Self-adaptive multi-objective harmony search for

Engineering Optimization

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Self-adaptive multi-objective harmony search for optimal design of water distribution networks Young Hwan Choi, Ho Min Lee, Do Guen Yoo & Joong Hoon Kim To cite this article: Young Hwan Choi, Ho Min Lee, Do Guen Yoo & Joong Hoon Kim (2017) Self-adaptive multi-objective harmony search for optimal design of water distribution networks, Engineering Optimization, 49:11, 1957-1977, DOI: 10.1080/0305215X.2016.1273910 To link to this article: http://dx.doi.org/10.1080/0305215X.2016.1273910

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Date: 03 September 2017, At: 18:09

ENGINEERING OPTIMIZATION, 2017 VOL. 49, NO. 11, 1957–1977 http://dx.doi.org/10.1080/0305215X.2016.1273910

Self-adaptive multi-objective harmony search for optimal design of water distribution networks Young Hwan Choia , Ho Min Leea , Do Guen Yoob and Joong Hoon Kim

a

a Department of Civil, Environmental and Architectural Engineering, Korea University, Seoul, Republic of Korea;

Downloaded by [Korea University] at 18:09 03 September 2017

b K-water Research Institute/Software Development & Engineering Department, Sofware Center (Concurrent Position), Korea Water Resources Corporation (K-water), Daejeon, Republic of Korea

ABSTRACT

ARTICLE HISTORY

In multi-objective optimization computing, it is important to assign suitable parameters to each optimization problem to obtain better solutions. In this study, a self-adaptive multi-objective harmony search (SaMOHS) algorithm is developed to apply the parameter-setting-free technique, which is an example of a self-adaptive methodology. The SaMOHS algorithm attempts to remove some of the inconvenience from parameter setting and selects the most adaptive parameters during the iterative solution search process. To verify the proposed algorithm, an optimal least cost water distribution network design problem is applied to three different target networks. The results are compared with other well-known algorithms such as multi-objective harmony search and the non-dominated sorting genetic algorithm-II. The efficiency of the proposed algorithm is quantified by suitable performance indices. The results indicate that SaMOHS can be efficiently applied to the search for Pareto-optimal solutions in a multiobjective solution space.

Received 25 November 2015 Accepted 7 December 2016 KEYWORDS

Optimal design of water distribution network; multi-objective harmony search; self-adaptive; parameter-setting-free

1. Introduction Many real-world problems can be formulated as optimization problems with multiple objectives. Since the first attempt to apply multi-objective optimization concepts using the genetic algorithm (Schaffer 1985), multi-objective optimization algorithms (MOOAs) have been researched deeply, and are widely applied to various engineering applications. In particular, for water distribution network (WDN) design, Gessler and Walski (1985) first suggested the application of a multi-objective optimal design of the WDN optimization model, called water distribution and system optimization (WADISO). Halhal et al. (1999) adopted multi-objective optimal design to solve the maintenance problem of WDN, in which the maximization of pressure head allowance at each node and minimization of cost were considered. Keedwell and Khu (2003) applied a hybrid evolutionary algorithm to the multi-objective optimal design of WDN. Prasad Nepal and Park (2004) solved the reliabilitymaximization and cost-minimization problems using a non-dominated sorting genetic algorithm (NSGA). Ostfeld, Oliker, and Salomons (2013) solved the multi-objective optimal design problem of WDNs using the split pipe method, where the minimal cost and maximal reliability are considered simultaneously. Likewise, MOOAs have the benefit of being able to handle various objectives that have a trade-off relationship, with each objective simultaneously satisfying its own constraints.

CONTACT Joong Hoon Kim

[email protected]

