An efficient decomposition and dual-stage multi-objective optimization method for water distribution systems with multiple supply sources

This paper proposes an efficient decomposition and dual-stage multi-objective optimization (DDMO) method for designing water distribution systems with multiple supply sources (WDS-MSSs). Three phases are involved in the proposed DDMO approach. In Phase 1, an optimal source partitioning cut-set is identified for a WDS-MSS, allowing the entire WDS-MSS to be decomposed into sub-networks. Then in Phase 2 a non-dominated sorting genetic algorithm (NSGA-II) is employed to optimize the sub-networks separately, thereby producing an optimal front for each sub-network. Finally in Phase 3, another NSGA-II implementation is used to drive the combined sub-network front (an approximate optimal front) towards the Pareto front for the original complete WDS-MSS. Four WDS-MSSs are used to demonstrate the effectiveness of the proposed approach. Results obtained show that the proposed DDMO significantly outperforms the NSGA-II that optimizes the entire network as a whole in terms of efficiently finding good quality optimal fronts.     

Comparative analysis of multi-objective evolutionary algorithms for QoS-aware web service composition

Web service composition combines available services to provide new functionality. The various available services have different quality-of-service (QoS) attributes. Building a QoS-optimal web service composition is a multi-criteria NP-hard problem. Most of the existing approaches reduce this problem to a single-criterion problem by aggregating different criteria into a unique global score (scalarization). However, scalarization has some significant drawbacks: the end user is supposed to have a complete a priori knowledge of its preferences/constraints about the desired solutions and there is no guarantee that the aggregated results match it. Moreover, non-convex parts of the Pareto set cannot be reached by optimizing a convex weighted sum. An alternative is to use Pareto-based approaches that enable a more accurate selection of the end-user solution. However, so far, only few solutions based on these approaches have been proposed and there exists no comparative study published to date. This motivated us to perform an analysis of several state-of-the-art multi-objective evolutionary algorithms. Multiple scenarios with different complexities are considered. Performance metrics are used to compare several evolutionary algorithms. Results indicate that GDE3 algorithm yields the best performances on this problem, also with the lowest time complexity.     

Convergence of multi-objective evolutionary algorithms to a uniformly distributed representation of the Pareto front

In evolutionary multi-objective optimization (EMO), the convergence to the Pareto set of a multi-objective optimization problem (MOP) and the diversity of the final approximation of the Pareto front are two important issues. In the existing definitions and analyses of convergence in multi-objective evolutionary algorithms (MOEAs), convergence with probability is easily obtained because diversity is not considered. However, diversity cannot be guaranteed. By combining the convergence with diversity, this paper presents a new definition for the finite representation of a Pareto set, the B-Pareto set, and a convergence metric for MOEAs. Based on a new archive-updating strategy, the convergence of one such MOEA to the B-Pareto sets of MOPs is proved. Numerical results show that the obtained B-Pareto front is uniformly distributed along the Pareto front when, according to the new definition of convergence, the algorithm is convergent.     

自适应混合多目标分布估计进化算法

针对多目标分布估计算法全局收敛性较弱的缺陷,提出了一种自适应混合多目标分布估计进化算法。其基本思想是:在多目标分布估计算法中引入全局收敛性较强的差分进化算法,当函数变化率较大时,用分布估计算法产生新种群;当函数变化率较小即算法可能陷入局部收敛时,用差分进化算法产生新种群。理论分析和数值实验结果表明,这种混合算法不仅具有良好的全局收敛性,而且解的分布性和均匀性较没有考虑目标函数变化率的混合多目标分布估计算法也有了一定程度的提高。     

Simultaneous topology and sizing optimization of a water distribution network using a hybrid multiobjective evolutionary algorithm

