We deal here with a scheduling problem GPPCSP (Generalized Parallelism and Preemption Constrained Scheduling Problem) which is an extension of both the well-known Resource Constrained Scheduling Problem and the Scheduling Problem with Disjunctive Constraints. We first propose a reformulation of GPPCSP: according to it, solving GPPCSP means finding a vertex of the Feasible Vertex Subset of an Antichain Polyhedron. Next, we state several theoretical results related to this reformulation process and to structural properties of this specific Feasible Vertex Subset (connectivity, ...). We end by focusing on the preemptive case of GPPCSP and by identifying specific instances of GPPCSP which are such that any vertex of the related Antichain Polyhedron may be projected on its related Feasible Vertex Subset without any deterioration of the makespan. For such an instance, the GPPCSP problem may be solved in a simple way through linear programming.
Mots-clés : partially ordered sets, hypergraphs, linear programming, polyhedra, multiprocessor scheduling, resource constrained project scheduling problem
@article{RO_2008__42_3_325_0,
author = {Damay, Jean and Quilliot, Alain and Sanlaville, Eric},
title = {Polyhedral reformulation of a scheduling problem and related theoretical results},
journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
pages = {325--359},
publisher = {EDP-Sciences},
volume = {42},
number = {3},
year = {2008},
doi = {10.1051/ro:2008024},
mrnumber = {2444491},
zbl = {1152.90437},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ro:2008024/}
}
TY - JOUR AU - Damay, Jean AU - Quilliot, Alain AU - Sanlaville, Eric TI - Polyhedral reformulation of a scheduling problem and related theoretical results JO - RAIRO - Operations Research - Recherche Opérationnelle PY - 2008 SP - 325 EP - 359 VL - 42 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ro:2008024/ DO - 10.1051/ro:2008024 LA - en ID - RO_2008__42_3_325_0 ER -
%0 Journal Article %A Damay, Jean %A Quilliot, Alain %A Sanlaville, Eric %T Polyhedral reformulation of a scheduling problem and related theoretical results %J RAIRO - Operations Research - Recherche Opérationnelle %D 2008 %P 325-359 %V 42 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ro:2008024/ %R 10.1051/ro:2008024 %G en %F RO_2008__42_3_325_0
Damay, Jean; Quilliot, Alain; Sanlaville, Eric. Polyhedral reformulation of a scheduling problem and related theoretical results. RAIRO - Operations Research - Recherche Opérationnelle, Tome 42 (2008) no. 3, pp. 325-359. doi : 10.1051/ro:2008024. http://www.numdam.org/articles/10.1051/ro:2008024/
[1] , Towards a general theory of action and time Art. Intel. 23 (1984) 123-154. | Zbl
[2] and , A polynomial activity insertion algorithm in a multiresource schedule with cumulative constraints and multiple nodes. EJOR 127-2 (2000) 297-316. | Zbl
[3] , and , Insertion techniques for static and dynamic resource constrained project scheduling. EJOR 149 (2003) 249-267. | MR | Zbl
[4] , Introduction to Sequencing and Scheduling. Wiley, NY (1974).
[5] , Resource constraints for preemptive and non preemptive scheduling. MSC Thesis, University Paris VI (1995).
[6] , Demassey, Tight LP bounds for resource constrained project scheduling. OR Spectrum 26 (2004) 11. | Zbl
[7] , On the topology of the genetic fine structure Proc. Acad. Sci. USA 45 (1959) 1607-1620.
[8] , Graphes et Hypergraphes. Dunod Ed., Paris (1975). | MR | Zbl
[9] , , and , Scheduling in computer and manufacturing systems. 2th edn, Springer-Verlag, Berlin (1993). | Zbl
[10] and , Testing for the consecutive ones property, interval graphs and graph planarity using PQ-tree algorithms. J. Comp. Sci. 13 (1976) 335-379. | MR | Zbl
[11] and , A linear programming and constraint propoagation based lower bound for the RCPSP. EJOR 127 (2000) 355-362. | Zbl
[12] , , and , A branch and bound algorithm for the resource constrained project scheduling problem. EJOR 107 (1998) 272-288. | Zbl
[13] and , Problèmes d'ordonnancements : modélisation, complexité et algorithmes. Masson Ed., Paris (1988).
