Set Partitioning Model Research Articles

Heterogeneous vehicle pickup and delivery problems: Formulation and exact solution

The heterogeneous green pickup and delivery problem (GPDP) aims to minimize carbon emissions of pickups and deliveries by a fleet of heterogeneous vehicles. We propose the first exact algorithm for solving the heterogeneous GPDP. This exact algorithm is based on a set partitioning model and the key characteristics of its optimal solution. We prove that the optimal solution is composed of the optimal trips of heterogeneous vehicles; thus, the solution space is significantly reduced by cutting the non-optimal trips. The proposed exact algorithm can rapidly find the optimal solution of the largest-scale instance from Li & Lim benchmarks and three types of vehicles. The exact algorithm is also validated by solving a practical case of heterogeneous GPDP in China. Based on the extensive experiments, we conclude an interesting managerial insight that heterogeneous vehicles can always effectively reduce carbon emissions except for two situations.

Exact Algorithms for the Vehicle Routing Problem with Time Windows and Combinatorial Auction

In this paper, we address an interesting variant of the vehicle routing problem with time windows (VRPTW) faced by the China National Petroleum Corporation (CNPC). The CNPC owns a limited number of tanker trucks for delivering the petroleum to oil stations within specific time windows in the regular seasons. However, during the peak seasons, some requests need to be outsourced to external third-party logistics companies. These companies provide several bids, each of which includes the oil stations to be served and the corresponding charges. The CNPC needs to select some bids and design routes for the self-owned trucks, so that all requests are satisfied and the total cost is minimized. To study this problem, we formulate it into an arc-flow model and a set-partitioning model, and propose six families of valid inequalities to strengthen the set-partitioning model. Based on the set-partitioning model, we propose a branch-and-price-and-cut algorithm and a branch-and-bound algorithm to solve the problem exactly. The proposed algorithms are tested on instances generated according to the well-known Solomon’s benchmark instances for the VRPTW and read-world data of the CNPC. The computational experiments demonstrate the effectiveness of the proposed algorithms. The online appendix is available at https://doi.org/10.1287/trsc.2018.0835 .

Re-planning In Home Healthcare: A Decomposition Approach To Minimize Idle Time For Workers While Ensuring Continuity Of Care

Abstract Scheduling and routing careworkers is a perpetual challenge in the home health care (HHC) industry: it is essential to ensure a good quality of care under reasonable expenses. HHC consists in providing care to fragile people, therefore the pool of patients often changes: the schedule must be computed again, but continuity of care must be preserved to keep the patients satisfied. The working environment in the HHC field is remarkably stressful therefore it causes a high turnover rate in the staff, and has a strong impact on service quality. It is crucial to provide schedules that satisty both patients and careworkers. In this paper, we address a weekly home health care re-planning problem with continuity constraints. A decomposition approach is proposed to offer a careworker-friendly solution: first a path-generation algorithm creates admissible tours, and then a set partitioning model selects the tours.

A set partitioning heuristic for the home health care routing and scheduling problem

The home health care routing and scheduling problem comprises the assignment and routing of a set of home care visits over the duration of a week. These services allow patients to remain in their own homes, thereby reducing governmental costs by decentralizing the care. In this work, we present a set partitioning heuristic which takes into account most of the industry’s practical constraints. The developed method is based on a set partitioning formulation and a large neighborhood search (LNS) framework. The algorithm solves a linear relaxation of a set partitioning model using the columns generated by the large neighborhood search. A constructive heuristic is then called to build an integer solution. This project is joint work with Alayacare, a start-up sited in Montral (Canada) developing an operations management platform for home health care agencies. They provide their clients with a flexible optimization module that solves real-life instances in no more than 10 minutes. Based on their real instances, the proposed method is able to provide a reduction in travel time by 37% and an increase by more than 16% in continuity of care. We also provide a public benchmark for this problem.