© 2017 Informa UK Limited, trading as Taylor & Francis Group


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However, in the case of metaheuristic algorithms, the importance of parameter settings is increasing because the settings for each optimization algorithm have a strong influence on the optimal solution. For this reason, some researchers have investigated self-adaptive methods to solve the challenge of parameter setting in metaheuristic algorithms. Geem and Sim (2010) proposed the parameter-setting-free (PSF) method, which is an example of a parameter self-adaptive technique. This enables the parameters [i.e. the harmony memory considering rate (HMCR) and pitch adjustment rate (PAR)] to be automatically set during the simulation. Mahdavi et al. (2007) proposed an improved harmony search (IHS) algorithm, which is the first variant since the advent of harmony search (HS). This proposed algorithm has an adjustable PAR parameter setting and dynamic bandwidth. It is obviously different from the basic HS, where the PAR parameter and bandwidth have fixed values. The PAR value is updated and increases linearly in each iteration, and the bandwidth value exponentially decreases. However, IHS is limited in that it only improves the local search ability. To overcome this limitation, the global-best harmony search (GHS) algorithm was developed by Omran and Mahdavi (2008). They introduced a new improvization mechanism by borrowing the concept of particle swarm optimization (PSO) and added a new operator, in which each particle represents a candidate solution to the optimization problem. Pan et al. (2012) introduced the self-adaptive global-best harmony search (SGHS). This method uses a self-adaptive parameter tuning mechanism that modifies the parameter setting according to self-consciousness and initializes harmony memory (HM) based on a low-discrepancy sequence. Chang-ming and Lin (2014) discussed the PAR parameter setting, suggesting three types of PAR strategy: convex differential growth, convex differential changes and concave differential growth. These strategies could boost the global search and be applied to different types of problem because each problem has a different PAR ratio. Although recent studies emphasize finding optimal parameters, these studies are limited in that they do not apply the multi-objective concept in WDN. Similarly to HS, various optimization algorithms have been developed to improve optimal design performance, applying self-adaptive techniques and automatic parameter tuning. Abbass, Sarker, and Newton (2001) applied the parameter self-adaptive algorithm to the Pareto differential evolution (PDE) algorithm to develop the self-adaptive PDE method. This allows the inadequate range of the crossover rate and mutation rate to be automatically set. Huang et al. (2009) applied PDE to multi-objective optimization. Through such applications, the self-adaptive algorithm has been widely used in many optimization studies on parameter setting and constraint adjustment (Deb 2001; Chen et al. 2006; Tessema and Yen 2006). However, the majority of previous studies related to self-adaptive methods consider a single objective function and apply either a genetic algorithm, or differential evolution or PSO. Therefore, in this study, the self-adaptive multi-objective harmony search (SaMOHS) algorithm is proposed. This algorithm uses not only the multi-objective harmony search (MOHS) algorithm, but also a self-adaptive method to solve multi-objective WDN design problems without requiring the burdensome tuning of adaptable parameters. To verify SaMOHS, this method is applied to well-known benchmark networks in WDN design. The results are compared with those obtained using MOHS and NSGA-II.

2. Self-adaptive multi-objective harmony search SaMOHS is an integrated optimization model that employs the HS algorithm (Geem, Kim, and Loganathan 2001; Kim, Geem, and Kim 2001), the PSF method, which is an example of a self-adaptive method that adopts non-dominated sorting (Fonseca and Fleming 1993), and the crowding-distance concept (Deb et al. 2000) to solve multi-objective problems. SaMOHS is defined as an optimization algorithm that allows the parameters of the optimization algorithm to be set up automatically and various objective functions to be considered simultaneously. This provides protection against mistakes when setting parameters and provides a reliable solution for decision makers by considering various objective functions. The pseudo-code describing the SaMOHS optimization process is provided in Table 1.

ENGINEERING OPTIMIZATION

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Table 1. Pseudo-code for the self-adaptive multi-objective harmony search (SaMOHS). Input: HMCRi , PARi , HMS Generate initial HM, OTM randomly Calculating fitness while Stopping criterion is not satisfied do if (rand < HMCRi ) Choosing an existing harmony randomly if (rand < PARi ) Adjusting the pitch randomly within the limits end if else Generate new harmonies via randomization end if Calculating the Pareto ranking of S considering HM Applying the crowding-distance method if New HM has a better diversity than the previous solution in HM then Updating HMend if Calculating HMCRi , PARi end whileOutput: Pareto optimal solutions from HM Note: HMCRi = harmony memory considering rate of the ith decision variable; PARi = pitch adjustment rate of the ith decision variable; HMS = harmony memory size; HM = harmony memory; OTM = operation type memory.

2.1. Step 1: Initializing harmony memory/operation type memory Initially, the HM is generated from random values in the search space. Operation type memory (OTM) is constituted depending on the HM generation type. The equations for HM and OTM are presented in Equation (1): ⎡ ⎢ ⎢ HM = ⎢ ⎣

x11

x21

···

x12 .. .

x22

···

x1HMS

x2HMS

xn1

⎤

⎡

⎢ xn2 ⎥ ⎢ ⎥ ⎥ , OTM = ⎢ ⎣ ⎦

··· · · · xnHMS

y11

y21

···

y12 .. .

y22

···

y1HMS

y2HMS

yn1

⎤

yn2 ⎥ ⎥ ⎥ ⎦

··· · · · ynHMS

(1)

where xn denotes a decision variable and yn denotes a generation method for the decision variable. Each yn represents one of three cases: (i) random selection, (ii) harmony memory consideration, and (iii) pitch adjustment. These three cases are chosen using the values of HMCR and PAR. If a uniformly generated value between 0 and 1 is greater than the HMCR, then the HS randomly chooses a new decision variable within the possible range. An HMCR of 0.95 means that the algorithm chooses a decision variable value considering the HM with a 95% probability. Following the same process, the PAR is compared with a uniformly generated random value. If the PAR is smaller, each variable is adjusted to its neighbouring values within a range of possible values. For example, when designing a WDN, the pipe diameter could be represented by the decision variable xn . The HM is then configured by selecting a suitable diameter from the range of commercial pipe diameters (i.e. 300, 450, 500, 600, 800 or 1000 mm). 2.2. Step 2: Calculating fitness If the optimization problem is given by Minimize f1 (x), Maximize f2 (x) Subject to xi ∈ Xi , i = 1, 2, . . . , N

(2)

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Y. H. CHOI ET AL.

then the objective functions are as follows: ⎡

f1 (x1 )

⎤

⎡

f1 (x1 ) f1 (x2 ) .. .

