# Resilience-based optimal design of water distribution network

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## Abstract

Optimal design of water distribution network is generally aimed to minimize the capital cost of the investments on tanks, pipes, pumps, and other appurtenances. Minimizing the cost of pipes is usually considered as a prime objective as its proportion in capital cost of the water distribution system project is very high. However, minimizing the capital cost of the pipeline alone may result in economical network configuration, but it may not be a promising solution in terms of resilience point of view. Resilience of the water distribution network has been considered as one of the popular surrogate measures to address ability of network to withstand failure scenarios. To improve the resiliency of the network, the pipe network optimization can be performed with two objectives, namely minimizing the capital cost as first objective and maximizing resilience measure of the configuration as secondary objective. In the present work, these two objectives are combined as single objective and optimization problem is solved by differential evolution technique. The paper illustrates the procedure for normalizing the objective functions having distinct metrics. Two of the existing resilience indices and power efficiency are considered for optimal design of water distribution network. The proposed normalized objective function is found to be efficient under weighted method of handling multi-objective water distribution design problem. The numerical results of the design indicate the importance of sizing pipe telescopically along shortest path of flow to have enhanced resiliency indices.

## Keywords

Capital cost Water distribution network Resilience Differential evolution## Introduction

An abundance of optimization techniques have been used for optimal design of water distribution with a main focus on cost minimization. Optimization of water distribution system either with cost or reliability or resilience as an objective requires a lot of computational effort and time and seldom converges directly to global optimal solution. As far as computational point of view, water distribution system problems are classified as highly complex and it is very unlikely that simple and efficient algorithms can be developed to obtain global optimal solution when design is subjected to numerous constraints and multi-objectiveness. Therefore, it is unfair to depend on a heuristic method which gives a reasonably good solution, if not the global one, with less computational effort. In many real-life problems, it is not necessary to obtain a global optimal solution. Any solution that satisfies constraints imposed to the problem and within a planned budget and close to optimal value can be treated as a satisfactory solution. Global minimum cost solution may not be right choice when the estimation of reliability of component or resilience of the system and uncertainty in the demand include errors and approximations (Loucks and Beek 2005). In general, resilience is defined as the ability of system to recover from its failure, setbacks, and to adapt well to change (e.g., climate change). Resilience of water distribution system strongly relates to the intrinsic capability of system to overcome failures. The water distribution network designed based on resilient point of view should have reduced failure probabilities and minimum or reduced failure consequences and able to recover quickly from failures. Recently, Cimellaro et al. (2015) defined resilience related to infrastructure facilities as the ability of the system to withstand, adapt to and rapidly recover from the effects of a disruptive event. The resilience of looped water distribution system can be enhanced by either providing redundancy at pipe capacity level or increasing component reliabilities or both. But such enhancement leads to increase in system cost. Thus, a trade-off between these two options is indispensable for budget-constrained design optimization. Generally, resilience measure depicts the additional capacity in the network and by increasing pipe sizes, resilience can be increased. However, it provides no quantification of performance under component failure condition as does the reliability measures. Least capital cost design for water distribution network using optimization technique usually results with a network configuration that can just satisfy the nodal demand under normal operating conditions with expected outlet pressure. The optimization model inherently selects the shortest possible path to the demand node and assigns a least dimensional pipe size to all en-route pipes to that node and other demand nodes too. There is lot of chances that the optimization model could have assigned smaller pipe sizes to almost all the redundant pipes in the looped network in view of cost minimization as a prime objective. Although loops ensure better connectivity to the nodes to the source, the undersized redundant pipes cannot satisfy the consumer’s demand those who depend on these pipes directly and also during abnormal demand scenarios. Todini (2000) presented the concept of “resilience index” as an indicator of the functioning of a network under failure circumstances. Prasad and Park (2004) arrived at a quantity termed “network resilience” that describes redundancy of links in the network, thereby broadening resilience index. Fu et al. (2012) pointed out that there is no universal hydraulic performance measure available for design of water distribution network.

