many-objective portfolio optimization approach for

Confidential manuscript submitted to Environmental Modelling & Software

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MANY-OBJECTIVE PORTFOLIO OPTIMIZATION APPROACH FOR STORMWATER MANAGEMENT PROJECT SELECTION ENCOURAGING DECISION MAKER BUY-IN

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M. Di Matteoa,b, H. R. Maiera , and G. C. Dandy a

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a

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b

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Corresponding author: Michael Di Matteo [email protected]

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School of Civil, Environmental and Mining Engineering, University of Adelaide, South Australia 5005 Water Technology Pty. Ltd., 1/198 Greenhill Road, Eastwood, South Australia 5063


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Abstract

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Although formal simulation-optimization approaches have been shown to be able to identify near-optimal outcomes for a range of stormwater management problems, stakeholder acceptance of these solutions can be problematic, especially if there is a lack of familiarity with the optimization processes and simulation model used to arrive at these solutions. To address this problem, a portfolio optimization problem formulation is introduced that allows stormwater best management practices (BMPs) to be evaluated by stakeholders before the portfolio selection process. This enables the search space to be constrained before the BMP optimization process, ensuring that model results are transparent and only represent solutions that are trusted by experienced practitioners. This has the effect of reducing reliance on simulation-optimiza tio n involving complex stormwater simulation models, and increasing buy-in to the optimiza tio n results. The portfolio optimization formulation is applied to a catchment management problem in Australia, using a typical many-objective optimization approach including visualiza tio n techniques.

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Keywords

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stormwater management; water sensitive urban design; portfolio optimization; many-objective optimization; visual analytics; multi-criteria decision analysis

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1 Introduction

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Recently, many-objective evolutionary algorithms have been developed and applied to identify Pareto optimal solutions to water resources planning problems (Kollat and Reed, 2007; Matrosov et al., 2015). These approaches have used visual analytics techniques to aid exploratio n and analysis of the typically large numbers (e.g. 1000s) of Pareto optimal solutions identified, and to select several suitable schemes to present to decision-makers. The many-objective optimization/visual analytics approaches have enabled trade-offs between four (4) or more planning objectives to be considered, which better reflects the number of planning objectives considered by practitioners in real-world water resources planning problems (i.e. better than typical single or bi-objective optimization problem formulations). As discussed in recent manyobjective optimization studies (Kasprzyk et al., 2012; Kasprzyk et al., 2015; Matrosov et al., 2015; Woodruff, 2016), optimizing planning solutions for a sub-problem of what is, in fact, a manyobjective problem can lead to ‘cognitive myopia’, which is a negative decision- making bias that arises due to drawing incorrect inferences and conclusions from limited problem information. In this light, consideration of a limited number of formal objectives in optimization studies can encourage the identification of solutions with sub-optimal performance with respect to criteria that are not included as formal objectives, but are important to contemporary water resources managers (Woodruff et al., 2013). It is therefore preferable to optimize with respect to all (relevant) formal objectives where possible.

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In the stormwater management optimization field, recent stormwater best manage me nt practice (BMP) optimization approaches have typically included an integrated stormwater simulation model (Bach et al., 2014) linked with an evolutionary algorithm (Maier et al., 2014) for the optimal sizing and placement of BMPs (Di Matteo et al., 2017a) within a watershed to achieve environmental benefits from treating stormwater runoff. However, for regional-scale stormwater management problems, formal objectives have been limited to two, including ecosystem health benefits (including water quality improvement) and cost (Chen et al., 2015; Chichakly et al., 2013;


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Lee et al., 2012; Zou et al., 2015). This is despite the fact that, in many cases, stormwater management optimization can be better represented as a many-objective optimization problem, considering a larger number of objectives. This is because stormwater managers must consider a range of performance criteria due to a number of socio-political drivers including: water supply security, public health protection, social amenity, urban flow regime improvement, environme nta l protection and flood mitigation (Askarizadeh et al., 2015; Marlow et al., 2013). In response to these drivers, BMPs have been developed to provide multiple functions in addition to water quality improvement, for example stormwater harvesting (Clark et al., 2015; Di Matteo et al., 2017a; Mitchell et al., 2007) and urban vegetation and amenity improvement (Sharma et al., 2016). Such BMPs may include structural and non-structural measures for detention, harvesting, infiltratio n, evaporation and transport of non-point source urban stormwater runoff (Lerer et al., 2015).

