Pickup And Delivery Problem With Time Windows Research Articles

Rolling Horizon Based Temporal Decomposition for the Offline Pickup and Delivery Problem with Time Windows

The offline pickup and delivery problem with time windows (PDPTW) is a classical combinatorial optimization problem in the transportation community, which has proven to be very challenging computationally. Due to the complexity of the problem, practical problem instances can be solved only via heuristics, which trade-off solution quality for computational tractability. Among the various heuristics, a common strategy is problem decomposition, that is, the reduction of a large-scale problem into a collection of smaller sub-problems, with spatial and temporal decompositions being two natural approaches. While spatial decomposition has been successful in certain settings, effective temporal decomposition has been challenging due to the difficulty of stitching together the sub-problem solutions across the decomposition boundaries. In this work, we introduce a novel temporal decomposition scheme for solving a class of PDPTWs that have narrow time windows, for which it is able to provide both fast and high-quality solutions. We utilize techniques that have been popularized recently in the context of online dial-a-ride problems along with the general idea of rolling horizon optimization. To the best of our knowledge, this is the first attempt to solve offline PDPTWs using such an approach. To show the performance and scalability of our framework, we use the optimization of paratransit services as a motivating example. Due to the lack of benchmark solvers similar to ours (i.e., temporal decomposition with an online solver), we compare our results with an offline heuristic algorithm using Google OR-Tools. In smaller problem instances (with an average of 129 requests per instance), the baseline approach is as competitive as our framework. However, in larger problem instances (approximately 2,500 requests per instance), our framework is more scalable and can provide good solutions to problem instances of varying degrees of difficulty, while the baseline algorithm often fails to find a feasible solution within comparable compute times.

The Pickup and Delivery Problem with Time Windows and Incompatibility Constraints in Cold Chain Transportation

This study investigates a new variant of the pickup and delivery problem with time windows (PDPTW) applied in cold chain transportation, which quantifies the effect of time on the quality of perishable products. Multiple commodities with incompatibility constraints are considered, where some types of products cannot be transported in a vehicle simultaneously because of their different properties and requirements for storage temperatures. The aim is to determine vehicles’ pickup and delivery routes as well as their departure times from the depot such that the travel cost and refrigeration cost of vehicles and the quality decay cost of products are minimized. We formulate this problem as a set partitioning model, which is solved exactly by a tailored branch-and-price (B&P) algorithm. To tackle the asymmetry issue arising from the pricing problem of the B&P framework, we develop a novel asymmetric bidirectional labeling algorithm. Benchmark instance sets based on real-world statistical data and classic PDPTW instance sets are first generated for this problem. Numerical results show that our B&P algorithm can solve most instances to optimality in an acceptable time frame. Moreover, our results demonstrate that integrating the refrigeration and quality decay costs into the objective function can significantly lower the total cost of cold chain transportation activities, compared with the widely adopted objective function minimizing only the travel cost. Funding: This work was supported by the National Key R&D Program of China [Grant 2018YFB1700600] and the National Natural Science Foundation of China [Grants 72101049, 71971090, 71821001]. Supplemental Material: The online supplement is available at https://doi.org/10.1287/trsc.2022.1167 .

Solving multi-objective pickup and delivery with time windows using mediocre evolutionary distributed microservices re-optimization algorithm

The pickup and delivery problem with time windows (PDPTW) is an important problem to be solved. This is due to its resemblance to real-life services such as food delivery, courier service, curbside pickup service, and dial-a-ride, among others. However, the research on multi-objective pickup and delivery problems with time windows (MOPDPTW) remains scarce even after many years. The MOPDPTW objective is to achieve the optimal solution on both the number of vehicle and total travelled distance. It is challenging to obtain optimal results, the recurring optimal results that has least difference in magnitude and broader Pareto sets at the same time. Hence, an algorithm that can fulfil these requirements are highly sought after. This paper presents a mediocre evolutionary distributed microservices re-optimization algorithm (MEDMRA) to solve the MOPDPTW. The combination of the re-optimize algorithm and mediocre evolutionary algorithm escape the local optimal, optimized results and produce broader Pareto value. We compare our results using Li & Lim instances. Our results outperform the latest published hybrid algorithms and competitive with the best-known solutions.

