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Water 2018, 10(3), 307; doi:10.3390/w10030307
Review
Lost in Optimisation of Water Distribution Systems? A Literature Review of System Design
^{1}
College of Engineering, Mathematics and Physical Sciences, University of Exeter, Streatham Campus, North Park Road, Exeter, Devon EX4 4QF, UK
^{2}
Faculty of Science and Technology, Federation University Australia, Mt Helen Campus, University Drive, Ballarat, Victoria 3350, Australia
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Author to whom correspondence should be addressed.
Received: 16 January 2018 / Accepted: 22 February 2018 / Published: 13 March 2018
Abstract
:Optimisation of water distribution system design is a wellestablished research field, which has been extremely productive since the end of the 1980s. Its primary focus is to minimise the cost of a proposed pipe network infrastructure. This paper reviews in a systematic manner articles published over the past three decades, which are relevant to the design of new water distribution systems, and the strengthening, expansion and rehabilitation of existing water distribution systems, inclusive of design timing, parameter uncertainty, water quality, and operational considerations. It identifies trends and limits in the field, and provides future research directions. Exclusively, this review paper also contains comprehensive information from over one hundred and twenty publications in a tabular form, including optimisation model formulations, solution methodologies used, and other important details.
Keywords:
water distribution systems; optimisation; literature review; design; rehabilitation; algorithms1. Introduction
Water distribution systems (WDSs) are one of the major infrastructure assets of the society, with new systems being continually developed reflecting the population growth, and existing systems being upgraded and extended due to raising water demands. Designing economically effective WDSs is a complex task, which involves solving a large number of simultaneous nonlinear network equations, and at the same time, optimising sizes, locations, and operational statuses of network components such as pipes, pumps, tanks and valves [1]. This task becomes even more complex when the optimisation problem involves a larger number of requirements for the designed system to comply with (e.g., water quality), includes additional objectives beside a leastcost economic measure (e.g., potential fire damage) and incorporates more reallife aspects (e.g., uncertainty, staging of construction).
The early research related to the design optimisation of WDSs can be dated from the 1890s to 1950s. It was based on the principle of economic velocity [2,3,4], which was gradually reviewed and replaced by establishing the minimum (annual) costs of the system (i.e., leastcost design) [5,6,7]. Due to lack of computational technology in that period, those previous studies involved manual calculations combined with graphical methods, often resulting in practical charts to derive economic pipe diameters. The development of the optimisation of WDS design, therefore, had been an incremental process over time and may have appeared to be “only too true that the design of the transmission and distribution system receives [at that period] little attention in spite of the great sums of money invested in such installations” [8].
A successive period from the 1960s to 1980s displays a more rapid progression, which was initiated by the introduction of digital computers to network analysis in 1957 [9]. The introduction of computers was subsequently followed by the development of iterative methods [10,11] and simulation packages [12,13] to solve simultaneous nonlinear network equations, and eventuated in the application of mathematical deterministic methods to solve WDS design optimisation problems. These methods, including linear programming (LP) [14], nonlinear programming (NLP) [15,16], and others [17], typically minimised the design or capital (and operational) costs of the system, which were combined into one economic measure.
Another significant advancement in the optimisation of WDSs represented an introduction of stochastic methods using principles of biological evolution [18] and natural genetics [19]. Nonetheless, it was not until the 1990s when these methods became more popular [20] due to their ability to solve complex, realworld problems for which deterministic methods incured difficulty or failed to tackle them at all [21,22], and to also control multiple objectives. The popularity of metaheuristics has resulted in a dramatic increase in the application [21,23] to optimal design of WDSs, with “the several hundred research papers written on the subject” by 2001 [24]. Optimisation of WDS design has also progressed from a costdriven singleobjective framework to multiobjective models, when various objectives that continually gain importance (e.g., environmental objectives, community objectives reflecting the level of service provided to customers) can be evaluated on more equal basis [25]. Some of the most recent developments include the use of an engineered (as opposed to a random) initial population to improve the algorithm convergence [26], application of online artificial neural networks (ANNs) to replace network simulations [27], analysis of the algorithm search behaviour [28] in relation to the WDS design problem features [29], and reduction of the search space [30] to increase computational efficiency.
2. Aim, Scope and Structure of the Paper
This paper aims to provide a comprehensive and systematic review of publications since the end of the 1980s to nowadays, which are relevant to the optimisation of WDS design, strengthening (i.e., pipe paralleling), expansion and rehabilitation. The purpose of the review is to enable one’s speedy familiarisation with the scope of the field, insight in the overwhelming amount of publications available and realisation of the future research directions. This paper contributes to and goes beyond the existing review literature for the optimisation of WDS design and rehabilitation [20,21,31,32,33,34,35,36,37,38,39] by not only identifying trends and limitations in the field, but also by providing comprehensive information from over one hundred and twenty publications in a tabular form, including optimisation model formulations, solution methodologies used, and other important details.
The paper consists of two parts: (i) the main review and (ii) an appendix in a tabular form (further referred to as the table), each having a different structure and purpose. The main review is structured according to publications’ design problems and general classification. The design problems cover application areas, such as new system design, existing system strengthening, expansion and rehabilitation, and time, uncertainty and performance considerations. The general classification captures all the main aspects of a design optimisation problem answering the questions: what is optimised (Section 4.1), how is the problem defined (Section 4.2), how is the problem solved (Section 4.3) and what is the application (Section 4.4)? The purpose of the main review is to provide the current status, analysis and synthesis of the current literature, and to suggest future research directions.
A significant portion of this review paper is represented by the table, which refers to over one hundred and twenty publications in a chronological order. Each paper is classified according to an optimisation model (i.e., objective functions, constraints, decision variables), water quality parameter(s), network analysis, optimisation method and test network(s) used. Obtained results as well as other relevant information are also included. The purpose of the table is to provide a representative list of publications on the topic detailing comprehensive information, so that it could be used as a primary reference point to identify one’s papers of interest in a timely manner. Hence, it presents a unique and integral contribution of this review.
3. Design Problems
Two types of a design problem have been identified based on the field progression as follows: (i) a traditional design (i.e., theoretical or static design) of a WDS with a single construction phase for an entire expected life cycle of the system usually considering fixed loading conditions reflecting maximum (and other) future demands (Section 3.1); (ii) an advanced design (i.e., reallife or dynamic design) of a WDS capturing the system modifications and growth (due to the development of the populated area) over multiple construction phases, including future uncertainties (e.g., in demands, pipe deterioration) and other performance considerations (Section 3.2).
3.1. Application Areas
3.1.1. New Systems: Design
Critical infrastructure, including water, energy and transport systems, is essential in ensuring the survival and wellbeing of populations worldwide. Since the ancient Greek civilisations, WDSs have been an important part of making human settlements sustainable, thus optimising these systems to meet various requirements has over time gained interest of researchers and practitioners alike. Generally, optimisation of WDS design involves determining sizes, locations and operational statuses of network components such as pipes, pumps, tanks and valves, while keeping the system design or capital (and operational) costs at their minimum. The problem scope is primarily dependent on a type of a WDS under consideration, which is either a branched or looped and gravity or pumped system.
A network topology, branched or looped, represents a fundamental distinction in the problem complexity at the network analysis stage due to a way of determining flows in pipes. In branched networks there is a unique flow distribution calculated directly using nodal demands, while in looped systems flows can undertake multiple and alternative paths from a source to a customer [40]. This possible variability results in iterative methods being required to solve pipe flows in looped networks, such as that described in [41].
Regarding gravity WDSs, a basic optimisation model minimises the design cost of the network subject to the nodal pressure requirements, with pipe sizes or diameters being the only decision variables [42,43,44,45,46,47,48]. Popular test networks used to solve such a problem are the twoloop network [14], Hanoi network [49] and Balerma irrigation network [50]. As far as pumped WDSs are concerned, the optimisation problem becomes more complex than in the case of gravity WDSs, because of the presence of pumps and tanks (see Section 3.1.3), which require selecting not only their sizes and locations [14,26,51,52], but also their operational statuses [14,29,53,54], as well as often running an extended period simulation (EPS) for multiple loading conditions. Unlike for gravity WDSs, there does not seem to be any test network that is frequently used by multiple authors for pumped WDSs.
Regarding test networks, nevertheless, study [26] comments that they are limited, in general, to simple transmission networks, socalled benchmark systems, excluding local distribution lines. This exclusion is mainly due to a dramatic increase in the problem dimension, thus computational time, if local pipes were included. A problem of excluding smaller distribution pipes from the optimisation is in oversizing the transmission mains, as local distribution networks provide alternative pathways and display significant capacity to carry when the transmission lines are out of service [26]. The lack of large and complex test networks has recently been addressed by a number of researchers [55,56,57] who developed methodologies for generating synthetic networks of varying sizes and complexity levels. Furthermore, several realworld networks have been used for the design competitions by international research teams working in the area of WDS design, including those that are described by [58,59].
The problem complexity further increases by considering multiple simultaneous objectives. Initially, singleobjective optimisation models were used to formulate WDS design problems, in which all objectives are combined into one economic (i.e., leastcost) measure (see, for example, [14,51,60,61,62]). A multiobjective optimisation approach was possibly first applied in the late 1990s (Figure 1), maximising the network benefit on one hand and minimising the system cost (of network rehabilitation) on the other hand [63]. In studies of newly designed WDSs, in addition to the economic measure, the other objectives considered were the pressure deficit [30,62,64,65,66,67] or excess [68,69] at network nodes, the penalty cost for violating the pressure constraint [70], greenhouse gas (GHG) emissions [71,72,73,74,75,76] or emission cost [77], water discolouration risk [68] and water quality [78]. A multiobjective optimisation approach is considered “very appealing for engineers as it provides a tool to investigate interesting tradeoffs”, for example, a marginal pressure deficit can be outweighed by a considerable cost reduction [67].
The singleobjective approach benefits from being able to identify one best solution, which is then easy to analyse and implement. Multiobjective methods, on the other hand, result in a set of tradeoff (Pareto, nondominated) solutions, which requires an additional step to select only one or a limited number of the promising solutions. Choosing such a reduced number of solutions from a potentially large (or even infinite) nondominated set is likely to be difficult for any decision maker. This task makes the multiobjective approach less desirable as there is often a requirement to make a clear decision to be implemented. The research question resulting from this challenge is how to select the best solution(s) from the Pareto set, which may involve providing the decision makers with a practical and representative subset of the nondominated set that is sufficiently small to be tractable [22]. For example, study [79] introduced gametheoretic bargaining models to take into account conflicting requirements and managed to reduce the solution sets to a reasonable size. Further investigation of the methodologies for identifying a handful of useful solutions, such as those where a small improvement in one objective would lead to a large deterioration in at least one other objective, is thus warranted. In addition to gametheoretic models, the approaches that are based on identifying ‘knees’ of the Pareto front or expected marginal utility, maximum convex bulge/distance from hyperplane, hypervolume contribution and local curvature [80] are all promising methods that require a thorough analysis on WDS problems.
3.1.2. Existing Systems
As a consequence of the development/growth and population density increase within urban areas, existing WDSs require to be upgraded to satisfy raising water demands. These upgrades involve system strengthening (i.e., pipe paralleling), rehabilitation (e.g., pipe cleaning and relining) and expansion. Even though these processes often take place within one WDS thus some of the research articles fall under all system strengthening, rehabilitation and expansion, they are divided into separate subsections in order to provide a systematic overview.
Strengthening
System strengthening represents a reinforcement of an existing WDS to meet future demands, through lying duplicated pipes in parallel to the existing water mains. It is also sometimes referred to as parallel network expansion [42] or pipe paralleling. The main objective and decision variables are, similar to the design of new WDSs, the minimisation of the design (or capital) cost and pipe diameters of duplicated pipes, respectively. Publically available test networks involving purely system strengthening include the New York City tunnels [81] and EXNET [82]. In addition, there are test networks considering system strengthening together with other design strategies (e.g., system expansion, rehabilitation), which include the 14pipe network with two supply sources [20,83] and Anytown network [84]. Of those publically available test networks, the most frequently applied is the New York City tunnels, which was often the only network used to test the proposed methodology. These studies used genetic algorithm (GA) [85,86], combined with ANNs [87], fast messy GA (fmGA) [88] and nondominated sorting genetic algorithm II (NSGAII) [89] as a solution algorithm.
The complexity of an optimisation problem involving exclusively system strengthening as a design strategy can be substantially increased by incorporating water quality considerations. Such applications include, apart from pipe sizes as decision variables, also water quality decision variables that can be in a form of disinfectant (i.e., chlorine) dosage rates [27,87]. In order to reduce computational effort of those problems, ANNs were implemented to replace network simulations to a large extent. Further increase in the complexity presents the use of a multiobjective approach, with additional objectives being system robustness [89] (uncertainty and system robustness are contained in Section 3.2.3), the pressure deficit at network nodes [62,65], and the number of demand nodes with pressure deficit [65,90]. In those studies, a conflicting relationship was identified between the economic (i.e., leastcost) objective and pressure deficit/the number of nodes with pressure deficit. Based on such information, the decision maker is able to “quantitatively evaluate the cost of pressure constraints attenuation which implies a reduction in the system service to its consumers.” Optimisation methods used in those studies were NSGAII [65,89,90], strength Pareto evolutionary algorithm 2 (SPEA2) [65] and cross entropy (CE) [62].
Rehabilitation
Due to aging water infrastructure, which causes a decreased level of service in terms of water quantity as well as quality for customers, increased operation costs and leakage, pipe breaks and other issues, existing WDSs require rehabilitation in a timely manner. Large investments are and will be needed in the future to rehabilitate ever deteriorating pipe networks [91] reaching the end of their lifecycle. Network rehabilitation consists of the replacement of pipes with the same or larger diameter, cleaning, or cleaning and lining of existing pipes; with the main objective to minimise the pipe rehabilitation cost. Within an optimisation model, pipe replacement options can be represented by binary [17] or integer [92] decision variables to identify the pipes selected for replacement, and continuous [17] or integer [92] diameters, respectively, of the replaced pipes. Pipe rehabilitation options are often binary decision variables (i.e., 1 = cleaning/lining, 0 = no action) [17,93]. If a pipe is not scheduled for rehabilitation, it is expected to be subject of break repair over a longer planning horizon. Hence, study [17] added the expected pipe repair costs to the rehabilitation cost of the network. Because a network rehabilitation strategy also has a direct impact on pump operating costs and GHG emissions due to pumping (i.e., they are reduced with an increased quantity of rehabilitated pipes) [94], pump energy costs have been added to the total leastcost objective [17,95].
Some studies consider only a single economic objective to formulate a network rehabilitation problem [17], while other investigations apply a multiobjective optimisation framework in order to incorporate measures affecting the level of service provided to customers (i.e., ‘community objectives’). Accordingly, additional objectives considered, beside the economic measure, include the network benefit [63], pressure violations at network nodes [68,95], velocity violations in pipes [95] causing potential sedimentation problems and subsequent water discolouration, water quality (i.e., disinfectant) deficiencies at network nodes [92], and potential fire damage expressed as lack of available fire flows [92]. To generate multiobjective optimal solutions, those studies use mainly metaheuristics or hyperheuristics, such as structured messy GA (SMGA) [63], NSGAII [95], nondominated sorting evolution strategy (NSES) [92], and evolution strategy (ES)/SPEA2 in a hyperheuristic framework with evolved mutation operators [68]. The resulting Pareto fronts can then serve decision makers in selecting a rehabilitation strategy that balances community objectives with a capital expenditure.
Note that publications included in this section belong to the category of static design, which involves a single network rehabilitation intervention for a near planning period, designed based on the current network status. Publications, which are concerned with staged rehabilitation interventions involving their timing over an extended planning horizon, are reviewed in Section 3.2.1.
Expansion
An expansion of a WDS means developing or expanding the existing system beyond its current boundary, with the main objective to minimise the total design (or capital) and operation cost. System expansion can be thought of as the following two interdependent design problems: (i) developing a new network that is connected to the existing one, and simultaneously (ii) strengthening, rehabilitating and upgrading the existing system in order to convey increased water demands. Hence, system expansion is the most complex WDS design problem as it can ultimately contain all aspects of designing new as well as existing systems. A typical example of the optimal network expansion is the Anytown network problem [84]. Essentially, the objective is to determine leastcost design and operation, using locations and sizes of new pipes (including duplicated pipes), pumps and tanks, as well as pipe rehabilitation options (i.e., cleaning and lining) as decision variables. Such extensive problems are often solved by combining a power of optimisation algorithms with “manual calculations and a good deal of engineering judgement” [84].
Although some studies solved the Anytown network problem as initially formulated [84], for example, study [83] by enumeration and [96] using GA, others included new aspects to the (original or modified) problem. Those aspects represent, for example, water quality [97] inclusive of the construction and operation costs of treatment facilities [53], new tank sizing approach (further discussed in Section 3.1.3) [93,98], and additional objectives, such as the network benefit incorporating multiple system performance criteria [93,99] or the hydraulic failure, fire flow deficit, leakage and water age with visual analytics used to explore the tradeoffs between numerous objectives [97]. These studies used SMGA [99], GA [53,93], and εNSGAII [97] to solve the problem. Study [93] combined GA with fuzzy reasoning, where system performance criteria are individually assessed by fuzzy membership functions and combined using fuzzy aggregation operators.
An example of large system expansion represents the battle of the water networks II (BWNII) optimisation problem, which involves the addition of new and parallel pipes, storage, operational controls for pumps and valves, and sizing of backup power supply, and includes the capital and operational costs, water quality, reliability and environmental considerations as performance measures [58]. This problem was solved by multiple authors within the Water Distribution Systems Analysis (WDSA) conference series [58]. Another example of large and realworld system expansion is presented in [100]. Apart from the decision variables for the BWNII, it also includes selections of pipe routes, expansions of water treatment plants (WTPs) and configurations of pressure zones. The common approach that is applied to solve both of those optimisation problems was the use of engineering judgement, which led to a reduction in the number and type of decision variables. In the case of the study of [100], some eliminated variables were included in separate optimisation problems. Study [58] demonstrates that “different combinations of engineering experience, computational power and problem formulation can give similar results”.
Despite the advances in optimisation methods developed for new system design, rehabilitation and/or expansion of WDS, most notably over the last three decades, the large, complex systems still represent a significant challenge to solve using a fully automated optimisation procedure. There are several reasons for that, including: (i) complexity resulting from a mixeddiscrete, nonlinear optimisation problem with often conflicting and difficult to assess objectives and performance measures; (ii) the large network sizes normally encountered in practice, which translates into large search spaces where a global optimum is almost impossible to find; (iii) the so called NoFreeLunch theorem [101], which says that not all of the optimisers are well suited to solving all problems, in other words, slow convergence of general populationbased optimisation methodologies that do not utilise some form of traditional engineering experience/heuristics; and (iv) the lack of computational efficiency of network simulators required by modern populationbased optimisation methods. A number of approaches have been developed to deal with these challenges, mainly aimed at increasing the computational efficiency of the optimisation process. Those improvements often include the division of a design problem into multiple phases [58] that can be solved separately, the involvement of engineering expertise and manual interventions [59] to reduce the search space, or the use of surrogate and metamodelling to speed up the simulation process [27]. The work that is still needed in the WDS design optimisation area is to understand the link between the performance of an algorithm (and its operators) and certain topological features of a WDS (e.g., existence of pumps/tanks, loops), as indicated in [29].
3.1.3. Problem Elements
Pipes
Unlike other network elements (e.g., pumps, tanks, valves), pipes are always included in the optimisation of WDS design, as the basic model is to determine such pipe sizes (or diameters) for which the design cost of the network is minimal, subject to the nodal pressure requirement. Even though pipe decision variables are incorporated in every optimisation model, they do not seem to have been unified. Assuming a given layout of the pipe network, there are two types of a decision variable, pipe sizes/diameters, and pipe segment lengths of a constant (known) diameter. Pipe sizes/diameters are discrete by nature of the problem, because they are to be selected from a set of commercially available sizes, however both discrete and continuous values are used mainly depending on the optimisation method. Discrete sizes are used mostly for stochastic algorithms (i.e., metaheuristics) [42,70,85,88,102,103,104,105,106,107,108,109], whereas continuous sizes for deterministic methods [16,110,111]. In regards to continuous sizes, the final solution can be modified by splitting a link into two pipes of closest upper and lowersized commercially available discrete diameter [16].
WDS design optimisation problems, which use pipe sizes/diameters as decision variables, can be referred to as a singlepipe design [112,113], while problems with pipe segment lengths of a constant (known) diameter as a splitpipe design [112,113]. Pipe segment lengths of a constant (known) diameter are predominantly used in conjunction with deterministic algorithms [14,114,115] or hybrid methods (i.e., combined deterministic and stochastic methods) [113,116,117]. Singlepipe design with discrete decision variables can provide, compared to splitpipe design and continuous diameters solutions, high quality [102], or good quality results without unnecessary restrictions imposed by splitpipe design [42]. Even if only pipe diameters are optimised, the design of WDS is a complex problem that requires a careful selection of decision variables as to minimise the search space. The choice between direct representation of discrete pipe diameters and splitpipe solutions has largely been resolved in favour of the former, but further improvements in decision variable coding might be possible.
In cases of an unspecified network layout (e.g., when designing a new or extending an existing WDS), additional decision variables are required in order to determine or select pipe routes [52,100]. These variables can be formulated, for example, as binary selecting a link which should be included into the pipe route [52]. Pipe routes can also be considered when strengthening an existing WDS, as “parallel pipes do not necessarily have to be laid in the same street”, they “may be laid in a parallel street or rightofway that may not have existed at previous construction times” [118]. Another possible type of a pipe decision variable are pipe closures/openings to adjust a pressure zone boundary within a WDS [100].
Pumps
There are two main aspects of including pumps into the optimisation of WDS design. First, the pump design or capital cost and second, the pump operating cost due to electricity consumption. Typically, electricity consumption is one of the largest marginal costs for water utilities, with the price of electricity rising globally making it a dominant cost in managing WDSs. Therefore, “the presence of pumps requires that both the design and the operation of the network should be considered in the optimisation” [99]. Accordingly, the minimisation of the pump design or capital cost as well as the pump operating cost to achieve minimal amount of electricity consumed by pumps ought to be included in an optimisation model. Pump operating cost is usually calculated on annual basis using the typical daily demand patterns (i.e., EPS), but a longer period can be considered depending on the planning horizon of a case study, for example, 20 years [17,119], 100 years [72,76,77]. Because this cost occurs at different times in the future, its present value is required to be included in the objective function. This conversion of future economic effects into the current time is undertaken via a present value analysis (PVA), described in detail in [71,72,77], using zero, constant or time varying discount rates.
In the model, pumps are controlled by three types of a decision variable. Firstly, a pump location, which are used when designing a new or extending and upgrading an existing WDS. Possible options to consider are, for example, to predetermine a limited number of potential pump locations [93,120], to evaluate network nodes as potential pump locations (yes/no) via binary variables [52] or to upgrade the current pump stations where new pumps are to be installed in parallel to existing ones [99]. Secondly, a pump size, which can be included as a pump capacity [14,121], pump type [75,76], pumping power [17], pump head/height [52,122], pump operation curve/headflow [93] or pump size in a combination with the number of pumps [26]. Thirdly, a pump schedule, which describes when the pump is on and off during a scheduling period (e.g., 24 h). It can be specified by a pumping power [53,54] or pump head [123] at each time step, the number of pumps in operation during 24 h [97], binary pump statuses [29], continuous options representing on/off times with a limit imposed on the number of pump switches [76], discrete options representing the time at which a pump is turned on/off using a predefined time step (e.g., 30 min) [75]. All of these decisions impact on the size of the search space and eventually on the computational efficiency of the optimisation algorithm used. A comparative study of various approaches would be useful to help determine what their advantages and disadvantages are and which one to use for a particular situation.
