Editorial Bilevel Programming, Equilibrium, and...

EditorialBilevel Programming, Equilibrium, and CombinatorialProblems with Applications to Engineering

Vyacheslav Kalashnikov,1,2,3 Timothy I. Matis,4

José Fernando Camacho Vallejo,5 and Sergii V. Kavun6

1Department of Systems & Industrial Engineering, Tecnologico de Monterrey (ITESM), Campus Monterrey,Avenida Eugenio Garza Sada 2501 Sur, 64849 Monterrey, NL, Mexico2Department of Social Modeling, Central Economics & Mathematics Institute (CEMI), Russian Academy of Sciences,Nakhimovsky pr. 47, Moscow 117418, Russia3Department of Computing, Sumy State University, Rimsky-Korsakov St. 2, Sumy 40007, Ukraine4Department of Industrial Engineering, Texas Tech University, P.O. Box 43061, Lubbock, TX 79409, USA5Facultad de Ciencias Fısico-Matematicas, Universidad Autonoma de Nuevo Leon, Avenida Universidad S/N,66450 San Nicolas de los Garza, NL, Mexico6Kharkiv Institute of Banking of the University of Banking of the National Bank of Ukraine,Peremogy Avenue 55, Kharkiv 61174, Ukraine

Correspondence should be addressed to Vyacheslav Kalashnikov; [email protected]

Received 16 September 2014; Accepted 16 September 2014

Copyright © 2015 Vyacheslav Kalashnikov et al.This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in anymedium, provided the originalwork is properly cited.

Although a wide range of applications fit the bilevel pro-gramming framework, real-life implementations are scarce,due mainly to the lack of efficient algorithms for tacklingmedium- and large-scale bilevel programming problems.Solving a bilevel (more generally, hierarchical) optimizationproblem, even in its simplest form, is a difficult task. A lotof different alternative methods may be used based on thestructure of the problem analyzed, but there is no generalmethod that guarantees convergence, performance, or opti-mality for every type of problem.

Many new ideas appeared and were discussed in worksof plenty of authors. Among them, we would name Dempe[1], Dempe et al. [2], Dempe and Dutta [3], Dewez et al. [4],Thi et al. [5], and Vicente and Calamai [6], whose works havedeveloped various ways of reducing original bilevel program-ming problems to equivalent single level ones, thus makingtheir solution somewhat an easier task for conventionalmathematical programming software packages.

Mixed-integer bilevel programming problems (with partof variables at the upper and/or lower level being integer/

Boolean ones) are even harder for the well-known conven-tional optimization techniques. For instance, a usual replace-ment of the lower level optimization problem with the cor-responding Karush-Kuhn-Tucker (KKT) conditions may notwork if some lower level variables are not continuous. There-fore, solid theoretical base is necessary to be found, in orderto propose efficient algorithmic procedures aimed at findinglocal or global solutions of such a problem.

A great amount of new applied problems in the area ofenergy networks have recently arisen that can be efficientlysolved only as mixed-integer bilevel programs. Among themare the natural gas cash-out problem, the deregulated electric-ity market equilibrium problem, biofuel problems, problemof designing coupled energy carrier networks, and so forth,if we mention only part of such applications. Bilevel modelsto describe migration processes are also in the running of themost popular new themes in the area of bilevel programming.

This special volume comprises papers dealing with threemain themes: bilevel programming, equilibriummodels, andcombinatorial (integer programming) problems and theirapplications to engineering.

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 490758, 3 pageshttp://dx.doi.org/10.1155/2015/490758

2 Mathematical Problems in Engineering

The special issue opens with a survey paper “Bilevelprogramming and applications” by S. Dempe et al. that sumsup some recent and new directions and results of the devel-opment of the mathematical methods aimed at the solutionof bilevel programs of different types and their applicationsto real-life problems.

One of such applications is the well-known bilevel pro-gramming problem arising in the natural gas industry andrelated to the solution of the important imbalance cash-out problem (a good part of the above-mentioned surveypaper is devoted to this area). Since the stochastic state-ment of the problem in question needs the generation ofscenario trees with a very large number of branches andarcs, the reduction of the quantity of common variables isan important challenge. This item is analyzed and solved byproposing newmethods of grouping the variables (classifyingand representing various groups of states with respect to thenatural gas prices) in the article “US natural gas market clas-sification using pooled regression” by V. V. Kalashnikov et al.The techniques proposed in this paper help classify regionsor states into groups or classes that share similar regressionparameters. Once obtained, these groups may be used tomake assumptions about corresponding natural gas prices infurther studies.

