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Mixed integer linear programming formulation for K-means clustering problem

CENTRAL EUROPEAN JOURNAL OF OPERATIONS RESEARCH 2024年第1期32卷 11-27页

作者： Agoston, Kolos Cs. E-Nagy, Marianna Corvinus Univ Budapest Inst Operat & Dec Sci Budapest Hungary Corvinus Univ Budapest Corvinus Ctr Operat Res Budapest Hungary

The minimum sum-of-squares clusering is the most widely used clustering method. The minimum sum-of-squares clustering is usually solved by the heuristic KMEANS algorithm, which converges to a local optimum. A lot of effort has been made to solve such kind of problems, but a mixed integer linear programming formulation (MILP) is still missing. In this paper, we formulate MILP models. The advantage of MILP formulation is that users can extend the original problem with arbitrary linear constraints. We also present numerical results, we solve these models up to sample size of 150.

关键词： Mathematical programming linear programming formulation Clustering K-means

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Workforce Scheduling linear programming formulation

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IFAC-PapersOnLine 2018年第11期51卷 264-269页

作者： Garaix, T. Gondran, M. Lacomme, P. Mura, E. Tchernev, N. Université Clermont-Auvergne LIMOS UMR CNRS 6158 Campus des Cézeaux Aubière Cedex63178 France Ecole des Mines de Saint-Etienne 158 cours Fauriel Saint-Etienne42000 France Université de Technologie de Troyes ICD-LOSI UMR CNRS 6281 12 rue Marie Curie CS 42060 Troyes Cedex10004 France

This paper introduces a linear programming formulation for a ternary-integration Workforce Scheduling and Routing Problem that incorporates scheduling of tasks, assigning of workers to the tasks according to their skills and the definition of the workers’ trips. Each task has a time window, and is related to a customer who has a preference list of the workers. Each worker has a cost, a preference list of tasks and a working time window. The objective is to perform the tasks and simultaneously minimizing the number of unassigned tasks, the traveling distance, the workers’ cost, and maximizing the customers and workers preference satisfaction. © 2018

关键词： linear programming Customer satisfaction Personnel Scheduling Integrated Problem linear programming formulation Minimizing the number of Preference lists Routing Ternary integrations Time windows Workforce scheduling

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Workforce Scheduling linear programming formulation

Workforce Scheduling Linear Programming Formulation

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16th IFAC Symposium on Information Control Problems in Manufacturing (INCOM)

作者： Garaix, T. Gondran, M. Lacomme, P. Mura, E. Tchernev, N. Univ Clermont Auvergne CNRS UMR 6158 LIMOS Campus Cezeaux F-63178 Aubiere France Ecole Mines St Etienne 158 Cours Fauriel F-42000 St Etienne France Univ Technol Troyes CNRS UMR 6281 ICDLOSI 12 Rue Marie CurieCS 42060 F-10004 Troyes France

This paper introduces a linear programming formulation for a ternary-integration Workforce Scheduling and Routing Problem that incorporates scheduling of tasks, assigning of workers to the tasks according to their skills and the definition of the workers' trips. Each task has a time window, and is related to a customer who has a preference list of the workers. Each worker has a cost, a preference list of tasks and a working time window. The objective is to perform the tasks and simultaneously minimizing the number of unassigned tasks, the traveling distance, the workers' cost, and maximizing the customers and workers preference satisfaction. (C) 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

关键词： Workforce Scheduling Routing linear programming formulation Integrated Problem

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Conditions for the Solvability of the linear programming formulation for Constrained Discounted Markov Decision Processes

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APPLIED MATHEMATICS AND OPTIMIZATION 2016年第1期74卷 27-51页

作者： Dufour, F. Prieto-Rumeau, T. INRIA Bordeaux Sud Ouest Team CQFD 200 Ave Vieille Tour F-33405 Talence France IMB 200 Ave Vieille Tour F-33405 Talence France UNED Dept Stat & Operat Res Calle Senda del Rey 9 Madrid 28040 Spain

We consider a discrete-time constrained discounted Markov decision process (MDP) with Borel state and action spaces, compact action sets, and lower semi-continuous cost functions. We introduce a set of hypotheses related to a positive weight function which allow us to consider cost functions that might not be bounded below by a constant, and which imply the solvability of the linear programming formulation of the constrained MDP. In particular, we establish the existence of a constrained optimal stationary policy. Our results are illustrated with an application to a fishery management problem.