This paper proposes a new direction for design optimization of a water distribution network (WDN). The new approach introduces an optimization process to the conceptual design stage of a WDN. The use of multiobjective evolutionary algorithms (MOEAs) for simultaneous topology and sizing design of piping networks is presented. The design problem includes both topological and sizing design variables while the objective functions are network cost and total head loss in pipes. The numerical technique, called a network repairing technique (NRT), is proposed to overcome difficulties in operating MOEAs for network topological design. The problem is then solved by using a number of established and newly developed MOEAs. Also, two new MOEAs namely multiobjective real code population-based incremental learning (RPBIL) and a hybrid algorithm of RPBIL with differential evolution (termed RPBIL–DE) are proposed to tackle the design problems. The optimum results obtained are illustrated and compared. It is shown that the proposed network repairing technique is an efficient and effective tool for topological design of WDNs. Based on the hypervolume indicator, the proposed RPBIL–DE is among the best MOEA performers.     

Improving multi-objective genetic algorithms with adaptive design of experiments and online metamodeling

Applications of multi-objective genetic algorithms (MOGAs) in engineering optimization problems often require numerous function calls. One way to reduce the number of function calls is to use an approximation in lieu of function calls. An approximation involves two steps: design of experiments (DOE) and metamodeling. This paper presents a new approach where both DOE and metamodeling are integrated with a MOGA. In particular, the DOE method reduces the number of generations in a MOGA, while the metamodeling reduces the number of function calls in each generation. In the present approach, the DOE locates a subset of design points that is estimated to better sample the design space, while the metamodeling assists in estimating the fitness of design points. Several numerical and engineering examples are used to demonstrate the applicability of this new approach. The results from these examples show that the proposed improved approach requires significantly fewer function calls and obtains similar solutions compared to a conventional MOGA and a recently developed metamodeling-assisted MOGA.     

Seeding the initial population of multi-objective evolutionary algorithms: A computational study

Most experimental studies initialize the population of evolutionary algorithms with random genotypes. In practice, however, optimizers are typically seeded with good candidate solutions either previously known or created according to some problem-specific method. This seeding has been studied extensively for single-objective problems. For multi-objective problems, however, very little literature is available on the approaches to seeding and their individual benefits and disadvantages. In this article, we are trying to narrow this gap via a comprehensive computational study on common real-valued test functions. We investigate the effect of two seeding techniques for five algorithms on 48 optimization problems with 2, 3, 4, 6, and 8 objectives. We observe that some functions (e.g., DTLZ4 and the LZ family) benefit significantly from seeding, while others (e.g., WFG) profit less. The advantage of seeding also depends on the examined algorithm.     

A hybrid multi-objective approach based on the genetic algorithm and neural network to design an incremental cellular manufacturing system

One important issue related to the implementation of cellular manufacturing systems (CMSs) is to decide whether to convert an existing job shop into a CMS comprehensively in a single run, or in stages incrementally by forming cells one after the other, taking the advantage of the experiences of implementation. This paper presents a new multi-objective nonlinear programming model in a dynamic environment. Furthermore, a novel hybrid multi-objective approach based on the genetic algorithm and artificial neural network is proposed to solve the presented model. From the computational analyses, the proposed algorithm is found much more efficient than the fast non-dominated sorting genetic algorithm (NSGA-II) in generating Pareto optimal fronts.     

Multi-objective evolutionary algorithms based on the summation of normalized objectives and diversified selection

Most multi-objective evolutionary algorithms (MOEAs) use the concept of dominance in the search process to select the top solutions as parents in an elitist manner. However, as MOEAs are probabilistic search methods, some useful information may be wasted, if the dominated solutions are completely disregarded. In addition, the diversity may be lost during the early stages of the search process leading to a locally optimal or partial Pareto-front. Beside this, the non-domination sorting process is complex and time consuming. To overcome these problems, this paper proposes multi-objective evolutionary algorithms based on Summation of normalized objective values and diversified selection (SNOV-DS). The performance of this algorithm is tested on a set of benchmark problems using both multi-objective evolutionary programming (MOEP) and multi-objective differential evolution (MODE). With the proposed method, the performance metric has improved significantly and the speed of the parent selection process has also increased when compared with the non-domination sorting. In addition, the proposed algorithm also outperforms ten other algorithms.     