⎤

⎥ ⎢ f1 (x2 ) ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ , Maximize F Minimize F1 = ⎢ = ⎥ ⎥ ⎢ 2 .. ⎦ ⎦ ⎣ ⎣ . HMS ) HMS (x f ) f1 (x 1

(3)


2.3. Step 3: Generating new harmony memory New HM elements can be generated through random selection, HM consideration or pitch adjustment. Random selection is used to generate HM elements using a randomly selected value in the allowed HM range (xi Lower , xi Upper ). Because, in this study, the set of commercial pipe diameters is used as the search space, a diameter is randomly selected from this set. HM consideration is a method that considers the existing HM to generate new memory. When generating a new element of memory (x1 New ), it is selected from a specific HM [x1 1 , x1 HMS ] as follows: xiNew

=

Upper xi ∈ [xiLower , xi ]

(i) 1 − HMCR

xi ∈ HM = [xi1 , xi2 , . . . ,

xiHMS ]

(ii) HMCR

(4)

Pitch adjustment is a method that generates a new memory element by adjusting the element according to Equation. The new memory element generated by considering HM is xiNew =

xiNew + bw

(i) PAR

xiNew

(ii) 1 − PAR

(5)

where new values are calculated for HMCR and PAR at every iteration. Because the solutions are generated by random selection with a high rate during the early iterations (1, 2, . . . , 100), HMCR and PAR are calculated after more than 100 iterations. In this study, 0.5 is selected for the initial HMCR and PAR values (HMCR1 = PAR1 = 0.5). 2.4. Step 4: Non-dominated sorting The non-dominated sorting concept (Fonseca and Fleming 1993) is determined using calculated objective functions to assign a rank to each solution. This method is applied to calculate the rank of a solution in an individual population in the case of a concave trade-off relationship and is based on the fact that the rank of higher priority solutions has a higher probability of being preserved in all populations. If one solution must be selected out of several solutions with the same rank, then the crowded-distance concept is adopted to preserve the solution that has a longer crowded-distance value. For example, Figure 1 illustrates the priority of eight solutions. A priority of 1 is given to units A, B, C, D and E, because these are non-dominated units. Because unit G is dominated by unit C, a priority of 2 is given to unit G. However, a priority of 3 is given to unit F, because it is dominated by units B and C. Similarly, if priority is assigned to all units, unit H is given a priority of 6. 2.5. Step 5: Updating harmony memory/operation type memory Units with a low non-dominated priority are preserved. If two units have the same priority, then the crowding distance is calculated and the unit with the higher value is preserved. Priorities are assigned to the solutions to determine a better diversity for the multi-objective optimization. After generating



1961

Figure 1. Concept of non-dominated sorting (Fonseca and Fleming 1993).

new solutions, the solution with the highest priority is preserved first. If it is necessary to select one solution for preservation, the crowding-distance concept proposed by Deb et al. (2000) is applied to increase the diversity of the preserved solutions. Figure 2 presents a schematic diagram of the crowding-distance concept, in which the nondominated solutions of the objective functions f 1 and f 2 are presented. It is calculated as follows: d(i) = |f1 (i) − f1 (k)| + |f2 (i) − f2 (k)|

(6)

where d(i) represents the crowding distance of the ith solution, and f 1 and f 2 are the objective functions.

Figure 2. Concept of crowding distance (Deb et al. 2000).

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2.6. Step 6: Calculating harmony memory considering rate/pitch adjustment rate The new HMCR and PAR are calculated using the OTM created by Steps 3–5. ⎤ ⎡ 1 y1 = Random y21 = Pitch ··· yn1 = Memory ⎢ y2 = Random y22 = Memory ··· yn2 = Pitch ⎥ ⎥ ⎢ 1 ⎥ ⎢ OTM = ⎢ ⎥ .. .. ⎦ ⎣ . ··· . HMS = Random yHMS = Random · · · yHMS = Memory y1 n 2

(7)

Information on which decision variable generation method was used to generate the new memory is given in the OTM in Equation (7), and the individual HMCR and PAR are determined as follows: j


HMCRi =

j

n(yi = Memory and Pitch) n(yi = Pitch) , PARi = j HMS n(yi = Memory and Pitch)

(8)

where n is a count function representing the number of specific operations (e.g. harmony memory considering or pitch adjusting) in HM. HMCRi and PARi represent the HMCR and PAR of the ith decision variable, respectively. As the number of iterations increases, the HMCR generally increases, but the PAR decreases. This trend enables us to increase the HMCR to 1 and decrease the PAR to 0. To prevent such a problem, noise value is set between 0 and 1 in the calculation of HMCRi and PARi . The equations governing this are as follows: HMCRi + × U(−1, 1) ∈ [0, 1] PARi + × U(−1, 1) ∈ [0, 1] HMCRi = , PARi = ∈ / [0, 1] ∈ / [0, 1] HMCRi PARi (9) where U(–1, 1) is a random value between [–1, 1]. 2.7. Step 7: Checking stopping criterion After checking the stopping criterion, the process either is terminated or returns to Step 3.

