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A two-echelon multi-trip vehicle routing problem with synchronization for an integrated water- and land-based transportation system
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This study considers an integrated water-and land-based transportation (IWLT) system for waste collection. Research on the issue is motivated by increased heavy street movements that damage quay walls as well as congestion. We present a novel two-echelon vehicle routing problem with satellite synchronization based on a two-index formulation and evaluate it on small-sized instances for 10 waste points and 4 hubs. We compare the proposed synchronized IWLT approach with three benchmarks that can reduce issues associated with heavy loads. It is shown that the proposed system can provide better solutions with less collection cost, reduced street movements and lightweight garbage vehicles.
Waste collection has become challenging for most cities due to the increasing population, lack of space, increased on-street traffic, and dangerous environmental and human health impacts. Amsterdam's municipality considers a hybrid over water and land waste collection system currently under testing that completely removes heavy fossil-fuelled garbage trucks from the equation to address these issues as well as the damages on the city's fragile quay walls. In this system, vessels serve as mobile depots for electric-powered cars that collect on-street garbage and need to regularly unload collected waste to capacitated vessels due to their smaller capacities than traditional heavy garbage trucks. Therefore, the on-street fleet operation produces a series of time-windowed unloading tasks, which, in turn, trigger waste transfer requests.These transfer requests constitute the demand for vessels. This study focuses on the operational fleet management of vessels for such an over-water collection system, assuming a solution for the on-street waste collection has already been determined to decide the transshipment locations, time windows, and expected waste loads of the unloading requests to be served by the vessels. Since we consider that vessels have relatively small capacities to facilitate sailing throughout the waterway network, they may have to carry out the last mile to a central waste facility multiple times. We propose a new model-based decision making strategy to the problem by formulating it as a multi-trip vehicle routing problem with time windows, service-dependent unloading times, and pickups (MVRPTW-SD). We present a two-index mathematical formulation that reduces the number of binary variables by at least 50% with respect to the state-of-the-art formulations in the literature. The proposed model can solve classical delivery test instances featuring 25 and 50 nodes, with average gaps of 0.1% and 7.9%, respectively. We introduce the MVRPTW-SD with collections to account for the operational differences of collection services from delivery services, which has not been studied in the literature. Finally, we conduct a sensitivity analysis to give insights into the design of hybrid water-and land-based waste collection systems in order to reduce overall transportation costs and propose further improvement directions. Our results indicate that the MVRPTW-SD with collections is highly dependent on the geographical distribution of the transshipment locations and the tightness of collection time windows besides the capacity of the vessels.
In the two-echelon vehicle routing problem (2E-VRP), the distribution network is split into two echelons. Different vehicles are operated on the first and second echelon to maintain economies of scale and adhere to any vehicle restrictions that may be present in either echelon. Intermediate facilities are located at the borders of the echelons to facilitate the consolidation and transshipment of goods between echelons. Examples of two-echelon distribution systems include express delivery, grocery and hypermarket products distribution, multi-modal freight transportation, city logistics, and e-commerce and home delivery services. In recent years, the body of literature on the 2E-VRP has expanded significantly. Over 60 research papers have appeared in the scientific literature so far, which underlines both the academic and practical relevance of 2E-VRPs. In this review, we structure and revise all literature on the 2E-VRP. Mathematical formulations, exact and heuristic solution methods, and benchmark datasets used to test and evaluate new algorithms are reviewed and discussed. This survey concludes with a selected list of open research areas on 2E-VRPs.
In this paper, we propose an exact branch-price-and-cut (BPC) algorithm for the two-echelon vehicle routing problem with time windows. This problem arises in city logistics when high-capacity and low-capacity vehicles are used to transport items from depots to satellites (first echelon) and from satellites to customers (second echelon), respectively. The aim is to determine a set of least-cost first- and second-echelon routes such that the load on the routes respect the capacity of the vehicles, each second-echelon route is supplied by exactly one first-echelon route, and each customer is visited by exactly one second-echelon route within its time window. We model the problem with a route-based formulation where first-echelon routes are enumerated a priori, and second-echelon routes are generated using column generation. The problem is solved using BPC. To generate second-echelon routes, one pricing problem per satellite is solved using a labeling algorithm which keeps track of the first-echelon route associated with each (partial) second-echelon route considered. Furthermore, to speed up the solution process, we introduce effective deep dual-optimal inequalities and apply known valid inequalities. We perform extensive computational experiments on benchmark instances and show that our method outperforms a state-of-the-art algorithm. We also conduct sensitivity analyses on the different components of our algorithm and derive managerial insights related to the structure of the first-echelon routes.
More and more studies have aimed to optimize the ground vehicle (GV) and unmanned aerial vehicle (UAV) system in which GVs function as mobile satellites and UAVs are dispatched from GVs for last-mile deliveries. From the two-echelon scheme perspective, GV routes originating at the depot are on one echelon, and UAV routes originating at satellites are on the other echelon. A change in a GV route may affect some UAV routes, which indicates the satellite synchronization. In the past decade, the optimization of vehicle routes in two-echelon networks has attracted increasing attention in the operations research community. We classify routing problems of two-echelon networks based on the modeling mechanism connecting the two echelons. Different formulations for describing connection mechanisms of the two-echelon scheme, especially constraints on capacitated satellites, satellite synchronization, vehicle coupling/decoupling at satellites, etc., are briefly introduced. There are several modeling challenges of optimizing delivery routes for a fleet of GV–UAV combinations that include new connection mechanisms between the two echelons. Some important variants, especially those involving mobile satellite synchronization, and GV–UAV flexibly coupling/decoupling, require new mathematical formulations and algorithms catering to more general and practical situations.
