Machine Scheduling with Resource Dependent Processing Times

We consider several parallel machine scheduling settings with the objective to minimize the schedule makespan. The most general of these settings is unrelated parallel machine scheduling. We assume that, in addition to its machine dependence, the processing time of any job is dependent on the usage of a scarce renewable resource. A given amount of that resource, e.g. workers, can be distributed over the jobs in process at any time, and the more of that resource is allocated to a job, the smaller is its processing time. This model generalizes classical machine scheduling problems, adding a time-resource tradeoff. It is also a natural variant of a generalized assignment problem studied previously by Shmoys and Tardos. On the basis of integer programming formulations for relaxations of the respective problems, we use LP rounding techniques to allocate resources to jobs, and to assign jobs to machines. Combined with Graham''s list scheduling, we thus prove the existence of constant factor approximation algorithms. Our performance guarantee is 6.83 for the most general case of unrelated parallel machine scheduling. We improve this bound for two special cases, namely to 5.83 whenever the jobs are assigned to machines beforehand, and to (5+e), e>0, whenever the processing times do not depend on the machine. Moreover, we discuss tightness of the relaxations, and derive inapproximability results.operations research and management science;

Scheduling unrelated parallel machines with resource-assignable sequence-dependent setup times

[EN] A novel scheduling problem that results from the addition of resource-assignable setups is presented in this paper. We consider an unrelated parallel machine problem with machine and job sequence-dependent setup times. The new characteristic is that the amount of setup time does not only depend on the machine and job sequence but also on the amount of resources assigned, which can vary between a minimum and a maximum. The aim is to give solution to real problems arising in several industries where frequent setup operations in production lines have to be carried out. These operations are indeed setups whose length can be reduced or extended according to the amount of resources assigned to them. The objective function considered is a linear combination of total completion time and the total amount of resources assigned. We present a mixed integer program (MIP) model and some fast dispatching heuristics. We carry out careful and comprehensive statistical analyses to study what characteristics of the problem affect the MIP model performance. We also study the effectiveness of the different heuristics proposed. © 2011 Springer-Verlag London Limited.The authors are indebted to the referees and editor for a close examination of the paper, which has increased its quality and presentation. This work is partially funded by the Spanish Ministry of Science and Innovation, under the project "SMPA-Advanced Parallel Multiobjective Sequencing: Practical and Theoretical Advances" with reference DPI2008-03511/DPI. The authors should also thank the IMPIVA-Institute for the Small and Medium Valencian Enterprise, for the project OSC with references IMIDIC/2008/137, IMIDIC/2009/198, and IMIDIC/2010/175.Ruiz García, R.; Andrés Romano, C. (2011). Scheduling unrelated parallel machines with resource-assignable sequence-dependent setup times. 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Eur J Oper Res 47(3):271–292Graham RL, Lawler EL, Lenstra JK, Rinnooy Kan AHG (1979) Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann Discrete Math 5:287–326Grigoriev E, Sviridenko M, Uetz M (2007) Unrelated parallel machine scheduling with resource dependent processing times. Math Program Ser A and B 110(1):209–228Guinet A (1991) Textile production systems: a succession of non-identical parallel processor shops. J Oper Res Soc 42(8):655–671Guinet A, Dussauchoy A (1993) Scheduling sequence dependent jobs on identical parallel machines to minimize completion time criteria. Int J Prod Res 31(7):1579–1594Horn WA (1973) Minimizing average flow time with parallel machines. Oper Res 21(3):846–847Kass GV (1980) An exploratory technique for investigating large quantities of categorical data. Appl Stat 29(2):119–127Kim DW, Kim KH, Jang W, Chen FF (2002) Unrelated parallel machine scheduling with setup times using simulated annealing. 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Discrete Appl Math 26(2–3):271–287Pinedo M (2002) Scheduling: theory, algorithms, and systems, 2nd edn. Prentice Hall, Upper SaddleRabadi G, Moraga RJ, Al-Salem A (2006) Heuristics for the unrelated parallel machine scheduling problem with setup times. J Intell Manuf 17(1):85–97Radhakrishnan S, Ventura JA (2000) Simulated annealing for parallel machine scheduling with earliness-tardiness penalties and sequence-dependent set-up times. Int J Prod Res 38(10):2233–2252Ruiz R, Sivrikaya Şerifoğlu F, Urlings T (2008) Modeling realistic hybrid flexible flowshop scheduling problems. Comput Oper Res 35(4):1151–1175Sivrikaya-Serifoglu F, Ulusoy G (1999) Parallel machine scheduling with earliness and tardiness penalties. Comput Oper Res 26(8):773–787Webster ST (1997) The complexity of scheduling job families about a common due date. Oper Res Lett 20(2):65–74Weng MX, Lu J, Ren H (2001) Unrelated parallel machines scheduling with setup consideration and a total weighted completion time objective. 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Hybrid Ant Colony Optimization For Fuzzy Unrelated Parallel Machine Scheduling Problems