3. Problem formulation In this study, a multi-objective optimal design model is developed for a WDN by considering the two design factors of construction cost and system resilience. The resilience is an index of facility deterioration and structural vulnerability. These two design factors appear to exhibit a trade-off relationship. 3.1. Objective functions 3.1.1. Least-cost design The majority of objective functions in WDN design are for minimizing cost. In this study, the cost estimation equation for network design proposed by Shamir and Howard (1968) is used. This cost can be estimated by multiplying the cost of each commercial pipe diameter and the length of each pipe. Thus, the sum of the costs of all pipes in the network is shown in Equation (10): MinCost =

N

C(Di )Li

(10)

i=1

where C(Di ) is the cost function of the ith pipe per unit length (m) of each pipe diameter, Li is the length (m) of the ith pipe, Di is the pipe diameter (mm) of the ith pipe, and N is the total number of pipes.


1963


3.1.2. Maximum resilience design The resilience of a WDN is an important index that conveys the current condition of the pipe network and has a significant role in its design, operation and maintenance. Resilience can be mainly classified into mechanical resilience and hydraulic resilience. Mechanical resilience represents the capability of the system to continuously supply water without the repair and replacement of system components such as pipes, pumps or valves. Hydraulic resilience describes the capability of the system to supply a sufficient amount of water to consumers at adequate pressure. In this study, the resilience index proposed by Todini (2000) is adopted, calculated as follows:

N ∗ j=1 qj (hj − hj ) Max Resilience = NR (11)

NP

N ∗ k=1 QK HK + i=1 Pi − j=1 qj hj where N, NR and NP are the number of nodes, reservoirs and pumps, respectively, hj is the pressure head at node j, hj * is the head requirement at node j, HK is the water level of reservoir K, QK is the water flow of reservoir K, and Pi is the power of pump i. The result of Equation (11) is in the range of 0–1, where a higher value indicates better resilience. 3.2. Hydraulic constraints and the penalty function In this study, water pressure and water velocity are used as design constraints. If the pressure at each node does not meet the pressure constraint or the water velocity at each pipe does not meet the velocity constraint during the hydraulic analysis of the optimal design, a large penalty point value is added to the penalty constant α. This penalty value affects the value of the objective function, meaning that

Figure 3. Self-adaptive multi-objective harmony search (SaMOHS) procedure for water distribution network design. HM = harmony memory; OTM = operation type memory; HMCR = memory considering rate; PAR = pitch adjustment rate.

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it will not be selected. If the system is close to meeting the hydraulic constraints, then this will not affect the selection of the optimal solution because the penalty point is small. To prevent such an occurrence, a large penalty point is generated by adding a large value to β. The penalty function P is calculated as follows: P=

α(|hi − hmin | or |hmax − hi |) + β α(|vj − vmin |or|vmax − vj |) + β

if, pressure constraint if, velocity constraint

(12)

where P is a penalty function, hi is the pressure head at node i (m), vj is the water velocity at pipe j (m/s), hmin and hmax are the minimum and maximum pressure heads (m), respectively, vmin and vmax and are the minimum and maximum water velocity (m/s), respectively, and α and β are the penalty constants.


3.3. Model flowchart This model is designed to obtain the optimal solution by iterating the pipe diameter setting, constraint checking and hydraulic analysis. Figure 3 illustrates the flow of this model. First, the pipe network information for the optimal design is acquired, and the optimal design is determined using SaMOHS. The hydraulic analysis program EPANET (Rossman 2000) runs simulations to verify that the hydraulic constraints are satisfied by the pipe network. If the network violates the hydraulic constraints, then the solutions are eliminated by imposing a penalty function. At every iteration, HMCR and PAR provide an adequate value for each decision variable. If this process reaches the predetermined number of maximum iterations, then it stops and the best solution is the optimal solution.

4. Application results To determine the optimal design method for a WDN, the results achieved by SaMOHS, MOHS and NSGA-II on three test problems (i.e. the two-loop, Hanoi and Saemangeum networks) are compared. 4.1. Performance indices In this study, the performance of the MOOAs is quantitatively evaluated with respect to these objectives: (1) to compare the convergence between two Pareto-optimal solutions (2) to maximize the distance between two extreme points of the optimal solutions determined by the MOOAs (3) to maximize the diffusion so that the optimal MOOA solutions have a smooth and homogeneous vector distribution. The MOOA results are compared using five evaluation criteria: spacing (SP), diversity index (DI), set coverage (CS), hypervolume (HV) and generational distance (GD). 4.1.1. Spacing Index SP is an evaluation index proposed by Schott (1995) to measure the homogeneity of the spatial distribution of optimal solutions, as well as how homogeneously consecutive solutions are distributed. SP can be calculated as shown in Equation (13): n 1 2 SP = (d¯ − di ) (13) n−1 i=1


1965

j j where di denotes minj (|f1i (x) − f1 (x)| + |f2i (x) − f2 (x)|), i, j = 1, . . . ,n; d¯ is the mean value of all di , and n is the number of non-dominated solutions. The values of SP are in the range of 0–1. Values closer to 0 indicate that the optimal solution is distributed more homogeneously.

