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I'm trying to schedule 30 teaching assistants to cover about 118 hours of office hours. Office hours at different times of day need different coverage (0, 1, 2, or 3 asssistants). People are scheduled on the half hour.

I've made an integer linear program such that I have a 0/1 variable indexed by worker and shift: 0 if not working then, 1 if working then. Coverage is easy, but it leads to some workers being scheduled for only a half-hour shift, which is not fair to them.

My second attempt was to have a richer set of indexed variables, by (worker, start time, length of shift). This is where the snags begin:

  • If I limit the number of shifts per worker to one shift per day, the IP solver grinds away for hours with no solution.

  • If I allow two shifts per worker per day, things work pretty well, except sometimes the solver schedules a single worker for two shifts that overlap. Which means the solver thinks I have 3 people on duty but I really only have 2.

  • My final attempt was to introduce constraints such that no worker can ever be scheduled for two shifts that overlap. At this point my tools grind for a while and then blow up with an out-of-memory error.

I'm using the RIMA optimization package with the COIN CPB solver. (Have also tried lpsolve.)

I feel like 30 workers into circa 150 slots should not be that difficult! So I think I must be formulating the problem in a stupid way. Thus my question: how can I learn how to formulate my scheduling problem in a way that solvers will do well with it?

(If it matters, the objective function I am trying to maximize is the total utility to all the workers of all the shifts they have been assigned. It's just a number indexed by worker and shift.)

Rodrigo de Azevedo
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Norman Ramsey
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  • (1) It's hard to digest anything here as there is no code and only unspecific-descriptions (2) Maybe MIP can work, but maybe SAT / CP / Metaheuristics or Hybrids of those are more suited for this problem. (3) *Not fair to them* often calls for some kind of regularization / objective term to handle this: often some kind of norm: l1/l2 (4) *grinds away for hours with no solution*: tune solver-params or try other alts for feas sol (5) Overlapping: many things are possible; but something i learned in the past: interval-graph = chordal = max-cliques calc in poly-time = convex-hull for IP = powerful – sascha Jan 14 '18 at 02:23
  • (6) We can't debug out of memory errors without code or analyzing your tools in use (7) Solver is called **CBC**. (8) *I feel like 30 workers into circa 150 slots should not be that difficult!* Well... if it's NP-hard, there will always be instances with asymptotics like 2*30 or worse. That's the basic message of NP-hardness. Of course real-world instances can look different, but we don't know your data. – sascha Jan 14 '18 at 02:25
  • Sometimes column generation approaches work very well for this type of models. – Erwin Kalvelagen Jan 14 '18 at 13:36
  • @sascha I wouldn't want to inflict my code on anyone. I want to learn how to formulate ILP problems, not to get help debugging my code. – Norman Ramsey Jan 15 '18 at 13:54

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This problem is almost exactly the set covering problem albeit with added constraints. The correct way to model this with integer programming is to use a column generation approach. This example of the cutting stock problem from IBM's CPLEX should be clear enough that you would most likely be able to adapt it to your problem.

Philippe Olivier
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