Evolutionary neural network for learning of scalable heuristics for pickup and delivery problems with time windows

In this paper, we address the pickup and delivery problem with time windows (PDP-TW) and heterogenous vehicles for minimisation of total tardiness by learning heuristics from a given set of solutions. In order to extract scalable heuristics from optimal or best feasible solutions, we propose a machine-learning (ML)-based approach called ENSIGHT (Evolutionary Neural network with Scalable Information for Generation of Heuristics for Transportation). ENSIGHT consists of three phases: solution generation, interpretation of solutions, and improvement of heuristics by an evolutionary neural network (ENN). First, a set of optimal or best feasible solutions for the training set of problem instances is acquired by using the proposed mathematical model. Second, as for the process interpreting those solutions, an approach for transforming them into training data by way of scalable input attributes as well as output discretisation is followed. Third, the ENN improves the learned heuristics by an evolutionary parameter optimisation process for minimization of total tardiness. To verify the performance of the proposed ENSIGHT, we conducted experiments and the results of which showed that it outperforms other ML techniques and the current dispatching rules (DRs). Moreover, the approach was demonstrated to be effective in learning scalable heuristics based on combined scalable inputs and discretisation as well as an evolutionary improvement process.

Analysis of pick-up and delivery and dial-a-ride problems dynamization methods and benchmark instances

As instâncias de benchmark disponíveis para as versões dinâmicas do problema de coleta e entrega com janelas de tempo (PDPTW - Pickup and Delivery Problem with Time Windows) e do problema dial-a-ride (DARP - Dial-A-Ride Problem) não compartilham as mesmas características e não necessariamente cobrem todas as características de situações reais. Analisa-se conjuntos de instâncias de PDPTW e DARP dinâmicos (DPDPTW e DDARP) atualmente disponíveis para uso e os métodos usados para gerá-los a partir de instâncias estáticas. Cada método de dinamização é aplicado a cada instância estática originalmente usada por eles. As instâncias dinâmicas resultantes são analisadas com as medidas de grau de dinamismo e urgência, bem como pelo número de pedidos estáticos e a correlação entre os limites inferiores das janelas de tempo de coleta e os instantes de chegada dos pedidos. Os resultados mostram que os conjuntos estudados apresentam baixa variabilidade de grau de dinamismo e urgência independentemente do método ou da instância estática usados para a dinamização.

Platform-Based Collaborative Routing using Dynamic Prices as Incentives

Over the last decade, platforms have emerged in numerous industries and often transformed them, posing new challenges for transportation research. Platform providers such as Uber, Uber Freight, Blackbuck, or Lyft mostly do not have immediate control over the physical resources needed to move people or goods. They often operate in a multi-sided market setting, where it is crucial to design clear incentives to motivate a third party to engage in collaboration. As a consequence, collaboration incentives become an integral part of decision support models for platform providers and they need to be developed at the operational level and applied dynamically. Naturally, this involves a trade-off between the interests of platform providers, shippers, and carriers. In this work, we investigate the real-world case of a platform provider operating as an intermediary between shippers and carriers in a less-than-truckload (LTL) business. We propose a new mixed-integer programming (MIP) formulation for the underlying collaborative pickup and delivery problem with time windows (PDPTW) that minimizes the price the platform pays to the carriers and enforces collaboration incentives for carriers through individual rationality constraints. This is facilitated by a dynamic pricing approach which ensures that carriers are better off collaborating than working on their own. The pricing is bounded by the costs and market conditions to keep the price range reasonable. We explore possible policies to be implemented by the platform and find that their business remains profitable when individual rationality is enforced and the platform could even guarantee increased profit margins to the carriers as incentives.

An Exact Algorithm for the Pickup and Delivery Problem with Time Windows and Last-in-First-out Loading

Applications of the pickup and delivery problem with time windows and last-in-first-out (LIFO) loading (PDPTWL) constraints can be found in the transportation of animals, heavyweight goods, and hazardous materials, where unloading vehicles requires more time and special handling. Examples include carrying livestock, cars, and chemical containers. Research on exact methods to solve the pickup and delivery problem with time windows (PDPTW) and its variants has mainly focused on branch-and-price-and-cut algorithms. In this paper, we propose a novel exact approach based on fragments—a series of pickup and delivery requests starting and ending with an empty vehicle. We use fragments to formulate a relaxed network flow model with side constraints. Lazy constraints are used to cut off any illegal solution that may occur while solving the integer program. Extensive computational experiments show that the proposed approach is superior to the current state-of-the-art method.