A branch-and-price-and-cut algorithm for a pickup and delivery problem in retailing

We investigate a real-world product distribution problem faced by a fast fashion retailer in Singapore. It is a generalization of the multi-commodity pickup and delivery traveling salesman problem (m-PDTSP) proposed by Hernández-Pérez and Salazar-González [15]. Each weekend the retailer forecasts the demand of each product for the next week. To ensure the inventory level equal to the forecast, the retailer schedules a fleet of vehicles to pick up or deliver products from/to each store at the beginning of each week. Shipping products from a store in surplus to another store in shortage is encouraged. For products that cannot be balanced among the stores, they are either picked up or delivered from/to the warehouse, which incurs a handling cost for each unit. The objective is to minimize the total travel distance as well as the total handling cost at the warehouse. To solve this problem, we propose a branch-and-price-and-cut algorithm based on a strong set-partitioning model, where an ad-hoc label-setting algorithm is designed to solve the pricing problem. The algorithm is tested on a set of instances generated according to a real-world database from the retailer and the m-PDTSP benchmark instances. Computational results show that our algorithm can optimally solve instances with 20 stores and over 100 products in one hour computational time. Moreover, it successfully solves several open m-PDTSP benchmark instances to optimality.

Solving a Large-Scale Multi-Depot Vehicle Scheduling Problem in Urban Bus Systems

This study proposes an improved model and algorithm for the large-scale multi-depot vehicle scheduling problem (MDVSP) with departure-duration restrictions. In this study, the time-space network is applied to model the large-scale MDVSP. Considering that crews usually change shifts in the depot, departure-duration restrictions are added to the classic set-partitioning model to ensure that buses return to the depot when crews reach their working time limits. By embedding a preliminary exploring tactic to the shortest path faster algorithm (SPFA), researchers developed an improved large neighborhood search (LNS) algorithm to solve large-scale instances of MDVSP with departure-duration restrictions. The proposed methodology is applied to a real-life case in China and several test instances. The results show that the improved LNS algorithm can achieve very good performance in computational efficiency without deteriorating solution quality, which is important for large-scale systems. More specifically, the total cost of the improved LNS algorithm is approximately equal to branch-and-price, but the computational time is much shorter in the case study. For test instances with different number of timetabled trips (500, 1000, 1500, and 2000), the Quality Gap (QG) is very small, approximately 0.35%, 0.38%, 0.63%, and 0.93%, while the Efficiency Ratio (ER) reaches up to 2.89, 2.98, 3.65, and 3.79, respectively.

A branch-and-price algorithm for scheduling of deteriorating jobs and flexible periodic maintenance on a single machine

In many production systems, maintenance activities including preventive maintenance, repairs and tool changes are periodically scheduled. The activities can revert the machine from a sub-normal processing rate to a normal one.In this paper, we study a single machine scheduling problem where deteriorating jobs and flexible periodic maintenance are considered. The single machine is operated to process a set of jobs with alternating processing periods and maintenance periods. In a processing period, a subset of jobs is sequentially processed and the completion time of the last job cannot exceed the allowed maximum duration. The actual processing time of each job grows at a linear job-specific deterioration rate and depends on its starting time within the period. Between two processing periods, a maintenance period with a fixed duration exists and the maintenance activities are carried out so that the processing rate of the machine is reverted to the normal rate. The objective is to schedule all the jobs to a set of processing periods and to minimize the makespan of the schedule.We formulate the problem using a set-partitioning model and, for a solution method, we make use of a branch-and-price algorithm. A label-setting algorithm with a dominance rule is designed to solve the pricing problem in column generation. Computational experiments are conducted on a set of randomly generated test instances to evaluate the performance of the proposed method.

