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Published by Basic Research Journal of Agricultural Science and Review

Basic Research Journal of Agricultural Science and Review ISSN 2315-6880 Vol. 3(12) pp. 131-145 December 2014 Available online http//www.basicresearchjournals.org Copyright ©2014 Basic Research Journal

Full Length Research Paper

Simultaneous layout and pipe size optimization of pressurized irrigation networks

1R. Rajabpour*, 2Naser Talebbeydokhti

1Department of Civil Engineering, Science and Research branch, Islamic Azad University, Tehran, Iran.

2Professor of Civil and Environmental Engineering, Head of Environmental Research and sustainable Development

Center, Shiraz University.

*Corresponding author email: [email protected]

Accepted 20 November, 2014

Abstract

The layout of irrigation water distribution networks is usually branched. In most of the previous studies pipe sizes of the network had been optimized assuming a predetermined layout of the distribution system. However, only few researches have focused on the simultaneous layout and pipe size optimization. In this study a hybrid approach is adapted for simultaneous layout and pipe size optimization of branched pipe networks as a cost minimization problem. This new approach is based on combination of a pipe size optimizer (LIDM) with a layout optimizer for joint layout and pipe size optimization. At each iteration, the layout optimizer algorithm acts as an outer loop and LIDM acts as an inner loop. Once all the solutions are developed (each solution is a specific layout), LIDM can be used to optimize pipe sizes of each developed branched layout (solution). Then solution costs can be calculated and according to them, layout optimizer rearranges the solutions and the process continues. Two different approaches are used for layout optimization. At the First approach by using the loop model each branched layout is encoded as a string of eliminating links and a Discrete Particle Swarm Optimization for combinatorial optimization, called JPSO is applied to select the best sting of eliminating links. To avoid the problems associated with the Loop Model, at the second approach instead of coding the solutions into strings, an especial form of Genetic Algorithm which includes a tree growing algorithm within its “reproduction” phase is used as the layout optimizer. Proposed methods are applied for simultaneous layout and pipe size optimization of a small benchmark example in the literature and the results are presented and compared to the existing results. The results showed that the developed methods have significant advantages compared to other methods used. Afterwards, another example is considered to illustrate the efficiency of the proposed Genetic Algorithm for layout and size optimization of real-world networks. Keywords: distribution networks; Optimization; branched layout; JPSO; Genetic Algorithm; Hybrid approach

INTRODUCTION The pressurized systems which were developed during the previous decades had considerable advantages compared to open canals. In fact, they guarantee better

services to the users and higher distribution efficiency. They overcome the topographical constraints and make it easier to measure the water volume delivered.


Operation, maintenance and management activities are more technical and easier to control (Lamaddalena and Sagardoy, 2000). In order to reduce the costs of the network, the layout of pressurized irrigation water distribution networks are branched. One of the important problems which should be considered in designing such networks is the simultaneous optimization of both the layout and the pipe sizes. The optimization of branched irrigation networks has attracted the attention of some researchers. Some researchers such as Labye (1981) only considered pipe size optimization of branched networks with a predetermined layout. He proposed an approach called Labye's Iterative Discontinuous Method (LIDM) for optimizing pipe sizes. Using LIDM enables the optimization process to allocate mixed pipe diameters.

Some researchers, on the other hand, have addressed the layout optimization of the branched pipe networks, neglecting the influence of the pipe sizes. Cembrowicz (1992) encoded the problem of branched layout optimization with Loop Model and applied GA for solving it. Walters and Smith (1995) used evolutionary algorithm (EA) for the selection of a branched network from a non-directed base graph. Geem et al. (2000) applied Harmony Search (HS) for the optimal design of branched networks.

Extensive research has been conducted on pipeline layouts in irrigation systems and has primarily focused on the optimization of the pipe networks. Optimization methods used include graph-theoretic methods (Xuezhen et al., 1995), linear programming (Throcharis et al., 2005; Lei and Luo, 2010), mixed integer linear programming (Shahi et al., 2012), analytic hierarchy processes (Liu et al., 1999), artificial neural networks, and genetic algorithms (Zhou and Lin, 2001); Reca, et al. (2008) evaluated the performance of several meta-heuristic techniques - genetic algorithms, simulated annealing, tabu search and iterated local search. He compared these techniques by applying them to medium-sized benchmark networks.

Some researchers have conducted comparative studies on the investment, energy consumption and operating costs associated with various types of pipeline sprinkler irrigation systems, including Rodríguez et al. (2009); Moreno et al. (2010); Theocharis et al. (2010,2005).

