Arc based ant colony optimization algorithm for solving ...scientiairanica.sharif.edu/article_4079_d486bc42109518f864fcbd1e0… · This particular behavior of ant colonies has inspired

Scientia Iranica A (2017) 24(3), 953{965

Sharif University of TechnologyScientia Iranica

Transactions A: Civil Engineeringwww.scientiairanica.com

Arc based ant colony optimization algorithm for solvingsewer network design optimization problem

R. Moeini�

Department of Civil Engineering, University of Isfahan, Postal Code: 81746-73441, Isfahan, Iran.

Received 31 August 2015; received in revised form 30 January 2016; accepted 10 May 2016

KEYWORDSArc based antcolony optimizationalgorithm;Sewer network;Exploration;Exploitation;Heuristic information;Optimal design.

Abstract. In this paper, Arc Based Ant Colony Optimization Algorithm (ABACOA)is used to solve sewer network design optimization problem with proposing two di�erentformulations. In both of the proposed formulations, i.e. UABAC and CABAC, the coverdepths of sewer network nodes are taken as decision variables of the problem. Theconstrained version of ABACOA (CABAC) is also proposed in the second formulationto optimally determine the cover depths of the sewer network nodes. The constrainedversion of ABACOA is proposed here to satisfy slope constraint explicitly leading toreduction of search space of the problem, which is compared with that by the unconstrainedarc based ACOA (UABAC). The ABACOA has two signi�cant advantages of e�cientimplementation of the exploration and exploitation features along with an easy andstraightforward de�nition of the heuristic information for the ants over the alternativeusual point based formulation. Two benchmark test examples are solved here using theproposed formulations, and the results are presented and compared with those obtainedby alternative point-based formulation and other existing methods. The results show thesuperiority of the proposed ABACOA formulation, especially the constrained version of it,to optimally solve the sewer network design optimization.© 2017 Sharif University of Technology. All rights reserved.

1. Introduction

Nowadays, optimization is one of the most important�elds of any engineering problems. In optimizationproblems, one should �nd the minimum/maximumof a function of many variables, called decision vari-ables, where the arguments may be subjected to someconstraints. Cost saving is a major goal of mostengineering problems, such as design and implemen-tation of a sewer network. Due to the high costassociated with the construction of a sewer network,many research studies over the last 50 years haveattempted to develop di�erent techniques in which theyare used to minimize the capital costs associated withsuch infrastructure. However, optimal design of a sewer

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network is a highly constrained mixed-integer nonlinearprogramming (MINLP) problem presenting a challengeeven to the modern heuristic search methods due tocontinues and discrete variables, nonlinear functions,and the high number of constraints involved [1].

Within the last decade, many researchers havefocused on the sewer network design optimization prob-lem and proposed di�erent methods from traditionaloptimization techniques to modern heuristic searchmethods. The methods used for sewer network designoptimization problem can be classi�ed as: 1) LinearProgramming (LP); 2) Nonlinear programming (NLP);3) Dynamic Programming (DP); 4) meta-heuristic; and5) hybrid methods. Each of these methods has itsown limitations. It is worth noting that, over thelast 30 years, a new kind of approximate method,called meta-heuristic, has been proposed that triesto combine basic heuristic methods in higher level

954 R. Moeini/Scientia Iranica, Transactions A: Civil Engineering 24 (2017) 953{965

structures aimed to explore a search space e�cientlyand e�ectively. This method is based on the factthat nature has the best built-in optimization tech-niques for maintaining the global systems. Sharingthe information between insects, such as �nding theshortest path of the food sources by ants, brings aboutthe idea of optimization techniques that have openeda wide new �eld for research. This class of methodincludes, but is not restricted to, Genetic Algorithms(GA), Simulated Annealing (SA), Tabu Search (TS),Particle Swarm Optimization (PSO) algorithm, AntColony Optimization Algorithm (ACOA), etc. In thelast three decades, the use of this method has extremelyreceived the attention of the researchers in di�erentengineering �elds. However, the main limitation of thismethod is problem dependence. In other words, noneof the proposed meta-heuristic methods is unique to alltypes of engineering problems. Also, the application ofthis method to real-world problem is full of complexi-ties.

In the �eld of sewer network design optimizationproblem with �xed layout, many di�erent researchstudies have been studied. Haestad [2] and Guoet al. [3] have reviewed the research studies of this�eld over the last 40 years. In general, traditionalmethods [4], LP [5-7], NLP [8,9], DP [10-16], meta-heuristic [17-32], and hybrid methods [33-38] have beenused in this �eld.