4.1.2. Diversity The DI, proposed by Zitzler (1999), is designed to evaluate the diversity of optimal MOOA solutions. It uses the minimum and maximum values of the objective function, and is calculated as follows: M 1 max fm − min fm 2

DI = (14) Max − F Min M Fm m m=1


where Fm Max and Fm Min are the maximum and minimum values of the Pareto fronts, fm is the mth objective function value, and M is the number of objective functions. 4.1.3. Set coverage The CS criterion, proposed by Zitzler, Deb, and Thiele (2000), is an index for comparing the nondominated degrees of optimal solutions from different iterations. The values taken by CS (X , X) are in the range of 0–1. If the value of CS is 1, this means that all X are dominated by X . To understand the exact non-dominated relationship between two different iterations, it is necessary to calculate both CS (X , X) and CS (X , X ) and analyse the resulting values. CS can be calculated as follows: CS(X , X ) =

|{a ∈ X ; ∃a ∈ X : a a }| |X|

(15)

where X and X denote the optimal solutions found in different iterations, and a and a represent each optimal solution. 4.1.4. Hypervolume The HV indicator (Zitzler 1999), which is a measure of the hypervolume of the multidimensional region enclosed by a front with respect to a reference point, represents the overall solution diversity of a search. It is calculated by Equation (16): HV =

|Q|

vi

(16)

i=1

where vi is a hypercube constructed using reference point W (found by constructing a vector of worst objective function values) at the ith solution, and Q is an objective space (the hatched region shown in Figure 4). Therefore, HV is calculated by a union of all hypercubes. 4.1.5. Generational distance The GD metric was first presented by Veldhuizen and Lamont (1998). The main objective of this criterion is to clarify the capability of the different algorithms to find a set of non-dominated solutions with the lowest Pareto-optimal front distance. This evaluation factor is defined in mathematical form as shown in Equation (17): ⎛ ⎞1/2 npf 1 GD = ⎝ di2 ⎠ (17) npf i=1

where npf is the number of members in the generated Pareto front (PFg ), di is the Euclidean distance between member i in PFg and the nearest member in PFoptimal , and Euclidean distance (d) is


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Figure 4. Concept of hypervolume.

calculated based on Equation (18): d(p, q) = d(q, p) =

n

1/2 2

(fqi − fpi )

(18)

i=1

where q = (fq1 , fq2 , . . . , fqn ) is a point on PFg , and p = (fp1 , fp2 , . . . , fpn ) is the nearest member to q in PFoptimal . The best possible value for the GD metric is 0, which corresponds to PFg exactly covering PFoptimal . Figure 5 illustrates the concept of GD. 4.2. Test problems The purpose of this study is to confirm the search efficiency and improvement in convenience provided by the application of the proposed self-adaptive method to the MOOA. Therefore, in this study, three algorithms (i.e. MOHS, NSGA-II and SaMOHS) are applied and fairly compared using the same size of HM/population number (Pn ) and each algorithm’s optimal parameters (e.g. MOHS: HMCR and PAR; NSGA-II: Pc and Pm ) depending on the test problem. This minimizes the effects of the parameter tuning and initial solution quality on the final solution quality. The initial solutions were generated using different numbers of HMS/Pn (i.e. two-loop: 30; Hanoi: 50; and Saemangeum: 50) by random search and this initial solution set was used in all iterative simulation runs. To determine the optimal parameters of MOHS and NSGA-II, parameter sensitivity analysis was performed. The HMCR/Pc was varied from 0.7 to 0.95 and the PAR/Pm was varied from 0.05 to 0.3 in 0.05 intervals. Therefore, a total of 36 parameter sets was used for 10 independent optimization runs. The 10 results for each test problem and optimization algorithm were compared using the performance indices described in Section 4.1 (i.e. CS, SP, DI, HV and GD), and the best parameter



1967

Figure 5. Concept of generational distance.

set was decided by the highest sum of indices value (SIV), calculated by Equation (19):

SIV =

n

0.2(CSi + SPi + DIi + HVi + GDi )

(19)

i=1

where CSi , SPi , DIi , HVi and GDi are the performance indices of the ith solution, and n is the number of total parameter sets ( = 36). Figure 6 shows the results of the parameter sensitivity analysis and Table 2 lists the best parameter set for each test problem. The HMCR/Pc of the three tests obtained a high value (above 0.9) except for the Saemangeum network ( = 0.7), and PAR/Pm gave a low value (under 0.2). This result confirms that high HMCR and low PAR values obtain good optimal solutions, as found in many previous studies. 4.2.1. Two-loop water distribution network The two-loop network, first proposed by Alperovits and Shamir (1977), is illustrated in Figure 7. This network is composed of two closed circuits, one reservoir, seven nodes and eight pipelines. The network conditions are as follows. The reservoir has a fixed water level of 210 m (689 ft), the total length of the pipelines is 1000 m (3281 ft), the Hazen–Williams roughness coefficient is set to 130, and the minimum head at each node is 30 m (98.4 ft). A total of 14 commercial pipe diameters was used for the two-loop network. The costs of the pipe diameters per unit length are shown in Table 3. 4.2.2. Hanoi network The Hanoi network in Vietnam was first proposed by Fujiwara and Khang (1990), and is illustrated in Figure 8. It is composed of two closed circuits, one reservoir, 32 nodes and 34 pipelines. The Hazen–Williams roughness coefficient is 130, and the minimum head at each node is 30 m. In total, six commercial pipe diameters were used for this network and their costs per unit length are shown in Table 4.


