Optimization of schedule robustness and stability under random machine breakdowns and processing time variability

In practice, scheduling systems are subject to considerable uncertainty in highly dynamic operating environments. The ability to cope with uncertainty in the scheduling process is becoming an increasingly important issue. This paper takes a proactive scheduling approach to study scheduling problems with two sources of uncertainty: processing time variability and machine breakdowns. Two robustness (expected total flow time and expected total tardiness) and three stability (the sum of the squared and absolute differences of the job completion times and the sum of the variances of the realized completion times) measures are defined. Special cases for which the measures can be easily optimized are identified. A dominance rule and two lower bounds for one of the robustness measures are developed and subseqently used in a branch-and-bound algorithm to solve the problem exactly. A beam search heuristic is also proposed to solve large problems for all five measures. The computational results show that the beam search heuristic is capable of generating robust schedules with little average deviation from the optimal objective function value (obtained via the branch-and-bound algorithm) and it performs significantly better than a number of heuristics available in the literature for all five measures.

[1]  Lars-Erik Lindgren.  Robustness and stability , 2007 .

[2]  Marshall L. Fisher,et al.  A dual algorithm for the one-machine scheduling problem , 1976, Math. Program..

[3]  Sheldon M. Ross,et al.  Introduction to probability models , 1975 .

[4]  Michael Pinedo,et al.  Scheduling: Theory, Algorithms, and Systems , 1994 .

[5]  Alexander H. G. Rinnooy Kan,et al.  Single machine flow-time scheduling with a single breakdown , 1989, Acta Informatica.

[6]  Robert H. Storer,et al.  Robustness Measures and Robust Scheduling for Job Shops , 1994 .

[7]  Frank Werner,et al.  Stability of an optimal schedule in a job shop , 1997 .

[8]  Sheldon M. Ross,et al.  Introduction to probability models , 1975 .

[9]  Willy Herroelen,et al.  Project scheduling under uncertainty: Survey and research potentials , 2005, Eur. J. Oper. Res..

[10]  Panagiotis Kouvelis,et al.  Robust scheduling to hedge against processing time uncertainty in single-stage production , 1995 .

[11]  Joseph Y.-T. Leung,et al.  Minimizing Total Tardiness on One Machine is NP-Hard , 1990, Math. Oper. Res..

[12]  Sanjay Mehta,et al.  Predictable scheduling of a single machine subject to breakdowns , 1999, Int. J. Comput. Integr. Manuf..

[13]  Averill M. Law,et al.  Simulation Modeling and Analysis , 1982 .

[14]  Reha Uzsoy,et al.  Predictable scheduling of a job shop subject to breakdowns , 1998, IEEE Trans. Robotics Autom..

[15]  Ihsan Sabuncuoglu,et al.  Robustness and stability measures for scheduling: single-machine environment , 2008 .

[16]  Joseph Y.-T. Leung,et al.  A note on scheduling parallel machines subject to breakdown and repair , 2004 .

[17]  C. Chu.  A branch-and-bound algorithm to minimize total tardiness with different release dates , 1992 .

[18]  Reha Uzsoy,et al.  Predictable scheduling of a single machine with breakdowns and sensitive jobs , 1999 .

[19]  Wei Li,et al.  On stochastic machine scheduling with general distributional assumptions , 1998, Eur. J. Oper. Res..

[20]  Ihsan Sabuncuoglu,et al.  A beam search-based algorithm and evaluation of scheduling approaches for flexible manufacturing systems , 1998 .

[21]  A.H.G. Rinnooy Kan,et al.  Scheduling on a single machine with a single breakdown to minimize stochastically the number of tardy jobs , 1991 .

[22]  Charles A. Holloway,et al.  Centralized Scheduling and Priority Implementation Heuristics for a Dynamic Job Shop Model , 1977 .

[23]  Sheldon M. Ross,et al.  Stochastic Processes , 2018, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics.

[24]  Pei-Chann Chang,et al.  One-machine rescheduling heuristics with efficiency and stability as criteria , 1993, Comput. Oper. Res..

[25]  A Gerodimos,et al.  Robust Discrete Optimization and its Applications , 1996, J. Oper. Res. Soc..

[26]  Roberto Tadei,et al.  A new decomposition approach for the single machine total tardiness scheduling problem , 1998, J. Oper. Res. Soc..

[27]  Robert H. Storer,et al.  A Graph-Theoretic Decomposition of the Job Shop Scheduling Problem to Achieve Scheduling Robustness , 1999, Oper. Res..

[28]  Ihsan Sabuncuoglu,et al.  Hedging production schedules against uncertainty in manufacturing environment with a review of robustness and stability research , 2009, Int. J. Comput. Integr. Manuf..

[29]  W. John Braun,et al.  Stochastic scheduling on a repairable machine with Erlang uptime distribution , 1998, Advances in Applied Probability.

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

[31]  Ari P. J. Vepsalainen.  Priority rules for job shops with weighted tardiness costs , 1987 .