Coordinated vehicle platooning on tree networks: Efficient time discretization and strengthened formulation

Introduction. The article discusses mathematical models of problems of constructing circular routes of vehicles in a multicommodity hierarchical network. As a rule, such networks consist of a decentralized backbone network and networks in the internal service areas of the backbone nodes (internal networks). In multicommodity networks, each node can exchange products (goods, cargo) with other nodes. In contrast to the distribution problems of a homogeneous interchangeable product, in multicommodity problems the flows of products are not interchangeable, the flow of each product must be delivered from a specific primary object to a specific customer. It is assumed that the multi-level structure of the transport network is defined and the geographical location of the main hubs and its internal service areas with a set of nodes for the delivery and collection of goods (customers) are known. Therefore, the problems of determining the main routes of vehicles and constructing circular routes of internal vehicles are considered independently of each other. The types of costs of real transport processes, which should be taken into account in the formation of the objective function of routing problems, are discussed and mathematical models of problems for constructing circular delivery routes with a heterogeneous fleet of vehicles are proposed. The possibility of solving the formulated problems with the help of well-known packages of mixed and integer linear programming is noted. Purpose. The aim of the article is to formulate new mathematical models of the problem of constructing circular routes of vehicles in the internal networks of servicing the main nodes, which take into account the real costs of transport processes and the geographical features of internal networks. The technique. The research methodology is based on the system analysis of many modern models, methods and algorithms for solving the problems of constructing circular routes for customer service in the internal zones of the main nodes of the hierarchical network. Results. On the basis of the review and analysis of known mathematical models, several new variants of mathematical formulation of problems of designing routes of vehicles for the transportation of discrete cargo in the internal zones of the central nodes of the network have been developed. To solve the problems, precise, heuristic and metaheuristic methods and algorithms can be used, implemented in many commercial and non-commercial packages of mixed and integer programming programs, for example, IBM ILOG CPLEX, GAMS, AIMMS, Gurobi Optimizer, ABACUS, COIN-OR, GLPK, lp_solve. Many of them are available for free on the NEOS server (https://neos-server.org/neos/). Scientific novelty and practical significance. The novelty of the work lies in the formulation of mathematical models of the problem of constructing circular routes of vehicles, which take into account the real costs of transport processes and geographical features of internal networks. The materials of the article form the methodological basis for the development of modern mathematical support for solving the problems of long-term, current and operational planning and management in the internal zones of the trunk nodes of the global hierarchical network. Keywords: problems of combinatorial optimization, mathematical models of circular routes of vehicles.

Home Healthcare (HHC) is an emerging and fast-expanding service sector that gives rise to challenging vehicle routing and scheduling problems. Each day, HHC structures must schedule the visits of caregivers to patients requiring specific medical and paramedical services at home. These operations have the potential to be unsuitable if the visits are not planned correctly, leading hence to high logistics costs and/or deteriorated service level. In this article, this issue is modeled as a vehicle routing problem where a set of routes has to be built to visit patients asking for one or more specific service within a given time window and during a fixed service time. Each patient has a preference value associated with each available caregiver. The problem addressed in this paper considers two objectives to optimize simultaneously: minimize the caregivers’ travel costs and maximize the patients’ preferences. In this paper, different methods based on the bi-objective non-dominated sorting algorithm are proposed to solve the vehicle routing problem with time windows, preferences, and timing constraints. Numerical results are presented for instances with up to 73 clients. Metrics such as the distance measure, hyper-volume, and the number of non-dominated solutions in the Pareto front are used to assess the quality of the proposed approaches.

