Using Heuristic Techniques to Account for Engineering 3 Aspects in Modularity-Based Water Distribution 4 Network Partitioning Algorithm

: This paper shows how heuristic techniques can be used to account for engineering aspects in the application of a water 7 distribution network (WDN) partitioning algorithm. In fact, being based on graph-theory concepts, most WDN partitioning algorithms 8 fail to consider explicitly such aspects as the number of boundary pipes and the similarity of district metered areas (DMAs) in terms of 9 number of nodes, total demand, and total pipe length, which are often considered by water utility managers to make their decisions. The 10 algorithm considered is the fast-greedy partitioning algorithm (FGPA), based on the original formulation of modularity as an indicator of 11 the strength of WDN partitioning. This algorithm operates by merging the elementary parts of the WDN in sequential steps until the 12 desired number of district metered areas is reached. Two heuristic optimization techniques were combined with FGPA to propose different 13 merging combinations: the former reproduces some specific features of the simulated annealing algorithm while the latter is based on the 14 multiobjective genetic algorithm. Applications were carried out on a real WDN considering the actual system of isolation valves. The 15 partitioning solutions obtained by the traditional FGPA without heuristics and by a literature algorithm based on spectral clustering were 16 taken as benchmark. The results proved that the former heuristic can help in obtaining numerous WDN partitioning solutions with high 17 modularity. The performance of these solutions can be evaluated in terms of practical engineering aspects to help WDN managers make 18 an informed choice about the ultimate solution. If the trade-off between engineering criteria needs to be thoroughly analyzed in the 19 context of WDN partitioning, the latter heuristic, in which FGPA creates DMAs through information encoded in proper weights, 20 can be effectively used. Compared to the benchmark solutions, the FGPA with the latter heuristic can yield solutions with fewer