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Conventional formulations of BMP selection problems are likely to contribute to a lack of acceptance of solutions obtained from optimization studies. This is because the practical relevance of the optimization solutions depends largely on how decision makers feel about the credibility of the evaluation techniques and data used in the decision-making process (Aumann, 2011). For example, stormwater management strategies developed by algorithms may not be trusted and adopted by decision makers who are unfamiliar with the optimization process and how the strategies are generated (Maier et al., 2014). In addition, stormwater simulation-optimiza tio n approaches (Maringanti et al., 2009; Srivastava et al., 2002) may not complement current practice for management of large regional catchments, which typically involves ad hoc selection and implementation of BMPs as funding becomes available.

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In order to develop trusted stormwater management strategies that are likely to be adopted in practice, decision maker engagement should be encouraged in all aspects of optimization studies applied to water resources problems (Maier et al., 2014; Voinov and Bousquet, 2010; Wu et al., 2016). Therefore, the problem formulation and system models used should incorporate existing modelling practice, and practitioners should aim to use optimization as a complementary tool to existing approaches where possible. Such an approach is likely to encourage the uptake of formal many-objective optimization approaches by decision makers, as it seeks to provide advice on the best course of action under the institutional and political constraints that exist in the real world . This would improve upon current practices.

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In order to address the shortcomings of existing optimization problem formulatio ns discussed above, the objectives of this paper are:

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i) To present a novel optimization problem formulation for selecting combinations of stormwater BMPs that a) can cater to a large number of performance criteria b) can handle a large number of decision options and potential strategies; ; c) can consider detailed interactions between interdependent parts of the systems; d) can enable the identification of solutions that represent the best possible trade-offs between performance criteria; e) enables trade-off information to be communicated in an easyto-understand fashion; and f) enables the development of solutions that are trusted by decision makers.

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ii) To demonstrate the utility of the optimization problem formulation by applying it as part of a generic optimization framework to a case study focused on the selection of stormwater BMPs for a major city in Australia. The generic optimization framework includes the novel portfolio optimization problem formulation (see objective i), a


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many-objective optimization technique to identify solutions to the problem, and a visual analytics package to explore, analyse and select portfolios of BMPs; and

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iii) To use the case study to a) investigate the possible many-objective trade-offs between lifecycle cost, water quality improvement, stormwater harvesting capacity and urban vegetation and amenity improvement; b) investigate the importance of a manyobjective approach compared with a bi-objective water quality-cost optimization, as has been undertaken in most previous studies; and c) demonstrate trends in the impact of particular BMP projects on Pareto optimal portfolio performance, and how these may influence decision-making.

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2 Proposed Stormwater BMP Selection Optimization Formulation

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2.1 Outline of Proposed BMP Selection Approach

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A conceptual outline of the proposed stormwater BMP selection optimization formula tio n is shown as items 1 and 2 within a generic many-objective optimization approach (Reed, 2007) (Figure 1). The portfolio optimization formulation presented allows a framework to satisfy desirable criteria for stormwater BMP selection methods, which is not the case with existing approaches (as outlined in the Introduction), as follows:

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i) The ability to develop solutions that are trusted by and have buy-in from decision makers is accounted for by formulating the problem as a portfolio optimiza tio n problem, as part of which only stormwater BMPs that are suggested by decision makers are considered as potential options and decision maker-driven evaluation of BMPs is used.

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ii) The ability to cater to a large number of performance criteria and options, as well as the ability to identify solutions that represent the best trade-offs between the performance criteria, is facilitated because the portfolio optimization formula tio n allows for the use of look-up tables for the evaluation of solution performance. As such, the optimization process allows detailed interactions between interdependent parts of a system to be considered without the need for a complex stormwater model that is linked with an optimization algorithm. The formulation handles the evaluation of performance of:

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(1) independently-functioning BMPs by calling on performance data stored in a lookup table, generated a priori (i.e. before the optimization process) by experienced practitioners using appropriate techniques to evaluate the performance of each individual BMP.