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(a)

(b) Figure 6. Results of parameter sensitivity analysis: (a) multi-objective harmony search (MOHS); (b) non-dominated sorting genetic algorithm-II (NSGA-II). PAR = pitch adjustment rate; HMCR = harmony memory considering rate; Pm = mutation rate; Pc = crossover rate.

4.2.3. Saemangeum network The Saemangeum network was first proposed by Yoo et al. (2015). It is composed of nine closed loops, one reservoir, 334 nodes and 356 pipes. The total length of the network is 41.38 km. Its layout is illustrated in Figure 9. The costs of the pipe diameters per unit length are presented in Table 5. In


1969

Table 2. Best parameter sets of multi-objective harmony search (MOHS) and non-dominated sorting genetic algorithm-II (NSGA-II). Parameter MOHS

HMS

HMCR

PAR

NI

NFE

DV

Two-loop network Hanoi network Saemangeum network

30 50 50

0.95 0.9 0.95

0.2 0.2 0.25

60,000 100,000 500,000

60,000 100,000 500,000

8 34 356

NSGA-II Two-loop network Hanoi network Saemangeum network

Pn 30 50 50

Pc 0.95 0.9 0.7

Pm 0.15 0.15 0.15

NI 2,000 2,000 10,000

NFE 60,000 100,000 500,000

DV 8 34 356


Note: HMS = harmony memory size; HMCR = harmony memory considering rate; PAR = pitch adjustment rate; Pn = population number; Pc = crossover rate; Pm = mutation rate; NI = number of iterations; NFE = number of function evaluations; DV = decision variable.

Figure 7. Layout of the two-loop network.

Table 3. Cost data for the two-loop network (Alperovits and Shamir 1977). Diameter (mm) 25.4 50.8 76.2 101.6 152.4 203.2 254.0

Cost ($/m)

Diameter (mm)

Cost ($/m)

2 5 8 11 16 23 32

304.8 355.6 406.4 457.2 508.0 558.8 609.6

50 60 90 130 170 300 550


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Figure 8. Layout of the Hanoi network.

Table 4. Cost data for the Hanoi network (Fujiwara and Khang 1990). Diameter (mm) 304.8 406.4 508.0 609.6 762.0 1,016.0

Figure 9. Layout of the Saemangeum network.

Cost ($/m) 45.7 70.4 98.4 129.3 180.7 278.3


1971

Table 5. Cost data for the Saemangeum network (Yoo et al., 2014). Cost (₩/m)


Pipe no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Pipe diameter (mm)

Construction costs

Material costs

Maintenance costs

80 100 150 200 250 300 350 400 450 500 600 700 800 900 1,000 1,100 1,200 1,350

65,000 65,999 76,410 86,028 96,135 105,325 113,818 126,797 136,250 147,792 171,991 211,413 307,640 359,048 415,702 482,074 576,736 687,390

15,000 27,583 40,686 58,716 81,160 103,231 125,107 148,836 155,522 181,823 211,396 273,528 339,740 384,619 451,932 547,224 606,962 716,075

6,500 6,600 7,641 8,603 9,614 10,533 11,382 12,680 13,625 14,779 17,199 21,141 30,764 35,905 41,570 48,207 57,674 68,739

total, 18 commercial pipe diameters were used, and the cost estimation is the same as the construction cost for each pipe diameter according to the K-water report (2010) on the cost estimation of the water supply facilities. The water pressure at each node and water velocity in the pipeline are considered to be the hydraulic constraints of the Saemangeum network. The constraints are 10 m and 35 m for the minimum and maximum head at each node, respectively, and 0.01 m/s and 2.5 m/s for the minimum and maximum water speed, respectively.

4.3. Optimization results In this section, the results of the optimal WDN design using the best parameter sets (Table 2) are presented. Twenty independent optimization runs were carried out for each test problem to obtain statistically significant results. Table 6 shows the results including the best, mean and worst performance index values for each test problem and optimization algorithm (i.e. NSGA-II, MOHS and SaMOHS). Figure 10 shows the initial and final configuration of the WDNs and each optimized network with its reduced total cost and increased system resilience. In the case of the Hanoi network, all pipe diameters in the initial condition were 1000 mm; however, the optimized pipe diameter is decreased to adapt to the system constraints. Therefore, the total cost and system resilience are improved while satisfying the nodal minimum pressure. As shown in Table 6, SaMOHS obtains better statistical solutions for almost all cases. In particular, CS and GD, which represent the convergence ability, for SaMOHS have the best performance for all three networks. Figure 11 and Table 6 show that SaMOHS generally achieves a better Pareto front than NSGAII and MOHS for all three networks. This figure also presents the best Pareto fronts reported by Wang et al. (2015), which were obtained from Pareto-optimal solutions using five different multiobjective evolutionary algorithms (MOEAs) (i.e. NSGA-II, ε-MOEA, ε-NSGA-II, AMALGAM and Borg), after applying several executions [10 trials with different population sizes (60, 120 and 240) and numbers of function evaluations (600,000)]. The optimal solutions from each set of simulation results were aggregated, and non-dominated sorting using the integrated solution was used to find the best Pareto front.

