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also that waiting times greater than 90 minutes are not counted. Therefore, it is possible that the routes produced are fragmented, which implies a long amplitude. While a long break in the day is not disruptive, it is not desirable, and as such having a compact route is also a criterion of staff satisfaction.

Table 1.1. Numerical results

Instance	Number of beneficiaries	Number of careworkers	Number of services	Beneficiary variations	Careworkers variations	Waiting time (h)	Travel time (h)	Execution time (s)
1	3	4	56	–	+1/-1	4. 2	3. 8	2. 2
2	11	5	85	+0/-1	+2/-3	4. 69	6. 49	2. 1
3	18	6	95	+2/-4	–	8. 41	5. 7	7. 6
4	26	8	156	+2/-7	+2/-3	1. 5	19.3	40.6
5	26	11	179	+6/-5	+2/-2	7. 15	6. 4	30.13
6	40	18	191	+8/-11	+8/-6	5. 83	1. 83	65.8
7	92	15	337	+7/-10	+3/-5	12. 0	21.3	101

1.7. Conclusion and perspectives

We have developed a method that solves real-world problems by taking into account the legal constraints resulting from French national collective agreements, internal policies specific to our case study, as well as changes to beneficiaries and staff. The objective of minimizing waiting times is to improve the working conditions of staff in order to limit turnover. These waiting times are calculated using Adomni-Quemera’s definition. However, while some periods of inactivity are perceived by careworkers as wasted time, others are viewed as real breaks and can be appreciated by staff. Proposing a new definition of waiting times, based on the experiences of careworkers, could result in new schedules that are even more satisfactory and adapted to each person. We can also consider a multi-objective approach which would also make it possible to limit the travel time and/or the amplitudes of the work shifts, which do not always go hand in hand with the minimization of waiting times.

      Time constraints turn out to be a scientific obstacle and can contribute to a substantial degradation of the schedule. A tool identifying the most impinging time constraints would make it possible to determine the services for which it would be advantageous to renegotiate the contractual hours with the beneficiaries.

      The proposed method is adapted to changes in the composition of staff and beneficiaries and to the strategic update; however, it is not necessarily adequate in the event of a one-off absence. Indeed, if a worker is temporarily unavailable, the organization does not always have a careworker available who is qualified to replace them at short notice. It is therefore necessary to overload the other careworkers and potentially shift the appointment times so as to be able to obtain a feasible solution. Respecting the continuity of care constraints becomes very restrictive and is no longer necessarily relevant. Moreover, as these absences are often unexpected, it is difficult in practice to upset the whole schedule at the last minute. Thus, it would be interesting to adapt our approach to the management of daily eventualities, by relaxing the constraints of continuity and by considering stability from a new angle: no longer that of continuity of care, but rather with the objective of impacting as few routes as possible, in order to meet operational requirements in the field.

1.8. References

      Béguin, F. (2018). Le nombre de candidats aux écoles d’aides-soignants en baisse. Le Monde [Online]. Available at: https://www.lemonde.fr/sante/article/2018/05/11/le-nombre-decandidats-aux-ecoles-d-aides-soignants-en-baisse_5297453_1651302.html.

      Begur, S.V., Miller, D.M., Weaver, J.R. (1997). An integrated spatial DSS for scheduling and routing home-health-care nurses. Interfaces, 27, 35–48.

Cappanera, P. and Scutellà, M.G. (2014). Joint assignment, scheduling, and routing models to home care optimization: A pattern-based approach. Transportation Science, 49, 830–852.

      Cappanera, P., Scutellà, M.G., Nervi, F., Galli, L. (2018). Demand uncertainty in robust home care optimization. Omega, 80, 95–110.

      Carello, G., Lanzarone, E., Mattia, S. (2018). Trade-off between stakeholders’ goals in the home care nurse-to-patient assignment problem. Operations Research for Health Care, 16, 29–40.

      Cheng, E. and Rich, J.L. (1998). A home health care routing and scheduling problem. Technical report [Online]. Available at: https://scholarship.rice.edu/handle/1911/101899.

      Cissé, M., Yalçındağ, S., Kergosien, Y., Şahin, E., Lenté, C., Matta, A. (2017). OR problems related to home health care: A review of relevant routing and scheduling problems. Operations Research for Health Care, 13–14, 1–22.

      Clarke, G. and Wright, J.W. (1964). Scheduling of vehicles from a central depot

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