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(2) interdependently-functioning BMPs by calling on performance data stored in a look-up table that are results of simulations, of smaller and localized BMP systems, conducted a priori.

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In the first step of the overall optimization framework (Figure 1), a list of potential stormwater management BMPs, p, is identified. These BMPs are then evaluated individually by practitioners and the interdependencies between them determined. All possible combinations of these individual projects make up the full portfolio solution space, which is expected to be too large to adequately evaluate by trial-and-error or enumeration. Therefore, in order to enable consideration of many performance criteria, F, and a wide exploration of the potential portfolios,


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P, a formal optimization approach is adopted. The best combinations of BMPs are represented as Pareto optimal solutions, P*, to a many-objective portfolio optimization problem formula tio n (Cruz et al., 2014). In order to analyze the large number of Pareto optimal solutions produced by the optimization process, and to present the optimal trade-offs to decision makers in a manner that is easy to understand, interactive visual analytics are used to explore trade-offs and impacts of BMPs on portfolio performance.

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The proposed formulation is in alignment with approaches using many-objective portfolio optimization problem formulations with decision maker-driven evaluation of objective functio n values. As pointed out by Maier et al. (2014), such a portfolio optimization formulation is likely to make “…many-objective optimization accessible to decision makers whose current level of decision making sophistication includes multi criteria decision analysis…”. This is because the options under consideration, as well as the final selection of the portfolio to be implemented, are based on the domain knowledge of individual practitioners. In contrast, there is likely to be less decision maker buy-in and trust when simulation-optimization approaches are used to determine optimal solutions, as interactions between complex systems of BMPs, and therefore the rationale behind the performance values of portfolios, are not transparent to decision makers who may not use complex simulation models to support decision-making.

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Having said this, it must be noted that, in practice, there is a trade-off between obtaining mathematically optimal solutions (which may be better approximated using a simulatio noptimization approach) versus obtaining solutions that encourage decision maker buy-in and will therefore more likely influence the final stormwater management strategy adopted (which may be better achieved by having decision maker-driven evaluation of portfolios). The proposed approach may not identify the mathematically optimal solutions, since the sizes of BMPs are not considered as decision variables and the evaluation of interactions between BMPs is performed a priori and/or informed by decision makers. However, as the approach balances competing desires to produce mathematically optimal solutions and solutions that are trusted by decision-makers, it makes the benefits of optimization more accessible to practitioners, and should encourage better stormwater management strategies to be adopted in practice.


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Figure 1 Conceptual outline of the proposed many-objective portfolio optimization formulation (1 and 2) within a generic optimization approach (1-4) (adapted from Kollat and Reed (2007)) for stormwater management best management practice (BMP) selection. pi is a project, P*j is a Pareto optimal portfolio of projects, and Fj is the set of objective function values corresponding to Pareto optimal portfolio P*j.n.

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The detailed steps for implementing the conceptual approach presented in Figure 1, including the proposed problem formulation, are given in Figure 2, which are explained in the following sections.


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F igure 2 Steps in adopted formal optimization framework for selecting portfolios of BMPs, including proposed problem formulation.

2.2 Problem Formulation The first part of the optimization framework consists of steps required to formulate a portfolio optimization problem that represents the stormwater management problem. To achieve multiple catchment benefits, numerous stormwater best management practices (BMPs) are typically considered to intercept and deal with runoff, at locations distributed throughout a catchment. Examples of BMPs may include: biofiltration systems (biofilters), which typically consist of a basin overlaying a filter medium; constructed wetlands, which are shallow, extensive ly vegetated basins that use enhanced sedimentation, fine filtration and pollutant uptake processes to remove runoff pollutants; and swales, which are vegetated channels. Appropriate types and locations of BMPs largely depend on site characteristics including soil type and properties, topography, infiltration rate, contributing connected impervious area, and the space available to access for maintenance. Site characteristics are typically assessed through on-site and geospatial studies (Inamdar, 2014). After site assessment, a short-list of feasible BMPs is agreed upon amongst decision makers, taking into account the potential to achieve desired performance criteria and other socio-political factors (Chichakly et al., 2013; Sharma et al., 2016).