Waterway transport is the most economical compared to other modes and it causes less social and environmental impacts. However, cargo transportation in urban centers is a challenge to be studied, as it generates problems such as congestion, accidents, and high delivery costs. This research aims to identify acceptable international practices in urban waterway logistics and, based on these examples, identify, and analyze modal use in Brazilian cities comparatively. Initially, from a literature review, practices were explored in countries where waterways are essential in the transport matrix and used in urban areas. Subsequently, the modal use in Brazilian cities was identified and analyzed according to available information and relationships found among the cases studied. A field survey was carried out in the city of Belém (PA) and a comparative analysis highlighted a great potential of this city for the use of waterways as an aid to urban logistics. Given the importance of goods distribution for cities' survival, the identified implementation possibilities can contribute to their development sustainably and to reduce cargo vehicles' impact on the quality of life.
Urban Freight Transport is a continuously growing market mainly based on the use of vehicles with combustion engines, whose environmental impact has become unsustainable. Because of the technological improvement of electric vehicles and their growing economic feasibility, the introduction of electric fleets for urban freight distribution is now a considerable opportunity. Cities are rapidly adapting, in need of tools to properly guide and manage these changes, as the rise of electric vehicles must be encouraged by an appropriate infrastructural system, from charging stations to dedicated areas. What is proposed in this work is an aggregate approach to the freight system, transport demand and supply, to support the design of a distribution system based on electric vehicles by means of an accessibility indicator that takes into account the supply of facilities, vehicle performances, and freight demand patterns. A study case regarding the Metropolitan City of Rome is also presented to interpret and understand the potentialities of this approach.
This paper studies the multi-commodity two-echelon capacitated vehicle routing problem with time windows. Specifically, we take customer-specific origin to destination, non-substitutable demands into account. The main components are: (i) first echelon tours, (ii) second echelon tours and (iii) movement of commodities from first echelon to second echelon, considering route connection and synchronization. Exploiting the structure of the problem, we propose a decomposition scheme which decouples the first and second echelon routing problems, by deleting the third component, and re-couples them by using constraints ensuring movement of commodities between the two echelons. Based on this decomposition, we present (i) a number of model formulations for the MC-2E-VRPTW, and (ii) develop an exact solution approach for this real-life problem. A computational study on a comprehensive set of instances shows the effectiveness in terms of computational effort and solution quality.
In this study, we investigate a routing problem in urban transportation which considers time-dependent travel time, multiple trips per vehicle, and loading time at the depot simultaneously. Its objective is to minimize the total travel distance while satisfying the time windows, vehicle capacity, and maximum trip duration constraints. We model the problem as a multi-trip time-dependent vehicle routing problem with time windows (MT-TDVRPTW). We formulate the time-dependent ready time function and duration function for any segment of consecutive nodes as piecewise linear functions and develop an iterative algorithm to derive them efficiently. Then, these two functions are embedded in the segment-based evaluation scheme to accelerate the local search operators. Based on them, we design a hybrid meta-heuristic algorithm to solve the problem, leveraging the adaptive large neighborhood search (ALNS) for guided exploration and the variable neighborhood descend (VND) for intensive exploitation. Moreover, we propose problem-specific local search operators and removal operators to enhance the effectiveness of the algorithm. Extensive experiments are conducted to assess the performance of the algorithm on instances of varied sizes. The algorithm is shown to be robust and efficient under different speed profiles and maximum trip duration limits. Finally, we evaluate the performance of the algorithm on a special case: the multi-trip vehicle routing problem with time windows.
We introduce a two-echelon urban delivery problem with second-level unmanned vehicles. This problem typically applies for delivering parcels or other small commodities to pedestrianized areas such as campuses or residential clusters. To model the proposed vehicle routing problem, we introduce a mixed-integer program. We further propose construction heuristics and a hybrid metaheuristic approach with backtracking for solving larger instances. A sensitivity analysis for vehicle speed combinations reveals that increasing small autonomous vehicle (SAV) speeds has only very limited effects on cost. We therefore recommend to keep SAV speeds rather low because of a more pedestrian friendly environment in practical implementations.
The demand for the transport of goods within the city is rising and with that the number of vans driving around. This has adverse effects on air quality, noise, safety and liveability in the city. LEFVs (Light Electric Freight Vehicles) offer a potential solution for this. There is already a lot of enthusiasm for the LEFVs and several companies have started offering these vehicles. Still many companies are hesitating to start and experiment. New knowledge is needed of logistics concepts for the application of LEFVs. This paper shows the outcomes of eight case studies about what is needed to successfully deploy LEFVs for city logistics.