Afshar (2006 and 2007) tackled the problem of joint layout and pipe size optimization of branched networks, by considering pipe sizes of maximum layout (a predefined layout that includes all possible links) as decision variables. The pipe sizes can be selected from a set of available pipe sizes. Adding a zero pipe diameter to the list of available pipe diameters would enable the optimization algorithm to remove if required, pipes from the network in its search towards an optimal tree layout. The search space of the optimization model

Rajabpour and Talebbeydokhti 132 in this formulation consists of all sub-networks of maximum layout. The search space rapidly grows with increasing network size and the number of available pipe sizes. For example the total number of possible combinations for a 3node ∗ 3node base graph of 12 links with 13 possible pipe sizes is 1412. A Large number of these possible combinations are, however, infeasible solutions which, if identified, could be excluded, thus leading to a much smaller search space. In another research Afshar (2005) applied Ant Algorithm (ACO) for simultaneous layout and size optimization of tree-like pipe networks. In this approach ants have to search for the optimal solution in the region which contains networks with feasible layout. The decision points of the problem are associated with the nodes of the maximum layout, and the solution components considered as the list of allowable pipe sizes of links in maximum layout.

In this study a new approach is introduced for simultaneous layout and pipe size optimization of branched pipe networks as a cost minimization problem. This new approach is based on combination of a pipe size optimizer (LIDM) with a layout optimizer. Adapting such hybrid approach to tackle the problem of joint layout and pipe size optimization of branched networks leads to much smaller search space in comparison with ACO application as reported by Afshar (2006); Labye (1981) proposed LIDM for optimizing pipe sizes in a branched irrigation network with a predetermined layout. Using LIDM enables the optimization process to allocate mixed pipe diameters to some links and therefore reduces the costs even more.

In this paper, two different approaches are used for layout optimization. At the First approach by using the loop model each branched layout is encoded as a string of eliminating links and a Discrete Particle Swarm Optimization for combinatorial optimization, called JPSO is applied to select the best sting of eliminating links. To avoid the problems associated with the Loop Model, at the second approach instead of coding the solutions into strings, an especial form of Genetic Algorithm which includes a tree growing algorithm within its “reproduction” phase is used as the layout optimizer. Discrete particle swarm optimization The standard PSO considers a swarm S containing s particles (S = 1, 2… s) in a d-dimensional continuous solution space (Kennedy and Eberhart, 2001). Each i-th particle of the swarm has a position

and a velocity

. The position represents a


133. Basic Res. J. Agric.Sci. Rev. solution to the problem, while the velocity gives the

rate of change for the position of particle i at the next iteration. The position of particle in each iteration is

adjusted according to: (1)

in which, ( ) indicates new values.

Each particle i of the swarm communicates with a social environment or neighborhood, ,

representing the group of particles with which it communicates, and which could change dynamically. In nature, a bird adjusts its position in order to find a better position, according to its own experience and the experience of its companions. In the same manner, in each iteration, each particle i updates its velocity reflecting the attractiveness of its best position so far

and the best position

of its social neighborhood N(i),

according to the equation:

(2)

The parameters , and are positive constant

weights representing the degrees of confidence of particle i in the different positions that influence its dynamics, while refers to a random number

with uniform distribution [0, 1] that is independently generated at each iteration.

The original PSO algorithm can only optimize problems in which the elements of the solution are continuous real numbers. Therefore several Discrete Particle Swarm Optimization (DPSO) methods have been proposed. In the DPSO proposed by Kennedy and Eberhart (1997) for problems with binary variables, the position of each particle is a vector

of the d-dimensional binary solution space, ,

but the velocity is still a vector , of the d-dimensional

continuous space, . A different way to update

the velocity was considered by Yang et al. (2004). A DPSO whose particles at each iteration are

affected alternatively by its best position and the best position among its neighbors was proposed by Al-kazemi and Mohan (2002). The multi-valued PSO (MVPSO) proposed by Pugh and Martinoli (2006) deals with variables with multiple discrete values. The position of each particle is a mono dimensional array in the case of a continuous PSO, a 2 dimensional array in the case of a DPSO, and a 3-dimensional array for a MVPSO. Indeed, the position of particle i in the MVPSO is expressed by the term x, representing the probability that the ijk i-th particle, in the j-th iteration, takes the k-th value.

