applying tabu search and simulated annealing to the optimal design of sewer networks

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Applying tabu search and simulatedannealing to the optimal design ofsewer networksShung-Fu Yeh a , Chien-Wei Chu a , Yao-Jen Chang a & Min-Der Lina

a Department of Environmental Engineering , National ChungHsing University , Republic of ChinaPublished online: 11 Oct 2010.

To cite this article: Shung-Fu Yeh , Chien-Wei Chu , Yao-Jen Chang & Min-Der Lin (2011) Applyingtabu search and simulated annealing to the optimal design of sewer networks, EngineeringOptimization, 43:2, 159-174, DOI: 10.1080/0305215X.2010.482989

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Engineering OptimizationVol. 43, No. 2, February 2011, 159–174

Applying tabu search and simulated annealing to the optimaldesign of sewer networks

Shung-Fu Yeh, Chien-Wei Chu, Yao-Jen Chang and Min-Der Lin*

Department of Environmental Engineering, National Chung Hsing University, Republic of China

(Received 11 November 2009; final version received 16 March 2010 )

Optimizations of sewer network designs create complicated and highly nonlinear problems wherein con-ventional optimization techniques often get easily bogged down in local optima and cannot successfullyaddress such problems. In the past decades, heuristic algorithms possessing robust and efficient globalsearch capabilities have helped to solve continuous and discrete optimization problems and have demon-strated considerable promise. This study applied tabu search (TS) and simulated annealing (SA) to theoptimization of sewer network designs. For a case study, this article used the sewer network design of acentral Taiwan township, which contains significantly varied elevations, and the optimal designs from TSand SA were compared with the original official design. The results show that, in contrast with the originaldesign’s failure to satisfy the minimum flow-velocity requirements, both TS and SA achieved least-costsolutions that also fulfilled all the constraints of the design criteria. According to the average performanceof 200 trials, SA outperformed TS in both robustness and efficiency for solving sewer network optimizationproblems.

Keywords: sewer network, optimization, tabu search, simulated annealing

1. Introduction

Since the sewer network (SN) is a basic component of urban infrastructures and requires hugeinvestments, optimization techniques for identifying the least-cost SN design in a given contexthave received considerable attention. SN optimization amounts to an NP-hard, discrete-variable,and highly nonlinear problem. Several studies have employed various conventional optimizationtechniques such as linear programming (Elimam et al. 1989), dynamic programming (Guptaet al. 1983, Walsh and Brown 1973), nonlinear programming (Gupta et al. 1976, Holland 1966),and the Lagrange multiplier (Swamee 2001) to yield optimal SN designs. However, most of theaforementioned conventional algorithms are either easily trapped in local optima or unsuitablefor solutions to discrete-variable problems.

In contrast, studies have proven that heuristic optimization algorithms such as the geneticalgorithm (GA), tabu search (TS), simulated annealing (SA), scatter search (SS), and ant algo-rithms (AA) can avoid local-optima entrapment and exhibit a greater likelihood of yielding the

*Corresponding author. Email: [email protected]

ISSN 0305-215X print/ISSN 1029-0273 online© 2011 Taylor & FrancisDOI: 10.1080/0305215X.2010.482989http://www.informaworld.com

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160 S.-F. Yeh et al.

global optimal solution for a given situation. Desher and Davis (1986) and Charalambous andElimam (1990) initially introduced heuristic optimization concepts into SN designs. Liang et al.(2004) used GA and TS to solve an SN optimization problem and respectively obtained 9% lowerand 16% lower construction costs than were obtained by means of traditional design methods.However, only four pipe diameters are selectable in their model, so there are legitimate doubtsabout whether or not such models can help solve problems featuring a wider variety of avail-able pipe diameters. Afshar et al. (2006) applied GA to the storm water management model(SWMM) for the optimal design of storm water networks. Guo et al. (2007) proposed an opti-mizer based on ‘cellular-automata (CA)’ for SN design, and this optimizer obtained near-optimalsolutions within a remarkably smaller number of computational steps than was the case with GA.Afshar et al. (2007) employed partially constraint ant colony optimization (ACO) algorithm tosolve a benchmark storm sewer network optimization problem. Izquierdo et al. (2008) solvedthe optimal design problems of wastewater collection networks through the particle swarm opti-mization (PSO) technique and obtained better solutions than dynamic programming. Pan andKao (2009) have integrated quadratic programming (QP) with GA to solve SN optimizationproblems. An example cited in Li and Matthew (1990) served as a case study for evaluatingboth the applicability and the efficiency of the GA-QP model. The results indicate that the GA-QP model could obtain various good design alternatives within an acceptable computationaltime.

