Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
SpringerLink
Log in

Capacitated ring arborescence problems with profits

  • Regular Article
  • Published:
OR Spectrum Aims and scope Submit manuscript

Abstract

In this work, we introduce profit-oriented capacitated ring arborescence problems and present exact and heuristic algorithms. These combinatorial network design problems ask for optimized bi-level networks taking into account arc costs and node profits. Solutions combine circuits on the inner level with arborescences on the outer level of the networks. We consider the prize-collecting, the budget-constrained and the target-profit models and develop corresponding exact branch-and-cut algorithms based on mixed-integer formulations and valid inequalities. Iterated local search heuristics based on the exploration of problem-specific neighborhoods are elaborated to strengthen the upper bounds. For a set of hard literature derived instances with up to 51 nodes, we provide computational results which give evidence for the efficiency of the proposed approaches. Furthermore, we extensively analyze the performance of our methods, the obtained solution networks and the impact of the cutting planes on the obtained lower bounds.

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

Access this article

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

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Notes

  1. The instances and the solutions can be requested from the corresponding author.

References

  • Abe FHN, Hoshino EA, Hill A (2015) The ring tree facility location problem. Electron Notes Discret Math 50:331–336

    Article  Google Scholar 

  • Archetti C, Feillet D, Hertz A, Speranza MG (2009) The capacitated team orienteering and profitable tour problems. J Oper Res Soc 60(6):831–842

    Article  Google Scholar 

  • Archetti C, Feillet D, Hertz A, Speranza MG (2010) The undirected capacitated arc routing problem with profits. Comput Oper Res 37(11):1860–1869

    Article  Google Scholar 

  • Archetti C, Bianchessi N, Speranza MG (2013) The capacitated team orienteering problem with incomplete service. Optim Lett 7(7):1405–1417

    Article  Google Scholar 

  • Archetti C, Speranza MG, Vigo D (2014) Vehicle routing problems with profits. Veh Routing Probl Methods Appl 18:273

    Article  Google Scholar 

  • Balas E (1989) The prize collecting traveling salesman problem. Networks 19(6):621–636

    Article  Google Scholar 

  • Balas E (1995) The prize collecting traveling salesman problem: II. Polyhedral results. Networks 25(4):199–216

    Article  Google Scholar 

  • Balas E, Fischetti M (1992) The fixed-outdegree 1-arborescence polytope. Math Oper Res 17(4):1001–1018

    Article  Google Scholar 

  • Balas E, Martin CH (1991) Combinatorial optimization in steel rolling. In: Workshop on combinatorial optimization in science and technology, COST

  • Baldacci R, Dell’Amico M (2010) Heuristic algorithms for the multi-depot ring-star problem. Eur J Oper Res 203(1):270–281

    Article  Google Scholar 

  • Baldacci R, Dell’Amico M, Salazar JJ (2007) The capacitated m-ring-star problem. Oper Res 55(6):1147–1162

    Article  Google Scholar 

  • Bautzer A, Gouveia L, Paias A, Pires JM (2016) Models for a Steiner multi-ring network design problem with revenues. TOP 24(2):360–380. https://doi.org/10.1007/s11750-015-0388-6

  • Chao I-M, Golden BL, Wasil EA (1996a) The team orienteering problem. Eur J Oper Res 88(3):464–474

    Article  Google Scholar 

  • Chao I-M, Golden BL, Wasil EA (1996b) A fast and effective heuristic for the orienteering problem. Eur J Oper Res 88(3):475–489

    Article  Google Scholar 

  • Chapovska O, Punnen AP (2006) Variations of the prize-collecting Steiner tree problem. Networks 47(4):199–205

    Article  Google Scholar 

  • Costa AM, Cordeau J-F, Laporte G (2006) Steiner tree problems with profits. INFOR Inf Syst Oper Res 44(2):99–115

    Google Scholar 

  • Costa AM, Cordeau J-F, Laporte G (2009) Models and branch-and-cut algorithms for the Steiner tree problem with revenues, budget and hop constraints. Networks 53(2):141–159

    Article  Google Scholar 

  • Duin C, Voß S (1997) Efficient path and vertex exchange in Steiner tree algorithms. Networks 29(2):89–105

    Article  Google Scholar 

  • Feillet D, Dejax P, Gendreau M (2005) Traveling salesman problems with profits. Transp Sci 39(2):188–205

    Article  Google Scholar 

  • Fu Z-H, Hao J-K (2014) Breakout local search for the Steiner tree problem with revenue, budget and hop constraints. Eur J Oper Res 232(1):209–220

    Article  Google Scholar 

  • Garey Michael R, Johnson David S (1979) Computers and intractability: a guide to the theory of NP-completeness. W. H. Freeman & Co., New York

    Google Scholar 

  • Gendreau M, Laporte G, Semet F (1997) The covering tour problem. Oper Res 45(4):568–576

    Article  Google Scholar 

  • Golden B, Raghavan S, Stanojević D (2008) The prize-collecting generalized minimum spanning tree problem. J Heurist 14(1):69–93

    Article  Google Scholar 

  • Gouveia L, Pires JM (2001) Models for a Steiner ring network design problem with revenues. Eur J Oper Res 133(1):21–31

    Article  Google Scholar 

  • Hachicha M, Hodgson MJ, Laporte G, Semet F (2000) Heuristics for the multi-vehicle covering tour problem. Comput Oper Res 27(1):29–42