In the paper, we propose a branch-cut-and-price algorithm for the two-echelon capacitated vehicle routing problem in which delivery of products from a depot to customers is performed using intermediate depots called satellites. Our algorithm incorporates significant improvements recently proposed in the literature for the standard capacitated vehicle routing problem such as bucket graph based labeling algorithm for the pricing problem, automatic stabilization, limited memory rank-1 cuts, and strong branching. In addition, we make some specific problem contributions. First, we introduce a new route based formulation for the problem which does not use variables to determine product flows in satellites. Second, we introduce a new branching strategy which significantly decreases the size of the branch-and-bound tree. Third, we introduce a new family of satellite supply inequalities, and we empirically show that it improves the quality of the dual bound at the root node of the branch-and-bound tree. Finally, extensive numerical experiments reveal that our algorithm can solve to optimality all literature instances with up to 200 customers and 10 satellites for the first time and thus double the size of instances which could be solved to optimality.
We present a multi-ob’jective two-echelon vehicle routing problem with vehicle synchronization and ‘grey zone’ customers arising in the context of urban freight deliveries. Inner-city center deliveries are performed by small vehicles due to access restrictions, while deliveries outside this area are carried out by conventional vehicles for economic reasons. Goods are transferred from the first to the second echelon by synchronized meetings between vehicles of the respective echelons. We investigate the assignment of customers to vehicles, i.e., to the first or second echelon, within a so-called ‘grey zone’ on the border of the inner city and the area around it. While doing this, the economic objective as well as negative external effects of transport, such as emissions and disturbance (negative impact on citizens due to noise and congestion), are taken into account to include objectives of companies as well as of citizens and municipal authorities. Our metaheuristic – a large neighborhood search embedded in a heuristic rectangle/cuboid splitting – addresses this problem efficiently. We investigate the impact of the free assignment of part of the customers (‘grey zone’) to echelons and of three different city layouts on the solution. Computational results show that the impact of a ‘grey zone’ and thus the assignment of these customers to echelons depend significantly on the layout of a city. Potentially pareto-optimal solutions for two and three objectives are illustrated to efficiently support decision makers in sustainable city logistics planning processes.
Urban population growth has permanently increased the commodity demands and freight flow within urban areas. The retail/postal industry is intent on finding appropriate internal approaches and a new business model to respond to the adverse impacts generated by urban freight activities. Usage of emerging transport modes is an efficient solution for these industries. Nevertheless, considerable research has paid less attention to the implementation status of distribution innovations, as well as to their suitability and application restrictions. Concurrently, a comprehensive consideration of various distribution innovations that operate together as a system is lacking. To this end, this paper adopted a literature review method and GE multifactorial analysis. Specifically, this paper reviewed the related articles that were published in the past six years (2013-2018) to define the concept of distribution innovations. In addition, we adopted the approach of GE multifactorial analysis to analyze the application status of distribution innovations from the perspective of academic research and company implementation. Following the suitability assessment and application of restriction analysis, we proposed the concept of sustainable inner-urban intermodal transport (SIUIT) for the retail/postal industry. This paper contributes to the sustainable urban freight literature by exploring possible future research directions of SIUIT.
We investigate the maximum induced matching problem (MIM), which is the problem of finding an induced matching having the largest cardinality on an undirected graph. The problem is known to be NP-hard for general graphs. We first propose a vertex-based integer programming formulation for MIM, which is more compact compared to an edge-based formulation found in the literature. We also introduce the maximum weight induced matching problem (MWIM), which generalizes MIM so that vertices and edges have weights. We adapt the edge-based formulation to MWIM, and propose a quadratic programming formulation of MWIM based on our vertex-based formulation. We then linearize our quadratic programming formulation, and devise a Benders decomposition algorithm that exploits a special structure of the linearized formulation. We also propose valid inequalities and formulation tightening procedures to improve the efficiency of our approach. Our computational tests on a large suite of randomly generated graphs show that our vertex-based formulation and decomposition approach significantly improve the solvability of MIM and MWIM, especially on dense graphs.
The online appendix and data are available at https://doi.org/10.1287/ijoc.2017.0764 .
The hospital location and service allocation is one of the most important aspects of healthcare systems. Due to lack of studies on covering location-allocation and scheduling problems with respect to the uncertain budget, this paper develops a bi-objective hybrid model to locate hospitals and allocate machines and services scheduled. The costs of establishing facilities are assumed to be uncertain, while a robust counterpart model is employed to overcome the uncertainty. Covering the demand of each service is limited as well. Moreover, hospitals have a limited space to the specialized equipment like CT scan and MRI machines, while there is a cost constraint on hospitals and the specialized equipment. The aim of this paper is to find a near-optimal solution including the number of hospitals and the specialized equipment, the location of hospitals, the assignment of demand of each service and the specialized equipment to hospitals, the determination of allowable number of each service of hospitals, the determination of demand that should be transferred from one hospital to another (patient transfer), and schedule services. As the proposed model, minimizing the total costs and the completion time of demand simultaneously, is an NP-hard problem, it is impossible to solve its large-scale version with exact methods in a reasonable time. Thus, a hybrid algorithm including simulated annealing optimization and the Benders decomposition is employed to solve it. The CPLEX optimizer verifies the presented algorithm to solve the proposed model. The sensitivity analysis is performed to validate the proposed robust model against of uncertain situations while the Monte Carlo simulation is used to analyze the quality and the robustness of solutions under uncertain situations. The results show that the uncertainty used in the proposed model properly formulates real-world situations compared to the deterministic case. Finally, the contributions and the future research are presented.