A constraint programming approach for the pickup and delivery problem with time windows

The pickup and delivery problem with time windows (PDPTW) is studied in this paper. It is referred to as the single-commodity capacitated vehicle routing problem with pickups and deliveries, in which a fleet of vehicles meet a group of customer demands. Each customer demand includes the usage of only one vehicle for both loading a specified quantity of one type of commodity at one place and delivering them to another place. All the demands of customers must be satisfied without exceeding the vehicle capacity and the pickup or delivery places time windows specified for each place. In this study, we introduce a novel Constraint Programming (CP) model for the PDPTW. CP is an exact solution approach that is popular for its performance to state complicated relationships and to achieve high-quality solutions within acceptable computational times for combinatorial optimization problems with complicated constraints such as the PDPTW. The performance of the proposed CP model is tested with well-known benchmark instances from literature. The results of computational analysis indicate that our CP model is very effective in finding high-quality solutions for even large size problems.

Vehicle Routing Problem of an Innovative B2C and O2O Joint Distribution Service

In respond to the problem of insufficient capacity in peak hours and idle capacity in off-peak hours of O2O delivery, the logistics industry has been challenged to minimize the total costs of both B2C and O2O delivery. Considering the high coincidence in geographical space of B2C and O2O orders, an innovative joint distribution service is proposed. This paper designs a modified C-W algorithm based on the merging of node-inserted paths to achieve the optimization of pickup and delivery problem with time windows (PDPTW). A numerical analysis is conducted to compare the independent distribution mode with the innovative joint distribution mode. The results highlight that the innovative joint distribution mode performs better than the independent distribution mode in both different order sizes and different lengths of O2O time windows, which can improve the delivery efficiency and lower the total costs.

A Comparative Study of the PSO and GA for the m-MDPDPTW

The m-MDPDPTW is the multi-vehicles, multi-depots pick-up and delivery problem with time windows. It is an optimization vehicles routing problem which must meet requests for transport between suppliers and customers for the purpose of satisfying precedence, capacity and time constraints. This problem is a very important class of operational research, which is part of the category of NP-hard problems. Its resolution therefore requires the use of evolutionary algorithms such as Genetic Algorithms (GA) or Particle Swarm Optimization (PSO). We present, in this sense, a comparative study between two approaches based respectively on the GA and the PSO for the optimization of m-MDPDPTW. We propose, in this paper, a literature review of the Vehicle Routing Problem (VRP) and the Pick-up and Delivery Problem with Time Windows (PDPTW), present our approaches, whose objective is to give a satisfying solution to the m-MDPDPTW minimizing the total distance travelled. The performance of both approaches is evaluated using various sets instances from [10] PDPTW benchmark data problems. From our study, in the case of m-MDPDPTW problem, the proposed GA reached to better results compared with the PSO algorithm and can be considered the most appropriate model to solve our m-MDPDPTW problem.

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.

Bidirectional labeling in column-generation algorithms for pickup-and-delivery problems

For the exact solution of many types of vehicle-routing problems, column-generation based algorithms have become predominant. The column-generation subproblems are then variants of the shortest-path problem with resource constraints which can be solved well with dynamic-programming labeling algorithms. For vehicle-routing problems with a pickup-and-delivery structure, the strongest known dominance between two labels requires the delivery triangle inequality (DTI) for reduced costs to hold. When the direction of labeling is altered from forward labeling to backward labeling, the DTI requirement becomes the pickup triangle inequality (PTI). DTI and PTI cannot be guaranteed at the same time. The consequence seemed to be that bidirectional labeling, one of the most successful acceleration techniques developed over the last years, is not applicable with the strong dominance on both directions for problems with a pickup-and-delivery structure. In this paper, we show that bidirectional labeling with the strong dominance rules in forward as well as backward direction is possible by using different cost matrices in the two directions. We adopt the both-sided strong bidirectional labeling approach to integrate the standard robust and non-robust cuts. Moreover, a recent acceleration technique that dynamically adjusts the bidirectional half-way point is implemented. Full-fledged branch-cut-and-price algorithms are tested on the pickup-and-delivery problem with time windows (PDPTW). In particular, an in-depth analysis of different mono- and bidirectional labeling algorithms is presented. Overall, compared to the standard pure forward labeling-based algorithm, we obtain average reductions in computation time of more than 40 percent when solving PDPTW instances exactly.

