Deterministic Integrated Optimization Model for SewageCollection Networks Using Tabu Search

Ali Haghighi1 and Amin E. Bakhshipour2

Abstract: This paper presents an integrated optimization model for designing sewage collection networks. The layout configuration isdesigned using the loop-by-loop cutting algorithm. Then, the network with a given layout is hydraulically designed to determine sewerdiameters, installation depths, and pump specifications. In both design steps, all technical constraints and criteria are systematically satisfied.Thereby, the optimization of sewer systems becomes totally unconstrained for the applied optimization solver. In this problem, the objectivefunction is the network’s construction cost and the decision variables are the parameters of layout generation and sewer specifications. Foroptimization of the cost function, the tabu search (TS) method as a deterministic combinatorial metaheuristic is developed and coupled to thedesign solvers. The proposed scheme is able to search adaptively in feasible parts of the problem’s decision space as well as to solve the twosubproblems of layout generation and sewer sizing simultaneously. Then the model is applied against a benchmark case study. It is found thatusing the integrated model, the design of sewer networks becomes computationally more efficient and systematic, and it is a very promisingapproach to attain the global optima. DOI: 10.1061/(ASCE)WR.1943-5452.0000435. © 2014 American Society of Civil Engineers.

Author keywords: Sewage collection networks; Layout; Sewer; Optimization; Tabu search (TS).

Introduction

Sewage collection networks are known as the most importantpart of the infrastructure of any modern city; they directly influ-ence public health and are essential for environmental protection.Annually, governments spend a lot of money on development andoperation of sewer systems, especially for those in flat areas. Insuch areas, neither significant changes in topography nor a dis-tinguished outlet location exists to help the designer sketch acost-effective sewer layout explicitly. In addition, due to the lackof suitable natural ground slopes, heavy excavation, and the useof large pipes and pumping facilities are inevitable in sewersconstructed in flat areas. These issues make the design of sewernetworks very expensive, not only for construction but also forcomputations. Hence, the development and application of optimi-zation models to sewers design seem quite necessary. If so, it wouldbe possible to gain a cost-effective design while all hydraulic andtechnical constraints associated with the sewer systems are system-atically met.

The design of a sewage collection network needs to solve twosuccessive subproblems: (1) generating the layout and (2) sizing thenetwork’s components. The latter involves sewer diameters and in-stallation depths, as well as the pumping facilities if required. Thesesubproblems are nonlinear and discrete in nature and include manycomplex constraints that come from the hydraulics principles, tech-nical criteria, and regional limitations. In general, three approachesmay be used to solve the aforementioned problems:

1. Full enumeration, in which all layout alternatives first aregenerated and then are hydraulically designed. The best of theexisting designs is finally chosen (Diogo et al. 2000; Diogoand Graveto 2006). This approach is a very promising way toreach the global optimum; however, it is practical only forsmall networks.

2. A separate design, in which the layout is designed manuallyor by defining a simplified objective function; in this, thetwo subproblems are disconnected and individually optimized(Liebman 1967; Bhave 1983; Tekeli and Belkaya 1986;Walters and Lohbeck 1993; Walters and Smith 1995; Pan andKao 2009; Afshar 2010; Haghighi 2013). In practice, this ap-proach is very useful, especially for large networks; however,it is difficult to determine the global optimum design.

3. Simultaneous design, through which the two subproblems,layout generation, and sewer sizing are implicitly optimizedtogether (Li and Matthew 1990). This approach, named as theintegrated optimization model in this study, is the only way toattain the global optimum design of large sewer systems.Nevertheless, integrating the two subproblems into a modeland coupling an optimization solver to that requires toughformulations and specific design algorithms.

As discussed later in this paper, the two subproblems of sewersystem design are very different from each other mathematically.In fact, the layout subproblem belongs to a difficult class of com-binatorial optimizations in graphs theory. Meanwhile, sewer sizingis a nonlinear discrete program that also can be viewed as a deci-sion-making problem. Both of these problems are nonlinear andhighly constrained, and they could be highly multimodal dependingon the cost function formulation.

For the optimization of sewer networks, this study introducesa comprehensive design model, shown schematically in Fig. 1.For this purpose, two adaptive algorithms are developed for solvingthe layout generation and sewer sizing subproblems. Using thesealgorithms, all constraints of the aforementioned subproblems aresystematically met, and there is no need for any constraint han-dling in the applied optimization solver. To minimize the problem’sconstruction costs, a deterministic optimization algorithm is also

1Assistant Professor, Dept. of Civil Engineering, Faculty of Engineer-ing, Shahid Chamran Univ. of Ahvaz, Ahvaz 61357-43337, Iran (corre-sponding author). E-mail: ali77h@gmail.com

2Graduated Student, Dept. of Civil Engineering, Faculty of Engineer-ing, Shahid Chamran Univ. of Ahvaz, Ahvaz 61357-43337, Iran.

Note. This manuscript was submitted on April 5, 2013; approved onJanuary 21, 2014; published online on January 22, 2014. Discussion periodopen until November 23, 2014; separate discussions must be submitted forindividual papers. This paper is part of the Journal of Water ResourcesPlanning and Management, © ASCE, ISSN 0733-9496/04014045(11)/$25.00.

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developed based on the tabu search (TS) method, which then iscombined with the design algorithms. Finally, the proposed schemeis applied to a benchmark example from the literature, and the re-sults are discussed and compared to the previous studies.

Layout Design

To solve the layout subproblem, an undirected looped base graphis initially created considering all sewers (edges) and possible con-nectivity manholes (vertices) in the area under design, as shown inFig. 2(a). A feasible sewer layout is a subgraph extracted from thebase graph which has the following characteristics:• It has no loop.• It is a spanning tree, meaning that all manholes in the base graph

must be kept in the extracted tree.• It includes all sewers in the base graph.• It has a root (i.e., a system’s outlet) toward which the sewers are

directed in such way that, except for the root, exactly one sewerleaves each manhole.These specifications form the constraints of the sewer layout

problem. These issues make the current problem more complexthan those conventionally known as spanning tree problems ingraphs theory.

One of the main parts of any sewer design optimization modelis an algorithm that can generate all possible layouts while satisfy-ing all the aforementioned constraints. In flat areas, the number offeasible layouts of the base graph exponentially grows with the net-work size and the final design is highly influenced by the layoutconfiguration. Hence, the efficiency and capability of the layoutgenerator play a major role in the whole design process. For thispurpose, several algorithms have been introduced and applied sofar, and in all of them, two concerns have proved problematic inaddition to the technical restrictions. First, the layout generatormust be able to overcome the nonlinearity of the problem’s objec-tive function and constraints. The use of combinatorial optimiza-tion methods and metaheuristics in recent years has solved thisproblem. Second, the weight of each sewer line and the fitness ofthe created layout can be estimated precisely only if the network isdesigned completely. On the other hand, all sewers and pumps mustbe sized first to make it possible to evaluate their construction costs

as the optimization weights. This implies that the two subproblemsof sewer systems must be solved instantaneously. However,alternatively, one may adopt some simplifications and accept lessreliability in results to separate the subproblems and solve themindividually. This approach definitely leads to designs that are cost-lier, but they involve fewer computations and easier optimization.

