On the Complexity of Some Common Geometric Location Problems

Megiddo, Nimrod; Supowit, Kenneth J.

doi:10.1137/0213014

Cited by 568 publications

(313 citation statements)

References 11 publications

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“…The p-median problem is known to be NP-hard (Megiddo and Supowit 1984). In fact, it is shown (Lin and Vitter 1992) that there is no polynomial-time algorithm with a constant worst-case bound for the p-median problem.…”

Section: Related Literature and Problem Complexitymentioning

confidence: 99%

Coil Batching to Improve Productivity and Energy Utilization in Steel Production

Tang

Meng

Chen

et al. 2016

M&SOM

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This paper investigates a practical batching decision problem that arises in the batch annealing operations in the cold rolling stage of steel production faced by most large iron and steel companies in the world. The problem is to select steel coils from a set of waiting coils to form batches to be annealed in available batch annealing furnaces and choose a median coil for each furnace. The objective is to maximize the total reward of the selected coils less the total coil-coil and coil-furnace mismatching cost. For a special case of the problem that arises frequently in practical settings where the coils are all similar and there is only one type of furnace available, we develop a polynomial-time dynamic programming algorithm to obtain an optimal solution. For the general case of the problem which is strongly NP-hard, an exact branch-and-price-and-cut solution algorithm is developed using a column and row generation framework. A variable reduction strategy is also proposed to accelerate the algorithm. The algorithm is capable of solving medium size instances to optimality within a reasonable computation time. In addition, a tabu search heuristic is proposed for solving larger instances. Three simple search neighborhoods as well as a sophisticated variable depth neighborhood are developed. This heuristic can generate near-optimal solutions for large instances within a short computation time.Using both randomly generated and real-world production data sets, it is shown that our algorithms are superior to a typical rule-based planning approach used by many steel plants. A decision support system that embeds our algorithms was developed and implemented at Baosteel to replace their rule-based planning method. The use of the system brings significant benefits to Baosteel, including an annual net profit increase of at least 1.76 million US Dollars, and a large reduction of standard coal consumption and carbon dioxide emissions.

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Section: Related Literature and Problem Complexitymentioning

confidence: 99%

Coil Batching to Improve Productivity and Energy Utilization in Steel Production

Tang

Meng

Chen

et al. 2016

M&SOM

View full text Add to dashboard Cite

show abstract

“…, p m ) are distinct) but not convex, thus making the solution of the continuous m-median problem hard in the general case. It is known (Agarwal and Sharir, 1998;Megiddo and Supowit, 1984) that the discrete version of the m-median problem is NP-hard for d 2. Gradient algorithms for the continuous m-median problems can be designed by means of the equality @H m (P, Q)…”

Section: A the Continuous Multi-median Problemmentioning

confidence: 99%

UAV Routing and Coordination in Stochastic, Dynamic Environments

Enright¹,

Pavone

Savla

2014

Handbook of Unmanned Aerial Vehicles

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Recent years have witnessed great advancements in the science and technology for unmanned aerial vehicles (UAVs), e.g., in terms of autonomy, sensing, and networking capabilities. This chapter surveys algorithms on task assignment and scheduling for one or multiple UAVs in a dynamic environment, in which targets arrive at random locations at random times, and remain active until one of the UAVs flies to the target's location and performs an on-site task. The objective is to minimize some measure of the targets' activity, e.g., the average amount of time during which a target remains active. The chapter focuses on a technical approach that relies upon methods from queueing theory, combinatorial optimization, and stochastic geometry. The main advantage of this approach is its ability to provide analytical estimates of the performance of the UAV system on a given problem, thus providing insight into how performance is affected by design and environmental parameters, such as the number of UAVs and the target distribution. In addition, the approach provides provable guarantees on the system's performance with respect to an ideal optimum. To illustrate this approach, a variety of scenarios are considered, ranging from the simplest case where one UAV moves along continuous paths and has unlimited sensing capabilities, to the case where the motion of the UAV is subject to curvature constraints, and finally to the case where the UAV has a finite sensor footprint. Finally, the problem of cooperative routing algorithms for multiple UAVs is considered, within the same queueing-theoretical framework, and with a focus on control decentralization.

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“…Megiddo and Supowit (1984) and Zemel (1984)). For a single vehicle, the average distance to a random point, generated according to a probability density function is given by the Weber or the continuous 1-median function, for which there exists a global minimizer as shown in Fekete et al (2005), termed as the median.…”

Section: Introductionmentioning

confidence: 99%

Vehicle placement to intercept moving targets

Bopardikar

Smith

Bullo

2010

Proceedings of the 2010 American Control Conference

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We address optimal placement of vehicles with simple motion, to intercept a mobile target that arrives stochastically on a line segment. The optimality of vehicle placement is measured through a cost function associated with intercepting the target. With a single vehicle, we assume that the target either moves with fixed speed and in a fixed direction or moves to maximize the vertical height or intercept time. We show that each of the corresponding cost functions is convex, has smooth gradient and has a unique minimizing location, and so the optimal vehicle placement is obtained by any standard gradient-based optimization technique. With multiple vehicles, we assume that the target moves with fixed speed and in fixed direction. We present a discrete time partitioning and gradient-based algorithm, and characterize conditions under which the algorithm asymptotically leads the vehicles to a set of critical configurations of the cost function.

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On the Complexity of Some Common Geometric Location Problems

Cited by 568 publications

References 11 publications

Coil Batching to Improve Productivity and Energy Utilization in Steel Production

Coil Batching to Improve Productivity and Energy Utilization in Steel Production

UAV Routing and Coordination in Stochastic, Dynamic Environments

Vehicle placement to intercept moving targets

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