Lost in Optimisation of Water Distribution Systems? A Literature Review of System Design
Abstract
:1. Introduction
2. Aim, Scope and Structure of the Paper
3. Design Problems
3.1. Application Areas
3.1.1. New Systems: Design
3.1.2. Existing Systems
Strengthening
Rehabilitation
Expansion
3.1.3. Problem Elements
Pipes
Pumps
Tanks
Valves
3.2. Time, Uncertainty and Performance Considerations
3.2.1. Staged Design
3.2.2. Flexible Design
3.2.3. Resilient, Reliable and Robust Design
3.2.4. Design for Water Quality
4. General Classification of Reviewed Publications
4.1. Application Area
 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.
4.2. Optimisation Model
 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.
4.2.1. General Optimisation Model
Objectives
Constraints
Decision Variables
4.3. Solution Methodology
4.3.1. Computational Efficiency
4.3.2. Recent Developments
4.4. Test Network
5. Future Research
6. Summary and Conclusion
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
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}. 

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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 
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MalaJetmarova, H.; Sultanova, N.; Savic, D. Lost in Optimisation of Water Distribution Systems? A Literature Review of System Design. Water 2018, 10, 307. https://doi.org/10.3390/w10030307
MalaJetmarova H, Sultanova N, Savic D. Lost in Optimisation of Water Distribution Systems? A Literature Review of System Design. Water. 2018; 10(3):307. https://doi.org/10.3390/w10030307
Chicago/Turabian StyleMalaJetmarova, Helena, Nargiz Sultanova, and Dragan Savic. 2018. "Lost in Optimisation of Water Distribution Systems? A Literature Review of System Design" Water 10, no. 3: 307. https://doi.org/10.3390/w10030307