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Solving tool indexing problem using harmony search algorithm with harmony refinement

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Abstract

The tool indexing problem (TIP) is the problem of allocating cutting tools to different slots in a tool magazine of Computer Numerically Controlled machine to reduce the processing time of jobs on the machine. This is one of the mostly encountered optimization problems in manufacturing systems. In TIP, the number of tools used by the machine is at most the number of slots available in the tool magazine. In this article, a customized harmony search (HS) algorithm, which utilizes a harmony refinement strategy for faster convergence, is presented to solve TIP. The harmony refinement method also helps to avoid getting stuck into local optima. The performance of the proposed method is tested on 27 instances taken from the literature and out of these it is found to improve the best known solutions for 16 instances. For the remaining instances, it gives the same results as found in the literature. Moreover, the performance of the proposed algorithm is tested on newly adapted 41 instances and for some of these instances the results obtained using the proposed algorithm are compared with that obtained using CPLEX.

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Funding

This work has been partially supported by DST-PURSE scheme, Government of India at University of Kalyani.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, University of Kalyani, Kalyani, West Bengal, 741235, India

    Soumen Atta, Priya Ranjan Sinha Mahapatra & Anirban Mukhopadhyay

Authors
  1. Soumen Atta
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  2. Priya Ranjan Sinha Mahapatra
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  3. Anirban Mukhopadhyay
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Correspondence to Anirban Mukhopadhyay.

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Appendices

Appendix A: Illustration of the mathematical formulation

Here, we illustrate the mathematical formulation mentioned in Sect. 2.2 using an example. Let us consider the values of the parameters first. Let \(m = 5\), \(n = 5\) and the frequency matrix as given below:

$$\begin{aligned} F = \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 0 &{} 2 &{} 10 &{} 10 &{} 14 \\ 2 &{} 0 &{} 16 &{} 6 &{} 8 \\ 10 &{} 16 &{} 0 &{} 2 &{} 10 \\ 10 &{} 6 &{} 2 &{} 0 &{} 14 \\ 14 &{} 8 &{} 10 &{} 14 &{} 0\end{array} \right) _{5\times 5} \end{aligned}$$

The optimal assignment matrix Y can be obtained by solving the above problem and the solution to this problem is given below:

$$\begin{aligned} Y = \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0\end{array} \right) _{5\times 5} \end{aligned}$$

The optimal value (cost) corresponding to this solution is 150, computed using (2). This solution Y represents the following assignments of tools to the slots: \(t_1 \rightarrow s_4\), \(t_2 \rightarrow s_1\), \(t_3 \rightarrow s_5\), \(t_4 \rightarrow s_2\) and \(t_5 \rightarrow s_3\), where \(s_i\) represents the ith slot and \(t_j\) denotes the jth tool. Note that the solution Y satisfies the constraints 3, 4 and 5.

Appendix B: CPLEX OPL Model

The CPLEX codes corresponding to the OPL model and data file are mentioned here.

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Atta, S., Sinha Mahapatra, P.R. & Mukhopadhyay, A. Solving tool indexing problem using harmony search algorithm with harmony refinement. Soft Comput 23, 7407–7423 (2019). https://doi.org/10.1007/s00500-018-3385-5

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