On the equivalences and differences of evolutionary algorithms

Evolutionary algorithms (EAs) are fast and robust computation methods for global optimization, and have been widely used in many real-world applications. We first conceptually discuss the equivalences of various popular EAs including genetic algorithm (GA), biogeography-based optimization (BBO), differential evolution (DE), evolution strategy (ES) and particle swarm optimization (PSO). We find that the basic versions of BBO, DE, ES and PSO are equal to the GA with global uniform recombination (GA/GUR) under certain conditions. Then we discuss their differences based on biological motivations and implementation details, and point out that their distinctions enhance the diversity of EA research and applications. To further study the characteristics of various EAs, we compare the basic versions and advanced versions of GA, BBO, DE, ES and PSO to explore their optimization ability on a set of real-world continuous optimization problems. Empirical results show that among the basic versions of the algorithms, BBO performs best on the benchmarks that we studied. Among the advanced versions of the algorithms, DE and ES perform best on the benchmarks that we studied. However, our main conclusion is that the conceptual equivalence of the algorithms is supported by the fact that algorithmic modifications result in very different performance levels.     

Embedding linear programming in multi objective genetic algorithms for reducing the size of the search space with application to leakage minimization in water distribution networks

This paper shows how embedding a local search algorithm, such as the iterated linear programming (LP), in the multi-objective genetic algorithms (MOGAs) can lead to a reduction in the search space and then to the improvement of the computational efficiency of the MOGAs. In fact, when the optimization problem features both continuous real variables and discrete integer variables, the search space can be subdivided into two sub-spaces, related to the two kinds of variables respectively. The problem can then be structured in such a way that MOGAs can be used for the search within the sub-space of the discrete integer variables. For each solution proposed by the MOGAs, the iterated LP can be used for the search within the sub-space of the continuous real variables. An example of this hybrid algorithm is provided herein as far as water distribution networks are concerned. In particular, the problem of the optimal location of control valves for leakage attenuation is considered. In this framework, the MOGA NSGAII is used to search for the optimal valve locations and for the identification of the isolation valves which have to be closed in the network in order to improve the effectiveness of the control valves whereas the iterated linear programming is used to search for the optimal settings of the control valves. The application to two case studies clearly proves the reduction in the MOGA search space size to render the hybrid algorithm more efficient than the MOGA without iterated linear programming embedded.     

An elitist non-dominated sorting genetic algorithm enhanced with a neural network applied to the multi-objective optimization of a polysiloxane synthesis process

This paper presents an original software implementation of the elitist non-dominated sorting genetic algorithm (NSGA-II) applied and adapted to the multi-objective optimization of a polysiloxane synthesis process. An optimized feed-forward neural network, modeling the variation in time of the main parameters of the process, was used to calculate the vectorial objective function of NSGA-II, as an enhancement to the multi-objective optimization procedure. An original technique was utilized in order to find the most appropriate parameters for maximizing the performance of NSGA-II. The algorithm provided the optimum reaction conditions (reaction temperature, reaction time, amount of catalyst, and amount of co-catalyst), which maximize the reaction conversion and minimize the difference between the obtained viscometric molecular weight and the desired molecular weight. The algorithm has proven to be able to find the entire non-dominated Pareto front and to quickly evolve optimal solutions as an acceptable compromise between objectives competing with each other. The use of the neural network makes it also suitable to the multi-objective optimization of processes for which the amount of knowledge is limited.     