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Table 6. Statistical optimization results obtained by the applied algorithm in this study. Best solution Index SP DI CS HV


GD

MOPs

Two-loop

Hanoi

NSGA-II 0.0144 0.0768 MOHS 0.0145 0.0259 SaMOHS 0.0131 0.0258 NSGA-II 0.8035 0.9042 MOHS 0.8238 0.9242 SaMOHS 0.8696 0.9094 NSGA-II 0.550 0.5685 MOHS 0.667 0.4633 SaMOHS 0.810 0.7011 NSGA-II 0.7657 0.9336 MOHS 0.7260 0.9258 SaMOHS 0.9757 0.9254 NSGA-II 2.51.E-05 5.91.E-06 MOHS 1.78.E-05 6.62.E-06 SaMOHS 1.34.E-05 4.58.E-06

Mean solution

Saemangeum Two-loop 0.0447 0.0198 0.0191 0.9068 0.9093 0.9380 0.2786 0.3446 0.4312 0.8327 0.8094 0.9315 1.52.E-05 1.20.E-05 8.81.E-06

0.0156 0.0149 0.0187 0.7045 0.6714 0.8146 0.3432 0.5643 0.6798 0.7465 0.7120 0.8075 4.72.E-05 4.97.E-05 3.99.E-05

Hanoi 0.487063 0.3761 0.2941 0.5670 0.5802 0.7911 0.49291 0.4225 0.6144 0.8181 0.8200 0.9058 6.39.E-06 6.16.E-06 4.48.E-06

Worst solution

Saemangeum Two-loop 0.2463 0.1916 0.1533 0.6230 0.6133 0.8358 0.4097 0.5325 0.6342 0.7667 0.7507 0.8395 3.43.E-05 2.55.E-05 1.49.E-05

Hanoi

0.1026 0.0768 0.0166 0.0259 0.0983 0.0358 0.6381 0.4475 0.4016 0.4714 0.7787 0.6324 0.326 0.3172 0.282 0.3662 0.392 0.3514 0.6771 0.5810 0.6117 0.6094 0.7236 0.7088 7.03.E-05 7.01.E-06 8.26.E-05 5.83.E-06 6.72.E-05 4.48.E-06

Saemangeum 0.0879 0.0208 0.0657 0.5319 0.4278 0.7404 0.3887 0.3764 0.4035 0.6165 0.5983 0.7019 5.41.E-05 3.96.E-05 2.13.E-05

Note: MOP = multi-objective problem; SP = spacing; DI = diversity index; CS = set coverage; HV = hypervolume; GD = generational distance; NSGA-II = non-dominated sorting genetic algorithm-II; MOHS = multi-objective harmony search; SaMOHS = self-adaptive multi-objective harmony search.

(a)

(b)

(c)

Figure 10. Initial and final configurations of the test problem: (a) two-loop network; (b) Hanoi network; (c) Saemangeum network.

This study uses five performance indices to compare the convergence and diversity of the optimal solutions of the three algorithms. The first performance, SP, is an index for evaluating the homogeneity of the spatial distribution among the multi-objective optimal solutions, and also evaluating how homogeneously consecutive solutions are distributed. The SP values for the three algorithms are similar, but SaMOHS achieves a more homogeneous distribution than the others. DI and HV measure solution diversity. The MOHS and SaMOHS algorithms generated wider Pareto fronts than NSGA-II. This is demonstrated visually in Figure 12 and Table 7, where the solutions of SaMOHS and MOHS are well spread on both sides of the objective function spaces, but for NSGA-II, they are concentrated on a certain part of the Pareto fronts. CS and GD measure solution convergence. CS compares the convergence of the total Pareto solution and each solution of the algorithm. The total Pareto solution is given by the final Pareto front after integrating each optimal solution of the three algorithms. As shown in Figure 11 and Table 6, SaMOHS achieves remarkably high convergence/non-domination compared with the other


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(a)

(b)

(c) Figure 11. Best Pareto front (PF): (a) two-loop network: (b) Hanoi network; (c) Saemangeum network. MOHS = multi-objective harmony search; NSGA-II = non-dominated sorting genetic algorithm-II; SaMOHS = self-adaptive multi-objective harmony search.

two algorithms in the two-loop and Hanoi networks. The results of the performance indices show that SaMOHS conclusively demonstrates the best convergence by achieving the highest CS value and lowest GD value. It also achieves similar results to the other two algorithms with respect to diversity and spatial distribution.