In terms of the model objectives, the pump design or capital and/or operating costs were mostly incorporated together with the costs of other network elements (e.g., pipes, tanks, valves) into one economic function (see, for example, [17,26,51,60,93,95,96,119]). Although a few studies, which considered the design and operating costs as part of separate objectives (e.g., [124]), reported on their conflicting tradeoff, this relationship was not confirmed for a higherdimensional space when required to balance numerous objectives [97]. Additionally, the pump maintenance cost (see, for example, [61,62,121]) as well as the pump replacement and refurbishment cost [71,72,77] were accounted for. More recently, GHG emission cost or GHG emissions due to the electricity that is consumed by pumps [71,72,73,74,75,76,77] were introduced as an environmental objective. Similar to the pump operating cost, a PVA can be used for the pump maintenance, replacement and refurbishment costs, as well as GHG emissions/cost. Even though there is a significant tradeoff between economic and environmental objectives (i.e., GHG emissions decrease with the increasing costs and vice versa), GHG emissions can be considerably reduced by a reasonable increase in the costs [71,72]. Additional results indicate that the price of carbon has no effect on the tradeoff [77], whereas the discount rates do [72], the use of variable speed pumps (VSPs) (rather than fixed speed pumps (FSPs)) leads to significant savings in both total costs and GHG emissions [74].
The mixedinteger nature of pumps as decision variables and their often significant impact in terms of hydraulic behaviour of the entire system, makes them a difficult element to include and control its impact during an optimisation run. Furthermore, the increased complexity of modelling VSPs and their incorporation into the optimisation problem pose another difficulty that has to be tackled by modern optimisation algorithms. Finally, the formulation of various objectives, including maintenance requirements (i.e., often surrogated by the number of times a pump is switched on during the optimisation period), represents another challenge for including pumps into overall WDS design studies.
Tanks
In spite of having a valuable role in WDSs contributing to their reliability and efficiency [125], storage tanks (further in the text referred simply to as tanks) are not often included in WDS design optimisation problems. Several types of a decision variable have been used in the literature to control tanks in the model, and a few objectives (or objective functions) have been developed to mainly evaluate tank performance. However, the use of those variables as well as objectives seems to vary across studies with no general framework on how to model tanks available. As far as decision variables are concerned, they include tank locations [71,72,96,97,98,99,120], tank volumes [16,53,93,96,98,99], minimum (and maximum) operational levels [93,96,98,99], tank heads [78], tank elevations [14], ratio between diameter and height [98], ratio between emergency volume and total volume [98]. Study [99] compared two approaches to model tanks in terms of operational levels, first of which calculates tank levels analytically during the network analysis, and second of which includes tank levels as independent variables. Although they yielded similar results, the former approach obtained more robust solutions.
In regard to objectives, the most frequently used account for the tank design or capital cost, which is normally part of the total system costs (i.e., pipes, pumps, etc.) [16,53,76,93,96,97,98,99,120]. Furthermore, additional objectives have been introduced evaluating, along with others, the tank performance. These objectives are the network benefit, including storage capacity difference [99], safety and operational volume capacities, and the filling capacity of the tank [93], and system hydraulic failure including tank failure index [97]. A positive relationship was identified between the total cost of the system and network benefit [93,99], whereas a negative relationship exists between the cost and failure index [97]. The effect of changing the tank balancing volume, so called tank reserve size (TRS), on the minimisation of system cost and GHG emissions was also investigated [76]. It was identified that a larger TRS could assist in reducing GHG emissions with no additional cost by modifying pumping schedules.
In addition to pumps, the presence or absence of a tank can also play a significant role in changing hydraulic behaviour of a WDS. This presents a large challenge for any optimisation approach as it creates a discontinuity (i.e., a large change in behaviour with or without a tank at a particular location), which has to be properly managed by the algorithm. Additionally, the setup of the tank (i.e., the link to the system, overflow valve operation, consideration of upper/lower level limits) within a simulation model can also play a significant role in the efficiency of the optimisation run.
Valves
The inclusion of valves in WDS design optimisation problems appears to be rather sporadic and descriptions related to their implementation are often very brief with not many details provided. Studies [14,26] accounted for valves in the overall costs of the system, based on optimal valve locations. The optimisation of a reallife scale WDS incorporating not only transmission pipelines, but also local distribution pipelines, concluded that optimal valve locations are to be affected by the presence of local lines which “provide alternative pathways when the main lines are out of service” [26]. As shutdown of valves used to isolate a portion of the WDS during an emergency (e.g., pipe break or a water quality incident) creates a change in hydraulic behaviour, the valve numbers and locations play part in the overall system design, particularly when the reliability or resilience of the system is considered. For example, study [126] presented a methodology for optimal system design accounting for valve shutdowns. Another application of valves is using their settings to influence the pressure distribution in the network (via pressure reducing valves (PRVs)) [16], or to determine timing of flows and flow rate values (either via flow control valves (FCVs) or PRVs) [127].
Valves were also used to incorporate a simpler model of VSPs into the multiobjective optimisation of WDS design including total economic cost of the system (i.e., design and operation) and GHG emissions [74]. In such an application, a pump power estimation method uses a FCV combined with an upstream reservoir to represent a pump in the system, with the aim to maintain the flows via the FCV into the downstream tanks as close as possible to the required flows. Hence, the determination of the most appropriate FCV setting for calculating pump power is formulated as a singleobjective minimisation problem that is subject to multiple flow constraints, which is implemented within a multiobjective GA (MOGA) framework [74].
A combined design of the isolating valve system and the pipe network presents a considerable challenge to optimisation methods. Not only that the number of decisions increases exponentially with the addition of valves, but also the consequences of various valve system designs can only be evaluated by investigating a large number of (probabilistic) scenarios making the whole process computationally inefficient. Furthermore, the location and status of isolating valves can form decision variables also when a WDS is to be divided into manageable subsystems. This is the case with the socalled district metering areas (DMAs), which have been first implemented in the UK primarily for leakage management purposes [128]. Due to the fact that the DMA optimal design is normally performed after a system has been constructed, this problem was deemed beyond the scope of this review paper.
3.2. Time, Uncertainty and Performance Considerations
3.2.1. Staged Design
A staged design represents an optimisation of a WDS over a long planning horizon divided into several construction phases, without considering future uncertainties (e.g., in demands, pipe deterioration). In other words, it is a deterministic dynamic design either over several prefixed time intervals or with timing decisions (i.e., year of action execution). The planning horizon can spread across a number of years to an expected life cycle of the system.
Initial work in the staged design is related to the development of multiquality water resources systems using a singleobjective approach, which minimised the costs of water allocation, facilities expansion, water transportation, and losses caused by insufficient supply [129]. It was formulated as a LP optimisation problem, into which nonlinear water quality equations were incorporated using a successive linear approximation iterative scheme. An advantage of using a staged design was demonstrated by realising linkages between certain management processes and variables, and a particular planning period (prefixed time interval).
Concerning WDSs, the staged design is often applied to rehabilitate an existing system as this problem inherently involves the timing of ongoing works over an extended planning horizon. Both single and multiobjective optimisation models were proposed to solve such problems. Singleobjective models included beside the network rehabilitation [130], also network strengthening [131] and expansion [124,132] combined into one leastcost objective, while multiobjective models incorporated the network benefit [131] or the system operating costs [124,132] as additional objectives. Optimisation methods used were GA [130], SMGA [131] and NSGAII [124,132]. As opposed to the study of [129], these models do not define prefixed time intervals, but include timing decision variables to schedule works, also referred to as eventbased coding [124,132]. This coding dramatically reduces the search space, thus the computing and memory requirements, because it eliminates unnecessary zero values of a traditional coding based on a timeinterval (e.g., yearly) basis [124]. A further search space reduction can be achieved by so called limited pipe representation introduced by [130], which involves placing an upper bound on the number of pipes considered for rehabilitation. These reductions in the search space and computing requirements are especially important for large size WDSs.
Moreover, the staged design was applied to extend and strengthen existing wastewater, recycled and drinking water systems applying an integrated optimisation scheme within a singleobjective framework using GA [127], and to plan a new WDS considering two objectives, the construction costs and network reliability, using NSGAII [118]. Both of these studies used prefixed time intervals to schedule the construction. In addition, study [118] analysed the effect of the scheduled construction on the network design using a set of scenarios reflecting different lengths of planning horizons (25–100 years), time intervals (25–100 years) and the number of construction phases (1–4). Both studies [118,127] confirmed that for long planning horizons, the staged design is cost effective. The system upgrades guarantee a predefined reliability and there is always opportunity to modify or redesign subsequent upgrades at the later stage, based on new uptodate predictions of potential future development [118].
By introducing staged design to WDSs, it is obvious that the search space increases almost exponentially to accommodate decisions at various times in the planning horizon. This is one of the key challenges for deterministic staged design, as computational efficiency of optimisation algorithms plays even more significant role than with static design. Another difficulty for achieving the optimised staged design is that even if an optimal solution can be found for each of the intermediate time steps, the algorithm has to ensure that contiguity among the staged decision is maintained, i.e., that the decisions selected in the previous stages are retained in the subsequent ones. An approach by [133] presents one way of obtaining that contiguity of decisions, starting from the solution at one extreme of the Pareto front. However, this issue is still an underresearched area, which requires more investigation. All of the above challenges apply even when the future is assumed to be perfectly known, which is unfortunately not the case.
3.2.2. Flexible Design
A flexible design represents one of the most recent developments in the design optimisation of WDSs. Similar to a staged design, a flexible design represents an optimisation of a WDS over a long planning horizon divided into several construction phases, but with the consideration of future uncertainties (e.g., in demands, pipe deterioration, urban expansion scenarios). Specifically, it is a probabilistic dynamic design over several prefixed time intervals and with the planning horizon ranging from a number of years to an expected life cycle of the system. Such a design allows for flexible and adaptive planning, which is favoured by water organisations that are often encouraged to include risk and uncertainty in their long term plans.
Uncertainties included in the flexible design are related to future demands [122,134,135,136] and future network expansions [137]. They are implemented using either a probabilistic demand assessment [135] or scenariobased approach via demand/expansion scenarios [122,134,136,137]. A decision tree has been introduced to combine the uncertain demands and intervention measures into optional decision paths [135]. Analogously, studies [122,137] have proposed the use of real options (ROs) approach, which is also based on decision trees that reflect future uncertainties. In ROs approach, a decision tree is formed by independent decision paths with assigned probabilities to each of the scenarios. This approach enables flexibility to be incorporated into the decision making process and to subsequently change the investment plan based on new circumstances [122].
The majority of the above studies apply multiobjective optimisation approach, including, besides an economic (leastcost) objective, the system resilience [135], reliability [136] or total pressure violations [137] as another objective. Stochastic optimisation algorithms, such as NSGAII [135,136], simulated annealing (SA), and multiobjective SA [122,137] have been employed to solve flexible design problems, except for [134] who applied integer LP (ILP) combined with preprocessing methods to reduce the dimensionality of the problem. These preprocessing methods included separating the (branched) network into subnetworks, reducing the number of decision variables (e.g., velocity constraints were used to eliminate unsuitable pipe diameters) and solving each subnetwork separately. As a consequence, the quality of the solution was improved and the proposed methodology [134] can be applied to large size WDSs.
While comparing to a traditional deterministic design, the results indicate that a flexible design has a higher initial cost (i.e., in the first construction phases) [122,136], which enables the system to adapt to various future conditions. However, it outperforms a traditional design in terms of the total cost over the entire planning horizon [122,135].
The application of flexible optimisation methodologies in WDS design that consider longterm uncertainty and management options, is yet to be explored to a larger extent in the literature. One of the key reasons is that it is not clear how various types of uncertainties, i.e., stochastic vs. deep uncertainty or aleatoric vs. epistemic uncertainty, are best represented in the optimisation process. The other possible reason is that the flexible design incurs additional computational costs that affects the overall computational efficiency of the optimisation algorithm. However, as the planning and design exercises are done sporadically, the additional computational burden and costs are often justified. Future uncertainties that might have an impact on WDS design, including climate change, population movements and economic development, make flexible design probably the most promising area of research over the next few decades.
3.2.3. Resilient, Reliable and Robust Design
System resilience, reliability and robustness present performance characteristics of a WDS in relation to current and most importantly future uncertain conditions. Although there is no universally agreed definition of any of these measures, the resilience can be defined in broadest terms as the ability of a WDS to adapt to or recover from a significant disturbance, which can be internal (e.g., pipe failure) or external (e.g., natural disaster) (adapted from [138]). Similarly, the reliability can be defined as the ability of a WDS to provide expected service, and can be expressed as the probability that the system will be in service over a specific period of time (adapted from [139]). The robustness represents the ability of a WDS to maintain its functionality under all circumstances (adapted from [138]), or over everyday fluctuations that have the potential to cause low to moderate (i.e., not catastrophic) loss of performance [89].
A robust design problem of a WDS is primarily concerned with uncertainties in model parameters. These uncertainties are related mainly to future demands [89,110,121,123,140,141], but can also consider pipe roughnesses [89,110,140,141], minimum nodal pressure requirements [110], network expansions [137] and others [142]. Study [89] states that “neglecting uncertainty in the design process may lead to serious underdesign of water distribution networks”.
Several approaches have been proposed in the literature to formulate a robust design problem for WDSs. Firstly, a redundant design approach which adds redundancy to the system to account for the uncertain parameters by assuming that those parameters are larger than expected [140]. Secondly, an integration approach where uncertainties are incorporated into the model formulation via either objective function [89] or constraints [140] sometimes referred to as a chanceconstrained model [110]. Both of those approaches assume a probabilistic distribution of uncertain parameters and convert an original stochastic optimisation problem into a deterministic one. Thirdly, a twophase optimisation approach that initially solves an optimisation problem with deterministic parameters (i.e., no uncertainties), and subsequently uses those obtained solutions as an initial population for a stochastic problem where future demands and pipe roughnesses are considered uncertain variables following a probability density function [141]. Fourthly, a scenariobased approach where the uncertainty is implemented via a set of scenarios, each assigned a probability [121]. Lastly and most recently, a robust counterpart (RC) approach which is nonprobabilistic and incorporates the uncertainty through an ellipsoidal uncertainty set constructed according to the userdefined protection level [123].
Despite a number of approaches to incorporate robustness into the design of WDSs, the measure has been defined fairly well and consistently in the literature, and consequently it has been used in optimisation studies. This may be due to the advances in robust optimisation in other fields and/or due to the focus on noncatastrophic loss of performance that is associated with robust operation. However, the other two measures, reliability and most notably resilience, have not been defined consistently in the WDS literature or have been considered seriously only fairly recently. Therefore, this section focuses on robust design of WDSs, with resilience and reliability being outside of the scope of this review paper. This also indicates that future research efforts could be directed toward a consistent and agreed definition of reliability and resilience, with optimisation methods being capable of solving WDS design considering them as objectives/performance measures.
3.2.4. Design for Water Quality
In the literature, water quality is incorporated into the WDS design optimisation problems in various ways concerning both an optimisation model and water quality measure used. In terms of optimisation models, singleobjective as well as multiobjective exist which include water quality considerations. In the former, water quality related expenditures, such as the cost of disinfection [27,120], cost of water treatment [53] or cost of losses incurred by insufficient quality [129], are combined with the system design/capital (and operation) costs into one objective. Alternatively, water quality is included as a constraint to a singleobjective model in a form of minimum (and maximum) disinfectant concentrations at the network nodes [87,143]. In the latter, water quality presents a sole objective, which is either water quality benefit (being maximised) [63,131], water quality deficiencies (being minimised) [92,97,144] or water quality reliability (being maximised) [78]. Regardless of an optimisation model used, study [120] highlighted an importance of incorporating water quality considerations with system design and operation in one optimisation framework, which enables promoting water quality in the design stage, rather than leaving potential water quality issues to be resolved during the system’s operational phase. Indeed, study [78] reports a significant tradeoff between water quality objective based on disinfectant residual and the system capital costs (i.e., the best quality solutions correspond to higher costs and vice versa), and demonstrates the sensitivity of the obtained solutions to a disinfectant dosage rate. Interestingly, there was not tradeoff found between water quality objective based on water age and the cost.
Regarding the water quality measure, it is dependent on the system specifics, its requirements, and also the optimisation model advancements progressively implementing water quality objectives useful to system operators. Basic water quality parameters that are used in optimisation models of drinking WDSs are chlorine [27,87,120,143] and chloramine [120], modelled as nonconservative applying first order decay kinetics, adjusted by a higher decay rate in parts of the system when needed [120]. In contrast, conservative water quality parameters are typically important for regional multiquality WDSs. These parameters are either specified within an optimisation model, such as salinity [129] or unspecified being modelled in conjunction with the operation of treatment facilities [53]. In multiobjective optimisation models, both specific parameters and surrogate measures are used to quantify water quality objectives. Water quality benefit is expressed as a function of the length of renewed and/or lined old pipes, as aged pipes are considered to cause the development of microorganisms and water discolouration [63,131]. Water quality deficiencies can be represented by a performance function on disinfectant residual reflecting governmental regulations [92], water age [97], or the risk of water discolouration due to the potential material after daily conditioning shear stress [144]. Water quality reliability is based on disinfectant residual [145] and/or water age [78].
Optimisation methods used to solve WDS design problems including water quality considerations were LP [129], GA [53,87,100,120,143] and differential evolution (DE) [27] for singleobjective models, and SMGA [63,131], NSES [92], εNSGAII [97], NSGAII and SPEA2 integrated with a heuristic Markovchain hyperheuristic (MCHH) [144] and ant colony optimisation (ACO) [78] for multiobjective models. These algorithms were mainly linked with a network simulator EPANET to solve network equations. Because these EPANET simulations, in particular water quality analyses, are very computationally demanding, they were replaced by ANNs [27,87,143] to reduce computational effort.
Unsurprisingly, introduction of water quality considerations increases the complexity of the quest for the optimal design considerably. This increased complexity is caused not only by the more complex simulations required to predict the temporal and spatial distribution of a variety of constituents within a distribution system, but also by the requirement to run shorter time step water quality computations [22]. Furthermore, computational efficiency is affected by the ability to model multiple constituents throughout the WDS via the EPANET MultiSpecies Extension, EPANETMSX [146].
4. General Classification of Reviewed Publications
Based on the selected literature analysis, the following are the four main criteria for the classification of design optimisation for WDSs: application area (Section 4.1), optimisation model (Section 4.2), solution methodology (Section 4.3) and test network (Section 4.4).
4.1. Application Area
As outlined in Section 3, there are four application areas: design of new systems (Section 3.1.1), strengthening, expansion and rehabilitation of existing systems (Section 3.1.2). Numerous publications do not deal only with those design optimisation problems, but also with the operational optimisation (see, for example, [14,26,53,71,120,135]), which is an equally important area if the total cost (i.e., including capital and operation expenditure) is considered. Hence, the system operation has been added to the following analysis. It represents papers optimising (mainly) the pump operation together with the system design, strengthening, expansion and/or rehabilitation. Figure 2 displays distribution of the application areas across the papers analysed and listed in Appendix A Table A1 as follows:
 Design of new systems is an application area with the highest representation (41%). Interestingly, an almost identical percentage (43%) totals applications for existing systems (i.e., strengthening, expansion and rehabilitation).
 An application area with the second highest representation (25%) is strengthening of existing systems.
 Expansion and rehabilitation of existing systems are both represented evenly by 9% of applications each.
 Optimisation of the system operation is represented by 16% of applications.
It is not surprising that design (and mostly using pipe diameters as decision variables) dominates the literature, which occurs mostly due to historical reasons. Namely, the sizing of pipes was addressed first in the literature, even before WDS simulation was possible. Other design variants, such as strengthening, expansion and rehabilitation, followed on, but use the same or quite similar performance measures and optimisation tools. The introduction of other network elements, such as pumps, tanks and valves, as well as various performance criteria, including water quality and operational considerations, appeared much later in the literature. Lately, more emphasis was put on robustness, reliability and resilience assessment of WDS design and operation, which seems to be the trend for the future.
4.2. Optimisation Model
An optimisation model is mathematically defined by three key components: objectives, constraints and decision variables. Figure 3 shows how many of these components are included in the optimisation models (of papers analysed in Appendix A Table A1), which indicates the degree of complexity of the formulation. Note that not all of the reviewed papers include mathematical formulations of an optimisation model used. Therefore, our assessment is limited to our interpretation of the provided information in the publications, where explicit formulation was partially presented or missing altogether.
 The number of objectives included in optimisation models ranges from one to six. The majority of models (69%) are singleobjective, determining the leastcost design. The second largest proportion (27%) represents twoobjective optimisation models. Multiobjective models including more than two objectives (i.e., 3–6 objectives) are very sparse as they represent only 4% of all formulations.
 The number of constraints incorporated in optimisation models ranges from zero to ten. Hydraulic constraints (such as conservation of mass of flow, conservation of energy and conservation of mass of constituent) are normally included as implicit constraints and are forced to be satisfied by a WDS modelling tool, such as EPANET, and thus are not included in these statistics. There are 5% of models with no constraints, which are mainly multiobjective optimisation models where the pressure requirement is defined as an objective rather than a constraint. Almost half models (48%) include only one constraint (mostly the minimum pressure requirement). A quarter of models (25%) incorporate two constraints. The proportion of optimisation models with exactly three or more (i.e., 4–10) constraints is 13% and 9%, respectively.
 The number of types of a decision (i.e., control) variable included in optimisation models ranges from one to 13. The majority of optimisation models (60%) uses one type of a decision variable, being a pipe diameter/size or pipe segment length of a constant (known) diameter. The use of more than one type of a decision variable is considerably less frequent and is represented by 16%, 11% and 13%, respectively, for two, three and more (i.e., 4–13) types of a decision variable.
Inspecting Figure 3, the question arises as to how many optimisation models there are, which include only one objective, one constraint and one type of a decision variable? There are, in total, 129 optimisation models formulated and solved in 124 papers listed in Appendix A Table A1. From those optimisation models, 30% (i.e., 39 models) consist of one objective (mostly design costs), one constraint (mostly the minimum pressure at nodes) and one type of a decision variable (mostly pipe diameters).
As indicated, the prevailing use of singleobjective optimisation is probably caused by the preference to arrive at a single solution, which can be implemented by decision makers. On the other hand, the preference for one constraint seems surprising as the number of constraints of the problem depends on the complexity of the system and the number of criteria expressed as constraints rather than objectives. Finally, the number and types of decision variables appearing in the literature is a function of historical developments in the field and the increasing trend is expected in the future. Research questions still remain as how to best formulate the optimisation model for a particular case, and what effect the model formulation has on obtained solution(s) [22,23].
4.2.1. General Optimisation Model
A general multiobjective optimisation model for optimal design of a WDS can be formulated as:
subject to:
where Equation (1) represents objective functions to be minimised (e.g., system capital costs) or maximised (e.g., system reliability), Equations (2)–(4) present three types of a constraint, with $x$ representing decision variables.
$$\mathrm{Minimise}/\mathrm{maximise}\left({f}_{1}\left(x\right),{f}_{2}\left(x\right),\dots ,{f}_{n}\left(x\right)\right)$$
$${a}_{i}\left(x\right)=0,\text{\hspace{1em}}i\in I=\left\{1,\dots ,m\right\},\text{\hspace{1em}}m\ge 0$$
$${b}_{j}\left(x\right)\le 0,\text{\hspace{1em}}j\in J=\left\{1,\dots ,n\right\},\text{\hspace{1em}}n\ge 0$$
$${c}_{k}\left(x\right)\le 0,\text{\hspace{1em}}k\in K=\left\{1,\dots ,p\right\},\text{\hspace{1em}}p\ge 0$$
Objectives
Objectives of a general optimisation model of WDS design are listed in Table 1. They can be divided into four distinct groups according to their type. The first group represents economic objectives such as capital and rehabilitation costs, and expected operation and maintenance costs of the system. The second group are community objectives, which report on the level of service provided to WDS customers, and which, if inadequate, could eventuate in water supply related issues for those customers. Those objectives include, for example, a benefit function (using various performance criteria), water quality deficiencies, pressure deficit at demand nodes, hydraulic failure of the system and potential fire damages. The third group presents performance objectives, reflecting the operation of a WDS, specifically system robustness, reliability and resilience. These objectives, although ultimately indicating the level of service for WDS customers, have separate classification, due to their primary purpose to report on the performance in relation to a WDS rather than to customers. The fourth group represents environmental objectives, namely GHG emissions, consisting of capital emissions due to manufacturing and installation of network components applicable at the WDS construction phase, and operating emissions due to electricity consumption occurring during the WDS life cycle.
Constraints