Speedup of water distribution simulation by domain decomposition

The Schur complement domain decomposition method is used for solution of large linear systems. The algorithm is based on the subdivision of the domain into smaller ones and the solution of those sub-domains independently. Regarding water distribution systems modeling, the hydraulic simulation could be formulated as a sequence of systems of linear equations. Therefore, this paper utilizes the domain decomposition method to accelerate the simulation process further. The method is evaluated using a large scale real-world system with 63,616 junctions and 64,200 pipes as case study. The case study shows that the methodology could improve the performance of hydraulic simulation app. by a factor of 8 without losing accuracy at a suitable level of domain decomposition. Although the optimal level of decomposition is case specific, considerable speedup might still be achievable by decomposing a large system into only a few subsystems.     

Investigating the run-time searching behavior of the differential evolution algorithm applied to water distribution system optimization

In recent years, the differential evolution algorithm (DEA) has frequently been used to tackle various water resource problems due to its powerful search ability. However, one challenge of using the DEA is the tedious effort required to fine-tune parameter values due to a lack of theoretical understanding of what governs its searching behavior. This study investigates DEA's search behavior as a function of its parameter values. A range of behavioral metrics are developed to measure run-time statistics about DEA's performance, with primary focus on the search quality, convergence properties and solution generation statistics. Water distribution system design problems are utilized to enable investigation of the behavioral analysis using the developed metrics. Results obtained offer an improved knowledge on how the control parameter values affect DEA's search behavior, thereby providing guidance for parameter-tuning and hence hopefully increasing appropriate take-up of the DEA within the industry in tackling water resource optimization problems.     

An integrated planning tool for design of recycled water distribution networks

Pipeline design of urban recycled water networks involves thousands of decisions to ensure delivery of water to multiple use locations with pipelines and pump stations correctly located, optimally sized, and compatible with existing infrastructure. Here, we introduce PRODOT, Pipeline ROuting and Design Optimization Tool, software that identifies near-minimum-cost pipeline routes; accounts for existing configurations, legal, environmental or safety concerns, and trade-offs in pipeline length, pipe installation methods, traffic congestion during construction; optimizes pump station locations, pumping energy, pipe diameters and pressure classes; and includes theoretical additional capacity of each pipe, facilitating future expansion. We illustrate the utility of PRODOT with a case study for a local utility comparing PRODOT-generated configurations to a configuration proposed by an experienced consulting firm. The comparison shows that PRODOT produces pipeline configurations similar to the consulting firm's proposal with improvements by effectively and more broadly incorporating options the consultant may not have considered.     

An analysis of the interface between evolutionary algorithm operators and problem features for water resources problems. A case study in water distribution network design

Evolutionary Algorithms (EAs) have been widely employed to solve water resources problems for nearly two decades with much success. However, recent research in hyperheuristics has raised the possibility of developing optimisers that adapt to the characteristics of the problem being solved. In order to select appropriate operators for such optimisers it is necessary to first understand the interaction between operator and problem. This paper explores the concept of EA operator behaviour in real world applications through the empirical study of performance using water distribution networks (WDN) as a case study. Artificial networks are created to embody specific WDN features which are then used to evaluate the impact of network features on operator performance. The method extracts key attributes of the problem which are encapsulated in the natural features of a WDN, such as topologies and assets, on which different EA operators can be tested. The method is demonstrated using small exemplar networks designed specifically so that they isolate individual features. A set of operators are tested on these artificial networks and their behaviour characterised. This process provides a systematic and quantitative approach to establishing detailed information about an algorithm's suitability to optimise certain types of problem. The experiment is then repeated on real-world inspired networks and the results are shown to fit with the expected results.     

A multi-objective evolutionary algorithm to exploit the similarities of resource allocation problems

The complexity of a resource allocation problem (RAP) is usually NP-complete, which makes an exact method inadequate to handle RAPs, and encourages heuristic techniques to this class of problems for obtaining approximate solutions in polynomial time. Different heuristic techniques have already been investigated for handling various RAPs. However, since the properties of an RAP can help in characterizing other RAPs, instead of individual solution techniques, the similarities of different RAPs might be exploited for developing a common solution technique for them. Two RAPs of quite different nature, namely university class timetabling and land-use management, are considered here for such a study. The similarities between the problems are first explored, and then a common multi-objective evolutionary algorithm (a kind of heuristic techniques) for them is developed by exploiting those similarities. The algorithm is problem-dependent to some extent and can easily be extended to other similar RAPs. In the present work, the algorithm is applied to two real instances of the considered problems, and its properties are derived from the obtained results.     

