# Deterministic Integrated Optimization Model for Sewage Collection Networks Using Tabu Search

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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 networks 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 problems 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 thenetworks 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 problemsconstruction 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.

ASCE 04014045-1 J. Water Resour. Plann. Manage.

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http://dx.doi.org/10.1061/(ASCE)WR.1943-5452.0000435http://dx.doi.org/10.1061/(ASCE)WR.1943-5452.0000435http://dx.doi.org/10.1061/(ASCE)WR.1943-5452.0000435http://dx.doi.org/10.1061/(ASCE)WR.1943-5452.0000435

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 systems 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 problems 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 networks 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 ( ) Sizing variables (d, s, P)

End

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

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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 problems 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 andNL 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 NL 1 aresewer-in-loop indicators that determine that a sewer is either in aloop (value 1) or not (value 0). Columns NL 2 and NL 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...

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