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Gradient-constrained discounted Steiner trees II: optimally locating a discounted Steiner point

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Abstract

A gradient-constrained discounted Steiner tree is a network interconnecting given set of nodes in Euclidean space where the gradients of the edges are all no more than an upper bound which defines the maximum gradient. In such a tree, the costs are associated with its edges and values are associated with nodes and are discounted over time. In this paper, we study the problem of optimally locating a single Steiner point in the presence of the gradient constraint in a tree so as to maximize the sum of all the discounted cash flows, known as the net present value (NPV). An edge in the tree is labelled as a b edge, or a m edge, or an f edge if the gradient between its endpoints is greater than, or equal to, or less than the maximum gradient respectively. The set of edge labels at a discounted Steiner point is called its labelling. The optimal location of the discounted Steiner point is obtained for the labellings that can occur in a gradient-constrained discounted Steiner tree. In this paper, we propose the gradient-constrained discounted Steiner point algorithm to optimally locate the discounted Steiner point in the presence of a gradient constraint in a network. This algorithm is applied to a case study. This problem occurs in underground mining, where we focus on the optimization of underground mine access to obtain maximum NPV in the presence of a gradient constraint. The gradient constraint defines the navigability conditions for trucks along the underground tunnels.

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References

  1. Brazil, M., Thomas, D.A., Weng, J.F.: Gradient-constrained minimum networks (II). Labelled or locally minimal Steiner points. J. Glob. Optim. 42(1), 23–37 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Sirinanda, K.G., Brazil, M., Grossman, P.A., Rubinstein, J.H., Thomas, D.A.: Maximizing the net present value of a Steiner tree. J. Glob. Optim. 62(2), 391–407 (2015)

    Article  MathSciNet  Google Scholar 

  3. Rahal, D., Smith, M., Van Hout, G., Von Johannides, A.: The use of mixed integer linear programming for long-term scheduling in block caving mines. In: Proceedings of the 31st International APCOM Symposium, Cape Town, South Africa, pp. 123–131 (2003)

  4. Trout, L.P.: Underground mine production scheduling using mixed integer programming. In: 25th International APCOM Symposium, Melbourne, Australia, pp. 395–400 (1995)

  5. 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 

  6. Smith, M., Sheppard, I., Karunatillake, G.: Using MIP for strategic life-of-mine planning of the lead/zinc stream at Mount Isa Mines. In: Proceedings of the 31st International APCOM Symposium, Cape Town, South Africa, pp. 465–474 (2003)

  7. Nehring, M., Topal, E.: Production schedule optimisation in underground hard rock mining using mixed integer programming. In: Project Evaluation Conference. The AusIMM, vol. 2007, No. 4, pp. 169–175 (2007)

  8. Brazil, M., Rubinstein, J.H., Thomas, D.A., Weng, J.F., Wormald, N.C.: Gradient-constrained minimum networks (I). Fundamentals. J. Glob. Optim. 21(2), 139–155 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Weng, J.F.: An approach to conic sections. Unpublished research note. (1998). http://people.eng.unimelb.edu.au/ksirinanda/

  11. Sirinanda, K.G., Brazil, M., Grossman, P.A., Rubinstein, J.H., Thomas, D.A.: Gradient-constrained discounted Steiner trees I—optimal tree configurations. J. Glob. Optim. (2015). doi:10.1007/s10898-015-0326-z

  12. Sirinanda, K.G., Brazil, M., Grossman, P.A., Rubinstein, J.H., Thomas, D.A.: Optimally locating a junction point for an underground mine to maximise the net present value. ANZIAM J. 55, C315–C328 (2014)

    Google Scholar 

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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
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  2. M. Brazil
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  3. P. A. Grossman
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  4. J. H. Rubinstein
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  5. D. A. Thomas
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Correspondence to K. G. Sirinanda.

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Sirinanda, K.G., Brazil, M., Grossman, P.A. et al. Gradient-constrained discounted Steiner trees II: optimally locating a discounted Steiner point. J Glob Optim 64, 515–532 (2016). https://doi.org/10.1007/s10898-015-0325-0

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  • DOI: https://doi.org/10.1007/s10898-015-0325-0

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