Constraints of a general optimisation model of WDS design are described in Table 2 and divided into three groups as follows. Hydraulic constraints are given by physical laws governing the fluid flow in a pipe network. These constraints are incorporated in an optimisation model either explicitly often in conjunction with deterministic [147] and hybrid optimisation techniques [116,117], or implicitly by a WDS modelling tool (e.g., EPANET) [26] and/or ANNs [27,87] normally in combination with stochastic optimisation algorithms. System constraints arise from limitations and operational requirements of a WDS, and include tank water level bounds, pressure/water quality requirements at demand nodes, etc. The ways to manage these constraints include an integration of EPANET (e.g., tank water levels), the augmented Lagrangian penalty method [17], a penalty function [26], a penalty function with a selfadaptive penalty multiplier [45,88], or a (modified) constraint tournament selection [148,149,150]. Constraints on decision variables, such as pipe diameters being limited to commercially available sizes and others, are handled explicitly by an optimisation algorithm.
Decision Variables
Decision variables of a general optimisation model of WDS design are listed in Table 3. They are grouped according to an element or aspect that drives the optimisation (i.e., pipes, pumps, tanks, valves, nodes, water quality and timing). In general, a pipe diameter/size is often the main (or the only) decision variable used in design optimisation of WDSs. Accordingly, a total of 60% optimisation models (of papers listed in Appendix A Table A1) use only one type of a decision variable (see Figure 3c), which is either a pipe diameter/size or the pipe segment length of a constant (known) diameter. As the complexity of an optimisation model increases, so does the number of types of a decision variable. An example of such an optimisation model could be an expansion and rehabilitation of an existing WDS with pumps, tanks and a treatment plant to meet future demands and water quality requirements.
Table 1, Table 2 and Table 3 provide a generic set of components used for formulating an optimisation problem involving initial design with subsequent operational management of a WDS. Particular circumstances being considered in different case studies may warrant only a portion of those components to be used.
4.3. Solution Methodology
An enormous effort has been dedicated to the application and development of optimisation methods to solve WDS design optimisation problems since the 1970s. Initially, deterministic methods namely LP [14,114,129], NLP [16,110] and mixedinteger NLP (MINLP) [17,115] were used. In the mid 1990s, after the first popular applications of a GA [20,151], there was a swing towards stochastic methods and they dominate the field since (see Figure 4). A great range of those methods has been applied to optimise design of WDSs to date, inclusive of (but not limited to) a GA [42,45,50,85,86,152,153,154], fmGA [88], noncrossover dither creeping mutationbased GA (CMBGA) [149], adaptive locally constrained GA (ALCOGA) [155], SA [60], shuffled frog leaping algorithm (SFLA) [103], ACO [104,156], shuffled complex evolution (SCE) [157], harmony search (HS) [105,158,159], particle swarm HS (PSHS) [160], parameter setting free HS (PSF HS) [161], combined cuckooHS algorithm (CSHS) [162], particle swarm optimisation (PSO) [106,153,154], improved PSO (IPSO) [163], accelerated momentum PSO (AMPSO) [164], integer discrete PSO (IDPSO) [165], newly developed swarmbased optimisation (DSO) algorithm [150], scatter search (SS) [166], CE [61,62], immune algorithm (IA) [167], heuristicbased algorithm (HBA) [168], memetic algorithm (MA) [107], genetic heritage evolution by stochastic transmission (GHEST) [169], honey bee mating optimisation (HBMO) [170], DE [46,153,154,171], combined PSO and DE method (PSODE) [172], selfadaptive DE method (SADE) [173], NSGAII [70], ES [68], NSES [92], cost gradientbased heuristic method [119], improved mine blast algorithm (IMBA) [174], discrete state transition algorithm (STA) [175], evolutionary algorithm (EA) [109], and convergencetrajectory controlled ACO (ACO_{CTC}) [176]. The vast majority of those studies solely solve a basic singleobjective leastcost design problem (i.e., pipe cost minimisation constrained by the nodal pressure requirement) and use a small number of available benchmark networks (e.g., Hanoi network [49], New York City tunnels [81], twoloop network [14]) to test the proposed optimisation method. The usual result obtained was a better or comparable optimal solution reached more efficiently than by algorithms previously used in the literature, without providing an explanation as to why the selected algorithm performed better for a particular test network. It seems, therefore, that research have been trapped, to some extent, in applying new metaheuristic optimisation methods to relatively simple (from an engineering perspective) design problems, without understanding the underlying principles behind algorithm performance. Moreover, study [177] stresses that there has been “little focus on understanding why certain algorithm variants perform better for certain case studies than others”.
Over the past decade, an increase in the use of deterministic and hybrid methods (i.e., a combined deterministic and stochastic method) can be observed from Figure 4. These methods, which are computationally more efficient when comparing to stochastic methods, thus more suitable for large realworld applications, include ILP [51,134], MINLP [147], a combined GA and LP method (GALP/GALP) [113,117], combined GA and ILP method (GAILP) [178], combined binary LP and DE method (BLPDE) [179], combined NLP and DE method (NLPDE) [111], hybrid discrete dynamically dimensioned search (HDDDS) [180], decompositionbased heuristic [52], optimal power use surface (OPUS) method paired with metaheuristic algorithms [47], and modified central force optimisation algorithm (CFOnet) [181]. However, WDS simulations may still be computationally prohibitive even with more efficient deterministic or hybrid optimisation methods, especially as the fidelity of the model and the number of decision variables increase [22].
The choice of the solution methodology depends on the type of problem being considered, the level of expertise of the analyst and the familiarity with the particular method/tool. Nonetheless, there is often no clear justification provided as to why a particular methodology has been selected over another and/or why an alternative methodology has not been tested. Quite often, this choice is based on the analyst’s preference, level of familiarity, and software availability, rather than on a comparison of the tests performed using two or more solution methodologies. This practice makes it difficult to progress towards the development of meaningful guidelines for the application of different optimisation methods [177]. An interesting research question for further studies would be how to characterise and select the best optimisation method for a particular WDS design problem.
However, that being stated, several attempts have been made to compare or evaluate algorithm performance for both single and multiobjective WDS design problems, but with an absence of a universally adopted method to date. A methodology for comparing the performance of various singleobjective algorithms involves assessing the best solution obtained (which is straightforward contrary to multiobjective optimisation), the convergence speed, and the spread and consistency of the solutions using a number of random starting seeds and evaluations [153,154]. A methodology has also been developed to evaluate the performance of a particular algorithm by assessing the effectiveness of its parameters (such as crossover and mutation) applying their different values [182]. In multiobjective optimisation, in general, performance metrics were proposed and are commonly used to compare performance of various algorithms in terms of the quality of the Pareto fronts obtained (see, for example, [183,184,185]). A comparison of solutions is substantially more complex than in singleobjective optimisation as there is no single performance metric both compatible and complete [186]. Possibly for that reason, some WDS design studies have limited their analysis to a visual comparison of solutions only (i.e., twoobjective Pareto fronts), which was criticised by [187]. Most recent research, progressively, evaluates the performance and search behaviour of multiobjective algorithms in relation to their parameters and/or WDS features [28] (more such studies are listed in Section 4.3.2).
4.3.1. Computational Efficiency
Numerous advancements have been reported in the literature to improve the computational efficiency of both optimisation algorithms and network simulators. These developments include methods for search space reduction [45,63,88,95,99,120,131,188,189], parallel programming techniques [109], hybridisation of the evolutionary search with machine learning techniques to limit the number of function evaluations [67], surrogate models (metamodels) to replace network simulations [27,43,67,87,143], approximation of the objective function by shortening the EPS [119], and enhanced methods for speedy network simulations for large size WDSs [190].
Various techniques for search space reduction have been proposed, which can be broadly classified as algorithmbased and networkbased methods. The algorithmbased techniques include the method for more efficient encoding of decision variables [63,99,131], a selfadaptive boundary search strategy for selection of the penalty factor within the optimisation algorithm to guide the search towards constraint boundaries [88], and the application of an artificial inducement mutation (AIM) to acceleratingly direct the search to the feasible region [95]. The networkbased techniques analyse either the network as a whole or individual pipes. The former include a network stratification into upper, middle and lower diameter sets using engineering judgment [188], and the critical path method [45,191]. The latter involve the elimination of certain pipes from the optimisation based on their preliminary capacity assessment [120], application of a pipe index vector (PIV), a measure of the relative importance of pipes regarding their hydraulic performance in the network, which assists in exclusion of impractical and infeasible regions from the search space [189], and introduction of upper/lower bounds on pipe diameters based on the initial analysis [30].
In terms of replacing time consuming network simulations with faster means, three types of a surrogate model have been applied to the design optimisation of WDSs to date. These models include a linear transfer function (LTF) [43], Kriging [67] and ANNs [27,87,143], which are used more frequently than two previous ones. The purpose of a surrogate model is to approximate network simulations (hydraulics and/or water quality), hence reduce the calls of the simulation model during the optimisation. Kriging uses solutions visited during the search to model the search landscape [192]. ANNs can be divided into two groups, offline ANNs, which are trained only once at the beginning of the optimisation, and recently proposed online ANNs, which are “retrained periodically during the optimisation in order to improve their approximation to the appropriate portion of the search space” [27]. ANN metamodels are often used in conjunction with water quality simulations [27,87,143].
All of those methods have shown promise on a limited number of test cases or a specific case study. It would be interesting to conduct a thorough comparison of all of those on a selection of benchmark cases of various sizes and complexity.
4.3.2. Recent Developments
More recently, a number of advancements, such as improving and understanding the algorithm performance and others, have been proposed in the literature indicating potential directions for future research. Some of those developments are a consequence of an appeal of [23,177] “to counteract potential repetition and stagnation in this field” to continually produce too many papers using “an ever increasing number of EA variants” and “theoretical or very simplistic case studies”.
Firstly, to improve the algorithm performance regarding the solution quality, an engineered initial population has been suggested [26,30,44,66,108]. Traditionally, a random (or naïve) initial population of solutions (expressed as pipe sizes) is used as a starting point for algorithms. An engineered initial population, in contrast, is created by taking into account the rules and hydraulics principles of water flow in a pipe network, or by performing preoptimisation runs under various parameter scenarios (e.g., [30]). Another way to achieve better algorithm convergence, particularly for design problems incorporating water quality, is to run the algorithm with hydraulic analysis only for several first generations, and subsequently add water quality analysis [120]. Furthermore, a postoptimisation technique can be used to refine the solutions that are found by an optimisation algorithm to get closer to the global optimum [193]. Secondly, a range of strategies have been introduced to eliminate the tedious and time demanding process of calibrating algorithm parameters (i.e., selecting the best performing combination of parameter values) for a particular test problem. These strategies involve the use of a statistical analysis [158], evolved heuristics (i.e., hyperheuristic approach) [68,144,194], and convergence trajectories [176]. Thirdly, several studies focused on analysing algorithm performance [195] and search behaviour [28,48,196] in relation to the WDS design problem features [29]. Lastly, methodological improvements using existing methods have been proposed rather than applying/developing new algorithms, with the aim to improve computational efficiency. Those improvements represent multiplephase optimisation concepts [30], which can be combined with graph decomposition [46,69] or clustering [90] techniques.
4.4. Test Network
An enormous diversity exists among test networks used in optimisation of WDS design. These networks vary in size, complexity, and the types of network components that they contain (i.e., nodes, pipes, pumps, tanks, reservoirs/sources and/or valves). The simplest networks are small gravity WDSs with one source, a few nodes and pipes (see, for example, [14,60]), or simplified pumped WDSs including only one source, one pump, one pipe and one tank (see, for example, [71]). An example of a large network represents EXNET [82], which is a realistic WDS comprising two sources, control valves and almost 2500 pipes. Figure 5 categorises test networks that are used (in the papers listed in Appendix A Table A1) by network size. In order to be consistent with the previous review [22], network size is expressed by the number of nodes within a network. Networks, for which the number of nodes cannot be identified from the reviewed paper or the references provided, are excluded from the analysis. Figure 5 reveals that nearly a half of the networks (49%) is limited in size to 20 nodes and the majority of the test networks (84%) contains up to 100 nodes. This finding is analogous to operational optimisation of WDSs, where networks with up to 100 nodes represent 80% of applications [22].
Figure 5 illustrates that in the large body of literature, various WDS design formulations and optimisation methods have usually been tested using small, computationally cheap networks. This prevalent usage of small networks is in contrast to the requirement to optimise design of realworld systems that contain hundreds of thousand elements (including pumping stations, tanks and valves) causing a single EPS simulation to take minutes or even hours to execute even on powerful desktop computers. Consequently, large networks are not often considered by optimal design studies. This situation can be remedied by using latest developments in methods capable of generating realistic WDS networks by [55,56,57], who have each developed their own automatic network generation software.
Realworld WDS design optimisation problems normally involve large size, complextopology networks, comprising a number of elements of various types. Such a problem is often solved by combining a sophisticated simulation model (to potentially analyse both hydraulics and water quality) with a nontrivial optimisation method. The approach ought to satisfy the requirements of a water utility and other stakeholders for objectives, constraints, decision variables, as well as model assumptions. Although studies exist that report on successful solutions to such problems [100,127,197,198,199], they are limited possibly due to the complexity associated with mathematically formulating objectives and constraints and/or finding the best solution. Study [200] even speculates that the realworld considerations need to be explicitly quantified, “if it is possible to do so at all”, otherwise the water industry will apply engineering judgment instead of any optimisation method to design WDSs.
Similar to network size, the frequency of use of test networks varies considerably, as some networks have been used only once, while others have been used repeatedly and by multiple authors. In particular, the prevalence of some networks attributes to their use as benchmark problems to test optimisation algorithms. These benchmark networks, all of which have been used (in the papers listed in Appendix A Table A1) 10 or more times, are listed in Table 4 in order of their usage count. They are, except for the Anytown network, gravityfed WDSs with the common objective to determine the most economical pipe design. The popularity of those benchmark networks contributed to high percentages of the first two categories in Figure 5, because the majority of them are limited in size to 20 and 100 nodes, respectively.
5. Future Research
Future research challenges for the optimisation of WDS design are illustrated in Figure 6 and divided into the following four groups: (i) model inputs, (ii) algorithm and solution methodology, (iii) search space and computational efficiency, and (iv) solution postprocessing. As far as model inputs are concerned, there is a requirement to explore how to best represent various types of uncertainties, i.e., stochastic vs. deep uncertainty or aleatoric vs. epistemic uncertainty, in the optimisation process. Additional future uncertainties, for example, climate change, population movements and economic development, might affect planning for optimal WDSs, and make flexible design one of the promising research areas over the next few decades. Another research challenge in regards to model inputs is to compare various approaches to pump decision variables, including VSPs and their coding, in order to determine their advantages, disadvantages and suitability for a particular case. Furthermore and overall, a research question remains how to best formulate the optimisation model for a particular case [22,23].
Concerning algorithm and solution methodology, a vast research area represents a progression towards better understanding of algorithm performance and its search behaviour. These aspects need to be further linked to the WDS design problem features including system topology (e.g., existence of pumps/tanks, loops) and initial population used. A related challenge is to eliminate a time consuming process of calibrating algorithm parameters to achieve a satisfactory performance, hence there is a question how to select the best performing combination of parameter values. Moreover, it is important to develop understanding related to the suitability of various optimisation methods for particular design problems and the incorporation of engineering judgement in the search. In relation to staged design, methods for ensuring contiguity among decisions, i.e., that the decisions selected in the previous stages are retained in the subsequent ones, are required.
Recently, there has been an observed increased interest in aspects of the search space and computational efficiency. Indeed, the reduction of the search space and an increase in the computational efficiency are significant particularly for realworld WDS optimisation problems as well as dynamic (i.e., staged and flexible) design, so they are expected to remain important and promising research areas into the future. The research community would benefit from a thorough comparison of existing methods for search space reduction and computational efficiency increase, which could use a selection of benchmark cases of various sizes and complexity. In addition to currently available methods for search space reduction, it might be possible to further improve decision variable coding.
Regarding solution postprocessing, an open question is how sensitive the obtained solution(s) is to the optimisation model used [22,23]. When multiobjective optimisation approach is used, a remaining challenge is to select a practical and representative subset of the nondominated solutions, which could be useful for the decision makers. Accordingly, there is a need for methods to identify a handful of effective solutions, such as those where a small improvement in one objective leads to a large deterioration in at least one other objective. The existing approaches, including maximum convex bulge/distance from hyperplane, hypervolume contribution, and local curvature [80] are all promising and require a thorough analysis on WDS design problems.
6. Summary and Conclusion
A systematic literature review of optimisation of water distribution system (WDS) design since the end of the 1980s to nowadays has been presented. The publications included in this review are relevant to the design of new WDSs, strengthening, expansion and rehabilitation of existing WDSs, and also consider design timing, parameter uncertainty, water quality and operational aspects. The value of this review paper is that it brings together a large number of publications for design optimisation of WDSs, just under three hundred in total, which have been published over the past three decades. Therefore, it may enable researchers to identify one’s articles of interest in a timely manner. The review analyses the current status, identifies trends and limits in the field, describes a general optimisation model, suggests future research directions. Exclusively, this review paper also contains comprehensive information for over one hundred and twenty publications in a tabular form, including optimisation model formulations (i.e., objectives, constraints, decision variables), solution methodologies used and other important details.
This review has identified the following main limits in the field and future research directions. It was demonstrated that there is still no agreement among researchers and practitioners on how to best formulate a WDS design optimisation model, how to include all relevant objectives and constraints, and whether and how to take into account various sources of uncertainty, while still allowing for an efficient search for the best solution to be achieved. Although a plethora of generic and problemspecific optimisation methods have been developed and applied over the years, there is no consensus on what optimisation method is best for a particular design problem, whether a single or multiplephase optimisation concept is to be used, and how engineering judgement can best be incorporated in the search. Therefore, a concerted effort by the research community is required to develop methods for objective comparison and validation of various optimisation algorithms and concepts on large, realworld problems. In addition, an analysis of available methods for reducing the search space, increasing computational efficiency, as well as selecting effective Pareto nondominated solutions representing a practical subset for decision makers, is needed using WDS design problems of various sizes and complexity. In spite of the overwhelming amount of literature that has been published over the past three decades, design optimisation of WDSs faces considerable research challenges in the years to come.
7. List of Terms
 Deterministic dynamic design = staged design over a long planning horizon divided into several construction phases, without considering future uncertainties.
 Deterministic static design = traditional design with a single construction phase for an entire expected life cycle of the system, without considering future uncertainties.
 Dynamic design = staged (i.e., reallife) design capturing the system modifications/growth over a long planning horizon divided into several construction phases (adopted from [118]).
 Hydraulic constraints = constraints arising from physical laws of fluid flow in a pipe network, such as conservation of mass of flow, conservation of energy, conservation of mass of constituent.
 Optimisation approach = singleobjective approach or multiobjective approach.
 Optimisation method = method, either deterministic or stochastic, used to solve an optimisation problem.
 Optimisation model = mathematical formulation of an optimisation problem inclusive of objective functions, constraints and decision variables.
 Probabilistic dynamic design = flexible design over a long planning horizon divided into several construction phases, with considering future uncertainties.
 Probabilistic static design = traditional design with a single construction phase for an entire expected life cycle of the system, with considering future uncertainties.
 Simulation model = mathematical model or software used to solve hydraulics and water quality network equations.
 Single pipe design = design which uses pipe sizes/diameters as decision variables (either discrete or continuous).
 Solution = result of optimisation, either from feasible or infeasible domain, so we refer to a ‘feasible solution’ or ‘infeasible solution’, respectively. In mathematical terms though an ‘infeasible solution’ is not classified as a solution.
 Splitpipe design = design which uses pipe segment lengths of a constant (known) diameter as decision variables.
 Static design = traditional (i.e., theoretical) design with a single construction phase for an entire expected life cycle of the system (adopted from [118]).
 System constraints = constraints arising from the limitations of a WDS or its operational requirements, such as water level limits at storage tanks, limits for nodal pressures or constituent concentrations, tank volume deficit etc.
Conflicts of Interest
The authors declare no conflict of interest.
Abbreviations
ACO  ant colony optimisation 
ACO_{CTC}  convergencetrajectory controlled ant colony optimisation 
ACS  ant colony system 
AEF  average emissions factor 
AIM  artificial inducement mutation 
ALCOGA  adaptive locally constrained genetic algorithm 
AMPSO  accelerated momentum particle swarm optimisation 
ANN  artificial neural network 
AS  ant system 
AS_{elite}  elitist ant system 
AS_{rank}  elitist rank ant system 
BB  branch and bound 
BBBC  big bangbig crunch 
BLIP  binary linear integer programming 
BLPDE  combined binary linear programming and differential evolution 
BWNII  battle of the water networks II (optimisation problem) 
CA  cellular automaton 
CAMOGA  cellular automaton and genetic approach to multiobjective optimisation 
CANDA  cellular automaton for network design algorithm 
CC  chance constraints 
CDGA  crossover dither creeping mutation genetic algorithm 
CE  cross entropy 
CFO  central force optimisation 
CGA  crossoverbased genetic algorithm with creeping mutation 
CMBGA  noncrossover dither creeping mutationbased genetic algorithm 
CR  crossover probability (parameter) 
CS  cuckoo search 
CSHS  combined cuckooharmony search 
CTM  cohesive transport model 
D  design 
dDE  dither differential evolution 
DDSM  demanddriven simulation method 
DE  differential evolution 
DMA  district metering area 
DPM  discoloration propensity model 
DSO  newly developed swarmbased optimisation algorithm 
EA  evolutionary algorithm 
EAWDND  evolutionary algorithm for solving water distribution network design 
EEA  embodied energy analysis 
EEF  estimated (24h) emissions factor (curve) 
EF  emissions factor 
EPANETpdd  pressuredriven demand extension of EPANET 
EPS  extended period simulation 
ES  evolution strategy 
F  mutation weighting factor (parameter) 
FCV  flow control valve 
fmGA  fast messy genetic algorithm 
FSP  fixed speed pump 
GA  genetic algorithm 
GAILP  combined genetic algorithm and integer linear programming 
GALP/GALP  combined genetic algorithm and linear programming 
GANEO  genetic algorithm network optimisation (program) 
GENOME  genetic algorithm pipe network optimisation model 
GHEST  genetic heritage evolution by stochastic transmission 
GHG  greenhouse gas (emissions) 
GOF  gradient of the objective function 
GP  genetic programming 
GRG2  generalised reduced gradient (solver) 
GUI  graphical user interface 
HBA  heuristicbased algorithm 
HBMO  honey bee mating optimisation 
HDDDS  hybrid discrete dynamically dimensioned search 
HDSM  headdriven simulation method 
HMCR  harmony memory considering rate (parameter) 
HMS  harmony memory size (parameter) 
HS  harmony search 
IA  immune algorithm 
IDPSO  integer discrete particle swarm optimisation 
ILP  integer linear programming 
IMBA  improved mine blast algorithm 
IPSO  improved particle swarm optimisation 
KLSM  Kang and Lansey’s sampling method [26] 
LCA  life cycle analysis 
LHS  Latin hypercube sampling 
LINDO  linear interactive discrete optimiser 
LM  Lagrange’s method 
LP  linear programming 
LTF  linear transfer function 
MA  memetic algorithm 
MBA  mine blast algorithm 
MBLP  mixed binary linear problem 
MCHH  Markovchain hyperheuristic 
MdDE  modified dither differential evolution 
MENOME  metaheuristic pipe network optimisation model 
mIA  modified immune algorithm 
MILP  mixed integer linear programming 
MINLP  mixed integer nonlinear programming 
MMAS  maxmin ant system 
MO  multiobjective 
MODE  multiobjective differential evolution 
MOEA  multiobjective evolutionary algorithm 
MOGA  multiobjective genetic algorithm 
MSATS  mixed simulated annealing and tabu search 
NBGA  noncrossover genetic algorithm with traditional bitwise mutation 
NFF  needed fire flow 
NLP  nonlinear programming 
NLPDE  combined nonlinear programming and differential evolution 
NSES  nondominated sorting evolution strategy 
NSGAII  nondominated sorting genetic algorithm II 
OP  operation 
OPTIMOGA  optimised multiobjective genetic algorithm 
OPUS  optimal power use surface 
PAR  pitch adjustment rate (parameter) 
PESAII  Pareto envelopebased selection algorithm II 
PHSM  prescreened heuristic sampling method 
PIV  pipe index vector 
PRV  pressure reducing valve 
PSF HS  parameter setting free harmony search 
PSHS  particle swarm harmony search 
PSO  particle swarm optimisation 
PSODE  combined particle swarm optimisation and differential evolution 
PVA  present value analysis 
RC  robust counterpart (approach) 
ROs  real options (approach) 
RS  random sampling 
RST  random search technique 
SA  simulated annealing 
SADE  selfadaptive differential evolution 
SAMODE  selfadaptive multiobjective differential evolution 
SCA  shuffled complex algorithm 
SCE  shuffled complex evolution 
SDE  standard differential evolution 
SE  search enforcement 
SFLA  shuffled frog leaping algorithm 
SGA  crossoverbased genetic algorithm with bitwise mutation 
SMGA  structured messy genetic algorithm 
SMODE  standard multiobjective differential evolution (i.e., optimising the whole network directly without decomposition into subnetworks) 
SMORO  scenariobased multiobjective robust optimisation 
SO  singleobjective 
SPEA2  strength Pareto evolutionary algorithm 2 
SS  scatter search 
SSSA  scatter search using simulated annealing as a local searcher 
STA  state transition algorithm 
TC  time cycle 
TRS  tank reserve size 
TS  tabu search 
VSP  variable speed pump 
WCEN  water distribution costemission nexus 
WDS  water distribution system 
WDSA  water distribution systems analysis (conference) 
WPP  water purification plant 
WSMGA  water system multiobjective genetic algorithm 
WTP  water treatment plant 
Appendix A
ID. Authors (Year) [Ref] SO/MO * Brief Description  Optimisation Model (Objective Functions ^{+}, Constraints **, Decision Variables ^{++})  Water Quality Network Analysis Optimisation Method  Notes 