Reclaimed water distribution network design under temporal and spatial growth and demand uncertainties

A significant—but underutilized—water resource is reclaimed water, i.e., treated wastewater that is reintroduced for various purposes. Especially in water scarce regions, reclaimed water is often the only remaining source of water to meet increasing population and water demands. In this paper, we develop a new model formulation for the cost-effective branched reclaimed water network design and solve it with an exact optimization method. We consider both construction and energy costs expended over a twenty-year period. Unlike other formulations, uncertain reclaimed water demands, temporal and spatial population changes are explicitly considered in our two-staged construction and expansion model. In order for the system to meet higher demands during the peak times and to evaluate energy use, we consider two pumping conditions: one with average demands, which is used to compute the average energy consumption, and the other with peak demands, which dominates pipe size and pump station capacity selection. By introducing binary variables that indicate discrete pipe and pump sizes, we linearize the nonlinear hydraulic equations and objective function terms. We develop methods to significantly reduce the problem dimension by exploiting the problem characteristics and network structure. Our computational results indicate that these methods are very effective. Finally, we apply our model to design a reclaimed water network for a realistic municipal system under estimated demand and population scenarios, and analyze the sensitivity of the system to model parameters.     

Solving a dial-a-ride problem with a hybrid evolutionary multi-objective approach: Application to demand responsive transport

Demand responsive transport allows customers to be carried to their destination as with a taxi service, provided that the customers are grouped in the same vehicles in order to reduce operational costs. This kind of service is related to the dial-a-ride problem. However, in order to improve the quality of service, demand responsive transport needs more flexibility. This paper tries to address this issue by proposing an original evolutionary approach. In order to propose a set of compromise solutions to the decision-maker, this approach optimizes three objectives concurrently. Moreover, in order to intensify the search process, this multi-objective evolutionary approach is hybridized with a local search. Results obtained on random and realistic problems are detailed to compare three state-of-the-art algorithms and discussed from an operational point of view.     

Simple heuristics-based selection of guides for multi-objective PSO with an application to electrical distribution system planning

In multi-objective particle swarm optimization (MOPSO), a proper selection of local guides significantly influences detection of non-dominated solutions in the objective/solution space and, hence, the convergence characteristics towards the Pareto-optimal set. This paper presents an algorithm based on simple heuristics for selection of local guides in MOPSO, named as HSG-MOPSO (Heuristics-based-Selection-of-Guides in MOPSO). In the HSG-MOPSO, the set of potential guides (in a PSO iteration) consists of the non-dominated solutions (which are normally stored in an elite archive) and some specifically chosen dominated solutions. Thus, there are two types of local guides in the HSG-MOPSO, i.e., non-dominated and dominated guides; they are named so as to signify whether the chosen guide is a non-dominated or a dominated solution. In any iteration, a guide, from the set of available guides, is suitably selected for each population member. Some specified proportion of the current population members follow their respective nearest non-dominated guides and the rest follow their respective nearest dominated guides. The proposed HSG-MOPSO is firstly evaluated on a number of multi-objective benchmark problems along with investigations on the controlling parameters of the guide selection algorithm. The performance of the proposed method is compared with those of two well-known guide selection methods for evolutionary multi-objective optimization, namely the Sigma method and the Strength Pareto Evolutionary Algorithm-2 (SPEA2) implemented in PSO framework. Finally, the HSG-MOPSO is evaluated on a more involved real world problem, i.e., multi-objective planning of electrical distribution system. Simulation results are reported and analyzed to illustrate the viability of the proposed guide selection method for MOPSO.