260 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Matheuristics:survey and synthesis

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    In integer programming and combinatorial optimisation, people use the term matheuristics to refer to methods that are heuristic in nature, but draw on concepts from the literature on exact methods. We survey the literature on this topic, with a particular emphasis on matheuristics that yield both primal and dual bounds (i.e., upper and lower bounds in the case of a minimisation problem). We also make some comments about possible future developments

    On a Vehicle Routing Problem with Customer Costs and Multi Depots

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    The Vehicle Routing Problem with Customer Costs (short VRPCC) was developed for railway maintenance scheduling. In detail, corrective maintenance jobs for unexpected occurring failures are planned to a short time horizon. These jobs are geographically distributed in the railway net. Furthermore, dependent on the severity of the failure, it can be necessary to reduce the top speed on the track section in order to avoid safety risks or a too fast deterioration. For fatal failures, it can even be necessary to close the track section. The resulting limitations on railway service lead to penalty costs for the maintenance operator. These must be paid until the track is repaired and the restrictions are removed. By scheduling the maintenance tasks, these penalty costs can be reduced by proceeding corresponding maintenance tasks earlier. However, this may in return lead to increased costs for moving the maintenance machines and crews. For this scheduling problem, the VRPCC was developed. With it, for each maintenance vehicle and crew, a route is defined that describes the order to proceed maintenance tasks. Two kinds of costs are considered: Firstly, travel costs for machinery and crew; and secondly, penalty costs for an unsafe track condition that have to be paid for each day from failure detection to maintenance completion. To model the penalties, the novel customer costs are defined. In detail, for each maintenance activity a customer cost coefficient is given which incur for each day between failure detection and failure repair. The objective function of this problem is defined by the sum of travel costs and time-dependent customer costs. With it, the priority of customers can be taken into account without losing the sight on travel costs. This new vehicle routing problem was introduced in this thesis by a non-linear partition and permutation model. In this model, a feasible solution is defined by a partition of the job set into subsets that represent the allocation of jobs to vehicles and a permutation for each subset that represent the order of processing the jobs. Then, the start times of the jobs were calculated based on the order given by the permutations. It was taken into account that work can only be done in eight hour shifts during the night. Based on the start times, the customer cost value of each job is computed which equals to the paid penalty costs. Then, the costs of a schedule are calculated via the sum of travel costs and customer costs. To solve the VRPCC by a commercial linear programming solver, different formulations of the VRPCC as mixed-integer linear program were developed. In doing so, the start times became decision variables. It turned out that including customer costs led to problems harder to solve than vehicle routing problems where only travel costs are minimized. Further, in the thesis several construction heuristics for the VRPCC were designed and investigated. Also two local search algorithms, first and best improvement, were applied. The computational experiments showed that the solutions generated by the local search algorithm were much better than the solutions of the construction heuristics. The main part of this thesis was to design a Branch-and-Bound algorithm for the VRPCC. For this purpose, new lower bounds for the customer cost part of the objective function were formulated. The computational experiments showed that a lower bound computed from the LP relaxation of a specific bin packing problem had the best trade-off between computational effort and bound quality. For the travel cost part of the objective function, several known lower bounds from the TSP were compared. To design a Branch-and-Bound algorithm, beside efficient lower bound, also suitable branching strategies are necessary to split the problem space into smaller subspaces. In this thesis two branching strategies were developed which are based on the non-linear partition and permutation model to take advantage from the problem structure. To be more precise, new branches are generated by appending or including a job to an uncompleted schedule. Consequently, the start times can be computed directly from the so far planned jobs and more tight lower bounds can be computed for the so far unplanned jobs. By means of computational experiments, the developed Branch-and-Bound algorithms were compared with the classical approach, which means solving a mixed-integer linear program of the VRPCC by a commercial solver. The results showed that both Branch-and-Bound algorithms solved the small instances faster than the classical approach

    A New Dynamic Programming Approach for Spanning Trees with Chain Constraints and Beyond