The Benders decomposition algorithm has been successfully applied to a wide range of difficult optimization problems. This paper presents a state-of-the-art survey of this algorithm, with an emphasis on its use in combinatorial optimization. We discuss the classical algorithm, the impact of the problem formulation on its convergence, and the relationship to other decomposition methods. We introduce a taxonomy of algorithmic enhancements and acceleration strategies based on the main components of the algorithm. The taxonomy provides the framework to synthesize the literature, and to identify shortcomings, trends and potential research directions. We also discuss the use of the Benders decomposition to develop efficient (meta-)heuristics, describe the limitations of the classical algorithm, and present extensions enabling its application to a broader range of problems.
Location-routing, inventory-routing, multi-echelon routing, routing problems with loading constraints are classes of problems that are receiving increasing attention in the scientific community. Problems in these classes generalize classical vehicle routing problems enlarging the decision space to optimize a broader system. The resulting problems are computationally harder to solve but offer opportunities to achieve remarkable additional savings. In this paper, we address the issue of quantifying the potential benefit deriving from tackling directly such complex problems instead of solving the individual problems in a not integrated manner. To this aim, we consider as a proof of concept the Capacitated Vehicle Routing Problem (CVRP) with Two-dimensional Loading constraints (2L-CVRP), a variant of the CVRP where rectangular-shaped items have to be delivered to customers and loading constraints have to be satisfied. We consider the 2L-CVRP in an integrated manner and compare the solutions with those obtained from three not integrated approaches based on addressing separately the routing and the loading problems. The importance of an integrated approach for the 2L-CVRP is validated through the study of the worst-case performance of the not integrated approaches, and conducting computational experiments on benchmark and new instances.
One of the significant side-effects of growing urbanization is the constantly increasing amount of freight transportation in cities. This is mainly performed by conventional vans and trucks and causes a variety of problems such as road congestion, noise nuisance and pollution. Yet delivering goods to residents is a necessity. Sustainable concepts of city distribution networks are one way of mitigating difficulties of freight services. In this paper we develop a two-echelon city distribution scheme with temporal and spatial synchronization between cargo bikes and vans. The resulting heuristic is based on a greedy randomized adaptive search procedure with path relinking. In our computational experiments we use artificial data as well as real-world data of the city of Vienna. Furthermore we compare three distribution policies. The results show the costs caused by temporal synchronization and can give companies decision-support in planning a sustainable city distribution concept.
Benders’ Decomposition (BD) is a prominent technique for tackling large mixed integer programming problems having a certain structure by iteratively solving a series of smaller master and subproblem instances. We apply a generalization of this technique called Logic-Based BD, which does not restrict the subproblems to have continuous variables only, to a bi-level vehicle routing problem originating in the timely distribution of printed newspapers to subscribers. When solving all master and subproblem instances exactly by CPLEX, it turns out that the scalability of the approach is quite limited. The situation can be dramatically improved when using a meaningful metaheuristic – in our case a variable neighborhood search – for approximately solving either only the subproblems or both, the master as well as the subproblem instances. More generally, it is shown that Logic-Based BD can be a highly promising framework also for hybrid metaheuristics.
This paper surveys the vehicle routing problems met in cities for good distribution. It applies the following methodology. First, it gives an overview of the literature devoted to vehicle route optimization in cities. Then, it classifies and analyses urban logistic flows. As a result, it identifies the principal scientific challenges that need to be addressed: time-dependency, multi-level and multi-trip organization of the distribution, dynamic information. Finally, it focuses on each one of these challenges, analyses the main difficulties they imply and how they are treated in the literature.
Urban freight transport is increasingly difficult to organize and manage: the demographic expansion of cities, the environmental issues, new logistical trends, scarcity of space and the evolving consumer habits are just some of the challenges. In most cities, road transportation is essentially the only mode used for performing urban distribution of goods. Inland navigation is used solely for transport of bulk materials over long distances bulk material (e.g. construction material and agricultural products). However, experiences in several European cities have demonstrated that the use of inland waterways can be a viable alternative for smaller-scale freight transport in urban areas. This present article aims at reviewing current experiences in the field of inland navigation for city logistics. Through the analysis of several implementation cases in Europe, we demonstrate that inland navigation can be used for various segments of the urban freight transport, including transportation of palletized goods (e.g. construction and industrial material), transportation of containerized goods (e.g. for the supply of the supermarkets), deliveries to local shops and restaurants, deliveries of parcels, transport of waste and recycled material and service trips. Furthermore, the article analyses the conditions in which these transport schemes were developed, highlights their key success factors, and investigates the particular role of local public authorities in organizing and managing these innovative transport schemes.
J. F. Benders devised a clever approach for exploiting the structure of mathematical programming problems withcomplicating variables (variables which, when temporarily fixed, render the remaining optimization problem considerably more tractable). For the class of problems specifically considered by Benders, fixing the values of the complicating variables reduces the given problem to an ordinary linear program, parameterized, of course, by the value of the complicating variables vector. The algorithm he proposed for finding the optimal value of this vector employs a cutting-plane approach for building up adequate representations of (i) the extremal value of the linear program as a function of the parameterizing vector and (ii) the set of values of the parameterizing vector for which the linear program is feasible. Linear programming duality theory was employed to derive the natural families ofcuts characterizing these representations, and the parameterized linear program itself is used to generate what are usuallydeepest cuts for building up the representations.In this paper, Benders'' approach is generalized to a broader class of programs in which the parametrized subproblem need no longer be a linear program. Nonlinear convex duality theory is employed to derive the natural families of cuts corresponding to those in Benders'' case. The conditions under which such a generalization is possible and appropriate are examined in detail. An illustrative specialization is made to the variable factor programming problem introduced by R. Wilson, where it offers an especially attractive approach. Preliminary computational experience is given.
