Analyse de sensibilité pour les problèmes linéaires en variables 0-1
RAIRO - Operations Research - Recherche Opérationnelle, Tome 37 (2003) no. 4, pp. 291-309.

Cet article est un travail de synthèse autour de l'analyse de sensibilité pour les problèmes linéaires en variables 0-1. De nombreux aspects sont ainsi abordés : historique et formes d'analyse de sensibilité, exemples d'application, complexité, conditions d'optimalité, algorithmes et approches. Nous dressons par ailleurs quelques perspectives de recherche actuelles dans ce domaine.

This paper is a state of the art on sensitivity analysis for 0-1 linear programming problems. Several aspects are considered: history and forms of sensitivity analysis, application examples, complexity, optimality conditions, existing algorithms and approaches.

DOI : https://doi.org/10.1051/ro:2004002
Mots clés : analyse de sensibilité, réoptimisation, rayon de stabilité, problèmes linéaires en 0-1
@article{RO_2003__37_4_291_0,
     author = {Thiongane, Babacar and Nagih, Anass and Plateau, G\'erad},
     title = {Analyse de sensibilit\'e pour les probl\`emes lin\'eaires en variables 0-1},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {291--309},
     publisher = {EDP-Sciences},
     volume = {37},
     number = {4},
     year = {2003},
     doi = {10.1051/ro:2004002},
     zbl = {1092.90031},
     mrnumber = {2065244},
     language = {fr},
     url = {http://archive.numdam.org/articles/10.1051/ro:2004002/}
}
Thiongane, Babacar; Nagih, Anass; Plateau, Gérad. Analyse de sensibilité pour les problèmes linéaires en variables 0-1. RAIRO - Operations Research - Recherche Opérationnelle, Tome 37 (2003) no. 4, pp. 291-309. doi : 10.1051/ro:2004002. http://archive.numdam.org/articles/10.1051/ro:2004002/

[1] E. Balas, An additive algorithm for solving linear programs with zero-one variables. Oper. Res. 13 (1965) 517-546. | MR 183535 | Zbl 0194.19903

[2] D.E Bell and J.F Shapiro, A convergent duality theory for integer programming. Oper. Res. 1 (1977) 467-477. | MR 444000 | Zbl 0355.90051

[3] C. Blair, Sensitivity analysis for knapsack problems: a negative result. Discrete Appl. Math. 81 (1998) 133-139. | MR 1492006 | Zbl 0895.90155

[4] P.J. Carstensen, Complexity of some parametric integer and network programming problems. Math. Program. 26 (1983) 64-75. | MR 696727 | Zbl 0585.90065

[5] N. Chakravarti and A.P.M. Wagelmans, Calculation of stability radius for combinatorial optimization problems. Oper. Res. Lett. 23 (1999) 1-7. | MR 1664225 | Zbl 0954.90037

[6] W. Cook, A.M.H. Gerards, A. Schrijver and E. Tardos, Sensitivity theorems in integer linear programming. Math. Program. 34 (1986) 251-264. | MR 839604 | Zbl 0648.90055

[7] A. Crema, An algorithm for the multiparametric 0-1 integer linear programming problem relative to the objective function. Eur. J. Oper. Res. 125 (2000) 18-24. | MR 1783192 | Zbl 0972.90049

[8] A. Crema, The multiparametric 0-1 integer linear programming problem: A unified approach. Eur. J. Oper. Res. 139 (2002) 511-520. | MR 1893597 | Zbl 1017.90064

[9] M. Desrochers and F. Soumis, A reoptimization algorithm for the shortest path problem with time windows. Eur. J. Oper. Res. 35 (1988) 242-254. | MR 939070 | Zbl 0677.90080

[10] M.L. Fisher, W.D. Northup and J.F. Shapiro, Using duality to solve discrete optimization problems: Theory and computational experience. Math. Prog. Study 3 (1975) 56-94. | MR 444006 | Zbl 0367.90087

[11] M.L. Fisher and J.F. Shapiro, Constructive duality in integer programming. SIAM J. Appl. Math. 27 (1974) 31-52. | MR 351446 | Zbl 0299.90010