The ACOA is one of the meta-heuristic methodsin which it mimics the natural foraging behavior of acolony of real ants to �nd the shortest path between thefood source and their nest. When ants are traveling,they deposit a substance called pheromone forminga pheromone trail, which is more attractive to otherants to follow them. Ants can smell the pheromoneand when choosing their path, they incline to choose,in all probability, paths marked by strong pheromoneconcentrations. It has been shown experimentally thatthis pheromone trail following behavior can give rise,once employed by a colony of ants, to the emergenceof the shortest paths. This particular behavior of antcolonies has inspired the ACOA in which a colonyof arti�cial ants cooperates to �nd good solutions fordiscrete optimization problems. The basic algorithm ofACOA is the Ant System (AS). Many other algorithms,such as Ant Colony System (ACS), elitist Ant System(ASelite), elitist-rank Ant System (ASrank), and Max-Min Ant System (MMAS), have been introduced toimprove the performance of the AS.

Due to the iterative nature of the solution gen-erations of meta-heuristic methods, the solution spaceuses the knowledge about solutions that have alreadybeen found to further guide the search. The searchingbehavior of the method can be described by twomain features, named exploration and exploitation.It is worth noting that the exploration is the ability

of the algorithm to search extensively through thesolution space, and the exploitation is the ability of themethod to search more comprehensively in the localneighborhood where good solutions have previouslybeen found. However, these features are in con ictwith each other. One of the most di�cult aspects topay attention to in meta-heuristic method is the trade-o� between these two features. To obtain good results,an agent should prefer actions tried in the past andfound to be e�ective in producing proper solutions, butto discover them, he has to try actions not previouslyselected [39].

In this paper, a new formulation, i.e. Arc BasedACOA (ABACOA) formulation, is used to solve sewernetwork design optimization problem. The ABACOAwas previously proposed for single and multi-reservoiroperation problems by Moeini and Afshar [39,40].The ABACOA has two signi�cant advantages overthe alternative point-based formulation as presentedin the following. First, the pheromone trails areassociated with the arcs of the graph leading to amore e�cient implementation of the exploration andexploitation features of the ACOA. Second, each arcof the proposed graph is representative of downstreamand upstream cover depths of each nodal pipe, andhence, average pipe cover depths lead to an easy andstraightforward de�nition of the heuristic informationfor the ants, so there will be a useful property thatis absent in the existing alternative formulations. It isworth noting that the performances and abilities of thisnew formulation, ABACOA, should be studied to solvedi�erent optimization problems, such as sewer networkdesign optimization problem, considered here.

To show the unique feature of the proposedABACOA, in this paper, this algorithm is used tosolve sewer network design optimization problem byproposing two di�erent formulations. In both formu-lations, the cover depths of sewer network nodes aretaken as decision variables of the problem. In the�rst formulation, named UABAC, the unconstrainedversion of ABACOA is used to determine the coverdepths of sewer network nodes. However, in the secondformulation, named CABAC, the characteristics of thearc based de�nition of the problem along with the serialfeatures of sewer network problem are then used todevelop a constrained version of the formulation. Theconstrained version of the ABACOA is proposed herefor the explicit satisfaction of slope constraint. Theconstrained version is used to recognize the infeasibleregions of the search space and remove them from thesearch space of the problem. Two benchmark testexamples are solved here using the proposed methods,and the results are presented and compared withunconstrained and constrained versions of point-basedACOA (PBACOA) and with those obtained by theother existing methods.

R. Moeini/Scientia Iranica, Transactions A: Civil Engineering 24 (2017) 953{965 955

2. Sewer network design optimization model

Sewer network is one of the most important infras-tructures of modern city which is designed to collectsewerages from city and transfer them to wastewatertreatment plants. A sewer network consists of man-holes, pipes, lifts and pumping stations, and otherappurtenances. Sewer network is a more expensive in-frastructure. Therefore, an optimization model shouldbe de�ned to design the least-cost sewer network. Thesewer network components can be optimally found bysolving this optimization model [1].

