A sequential GRASP for the therapist routing and scheduling problem

This paper presents a new model and solution methodology for the problem faced by companies that provide rehabilitative services to clinic and home-bound patients. Given a set of multi-skilled therapists and a group of geographically dispersed patients, the objective is to construct weekly tours for the therapists that minimize the travel, treatment, and administrative costs while ensuring that all patients are seen within their time windows and that a host of labor laws and contractual agreements are observed. The problem is complicated by three factors that prevent a daily decomposition: (i) overtime rates kick in only after 40 regular hours are worked during the week, (ii) new patients must be seen by a licensed therapist on their first visit, and (iii) for some patients only the frequency and not the actual days on which they are to be seen is specified. The problem is formulated as a mixed-integer linear program but after repeated attempts to solve small instances with commercial software failed, we developed an adaptive sequential greedy randomized adaptive search procedure. The phase I logic of the procedure builds one daily schedule at a time for each therapist until all patients are routed. In phase II, several neighborhoods are explored to arrive at a local optimum. Extensive testing with both real data provided by a U.S. rehab company and datasets derived from them demonstrated the value of the purposed procedure with respect to current practice. The results indicated that cost reductions averaging over 18.09 % are possible.