In this context, Bhave (1983) gave a same weight to all pipesin the network and then found the network’s shortest path spanningtree as its best layout. In that approach the role of pipe diametersand installation depths in the optimization of cost function wasmissed. To compensate for this deficiency, Takeli and Belkaya(1986) introduced three strategies for weighting the pipes in thebase graph. These weights were the reciprocal of ground surfacegradient, pipe length, and excavation corresponding to the mini-mum hydraulic gradient required for self-cleaning velocity. Foreach weight, the spanning tree with the shortest path was appliedto find the best layout. Li and Matthew (1990) proposed the search-ing direction method, which exploited the Dijkstra algorithm (fromthe graphs theory; Minieka 1978) to generate the shortest-pathspanning tree of the base graph. That method was combined withdiscrete differential dynamic programming (DDDP), which utilizedan iterative procedure to generate the layout while keeping the pipeweights constant, and then to size the sewers and pumps whilekeeping the layout fixed. That study was a pioneer that managesto solve the sewer subproblems implicitly.

Over the past two decades, as computers tremendously pro-gressed in terms of speed and memory, metaheuristic algorithmswere highly developed and widely used in many complex engineer-ing problems. Metaheuristics can overcome the nonlinearity anddiscreteness of the layout design subproblem efficiently. However,they are fundamentally weak at constraint handling and often needspecial tricks for this purpose. Walters and Lohbeck (1993) usedtwo types of genetic algorithms (GAs) to find the optimum layoutof pipe networks from an initially directed base graph. It was shownthat the initial directions play a significant role in the characteristics

Start

Tabu search

Generate the layoutSize the sewers and pumps

Evaluate the cost function Check the

convergence

No

Yes

Layout variables (α β ) Sizing variables (d, s, P)

End

Fig. 1. Conceptual plan of the integrated optimization model forsewer networks

1

2 3 4

5

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[6] [7]

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Loo

p 1

Loo

p 2

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p 3

Up

node

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de

Fig. 2. (a) Example of a base graph and its representative matrix B;(b) the resulting layout

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of the raised spanning trees and the efficiency of the applied opti-mization method. Despite this fact, the approach needed great con-sideration for defining the sewer directions in the initial base graphto maintain all possible connectivity and other restrictions. Waltersand Smith (1995) also introduced a new GA for the layout optimi-zation. A layout generator (namely, the tree-growing algorithm)was combined with the GA. The new scheme was able to generatespanning trees of undirected base graphs on the basis of randomchromosomes from the GA. Diogo and Graveto (2006) proposeda deterministic model for the full enumeration of all possible layoutalternatives. Then the optimum layout was determined by the eco-nomical comparison of all solutions with optimized designs. Theyshowed that if the layout constraints are appropriately manipulated,many unacceptable trees of the base graph can be kept out of theenumeration. This approach was practicable in fact only for smallto medium-sized networks. For large sewer systems, those inves-tigators proposed the simulated annealing (SA) method with thelayout generator algorithm. Later, with a special focus on the con-straint handling of the layout subproblem, Haghighi (2013) intro-duced an adaptive layout generator called the loop-by-loop cuttingalgorithm. Using this algorithm, the undirected base graph isopened with a step-by-step procedure while the layout constraintsare systematically met. This algorithm is simply implemented andcan solve the problem’s complexities efficiently. By this approach,the problem becomes quite unconstrained and possible to connectto any metaheuristic easily. Because of these benefits, this algo-rithm is adopted here and exploited in the integrated optimizationmodel. In what follows, the applied algorithm is introduced briefly,while more detailed explanations are found in Haghighi (2013).

Loop-by-Loop Cutting Algorithm

To generate possible layouts of a network, an undirected base graphis initially provided for the network under design, as demonstratedin Fig. 2(a). Every pipe, manhole, and loop in the base graph isgiven a number. The base graph is then mathematically representedby a matrix named B. The B matrix is consisting of m rows andNLþ 3 columns, where m and NL are the number of sewers (net-work size) and the number of loops, respectively. In this matrix,column 1 contains the sewer names, columns 2 to NLþ 1 aresewer-in-loop indicators that determine that a sewer is either in aloop (value 1) or not (value 0). Columns NLþ 2 and NLþ 3 alsoinclude the names of sewer ends that are arbitrarily assigned sincethe graph is undirected. For example, the matrix of the base graphof Fig. 2(a) has been presented next to it.

To create a feasible layout from the base graph, all loops mustbe opened. To this end, one pipe from each loop must be cut,which may be done either from its upstream or its downstreammanhole. Thus, there are two decision variables for opening eachloop, including the name of the selected pipe to be cut and thename of its truncation end. These variables are characterized byα and β, respectively, in the algorithm. Here, α is a real-valuednumber in the interval (0, 1) that indirectly points to the pipe to becut, whereas β is a binary number (0 or 1) that shows where theselected pipe is cut, with 0 meaning upstream and 1 meaningdownstream.

For each loop, the variable β explicitly determines the cutlocation, while α is a coded variable that needs to be decoded first.On the basis of a given real-valued α on (0, 1), the pipe to be cut forloop i is determined by the following relationship:

μi ¼ round

�1þ αi

�Xmj¼1

Bj;iþ1 − 1

��ð1Þ

where subscript i indicates the loop number in the base graph; m isthe number of sewers;

Pmj¼1 Bj;iþ1 is the total number of pipes

in loop i, which are possible to be cut; and round rounds theparenthesis to its integer value μ. For the loop at hand, loop i,the corresponding column in matrix B, column iþ 1, is taken intoaccount. From the top of this column, the row of the μth nonzeromember indicates the pipe to be cut. For example, suppose that foropening loop 1 I n the base graph of Fig. 2(a), random values of0.6 and 1 are given to α1 and β1 respectively. Column 2 in matrix Brepresents the pipes that are possible to be cut for opening loop 1,which are in the rows with nonzero members. As seen, thereare three pipes in loop 1, and it is found from matrix B thatP

7j¼1 Bj;2 ¼ 4. Consequently, α1 ¼ 0.6 results in μ1 ¼ round½1þ

0.6ð4 − 1Þ� ¼ 3, meaning that pipe 6 (as the third pipe of loop 1in matrix B) is chosen to be cut. Also, β1 ¼ 1 determines thatpipe 6 must be cut from its downstream end, which is node 2 inmatrix B [Fig. 2(b)].

