IDEAS home Printed from https://ideas.repec.org/a/spr/mathme/v96y2022i1d10.1007_s00186-022-00776-y.html
   My bibliography  Save this article

Solving continuous set covering problems by means of semi-infinite optimization

Author

Listed:
  • Helene Krieg

    (Fraunhofer Institute for Industrial Mathematics ITWM)

  • Tobias Seidel

    (Fraunhofer Institute for Industrial Mathematics ITWM)

  • Jan Schwientek

    (Fraunhofer Institute for Industrial Mathematics ITWM)

  • Karl-Heinz Küfer

    (Fraunhofer Institute for Industrial Mathematics ITWM)

Abstract

This article introduces the new class of continuous set covering problems. These optimization problems result, among others, from product portfolio design tasks with products depending continuously on design parameters and the requirement that the product portfolio satisfies customer specifications that are provided as a compact set. We show that the problem can be formulated as semi-infinite optimization problem (SIP). Yet, the inherent non-smoothness of the semi-infinite constraint function hinders the straightforward application of standard methods from semi-infinite programming. We suggest an algorithm combining adaptive discretization of the infinite index set and replacement of the non-smooth constraint function by a two-parametric smoothing function. Under few requirements, the algorithm converges and the distance of a current iterate can be bounded in terms of the discretization and smoothing error. By means of a numerical example from product portfolio optimization, we demonstrate that the proposed algorithm only needs relatively few discretization points and thus keeps the problem dimensions small.