A new DPSO proposed in (Moreno-Perez et al., 2007) and (Martinez-Garcia and Moreno-Perez, 2008)

Does not consider any velocity since, from the lack of continuity of the movement in a discrete space, the notion of velocity loses sense; however they kept the attraction of the best positions. They interpret the weights of the updating equation as probabilities that, at each iteration, each particle has a random behavior, or acts in away guided by the effect of an attraction. The moves in a discrete or combinatorial space are jumps from one solution to another. The attraction causes the given particle to move towards this attractor if it results in an improved solution. An inspiration from the nature for this process is found in frogs, which jump from a lily pad to a pad in a pool. Thus, this new discrete PSO is called Jumping Particle Swarm Optimization (JPSO). Consoli et al. tested capabilities of JPSO by solving minimum labeling Steiner tree problem, an NP-hard graph problem. Based on their computational analysis, JPSO clearly outperformed all the other procedures, obtaining high-quality solutions in short computational running times. This confirms the ability of JPSO method to deal with NP-hard combinatorial problems.

Genetic algorithm Genetic algorithms are stochastic search procedures based on the evolutionary mechanisms of natural selection and genetics (Holland, 1975). Genetic algorithms mimic the very effective optimization model that has evolved naturally for dealing with large, highly complex systems. Genetic algorithms do not optimize, evolution uses whatever material is at its disposal to produce above-average individuals [Charbonneau and Knapp, 1996]. Because of their flexibility and robustness, genetic algorithms have successfully been used to solve NP hard problems arising in many sciences and engineering branches. Good descriptions of genetic algorithms are given by Goldberg (1989); Michalewicz (1992).

In genetic algorithm, each solution is codified into a chromosome-like structure. Each chromosome has an objective function value, called the fitness. A set of chromosomes together with their associated fitness is called the population. The genetic algorithm starts with a random initial population and later follows an iterative reproductive cycle.

The population size and the number of generations through which this population is to evolve are both input parameters in the model. Once the initial population is created and evaluated, the genetic operators are applied iteratively though successive generations in order to progressively improve the population. The breeding process comprises three simple operators: selection, crossover and mutation. The selection operator chooses parents from the population in such a way that favors the fitter individuals. The parents are


Rajabpour and Talebbeydokhti 134

Figure 1. Elementary network scheme [2].

combined to form offspring individuals that inherit the characteristics of their parents using the crossover operator. Later, the mutation operator is applied to promote population variability. Labye’s Iterative discontinuous method (LIDM) The approach proposed by Labye (1981), called Labye's Iterative Discontinuous Method (LIDM), for optimizing pipe sizes in an irrigation network is described in this section. This method is developed in two stages. In the first stage, an initial solution is constructed giving, for each section k of the network, the minimum commercial diameter (Dmin) according to the maximum allowable flow velocity (vmax) in a pipe, when the pipe conveys the calculated discharge (Ql). After knowing the initial diameters, it is possible to calculate the piezometric elevation Z 0,in at the upstream end of the network, which satisfies the minimum head (Hj, min) required at the most unfavorable hydrant j:

(3)

Here hj is the total head losses along the pathway connecting the hydrant j to the upstream end of the network. Head losses are computed with Darcy-Weisbach equation. The diameters Dl, min and the initial piezometric elevation Z 0,in constitute the initial set of parameters for the optimization of the pipe diameters. This is performed by an iterative procedure. At each iteration, only two pipes (p and p+1) are considered for each branch. Their diameters, are Dp and Dp+1 where Dp+1 > Dp. Afterward it is possible to compute the coefficient β as follows:

(4)

where C and J are the cost and the friction loss per unit length of pipe, respectively.. Considering any branching sub-network (SN) at the end of the section l, it is possible to minimize the variation of costs ∆C of the network SN

* (Figure 1) using linear programming of

equations 5 and 6:

(5)

Subject to: (6)

where ∆Z is the variation of the head in the upstream head of (SN), and ∆hl is the variation in the head losses at section l due to the changes of pipe diameters. The optimal solution of equations (5) and (6) produces ∆Z=∆Z′ and ∆hl=0 when βp, SN<βp, l, and ∆Z=0 and ∆hl=∆Z′ when βp, SN>βp, l. As a result the minimum value for ∆C is

(7)

where β* = min ( βp,SN, βp,l ). The slope of βp,SN is

(8)

when SN is constituted by two sections (1 and 2) in derivation (Figure 1), and

(9)

when sections 1 and 2 are in series; βp,1 and βp,2 are the coefficients defined in equation (4) for the branch 1 and 2 (Figure. 1), with βp,l = 0 at the terminal sections having excess of head (at the downstream end node). The procedure is performed starting the optimization from the downstream sections. The magnitude of ∆Ziter

for each iteration is:

