Six Constables, Three Posts, Two Shifts
Six constables-A, B, C, D, E and F-must staff three posts (Gate, Yard, Control Room) on each of two shifts, day and night. Each shift needs exactly one person per post, so all six are deployed daily. Posting rules: (i) A can work only the day shift; (ii) B must be at the Gate; (iii) C and D must be on different shifts; (iv) E can work only the night shift; (v) F cannot be in the Control Room. The draft roster is: Day-B at Gate, C at Yard, A at Control Room; Night-D at Gate, E at Yard, F at Control Room. The warden also says that if B is on the day shift, D must be on the night shift; and no constable may work both shifts. A constable reports sick at the last minute, leaving five persons for six posts. The reserve constable G can cover any post but only for four hours, and a shift is eight hours.
1. Audit the draft roster against all rules. Identify every violation and propose the smallest change that makes it valid while keeping B on the day shift at the Gate.
Read the worked answer
Check each rule. A is on day-allowed. B is at Gate-allowed. C is day and D is night-different, allowed. E is night-allowed. B is day, so D must be night-satisfied. The only violation is F in the Control Room, which rule (v) forbids.
The smallest change is to swap E and F. Night becomes D-Gate, F-Yard, E-Control Room. E stays on night, F leaves the Control Room, and no other rule is disturbed. A swap of F with C would put C on night with D, violating rule (iii), so it is not valid.
2. With one constable sick, can the six posts be covered for the full eight-hour shift using G? Explain with person-hours.
Read the worked answer
No. Six posts for eight hours require 6 × 8 = 48 person-hours. Five healthy constables provide 5 × 8 = 40 person-hours. G adds only 4 hours, giving 44 person-hours. The shortfall is 4 hours, so one post would be uncovered for half the shift.
G cannot cover a full post and also make up the missing half without exceeding four hours. No constable may work both shifts. The warden must call another relief constable, authorise overtime under the rules, or reduce the number of posts only if security instructions permit it. The roster arithmetic alone shows that five-plus-part-time-G is insufficient.
3. The warden now wants A in the Control Room and D on the night shift. Is there a valid roster? If yes, give one; if no, explain why.
Read the worked answer
Yes. A is day, so A can be the day Control Room. D is night. Rule (iii) then forces C to be day. B must be at the Gate. E must be night. F is the remaining person.
Day: A-Control Room, B-Gate, C-Yard. Night: D, E and F fill Gate, Yard and Control Room. F cannot be in Control Room, so F must be Gate or Yard. One valid night line-up is D-Gate, F-Yard, E-Control Room. This satisfies A day, B Gate, C day with D night, E night, F not in Control, and B day implies D night.