This study extends the best hybrid ant colony optimization variant developed by Liao et al. (2014) for crisp unrelated parallel machine scheduling problems to solve fuzzy unrelated parallel machine scheduling problems in consideration of trapezoidal fuzzy processing times, trapezoidal fuzzy sequencing dependent setup times and trapezoidal fuzzy release times. The objective is to find the best schedule taking minimum fuzzy makespan in completing all jobs. In this study, fuzzy arithmetic is used to determine fuzzy completion times of jobs and defuzzification function is used to convert fuzzy numbers back to crisp numbers for ranking. Eight fuzzy ranking methods are tested to find the most feasible one to be employed in this study. The fuzzy arithmetic testing includes four different cases and each case with the following operations separately, i.e., addition, subtraction, multiplication and division, to investigate the spread of fuzziness as fuzzy numbers are subject to more and more number of operations. The effect of fuzzy ranking methods on hybrid ant colony optimization (hACO) is investigated. To prove the correctness of our methodology and coding, unrelated parallel machine scheduling with fuzzy numbers and crisp numbers are compared based on scheduling problems up to 15 machines and 200 jobs. Relative percentage deviation (RPD) is used to evaluate the performance of hACO in solving fuzzy unrelated parallel machine scheduling problems. A numerical study on large-scale scheduling problems up to 20 machines and 200 jobs is conducted to assess the performance of the hACO algorithm. For comparison, a discrete particle swarm optimization (dPSO) algorithm is implemented for fuzzy unrelated parallel machine scheduling problem as well. The results show that the hACO has better performance than dPSO not only in solution quality in terms of RPD value, but also in computational time

Optimizing Production Schedule with Energy Consumption and Demand Charges in Parallel Machine Setting

Environmental sustainability concerns, along with the growing need for electricity and associated costs, make energy-cost reduction an inevitable decision-making criterion in production scheduling. In this research, we study the problem of production scheduling on nonidentical parallel machines with machine-dependent processing times and known job release dates to minimize total completion time and energy costs. The energy costs in this study include demand and consumption charges. We present a mixed-integer nonlinear model to formulate the problem. The model is then linearized and its performance is tested through numerical experiments

Approximation Schemes for Machine Scheduling

In the classical problem of makespan minimization on identical parallel machines, or machine scheduling for short, a set of jobs has to be assigned to a set of machines. The jobs have a processing time and the goal is to minimize the latest finishing time of the jobs. Machine scheduling is well known to be NP-hard and thus there is no polynomial time algorithm for this problem that is guaranteed to find an optimal solution unless P=NP. There is, however, a polynomial time approximation scheme (PTAS) for machine scheduling, that is, a family of approximation algorithms with ratios arbitrarily close to one. Whether a problem admits an approximation scheme or not is a fundamental question in approximation theory. In the present work, we consider this question for several variants of machine scheduling. We study the problem where the machines are partitioned into a constant number of types and the processing time of the jobs is also dependent on the machine type. We present so called efficient PTAS (EPTAS) results for this problem and variants thereof. We show that certain cases of machine scheduling with assignment restrictions do not admit a PTAS unless P=NP. Moreover, we introduce a graph framework based on the restrictions of the jobs and use it in the design of approximation schemes for other variants. We introduce an enhanced integer programming formulation for assignment problems, show that it can be efficiently solved, and use it in the EPTAS design for variants of machine scheduling with setup times. For one of the problems, we show that there is also a PTAS in the case with uniform machines, where machines have speeds influencing the processing times of the jobs. We consider cases in which each job requires a certain amount of a shared renewable resource and the processing time is depended on the amount of resource it receives or not. We present so called asymptotic fully polynomial time approximation schemes (AFPTAS) for the problems

A Dynamic Heuristic for the Stochastic Unrelated Parallel Machine Scheduling Problem

This paper addresses the problem of batch scheduling in an unrelated parallel machine environment with sequence dependent setup times and an objective of minimizing the total weighted mean completion time. The jobs\u27 processing times and setup times are stochastic for better depiction of the real world. This is a NP-hard problem and in this paper, new heuristics are developed and compared to existing ones using simulation. The results and analysis obtained from the computational experiments proved the superiority of the proposed algorithm PMWP over the other algorithms presented

A statistical comparison of metaheuristics for unrelated parallel machine scheduling problems with setup times

Manufacturing scheduling aims to optimize one or more performance measures by allocating a set of resources to a set of jobs or tasks over a given period of time. It is an area that considers a very important decision-making process for manufacturing and production systems. In this paper, the unrelated parallel machine scheduling problem with machine-dependent and job-sequence-dependent setup times is addressed. This problem involves the scheduling of tasks on unrelated machines with setup times in order to minimize the makespan. The genetic algorithm is used to solve small and large instances of this problem when processing and setup times are balanced (Balanced problems), when processing times are dominant (Dominant P problems), and when setup times are dominant (Dominant S problems). For small instances, most of the values achieved the optimal makespan value, and, when compared to the metaheuristic ant colony optimization (ACOII) algorithm referred to in the literature, it was found that there were no significant differences between the two methods. However, in terms of large instances, there were significant differences between the optimal makespan obtained by the two methods, revealing overall better performance by the genetic algorithm for Dominant S and Dominant P problems.FCT—Fundação para a Ciência e Tecnologia through the R&D Units Project Scope UIDB/00319/2020 and EXPL/EME-SIS/1224/2021 and PhD grant UI/BD/150936/2021

Predictive/reactive scheduling with controllable processing times and earliness-tardiness penalties

In this study, a machine scheduling problem with controllable processing times in a parallel-machine environment is considered. The objectives are the minimization of manufacturing cost, which is a convex function of processing time, and total weighted earliness and tardiness. It is assumed that parts have job-dependent earliness and tardiness penalties and distinct due dates, and idle time is allowed. The problem is formulated as a time-indexed integer programming model with discrete processing time alternatives for each part. A linear-relaxation-based algorithm is used to assign the parts to the machines and to find a sequence on each machine. A non-linear programming model is proposed to find the optimal starting and processing times of the parts for a given sequence. The proposed non-linear programming model is converted to a minimum-cost network flow model by piecewise linearization of the convex manufacturing cost in the objective function. The proposed method is used to find initial schedules in predictive scheduling. The proposed models are revised to incorporate a stability measure for reacting to unexpected disruptions such as machine breakdown, arrival of a new job, delay in the arrival or the shortage of materials in reactive scheduling