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Two-echelon distribution systems are often considered in city logistics to maintain economies of scale and satisfy the emission zone requirements in the cities. In this work, we formulate the two-echelon vehicle routing problem with stochastic demands as a chance-constrained stochastic optimization problem, where the total demand of the customers in each second-echelon route should fit within the second-echelon vehicle capacity with a high probability. We propose two efficient solution procedures based on column generation. Key to the efficiency of these procedures is the underlying labeling algorithm to generate new columns. We propose a novel labeling algorithm based on simultaneous construction of second-echelon routes and a labeling algorithm that builds second-echelon routes sequentially. To further enhance the performance of the solution procedure, we use statistical inference tests to ensure that the chance constraints are met. We reduce the number of customer combinations for which the chance constraint needs to be verified by imposing feasibility bounds on the stochastic customer demands. With these bounds, the runtimes of the labeling algorithms are reduced significantly. The novel labeling algorithm, statistical inference, and feasibility bounds can also be applied to dependent, correlated, and data-driven (scenario-based) demand distributions. Finally, we show the value of the stochastic formulation in terms of improved solution cost and guaranteed feasibility of second-echelon routes. Funding: This work was funded by the Dutch Research Council (NWO) DAREFUL project [Grant 629.002.211] and was carried out on the Dutch national e-infrastructure with the support of SURF Cooperative. Supplemental Material: The online appendices are available at https://doi.org/10.1287/trsc.2022.1162 .

We consider the vehicle routing problem with stochastic demands (VRPSD), a problem in which customer demands are known in distribution at the route planning stage and revealed during route execution upon arrival at each customer. A long-standing open question on the VRPSD concerns the benefits of allowing, during route execution, partial reordering of the planned customer visits. Given the practical importance of this question and the growing interest on the VRPSD under optimal restocking, we study the VRPSD under a recourse policy known as the switch policy. The switch policy is a canonical reoptimization policy that permits only pairs of successive customers to be reordered. We consider this policy jointly with optimal preventive restocking and introduce a branch-cut-and-price algorithm to compute optimal a priori routing plans. This algorithm features pricing routines where value functions represent the expected cost-to-go along planned routes for all possible states and reordering decisions. To ensure pricing tractability, we adopt a strategy that combines elementary pricing with completion bounds of varying complexity, and solve the pricing problem without relying on dominance rules. Our numerical experiments demonstrate the effectiveness of the algorithm for solving instances with up to 50 customers. Notably, they also give us new insights into the value of reoptimization. The switch policy enables significant cost savings over optimal restocking when the planned routes come from an algorithm built on a deterministic approximation of the data, an important scenario given the difficulty of finding optimal VRPSD solutions. The benefits are smaller when comparing optimal a priori VRPSD solutions obtained for both recourse policies. As it appears, further cost savings may require joint reordering and reassignment of customer visits among vehicles when the context permits., Comment: 28 pages

Competition between one-way car-sharing operators is currently increasing. Fleet relocation as a means to compensate demand imbalances constitutes a major cost factor in a business with low profit margins. Existing decision support models have so far ignored the aspect of a competitor when the fleet is rebalanced for better availability. We present mixed-integer linear programming formulations for a pickup and delivery orienteering problem under different business models with multiple (competing) operators. Structural solution properties, including existence of equilibria and bounds on losses as a result of competition, of the competitive pickup and delivery problem under the restrictions of unit-demand stations, homogeneous payoffs, and indifferent customers based on results for congestion games are derived. Two algorithms to find a Nash equilibrium for real-life instances are proposed. One can find equilibria in the most general case; the other can only be applied if the game can be represented as a congestion game, that is, under the restrictions of homogeneous payoffs, unit-demand stations, and indifferent customers. In a numerical study, we compare different business models for car-sharing operations, including a merger between operators and outsourcing relocation operations to a common service provider (coopetition). Gross profit improvements achieved by explicitly incorporating competitor decisions are substantial, and the presence of competition decreases gross profits for all operators (compared with a merger). Using a Munich, Germany, case study, we quantify the gross profit gains resulting from considering competition as approximately 35% (over assuming absence of competition) and 12% (over assuming that the competitor is omnipresence) and the losses because of the presence of competition to be approximately 10%.

