Batch delivery scheduling with batch delivery cost on a single machine

Abstract We consider a scheduling problem in which n independent and simultaneously available jobs are to be processed on a single machine. The jobs are delivered in batches and the delivery date of a batch equals the completion time of the last job in the batch. The delivery cost depends on the number of deliveries. The objective is to minimize the sum of the total weighted flow time and delivery cost. We first show that the problem is strongly NP-hard. Then we show that, if the number of batches is B , the problem remains strongly NP-hard when B  ⩽  U for a variable U  ⩾ 2 or B  ⩾  U for any constant U  ⩾ 2. For the case of B  ⩽  U , we present a dynamic programming algorithm that runs in pseudo-polynomial time for any constant U  ⩾ 2. Furthermore, optimal algorithms are provided for two special cases: (i) jobs have a linear precedence constraint, and (ii) jobs satisfy the agreeable ratio assumption, which is valid, for example, when all the weights or all the processing times are equal.

[1]  David S. Johnson,et al.  Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran , 1979 .

[2]  Mihalis Yannakakis,et al.  Optimal Scheduling of Products with Two Subassemblies on a Single Machine , 1989, Oper. Res..

[3]  J. M. Moore,et al.  A Functional Equation and its Application to Resource Allocation and Sequencing Problems , 1969 .

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

[5]  Denis Naddef,et al.  One-pass batching algorithms for the one-machine problem , 1988, Discret. Appl. Math..

[6]  David Shallcross,et al.  A polynomial algorithm for a one machine batching problem , 1992, Oper. Res. Lett..

[7]  Susanne Albers,et al.  The Complexity of One-Machine Batching Problems , 1993, Discret. Appl. Math..

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

[9]  Bertrand M. T. Lin,et al.  Single Machine Scheduling to Minimize Batch Delivery and Job Earliness Penalties , 1997, SIAM J. Optim..

[10]  Guoqing Wang,et al.  Parallel machine scheduling with batch delivery costs , 2000 .

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

[12]  Mihalis Yannakakis,et al.  Batch sizing and job sequencing on a single machine , 1991 .

[13]  T. C. Edwin Cheng,et al.  Single machine scheduling with batch deliveries , 1996 .

[14]  Michael J. Magazine,et al.  Batching in single operation manufacturing systems , 1985 .

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

[16]  T.C.E. Cheng,et al.  Batch Delivery Scheduling on a Single Machine , 1994 .