**1. Introduction and Literature Review**

Optimal sensor placement (OSP) for leakage detection in water distribution networks (WDNs) currently represents an exciting and lively field of research, aimed at optimising processes of network control, management, and maintenance [1].

With the explosion of the sensors market, and the consequent access to pressure sensors, various technical questions appear for water utilities. The most important are how many sensors to install, and, given a number of sensors, where to install them. There is a clear trade-off analysis to be performed that aims to answer the first question. Having more sensors in the network means more data that can be used to get more complete knowledge about the system. However, having more sensors also means more money spent. As a result, economical reasons make the choice of a monitoring strategy a crucial decision. The selection of good monitoring points may bring more and better information about the system, for less money.

Considering the complex task of defining the position and number of sensors in a water network, it is reasonable that there are many works in the literature presenting different approaches for the OSP

problem. In this introduction we just focus on some of those most directly related to the approach we present in this paper.

With the aim of finding leaks in water systems, a methodology for pressure sensor placement based on sensitivity analysis is presented in [2]. The authors generate a sensitivity matrix based on simulations of leaks, from which a signature matrix may be extracted. Genetic algorithms are then used to maximise the isolation of leaks. In [3], a multi objective technique to find the optimal monitoring points for leak detection, given a number of sensors, is applied. The sensitivity matrix is calculated based on the percentage variation of pressure from the normal scenario to an abnormal one.

Hydraulic simulations are performed based on a body of input information, including pipe roughness, nodal water demand, etc. The accuracy of the hydraulic state derived depends on the quality of the input information. Since many inputs are not directly measured, uncertainty analysis is often needed. For example, the authors of [4] modify the work of [2], by adding nodal demand uncertainty analysis to build the sensitivity matrix.

Additionally, in [5], uncertainties coming from water demand in the network are included. The authors use genetic algorithms to investigate optimal sensor placement based on the sensitivity matrix and residual vectors.

To simulate small leaks in water distribution systems, a demand driven approach (DDA) can be used, modelling the leak as a function of only the pressure. Of course, for large leaks and pipe bursts, a DDA has a set of limitations. If the anomaly leads the system's pressure under the minimal operational pressure, the total demand cannot be supplied. In [6], an optimal pressure sensor placement methodology based on nodal entropy is presented. The authors simulate anomalies using a pressure driven approach (PDA) and compare the results with DDA simulations. Using the entropy method, the authors rank the nodes with high entropy as the best monitoring points.

The sensitivity matrix is widely used in works of OSP. However, the use of that matrix without considering other parameters can lead to the concentration of sensors in a reduced region of the network, which leads to significant coverage reduction. To cope with it, in [7], it is combined the sensitivity matrix with the maximisation of the entropy related to the sensor network. The entropy's maximisation guarantees better spread of the sensors, according to the authors. Optimisation approaches are also widely applied to locate optimal monitoring points. In [8], an approach to minimise the distance of localised leaks based on sensors' data is developed. A pre-defined number of sensors is used as a constraint for the optimisation problem. A genetic algorithm is used to find the optimal number of sensors and their strategic monitoring position. Aiming to maximise the number of failures detected, the authors in [9] present an optimal sensor placement using a minimum test cover (MTC) with approximated solutions. The authors develop a new augmented greedy algorithm for solving the MTC problem.

The main purpose of this paper is to evaluate relationships among pressure sensors in a network to get as complete as possible an understanding of their mutual influence, and thus identify those candidate nodes to host sensors that may have bigger impact on the network information. The ultimate aim is to guarantee better network control by optimising the number of sensors and their location in the network. Identifying those network nodes that capture bigger influence can be strategic, since variations on those nodes may directly reflect variations on other nodes in the system, and this will eventually reduce the investment in sensors.

We claim that a multi-criteria decision-making (MCDM) approach may effectively support the problem being faced. A methodology that appears to be best suited to such an aim is the decision making trial and evaluation laboratory (DEMATEL), first implemented by Fontela and Gabus [10,11].