A review of the literature reveals that studies on the application of heuristic algorithms to SN-optimization problems have not been satisfactorily fruitful and are worth further investigation.Although GA is the most popular and widely used heuristic algorithm, the global optimizationperformance of TS and SA are also promising in many combinatorial problems (Cunha andSousa 1999, 2001, Cunha and Ribeiro 2004, Gupta 2002, Jeon and Kim 2004, Kolahan and Liang1996). Therefore, the present study develops an optimization procedure that, based on TS and SA,can help solve SN optimization problems and can further expand the conceptual framework andimplementation of TS and SA. The present study investigates a sewer network designs belongingto central Taiwan townships as case studies, and the optimal designs from TS and SA are comparedwith the original official designs.

2. Optimization model formulation

This article’s proposed SN optimization model is formulated as a least-cost problem for identifyingthe pipe sizes and the excavation depths that yield the minimum cost for a given layout. The pipelayout and the sewerage amount are assumed to be known factors, and no pumps are consideredin the networks. The objective function expressing the network cost is formulated as a functionof the pipe diameters, pipe lengths, and excavation depths. The optimal design problem for theSN discussed in this study can be shown as the following mathematical statement:

Minimize f (Di, Li, Si) =m∑

i=1

Costi(Di, Li, Hi(Si)) (1)

where m is the total number of pipes in the system; Di is the diameter of pipe i selected fromthe set of commercial pipe sizes {D}; Li is the length of pipe i; Hi(Si) is the excavation depth ofpipe i, which is a function of slope Si ; and Costi (Di , Li , Hi(Si)) the construction cost of pipe i,which is a function of the slope of the pipe diameter, pipe length, and excavation depth.

Many design criteria act as constraints on SN designs. For example, the requirements of flowvelocity, of pipes’ cover depth, of water-depth ratios, and of pipe diameter must be considered foreach pipeline. A detailed explanation of these requirements is presented below.

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Engineering Optimization 161

2.1. Flow-velocity constraint

Since an SN conveys sewage to waste water treatment plants via gravitational force, the flowvelocity should be maintained within the minimum and maximum requirements in each pipe:

0.6 m/sec ≤ Vi ≤ 3.0 m/sec i = 1, 2, . . . , m (2)

where Vi represents the wastewater flow velocity of pipe i. The flow velocity can be calculatedaccording to Manning’s equation:

V = 1

nR

23 S

12 , Q = V × A (3)

where V is flow velocity (m/sec), n is the roughness coefficient, R is the hydraulic radius (m), S isthe hydraulic gradient, Q is the discharge volume per unit time (m3/sec), and A is the cross-sectionarea (m2) of the pipe.

Constraint Equation (2) ensures that the minimum velocity is large enough to prevent solidwastewater-deposit substances in the pipes, and the maximum flow velocity is not so high as toabrade and damage the pipe walls.

2.2. Cover-depth constraint

For each pipe in an SN, the minimum cover-depth requirement is another constraint:

Ei ≥ Emin i = 1, 2, . . . , m (4)

where Ei represents the thickness of overlaying soil for burying pipe i, and Emin represents theminimum required thickness of overlaying soil. The minimum thickness of overlaying soil is 2metres in this study.

2.3. ‘Water-depth ratio’ constraint

The next set of constraints expresses the ‘water-depth ratio’ requirements at each pipe:⎧⎪⎪⎨⎪⎪⎩

for Di ≤ 0.5 m,di

Di

≤ 0.5

for Di ≥ 0.6 m, 0.7 ≤ di

Di

≤ 0.8i = 1, . . . , m (5)

wheredi

Di

represents the water-depth ratio of pipe i, and di is the depth of the sewer in pipe i. Restrictionsof the water-depth ratio ensure (1) a correct relationship between the sewage pipes and gravityflow, (2) adequate ventilation of the sewage pipes to prevent hydrogen-sulphide erosion of pipewalls, and (3) appropriate spaces in cases where the amount of waste water increases abruptly.