There are few attempts in which cost and resilience considered as twin objectives to optimize network with the aim of arriving least cost network with best possible performance. Although the formulation of multi-objective problem for water distribution network is simple, solving and arriving Pareto front is found to be challenging. Bolognesi et al. (2014) developed pseudo-front in the hydraulically feasible region in the multi-objective design of water distribution network by combining any two objectives (i.e., cost and performance measures such as deviation from minimum of pressure deficit, deviation of velocity and resilience measure). The formulated problem in three different objectives was solved by genetic heritage and stochastic evolution transmission (GHEST) algorithm. Creaco et al. (2014) formulated a multi-objective optimization problem for phasing design of water distribution system, which has two objectives, one as minimizing the costs of the upgrades converted into value at the initial time and other as maximizing the minimum pressure surplus over time. Ostfeld et al. (2014) used genetic algorithm for the multi-objective optimization of water distribution network considering cost and resilience index as twin objectives of the design and utilized split pipe size method, which assign more than one diameter for each pipe in the network. Piratla (2015) presented a three-objective optimization model to obtain various sustainable and resilient design alternatives. The trade-off among three objectives can help to identify most beneficial solution from Pareto optimal set of solutions. Recently, Wang et al. (2014) applied various multi-objective evolutionary algorithms to generate true Pareto front for two-objective model that belongs to optimal design of water distribution network. The various algorithms were tested using 12 benchmark networks available in the literature. The results of study indicate that non-dominated sorting genetic algorithm-II (NSGA-II) is found to be most promising approach for solving two-objective water distribution network problems. Creaco et al. (2016) proposed three-objective optimization model to minimize the costs and maximize both the resilience and the loop-based diameter uniformity indexes. The study provides an improvement in the NSGA-II algorithm to overcome the difficulties that arises with more than two objectives. Most of the previous studies focused more on generating true Pareto front to the problem in an efficient manner. The aim of the present work is to provide a simple way to decision makers for obtaining best solution while considering both objectives simultaneously with or without due weightage to particular objective function. Further, it brings out the way in which the resiliency of network gets increased at the expense of cost in optimal manner. Hence, minimizing the cost and maximizing the resilience index are dealt in two different ways as a single objective optimization problem and illustrated with widely used Hanoi water distribution networks.

## Resilience indices

Water distribution network is a prime hydraulic infrastructure to water supply system. Design of such system economically and efficiently is found to be one of the important research topics among researcher even in the present day. Todini (2000) perhaps is the first researcher who proposed a resilience index for water distribution network by relating nodal pressure and demand to address the intrinsic capability of system to overcome failures. Further, Todini (2000) pointed out that proposed resilience measure is a surrogate to the reliability of the system. The calculation of reliability of the system requires pipe failure data and its consequence in the supplying water to the consumers. Although the resilient measure does not take into account of the failure data, network designed based on resilient measure can sustain at failure of its components. It is clear that resilient design of water distribution network is strongly related to the intrinsic capability of system that will have reduced failure probabilities and minimum or reduced failure consequences and able to recover quickly from failure. Resilience of the system is also viewed as a measure of capability of the system to absorb the shocks or to perform under perturbation. Howard and Bartram (2010) defined the resilience of a piped water supply as a function of the resilience of individual components of the system, namely, the source, treatment, and distribution through primary, secondary, and tertiary pipes and in system storage infrastructure. Wu et al. (2011) developed a surplus power factor as resilience measure for optimization of water main transmission system. Yazdani and Jeffray (2012) used robustness and redundancy to define the resilience of water distribution system. Liu et al. (2012) provided an overall definition for resilience as capacity of water resources system to maintain it essential functions as before during an event of unexpected stresses and disturbances. Resilience-based design of various infrastructures facilities is also gaining importance. Bruneau et al. (2003) presented a conceptual framework for defining seismic resilience by integrating four dimensions of community resilience, namely technical, organization, social, and economic which can be used to quantify measures of resilience for various types of physical and organizational systems. Wang et al. (2009) illustrated that the resilient infrastructure is one that shows (a) reduced failure probabilities, (b) reduced consequence of failure, and (c) reduced time to recovery. Cimellaro et al. (2010) presented concepts of disaster resilience and procedure for its quantitative evaluation and the same is illustrated using two examples through hospital network subjected to earthquakes. Saldarriaga et al. (2010) presented a new prioritization approach based on the resilience index and dissipated hydraulic power in the pipe to arrive hydraulically and economically near optimal solution to the rehabilitation problems. Pandit and Crittenden (2012) proposed an index to address the resilience using six network attributes. Davis (2014) demonstrated the relation between resilience and water system serviceability, and dependence of community resilience on water system resilience and defined five water service categories namely water delivery, quality, quantity, fire protection, and functionality to address the water system resilience. These five service categories were applied to Los Angeles water system after earthquake condition to find the means of improving the system for quick restoration. Cimellaro et al. (2015) proposed a new resilience index for water distribution network by combining three issues based on water availability to the consumer, storage level of water tank, and water quality. The developed index was applied to Calascibetta town water supply system located in Italy under different disruptive scenarios.

