An experimental and analytical study of order constraints for single machine scheduling with quadratic cost

Höhn, Wiebke; Jacobs, Tobias

doi:10.1137/1.9781611972924.11

Cited by 15 publications

(27 citation statements)

References 1 publication

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“…As we showed in Höhn and Jacobs [2012a], the local α-gap constraint is observed by polynomial cost functions with nonnegative coefficients for α being the cost function's degree. This leads directly to the following corollary.…”

Section: Combinatorial Analysis Of Smith's Rulementioning

confidence: 89%

“…Refinements of this method have been described in a long series of articles [Bagga and Kalra 1980;Gupta and Sen 1984;Sen et al 1990;Della Croce et al 1995;Mondal and Sen 2000]. In a related article we combine and improve the methods of these articles, and compare them in an extensive computational study [Höhn and Jacobs 2012a]. Also, for tardiness cost g j : t → max{0, t − d j } with job-specific deadlines d j , there are several articles aiming at the design of exact algorithms (see, e.g., the survey by Potts and Strusevich [2009]).…”

Section: Relatedmentioning

confidence: 99%

“…In Höhn and Jacobs [2012a], we have discussed the general concept of order constraints in the context of exact solution methods. Local α-gap constraints are a special kind of such constraints.…”

Section: Combinatorial Analysis Of Smith's Rulementioning

confidence: 99%

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On the Performance of Smith’s Rule in Single-Machine Scheduling with Nonlinear Cost

Höhn

Jacobs²

2015

ACM Trans. Algorithms

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We consider a single-machine scheduling problem. Given some continuous, nondecreasing cost function, we aim to compute a schedule minimizing the weighted total cost, where the cost of each job is determined by the cost function value at its completion time. This problem is closely related to scheduling a single machine with nonuniform processing speed. We show that for piecewise linear cost functions it is strongly NP-hard. The main contribution of this article is a tight analysis of the approximation guarantee of Smith's rule under any convex or concave cost function. More specifically, for these wide classes of cost functions we reduce the task of determining a worst-case problem instance to a continuous optimization problem, which can be solved by standard algebraic or numerical methods. For polynomial cost functions with positive coefficients, it turns out that the tight approximation ratio can be calculated as the root of a univariate polynomial. We show that this approximation ratio is asymptotically equal to k (k−1)/(k+1) , denoting by k the degree of the cost function. To overcome unrealistic worst-case instances, we also give tight bounds for the case of integral processing times that are parameterized by the maximum and total processing time. ACM Reference Format:Wiebke Höhn and Tobias Jacobs. 2015. On the performance of Smith's rule in single-machine scheduling with nonlinear cost. ACM Trans.

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Section: Combinatorial Analysis Of Smith's Rulementioning

confidence: 89%

Section: Relatedmentioning

confidence: 99%

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On the Performance of Smith’s Rule in Single-Machine Scheduling with Nonlinear Cost

Höhn

Jacobs²

2015

ACM Trans. Algorithms

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“…Mainly, the branch-and-bound approaches with pruning rules implying order properties have been focused on the quadratic penalty function f (t) := t 2 (see [1][2][3]9,12,15,17,18]). …”

Section: Related Workmentioning

confidence: 99%

“…This approach was initiated by [14] and developed further in [4,9] for a related scheduling problem. Formally, we model of scheduling problem as a directed acyclic graph consisting of all subsets S ⊆ {1, .…”

Section: Corollary 1 (Our Rules) For Any Two Jobs I J With W I > W Jmentioning

confidence: 99%

For the airplane refueling problem local precedence implies global precedence

Vásquez

2014

Optim Lett

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We are given n airplanes, which can refuel one another during the flight. Each airplane has a specific tank volume and gas consumption rate. The goal of the airplane refueling problem is to find a drop out permutation for the planes that maximizes the distance traveled by the last plane to drop out. This paper studies some structural properties of the problem and proposes pruning rules for an exact resolution.

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A Fast Exact Algorithm for Airplane Refueling Problem

Luo

et al. 2019

Combinatorial Optimization and Applications

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We consider the airplane refueling problem, where we have a fleet of airplanes that can refuel each other. Each airplane is characterized by specific fuel tank volume and fuel consumption rate, and the goal is to find a drop out order of airplanes that last airplane in the air can reach as far as possible. This problem is equivalent to the scheduling problem 1|| wj (− 1 C j ). Based on the dominance properties among jobs, we reveal some structural properties of the problem and propose a recursive algorithm to solve the problem exactly. The running time of our algorithm is directly related to the number of schedules that do not violate the dominance properties. An experimental study shows our algorithm outperforms state of the art exact algorithms and is efficient on larger instances.

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An experimental and analytical study of order constraints for single machine scheduling with quadratic cost

Cited by 15 publications

References 1 publication

On the Performance of Smith’s Rule in Single-Machine Scheduling with Nonlinear Cost

On the Performance of Smith’s Rule in Single-Machine Scheduling with Nonlinear Cost

For the airplane refueling problem local precedence implies global precedence

A Fast Exact Algorithm for Airplane Refueling Problem

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