Applying Artificial Immune System to Minimize Construction Cost of Water

8/8/2019 Applying Artificial Immune System to Minimize Construction Cost of Water

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Applying Artificial Immune System to Minimize Construction Cost of WaterDistribution Networks

Min-Der Lin, Chien-Wei Chu Department of Environmental Engineering, National Chung Hsing University, Taiwan, R.O.C.

[email protected]

Abstract

Recent studies increasingly indicate that heuristicalgorithms are powerful and effective for solving water distribution networks (WDN) optimization problems. This

paper employs the artificial immune system (AIS), to solvethe least-cost design problems of WDN. The well-knownWDN problem, New York City tunnel problem, is used asa case study. The results reveal that the computational efficiency and quality obtained by AIS are as good as or

better than the results reported in literatures. Furthermore, AIS significantly retrenches the number of iterations for searching the optimal solution. AIS is shownto be an evolutionary technique capable of solving complex combination optimization problems.

1. Introduction

Since water distribution networks (WDN) are vitalcomponents in urban infrastructure and requireconsiderable investment, optimization of WDN hasreceived considerable attention over the past 30 years.Such optimization problems consider various aspects suchas capital, operation and maintenance costs, layout design,hydraulics, reliability, material availability, and demand

pattern. Various traditional optimization techniques, suchas linear programming gradient [1-3], dynamic

programming [4], and nonlinear programming [5], have been developed principally for solving minimal-cost problems. However, most of these models are either linearized or simplified to make application possible.Additionally, selection of commercially availablediscrete-value pipe diameters to generate a least-costwater supply network is an NP-hard problem that isextremely difficult to solve using traditional optimizationtechniques [6]. Recently, probabilistic heuristicalgorithms such as genetic algorithms (GA), simulatedannealing (SA), tabu search (TS), and ant colonyoptimization (ACO), have been utilized to solve water distribution networks (WDN) optimization problems [7-11]. Another important heuristic algorithm in simulating

biological processes is the artificial immune system (AIS)which is a relatively new optimization algorithm thatimitates the immune system defense process againstinvaders in a biological body [12]. This work is the first toapply AIS to a WDN optimization problem.

2. Optimization model formulation

The WDN optimization model proposed in this studyis a least-cost problem for identifying the pipe size thatgenerates minimum cost for a given layout. We assumethat pipe layout, nodal demands, head and velocityrequirements are all known. The networks have no pumpsand the reservoirs are considered water source nodes withfixed heads. The objective function for network cost is

formulated a function of pipe diameters. The optimaldesign problem for a general water distribution network can be shown as the following mathematical statement [8,13-14]:

∑=

=n

iiin L DC D D f Minimize

11 ),(),,( … (1)

where n is the total number of pipes in the system, D i isthe diameter of pipe i selected from the set of commercial

pipe sizes {D}, and C i( D i , L i) is the cost of pipe i with thediameter D i and length Li. This objective function isconditioned using the following constraints.

For each junction node, the mass conservation lawshould be satisfied:

NN mS QQ mmout min∈∀=−∑ ∑ ,,, (2)

where Q in,m and Qout,m are the inflow and outflow of nodem, respectively, S m is the external inflow or demand atnode m, and NN is the node set.

For each basic loop in the network, the energyconservation law is utilized as another constraint set:

NLk H Loopl k

k ∈∀=Δ∑

∈

,0 (3)

where Δ H k is head loss in pipe k , and NL is loop set. Thehead loss in each pipe is the head difference between itsconnected nodes, and can be formulated using the Hazen-Williams equation:

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NP k

QQ DC

Lw H H H k k

k k

k k k k

∈∀

=−=Δ−1

,2,1α

β α (4)

where H 1 ,k and H 2 ,k are head of both ends of a pipe, k , w isthe numerical conservation constant (depending on units),

C k is the roughness coefficient of pipe k (depending onthe material), α and β are regression coefficients, k Q is theflow of pipe k ; and NP is the pipe set. Generally, the pipeswith large values of w result in increased head loss and,therefore, require increased diameters to deliver the sameamount of water because WDN may violate minimum

pressure requirements, whereas small w values may justmeet the constraints. When a WDN is designed usingdifferent w values, the designs with high w values willhave unfavorable hydraulic conditions and result inexpensive alternatives.