*q*

_{ i }represents demand at node

*i*;

*h*

_{avl,j }, available pressure head at node

*j*;

*h*

_{min,j }, minimum pressure head at node

*j*;

*Q*

_{r}, flow from reservoir

*i*;

*h*

_{res,i }, sum of reservoir elevation and its water level of reservoir

*i*;

*P*

_{ b }, capacity of pump

*b*; and

*ν*is the specific weight of the liquid.

*Q*

_{s}and

*h*

_{res}are the discharge and the head, respectively, available at reservoir

*i*;

*R*, the number of reservoirs;

*Q*

_{ j }and

*h*

_{ j }, the demand and the available head at node

*j*; and

*N*is the number of demand nodes. The power efficiency \( \eta \) theoretically ranges between 0 and 100% (poor and good). Recently, Suribabu et al. (2016) illustrated methods for improving the resilience of existing water distribution network, and analysis of the results has shown that marginal increase in cost can fetch significant increase in the resilience of the network.

It is evident that selection of higher dimension pipe than actually required as far as economical point of view can increase the resilience of network. Hence, minimizing the cost and maximizing the resilience of the network will be more appropriate if the network is to be designed based on not only economical point of view but also resilience of the system. Reliability of the system can be increased by providing stand-by pumps and storage facilities if water source is reliable. But in the case of components such as pipes and valves, economical and technical feasibility of providing stand-by pipes and valves is herculean task. It is difficult and too costly to have parallel pipe for each pipe or even for some critical pipes. In case of possible situations to have parallel pipes, the house connections taken from the main pipe cannot be shifted to parallel pipe at the time of failure or abnormal operating condition. Hence, it is wise to size the pipe to have additional capacity at the design stage itself. The question of extending capacity of pipe can be achieved by formulating design problem as a multi-objective problem. This paper uses these three existing measures which address the resilience of water distribution network.

## Weighted diameter

*L*

_{ i }is the length of the pipe

*i*and

*D*

_{ i }is the diameter of the pipe

*i*.

This simple expression will be useful to assess the increase in overall pipe size due to increase in overall cost. Further, it provides an idea about how the additional space is created to increase the resilience of the network.

## Optimization model

*d*(

*j*), the diameter selected for link

*j*; \( P_{d(j)}, \) the pipe cost per unit length for the link

*j*with diameter

*d*(

*j*); and \( l_{j} \) is the length of link. RM is a resilience measure according to selected index from expressions (1), (2), and (3).

This objective function needs to be minimized subject to a set of hydraulic constraints as follows:

*Q*is the pipe flow; ND

_{ n }, the demand at node

*n*; in,

*n*, the set of pipes entering to the node

*n*; and out,

*n,*the set of pipes emerging from node

*n*; and NN is the node set.

_{ i }is the head loss due to friction in pipe

*i*; NL, the loop set; Δ

*H*, the difference between nodal heads at both ends; and Δ

*H*= 0, if the path is closed.

