Nature of real-world multi-objective vehicle routing with evolutionary algorithms

The Vehicle Routing Problem with Time Windows VRPTW) is an important logistics problem which in the realworld appears to be multi-objective. Most research in this area has been carried out using classic datasets designed for the single-objective case, like the well-known Solomon's problem instances. Some unrealistic assumptions are usually made when using these datasets in the multi-objective case (e.g. assuming that one unit of travel time corresponds to one unit of travel distance). Additionally, there is no common VRPTW multiobjective oriented framework to compare the performance of algorithms because different implementations in the literature tackle different sets of objectives. In this work, we investigate the conflicting (or not) nature of various objectives in the VRPTW and show that some of the classic test instances are not suitable for conducting a proper multi-objective study. The insights of this study have led us to generate some problem instances using d ata from a real-world distribution company. Experiments in these new dataset using a standard evolutionary algorithm NSGA-II) show stronger evidence of multi-objective features. Our contribution focuses on achieving a better understanding about the multi-objective nature of the VRPTW, in particular the conflicting relationships between 5 objectives: number of vehicles, total travel distance, makespan, total waiting time, and total delay time

Algorithms for the multi-objective vehicle routing problem with hard time windows and stochastic travel time and service time

This paper introduces a multi-objective vehicle routing problem with hard time windows and stochastic travel and service times. This problem has two practical objectives: minimizing the operational costs, and maximizing the service level. These objectives are usually conflicting. Thus, we follow a multi-objective approach, aiming to compute a set of Pareto-optimal alternatives with different trade-offs for a decision maker to choose from. We propose two algorithms (a Multi-Objective Memetic Algorithm and a Multi-Objective Iterated Local Search) and compare them to an evolutionary multi-objective optimizer from the literature. We also propose a modified statistical method for the service level calculation. Experiments based on an adapted version of the 56 Solomon instances demonstrate the effectiveness of the proposed algorithms

Estudio del problema de ruteo de vehículos con balance de carga :Aplicación de la meta-heurística Búsqueda Tabú.

92 páginasEl Problema de Ruteo de Vehículos (VRP – por su sigla en inglés) es uno de los problemas de optimización combinatoria más estudiados en las últimas décadas. Este consiste en determinar un conjunto de rutas para una flota de vehículos que parte de uno o más depósitos para satisfacer la demanda de clientes dispersos geográficamente. El enfoque tradicionalmente utilizado ha sido la optimización de un solo objetivo; sin embargo, en la realidad organizacional optimizar más de un objetivo permite la toma de decisiones con una visión de negocio más integral. El presente trabajo estudia el problema de ruteo de vehículos bajo un enfoque multi-objetivo, en el cual se incorpora además de la minimización de la distancia, el balance de carga como objetivo de optimización. Al hacer una exhaustiva revisión de la literatura del problema de ruteo de vehículos multi-objetivo se evidenció que el balance de carga es un objetivo que se ha estudiado poco y en la mayoría de los trabajos analizados, se ha considerado el balance de carga desde la perspectiva de la longitud de las rutas. Como consecuencia, en este trabajo se definió el balance de carga como la diferencia de carga entre los vehículos con mayor y menor cantidad de producto a transportar hacia los clientes. Para la caracterización del problema de ruteo de vehículos multi-objetivo, mono-depósito con balance de cargas, se desarrolló un modelo de programación entera mixta el cual se implementó en GAMS y se probó con las primeras siete instancias de Augerat et al. (1998) obteniendo resultados prometedores tanto en el enfoque mono-objetivo como en el multi-objetivo. Por otra parte, teniendo en cuenta la complejidad del problema estudiado, se desarrolló un algoritmo de Búsqueda Tabú con tamaños de lista tabú fija y dependiente del número de nodos, el cual se probó con todas las instancias de Augerat et al