    Article  Google Scholar 

  • Hill A (2015) Multi-exchange neighborhoods for the capacitated ring tree problem. LNCS 8962:85–94

    Google Scholar 

  • Hill A, Schwarze S (2018) Exact algorithms for bi-objective ring tree problems with reliability measures. Comput Oper Res 94:38–51

    Article  Google Scholar 

  • Hill A, Voß S (2016) Optimal capacitated ring trees. EURO J Comput Optim 4(2):137–166

    Article  Google Scholar 

  • Hill A, Voß S (2018) Generalized local branching heuristics and the capacitated ring tree problem. Discret Appl Math 242:34–52

    Article  Google Scholar 

  • IBM (2013) IBM ILOG CPLEX Optimization Studio documentation V12.6.0

  • Johnson DS, Minkoff M, Phillips S (2000) The prize collecting Steiner tree problem: theory and practice. In: Proceedings of the 11th annual ACM-SIAM symposium on discrete algorithms, SODA ’00, Philadelphia, PA, USA. Society for Industrial and Applied Mathematics, pp 760–769

  • Klincewicz JG (1998) Hub location in backbone/tributary network design: a review. Locat Sci 6(14):307–335

    Article  Google Scholar 

  • Letchford AN, Eglese RW, Lysgaard J (2002) Multistars, partial multistars and the capacitated vehicle routing problem. Math Progr 94(1):21–40

    Article  Google Scholar 

  • Ljubić I, Weiskircher R, Pferschy U, Klau GW, Mutzel P, Fischetti M (2006) An algorithmic framework for the exact solution of the prize-collecting Steiner tree problem. Math Progr 105(2–3):427–449

    Article  Google Scholar 

  • Lourenço HR, Martin OC, Stützle T (2003) Iterated local search. In: Handbook of metaheuristics. Springer, Berlin, pp 320–353

  • Park J, Kim B-I (2010) The school bus routing problem: a review. Eur J Oper Res 202(2):311–319

    Article  Google Scholar 

  • Polzin T, Daneshmand SV (2001) A comparison of Steiner tree relaxations. Discret Appl Math 112(1): 241–261. ISSN 0166-218X. Combinatorial Optimization Symposium, Selected Papers

  • Schittekat P, Kinable J, Sörensen K, Sevaux M, Spieksma F, Springael J (2013) A metaheuristic for the school bus routing problem with bus stop selection. Eur J Oper Res 229(2):518–528

    Article  Google Scholar 

  • Vansteenwegen P, Souffriau W, Berghe GV, Van Oudheusden D (2009) Iterated local search for the team orienteering problem with time windows. Comput Oper Res 36(12):3281–3290

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Wagner D, Raidl GR, Pferschy U, Mutzel P, Bachhiesl P (2007) A multi-commodity flow approach for the design of the last mile in real-world fiber optic networks. In: Waldmann K-H, Stocker UM (eds) Operations Research Proceedings 2006: Selected Papers of the Annual International Conference of the German Operations Research Society (GOR). Springer, Berlin, pp 197–202

  • Wong RT (1984) A dual ascent approach for Steiner tree problems on a directed graph. Math Progr 28(3):271–287

    Article  Google Scholar 

Download references

Acknowledgements

We are indebted to the anonymous referees who helped to notably improve this paper through their valuable comments.

Author information

Authors and Affiliations

  1. Department of Industrial and Manufacturing Engineering, California Polytechnic State University, 1 Grand Avenue, San Luis Obispo, CA, 93407, USA

    Alessandro Hill

  2. Department of Electrical, Electronic, and Information Engineering “Guglielmo Marconi”, University of Bologna, Via Venezia 52, 47521, Cesena, Italy

    Roberto Baldacci

  3. Faculty of Computing, University of Mato Grosso do Sul, Campo Grande, MS, Brazil

    Edna Ayako Hoshino

Authors
  1. Alessandro Hill
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Roberto Baldacci
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Edna Ayako Hoshino
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alessandro Hill.

Appendix A

Appendix A

In the following, we present the detailed results obtained by our algorithms for the used set of test instances. Each line of Table 3 is associated with a PC-RAP instance, whose number is given in column P. The numbers of given nodes (\(|V'|\),\(|U_2|\), \(|U_1|\), |W|) and the capacity bounds (q, m) are given in the corresponding columns. The bounds found by the branch-and-cut method within a one hour time limit are given in columns lb and ub. The initial objective value that was found by our heuristic is given in heu. Note that for the maximization problems (PC-RAP, BC-RAP), the start solution, obtained heuristically, gives a lower bound, whereas for TP-RAP, we obtain an upper bound. Column \(\delta \) contains the optimality gap (\(100(ub-lb)/lb\)) and the collected profit is given in a dedicated column. Whenever \(\delta =0\) the corresponding instance has been solved to optimality. The number of branching nodes and run time in seconds is given in column \(\#\) and column t, respectively. Results for BC-RAP and TP-RAP are listed in Table 4. In addition to the information described above, the budget (B), target profit (T), and the sum of the arc costs in the best integer feasible solution (costs) are provided.

Table 3 Instance properties, and results for PC-RAP
Full size table
Table 4 Results for BC-RAP and TP-RAP instances
Full size table

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hill, A., Baldacci, R. & Hoshino, E.A. Capacitated ring arborescence problems with profits. OR Spectrum 41, 357–389 (2019). https://doi.org/10.1007/s00291-018-0539-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00291-018-0539-x

Keywords

Search

Navigation