(10)

where EZiter is the minimum observed for the excess of charge [m] in the nodes where the head changes at each iteration, ∆hiter is the minimum value of the head losses variation [m] in those sections where diameters change during the same iteration, and (Z0,iter – Z0) is the difference between computed and actual piezometric heads [m] at the upstream end. The iterative procedure continues until Z0 is satisfied. Hybrid Approach Hybrid approach is based on combination of a pipe size optimizer (LIDM) with a layout optimizer for joint layout and pipe size optimization. In this approach, in each iteration, the layout optimizer algorithm acts as an outer


135. Basic Res. J. Agric.Sci. Rev.

Figure 1. coding a spanning tree through maximum layout with Loop Model

Figure 2. Jump toward an attractor (Particles encoded with Loop Model)

loop and LIDM acts as an inner loop. Once all the solutions are developed (each solution is a specific layout), Labye's Iterative Discontinuous Method can be used to optimize pipe sizes of each developed branched layout (solution). Then solutions costs can be calculated and according to them, layout optimizer rearranges the solutions and the process continues. The basic algorithm of this approach is outlined below: 1- Form an Initial population of a random Tree layouts 2- Optimize the pipe sizes in each layout (pipe size optimization algorithm) 3- Evaluate cost of each layout 4- Generate a new set of tree layouts (using layout optimization algorithm) 5- Repeat from (2), until some termination criterion is reached. Using LIDM as a pipe size optimizer enables the optimization process to allocate mixed pipe diameters to some links. LIDM chooses optimum diameters for network pipes in such a way following constrains hold satisfied: Hydraulic constrains; Nodal head and pipe flow velocity constrains a Pipe size availability constrains. On the other hand layout optimization algorithm (layout optimizer) selects the least cost spanning tree through a maximum layout, which includes all possible links.

Loop model Cembrowicz (1992) encoded the problem of branched layout optimization with Loop Model and applied GA for solving it. Cembrowicz (1992) uses graph theory in describing his evolutionary design approach to the problem of branched layout optimization. First he identifies loops within the base graph, and then points out that a tree network can be formed if one link is removed from each loop. The links to be removed (referred to as chords) then become the design variables of the layout problem, which is now one of selecting the set of chords (the co-tree) that, when removed from the base graph, results in the minimum cost spanning tree. He called this approach Loop Model.

The main draw-back of Loop Model is that the removal of an arbitrary link from each loop does not necessarily result in a tree, as illustrated in Figure 3, so a check is necessary on the resulting graph to ensure feasibility. As the network size increases, so does the chance of forming an infeasible network. Moreover water distribution networks with two or more recourses cannot be encoded by Loop Model. Cembrowicz has used the model with success one base graph with 34 nodes, 48 links and 15 loops. Figure 2 shows coding a spanning tree through maximum layout with Loop Model.


Using JPSO for branched layout optimization encoded with lmodel The JPSO used for branched layout optimization encoded with Loop Model, considers a swarm S containing s particles (S = 1, 2 … s) whose positions

evolve in the solution space,

jumping from one solution to another. The position of each particle is encoded as a feasible solution to the branched layout optimization problem. At each iteration, there are four different forms of selectable jumps for each particle, from which one should be selected. First one is a random jump (Type one). This kind of jump consists of selecting at random a feature of the solution and changing its value. Second is a jump toward particle`s best position so far

(Type one). Third is a jump toward the best position of particle`s local neighborhood

(Type there) and forth is a jump toward the best particle in the current iteration, which is called global best

(Type four). Bi, Gi and G* are so

called attractors. A jump approaching an attractor consists of changing a feature of the current solution by a feature of the attractor. The process of position updating can be described with eq 11.

(11)

Equation 11 shows that random jump take place with the probability c1 and jumps toward attractors Bi, Gi and G

* take place with probabilities , and

, respectively. At each iteration

in order to update the position with this operation, the unit interval [0, 1] is divided into four intervals [0, c1), [c1, c1+c2), [c1+c2, c1+c2+ c3) and [c1+c2+ c3, 1], which are representatives of jumps type one, two, there and four, respectively. Then a random number is generated with uniform distribution in [0, 1] and based on the interval to which the generated random number belongs, the corresponding type of jump would be selected. As the factors c1, c2, c3 and c4 control the balance between global and local search, we suggest to have c1 decrease and c2, c3 and c4 increase linearly with time, in a way to first emphasize global search and then, with each cycle of the iteration, to prioritize local search.