*i*of connecting nodes

*j*and

*k*:

*C*

_{HW}, the Hazen–Williams coefficient;

*D*

_{ i }, the diameter of the pipe

*i*;

*L*

_{ i }, the length of the pipe

*i*;

*α*is the conversion factor which depend on the units, different values of

*α*are found in the literature—as low as 10.4516 to as high as 10.9031 (Savic and Walters 1997).

_{ j }is the pressure head at node,

*j*and

*h*

_{min,j }is the minimum required pressure head.

## Normalized objective function

*C*is the cost of the network;

*C*

_{min}, optimal cost of the network;

*C*

_{max}, maximum cost of the network;

*W*, weightage factor; RM, resilience measure of the network; RM

_{min}, resilience measure of the network corresponding to the optimal solution; RM

_{max}is the resilience measure of the network corresponding to the maximum cost solution.

Minimum cost solution denotes the optimal solution obtained without considering resilience measure and RM_{min} is its resilience measure value. *C* _{max} is obtained by configuring all the pipes of network with maximum commercial diameter. RM_{max} denotes the resilience measure obtained for *C* _{max} network.

## Differential evolution algorithm

- 1.Initial candidate solutions are generated randomly for chosen population size (pop_size) to form initial population and accounting this as a first generation (
*G*= 1). The expression for creating random solution is as follows:where \( r_{i,j}^{G} \) denotes a uniformly distributed random value within the range from 0.0 to 1.0; \( d_{j}^{(U)} \) and \( d_{j}^{(L)} \) are upper and lower limits of variable \( d_{j}^{{}} \); and$$ d_{i,j}^{0} = d_{j}^{(L)} \, + r_{i,j}^{G} (d_{j}^{(U)} - d_{j}^{(L)} )\,,\quad \forall \,i = 1\;{\text{to}}\;s,\quad \forall j = 1\;{\text{to}}\;n, $$(12)*s*and*n*denotes population size and number of variables, respectively. - 2.In the next step, weighted vector is calculated by multiply mutation factor
*F*with differential vector obtained by finding the difference between two randomly selected vectors from population:$$ \begin{array}{*{20}c} {w_{j}^{G} = F*\left( {d_{{A,j}}^{G} - d_{{B,j}}^{G} } \right)} & {\forall _{j} = 1\;{\text{to}}\;n} \\ \end{array} . $$(13)The weighing factor (

*F*) is usually selected between 0.4 and 1.0. - 3.The population of trial vectors \( P^{(G + 1)} \) is generated as follows:where \(i = 1, \ldots ,{\text{pop}}\_{\text{size}}\), \(A \in [1, \ldots ,{\text{pop}}\_{\text{size}}]\), \(B \in [1, \ldots ,{\text{pop}}\_{\text{size}}]\), \(C \in [1, \ldots ,{\text{pop}}\_{\text{size}}]\), \(A \ne B \ne C \ne i\), \(C_{r} \in [0\;{\text{to}}\;1]\), \(F \in [0\;{\text{to}}\;1]\), \(r \in [0\;{\text{to}}\;1]\).$$ d_{i,j}^{G + 1} = \left\{ {\begin{array}{*{20}l} {d_{C,j}^{(G)} + w_{j}^{G} } \hfill & {{\text{if}}\;r_{i,j}^{{}} \le C_{r} ;\;\;\;{\mkern 1mu} {\kern 1pt} \forall j = 1\;{\text{to}}\;n} \hfill \\ {d_{i,j}^{(G)} } \hfill & {\text{otherwise}} \hfill \\ \end{array} } \right., $$(14)
*C*_{r}is crossover constant, which assists for differential perturbation in order to select the pipe diameter either from noisy vector or target vector to form a new population for next generation. - 4.
The population of next generation \( P^{(G + 1)} \) is created as follows (Selection):

*i*th individual in

*G*th generation.

Further details on implementation of DE to the design of water distribution are available in Suribabu (2010). The recent studies on optimal design of water distribution network using DE algorithm (Suribabu 2010; Vasan and Simonovic 2010; Zheng et al. 2011a, b, 2012a, b, Dong et al. 2012; Marchi et al. 2014) have shown as a most promising population-based stochastic search technique. In the present study, the combined computer coding of DE algorithm and functions of EPANET (Rossman 2000) Toolkits for hydraulic simulation has been written in Visual Basic language.