When a loop was opened, the base graph changes; hence, theprevious matrix B needs to be updated before other loops areopened. Subject to the previously addressed constraints of the lay-out subproblem, the updates of matrix B are applied step by step asthe network is opened loop by loop. For this purpose, matrix B ismodified after each loop opening to meet the following criteria:• The new manhole: As a pipe is cut for opening loop i, a new

manhole appears at the truncation end, which is named as nþ i,where n is the number of manholes in the base graph. The newmanhole is located in this matrix instead of the previous one.For example, in Fig. 2(b), after loop 1 was opened by cuttingpipe 6 from manhole 2, the new manhole 6 appears in the graphand is substituted for manhole 2 in B6;6.

• The flow direction: The flow direction of the cut pipe is from itsnew manhole, nþ i, toward its next end. This update is done bychanging the location of the cut pipe’s ends in matrix B so thatthe new manhole nþ i is placed on column NLþ 2 if it is notalready there.

• Once pipe is cut: If a pipe has been cut to open a loop, it is nolonger possible to be cut for other loops. This constraint is metby switching all nonzero sewer-in-loop members in the row ofthe cut pipe to zero. For example, in Fig. 2(b), since pipe 6 is cutfor opening loop 1, all members of B6;2 to NLþ 1 become zero.This update does not let pipe 6 be selected again for the remain-ing loops in the rest of the process.

• Network integrity: Suppose all pipes in a manhole can be cut fordifferent loops, like sewers 4, 5, and 2 in Fig. 2(b), which arecommon in manhole 3. In such cases, at least one link must notbe cut to drain the common manhole and to keep the networkintegrity. To satisfy this issue, the downstream manhole of thecut pipe is checked. If there is exactly one intact pipe (i.e., notpreviously cut) connected to this manhole, it must not be cut forthe next loops. For this purpose, all the sewer-in-loop membersof matrix B in the row of the aforementioned intact pipe arechanged to zero.When these updates complete, matrix B is ready to use for

opening the next loops on the basis of the given α and β variables.This procedure is continued until all NL loops are opened and afeasible layout results, which has m pipes and nþ NL manholes,whilem ¼ nþ NL − 1. Then the sewers in the obtained layout canbe easily directed toward the outlet node following the principlethat except for the outlet, exactly one sewer leaves every manhole[Fig. 2(b)]. The sewer directions are also introduced to the finalversion of matrix B by changing the location of the sewer ends incolumns NLþ 2 and NLþ 3 if they are not already correct. Afterthe direction of all pipes are updated, every manhole appears onlyonce in column NLþ 2.

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At this point, box A as the layout generator in Fig. 1 is fulfilled.Using the loop-by-loop cutting algorithm, every arbitrary input(2NL parameters of α and β) always results in a feasible layoutsince all constraints of the layout subproblem satisfy the algorithmautomatically.

Sizing the Sewers

After generating a feasible layout, the second subproblem is solvedto size the sewer diameters, installation depths, and pump stations.This subproblem is a consequence of decision making that candeterministically be solved by the method of discrete differentialdynamic programming (DDDP) as an extension of classical dy-namic programming (Heidari et al. 1971; Mays and Yen 1975;Mays et al. 1976; Li and Matthew 1990). Subject to the problem’sconstraints and decision variable bounds, the user needs to definethe DDDP’s stages manually. Then, the optimum design is soughtamong the discrete solutions appear in successive stages. For manyyears, DDDP was the best deterministic approach for solving manycomplex decision-making problems, such as the sewer networkdesign. However, it is computationally inefficient, especially in thecase of large problems, in which the curse of dimensionality sig-nificantly restricts the method in practice. In addition, establishingthe DDDP stages needs great attention to ensure that the entiredecision space is explored, while all constraints are met precisely.In fact, there is a tradeoff between the number of stages (computa-tional load) and the reliability of the results, which often leads tomaximizing the optimization speed, but missing some solutions.For these reasons, the DDDP cannot guarantee the global optima,as well as being computationally slow and hard to implement.Later, nonlinear optimization methods, especially nature-inspired

metaheuristics, were increasingly utilized. For instance, ant colonyoptimization, particle swarm optimization, and GAs are widelyused approaches in the fields of water and wastewater engineering.These methods are simple to implement and very promising in find-ing the global optima, and they can be coupled easily with any com-plex problem. Nevertheless, metaheuristics are mostly stochastic,with some special random-based operators that need to be care-fully calibrated in the bingeing. Metaheuristics are also inher-ently unconstrained methods. For solving constrained problems,they must be equipped with special constraint-handling strategiessuch as penalty functions. For sewer network design, this issue isseriously problematic. As addressed later in this paper, the sewer-sizing problem is full of constraints that are nonlinear, mixeddiscrete, continuous and sequential. In particular, the latter limitsthe use of common penalizing strategies. In recent years, hybridmathematical-metaheuristic approaches were found very useful insuch problems (Tu et al. 2005; Cisty 2010; Haghighi et al. 2011).For sewer systems, Pan and Kao (2009) hybridized a GA and themathematical optimization method of quadratic programming (QP)and named their scheme GA-QP. Most nonlinear continuous con-straints were masterly formulated in QP, while a few others, like theselection of discrete pipe diameters, were handled in GA. In thathybrid model, QP acts as the interior solver and forms the mainskeleton of the optimization. It was shown that QP speeds up theoptimization and precisely meets a main part of the constraints.In spite of all these merits (especially in terms of the constrainthandling), GA-QP is obviously more complicated than the previousapproaches. From the standpoint of formulation, implementation,and generalization, GA-QP is indeed more complex than eitherDDDP and GA alone. Haghighi and Bakhshipour (2012) proposedan adaptive GA, and they specifically focused on the constrainthandling issues and their role in the efficiency (speed) and reliabil-ity (accuracy) of the optimization, as well as their ability to solvelarge problems. There, the sewer design constraints were incorpo-rated into the GA operators in such a way that every randomchromosome was adaptively decoded to a real feasible design.It was found that the adaptive GA is computationally very fast andeasy to implement compared to the previous methods.

Inspired by this investigation, the current work also introducesan adaptive algorithm for sizing the components of a sewer networkwith a given layout. This algorithm is then allocated in box Bin Fig. 1.