After selecting jump type i with the probability ci, we choose one of the decision variables randomly with uniform probability. If the selected jump was type one then we change the value of chosen decision variable randomly, else, substitute value of chosen decision variable with corresponding one in the attractor. At the second step we choose a random number ζ with uniform distribution [0, 1]. If ζ < ci then we choose another decision variable randomly and change its


value again and repeat this step; otherwise the jump stops. Figure 3 illustrates Jump toward an attractor for particles encoded with Loop Model. JPSO algorithm is described below: Note: Each branched layout is encoded as a string of eliminating links. 1. Initial generation of a random population of m particles 2. Objective function evaluation 2.1. Set i =1 2.2. If i>m then go to 3 2.3. If the i-th particle was infeasible then set its cost equal to P (penalty cost) and go to 2.5 else go to 2.4 2.4. Using LIDM determine optimal diameters for links of i-th particle and calculate its cost 2.5. Set i=i+1 and go to 2-2 3. Update Bi , Gi and G

*

4. Update c1, c2 , c3 and c4 (equation 11) (Note: during the iterations c1 linearly decreases and 5. Update particle positions 5.1. Set i=1 5.2. If i>m then go to 6 5.3. Select an attractor for i-th particle Generate random number R (0<R<1) If 0<R<c1 then jumps take place randomly and (ci←c1) If c1<R<c1+c2 then (attractor←Bi) and (ci←c2) If c1+c2<R<c1+c2 +c3 then (attractor←Gi) and (ci←c3) If c1+c2 +c3 <R<1 then (attractor←G

*) and (ci←c4)

5.4. Perform jumps in i-th particle 5.4.1. Generate random number ζ (0 ≤ ζ ≤ 1) 5.4.2. If ζ<ci then perform jump and return to 5.4.1, else go to 5.5 5.5. Set i=i+1 and go to 5.2 6. If termination criteria was met, then algorithm stops else return to 2 JPSO Parameters Swarm size (s), factors c1, c2, c3 and c4, maximum number of iterations (m) and neighborhood status should be determined as input parameters. Neighborhood status considered to be circular with radius one. So that each particle has one right neighbor and one left neighbor. Factors c1, c2, c3 and c4 considered to vary linearly during the iterations. The value of c1 starts from an initial value, c1

i and decreases

constantly so that at the last iteration (m-th iteration) reaches to its final value c1

f. The rate of decrease in c1,

, is equal to . The initial values of

Factors c2, c3 and c4 (c2i, c3

i and c4

i) increase with a

constant rate of during the iterations.



Figure 3. Flow Chart of Genetic Algorithm

Figure 4. (a) Construction Graph for a base layout with 6 nodes and 11 pipes (b) Sequential Solution construction by Tree Growing Algorithm

Using genetic algorithm for branched layout optimization In this paper an especial form of Genetic Algorithm is used to deal with the problem of the layout optimization. To avoid the problems associated with formation of infeasible solutions, according to methodology proposed in Walters and Smith (1995), instead of coding the solutions into strings the proposed algorithm

deals directly with branched layouts (figure 5). A major innovative element is in the way design information is passed down from parents to offspring. The total genetic information of the parents is pooled, so the pool contains more information than is required to form the offspring, guaranteeing that feasible offspring can be formed by suitable growth algorithm. The flowchart of this algorithm is illustrated in Figure 4 and its particularities are discussed in the following


sections. Coding scheme Conventional Genetic Algorithms map the values of the design variables onto a code (chromosome) using a binary or non-binary alphabet. The single chromosome forms a long string of digits and completely defines a design. Subsequently the optimization algorithm manipulates the chromosomes (corresponding to a population of designs) using the processes of selection, crossover and mutation. However, using conventional coding scheme in the problem of selecting a tree network from a non-directed base graph presents significant difficulties. These arise from the impossibility of defining a way of mapping the design variables for this problem onto a chromosome such that the conventional crossover and mutation processes necessarily produce feasible solutions. To avoid these problems, instead of coding the solutions into strings, the algorithm deals directly with layouts. An efficient Tree Growing Algorithm is used for solution generation and therefore despite loop model, Genetic algorithm only generates feasible solutions and there is no need to check the feasibility of solutions. Tree growing algorithm The tree growing algorithm is used to produce spanning trees on a random basis from a base graph. The trees produced will on average be biased towards those which diverge from a root node specified on the base graph (Walter and Smith, 1995). To grow a tree, the following algorithm was devised, in which: C is the set of nodes contained within the growing tree.