Benders decomposition uses a strategy of ``learning from one's mistakes.'' The aim of this paper is to extend this strategy
to a much larger class of problems. The key is to generalize the linear programming dual used in the classical method to an
``inference dual.'' Solution of the inference dual takes the form of a logical deduction that yields Benders cuts. The dual
is therefore very different from other generalized duals that have been proposed. The approach is illustrated by working out
the details for propositional satisfiability and 0-1 programming problems. Computational tests are carried out for the latter,
but the most promising contribution of logic-based Benders may be to provide a framework for combining optimization and constraint
programming methods.
In this paper, we address the Two-Echelon Vehicle Routing Problem (2E-VRP) in which freight is delivered from depots to intermediary satellites, from where it is delivered to customers. Different from other multi-level problems, the focus in 2E-VRP is on the global routing costs of the overall network. 2E-VRP has been recently introduced and, to our knowledge, there are no studies of the impact of instance parameters on the global cost. We present the results of a broad experimental study aimed at analyzing the impact on the total distribution cost of several parameters including customer distribution, satellites-location rules, depot location, number of satellites, mean accessibility of the satellites, and mean transportation cost between the satellites and the customers.
City Logistics aims to reduce the nuisances associated to freight transportation in urban areas while supporting the economic and social development of the cities. The fundamental idea is to view individual stakeholders and decisions as components of an integrated logistics system. This implies the coordination of shippers, carriers, and movements as well as the consolidation of loads of several customers and carriers into the same environment-friendly vehicles. City Logistics explicitly aims to optimize such advanced urban transportation systems. We focus on a challenging City Logistics planning issue, the integrated short-term scheduling of operations and management of resources, for the general case involving a two-tier distribution structure. We investigate the main issues related to the problem, introduce a new problem class, propose both a general model and formulations for the main system components, and identify promising solution avenues.
It is well-known that the Lagrangian dual of an Integer Linear Program
(ILP) provides the same
bound as a continuous relaxation involving the convex hull of all
the optimal solutions of the Lagrangian
relaxation. It is less often realized that this equivalence is effective,
in that basically all known algorithms for
solving the Lagrangian dual either naturally compute an (approximate)
optimal solution of the convexified
relaxation, or can be modified to do so. After recalling these results
we elaborate on the importance of
the availability of primal information produced by the Lagrangian
dual within both exact and approximate
approaches to the original (ILP), using three optimization problems
with different structure to illustrate some
of the main points.
We present Branch-and-Check, a hybrid framework integrating Mixed Integer Programming and Constraint Logic Programming, which encapsulates the traditional Benders Decomposition and Branch-and-Bound as special cases. In particular we describe its relation to Benders and the use of nogoods and linear relaxations. We give two examples of how problems can be modelled and solved using Branch-and-Check and present computational results demonstrating more than order-of-magnitude speedup compared to previous approaches. We also mention important future research issues such as hierarchical, dynamic and adjustable linear relaxations.
The logistics companies executing the last mile delivery of goods in urban areas deal every day with the problem of routing their vehicles, while taking into account multiple trips per vehicle, time-dependent travel time, customers’ time windows and loading time at the depot simultaneously. This paper addresses this problem, known as Multi-Trip Time-Dependent Vehicle Routing Problem with Time Windows, aiming at its exact solution, minimizing the total travelled distance of a company's fleet. Based on a literature model, a new reformulation with reduced size is suggested. This reformulation is decomposed by applying the Benders method in an effective way, resulting in a subproblem with no duality gap. By exploiting the special features of the problem and the particular structure of the decomposition made, several novel valid inequalities are introduced, in order to both tighten the non-decomposed formulations and warm start the relaxed master problem to achieve less infeasible solutions and higher lower bounds. For the solution of the problem, an innovative algorithm is proposed, including suboptimal master solutions and a multi-cut generation procedure, which is based on the careful observation of the values of the Benders dual subproblem variables. The impact of the valid inequalities as well as two variants of the suggested algorithm are tested on benchmark data and they are compared with the non-decomposed models and a heuristic introduced in the literature. The computational results indicate improved efficiency and stronger bounds for the proposed algorithm.
Last-mile collection and delivery services often rely on multi-echelon logistic systems with many types of practical spatial, temporal, and resource constraints. We consider three extensions of the basic 2-echelon vehicle routing problem that are of practical interest: First, second-echelon vehicles need to simultaneously deliver and collect goods at customers within their specified time window. Second, first-echelon vehicles are allowed to perform multiple trips during the planning horizon. Third, the intermediate facilities, called satellites, allow temporary storage of goods, but the quantity that can be stored at a time is limited. This paper integrates these complicating features in a single mathematical model. To solve the problem, we design a decomposition-based matheuristic. It employs a reduced and refined mixed-integer programming formulation and two echelon-specific large neighborhood searches (LNS) to produce improving routes for the respective echelon. The most important algorithmic component is the feasibility check of LNS that relies on a sequence of constant-time and low-complexity tests. The final test allows re-scheduling of the operations taking place at a satellite. Extensive computational experiments systematically evaluate the new components of the matheuristic and benchmark it against a recent exact method for a related problem. Moreover, the impact of the main problem features such as the number and capacity of satellites as well as the integration of forward and reverse flows is analyzed.