关键词： Markov decision processes linear programming formulation Constrained problems

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Interpolating Constrained Control of Interconnected Systems

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IFAC-PapersOnLine 2018年第9期51卷 7-12页

作者： Scialanga, Sheila Ampountolas, Konstantinos School of Engineering University of Glasgow GlasgowG12 8QQ United Kingdom

This paper presents a decentralised interpolating control scheme for the control of linear discrete-time interconnected systems with local state and control constraints. The control law of each distinct subsystem relies on the gentle interpolation between a local high-gain controller, which satisfies some user-desired performance specifications, with a global low-gain controller. For each subsystem both low- and high-gain controllers can be efficiently determined off-line, while the inexpensive interpolation between them is performed on-line. For the interpolation, a new low-dimensional linear programming formulation is developed, which is computationally less expensive compared to previous works. Therefore, it is appropriate for realtime control of large-scale interconnected systems. Proofs of recursive feasibility and asymptotic stability of the interpolating scheme are given for decoupled as well as interconnected subsystems with coupled state constraints. Two numerical examples show that the proposed decentralised interpolating control outperforms previously proposed interpolating schemes. Finally, it is faster than model predictive control, while their control behaviour and performance is almost identical. © 2018

关键词： Discrete time control systems Asymptotic stability Controllers Decentralized control Gain control Interpolation Large scale systems linear programming Model predictive control Constrained controls Decentralised control Interconnected subsystems Invariant set Large scale interconnected systems linear programming formulation Low and high gain Performance specifications

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A Fix-and-Optimize Variable Neighborhood Search for the Biomedical Sample Transportation Problem

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IFAC-PapersOnLine 2018年第11期51卷 992-997页

作者： Toschi, Marta Lanzarone, Ettore Anaya-Arenas, Ana Maria Bélanger, Valérie Nicoletta, Vittorio Ruiz, Angel Milan Italy Department of Logistics and Operations Management HEC Montréal Montréal Canada Montréal Canada Department of Operations and Decision Systems Université Laval Québec Canada Centre interuniversitaire de recherche sur les reseaux d'entreprise la logistique et le transport Montréal and Québec Canada

Biomedical tests play a crucial role in helping physicians to make accurate diagnoses. To perform these tests, thousands of samples are daily transported from several healthcare facilities, where they are collected from patients, to laboratories, where they are analyzed. We consider the challenging Biomedical Sample Transportation Problem (BSTP), which is a complex variant of the vehicle routing problem with time windows, where both the number of visits and the opening and closing hours of the collection centers are decision variables, while the objective is to minimize the total duration of routes. We propose a linear programming formulation for the BSTP, and we develop a matheuristics to solve the problem in real-size problem instances, which consists of a decomposition coupled with a Variable Neighborhood Search (VNS) algorithm. The decomposition is based on a spatio-temporal clustering method, which takes into account both the travel times between the centers and their collection periods; then, a Fix-and-Optimize VNS is applied to improve the decomposed solution. The performance of the proposed method is assessed over a large number of realistic instances, which are based on the laboratory network in the Province of Québec, Canada. Results show good quality solutions and the capability of the matheuristics to solve real-size problem instances within an adequate time. © 2018

关键词： Problem solving Diagnosis linear programming Plant shutdowns Travel time Vehicle routing Decision variables Healthcare facility linear programming formulation Spatio temporal clustering Transportation problem Variable neighborhood search Vehicle routing problem with time windows Vehicle Routing Problems

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A Model to Optimize the Reference Storage Assignment in a Supermarket to Expedite the Part Feeding Activities

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IFAC-PapersOnLine 2018年第11期51卷 1470-1475页

作者： Zangaro, Francesco Battini, Daria Calzavara, Martina Persona, Alessandro Sgarbossa, Fabio Department of Management and Engineering University of Padua Stradella San Nicola 3 Vicenza36100 Italy

In many industrial environments, picking operations are performed with tow trains in order to efficiently feed a production or an assembly system. This operation is often time-consuming, especially for the automotive environment, which employs the supermarket concept very often to manage thousands of different parts. This research aims at improving the picking operations by optimizing the storage assignment of the references in a supermarket. This is a process in which the references are placed in specific racks of the warehouse. A linear programming formulation is provided so that the picking operations can be performed in the minimum amount of time. The model considers some constraints linked to the specific kind of equipment used in a warehouse, and the ergonomic energy expenditure. This approach has been implemented in a numerical case inspired by a real industrial application. © 2018

关键词： Retail stores linear programming Optimization Warehouses Assembly systems Automotive environment Energy expenditure Industrial environments linear programming formulation Picking operations Reference placement Storage assignment

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Optimal Routing of Energy-aware Vehicles in Networks with Inhomogeneous Charging Nodes 22

Optimal Routing of Energy-aware Vehicles in Networks with In...

