Constant-factor approximations for Capacitated Arc Routing without triangle inequality

Given an undirected graph with edge costs and edge demands, the Capacitated Arc Routing problem (CARP) asks for minimum-cost routes for equal-capacity vehicles so as to satisfy all demands. Constant-factor polynomial-time approximation algorithms were proposed for CARP with triangle inequality, while CARP was claimed to be NP-hard to approximate within any constant factor in general. Correcting this claim, we show that any factor {\alpha} approximation for CARP with triangle inequality yields a factor {\alpha} approximation for the general CARP

Approximation Algorithms for Mixed, Windy, and Capacitated Arc Routing Problems

We show that any alpha(n)-approximation algorithm for the n-vertex metric asymmetric Traveling Salesperson problem yields O(alpha(C))-approximation algorithms for various mixed, windy, and capacitated arc routing problems. Herein, C is the number of weakly-connected components in the subgraph induced by the positive-demand arcs, a number that can be expected to be small in applications. In conjunction with known results, we derive constant-factor approximations if C is in O(log n) and O(log(C)/log(log(C)))-approximations in general

Approximation Algorithms for Capacitated Location Routing

An approximation algorithm for an optimization problem runs in polynomial time for all instances and is guaranteed to deliver solutions with bounded optimality gap. We derive such algorithms for different variants of capacitated location routing, an important generalization of vehicle routing where the cost of opening the depots from which vehicles operate is taken into account. Our results originate from combining algorithms and lower bounds for different relaxations of the original problem, and besides location routing we also obtain approximation algorithms for multi-depot capacitated vehicle routing by this framework. Moreover, we extend our results to further generalizations of both problems, including a prize-collecting variant, a group version, and a variant where cross-docking is allowed. We finally present a computational study of our approximation algorithm for capacitated location routing on benchmark instances and large-scale randomly generated instances. Our study reveals that the quality of the computed solutions is much closer to optimality than the provable approximation factor

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Time and multiple objectives in scheduling and routing problems

Many optimization problems encountered in practice are multi-objective by nature, i.e., different objectives are conflicting and equally important. Many times, it is not desirable to drop some of them or to optimize them in a composite single objective or hierarchical manner. Furthermore, cost parameters change over time which makes optimization problems harder. For instance, in the transport sector, travel costs are a function of travel time which changes depending on the time of the day a vehicle is travelling (e.g., due to road congestion). Road congestion results in tremendous delays which lead to a decrease in the service quality and the responsiveness of logistic service providers. In Chapter 2, we develop a generic approach to deal with Multi-Objective Scheduling Problems (MOSPs) with State-Dependent Cost Parameters. The aim is to determine the set of Pareto solutions that capture the trade offs between the different conflicting objectives. Due to the complexity of MOSPs, an efficient approximation based on dynamic programming is developed. The approximation has a provable worse case performance guarantee. Even though the generated approximate Pareto front consist of fewer solutions, it still represents a good coverage of the true Pareto front. Furthermore, considerable gains in computation times are achieved. In Chapter 3, the developed methodology is validated on the multi-objective timedependent knapsack problem. In the classical knapsack problem, the input consists of a knapsack with a finite capacity and a set of items, each with a certain weight and a cost. A feasible solution to the knapsack problem is a selection of items such that their total weight does not exceed the knapsack capacity. The goal is to maximize the single objective function consisting of the total pro t of the selected items. We extend the classical knapsack problem in two ways. First, we consider time-dependent profits (e.g., in a retail environment profit depends on whether it is Christmas or not)

A Special Case of the Multiple Traveling Salesmen Problem in End-of-Aisle Picking Systems

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