Problem Solving In Operation Management pp 117-133 | Cite as

# A Service Location Model in a Bi-level Structure

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## Abstract

The principal aim of this chapter is to show a network location services model for a specific problem, which has originally been formulated as one with one objective. The multi-objective strategy has been useful in situations where there is more than one objective and where in many cases they may be contradictory. Such approach does not consider interdependence among each other. Multilevel programming, on the other hand, does take it into consideration, which allows for a hierarchical organization of the objectives and the consideration of relationships among them. The proposed model was applied for a drug distribution network in the State of Mexico, for which optimum storage location is suggested.

## Keywords

Bi-level programming Location Networks Distribution## References and Bibliography

- E. Aiyoshi, Shimizu, Hierarchical decentralized systems and its new solution by barrier method. IEEE Trans. Syst. Man Cybern.
**11**, 444–448 (1981)MathSciNetCrossRefGoogle Scholar - G. Savard, J. Gauvin, The steepest descent direction for the nonlinear bilevel programming problem. Technical Report G-90-37, Groupe d’ Études et de Recherche en Analyse des Décisions (1990)Google Scholar
- G. M. Roodman, Postoptimality analysis in zero-one programming by implicit enumeration. Naval Res. Logist. Quarterly.
**19**(3), 435–447 (1972)Google Scholar - H. Benson, On the structure and properties of a linear multilevel programming problem. J. Optim. Theory Appl.
**60**, 353–373 (1989)MathSciNetCrossRefGoogle Scholar - H. Stackelberg,
*Market Structure and Equilibrium*(Springer-Verlag Wien, New York, 1934)zbMATHGoogle Scholar - J. Bard, Optimality conditions for the bilevel programming problem. Naval Research Logistics Quarterly
**31**, 13–26 (1984)MathSciNetCrossRefGoogle Scholar - J. Bard, Some properties of the bilevel programming problem. J. Optim. Theory Appl.
**68**, Technical note, 371–378 (1991)MathSciNetCrossRefGoogle Scholar - J. Bard,
*Practical Bilevel Optimization. Algorithms and Applications*(Kluwer Academic Publishers, Boston, 1998)CrossRefGoogle Scholar - J. F. Bard, J.E. Falk, An explicit solution to the multi-level programming problem. Comput. Oper. Res.
**9**(1), 77–100 (1982)MathSciNetCrossRefGoogle Scholar - J. Bard, J. Moore, An algorithm for the discrete bilevel programming problem. Nav. Res. Logist.
**39**, 419–435 (1992)MathSciNetCrossRefGoogle Scholar - J. Braken, J. McGill, Mathematical programs with optimization problems in the constraints. Oper. Res.
**21**, 21–37 (1973)MathSciNetGoogle Scholar - J. Outrata, Necessary optimality conditions for Stackelberg problems. J. Optim. Theory Appl.
**76**, 305–320 (1993)MathSciNetCrossRefGoogle Scholar - J. Ye, D. Zhu, Optimality conditions for bi-level programming problems. Technical Report DMS-618-IR, Department of Mathematics and Statistics, University of Victoria (1993)Google Scholar
- L. Vicente, P. Calamai, Geometry and local optimality conditions for bi-level programs with quadratic strictly convex lower level. Technical Report #198-O-150294, Department of Systems Design Engineering, University of Waterloo (1994)Google Scholar
- O. Ben-Ayed, C. Blair, Computational difficulties of bilevel linear programming. Oper. Res.
**38**, 556–560 (1990)MathSciNetCrossRefGoogle Scholar - P. Hansen, B. Jaumard, G. Savard, New branch and bound rules for linear bilevel programming. SIAM J. Sci. Stat. Comput.
**13**, 1194–1217 (1992)MathSciNetCrossRefGoogle Scholar - R. Jeroslow, The polynomial hierarchy and simple model for competitive analysis for competitive analysis. Math. Program. 32, 146–164 (1985)Google Scholar
- S. Dempe, A necessary and sufficient optimality condition for bilevel programming prob- lems. Optimization
**25**, 341–354 (1992)MathSciNetCrossRefGoogle Scholar - S. Dempe,
*Foundations of Bilevel Programming*(Kluwer Academic Publishers, United States of America, 2002)zbMATHGoogle Scholar - U. Wen, Mathematical methods for multilevel linear programming. PhD thesis, Department of Industrial Engineering, State University of New York at Buffalo (1981)Google Scholar
- W. Bialas, M. Karwan, J. Shaw, “A parametric complementary pivot approach for two-level linear programming.” State University of New York at Buffalo.
**57**(1980)Google Scholar - W. Candler, R. Norton, Multilevel programming. Technical Report 20, World Bank Development Research Center, Washington D.C. (1977)Google Scholar
- Y. Chen, M. Florian, The nonlinear bilevel programming problem: A general formulation and optimality conditions. Technical Report CRT-794, Centre de Recherché sur les Transports (1991)Google Scholar
- Y. Ishisuka, Optimality conditions for quasi-differentiable programs with applications to two-level optimization. SIAM J. Control. Optim.
**26**, 1388–1398 (1988)MathSciNetCrossRefGoogle Scholar - Z. Bi, P. Calami, Optimality conditions for a class of bilevel programming problems. Technical Report #191-O-191291, Department of Systems Design Engineering, University of Waterloo (1991)Google Scholar