Sewer Hydraulics

The steady flow in sewer lines is expressed by Manning’s equation:

V ¼ 1

nR2=3S1=2 ð2Þ

where V = flow velocity, A = flow cross-sectional area,n = Manning’s coefficient, R = hydraulic radius, and S = sewerslope. The geometrical specifications of circular sectionscommonly used in sewer design are obtained from the followingequations:

ðh=DÞ ¼ 1

2×

�1 − cos

θ2

�ð3Þ

AA0

¼�θ − sin θ

2π

�ð4Þ

R ¼ D4×

�θ − sin θ

θ

�ð5Þ

[9][10][11]

[13] [15]

[12]

[5]

[2]

[4]

[6]

[7]

[8]

[14]

[16][17]

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[24]

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[27] [28]

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[35] [36][37]

[38][39]

[40]

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[43] [44]

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[53]

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54535251

4645

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2.62.8

2.4

2.2

2.0

2.0

2.2

Fig. 3. Base graph of the case study (data from Li and Matthew 1990)

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where D = pipe diameter, A0 = pipe cross-sectional area, (h=D) =proportional water depth, and θ = central angle from the center ofthe section to the water surface.

Adaptive Algorithm for Sewer Design

For a network having NP pipes and a given fixed layout, the designvariables of the second subproblem are NP pipe diameters [D], NPpipe slopes [S], and NP pump indicators [P]. Herein, a vector hav-ing 3NPmembers in the range of (0, 1) named as the normal designvector is introduced to the model. This vector indirectly representspipe diameters (the first NP members) and slopes (the second NPmembers) and directly gives the pump indicators (the third NPmembers). If a pipe’s pump indicator P ¼ 1, there is a lift stationat its upstream manhole; otherwise, if P ¼ 0, there is no pump.

The normal pipe diameters and slopes in the normal design vec-tor are denoted by [d] and [s], respectively. These variables needto be decoded to obtain the meaningful design values [D] and [S],such that the following criteria are met.

For every sewer:• The flow velocity must be kept within a permissible limit for

self-cleaning capability (minimum velocity) and prevented fromscouring (maximum velocity).

• The proportional water depth must be kept below a specifiedmaximum level.

• Pipe diameters must be chosen from a commercially avail-able list.

• A minimum buried depth needs to be kept to prevent damagesfrom the surface activates.For every manhole:

• Inlet pipes need to be no lower than the outlet pipe.• The outlet pipe’s diameter must be equal to or greater than the

upstream inlet pipes.A normal design vector can be either randomly or systematically

produced on interval (0, 1) by an optimization solver. To extracta feasible design from a given normal alternative subject to theseconstraints, the following adaptive algorithm is proposed.

Step 1: For every sewer diameter, there is now a normal realvalue d between 0 and 1. The corresponding design diameter D isobtained as the following:

D ¼ Dmin þ ðDmax −DminÞ × d ð6Þ

where Dmax is the largest commercially available size and Dmin isthe minimum allowable size for the pipe under consideration. Dminis determined with respect to two limitations. First, the pipe must becapable of conveying the design flow rate Q and consequently betrue for the following constraint (Pan and Kao 2009):

QVmax

≤�AA0

�ðh=DÞmax

×

�D2

4× π

�ð7Þ

where Vmax = maximum permissible flow velocity and ðh=DÞmax =maximum permissible proportional water depth. Substitutingðh=DÞmax in Eq. (3), θmax is explicitly obtained as follows:

θmax ¼ 2 cos−1ð1 − 2 × ðh=DÞmaxÞ ð8Þ

Substituting Eq. (8) in Eq. (4), the proportional area (A=A0) isalso calculated. Then, a Dmin is obtained from Eq. (8).

Second, the diameter of every pipe must be equal to or greaterthan that of its upstream pipes, which means that

D ≥ max½DU� ð9Þ

where [DU] contains pipe diameters connected to the upstream endof the pipe at hand. It is worth noting that [DU] is available fromthe previous calculations. Between the diameters obtained byEqs. (8) and (9), the greater size is assigned to Dmin in Eq. (6). Thisequation results in a real diameter size that needs to be finallyrounded up in the commercial list for assigning the design diameterD to the pipe. The described procedure is applied step by step toall pipes in such a way that at first, the network’s upstream pipes’so-called chord links (which have no pipe at their upstream) areconsidered. Then the remaining upstream pipes, which have beenalready sized using Eq. (6), are taken into account. This continuesuntil the normal vector [d] is interpreted in the design diametervector [D] for the whole network.

Step 2: For every sewer slope, there is also a normal real value sbetween 0 and 1. Similar to the previous step, the correspondingdesign slope S is obtained as follows:

S ¼ Smin þ ðSmax − SminÞ × s ð10Þwhere Smin and Smax are the minimum and maximum permissibleslopes for the pipe, respectively. Smin is determined so that threeconstraints are met. First, the sewer slope needs to be greater thanthe minimum constructional value of Sc, and hence

S ≥ Sc ð11Þ

Second, associated with the pipe discharge and its diameter sizefrom the previous step, the pipe slope must satisfy the constraint ofthe maximum proportional water depth. For the pipe at hand, θmaxwas already obtained from Eq. (8). Therefore, it is expected that

S ≥ ðnQÞ2 × ½ðAR2=3Þθmax�−2 ð12Þ

Third, the pipe flow velocity always must be greater than theminimum velocity Vmin which leads to the following by substitut-ing Vmin into Manning’s equation:

S ≥ ðnVminÞ2 × ½ðR2=3Þθmin�−2 ð13Þ

Thereafter, each of these three constraints, Eqs. (11)–(13),returns the greatest value selected as the lower bound of pipe slope,Smin, in Eq. (10). Meanwhile there is only one restriction for Smax,which is associated with the maximum velocity Vmax. This con-straint is also taken into account as follows:

S ≤ ðnVmaxÞ2 × ½ðR2=3Þθ 0max�−2 ð14Þ

where θ 0max is obtained by substituting Vmax into Manning’s equa-

tion, similar to what was done to obtain θmin in Eq. (13). The rightside of Eq. (14) is used as the upper bound of pipe slope Smax inEq. (10). Step 2 is repeated for all pipes, eventually resulting in thedesign pipe slopes [S] from the initial normal vector [s].

Step 3: At this point, the installation depths of the pipes aredetermined with respect to their slope, diameter size, and pumpstation status, as well as the least cover depth that the pipe requires.The least cover depth, Cmin, is a user-defined constraint that is de-cided based on the mechanical properties of soil and pipe material,as well as on the surface activities and traffic loads. This constraintis satisfied by placing the sewers, which now have a design diam-eter, slope, and pump indicator, at the proper elevations. To thisend, initially, the installation elevations of upstream branches in thenetwork, like cut pipes 1, 3, and 6 in Fig. 2(b), are determined. Foran upstream branch, the minimum cover depth is assigned to itsupstream end as

EU ¼ GU − Cmin ð15Þ

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in which EU and GU are the pipe’s upstream crown and groundsurface elevations, respectively. Then, the downstream crown ele-vation ED is obtained from

ED ¼ EU − S × L ð16Þwhere L is the pipe length. The least cover depth is also checked atthe pipe’s downstream end so that if GD − ED < Cmin, then

ED ¼ GD − Cmin ð17ÞAccordingly

EU ¼ EDþ L × S ð18ÞAfter that, the installation elevations were computed for all

upstream branches, the remaining upstream inlet pipes, which werealready placed, are taken into account. For a pipe in this group, thefollowing is the case:• If the pump indicator P ¼ 1, a lift station is allocated at the

pipe’s upstream manhole. If so, the installation elevations areobtained using Eqs. (15)–(18). On the other hand, the lift stationon a pipe makes that an upstream branch.