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The performance of each BMP is then evaluated independently against multiple criteria, using accepted models based on the contributing sub-watershed for each BMP, and in consultatio n with experienced local experts (Inamdar, 2014). In the absence of an adequate regional-sca le integrated model to evaluate the downstream impact of BMPs, interactions between BMPs that influence individual BMP performance are evaluated based on expert judgment and modelling of BMPs and multiple contributing sub-watersheds, to determine decision-making rules or performance models for interdependent projects (for examples of formulating interactions in portfolio optimization see Section 2 ‘Description and formalization of the problem’ in Cruz et al. (2014)). The individual projects, their performance, interdependencies and practical limitations on portfolio size are then formulated as the decision variables, objectives and constraints of a mathematical optimization problem. 2.2.1 A priori evaluation of independent and interdependent BMPs

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As part of the proposed formulation, the performance of independent and interdepende nt BMPs can be evaluated a priori, to eliminate the need for a complex stormwater model to be called during the optimization process. In the case where a small number of BMPs are interdepende nt, an a priori analysis of interactions between those particular BMPs can be used to estimate their objective function values when adopted together in different conbination. When one or more interdependent BMPs appear together in a potential solution, alternate objective function values can be used (i.e. by the algorithm, by referring to a look-up table of simulation results) to reflect the interdependency, compared to where they appear individually.

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To illustrate how dependency between BMPs can be handled in the optimiza tio n formulation, consider the regional stormwater BMP system in Figure 3, where BMPs treat stormwater runoff from upstream catchments, as described below.

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The individual BMPs to the left can be considered to operate independently from all others. This is because 1) inflows from these BMPs cannot consist of treated outflows from another BMP, and 2) outflows from these BMPs cannot be inflows into another BMP (as there are no BMPs located at the downstream end of the 1st order stream). This means where BMPs are assumed not to be implemented (e.g. as an outcome of the optimization process), the performance of an independent BMP, if it is present in the system, will be independent of any other system configuration, as it will have the same inflow characteristics. Therefore, the performance of each independent BMP can be simulated using a model with one node representing the BMP, a priori of the optimization process, rather than needing a complex model consisting of all BMPs to be called during the optimization process (as is typical of previous studies).

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The BMPs to the right are interdependent, since removing A and/or B will affect inflow characteristics and therefore the performance of C. The performance of the local sub-system can be simulated using a stormwater model with three nodes and two drainage links, which is still simpler than a model of the whole system. Solutions to the sub-system of BMPs can be enumerated a priori. The enumeration may include several sizes for each BMP and several combinations of BMPs. The objective function values of Pareto optimal configurations of the sub-system can be kept in a look-up table to be called upon by the optimization process to efficiently calculate the whole portfolio objective function values. The appropriate sizes for the configuration selected by the optimization algorithm can be made available for further analysis. This removes the requirement for a large and complex stormwater model to be called to simulate every solution (again, as is typical of previous studies). Importantly, this approach is effective where there is a


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large number of BMPs in the regional system and the sub-systems consist of small numbers of BMPs (e.g. 2 to 5), such that the computational time to enumerate the sub-system using a simple model is short enough to be appropriate for the decision-making task. All sub-systems within a regional system can be enumerated separately using simple models to provide performance data for the look-up table.

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F igure 3 Schematic diagram of a regional stormwater BMP system showing a) independent BMPs and b) a local system of interdependent BMPs. Numbers indicate classic stream order, with catchment outlet at ‘O’. 2) Example look-up table based on enumeration of possible configurations of the interdependent system.