[12] T. Gal and H.J. Greenberg, Advances in sensitivity analysis and parametric programming. Kluwer Academic Publishers, Boston-Dordrecht-London (1997). | Zbl 0881.00025

[13] R.S Garfinkel and G.L. Nemhauser, Integer programming. Wiley, New York (1972). | MR 381688 | Zbl 0259.90022

[14] A.M. Geoffrion and K. Nauss, Parametric and postoptimality analysis in integer linear programming. Manage. Sci. 23 (1977) 453-466. | Zbl 0358.90041

[15] E.N. Gordeev, The complexity of stability study in discrete optimization problems. Cybernetics questions 133, computer center of the USSR academy of sciences, Moscow (1989) 41-77 (en russe). | Zbl 0701.90077

[16] E.N. Gordeev, Solution stability in a shortest path problem. Discrete Math. 1 (1989) 45-56 (en russe). | MR 1044234

[17] H.J. Greenberg, An annoted bibliography for post-solution analysis in mixed integer programming and combinatorial optimization, in Advances in Computational and Stochastic Optimization, Logic Programming and Heuristic Search, edited by D.L. Woodruff. Kluwer Academic Publishers, Boston, MA (1998) http://carbon.cudenver.edu/ hgreenbe/aboutme/pubrec.html | MR 1602329 | Zbl 0914.90204

[18] D. Gusfield, Sensitivity analysis for combinatorial optimization. Memo. No. UCB/ERL M80/22, Electronics Research Laboratory, Univ. of California, Berkeley, California (1980).

[19] D. Gusfield, Parametric combinatorial computing and a problem of program module distribution. J. Association for Computing Machinery 30 (1983) 551-563. | MR 709832 | Zbl 0628.68035

[20] S.V. Hoesel and A. Wagelmans, Sensitivity analysis of the economic lot-sizing problem. Discrete Appl. Math. 45 (1993) 291-312. | MR 1236732 | Zbl 0790.90028

[21] S.V. Hoesel and A. Wagelmans, On the complexity of postoptimality analysis of 0/1 programs. Discrete Appl. Math. 91 (1999) 251-263. | MR 1670187 | Zbl 0917.90250

[22] S. Holm and D. Klein, Three methods for postoptimal analysis in integer linear programming. Math. Prog. Study 21 (1984) 97-109. | MR 751245 | Zbl 0543.90083

[23] L. Jenkins, Parametric-objective integer programming using knapsack facets and Gomory cutting planes. Eur. J. Oper. Res. 31 (1987) 102-109. | MR 894602 | Zbl 0624.90072

[24] L. Jenkins, Parametric methods in integer linear programming. Ann. Oper. Res. 27 (1990) 77-96. | MR 1088988 | Zbl 0718.90088

[25] D. Klein and S. Holm, Discrete right hand-side parametrization for linear integer programs. Eur. J. Oper. Res. 2 (1978) 50-53. | Zbl 0384.90090

[26] D. Klein and S. Holm, Integer programming postoptimal analysis with cutting planes. Manage. Sci. 25 (1979) 64-72. | MR 530939 | Zbl 0442.90067

[27] M.Y. Kovalyev and Y.N Sotskov, ϵ-approximate solution stability of boolean linear form minimization. Vesti Akad. Navuk BSSR, Ser. Fiz.-Mat. Navuk 2 2 (1990) 111-116 1990 (en russe). | Zbl 0699.49030

[28] V.K. Leontev, Stability in traveling salesman problem. At. i Mat. Fiz. 15 (1975) 1298-1309 (en russe). | MR 406446

[29] V.K. Leontev, Stability in combinatorial choice problems. Dokl. Akad. Nauk SSSR 1 (1976) 23-25 (en russe). | MR 401489 | Zbl 0356.90043

[30] V.K. Leontev, Stability in linear discrete problems. Cybernet. Probl. 35 (1979) 169-184 (en russe). | MR 539891 | Zbl 0439.93040

[31] M. Libura, Sensitivity analysis for integer knapsack problem. Archiwum Automatyki i Telemechanik 22 (1977) 313-322 (en polonais). | MR 456461 | Zbl 0365.90097