• Otherwise (P ¼ 0), the pipe’s upstream crown is placed at thelowest downstream crown of its inlet pipes. Then Eqs. (16)–(18)are used to calculate the downstream crown elevation, as well asto control the minimum cover depth.This procedure is continued until for all pipes and manholes, the

installation depths are determined.It is worth mentioning that this approach assumes that the

ground slope is constant along a sewer line between its nodes.In reality, there may be local high and low points along a sewerline, which means that the ground slope is not necessarily constantalong a sewer. Although this issue can be neglected in many flatcities, but in some projects, it might be significant. To solve thisproblem, it is required to add more manholes to the initial basegraph and break long pipes into shorter pieces.

By means of the proposed algorithm, every random normal de-sign alternative is acceptable since all hydraulic and technical con-straints are systematically met. Similar to the loop-by-loop cuttingalgorithm, the sewer sizing algorithm does not need any constrainthandling trick into the optimization solver and always returns a fea-sible design. Hence, this algorithm would be very well suited to becombined with many metaheuristics. At this point, box B in Fig. 1is also completed, and this plan is one step closer to its goal.

Tabu Search Optimization

When the two subproblems of sewer networks design are integratedinto a model, a large and hard class of nonlinear combinatorial op-timization is formed. Fortunately, the adaptive design algorithmsfor layout generation and sewer sizing make the problem easier soit can be solved efficiently by many heuristics approaches. For theoptimization of a sewer network design, a Nd-size problem isdefined as follows:

Cðα; β; d; s;PÞ ¼XNP

i¼1

ðCPi þ Pi × CLiÞ þXNPþ1

i¼1

CMi ð19Þ

It is subject to the following conditions:• For i ¼ 1 to NL, 0 ≤ αi ≤ 1• For i ¼ 1 to NL, βi ∈ f0; 1g• For i ¼ 1 to NP, 0 ≤ di ≤ 1 and 0 ≤ si ≤ 1• For i ¼ 1 to NP, Pi ∈ f0; 1g

where C is the cost function that evaluates design alternativesin box D in Fig. 1; CP and CM are the sewer and manhole

construction costs estimated as a function of pipe diameter D andlength L, respectively, as well as its buried depth H; and CL are theconstruction costs of the pump station, which are generally esti-mated with respect to the pumping head and flow rate. In this prob-lem, the number of decision variables is Nd ¼ 2NLþ 3NP, amongwhich 2NL variables (α, β) are for layout generation and 3NPvariables (d, s, P) are for sewer sizing.

In this study, TS is exploited to solve this programming prob-lem. TS was originally introduced by Hertz and Werra (1987) andalso Glover (1989). It was then developed by Glover (1990, 1995)and popularized by Hertz and Werra (1990). Later, this methodwas applied to solve many combinatorial optimization problems invarious fields of engineering and economics. The distinguishedadvantages of TS compared to other metaheuristics is that TS is adeterministic search, meaning that there is no randomness in it. Asa consequence, TS is generally found to be computationally veryefficient. TS as a metaheuristic is nothing but a simple local search,which is equipped with some features for solving combinatorialoptimizations, as well as for escaping local optima.

Simply phrased, TS initiates the exploration of a decision spacewith an arbitrary starting point. In the vicinity of the current point,the best solution is sought with respect to the given objective func-tion. Whether the best neighbor point is better than the current pointor not, it is adopted as the new solution and the search is continued.This is the main difference between the TS and local search algo-rithms, in which worse neighbors are unacceptable, and this is whythey quickly fall into a local optima. This feature helps the TS be ametaheuristic and not be easily stopped by local optima; however,there are two concerns with this issue. First, it may significantlyincrease the number of objective function evaluations and takesa lot of time to complete the search of the decision space. Second,it is very probable that the search becomes trapped in a loop ofsuccessive solutions that periodically leads to the same results. Toovercome these issues, TS systematically uses the memory of thesearch in such a way that the exploration route so far traveled isremembered and exploited to decide new points and search direc-tions. For this purpose, the best solution at each iteration or themove toward it is sent to a tabu list. The length of the tabu listis a user-defined parameter in TS upon which a long or shortmemory search is formed. Throughout the search, TS is forbiddento pick points from the tabu list, even if those points are superiorto other neighbors. With this idea, no visited point is revisitedagain; consequently, the search does not fall in a loop of solutionsanymore.

Furthermore, there are some other features in TS that control asearch’s accuracy and efficiency. For instance, the diversification,intensification, and aspiration are the most common phases in aconventional TS. If all these features are appropriately programmedand exploited in a search, TS would be a serious rival to stochasticmetaheuristics. Herein, a TS algorithm is introduced as followsto be coupled to the developing optimization model. The TS isallocated in box C in Fig. 1 and simultaneously solves the twosubproblems of sewer network design:1. An initial design alternative (α, β, d, s, P) is arbitrarily gen-

erated with respect to the decision variable limits on interval(0, 1). This solution is termed as K, and let K� ¼ K, where K�is the best answer so far visited.

2. Set the cycle number j ¼ 1.3. Set the iteration number i ¼ 1.4. Solution K is sent to the tabu list T with a user-specified

length jTj.5. For solution K, a neighborhood zone NðKÞ in the problem’s

decision space is generated. The structure and production strat-egy of NðKÞ play a great role in the efficiency and accuracy

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of a TS run. For the present problem, the univariate searchdirection method is used. For this purpose, at each iteration i,a univariate move ti is defined with an Nd-length vector whoseall members are zero, except the ith member that is one, whichis a unit. At iteration i, the TS searches through the corre-sponding direction by producing neighbor points for thecurrent solution K as the following:• i ≤ NL, K1;2 ¼ K � λti (for variable α)• NL < i ≤ 2NL, K1 ¼ 0 if K ¼ 1 and K1 ¼ 1 otherwise.