The objective functions and constraints of theoptimization model should be de�ned and be solvedby e�ective methods in order to �nd sewer networkparameters such as pipe diameters, slopes, average pipecover depths, drops and pumping station locations andheights. The objective function of this optimizationmodel is minimization of sewer network constructioncost. In the absence of any pumps and drops, itcan be formulated for gravitational sewer network asfollows [37]:

Minimize C=NPXl=1

LlKpip(dl; El)+NMXnm=1

Kman(hnm);(1)

where:C Construction cost function of sewer

networkNP Total number of sewer pipesNM Total number of manholesLl Length of pipe l (l = 1; :::; NP )Kpip The unit cost of sewer pipe provision

and installation de�ned as a functionof its diameter (dl) and average coverdepth (El)

Kman The construction cost of manhole as afunction of manhole height (hnm)

Furthermore, the constraints of the sewer networkdesign optimization model with a pre-speci�ed layoutare hydraulic, operational, and availability constraintswhich are formulated as:

Vmin � Vl � Vmax 8l = 1; :::; NP; (2)

Sl � Smin 8l = 1; :::; NP; (3)

Emin � El � Emax 8l = 1; :::; NP; (4)

�min � �l � �max 8l = 1; :::; NP; (5)

�l =�yd

�l

8l = 1; :::; NP; (6)

Ql =1nalr

2=3l S1=2

l 8l = 1; :::; NP; (7)

dl 2 D 8l = 1; :::; NP; (8)

dl � dl 8l = 1; :::; NP; (9)

where:Vl Flow velocity of pipe l at the design

owVmax Maximum allowable velocity of sewer

owVmin Minimum allowable velocity of sewer

owSl Slope of the sewer pipeSmin Minimum sewer pipe slopeEmin Minimum cover depth of sewer pipeEmax Maximum cover depth of sewer pipeEl Average cover depth of pipe ldl Diameter of sewer pipe lyl Sewer ow depth in pipe l�max Maximum allowable relative ow depth�min Minimum allowable relative ow depth�l Relative ow depth of pipe lQl The discharge of sewer pipe lal Wetted cross-section area of sewer pipe

l at sewer ow depth of ylrl Hydraulic radius of the sewer pipe l at

sewer ow depth of yln Manning coe�cientD Discrete set of commercially available

sewer pipe diameters assumed equalfor all pipes of the network

dl Set of downstream pipe diameters ofpipe l

This optimization model of the sewer networkdesign with a pre-speci�ed layout (Eqs. (1) to (9)) isa highly constrained mixed-integer nonlinear problem(MINLP) in which the complexity of the problemrequires that an e�ective method be proposed to solveit.

3. The proposed methods for solvingoptimization model

Herein, ACOA is used to solve sewer network designoptimization problem. The basic steps of the ACOAfor solving optimization problem were presented byAfshar and Moeini [41], and therefore, will not bepresented here. However, Figure 1 shows the basicsteps of ACOA procedure brie y to solve optimizationproblem. Formulation of an optimization problemby ACOA requires that the problem be de�ned asgraph G = (DP;OPi;CO). This graph is de�ned bya set of nodes referred to as decision points, DP =


Figure 1. ACOA procedure to solve optimization problem.

fdp1; dp2; :::; dpIg, and edges referred to as optionsavailable at each decision point i, OPi = fopijg withcorresponding costs CO = fcoijg, [1].

Here, an arc based formulation of ACOA isproposed by de�ning another form of the graph forits e�cient application to solve sewer network designoptimization problem. This formulation has two sig-ni�cant advantages over the alternative usual pointbased formulation. The signi�cance of the proposedformulation is that each arc of the new de�ned graphis representative of some parameters leading to aneasy and straightforward de�nition of the heuristicinformation for the ants, which is a useful propertyabsent in the existing alternative point-based formula-

tion. Furthermore, the pheromone trails are associatedwith the arcs of the new de�ned graph leading tomore e�cient implementation of the exploration andexploitation features of the ACOA. The characteristicsof the arc based ACOA along with the serial featureof sewer network optimization problem are also usedto develop a constrained version of arc based ACOA.Furthermore, usual form of the graph for ACOA is alsoused to propose unconstrained and constrained point-based ACOA formulations, i.e. UPBAC and CPBAC,respectively, for comparison purposes. It should benoted that these formulations are the modi�ed formsof Afshar's [21] formulations with minor corrections.

It is worth noting that the problem graph is


very much dependent on the decision variables of theproblem. Here, in the absence of pumps and drops, thecover depths of the sewer network nodes are consideredas the decision variables of the problem leading to aneasy de�nition of the problem graph. Furthermore,an assumption used here is that the gravitationalsewer network is considered. This assumption leadsto the fact that, for each node, the downstream nodalelevations of entering pipes are the same as upstreamnodal elevations of outgoing pipe.