1. Alperovits and Shamir (1977) [14] SO Optimal water distribution system (WDS) design and operation with split pipes considering multiple loading conditions using linear programming (LP) with a twophase procedure.  Objective (1): Minimise (a) the overall capital cost of the network including pipes, pumps, reservoirs and valves, (b) present value of operating costs (pumps, penalties on operating the dummy valves). Constraints: (1) Min/max pressure limits, (2) sum of lengths of pipe segments within an arc equals to the length of this arc, (3) nonnegativity requirement for the length of pipe segments. Decision variables: (1) Flows in pipes as primary variables, (2) length of pipe segments of constant pipe diameter (so called splitpipe decision variables), (3) dummy valve variables to represent multiple loadings (demands), (4) pump locations and capacities, (5) valve locations, (6) reservoir elevations, (7) pump operation statuses, (8) valve settings.  Water quality: N/A. Network analysis: Initial flow distribution is to be specified, flows are then redistributed using a gradient method within an optimisation process. Optimisation method: LP gradient method. 

2. Schwarz et al. (1985) [129] SO Optimal development of a regional multiquality water resources system over a planning horizon (e.g., several years) using LP.  Objective (1): Minimise the costs of (a) water supply (water), (b) temporary curtailment of water supply, (c) network expansion, (d) conveying water, (e) excess salination. Major constraints: (1) Water quantity bounds, (2) water quality bounds, (3) regional water balance (quantity), (4) capacity expansion of the network, (5) annual source water balance (quantity), (6) annual source mass balance (salinity), (7) node mass balance (salinity). Decision variables: (1) Target water supply (m^{3}/year), (2) temporary curtailment of water supply (m^{3}/year), (3) capacity expansion (m^{3}/day), (4) conveyance of water (m^{3}/day), (5) amount of water used from storage (m^{3}/day), (6) salinity (mg/L).  Water quality: Salinity. Network analysis: TEKUMA model [202,203]. Optimisation method: TEKUMA model [202,203] using LP. 