The next set of constraints express the pressure headrequirements at each node:

NN m H H mm∈∀≥ ,min (5)

where minm H is the minimum pressure head requirement at

each node. Moreover, the diameter of each pipe must belong to a commercial size set:

NP k D Dk ∈∀∈ },{ (6)

3. Methodology for artificial immune system

The immune system discriminates between self cellsand foreign non-self pathogens, and is the first line of adefense system against foreign pathogens. Immuneresponses include natural immunity that quicklyeliminates foreign non-self pathogens and adaptiveimmunity that targets particular pathogens.

The second line of defense, humoral immunity andcell-mediated immunity, comprise the immune responseof immunocompetent cells, which include B lymphocytes(or B cells) and T lymphocytes (or T cell) [15]. Both celltypes have surface receptor molecules (the B cell receptor molecule is also called an antibody). Foreign pathogens,also called antigens, are recognized by antibodies. After an antigen is recognized by immune cell receptors, theantigen stimulates B cells to proliferate cells that secreteantibodies that are plasma cells. Proliferation of B cells inthe immune system is followed by mitosis. Once

proliferate B cells, the system has a clone of cells thatcopies stimulative B cell [16]. This function, called aclonal selection state, occurs when a pathogen invades anorganism. During immune response, some of the immunecells that recognize the pathogens proliferate become

plasma cells, whereas others are maintained as memory

cells [17]. When exposed to a second antigenic stimulus,these cells begin differentiating into plasma cells capableof producing high-affinity antibodies, preselected for thespecific antigen that stimulated the primary response.From this viewpoint, antibodies and antigen can belooked as through operation generate feasible solutionsand global best solution, respectively, for an objective

function when solving WDN optimization problems.Figure 1 presents the computational procedure of AIS.For further information, AIS theory and processes aredescribed in detail by [18] and [12].

Figure 1. Flowchart of artificial immune system

Step 1 : Define antigen .When using AIS to evaluate WDN optimization,

representation of the objective function and constraintsare regarded as antigens.

Step 2: Generate an initial population of antibodies.The initial population of antibodies is generated, as in

the genetic algorithm procedure, via random coding.Binary and real number coding are the two most commoncoding techniques used. In this study, decision variablesare the commercial pipe diameters in each segment of the

WDN. Step 3: Evaluate the fitness of each antibody in current population.

The fitness of each antibody in the current generationis calculated based on its objective function value and

potential constraint violations. In evaluating the fitness of individual antibodies, the constraint requirements werealso examined. When a constraint is violated, the degreeof violation is weighted to penalize its fitness. Antibodies

629629629629



with high fitness ( fit i) represent good individuals. Insolving the minimum cost problem for WDN design, the

fit i of each (antibody) i is calculate by

)1/(1 ii f fit += (7)

where f i is the cost of the WDN, as indicated in Eq. (1). Step 4: Select the n best antibodies in the initial population based on their fitness. Step 5: Clone these n best antibodies to generate atemporary population of clone set (C).

Generate the clone set ( C ) for the best n individuals(antibodies) selected in step 4. The clone set ( C ) has

possession of better antibodies, thereby increasing thefunction of fitness with the antigen [18].

Step 6: Clone set (C) executes genetic operations for generating new antibodies.

In this step, genetic operations, such as crossover andmutation, similar to those in a GA, are performed by the

clone set (C) to generate new and generally improvedantibodies. The crossover operation produces newantibodies by mixing the genetic material in chromosomefrom the original antibodies in the current population.

Step 7 : Evaluate the new fitness of new antibodiesgenerated by genetic operations. Re-select improvedindividuals that are superior to individuals in the memoryset. When the memory set is updated, improvedindividuals are replaced by inferior individuals in thememory set, thereby generating a new memory set [19].

Step 8: When a termination criterion is satisfied, thealgorithm is stopped; otherwise, return to Step 3. Thetermination criteria used in this work is the maximumnumber of iterations.