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    Short spanning trees subject to additional constraints are important building blocks in various approximation algorithms. Especially in the context of the Traveling Salesman Problem (TSP), new techniques for finding spanning trees with well-defined properties have been crucial in recent progress. We consider the problem of finding a spanning tree subject to constraints on the edges in cuts forming a laminar family of small width. Our main contribution is a new dynamic programming approach where the value of a table entry does not only depend on the values of previous table entries, as it is usually the case, but also on a specific representative solution saved together with each table entry. This allows for handling a broad range of constraint types. In combination with other techniques -- including negatively correlated rounding and a polyhedral approach that, in the problems we consider, allows for avoiding potential losses in the objective through the randomized rounding -- we obtain several new results. We first present a quasi-polynomial time algorithm for the Minimum Chain-Constrained Spanning Tree Problem with an essentially optimal guarantee. More precisely, each chain constraint is violated by a factor of at most 1+ε1+\varepsilon, and the cost is no larger than that of an optimal solution not violating any chain constraint. The best previous procedure is a bicriteria approximation violating each chain constraint by up to a constant factor and losing another factor in the objective. Moreover, our approach can naturally handle lower bounds on the chain constraints, and it can be extended to constraints on cuts forming a laminar family of constant width. Furthermore, we show how our approach can also handle parity constraints (or, more precisely, a proxy thereof) as used in the context of (Path) TSP and one of its generalizations, and discuss implications in this context.Comment: A short version of this work appeared in the proceedings of the 30th annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2019

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization

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    Many real-life decision problems are discrete in nature. To solve such problems as mathematical optimization problems, integrality constraints are commonly incorporated in the model to reflect the choice of finitely many alternatives. At the same time, it is known that semidefinite programming is very suitable for obtaining strong relaxations of combinatorial optimization problems. In this dissertation, we study the interplay between semidefinite programming and integrality, where a special focus is put on the use of cutting-plane methods. Although the notions of integrality and cutting planes are well-studied in linear programming, integer semidefinite programs (ISDPs) are considered only recently. We show that manycombinatorial optimization problems can be modeled as ISDPs. Several theoretical concepts, such as the Chvátal-Gomory closure, total dual integrality and integer Lagrangian duality, are studied for the case of integer semidefinite programming. On the practical side, we introduce an improved branch-and-cut approach for ISDPs and a cutting-plane augmented Lagrangian method for solving semidefinite programs with a large number of cutting planes. Throughout the thesis, we apply our results to a wide range of combinatorial optimization problems, among which the quadratic cycle cover problem, the quadratic traveling salesman problem and the graph partition problem. Our approaches lead to novel, strong and efficient solution strategies for these problems, with the potential to be extended to other problem classes

    Operational Research: methods and applications

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    This is the final version. Available on open access from Taylor & Francis via the DOI in this recordThroughout its history, Operational Research has evolved to include methods, models and algorithms that have been applied to a wide range of contexts. This encyclopedic article consists of two main sections: methods and applications. The first summarises the up-to-date knowledge and provides an overview of the state-of-the-art methods and key developments in the various subdomains of the field. The second offers a wide-ranging list of areas where Operational Research has been applied. The article is meant to be read in a nonlinear fashion and used as a point of reference by a diverse pool of readers: academics, researchers, students, and practitioners. The entries within the methods and applications sections are presented in alphabetical order. The authors dedicate this paper to the 2023 Turkey/Syria earthquake victims. We sincerely hope that advances in OR will play a role towards minimising the pain and suffering caused by this and future catastrophes

    Sine Cosine Algorithm for Optimization

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    This open access book serves as a compact source of information on sine cosine algorithm (SCA) and a foundation for developing and advancing SCA and its applications. SCA is an easy, user-friendly, and strong candidate in the field of metaheuristics algorithms. Despite being a relatively new metaheuristic algorithm, it has achieved widespread acceptance among researchers due to its easy implementation and robust optimization capabilities. Its effectiveness and advantages have been demonstrated in various applications ranging from machine learning, engineering design, and wireless sensor network to environmental modeling. The book provides a comprehensive account of the SCA, including details of the underlying ideas, the modified versions, various applications, and a working MATLAB code for the basic SCA

    Operational research:methods and applications

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    Throughout its history, Operational Research has evolved to include a variety of methods, models and algorithms that have been applied to a diverse and wide range of contexts. This encyclopedic article consists of two main sections: methods and applications. The first aims to summarise the up-to-date knowledge and provide an overview of the state-of-the-art methods and key developments in the various subdomains of the field. The second offers a wide-ranging list of areas where Operational Research has been applied. The article is meant to be read in a nonlinear fashion. It should be used as a point of reference or first-port-of-call for a diverse pool of readers: academics, researchers, students, and practitioners. The entries within the methods and applications sections are presented in alphabetical order

    Two-Stage Vehicle Routing Problems with Profits and Buffers: Analysis and Metaheuristic Optimization Algorithms

