On complexity of the bilevel location and pricing problems

We consider the bilevel mixed integer location and pricing problems. Each problem is determined by the optimization problems of the upper and lower levels of which the first describes the choice of location and pricing, while the second models the reaction of the customers on the upper-level solutio...

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Published in:	Journal of applied and industrial mathematics Vol. 8; no. 4; pp. 574 - 581
Main Authors:	Panin, A. A., Plyasunov, A. V.
Format:	Journal Article
Language:	English
Published:	Moscow Pleiades Publishing 01.10.2014 Springer Nature B.V
Subjects:	Complexity Customers Integer programming Manufacturers Mathematical analysis Mathematical models Mathematics Mathematics and Statistics Mills Mixed integer Optimization Prices Pricing Pricing policies Strategy Studies Poly-APX-completeness AP-reducibility bilevel problem computational complexity and approximation complexity location NP-hardness in the strong sense pricing
ISSN:	1990-4789, 1990-4797
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Abstract	We consider the bilevel mixed integer location and pricing problems. Each problem is determined by the optimization problems of the upper and lower levels of which the first describes the choice of location and pricing, while the second models the reaction of the customers on the upper-level solution. The article focuses on studying the computational complexity of bilevel problems with various pricing strategies: uniform, mill, and discriminatory pricing. We show that, for an arbitrary pricing strategy, the corresponding optimization problem is NP-hard in the strong sense, belongs to the class Poly-APX, and is complete in it with respect to AP-reducibility.
AbstractList	We consider the bilevel mixed integer location and pricing problems. Each problem is determined by the optimization problems of the upper and lower levels of which the first describes the choice of location and pricing, while the second models the reaction of the customers on the upper-level solution. The article focuses on studying the computational complexity of bilevel problems with various pricing strategies: uniform, mill, and discriminatory pricing. We show that, for an arbitrary pricing strategy, the corresponding optimization problem is NP-hard in the strong sense, belongs to the class Poly-APX, and is complete in it with respect to AP-reducibility. We consider the bilevel mixed integer location and pricing problems. Each problem is determined by the optimization problems of the upper and lower levels of which the first describes the choice of location and pricing, while the second models the reaction of the customers on the upper-level solution. The article focuses on studying the computational complexity of bilevel problems with various pricing strategies: uniform, mill, and discriminatory pricing. We show that, for an arbitrary pricing strategy, the corresponding optimization problem is NP-hard in the strong sense, belongs to the class Poly-APX, and is complete in it with respect to AP-reducibility.[PUBLICATION ABSTRACT]
Author	Panin, A. A. Plyasunov, A. V.
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Keywords	Poly-APX-completeness AP-reducibility bilevel problem computational complexity and approximation complexity location NP-hardness in the strong sense pricing
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References_xml	– reference: AboolianRBermanOKrassDOptimizing Pricing and Location Decisions for Competitive Service Facilities Charging Uniform PriceJ. Oper. Res. Soc.2008591506151910.1057/palgrave.jors.26024931168.90607 – reference: BouhtouMGrigorievAVan Der KraaijA FVan HoeselSSpieksmaFUetzMPricing Bridges to Cross a RiverNaval Res. Logistics.200754441142010.1002/nav.202161210.91024 – reference: SerraDReVelleCCompetitive Locations and Pricing on NetworksGeograph. Anal.19993110912910.1111/j.1538-4632.1999.tb00972.x – reference: AboolianRBermanOKrassDCompetitive Facility Location Model with Concave DemandEuropean J. Oper. Res.200718159861910.1016/j.ejor.2005.10.0751131.900272308701 – reference: BazganCEscofferBPaschosV ThCompleteness in Standard and