(Partial solution) is the set of arcs within the growing

tree.

N( ) is the set of arcs adjacent to the growing tree,

any one of which has an equal chance of forming the next branch. 1. Identify the root node Nr 2. Initialize C = [Nr] 3. Initialize = [ ]

4. Initialize N( ) = [arcs in base graph connected to

root node] 5. Choose arc, a, at random from N( )

6.

7. Identify newly connected node, N 8.

9. ac = [arcs connected to N in base graph (excluding arc a)]

Rajabpour and Talebbeydokhti 138 Update N( ), by removing arc a and any newly

infeasible arcs, and adding any of arcs of ac, which are 10. feasible candidates. N( ) now contains all

feasible choices for the next arc of the tree. 10.1. (remove newly connected arc

from list) 10.2. For each arc ac(i) in ac, Is ac(i) in N( ) already?

Yes: (remove ac(i) from list, as

tree is now such that adding ac(i) would cause a loop) No: Are both end nodes of ac(i) in C ? Yes: (leave list unaltered, as adding ac(i)

would cause a loop in the tree) No: (add ac(i) to list)

Initial population The evolutionary process starts with an initial population of random feasible solutions for the problem. This population is created by applying tree growing algorithm on predefined maximum layout as base graph. The maximum layout should include all the possible and useful links which could contribute to the optimum layout of the network. The complete set of all candidate links can be theoretically very large, but physical conditions such as plot boundaries, gravel roads, etc., generally restrict the actual number of candidate links to a considerably smaller subset. Selection technique The basic mechanism in evolution is the survival of the fittest individuals. Therefore a procedure that encourages the selection of the best parents is required. In this work, a stochastic sampling mechanism has been used, based on the ‘‘roulette wheel algorithm’’ technique. The proposed selection function lays out a line in which each parent corresponds to a section of the line of length proportional to its expectation. The algorithm moves along the line in steps of equal size, one step for each parent. At each step, the algorithm allocates a parent from the section it lands on. The first step is a uniform random number less than the step size. Directly using fitness as a measure of breeding probability entails a number of shortcomings. (Goldberg, 1989). To overcome these drawbacks, a relative fitness is used as a measure of selection probability in the roulette wheel algorithm, instead of the true fitness. The relative fitness is calculated as a function of the individuals’ ranking. The rank of an individual is its position in the sorted scores. The rank of the fittest individual is 1, the next fittest is 2, and so on.



Figure 5. Crossover process in Genetic Algorithm

Crossover operator The crossover operator is another basic operator in genetic algorithms. Crossover combines two individuals, or parents, to form a new individual, or child, for the next generation. Figure 6 illustrates the proposed crossover process. As illustrated in figure 6, first, the genetic information from the two selected parents are combined into a common pool, by superimposing the two parental tree graphs to form a base graph. At the second step, a new spanning tree (a new child) is produced by applying the tree growing algorithm on the base graph. Mutation operator Mutation functions make random changes in the individuals in the population, which provide genetic diversity and enable the Genetic Algorithm to search a broader space. In this work, the algorithm combines a selected individual with a random sub set of maximum layout to form a new base graph and uses the tree growing algorithm to produce a new spanning tree (a new individual) from this base graph.

Reproduction options These options determine how the genetic algorithm creates children at each new generation. Elite count specifies the number of individuals that are guaranteed to survive to the next generation. Crossover fraction specifies the fraction of the next generation, other than elite individuals, that are produced by crossover. The remaining individuals, other than elite individuals, in the next generation are produced by mutation. Model application The proposed methods are applied for simultaneous layout and pipe size optimization of two benchmark examples in the literature. Example 1 The first example to be considered is that of a simple network shown if Figure 7(a). The network consists of nine nodes, twelve links, and a source located at node number 9. This example has been considered as a test network by Geem et al. (2000) to test the performance



Figure 6 (a) Maximum layout of Network1, (b) Optimal Layout Obtained for Network 1, Afshar (2005), (c) Optimal Layout Obtained for Network 1 with Proposed Methods

Table 1. Nodal Demand for Network 1.

Node 1 2 3 4 5 5 7 8 9

Demand (l/s) 10 20 10 20 10 20 10 20 …

Table 2. Cost Data for Network 1

Diameter(mm) 1 2 3 4 6 8 10 12 14 16 18 20 22

Cost($/m) 2 5 8 11 16 23 32 50 60 90 130 170 300

Table 3. Optimal Layout and Pipe Sizes for Network 1.