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2.2.2 Limiting the solution search space In the proposed formulation, to further reduce the solution search space, and complexity of analysis of solutions, additional considerations, including the size, type and location of BMPs, can be handled without them being decision variables, as follows: •

The size of BMPs is often largely dependent on the water quality improve me nt required by regulators. Where there are regulatory water quality improve me nt targets, to minimize costs, the size of BMPs is set such that these targets are ‘just’ met. For example, where Total Nitrogen (TN) is a limiting pollutant, and the target reduction of TN from increased runoff from a development is 45% removal, the size of a BMP is increased until it just achieves 45% reduction. For an individ ua l BMP, this is a simple optimization that can be undertaken a priori. For small subsystems, the target may apply as a constraint for the system, rather than individ ua l BMPs, when considering the Pareto optimal configurations to include in the lookup table.

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The type of BMP (e.g. biofilter, wetland, swale) is often highly dependent on the site characteristics, design objectives and preferences of planners. Therefore, often a small number of BMP types is available for each particular location. The proposed approach can handle more than one BMP type at a location. Where two or more BMP projects exist in the same location, this would require a simple mutual exclusivity constraint on the decision variables for these projects (e.g. a constraint


could be: if Project A at Location 1 is selected for the portfolio, then Project B at Location 1 cannot be selected).

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In the proposed approach, the project decision variables act as a surrogate for the location of BMPs. For example, Project C may exist at Location 2 in a catchment, so by selecting Project C in a portfolio, this means Location 2 is also selected. Impacts on BMP performance specific to a location can be reflected in the costs and benefits associated with the project at the location and in terms of the objective functions representing interactions with other projects.

2.2.3 Encouraging stakeholder buy-in

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The proposed formulation assists with building trust in the solutions generated by the optimization process by only considering solutions proposed by end users. This aspect of the optimization approach aligns with existing multi-criteria analysis (MCA) approaches currently used in practice, without introducing additional complexity through typical optimiza tio n approaches. In practice, typically individual projects are identified by consultants and stakeholders through a consultation process. Then a guided scoring of individual projects is carried out. Typically, the stakeholders provide weighted scores reflecting their preferences for differe nt criteria, and a final short-list of solutions is selected based on these preferences. One of the benefits of this approach is that the scoring is transparent to stakeholders, and therefore stakeholders are likely to buy-into the results of the MCA. However, often stakeholder preferences change once a diverse set of solutions and trade-offs between the set of solutions’ objectives are visualized. Traditional optimization problems can provide diverse solution sets and trade-offs, but the solutions are typically assessed using a simulation model, the results of which may not be trusted by all stakeholders, especially where stakeholders have limited familiarity with modelling techniques and where systems are complex and their performance is not easily traceable to design decision changes. In addition, the proposed solutions are likely to be unfamiliar to stakeholders, making them more difficult to be trusted.

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However, with the proposed formulation, the scores attributed to individual BMPs by stakeholders can be used as objective function values for a portfolio optimization analysis. Therefore, the scores attributed by stakeholders are reflected in the diverse trade-offs across a range of preferences identified in the optimization process that can be explored and analyzed, and reveal optimal combinations of projects not possible with MCA techniques. With the direct link between the scores attributed by stakeholders and the optimization results, there is arguably a better chance that optimization results will be used to support decision-making, especially where decision-makers have not been trained extensively in decision support approaches or BMP systems modelling. With the proposed formulation, a variety of expertise can be accommodated, whilst maintaining a diverse set of solutions to select from. It is acknowledged there are further issues related to trust at other stages of the decision-making process that have not been considered here and require further work. However, being able to use the results generated through existing analysis techniques improves the likelihood that optimization will be adopted for BMP selection problems in practice, especially since the individual components of solutions identified by the optimization process were proposed by, and are therefore familiar to, stakeholders.