Modularity was first formulated for unspecific unweighted and weighted networks in the studies of Newman (2004a, b).It is a topological index that describes the possibility of identifying communities in a network.If the focus is just modularity, then the higher the modularity the better the identification of communities.
The original formulation of modularity was used in some studies (e.g., Diao et Clauset et al. (2004).Starting from a configuration in which each node is a DMA of its own, this algorithm operates by aggregating nodes sequentially, while maximizing the increment of modularity at each step, until the target number of DMAs has been reached.However, a limit of this formulation (FGPA applied to the original formulation of modularity) lies in the fact that it does not account directly for engineering aspects related to WDNs, such as the number of boundary pipes and the uniformity of DMAs in terms of demands and ground elevations.Furthermore, it neglects the fact that in WDNs, isolation valves are usually available at pipe ends rather than in the middle of pipes [an assumption considered by Diao et al. (2013) and by Ciaponi et al. (2016)].The presence of these limits undermines the applicability of FGPA to real case studies.In fact, water utility managers always make their WDN partitioning decisions based on such engineering aspects as those mentioned previously.
Bearing these limits in mind, Giustolisi  In the following sections, first the methodology, made up of FGPA and of the two heuristic techniques, is described.The applications to a real WDN follow.Finally, the primary findings of the study are summarized in the conclusions.Eq. ( 5), can be easily identified.Obviously, when a random number sufficiently close to 0 is generated, index ¼ 1 is sampled from FðindexÞ in Eq. ( 5), corresponding to the DMA merging combination with the highest value of ΔM, which is the same merging combination that would be given by the traditional FGPA.As for base in Eq. ( 5), preliminary calculations were done to understand which values to assign to this variable as the steps of the FGPA proceed.Specifically, three options were explored: 1.An even value of base within the range [0, 1]; 2. Growing values of base; and 3. Decreasing values of base.
Finally, Option 2 proved successful and the following expression was adopted, which yields a value equal to 0 at the initial step of FGPA and gradually larger values at the following steps:   4)] are encoded in the genes.To obtain the modulation mentioned previously, coefficient α is added in Eq. ( 4) as multiplying factor of k i k j =2, yielding the following expression: To obtain the variation of the weights of the WDN pipes and the modulation of α, each individual is made up of N disstart þ 1 genes.
The first N disstart genes are multiplicative factors ω s;j of the weights ω of the WDN pipes (initially set at 1=n p ), to be defined within the range ½0; þ∞.If a pipe belongs to the generic jth segment, its weight ω is multiplied by ω s;j .If a pipe is at the boundary between the jth and the kth segment, its weight ω is multiplied by 0.5ðω s;j þ ω s;k Þ.Then, the weights ω of the WDN pipes can be rescaled to reobtain Σω ¼ 1, to be used to assess modularity.The last gene, ranging from 0 to 1, is used for α.         24.This query was generated by an automatic reference checking system.This reference could not be located in the databases used by the system.While the reference may be correct, we ask that you check it so we can provide as many links to the referenced articles as possible.
water distribution network (WDN) into district 27 metered areas (DMAs) has become a very common practice.In 28 fact, it is very beneficial, in that it facilitates demand management, 29 leakage detection, and abatement through service pressure control, 30 model calibration, and so forth (Walski et al. 2003).The separation 31 of each DMA from the rest of the WDN is carried out following the 32 definition of boundaries.At each boundary pipe, the DMA can be 33 physically or virtually separated from the remaining WDN, by clos-34 ing an isolation valve or installing a flow meter, respectively.The final goal of WDN partitioning is the possibility of monitoring and controlling the exchange of flow between WDN DMAs, which is null in the case of physical separation.Examples of WDN partitioning into DMAs are available starting from the early 2000s (Farley 2001; Morrison 2004; Giugni et al. 2008).Numerous algorithms have been proposed for WDN partitioning.Some of the algorithms were developed based on graph and spectral theories (e.g., Deuerlein 2008; Perelman and Ostfeld 2011; Zheng et al. 2013; Candelieri et al. 2014; Di Nardo et al. 2016; Galdiero et al. 2016; Hajebi et al. 2016; Herrera et al. 2016; Di Nardo et al. 2017; Zhang 2017; Liu and Han 2018).Others combine graph theory-based techniques and engineering principles, such as the algorithms proposed by Alvisi and Franchini (2013) and Ferrari et al. (2014).A further group of algorithms uses the concept of modularity (Diao et al. 2013; Giustolisi and Ridolfi 2014a, b; Perelman et al. 2015; Campbell et al. 2016; Ciaponi et al. 2016; Laucelli et al. 2017).
al. 2013; Ciaponi et al. 2016) for WDN partitioning into DMAs.Specifically, Diao et al. (2013) and Ciaponi et al. (2016) made use of the fast-greedy partitioning algorithm (FGPA), which was based on modularity and developed through the graph 1 Associate Professor, Dipartimento di Ingegneria Civile e Architettura, Univ. of Pavia, Via Ferrata 3, Pavia 27100, Italy; Honorary Senior Research Fellow, College of Engineering, Physical and Mathematical Sciences, Univ. of Exeter, Exeter EX4, UK; Adjunct Senior Lecturer in the School of Civil, Environmental and Mining Engineering, Univ. of Adelaide, Adelaide 5005, Australia (corresponding author).ORCID: https://orcid.org/0000-0003-4422-2417.Email: creaco@unipv.it 2 and Ridolfi (2014a, b) modified the original formulation of modularity to obtain a WDNoriented modularity index.A further contribution of Giustolisi and Ridolfi (2014a, b) was to present a modularity-based multiobjective approach for WDN partitioning.The modified modularity index by Giustolisi and Ridolfi (2014a, b) is expressed as the sum of two contributions: the former is a decreasing function of the number of boundary pipes separating DMAs while the latter is a growing function of the similarity of DMAs in terms of a preassigned criterion, such as demand or pipe length distribution across DMAs.As Laucelli et al. (2017) showed, the modified modularity can be inserted in a multiobjective context, where it is maximized while the number of boundary pipes between DMAs is minimized, thus yielding a Pareto front of optimal trade-off solutions among which WDN managers can choose the ultimate partitioning solution.An alternative approach is presented in this paper to account for engineering aspects in the application of WDN partitioning algorithms based on modularity.Unlike the approach proposed by Giustolisi and Ridolfi (2014a, b), based on a modified formulation of modularity, the novel approach presented in this paper is based on the application of heuristic techniques to the FGPA developed by Clauset et al. (2004) starting from the original formulation of modularity (Newman 2004a, b).To this end, two heuristic techniques were used, the former inspired by the simulated annealing optimization and the latter made up of a multiobjective genetic algorithm.By adding some randomness to the DMA merging in FGPA, Heuristic 1 obtains numerous WDN partitioning solutions featuring high modularity values, some of which are even larger than those obtained through the traditional FGPA.Besides proving the suboptimality of the solutions yielded by the traditional FGPA, Heuristic 1 offers the possibility of accounting for engineering aspects in the postprocessing.In fact, the solutions generated by FGPA modified with Heuristic 1 can be evaluated in terms of various engineering aspects (e.g., number of boundary pipes, demand and pipe length uniformity across DMAs, and so forth), thus enabling an informed choice of the ultimate partitioning solution.Subsequently, Heuristic 2 was developed to show that, if DMA merging in FGPA is driven by proper weights encoded in the individual genes of a multiobjective genetic algorithm, the trade-off between various engineering aspects to be simultaneously optimized can be easily considered directly in the optimization phase.
modularity, reference is made hereinafter to a WDN with 129 nn nodes and n p pipes, including fictitious pipes representative 130 of the n valve present isolation valves (e.g., Creaco et al. 2010; 131 Giustolisi and Savic 2010).First the weight ω of the WDN pipes 132 must be set, in such a way as to have Σω ¼ 1.In the case of un-133 weighted network, the weight ω of the generic pipe can be set at 134 1=n p (leading to an identical weight for all pipes).If the pipes 135 are weighted as a function of supplied demands, the weight ω of 136 the generic pipe can be set at Dem/Demt, where Dem and Demt 137 are the demand supplied along the pipe and the overall demand 138 of the WDN, respectively.Otherwise, if the pipes are weighted 139 as a function of pipe lengths, ω can be set at Lp=Lt, where Lp 140 and Lt are the pipe length and the overall length of the WDN, 141 respectively.This can be extended to whatever kind of weight.Then, 142 the incidence topological matrix A, with size n p xnn, can be con-143 structed.In the generic row of A, associated with the generic net-144 work pipe, the generic element can take on the values 0, − ffiffiffi ω p or 145 ffiffiffi ω p , whether the node corresponding to the matrix element is not at 146 the ends 6 of the pipe, it is the initial node of the pipe, or the final 147 node of the pipe, respectively.Starting from A, the vector K 148 (nnx1) and matrix D (nnxnn) can be calculated through the follow-149 ing expressions: 150 where diagonal( ), diag( ), and | | indicate the vector extracted from 151 the diagonal of a square matrix, the diagonal square matrix con-152 structed starting from a vector, and the absolute value, respectively.153 The K and D have an important topological meaning.In fact, the 154 element k i of vector K represents the total weight associated with 155 the pipes connected to the ith node.The generic element D ij of D 156 represents the pipe weight connecting the ith and jth node.Follow-157 ing the definition of K and D, the WDN modularity M can be for-158 mulated as (Newman 2004a, b)

