Analytics Problem

[Bannana]

A city authority is considering placing a toll booth on its new bridge. The beginning
times for the shifts are 8am, noon, 4pm, 8pm, midnight and 4am. A collector
beginning a shift at one of the above times works for the next 8 hours.
The following staffing levels during each of the 24-hour periods have been estimated

Hour.................... Minimum Collectors Needed
8am - Noon.............. . 5
Noon – 4pm ................6
4pm – 8pm .................10
8pm - Midnight........... .7
Midnight – 4am ...........4
4am – 8am .................6

Find the minimum number of collectors that need to be hired to begin the 8 hour shifts
at each of the six times.

Help would be fantastic!!!

[cavedave]

One way to solve it is to use linear programming.

This problem is a version of the post office problem described and solved here. So if you want to learn analytics of this type this tutorial (starting at its first part) is a good start.

If you want someone to solve it for you. Do the coding and such that is different though.

[cavedave]

Turns out you can do this with 19 people

[hivizman]

On the numbers given, it is possible to arrange the starting times of the 19 employees so that, in each of the six time slots, all the employees working during that time slot are fully utilised. The total number of time slots that need to be filled is 38, which is 19 x 2, so 19 collectors must be the minimal solution, since each collector can work for two shifts.

However, if you change the numbers required in each time slot, then it may not be possible to ensure that all employees work for their entire shift. For example, if you increase the number of employees needed in the 8AM-noon period to 10, then you will need an additional five employees to cover that time slot, but it is no longer, I believe, possible to arrange the start times so that every employee is fully employed for their full eight-hour shift. Hence unless you can recruit five part-timers, some of the collectors will not be required to work for part of their shifts.

[cavedave]

This program will work this out

var x1 >= 0, integer;
var x2 >= 0, integer;
var x3 >= 0, integer;
var x4 >= 0, integer;
var x5 >= 0, integer;
var x6 >= 0, integer;
/* objective function */
minimize z: x1+x2+x3+x4+x5+x6;

/* Constraints */
s.t. ctr1:x1 + x6 >= 5;
s.t. ctr2:x1+x2>= 6;
s.t. ctr3:x2+x3>= 10;
s.t. ctr4:x3+x4>= 7;
s.t. ctr5:x4+x5>= 4;
s.t. ctr6:x5+x6>= 6;
data;
end;

In a real system weekends, bank holidays, overtime and all sorts of other things also need to be juggled.
No. Activity(new employees)
------ -------------
1 * 0
2 * 6
3 * 4
4 * 3
5 * 1
6 * 5