Note—A Node Elimination Procedure for Townsend's Algorithm for Solving the Single Machine Quadratic Penalty Function Scheduling Problem

Bagga, Pallavi; Kalra, K.R.

doi:10.1287/mnsc.26.6.633

Cited by 35 publications

(16 citation statements)

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“…Townsend [1978] was the first to propose a practically efficient branch-and-bound scheme for the same problem. 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].…”

Section: Relatedmentioning

confidence: 99%

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: Relatedmentioning

confidence: 99%

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%

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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“…β = 2. Branch-and-bound approaches with pruning rules implying order properties have been proposed, see [16,2,13,1,5,15]. In 2000, Mondal and Sen [12] conjectured that β = 2, (w i ≥ w j ) ∧ (w i /p i > w j /p j ) implies the global order property i ≺ g j, and provided experimental evidence that this constraint would significantly improve the runtime of a branch-and-prune search.…”

Section: Related Workmentioning

confidence: 99%

“…Most research focused on the quadratic objective function, i.e. for β = 2, and exact algorithms have been proposed [12,5,13,2,16].…”

Section: Introductionmentioning

confidence: 99%

Order constraints for single machine scheduling with non-linear cost

Dürr¹,

Vásquez²

2013

2014 Proceedings of the Sixteenth Workshop on Algorithm Engineering and Experiments (ALENEX)

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International audienceTypically in a scheduling problem we are given jobs of different processing times $p_j$ and different priority weights $w_j$, and need to schedule them on a single machine in order to minimize a specific cost function. In this paper we consider the non-linear objective function $\sum w_j C_j^\beta$, where $C_j$ is the completion time of job $j$ and $\beta>0$ is some arbitrary real constant. Except for $\beta=1$ the complexity status of this problem is open. Past research mainly focused on the quadratic case ($\beta=2$) and proposed different techniques to speed up exact algorithms. This paper proposes new pruning rules and generalizations of existing rules to non-integral $\beta$. An experimental study evaluates the impact of the proposed rules on the exact algorithm A*

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Note—A Node Elimination Procedure for Townsend's Algorithm for Solving the Single Machine Quadratic Penalty Function Scheduling Problem

Cited by 35 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

Order constraints for single machine scheduling with non-linear cost

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