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

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

Optimisation models and algorithms for workforce scheduling and routing

This thesis investigates the problem of scheduling and routing employees that are required to perform activities at clients’ locations. Clients request the activities to be performed during a time period. Employees are required to have the skills and qualifications necessary to perform their designated activities. The working time of employees must be respected. Activities could require more than one employee. Additionally, an activity might have time-dependent constraints with other activities. Time-dependent activities constraints include: synchronisation, when two activities need to start at the same time; overlap, if at any time two activities are being performed simultaneously; and with a time difference between the start of the two activities. Such time difference can be given as a minimum time difference, maximum time difference, or a combination of both (min-max). The applicability of such workforce scheduling and routing problem (WSRP) is found in many industries e.g. home health care provision, midwives visiting future mothers, technicians performing installations and repairs, estate agents showing residences for sale, security guards patrolling different locations, etc. Such diversity makes the WSRP an important combinatorial optimisation problem to study. Five data sets, obtained from the literature, were normalised and used to investigate the problem. A total of 375 instances were derived from these data sets. Two mathematical models, an integer and a mixed integer, are used. The integer model does not consider the case when the number of employees is not enough to perform all activities. The mixed integer model can leave activities unassigned. A mathematical solver is used to obtain feasible solutions for the instances. The solver provides optimal solutions for small instances, but it cannot provide feasible solutions for medium and large instances. This thesis presents the gradual development of a greedy heuristic that is designed to tackle medium and large instances. Five versions of the greedy heuristic are presented, each of them obtains better results than the previous one. All versions are compared to the results obtained by the mathematical solver when using the mixed integer model. The greedy heuristic exploits domain information to speed the search and discard infeasible solutions. It uses tailored functions to deal with each of the time-dependent activity constraints. These constraints make more difficult the solution process. Further improvements are obtained by using tabu search. It provides moves based on the tailored functions of the greedy heuristic. Overall, the greedy heuristic and the tabu search, maintain feasible solutions at all times. The main contributions of this thesis are: the definition of WSRP; the introduction of 375 instances based on five data sets; the adaptation of two mathematical models; the introduction of a greedy heuristic capable of obtaining better results than the solver; and, the implementation of a tabu search to further improve the results

Optimisation models and algorithms for workforce scheduling and routing

This thesis investigates the problem of scheduling and routing employees that are required to perform activities at clients’ locations. Clients request the activities to be performed during a time period. Employees are required to have the skills and qualifications necessary to perform their designated activities. The working time of employees must be respected. Activities could require more than one employee. Additionally, an activity might have time-dependent constraints with other activities. Time-dependent activities constraints include: synchronisation, when two activities need to start at the same time; overlap, if at any time two activities are being performed simultaneously; and with a time difference between the start of the two activities. Such time difference can be given as a minimum time difference, maximum time difference, or a combination of both (min-max). The applicability of such workforce scheduling and routing problem (WSRP) is found in many industries e.g. home health care provision, midwives visiting future mothers, technicians performing installations and repairs, estate agents showing residences for sale, security guards patrolling different locations, etc. Such diversity makes the WSRP an important combinatorial optimisation problem to study. Five data sets, obtained from the literature, were normalised and used to investigate the problem. A total of 375 instances were derived from these data sets. Two mathematical models, an integer and a mixed integer, are used. The integer model does not consider the case when the number of employees is not enough to perform all activities. The mixed integer model can leave activities unassigned. A mathematical solver is used to obtain feasible solutions for the instances. The solver provides optimal solutions for small instances, but it cannot provide feasible solutions for medium and large instances. This thesis presents the gradual development of a greedy heuristic that is designed to tackle medium and large instances. Five versions of the greedy heuristic are presented, each of them obtains better results than the previous one. All versions are compared to the results obtained by the mathematical solver when using the mixed integer model. The greedy heuristic exploits domain information to speed the search and discard infeasible solutions. It uses tailored functions to deal with each of the time-dependent activity constraints. These constraints make more difficult the solution process. Further improvements are obtained by using tabu search. It provides moves based on the tailored functions of the greedy heuristic. Overall, the greedy heuristic and the tabu search, maintain feasible solutions at all times. The main contributions of this thesis are: the definition of WSRP; the introduction of 375 instances based on five data sets; the adaptation of two mathematical models; the introduction of a greedy heuristic capable of obtaining better results than the solver; and, the implementation of a tabu search to further improve the results