1
172 pipes.The subtraction of term k i k j =2 contributes to the uniform 173 distribution of Σω over the DMAs.While the original formulation 174 of Newman (2004a, b) gives the same relevance to the two terms 175 inside the bracket, Giustolisi and Ridolfi (2014b) argued that the two terms can be modulated by introducing a multiplying factor in the second term.The objective of the FGPA lies in obtaining a WDN partitioning featuring a high value of M. After the target number N dis of DMAs has been set, the algorithm considers a starting partitioning of the WDN into N disstart DMAs.A suitable starting partitioning is made up of the segments, i.e., the smallest WDN pieces that can be disconnected, through closure of present isolation valves, while avoiding service disruptions throughout the whole network or in large portions (Creaco et al. 2010).For segment identification, suitable algorithms can be used, such as those proposed by Jun and Loganathan (2007), Giustolisi and Savic (2010), and Creaco et al. (2010), based on the real positions of the isolation valves in the WDN.Therefore, at the initial step, the number of DMAs in the WDN is equal to N disstart .At the second step, two DMAs are merged, and the number of DMAs becomes N disstart − 1.The aggregation process is repeated in the following steps until the network merges to N dis DMAs.At the generic step, the choice of the two DMAs to merge is made to obtain the highest ΔM, where ΔM is a variation in M.An explicative example of the FGPA is shown in Fig. 1 for a simple WDN, in which n valve ¼ 6 isolation valves are present [Fig.1(a)].After replacing the valves with fictitious pipes, the 7 layout in Fig. 1(b) is obtained, made up of nn ¼ 13 nodes and n p ¼ 14 pipes.The application of the algorithm of Creaco et al. (2010) for segment identification detects four segments in the WDN.The application of FGPA starting from N disstart ¼ 4 to obtain N dis ¼ 2 DMAs produces the merging of Segment 1 with Segment 3 and of Segment 2 with Segment 4, in two sequential steps [Fig.1(c)].As an explicative example, Fig. 1 shows that the merging of the DMAs takes place while keeping some fictitious pipes representative of isolation valves at DMA boundaries.Others, instead, are incorporated into DMAs.However, as Fig. 1 shows, note that the use of the configurations of N disstart segments as the starting condition for the propagation of FGPA guarantees that, in the aggregation of DMAs, there are always valve-fitted pipes at the boundaries, without any artificial tuning of pipe weights in the modularity function.Because FGPA is modularity M driven, a remark must be made about how the presence of the fictitious pipes representative of isolation valves impacts on M. In the case of the unweighted graph, ω ¼ 1=n p , for both the fictitious pipes and the other pipes of the WDN.Therefore, the value of M is influenced both by the uniform distribution of the total number of pipes over the DMAs and by the number of boundary pipes that are left outside DMAs.In the case of weighted graph (e.g., ω ¼ Dem=Demt or ω ¼ Lp=Lt), instead, the fictitious pipes representative of the isolation valves have weight ω ≈ 0. In fact, Dem and Lp are close to 0 for these pipes.Therefore, the number of fictitious pipes at the boundaries has reduced impact on M, which is then affected only by the uniform distribution of the sum Σω over the DMAs.In this implementation where FGPA starts propagating from the N disstart segments present in the WDN, the computational complexity of the algorithm (number of logical operations) is O½n valve • d • logðN disstart Þ, where d is the depth of the dendrogram describing the community structure of the WDN.This means that the running time grows linearly with n valve , d, and logðN disstart Þ.The 8 logic structure of FGPA can be summarized in the pseudocode in Fig. 2(a).Heuristic Unlike the original FGPA, the possibility of merging two DMAs with a lower ΔM than the highest value mentioned previously 238 is considered in Heuristic 1 [pseudocode in Fig. 2(b)].This was 239 done to insert some randomness in DMA merging, which obtains 240 various WDN partitioning solutions rather than the single determin-241 istic solution of the traditional FGPA.Furthermore, it is not 242 guaranteed that the DMA merging that produces the highest pos-243 itive ΔM at the generic step is the most effective choice to obtain 244 the best WDN partitioning into N dis DMAs, i.e., the solution with 245 the highest value of M. At the generic step of the partitioning algo-246 rithm, let us assume N comb possible combinations of DMAs for the 247 merging, each of which features its value of ΔM.These values can 248 then be sorted in descending order and then associated with an 249 index.A probability function F can then be calculated as F ¼ base þ ð1 − baseÞ index N comb expo ð5Þ 250 where base and expo are two parameters, to be set within the range 251 [0, 1] and [0, þ∞].Basically, F is a monotonic growing function of 252 index, ranging from 0 to 1 and yielding the probability of nonex-253 ceedance of the generic value of ΔM.The generic combination of 254 merging can be easily sampled from F. In fact, if a random number 255 is generated between 0 and 1, the closest among the values of F 256 larger than the random number, and its associated index through 257