(for variable β)• 2NL < i ≤ 2NLþ NP, K1;2 ¼ K � λti (for variable d)• 2NLþNP< i≤ 2NLþ2NP, K1;2 ¼K�λti (for vari-

able s)• 2NLþ NP < i ≤ Nd, K1 ¼ 0 if K ¼ 1 and K1 ¼ 1 other-

wise. (for variable P)where K1;2 stands for two neighbors of K that constituteNðKÞ at iteration i and λ is an increment that controls thediversification and intensification of search for real parame-ters. In this study, for the layout design parameters α, theincrement λ is determined with respect to the loop withthe most pipes in the base graph. This results in λ ¼ 1=ðthe number of pipes in that loop). For normal pipe diam-eters d, λ is set to be 1=ðnumber of commerciallyavailable diametersÞ; and for normal slopes s, a smallenough increment is defined by engineering judgments.However, in flat areas, the normal slope values are rationallyzero or very close to that. This assumption results in the leastpossible design slopes (S) through the introduced sizingalgorithm.

6. The allowable neighborhood, named V�, is obtained by ex-tracting the tabu solutions from the generated neighbors suchthat V� ¼ NðKÞ − NðKÞ ∩ T.

7. For all points in V�, the problem is solved and the cost functionis evaluated. With no comparison with the current solution, thebest solution in V� is found and termed as U. Set K ¼ U, andif the cost function CðKÞ < CðK�Þ, let K� ¼ K. Also, let i ¼iþ 1 and if i ≤ Nd go to step 4.

8. Set the cycle number j ¼ jþ 1. At this point, TS has partiallysought the decision space in all directions. Now, it is said thata search cycle has been done. The algorithm now encountersthree user-defined conditions:a. The stop criterion: If the cycle number j > jmax the search

is terminated. jmax is the maximum cycle number.b. The diversification: To make it possible to scape local

optima, TS uses a feature to diversify the search engine inproblem’s decision space. In stochastic metaheuristics thisis simply done by means of random operators like the mu-tation in GA. By contrast, there is no randomness in TS.Hence, this method relies on deterministic diversificationschemes that need to be planned carefully for each specialproblem. The diversification phase periodically appearsthrough the search cycles. For this purpose, a user-definedperiod number τ is used which determines that the TS isdiversified at cycles div ¼ fτ ; 2τ ; 3τ ; : : : g. At each diver-sification cycle, when j ∈ div, the memory of search ineach individual direction (decision variable) is inquired.A few number of decision variables (for example, 20% ofNd) that have the fewest variations through the past searchiterations are banished to newpositions very far fromwherethey have relaxed. Simply speaking, at each diversificationphase, the standard deviation of all decision variables areevaluated and less fluctuating directions are found. Thecorresponding decision variables are then shot to newpositions associated with their current positions and limits.

c. The intensification: TS is inherently a combinatorial tech-nique which is well suited to discrete problems. For con-tinuous decision spaces there is a trade-off between theintensification of search and the computations load. Forcontinuous directions, like the sewer slope variables in thisstudy, this challenge is handled by the neighborhood in-crement λ. In intensification phase, the pre-defined λ forcontinuous variables is decreased to a smaller value λintthereby, the search is intensified in corresponding direc-tions. In the developed algorithm, the intensification isperiodically applied to the search in one cycle beforethe diversification phase i.e., when j ∈ fτ − 1; 2τ − 1;3τ − 1; : : : g. After that, the increment λint is again in-creased to the previous value λ.

9. If one of the stopping criteria was met the optimization isterminated. Otherwise, it goes to step 3.

Now, all elements of the integrated optimization model forsewer networks design are available and the proposed schemecan be utilized in practice. The next section deals with this issue,in which a benchmark example from the literature is introduced andoptimized.

Illustrative Example

This example introduces a sewage collection system with 79 pipes,57mainmanholes, and 23 loops in its base graph, as shown in Fig. 3.The sewer network is constructed in a 260-hectare residential areawith a very flat topography. This problem was originally presentedand solved by Li andMatthew (1990) and then was used as a bench-mark in later investigations, as discussed later in this paper. Thenetwork’s outlet is manhole 56 connected to pipe 59, toward whichall pipes in the resultant layout are directed. Other physical infor-mation about the example, including pipe lengths and individualdischarges and manhole elevations, are found in Li and Matthew(1990), as well as in Haghighi and Bakhshipour (2012).

The construction cost of the sewer network is estimated usingEq. (19) with respect to the cost components presented in Table 1.For designing the sewers, a commercially available list of pipe

Table 1. Cost Function Components (in Yuan) for the Example (Data fromLi and Matthew 1990)

Cost functionComponentparameters

CP (Sewer cost)ð4.27þ 93.59D2 þ 2.86D ×Ha þ 2.39H2

aÞ × L D ≤ 1 m,Ha ≤ 3 m

ð36.47þ 88.96D2 þ 8.70D ×Ha þ 1.78H2aÞ × L D ≤ 1 m,

Ha > 3 mð20.50þ 149.27D2 − 58.96D ×Ha þ 17.75H2

aÞ × L D > 1 m,Ha ≤ 4 m

ð78.44þ 29.25D2 þ 31.80D ×Ha − 2.32H2aÞ × L D > 1 m,

Ha > 4 mCM (manhole cost)136.67þ 166.19D2 þ 3.50D ×H þ 16.22H2 D ≤ 1 m,

H ≤ 3 m132.67þ 790.94D2 − 280.23D ×H þ 34.97H2 D ≤ 1 m,

H > 3 m209.04þ 57.53D2 þ 10.93D ×H þ 19.88H2 D > 1 m,

H ≤ 4 m210.66 − 113.04D2 þ 126.43D ×H − 0.60H2 D > 1 m,

H > 4 mCL (pump cost)270,021þ 316.42Q − 0.1663Q2 QðL=sÞHa = averageburieddepth —

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diameters, including 0.2, 0.25, 0.30, 0.35, 0.38, 0.40, 0.45, 0.50,0.53, 0.60, 0.70, 0.80, 0.90, 1.00, 1.05, 1.20, 1.35, 1.40, 1.50,1.60, 1.80, 2.00, 2.20, and 2.40 m, is used. Manning’s coeffi-cient for all pipes is considered to be 0.014. Other technical cri-teria and constraints in this example are also presented in Table 2.It is worth noting that values in Table 2 are decided based on thenational standards or other technical references and suggestions.Some values in Table 2 (e.g., the minimum slope of 0.0005 forQ > 15 L=s) seem to be insufficient for real-life situations.However, to make it possible to compare the results of this workwith previous studies, the values in Table 2 are adopted here asthey are.

For optimization of the system, matrix B of the base graph isprepared and introduced to the integrated model (Fig. 1). TS startsto solve the problem at a particular point, which in this case is adesign alternative including 46 α and β parameters for layout gen-eration and 237 d, s, and P parameters for sewer sizing. In total,there are 283 decision variables in this example; of them, β and Pare binary and the others are real-valued on interval (0, 1). Also, thenormal pipe slopes, which are inherently continuous, need to bediscretized first.

