Complexity of problem TF 2

In this paper, we study a flow shop scheduling problem with an interstage transporter, in which there are a set of n jobs, two machines A and B, and one transporter located at machine A initially. Each job has to be processed on machine A first, then transported by a vehicle to machine B, finally processed on machine B. The interstage transporter can carry at most two jobs between the machines at a time. When the transporter carries items from machine A to machine B, it will take time ?; when it is back from machine B to machine A, it will take zero time. And there is an unlimited buffer on each machine to store jobs that will be processed. The objective is to minimize the completion time on machine B. The complexity of the problem is open in Journal of Scheduling 2001 5. We prove the problem is NP-hard in the strong sense by 4-partition problem. We study a flow shop scheduling problem with two machines and one transporter.Each time, the transporter can carry at most 2 jobs between two machines.We prove the problem is strongly NP-hard in by 4-partition problem.

[1]  Parkash Lal Maggu,et al.  On 3xn sequencing problem involving equivalent-job for a job-block with transportation-times of jobs , 1981 .

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

[3]  Vitaly A. Strusevich,et al.  An Improved Approximation Algorithm for the Two-Machine Flow Shop Scheduling Problem with an interstage transporter , 2007, Int. J. Found. Comput. Sci..

[4]  T. C. Edwin Cheng,et al.  Complexity Results for Flow-Shop and Open-Shop Scheduling Problems with Transportation Delays , 2004, Ann. Oper. Res..

[5]  Zhi-Long Chen,et al.  Machine scheduling with transportation considerations , 2001 .

[6]  D Rebaine,et al.  Two-machine open shop scheduling with special transportation times , 1999, J. Oper. Res. Soc..

[7]  Victor J. Rayward-Smith,et al.  Open shop scheduling with delays , 1992, RAIRO Theor. Informatics Appl..

[8]  Sigrid Knust,et al.  Makespan minimization for flow-shop problems with transportation times and a single robot , 2001, Discret. Appl. Math..

[9]  S. M. Johnson,et al.  Optimal two- and three-stage production schedules with setup times included , 1954 .

[10]  Vitaly A. Strusevich,et al.  Transporting jobs through a two-machine open shop , 2009 .

[11]  Leslie A. Hall.  Approximability of flow shop scheduling , 1998, Math. Program..

[12]  Vitaly A. Strusevich,et al.  A Heuristic for the Two-machine Open-shop Scheduling Problem with Transportation Times , 1999, Discret. Appl. Math..

[13]  Chung Yee Lee,et al.  Two-machine shop scheduling with an uncapacitated interstage transporter , 2005 .

[14]  Mauro Dell'Amico,et al.  Shop Problems With Two Machines and Time Lags , 1996, Oper. Res..

[15]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[16]  Hiroshi Kise,et al.  ON AN AUTOMATED TWO-MACHINE FLOWSHOP SCHEDULING PROBLEM WITH INFINITE BUFFER , 1991 .