Daily scheduling of caregivers with stochastic times

This paper addresses a daily caregiver scheduling and routing problem arising in home health care or home care service providers. The problem is quite challenging due to its uncertainties in terms of travel and service times derived from changes in road traffic conditions and customer health status in practice. We first model the problem as a stochastic programme with recourse, where the recourse action is to skip customers without services if the caregiver arrives later than their latest starting service time (i.e. hard time window requirements). Then, we formulate the problem as a set partitioning model and solve it with a branch-and-price (B&P) algorithm. Specifically, we devise an effective discrete approximation method to calculate the arrival time distribution of caregivers, incorporate it into a problem-specific label algorithm, and use a removal-and-insertion-based heuristic and the decremental state-space relaxation technique to accelerate the pricing process. Finally, we conduct numerical experiments on randomly generated instances to validate the effectiveness of the discrete approximation method and the proposed B&P algorithm.

An efficient column-generation algorithm for a new fleet size and mix pickup and delivery problem with Time windows

In this paper, a new variant of the pickup and delivery problem with time windows (PDPTW) named the Fleet Size and Mix Pickup and Delivery Problem with Time Windows (FSMPDPTW) is addressed. This work is motivated by fleet sizing for a daily route planning arising at a Hospital center. In fact, a fleet of heterogeneous rented vehicles is used every day to pick up goods to locations and to deliver it to other locations. The heterogeneous aspect of the fleet is in term of capacity, fixed cost and fuel mileage. The objective function is the minimization of the total fixed cost of vehicles used and the minimization of the total routing cost. A set partitioning model is proposed to model the problem, and an efficient column-generation algorithm is used to solve it. In order to test our method, we propose a new set of benchmarks based on Li and Lim’s benchmark (altered Solomon’s benchmark) for demands and from Lui and Shen’s benchmark for types of vehicles. In the propounded column-generation algorithm, the pricing problem is divided in sub-problems such that each vehicle type have its own pricing problem. A mixed integer linear program is proposed to model and solve the pricing sub-problems. Regret heuristics are proposed to speed-up the resolution of pricing sub-problems. Computational experiments are done on 56 (with up to 100 customers) new proposed instances.

A based-on-set-partitioning exact approach to multi-trip of picking up and delivering customers to airports

Picking up and delivering customers to airports (PDCA) is a new service provided in China. The multi-trip mode of PDCA (MTM-PDCA) service is a promising measure to reduce operation costs. To obtain the exact solution, we propose a novel modelling approach including two stages. In the first stage, all feasible trips of each subset of the customer point set are produced, and then the two local optimum trips of each subset can be obtained easily. Subsequently, using the local optimum trips obtained in the first stage, we establish the novel trip-oriented set-partitioning (TO-SP) model to formulate MTM-PDCA. The MTM-PDCA based on the TO-SP model can be solved exactly by CPLEX. By testing extensive instances, we summarise several managerial insights that can be used to successfully reduce the costs of PDCA by using multi-trip mode.

A based-on-set-partitioning exact approach to multi-trip of picking up and delivering customers to airport

Picking up and delivering customers to airports (PDCA) is a new service provided in China. The multi-trip mode of PDCA (MTM-PDCA) service is a promising measure to reduce operation costs. To obtain the exact solution, we propose a novel modelling approach including two stages. In the first stage, all feasible trips of each subset of the customer point set are produced, and then the two local optimum trips of each subset can be obtained easily. Subsequently, using the local optimum trips obtained in the first stage, we establish the novel trip-oriented set-partitioning (TO-SP) model to formulate MTM-PDCA. The MTM-PDCA based on the TO-SP model can be solved exactly by CPLEX. By testing extensive instances, we summarise several managerial insights that can be used to successfully reduce the costs of PDCA by using multi-trip mode.

Integrated Vehicle Scheduling and Vehicle Assignment

The vehicle scheduling problem has been extensively studied in the past decades. Yet, most models and methods given in the literature consider only a theoretical scenario where vehicles just have to service the timetabled trips of the input. However, schedules created this way cannot be used in real life, as they should also consider constraints such as refueling, parking, and maintenance, which are all connected to the vehicle servicing the trips. In this paper, we give a set partitioning model for the multi-depot integrated vehicle scheduling and vehicle assignment problem. This model can also be used as a general framework, which can integrate multiple activities based on the rules or regulation of the different possible input scenarios. We give a column generation-based solution method, and demonstrate its efficiency on randomly generated test instances, which treat the refueling of vehicles with two different fuel types as the vehicle-specific activity.

