Polyhedral reformulation of a scheduling problem and related theoretical results
RAIRO - Operations Research - Recherche Opérationnelle, Tome 42 (2008) no. 3, pp. 325-359.

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.

DOI : 10.1051/ro:2008024
Classification : 90B35
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] J.F. Allen, Towards a general theory of action and time Art. Intel. 23 (1984) 123-154. | Zbl

[2] C. Artigues and F. Roubellat, A polynomial activity insertion algorithm in a multiresource schedule with cumulative constraints and multiple nodes. EJOR 127-2 (2000) 297-316. | Zbl

[3] C. Artigues, P. Michelon and S. Reusser, Insertion techniques for static and dynamic resource constrained project scheduling. EJOR 149 (2003) 249-267. | MR | Zbl

[4] K. R. Baker, Introduction to Sequencing and Scheduling. Wiley, NY (1974).

[5] P. Baptiste, Resource constraints for preemptive and non preemptive scheduling. MSC Thesis, University Paris VI (1995).

[6] P. Baptiste, Demassey, Tight LP bounds for resource constrained project scheduling. OR Spectrum 26 (2004) 11. | Zbl

[7] S. Benzer, On the topology of the genetic fine structure Proc. Acad. Sci. USA 45 (1959) 1607-1620.

[8] C. Berge, Graphes et Hypergraphes. Dunod Ed., Paris (1975). | MR | Zbl

[9] J. Blazewiecz, K.H. Ecker, G. Schmlidt and J. Weglarcz, Scheduling in computer and manufacturing systems. 2th edn, Springer-Verlag, Berlin (1993). | Zbl

[10] K.S. Booth and J.S. Lueker, 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] P. Brucker and S. Knust, A linear programming and constraint propoagation based lower bound for the RCPSP. EJOR 127 (2000) 355-362. | Zbl

[12] P. Brucker, S. Knust, A. Schoo and O. Thiele, A branch and bound algorithm for the resource constrained project scheduling problem. EJOR 107 (1998) 272-288. | Zbl

[13] J. Carlier and P. Chretienne, Problèmes d'ordonnancements : modélisation, complexité et algorithmes. Masson Ed., Paris (1988).

[14] M. Carter, A survey on practical applications of examination timetabling algorithms. Oper. Res. 34 (1986) 193-202. | MR

[15] M. Chein and M. Habib, The jump number of Dags and posets. Ann. Discrete Math. 9 (1980) 189-194. | MR | Zbl

[16] E. Demeulemeester and W. Herroelen, New benchmark results for the multiple RCPSP. Manage. Sci. 43 (1997) 1485-1492. | Zbl

[17] J. Damay, Techniques de resolution basées sur la programmation linéaire pour l'ordonnancement de projet. Ph.D. Thesis, Université de Clermont-Ferrand, (2005).

[18] J. Damay, A. Quilliot and E. Sanlaville, Linear programming based algorithms for preemptive and non preemptive RCPSP. EJOR 182 (2007) 1012-1022. | Zbl

[19] K. Djellab, Scheduling preemptive jobs with precedence constraints on parallel machines. EJOR 117 (1999) 355-367. | Zbl

[20] D. Dolev and M.K. Warmuth, Scheduling DAGs of bounded heights. J. Algor. 5 (1984) 48-59. | MR | Zbl

[21] P. Duchet, Problèmes de représentations et noyaux. Thèse d'Etat Paris VI (1981).

[22] B. Dushnik and W. Miller, Partially ordered sets. Amer. J. Math. 63 (1941) 600-610. | JFM | MR

[23] D.R. Fulkerson and J.R. Gross, Incidence matrices and interval graphs. Pac. J. Math. 15 (1965) 835-855. | MR | Zbl

[24] S.P. Ghosh, File organization: the consecutive retrieval property. Comm. ACM 9 (1975) 802-808. | Zbl

[25] R.L. Grahamson, E.L. Lawler, J.K. Lenstra and A.H.G. Rinnoy-Khan, Optimization and approximation in deterministic scheduling: a survey. Ann. Discrete Math. 5 (1979) 287-326. | MR | Zbl

[26] J. Josefowska, M. Mika, R. Rozycki, G. Waligora and J. Weglarcz, 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] W. Herroelen, E. Demeulemeester and B. De Reyck, A classification scheme for project scheduling, in Project Scheduling: recent models, algorithms and applications. Kluwer Acad Publ. (1999) 1-26.

[28] D.G. Kindall, Incidence matrices, interval graphs and seriation in archaeology, Pac. J. Math. 28 (1969) 565-570. | MR | Zbl

[29] R. Kolisch, A. Sprecher and A. Drexel, Characterization and generation of a general class of resource constrained project scheduling problems, Manage. Sci. 41, (10), (1995) 1693-1703. | Zbl

[30] L.T. Kou, Polynomial complete consecutive information retrieval problems. SIAM J. Comput. 6 (1992) 67-75. | MR | Zbl

[31] E.L. Lawler, K.J. Lenstra, A.H.G. Rinnoy-Kan and D.B. Schmoys, 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] F. Luccio and F. Preparata, Storage for consecutive retrieval. Inform. Processing Lett. 5 (1976) 68-71. | MR | Zbl

[33] A. Mingozzi, V. Maniezzo, S. Ricciardelli and L. Bianco, An exact algorithm for project scheduling with resource constraints based on a new mathematical formulation. Manage. Sci. 44 (1998) 714-729. | Zbl

[34] R.H. Mohring and F.J. Rademacher, Scheduling problems with resource duration interactions. Methods Oper. Res. 48 (1984) 423-452. | Zbl

[35] A. Moukrim and A. Quilliot, Optimal preemptive scheduling on a fixed number of identical parallel machines. Oper. Res. Lett. 33 (2005) 143-151. | MR | Zbl

[36] A. Moukrim and A. Quilliot, A relation between multiprocessor scheduling and linear programming. Order 14 (1997) 269-278. | MR | Zbl

[37] R.R. Muntz and E.G. Coffman, Preemptive scheduling of real time tasks on multiprocessor systems. J.A.C.M. 17 (1970) 324-338. | MR | Zbl

[38] C.H. Papadimitriou and M. Yannanakis, Scheduling interval ordered tasks. SIAM J. Comput. 8 (1979) 405-409. | MR | Zbl

[39] J.H. Patterson, A comparizon of exact approaches for solving the multiple constrained resource project scheduling problem. Manage. Sci. 30 (1984) 854-867.

[40] A. Quilliot and S. Xiao, Algorithmic characterization of interval ordered hypergraphs and applications. Discrete Appl. Math. 51 (1994) 159-173. | MR | Zbl

[41] N. Sauer and M.G. Stone, Rational preemptive scheduling. Order 4 (1987) 195-206. | MR | Zbl

[42] N. Sauer and M.G. Stone, Preemptive scheduling of interval orders is polynomial. Order 5 (1989) 345-348. | MR | Zbl

[43] A. Schrijver, Theory of Linear and Integer Programming. Wiley, NY (1986). | MR | Zbl

[44] P. Van Hentenryk, Constraint Programming. North Holland (1997).

Cité par Sources :