4. Case study

The New York City Tunnel (NYCT) problem, whichwas first considered by Schaake and Lai in 1969[4], andhas since been the subject of numerous studies [5, 14, 20-22], was used to demonstrate the utility of the proposedapproach. The layout of the NYCT system consists of 20nodes connected by 21 tunnels in Figure 2. For eachduplicate tunnel, 16 possible decisions, including 15available diameters, and the “do nothing” option, exist;therefore, the search space for this optimization problemcontains 16 21 = 1.93 × 1025 possible designs. For summaries of system data and unit costs of tunnelnetwork, please refer to [14].

1

R E S E R V O I R 3 0 0 f t

2

10

7

6

12

20

16

11

13

1 8 1 9

15

14

98

17

5

4

3

1 15

14

1 3

12

1 7 1 8

19

11

21

10

20

16

9

8

4

5

6

7

2

3

Figure 2. Layout of New York City Tunnel System

Tables 1 and 2 present summaries of optimal network designs identified with the AIS, and those from previousstudies. The figures in Table 1 represent new pipediameters to be added in parallel to existing lines. For the

NYCT problem, the optimization model was run twice,each time using different coefficients in the Hazen-Williams Eq. (4). The values w = 10.5088, α = 1.85, and β = 4.87 were used for the first run (AIS run 1), whereas w= 10.9031, α = 1.85, and β = 4.87 were used in the secondrun (AIS run 2). Computational results were based on 100runs of the AIS using different random starting points.The optimal costs of $37.13 million (for w =10.5088) and$40.42 million (for w =10.9031) obtained by the AIS,respectively, are as good as those obtained using the GA(for w =10.5088 and 10.9031) and TS (for w =10.5088)[9].

As the computational results indicated, optimalsolutions were found by the AIS in a minimum of 13,700and 14,500 evaluations for w =10.5088 and w =10.9031,respectively, (Table 1 and Figure 3) compared with37,186–1,000,000 evaluations required by the GA andfast messy genetic algorithm (fmGA). The computationalresult indicated that AIS’s evaluations retrenched over 63% of iterations for previous solutions [8, 21]. Table 3

presents the success rate for AIS based on the percentageof algorithms achieving a global optimum in 100 runs.The computational result reveals that 100% of solutionsachieved using AIS was within 5% of the globalminimum cost.

630630630630



Table 1. Optimal Solutions of New York City TunnelSystem Obtained by Different Techniques

fmGAa

fast messy Genetic Algorithm NA

b

not availableAIS

c artificial immune system

Table 2 . Nodal Pressures for Optimal Solutions of NewYork City Tunnel System Obtained by Different

Techniques

Table 3. Success Rates of IA for Searching OptimalDesigns of New York City Tunnel System

0 4000 8000 12000 16000Evaluation Number

20

40

60

80

100

120

P i p e

C o s

t ( m i l l i o n

$ )

IA

Figure 3. Evolution of pipe cost for New York network using AIS ( w=10.5088)

5. Conclusion

This work is the first attempt to apply AIS for solvingthe least-cost design problem of WDNs, with the goal of generating insights that may prove useful. By combininga large number of solutions, the AIS investigates differentregions of the solution space to generate solutions andsearches for the global optimum solution.

The AIS developed in this study was evaluated bysolving the NYCT WDN optimization problem, whichemploys two different Hazen-Williams coefficient valuesin previous studies. The AIS provides solutions as good asthose obtained by other studies in terms of ability to findthe global optimum solution and computationalefficiency. The AIS obtained the optimal solution infewer iterations than some of the most competitivealgorithms, such as the GA, fmGA and TS. The successrates based on 100 runs of the AIS using different randomstarting points demonstrate that AIS is a promisingtechnique for solving the WDN optimization problem.

Application of AIS to the WDN optimization problemis still in its infancy and further improvements is

necessary. For example, sensitivity analysis of AIS for parameters utilized in the model, and development of algorithmic strategies to improve computationalefficiency and quality, is likely required for solving

practical or large WDN optimization problems.

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Acknowledgement

The authors would like to thank the anonymousreviewers for their expertise and auditing which helpimprove this work.

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