The Vehicle Rescheduling Problem

The capacitated vehicle routing problem is to find a routing schedule describing the order in which geographically dispersed customers are visited to satisfy demand by supplying goods stored at the depot, such that the traveling costs are minimized. In many practical applications, a long term routing schedule has to be made for operational purposes, often based on average demand estimates. When demand substantially differs, constructing a new schedule is beneficial. The vehicle rescheduling problem is to find a new schedule that not only minimizes the total traveling costs but also minimizes the costs of deviating from the original schedule. In this paper two mathematical programming formulations of the rescheduling problem are presented as well as two heuristic methods, a two-phase heuristic and a modified savings heuristic. Computational and analytical results show that for sufficiently high deviation costs, the two-phase heuristic generates a schedule that is on average close to optimal or even guaranteed optimal, for all considered problem instances. The modified savings heuristic generates schedules of constant quality, however the two-phase heuristic produces schedules that are on average closer to the optimum.vehicle routing;operational planning;vehicle rescheduling problem

The case for limited-preemptive scheduling in GPUs for real-time systems

Many emerging cyber-physical systems, such as autonomous vehicles, have both extreme computation and hard latency requirements. GPUs are being touted as the ideal platform for such applications due to their highly parallel organisation. Unfortunately, while offering the necessary performance, GPUs are currently designed to maximise throughput and fail to offer the necessary hard real-time (HRT) guarantees. In this work we discuss three additions to GPUs that enable them to better meet real-time constraints. Firstly, we provide a quantitative argument for exposing the non-preemptive GPU scheduler to software. We show that current GPUs perform hardware context switches for non-preemptive scheduling in 20-26.5μs on average, while swapping out 60-270KiB of state. Although high, these overheads do not forbid non-preemptive HRT scheduling of real-time task sets. Secondly, we argue that limited-preemption support can deliver large benefits in schedulability with very minor impact on the context switching overhead. Finally, we demonstrate the need for a more predictable DRAM request arbiter to reduce interference caused by processes running on the GPU in parallel

A branch-and-cut algorithm for the Time Window Assignment Vehicle Routing Problem

This paper presents a branch-and-cut algorithm for the Time Window Assignment Vehicle Routing Problem (TWAVRP), the problem of assigning time windows for delivery before demand volume becomes known. A novel set of valid inequalities, the precedence inequalities, is introduced and multiple separation heuristics are presented. In our numerical experiments the branch-and-cut algorithm is 3.8 times faster when separating precedence inequalities. Furthermore, in our experiments, the branch-and-cut algorithm is 193.9 times faster than the best known algorithm in the literature. Finally, using our algorithm, instances of the TWAVRP are solved which are larger than the small scale instances previously presented in the literature

The Time Window Assignment Vehicle Routing Problem

In many distribution networks, it is vital that time windows in which deliveries are made are assigned to customers for the long term. However, at the moment of assigning time windows demand is not known. In this paper we introduce the time window assignment vehicle routing problem, the TWAVRP. In this problem time windows have to be assigned before demand is known. Next the realization of demand is revealed and an optimal vehicle routing schedule has to be made that satisfies the time window constraints. We assume that different scenarios of demand realizations are known, as well as their probability distribution. The TWAVRP is the problem of assigning time windows such that the expected traveling costs are minimized. We propose a formulation of the TWAVRP and develop two variants of a column generation algorithm to solve the LP relaxation of this formulation. Numerical experiments show that these algorithms provide us with very tight LP-bounds to instances of moderate size in reasonable computation time. We incorporate the column generation algorithm in a branch and price algorithm and find optimal integer solutions to small instances of the TWAVRP. In our numerical experiments, the branch and price algorithm typically finds the optimal solution very early in the branching procedure and spends most time on proving optimality

