**5. Conclusions**

Given the fundamental role played by water networks in daily life and business activities, the present research focuses on suitably operating, managing, and maintaining these assets. In this regard, the problem of optimal sensor placement has been faced with the aim of improving the operations of monitoring and the control of the networks.

Most used methods calculate the sensitivity matrix for a specific simulation time, usually the highest consumption time. The fuzzy DEMATEL handles extended period simulations, suitably transformed into fuzzy numbers. The approach enables us to have information for many horizon simulations, making sensor placement more robust. Other possible applications of fuzzy DEMATEL are the simulations of several leakage scenarios with different emitter coefficients. In that case, the sensitivity matrix could be built considering from small to large leaks.

The use of the sensitivity matrix to generate a conditional entropy helps guarantee the spread of sensors in the network. A clear improvement is found, as observed from the comparisons between results from global sensitivity and global entropy of the sensor network, as shown in the case study.

The use of fuzzy DEMATEL for optimal sensor placement helps water companies identify the most suitable monitoring points. Using conditional entropy, the spread of sensors is guaranteed by using the last positions of the DEMATEL ranking. One important positive point of this approach is the absence of optimisation, which usually requires prior knowledge of the number of sensors to be installed. With the presented fuzzy DEMATEL approach, the sensors' network can be implemented in steps, without requiring new simulations.

The fuzzy DEMATEL approach presented in this paper produces similar results to the ones obtained with the optimisation algorithm in [7]. Both methodologies use sensitivity and entropy to place sensors in the network. The main advantage of fuzzy DEMATEL hinges on the final rank obtained: this rank enables placing new sensors without performing new simulations. In optimisation-based approaches, adding new sensors requires new simulations. Incidentally, the approach of WaterIng© finds a solution with better results for both sensitivity and entropy.

Possible future developments of the presented research may regard further investigations about how to choose other quantitative parameters to collect input data. For example, the proposed modified version of the fuzzy DEMATEL may be integrated with other MCDM methodologies to identify a suitable set of parameters, all related to relationships of influence among the considered factors. The selection of the number of sensors should also be further investigated, so as to provide the utilities with a Pareto-like solution enabling them to select the most appropriate number, an aspect not treated in this paper. Such a joint process made of (multi-objective) optimisation plus MCDM methods can be a way of identifying the optimal number of sensors, using the ranking provided for the fuzzy DEMATEL methodology herein developed.

**Author Contributions:** Conceptualisation, S.C. and J.F.-C.; methodology, B.M.B. and S.C.; software, B.M.B. and I.M.; validation, S.C. and J.F.-C.; formal analysis, J.I. and I.M.; writing—original draft preparation, J.F.-C., B.M.B., and S.C.; writing—review and editing, J.I.; supervision, J.I. and I.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research has been partially supported by the CNPq grant with number 156213/2018-4.

**Conflicts of Interest:** The authors declare no conflict of interest.