3. Kessler and Shamir (1989) [114] SO Optimal WDS design with split pipes using LP with a twophase procedure.  Objective (1): Minimise (a) the design cost of the network (pipes). Constraints: (1) Pressure limitations at selected nodes, (2) sum of lengths of pipe segments within an arc equals to the length of this arc. Decision variables: (1) Lengths of pipe segments of constant (all available) pipe diameters (so called splitpipe decision variables), (2) flows in pipes.  Water quality: N/A. Network analysis: Flow in pipes is calculated using projected gradient method within an optimisation process. Optimisation method: LP gradient method. 

4. Lansey and Mays (1989) [16] SO Optimal WDS design, rehabilitation and operation considering multiple loading conditions using nonlinear programming (NLP) with a twophase procedure.  Objective (1): Minimise (a) the design cost of the network including pipes, pumps and tanks, (b) penalty cost for violating nodal pressure heads. Constraints: (1) Lower and upper pressure bounds at nodes, (2) design constraints (i.e., storage requirements), (3) general constraints. Decision variables: (1) Pipe diameters (continuous), (2) pump sizes (horsepower or headflow), (3) valve settings, (4) tank volumes.  Water quality: N/A. Network analysis: KYPIPE [12]. Optimisation method: NLP solver generalised reduced gradient (GRG2) [206]. 

5. Lansey et al. (1989) [110] SO Optimal WDS design including uncertainties in demands, minimum pressure requirements and pipe roughnesses using NLP.  Objective (1): Minimise (a) the design cost of the network (pipes). Constraints: (1) Conservation of mass of flow and energy, (2) min pressure at the nodes, (3) pipe diameter bigger than or equal to zero. Decision variables: (1) Pipe diameters (continuous), (2) pressure head at nodes.  Water quality: N/A. Network analysis: Network hydraulics is included as a constraint to the optimisation model. Optimisation method: NLP solver GRG2 [206]. 

6. Eiger et al. (1994) [115] SO Optimal WDS design with split pipes using mixedinteger NLP (MINLP) with a twophase procedure.  Objective (1): Minimise (a) the design cost of the network (pipes). Constraints: (1) Pressure limitations at selected nodes, (2) sum of lengths of pipe segments within an arc equals to the length of this arc. Decision variables: (1) Lengths of pipe segments of constant (all available) pipe diameters (so called splitpipe decision variables), (2) flows in pipes. Note: Same formulation as in Kessler and Shamir (1989).  Water quality: N/A. Network analysis: Flow in pipes is calculated using projected gradient method within an optimisation process. Optimisation method: Branch and bound (BB) method. 

7. Kim and Mays (1994) [17] SO Optimal WDS rehabilitation and operation over a planning horizon (e.g., 20 years) using MINLP with a twophase procedure.  Objective (1): Minimise the sum of the present value of the (a) pipe replacement cost, (b) pipe rehabilitation cost, (c) expected pipe repair (i.e., break repair) cost, (d) pump energy cost. Constraints: (1) Demand supplied to each node should be greater or equal to the required demand, (2) min/max pressures at demand nodes, (3) constraints on binary decision variables representing pipe replacement and rehabilitation options, (4) constraints on continuous decision variables representing the diameter of the replaced pipe and pump horsepower. Decision variables: (1) Pipe replacement option (binary), (2) pipe rehabilitation option (binary), (3) pipe diameters of the replaced pipes (continuous), (4) pump horsepower (continuous).  Water quality: N/A. Network analysis: KYPIPE [12]. Optimisation method: BB method combined with GRG2 [206]. 

8. Murphy et al. (1994) [96] SO Optimal WDS strengthening, expansion, rehabilitation and operation considering multiple loading conditions using genetic algorithm (GA).  Objective (1): Minimise the design cost of the network including (a) pipes, (b) pumps, (c) tanks, and (d) the pump energy costs. Constraints: (1) Limits for nodal pressure heads, (2) limits for tank water levels. Decision variables: Options for (1) new pipes, (2) duplicated pipes, (3) cleaned/lined pipes, (4) pumps, (5) tanks.  Water quality: N/A. Network analysis: Unspecified steady state hydraulic solver. Optimisation method: GA. 

9. Loganathan et al. (1995) [116] SO Optimal WDS design and strengthening with split pipes using a combination of LP, mutlistart local search and simulated annealing (SA) in a twophase procedure.  Objective (1): Minimise (a) the design cost of the network (pipes). Constraints: (1) Min pressure at the nodes, (2) sum of pipe segment lengths must be equal to the link length, (3) nonnegativity of segment lengths. Decision variables: (1) Lengths of pipe segments of known diameters (so called splitpipe decision variables).  Water quality: N/A. Network analysis: Explicit mathematical formulation (steady state). Optimisation method: Combined LP, mutlistart local search and SA. 

10. Dandy et al. (1996) [85] SO Optimal WDS strengthening using GA.  Objective (1): Minimise (a) the sum of material and construction costs of pipes, (b) the penalty cost for violating the pressure constraints. Constraints: (1) Min/max pressure limits at certain network nodes, (2) min diameters for certain pipes in the network. Decision variables: (1) Pipe diameters (discrete diameters are coded using binary substrings).  Water quality: N/A. Network analysis: KYPIPE [12] and another hydraulic solver developed for the paper. Optimisation method: GA. 

11. Halhal et al. (1997) [63] MO Optimal WDS rehabilitation and strengthening over a planning horizon (e.g., several years) using structured messy GA (SMGA).  Objective (1): Maximise the weighted sum of the following benefits of the network: (a) hydraulic performance, (b) physical integrity of the pipes, (c) system flexibility, (d) water quality. Objective (2): Minimise the cost (supply and installation) of the network including (a) new parallel pipes (i.e., duplication), (b) cleaning and lining existing pipes, (c) replacing existing pipes. Constraints: (1) Costs cannot exceed the available budget. Decision variables: String comprising 2 substrings: (1) substring consisting of pipe numbers, (2) substring consisting of decisions associated with those pipes (8 possible decisions). Note: One MO model including both objectives.  Water quality: A general water quality consideration. Network analysis: EPANET. Optimisation method: SMGA. 

12. Savic and Walters (1997) [42] SO Optimal WDS design and strengthening using GA.  Objective (1): Minimise (a) the design cost of the network (pipes). Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: Network solver based on the EPANET. Optimisation method: GANET [208] using GA. 

13. Cunha and Sousa (1999) [102] SO Optimal WDS design using SA.  Objective (1): Minimise (a) the design cost of the network (pipes). Constraints: (1) Min pressure at the nodes, (2) min pipe diameter. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: Newton method to solve hydraulic equations to obtain flows and heads. Optimisation method: SA. 

14. Gupta et al. (1999) [188] SO Optimal WDS strengthening and expansion using GA with search space reduction.  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty for violating minimum residual head. Constraints: (1) Min residual head, (2) min desirable velocity in a pipe. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: ANALIS [212]. Optimisation method: GA. 

15. Halhal et al. (1999) [131] MO Optimal WDS rehabilitation and strengthening over a planning horizon (i.e., 10 years) using SMGA.  Objective (1): Maximise the present value of the benefit of the network rehabilitation over the planning period (incorporating the welfare index), using the following performance criteria: (a) carrying capacity, (b) physical integrity of the pipes, (c) system flexibility, (d) water quality. Objective (2): Minimise (a) the present value of the rehabilitation costs over the planning period. Constraints: (1) Rehabilitation costs less than or equal to the budget. Decision variables: String comprising 3 substrings (1) location substring: pipe numbers of pipes scheduled for rehabilitation (integer), (2) decision substring: rehabilitation option (integer), (3) timing substring: year of rehabilitation execution (integer). Note: One MO model including both objectives.  Water quality: A general water quality consideration. Network analysis: Unspecified solver (steady state). Optimisation method: SMGA. 

16. Montesinos et al. (1999) [86] SO Optimal WDS strengthening using GA.  Objective (1): Minimise (a) the design cost of the network (pipes). Constraints: (1) Min pressure at the nodes, (2) max velocity in the pipes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: NewtonRaphson method [10]. Optimisation method: GA. 

17. Walters et al. (1999) [99] MO Optimal WDS strengthening, expansion, rehabilitation and operation with multiple loading conditions and two approaches to model tanks using SMGA. Note: Discussion: [213], Erratum to Discussion: [214]  Objective (1): Maximise the weighted sum of the benefits of the network rehabilitation, using the following performance criteria: (a) nodal pressure shortfall, (b) storage capacity difference, (c) tank operating level difference or tank flow difference. Objective (2): Minimise (a) the capital cost of the network including pipes, pumps, tanks, (b) present value of the energy consumed during a specified period. Constraints: (1) Pressure constraints for different loading patterns, (2) flow constraints into and out of the tanks. Decision variables: String comprising 2 substrings (1) location substring: pipes, pumps, tanks (integer of 1 or 2 digits), (2) decision substring (expansion/rehabilitation options): pipes, pumps, tanks (integer of 1, 2 or 5 digits). Note: One MO model including both objectives.  Water quality: N/A. Network analysis: Unspecified solver (steady state). Optimisation method: SMGA. 

18. Costa et al. (2000) [60] SO Optimal WDS design and operation using SA.  Objective (1): Minimise the capital cost of the network including (a) pipes, (b) pumps, (c) present value of pump energy costs. Constraints: (1) Min head bound on demand nodes. Decision variables: (1) Pipe diameters (discrete), (2) pump sizes (discrete).  Water quality: N/A. Network analysis: NewtonRaphson method [10]. Optimisation method: SA. 

19. Dandy and Hewitson (2000) [120] SO Optimal WDS design, strengthening and operation including water quality considerations using GA with search space reduction.  Objective (1): Minimise (a) the capital cost of new pipes, pumps and tanks, present value of (b) pump energy costs, (c) likely cost to the community due to waterborne diseases, (d) likely community cost due to disinfection byproducts, (e) community cost of chlorine levels that exceed acceptable limits, (f) cost of disinfection, (g) penalty cost for violating constraints. Constraints: (1) Min pressure at the demand nodes, (2) tanks must refill at the end of the cycle. Decision variables: (1) Sizes of new and duplicate pipes, (2) sizes of new pumps and tanks, (3) locations of new pumps and tanks, (4) decision rules for operating the system, (5) dosing rates of chloramine/chlorine at selected points.  Water quality: Chloramine, chlorine. Network analysis: EPANET (extended period simulation (EPS)). Optimisation method: GA. 

20. Vairavamoorthy and Ali (2000) [43] SO Optimal WDS design and strengthening incorporating a linear transfer function (LTF) model to approximate network hydraulics using GA.  Objective (1): Minimise (a) the capital cost of the network (pipes), (b) penalty for violating the pressure constraints. Constraints: (1) Min/max pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: Steady state hydraulic solver based on the gradient method [215]. Optimisation method: GA. 

21. Dandy and Engelhardt (2001) [130] SO Optimal WDS rehabilitation (considering only pipe replacement) over a planning horizon (i.e., 20 years) using GA.  Objective (1): Minimise (a) the system cost of the rehabilitated network (pipes)—present values of pipe failure costs (i.e., repair costs of existing and new pipes) and pipe replacement costs are considered. Constraints (case 1): N/A. Constraints (case 2): (1) Allowable budget for each time step (i.e., 5year block). Constraints (case 3): (1) As above in the case 2, (2) min pressure at the nodes, (3) max velocity in the pipes. Decision variables (case 1): (1) Replacement decision (0 = no replacement, 1 = replace). Decision variables (case 2): (1) Timing of the replacement (integer) (“all pipe representation”); or (1) pipe to be replaced (integer), (2) timing of the replacement (integer) (“limited pipe representation”). Decision variables (case 3): (1)–(2) as above in the case 2, (3) diameter of the new pipe (integer).  Water quality: N/A. Network analysis: EPANET (Case 3 only). Optimisation method: GA. 

22. Wu and Simpson (2002) [88] SO Optimal WDS strengthening using fast messy GA (fmGA).  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty for violating the pressure constraint. Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: fmGA. 

23. Eusuff and Lansey (2003) [103] SO Optimal WDS design and strengthening using shuffled frog leaping algorithm (SFLA).  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty cost for pressure head violations. Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: SFLA. 

24. Maier et al. (2003) [104] SO Optimal WDS strengthening, expansion and rehabilitation using ant colony optimisation (ACO).  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty for violating the pressure constraint. Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete), (2) pipe rehabilitation options (binary).  Water quality: N/A. Network analysis: WADISO [217], final solutions checked by EPANET. Optimisation method: ACO. 

25. Liong and Atiquzzaman (2004) [157] SO Optimal WDS design using SCE.  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty cost for violating the pressure head bound. Constraints: (1) Min nodal pressure head bound, (2) min/max bound on pipe sizes. Decision variables: (1) Pipe sizes (converted to commercially available diameters).  Water quality: N/A. Network analysis: EPANET. Optimisation method: SCE [218]. 

26. Broad et al. (2005) [87] SO Optimal WDS strengthening including water quality considerations using offline artificial neural networks (ANNs) and GA.  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty cost for violating pressure head, (c) penalty cost for violating chlorine residual. Constraints: (1) Min/max pressure at the nodes, (2) min/max chlorine residual at the nodes. Decision variables: (1) Pipe diameters, (2) chlorine dosing rates.  Water quality: Chlorine. Network analysis: Offline ANN. Optimisation method: GA. 

27. Farmani et al. (2005) [65] MO Optimal WDS design and strengthening using nondominated sorting genetic algorithm II (NSGAII) and strength Pareto evolutionary algorithm 2 (SPEA2).  Objective (1): Minimise (a) the design cost of the network (pipes). Objective (2): Minimise (a) the maximum individual head deficiency at the network nodes. Objective (3) (only for the EXNET test network): Minimise (a) the number of demand nodes with head deficiency. Constraints: N/A. Decision variables: (1) Pipe diameters (discrete). Note: Two MO models, the first including objectives (1) and (2) (applied to the New York City tunnels and Hanoi network); the second objectives (1), (2) and (3) (applied to the EXNET network).  Water quality: N/A. Network analysis: EPANET. Optimisation method: NSGAII and SPEA2 are compared. 

28. Keedwell and Khu (2005) [44] SO Optimal WDS design using a combined cellular automaton for network design algorithm (CANDA) and GA (CANDAGA) including an engineered initial population.  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty for violating the pressure constraint. Constraints: (1) Min/max pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: CANDAGA. 

29. Ostfeld (2005) [53] SO Optimal design and operation of multiquality WDSs including multiple loading conditions and water quality considerations using GA.  Objective (1): Minimise (aD ^{?}) the construction costs of pipes, tanks, pump stations and treatment facilities, (bOP ^{??}) annual operation costs of pump stations and treatment facilities. Constraints: (1) Min/max heads at consumer nodes, (2) max permitted amounts of water withdrawals at sources, (3) tank volume deficit at the end of the simulation period, (4) min/max concentrations at consumer nodes, (5) removal ratio constraints. Decision variables: D: (1) Pipe diameters, (2) tank max storage, (3) max pumping unit power, (4) max removal ratios at treatment facilities, OP: (5) scheduling of pumping units, (6) treatment removal ratios.  Water quality: Unspecified conservative parameters. Network analysis: EPANET (EPS). Optimisation method: GA. 

30. Vairavamoorthy and Ali (2005) [189] SO Optimal WDS design using GA with a pipe index vector (PIV) and search space reduction in a threephase procedure.  Objective (1): Minimise (a) the design cost of the network (pipes). Constraints: (1) Min/max pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: Explicit mathematical formulation (steady state for peak demands). Optimisation method: GA with PIV. 

31. VamvakeridouLyroudia et al. (2005) [93] MO Optimal WDS strengthening, expansion, rehabilitation and operation considering multiple loading conditions using GA with fuzzy reasoning.  Objective (1): Minimise (a) the design cost of the network including pipes, pumps and tanks. Objective (2): Maximise the benefit/quality of the solution, using the following system performance criteria (constraints): (a) min pressure at the nodes, (b) max velocity in the pipes, (c) safety volume capacity for tanks, (d) safety volume capacity for the network as a whole, (e) pump operational capacity, (f) operational volume capacity for tanks, (g) filling capacity for tanks, (h) operational volume capacity for the network as a whole, (i) filling capacity for the network as a whole. Constraints: N/A. Decision variables: (1) Commercially available pipe diameters (integer), (2) cleaning/lining of existing pipes (binary: 0 = no action, 1 = cleaning/lining), (3) the number of new pumps (integer) with predefined operation curve, (4) volume of a new tank (integer, 0 = no tank), (5) min operational level of this tank (integer). Note: One MO model including both objectives.  Water quality: N/A. Network analysis: EPANET. Optimisation method: GA combined with fuzzy reasoning. 

32. Atiquzzaman et al. (2006) [64] MO Optimal WDS design using NSGAII.  Objective (1): Minimise (a) the design cost of the network (pipes). Objective (2): Minimise (a) the total pressure deficit at the network nodes. Constraints: (1) Pipe diameters limited to commercially available sizes, (2) min pressure at the nodes, (3) lower and upper limit of total pressure deficit, (4) lower and upper limit of total network cost. Decision variables: (1) Commercially available pipe diameters (integer). Note: One MO model including both objectives.  Water quality: N/A. Network analysis: EPANET. Optimisation method: NSGAII. 

33. Geem (2006) [105] SO Optimal WDS design and strengthening using harmony search (HS).  Objective (1): Minimise (a) the design cost of the network (pipes), (b) the penalty cost for violating the pressure constraint. Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: HS. 

34. Keedwell and Khu (2006) [66] MO Optimal WDS design using cellular automaton and genetic approach to multiobjective optimisation (CAMOGA) and NSGAII including an engineered initial population.  Objective (1): Minimise (a) the design cost of the network (pipes). Objective (2): Minimise (b) the total head deficit at the network nodes. Constraints: (1) Max total head deficit. Decision variables: (1) Pipe diameters (discrete). Note: One MO model including both objectives.  Water quality: N/A. Network analysis: EPANET. Optimisation method: CAMOGA and NSGAII are compared. 