Metodología para crear rutas alimentadoras en sistemas de transporte masivo

Se propone una metodología para identificar rutas alimentadoras en zonas no conectadas para un sistema de transporte masivo, con el fin de aumentar la cobertura del servicio y mejorar el nivel de ocupación del sistema. La metodología propuesta consta de dos etapas: 1) estructurar escenarios de áreas no conectadas al sistema de transporte y 2) combinar técnicas heurísticas y exactas para resolver el problema de rutas alimentadoras. La metodología considera dentro de sus restricciones la duración de la ruta y la capacidad del vehículo alimentador. Para su modelamiento se establece una analogía entre los problemas del transporte de pasajeros y el problema de localización y ruteo, Location Routing Problem (LRP), que usualmente es aplicado a problemas de transporte de mercancías. La metodología de solución propuesta es una matheurística que combina las heurísticas Lin-Kernighan-Helsgaun (LKH) y ahorros con el algoritmo de ramificación y corte, Branch-and-Cut, aplicado sobre un modelo lineal entero mixto de partición de conjuntos (Set Partitioning) para LRP. Esta propuesta metodológica es validada con casos de prueba reales del sistema de transporte masivo de la ciudad de Pereira (Megabús), donde se consideran algunas zonas no conectadas del Área Metropolitana Centro Occidente, localizada en el eje cafetero colombiano.

The Vehicle Routing Problem with Transhipment Facilities

This paper proposes an exact method for solving an optimization problem arising in several distribution networks where customers can be served directly, using vehicle routes from a central depot, or through transhipment facilities. The problem consists of optimizing the following inter-dependent decisions: selecting transhipment facilities, allocating customers to these facilities, and designing vehicle routes emanating from a central depot to minimize the total distribution cost. This problem is called the Vehicle Routing Problem with Transhipment Facilities (vrptf). The paper describes two integer-programming formulations for the vrptf, i.e., an edge-flow based formulation and a Set Partitioning (SP) based formulation. The LP-relaxation of the two formulations are further strengthened by the addition of different valid inequalities. We also describe two new route relaxations used by dual ascent heuristics to find near-optimal dual solutions of LP-relaxation of the SP model. The valid inequalities and the route relaxations are used in a branch-and-cut-and-price approach to solve the problem to optimality. The proposed method is tested on a large family of instances, including real-world examples. The computational results obtained indicate the effectiveness of the proposed method.

Declarative Representation and Solution of Vehicle Routing with Pickup and Delivery Problem

Recently we have proposed a multi-agent system that provides an intelligent logistics brokerage service focusing on the transport activity for the efficient allocation of transport resources (vehicles or trucks) to the transport applications. The freight broker agent has a major role to coordinate transportation arrangements of transport customers (usually shippers and consignees) with transport resource providers or carriers, following the freight broker business model. We focus on the fundamental function of this business that aims to find available trucks and to define their feasible routes for transporting requested customer loads. The main contribution of this paper is on formulating our scheduling problem as a special type of vehicle routing with pickup and delivery problem. We propose a new set partitioning model of our specific problem. Vehicle routes are defined on the graph of cities, rather than on the graph of customer orders, as typically proposed by set partitioning formulations. This approach is particularly useful when a large number of customer orders sharing a significantly lower number of pickup and delivery points must be scheduled. Our achievement is the declarative representation and solution of the model using ECLiPSe state-of-the-art constraint logic programming system.