2.4. Pipe-diameter constraint

The diameter of each pipe must correspond to a commercial size set, and the downstream pipelinediameter must be greater than or equal to the upstream pipeline diameter.

Available in this study are 16 commercial-pipe sizes whose diameters range from 0.25 metreto 1.65 metres; the cost functions are mentioned in Yeh et al. (2008).

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3. Tabu search

Tabu search (TS) is a meta-heuristic approach proposed by Glover (1986). It is a powerful opti-mization technique that has been successfully applied to a variety of combinatorial optimizationproblems such as vehicle routing problems, travelling salesman problems, water network prob-lems and problems concerning groundwater restoration and management (Chao 2002, Cunha andRibeiro 2004, Gupta 2002, Tsubakitani and Evans 1998, Rasiah et al. 2005). TS stores the tracksof past searches in an ‘adaptive memory’ system to prevent cycling of previous moves, and uses‘effective searching’ strategies that accept even inferior solutions to avoid confining the searchwithin a particular area; that is, to avoid entrapment in local optima. Basic versions of TS can befound in Glover and Laguna (1997).

The computational procedures of the proposed TS for solving SN optimization problems aresummarized in the following section, and the flow chart is presented in Figure 1.

Step 1: Move

The ‘move’ mechanism represents the process of pointing the current solution state towardanother one. This study integrates perturbations into the current solution to generate adjacent

Generate Random Initial Solution

Move From Current Solution

Candidate List

Tabu List

Aspiration Criterion

Accept and Update Tabu List

Termination criterion

Stop

NO

YES

YES

Found Next Solution

NO

Intensification and Diversification Strategy

YES

YES

NO

NO

Start

Figure 1. TS flowchart.

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solutions:

Dt+1i = Dt

i ± �DN (6)

St+1i = St

i ± �SL (7)

where Dti is the diameter of pipe i of the t th iteration, St

1 is the slope of pipe i of the t th iteration,and �DN and �SL are the perturbations of pipe diameter and slope, respectively.

Step 2: Tabu list

Tabu list (TL) is the most distinct feature of TS. It records previously encountered moves (solu-tions) to prevent cycling back to the previously visited solutions. The searching strategy is affectedby the size of TL, which is often determined by experience. Several studies investigated TL

Figure 2. SA flowchart.

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issues. Liang et al. (2004) designed TS-based sewer networks and found that larger TL does notnecessarily yield better solutions. Glover (1986) suggested seven as the best TL size.

Step 3: Aspiration criterion

Activation of an aspiration criterion serves to overcome the tabu status of a move whenever thesolution then produced is better than the best historical solution achieved. An aspiration criterionallows a move, even when the move is a tabu, if new move results in a solution better than thecurrent best-known solution.

Step 4: Intensification and diversification strategy

Intensification strategy rests on modifications of the choice rule that promote move combinationsand solution features historically found good. They may also initiate a return to attractive regions

200,L=17.5M

200,L=12M

200,L=11.48M

200,L=14.4M

200,L=20.6M

Ha0205H=1.68M

200,L=12.8M

Ha0202B1H=1.83M

H a0202A1H=2.4M

Ha0202A2H =1.74M

Ha0203A2H=1.67M

200L

=7.

3M

200,L=12.8M

Ha0203A1H=1.71M

Ha0203H=2.14M

Ha0202B4H=1.74M

Ha0202B3H=1.81M

Ha0202B3-1H=1.05M

200,

L=

27.1

M

H=1.74MHa0202A1

H=1.42M

200,L=28.5M

H=1.74M

H=1.63M

Ha0202

Ha0202B1

H a0202B1-1

200,L=5.1M

L :Pipe LengthPipe Diameter

200L

=23

.4M

H:Manhole Depth

Figure 3. The geographic location of Nanjuang township.

