MONDAY QUARTERBACKING: How many policemen do we actually need?
By
Mobolaji E. Aluko, PhD
maluko@scs.howard.edu
I am a firm believer in the principle of applying cold mathematical calculations before applying hot, political considerations. So, here goes, but if you do not like mathematics, please skip to the next article:
Today, I turn on my mathematical thinking cap to answer a rather simple question: How many policemen do we really need in Nigeria? I am also particularly keen to answer those critics of mine who worry about where all my scientific training has gone in all my socio-political analysis of Nigeria.
So what should determine the number N of police?
Thus, at the outset, we have a yet-to-be-determined mathematical function:
N = f (A, P, S, T, H, C)
How do we determine the exact functionality?
Very simple. Let us start with the implications of S and T. During a period T, a policeman would have moved around a linear distance of S*T miles. But what area would he have covered?
For the purpose of this write-up, let us draw a square of side X, and assumes that it is sufficient for a policeman to walk the perimeter of the square within the time T. Then we must have:
ST = 4X
Thus the area which he has walked in this time (which is X squared, or X^2) is as follows:
X^2 = [ST/4]^2 = [ST]^2/16
Therefore, so far, the total number of policemen required to police a total atlas area of A and livable fraction v (which excludes rivers, lakes, mountain tops, etc.) is (for a minimum number of 1):
N = 1 + 16Av/ [ST]^2
Now, typically, the police will be able to use four kinds of transport: by foot, on a bicycle, or motorbike, or by a cruise car. Then, we can have that:
1/S = L1/S1 + L2/S2 + L3/S3 + L4/S4
where
L1 + L 2 + L3 + L4 = 1 0 < = Li < = 1 ; I =1,2,3,4
S1 = speed of walking on foot ~ 4 mph; L1 = fraction of all policemen on foot
S2 = speed of bicycle ~ 10 mph; L2 = fraction on bicycle
S3 = speed of motorbike ~ 20 mph; L3 = fraction on motorbike
S4 = speed of cruise car ~ 40 mph; L4 = fraction in cruise cars
Thus, it will take 5 policeman walking on foot to patrol a livable area of 1 square mile (each will patrol the perimeters of 0.2 square miles area) if they are required to make each complete round in 30 minutes. On the other hand, were they in a cruiser car (e.g. at 40 mph), only 1 policeman (the minimum) would be needed.
Now, the number of policemen should increase in some manner relative to the number of people within the area that they patrol, but certainly not DIRECTLY in proportion. If the above number of policemen is to be modified by 10% relative to each Z number of residents per policeman within the patrolled area, then we will have that:
N = {1 + 16Av / [ST]^2} {1 + 0.1*[ P/A * [ST]^2/(16*Z)]}
The UN recommendation is Z = 400.
To account for criminality, let us assume that a total of Q governing units (e.g. countries) are ranked according to the average numbers of murders, burglaries and rapes for the previous five years. If a particular governing unit's (or country's) ranking is C, and a ranking of Q out of Q would lead to a 10% increase of policemen required over that with a ranking of 1 (least criminality), and if a one percent increase in policemen is allowed as each rank increases through each tenth of Q rankings, then we will have:
N = {1 + 16Av/ [ST]^2} {1 + 0.1*[P/A * [ST]^2/(16Z)] + 0.1*[C/Q]}
If we do not expect the
policemen to do more than 24/H-hour shifts, and
if we expect a 24-hour patrol, then we will have:
N = H* {1 + 16Av/ [ST]^2} {1 + 0.1*[P/A * [ST]^2/(16Z)] + 0.1*[C/Q]}
Finally, since the number of policemen must be an integer, we must have that:
N = INTEGER[H* {1 + 16Av/[ST]^2} {1 + 0.1*[P/A * [ST]^2/(16Z)]+0.1*[C/Q]}]
Thus we have the final functional form:
N = f (A, v, P, H, Z,C/Q, S[L1, L2, L3, (L4, S1, S2, S3, S4)], T)
This formula can be used for THE WHOLE COUNTRY or for a subset thereof (eg a state or local government area) provided the area A is appropriately defined and the relevant parameters for that particular area are used. Note that in any case, the ACTUAL choice of some of these values, viz. H, Z, S and T would be POLITICAL. Some of the numbers can also be adjusted during a security emergency.
The equation can also be taken as a POLICY IMPLICATION formula: if you decide on a number of policemen without applying the formula, then for a fixed set of variables, one can determine the values of the remainder of the variables to determine what the chosen number implies.
Sample Calculations For Nigeria:
It will be found that the number of policemen required to secure a given
area is most sensitive to S and T. So, suppose we fix, for Nigeria:
A = 360,000 square miles
V = 1 (for maximum policemen for this calculation)
P = 120,000,000
H = 2
Z = 400 (UN-proposed ratio of persons per policeman, which is
independent of area size of country)
C/Q = 1 (ie assuming most corrupt nation, eg according to Transparency
International)
Note that for a country of population 120,000,000, the UN-proposed size of
police force is 300,000.
Then we will have the following results:
Table 1: Number of Policemen for various T and S values for Nigeria
T = 1 hour T = 0.5 hr
S = S1 (L1 = 1; all policemen patrol on foot) N = 852,000 3.2 million
S = S4 (L4 = 1; patrol by cruise car only) N = 68,000 92,000
L1 = L2 = L3 = L4 = 0.25 N = 196,000 606,000
ie there are equal number of all types of policemen:
L1 = 0.5, L2 = 0.25; L3 = 0.15; L4 = 0.1 N = 378,000 1.3 million
L1 = 0.25, L2 = 0.5, L3 = 0.10, L4 = 0.15 N = 243,000 793,000
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Table 2: Sample Calculation for Lagos State
A = 1,400 square miles
V = 1 (for maximum policemen for this calculation)
P = 10,000,000
H = 2
Z = 400 (UN-proposed ratio of persons per policeman, which is
independent of area size of country)
C/Q = 0.5 (ie roughly half of Nigeria's criminality due to Lagos!)
Note that for a state like Lagos of population 10,000,000, the UN-proposed
size of police force is 25,000.
T = 1 hour T = 0.5 hr
S = S1 (L1 = 1; all policemen patrol on foot) N = 8,000 16,800
S = S4 (L4 = 1; patrol by cruise car only) N = 5,400 5,200
L1 = L2 = L3 = L4 = 0.25 N = 5,500 7,000
L1 = 0.5, L2 = 0.25; L3 = 0.15; L4 = 0.1 N = 6,200 9,700
L1 = 0.25, L2 = 0.5, L3 = 0.10, L4 = 0.15 N = 5,700 7,700
Note: S1 = 4; S2 = 10; S3 = 25; S4 = 40
1/S = L1/S1 + L2/S2 + L3/S3 + L4/S4; L1 + L2 + L3 + L4 = 1 0 < = Li < = 1 i=1,2,3,4
Conclusion
We find that through mathematical reasoning, a reasonable formula can be derived
for the number of policemen required to secure an area. Thus, we find that with
some investments in increasing the mobility of our police force, we may be able
to substantially cut down on their numbers and address the security concerns of
our citizens.
Again, we emphasize that this formula need not be applied to the whole nation: it can be applied to various localities around the countries (e.g. local government areas or states) since some of the parameters will vary across the country.