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2.2.3 Portfolio optimization problem To identify portfolios that represent the best trade-off between many objectives, the project portfolio selection problem is defined as the optimization of vector F(P), composed of n objective functions: F(P) = [f 1 ,f 2 ,…,f n ]

(1)

where P is a portfolio of projects, and F is a vector of the associated costs and benefits of a portfolio. For a more in-depth description of a generic portfolio optimization formulation, the reader is referred to Cruz et al. (2014). The generic decision variables, objectives, and constraints particular to the stormwater management portfolio selection problem are as follows. Decision Variables It is assumed that each BMP project has a pre-determined size, type and location. As such, each decision variable is a binary variable, di, that represents the decision whether or not to adopt project, pi. There are N p possible projects, and thus N p decision variables, given by: d = d1, d2 ,…,dNp , where di ∈0,1, for all i ∈ N>0, 1 ≤ i ≤ Np . A portfolio, P, is defined as the set of projects pi for all i where di = 1. Objectives

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Although objectives depend on decision maker interests, four formal objectives addressing one or more economic, social, or environmental stormwater management goals are included in the proposed generic formulation: economic cost, water quality improvement, stormwater harvesting capacity, and combined urban vegetation and amenity improvement. The objectives for lifec yc le cost, water quality improvement and stormwater harvesting are adapted from general objectives presented in (Di Matteo et al., 2017a), a generic optimization approach for distributed stormwater harvesting systems. The green score objective formulation is provided as an example of adapting multi-criteria analysis results (here, scores for multiple sub-criteria including tree cover, amenity) within a single overarching criterion (here, urban amenity) for use as a formal optimiza tio n objective.

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Economic cost is a primary concern for decision makers responsible for maximizing return on investment, including capital, maintenance and operating costs. Water quality improvement is a key environmental objective considered by stormwater management authorities (Chichakly et al., 2013; Yang and Best, 2015). Maximizing stormwater harvesting (SWH) volume is a primary motivation for implementing projects with SWH capacity in order to reliably meet irrigatio n demand, and can also contribute to runoff volume reduction and groundwater recharge known to produce ecosystem health benefits (Askarizadeh et al., 2015). An amenity improvement score is proposed as the social criterion, as BMPs are typically located in public open spaces and are maintained using public resources and urban vegetation and amenity improvement is often an important criterion for evaluating BMPs.

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In the proposed formulation, the economic cost of a portfolio of projects is represented as a life cycle cost LCC [$] (Eq. 2) (Di Matteo et al., 2017a), which is a discounted sum of expected future costs for stormwater management assets, including BMPs and transfer infrastruc ture


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required to harvest stormwater (Taylor and Wong, 2002). The life cycle cost objective function for each candidate portfolio of BMPs is given by: MINIMIZE: 𝑓𝑐𝑜𝑠𝑡 = 𝐿𝐶𝐶𝐵𝑀𝑃 + 𝐿𝐶𝐶𝑆𝑊𝐻

(2)

where 𝐿𝐶𝐶𝐵𝑀𝑃 = ∑𝑁 𝑖 =1 { (𝑇𝐴𝐶𝐵𝑀𝑃𝑖 ) + 𝑃𝑊𝐹𝑒𝑠𝑡𝑎𝑏 ,𝐵𝑀𝑃𝑖 (𝑆𝐴 𝐵𝑀𝑃𝑖 × 𝐸𝐶𝐹𝐵𝑀𝑃𝑖 × 𝑀𝐵𝑀𝑃𝑖 ) + 𝑃𝑊𝐹𝑚𝑎𝑖𝑛𝑡,𝐵𝑀𝑃𝑖 (𝑆𝐴𝐵𝑀𝑃𝑖 × 𝑀𝐵𝑀𝑃𝑖 )}

(3)

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𝐿𝐶𝐶𝑆𝑊𝐻 = 𝐶𝐶𝑎𝑝𝑇𝑎𝑛𝑘 + 𝐶𝐶𝑎𝑝𝑃𝑖𝑝𝑒 + 𝐶𝐶𝑎𝑝𝐶𝑜𝑛𝑡𝑟𝑜𝑙 + 𝐶𝐶𝑎𝑝𝑃𝑢𝑚𝑝 + 𝑃𝑊𝐹𝑚𝑎𝑖𝑛𝑡 (𝐶𝑚𝑇𝑎𝑛𝑘 + 𝐶𝑚𝑃𝑖𝑝𝑒 + 𝐶𝑚𝐶𝑜𝑛𝑡𝑟𝑜𝑙 + 𝐶𝑚𝑃𝑢𝑚𝑝 ) (4)

