A branch-and-bound algorithm for identical parallel machine total tardiness scheduling problem with preemption

This article examines the problem of scheduling preemptive jobs on identical parallel machines to minimize total tardiness. The problem is known to be NP-hard. An efficient heuristic is developed for solving large-sized problems. A lower bound scheme is also presented. Both of the proposed heuristic and lower bound are incorporated into a branch-and-bound algorithm to optimally solve small-sized problems. Computational results are reported. The branch-and-bound algorithm can handle problems of up to 16 jobs and 5 machines in size within a reasonable amount of time. The solution obtained by the heuristic has an average percentage deviation of 4.01% from the optimal value, while the initial lower bound has an average percentage deviation of 41.55% from the optimal value. Moreover, the heuristic finds approved optimal solutions for more than 45% of the problem instances tested.

[1]  Svetlana A. Kravchenko,et al.  Scheduling Jobs with Equal Processing Times , 2009 .

[2]  Jeffrey E. Schaller,et al.  Note on Shim and Kim's lower bounds for scheduling on identical parallel machines to minimize total tardiness , 2009, Eur. J. Oper. Res..

[3]  E.L. Lawler,et al.  Optimization and Approximation in Deterministic Sequencing and Scheduling: a Survey , 1977 .

[4]  Seiki Kyan,et al.  Worst Case Bound of an LRF Schedule for the Mean Weighted Flow-Time Problem , 1986, SIAM J. Comput..

[5]  Nasser Salmasi,et al.  Parallel machine scheduling problem with preemptive jobs and transportation delay , 2014, Comput. Oper. Res..

[6]  Chengbin Chu,et al.  Parallel machine scheduling to minimize total tardiness , 2002 .

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

[8]  Fuh-Der Chou,et al.  A minimum-cost network flow approach to preemptive parallel-machine scheduling , 2013, Comput. Ind. Eng..

[9]  Marek Chrobak,et al.  The complexity of mean flow time scheduling problems with release times , 2006, J. Sched..

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

[11]  Frank Werner,et al.  Parallel machine problems with equal processing times: a survey , 2011, J. Sched..

[12]  Yeong-Dae Kim,et al.  Scheduling on parallel identical machines to minimize total tardiness , 2007, Eur. J. Oper. Res..

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

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

[15]  Reha Uzsoy,et al.  A new dynamic programming algorithm for the parallel machines total weighted completion time problem , 1992, Oper. Res. Lett..

[16]  S. Sarin,et al.  An improved branching scheme for the branch and bound procedure of scheduling n jobs on m parallel machines to minimize total weighted flowtime , 1988 .

[17]  Eugene L. Lawler,et al.  On Preemptive Scheduling of Unrelated Parallel Processors by Linear Programming , 1978, JACM.

[18]  Frank Werner,et al.  Erratum to: Minimizing total tardiness on parallel machines with preemptions , 2013, J. Sched..

[19]  Joseph Y.-T. Leung,et al.  Minimizing Mean Flow Time with Release Time Constraint , 1990, Theor. Comput. Sci..

[20]  Jeffrey E. Schaller,et al.  Minimizing total tardiness for scheduling identical parallel machines with family setups , 2014, Comput. Ind. Eng..

[21]  Odile Bellenguez-Morineau,et al.  New complexity results for parallel identical machine scheduling problems with preemption, release dates and regular criteria , 2011, Eur. J. Oper. Res..

[22]  Salah E. Elmaghraby,et al.  On the Scheduling of Jobs on a Number of Identical Machines. , 1972 .

[23]  M. Azizoglu,et al.  Tardiness minimization on parallel machines , 1998 .

[24]  Frank Werner,et al.  Minimizing total tardiness on parallel machines with preemptions , 2012, J. Sched..

[25]  Peter Brucker,et al.  Scheduling jobs with equal processing times and time windows on identical parallel machines , 2008, J. Sched..

[26]  Meral Azizoglu.  Preemptive scheduling on identical parallel machines subject to deadlines , 2003, Eur. J. Oper. Res..

[27]  Meral Azizoglu,et al.  On the minimization of total weighted flow time with identical and uniform parallel machines , 1999, Eur. J. Oper. Res..

[28]  Jatinder N. D. Gupta,et al.  Scheduling identical parallel machines with fixed delivery dates to minimize total tardiness , 2015, Eur. J. Oper. Res..

[29]  Shunji Tanaka,et al.  A branch-and-bound algorithm with Lagrangian relaxation to minimize total tardiness on identical parallel machines , 2008 .

[30]  Joseph Y.-T. Leung,et al.  Preemptive Scheduling to Minimize Mean Weighted Flow Time , 1990, Inf. Process. Lett..

[31]  Berit Johannes,et al.  Scheduling parallel jobs to minimize the makespan , 2006, J. Sched..

[32]  Ameur Soukhal,et al.  Scheduling on parallel machines with preemption and transportation delays , 2012, Comput. Oper. Res..