Complexity of single machine scheduling problems under scenario-based uncertainty

Aloulou, Mohamed Ali; Croce, Federico Della

doi:10.1016/j.orl.2007.11.005

Cited by 70 publications

(43 citation statements)

References 8 publications

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“…Let p 1 = 3, p 2 = 1, p 3 = 2,p 1 = 1,p 2 = 10,p 3 = 5. By definition we have U Γ p = {(3, 1, 2), (4, 1, 2), (3,11,2), (3, 1, 7)}. Notice that a schedule is completely caracterized by a permutation of the job.…”

Section: Example For 1||umentioning

confidence: 99%

“…We prove that 1||U Γ p | C j is polynomial by extending Theorem 1. Comparing with [2,6,23], the result illustrates how U Γ -robust scheduling can lead to more tractable problems than robust scheduling with arbitrary uncertainty sets. We show then that 1||U Γ p | w j C j is weakly N P-hard if Γ = 1 and strongly N P-hard if Γ > 1.…”

mentioning

confidence: 91%

“…In spite of its practical relevance, robust scheduling has hardly become a practical tool since differents papers [2,6,23] have shown that very simple scheduling problems become N P-hard as soon as U contains more than one scenario. This is the case, for instance, for minimizing the sum of completion times on a single machine, which does not admit a Polynomial Time Approximation Scheme (PTAS) either, unless P = N P (see [17]).…”

Section: Introductionmentioning

confidence: 99%

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Robust scheduling with budgeted uncertainty

Bougeret¹,

Pessoa

Poss³

2019

Discrete Applied Mathematics

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In this work we study min max robust scheduling problems assuming that the processing times can take any value in the budgeted uncertainty set introduced by Sim (2003,2004). We consider problems on a single machine that minimize the (weighted and unweighted) sum of completion times and problems that minimize the makespan on parallel and unrelated machines. We provide polynomial algorithms and approximation algorithms: constant factor, average non-constant factor, (fully or not) polynomial time approximation schemes. In addition, we prove that the robust version of minimizing the weighted completion time on a single machine is N P-hard in the strong sense.Keywords approximation algorithms, robust optimization, scheduling IntroductionScheduling is a very wide topic in combinatorial optimization with applications ranging from production and manufacturing systems to transportation and logistics systems. Stated generally, the objective of scheduling is to allocate optimally scarce resources to activities over time. The practical relevance and the difficulty of solving the general scheduling problem have motivated an intense research activity in a large variety of scheduling environments. Scheduling problems are usually defined in the following way: given a set of n jobs represented by J , a set of m machines represented by M, and processing times represented by the tuple p, we look for a schedule σ of the jobs on the machines that satisfies the side constraints, represented by the set S of feasible schedules, and minimize objective function f (σ, p). Formally, this amounts to solve optimization problem min σ∈S f (σ, p).Various sources of uncertainty affect real scheduling problems, among which machine breakdowns, working environment changes, worker performance instabilities, tool quality variations and unavailability. Ignoring these uncertainties usually yields schedules that perform poorly under real conditions. Hence, researchers have introduced frameworks where the uncertainty is directly taken into account either by considering random variables as input or in a worst-case approach where the uncertainty parameters are constrained in a set. These frameworks are respectively denoted by Stochastic Programming and Robust Optimization (RO). We disregard the former in this paper because of its requirement for a probabilistic distribution of the random inputs, which is very difficult to obtain in practice. We focus instead on Robust Scheduling, which models the uncertainty on the processing times by a finite set U ⊂ N n . 1 In the robust problem, the maximum value of f (σ, p) over all p ∈ U should be minimized. Formally, this amounts to solve optimization problem min σ∈S max p∈U f (σ, p), or equivalently, min σ∈S F (σ, U ) where F (σ, U ) = max p∈U f (σ, p) represents the robust objective function. We say that a schedule σ * ∈ S is robust if it solves the associated scheduling problem min σ∈S F (σ, U ).Robust schedules are desirable from a practical perspective because they hedge against adverse conditions of t...

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Section: Example For 1||umentioning

confidence: 99%

mentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

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Robust scheduling with budgeted uncertainty

Bougeret¹,

Pessoa

Poss³

2019

Discrete Applied Mathematics

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“…de Farias et al (2010) identify a family of valid inequalities to strengthen the mixed-integer formulation of their problem. Furthermore, Aloulou and Croce (2008) provide several complexity results in the domain of robust single-machine scheduling. There also exist a few recent papers that propose scenario-based models for scheduling problems with multiple machines (see, e.g., Alonso-Ayuso et al, 2007;Kasperski et al, 2012); however, these also rely on risk-neutral or robust approaches.…”

Section: Machine Scheduling Under Uncertaintymentioning

confidence: 99%

Minimizing value-at-risk in single-machine scheduling

2016

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The vast majority of the machine scheduling literature focuses on deterministic problems in which all data is known with certainty a priori. In practice, this assumption implies that the random parameters in the problem are represented by their point estimates in the scheduling model. The resulting schedules may perform well if the variability in the problem parameters is low. However, as variability increases accounting for this randomness explicitly in the model becomes crucial in order to counteract the ill effects of the variability on the system performance. In this paper, we consider single-machine scheduling problems in the presence of uncertain parameters. We impose a probabilistic constraint on the random performance measure of interest, such as the total weighted completion time or the total weighted tardiness, and introduce a generic risk-averse stochastic programming model. In particular, the objective of the proposed model is to find a non-preemptive static job processing sequence that minimizes the value-at-risk (VaR) of the random performance measure at a specified confidence level. We propose a Lagrangian relaxation-based scenario decomposition method to obtain lower bounds on the optimal VaR and provide a stabilized cut generation algorithm to solve the Lagrangian dual problem. Furthermore, we identify promising schedules for the original problem by a simple primal heuristic. An extensive computational study on two selected performance measures is presented to demonstrate the value of the proposed model and the effectiveness of our solution method.

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“…Yang and Yu (2002) showed that a min-max version of the same problem is NP-hard and presented a dynamic programming method and two heuristic algorithms where min-max version minimizes maximum cost over all scenarios. Aloulou and Croce (2008) considered a min-max version of the single-machine scheduling problem with uncertain due dates, weights, and processing times. The performance measures considered were total weighted completion times, the general maximum cost function, and the number of late jobs.…”

Section: Literature Reviewmentioning

confidence: 99%

Min-Max Regret Version of an m-Machine Ordered Flow Shop with Uncertain Processing Times

Park¹,

Choi²

2015

Management Science and Financial Engineering

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We consider an m-machine flow shop scheduling problem to minimize the latest completion time, where processing times are uncertain. Processing time uncertainty is described through a finite set of processing time vectors. The objective is to minimize maximum deviation from optimality for all scenarios. Since this problem is known to be NPhard, we consider it with an ordered property. We discuss optimality properties and develop a pseudo-polynomial time approach for the problem with a fixed number of machines and scenarios. Furthermore, we find two special structures for processing time uncertainty that keep the problem NP-hard, even for two machines and two scenarios. Finally, we investigate a special structure for uncertain processing times that makes the problem polynomially solvable.

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Complexity of single machine scheduling problems under scenario-based uncertainty

Cited by 70 publications

References 8 publications

Robust scheduling with budgeted uncertainty

Robust scheduling with budgeted uncertainty

Minimizing value-at-risk in single-machine scheduling

Min-Max Regret Version of an m-Machine Ordered Flow Shop with Uncertain Processing Times

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