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    This thesis considers the Two-Stage Vehicle Routing Problem (VRP) with Profits and Buffers, which generalizes various optimization problems that are relevant for practical applications, such as the Two-Machine Flow Shop with Buffers and the Orienteering Problem. Two optimization problems are considered for the Two-Stage VRP with Profits and Buffers, namely the minimization of total time while respecting a profit constraint and the maximization of total profit under a budget constraint. The former generalizes the makespan minimization problem for the Two-Machine Flow Shop with Buffers, whereas the latter is comparable to the problem of maximizing score in the Orienteering Problem. For the three problems, a theoretical analysis is performed regarding computational complexity, existence of optimal permutation schedules (where all vehicles traverse the same nodes in the same order) and potential gaps in attainable solution quality between permutation schedules and non-permutation schedules. The obtained theoretical results are visualized in a table that gives an overview of various subproblems belonging to the Two-Stage VRP with Profits and Buffers, their theoretical properties and how they are connected. For the Two-Machine Flow Shop with Buffers and the Orienteering Problem, two metaheuristics 2BF-ILS and VNSOP are presented that obtain favorable results in computational experiments when compared to other state-of-the-art algorithms. For the Two-Stage VRP with Profits and Buffers, an algorithmic framework for Iterative Search Algorithms with Variable Neighborhoods (ISAVaN) is proposed that generalizes aspects from 2BF-ILS as well as VNSOP. Various algorithms derived from that framework are evaluated in an experimental study. The evaluation methodology used for all computational experiments in this thesis takes the performance during the run time into account and demonstrates that algorithms for structurally different problems, which are encompassed by the Two-Stage VRP with Profits and Buffers, can be evaluated with similar methods. The results show that the most suitable choice for the components in these algorithms is dependent on the properties of the problem and the considered evaluation criteria. However, a number of similarities to algorithms that perform well for the Two-Machine Flow Shop with Buffers and the Orienteering Problem can be identified. The framework unifies these characteristics, providing a spectrum of algorithms that can be adapted to the specifics of the considered Vehicle Routing Problem.:1 Introduction 2 Background 2.1 Problem Motivation 2.2 Formal Definition of the Two-Stage VRP with Profits and Buffers 2.3 Review of Literature on Related Vehicle Routing Problems 2.3.1 Two-Stage Vehicle Routing Problems 2.3.2 Vehicle Routing Problems with Profits 2.3.3 Vehicle Routing Problems with Capacity- or Resource-based Restrictions 2.4 Preliminary Remarks on Subsequent Chapters 3 The Two-Machine Flow Shop Problem with Buffers 3.1 Review of Literature on Flow Shop Problems with Buffers 3.1.1 Algorithms and Metaheuristics for Flow Shops with Buffers 3.1.2 Two-Machine Flow Shop Problems with Buffers 3.1.3 Blocking Flow Shops 3.1.4 Non-Permutation Schedules 3.1.5 Other Extensions and Variations of Flow Shop Problems 3.2 Theoretical Properties 3.2.1 Computational Complexity 3.2.2 The Existence of Optimal Permutation Schedules 3.2.3 The Gap Between Permutation Schedules an Non-Permutation 3.3 A Modification of the NEH Heuristic 3.4 An Iterated Local Search for the Two-Machine Flow Shop Problem with Buffers 3.5 Computational Evaluation 3.5.1 Algorithms for Comparison 3.5.2 Generation of Problem Instances 3.5.3 Parameter Values 3.5.4 Comparison of 2BF-ILS with other Metaheuristics 3.5.5 Comparison of 2BF-OPT with NEH 3.6 Summary 4 The Orienteering Problem 4.1 Review of Literature on Orienteering Problems 4.2 Theoretical Properties 4.3 A Variable Neighborhood Search for the Orienteering Problem 4.4 Computational Evaluation 4.4.1 Measurement of Algorithm Performance 4.4.2 Choice of Algorithms for Comparison 4.4.3 Problem Instances 4.4.4 Parameter Values 4.4.5 Experimental Setup 4.4.6 Comparison of VNSOP with other Metaheuristics 4.5 Summary 5 The Two-Stage Vehicle Routing Problem with Profits and Buffers 5.1 Theoretical Properties of the Two-Stage VRP with Profits and Buffers 5.1.1 Computational Complexity of the General Problem 5.1.2 Existence of Permutation Schedules in the Set of Optimal Solutions 5.1.3 The Gap Between Permutation Schedules an Non-Permutation Schedules 5.1.4 Remarks on Restricted Cases 5.1.5 Overview of Theoretical Results 5.2 A Metaheuristic Framework for the Two-Stage VRP with Profits and Buffers 5.3 Experimental Results 5.3.1 Problem Instances 5.3.2 Experimental Results for O_{max R, Cmax≤B} 5.3.3 Experimental Results for O_{min Cmax, R≥Q} 5.4 Summary Bibliography List of Figures List of Tables List of Algorithm
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