A pick-up and delivery problem with time windows by electric vehicles

In the pick-up and delivery problem with time windows (PDPTW), each transportation service is delivered, from an origin to a destination, satisfying both the time windows and the precedence constraints. This paper addresses the related vehicle routing problem by using only electric vehicles (EVs) and by introducing the recharging stations (RSs). The problem is formulated as a multi-objective mixed integer linear model for minimising the total travel distance, the total cost for the EVs used and the total penalty cost for the unsatisfied time windows. In addition, length constraints on the routes are imposed in order to include several aspects such as the limited availability of the RSs. The weighted sum method is adopted and, to properly set the weights, three methods, derived from the analytical hierarchical process, are compared. Computational experiments on some instances are carried out, in order to assess the behaviour of our approach in terms of solution quality.

Branch-Price-and-Cut Algorithms for the Pickup and Delivery Problem with Time Windows and Last-in-First-Out Loading

This paper proposes models and algorithms for the pickup and delivery vehicle routing problem with time windows and last-in-first-out (LIFO) loading constraints (PDPTWL). The LIFO loading rule ensures that no handling is required prior to unloading an item from a vehicle: a linear stack loading structure is maintained and an item can only be delivered if it is the last one in the stack. Three exact branch-price-and-cut algorithms are proposed for this problem. The first incorporates the LIFO constraints in the master problem. The second one handles the LIFO constraints directly in the shortest path pricing problem. It applies a dynamic programming algorithm relying on an ad hoc dominance criterion. The third algorithm is a hybrid between the first two methods. Known valid inequalities are adapted to the PDPTWL and the impact of different path relaxations on the total computation time is investigated. Computational results obtained on instances derived from known instances of the pickup and delivery problem with time windows (PDPTW) are reported.

Two solution representations for solving multi-depot vehicle routing problem with multiple pickup and delivery requests via PSO

Two solution representations for solving the generalized multi-depot vehicle routing problem with multiple pickup and delivery requests (GVRP-MDMPDR) is presented in this paper. The representations are used in conjunction with GLNPSO, a variant of PSO with multiple social learning terms. The computational experiments are carried out using benchmark test instances for pickup and delivery problem with time windows (PDPTW) and the generalized vehicle routing problem for multi-depot with multiple pickup and delivery requests (GVRP-MDMPDR). The preliminary results illustrate that the proposed method is capable of providing good solutions for most of the test problems.

Multi-objective Optimization in Dial-a-ride Public Transportation

In the Dial-a-Ride public transportation systems, each customer requirement is specified in terms of a pickup (origin), of a delivery (destination) and of a time window within it has to be satisfied. The aim is to find a set of routes, each assigned to a vehicle, in order to satisfy the set of requests, under capacity, time windows, precedence and pairing conditions. It is usually assumed that the demand of a request, picked up at its origin, is exactly delivered at its destination (one-to-one service) and that the fleet of the vehicles is based at a single depot. From a modelling point of view, the problem could be addressed as a one-to-one capacitated Pickup and Delivery Problem with Time Windows (PDPTW) and therefore, the mathematical formulation presents, beyond the traditional capacity constraints on the vehicles, also the pairing, the precedence and the time windows conditions. In particular, the pairing conditions guarantee that each couple (pickup, delivery) has to belong to the same route while the precedence constraints impose that each pickup has to be served before the associated delivery. This paper addresses the problem with the aim of optimizing, at the same time, the maximum total ride time and the total waiting time. Then, a bi-objective PDPTW with a constraint on the maximum duration of each route is proposed and solved by a two-step approach. In particular, the first step determines a set of feasible routes by meta-heuristics. These routes are used in second step in a bi-objective set partitioning formulation solved by the epsilon-constraint method to generate efficient solutions. The parameters of the meta-heuristics are properly set by a racing procedure. Computational experiments on some benchmark instances are carried out to assess the performance of the proposed approach.