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where there can be a more thorough search. The best solution should both record and store so itcan examine their immediate neighbourhoods; explicit memory is closely related to the basis ofpast iterative operations to constitute an elite-candidate list. Modification of choice rules helpsdiversify strategies and helps drive the search into unvisited regions and generate solutions thatdiffer in various significant ways from those searched before. Intensification and diversificationare fundamental cornerstones of longer-term memory in TS and reinforce each other.

Step 5: Termination criterion

The proposed algorithm has two termination criteria: the first criterion is terminate after a max-imum number of iterations; and the second criterion is terminate after a maximum number ofiterations without improvement of solution.

Figure 4. Pipeline layout of the Nanjuang SN.

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166S.-F.Yeh

etal.

Table 1. Basic data of the Nanjuang SN.

Node Ground elevation(m) Node Ground elevation(m)Pipe Pipe Pipe Pipeno. From To Qp (cms) length (m) Upstream Upstream no. From To Qp (cms) length (m) Upstream Upstream

1 A28 A27 0.00003 75 221.60 220.25 32 B5 B4 0.0060 52 220.97 219.502 A27 A26 0.00005 60 220.25 217.79 33 B4 B3 0.0070 34 219.50 216.653 A26 A25 0.0010 47 217.79 215.86 34 B3 B2 0.0070 48 216.65 215.334 A25 A24 0.0010 55 215.86 214.24 35 B2 B1 0.0080 55 215.33 213.345 A24 A23 0.0020 64 214.24 212.26 36 B1 A15 0.0080 50 213.50 213.346 A23 A22 0.0020 16 212.26 212.13 37 C9 C8 0.0010 45 213.34 214.837 A22 A21 0.0020 30 212.13 211.93 38 C8 C7 0.0010 50 214.83 214.448 A21 A20 0.0030 23 211.93 211.74 39 C7 C6 0.0020 70 214.44 213.299 A20 A19 0.0040 38 211.74 212.15 40 C6 C5 0.0020 36 213.29 212.72

10 A19 A18 0.0050 42 212.15 213.07 41 C5 C4 0.0020 78 212.72 212.711 A18 A17 0.0070 50 213.07 212.13 42 C4 C3 0.0020 70 224.25 220.3712 A17 A16 0.0070 41 212.13 211.15 43 C3 C2 0.0020 54 220.37 218.9513 A16 A15 0.0100 87 211.15 212.70 44 C2 C1 0.0020 26 218.95 218.2214 A15 A14 0.0180 37 212.70 212.69 45 C1 B5 0.0030 51 218.22 221.9315 A14 A13 0.0190 70 212.69 211.14 46 D13 D12 0.0020 38 221.93 217.4916 A13 A12 0.0250 21 212.17 211.14 47 D12 D11 0.0020 56 217.49 215.7317 A12 A11 0.0270 58 211.14 211.07 48 D11 D10 0.0030 51 215.73 214.4018 A11 A10 0.0280 70 211.07 209.41 49 D10 D9 0.0030 23 214.40 213.5019 A10 A9 0.0290 66 209.41 208.79 50 D9 D8 0.0030 48 230.64 225.2420 A9 A8 0.0330 64 208.79 207.28 51 D8 D7 0.0030 20 225.24 226.2721 A8 A7 0.0360 92 207.28 205.89 52 D7 D6 0.0030 29 226.27 225.9322 A7 A6 0.0380 92 205.89 204.25 53 D6 D5 0.0050 40 225.93 225.5223 A6 A5 0.0390 30 204.25 202.62 54 D5 D4 0.0060 62 225.52 224.8824 A5 A4 0.0400 90 202.62 202.1 55 D4 D3 0.0060 18 224.88 221.1925 A4 A3 0.0460 86 202.10 200.78 56 D3 D2 0.0070 50 221.19 215.8526 A3 A2 0.0500 125 200.78 198.61 57 D2 D1 0.0070 70 215.85 217.0627 A2 A1 0.0500 350 203.07 201.48 58 D1 A13 0.0070 11 217.06 220.3528 B9 B8 0.0020 65 201.48 199.32 59 E3 E2 0.0040 100 220.35 220.8529 B8 B7 0.0030 64 199.32 198.61 60 E2 E1 0.0040 100 220.85 218.9630 B7 B6 0.0030 56 198.61 195.58 61 E1 A3 0.0040 76 218.96 212.1731 B6 B5 0.0030 45 195.58 189.73

Note: Qp is design discharge in cubic metres per second.