## Illustration with Hanoi network

Node and link data for Hanoi network

Node No. | Demand (m | Link index | Arc | Length (m) |
---|---|---|---|---|

1 | −19,940 | 1 | (1,2) | 100 |

2 | 890 | 2 | (2,3) | 1350 |

3 | 850 | 3 | (3,4) | 900 |

4 | 130 | 4 | (4,5) | 1150 |

5 | 725 | 5 | (5,6) | 1450 |

6 | 1005 | 6 | (6,7) | 450 |

7 | 1350 | 7 | (7,8) | 850 |

8 | 550 | 8 | (8,9) | 850 |

9 | 525 | 9 | (9,10) | 800 |

10 | 525 | 10 | (10,11) | 950 |

11 | 500 | 11 | (11,12) | 1200 |

12 | 560 | 12 | (12,13) | 3500 |

13 | 940 | 13 | (10,14) | 800 |

14 | 615 | 14 | (14,15) | 500 |

15 | 280 | 15 | (15,16) | 550 |

16 | 310 | 16 | (16,17) | 2730 |

17 | 865 | 17 | (17,18) | 1750 |

18 | 1345 | 18 | (18,19) | 800 |

19 | 60 | 19 | (19,3) | 400 |

20 | 1275 | 20 | (3,20) | 2200 |

21 | 930 | 21 | (20,21) | 1500 |

22 | 485 | 22 | (21,22) | 500 |

23 | 1045 | 23 | (20,23) | 2650 |

24 | 820 | 24 | (23,24) | 1230 |

25 | 170 | 25 | (24,25) | 1300 |

26 | 900 | 26 | (25,26) | 850 |

27 | 370 | 27 | (26,27) | 300 |

28 | 290 | 28 | (27,16) | 750 |

29 | 360 | 29 | (23,28) | 1500 |

30 | 360 | 30 | (28,29) | 2000 |

31 | 105 | 31 | (29,30) | 1600 |

32 | 805 | 32 | (30,31) | 150 |

33 | (31,32) | 860 | ||

34 | (32,25) | 950 |

Cost data for pipes for Hanoi network

Diameter (in) | Diameter (mm) | Cost (units) |
---|---|---|

12 | 304.8 | 45.73 |

16 | 406.4 | 70.40 |

20 | 508.0 | 98.38 |

24 | 609.6 | 129.333 |

30 | 762.0 | 180.8 |

40 | 1016.0 | 278.3 |

## Results and discussion

Optimal solutions for the network based on first objective function

Pipe ID | Cost/RI | Cost/MRI and cost/ | Optimal cost |
---|---|---|---|

1 | 1016 | 1016 | 1016 |

2 | 1016 | 1016 | 1016 |

3 | 1016 | 1016 | 1016 |

4 | 1016 | 1016 | 1016 |

5 | 1016 | 1016 | 1016 |

6 | 1016 | 1016 | 1016 |

7 | 762 | 762 | 1016 |

8 | 762 | 762 | 1016 |

9 | 609.6 | 762 | 1016 |

10 | 762 | 762 | 762 |

11 | 762 | 762 | 609.6 |

12 | 508 | 508 | 609.6 |

13 | 609.6 | 406.4 | 508 |

14 | 762 | 508 | 406.4 |

15 | 762 | 609.6 | 304.8 |

16 | 1016 | 1016 | 304.8 |

17 | 1016 | 1016 | 406.4 |

18 | 1016 | 1016 | 609.6 |

19 | 1016 | 1016 | 508 |

20 | 1016 | 1016 | 1016 |

21 | 609.6 | 609.6 | 508 |

22 | 406.4 | 406.4 | 304.8 |

23 | 1016 | 762 | 1016 |

24 | 762 | 508 | 762 |

25 | 609.6 | 304.8 | 762 |

26 | 304.8 | 508 | 508 |

27 | 609.6 | 762 | 304.8 |

28 | 609.6 | 762 | 304.8 |

29 | 406.4 | 406.4 | 406.4 |

