Abstract
The problem of scheduling n nonpreemptive jobs having a common due date d on m, m ≥ 2, parallel identical machines to minimize total tardiness is studied. Approximability issues are discussed and two families of algorithms {A ε} and {B ε} are presented such that (T 0 − T*)/(T* + d) ≤ ε holds for any problem instance and any given ε > 0, where T* is the optimal solution value and T 0 is the value of the solution delivered by A ε or B ε. Algorithms A ε and B ε run in O(n 2m/εm−1) and O(n m+1/εm) time, respectively, if m is a constant. For m = 2, algorithm A ε can be improved to run in O(n 3/ε) time.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Garey, M.R. and D.J. Johnson. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. San Francisco: Freeman.
Garey, M.R. and D.S. Johnson. (1976). “Approximation Algorithms for Combinatorial Problems: An Annotated Bibliography.” In Algorithms and Complexity. New Directions and Recent Results.–Proc. Symp. Cornegie-Mellon University, New York, pp. 41–52.
Gens, G.V. and E.V. Levner. (1981). “Efficient Approximate Algorithms for Combinatorial Problems.” Preprint, Central Economical and Mathematical Institute of the USSR Academy of Sciences, Moscow (in Russian).
Graham, R.L., E.L. Lawler, J.K. Lenstra, and A.H.G. Rinnooy Kan. (1979). “Optimization and Approximation in Deterministic Sequencing and Scheduling: A Survey.” Annals of Discrete Mathematics 5, 287–326.
Horowitz, E. and S. Sahni. (1976). “Exact and Approximate Algorithms for Scheduling Nonidentical Processors.” Journal of the ACM 23, 317–327.
Ibarra, O. and C.E. Kim. (1975). “Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems.” Journal of the ACM 22, 463–468.
Korte, B. and R. Schrader. (1981). “On the Existence of Fast Approximation Schemes.” In Nonlinear Programming, Vol. 4. Proc. 4th Symp., Madison, Wisc., July 14–16, 1980. New York, pp. 415–437.
Kovalyov, M.Y. and W. Kubiak. (1998). “A Fully Polynomial Approximation Scheme for Minimizing Makespan of Deteriorating Jobs.” Journal of Heuristics 3, 287–297.
Kovalyov, M.Y. and Y.M. Shafransky. (1986). “The Construction of ε-Approximate Algorithms in Successively Constructed Sets.” U.S.S.R. Comput. Maths. Math. Phys. 26, 30–38.
Root, J.G. (1965). “Scheduling with Deadlines and Loss Functions on κ Parallel Machines.” Management Science 11, 460–475.
Sahni, S. (1976). “Algorithms for Scheduling Independent Tasks.” Journal ACM 23, 116–127.
Sahni, S. (1977). “General Techniques for Combinatorial Approximation.” Operations Research 25, 920–936.
Woeginger, G.J. (1998). “When Does a Dynamic Programming Formulation Guarantee the Existence of an FPTAS?” Report Woe-27, TU Graz, Austria.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kovalyov, M.Y., Werner, F. Approximation Schemes for Scheduling Jobs with Common Due Date on Parallel Machines to Minimize Total Tardiness. Journal of Heuristics 8, 415–428 (2002). https://doi.org/10.1023/A:1015487829051
Issue Date:
DOI: https://doi.org/10.1023/A:1015487829051