F1: 1 Fig. 1 .
(a) Network with isolation valves installed; (b) network with F1:2 fictitious pipes installed instead of isolation valves to enable segment F1:3 identification; and (c) joining of segments for the construction of two F1is provided hereinafter to clarify this concept, con-259 sidering N comb ¼ 20 possible combination of DMAs for the merg-260 ing of N, producing ΔM ranging from 0.0001 to 0.01.These values 261 are sorted in descending order and associated with index [Fig.3(a)].262 Then, function F is calculated as a function of index for three pairs 263 of values of base and expo.Fig. 3(b) shows FðindexÞ, from which 264 the sampling of the merging combination is carried out.Fig. 3(b) 265 shows that the pair base ¼ 0 − expo ¼ 1 gives an even probability to all the indexes, and therefore to all the values of ΔM.The growth of base and the drop of expo increase the probability of selection for the lower indexes, and therefore for the higher values of ΔM.

Heuristic 1
is embedded in the traditional sequence of FGPA steps from N disstart to N dis .At the generic step, a random number is generated from 0 to 1 to sample the merging combination of DMAs available at that step from FðindexÞ in Eq. (5).Heuristic 1 can be repeated using different sequences of random numbers, producing different values of M for a number of DMAs ranging from N disstart to N dis .Some of these values may result larger than those produced by the traditional FGPA with no heuristic.
This enables DMA merging combinations with lower values of ΔM than the maximum possible value to be selected especially at the initial steps.When N is far from N disstart , that is, at the final steps of FGPA, the merging combinations associated with very high M increment are privileged instead.This brings Heuristic 1 close to the simulated annealing technique(Kirkpatrick and Gelatt 1983) where directions different from that where the objective function experiences the steepest ascent are facilitated at the initial steps, in an attempt to find a global optimum.