35. Reca and Martínez (2006) [50] SO Optimal WDS and irrigation network design using GA.  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty for violating the pressure constraint. Constraints: (1) Min pressure at the nodes, (2) min/max flow velocities. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: Genetic algorithm pipe network optimisation model (GENOME) using GA. 

36. Samani and Mottaghi (2006) [51] SO Optimal WDS design, operation and maintenance using integer LP (ILP). Note: Discussion: [229]  Objective (1): Minimise (a) the capital cost of the network (pipes), (b) capital, operation and maintenance costs of pumps and reservoirs. Constraints: (1) Only one pipe diameter per network branch, (2) only one pump or reservoir per network location, (3) min/max pressure at the nodes, (4) min/max velocity in the pipes. Decision variables: (1) Integer variables related to pipe diameters and pumps/reservoirs.  Water quality: N/A. Network analysis: Unspecified hydraulic solver (a single loading condition). Optimisation method: Linear interactive discrete optimiser (LINDO) program using BB method. 

37. Suribabu and Neelakantan (2006) [106] SO Optimal WDS design using PSO.  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty for violating the pressure constraint. Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: PSONET program using PSO. 

38. Babayan et al. (2007) [89] MO Optimal robust WDS strengthening considering uncertainties in future demands and pipe roughnesses using NSGAII.  Objective (1): Minimise (a) the design cost of the network/rehabilitation. Objective (2): Maximise (a) the level of network robustness. Constraints: (1) Design/rehabilitation options are limited to the discrete set of available options. Decision variables: (1) Design/rehabilitation option index (discrete). Note: One MO model including both objectives.  Water quality: N/A. Network analysis: EPANET. Optimisation method: NSGAII. 

39. Lin et al. (2007) [166] SO Optimal WDS design and strengthening using scatter search (SS).  Objective (1): Minimise (a) the design cost of the network (pipes). Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: SS. 

40. Perelman and Ostfeld (2007) [61] SO Optimal WDS design, operation and maintenance using cross entropy (CE).  Objective (1): Minimise (a) (all test networks) the design cost of the network (pipes), (b) (test network (3) only) construction costs of pumps and tanks, (c) (test network (3) only) operation and maintenance costs of pumps. Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: CE for combinatorial optimisation [230]. 

41. Tospornsampan et al. (2007) [112] SO Optimal WDS design and strengthening with split pipes using SA.  Objective (1): Minimise (a) the design cost of the network (pipes). Constraints: (1) Min/max pressure at the nodes, (2) min/max diameter for the pipes, (3) min discharge for the pipes, (4) the total length of pipe segments equal to the length of the corresponding link, (5) nonnegativity for pipe segment lengths. Decision variables: (1) Two pipe diameters for each link (discrete), (2) pipe segment lengths (continuous) for the first diameter.  Water quality: N/A. Network analysis: Not specified. Optimisation method: SA. 

42. Zecchin et al. (2007) [156] SO Optimal WDS design and strengthening using ACO.  Objective (1): Minimise (a) the design cost of the network (pipes), (b) the penalty cost for violating the pressure constraint. Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete). Note: Formulated in [201].  Water quality: N/A. Network analysis: EPANET. Optimisation method: ACO (5 algorithms). 

43. Chu et al. (2008) [167] SO Optimal WDS strengthening using immune algorithm (IA).  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty for violating the pressure constraint. Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: Not specified. Optimisation method: IA and modified IA (mIA) are compared. 

44. Jin et al. (2008) [95] MO Optimal WDS rehabilitation and operation using NSGAII with artificial inducement mutation (AIM) to accelerate algorithm convergence.  Objective (1): Minimise (a) the rehabilitation cost of the network involving pipe replacement, (b) energy cost for pumping. Objective (2): Minimise (a) the sum of the velocity violations (shortfalls or excesses) weighted by the pipe flow. Objective (3): Minimise (a) the sum of pressure violations (excesses) weighted by the node demand. Constraints: (1) Pipe diameters limited to available standard diameter set. Decision variables: (1) Pipe diameters (real). Note: One MO model including all objectives.  Water quality: N/A. Network analysis: EPANET. Optimisation method: NSGAII with AIM. 

45. Kadu et al. (2008) [45] SO Optimal WDS design using GA with search space reduction.  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty for violating the pressure constraint. Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: GRANET, a hydraulic solver based on gradient method [215]. Optimisation method: GAWAT program using GA. 

46. Ostfeld and Tubaltzev (2008) [54] SO Optimal WDS design and operation considering multiple loading conditions using ACO.  Objective (1): Minimise (a) the pipe construction costs, (b) annual pump operation costs, (c) pump construction costs, (d) tank construction costs, (e) penalty function for violating pressure at nodes. Constraints: (1) Min/max pressure at consumer nodes, (2) max water withdrawals from sources, (3) tank volume deficit at the end of the simulation period. Decision variables: (1) Pipe diameters, (2) pump power at each time interval.  Water quality: N/A. Network analysis: EPANET (EPS). Optimisation method: ACO, compared to the previous study also using ACO [104]. 

47. Perelman et al. (2008) [62] MO Optimal WDS design, strengthening, operation and maintenance using CE.  Objective (1): Minimise (a) (both test networks) the design cost of the network (pipes), (b) (test network (2) only) construction costs of pumps and tanks, (c) (test network (2) only) operation and maintenance costs of pumps. Objective (2): Minimise (a) the maximum pressure deficit of the network demand nodes. Constraints: N/A. Decision variables: (1) Pipe diameters (discrete). Note: One MO model including both objectives.  Water quality: N/A. Network analysis: EPANET. Optimisation method: CE for combinatorial optimisation [230]. 

48. Van Dijk et al. (2008) [152] SO Optimal WDS design and strengthening using GA with an improved convergence.  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty for violating the pressure constraint. Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: Genetic algorithm network optimisation (GANEO) program using GA. 

49. Wu et al. (2008) [71] MO, SO Optimal WDS design and operation including greenhouse gas (GHG) emissions using multiobjective GA (MOGA).  Objective (1): Minimise (a) the capital cost of the network including pipes and pumps, (b) present value of pump replacement costs, (c) present value of pump operating costs (due to electricity consumption). Objective (2): Minimise GHG emissions including (a) capital GHG emissions (due to manufacturing), (b) present value of operating GHG emissions (due to electricity consumption). Constraints: (1) Min flowrate in pipes. Decision variables: (1) Pipe sizes (discrete), (2) pump sizes (discrete), (3) tank locations (discrete). Note: One MO model including both objectives, one SO model including objective (1).  Water quality: N/A. Network analysis: EPANET. Optimisation method: MOGA (based on NSGAII). 

50. Dandy et al. (2009) [127] SO Optimal expansion, strengthening and operation of wastewater, recycled and potable water systems for planning purposes using GA.  Objective (1): Minimise the total design cost of (a) wastewater, (b) recycled and potable networks. Constraints: Wastewater system: (1) Max surcharge in gravity sewers, (2) min/max velocity in rising mains, (3) treatment plant capacity. Potable/recycled systems: (4) Min pressure at the nodes. Decision variables: Wastewater system: Options for (1) trunk sewers upgrades, (2) new diversion sewers, (3) pump stations upgrades, (4) new pump stations, (5) new storage facilities, (6) new treatment plants. Potable/recycled systems: Options for (7) new/duplicate pipelines, (8) new/expanded pump stations, (9) new storages, (10) valve settings, (11) pump controls, (12) potable topups, (13) flowrates from sources.  Water quality: Not specified. Network analysis: Not specified. Optimisation method: GA. 

51. di Pierro et al. (2009) [67] MO Optimal WDS design using ParEGO and LEMMO with a limited number of function evaluations.  Objective (1): Minimise (a) the total cost of the network (pipes). Objective (2): Minimise (a) the head deficit. Constraints: (1) Min head at the nodes. Decision variables: (1) Pipe diameters (discrete). Note: One MO model including both objectives.  Water quality: N/A. Network analysis: EPANET (EPS for the test network (2)). Optimisation method: Hybrid algorithms ParEGO [192] and LEMMO [242]. 

52. Geem (2009) [160] SO Optimal WDS design and strengthening using particle swarm HS (PSHS).  Objective (1): Minimise (a) the design cost of the network (pipes). Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: PSHS. 

53. Giustolisi et al. (2009) [141] SO, MO Optimal robust WDS design considering uncertainties in demands and pipe roughness using optimised multiobjective GA (OPTIMOGA) with a twophase procedure.  Objective (1) (for a deterministic phase): Minimise (a) the design cost of the network (pipes), (b) pressure deficit at the critical node (i.e., the worstperforming node). Objective (2) (for a stochastic phase): Minimise (a) the design cost of the network (pipes). Objective (3) (for a stochastic phase): Maximise (a) the robustness of the network. Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete) (for both deterministic and stochastic problems), (2) future nodal demands (for stochastic problem only), (3) future pipe roughnesses (for stochastic problem only). Note: One SO model (i.e., deterministic) including objective (1); one MO model (i.e., stochastic) including objectives (2) and (3).  Water quality: N/A. Network analysis: Demanddriven analysis [11]. Optimisation method: OPTIMOGA [245]. 

54. Krapivka and Ostfeld (2009) [117] SO Optimal WDS design with split pipes using a combination of GA and LP (GALP) in a twophase procedure.  Objective (1): Minimise (a) the design cost of the network (pipes). Constraints: (1) Min pressure at the nodes, (2) sum of pipe segment lengths must be equal to the link length, (3) nonnegativity of segment lengths. Decision variables: (1) Lengths of pipe segments of known diameters (so called splitpipe decision variables). Note: Formulated in [116].  Water quality: N/A. Network analysis: Explicit mathematical formulation (steady state). Optimisation method: Combined GALP. 

55. Mohan and Babu (2009) [168] SO Optimal WDS design using heuristicbased algorithm (HBA).  Objective (1): Minimise (a) the design cost of the network (pipes). Constraints: (1) Min head at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: HBA. 

56. Mora et al. (2009) [158] SO Optimal WDS design using HS with optimised algorithm parameters.  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty for violating the pressure constraint. Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: Not specified. Optimisation method: HS. 

57. Rogers et al. (2009) [100] SO Optimal WDS expansion, operation and maintenance planning with reliability and water quality considerations over a planning horizon (i.e., 25 years) using GA.  Objective (1): Minimise the life cycle cost of the network including (a) capital costs, (b) energy costs, (c) operation costs, (d) maintenance costs, (e) penalty cost for violating constraints. Constraints: (1) Min pressure at the nodes, (2) min/max storage facility levels, (3) min/max watermain velocities. Decision variables: Options for (1) watermains (pipe sizing and routes), (2) new pump stations, (3) pump station expansions, (4) elevated storage facilities, (5) reservoir expansions, (6) control valves, (7) expansions at the two existing water purification plants (WPPs), (8) pressure zone configurations (pressure zone boundaries).  Water quality: Water age (as a surrogate measure for water quality). Network analysis: EPANET. Optimisation method: GANET using GA, and a heuristic solver for postprocessing. 

58. Tolson et al. (2009) [180] SO Optimal WDS design and strengthening using hybrid discrete dynamically dimensioned search (HDDDS).  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty for violating the pressure constraint. Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameter option numbers (integer).  Water quality: N/A. Network analysis: EPANET. Optimisation method: HDDDS. 

59. Banos et al. (2010) [107] SO Optimal WDS design using memetic algorithm (MA).  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty for violating the pressure constraint. Constraints: (1) Min pressure at the nodes, (2) min/max flow velocities. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: MA. 

60. Bolognesi et al. (2010) [169] SO Optimal WDS design and strengthening using genetic heritage evolution by stochastic transmission (GHEST).  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty for violating the head constraint. Constraints: (1) Min head at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: GHEST. 

61. Cisty (2010) [113] SO Optimal WDS design with split pipes using a combined GA and LP method (GALP) in a twophase procedure.  Objective (1): Minimise (a) the design cost of the network (pipes). Constraints: (1) The sum of the unknown lengths of the individual diameters in each section has to be equal to its total length, (2) total pressure losses in a hydraulic path between a pump/tank and every critical node should be equal to or less than the known value (based on the minimum pressure requirements), (3) the lengths are positive (and greater than a nominated minimum value). Decision variables: (1) Lengths of selected pipe diameters for each section.  Water quality: N/A. Network analysis: Explicit mathematical formulation, EPANET used only for the computation of friction headlosses. Optimisation method: GALP. 

62. Filion and Jung (2010) [142] SO Optimal WDS design including fire flow protection using PSO.  Objective (1): Minimise (a) the design cost of the network (pipes), (b) cost of potential economic damages by the fire (expected conditional fire damages). Constraints: (1) Max velocity in the pipes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: PSO. 

63. Mohan and Babu (2010) [170] SO Optimal WDS design using honey bee mating optimisation (HBMO).  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty for violating the head constraint. Constraints: (1) Min head at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: HBMO [249]. 

64. Prasad (2010) [98] SO Optimal WDS strengthening, expansion, rehabilitation and operation with a new approach for tank sizing considering multiple loading conditions using GA.  Objective (1): Minimise the capital cost of the network including (a) pipes, (b) pumps, (c) tanks, (d) present value of the energy cost. Constraints: (1) Min pressure at the nodes, (2) max velocity in the pipes, (3) volume of water pumped greater than or equal to the system daily demand, (4) tanks recover their levels by the end of the simulation period, (5) total tank inflows greater than or equal to total tank outflows, (6) bounds on decision variables. Decision variables: For pipes: (1) New/duplicate diameters (integer), (2) options for existing pipes (0 = no change, 1 = clean and line). For pumps: (3) the number of pumps (integer). For tanks: (4) Location (integer), (5) total volume (real), (6) min operational level (real), (7) ratio between diameter and height (real), (8) ratio between emergency volume and total volume (real).  Water quality: N/A. Network analysis: EPANET (EPS). Optimisation method: GA. 

65. Suribabu (2010) [171] SO Optimal WDS design, strengthening, expansion and rehabilitation using differential evolution (DE).  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty for violating the pressure constraint. Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete at the initialisation, converted to continuous in the DE process and back to discrete before the selection for the next generation), (2) pipe rehabilitation options.  Water quality: N/A. Network analysis: EPANET. Optimisation method: DE [250]. 

66. Wu et al. (2010) [77] MO, SO Optimal WDS design and operation including GHG emissions over a planning horizon (i.e., 100 years) using water system multiobjective GA (WSMGA).  Objective (1): Minimise (a) the capital cost of the network including pipes and pumps (i.e., purchase and installation of pipes and pumps, and construction of pump stations), (b) present value of pump replacement/refurbishment costs, (c) present value of pump operating costs (i.e., electricity consumption). Objective (2): Minimise GHG emission cost including (a) capital GHG emissions (i.e., manufacturing and installation of pipes), (b) present value of operating GHG emissions (i.e., electricity consumption). Constraints: (1) System must be able to deliver at least the average flow(s) on the peak day to the tank(s). Decision variables: (1) Pipe sizes (discrete), (2) pump sizes (discrete). Note: One MO model including both objectives; one SO model summing up objectives (1) and (2).  Water quality: N/A. Network analysis: Not specified. Optimisation method: WSMGA (used for both singleobjective and multiobjective problems, based on NSGAII). 

67. Wu et al. (2010) [72] MO Optimal WDS design and operation including GHG emissions over a planning horizon (i.e., 100 years) using WSMGA.  Objective (1): Minimise (a) the capital cost of the network including pipes and pumps (i.e., purchase and installation of pipes and pumps, and construction of pump stations), (b) present value of pump replacement/refurbishment costs, (c) present value of pump operating costs (i.e., electricity consumption). Objective (2): Minimise GHG emissions including (a) capital GHG emissions (i.e., manufacturing and installation of pipes), (b) present value of operating GHG emissions (i.e., electricity consumption). Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe sizes (discrete), (2) pump selection (discrete), (3) tank location selection (discrete). Note: One MO model including both objectives.  Water quality: N/A. Network analysis: EPANET. Optimisation method: WSMGA (based on NSGAII with several modifications). 

68. Geem and Cho (2011) [161] SO Optimal WDS design using parameter setting free HS (PSF HS).  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty cost for violating the pressure constraint. Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: PSF HS. 

69. Geem et al. (2011) [159] SO Optimal WDS design using HS.  Objective (1): Minimise (a) the design cost of the network (pipes). Constraints: (1) Min/max pressure at the nodes, (2) min/max velocity in the pipes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: HS. 

70. Goncalves et al. (2011) [52] SO Optimal WDS design and operation using a decompositionbased heuristic with a threephase procedure.  Objective (1): Minimise (a) the investment cost of pipes, (b) investment cost of pumps and the power cost, (c) energy cost of the system. Constraints: (1) Each hydrant visited by exactly one path, (2) each junction/withdrawal visited at the most by one path, (3) a single diameter selected for an arc, (4) one pressure class selected for an arc, (5) min/max velocity in arcs, (6) max pressure in arcs, (7) min pressure at the hydrants, (8) min/max height for a pump at the nodes, (9) min/max land area to irrigate downstream the arcs, (10) binary and nonnegativity constraints. Decision variables: (1) Arc included into the route (0 = no, 1 = yes), (2) diameter assigned to the arc (0 = no, 1 = yes), (3) pressure class assigned to the arc (0 = no, 1 = yes), (4) pump installed at the node (0 = no, 1 = yes), (5) pumping height of installed pumps, (6) water flow in arcs, (7) land area to irrigate downstream the arcs.  Water quality: N/A. Network analysis: Explicit mathematical formulation. Optimisation method: Steiner tree constructivebased heuristic followed by improved local search heuristic (first subproblem), simple calculation of flows and irrigated areas (second subproblem), CPLEX [207] (third subproblem). 

71. Haghighi et al. (2011) [178] SO Optimal WDS design using a combined GA and ILP method (GAILP) in a twophase procedure.  Objective (1): Minimise (a) the design cost of the network (pipes). Constraints: (1) Min/max pressure limits, (2) min/max velocity in the pipes, (3) only one diameter for each pipe can be assigned. Decision variables: (1) Zerounity variables related to the pipe diameters.  Water quality: N/A. Network analysis: EPANET. Optimisation method: GAILP. 

72. Qiao et al. (2011) [163] SO Optimal WDS design using improved PSO (IPSO).  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty cost for violating the pressure constraints. Constraints: (1) Min/(max) pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: Not specified. Optimisation method: IPSO. 

73. Wu et al. (2011) [73] MO Optimal WDS design and operation including GHG emissions over a planning horizon (i.e., 100 years), analysing sensitivity of tradeoffs between economic costs and GHG emissions, using WSMGA.  Objective (1): Minimise (a) the capital cost of the network including pipes and pumps (i.e., purchase and installation of pipes and pumps, and construction of pump stations), (b) present value of pump replacement/refurbishment costs, (c) present value of pump operating costs (i.e., electricity consumption). Objective (2): Minimise GHG emissions including (a) capital GHG emissions (i.e., manufacturing and installation of pipes), (b) present value of operating GHG emissions (i.e., electricity consumption). Constraints: (1) Min pressure at the nodes, (2) min flowrates within the system. Decision variables: (1) Pipe sizes (discrete). Note: One MO model including both objectives.  Water quality: N/A. Network analysis: EPANET. Optimisation method: WSMGA (based on NSGAII with several modifications). 