It should be noted that the proposed formulationsare used here for sewer network with prede�ned layout,and therefore, the sewer discharge is known. In otherwords, the steady-state condition is considered for hy-draulic model. Furthermore, the proposed formulationsare outlined here using MMAS for sewer network designoptimization problem.

3.1. Unconstrained Point-Based ACOA(UPBAC)

In this formulation, named UPBAC, the conventionalapplication of ACOA based on the usual form ofthe graph is used to determine decision variables.By considering the sewer network nodes as decisionpoints, the options available at each decision pointare represented by all �nite numbers of discrete coverdepths of sewer network nodes. The graph representa-tion of this formulation is shown in Figure 2, wherevertical lines represent the decision points (nodes),circles represent the components of nodal cover depths(j = 1; :::; J) at each decision point i (i = 1; :::; I), thedash lines represent potential solutions on the graph,bold circles represent nodal cover depths selected byarbitrary ant, and �nally, the bold lines represent atrial solution on the graph constructed by an arbi-trary ant. In this formulation, each ant starts itsmovement from an arbitrary decision point and selectsa cover depth from the set of available cover depthsfor each node. It should be noted that based on themethodology of UPBAC formulation, the constructionof layout with negative slope is possible in which itis an infeasible solution. Here, penalty method isused to avoid infeasible solution, which is explainedlater.

By determining the nodal cover depths, nodalelevations and pipe slopes of sewer network, the sewer

Figure 2. Problem graph of unconstrained version ofpoint based ACOA.

network pipes' diameters should be determined tocomplete the design process. Di�erent methods canbe used to calculate the pipe diameters when the pipeslopes are known at steady-state condition. Here, thepipe diameters are calculated explicitly, such that allthe constraints are fully satis�ed, if possible. Therefore,by starting the design processes from upstream pipes,the smallest commercially available diameter fully sat-isfying Constraints (2), (5), (6), (8), and (9) is takenas sewer pipe diameter.

3.2. Constrained Point-Based ACOA(CPBAC)

In CPBAC formulation, the constrained version ofACOA based on the usual form of the graph is used todetermine decision variables. The constrained versionof point-based ACOA is proposed for the explicitenforcement of the problem constraints by limiting theant's options to feasible ones at each decision point ofthe problem. This leads to limiting the search spaceof the problem to feasible region, and therefore willbe shown to improve convergence characteristics andobtain better results.

By considering the sewer network nodes as deci-sion points, in this formulation, the options available ateach decision point are represented by a �nite numberof discrete cover depths of sewer network nodes, whichhave satis�ed minimum slope constraint. Here, for asewer network with prede�ned layout, each ant startsits movement form inlets and selects an option, coverdepth, from the available feasible options' list. Inother words, the feasible cover depths of downstreamnodes can be easily de�ned using known upstream coverdepths and minimum slope constraint. Figure 3 repre-sents the graph of this formulation, where vertical linesrepresent the decision points (sewer network nodes),circles represent the components of nodal cover depths(j = 1; :::; J) at each decision point i (i = 1; :::; I)with the dashed ones representing the infeasible andthe solid one representing feasible cover depths, thedash lines represent potential solutions on the graph,bold circles represent nodal cover depths selected byarbitrary ant, and �nally, the bold lines represent atrial solution on the graph constructed by an arbitraryant. It is worth noting that, in contrast to UPBACformulation, in this formulation, each ant starts its

Figure 3. Problem graph of constrained version of pointbased ACOA.


movement from inlets of sewer network and selects acover depth from feasible cover depths' list, which ismuch smaller than original list. By determining thenodal cover depths, other parameters are determinedsuch as UPBAC formulation.

3.3. Unconstrained Arc Based ACOA(UABAC)

In UABAC formulation, the arc based ACOA is usedto determine decision variables. With the nodalcover depths taken as the decision variables, in thisformulation, the allowable range of the nodal coverdepths is discretised into a �xed number of J discretevalues, and each discrete point j (j = 1; 2; :::; J) istaken as the decision point of the problem. Theset of options, OPi, available at a decision point iwill then be represented by a total number of J arcsjoining the decision point, i, representing the jthdiscrete upstream nodal cover depth to all discretedownstream nodal cover depths of corresponding pipe.Figure 4 represents the graph of this formulation, wherevertical lines represent the sewer network nodes, circlesrepresent the components of discretized nodal coverdepths (j = 1; ::; J) representing decision points dpi(i = 1; :::; I), the dash lines represent arcs connectingtwo decision points, bold circles represent nodal coverdepths selected by arbitrary ant, and �nally, the boldlines represent a trial solution on the graph constructedby an arbitrary ant.

