3 political redistricting by computer, Operations Res. 13 (1965) 993-1006. [5] R.C. Larson, Urban Police Patrol Analysis (M.I.T. Press, Cambridge, MA, 1971). [6] J.D.C. Little, K.G. Murthy, F.W. Sweeney and C. Katel, An algorithm for the travellingsalesman problem, Operations Res. 11 0963) 972-989. [7] M.M. Malts, Measure of effectiveness in crime reduction. Ope"ations Res. 23 (3) (1975).