74. Zheng et al. (2011) [111] SO Optimal WDS design and strengthening using a combined NLP and DE method (NLPDE) in a threephase procedure.  Objective (1): Minimise (a) the design cost of the network (pipes). Constraints: (1) Min pressure at the nodes, (2) min/max diameter of pipes. Decision variables: (1) Pipe diameters (continuous for NLP, discrete for DE where continuous diameters are rounded to the nearest commercial pipe sizes after the mutation process).  Water quality: N/A. Network analysis: Explicit mathematical formulation for NLP, EPANET for DE. Optimisation method: NLPDE. 

75. Artina et al. (2012) [70] MO Optimal WDS design using parallel NSGAII.  Objective (1): Minimise (a) the design cost of the network (pipes). Objective (2): Minimise (a) the penalty cost for violating the pressure constraint. Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete). Note: One MO model including both objectives.  Water quality: N/A. Network analysis: EPANET. Optimisation method: Parallel NSGAII. 

76. Bragalli et al. (2012) [147] SO Optimal WDS design and strengthening using MINLP.  Objective (1): Minimise (a) the design cost of the network (pipes). Constraints: (1) Min/max pipe diameters/pipe cross sectional areas, (2) min/max hydraulic heads, (3) flow bounds. Decision variables: (1) Pipe flows, (2) pipe diameters/pipe cross sectional areas, (3) hydraulic heads at junctions.  Water quality: N/A. Network analysis: Explicit mathematical formulation. Optimisation method: BONMIN (an open source MINLP code) [255] using BB method. 

77. Kang and Lansey (2012) [26] SO Optimal WDS design and operation including the integrated transmissiondistribution network considering multiple loading conditions using GA with an engineered initial population.  Objective (1): Minimise (a) the pipe construction (the sum of the base installation cost, trenching and excavation, embedment, backfill and compaction costs, and valve, fitting, and hydrant cost), (b) pump construction cost, (c) pump operation cost (energy consumed by pumps), (d) penalty for violating the pressure constraint. Constraints: (1) Min pressure at the nodes for three demand loading conditions (average, instantaneous peak and fire flows). Decision variables: (1) Pipe sizes, pump station capacity including (2) pump sizes and (3) the number of pumps.  Water quality: N/A. Network analysis: EPANET. Optimisation method: GA (for optimisation), a new heuristic (for generating an engineered initial population to improve the GA convergence). 

78. Kanta et al. (2012) [92] MO Optimal WDS redesign/rehabilitation (pipe replacement) including fire damage and water quality objectives using nondominated sorting evolution strategy (NSES).  Objective (1): Minimise (a) the potential fire damage, calculated as lack of available fire flows at selected hydrant nodes taking into account the importance of a hydrant location. Objective (2): Minimise (a) the water quality deficiencies, represented by a performance function on chlorine residual at selected monitoring nodes reflecting governmental regulations for drinking water quality. Objective (3): Minimise (a) the system redesign cost, expressed as a ratio of actual redesign cost over maximum expected redesign cost. Constraints: (1) Min pressure at the hydrant nodes, (2) pipe diameters limited to commercially available sizes, (3) max number of pipe decision variables (i.e., pipes to be replaced). Decision variables: (1) Pipes selected for replacement (integer), (2) diameters of replaced pipes (integer). Note: One MO model including all objectives.  Water quality: Disinfectant (i.e., chlorine). Network analysis: EPANET (demanddriven analysis to calculate the fire flows, using a hydrant lifting technique to satisfy the pressure constraint). Optimisation method: NSES. 

79. McClymont et al. (2012) [194] SO Optimal WDS rehabilitation (pipe resizing) using ES with evolved mutation heuristics.  Objective (1): Minimise (a) the design cost of the network (pipes). Constraints: (1) Min/max pressure at the nodes, (2) max velocity in the pipes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: Not specified. Optimisation method: ES. 

80. Sedki and Ouazar (2012) [172] SO Optimal WDS design and strengthening using a combined PSO and DE method (PSODE).  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty cost for violating the pressure constraint. Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: PSODE. 

81. Wu et al. (2012) [74] MO Optimal WDS design, operation and maintenance including GHG emissions, incorporating variable speed pumps (VSPs) using MOGA.  Objective (1): Minimise the total economic cost of the system including (a) capital cost (i.e., purchase, installation and construction of network components), (b) present value of operating costs (i.e., electricity consumption due to pumping), (c) present value of maintenance and endoflife costs. Objective (2): Minimise the total GHG emissions of the system including (a) capital GHG emissions (i.e., manufacturing and installation of network components), (b) present value of operating GHG emissions (i.e., electricity consumption due to pumping), (c) present value of maintenance and endoflife emissions. Constraints: (1) Min flowrates within the system. Decision variables: (1) Pipe sizes (discrete). Note: One MO model including both objectives.  Water quality: N/A. Network analysis: EPANET. Optimisation method: MOGA. 

82. Fu et al. (2013) [97] MO Optimal WDS strengthening, expansion, rehabilitation and operation including multiple loading conditions and water quality objective applying manyobjective visual analytics using εNSGAII.  Objective (1): Minimise the capital cost for network expansion/rehabilitation including (a) pipes, (b) storage tanks, (c) pumps. Objective (2): Minimise (a) the operating cost of the system (i.e., energy cost for pump operation) during a design period. Objective (3): Minimise hydraulic failure of the system, expressed by the total system failure index (SFI) combining (a) nodal failure index and (b) tank failure index. Objective (4): Minimise (a) the fire flow deficit, representing the potential fire damage. Objective (5): Minimise (a) the total leakage of the system, considering background leakage from pipes only (calculated based on the pipe pressure). Objective (6): Minimise (a) the water age. Constraints: N/A. Decision variables: (1) Pipe diameters for new pipes (integer), (2) options for existing pipes including cleaning and lining or duplicating with a parallel pipe (integer), (3) tank locations (integer), (4) the number of pumps in operation during 24 hours (integer). Note: One MO model including all objectives.  Water quality: Water age (as a surrogate measure for water quality). Network analysis: Pressuredriven demand extension of EPANET (EPANETpdd) (EPS). Optimisation method: εNSGAII. 

83. Kang and Lansey (2013) [121] MO Scenariobased robust optimal planning of an integrated water and wastewater system considering demand uncertainties using NSGAII.  Objective (1): Minimise (a) the systems initial construction cost (pipes, pumps, tanks, wastewater plants), (b) expected operation and maintenance costs, (c) adaptive construction cost to expand the system if needed, (d) penalty cost for violating constraints. Objective (2): Minimise (a) the variability of actual costs across scenarios for the design solution, calculated as the standard deviation. Constraints: (1) Min pressure at the nodes, (2) min velocity in the sewer pipes, (3) max pump station capacities, (4) max storage tank sizes. Decision variables: (1) Pipe sizes (discrete), (2) pump station capacities (discrete), (3) wastewater treatment plant capacities (discrete). Note: One MO model including both objectives.  Water quality: N/A. Network analysis: Not specified. Optimisation method: NSGAII. 

84. McClymont et al. (2013) [144] MO Optimal WDS design and rehabilitation including the water discolouration risk using NSGAII and SPEA2 integrated with a new heuristic Markovchain hyperheuristic (MCHH).  Objective (1): Minimise (a) the cost of network infrastructure (pipes), (b) penalty for violating the pressure constraint, (c) penalty for violating the velocity constraint. Objective (2): Minimise (a) the water discolouration risk expressed as the sum of cumulative potential material after daily conditioning shear stress for all pipes in the network, (b) penalty for violating the pressure constraint, (c) penalty for violating the velocity constraint. Objective (3): Minimise (a) the sum of the cumulative head excess, (b) penalty for violating the pressure constraint, (c) penalty for violating the velocity constraint. Constraints: (1) Min head at the nodes, (2) max velocity in the pipes. Decision variables: (1) Pipe diameters. Note: One MO model including all objectives.  Water quality: Water discolouration. Network analysis: EPANET, discoloration propensity model (DPM). Optimisation method: NSGAII and SPEA2 integrated with MCHH. 

85. Zhang et al. (2013) [134] SO Optimal design, strengthening, expansion and operation of a reclaimed WDS considering demand uncertainty with the timestaged construction over a planning horizon (i.e., 20 years) using ILP.  Objective (1): Minimise (a) the cost of installing pipes, (b) cost of constructing pump stations, (c) pump energy cost of operating the system, at the stage one (time horizon 0–10 years), (d) expected cost of installing additional pipes, pumps and operating the system, at the stage two (time horizon 10–20 years). Constraints: (1) Min pressure at the nodes for peak demands, (2) min pressure at the nodes for average demands, (3) only one pipe size selected for each link, (4) only one pump size selected for average demands, (5) only one pump size selected for peak demands, (6) ensuring that the existing pump station is either expanded or a new one constructed at the stage two, (7) binary constraints. Decision variables (stage 1): (1) Pipe of size j installed in link i, (2) pump size p installed at station s for peak demands, (3) same as (2) for average demands. Decision variables (stage 2): (4) Additional pipe of size k installed for link i, (5) if no pump installed at stage 1, pump size p installed at station s for peak demands, (6) if pump installed at stage 1, additional pump of size p installed at station s for peak demands, (7) pump size p installed at station s for average demands. Note: All decision variables are binary (0 = no, 1 = yes).  Water quality: N/A. Network analysis: Explicit mathematical formulation. Optimisation method: GAMS CPLEX solver [207] using branch and cut method. 

86. Zheng et al. (2013) [46] SO Optimal design of a multisource WDS using network decomposition and DE in a twophase procedure.  Objective (1): Minimise (a) the design cost of the network (pipes). Constraints: (1) Min/max pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: DE (the modification based on the approach of [171] to manage a discrete problem). 

87. Zheng et al. (2013) [173] SO Optimal WDS design and strengthening using a selfadaptive DE method (SADE).  Objective (1): Minimise (a) the design cost of the network (pipes). Constraints: (1) Min head at the nodes. Decision variables: (1) Pipe diameters (integer, with continuous values created during the mutation process which are then truncated to the nearest integer size).  Water quality: N/A. Network analysis: EPANET. Optimisation method: SADE. 

88. Zheng et al. (2013) [149] SO Optimal WDS design and strengthening using noncrossover dither creeping mutationbased GA (CMBGA).  Objective (1): Minimise (a) the design cost of the network (pipes). Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: CMBGA. 

89. Aghdam et al. (2014) [164] SO Optimal WDS design and strengthening using accelerated momentum PSO (AMPSO).  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty for violating the pressure constraint. Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: AMPSO. 

90. Bi and Dandy (2014) [27] SO Optimal WDS design and strengthening including water quality considerations using online ANN and DE.  Objective (1): Minimise (a) the design cost of the network (pipes), (b) the net present value of chlorine cost over a planning horizon. Constraints: (1) Min head at the nodes, (2) min chlorine concentration at the nodes. Decision variables: (1) Pipe diameters (discrete), (2) chlorine dosage rates at the WTPs.  Water quality: Chlorine. Network analysis: Online ANN. Optimisation method: DE. 

91. Creaco et al. (2014) [118] MO Optimal WDS design, strengthening and expansion accounting for construction phasing in prefixed time intervals (i.e., 25 years) over a planning horizon (i.e., 100 years) using NSGAII.  Objective (1): Minimise (a) the total present worth construction cost of the network (pipes), calculated as the sum of the present worth costs of the n upgrades, (b) penalty for violating the pressure surplus constraint. Objective (2): Maximise (a) the network reliability, calculated as the minimum pressure surplus over the whole construction time. Constraints: (1) Pressure surplus bigger or equal to zero. Decision variables: (1) Pipe diameters (coded as integer numbers), with the genes consistently ordered (within each individual) according to the construction phases. Note: One MO model including both objectives.  Water quality: N/A. Network analysis: Demanddriven analysis [11]. Optimisation method: Modified NSGAII. 

92. Ezzeldin et al. (2014) [165] SO Optimal WDS design using integer discrete PSO (IDPSO).  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty cost for violating the pressure constraint. Constraints: (1) Min pressure at the nodes, (2) min/max pipe diameters. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: NewtonRaphson method [10]. Optimisation method: IDPSONET program using IDPSO. 

93. Johns et al. (2014) [155] SO Optimal WDS design, strengthening and operation using adaptive locally constrained GA (ALCOGA).  Objective (1): Minimise (a) (all test networks) the design cost of the network (pipes), (b) (test network (4) only) cost of tanks, (c) (test network (4) only) pump energy cost. Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete), (2) (test network (4) only) tank locations (binary), (3) (test network (4) only) the number of pumps in operation during 24 h at every 1h time step (binary).  Water quality: N/A. Network analysis: Not specified. Optimisation method: ALCOGA. 

94. McClymont et al. (2014) [68] MO Optimal WDS rehabilitation (pipe resizing) using ES with evolved mutation operators in a threephase procedure.  Objective (1): Minimise (a) the design cost of the network (pipes). Objective (2): Minimise (a) the total head deficit at the nodes. Constraints: N/A. Decision variables: (1) Pipe diameters (discrete). Note: One MO model including both objectives.  Water quality: N/A. Network analysis: Not specified. Optimisation method: ES. 

95. Roshani and Filion (2014) [124] MO Optimal WDS rehabilitation, strengthening, expansion and operation with asset management strategies over a planning horizon (i.e., 20 years) using NSGAII with eventbased coding.  Objective (1): Minimise the present value of the capital costs of the network including (a) pipe replacement, (b) pipe duplication, (c) pipe lining, (d) installation of new pipes. Objective (2): Minimise the present value of the operating costs including (a) lost water to leakage, (b) break repair, (c) electricity to pump water. Constraints: (1) Max yearly annual budget for the total of all costs (excluding leakage), (2) min pressure at the nodes, (3) max velocity in the pipes. Decision variables: (1) Time of rehabilitation, (2) place of rehabilitation, type of rehabilitation including (3) the diameter of a pipe being replaced/duplicated and (4) the diameter of a new pipe in an area slated for future growth, (5) the type of lining technology used. Note: One MO model including both objectives.  Water quality: N/A. Network analysis: EPANET. Optimisation method: NSGAII. 

96. Zheng et al. (2014) [179] SO Optimal WDS design and strengthening using a combined binary LP and DE method (BLPDE) in a threephase procedure.  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty cost for violating the nodal head requirement. Constraints: (1) Total head loss used by the pipes (from the source to a node) should be less than the value of the head at the source minus the head requirement at a node, (2) only one pipe diameter selected for each link. Decision variables: (1) Pipe diameters (binary for BLP, continuous for DE rounded to the nearest commercially available discrete diameters after the mutation process).  Water quality: N/A. Network analysis: EPANET Optimisation method: BLPDE. 

97. Basupi and Kapelan (2015) [135] MO Optimal flexible WDS strengthening, expansion, rehabilitation and operation considering demand uncertainty and optional intervention paths in prefixed time intervals (i.e., 25 years) over a planning horizon (i.e., 50 years) using NSGAII.  Objective (1): Minimise the total intervention cost including (a) capital cost of rehabilitation intervention, (b) pump energy consumption cost. Objective (2): Maximise (a) the endofplanning horizon system resilience, using a resilience index [266]. Constraints: (1) Min head requirement at the nodes. Decision variables: Intervention options (discrete) including (1) addition of new pipes, (2) duplication/cleaning/lining of existing pipes, (3) addition and (4) sizing of new tanks, (5) pump schedules, (6) threshold demands (discrete). Note: One MO model including both objectives.  Water quality: N/A. Network analysis: EPANET. Optimisation method: NSGAII. 

98. Bi et al. (2015) [108] SO Optimal WDS design using GA with an engineered initial population.  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty cost for violating the pressure constraints. Constraints: (1) Min/(max) pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: GA (for optimisation), a new heuristic called prescreened heuristic sampling method (PHSM) (for generating an engineered initial population to improve the GA convergence). 

99. Creaco et al. (2015) [136] MO Optimal WDS design, strengthening and expansion accounting for demand uncertainty and construction phasing in prefixed time intervals (i.e., 25 years) over a planning horizon (i.e., 100 years) using NSGAII.  Objective (1): Minimise (a) the total present worth construction cost of the network including (a) the cost of installing pipes at new sites, (b) cost of installing pipes in parallel to existing pipes. Objective (2): Maximise (a) the network reliability, calculated as the minimum pressure surplus over the whole construction time. Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (coded as integer numbers). Note: One MO model including both objectives.  Water quality: N/A. Network analysis: Demanddriven analysis [11]. Optimisation method: Modified NSGAII. 

100. Dziedzic and Karney (2015) [119] SO Optimal WDS design, strengthening and operation considering multiple loading conditions over a planning horizon (i.e., 20 years) using cost gradientbased heuristic method with computational time savings.  Objective (1): Minimise (a) the pump energy cost, (b) damage cost, (c) capital cost of the network (pipes). Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: Cost gradientbased heuristic method. 

101. Marques et al. (2015) [122] SO Optimal WDS design, strengthening, expansion and operation with a real options (ROs) concept and demand uncertainty, accounting for construction phasing in prefixed time intervals (10–20 years) over a planning horizon (60 years) using SA.  Objective (1): Minimise (a) the cost of the initial solution to be implemented in year zero (for interval 0–20 years) incl. pipes, pumps and pump energy costs, (b) cost of the future conditions incl. pipes, pumps and pump energy costs (cost of all scenarios weighted by the corresponding probability of each scenario), (c) regret term incl. pipes, pumps and pump energy costs (squared differences between the cost of the solution to implement and the optimal cost for each scenario). Constraints: (1) Min/max pressure at the nodes, (2) min pipe diameter, (3) only one commercial diameter assigned to a pipe. Decision variables: (1) Pipe diameters (discrete), (2) pump heads.  Water quality: N/A. Network analysis: EPANET. Optimisation method: SA. 

102. Marques et al. (2015) [137] MO Optimal WDS design, expansion and operation with a ROs concept and network expansion uncertainty, accounting for construction phasing in prefixed time intervals (20 years) over a planning horizon (60 years) using multiobjective SA.  Objective (1): Minimise (a) the cost of the initial solution to be implemented in year zero (for interval 0–20 years) incl. pipes, pumps, pump energy costs, carbon emissions cost for pipes and energy (b) cost of the future conditions incl. pipes, pumps, pump energy costs, carbon emissions cost for pipes and energy (cost of all scenarios weighted by the corresponding probability of each scenario). Objective (2): Minimise (a) total pressure violations for future scenarios (the sum of pressure violations for each scenario, each interval (starting from T = 2), each demand condition and each network node). Constraints: (1) Min pressure at the nodes, (2) min pipe diameter, (3) only one commercial diameter assigned to a pipe. Decision variables: (1) Pipe diameters (discrete). Note: One MO model including both objectives.  Water quality: N/A. Network analysis: EPANET. Optimisation method: Multiobjective SA. 