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(a)

(b)

(c) Figure 12. Best performance indices: (a) two-loop network; (b) Hanoi network; (c) Saemangeum network. NSGA-II = nondominated sorting genetic algorithm-II; MOHS = multi-objective harmony search; SaMOHS = self-adaptive multi-objective harmony search.

Unlike the other two algorithms, SaMOHS uses parameter sensitivity analysis to set up the parameters automatically for each decision variable and hence does not require adequate parameter setting. Figure 13 and Table 8 show the changes in parameter values (HMCR and PAR) over the iterations. The decision variable in the optimal WDN design problem represents pipe diameter. Thus, the parameters alter with the pipe, which is the decision variable in the Hanoi network. For example, in the case of the HMCR and PAR of pipe 1 in the Hanoi network, the HMCR gradually increases to nearly 1 as the number of iterations increases, while PAR gets closer to 0. However, in the case of pipe 3, after 5000 iterations, the PAR becomes close to 0 while the HMCR decreases and finally converges to 0.4. Therefore, the results are different for pipes 1 and 2. This means that the value of 0.4 is an adequate HMCR value for pipe 3 during the optimization. As mentioned above, SaMOHS increases the efficiency and


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Table 7. Results of optimization. Problem Two-loop network Hanoi network Saemangeum network

Cost (M $) Resilience Cost (M $) Resilience Cost (10 B KRW) Resilience

NSGA-II

MOHS

SaMOHS

0.530–4.150 0.299–0.891 6.352–9.947 0.231–0.347 1.074–1.347 0.313–0.505

0.450–4.120 0.259–0.884 6.262–9.794 0.220–0.348 1.076–1.311 0.316–0.513

0.430–4.150 0.253–0.895 6.348–10.399 0.231–0.351 1.073–1.321 0.314–0.514


Note: NSGA-II = non-dominated sorting genetic algorithm-II; MOHS = multi-objective harmony search; SaMOHS = self-adaptive multi-objective harmony search.

Figure 13. History of the harmony memory considering rate (HMCR) and pitch adjustment rate (PAR) for a Hanoi network.

Table 8. Results of optimization. Iteration

Pipe1_PAR

Pipe1_PAR

Pipe1_PAR

Pipe1_HMCR

Pipe1_HMCR

Pipe1_HMCR

1 2 99 100 101 102 693 694 3001 3002 8999 9000 9999 10,000

0.5 0.5 0.5 0.5 0.408796 0.40979 0.209945 0.21447 5.71E-02 4.40E-02 4.27E-02 0.053508 5.34E-02 5.34E-02

0.5 0.5 0.5 0.5 0.440471 0.444375 0.098495 0.103021 5.73E-02 0.045931 5.06E-02 0.056793 5.11E-02 5.11E-02

0.5 0.5 0.5 0.5 0.459422 0.459907 0.276949 0.281474 5.67E-02 0.054852 4.15E-02 4.16E-02 5.81E-02 5.81E-02

0.5 0.5 0.5 0.5 0.408796 0.40979 0.209945 0.21447 5.71E-02 4.40E-02 4.27E-02 0.053508 5.34E-02 5.34E-02

0.5 0.5 0.5 0.5 0.440471 0.444375 0.098495 0.103021 5.73E-02 0.045931 5.06E-02 0.056793 5.11E-02 5.11E-02

0.5 0.5 0.5 0.5 0.459422 0.459907 0.276949 0.281474 5.67E-02 0.054852 4.15E-02 4.16E-02 5.81E-02 5.81E-02

Note: PAR = pitch adjustment rate; HMCR = harmony memory considering rate.

convenience when solving an optimization problem because SaMOHS uses a self-adaptive method to produce results similar to or better than NSGA-II and MOHS, despite the fact that no parameter checking or correction processes are performed, which is an essential part of solving an optimization problem.

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5. Conclusions


In this study, the SaMOHS algorithm was developed to consider various design factors and enhance the efficiency of the optimal design for a WDN. The SaMOHS model proposed in this study was applied to three well-known benchmark networks for optimal WDN design. Finally, performance indices were compared to evaluate the convergence and diversity of the solutions as well as the homogeneous spatial distributions of consecutive solutions. The evaluation of the SaMOHS Pareto fronts for three different WDN design problems indicates that the proposed method generally performed better in terms of both convergence and diversity than NSGA-II and extended versions of MOHS. Furthermore, SaMOHS achieved similar results in terms of convergence and diversity even if no parameter sensitivity analysis was performed. Such a characteristic enables the inconvenience of setting parameters to be eliminated. Moreover, it increases search efficiency by automatically setting up the optimal parameters during the iterations of the solution search and considering various design factors simultaneously.

Disclosure statement No potential conflict of interest was reported by the authors.

Funding This work was supported by a grant from the National Research Foundation (NRF) of Korea, funded by the Korean Government (MSIP) [grant number NRF-2016R1A2A1A05005306].

ORCID Joong Hoon Kim

http://orcid.org/0000-0002-3729-7560

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