Time-Dependent Vehicle Routing of Free Pickup and Delivery Service in Flight Ticket Sales Companies Based on Carbon Emissions

The time-dependent pollution-routing problem of free pickup and delivery of passengers to the airport service (TDFPDS) is an extension of the time-dependent pollution-routing problems, arising in flight ticket sales companies for the service of free pickup and delivery of airline passengers to the airport. The problem consists of routing a fleet of vehicles in order to deliver a set of customers to the airport considering the traffic congestion, time window constraints, and maximum ride time constraints. The cost function includes fuel consumption and driver costs. We provide an analytical characterization of the optimal solutions for a fixed route and propose a novel heuristic for a given route based on the analysis of the illustrative examples. The heuristic algorithm is embedded into a set-partitioning model to produce high-quality routing plans. Finally, using wide variety of random instances, we present results on the computational performance of the heuristic and also on the impact of the congestion and the maximum ride time constraints.

A Hybrid Iterative Local Search Algorithm for The Electric Fleet Size and Mix Vehicle Routing Problem with Time Windows and Recharging Stations

As the cities around the world become larger, quality of life of the citizens is more and more threatened due to the traffic congestion, energy consumption, noise disturbance and carbon emissions because of of increasing transport. Electrical vehicles present an opportunity to reduce greenhouse gas emissions. But limited driving range and long recharging times are among the challenges that the research community has to face. This paper proposes an iterative local search algorithm coupled with a set partitioning model to solve The Electric Fleet Size and Mix Vehicle Routing Problem with Time Windows and Recharging Stations. Several types of electrical vehicles with varying driving range, capacity and fixed cost should service a set of customers within their time limits and during their tours each vehicle can be recharged in stations. We show the efficiency and the quality of our method by solving benchmark instances of the Heterogeneous Fleet Electric Vehicle Problems with Time Windows.

A column generation approach for aircraft sequencing problems: a computational study

This paper investigates the computational tractability of aircraft sequencing problems over multiple runways under mixed mode operations, contrasting an enhanced mixed-integer programme (MIP) and an accelerated column generation approach. First, we examine the benefit of augmenting a base MIP with valid inequalities, preprocessing routines, and symmetry-defeating hierarchical constraints in order to improve the performance of branch-and-bound (B&B)/cut techniques as implemented in commercial solvers. Second, we alternatively reformulate the problem as a set partitioning model that prompts the development of a specialized column generation approach. The latter is accelerated by incorporating an interior point dual stabilization scheme and a complementary column generation routine. Empirical results using a set of new, computationally challenging instances and classical instances in the OR Library reveal the potential and limitations of the two methodologies.

Mathematical formulations and exact algorithm for the multitrip cumulative capacitated single-vehicle routing problem

This paper addresses the multitrip Cumulative Capacitated Single-Vehicle Routing Problem (mt-CCSVRP). In this problem inspired by disaster logistics, a single vehicle can perform successive trips to serve a set of affected sites and minimize an emergency criterion, the sum of arrival times. Two mixed integer linear programs, a flow-based model and a set partitioning model, are proposed for small instances with 20 sites. An exact algorithm for larger cases transforms the mt-CCSVRP into a resource-constrained shortest path problem where each node corresponds to one trip and the sites to visit become resources. The resulting problem can be solved via an adaptation of Bellman–Ford algorithm to a directed acyclic graph with resource constraints and a cumulative objective function. Seven dominance rules, two upper bounds and five lower bounds speed up the procedure. Computational results on instances derived from classical benchmark problems for the capacitated VRP indicate that the exact algorithm outperforms a commercial MIP solver on small instances and can solve cases with 40 sites to optimality.

Integrated Berth Allocation and Quay Crane Assignment Problem: Set partitioning models and computational results

Most of the operational problems in container terminals are strongly interconnected. In this paper, we study the integrated Berth Allocation and Quay Crane Assignment Problem in seaport container terminals. We will extend the current state-of-the-art by proposing novel set partitioning models. To improve the performance of the set partitioning formulations, a number of variable reduction techniques are proposed. Furthermore, we analyze the effects of different discretization schemes and the impact of using a time-variant/invariant quay crane allocation policy. Computational experiments show that the proposed models significantly improve the benchmark solutions of the current state-of-art optimal approaches.