Link D1(cm) L 1(m) D2(cm) L2(m) Node P(m) 1 10 100 … … 1 30.02 2 … … … … 2 31.93 3 14 100 … … 3 30.62 4 … … … … 4 34.76 5 8 26.13 6 73.87 5 32.9 6 … … … … 6 30.02 7 14 100 … … 7 42.05 8 … … … … 8 39.78 9 10 100 … … 9 … 10 8 21.34 6 78.66

11 14 53.62 12 46.38

12 14 100 … …

Cost 37764.1

of the harmony search method proposed for layout optimization. Afshar (2005, 2006 and 2007) used this example to emphasize the necessity of joint layout pipe size optimization. The water demand at each node of the network is shown in Table 1. Table 2 shows costs of different pipe sizes. The Hazen-Williams coefficient is assumed equal to 130 for all the pipes. The elevation of all the demand nodes is set equal to zero and that of

the source node is assumed to be 50 m. Minimum pressure requirement of 30 meter is used at all the demand nodes.

Figure 7(c) and table 3 show the resulting layout, pipe

diameters and nodal heads obtained using the following values for JPSO parameters: s= 10, m=30, c1

i =0.55,

c1f=0.2, c2

i= 0.15, c3

i =0.15 and c4

i=0.15. It should be

remarked that the method was able to find the solution



Figure 7. Maximum layout of Network2

Table 4. Cost Data for Network 2

Diameter(mm) 125 150 200 250 300 350 400 450 500 550 600 650 700 Cost($/m) 58 62 71.7 88.9 112.3 138.7 169 207 248 297 347 405 470

of $37,764 within 160 network evaluations. On the other hand, the Genetic Algorithm was able to find a solution same as the solution found by JPSO faster and within only 30 network evaluations. This solution is obtained considering Population Size= 10, Elite count=1 and Crossover fraction: 0.8. Example 2 The second example is considered to demonstrate the applicability and efficiency of the method for the layout and pipe size design of real-world tree networks. The network, shown in Figure 8, is the small part of the Winnipeg system with 2 sources, 20 nodes, and 37 possible links [3]. The costs of different pipes are shown in Table 4. The length of the 37 possible links included in the maximum layout of the network along with the nodal demands and minimum nodal head requirements are given in Table 5.

Figure 9 and Table 6 show the resulting layout, pipe diameters, and nodal heads obtained using the proposed Genetic Algorithm method. Progress of evolution process is shown in Figure 10. The method was able to find the solution of $1,599,013 within 440

function evaluation. This solution is obtained considering Population Size= 20, Elite count=2 and Crossover fraction: 0.95. DISCUSSION Tables 7 and 8 show using hybrid approach leads to better solutions in too much smaller numbers of function evaluations for both of considered examples. Here it should be noted that CPU time for each function evaluation at non hybrid approaches (Afshar, 2005, 2006 and 2007) is smaller than corresponding one at hybrid approach. Because, function evaluation at non hybrid approaches involves network hydraulic analysis, but calculating objective function at hybrid approach consists of applying LIDM to optimize pipe sizes which takes more time. In example 1, it should be mentioned that greater performance of Hybrid approach is not only because of LIDM's abilities in pipe sizing but also the obtained layout is better. To prove this, we optimized pipe sizes of layout shown in figure 7(b) (Layout of best solution among non hybrid methods) by LIDM which resulted in solution of $38,301which is still more expensive in compare to $37,761 (Cost of solution



Table 5. Details of Network 2.

Link From To Length(m) Node Q(l/s) Min.Head(m)

1 1 2 760 1 165 75 2 1 4 520 2 220 74 3 1 6 890 3 145 73 4 2 3 1120 4 165 72 5 2 5 610 5 … 102 6 2 6 680 6 140 73 7 3 5 680 7 175 67 8 3 7 870 8 180 72 9 4 8 860 9 140 70 10 4 9 980 10 160 69 11 5 7 890 11 170 71 12 5 10 750 12 160 70 13 6 9 620 13 190 64 14 6 10 800 14 200 73 15 7 12 730 15 150 73 16 7 13 680 16 … 96 17 8 9 480 17 165 67 18 8 15 860 18 140 70 19 9 11 800 19 185 70 20 9 14 770 20 165 67 21 10 11 350 22 10 12 620 23 11 12 670 24 11 16 790 25 11 18 1150 26 12 13 750 27 12 17 550 28 13 17 700 29 14 15 500 30 14 16 450 31 14 19 750 32 15 19 720 33 16 18 540 34 16 19 700 35 17 18 850 36 18 20 750 37 19 20 970

Figure 8. Optimal Layout Obtained for Network 2 with Proposed Method.



