O.R. Applications
Tree knapsack approaches for local access network design

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Abstract

In the process of solving many forms of the local access network design problem, the basic model of the tree knapsack problem (TKP) is used as a building block for the search engine of the solution strategy. Various solution strategies can be used to solve this problem. An approach that use standard software coupled with enhanced modelling is presented for the TKP. Enhanced modelling is used to partition the TKP into sub-problems that is easier to solve using standard off the shelve software. The basic approach is described and empirical work is presented. Empirical comparisons are also given relating this approach with some algorithms suggested by other authors.

Section snippets

Local access telecommunication networks

As an introduction to this class of problems we quote Balakrishnan et al. (1991, p. 240): “Because modernizing and expanding switching and transmission facilities requires enormous investments, telephone companies emphasize cost effectiveness in implementing selected expansion problems with high demand growth potential. For each project, network planners face complex choices concerning where and when to expand capacity or replace current technology in order to meet the increasing demand for

Solution methods

This section will deal with specific solution methods for the TKP problem. The TKP problem has the zero–one knapsack problem (KP) as special case. Martello and Toth (1987, p. 214) state that a lot of work has been done on the KP as they arise as sub-problems in many integer programming applications and may represent various practical situations. Martello et al. presented a paper on the trends in exact algorithms for the 0–1 KP in Martello et al. (2000). For the 0–1 KP branch and bound

Proposed solution method

The solution method proposed in this paper is one that uses standard software and aims to use standard, off the shelf software more efficiently. The main idea is to use enhanced modelling in order to obtain exact solutions efficiently for large instances of TKP problems. The enhanced modelling will be discussed in detail in subsequent sections. First it is necessary to introduce certain concepts used in the algorithm.

The partitioning TKP algorithm

Introduce a variable CLB to store the current best solution obtained. The current lower bound is continuously updated and gives the optimal solution value when the algorithm terminates.

  • procedure PART_TKP

  • begin

  • CLB = −∞

  • Set ZTKPR = solution to ILPR(TKP) and identify the solution values xj for j = 0, 1, 2, …, n

  • Define P = {1, 2, 3, …, n}

  • Solve parametrically ILPR(TKP, p) for p  P to obtain associated objective function values of ILPR(TKP, p) denoted by ZTKPR(p). If ILPR(TKP, p) is infeasible, set ZTKPR(p) = −∞.

  • while P  do

Empirical work

Empirical work was undertaken to test the basic performance of the algorithm presented. In order to test the algorithm data had to be generated. This was due to the fact that no real world data could be obtained in order to test the partitioning TKP algorithm.

For the purpose of the empirical work presented, it might be helpful to point out that the partitioning TKP algorithm tries to use enhanced modelling to solve the problem instances more efficiently. Hence the use of the terms enhanced

Conclusion

By using standard software coupled with enhanced modelling a considerable speedup was obtained when solving TKP instances. This was particularly true for larger problem instances. For small instances the overhead generated by the partitioning scheme caused it to solve instances slower than other approaches. For larger instances the partitioning algorithm solved the problems very efficiently and consistently. At a first glance it may seem a rather surprising result that the relatively simple

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This work forms part of the research done at North-West University within the TELKOM CoE research programme, funded by TELKOM, GRINTEK TELECOM and THRIP.

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