It is worth noting that, in this formulation, eachant starts its movement from inlets and it only hasto choose one arc out of the total number of Jarcs joining the decision point under consideration tothe decision points of the corresponding pipe whichrepresent upstream and downstream nodal elevationsof each pipe, respectively. Once an option, an arc,is chosen by an ant, the next decision point to moveto is known. This process continues when all decisionpoints of the problem are covered by ant. Finally, bydetermining the nodal cover depths, other parametersare determined such as UPBAC formulation. It shouldbe noted that based on the methodology of thisformulation, the construction of layout with negativeslope is possible in which it is an infeasible solution.

Figure 4. Problem graph of unconstrained version of arcbased ACOA.

Here, penalty method is also used to avoid infeasiblesolution, which is explained later.

The arc based formulation has signi�cant ad-vantages over the alternative point-based formulation,which are presented before. Heuristic information,�ij , is a useful component of the ACOA in which theproper de�nition of it is required for the best algorithmperformance. Generally, heuristic value represents thecosts of choosing option j at point i, and therefore,related to the cost function equation. Here, basedon the sewer network cost function of Eq. (1), thecost function of sewer network is only related to theaverage cover depths and diameters of sewer pipe,and therefore, the heuristic information can be easilycomputed in the proposed arc based formulation. Asnoted earlier, in the proposed arc based formulation,each arc will uniquely de�ne the value of the upstreamand downstream nodal cover depths of each pipe. Thisin turn leads to the possibility of de�ning average coverdepth of each pipe, El, leading to the possibility ofde�ning heuristic information. In other words, this inturn leads to the possibility of de�ning some part ofcost function, and consequently heuristic informationfor each arc. This possibility does not exist in thealternative usual point-based formulation in which eachoption only de�nes the value of the cover depthsof corresponding nodes with which no cost can beassociated. The heuristic information of the problemcan, therefore, be de�ned as:

�ij =1

kpip(El);

El =Ei + Ej

2; (10)

where all parameters have been de�ned before. Fur-thermore, in the proposed arc based formulation, thepheromone trails are now associated with the arcs ofthe graph leading to more e�cient implementation ofthe exploration and exploitation mechanisms of theACOAs. These points will be veri�ed later whenconsidering numerical examples.

3.4. Constrained Arc Based ACOA (CABAC)In this formulation, named CABAC, the proposed arcbased ACOA formulation is augmented with the inter-racial building capability of ACOA by proposing theconstrained version of arc based ACOA formulation.The constrained version of arc based ACOA is proposedfor the explicit enforcement of the minimum slopeconstraint by limiting the ant's options to feasible onesat each decision point of the problem.

With the nodal cover depths also taken as thedecision variables, in this formulation, the allowablerange of the nodal cover depths is discretized into a�xed number of J discrete value, and each discrete


Figure 5. Problem graph of constrained version of arcbased ACOA.

point j (j = 1; 2; :::; J) is taken as the decision pointof the problem. The set of options, OPi, availableat a decision point i, will then be represented bya number of J arcs joining the decision point i,representing the jth discrete upstream nodal coverdepth, to some discrete downstream nodal cover depthsof the corresponding pipe satisfying minimum slopeconstraint. In other words, in the proposed CABAC,the ant is forced to choose an option only from thosearcs which have satis�ed minimum slope constraint.The modi�ed graph representation of the problem forthe application of the proposed CABAC is shown inFigure 5, in which, circles represent the componentsof discretized nodal cover depths (j = 1; :::; J) repre-senting decision points, dpi (i = 1; :::; I), in which thedash circle represents the infeasible ones; the dash linesrepresent arcs connecting two decision points satisfyingminimum pipe slope constraint; bold circles representnodal cover depths selected by an arbitrary ant; �nally,the bold lines represent a trial solution on the graphconstructed by an arbitrary ant. It can be clearlyseen from Figure 5 that the resulting search spaceis much smaller than the original search space, asshown in Figure 4. Therefore, it is expected thatthe resulting formulation will perform better than theUABAC.

