β -Robust Parallel Machine Scheduling with Uncertain Durations

Uncertainty is a main characteristic of many real-world scheduling problems, and scheduling solutions should be robust against changes. In this paper, we consider parallel machines that the processing time of each task follows a normal distribution and total completion times as main optimization criterion. We propose a mixed-integer nonlinear programming (MINLP) model to minimize the risk of the completion time exceeding a fixed value. The objective is to find a β-robust schedule. Additionally, we use a proper approximation to convert the MINLP model to an integer programming one. The computational results on a small example are presented to demonstrate the effectiveness of the proposed approach. Furthermore, we compare and analyze the associated results of these models. Finally, the conclusion is provided.

[1]  Chengbin Chu,et al.  Identical parallel-machine scheduling under availability constraints to minimize the sum of completion times , 2009, Eur. J. Oper. Res..

[2]  Shanlin Yang,et al.  Non-identical parallel-machine scheduling research with minimizing total weighted completion times: Models, relaxations and algorithms ☆ , 2009 .

[3]  Michael Pinedo,et al.  Current trends in deterministic scheduling , 1997, Ann. Oper. Res..

[4]  Ming Zhao,et al.  A family of inequalities valid for the robust single machine scheduling polyhedron , 2010, Comput. Oper. Res..

[5]  Warren B. Powell,et al.  Solving Parallel Machine Scheduling Problems by Column Generation , 1999, INFORMS J. Comput..

[6]  Edward G. Coffman,et al.  Scheduling independent tasks to reduce mean finishing time , 1974, CACM.

[7]  Kenneth N. Brown,et al.  Scheduling with uncertain durations: Modeling beta-robust scheduling with constraints , 2009, Comput. Oper. Res..

[8]  Willy Herroelen,et al.  Scheduling for stability in single-machine production systems , 2007, J. Sched..

[9]  Mohammad Ranjbar,et al.  Two branch-and-bound algorithms for the robust parallel machine scheduling problem , 2012, Comput. Oper. Res..

[10]  J. Christopher Beck,et al.  Proactive Algorithms for Job Shop Scheduling with Probabilistic Durations , 2011, J. Artif. Intell. Res..

[11]  Roberto Musmanno,et al.  Robust scheduling of parallel machines with sequence-dependent set-up costs , 2005, Eur. J. Oper. Res..

[12]  Yugeng Xi,et al.  Robust and stable scheduling of a single machine with random machine breakdowns , 2006 .

[13]  Yeong-Dae Kim,et al.  A branch and bound algorithm for an identical parallel machine scheduling problem with a job splitting property , 2008, Comput. Oper. Res..

[14]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[15]  W. A. Horn.  Technical Note - Minimizing Average Flow Time with Parallel Machines , 1973, Oper. Res..

[16]  Christodoulos A. Floudas,et al.  A new robust optimization approach for scheduling under uncertainty: II. Uncertainty with known probability distribution , 2007, Comput. Chem. Eng..

[17]  George L. Vairaktarakis,et al.  Robust scheduling of a two-machine flow shop with uncertain processing times , 2000 .

[18]  Eugene L. Lawler,et al.  Sequencing and scheduling: algorithms and complexity , 1989 .

[19]  Christodoulos A. Floudas,et al.  Advances in robust optimization approaches for scheduling under uncertainty , 2005 .

[20]  Jatinder N. D. Gupta,et al.  Scheduling identical parallel machines to minimize total tardiness , 2008 .

[21]  Chung-Cheng Lu,et al.  Robust scheduling on a single machine to minimize total flow time , 2012, Comput. Oper. Res..

[22]  Mauro Dell'Amico,et al.  Heuristic and Exact Algorithms for the Identical Parallel Machine Scheduling Problem , 2008, INFORMS J. Comput..

[23]  Joseph Y.-T. Leung,et al.  Minimizing Mean Flow Time in Two-Machine Open Shops and Flow Shops , 1993, J. Algorithms.

[24]  Cynthia A. Phillips,et al.  Minimizing average completion time in the presence of release dates , 1998, Math. Program..

[25]  R. L. Daniels,et al.  β-Robust scheduling for single-machine systems with uncertain processing times , 1997 .

[26]  Philippe Baptiste,et al.  Lower bounds for parallel machine scheduling problems , 2008 .

[27]  Robert McNaughton,et al.  Scheduling with Deadlines and Loss Functions , 1959 .

[28]  Christodoulos A. Floudas,et al.  A new robust optimization approach for scheduling under uncertainty: : I. Bounded uncertainty , 2004, Comput. Chem. Eng..