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where the sum of the cost of BMPs to capture and treat stormwater runoff, LCCBMP [$] (Eq. 3), and to transfer harvested water to a balancing storage for further treatment and distributio n, LCCSWH [$] (Eq. 4), is applied, with BMPi representing the ith BMP in the candidate portfolio, N [integer] is the number of projects in the portfolio, TAC [$] is the total acquisition cost as a functio n of SA, the surface area of BMPi. and N [integer] is the number of projects in a portfolio.

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PFWestab [fraction], for the establishment period, and PWFmaint, for the remaining design life of system components, are the present worth factor for a series of annual costs computed using a discount rate. ECF [fraction] is the establishment cost factor (i.e., multiplier) for the annual maintenance cost M [$] during the establishment period (typically 1-2 years) for each BMP. For BMPs with a stormwater harvesting function, CCapTank [$], CCapPipe [$], CCapControl [$], and CCapPump [$] are the capital costs for required storage tank, control systems, pipes, and pump stations, and CmTank [$], CmPipe [$], CmControl [$], and CmPump [$] are the annual maintenance costs for the tank, pipes, control systems and pumps, and operating costs, respectively.

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The water quality improvement indicator adopted in the proposed framework is the total average annual pollutant load reduction of one target pollutant (Eq. 5). Only one target polluta nt is adopted to limit the number of objectives and therefore limit the difficulty in identifying optimal solutions, however, if the trade-offs between multiple water quality indicators need to be known, then these can be added as objectives. This indicator is widely adopted to assess the performance of WSUD approaches, including SWH systems (Browne et al., 2012). The target pollutant(s) will depend on decision maker interests. The water quality improvement objective function is:

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MAXIMIZE: 𝑓𝑞𝑢𝑎𝑙𝑖𝑡𝑦 = ∑N i=1 𝑆𝑜𝑢𝑟𝑐𝑒𝑖 − 𝑅𝑒𝑠𝑖𝑑𝑖

(5)

where, 𝑓𝑞𝑢𝑎𝑙𝑖𝑡𝑦 [mass year-1 ] is the mean annual pollutant mass retained by BMPs in each candidate portfolio, N is the number of BMPs in a portfolio, Residi [mass year-1 ] is the mean annual mass of pollutant leaving the ith BMP’s contributing catchment area, and Source [mass year-1 ] is the mean annual mass of pollutant that reaches the ith BMP’s catchment outlet in a postdevelopment catchment baseline scenario without intervention. Resid and Source should be determined using a stormwater quality assessment model accepted by the stormwater manageme nt authority (Bach et al., 2014; Coombes et al., 2002).


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Average annual supply capacity (Eq. 6) is adopted as an indicator of stormwater harvesting performance (Mitchell et al., 2008). This metric is proposed because it can be determined from generic storage-yield-reliability curves for a catchment at the project screening phase of stormwater management (Browne et al., 2012; Hanson and Vogel, 2014), or other techniques (Inamdar, 2014). In addition, the average annual capacity approximates runoff volume reduction due to harvesting, which has ecosystem health benefits (Askarizadeh et al., 2015). The supply stormwater harvesting objective function is:

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MAXIMIZE: 𝑓𝑠𝑢𝑝𝑝𝑙𝑦 = ∑ni=1 𝑆𝑢𝑝𝑝𝑙𝑦𝑖

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where Supplyi [volume] is the average annual stormwater harvesting supply capacity for the ith BMP in a portfolio.

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The urban vegetation and amenity improvement indicator depends on decision maker interests, which may include maximizing vegetation and tree coverage and quality of recreation spaces. Each project should be appraised and evaluated (scored) by vegetation experts. The cumulative urban vegetation improvement objective function is:

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MAXIMIZE: 𝑓𝑔𝑟𝑒𝑒𝑛 = ∑ni=1 𝐺𝑟𝑒𝑒𝑛𝑖

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where Greeni [integer] is a score, determined by expert assessment, attributed to the ith project in a portfolio.