It is worth noting that the proposed formulations,UPBAC, CPBAC, UABAC, and CABAC, however,may lead to trial solutions that may violate some of theproblem constraints. To discourage the ants to makedecisions which constitute an infeasible solution, highercosts are associated with the solutions that violate theproblem constraints. This is achieved via the use of apenalty method in which the total cost of the problemsis considered as the sum of the problems cost andpenalty cost as follows:

Cp = C + �p �KXk=1

CSVk; (11)

where C is the cost function associated with theobjective functions (Eq. (1)); Cp is the penalized costfunction; CSVk is the value of the kth constraintviolation; and �p represents the penalty parameter.

4. Test examples, results, and discussions

In this section, the performance of the proposed formu-lations is tested against two benchmark examples fromthe literature. The �rst one (test example I) is the caseof a network originally proposed by Mays and Yen [42],then considered again by Mays and Wenzel [43], andalso solved by other researchers. This sewer networkhas 21 nodes and 20 pipes (Figure 6). This networkis constrained to a maximum ow velocity of 12 fps,minimum ow velocity of 2 fps, maximum relative owdepth of 0.9, minimum relative ow depth of 0.1, andminimum cover depth of 8 ft. It is assumed here thatthe pipes have the variable Manning coe�cient withthe value of 0.013 at full condition. Details of thenetwork, including nodal ground elevations and lengthand design discharge of each pipe, are given in Table 1.Commercial pipe diameters are 12, 15, 18, 21, 24, 30,36, 42, and 48 inches, and other problem data of thistest example are presented in the paper of Mays andWenzel [43]. Here, the following relation is used forpipe installation and manhole costs:

Kpip =

8>>>>>>>>>>><>>>>>>>>>>>:

10:98dl + 0:8El � 5:98if dl � 30 and El � 100

5:94dl + 1:166El + 0:504dlEl � 9:64if dl � 30 and El � 100

30:0dl + 4:9El � 105:9if dl > 30

Kman = 250 + h2m: (12)

The second one (test example II) is the case of a part ofthe network designed by Mansoury and Khanjani [44]for `Kerman' city, Iran. This sewer network has 21nodes and 20 pipes (Figure 7). This network isconstrained to a maximum ow velocity of 3 m/s,minimum ow velocity of 0.6 m/s, maximum relative ow depth of 0.82, minimum relative ow depth of0.1, and minimum cover depth of 2.45 m. Constant

Figure 6. Test example I sewer network layout.


Table 1. Data of test example I.

Pipeno.

Nodeno.

Groundelevation

(ft)Length

(ft)

Designdischarge

(cfs)Up Down Up Down

1 11 22 500 495 350 42 22 33 495 487 400 33 33 42 487 480 350 24 12 32 490 485 400 45 32 42 485 480 430 46 42 52 480 470 550 57 23 34 490 485 500 88 34 43 485 475 450 49 43 52 475 470 350 410 52 61 470 465 500 611 31 41 485 475 500 912 41 51 475 470 350 713 51 61 470 465 350 414 61 71 465 455 565 715 44 53 460 464 400 416 53 62 464 460 300 217 62 71 460 455 345 318 71 81 455 451 400 719 81 91 451 448 500 220 91 10 448 445 612 5

Figure 7. Test example II sewer network layout.

Manning coe�cient is considered as 0.013. Details ofthe network, including nodal ground elevations andlength and design discharge of each pipe, are given inTable 2. Commercial pipe diameters are 200, 250, 300,400, 500, 600, and 700 millimetres, and other problemdata of this test example are presented in the paperof Mansoury and Khanjani [44]. Here, the followingrelation is used for pipe installation and manhole costs:

Kpip = 1:93e3:43dl + 0:812El + 0:437E1:53l dl;

Kman = 41:46hm: (13)

Table 2. Data of test example II.

Pipeno.

Nodeno.