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SA (T0=9.4912)

42,000,000

42,500,000

43,000,000

43,500,000

44,000,000

44,500,000

45,000,000

50N 100N 150N 200N 250N 300N

Markov chain length

Cos

t (N

T$)

=0.8

=0.85

=0.9

=0.95

Figure 5. Costs of the SN obtained via different SA frameworks in 100 trials.

Table 2. Pre-optimality analysis of SA parameter settings.

Markov chain length 300N∗ 250N 200N 150N 100N 50N

Temperature decreasing rate (α) = 0.95, T0 = 9.4912Max cost (NT$) 44,924,249 45,130,564 44,786,706 44,593,926 44,916,561 44,948,628Min cost (NT$) 43,297,764 43,297,764 43,636,548 43,636,548 43,595,669 43,607,932Average cost (NT$) 44,473,647 44,451,722 44,474,713 44,468,335 44,508,228 44,551,287Standard deviation 285,110 310,615 256,218 265,979 245,257 165,069




∗N is the number of decision variables.

Table 3. Pre-optimality analysis of TS parameter settings.

Tabu list (TL) 7 24 (20%N∗) 49 (40%N) 73 (60%N) 97 (80%N)

Max cost (NT$) 46,385,744 45,930,959 44,849,894 46,291,886 46,501,951Min cost (NT$) 43,427,703 43,435,899 43,445,933 43,443,100 43,439,412Average cost (NT$) 43,849,926 43,800,238 43,778,478 43,895,756 43,841,681Standard deviation 481,013 404,945 320,779 495,237 454,385

∗N is the number of decision variables.

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4. Simulated annealing

SA is a stochastic global-optimization method that was proposed by Kirkpatrick et al. (1983) andthat is based on a strong analogy between the physical annealing process of solids and the problemof solving combinatorial optimization problems. The basic concept gets its name from the physicalprocess called annealing, which involves heating up a solid to a sufficiently high temperature andthen slowly cooling the material into one possessing a minimum-energy crystalline structure.The technique reflects an attempt, in the formation of solid-material arrangements, to modelthe behaviour of atoms during annealing. The technique starts from an initial solution at a hightemperature, and makes a series of moves according to an annealing schedule. The temperature isgradually decreased as SA proceeds until the stopping condition is met. Figure 2 presents a flowchart of the computational procedure of the proposed SA.

The computational procedure of the proposed SA for solving SN optimization problems issummarized in the following steps.

Step 1: Setting the initial parameters

The setting of initial parameters includes defining the initial temperature T0, the temperature-decreasing rate α, and the Markov chain length (MCL). The initial solution X0 is randomlygenerated.

Step 2: Generate neighbour solution Xi+1

Similar to the process of TS, SA introduces perturbations to the current solution Xi to generateneighbour solution Xi+1.

Step 3: Calculate the change of object function �F = F(Xi+1) − F(Xi)

Step 4: Determine the acceptance of Xi+1

If �F < 0, the neighbour solution Xi+1 is unconditionally accepted. However, if �F > 0, Xi+1

is only accepted with a probability P , which is defined as

P = exp

(−�F

kT

)(8)

where k is the Boltzmann constant and T is the current temperature. If the Xi+1 is accepted,the current solution Xi is replaced by Xi+1 and proceed to Step 5. Otherwise, go back toStep 2.

Table 4. Evaluation statistics concerning TS and SA in the 200 trials.

Test item TS SA

Minimum construction cost (NT$) 43,230,006 43,297,764Maximum iteration 20,309,482 1,050,257Minimum iteration 2,935,690 722,914Average iteration 5,758,189 873,640Average computational time (sec) 2,380 174

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Figure 6. Evolution of the Nanjuang SN’s minimum costs relative to SA and TS.

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Step 5: Check the termination criterion.

If the termination conditions of the cooling schedule are satisfied, end the algorithm; otherwisego to Step 6.

Step 6: Reduce the temperature

If the current temperature Ti decreases at the rate of α, then go to Step 2.