In this paper, we explore three efficient time discretization techniques for the local discontinuous Galerkin (LDG) methods to solve partial differential equations (PDEs) with higher order spatial derivatives. The main difficulty is the stiffness of the LDG spatial discretization operator, which would require a unreasonably small time step for an explicit local time stepping method. We focus our discussion on the semi-implicit spectral deferred correction (SDC) method, and study its stability and accuracy when coupled with the LDG spatial discretization. We also discuss two other time discretization techniques, namely the additive Runge-Kutta (ARK) method and the exponential time differencing (ETD) method, coupled with the LDG spatial discretization. A comparison is made among these three time discretization techniques, to conclude that all three methods are efficient when coupled with the LDG spatial discretization for solving PDEs containing higher order spatial derivatives. In particular, the SDC method has the advantage of easy implementation for arbitrary order of accuracy, and the ARK method has the smallest CPU cost in our implementation.

An efficient and robust time discretization procedure of theta type is proposed in the frame of the finite element‐circuit equation coupling for electronic circuits with switches, i.e. the use of diodes, thyristors and transistors. This procedure enables the use of the Crank‐Nicolson scheme whatever the circuit and its working conditions by eliminating the undesirable oscillations of the solution peculiar to this scheme. It is based on the accurate determination of the switching instants and on a local modification of the theta parameter.

In this paper a model‐based heuristic approach for a typical distribution problem is presented. In order to be cost‐effective, the distribution process for many customers, each of them with orders of considerable volume, should be dealt with like a combination of two well‐known problems: the vehicle routing problem (VRP) and the container loading problem (CLP). This paper studies a particular integration of these two problems called the vehicle routing and loading problem (VRLP). The VRLP is an operational problem that must be solved daily by many production and distribution companies. Like the two original problems (VRP and 3D‐CLP), the VRLP is NP‐hard. In this work, regardless of the complexity of this problem, a mixed integer linear programming (MILP) model that characterizes the VRLP with time windows is presented and it is also used to solve the problem optimally. Then, a model‐based heuristic that improves the computational time, when bigger instances need to be solved, is also presented. In order to prove the viability of the approach and the developed MILP model, tests with the benchmark instances of the VRLP were made and the results compared with other published works. Despite the long computational time needed to solve bigger instances, the VRLP model could be used to compute optimal solutions or at least good lower bounds, in order to have a base of comparison when nonexact methods are applied to the VRLP.