DEMATEL is helpful when dealing with complex systems, such as water networks, since they are characterised by many aspects/elements directly or indirectly interdependent with each other, and this condition makes hard many decision-making tasks. As asserted, for example, in [12], the use of DEMATEL supports the visualisation of interferences existing among the relevant aspects of a given problem, thereby helping comprehensive understanding of the intensity and direction of direct

and indirect relations for each pair of factors under study. This technique deals with interactions through a step-by-step approach [13]; it has been widely applied in the literature for management problems characterised by the presence of heavy interdependence among elements [14–17]; and many developments of its application have been proposed in a wide number of fields (see [18–20], among others).

To address the stated problem, we herein propose a new approach within the framework of the fuzzy DEMATEL method. The fuzzy DEMATEL represents a development of the traditional crisp DEMATEL, extended by Wu and Lee [21], and makes use of elements of the fuzzy set theory [22] for better managing uncertainty affecting input evaluations.

As stated in [23], criteria should be analysed under uncertain conditions when working in vague contexts. Additionally, after stating that decision-making processes are human activities mainly accomplished in uncertain environments, in [24] it is emphasised as the crisp DEMATEL can reflect information only in a partial way. The authors consider the usefulness of applying fuzzy theory to extend the traditional method, so that judgements of preference can be translated into fuzzy numbers, after having been expressed by decision-makers through the adoption of a specific fuzzy linguistic scale.

From that angle, the author of [25] agrees with the fact that making use of fuzzy numbers minimises subjective bias, and for this reason, the fuzzy DEMATEL has to be preferred to the traditional crisp version when it comes to real-world applications. After presenting a literature review related to the various fields of fuzzy DEMATEL application, the author applies this methodology to determine those critical aspects having a major impact on local sustainable development through adaptive reuse projects. The authors in [26] also highlight difficulties in making decisions in a fuzzy environment, especially when complex selection criteria are involved. The authors propose fuzzy DEMATEL to determine the most influential factors when evaluating/selecting suppliers, finding that the aspect of financial stability has the highest impact on project implementation. Additionally, in [27], use is made of fuzzy DEMATEL to design a formal framework to use as a driver during the process of business strategy formulation. The authors also stress that the integration with other methodologies is useful to overcome subjectivity of evaluations, and to generally optimise final results of analyses. With regard to the field of critical infrastructures, in [28], it is proposed a hybrid MCDM approach based on fuzzy DEMATEL for failure risk assessment to capture the dynamic nature of opinions provided by a team of experts.

To the best of the authors' knowledge, fuzzy DEMATEL has been scarcely applied in the sector of water network management so far. Water networks are really complex systems made of many interconnected elements, such as tanks, pumps, valves, treatment facilities, and hundreds or even thousands of kilometres of underground pipes [29]. Critical components of networks are characterised by the presence of strong degrees of interdependence, which have a huge impact on the quality of the final service, especially when it comes to minimising operation failures. For this main reason, not only does a fuzzy DEMATEL-based application appear suitable to dealing with the type of stated problem, but it also may represent a powerful approach to fuel the process of OSP.

This paper suggests a novel way to face the OSP problem, based on a new modified version of the traditional fuzzy DEMATEL. Our proposal addresses two main issues: (1) Reducing the huge amount of time often spent during the stage of collection of expert evaluations; and (2) making evaluations as objective as possible, despite that they are represented by fuzzy numbers. To pursue this twofold objective, we herein propose to replace the input matrices of expert linguistic assessments with a single input matrix of linguistic assessments related to a suitable quantitative parameter that expresses the degree of influence between pairs of elements. Even though such a new development is herein applied to the OSP process in a WDN, we claim that it can be extended to other kinds of complex problems.

The paper is structured as follows. After this introduction, including some literature reviewing and stating of the significance of the problem for the water supply field, Section 2, devoted to materials and methods, provides a concise description of the elements involved in the problem under analysis and presents the novel approach we propose, aimed at getting the final ranking of nodes showing those most convenient for hosting sensors. Section 3 provides a numerical example in which the proposed approach is applied, for exemplification purposes, to a very small WDN of the benchmark literature, whereas Section 4 shows the results for a larger network, including comparisons with two optimisation-based OSP methods. Lastly, Section 5 gives the conclusions and raises likely future developments of research.