5. An illustrative example of SN

To evaluate the optimization performance of TS and SA, an illustrative example, the SN ofNanjuang Township, was used as a case study. Nanjuang Township, located in central Taiwan andhaving a total area of about 7.64 × 105 square kilometres, is composed of nine villages and hasa population of about 4500. The total length of the pipelines is approximately 3620 metres andconveys waste water from households to a waste water treatment plant whose designed capacity is4900 CMD. The layout of the SN contains one main link and four junction links and consists of 62nodes connected by 61 pipes. There are 16 diameter sizes available for each pipe. Figures 3 and 4demonstrate the geographical information and the SN layout of Nanjuang Township, respectively.The basic data of the SN is summarized in Table 1.

6. Results and discussion

6.1. Parameter analysis of TS and SA

To achieve good problem-solving performances, it is usually necessary to testify and to fine tuneuser-defined parameters of heuristic algorithms. This study varied the parameters of TS and SA toidentify the most beneficial corresponding values; that is, values exhibiting the best optimizationperformance. The parameters evaluated in SA are temperature-decreasing rate (α) and Markovchain length (MCL); the size of tabu list (TL) was tested in relation to TS. Several values foreach parameter were evaluated while the other parameters remained constant. For each setting ofparameters, there were solutions for 100 trials so that a statistically meaningful average evolutioncould be arrived at. The implementation of each trial rested on theVisual Fortran 90 in combinationwith an Intel Xeon Processor 3 GHz PC.

Figure 5 presents the minimum costs of the Nanjuang SN obtained by 24 parameter settings ofSA with α ranging from 0.8 to 0.95 and MCL ranging from 50N to 300N , where N is the numberof decision variables. For each parameter setting, Table 2 summarizes the maximum, minimum,

Table 5. Success rates of TS and SA for optimizing the Nanjuang-SNdesign in the 200 trials.

TS SA

Success rate of achieving solutions within a3% deviation of the global minimum

87.0% 90.5%


89.0% 93.5%


89.5% 94.0%

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ization171

Table 6. Detailed ‘SN pipeline’-design results stemming from the original official design, the TS-based design, and the SA-based design.

Original official design TS SA Original official design TS SAPipe Pipeno. D (m) V(m/s) d/D D (m) V(m/s) d/D D (m) V(m/s) d/D no. D(m) V(m/s) d/D D (m) V(m/s) d/D D (m) V(m/s) d/D