Table 6. Optimal Pipe Size Solutions for Network 2

Link D1(cm) L 1(m) D2(cm) L2(m) Node P(m)

1 40 760 45 … 1 79.4 2 30 387.4 35 132.6 2 90.3 5 50 610 55 … 3 73.0 6 25 533.6 30 146.4 4 72.3 7 20 128 25 552 5 … 11 35 506.8 40 383.2 6 73.1 16 30 680 35 … 7 78.4 12 40 741.8 45 8.2 8 72.1 21 25 102.1 30 247.9 9 77.4 22 30 620 35 … 10 79.6 17 30 97.4 35 382.6 11 71.1 20 40 770 45 … 12 70.1 29 25 411.2 30 88.8 13 64.1 30 50 450 55 … 14 87.9 33 40 540 45 … 15 73.1 34 25 5.1 30 844.9 16 … 35 25 75 30 675 17 67.1 36 25 414.2 30 285.8 18 81.0

Cost($) 1599013.02 19 70.1 20 67.1

Figure 9. Progress of evolution of example 2 (Genetic Algorithm).

Table 7. Summary of results obtained by different researchers for Network 1

Method Cost of best solution ($)

No. of function evaluations

1 Separate Layout-Size Optimization (Afshar , 2007) 39,800 … 2 GA (Afshar , 2007) 39,400 7,500 3 ACO (Afshar , 2006) 39,500 8,900 4 ACO (Afshar , 2005) 38,600 3,500 5 Hybrid ACO/LIDM (Kashkooli and Monem, 2009) 37,761 20 6 JPSO 37,761 160 7 Genetic Algorithm 37,761 30

obtained by all of hybrid methods for example 1).

According to Trent (1954), the total number of spanning trees that can be formed from maximum layout of example 1 is 192 which is search space size of hybrid approach. As shown in table 9, adapting hybrid approach to tackle the problem of joint layout

and pipe size optimization leads to much smaller search space for layout optimizer in comparison with other approaches (Afshar, 2005, 2006 and 2007). Greater performance of hybrid methods is mainly due to smaller size of search space and strong pipe sizing algorithm. Table 7, 8, 9.



Table 8. Summary of results obtained by different researchers for Network 2

Method Cost of best solution ($)

No. of function evaluations

1 GA (Afshar , 2007) 1,783,086 100,000 2 ACO (Afshar , 2006) 1,713,741 17,800 3 ACO (Afshar , 2005) 1,696,031 12,840 4 Hybrid ACO/LIDM (Kashkooli and Monem, 2009) 1,599,013 210 5 Genetic Algorithm 1,599,013 440

Table 9. Search space dimensions considered at different approaches for simultaneous layout and pipe size optimization of example 1

Method Search space dimensions Relative dimensions

GA (Afshar , 2007) and ACO (Afshar , 2006) (13+1)12

2.95*1011

ACO (Afshar , 2005) 192*138

138

Hybrid methods 192 1

CONCLUSION The application of the hybrid JPSO/LIDM and GA/LIDM method is presented for the simultaneous layout and pipe size optimization of water distribution networks. The JPSO and GA algorithm acts as an outer loop and LIDM acts as an inner loop. Then solution costs can be calculated and JPSO and GA procedure continues. Appling LIDM as inner loop for calculating fitness value of each solution significantly reduces the numbers of JPSO and GA solution component, because there is no need any more to considered solution components as the list of all allowable pipe sizes of links in maximum layout, and merely the links of maximum layout can be considered as solution components. On the other hand according to abilities of LIDM, the proposed model is able allocate mixed pipe diameters to some links and reduce the costs even more. The great reduction in search space size and the ability to allocate mixed pipe diameters are significant advantages of the proposed method.

Proposed methods are applied for simultaneous layout and pipe size optimization of a small benchmark example in the literature and the results are presented and compared to the existing results. The results showed that the developed methods have significant advantages compared to other methods used. Afterwards, another example is considered to illustrate the efficiency of the proposed Genetic Algorithm for layout and size optimization of real-world networks. REFERENCES Afshar MH (2005). Layout and Size Optimization of TREE-LIKE Pipe

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