Heuristic decomposition and mathematical programming for workforce scheduling and routing problems

This thesis presents a PhD research project using a mathematical programming approach to solve a home healthcare problem (HHC) as well as general workforce scheduling and routing problems (WSRPs). In general, the workforce scheduling and routing problem consists of producing a schedule for mobile workers to make visits at different locations in order to perform some tasks. In some cases, visits may have time-wise dependencies in which a visit must be made within a time period depending on the other visit. A home healthcare problem is a variant of workforce scheduling and routing problems, which consists in producing a daily schedule for nurses or care workers to visit patients at their home. The scheduler must select qualified workers to make visits and route them throughout the time horizon. We implement a mixed integer programming model to solve the HHC. The model is an adaptation of the WSRP from the literature. However, the MIP solver cannot solve a large-scale real-world problem defined in this model form because the problem requires large amounts of memory and computational time. To tackle the problem, we propose heuristic decomposition approaches which split a main problem into sub-problems heuristically and each sub-problem is solved to optimality by the MIP solver. The first decomposition approach is a geographical decomposition with conflict avoidance (GDCA). The algorithm avoids conflicting assignments by solving sub-problems in a sequence in which worker's availabilities are updated after a sub-problem is solved. The approach can find a feasible solution for every HHC problem instance tackled in this thesis. The second approach is a decomposition with conflict repair and we propose two variants: geographical decomposition with conflict repair (GDCR) and repeated decomposition and conflict repair (RDCR). The GDCR works in the same way as GDCA but instead of solving sub-problems in a given sequence, they are solved with no specific order and conflicting assignments are allowed. Later on, the conflicting assignments are resolved by a conflicting assignments repair process. The remaining unassigned visits are allocated by a heuristic assignment algorithm. The second variant, RDCR, tackles the unassigned visits by repeating the decomposition and conflict repair until no further improvement has been found. We also conduct an experiment to use different decomposition rules for RDCR. Based on computational experiments conducted in this thesis, the RDCR is found to be the best of the heuristic decomposition approaches. Therefore, the RDCR is extended to solve a WSRP with time-dependent activities constraints. The approach requires modification to accommodate the time-dependent activities constraints which means that two visits may have time-wise requirements such as synchronisation, time overlapped, etc. In addition, we propose a reformulated MIP model to solve the HHC problem. The new model is considered to be a compact model because it has significantly fewer constraints. The aim of the reformulation is to reduce the solver requirements for memory and computational time. The MIP solver can solve all the HHC instances formulated in a compact model. Most of solutions obtained with this approach are the best known solutions so far except for those the instances for which the optimal solution can be found using the full MIP model. Typically, this approach requires computational time below one hour per instance. This problem reformulation is so far the best approach to solve the HHC instances considered in this thesis. The heuristic decomposition and model reformulation proposed in this thesis can find solutions to the real-world home healthcare problem. The main achievement is the reduction of computational memory and computational time which are required by the optimisation solver. Our studies show the best way to control the use of solver memory is the heuristic decomposition approach, particularly the RDCR method. The RDCR method can find a solution for every instance used throughout this thesis and keep the memory usage within personal computer memory ranges. Also, the computational time required to solve an instance being less than 8 minutes, for which the solution gap to the optimal solution is on average 12%. In contrast, the strong point of the model reformulation approach over the heuristic decomposition is that the model reformulation provides higher quality solutions. The relative gaps of solutions between the solution for solving the reformulated model and the solution from solving the full model is less than 1% whilst its the computational time could be up to one hour and its computational memory could require up to 100 GB. Therefore, the heuristic decomposition approach is a method for finding a solution using restricted resources while the model reformulation is an approach for when a high solution quality is required. Hence, two mathematical programming based heuristic approaches are each more suitable in different circumstances in which both find high quality solutions within an acceptable time limit.