1 0.25 0.26 0.05 0.35 0.60 0.04 0.35 0.60 0.04 32 0.25 0.84 0.33 0.25 0.61 0.27 0.25 0.60 0.272 0.25 0.36 0.08 0.35 0.60 0.04 0.35 0.60 0.04 33 0.25 0.85 0.34 0.25 0.63 0.29 0.25 0.60 0.303 0.25 0.41 0.10 0.35 0.60 0.05 0.35 0.60 0.05 34 0.25 0.85 0.34 0.25 0.63 0.29 0.25 0.60 0.304 0.25 0.48 0.13 0.35 0.60 0.05 0.35 0.60 0.05 35 0.25 0.86 0.35 0.25 0.60 0.32 0.25 0.60 0.325 0.25 0.48 0.13 0.35 0.60 0.06 0.35 0.60 0.06 36 0.25 0.86 0.35 0.25 0.64 0.31 0.25 0.60 0.326 0.25 0.55 0.16 0.35 0.60 0.06 0.35 0.60 0.06 37 0.25 0.41 0.10 0.25 0.84 0.05 0.25 0.84 0.057 0.25 0.55 0.16 0.35 0.60 0.06 0.35 0.60 0.06 38 0.25 0.41 0.10 0.25 0.60 0.06 0.25 0.60 0.068 0.25 0.61 0.19 0.35 0.61 0.08 0.35 0.60 0.08 39 0.25 0.46 0.12 0.25 0.60 0.10 0.25 0.60 0.109 0.25 0.67 0.22 0.35 0.60 0.10 0.35 0.60 0.10 40 0.25 0.46 0.12 0.25 0.61 0.10 0.25 0.60 0.1010 0.25 0.68 0.23 0.35 0.61 0.12 0.35 0.60 0.12 41 0.25 0.70 0.24 0.25 0.61 0.10 0.25 0.60 0.1011 0.25 0.75 0.27 0.35 0.60 0.17 0.35 0.60 0.17 42 0.25 0.70 0.24 0.25 0.61 0.10 0.25 0.60 0.1012 0.25 0.76 0.28 0.35 0.63 0.16 0.35 0.60 0.17 43 0.25 0.70 0.24 0.25 0.61 0.10 0.25 0.60 0.1013 0.25 0.85 0.34 0.35 0.62 0.23 0.35 0.60 0.24 44 0.25 0.72 0.25 0.25 0.61 0.10 0.25 0.60 0.1014 0.25 1.01 0.49 0.35 0.60 0.35 0.35 0.61 0.35 45 0.25 0.73 0.26 0.25 0.60 0.14 0.25 0.60 0.1415 0.25 1.03 0.50 0.35 0.66 0.34 0.35 0.61 0.36 46 0.25 0.51 0.14 0.25 1.07 0.07 0.25 1.08 0.0716 0.30 1.00 0.47 0.35 0.62 0.43 0.35 0.61 0.44 47 0.25 0.51 0.14 0.25 0.60 0.10 0.25 0.60 0.1017 0.30 1.02 0.49 0.35 0.68 0.42 0.35 0.61 0.46 48 0.25 0.61 0.19 0.25 0.61 0.14 0.25 0.60 0.1418 0.30 1.03 0.50 0.35 0.67 0.44 0.35 0.61 0.48 49 0.25 0.61 0.19 0.25 0.61 0.14 0.25 0.60 0.1419 0.30 1.04 0.50 0.35 0.70 0.44 0.35 0.60 0.50 50 0.25 0.61 0.19 0.25 0.60 0.14 0.25 0.60 0.1420 0.30 1.16 0.50 0.35 1.32 0.32 0.35 1.28 0.32 51 0.25 0.61 0.19 0.25 1.00 0.09 0.25 1.02 0.0921 0.30 1.26 0.50 0.35 1.39 0.32 0.35 1.41 0.32 52 0.25 0.61 0.19 0.25 1.20 0.08 0.25 1.21 0.0822 0.30 1.35 0.50 0.35 1.41 0.33 0.35 1.42 0.33 53 0.25 0.67 0.22 0.25 0.60 0.23 0.25 0.60 0.2323 0.30 1.36 0.50 0.35 1.41 0.34 0.35 1.42 0.33 54 0.25 0.70 0.24 0.25 0.62 0.26 0.25 0.60 0.2724 0.30 1.38 0.50 0.35 1.35 0.35 0.35 1.34 0.35 55 0.25 0.73 0.26 0.25 0.62 0.26 0.25 0.60 0.2725 0.30 1.51 0.50 0.35 1.68 0.33 0.35 1.70 0.33 56 0.25 0.75 0.27 0.25 0.61 0.29 0.25 0.60 0.3026 0.40 1.60 0.50 0.35 1.71 0.35 0.35 1.71 0.35 57 0.25 0.76 0.28 0.25 1.14 0.18 0.25 1.15 0.1727 0.40 0.99 0.47 0.35 1.50 0.38 0.35 1.49 0.38 58 0.25 0.76 0.28 0.25 1.53 0.13 0.25 1.54 0.1328 0.25 0.53 0.15 0.25 0.62 0.10 0.25 0.62 0.10 59 0.25 0.63 0.20 0.25 0.71 0.16 0.25 0.72 0.1629 0.25 0.57 0.17 0.25 0.89 0.10 0.25 0.89 0.10 60 0.25 0.63 0.20 0.25 0.78 0.15 0.25 0.79 0.1530 0.25 0.57 0.17 0.25 0.73 0.12 0.25 0.73 0.12 61 0.25 0.63 0.20 0.25 0.60 0.19 0.25 0.60 0.1931 0.25 0.61 0.19 0.25 0.88 0.10 0.25 0.89 0.10

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and average of the best solutions found in 100 trials as well as the standard deviations of the 100solutions. The standard deviation of the best solutions obtained by different trials is an importantmeasure of the optimization algorithm’s robustness; Table 2 shows that the standard deviationsare all less than 1% (in contrast with the average best solution). The lowest standard deviationis found when α = 0.95 and MLC = 50N , and therefore, the current study’s SA employs thisparameter setting.

