On the exact solution of a large class of parallel machine scheduling problems

Bulhões, Teobaldo; Sadykov, Ruslan; Subramanian, Anand; Uchôa, Eduardo

doi:10.1007/s10951-020-00640-z

Cited by 13 publications

(8 citation statements)

References 41 publications

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“…The latter authors proposed new branching schemes and suppression mechanisms for the pricing sub-problem. The generic work of [12] studied a larger class of parallel machines with identical, uniform and unrelated machines with regular and non-regular objectives. They proposed a B&P algorithm with new cuts and proved optimality (previously unknown) for a number of classical instances.…”

Section: Related Workmentioning

confidence: 99%

“…The restrictions below the line can be used to incorporate the knowledge that jobs arriving not earlier than d must follow the WSPT rule as per Lemma 1. Constraint sets ( 8) and ( 9) relate the release date variables r j with the binary variable rk used to indicate that a job arrives exactly at time d. Constraint sets (10) and (11) ensure that the jobs on each machine arriving not earlier than d. The last constraint set (12) makes rj a binary variable.…”

Section: Milp Approachmentioning

confidence: 99%

“….5% (8.5%) 1 x 300 0.0% (0.0%) 14.9% (21.0%) 1.2% (2.8%) 0.0% (0.0%) 0.0% (0.0%) 0.0% (0.0%) 0.0% (0.0%) 12.7% (23.3%) 1 x 400 0.0% (0.0%) 17.6% (25.2%) 1.6% (2.9%) 0.0% (0.0%) 0.0% (0.0%) 0.0% (0.0%) 0.0% (0.0%) 21.9% (30.7%) 1 x 500 0.0% (0.0%) 18.3% (25.9%) 2.0% (3.3%) 0.0% (0.0%) 0.0% (0.0%) 0.0% (0.0%) 0.0% (0.0%)…”

Section: Sizementioning

confidence: 99%

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Single and Parallel Machine Scheduling with Variable Release Dates

Mohr,

Mejía,

Yuraszeck

2021

Preprint

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In this paper we study a simple extension of the total weighted flowtime minimization problem for single and identical parallel machines. While the standard problem simply defines a set of jobs with their processing times and weights and assumes that all jobs have release date 0 and have no deadline, we assume that the release date of each job is a decision variable that is only constrained by a single global latest arrival deadline. To our knowledge, this simple yet practically highly relevant extension has never been studied. Our main contribution is that we show the NPcompleteness of the problem even for the single machine case and provide an exhaustive empirical study of different typical approaches including genetic algorithms, tree search, and constraint programming.

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Section: Related Workmentioning

confidence: 99%

Section: Milp Approachmentioning

confidence: 99%

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Single and Parallel Machine Scheduling with Variable Release Dates

Mohr,

Mejía,

Yuraszeck

2021

Preprint

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“…Their algorithm is based on an arc-time-indexed formulation and is improved with a number of combinatorial techniques, including variable fixing by reduced costs, extended capacity cuts, dual stabilization, and the direct solution of the formulation by a MILP solver if the fixing procedure had consistently reduced the number of variables. The method in [78] was later extended by Bulhões et al [14], who proposed a branch-and-cut-and-price algorithm to solve a path formulation for parallel machine scheduling in which the branching is based on the variables of the arc flow model.…”

Section: Scheduling Problemsmentioning

confidence: 99%

Arc Flow Formulations Based on Dynamic Programming: Theoretical Foundations and Applications

de Lima,

Alves,

Clautiaux

et al. 2020

Preprint

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Network flow formulations are among the most successful tools to solve optimization problems. One of such formulations is the arc flow, where variables represent flows on individual arcs of the network. For NP-hard problems, polynomial-sized arc flow models typically provide weak linear relaxations and may have too much symmetry to be efficient in practice. Instead, arc flow models with a pseudo-polynomial size usually provide strong relaxations and are efficient in practice. The interest in pseudo-polynomial arc flow formulations has grown considerably in the last twenty years, in which they have been used to solve many open instances of hard problems. A remarkable advantage of arc flow models is the possibility to solve practical-sized instances directly by a Mixed Integer Linear Programming solver, avoiding the implementation of complex methods based on column generation.In this survey, we present theoretical foundations of pseudo-polynomial arc flow formulations, by showing a relation between their network and Dynamic Programming (DP). This relation allows a better understanding of the strength of these formulations, through a link with models obtained by Dantzig-Wolfe decomposition. The relation with DP also allows a new perspective to relate state-space relaxation methods for DP with arc flow models. We also present a dual point of view to contrast the linear relaxation of arc flow models with that of models based on paths and cycles. To conclude, we review the main solution methods and applications of arc flow models based on DP in several domains such as cutting, packing, scheduling, and routing.

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“…Further research in this area may include the study of different objective functions, such as makespan minimization or the sum of weighted completion times; to this respect, it should be noted that the current paper also contributes to the development of a solution method for a corresponding robust variant of P ||C max , which could be solved using RMAP as a subproblem in a binary search procedure, similarly to Dell'Amico et al (2008). There are also possible branching schemes for RMAP that could preserve a certain structure for the pricing subproblem such that that problem can still be solved by a DP algorithm after branching (e.g., Chen and Powell, 1999;Bulhões et al, 2017), which deserve further investigation. Other opportunities for studying richer scheduling models are legion, such as the introduction of precedence constraints or the inclusion of multiple resource categories.…”

Section: Sensitivity Analysismentioning

confidence: 99%

The robust machine availability problem – bin packing under uncertainty

2018

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We define and solve the robust machine availability problem in a parallel machine environment, which aims to minimize the number of identical machines required while completing all the jobs before a given deadline. The deterministic version of this problem essentially coincides with the bin packing problem. Our formulation preserves a user-defined robustness level regarding possible deviations in the job durations. For better computational performance, a branch-andprice procedure is proposed based on a set covering reformulation. We use zero-suppressed binary decision diagrams for solving the pricing problem, which enable us to manage the difficulty entailed by the robustness considerations as well as by extra constraints imposed by branching decisions. Computational results are reported that show the effectiveness of a pricing solver with zero-suppressed binary decision diagrams compared with a mixed integer programming solver.

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On the exact solution of a large class of parallel machine scheduling problems

Cited by 13 publications

References 41 publications

Single and Parallel Machine Scheduling with Variable Release Dates

Single and Parallel Machine Scheduling with Variable Release Dates

Arc Flow Formulations Based on Dynamic Programming: Theoretical Foundations and Applications

The robust machine availability problem – bin packing under uncertainty

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