103. McClymont et al. (2015) [29] SO Optimal WDS design and operation, investigating linkages between algorithm search operators and the WDS design problem features, using elitist EA.  Objective (1): Minimise (a) the design cost of the network (pipes), (b) the energy cost of running pumps. Constraints: (1) Min/max pressure at the nodes, (2) max velocity in the pipes. Decision variables: (1) Pipe diameters (discrete), (2) pump statuses (binary).  Water quality: N/A. Network analysis: EPANET. Optimisation method: Elitist EA. 

104. Roshani and Filion (2015) [132] MO Optimal WDS rehabilitation, strengthening, expansion and operation with GHG emissions over a planning horizon (i.e., 20 years) using NSGAII with eventbased coding.  Objective (1): Minimise the present value of the capital costs of the network including (a) pipe replacement, (b) pipe duplication, (c) pipe lining, (d) installation of new pipes. Objective (2): Minimise the present value of the operating costs including (a) lost water to leakage, (b) break repair, (c) electricity to pump water, (d) carbon cost associated with electricity use. Constraints: (1) Min pressure at the nodes, (2) max velocity in the pipes. Decision variables: (1) Time of rehabilitation, (2) place of rehabilitation, type of rehabilitation including (3) the diameter of a pipe being replaced/duplicated and (4) the diameter of a new pipe in an area slated for future growth, (5) the type of lining technology used. Note: One MO model including both objectives.  Water quality: N/A. Network analysis: EPANET. Optimisation method: NSGAII. 

105. Sadollah et al. (2015) [174] SO Optimal WDS design and strengthening using improved mine blast algorithm (IMBA).  Objective (1): Minimise (a) the design cost of the network (pipes). Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: IMBA. 

106. Saldarriaga et al. (2015) [47] SO Optimal WDS design using optimal power use surface (OPUS) method paired with metaheuristic algorithms.  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty for violating the pressure constraint. Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: Not specified. Optimisation method: OPUS combined with: GA in REDES, GA in GANETXL, GA in MATLAB, HS in REDES, SA in MATLAB, greedy algorithm in REDES. 

107. Stokes et al. (2015) [76] MO Optimal WDS design and operation including GHG emissions over a planning horizon (i.e., 100 years), investigating the effect of changing tank reserve size (TRS), using Borg multiobjective EA (MOEA).  Objective (1): Minimise (a) the construction costs of the network (pipes, pumps, tanks), (b) operating costs (electricity consumed by pumps). Objective (2): Minimise GHG emissions associated with the system (a) construction, (b) operation (electricity consumed by pumps). Constraints: (1) Min pressure at the nodes, (2) the total volume pumped equal to or greater than the total demand during the EPS. Decision variables: (1) Pipe diameters (discrete), (2) pump types (discrete), (3) pump scheduling decision variable (continuous). Note: One MO model including both objectives. For the test network (1), both design and operation components are included; for the test network (2) (Dtown), only operation components are included.  Water quality: N/A. Network analysis: EPANET (EPS). Optimisation method: Borg MOEA [273]. 

108. Stokes et al. (2015) [75] MO Optimal WDS design and operation including GHG emissions considering varying emission factors, electricity tariffs and water demands using NSGAII.  Objective (1): Minimise (a) the design costs of the network (pipes and pumps), (b) operating costs (electricity consumed by pumps). Objective (2): Minimise GHG emissions associated with the system (a) design (pipes), (b) operation (electricity consumed by pumps). Constraints: (1) Min pressure at the nodes, (2) the sum of the instantaneous pump supply equal to or greater than the sum of the instantaneous water demands. Decision variables: (1) Pipe diameters (discrete), (2) pump types (discrete), (3) pump schedules (discrete options representing the time at which a pump is turned on/off, using a time step of 30 minutes). Note: One MO model including both objectives.  Water quality: N/A. Network analysis: EPANET (EPS). Optimisation method: NSGAII. 

109. Wang et al. (2015) [195] MO Optimal WDS design, strengthening and rehabilitation of wellknown benchmark problems with the aim to obtain the bestknown approximation of the true Pareto front using various MOEAs.  Objective (1): Minimise (a) the design costs of the network (pipes). Objective (2): Maximise (a) the network resilience [275]. Constraints: (1) Min/max pressure at the nodes (max pressure only for some test networks), (2) max velocity in the pipes (only for some test networks). Decision variables: (1) Diameters of new or duplicate pipes (integer) (duplicate pipes only for some test networks), (2) cleaning of existing pipes or donothing option (integer) (only for some test networks). Note: One MO model including both objectives.  Water quality: N/A. Network analysis: EPANET. Optimisation method: Five stateoftheart MOEAs are used including AMALGAM [276], Borg [273], NSGAII [258], εMOEA [277], εNSGAII [278]. 

110. Zheng (2015) [196] SO Optimal WDS design and strengthening using four DE variants with a comparison of their searching behaviour.  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty cost for violating the pressure constraint. Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete, with continuous values adjusted to the closest discrete sizes according to [111]).  Water quality: N/A. Network analysis: EPANET. Optimisation method: DE (4 variants). 

111. Zheng et al. (2015) [69] MO Optimal WDS design considering multiple loading conditions using multiobjective DE algorithm (MODE) with a graph decomposition technique.  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty for violating the pressure head constraint. Objective (2): Maximise (a) the minimum head excess across the network of multiple demand loading cases, (b) penalty for violating the pressure head constraint. Constraints: (1) Min/max allowable pipe diameters. Decision variables: (1) Pipe diameters (discrete). Note: One MO model including both objectives.  Water quality: N/A. Network analysis: EPANET (EPS for the second objective). Optimisation method: MODE. 

112. Zheng et al. (2015) [48] SO Optimal WDS design using DE, analysing impact of algorithm parameters on its search behaviour.  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty cost for violating the pressure constraint. Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete, with continuous values produced in the initialisation and mutation processes of DE converted to the nearest discrete pipe diameters).  Water quality: N/A. Network analysis: EPANET. Optimisation method: DE. 

113. Zhou et al. (2015) [175] SO Optimal WDS design and strengthening using discrete state transition algorithm (STA).  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty for violating the pressure constraint. Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: NewtonRaphson method [10]. Optimisation method: Discrete STA [282]. Note: Continuous STA [283]. 

114. Andrade et al. (2016) [143] SO Optimal WDS design with improved offline ANNs to replace water quality simulations and the probabilistic approach to generate training data sets, using GA.  Objective (1): Minimise (a) the system cost of the network (the pipe and installation costs). Constraints: (1) Min pressure at the nodes, (2) min chlorine concentration at the nodes. Decision variables: (1) Pipe diameters (discrete), (2) chlorine dosages at the water source (discrete).  Water quality: Chlorine. Network analysis: EPANET, offline ANN (for water quality analyses). Optimisation method: GA. 

115. Jabbary et al. (2016) [181] SO Optimal WDS design using a modified central force optimisation algorithm (CFOnet).  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty cost of violating the pressure constraint, (c) penalty cost of violating the velocity constraint. Constraints: (1) Min/max commercial pipe diameters, (2) min/max velocity in the pipes, (3) min/max pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: CFOnet. 

116. Schwartz et al. (2016) [123] SO Optimal robust WDS design and operation considering multiple loading conditions and demand uncertainty using the robust counterpart (RC) approach and CE.  Objective (1): Minimise the construction and operation costs of the network including (a) pipe capital costs, (b) tank capital costs, (c) pump station capital cost, (d) energy costs related to the operation of the system during a TC of operation. Constraints: (1) Min/max tank water volumes at the last time period of the cycle, (2) min desired nodal heads, (3) tank closure constraints defined by the difference between the tank water level at the start and end of the TC. Decision variables: (1) Pipe diameters (discrete), (2) pump station heads at all time periods reflecting the pump curve needed for the system.  Water quality: N/A. Network analysis: Explicit mathematical formulation (nonlinear equations are linearised). Optimisation method: CE for combinatorial optimisation [230,285]. 

117. Sheikholeslami and Talatahari (2016) [150] SO Optimal WDS design using a newly developed swarmbased optimisation(DSO) algorithm.  Objective (1): Minimise (a) the design cost of the network (pipes). Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: DSO algorithm. 

118. Sheikholeslami et al. (2016) [162] SO Optimal WDS design using a combined cuckooHS algorithm (CSHS) in a twophase procedure.  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty for violating the pressure constraint. Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: CSHS algorithm. 

119. Zheng et al. (2016) [28] MO Optimal WDS design and strengthening, analysis and comparison of the searching behaviour of NSGAII, selfadaptive multiobjective DE (SAMODE) and Borg.  Objective (1): Minimise (a) the total network cost, including pipe material and construction costs. Objective (2): Maximise (a) the network resilience. Constraints: (1) Min/max pressure at the nodes, (2) min/max velocity in the pipes. Decision variables: (1) Pipe diameters (discrete). Note: One MO model including both objectives.  Water quality: N/A. Network analysis: EPANET. Optimisation method: NSGAII, SAMODE, and Borg are compared. 

120. AvilaMelgar et al. (2017) [109] SO Optimal WDS design using EA in a grid computing environment.  Objective (1): Minimise (a) the design cost of the network (pipes). Constraints: (1) Min/max pressure at the nodes, (2) min/max velocity in the pipes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: EA. 

121. Cisty et al. (2017) [30] MO Optimal WDS design using NSGAII with a twophase procedure and search space reduction.  Objective (1): Minimise (a) the design cost of the network (pipes). Objective (2): Minimise (a) the total head deficit in the network. Constraints: N/A. Decision variables: (1) Pipe diameters (discrete). Note: One MO model including both objectives.  Water quality: N/A Network analysis: Not specified. Optimisation method: NSGAII (for both phases of the optimisation procedure). 

122. Muhammed et al. (2017) [90] MO Optimal WDS strengthening using a clusterbased technique and NSGAII in a twophase procedure.  Objective (1): Minimise (a) the total capital cost of duplicated pipes. Objective (2): Minimise (a) the total number of demand nodes with pressure below the minimum pressure requirement. Constraints: (1) The sum of the pressure deficiencies in all the nodes with negative pressure. Decision variables: (1) Pipe diameters (discrete). Note: One MO model including both objectives.  Water quality: N/A. Network analysis: EPANET. Optimisation method: GANETXL [208] using NSGAII. 

123. Shokoohi et al. (2017) [78] MO Optimal WDS design including water quality objective using ACO.  Objective (1): Minimise (a) the construction cost of the network (pipe cost, excavation, demolition etc.), (b) chlorine cost calculated as oneyear chlorine usage (applied in the tanks). Objective (2A): Maximise (a) water quality reliability based on chlorine residual [145]. Objective (2B): Maximise (a) water quality reliability based on water age. Objective (2C): Maximise (a) combined water quality reliability based on both chlorine residual and water age. Constraints: (1) Min/max pressure at the nodes, (2) max velocity in the pipes. Decision variables: (1) Pipe diameters (discrete), (2) tank heads (discrete), (3) chlorine injection dosages in the tanks (discrete). Note: Three twoobjective optimisation models, where the objective (1) is combined with either objective (2A), (2B) or (2C).  Water quality: Chlorine, water age. Network analysis: EPANET (EPS). Optimisation method: ACO. 

124. Zheng et al. (2017) [176] SO Optimal WDS design and strengthening using convergencetrajectory controlled ACO (ACO_{CTC}) algorithm with parameteradaptive strategy.  Objective (1): Minimise (a) the design cost of the network (pipes), (b) penalty for violating the pressure constraint. Constraints: (1) Min pressure at the nodes. Decision variables: (1) Pipe diameters (discrete).  Water quality: N/A. Network analysis: EPANET. Optimisation method: ACO_{CTC}. 

Notes: * SO = Singleobjective (approach/model), MO = Multiobjective (approach/model). ^{+} Objective function is referred to as ‘objective’ in the column below due to space savings. ** Conservation of mass of flow, conservation of energy, and conservation of mass of constituent (for water quality network analysis) are not listed. ^{++} Control variables are listed, state variables resulting from network hydraulics are not necessarily listed. ^{?} D = Design. ^{??} OP = Operation.
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Figure 1.
Papers (from Appendix A Table A1) by year and optimisation approach.
Figure 2.
Application areas (of papers from Appendix A Table A1).
Figure 3.
Optimisation models (of papers from Appendix A Table A1) by: (a) number of objectives, (b) number of constraints, (c) number of types of a decision variable, in an optimisation model.
Figure 4.
Optimisation methods (of papers from Appendix A Table A1) by year.
Figure 5.
Test networks (of papers from Appendix A Table A1) by network size.
Objective Type  Objectives  Reference (An Example) 

Economic  Capital costs of the system, including purchase, installation and construction of network components (pipes, pumps, tanks, treatment plants, valves, etc.)  [53,74,121] 
Rehabilitation costs of the system, including pipe/pump replacement, pipe cleaning/lining, pipe break repair  [17,124] (pipes), [77] (pumps)  
Expected operation costs of the system, including pump stations, treatment plants and disinfectant dosage  [53] (pump stations and treatment plants), [27] (disinfectant dosage)  
Expected maintenance costs of the system  [121]  
Community  Benefit/benefit of the solution (i.e., rehabilitation, expansion and strengthening) using various performance criteria by authors  [131] (welfare index to place greater importance on early improvements), [99] (quantity shortfalls as criteria), [93] (e.g., safety volumes and operational capacities as criteria) 
Water quality (e.g., disinfectant, sedimentation, discolouration) deficiencies or water age at customer demand nodes, water discolouration risk, velocity violations (causing sedimentation/discolouration)  [92,120] (water quality deficiencies), [97] (water age), [144] (water discolouration), [95] (velocity violations)  
Pressure deficit at customer demand nodes (maximum individual or total), or the number of demand nodes with the pressure deficit  [65] (maximum individual deficit), [66,68] (total deficit), [90] (the number of demand nodes)  
Hydraulic failure of the system expressed by the failure index  [97]  
Potential fire damages using either expected conditional fire damages or fire flow deficit  [142] (expected conditional fire damages), [92] (fire flow deficit)  
Performance  System robustness using either a redundant design approach, integration approach (via objective function or constraints), twophase optimisation approach, scenariobased approach or RC approach  [140] (redundant design), [89] (integration via objective function), [110,140] (integration via constraints), [141] (twophase optimisation), [121] (scenariobased), [123] (RC) 
System reliability  [118]  
System resilience  [135]  
Environmental  GHG emissions or emission costs consisting of capital emissions (due to manufacturing and installation of network components) and operating emissions (due to electricity consumption)  [77] (capital and operating GHG emission costs), [73,75] (capital and operating GHG emissions), [132] (operating GHG emission cost) 
Constraints  Reference (An Example) 

Hydraulic constraints given by physical laws of fluid flow in a pipe network: (i) conservation of mass of flow, (ii) conservation of energy, (iii) conservation of mass of constituent  [41] 
System constraints given by limitations and operational requirements of a WDS, for example, minimum/maximum pressure at (demand) nodes and flow velocity in pipes, water deficit/surplus at storage tanks at the end of the simulation period, maximum water withdrawals from sources  [54] (limits on nodal pressure, storage tank deficit and water withdrawals from sources), [127] (limits on pipe velocity) 
Constraints on decision variables $x$, for example, limits on pipe diameters, pipe segment lengths (so called splitpipe design), pump station capacities  [92] (limits on pipe diameters), [117] (limits on pipe segments), [121] (limits on pump stations) 
Decision Variables  Reference (An Example) 

Pipes: pipe diameters/sizes, pipe duplications, pipe rehabilitation options (pipe replacement, pipe cleaning/lining), pipe break repair, pipe segment lengths (so called splitpipe design), future pipe roughnesses, pipe routes, pipe closures/openings (to adjust a pressure zone boundary)  [75] (diameters), [132] (duplications, replacement, lining and break repair), [117] (segments), [141] (roughnesses), [52] (routes), [100] (routes and closures/openings) 
Pumps: pump locations, pump sizes (pump capacities, pump types, pumping power, pump head/height or headflow), the number of pumps, pump schedules (pumping power or pump head at each time step, the number of pumps in operation during 24 h, binary statuses at time steps, on/off times)  [52,99] (locations), [14] (locations and capacities), [75] (types), [17] (power), [52,122] (head/height), [93] (headflow), [26] (sizes and the number of pumps), [53,123] (power or head at each time step), [97] (the number of pumps in operation), [29] (binary statuses), [75] (on/off times) 
Tanks: tank locations, tank sizes/volumes, minimum operational level, ratio between diameter and height, ratio between emergency volume and total volume, tank heads  [98] (locations, sizes/volumes, minimum operational level, ratios), [78] (heads) 
Valves: valve locations, valve settings (headlosses or flows)  [14] (locations), [16] (headlosses via a roughness coefficient), [127] (headlosses and flows) 
Nodes: flowrates from sources, future nodal demands, threshold demands, hydraulic heads at junctions  [127] (flowrates), [135,141] (demands), [147] (heads) 
Water quality: disinfectant dosage rates (at the sources, at the treatment plants, in the tanks), treatment removal ratios, treatment plant capacities  [143] (dosage at the sources), [27] (dosage at the treatment plants), [78] (dosage in the tanks), [53] (removal ratios), [121] (capacities) 
Timing: year of action (e.g., network expansion, rehabilitation, pipe replacements) execution  [131] (network expansion and rehabilitation), [130] (pipe replacements) 
Test Network Name  No. of Nodes  Network Description  Optimisation Problem  Network Modified Versions  Network Usage Count * 

Hanoi network ^{++} [49]  32  Network organised in three loops supplied by gravity from a single source  New system design (pipes)  Double Hanoi network, triple Hanoi network (both [113])  55 
New York City tunnels ^{++} [81]  20  Tunnel system supplied by gravity from a single source, constituting the primary WDS of the New York city  Existing system strengthening (i.e., pipe paralleling) to meet projected demands  Double New York City tunnels [201]  42 
Twoloop network ^{++} [14]  7  Small network with two loops supplied by gravity from a single source  New system design (pipes)  Adapted to system strengthening and expansion over a planning horizon [118]  40 
Balerma irrigation network ^{++} [50]  447  Large looped network supplied by gravity from four sources, adapted from the existing irrigation network in Balerma, Spain  New system design (pipes)  N/A  20 
Anytown network [84]  19  Hypothetical looped system supplied by three parallel pumps from a single source  Existing system strengthening, expansion and rehabilitation (pipes, pumps, tanks) to meet projected demands  ** With additional source and tank [53], with additional tank [119] proposed by [83]  15 
Notes: * The count of the network usage as in the papers listed in Appendix A Table A1 inclusive of networks’ modified versions. ^{++} Detailed overview of results to a leastcost design problem obtained by various authors is presented in [39]. ** Other modified versions of the Anytown network exist.
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