F2: 1 Fig. 2 . 1 Fig. 3 .Y 2
Pseudocodes of (a) FGPA; (b) FGPA with Heuristic 1; and (c) FGPA with Heuristic 2. F3:Preparatory steps for the heuristic merging of DMAs: (a) asso-F3:2 ciation of each value of ΔM with an index; and (b) association of index F3:3 with the probability of nonexceedance of ΔM.The application of Heuristic 1 involves running FGPA for a certain number of times (N times ).The computational complexity of FGPA with Heuristic 1 is then N times larger than that of the traditional FGPA.Heuristic In the framework of WDN partitioning, different objectives from the maximization of M are usually pursued, which include maximization of the uniformity of supplied demands over DMAs, service pressure inside DMAs or of other variables.A further practical objective is the minimization of the number of inter-DMA boundary pipes, at each of which either an isolation valve will be closed or a flow-meter will be installed, thus causing undesirably the loss of reliability or the disbursement of funds, respectively.In Heuristic 2, the possibility of considering some of the engineering aspects mentioned previously is accounted for by means of the multiobjective genetic algorithm NSGAII (Deb et al. 2002).Specifically, the objective functions considered include the coefficient of variation (ratio of the standard deviation to the mean value) of the total demands delivered to the DMAs, which is an inverse function of the uniformity of supplied demands, and the number of boundary pipes.Both objective functions are simultaneously minimized.These objective functions are in line with those considered by other authors in the scientific literature (e.g., Giustolisi and Ridolfi 2014a, b; Di Nardo et al. 2016; Liu and Han 2018).In fact, the minimization of the number of boundary pipes is considered in almost all the WDN partitioning algorithms, including those based on spectral clustering (Di Nardo et al. 2016; Liu and Han 2018), which aim to solve a relaxed version of the minimum cut problem for the graph.The issue of DMA uniformity, expressed in different forms including demand distribution, was also considered as design criterion by various authors (Giustolisi and Ridolfi 2014a, b; Di Nardo et al. 2016; Liu and Han 2018).The coefficient of variation of demands across DMAs can be related to the second term of the modified index of modularity of Giustolisi and Ridolfi (2014a, b), when pipe weights in the index are expressed as a function of allocated user demands.Furthermore, Liu and Han (2018) presented a design criterion based on a similar formulation to that used in this paper as the second objective function.The genes of each individual in Heuristic 2 are used to drive the aggregation of DMAs in the traditional FGPA, applied to the initially unweighted graph, to obtain optimal solutions in the expected trade-off.To influence the sequential aggregation of DMAs with the aim to pursue this trade-off, the variation of the weights of the WDN pipes, initially all set at 1, and the modulation of the effects of the two terms present in the original formulation of modularity [Eq.(

F4: 1 Fig. 4 .
Figs. 5(a and b) in terms of MðN dis Þ.The graph in Fig. 5(a) reports a value of M equal to about 0.46 in correspondence to N distart .When N dis decreases due to the merging of DMAs, M grows up to a maximum value of about 0.89 in correspondence to N dis ¼ 23.This means that at N dis ¼ 23 it is possible to obtain the most modular WDN partitioning into DMAs, with a uniform distribution of pipes over the DMAs and with a low number of boundary pipes left out of the DMAs.To the left of this value, M falls to 0 for N dis ¼ 1.In fact, at N dis ¼ 1 all the pipes belong to a single DMA and there is no WDN partitioning.The graph in Fig. 5(b) shows a different pattern MðN dis Þ.In fact, the maximum of M lies in correspondence to the highest value of N dis ¼ N distart ¼ 682.To the left of this value, M decreases toward 0 at N dis ¼ 1.The different behavior in the two graphs in Fig. 5 is because, as mentioned previously, in the case of weighted graph, M is affected only by the distribution of Demt over the DMAs while the number of boundary pipes has no impact on M. As an example, Fig. 6(a) reports, for N dis ¼ 5, the WDN partitioning results of FGPA-unweighted graph.Though being slightly small in light of the WDN total size and demand, a total number N dis ¼ 5 of DMAs was chosen in this context because it enables easy visualization of the results of WDN partitioning.In the graph, to make distinction between the DMAs, a different color is used to characterize the pipes of each DMA.Boundary pipes, the end nodes of which belong to two different D 10

