## 1 Introduction  
A water distribution network (WDN) is an essential infrastructure supplying water to maintain daily and economic activities of the users connected to it. The WDN itself is composed of many elements that are interconnected and state a certain degree of functionality for the network. A disruption event upon the network will result in a loss of network functionality, which will consequently affect the customers to a certain degree. Among the many elements in a WDN, pipes are the primary components connecting each other. Each of these pipes can be afflicted by failure (i.e., breakage), which renders the pipe unusable. In the event of a pipe failure, it has to be isolated from the network, and the network functionality will decrease depending on the importance of the removed pipe. By calculating the functionality decrement of individual pipes, the criticality of each pipe can be estimated. The criticality of the pipe is valuable information that can be used to determine the proper maintenance priority accordingly [1]. Since a WDN is an infrastructure with hydraulic capability that serves many users, a failure event will impact both the inner workings and the receiving side. [PERSON] et al. [2] introduced the primary concepts of WDN design reliability into mechanical and hydraulic reliability. According to [PERSON] [3], the mechanical reliability focuses on the network topology, evaluating system connectivity given the failure conditions, and the hydraulic reliability refers to the ability of a system to meet the requirements of water flow and pressure. In the context of mechanical reliability, [PERSON] et al. [4] suggested a mechanical reliability indicator using a concept of pipe availability based on pipe failure statistics. Later, [PERSON] and [PERSON] [5] advanced this pipe availability indicator by estimating the failure probability according to a pipe size variation. [PERSON] et al. [6] also advanced the pipe availability by considering the water availability under a simulation of pipe failure and repair events. On the other hand, in the context of hydraulic reliability, [PERSON] and [PERSON] [7] estimated system reliability based on the nodal pressure simulated, and later [PERSON] and [PERSON] [8] suggested a reliability index using actual demand and pressure at demand nodes. The available pressure and supplied flows can be considered simultaneously as an energy term. The resilience index (RI) is a well-known measurement of WDN reliability based on nodal flow and excessive pressure [9]. The RI has been used to evaluate system performance and contributed to the development of various post-indexes. For example, [PERSON] and [PERSON] [10] developed the network resilience index (NRI) by considering network connectivity. The NRI is known as a representative hydraulic-topological combined reliability index. [PERSON] and [PERSON] [11] developed the modified resilience index (MRI), which calculates redundancy of WDN using surplus energy and minimum required energy. Recently, the resilience indexes were revisited by [PERSON] et al. [12] by suggesting a new definition of surplus and the minimum required head at a node considering the flow direction in pipes and the hydraulic gradient of the network. Besides the resilience-type indexes, there are many other indicators developed using the energy-based concepts. [PERSON] et al. [13] proposed five energy-efficiency indicators to evaluate the WDN capacity, and later [PERSON] et al. [14] suggested a new method for assessing the energy states in WDNs. Similarly, [PERSON] and [PERSON] [15] analyzed water distribution sustainability using different energy classifications in water networks, such as supplied, delivered, lost, and dissipated energy. Moreover, [PERSON] et al. [16] proposed five energy metrics based on six energy categories, including supplied, delivered, required, downstream flow, leakage, and friction energy in WDNs. Recently, several water quality indicators were developed. [PERSON] et al.