For this purpose, two increments termed as the normal λ andthe intensification λint are used in the TS. For this case study, it isconsidered that λ ¼ 0.01 and λint ¼ 0.001. However, in flat areaslike this case study, it is rationally expected that the optimum nor-mal slopes finally are obtained at around zero. This means that thepipes are given the minimum allowable slopes with respect to thehydraulic constraints introduced through Eqs. (10)–(13).

The problem was optimized several times by introducing dif-ferent starting points, among which two runs are presented in Fig. 4.In these two instances, the starting points are the ones respectivelywith the zero- and random-valued variables. Also, the mean com-putational time for solving the case study is about 50 min using a

PC, Intel Core i5, with 1.7 GHz central processing unit (CPU) and4 GB random access memory (RAM).

The semilogarithmic graphs in Fig. 4 demonstrate the trend ofcost function minimization versus the iteration number through TSoptimization. It shows that both runs have finally converged to asame optimum result, but not at the same speed. In this example,TS has clearly manifested the advantages of a metaheuristic that isnot seriously influenced and restricted by starting points. Never-theless, as found in Fig. 4, the zero-value starting point has resultedin a faster convergence, especially in primary iterations. A mainreason for this issue is that the sewer network is constructed in avery flat area, with the result that most normal sewer slopes [s] inthe optimum design are zero, which leads to the least acceptabledesign slopes [S]. Another reason is that, for a cost-effective design,pump stations are rationally avoided as much as possible. This alsomeans that most of the pump indicators [P] in the optimum designare zero. As a consequence, the run with the zero-valued startingpoint has presented a more efficient optimization, inasmuch asmany of the decision variables have been initially well quantifiedin its starting point. On the other hand, it is expected that the start-ing point computationally affects the efficiency of TS; but, hope-fully, this would not cause a serious detriment to the reliability ofthe final results.

Focusing on the run with the zero-valued starting point in Fig. 4shows that the TS starts to optimize the sewer system with theinitial cost of 3.78 × 108 units. After 123 iterations, the TS findsa design with a cost function of 1.67 × 106 units, which is veryclose to the best scheme previously optimized using DDDP byLi and Matthew (1990). That design is shown in Fig. 5(a) andits layout is named layout I. Later, Pan and Kao (2009), usingGA-QP, and Haghighi and Bakhsipour (2012), using adaptiveGA, adopted layout I, fixed in the plan, and optimized the prob-lem for sizing the sewers and pumps. Their optimum designshad construction costs of 1.74 × 106 [Fig. 5(b)] and 1.69 × 106

[Fig. 5(c)] units, respectively, neither of which was better than theprevious design.

This search is still at iteration 123, and it continues until iteration157, at which a new design with a cost function of 1.59 × 106 unitsis obtained. This design is also very close to the best scheme sofar, which was found by Haghighi (2012) for this example. Theconfiguration of that design is shown in Fig. 5(d), and its layoutis named layout II. In that configuration, the layout was optimizedusing a GA and based on a simplified objective function that iso-lated the layout subproblem from the sewer sizing one. Then thesewers and pumps were optimized using DDDP.

The search continues while optimization of the cost functionvery slowly progresses. The best design is finally found at iteration415, with a cost of 1.43 × 106 units. Afterward, the TS is allowedto continue until iteration 800, when it is found that the results donot improve, so then the search is terminated. The currently foundoptimum design is also presented in Fig. 5(e), which is comparableto the other designs depicted in Fig. 5. The new layout is namedlayout III.

As seen in Fig. 5, the three designs of (a), (b), and (c), which areon the basis of layout I, have a pump station in their configuration.The designs of (d) and the current work, (e), which are on the basisof layout II and III, respectively, not only are cheaper, but alsohave no pump in their schemes. Consequently, they are morefavorable in their operation.

Table 3 summarizes this discussion and presents more infor-mation about the different approaches so far applied to the casestudy. As concluded from this table, the previous designs are about11%–22% costlier than the current design.

Table 2. Design Constraints for the Example (Data from Li andMatthew 1990)

Description Constraint

Maximum velocity Vmax 5.0 m=sMinimum velocity Vmin 0.7 m=s (if D ≤ 500 mm, Q > 15 L=s)

0.8 m=s (if D > 500 mm, Q > 15 L=s)Minimum slope Smin 0.003 (if Q ≤ 15 L=s)

0.0005 (if Q > 15 L=s)Maximum proportionalwater depth (h=DÞmax

0.6 (if D ≤ 300 mm)0.7 (if D ¼ 350–450 mm)0.75 (if D ¼ 500–900 mm)0.8 (if D ≥ 1,000 mm)

Minimum cover depth Cmin 1 m

1.00E+06

1.00E+07

1.00E+08

1.00E+09

0 200 400 600 800

Cos

t

Iteration

Zero-valued starting point

Random-valued starting point

Fig. 4. Cost function minimization in TS

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P

3.023.023.00(a) (b)

(c) (d)

(e)