[14] , A survey on practical applications of examination timetabling algorithms. Oper. Res. 34 (1986) 193-202. | MR
[15] and , The jump number of Dags and posets. Ann. Discrete Math. 9 (1980) 189-194. | MR | Zbl
[16] and , New benchmark results for the multiple RCPSP. Manage. Sci. 43 (1997) 1485-1492. | Zbl
[17] , Techniques de resolution basées sur la programmation linéaire pour l'ordonnancement de projet. Ph.D. Thesis, Université de Clermont-Ferrand, (2005).
[18] , and , Linear programming based algorithms for preemptive and non preemptive RCPSP. EJOR 182 (2007) 1012-1022. | Zbl
[19] , Scheduling preemptive jobs with precedence constraints on parallel machines. EJOR 117 (1999) 355-367. | Zbl
[20] and , Scheduling DAGs of bounded heights. J. Algor. 5 (1984) 48-59. | MR | Zbl
[21] , Problèmes de représentations et noyaux. Thèse d'Etat Paris VI (1981).
[22] and , Partially ordered sets. Amer. J. Math. 63 (1941) 600-610. | JFM | MR
[23] and , Incidence matrices and interval graphs. Pac. J. Math. 15 (1965) 835-855. | MR | Zbl
[24] , File organization: the consecutive retrieval property. Comm. ACM 9 (1975) 802-808. | Zbl
[25] , , and , Optimization and approximation in deterministic scheduling: a survey. Ann. Discrete Math. 5 (1979) 287-326. | MR | Zbl
[26] , , , and , An almost optimal heuristic for preemptive Cmax scheduling of dependant task on parallel identical machines. Annals Oper. Res. 129 (2004) 205-216. | MR | Zbl
[27] , and , A classification scheme for project scheduling, in Project Scheduling: recent models, algorithms and applications. Kluwer Acad Publ. (1999) 1-26.
[28] , Incidence matrices, interval graphs and seriation in archaeology, Pac. J. Math. 28 (1969) 565-570. | MR | Zbl
[29] , and , Characterization and generation of a general class of resource constrained project scheduling problems, Manage. Sci. 41, (10), (1995) 1693-1703. | Zbl
[30] , Polynomial complete consecutive information retrieval problems. SIAM J. Comput. 6 (1992) 67-75. | MR | Zbl
[31] , , and , Sequencing and scheduling: algorithms and complexity, in Handbook of Operation Research and Management Sciences, Vol 4: Logistics of Production and Inventory, edited by S.C. Graves, A.H.G. Rinnoy-Kan and P.H. Zipkin, North-Holland, (1993) 445-522.
[32] and , Storage for consecutive retrieval. Inform. Processing Lett. 5 (1976) 68-71. | MR | Zbl
[33] , , and , An exact algorithm for project scheduling with resource constraints based on a new mathematical formulation. Manage. Sci. 44 (1998) 714-729. | Zbl
[34] and , Scheduling problems with resource duration interactions. Methods Oper. Res. 48 (1984) 423-452. | Zbl
[35] and , Optimal preemptive scheduling on a fixed number of identical parallel machines. Oper. Res. Lett. 33 (2005) 143-151. | MR | Zbl
[36] and , A relation between multiprocessor scheduling and linear programming. Order 14 (1997) 269-278. | MR | Zbl
[37] and , Preemptive scheduling of real time tasks on multiprocessor systems. J.A.C.M. 17 (1970) 324-338. | MR | Zbl
[38] and , Scheduling interval ordered tasks. SIAM J. Comput. 8 (1979) 405-409. | MR | Zbl
[39] , A comparizon of exact approaches for solving the multiple constrained resource project scheduling problem. Manage. Sci. 30 (1984) 854-867.
[40] and , Algorithmic characterization of interval ordered hypergraphs and applications. Discrete Appl. Math. 51 (1994) 159-173. | MR | Zbl
[41] and , Rational preemptive scheduling. Order 4 (1987) 195-206. | MR | Zbl
[42] and , Preemptive scheduling of interval orders is polynomial. Order 5 (1989) 345-348. | MR | Zbl
[43] , Theory of Linear and Integer Programming. Wiley, NY (1986). | MR | Zbl
[44] , Constraint Programming. North Holland (1997).
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