Vehicle Routing with Uncertain Demand

In distribution networks a supplier transports goods from a distribution center to customers by means of vehicles with limited capacity. Drivers will drive routes on which they visit multiple customers to make deliveries. Typically, deliveries are made regularly and a fixed schedule is maintained. A fixed schedule is beneficial for many operational purposes, as it for instance allows for easy planning of the packing of the vehicles at the distribution center, or it allows the customer to roster the delivery handling personnel. A fixed schedule is often reused to make weekly deliveries for a period of a year or longer. However, at the moment of designing a schedule, the demand of the customers is usually unknown. Moreover, in most cases, demand of a customer will be different for each delivery. Therefore, it will be necessary to construct or adapt vehicle routes for each day of delivery, without deviating too much from the fixed schedule. In this thesis several different views on a fixed schedule are explored. It addresses the need from practice to incorporate the uncertainty of demand in transportation models to increase the efficiency of transport. Innovative vehicle routing models are presented taking uncertain or varying demand into account. New algorithms using state-of-the-art methods are presented based on these models, to construct fixed schedules and vehicle routes. The algorithms make use of recent scientific advances in mathematical programming, specifically in the domain of vehicle routing

The Vehicle Rescheduling Problem

The capacitated vehicle routing problem is to find a routing schedule describing the order in which geographically dispersed customers are visited to satisfy demand by supplying goods stored at the depot, such that the traveling costs are minimized. In many practical applications, a long term routing schedule has to be made for operational purposes, often based on average demand estimates. When demand substantially differs, constructing a new schedule is beneficial. The vehicle rescheduling problem is to find a new schedule that not only minimizes the total traveling costs but also minimizes the costs of deviating from the original schedule. In this paper two mathematical programming formulations of the rescheduling problem are presented as well as two heuristic methods, a two-phase heuristic and a modified savings heuristic. Computational and analytical results show that for sufficiently high deviation costs, the two-phase heuristic generates a schedule that is on average close to o

A Branch-and-Price Approach for a Ship Routing Problem with Multiple Products and Inventory Constraints

In the oil industry, different oil components are blended in a refinery to fuel products. These products are transported to different harbors by ship. Due to the limited storage capacity at the harbors and the undesirability of a stock-out, inventory levels at the harbors have to be taken into account during the construction of the ship routes. In this paper, we give a detailed description of this problem, which we call the ship routing problem with multiple products and inventory constraints. Furthermore, we formulate this problem as a generalized set-covering problem, and we present a Branch-and-Price algorithm to solve it. The pricing problems have a very complex nature. We discuss a dynamic programming algorithm to solve them to optimality

A p-step formulation for the capacitated vehicle routing problem

We introduce a _p_-step formulation for the capacitated vehicle routing problem (CVRP). The parameter _p_ indicates the length of partial paths corresponding to the used variables. This provides a family of formulations including both the traditional arc-based and path-based formulations. Hence, it is a generalization which unifies arc-based and path-based formulations, while also providing new formulations. We show that the LP bound of the _p_-step formulation is increasing in _p_, although not monotonically. Furthermore, we prove that computing the set partitioning bound is NP-hard. This is a meaningful result in itself, but combined with the _p_-step formulation this also allows us to show that there does not exist a strongest compact formulation for the CVRP, if _P ≠ NP_. While ending the search for a strongest compact formulation, we propose th

The path programming problem and a partial path relaxation

We introduce the class of path programming problems, which can be used to model many known optimization problems. A path programming problem can be formulated as a binary programming problem, for which the pricing problem can be modeled as a shortest path problem with resource constraints when column generation is used to solve its linear programming relaxation. Many optimization problems found in the literature belong to this class. We provide a framework for obtaining a partial path relaxation of a path programming problem. Like traditional path relaxations, the partial path relaxation allows the computational complexity of the pricing problem to be reduced, at the expense of a weaker linear programming bound. We demonstrate the versatility of this framework by providing different examples of partial path relaxations for a crew scheduling problem and vehicle routing problem

Shared Capacity Routing Problem – An Omni-channel Retail Study

More and more retailers allow customers to order goods online and then pick them up in a store. In this setting, these orders are typically served from a dedicated warehouse. This often means that the stores are visited by different vehicles to replenish the store inventory and to supply the pick-up points. Motivated by a collaboration with an omni- channel grocery retailer in the Netherlands, we study how to best share capacity between the routes associated with these different sales cha