Table 3 summarizes the evaluation results of TS, with TL sizes ranging from 7 to 80% N . Thestandard deviations are all less than 1.2%, and the lowest standard deviation is found where TLsize equals 40% N , which this study uses in TS.

6.2. Comparison of the optimization results of TS and SA

This study implemented both of the algorithms in order to compare the optimization performanceof TS and SA, and to optimize the Nanjuang SN for 200 trials. As Table 4 indicates, the minimumcosts obtained through TS and those obtained through SA are very close to each other (43,230,006vs. 43,297,764 NT$). However, the average number of iterations for TS and SA are 5,758,189 and873,640, respectively, indicating that the optimization performance of SA is much more efficientthan that of TS. Figure 6 shows that SA converged on and located the minimum cost much fasterthan TS. This rapidity may reflect the need of TS to execute more iteration to check the activationthresholds for the aspiration-criterion strategy, the intensification strategy, and the diversificationstrategy.

To evaluate the performance of SA and TS searches for the optimal SN designs, this studyhas investigated the variable ‘success rate,’ which represents the percentage of trials successfullyleading to the global optimum (or neighbourhoods of the global optimum) among a given numberof optimization trials. Table 5 presents the success rates of TS and SA based on 200 trials. It showsthat the success rates of SA and TS for obtaining solutions (with a possible 3% deviation fromthe global optimum) were over 90% and 85%, respectively. It can be concluded that SA is morereliable than TS.

Because Nanjuang’s variation of elevations is significant, maintaining the required minimumdesign flow velocity becomes a critical issue, especially when only a small amount of upstreamwaste water is collected. These types of situations frequently result in general SN designs’ notsatisfying the minimum flow-velocity requirement, insofar as the designs usually employ graduallysloping invert elevations to reduce the construction (excavation) cost. Table 6 compares three setsof the Nanjuang SN designs: the original official design, the TS-based design, and the SA-baseddesign. It is noteworthy that, for the original design, there are 16 pipelines whose flow velocitiesviolate the minimum requirement (V ≥ 0.6m/s); both TS and SA can yield designs satisfying allof the design criteria. As shown in Table 7, both the average slopes of the TS-based and SA-based

Table 7. The SN link slopes and costs stemming from the original official design, the TS-based design, and the SA-baseddesign.

Original official design TS SA

Link Average slope Cost (NT$) Average slope Cost (NT$) Average slope Cost (NT$)

A 1.00% 21,214,130 1.63% 22,033,181 1.63% 22,154,172B 1.00% 6,261,089 1.86% 5,707,445 1.80% 5,679,292C 0.84% 4,925,312 2.91% 5,842,409 2.87% 5,825,621D 0.95% 6,544,894 3.84% 6,707,836 3.86% 6,700,114E 0.95% 2,935,729 1.54% 2,939,135 1.56% 2,938,565

Total construction cost 41,881,154 43,230,006 43,297,764

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SN-design links are greater than those of the original official design. Therefore, TS and SA cansatisfy the minimum velocity requirement, although they resulted in construction costs slightlyhigher (by 3.2% and 3.4%, respectively) than the construction costs of the original design.

7. Conclusion

This study successfully employed TS and SA to solve the least-cost design problem of an SN whoseelevations are significantly varied. This study also conducted a pre-optimal analysis concerningthe parameter settings of TS and SA to identify the most beneficial parameter values (that is, thevalues that corresponded to the best optimization performance).

In contrast to the original official design, which was found to violate the minimum flow-velocity requirements, both TS and SA provided the least-cost designs that completely satisfiedall the design-criteria constraints. Notably, SA was found to be more reliable and efficient thanTS for optimal-design solutions to SN problems.

Acknowledgements

This research was funded by a grant from the National Science Council of the Republic of China (NSC 96-2628-E-005-001-MY3). The authors gratefully appreciate this support.

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applying tabu search and simulated annealing to the optimal design of sewer networks

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