[32] M. Libura, Optimality conditions and sensitivity analysis for combinatorial optimization problems. Control Cybernet. 25 (1996) 1165-1180. | MR 1454711 | Zbl 0865.90110

[33] M. Libura, E.S. Van Der Poort, G. Sierksma and J.A.A. Van Der Veen, Stability aspects of the traveling salesman problem based on k-best solutions. Discrete Appl. Math. 87 (1998) 159-185. | MR 1650435 | Zbl 0910.90264

[34] E. Loukakis and A.P. Muhlemann, Parametrisation algorithms for the integer linear programs in binary variables. Eur. J. Oper. Res. 17 (1984) 104-115. | MR 749142 | Zbl 0542.90063

[35] K. Nauss, Parametric integer programming. Ph.D. thesis, Western Management Science Institute, UCLA (1975).

[36] C.J. Piper and A. Zoltners, Implicit enumeration based algorithms for post-optimising zero-one programs. Naval Res. Logist. Quarterly 22 (1975) 791-809. | MR 434461 | Zbl 0354.90057

[37] C.J. Piper and A. Zoltners, Some easy postoptimality analysis for zero-one programming. Manage. Sci. 22 (1976) 759-765. | Zbl 0325.90048

[38] G.M. Roodman, Postoptimality analysis in zero-one programming by implicit enumeration. Naval Res. Logist. Quarterly 19 (1972) 435-447. | MR 316089 | Zbl 0262.90047

[39] G.M. Roodman, Postoptimality analysis in zero-one programming by implicit enumeration. The mixed integer case. Naval Res. Logist. Quarterly 21 (1974) 595-607. | MR 368756 | Zbl 0304.90078

[40] L. Schrage and L. Wolsey, Sensitivity analysis for branch and bound integer programming. Oper. Res. 33 (1985) 1008-1023. | MR 806917 | Zbl 0583.90074

[41] J.F. Shapiro, Generalized lagrange multipliers in integer programming. Oper. Res. 19 (1971) 68-76. | MR 275877 | Zbl 0226.90031

[42] Y.N. Sotskov, Stability of an optimal schedule. Eur. J. Oper. Res. 55 (1991) 91-102. | Zbl 0755.90048

[43] Y.N Sotskov, The stability of the appoximate boolean minimization of a linear form. USSR Comput. Math. Math. Phys. 33 (1993) 699-707. | MR 1218872 | Zbl 0799.90110

[44] Y.N. Sotskov, V.K. Leontev and E.N. Gordeev, Some concepts of stability analysis in combinatorial optimization. Discrete Appl. Math. 58 (1995) 169-190. | MR 1331170 | Zbl 0833.90098

[45] Y.N Sotskov, A.P.M. Wagelmans and F. Werner, On the calculation of the stability radius of an optimal or an approximate schedule. Ann. Oper. Res. 83 (1998) 213-252. | MR 1661683 | Zbl 0911.90222

[46] R.E. Tarjan, Sensitivity analysis of minimum spanning trees and shortest path trees. Inform. Proc. Lett. 14 (1982) 30-33. | MR 654072

[47] B. Thiongane, Réoptimisation dans le dual lagrangien du biknapsack en variables 0-1. Thèse de Doctorat en Informatique, Université de Paris 13 (2003).

[48] B. Thiongane, A. Nagih and G. Plateau, Adapted step size in a 0-1 biknapsack lagrangean dual solving algorithm. En révision pour Ann. Oper. Res. (2002). | MR 2176842 | Zbl 1091.90045

[49] B. Thiongane, A. Nagih, and G. Plateau, Lagrangean heuristics combined with reoptimization for the 0-1 biknapsack problem. En révision pour Discrete Appl. Math. (2002). | Zbl 1111.90096

[50] A.P.M Wagelmans, Sensitivity analysis in combinatorial optimization. Ph.D. thesis, Eramsus University, Rotterdam (1990).

[51] P. Winter, Steiner problem in networks : A survey. Networks 17 (1987) 129-167. | MR 883025 | Zbl 0646.90028