A Flexible Transport Service for Passengers

The realization of innovative passengers transport services requires more and more often a greater flexibility and inexpensiveness of the service. To answer this request in many cases the physical solution is to realize a demand responsive transportation system (DRTS). A DRTS require the planning of travel paths (routing) and customers pick-up and drop-off times (scheduling) according to received requests, respecting the limited capacity of the fleet and time constraints (hard time windows) for each network's node, and the service time of the system. By the modelling point of view a DRTS can be effectively represented with a Dial-a-ride problem (DaRP). A DaRP derives from the Pick-up and Delivery Problem with Time Windows (PDPTW) and may operate according to a static or to a dynamic mode. In the static setting, all customers’ requests are known beforehand and the DaRP returns the vehicles routing and the passengers pick up and drop off time scheduling. The static setting may be representative of a phase of reservation occurred the day before the execution of the service. But, if the reservation requests must be processed on-line, even during the booking process there may be a certain level ad dynamism. In fact, if the algorithm works online, it manages each and every incoming request separately, and accepts or refuses it immediately, without knowing anything about the following. The operative program is constantly updated after each received request without refusal to carry out previous accepted services. In the dynamic mode, customers’ requests arrive when the service is already running and, consequently, the solution may change whilst the vehicle is already travelling. In this mode it is necessary that the schedule is updated when each new request arrives and that this is done in a short time to ensure that the potential customer will not leave the system before a possible answer. In this work, we describe a flexible people transport system capable of managing incoming transport demand in dynamic mode, using a solution architecture based on a two-stage algorithm to solve Dial-a-Ride Problem instances. In the first stage, a constructive heuristic algorithm quickly provides a feasible solution to accept the incoming demand. The algorithm in the second stage try to improve the solution evaluated at the first stage by using the time between two consecutive transportation events. The algorithm, unlike most of the works in the literature, use an objective function that optimizes the service punctuality.

A branch-and-cut-and-price approach for the pickup and delivery problem with shuttle routes

The Pickup and Delivery Problem with Shuttle routes (PDPS) is a special case of the Pickup and Delivery Problem with Time Windows (PDPTW) where the trips between the pickup points and the delivery points can be decomposed into two legs. The first leg visits only pickup points and ends at some delivery point. The second leg is a direct trip – called a shuttle – between two delivery points. This optimization problem has practical applications in the transportation of people between a large set of pickup points and a restricted set of delivery points.This paper proposes three mathematical models for the PDPS and a branch-and-cut-and-price algorithm to solve it. The pricing sub-problem, an Elementary Shortest Path Problem with Resource Constraints (ESPPRC), is solved with a labeling algorithm enhanced with efficient dominance rules. Three families of valid inequalities are used to strengthen the quality of linear relaxations. The method is evaluated on generated and real-world instances containing up to 193 transportation requests. Instances with up to 87 customers are solved to optimality within a computation time of one hour.

A Hybrid Particle Swarm Optimization Algorithm for Multi-Objective Pickup and Delivery Problem with Time Windows

This paper studies the multi-objective pickup and delivery problem with time windows (PDPTW), in which a fleet of homogeneous vehicles with the same capacities located in a depot serve a collection of given transportation requests. Each request is composed of a pickup location, a delivery location and a given load. The PDPTW is to determine a vehicle scheduling strategy with the objectives of minimizing the number of vehicles utilized, the total travel distances and the total waiting times. A mixed integer programming model is built to formulate this multi-objective PDPTW. Then a novel hybrid particle swarm optimization (HPSO) is proposed to solve this problem. This algorithm adds particles neighbor information to diversify the particle swarm and use the variable neighborhood search (VNS) to enhance the convergence speed. Finally, some numerical experiments based on existing benchmark instances are given to show the effectiveness and feasibility of the algorithm.

An Exact Algorithm for the Pickup and Delivery Problem with Time Windows

The pickup and delivery problem with time windows (PDPTW) is a generalization of the vehicle routing problem with time windows. In the PDPTW, a set of identical vehicles located at a central depot must be optimally routed to service a set of transportation requests subject to capacity, time window, pairing, and precedence constraints. In this paper, we present a new exact algorithm for the PDPTW based on a set-partitioning–like integer formulation, and we describe a bounding procedure that finds a near-optimal dual solution of the LP-relaxation of the formulation by combining two dual ascent heuristics and a cut-and-column generation procedure. The final dual solution is used to generate a reduced problem containing only the routes whose reduced costs are smaller than the gap between a known upper bound and the lower bound achieved. If the resulting problem has moderate size, it is solved by an integer programming solver; otherwise, a branch-and-cut-and-price algorithm is used to close the integrality gap. Extensive computational results over the main instances from the literature show the effectiveness of the proposed exact method.