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This is my basic setup: I have five time slots (A, B, C, D, E; each day has these five time slots) over 60 days that all need to be filled with n people each (n must remain a variable). However, each person has a different availability (ex. ABDE, BCDE, AE, BCD, etc.). I want this to be distributed equally so all workers get the same number of shifts, but some workers should be able to input in a number of shifts that they want to do.

I recognize that this is a subproblem of the nurse scheduling problem, and I have spend a few hours trying to grind through some academic papers on it. However, it gets very convoluted very quickly and I get lost somewhere in the mix, as I do not have a knowledge of these kinds of algorithms. This seems to be a simpler version with fewer constraints, so I was wondering what a simple yet effective algorithm would be to solve this problem and if you could provide details on it.

I have come up with a two crude algorithm ideas, and I don't know which of these (if any at all) would be the most effective:

  • First distribute work to the person with (1) the least number of shifts assigned so far and then (2) with the least number of availabilities.

  • Distribute all work days to those with the least number of availabilities, but fill a person's entire schedule before progressing on to the next person. Shifts get assigned by determining which time slot (A/B/C/D/E) has the least number of people working it.

The thing that I don't like about each of these algorithms is that it seems very easy to get stuck at any given point, and from there I really wouldn't know what to do.

Thanks.

D.W.
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louie mcconnell
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2 Answers2

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The simplest solution (in terms of saving you the time of understanding the literature) is probably going to be to use integer linear programming (ILP / MILP). You can formulate it as an ILP instance, then apply an off-the-shelf ILP solver.

Introduce zero-or-one variables $x_{i,j}$, with the goal that $x_{i,j}=1$ means that the $i$th person is assigned to $j$th time slot, and $x_{i,j}=0$ means that they're not assigned to that time slot. Now you can write down some linear inequalities/equations constraining these, e.g.,

$\sum_{i} x_{i,j} = n$ for each $i$ (each time slot needs to be filled with $n$ people)

$\sum_j x_{i,j} = c_i$ for each $i$, where $c_i = $ the number of shifts the $i$th worker should be assigned

Also if the $i$th worker isn't available for the $j$th slot, then add a constraint that $x_{i,j}=0$.

Also, $0 \le x_{i,j} \le 1$ (to force each $x_{i,j}$ to be zero or one).

The combination of all of these constraints gives you an integer linear program. Ask the ILP solver to find a feasible solution, and boom, you're done. This might not be the most efficient possible algorithm, but it will be simple to implement and play around with, and I expect that it'll give high-quality solutions and be fast enough for the size of problem you're likely to encounter in practice.

D.W.
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Given my understanding of your problem, you should be able to model it as a Maximum Flow Problem for which some polynomial algorithm exists.

Each of your workers would be represented by a node and each of your (60*5) time slot would be represented by a node. An edge of capacity 1 would exist between a worker and a time slot if that worker has the possibility of working that time slot.

You would then introduce an edge of capacity n between each of your time slot nodes and your sink node.

You would introduce edges between the source nodes and the worker nodes having a capacity of number of slots that the worker wants to work (if this is specified) otherwise it would be the ceiled value of (60*5*n - number of shifts that all workers specified that they wanted to work)/(number of worked having not specified a number of shifts to work).

If your total flow has a value of 60*5*n you have a feasible solution to your problem.

Renaud M.
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