Groundelevation

(m)Length

(m)

Designdischarge

(m3/s)Up Down Up Down

1 1 4 74.59 73.66 260 27.92 2 9 70.7 69.9 300 54.93 3 15 73 71.50 400 21.14 4 5 73.66 72.1 460 30.45 5 6 72.1 71.99 260 32.46 6 7 71.19 69.85 300 347 7 8 69.85 68.24 450 36.68 8 12 68.24 67.82 400 38.79 9 10 69.9 69.3 270 56.210 10 11 69.3 68.4 310 5811 11 12 68.4 67.28 440 59.612 12 13 67.28 66.22 470 96.713 13 14 66.22 65.82 350 101.214 14 20 65.82 65.42 340 104.715 15 16 71.5 70.1 400 26.416 16 17 70.1 68.6 400 3017 17 18 68.6 66.8 500 31.918 18 19 66.8 66.1 400 40.319 19 20 66.1 65.42 590 44.620 20 21 65.42 64.5 320 165.9

A set of preliminary runs is �rst done to �nd the propervalues of MMAS parameters as shown in Table 3 forthe proposed formulations. All the results presentedhereafter are based on a uniform discretisation of theallowable range of cover depths into 30 intervals for allthe proposed formulations and test examples. It shouldbe noted that in PBACOA formulations, no heuristicinformation can be de�ned, and therefore, the value of� = 0:0 is used.

The results of 10 runs carried out using di�er-ent randomly generated initial guesses for the testexamples along with the scaled standard deviation, thenumber of �nal feasible solutions, and the number offunction evaluations and CPU time to get minimumsolution cost are presented in Table 4. It is clearly seenfrom Table 4 that all measures of the quality of the�nal solutions, such as the minimum, maximum andaverage costs, the number of �nal feasible solutions,and the numbers of function evaluations to get mini-mum solution cost, are improved when using ABACOAcompared to PBACOA. Furthermore, it is seen thatconstrained versions of each PBACOA and ABACOAhave been able to outperform the corresponding un-constrained version of these formulations regarding thequality of the solution due to the fact that the search


Table 3. Values of MMAS parameters.

Testexample

Formulation Iteration Ant Functionevaluation

� � � pbest

I PBACOA 1000 200 200000 1 0 0.95 0.2ABACOA 1000 200 200000 1 0.2 0.95 0.2

II PBACOA 1000 200 200000 1 0 0.95 0.2ABACOA 1000 200 200000 1 0.1 0.95 0.2

Table 4. Maximum, minimum, and average solution cost values over 10 runs obtained with the proposed formulations.

Tes

tex

ample

Form

ula

tion

Solution cost value Scaledstandarddeviation

No. of runswith �nalfeasiblesolution

No. of functionevaluations toget minimumsolution cost

CPU timeto get

minimumsolution cost

Minimum Maximum Average value value (sec)

I

UPBAC 237200 245751 240072 0.0106 10 158800 97CPBAC 236287 239113 237757 0.0048 10 82400 58UABAC 236287 242390 238497 0.0083 10 47200 100CABAC 236287 237978 236658 0.0022 10 42800 63

II

UPBAC 78362.9 90725.7 82063.4 0.0561 10 147000 88CPBAC 78190.4 79402.7 78780.7 0.0079 10 25600 14UABAC 77674.1 90143.7 81191.2 0.0429 10 124000 98CABAC 77674.1 77674.1 77674.1 0 10 15800 18

Figure 8. Variation of average solution cost values of testexample I using unconstrained and constrained versions ofABACOA.

space size of the problem is much smaller than those ofthe unconstrained formulations.

Figures 8 and 9 show convergence curves of theaverage solution costs obtained in ten runs using uncon-strained version of ABACOA compared to constrainedversion of it for test examples I and II, respectively.These �gures indicate the superior performance ofthe constrained version of this formulation comparedto unconstrained version of it in which the proposedconstrained version leads to lower cost of the �nalsolution during the evolution process. Furthermore,Figures 10 and 11 show convergence curves of theaverage solution costs obtained in ten runs usingCPBAC and CABAC formulations for test examples

Figure 9. Variation of average solution cost values of testexample II using unconstrained and constrained versionsof ABACOA.

I and II, respectively. These �gures indicate superiorperformance of the CABAC compared to CPBAC.Finally, Figures 12 and 13 show the characteristics ofthe optimal solution obtained for test examples I andII, respectively, using CABAC in which the numbers inparentheses are nodal cover depths.

Test example I was solved by Mays and Wenzel us-ing Di�erential Dynamic Programming (DDP) and theoptimal solution of 265775 was reported [43]. Robinsonand Labadie solved this problem later using DP, andthe optimal solution of 275218 was reported [45]. Thisproblem was later solved by Miles and Heaney usinga spreadsheet template and 245874 was reported [46].Furthermore, this problem was also solved by Afshar


Figure 10. Variation of average solution cost values oftest example I using CPBAC and CABAC.