This study focuses on the problem of sequencing requests for an end-of-aisle automated storage and retrieval system in which each retrieved load must be returned to its earlier storage location after a worker has picked some products from the load. At the picking station, a buffer is maintained to absorb any fluctuations in speed between the worker and the storage/retrieval machine. We show that, under conditions, the problem of optimally sequencing the requests in this system with a buffer size of m loads forms a special case of the multiple traveling salesmen problem in which each salesman visits the same number of cities. Several interesting structural properties for the problem are mathematically shown. In addition, a branch-and-cut method and heuristics are proposed. Experimental results show that the proposed simulated annealing-based heuristic performs well in all circumstances and significantly outperforms benchmark heuristics. For instances with negligible picking times for the worker, we show that this heuristic provides solutions that are, on average, within 1.8% from the optimal value.

This paper introduces the vehicle routing problem with time windows and shifts (VRPTWS). At the depot, several shifts with nonoverlapping operating periods are available to load the planned trucks. Each shift has a limited loading capacity. We solve the VRPTWS exactly by a branch-and-cut-and-price algorithm. The master problem is a set partitioning with an additional constraint for every shift. Each constraint requires the total quantity loaded in a shift to be less than its loading capacity. For every shift, a pricing subproblem is solved by a label-setting algorithm. Shift capacity constraints define knapsack inequalities; hence we use valid inequalities inspired from knapsack inequalities to strengthen the linear programming relaxation of the master problem when solved by column generation. In particular, we use a family of tailored robust cover inequalities and a family of new nonrobust cover inequalities. Numerical results show that nonrobust cover inequalities significantly improve the algorithm.

Transport companies often have a published timetable. To maintain timetable reliability despite delays, companies include buffer times during timetable development and adjust the traveling speed during timetable execution. We develop an approach that integrates timetable development and execution. We model execution of the timetable as a stochastic dynamic program (SDP). An SDP is a natural framework to model random events causing (additional) delay, propagation of delays, and real-time optimal speed adjustments. However, SDPs alone cannot incorporate the buffer allocation during timetable development, as buffer allocation requires choosing the same action in different states of the SDP. Motivated by the practical need for timetables that operate well during timetable execution, our model seeks the buffer allocation that yields the SDP that has minimal long-run average costs. We derive several analytical insights into the model. We prove that costs are joint convex in the buffer times, and we develop theory to compute subgradients. Our fast and exact algorithm for buffer time allocation is based on these results. Our case study considers container vessels sailing a round tour consisting of 14 ports based on Maersk data. The algorithm finds the optimal timetable in roughly 70 seconds for realistic problem instances. The optimal timetable yields cost reductions of about four to 10 million U.S. dollars per route per year in comparison with the current timetable. Finally, we show the robustness of our solution approach for different parameter settings using a sensitivity analysis. The online appendices are available at https://doi.org/10.1287/trsc.2018.0842 .

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 .

Freight distribution with electric vehicles (EVs) is a promising alternative to reduce the carbon footprint associated with city logistics. Algorithms for planning routes for EVs should take into account their relatively short driving range and the effects of traffic congestion on the battery consumption. This paper proposes new methodology and illustrates how it can be applied to solve an electric vehicle routing problem with stochastic and time-dependent travel times where battery recharging along routes is not allowed. First, a new method for generating network-consistent (correlated in time and space) and time-dependent speed scenarios is introduced. Second, a new technique for applying branch and price on instances defined on real street networks is developed. Computational experiments demonstrate the effectiveness of the approach for finding optimal or near-optimal solutions in instances with up to 133 customers and almost 1,500 road links. With a high probability, the routes in the obtained solutions can be performed by EVs without requiring intermediate recharging stops. An execution time control policy to further reduce the chances of stranded EVs is also presented. In addition, we measure the cost of independence, which is the impact on solution feasibility when travel times are assumed statistically independent. Last, we give directions on how to extend the proposed framework to handle recourse actions.

The pollution-routing problem (PRP) is a recently introduced green vehicle routing problem in the field of green logistics. It concerns routing a number of vehicles to serve a set of geographically dispersed customers within their time windows, jointly with determining their speed on each arc so as to minimize fuel and driving costs. Because of its complexity, all known solution methods are based on (meta-)heuristics. This paper presents an exact solution based on a branch-and-price algorithm for a variant of the PRP. The master problem is a set-partitioning problem, and the pricing problem is a speed- and start-time elementary shortest path problem with resource constraints, in which the speed and start time at the depot needs to be decided on for each individual route. The master problem is solved by means of column generation, and a tailored labeling algorithm is used to solve the pricing problem. New dominance criteria are developed to discard unpromising labels by exploiting the structure of the ready time and the fuel consumption functions. Extensive computational experiments show the value of the proposed algorithm. Keywords: green vehicle routing; fuel consumption; vehicle routing problem; branch and price

On a daily basis, large-scale disruptions require infrastructure managers and railway operators to reschedule their railway timetables together with their rolling stock and crew schedules. This research focuses on timetable rescheduling for passenger train services on a macroscopic level in a railway network. An integer linear programming model is formulated for solving the timetable rescheduling problem, which minimizes the number of cancelled and delayed train services while adhering to infrastructure and rolling stock capacity constraints. The possibility of rerouting train services to reduce the number of cancelled and delayed train services is also considered. In addition, all stages of the disruption management process (from the start of the disruption to the time the normal situation is restored) are taken into account. Computational tests of the described model on a heavily used part of the Dutch railway network show that the model is able to find optimal solutions in short computation times. This makes the approach applicable for use in practice.