The distribution of goods to a set of geographically dispersed customers is a common problem faced by carrier companies, well-known as the Vehicle Routing Problem (VRP). The VRP consists of finding an optimal set of routes that minimizes total travel times for a given number of vehicles with a fixed capacity. Given the demand of each customer and a depot, the optimal set of routes should adhere to the following conditions: ?? Each customer is visited exactly once by exactly one vehicle. ?? All vehicle routes start and end at the depot. ?? Every route has a total demand not exceeding the vehicle capacity. The travel times between any two potential locations are given as input to the problem. Consequently, the total travel is computed by summing up the travel time over the chosen routes. In reality, carrier companies are faced with a number of other issues not conveyed in the VRP. The research in this thesis introduces a number of realistic variants of the VRP. These variants consider the VRP as a core component and incorporate additional features. By definition the VRP is NP-hard. Throughout the years a vast amount of research was aimed at developing both exact and heuristic solution procedures. Building on this established literature, solution procedures are developed to fit the variants proposed in this thesis. The standard VRP considers that the travel time between any pair of locations is constant throughout the day. However, congestion is present in most road networks. Considering traffic congestion results in time-dependent travel times, where the travel time between two location depends not only on the distance between them but also on the time of day one chooses to traverse this distance. Time-dependent travel times are considered in Chapters 2 and 3 of this thesis. Thus, in these Chapters we incorporate the time dimension into the VRP. The standard VRP does not take into account any customer service aspect. The customers are presumed to be available to receive their goods upon arrival of the vehicles. However, a number of carrier companies quote their expected arrival time to their customers. We introduce the concept of self-imposed time windows (SITW). SITW reflect the fact that the carrier company decides on when to visit the customer and communicates this to the customer. Once a time window is quoted to a customer the carrier company strives to provide service within this time window. SITW differ from time windows in the widely studied VRP with time windows (VRPTW), as the latter are exogenous constraints. In Chapters 4 and 5 SITW are endogenous decisions in stochastic environments. Thus, in addition to the sequencings decisions required by the VRP further timing decisions are needed. This thesis extends the VRP in two major dimensions: time-dependent travel times and self-imposed time windows. In reality carrier companies are faced with various uncertainties. The presented models incorporated some of these uncertainties by addressing three stochastic aspects: (I) In Chapter 3 stochastic service times are considered. (II) In Chapter 4, stochasticity in travel time is modeled to describes variability caused by random events such as car accidents or vehicle break down. (III) Finally, in Chapter 5 the objective was to construct a long term plan for providing consistent service to reoccurring customers. Stochasticity in this thesis is treated in an a priori manner. The plan, consisting of routes and timing decisions where necessary, is determined beforehand and is not modified according to the realization of the random events. Chapter 2 addresses environmental concerns by studying CO2 emissions in a timedependent vehicle routing problem environment. In addition to the decisions required for the assignment and scheduling of customers to vehicles, the vehicle speed limit is considered. The emissions per kilometer as a function of speed, is a function with a unique minimum speed v*. However, we show that limiting vehicle speed to this v* might be sub-optimal, in terms of total emissions. We adapted a Tabu search procedure for the proposed model. Furthermore, upper and lower bounds on the total amount of emissions that may be saved are presented. Quantifying the tradeoff between minimizing travel time as opposed to CO2 emissions is an important contribution. Another important contribution lies in incorporating fuel costs in the optimization. As fuel costs are correlated with CO2 emissions, Chapter 2 shows that even in today’s cost structure limiting vehicle speeds is beneficial. Chapter 3 defines the perturbed time-dependent VRP (P-TDVRP) model which is designed to handle unexpected delays at the various customer locations. A solution method that combines disruptions in a Tabu Search procedure is proposed. In Chapter 3 we identify situations capable of absorbing delays. i.e. where inserting a delay will lead to an increase in travel time that is less than the delay length itself. Based on this, assumptions with respect to the solution structure of P-TDVRP are formulated and validated. Furthermore, most experiments showed that the additional travel time required by the P-TDVRP, when compared to the travel time required by the TDVRP, was justified. In Chapter 4 the notion of self imposed time windows is defined and embedded in the VRP-SITW model. The objective of this problem is to minimize delay costs (caused by late arrivals at customers) as well as traveling time. The problem is optimized under various disruptions in travel times. The basic mechanism of dealing with these disruptions is allocating time buffers throughout the routes. Thus, additional timing decisions are taken. The time buffers attempt to reduce potential damage of disruptions. The solution approach combines a linear programming model with a local search heuristic. In Chapter 4, two main types of experiments were conducted: one compares the VRP with VRP-SITW while the other compares VRPTW with VRPSITW. The first set of experiments assessed the increase in operational costs caused by incorporating SITW in the VRP. The second set of experiments enabled evaluating the savings in operational costs by using flexible time windows, when compared to the VRPTW. Chapter 5 extends the customer service dimension by considering the consistent vehicle routing problem. Consistency is defined by having the same driver visiting the same customers at roughly the same time. As such, two main dimensions of consistency are identified in the literature, driver- and temporal consistency. In Chapter 5, driver consistency is imposed by having the same driver visit the same customers. Furthermore, we impose temporal consistency by SITW. A stochastic programming formulation is presented for the consistent VRP with stochastic customers. An exact solution method is proposed by adapting the 0-1 integer L- shaped algorithm to the problem. The method was able to solve the majority of test instances to optimality.

