Skip to main content
SpringerLink
Log in

Maximizing the net present value of a Steiner tree

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

The theory of Steiner trees has been extensively applied in physical network design problems to locate a Steiner point that minimizes the total length of a tree. However, maximizing the total generated cash flows of a tree has not been investigated. Such a tree has costs associated with its edges and values associated with nodes. In order to reach the nodes in the tree, the edges need to be constructed. The edges are constructed in a particular order and the costs of constructing the edges and the values at the nodes are discounted over time. These discounted costs and values generate cash flows. In this paper, we study the problem of optimally locating a single Steiner point so as to maximize the sum of all the discounted cash flows, known as the net present value (NPV). An application of this problem occurs in underground mining where, we want to optimally locate a junction point in the underground access network to maximize the NPV. We propose an efficient iterative algorithm to optimally locate a single degree-3 Steiner point. We show this algorithm converges quickly and the Steiner point is unique subject to realistic design parameters.

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

Access this article

Log in via an institution

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

Similar content being viewed by others

References

  1. Hwang, F.K., Richards, D.S., Winter, P.: The Steiner Tree Problem. Elsevier Science Publishers, Amsterdam (1992)

    MATH  Google Scholar 

  2. Brazil, M., Rubinstein, J.H., Thomas, D.A., Weng, J.F., Wormald, N.C.: Gradient constrained minimum networks (I). Fundamentals. Journal of Global Optimization 21(2), 139–155 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Brazil, M., Thomas, D.A.: Network optimisation for the design of underground mines. Networks 49(1), 40–50 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Lane, K.F.: The Economic Definition of Ore—Cutoff Grade in Theory and Practice. Mining Journal Books Limited, London (1988)

    Google Scholar 

  5. Nehring, M., Topal, E.: Production Schedule Optimisation in Underground Hard Rock Mining Using Mixed Integer Programming. Project Evaluation Conference, The Australasian Institute of Mining and Metallurgy, vol. 4, pp. 169–175 (2007)

  6. Nehring, M., Topal, E., Little, J.: A new mathematical programming model for production schedule optimisation in underground mining operations. J. S. Afr. Inst. Min. Metall. 110(8), 437–447 (2010)

    Google Scholar 

  7. Newman, A.M., Kuchta, M.: Using aggregation to optimize long-term production planning at an underground mine. Eur. J. Oper. Res. 176(2), 1205–1218 (2007)

    Article  MATH  Google Scholar 

  8. Newman, A.M., Rubio, E., Caro, R., Weintraub, A., Eurek, K.: A review of operations research in mine planning. Interfaces 40(3), 222–245 (2010)

    Article  Google Scholar 

  9. Montalant, F., et al.: Analyse des infiniment petits, pour lintelligence des lignes courbes. chez Francois Montalant (1716)

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, The University of Melbourne, Melbourne, VIC, 3010, Australia

    K. G. Sirinanda, P. A. Grossman & D. A. Thomas

  2. Department of Electrical and Electronic Engineering, The University of Melbourne, Melbourne, VIC, 3010, Australia

    M. Brazil

  3. Department of Mathematics and Statistics, The University of Melbourne, Melbourne, VIC, 3010, Australia

    J. H. Rubinstein

Authors
  1. K. G. Sirinanda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Brazil
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. A. Grossman
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. J. H. Rubinstein
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. D. A. Thomas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. G. Sirinanda.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sirinanda, K.G., Brazil, M., Grossman, P.A. et al. Maximizing the net present value of a Steiner tree. J Glob Optim 62, 391–407 (2015). https://doi.org/10.1007/s10898-014-0246-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-014-0246-3

Keywords

Search

Navigation