Figure 11. Variation of average solution cost values oftest example II using CPBAC and CABAC.

proposing ACOA-AR (Adaptive Re�nement processfor ACOA), PCACOA, RPSO (Rebirthing ParticleSwarm Optimization), and CACOA methods; the op-timal solutions of 241496, 242539, 242162, and 242119were respectively reported requiring 29900, 13900,30000, and 20000 function evaluations [21,23,47,48]. Inaddition, Afshar et al. used Cellular Automata (CA)

method in which it required 72 function evaluations toget the optimal solution of 253483 for this problem [24].Later, Afshar and Rohani proposed discrete and con-tinuous CA based hybrid methods to solve this prob-lem, and the optimal solutions of 247412 and 248100were reported requiring 69 and 7 function evaluations,respectively [35]. A GA-based model was proposed byPalumbo et al. to get the optimal solution of 251971for this problem [26]. Finally, Yeh et al. used TS andSA to solve this problem, and the optimal solutions of241770 and 244571 were respectively reported requiring1034809 and 15932235 function evaluations [27]. Theseresults can be compared with the cost value of 236287obtained with 42800 function evaluations using theproposed CABAC for test example I, indicating thesuperiority of the proposed formulation with propercomputational e�ort.

Test example II was solved, �rstly, by Mansouriand Khanjani using GA and NLP, and the optimalsolution of 83116 was reported [44]. NLP-BFGS,NLP-Fletcher-Reeves, and GA were used later bySetoodeh to solve this test example, and he reportedthe optimal solutions of 82732, 81553, and 77736,respectively [49]. Afshar et al. proposed CA methodto solve this problem in which it required 15 functionevaluations to get the optimal solution of 80879 [24].Finally, this problem was solved by Afshar and Rohaniwho proposed discrete and continuous CA based hybridmethods, and the optimal solutions of 77327 and 77433were reported requiring 45 and 38 function evaluations,respectively [35]. These results can be compared withthe cost value of 77674.1 obtained with 15800 functionevaluations using the proposed CABAC for this testexample, indicating the superiority of the proposedformulation with proper computational e�ort.

Figure 12. Characteristics of the optimal solution of example I using CABAC.


Figure 13. Characteristics of the optimal solution of test example II using CABAC.

5. Conclusions

Here, two di�erent formulations of Arc Based AntColony Optimization Algorithm (ABACOA) were pro-posed to solve sewer network design optimization prob-lem considering the cover depths of sewer networknodes as decision variables of the problem. Here,in the second formulation, the constrained version ofABACOA was proposed to satisfy slope constraintexplicitly leading to reduction of search space of theproblem. Two benchmark test examples were solvedhere using the proposed formulations, and the resultswere presented and compared with those obtained withalternative usual point-based formulation and otherexisting methods. Comparison of the results showedsuperiority of the proposed ABACOA to optimallysolve the sewer network design optimization in which ithas two distinct advantages of e�cient implementationof the exploration and exploitation features and alsoan easy de�nition of the heuristic information for theants over the alternative usual point-based formulation.Furthermore, the results indicate the ability of theconstrained version of ABACOA to e�ciently ande�ectively solve this problem in which it was shownto produce better results with smaller computationale�ort and less sensitivity to the randomly generatedinitial guess represented by the scaled standard devi-ation of the solutions produced in ten di�erent runsin comparison with all other proposed formulations. Inother words, the average solution costs of test examplesI and II were improved 0.8% and 4.5%, respectively,using CABAC in comparison with UABAC. In addi-tion, the average solution costs of test examples I andII were improved 0.5% and 1.5%, respectively, usingarc based ACOA in comparison with point-based ones.

Finally, the obtained results of the proposed CABACformulation for both of the test examples were betterthan all available results.

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Biography

Ramtin Moeini was born in Isfahan, Iran in 1981.He received his BS degree from Civil EngineeringDepartment of Isfahan University of Technology (IUT)in 2002 and his MS degree (honors), and his PhDdegrees (honors) from Civil Engineering Departmentof Iran University of Science and Technology (IUST) in2007 and 2013, respectively. He has been an AssistantProfessor of University of Isfahan since 2013. He isauthor/coauthor of 19 research articles in internationaland national journals and 20 conference papers. Hisresearch interests include optimization, evolutionaryalgorithms, water resource engineering, and water andsewer networks.

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