General stochastic single-machine scheduling with regular cost functions

We examine a stochastic scheduling model with n jobs and a single machine, where the processing times of the jobs are random variables with arbitrary distributions, and the machine is subject to stochastic breakdowns. Two types of regular performance measures, the total cost and the maximum cost, with general cost functions are investigated. We derive conditions under which a sequence in nondecreasing stochastic order of processing times minimizes the total expected cost, whereas a sequence in nondecreasing stochastic order of due dates minimizes the maximum expected cost. In addition, results on almost sure minimization of the total stochastic cost or the maximum stochastic cost are also provided. Applications of the general results to various special cases with commonly studied measures like mean weighted flowtime, mean weighted tardiness, weighted number of tardy jobs, maximum tardiness, etc., are discussed, which extend the previous results on these cases to more general situations.

[1]  Ali Allahverdi,et al.  Scheduling on M parallel machines subject to random breakdowns to minimize expected mean flow time , 1994 .

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

[3]  F. G. Forst.  Bicriterion stochastic scheduling on one or more machines , 1995 .

[4]  Subhash C. Sarin,et al.  Scheduling independent jobs with stochastic processing times and a common due date on parallel and identical machines , 1989 .

[5]  Hiroshi Kise,et al.  Note-On Baluts Algorithm and NP-Completeness for a Chance-Constrained Scheduling Problem , 1983 .

[6]  Michael Pinedo,et al.  Scheduling tasks with exponential service times on non-identical processors to minimize various cost functions , 1980, Journal of Applied Probability.

[7]  E. Lawler.  Sequencing Jobs to Minimize Total Weighted Completion Time Subject to Precedence Constraints , 1978 .

[8]  G. Rand.  Sequencing and Scheduling: An Introduction to the Mathematics of the Job-Shop , 1982 .

[9]  Michael Pinedo,et al.  Scheduling stochastic jobs with due dates on parallel machines , 1990 .

[10]  Stephen J. Balut,et al.  Scheduling to Minimize the Number of Late Jobs When Set-Up and Processing Times are Uncertain , 1973 .

[11]  Michael H. Rothkopf.  Scheduling with Random Service Times , 1966 .

[12]  K. Glazebrook.  Scheduling tasks with exponential service times on parallel processors , 1979 .

[13]  S. Ross,et al.  Scheduling Jobs Subject to Nonhomogeneous Poisson Shocks , 1980 .

[14]  J. Walrand,et al.  Scheduling jobs with stochastically ordered processing times on parallel machines to minimize expected flowtime , 1986, Journal of Applied Probability.

[15]  O. J. Boxma,et al.  Minimizing the expected weighted number of tardy jobs in stochastic flow shops , 1986 .

[16]  Wayne E. Smith.  Various optimizers for single‐stage production , 1956 .

[17]  A.H.G. Rinnooy Kan,et al.  Single‐machine scheduling subject to stochastic breakdowns , 1990 .

[18]  W. Townsend.  The Single Machine Problem with Quadratic Penalty Function of Completion Times: A Branch-and-Bound Solution , 1978 .

[19]  Michael H. Rothkopf,et al.  Scheduling Independent Tasks on Parallel Processors , 1966 .

[20]  William L. Maxwell,et al.  Single machine sequencing with random processing times and random due-dates† , 1969 .

[21]  E. Lawler.  A “Pseudopolynomial” Algorithm for Sequencing Jobs to Minimize Total Tardiness , 1977 .

[22]  T. Ibaraki,et al.  A polynomial time algorithm for a chance-constrained single machine scheduling problem , 1983 .

[23]  Kevin Mahon,et al.  Deterministic and Stochastic Scheduling , 1983 .

[24]  Jan Karel Lenstra,et al.  Recent developments in deterministic sequencing and scheduling: a survey : (preprint) , 1981 .

[25]  R. A. Blau.  N-Job, One Machine Sequencing Problems Under Uncertainty , 1973 .

[26]  Kevin D. Glazebrook,et al.  Scheduling stochastic jobs on a single machine subject to breakdowns , 1984 .

[27]  Subhash C. Sarin,et al.  Sequencing jobs on a single machine with a common due date and stochastic processing times , 1991 .

[28]  B. P. Banerjee,et al.  Single Facility Sequencing with Random Execution Times , 1965 .

[29]  Raymond E. Miller,et al.  Complexity of Computer Computations , 1972 .

[30]  Michael Pinedo,et al.  INEQUALITIES AND BOUNDS FOR THE SCHEDULING OF STOCHASTIC JOBS ON PARALLEL MACHINES , 1985 .

[31]  C. Derman,et al.  A Renewal Decision Problem , 1978 .

[32]  J. R. Jackson.  SCHEDULING A PRODUCTION LINE TO MINIMIZE MAXIMUM TARDINESS , 1955 .

[33]  R. Weber.  Scheduling jobs by stochastic processing requirements on parallel machines to minimize makespan or flowtime , 1982, Journal of Applied Probability.

[34]  H. Kise,et al.  AN EFFICIENT ALGORITHM FOR A CHANCE-CONSTRAINED SCHEDULING PROBLEM , 1982 .

[35]  Michael Pinedo,et al.  Stochastic Scheduling with Release Dates and Due Dates , 1983, Oper. Res..

[36]  Thomas Kämpke.  On the optimality of static priority policies in stochastic scheduling on parallel machines , 1987 .