One of the principal management problems dealt within numerous organizations and institutions in the publicsector is to decide how to invest and manage funds availableto support potential research and development projectsin various areas. A classification of the portfolio problemdistinguishes two types: (a) static and (b) dynamic. In (a) onlyprojects proposals for funding are involved, while in (b) atcertain moments some active projects are withdrawn fromthe portfolio and some inactive projects are activated.Withinthis framework, a social project is a group of tasks or activitiesconsuming funds, carried out during a given period of timein one or more regions, with an impact on the objectivesset by an organization, and focused on providing solutionto problems or needs of the society. The paper “Selectinglarge portfolios of social projects in public organizations” by I.Litvinchev et al. models the task as a two-objective mixed-integer linear programming problem. The model supportsboth complete (all or nothing) and partial (a certain amountfrom a given interval of funding) resource allocation policies.Numerical results for large scale problem instances are alsopresented. The computational experiment demonstrates thatincreasing the number of projects subject to funds reductionresults in a decrease in both objectives. However, if thenumber of projects subject to funds reduction is fixed but thestrength of funds reduction is increased, one of the objectivesmay grow at the expense of a larger drop in another.

A capacitated fixed-cost facility location problem withtransportation choices (CFCLP-TC) is studied in the paper“Variations in the flow approach to CFCLP-TC for multi-objective supply chain design” by M. P. Hertwin et al. Theproblem based on a production network of two echelons withmultiple plants, a set of potential distribution centers andcustomers, is formulated as an optimization model with twoobjective functions with time and cost as variables. The maincontribution of this paper is in an extension of the existing

approaches to the supply of products to customers throughmultiple sources, the direct flow between plants and cus-tomers, and the product flow between distribution centers.Based upon these approaches, the authors generate mathe-matical programming models and propose solution methodsby the 𝜀-constraint approach. Namely, they generate Paretoboundaries and thus compare each of those approaches withthe original model. The models are implemented in GAMSand solved with CPLEX.

Another work presenting a novel capacitated modelfor supply chain network design is the paper “Design andoptimization of capacitated supply chain networks includingquality measures” by K. K. Castillo-Villar et al. The modelin question (abbreviated as SCND-COQ) is described as amixed-integer nonlinear programming problem, which canbe used at a strategic planning level to design a supply chainnetwork that maximizes the total profit subject to meetingan overall quality level of the final product at the minimumcosts. To process the problem, the authors propose fivecombinatorial optimization algorithms based on nonlinearoptimization, heuristic, andmetaheuristic approaches, whichare used to solve realistic instances of practical size.The com-putational results demonstrate that the state and individualrepresentation in the simulated annealing and the geneticalgorithm, respectively, has a significant impact on theimprovement of the solution quality.

A problem of packing a limited number of unequal circlesin a fixed size rectangular container is considered in thepaper “Integer programming formulations for approximatepacking circles in a rectangular container” by I. Litvinchevand E. L. O. Espinosa. The aim of this research is tomaximize the (weighted) number of circles placed into thecontainer orminimize the waste.This problem has numerousapplications in logistics, including production and packingfor the textile, apparel, naval, automobile, aerospace, and foodindustries. Frequently the problem is formulated as a non-convex continuous optimization problem, which is solved byheuristic techniques combined with local search procedures.The authors propose new formulations for an approximatesolution of the packing problem. The container is approx-imated by a regular grid and the nodes of the grid areconsidered as potential positions for assigning centers ofthe circles. The packing problem is then stated as a largescale linear 0-1 optimization problem. The binary variablesrepresent the assignment of centers to the nodes of the grid.Nesting circles inside one another are also considered. Theresulting binary problem is then solved by commercial soft-ware. Numerical results are presented to demonstrate the effi-ciency of the proposed approach and compared with knownresults.

The paper “A heuristic procedure for a ship routing andscheduling problem with variable speed and discretized timewindows” by K. K. Castillo-Villar et al. develops a heuristicalgorithm solving a routing and scheduling problem fortramp shipping with discretized time windows. The problemconsists of determining the set of cargoes that should beserved by each ship, as well as the arrival, departure, andwaiting times at each port, while minimizing the total costs.The heuristic proposed is based on a variable neighborhoodsearch, considering a number of neighborhood structures

Mathematical Problems in Engineering 3

to find a solution to the problem. The authors presentcomputational results, and, for comparison purposes, theyconsider instances that can be solved directly by CPLEX totest the performance of the proposed heuristic. The heuristicachieves good solution qualities within reasonable computa-tional times. The obtained numerical results are encouragingandmake the presented heuristic quite promising to be usefulwhen solving large real-size examples.