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22nd Mediterranean Conference on Control and Automation (MED)

作者： Pourazarm, S. Cassandras, C. G. Boston Univ Div Syst Engn Boston MA 02215 USA Boston Univ Ctr Informat & Syst Engn Boston MA 02215 USA

ISBN: (纸本)9781479959013

We study the routing problem for vehicles with limited energy through a network of inhomogeneous charging nodes. This is substantially more complicated than the homogeneous node case studied in [1]. We seek to minimize the total elapsed time for vehicles to reach their destinations considering both traveling and recharging times at nodes when the vehicles do not have adequate energy for the entire journey. We study two versions of the problem. In the single vehicle routing problem, we formulate a mixed-integer nonlinear programming (MINLP) problem and show that it can be reduced to a lower dimensionality problem by exploiting properties of an optimal solution. We also obtain a linear programming (LP) formulation allowing us to decompose it into two simpler problems yielding near-optimal solutions. For a multi-vehicle problem, where traffic congestion effects are included, we use a similar approach by grouping vehicles into "subflows". We also provide an alternative flow optimization formulation leading to a computationally simpler problem solution with minimal loss in accuracy.

关键词： battery powered vehicles energy conservation integer programming network theory (graphs) nonlinear programming vehicle routing MINLP problem inhomogeneous charging nodes linear programming formulation lower dimensionality problem mixed-integer nonlinear programming problem optimal energy-aware vehicle routing recharging times total elapsed time minimization traffic congestion effects traveling times Artificial neural networks Charging stations Energy consumption linear programming Nonhomogeneous media Routing Vehicles

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DISCOUNTED CONTINUOUS-TIME CONSTRAINED MARKOV DECISION PROCESSES IN POLISH SPACES

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ANNALS OF APPLIED PROBABILITY 2011年第5期21卷 2016-2049页

作者： Guo, Xianping Song, Xinyuan Zhongshan Univ Sch Math & Computat Sci Guangzhou 510275 Guangdong Peoples R China Chinese Univ Hong Kong Dept Stat Shatin Hong Kong Peoples R China

This paper is devoted to studying constrained continuous-time Markov decision processes (MDPs) in the class of randomized policies depending on state histories. The transition rates may be unbounded, the reward and costs are admitted to be unbounded from above and from below, and the state and action spaces are Polish spaces. The optimality criterion to be maximized is the expected discounted rewards, and the constraints can be imposed on the expected discounted costs. First, we give conditions for the nonexplosion of underlying processes and the finiteness of the expected discounted rewards/costs. Second, using a technique of occupation measures, we prove that the constrained optimality of continuous-time MDPs can be transformed to an equivalent (optimality) problem over a class of probability measures. Based on the equivalent problem and a so-called (w) over bar -weak convergence of probability measures developed in this paper, we show the existence of a constrained optimal policy. Third, by providing a linear programming formulation of the equivalent problem, we show the solvability of constrained optimal policies. Finally, we use two computable examples to illustrate our main results.

关键词： Continuous-time Markov decision process unbounded transition rates occupation measure linear programming formulation constrained optimal policy

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Optimal Routing Strategy by Hose Model with Link-Traffic Bounds

Optimal Routing Strategy by Hose Model with Link-Traffic Bou...

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54th Annual IEEE Global Telecommunications Conference (GLOBECOM)

作者： Kitahara, Yoshiki Oki, Eiji Univ Electrocommun Dept Informat & Commun Tokyo Japan

ISBN: (纸本)9781424492688

This paper presents an optimal routing strategy based on the Hose model with bounds of Link Traffic (HLT), which we introduce. HLT is specified by the total traffic passing through each link in addition to the traffic bounds described in the hose model. The pipe model, which is specified by the exact traffic matrix, provides the best routing performance, but the traffic matrix is difficult to measure and predict accurately. While the hose model employs just the total outgoing/incoming traffic from/to each node, it offers lower routing performance than the pipe model, due to insufficient traffic information. The Hose model with bounds of Source-Destination Traffic (HSDT), where the upper and lower bounds of traffic demands for source-destination pairs are added as constraints, is a construction that lies between the pipe and hose models, but determining additional bounds is not easy for the network operators to specify. HLT, which lightens the difficulty of the pipe model, but narrows the range of traffic conditions specified by the hose model, offers better routing performance than the hose model. In addition, the HLT model resolves the difficulty of the HSDT model with regard to determining appropriate additional bounds. An optimal-routing formulation extended from the pipe model to the HLT model can not be solved as a regular linear programming (LP) problem. Our solution, the introduction of a duality theorem, turns this problem into an LP formulation that can be easily solved. Numerical results via simulations show that HLT offers 20-35% lower network congestion ratios than the hose model. In addition, the congestion ratios of the pipe and HLT models differ by less than 0.1.

关键词： Hose model Hoses IEEE Communications Society IP networks Numerical models Peer to peer computing Robustness Routing duality (mathematics) duality theorem linear programming linear programming formulation link-traffic bounds optimal routing strategy pipe model source-destination traffic telecommunication network routing telecommunication traffic traffic matrix

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