This paper considers the problem of delivering a certain number of packages to a set of customers using multiple drones dispatched from a single truck. Once a drone delivers a package, it returns to the truck to pick up another package while the truck waits at the launch location. The aim is to determine the truck route and the sequence of drone trips from the truck such that all the customer demands are served, either by the truck or drones, to minimize the sum of customer waiting times. We propose a new formulation of this problem based on an extended graph network representation. A new logic-based Benders decomposition method enhanced with relaxations is also developed and evaluated on several benchmark instances. The numerical results show that the proposed exact method obtains optimal solutions to the majority of the problem instances within the time limit of one hour.
We propose a general logic-based Benders decomposition (LBBD) for production planning problems with process configuration decisions. This family of problems appears in contexts where the machines are set up according to specific patterns, templates, or, in general, process configurations that allow for simultaneously producing products of different types. The problem requires determining feasible configurations for the machines and their corresponding production levels to fulfill the demand at the minimum total cost. The structure of this problem contains nonlinear constraints that link the number of units produced of each product with the used configurations and their production levels. We decompose the original problem into a master problem, where the configurations are determined, and a subproblem, where the production amounts are determined. This allows us to apply the LBBD technique to solve the problem using a standard LBBD implementation and a branch-and-check algorithm. LBBD enhancements through logic-based inequalities generated for subsets of products with common characteristics are proposed. Such inequalities represent a form of the subproblem relaxation added to the master problem during its resolution. In our computational experiments, we apply the proposed LBBD approaches to two different applications from the literature: cutting stock problems in the steel industry and a printing problem. Results show that the LBBD methods find optimal solutions much faster than the solution approaches in the literature and have a superior performance with respect to the number of instances solved to optimality and the solution quality.
Summary of Contribution: In this work, we introduce a unified exact solution algorithm based on logic-based Benders decomposition to solve a class of integrated production planning problems that include process configuration decisions. We propose a general mathematical representation of the original integrated planning problem and logic-based Benders reformulations that can be applied to solve several problems within the studied class. Our implementation frameworks provide guidelines to practitioners in the field. The solution approaches in this paper together with the proposed methodological enhancements can be adapted to solve other integrated planning problems in a similar context, including the case when the original problem has a complex combinatorial and nonlinear structure.
The European Commission has dedicated itself steadily to multimodal freight transport to reduce problems such as air pollution or congestion and to reach the goals set by the Paris Agreement. Despite the political efforts to promote multimodality, the majority of freight transport is still carried out by truck. The aim of this paper is to capture the reasons for the small share of multimodal transport and suggest measures to promote multimodal transport. To collect data, a multiple-case study was conducted involving ten logistics service providers. The barriers to multimodal freight transport are analyzed in a holistic manner using interpretive structural modeling. Overall, fifteen barriers are presented which are classified as demand-related barriers, shipment characteristics, infrastructural/supply-related barriers, organizational barriers and legal / political barriers. Based on that, a bottom-up approach involving the ten logistics service providers (LSPs) is used to develop user-centered policy measures for multimodal transport. The direct involvement of LSPs facilitates acceptance of the proposed measures. The internalization of external costs, efficient information provision and education/training/awareness raising are rated as high impact measures to promote multimodal transport.
Motivated by the practice of urban waste collection, we introduce a new variant of the multi-trip vehicle routing problem with time windows, where vehicles unload cargos collected from customers at a depot that owns a limited unloading capacity, and therefore some vehicles need to wait at a queue once the unloading capacity is fully occupied. Unloading queue at depot significantly compicates this problem since it makes a trip involve traveling, waiting, and unloading stages. We first formulate this problem into a trip-based set partitioning model, which has two sets of mutual exclusion constraints. To solve the model, we propose a branch-and-price-and-cut algorithm (BPC), in which the linear relaxation of the model is solved by column generation, and moreover, the relaxation gap is tightened by incorporating rounded capacity inequalities. Particularly, the mutual exclusion constraints induce a new and challening pricing problem, and we design a novel label-setting algorithm with tailored label structure to solve it. We conduct compuational experiments on a set of random instances generated from the Solomon’s benchmark, and the results demonstrate the effectiveness of the proposed model and algorithm.
Integrated operating room planning and scheduling (IORPS) allocates patients optimally to different days in a planning horizon, assigns the allocated set of patients to ORs, and sequences/schedules these patients within the list of ORs and surgeons to maximize the total scheduled surgical time. The state-of-the-art model in the IORPS literature is a hybrid constraint programming (CP) and integer programming (IP) technique that is efficiently solved by a multi-featured Branch-Price-&Cut (BP&C) algorithm. We extend the IORPS literature in two ways: (i) we develop new mixed-integer programming (MIP) and CP models that improve the existing CP-IP model and (ii) we develop various combinatorial Benders decomposition algorithms that outperform the existing BP&C algorithm. Using the same dataset as used for the existing methods, we show that our MIP model achieves an average optimality gap of 3.84%, outperforming the existing CP-IP model that achieves an average optimality gap of 11.84%. Furthermore, our MIP model is 54 - 92 times faster than the CP-IP model in some of the optimally solved instances of the problem. We demonstrate that our best Benders decomposition approach achieves an average optimality gap of 0.58%, whereas the existing BP&C algorithm achieves an average optimality gap of 2.81%.