It is well-known that modern approaches to the devel-opment of a national economy are often characterized withan imbalanced inflation of some economic branches leadingto a disproportional socioeconomic territories development(SETD). Such disproportions, together with other similarfactors, frequently result in a lack of economic integrity,various regional crises, and a low rate of the economic andterritorial growth.Those disproportions may also conduce toan inadequate degree of the interregional collaboration. Thepaper “Simulation of territorial development based on fiscalpolicy tools” by R. Brumnik et al. proposes some new ways ofregulating imbalances in the territorial development makinguse of the fiscal policy tools. The latter can immediatelyreduce the amplitude of economic cycle fluctuations andprovide a stable development of the economic state system.The same approach is applied to control the processes oftransformation of the tax legislation and tax relations, aswell as the levying and redistribution of the recollected taxesamong the territories’ budgets (this approach is also known asa tax policy). To resume, this paper describes comprehensivemodels of financial regulation of the socioeconomic territo-rial development that can help in estimating and choosingthe right financial policy parameters. The research highlightsthe following conclusions: analysis of the predictive dynamicsof socioeconomic development territories, in the case of theimplementation of an optimistic scenario of tax revenue,demonstrates the effectiveness of the adopted stabilizationpolicy.

Though a bit off the special issue’s general stream, stillinteresting results of research are provided by the article“The solution of fourth order boundary value problem arisingout of the beam-column theory using Adomian decompositionmethod” by O. Kelesoglu. In this study, Adomian decomposi-tion method (ADM) is applied to a linear nonhomogeneousboundary value problem arising from the beam-columntheory. The obtained results of numerical experiments arepresented in tables and illustrated with graphs.The proposedapproach provides a rapidly converging method, with theapproximate solutions tending to the exact solution of theproblem in question. This fact characterizes the method asappropriate and reliable for such kind of problems.

Wehope that readers of this special issuewill findnot onlynew ideas and algorithms dealing with the difficult appliedproblems like supply chain design, assignment problems,bilevel programming models, and so forth, but also someinteresting results related to regression analysis, as well as themodern simulation techniques.

Acknowledgments

A special gratitude is also due to the long list of referees,whose valuable comments and suggestions have helped us a

lot in improving our results and/or presentation style. Theresearch activity of Vyacheslav Kalashnikov was financiallysupported by Tecnologico de Monterrey (ITESM), CampusMonterrey, and by the SEP-CONACYT Project CB-2013-01-221676, Mexico. The research activity of Jose FernandoCamacho Vallejo was financially supported by the ProjectPROMEP/103.5/10/3889, the Academic Groups ResearchProject PROMEP/103.5/12/4953, and the Autonomous Uni-versity of Nuevo Leon (UANL) within the Support Programfor Scientific Research and Technology (PAICYT) within theProject CE960-11.

Vyacheslav KalashnikovTimothy I. Matis

Jose Fernando Camacho VallejoSergii V. Kavun

References

[1] S. Dempe, Foundations of Bilevel Programming, Springer, Hei-delberg, Germany, 2002.

[2] S. Dempe, B. S. Mordukhovich, and A. B. Zemkoho, “Necessaryoptimality conditions in pessimistic bilevel programming,”Optimization, vol. 63, no. 4, pp. 505–533, 2014.

[3] S. Dempe and J. Dutta, “Is bilevel programming a special caseof amathematical programwith complementarity constraints?”Mathematical Programming A, vol. 131, pp. 37–48, 2012.

[4] S. Dewez, M. Labbe, P. Marcotte, and G. Savard, “New for-mulations and valid inequalities for a bilevel pricing problem,”Operations Research Letters, vol. 36, no. 2, pp. 141–149, 2008.

[5] L. Thi, T. Duc, and P. Dinh, “A DC programming approach fora class of bilevel programming problems and its application inportfolio selection,” Numerical Algebra, Control and Optimiza-tion, vol. 2, no. 1, pp. 167–185, 2012.

[6] L. N. Vicente and P. H. Calamai, “Bilevel and multilevelprogramming: a bibliography review,” Journal of Global Opti-mization, vol. 5, no. 3, pp. 291–306, 1994.

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of

Editorial Bilevel Programming, Equilibrium, and...

Documents

Transcript of Editorial Bilevel Programming, Equilibrium, and...