Differential Approximation Classes: Poly-(D)APX- and (D)PTAS-CompletenessTheor. Comput. Sci.200533927229210.1016/j.tcs.2005.03.0071068.68063 – reference: VivesXOligopoly Pricing: Old Ideas and New Tools1999Cambridge, AMIT Press – reference: HanjoulPHansenPPeetersDThisseJ-FUncapacitated Plant Location under Alternative Spatial Price PoliciesMarket Sci.19903641570694.900451038215 – reference: GareyM RJohnsonD SComputers and Intractability: A Guide to the Theory of NP-Completeness1979FreemanSan Francisco0411.68039 – reference: PaninA APlyasunovAVThe Pricing Problem. II: Computational ComplexityDiskret. Anal. Issled. Oper.20121965671[J. Appl. Industr. Math. 7 (3), 420–430 (2013)] – reference: PaninA APlyasunovA VThe Pricing Problem. I: Exact and Approximate AlgorithmsDiskret. Anal. Issled. Oper.20121923140[J. Appl. Industr. Math. 7 (2), 241–251 (2013)] – reference: AusielloGCrescenziPGambosiGKannVMarchetti-SpaccamelaAProtasiMComplexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties1999BerlinSpringer10.1007/978-3-642-58412-10937.68002 – reference: EiseltH AMarianovVFoundations of Location Analysis2011New YorkSpringer10.1007/978-1-4419-7572-0 – reference: DasciALaporteGLocation and Pricing Decisions of a Multistore Monopoly in a Spatial MarketJ. Region. Sci.20044448951510.1111/j.0022-4146.2004.00346.x – reference: DempeS JFoundations of Bilevel Programming2002DordrechtKluwer Acad. Publ.1038.90097 – reference: GrigorievAvan LoonJSviridenkoMUetzMVredeveldTOptimal Bundle Pricing with Monotonicity ConstraintOper. Res. Lett.200836560961410.1016/j.orl.2008.04.0081153.914372459512 – reference: KochetovYu APlyasunovA VPolynomially Solvable Class of the Problems of Bilevel Linear ProgrammingDiskret. Anal. Issled. Oper. Ser. 219971223331659028 – reference: CrescenziPKannVSilvestriRTrevisanLStructure in Approximation ClassesSIAM J. Comput.19992851759178210.1137/S00975397963042200932.681371694160 – volume-title: Oligopoly Pricing: Old Ideas and New Tools year: 1999 ident: 4375_CR17 – volume: 31 start-page: 109 year: 1999 ident: 4375_CR16 publication-title: Geograph. Anal. doi: 10.1111/j.1538-4632.1999.tb00972.x – volume: 36 start-page: 609 issue: 5 year: 2008 ident: 4375_CR14 publication-title: Oper. Res. Lett. doi: 10.1016/j.orl.2008.04.008 – volume: 19 start-page: 31 issue: 2 year: 2012 ident: 4375_CR3 publication-title: Diskret. Anal. Issled. Oper. – volume: 339 start-page: 272 year: 2005 ident: 4375_CR8 publication-title: Theor. Comput. Sci. doi: 10.1016/j.tcs.2005.03.007 – volume-title: Foundations of Location Analysis year: 2011 ident: 4375_CR13 doi: 10.1007/978-1-4419-7572-0 – volume: 28 start-page: 1759 issue: 5 year: 1999 ident: 4375_CR10 publication-title: SIAM J. Comput. doi: 10.1137/S0097539796304220 – volume: 59 start-page: 1506 year: 2008 ident: 4375_CR6 publication-title: J. Oper. Res. Soc. doi: 10.1057/palgrave.jors.2602493 – volume-title: Computers and Intractability: A Guide to the Theory of NP-Completeness year: 1979 ident: 4375_CR1 – volume: 19 start-page: 56 issue: 6 year: 2012 ident: 4375_CR4 publication-title: Diskret. Anal. Issled. Oper. – volume-title: Foundations of Bilevel Programming year: 2002 ident: 4375_CR12 – volume: 36 start-page: 41 year: 1990 ident: 4375_CR15 publication-title: Market Sci. – volume: 54 start-page: 411 issue: 4 year: 2007 ident: 4375_CR9 publication-title: Naval Res. Logistics. doi: 10.1002/nav.20216 – volume: 44 start-page: 489 year: 2004 ident: 4375_CR11 publication-title: J. Region. Sci. doi: 10.1111/j.0022-4146.2004.00346.x – volume: 181 start-page: 598 year: 2007 ident: 4375_CR5 publication-title: European J. Oper. Res. doi: 10.1016/j.ejor.2005.10.075 – volume: 1 start-page: 23 issue: 2 year: 1997 ident: 4375_CR2 publication-title: Diskret. Anal. Issled. Oper. Ser. 2 – volume-title: Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties year: 1999 ident: 4375_CR7 doi: 10.1007/978-3-642-58412-1
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SubjectTerms	Complexity Customers Integer programming Manufacturers Mathematical analysis Mathematical models Mathematics Mathematics and Statistics Mills Mixed integer Optimization Prices Pricing Pricing policies Strategy Studies
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