A variable neighborhood search based matheuristic for nurse rostering problems

Abstract

A practical nurse rostering problem, which arises at a ward of an Italian private hospital, is considered. In this problem, it is required each month to assign shifts to the nursing staff subject to various requirements. A matheuristic approach is introduced, based on a set of neighborhoods iteratively searched by a commercial integer programming solver within a defined global time limit, relying on a starting solution generated by the solver running on the general integer programming formulation of the problem. Generally speaking, a matheuristic algorithm is a heuristic algorithm that uses non trivial optimization and mathematical programming tools to explore the solutions space with the aim of analyzing large scale neighborhoods. Randomly generated instances, based on the considered nurse rostering problem, were solved and solutions computed by the proposed procedure are compared to the solutions achieved by pure solvers within the same time limit. The results show that the proposed solution approach outperforms the solvers in terms of solution quality. The proposed approach has also been tested on the well known Nurse Rostering Competition instances where several new best results were reached.

Appendix: ILP model formulation

Hereafter is presented the complete ILP model, for each constraint an explanatory comment is added below the constraint. We indicate between brackets the related item of Sect. 2 whenever it applies.

Let we represent the number of weekends of the current month, min_pers _j,k the minimum number of personnel required in each shift for each day and requests _i,j,k the Off and Rest shifts requested by nurses.

Objective function:

Each nurse must be assigned to exactly one shift per day.

(C1) Minimum number of personnel required in day j for shift k

(C2) For each nurse, no more than five consecutive working days

(C3) Forbidden sequence: N-R-N

(C3) Forbidden sequence: N-A

(C3) Forbidden sequence: N-O

(C3) Forbidden sequence: N-M

(C3) Forbidden sequence: A-M

(C3) For each nurse, no more than two consecutive Night shifts

(C3) For each nurse, no more than two consecutive Rest shifts

(C3) For each nurse, after a sequence of Night shifts, two Off-duty shifts must be scheduled

(C3) For each nurse, no more than four consecutive Afternoon shifts

(C4) For each nurse, the total number of Rest shifts must be equal to wk _ends ±1.

(C5) For each nurse, all requests of Off and Rest shifts must be satisfied

(C6) For each nurse, the total number of Afternoon shifts must be less than or equal to the number of Morning shifts

For each nurse, the variable ww (working weekend) is introduced

(C7) For each nurse, at least we−bu free weekends, where we is the number of weekends of the current month and bu is the number of busy (working) weekends

(C8) For each nurse, no more than two R shifts in four consecutive days

(C9) For each nurse, no more than two N shifts in five consecutive days

(C10) For each nurse, the total number of shift type T must be between MIN and MAX as described in Table 2

(C11) For each nurse needing a shadowing period assign the same schedule of a trained nurse

(C11) For each nurse, set possible incompatibility with other nurses