2.91 3.02

3.76

2.76

3.01

3.01

3.00

1.89

1.98

3.02

3.443.03

3.002.23

3.003.02

3.013.01

3.01

3.02

3.64

4.17 3.00

1.481.81

2.39

2.083.01

3.01

3.01 3.013.01

3.032.07

2.67

3.023.00

3.01 3.00

3.011.86

1.79

2.721.20

3.62

3.003.00

3.02

3.0

4.15

1.06

1.96

1.77

1.86

3.04

3.02

3.02

3.00

1.29

3.01

3.01

3.68

1.23

1.23

1.07

1.39

3.03

3.03

3.00 3.003.02

3.44

1.82

3.02

3.57 1.98

3.003.003.00

2.95 3.00

3.81

2.62

3.00

3.00

3.00

1.25

1.99

3.00

3.443.00

3.002.23

3.003.00

3.003.00

3.00

3.00

3.64

4.36 3.00

1.981.79

1.21

1.233.00

3.00

3.00 3.003.00

3.002.06

2.65

3.003.00

3.00 3.00

3.001.83

1.52

2.711.21

3.643.00

3.003.00

3.00

4.31

1.06

1.46

0.86

0.91

3.00

3.00

3.00

3.00

2.33

3.00

3.00

3.72

2.92

3.19

1.06

1.09

3.00

3.00

3.00

3.003.00

3.43

1.80

3.00

3.56

P

1.98

P

3.003.003.00

3.00 3.00

3.89

2.66

3.00

3.00

3.00

2.673.51

2.01

3.00

3.453.00

3.002.29

3.003.00

3.003.00

3.00

3.00

3.68

4.23 3.00

1.421.48

1.60

1.233.00

3.00

3.00 3.003.00

3.092.06

2.723.00

3.00

3.00 3.00

3.001.86

1.59

2.791.22

3.67

3.003.00

3.00

3.00

4.24

1.06

1.90

1.79

1.88

3.90

3.00

3.00

3.00

1.27

3.00

3.00

3.69

1.23

1.21

1.08

1.08

3.00

3.00

3.00 3.003.00

3.43

1.81

4.10

3.56

3.003.003.00

3.00 2.23

3.81

3.00

3.00

2.39

3.00

3.00 3.00

3.00

2.012.37

3.003.00

3.003.00

3.003.00

3.32

3.41

2.06

3.00 3.00

3.003.00

1.51

3.003.00

3.00

1.28 3.003.00

2.581.11

1.31

3.003.00

3.00 3.88

0.981.10

3.01

3.243.93

3.12

3.003.00

3.00

3.4

3.00

3.00

0.81

0.79

0.80

3.00

3.00

3.00

3.00

3.00

3.21

3.21

3.00

3.00

3.00

3.00

3.00

3.94

3.00

2.76

1.003.12

0.91

0.79

3.0

3.59

3.00

< 0.3 m

0.3 - 0.45 m

0.45 - 0.6 m

> 0.6 m

> 1.0 m

< -3.5 m

> -2.0 m

> -3.5 m

Pipe DiameterManhole Bottom Elevation

Slope (10 )-3

P

Outlet

Pump station

3.003.003.00

3.00 3.67

3.84

3.00

3.00

3.00

3.14

1.21

1.42

1.96

3.003.00

3.002.95

3.003.00

3.003.00

3.00

4.14

4.04

3.00 3.00

3.003.00

2.46

1.611.11

1.23

2.68 3.003.00

3.003.00

3.00

3.003.00

3.00 3.00

3.003.00

3.00

3.261.80

3.48

3.003.00

3.00

3.00

3.00

3.00

0.79

0.80

0.82

3.00

3.00

3.00

3.00

3.00

0.97

1.01

3.00

3.00

3.00

3.00

3.00

3.00

3.98

0.92

3.003.00

0.89

0.78

3.00

3.67

3.00

Fig. 5. Optimum designs for the case study: (a) Li and Matthew (1990), C ¼ 1.67 × 106; (b) Pan and Kao (2009), C ¼ 1.74 × 106; (c) Haghighi andBakhshipour (2012), C ¼ 1.69 × 106; (d) Haghighi (2013), C ¼ 1.59 × 106; (e) the integrated model (in the current study), C ¼ 1.43 × 106

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Conclusions

For designing a sewage collection network, two subproblems mustbe successively solved to generate the network layout and to sizethe pipes and pumps. The layout configuration, especially in flatareas, greatly influences the final design costs and performance.In such areas, the number of possible layout alternatives increasesexponentially with the number of pipes in the network. Meanwhile,in flat areas, there are neither significant changes in the land topog-raphy nor a distinguished location for the system’s outlet. Hence, itis practically impossible to find an optimum layout that is explicitlyindependent of the sewer and pump specifications and costs. Whenthe sewer subproblems are intended to be implicitly optimized forminimal cost, a hard class of combinatorial optimization is formedthat is nonlinear, mixed integer-real, highly constrained, and inmany cases, large-scale and multimodal.

This study introduced an integrated optimization model forsolving the two subproblems simultaneously. An adaptive lay-out generator (namely, the loop-by-loop cutting algorithm) wasadopted and developed into the model. By using this algorithm,it is possible to generate feasible layouts of a network from an ini-tially looped and undirected base graph. For each loop in the basegraph, two real normal- and binary-valued variables are randomlydefined. Then, through a step-by-step procedure, all loops are cut,while all constraints associated with the sewer layout are system-atically satisfied. In this work, the location of the outlet point wasconsidered fixed. However, this point can also be added to the prob-lem’s decision variables and optimized by the model. Afterward,for the sewer network with a given layout, an adaptive designalgorithm was developed to size the pipes and pumps.

Similar to the layout generator algorithm, the sewer sizing al-gorithm works on the basis of some real normal- and binary-valuedparameters. Through an adaptive analysis-design procedure, all hy-draulic and technical constraints of the sewer sizing subproblem areautomatically met and a feasible design is derived. For optimizationof the network, a deterministic search engine based on the TSmethod was developed and coupled to the design algorithms. Sincethe constraints of both subproblems are met completely in theircorresponding solution algorithms, there is no need to use any con-straint handling in the tabu search. On the other hand, the sewernetwork optimization problem becomes totally unconstrained forthe applied solver. This makes TS and any other applied metaheur-istic computationally more efficient and promising approaches tofind the global optima of the problem. The combination of layoutgeneration, sewer sizing, and TS optimization was designated theintegrated model in this study. Then the model was applied againsta benchmark example from the literature. In comparison with pre-vious studies, the proposed model found a better design configu-ration in terms of the construction costs; however, the improvementwas not very dramatic. The previous methods isolate the two sub-problems by some simplifications, and engineering assumptions

also can approach the global optima very well, producing goodresults. The integrated model makes it possible to apply everymetaheuristic to the design of sewer systems easily, and it introdu-ces a comprehensive design package that can be developed and pro-fessionalized for more realistic design conditions. In the case study,the design is completely controlled by a single “design flow,” andall constraints are satisfied by that. Meanwhile, in reality, the sewerflows continuously vary on several time scales. Real designs mustintroduce different flows to different constraints with respect tothe flow variations. For instance, the constraints associated withthe maximum velocity and sewer sizes need to be met based on themaximum hourly flows at the end of the design period, while theminimum velocities must be controlled by flows in the first yearsof operation. Furthermore, the design of sewer systems, like otherengineering problems, include significant uncertainties in designparameters and assumptions. The proposed model is unable to han-dle such uncertainties; however, it can be used as an efficient sub-routine in uncertainty analysis models that utilize approaches suchas the Monte Carlo simulation method or fuzzy set theory.

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Table 3. Different Approaches so far Applied to the Case Study

Investigators Layout generation method Design approach Optimization method Cost (106 units)

Li and Matthew (1990) Dijkstra algorithm (layout I) Iterative DDDP 1.67ðPÞ

Pan and Kao (2009) Layout I was adopted Separated GA-QP 1.74ðPÞ

Haghighi and Bakhshipor (2012) Layout I was adopted Separated GA 1.69ðPÞ

Haghighi (2012) Layout I was adopted Separated SA 1.69ðPÞ

Haghighi (2013) Loop by loop cutting algorithm (layout II) Separated GA-DDDP 1.59The current work Loop by loop cutting algorithm (layout III) Integrated TS 1.43

Note: (P) = having a pump station.

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