Skip to main content
Springer Nature Link
Log in

A Compact Formulation for the mDmSOP: Theoretical and Computational Time Analysis

  • Conference paper
  • First Online:
Evolution in Computational Intelligence (FICTA 2023)

Part of the book series: Smart Innovation, Systems and Technologies ((SIST,volume 370))

  • 256 Accesses

Abstract

The multi-Depot multiple Set Orienteering Problem (mDmSOP) is one of the recently proposed variants of the Set Orienteering Problem (SOP), which has applicability in different real-life applications such as delivering products and mobile crowd-sensing. The objective of the problem is to collect maximum profit from clusters within a given budget. In this paper, we propose an improved integer linear programming (ILP) formulation of the mDmSOP and conduct a time analysis of the results. We solved it using GAMS 39.2.0 and found that we can reduce a large number of constraints while changing sub-tour elimination constraints only. In the case of small instances, the improved mathematical formulation gives better results in all of the test cases for small instances up to 76 vertices except one instance of 16eil76 when \(w<0.5\), and it gives better results in 93.33% of cases for small instances and 88.23% of cases while simulating on mid-size instances up to 198 nodes when \(w=0.5\).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Angelelli, E., Archetti, C., Vindigni, M.: The clustered orienteering problem. Eur. J. Oper. Res. 238(2), 404–414 (2014)

    Article  MathSciNet  Google Scholar 

  2. Bazrafshan, R., Hashemkhani Zolfani, S., Mirzapour Al-e Hashem, S.M.J.: Comparison of the sub-tour elimination methods for the asymmetric traveling salesman problem applying the SECA method. Axioms 10(1), 19 (2021)

    Google Scholar 

  3. Bektas, T.: The multiple traveling salesman problem: an overview of formulations and solution procedures. Omega 34(3), 209–219 (2006)

    Google Scholar 

  4. Bektaş, T., Gouveia, L.: Requiem for the Miller–Tucker–Zemlin subtour elimination constraints? Eur. J. Oper. Res. 236(3), 820–832 (2014)

    Article  MathSciNet  Google Scholar 

  5. Campuzano, G., Obreque, C., Aguayo, M.M.: Accelerating the Miller–Tucker–Zemlin model for the asymmetric traveling salesman problem. Expert Syst. Appl. 148, 113229 (2020)

    Article  Google Scholar 

  6. Dantzig, G., Fulkerson, R., Johnson, S.: Solution of a large-scale traveling-salesman problem. J. Oper. Res. Soc. Am. 2(4), 393–410 (1954)

    MathSciNet  Google Scholar 

  7. Desrochers, M., Laporte, G.: Improvements and extensions to the Miller–Tucker–Zemlin subtour elimination constraints. Oper. Res. Lett. 10(1), 27–36 (1991)

    Article  MathSciNet  Google Scholar 

  8. Fischetti, M., Salazar González, J.J., Toth, P.: A branch-and-cut algorithm for the symmetric generalized traveling salesman problem. Oper. Res. 45(3), 378–394 (1997)

    Article  MathSciNet  Google Scholar 

  9. Hamilton, W.R.: Problema del viajante

    Google Scholar 

  10. Kant, R., Mishra, A.: The multi Depot multiple set orienteering problem. Unpublished Manuscript (2022)

    Google Scholar 

  11. Miller, C.E., Tucker, A.W., Zemlin, R.A.: Integer programming formulation of traveling salesman problems. J. ACM (JACM) 7(4), 326–329 (1960)

    Article  MathSciNet  Google Scholar 

  12. Noon, C.E.: The generalized traveling salesman problem. Ph.D. thesis. University of Michigan (1988)

    Google Scholar 

  13. Öncan, T., Altınel, I.K., Laporte, G.: A comparative analysis of several asymmetric traveling salesman problem formulations. Comput. Oper. Res. 36(3), 637–654 (2009)

    Article  MathSciNet  Google Scholar 

  14. Sawik, T.: A note on the Miller–Tucker–Zemlin model for the asymmetric traveling salesman problem. Bull. Pol. Acad. Sci. Tech. Sci. 3 (2016)

    Google Scholar 

  15. Sherali, H.D., Driscoll, P.J.: On tightening the relaxations of Miller–Tucker–Zemlin formulations for asymmetric traveling salesman problems. Oper. Res. 50(4), 656–669 (2002)

    Article  MathSciNet  Google Scholar 

  16. Vansteenwegen, P., Souffriau, W., Van Oudheusden, D.: The orienteering problem: a survey. Eur. J. Oper. Res. 209(1), 1–10 (2011)

    Article  MathSciNet  Google Scholar 

  17. Velednitsky, M.: Short combinatorial proof that the DFJ polytope is contained in the MTZ polytope for the asymmetric traveling salesman problem. arXiv preprint arXiv:1805.06997 (2018)

  18. Yuan, Y., Cattaruzza, D., Ogier, M., Semet, F.: A note on the lifted Miller–Tucker–Zemlin subtour elimination constraints for routing problems with time windows. Oper. Res. Lett. 48(2), 167–169 (2020)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of CSIS, Birla Institute of Technology and Science, Pilani, 333031, India

    Ravi Kant & Abhishek Mishra

Authors
  1. Ravi Kant
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abhishek Mishra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ravi Kant .

Editor information

Editors and Affiliations

  1. Department of Electronics Engineering, Faculty of Engineering and Technology (UNSIET), Veer Bahadur Singh Purvanchal University, Jaunpur, Uttar Pradesh, India

    Vikrant Bhateja

  2. Middlesex University, London, UK

    Xin-She Yang

  3. Faculty of Engineering, University of Porto, Porto, Portugal

    Marta Campos Ferreira

  4. Cardiff Metropolitan University, Cardiff, Warwickshire, UK

    Sandeep Singh Sengar

  5. Institute for Technological Development and Innovation in Communications, University of Las Palmas de Gran Canaria, Las Palmas de Gran Canaria, Spain

    Carlos M. Travieso-Gonzalez

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Kant, R., Mishra, A. (2023). A Compact Formulation for the mDmSOP: Theoretical and Computational Time Analysis. In: Bhateja, V., Yang, XS., Ferreira, M.C., Sengar, S.S., Travieso-Gonzalez, C.M. (eds) Evolution in Computational Intelligence. FICTA 2023. Smart Innovation, Systems and Technologies, vol 370. Springer, Singapore. https://doi.org/10.1007/978-981-99-6702-5_9

Download citation

Publish with us

Policies and ethics