423
As a benchmark, Fig. 6(b) reports the WDN partitioning solu-424 tion obtained with the spectral clustering algorithm of Di Nardo 425 et al. (2016), applied with the constraint of having boundary pipes 426 at valve-fitted pipes.Though being obtained without considering 427 modularity explicitly, this solution features a quite high value of 428 M, equal to 0.749 evaluated in the case of unweighted graph.In 429 fact, the key ingredients of modularity, namely the balancing be-430 tween DMAs and the low number of boundary pipes, are also the 431 objectives of spectral clustering partitioning methods.However, the 432 fact that the M value for Fig. 6(b) is smaller than that for Fig. 6(a) is 433 due to the larger number of boundary pipes (57) provided by the 434 algorithm of Di Nardo et al. (2016).

435FGPA with Heuristic 1 436 1 Fig. 5 .F6: 1 Fig. 6 .
This subsection aims to prove how the results of FGPA with 437 Heursitic 1 can be evaluated in terms of engineering aspects.The 438 FGPA with Heuristic 1 was run N times ¼ 100 in the unweighted 439 graph.Each time, a pattern MðN dis Þ similar to Fig. 5(a) was 440 obtained.Due to the stochastic nature of this algorithm, the results 441 were different from one run to the other.Then, the pattern r M ðN dis Þ 442 was calculated in each run, where r M is the ratio of the M value 443 obtained in the run for the generic value of N dis to the correspond-444 ing M obtained in the traditional FGPA.The graph in Fig. 7(a) 445 shows that r M ðN dis Þ is always around 1, highlighting that the FGPA 446 with Heuristic 1 is always able to yield WDN partitioning solutions 447 with high modularity.Furthermore, it must be remarked that the F5:Application of FGPA with modularity M expressed for (a) un-F5:2 weighted graph; and (b) graph weighted based on pipe demands.(a) Application of FGPA with modularity M expressed for un-F6:2 weighted graph; and (b) benchmark solution obtained with the spectral F6:3 clustering algorithm of Di Nardo et al. (2016).r M ðN dis Þ is always larger than 1.This proves both the suboptimality of the traditional FGPA and the possibility to explore solutions with higher modularity thanks to adoption of Heuristic 1.The best gain obtainable with the FGPA with Heuristic 1 is for N dis < 10, with a maximum value of r M close to 1.05.The FGPA with Heuristic 1 was run 100 times also in the weighted graph (ω ¼ Dem=Demt).Similar calculations to those of the weighted graph led to the graph reporting r M ðN dis Þ in Fig. 7(b).This graph only reports the values of r M for N dis < 200 because the others are almost coincident with 1. Similar remarks to the application with the unweighted graph can be made also in this case.The subsequent results in this subsection are shown for N dis ¼ 5, that is considering WDN partitioning into five DMAs.

F7: 1 Fig. 7 . 1 Fig. 8 . 14 .
Application of FGPA with Heuristic 1. Ratio r M as a function of F7:2 N dis for (a) unweighted graph; and (b) weighted graph based on pipe F7:3 demands.Each color indicates a different run.F8:Weibul fre 13 quency F of the modularity function M obtained F8:2 through FGPA with Heuristic 1 for N dis ¼ 5 in the (a) unweighted F8:3 graph; and (b) weighted graph based on pipe demands.NSGAII was run with a population pop ¼ 508 100 individuals and for n gen ¼ 100 generations to search for 509 optimal WDN partitioning into five DMAs, in the trade-off be-510 tween N bp and C v;Dd , to be simultaneously minimized.Coefficient C v;Dd , was calculated starting from peak hour demands distributedalong pipes.The gene ω s;1 was set to a fixed value (i.e., 1) whereas ω s;j , with j ¼ 2; : : : ; N disstart , were allowed to range.This was done to prevent the gene rescaling (performed to guarantee

F9: 1 Fig. 9 .
Relationship between M and (a) N bp ; (b) C v;Nnd ; (c) C v;L ; and (d) C v;Dd for the solutions obtained through FGPA with Heuristic 1 in the F9:2 unweighted graph.Comparison shown with the values of the traditional FGPA.

F13: 1 Fig. 13 .
Pareto front of optimal trade-off solution between number F13:2N civ of closed isolation valves and generalized resilience/failure F13:3 GRF index under peak demand conditions for the configuration of five F13:4 DMAs shown in Fig.12.Postprocessing of the solutions under average F13