Coordinated Vehicle Platooning with Fixed Routes: Adaptive Time Discretization, Strengthened Formulations and Approximation Algorithms

For many variants of vehicle routing and scheduling problems solved by a branch-price-and-cut (BPC) algorithm, the pricing subproblem is an elementary shortest-path problem with resource constraints (SPPRC) typically solved by a dynamic-programming labeling algorithm. Solving the SPPRC subproblems consumes most of the total BPC computation time. Critical to the performance of the labeling algorithms and thus the BPC algorithm as a whole is the use of effective dominance rules. Classical dominance rules rely on a pairwise comparison of labels and have been used in many labeling algorithms. In contrast, partial dominance describes situations where several labels together are needed to dominate another label, which can then be safely discarded. In this work, we consider SPPRCs, where a linear tradeoff describes the relationship between two resources. We derive a unified partial dominance rule to be used in ad hoc labeling algorithms for solving such SPPRCs as well as insights into its practical implementation. We introduce partial dominance for two important variants of the vehicle routing problem, namely the electric vehicle routing problem with time windows with a partial recharge policy and the split-delivery vehicle routing problem with time windows (SDVRPTW). Computational experiments show the effectiveness of the approach, in particular for the SDVRPTW, leading to an average reduction of 20% of the total BPC computation time, with savings of 30% for the more difficult instances requiring more than 600 s of computation time.

In this research, author reviews references related to the topic of multi-criterion (goal programming, multiple objective linear and nonlinear programming, bi-criterion programming, Multi-Attribute Decision-Making, Compromise Programming, Surrogate Worth Trade-off Method) and various versions of vehicle routing problem (VRP), Multi-Depot VRP (MDVRP), VRP with time windows (VRPWTW), Stochastic VRP (SVRP), Capacitated VRP (CVRP), Fuzzy VRP (FVRP), Location VRP (LVRP), Backhauling VRP(BHVRP), Facility Location VRP (FLVRP), and Inventory control VRP (ICVRP). Although VRP is a research area with rich research works and powerful researchers, only 81 articles are found that relate various vehicle routing type problems with various multiple objectives techniques. This author found that there is no research done in some areas of VRP (i.e., FVRP, ICVRP, LRP and CVRP). It is interesting to see that this research area was completely unattractive to master students (with zero research reported) and somewhat attractive to doctoral students (with 6 researches reported). Among the many multi-criterion programming techniques available, only three of them (goal programming, bi-criterion programming, linear and nonlinear multi-objective programming) are being employed to solve the problem.

Incompressible two-phase flow: Diffuse interface approach for large density ratios, grid resolution study, and 3D patterned substrate wetting problem

Predicting the unsteady flow behaviour in open channel systems is a complex computational problem. It is generally recognized that the Method of Characteristics (MOC) best represents the physical nature of transient flow phenonmena, but difficulties associated with discretization and interpolation in a fixed grid have made this scheme less popular than other explicit methods. An alternative approach to interpolation involves adjusting the slope of the characteristic lines and integrating an “equivalent partial differential equation” along the adjusted characteristics. This recently developed Modified Characteristics Scheme now makes MOC more computationally competitive with other finite difference techniques. The modified characteristics approach uses a new transformation of the original partial differential dynamic and continuity equations governing unsteady flow into characteristic form based on the total derivative concept. The approach is physically more intuitive and simpler than the purely mathematical lambda or eigenvalue transformation procedures. In applying the technique to transient flow studies, the analyst is allowed to distort the physical problem by setting the Courant number of each computational cell to one, thus exactly satisfying the necessary stability criterion. This distortion is corrected for by the presence of additional partial differential terms that arise from the transformation procedure, which can be integrated using Taylor’s series expansions around nodal values. Depending on whether the transformation commences with the dynamic or continuity relation, the mechanical process of substitution and elimination will reveal whether or not temporal or spatial approximations are required for the additional terms.