30 | 304.8 | 304.8 | 304.8 |

31 | 304.8 | 304.8 | 304.8 |

32 | 406.4 | 304.8 | 406.4 |

33 | 406.4 | 406.4 | 406.4 |

34 | 609.6 | 508 | 609.6 |

Cost $ | 7,128,424.4 | 6,650,114.42 | 6,081,087 |

Weighted diameter (mm) | 738.58 | 702.76 | 655.63 |

RI | 0.317 | 0.289 | 0.192 |

MRI | 1.739 | 1.675 | 1.447 |

PE | 0.522 | 0.503 | 0.434 |

_{min}value. RM value of infeasible solution will be less than RM

_{min}. If it is used as it is then second term becomes negative and such a solution will be selected as a best solution. To avoid and eliminate such a condition, the RM value of such solution is increased 0.000001 from RM

_{min}so that difference between RM and RM

_{min}will be always equal to 0.000001. For each weightage pair [i.e.,

*w*and (1 −

*w*)], ten trial runs are carried out by changing random seed value. The least value obtained as per normalized objective function is selected for every pair of weights and Pareto curve is generated and presented in Figs. 2, 3, 4. It can be seen from each curve that slope of the curve is steep between 6.25 and 7.5 millions and mild slope appears above 7.5 millions. This depicts that there is a drastically improvement of the network performance when the network is designed optimally within the range of cost from 6.25 to 7.5 millions. It is found that the proposed normalized objective function is fully capable of arriving an optimal solution with minimum number of trial runs under population size of 50 and maximum number of generation 2000. The success rate of getting optimal solution irrespective of RM used for optimization is around 50%. The main observation in the solution configuration is that the optimized solution at the higher level resilience measures follows dimensioning the pipe telescopically along the shortest path to the demand nodes. It is considered generally that transporting demand along shortest path provides cheapest mode of transport (Kadu et al. 2008). With an increase in solution cost, the diameter of the pipes in certain routes which fall in the shortest path to the nodes also increases to next commercial diameter.

Optimal solutions for the network-based second objective function

Pipe ID | Maximum diameter limited to 1016 mm | Maximum diameter limited to 1524 mm | ||||||
---|---|---|---|---|---|---|---|---|

| | | | | | | | |

Solution 1 | Solution 2 | Solution 3 | Solution 4 | Solution 5 | Solution 6 | Solution 7 | Solution 8 | |