As one of the most necessary infrastructures for two-echelon distribution with cross-docking systems, satellites enable transshipment and consolidation for cargo deliveries. Considering specially satellites’ real-time transshipment capacity (RTC) varying with transshipment and consolidation operations, we introduce the two-echelon distribution system considering the real-time transshipment capacity varying (called the 2E-DS-RTC). The 2E-DS-RTC adopts RTC constraints and time constraints to make routings of the two echelons interacting. Of each satellite, the RTC is constrained by the maximal transshipment capacity (MTC) and the occupied transshipment capacity. A mixed integer linear programming model for the 2E-DS-RTC is proposed. The savings-based algorithm followed by the variable neighborhood search phase is provided. The mathematical formulation and the two-stage heuristic are tested by using 20 randomly-generated small-scale instances and 99 realistic instances with up to 30 satellites and 900 customers. Some small-scale instances can be solved directly by CPLEX to find exact solutions. The computational results of realistic instances indicate that the heuristic can solve various scale instances of the 2E-DS-RTC such that the solution quality and the computation time are acceptable.
In considering route optimization at a series of express stages from pickup to delivery via the intercity linehaul, we introduce the two-echelon vehicle routing problem with satellite bi-synchronization (2E-VRP-SBS) from the perspective of modeling the routing problems of two-echelon networks. The 2E-VRP-SBS involves the inter-satellite linehaul on the first echelon, and the pickups from senders to origin satellites (i.e., satellites for cargo collection) and deliveries from destination satellites (i.e., satellites for cargo deliveries) to receivers on the second echelon. The 2E-VRP-SBS integrates satellite bi-synchronization constraints, multiple vehicles, and time window constraints on the two-echelon network and aims to find cost-minimizing routes for various types of trucks. Satellite bi-synchronization constraints, which synchronously guarantee the synchronization at origin satellites and the synchronization at destination satellites, provide an innovative method to formulate the two-echelon routing problem. In this study, we develop a mixed-integer programming model for the 2E-VRP-SBS. An exact method using CPLEX solver is presented and a modified adaptive large neighborhood search is conducted. Furthermore, the effectiveness of the 2E-VRP-SBS formulation and the applicability of the heuristic for various instances are experimentally evaluated.
The need to reduce pollution and traffic in city centers requires the use of small vans, electric vehicles, and drones to distribute goods. Because of autonomy and capacity issues, these vehicles need to perform multiple trips from/to the depot during the day. The category of decision-making problems modeling such distribution problems are known as multitrip vehicle-routing problems (MTVRPs), which generalize the well-known vehicle-routing problem by allowing vehicles to perform multiple trips per day. Several MTVRPs are solved in the literature with different mathematical models and algorithms. In “An Exact Solution Framework for Multitrip Vehicle-Routing Problems with Time Windows,” R. Paradiso, R. Roberti, D. Laganà, and W. Dullaert propose a single algorithm that can solve, to optimality, the MTVRP with capacity and time windows constraints and four variants of this problem featuring additional operational constraints. The proposed framework significantly outperforms the state-of-the-art algorithms from the literature.
The harvesting and transportation system involves a harvest scheduling and a transportation plan. The grain, harvested by combine-harvesters, is then transported by transporters from disperse farmlands to the depot. The spot where combine-harvesters transfer wheat to transporters is dynamic because the location of these spots correspond with combine-harvesters’ work. In this paper, the harvesting and transportation problem is considered as a two-echelon multi-trip vehicle routing problem with a dynamic satellite (2E-MTVRPDS) because the combine-harvester is used multiple times in the planning horizon and the transporter is used multiple times in a work day. The mixed integer linear programming model is proposed based on the features of the problem. This work presents an optimum solution with a heuristic algorithm. The dynamic satellite is transferred as the static case in the heuristic. The computational experiments are constructed to test the performances of the proposed algorithm. Five instances with different sizes are adopted to test the stability of the algorithm. The calculation deviation of testing instances is acceptable. On one hand, the optimal effectiveness can be achieved when the number of instances is less than 200. With the increase in the number of instances, the optimal efficiency declines. On the other hand, the optimal solution appears to have a time window of 0.2 h in all instances with different sizes. This study provides a decision model for agricultural production to implement optimal harvesting operations.
This paper studies the two-echelon capacitated vehicle routing problem with time windows. The first echelon consists of transferring freight from depots to intermediate facilities (i.e., satellites), whereas the second echelon consists of transferring freight from these facilities to the final customers, within their time windows. We propose two path-based mathematical formulations for our problem: (1) in one formulation, paths are defined over both first- and second-echelon tours, and (2) in the other one, the first- and second-echelon paths are decomposed. Branch-and-price–based algorithms are developed for both formulations. We compare both formulations and solution methods on a comprehensive set of instances and are able to solve instances up to five satellites and 100 customers to optimality. This paper is the first paper in the literature that solves such large instance sizes.