In this work, we present a novel Lagrange–Galerkin method for the resolution of scalar hyperbolic conservation laws. The scheme considers: (i) a conservative, weak, Lagrangian formulation which is formally discretized in space and in time with arbitrary order of accuracy, (ii) a forward-in-time integration of the fluid trajectories to allow for more stable and efficient time discretizations, (iii) nodal space-discretizations on unstructured triangular meshes and (iv) a novel and simple operator which employs the values of the fine-scale term of the solution to detect and capture the discontinuities. The method has been tested on several benchmark problems –including a hard case of non-convex flux– using a third-order time-integration formula and up to fourth-order finite elements, yielding the expected convergence rates both for smooth and discontinuous solutions. To the best of our knowledge, this is the first Lagrange–Galerkin method for hyperbolic conservation laws in the literature that allows for discontinuous solutions.

The present study is concerned with impact processes that appear in civil and military security technology, dynamic soil compaction, vehicle crash or fastening and demolition technology. They are characterized by varying non-linearities, as e.g. large deformations and strains, highly non-linear material behavior, frictional contact between multiple bodies and stress wave propagation. A combination of different methods in adaptivity, constitutive modeling, element technology, efficient time discretization and contact are essential for the reliable computation of practical relevant engineering tasks and for predictions in industrial applications. Accuracy, robustness and efficiency are the authoritative requirements for the solution of those complex problems.

Modeling rolling batch planning as vehicle routing problem with time windows

Introduction. The Vehicle Routing Problem (VRP), first formulated by Danzig and Ramseur in 1959, has remained one of the most popular research subjects to date. This popularity stems from numerous factors, including its wide applicability across various economic sectors. VRP belongs to the class of NP-hard problems, implying high computational complexity in finding optimal solutions, especially for large-scale variations. Over the past 25 years, approaches to its classification and solution have evolved significantly, driven by real-world requirements and constraints, as well as advancements in optimization methods and computational power. This article analyzes research findings from studies focused on VRP, confirming a substantial shift in researchers' attention towards metaheuristic approaches. It examines application of the most popular swarm intelligence algorithms and their variations, including hybrids, for solving VRP, and what makes them successful. Furthermore, the study investigates the correlation between sets of algorithm parameters. The purpose of the paper is to investigate usage of swarm intelligence algorithms for solving the Vehicle Routing Problem. Paper attempts to determine what makes them effective for solving VRPs (if such) and how this is related to their parameter set. In addition, the study explores whether there is a correlation between the parameter sets of SI algorithms considered effective for VRPs. Results. An analysis of the results of research articles on VRP was conducted, which made it possible to identify the most popular variations of VRP and rank the methods for solving them. A comprehensive analysis of the most popular SI algorithms, including their variations and hybrids, for solving VRP was conducted. Their strengths and weaknesses were analyzed, and algorithmic features that make them effective in solving VRP were identified. A correlation analysis was conducted between the optimal parameter sets of algorithms and a strong dependence of the optimal parameter sets on the specific variation of VRP being solved was revealed. Conclusions. The analysis of the literature reveals that the Capacitated Vehicle Routing Problem (CVRP) remains the most prevalent VRP variant among researchers. Another popular variant is the Vehicle Routing Problem with Time Windows (VRPTW). Overall, there is an increasing trend in the popularity of VRP variants that incorporate real-world assumptions: Open VRP (OVRP), Dynamic VRP (DVRP), and Time-Dependent VRP (TDVRP). Often, real-life parameters such as cash transportation, small parcel delivery, waste collection, or social legislation regarding drivers' working hours, prompt researchers to develop narrow mathematical models. Unfortunately, these models are typically hard-wired to a specific problem, and some are even specifically adapted to particular test instances. The most popular methods studied in the literature are metaheuristic methods, classical heuristic methods, and exact methods. Various Swarm Intelligence (SI) algorithms were analyzed. Their shared properties of exploration, exploitation, and resistance to local optima make them well-suited for the complex combinatorial nature of VRP. However, the choice of algorithm and its parameters is strongly interrelated with the specific VRP variation, emphasizing the need for integrated approaches to their selection and tuning. Despite significant progress, challenges remain in effectively solving large-scale real-world VRPs. Keywords: Vehicle Routing Problem, swarm intelligence, metaheuristic methods, logistics.