Author's Personal Copy Applied Soft Computing Parallel Machine Scheduling Problem to Minimize the Makespan with Resource Dependent Processing Times

This paper considers the identical parallel machine scheduling problem to minimize the makespan with controllable processing times, in which the processing times are linear decreasing functions of the consumed resource. The total resource consumption is limited. This problem is NP-hard even if the total resource consumption equals to zero. Two kinds of machines, critical machine and non-critical machine, are defined. Some theoretical results are provided. And then, a simulated annealing algorithm is designed to obtain the near-optimal solutions with high quality. To evaluate the performance of the proposed algorithm, we generate the random test data in our experiment to simulate the ingot preheating before hot-rolling process in steel mills. The accuracy and efficiency of the simulated annealing algorithm is tested based on the data with problem size varying from 200 jobs to 1000 jobs. By examining 10,000 randomly generated instances, the proposed simulated annealing algorithm shows an excellent performance in not only the solution quality but also the computation time.

[1]  Raymond G. Vickson,et al.  Two Single Machine Sequencing Problems Involving Controllable Job Processing Times , 1980 .

[2]  T.C.E. Cheng,et al.  Resource optimal control in some single-machine scheduling problems , 1994, IEEE Trans. Autom. Control..

[3]  Dvir Shabtay,et al.  Parallel machine scheduling with a convex resource consumption function , 2006, Eur. J. Oper. Res..

[4]  B. Earl Wells,et al.  Task Scheduling in a Finite-Resource, Reconfigurable Hardware/Software Codesign Environment , 2006, INFORMS J. Comput..

[5]  Ji-Bo Wang,et al.  Single machine scheduling problems with controllable processing times and total absolute differences penalties , 2005, Eur. J. Oper. Res..

[6]  Rui Zhang,et al.  A hybrid immune simulated annealing algorithm for the job shop scheduling problem , 2010, Appl. Soft Comput..

[7]  Adam Janiak,et al.  Resource Management in Machine Scheduling Problems: A Survey , 2007 .

[8]  R. G. Vickson,et al.  Choosing the Job Sequence and Processing Times to Minimize Total Processing Plus Flow Cost on a Single Machine , 1980, Oper. Res..

[9]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[10]  Jan Karel Lenstra,et al.  Complexity of machine scheduling problems , 1975 .

[11]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[12]  T. C. Edwin Cheng,et al.  Single machine scheduling with resource dependent release times and processing times , 2005, Eur. J. Oper. Res..

[13]  Chung-Lun Li,et al.  Single-machine scheduling with trade-off between number of tardy jobs and resource allocation , 1996, Oper. Res. Lett..

[14]  Eugeniusz Nowicki,et al.  A Bicriterion Approach to Preemptive Scheduling of Parallel Machines with Controllable Job Processing Times , 1995, Discret. Appl. Math..

[15]  Dvir Shabtay,et al.  A bicriterion approach to time/cost trade-offs in scheduling with convex resource-dependent job processing times and release dates , 2006, Comput. Oper. Res..

[16]  T. C. Edwin Cheng,et al.  Scheduling with controllable release dates and processing times: Makespan minimization , 2006, Eur. J. Oper. Res..

[17]  Rakesh K. Sarin,et al.  Technical Note - Single Machine Scheduling with Controllable Processing Times and Number of Jobs Tardy , 1989, Oper. Res..

[18]  K. R. Baker,et al.  A bicriterion approach to time/cost trade-offs in sequencing , 1982 .

[19]  Chung-Lun Li,et al.  Single machine scheduling to minimize total compression plus weighted flow cost is NP-hard , 2001, Inf. Process. Lett..

[20]  T. C. Edwin Cheng,et al.  Single machine scheduling with a variable common due date and resource-dependent processing times , 2003, Comput. Oper. Res..

[21]  Ewa Figielska,et al.  A genetic algorithm and a simulated annealing algorithm combined with column generation technique for solving the problem of scheduling in the hybrid flowshop with additional resources , 2009, Comput. Ind. Eng..

[22]  Jan Karel Lenstra,et al.  Perspectives on parallel computing , 1989 .

[23]  Dvir Shabtay,et al.  Minimizing the total weighted flow time in a single machine with controllable processing times , 2004, Comput. Oper. Res..

[24]  Wieslaw Kubiak,et al.  Positive half-products and scheduling with controllable processing times , 2005, Eur. J. Oper. Res..

[25]  Dvir Shabtay,et al.  A survey of scheduling with controllable processing times , 2007, Discret. Appl. Math..

[26]  Reza Tavakkoli-Moghaddam,et al.  Electromagnetism-like mechanism and simulated annealing algorithms for flowshop scheduling problems minimizing the total weighted tardiness and makespan , 2010, Knowl. Based Syst..

[27]  Klaus Jansen,et al.  Parallel Machine Scheduling Problems with Controllable Processing Times , 2000, ICALP Satellite Workshops.

[28]  Dvir Shabtay,et al.  Convex resource allocation for minimizing the makespan in a single machine with job release dates , 2004, Comput. Oper. Res..

[29]  Byung-Cheon Choi,et al.  Single machine scheduling problem with controllable processing times and resource dependent release times , 2007, Eur. J. Oper. Res..

[30]  Adam Janiak,et al.  Single machine scheduling subject to deadlines and resource dependent processing times , 1996 .

[31]  T. C. Edwin Cheng,et al.  Scheduling with controllable release dates and processing times: Total completion time minimization , 2006, Eur. J. Oper. Res..

[32]  Eugeniusz Nowicki,et al.  A survey of results for sequencing problems with controllable processing times , 1990, Discret. Appl. Math..

[33]  Adam Janiak,et al.  Single machine scheduling problem with a common deadline and resource dependent release dates , 1991 .