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The intelligent water drops algorithm: why it cannot be considered a novel algorithm

A brief discussion on the use of metaphors in optimization

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Abstract

In this article, we rigorously analyze the intelligent water drops (IWD) algorithm, a metaphor-based approach for the approximate solution of discrete optimization problems proposed by Shah-Hosseini (in: Proceedings of the 2007 congress on evolutionary computation (CEC 2007), IEEE Press, Piscataway, NJ, pp 3226–3231, 2007). We demonstrate that all main algorithmic components of IWD are simplifications or special cases of ant colony optimization (ACO), and therefore, IWD is simply a particular instantiation of ACO. We show that the natural metaphor of “water drops flowing in rivers removing the soil from the riverbed”, the source of inspiration of IWD, is unnecessary, misleading and based on unconvincing assumptions of river dynamics and soil erosion that lack a real scientific rationale. We carry out a detailed review of modifications and extensions proposed to IWD since its first publication in 2007. We find that research on IWD is for the most part misguided and that the vast majority of the ideas explored in the literature on IWD have been studied many years before in the context of ACO. Finally, we discuss the use of natural metaphors as a source of inspiration for optimization algorithms, which has become an extremely popular trend in the last 15 years, and propose some criteria to limit their usage to the cases in which the metaphor is indeed useful.

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Notes

  1. As explained in Dorigo et al. (1991a), positive feedback allows ants to generate a process that reinforces itself, that is, the higher the number of ants following a trail, the more attractive that trail becomes for being followed.

  2. The rule for selecting solution components, called transition rule, implemented during the stochastic solution construction varies among ACO variants.

  3. In some ACO implementations, the pheromone update can be interleaved with the solution construction (e.g., see (Gambardella and Dorigo 1995; Dorigo and Gambardella 1997b)), an example being the offline pheromone update implemented in ACS (Dorigo and Gambardella 1997b).

  4. Note that, even though this is highly counter-intuitive, the amount of soil that a water drop collects when adding a new solution component to the partial solution it is building is different from the amount of soil that is removed from the soil variable associated with the added component. For a detailed example, see the online supplementary material.

  5. The author calls heuristic undesirability to the inverse of the heuristic information used in ACO. For example, in the traveling salesman problem, ACO’s heuristic information is commonly defined as \(\eta _{ij} =1/ d_{ij}\), where \(d_{ij}\) indicates the distance between city i and city j. In IWD, the heuristic undesirability is, for the same problem, defined as \(\mathrm{HUD}_{ij}=d_{ij}\).

  6. Finding values for the parameters of stochastic algorithms that guarantee a good algorithm performance is known to be a non-trivial task. See Stützle et al. (2012) for a comprehensive review of how this problem has been studied in the ACO literature.

  7. ACS is one of the oldest and best performing ACO algorithms (Dorigo and Gambardella 1997b); its global update rule is called global pheromone trail updating rule.

  8. There are two versions of this component in IWD. In the first one (Shah-Hosseini 2007), the \(\rho \) parameter was defined in [0, 1], making Eqs. 10 and 9 identical. However, for unknown reasons, in a later publication (Shah-Hosseini 2009) the interval of variability of parameter \(\rho \) was changed to \([-1,0]\), leading to a somewhat different behavior of the global update procedure, as explained here.

  9. The GA’s community realized that proportionate selection is not best suited for optimization because it assumes only positive fitness values and cannot differentiate between small fitness differences. The linear (Baker 1987) and nonlinear (Goldberg and Deb 1991; Michalewicz 1992) ranking selection methods were introduced to alleviate these problems.

  10. For example, consider the metaphoric explanation of gravity by Einstein in terms of space–time grid warping caused by an object’s mass, or selfish genes in genetics to introduce the idea of organism differentiation within same species, or the terminology adopted in graph theory that includes terms such as tree, root, leaf, forest.

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Acknowledgements

Marco Dorigo and Thomas Stützle acknowledge support from the Belgian FRS-FNRS, of which they are Research Directors.

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  1. IRIDIA, Université Libre de Bruxelles, Brussels, Belgium

    Christian Leonardo Camacho-Villalón, Marco Dorigo & Thomas Stützle

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  1. Christian Leonardo Camacho-Villalón
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Camacho-Villalón, C.L., Dorigo, M. & Stützle, T. The intelligent water drops algorithm: why it cannot be considered a novel algorithm. Swarm Intell 13, 173–192 (2019). https://doi.org/10.1007/s11721-019-00165-y

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