Suggested Citation

  • Helene Krieg & Tobias Seidel & Jan Schwientek & Karl-Heinz Küfer, 2022. "Solving continuous set covering problems by means of semi-infinite optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(1), pages 39-82, August.
    • Handle: RePEc:spr:mathme:v:96:y:2022:i:1:d:10.1007_s00186-022-00776-y
    DOI: 10.1007/s00186-022-00776-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00186-022-00776-y
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s00186-022-00776-y?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Van den Broeke, Maud & Boute, Robert & Cardoen, Brecht & Samii, Behzad, 2017. "An efficient solution method to design the cost-minimizing platform portfolio," European Journal of Operational Research, Elsevier, vol. 259(1), pages 236-250.
    2. Lopez, Marco & Still, Georg, 2007. "Semi-infinite programming," European Journal of Operational Research, Elsevier, vol. 180(2), pages 491-518, July.
    3. J. Bennell & G. Scheithauer & Y. Stoyan & T. Romanova, 2010. "Tools of mathematical modeling of arbitrary object packing problems," Annals of Operations Research, Springer, vol. 179(1), pages 343-368, September.
    4. Y. Stoyan & T. Romanova & G. Scheithauer & A. Krivulya, 2011. "Covering a polygonal region by rectangles," Computational Optimization and Applications, Springer, vol. 48(3), pages 675-695, April.
    5. Gang Du & Yi Xia & Roger J. Jiao & Xiaojie Liu, 2019. "Leader-follower joint optimization problems in product family design," Journal of Intelligent Manufacturing, Springer, vol. 30(3), pages 1387-1405, March.
    6. Hatim Djelassi & Alexander Mitsos, 2017. "A hybrid discretization algorithm with guaranteed feasibility for the global solution of semi-infinite programs," Journal of Global Optimization, Springer, vol. 68(2), pages 227-253, June.
    7. Alexander Mitsos & Angelos Tsoukalas, 2015. "Global optimization of generalized semi-infinite programs via restriction of the right hand side," Journal of Global Optimization, Springer, vol. 61(1), pages 1-17, January.
    8. Xing-Si Li & Shu-Cherng Fang, 1997. "On the entropic regularization method for solving min-max problems with applications," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 46(1), pages 119-130, February.
    9. Du, Gang & Jiao, Roger J. & Chen, Mo, 2014. "Joint optimization of product family configuration and scaling design by Stackelberg game," European Journal of Operational Research, Elsevier, vol. 232(2), pages 330-341.
    10. Kohli, Rajeev & Krishnamurti, Ramesh, 1989. "Optimal product design using conjoint analysis: Computational complexity and algorithms," European Journal of Operational Research, Elsevier, vol. 40(2), pages 186-195, May.
    11. Paul E. Green & Abba M. Krieger, 1985. "Models and Heuristics for Product Line Selection," Marketing Science, INFORMS, vol. 4(1), pages 1-19.
    12. E. Polak & J. O. Royset & R. S. Womersley, 2003. "Algorithms with Adaptive Smoothing for Finite Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 119(3), pages 459-484, December.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Daniel Jungen & Hatim Djelassi & Alexander Mitsos, 2022. "Adaptive discretization-based algorithms for semi-infinite programs with unbounded variables," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(1), pages 83-112, August.
    2. Daria Dzyabura & Srikanth Jagabathula, 2018. "Offline Assortment Optimization in the Presence of an Online Channel," Management Science, INFORMS, vol. 64(6), pages 2767-2786, June.
    3. Robert N. Boute & Maud M. Van den Broeke & Kristof A. Deneire, 2017. "Barco Implements Platform-Based Product Development in Its Healthcare Division," Decision Analysis, INFORMS, vol. 48(01), pages 35-44, February.
    4. Duarte, Belmiro P.M. & Sagnol, Guillaume & Wong, Weng Kee, 2018. "An algorithm based on semidefinite programming for finding minimax optimal designs," Computational Statistics & Data Analysis, Elsevier, vol. 119(C), pages 99-117.
    5. Van den Broeke, Maud & Boute, Robert & Cardoen, Brecht & Samii, Behzad, 2017. "An efficient solution method to design the cost-minimizing platform portfolio," European Journal of Operational Research, Elsevier, vol. 259(1), pages 236-250.
    6. Xinfang (Jocelyn) Wang & Jeffrey D. Camm & David J. Curry, 2009. "A Branch-and-Price Approach to the Share-of-Choice Product Line Design Problem," Management Science, INFORMS, vol. 55(10), pages 1718-1728, October.
    7. G. E. Fruchter & A. Fligler & R. S. Winer, 2006. "Optimal Product Line Design: Genetic Algorithm Approach to Mitigate Cannibalization," Journal of Optimization Theory and Applications, Springer, vol. 131(2), pages 227-244, November.
    8. Evan Rash & Karl Kempf, 2012. "Product Line Design and Scheduling at Intel," Interfaces, INFORMS, vol. 42(5), pages 425-436, October.
    9. Schön, Cornelia, 2010. "On the product line selection problem under attraction choice models of consumer behavior," European Journal of Operational Research, Elsevier, vol. 206(1), pages 260-264, October.
    10. Dimitris Bertsimas & Velibor V. Mišić, 2019. "Exact First-Choice Product Line Optimization," Operations Research, INFORMS, vol. 67(3), pages 651-670, May.
    11. Winfried J. Steiner & Harald Hruschka, 2002. "Produktliniengestaltung mit Genetischen Algorithmen," Schmalenbach Journal of Business Research, Springer, vol. 54(7), pages 575-601, November.
    12. Jan Schwientek & Tobias Seidel & Karl-Heinz Küfer, 2021. "A transformation-based discretization method for solving general semi-infinite optimization problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 93(1), pages 83-114, February.
    13. Alexouda, Georgia & Paparrizos, Konstantinos, 2001. "A genetic algorithm approach to the product line design problem using the seller's return criterion: An extensive comparative computational study," European Journal of Operational Research, Elsevier, vol. 134(1), pages 165-178, October.
    14. Demiröz, Barış Evrim & Altınel, İ. Kuban & Akarun, Lale, 2019. "Rectangle blanket problem: Binary integer linear programming formulation and solution algorithms," European Journal of Operational Research, Elsevier, vol. 277(1), pages 62-83.
    15. J. O. Royset & E. Y. Pee, 2012. "Rate of Convergence Analysis of Discretization and Smoothing Algorithms for Semiinfinite Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 155(3), pages 855-882, December.
    16. Alexandre Belloni & Robert Freund & Matthew Selove & Duncan Simester, 2008. "Optimizing Product Line Designs: Efficient Methods and Comparisons," Management Science, INFORMS, vol. 54(9), pages 1544-1552, September.
    17. Hatim Djelassi & Alexander Mitsos, 2021. "Global Solution of Semi-infinite Programs with Existence Constraints," Journal of Optimization Theory and Applications, Springer, vol. 188(3), pages 863-881, March.
    18. Pantourakis, Michail & Tsafarakis, Stelios & Zervoudakis, Konstantinos & Altsitsiadis, Efthymios & Andronikidis, Andreas & Ntamadaki, Vasiliki, 2022. "Clonal selection algorithms for optimal product line design: A comparative study," European Journal of Operational Research, Elsevier, vol. 298(2), pages 585-595.
    19. Tsafarakis, Stelios & Zervoudakis, Konstantinos & Andronikidis, Andreas & Altsitsiadis, Efthymios, 2020. "Fuzzy self-tuning differential evolution for optimal product line design," European Journal of Operational Research, Elsevier, vol. 287(3), pages 1161-1169.
    20. Tan Wang & Genaro Gutierrez, 2022. "Robust Product Line Design by Protecting the Downside While Minding the Upside," Production and Operations Management, Production and Operations Management Society, vol. 31(1), pages 194-217, January.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:mathme:v:96:y:2022:i:1:d:10.1007_s00186-022-00776-y. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.