This paper studies the real-time crew rescheduling problem in case of large-scale disruptions. One of the greatest challenges of real-time disruption management is the unknown duration of the disruption. In this paper we present a novel approach for crew rescheduling where we deal with this uncertainty by considering several scenarios for the duration of the disruption. The rescheduling problem is similar to a two-stage optimization problem. In the first stage, at the start of the disruption, we reschedule the plan based on the optimistic scenario (i.e., assuming the shortest possible duration of the disruption), while taking into account the possibility that another scenario will be realized. We require a prescribed number of the rescheduled crew duties (a sequential list of tasks which have to be performed by a single crew member) to be recoverable. The true duration of the disruption is revealed in the second stage. By the recoverability of the duties, we expect that the first stage solution can easily be turned into a schedule that is feasible for the realized scenario. We demonstrate the effectiveness of our approach by an application in real-time railway crew rescheduling. The ideas of this paper generalize to certain vehicle rescheduling and manufacturing problems where timetabled tasks which have a fixed start and end location are to be carried out by a given number of servers. We test our approach on a number of instances of Netherlands Railways (NS), the main operator of passenger trains in the Netherlands. The numerical experiments show that the approach indeed finds schedules which are easier to adjust if it turns out that another scenario than the optimistic one is realized for the duration of the disruption.

Planning the transport of maritime containers from the seaport to final inland destinations is challenged by uncertainties regarding the time the container is released for further transport and the transit time from the port to its final destination. This paper assesses the value of information in container transport in terms of efficiency and reliability. The analysis uses a stylized single period model where a decision maker allocates released containers to two transport modes (slow, low price, no flexible departure times versus fast, high price, flexible departure times) and plans the departure time of the inflexible mode. We construct Pareto frontiers and the corresponding Pareto optimal decisions under various information scenarios and show that the Pareto frontiers move in a favorable direction when the level of information increases. The mathematical results are explained and illustrated by means of a numerical example involving barge transport. We also perform a sensitivity analysis and study the impact of erroneous information on efficiency and reliability based on a numerical analysis.

The transport sector is the second-largest carbon emissions contributor in Europe, and its emissions continue to increase. Many producers are committing themselves to reducing transport emissions voluntarily, possibly in anticipation of increasing transport prices. In this paper we study a producer that has outsourced transport and has decided to cap its carbon emissions from outbound logistics for a group of customers. Setting an emission constraint for a group of customers allows for taking advantage of the portfolio effect. We focus on reducing emissions by switching transport modes within an existing network, because this has a large impact on emissions. In addition, the company sets the sales prices, which influence demand. The problem is solved by decomposing the multiproduct problem into several single-product problems, which we then analyze separately. Using the single-product solutions, we create an efficient frontier that reflects the trade-off between total carbon emissions and the total profit. It is observed that a diminishing rate of return applies in reducing emissions by switching transport modes. In a case study, we apply our method to a producer of bulk liquids and find that emissions can be reduced by 10% at only a 0.7% increase in total logistics cost.

This paper presents a branch-and-price algorithm for the time-dependent vehicle routing problem with time windows (TDVRPTW). We capture road congestion by considering time-dependent travel times, i.e., depending on the departure time to a customer, a different travel time is incurred. We consider the variant of the TDVRPTW where the objective is to minimize total route duration and denote this variant the duration minimizing TDVRPTW (DM-TDVRPTW). Because of time dependency, vehicles' dispatch times at the depot are crucial as road congestion might be avoided. Because of its complexity, all known solution methods to the DM-TDVRPTW are based on (meta-)heuristics. The decomposition of an arc-based formulation leads to a set-partitioning problem as the master problem, and a time-dependent shortest path problem with resource constraints as the pricing problem. The master problem is solved by means of column generation, and a tailored labeling algorithm is used to solve the pricing problem. We introduce new dominance criteria that allow more label dominance. For our numerical results, we modified Solomon's data sets by adding time dependency. Our algorithm is able to solve about 63% of the instances with 25 customers, 38% of the instances with 50 customers, and 15% of the instances with 100 customers.