A branch and bound algorithm to minimize the total flow time for m-machine permutation flowshop problems

Abstract The m -machine permutation flowshop problem with the total flow-time objective is a common scheduling problem, which is known to be NP-hard for m ⩾2. In this article, we develop a branch and bound algorithm to solve both the weighted and unweighted version of this problem. Our algorithm incorporates a new machine-based lower bound and a dominance test for pruning nodes. Computational experiments suggest that the algorithm can handle test problems with n ⩽15. It also seems capable of dealing with larger problems for the unweighted objective, especially when the processing times are correlated.

[1]  Wlodzimierz Szwarc,et al.  Solvable cases of the flow‐shop problem without interruptions in job processing , 1983 .

[2]  Neta Amit,et al.  Openshop and flowshop scheduling to minimize sum of completion times , 1984, Comput. Oper. Res..

[3]  L. V. Wassenhove,et al.  An algorithm for single machine sequencing with deadlines to minimize total weighted completion time , 1983 .

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

[5]  A. J. Clewett,et al.  Introduction to sequencing and scheduling , 1974 .

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

[7]  E. Ignall,et al.  Application of the Branch and Bound Technique to Some Flow-Shop Scheduling Problems , 1965 .

[8]  S. L. Velde.  Minimizing the sum of the job completion times in the two-machine flow shop by Lagrangian relaxation , 1991 .

[9]  William L. Maxwell,et al.  Theory of scheduling , 1967 .

[10]  Eugene L. Lawler,et al.  Chapter 9 Sequencing and scheduling: Algorithms and complexity , 1993, Logistics of Production and Inventory.

[11]  Jeffrey B. Sidney,et al.  Decomposition Algorithms for Single-Machine Sequencing with Precedence Relations and Deferral Costs , 1975, Oper. Res..

[12]  Han Hoogeveen,et al.  Stronger Lagrangian bounds by use of slack variables: Applications to machine scheduling problems , 1992, Math. Program..

[13]  Lucio Bianco,et al.  Scheduling of a single machine to minimize total weighted completion time subject to release dates , 1982 .

[14]  Teofilo F. Gonzalez,et al.  Flowshop and Jobshop Schedules: Complexity and Approximation , 1978, Oper. Res..

[15]  S. Bansal.  Minimizing the Sum of Completion Times of n Jobs over m Machines in a Flowshop—A Branch and Bound Approach , 1977 .

[16]  S. S. Panwalkar,et al.  An ordered flow-shop sequencing problem with mean completion time criterion† , 1976 .

[17]  R. Ahmadi,et al.  Improved lower bounds for minimizing the sum of completion times of n jobs over m machines in a flow shop , 1990 .

[18]  Rodney Dean Traub.  Single machine scheduling with dynamic arrivals , 1994 .

[19]  Chengbin Chu.  A branch‐and‐bound algorithm to minimize total flow time with unequal release dates , 1992 .

[20]  S. Bansal.  Single machine scheduling to minimize weighted sum of completion times with secondary criterion -- A branch and bound approach , 1980 .

[21]  C. Rajendran.  Heuristic algorithm for scheduling in a flowshop to minimize total flowtime , 1993 .

[22]  F. D. Croce,et al.  The two-machine total completion time flow shop problem , 1996 .

[23]  B. J. Lageweg,et al.  A General Bounding Scheme for the Permutation Flow-Shop Problem , 1978, Oper. Res..

[24]  Yih-Long Chang,et al.  A new heuristic for the n-job, M-machine flow-shop problem , 1991 .

[25]  C. Potts.  A Lagrangean Based Branch and Bound Algorithm for Single Machine Sequencing with Precedence Constraints to Minimize Total Weighted Completion Time , 1985 .

[26]  Reha Uzsoy,et al.  Single-machine scheduling with dynamic arrivals : Decomposition results and an improved algorithm , 1996 .

[27]  Chris N. Potts,et al.  On-line scheduling of a single machine to minimize total weighted completion time , 2002, SODA '02.

[28]  Chris N. Potts,et al.  An adaptive branching rule for the permutation flow-shop problem , 1980 .

[29]  Edward Anderson,et al.  Minimizing flow time on a single machine with job classes and setup times , 1991 .