The online appendix is available at https://doi.org/10.1287/trsc.2018.0844 .
Two-echelon distribution systems are attractive from an economical standpoint and help keeping large vehicles out of city centers. Large trucks can be used to deliver goods to intermediate facilities in accessible locations, whereas smaller vehicles allow to reach the final customers. Due to their reduced size and emissions, companies have adopted an electric fleet for last-mile deliveries. Route planning in multi-tier logistics leads to notoriously difficult problems. This difficulty is accrued in the presence of an electric fleet, since each vehicle must operate on a smaller range, and may require visits to charging stations. To study these challenges, we introduce the Electric Two-echelon Vehicle Routing Problem (E2EVRP) as a prototypical problem. We propose a large neighbourhood search metaheuristic as well as an exact mathematical programming algorithm, which uses decomposition techniques to enumerate promising first-level solutions, in conjunction with bounding functions and route enumeration for the second-level routes. These sophisticated algorithms produce optimal or near-optimal solutions for the problem, and allow us to carefully evaluate the impact of several defining features of optimized battery-powered city distribution networks. In particular, through a simple simulation of a metropolitan area via representative of E2EVRPs benchmark instances, we observe that the detour miles due to recharging decrease proportionally to with as a function of the charging stations density ; e.g., in a scenario where the density of charging stations is doubled, recharging detours are reduced by 58\%. The range of the electric vehicles has an even bigger impact, as in our experiments, an increase of battery capacity to a range of 150km helps performing the majority of suburban delivery tours without need for en-route recharging.
The two-echelon vehicle routing problem (2E-VRP) consists in making deliveries to a set of customers using two distinct fleets of vehicles. First-level vehicles pick up requests at a distribution center and bring them to intermediate sites. At these locations, the requests are transferred to second-level vehicles, which deliver them. This paper addresses a variant of the 2E-VRP that integrates constraints arising in city logistics such as time window constraints, synchronization constraints, and multiple trips at the second level. The corresponding problem is called the two-echelon multiple-trip vehicle routing problem with satellite synchronization (2E-MTVRP-SS). We propose an adaptive large neighborhood search to solve this problem. Custom destruction and repair heuristics and an efficient feasibility check for moves have been designed and evaluated on modified benchmarks for the VRP with time windows.
In this paper, we propose an Integer Programming formulation and two branch-and-price implementations for the Two-Echelon Capacitated Vehicle Routing Problem. One algorithm considers routes that satisfy the elementarity condition, while the other relaxes such constraint when pricing routes. For instances that could not be solved to proven optimality within a given time limit, our computational experience suggests that the former provides sharper upper bounds while the latter offers tighter lower bounds. As a by-product, ten new best upper bounds and two new optimality certificates are provided here.
Collaborative hub networks can provide an answer to the need to decrease logistics cost and maintain logistics service levels by shifting consolidated flows to modes that are better suited for handling large volumes (rail, barge, coastal shipping), so economies of scale can be obtained. This necessity has been increased by the tendency of globalization of industries, smaller shipments sizes, high frequencies, and the fragmentation of flows. Through collaboration the necessary synchronization between expensive but fast and flexible means of transport and inexpensive, but slow and inflexible means can be combined in an intermodal hub network. This paper shows the rationale behind these collaborative hub networks, based on the literature on the design of many-to-many hub networks. The resulting methodology is explained through presenting the results of the design and implementation of collaborative hub network for the distribution of fast moving consumer goods using a combination of trucking and inland barges. This concept, first proposed by Vermunt [Vermunt, A.J.M., 1999. Multilognet, the intelligent multimodal logistics network, an important node in the worldwide logistics net, Vermunt Logistiek Advies v.o.f., working paper (in Dutch)], won the European Intermodal Award of the European Intermodal Association in 2003, and after extensive research was launched in The Netherlands as a commercial pilot by logistics service provider Vos Logistics and barge operator Riverhopper in January 2004.
This paper considers the design and analysis of algorithms for vehicle routing and scheduling problems with time window constraints. Given the intrinsic difficulty of this problem class, approximation methods seem to offer the most promise for practical size problems. After describing a variety of heuristics, we conduct an extensive computational study of their performance. The problem set includes routing and scheduling environments that differ in terms of the type of data used to generate the problems, the percentage of customers with time windows, their tightness and positioning, and the scheduling horizon. We found that several heuristics performed well in different problem environments; in particular an insertion-type heuristic consistently gave very good results.
(This article originally appeared in Management Science, January 1981, Volume 27, Number 1, pp. 1–18, published by The Institute of Management Sciences.)
One of the most computationally useful ideas of the 1970s is the observation that many hard integer programming problems can be viewed as easy problems complicated by a relatively small set of side constraints. Dualizing the side constraints produces a Lagrangian problem that is easy to solve and whose optimal value is a lower bound (for minimization problems) on the optimal value of the original problem. The Lagrangian problem can thus be used in place of a linear programming relaxation to provide bounds in a branch and bound algorithm. This approach has led to dramatically improved algorithms for a number of important problems in the areas of routing, location, scheduling, assignment and set covering. This paper is a review of Lagrangian relaxation based on what has been learned in the last decade.
A branch-and-cut algorithm for the symmetric two-echelon capacitated vehicle routing problem
- Jepsen