1 | 1016 | 1016 | 1016 | 1016 | 1524 | 1524 | 1524 | 1524 |

2 | 1016 | 1016 | 1016 | 1016 | 1524 | 1524 | 1524 | 1524 |

3 | 1016 | 1016 | 1016 | 1016 | 1270 | 1524 | 1524 | 1524 |

4 | 1016 | 1016 | 1016 | 1016 | 1270 | 1524 | 1524 | 1524 |

5 | 1016 | 1016 | 1016 | 1016 | 1016 | 1524 | 1524 | 1524 |

6 | 1016 | 1016 | 1016 | 1016 | 1016 | 1270 | 1524 | 1524 |

7 | 1016 | 1016 | 1016 | 1016 | 1016 | 1270 | 1270 | 1270 |

8 | 1016 | 762 | 1016 | 1016 | 762 | 1270 | 1270 | 1270 |

9 | 762 | 762 | 762 | 1016 | 762 | 1016 | 1270 | 1016 |

10 | 762 | 762 | 762 | 1016 | 762 | 762 | 762 | 1016 |

11 | 609.6 | 762 | 762 | 1016 | 609.6 | 609.6 | 762 | 762 |

12 | 508 | 609.6 | 609.6 | 762 | 508 | 508 | 609.6 | 762 |

13 | 304.8 | 508 | 609.6 | 609.6 | 304.8 | 609.6 | 762 | 304.8 |

14 | 508 | 609.6 | 762 | 762 | 406.4 | 508 | 406.4 | 609.6 |

15 | 609.6 | 762 | 762 | 1016 | 406.4 | 304.8 | 304.8 | 762 |

16 | 1016 | 1016 | 1016 | 1016 | 762 | 304.8 | 304.8 | 1524 |

17 | 1016 | 1016 | 1016 | 1016 | 762 | 508 | 609.6 | 1524 |

18 | 1016 | 1016 | 1016 | 1016 | 1016 | 762 | 1016 | 1524 |

19 | 1016 | 1016 | 1016 | 1016 | 1016 | 762 | 1016 | 1524 |

20 | 1016 | 1016 | 1016 | 1016 | 1270 | 1524 | 1524 | 1524 |

21 | 609.6 | 609.6 | 762 | 762 | 609.6 | 609.6 | 762 | 762 |

22 | 406.4 | 406.4 | 508 | 508 | 304.8 | 406.4 | 406.4 | 508 |

23 | 762 | 1016 | 1016 | 1016 | 1016 | 1270 | 1524 | 1016 |

24 | 508 | 762 | 762 | 1016 | 762 | 1016 | 1016 | 609.6 |

25 | 304.8 | 508 | 609.6 | 762 | 609.6 | 762 | 1016 | 304.8 |

26 | 609.6 | 406.4 | 508 | 508 | 406.4 | 609.6 | 609.6 | 762 |

27 | 762 | 609.6 | 762 | 762 | 406.4 | 304.8 | 406.4 | 1016 |

28 | 762 | 762 | 762 | 1016 | 508 | 304.8 | 304.8 | 1270 |

29 | 406.4 | 508 | 508 | 609.6 | 406.4 | 508 | 609.6 | 609.6 |

30 | 304.8 | 406.4 | 406.4 | 304.8 | 304.8 | 406.4 | 609.6 | 406.4 |

31 | 304.8 | 304.8 | 304.8 | 406.4 | 304.8 | 304.8 | 304.8 | 304.8 |

32 | 406.4 | 304.8 | 406.4 | 406.4 | 508 | 304.8 | 304.8 | 406.4 |

33 | 406.4 | 406.4 | 508 | 508 | 609.6 | 304.8 | 304.8 | 508 |

34 | 609.6 | 609.6 | 762 | 762 | 609.6 | 609.6 | 609.6 | 762 |

RI | 0.2980 | 0.3281 | 0.3384 | 0.3487 | 0.7798 | 0.8460 | 0.8763 | 0.8969 |

Cost ($) | 67,10,999 | 74,17,236 | 77,97,775 | 86,81,431 | 71,47,182 | 80,22,887 | 91,66,292 | 106,60,762 |

| 712.04 | 762.55 | 792.09 | 848.79 | 746.30 | 804.90 | 886.86 | 997.00 |

The performance measure of the Hanoi network cannot be increased further as two of its pipes from the source have velocity greater than 6 m/s which makes large pressure head loss in the network. The velocity of these pipes can be reduced only by increasing its size else parallel pipes need to be provided. Since optimization of the network needs to be done with the options tabulated in Table 1 according to Fujiwara and Khang (1990), all the researchers used same options and it is optimized. These two pipes are always assigned as 1016 mm diameter what so ever the cost of the network obtained through several approaches and algorithms. In this study, two more options have been added in the list. These two new pipes will have cost of 347.13 units and 458.2 units, respectively. The network is again optimized for revised set of pipe sizes for pressure head limit 30 m. As two more options are incorporated in the optimization, the search space size has expanded and which warranted more number of generations. For revised options of diameter, population size of 40 and number of generation is fixed as 20,000.

