Polynomial subcases of the home health care routing and scheduling problem with fixed services
Résumé
In home services agencies, planners are in charge of scheduling and routing caregivers to clients’ homes to provide services. This problem is called the Home Health Care Routing and Scheduling Problem (Cheng and Rich, 1998) and it is usually treated as a variant of a VRP with specific constraints such as time windows, legal regulations or skill requirements. Obviously, this problem is NP-hard so most papers address it with matheuristics (Decerle et al., 2016) or other optimization methods such as branch-price-and-cut algorithms (Trautsamwieser and Hirsch, 2011). An exhaustive review of the topic can be found in Cisse et al. (2017).
We will focus on two polynomial subcases of the HHCRSP, and use tools from graph theory to solve them. We study a specific case inspired by a real case and by the work of Di Mascolo et al. (2017) where the starting times and ending times of the services are fixed. We simplify the problem by considering that all caregivers start from the same depot, and we do not take into account any working time regulation or skill requirement. Every service needs to be performed by one careworker and our goal is to minimize the number of caregivers needed to provide all the services.
In our first subcase, we consider that traveling times are included in the duration of the task, as it is often the case for home services in France, especially in urban areas. Let’s consider the interval graph G(V,E) of the problem, where the vertices represent the services, and the edges join overlapping services. We notice that any independant set is an admissible tour for a careworker. Thus, we are looking for a minimum independent set partition. Since interval graphs are chordal, we can find a perfect elimination ordering of V in polynomial time. By applying a greedy coloring algorithm to this ordering we obtain a perfect coloring of the graph, that is to say a solution with the optimal number of careworkers such that every set of vertices with the same color is an admissible tour.
For the second problem, traveling times are not included in the tasks anymore, and we make the assumption that they follow the triangle inequality. We consider the complementary of the interval graph which is a comparability graph, and we delete the edges between incompatible services in regards with traveling times. We can easily check that we still have a graph of comparability, and we can find a transitive orientation in linear time. Then, Dilworth’s chain decomposition theorem gives us the width of the set of vertices, that is to say the minimum number of caregivers needed to perform all the services. We transform the graph into a circulation network by adding a source and a sink, lower bounds on vertices, weights and upper bounds on some arcs. Then, we can solve a minimum-cost circulation problem with the strongly polynomial Tardos’ algorithm and we get the solution which requires the minimum number of careworkers, and which minimizes traveling times in a second part.
This method can lead to unbalanced tours between caregivers, so it would be interesting to take into account constraints related to work regulation in future work.