The minimum cost solution obtained for 30 m pressure head limit is $5,275,863. It is to be noted that this cost is lesser than the minimum cost solution obtained while maximum diameter size is kept as 1016 mm for same pressure limit. RI, MRI, and PE values of minimum cost network are 0.364, 0.849, and 0.555, respectively. It is very clear that though the cost of this network ($5,275,863) is lesser than optimal cost ($6,081,087) corresponding to maximum pipe diameter option 1016 mm, respective performance measure values are higher. Hence, increasing pipe size options at higher side is justifiable. Table 4 shows the optimal solution obtained for four different combinations of weights adopted in the normalized objective function. It is to be noted that the network is configured along shortest path concept as similar to the previous case. In certain branches to the nodes, the telescopically decreasing pipes may not be the case due to selection of commercial diameters which will provide extra-annular area than actually required. This brings certain benefits to the network configuration in view of cost minimization. The water supposed to be delivered via shortest path to certain nodes will be directed along a path distinct from shortest path which may be longest, but that brings the cost economy in view of additional annular area available due to discrete selection of pipe sizes than actually required if continuous size is permitted. According to minimum spanning tree of Honai network, pipe identity numbers 14, 25, and 31 are acting as the redundant members. The shortest path for node 13 falls in the route connecting pipe ID 1–12. Table 4 shows that all the presented solution here follows this path and pipe sizes are telescopically reducing along this path. For node ID 14, the shortest path from source is along pipe identity numbers 1, 2, 19, 18, 17, 16, 15, and 14. Except solution 7, remaining solutions fall in this path. In solution 7, pipes close to redundant pipe 14 is having higher diameter than its neighboring pipes, which indicate that the path selected for node 15 is 1–9, 13, and 14. But up to node 15, it follows along 1, 2, 19, 18, 17, 16, and 15. Similar observation can be made for node 26, except solution 7, the remaining solutions adhere to the same path in which size of the pipe is reducing telescopically. In case of node 31, the shortest path appears along pipes ID numbers 1, 2, 20, 23, 24, 25, 34, and 33. All the solution follows this path in conveying water to meet demand for node 31. Hence, it is very clear from this analysis that water is transported along the shortest path when network is optimized for higher performance. Increase in the cost of network is attributed with an increase in weighted diameter. There is a drastic increase in the performance value when two higher diameter options are made in the search space. Selection of lower and upper bound in diameter options is another crucial factor in enhancing performance measures. It is to be noted that flows are usually concentrated on the shortest path to minimize the cost. However, this may not be true always (Maidamwar et al. 2000); for an example, in the case of layout and difference in elevations between the nodes along the path favors to take up circuitous route due to availability of gravity.

## Conclusion

Getting an optimal design solution for water distribution network (WDN) is the most explored research area with a main objective of minimizing the cost of networks. The least cost design often fails under changes in nodal demands and pipe roughness. Reliability, robustness, or resilience measures are required to be considered simultaneously with economic cost in the optimal WDN design. Exact quantification of uncertainty in the demand and change in the roughness of the pipe due to age is normally addressed as a difficult task in the design. Hence, designing the network with some consideration to overcome these potential problems in terms of measures such as resilience or reliability is being worth adding design parameters. This paper attempts to minimize the cost of the network per resilience measure subject to pressure head constraint and limited to select within available commercial sizes of pipe. And this paper also presented the simple way to integrate the cost of the network and resilience measure as a single objective function through normalization approach and its capability is addressed using benchmark network. Honai network is considered to optimize the network with the proposed objective functions using differential evolution (DE) algorithm. In the present approach, the multi-objective nature of problem is converted into single objective (normalized objective function) and Pareto front is developed by changing the weights for both cost and resilience measure. Hence, it warranted several runs and trial runs in each selected weights. This also can be solved directly using non-dominated sorting approach which can produce Pareto front. It is evident from the study that whenever the performance measure is given prime importance than cost, then network’s pipes are configured as telescopically decreasing size along shortest path. Although sizing the pipes for water distribution network based on shortest flow path results higher than the least cost solution (global least cost), analysis of results indicates that such a solution is found to be promising as far as resilience point of view. Handling two different units of objective functions as single objective formulation with one metric requires normalization. The proposed way of normalization and combining through weights provided an easy way for decision maker to select a best solution. Real water distribution network may contain a different set of objectives to address the different stakeholders concerns. Although considered objective functions cannot fully represent the complexity real system, proposed formulation provides an easy way of combining several objectives and generates the Pareto front with minimum number of solutions.

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