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(6)

(7)

Constraints Strategic and logical constraints on the selection of projects and performance of portfolios could be considered, and are case specific (Cruz et al., 2014). For example, where multiple subregion catchment institutions fund an integrated catchment strategy, constraints on the selection of projects could (1) ensure equitable distribution of projects amongst constituent stormwater management sub-regions, (2) limit the maximum number of projects in a portfolio, N max , and projects within each sub-region, (3) prevent the presence of mutually-exclusive projects, as some BMPs may be redundant in the same portfolio, and (4) limit the budget allocated to projects within each sub-region. Additional considerations for portfolio-based constraints are discussed in Cruz et al. (2014). 2.3 Optimization Process The second part of the optimization framework (Figure 2) describes the algorithmic processes used to solve the optimization problem. Only portfolios that are non-dominated (i.e. none of the objective functions can be improved in value without degrading one or more of the other objective function values) can be considered as portfolios that represent the best trade-off between objectives. To identify the non-dominated, or ‘Pareto optimal’ solutions to the mathematical optimization formulation, use of a many-objective metaheuristic algorithm is suggested. Metaheuristic algorithms have several advantages over traditional optimiza tio n approaches (such as linear programming). They can deal with multiple objectives simultaneo us ly (Maier et al., 2014) and have been successful in recent planning and design optimization studies considering urban water (Beh et al., 2014; Beh et al., 2017; Blinco et al., 2017; Marchi et al., 2016;


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Newman et al., 2014; Paton et al., 2014; Wu et al., 2017) and distributed BMP systems (Chichakly et al., 2013; Di Matteo et al., 2017a).

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As part of the generic optimization process, a number of solutions is generated with the aid of a many-objective metaheuristic algorithm. Each solution represents a set of binary decisions on whether or not to adopt each available project in a portfolio. In the construction of a solution, projects are added to a portfolio until a constraint on the maximum number of projects is reached, or all projects have been considered i.e. a portfolio can consist of fewer than the maximum number of projects. Then, portfolios are evaluated against logical and strategic conditions (for, example mutual exclusivity of projects). If a portfolio violates these conditions, the objective functio n values are set to a penalty value. The penalty value will depend on the optimization problem considered. Next, the performance of valid portfolios is evaluated by calculating objective functions, including interactions (see Section 2.2.1). After evaluation, final penalties are applied to objective function values of solutions that fail to meet defined constraints. The metaheuris tic algorithm uses objective function values to assess the fitness of solutions and to iteratively modify solutions. Over a number of iterations, solutions converge towards the set of Pareto optimal portfolios, which are non-dominated in the set of all feasible portfolios. The metaheuristic iterative approach continues until specific termination criteria are met (for example, a maximum number of iterations). The non-dominated solutions identified by the optimization process are Pareto optimal (or “near-optimal” as one can never prove that the true Pareto front has been found when using evolutionary algorithms) stormwater management portfolios.

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2.4 Visual Analysis of Pareto Optimal Portfolios An interactive visual analytics package (Hadka et al., 2015; Kollat and Reed, 2007) is suggested to assist decision makers to explore, analyze and ultimately select appropriate portfolios that represent a desired compromise between performance criteria and practical stormwater management strategies (Maier et al., 2014). Firstly, the Pareto optimal portfolio performance and decision data, as well as alternative data that may be useful for decision-making (e.g. average contributing catchment size, BMP type, number of projects) are uploaded into the visual analytics package. Next, high-dimensional coordinate plots or parallel coordinate plots (Inselberg, 2009) are used to visualize the performance of the large number of Pareto optimal portfolios in manyobjective space. Then, in order to reduce the number of portfolios considered for further analysis, dynamic filtering to eliminate undesirable solutions can be carried out by analysts based on the decision maker’s budget constraints and minimum preferences for each benefit, and eliminate apparently undesirable combinations of BMPs not anticipated a priori (Piscopo et al., 2015). Within the reduced set, decision makers and analysts can use brushing to highlight sub-sets of interesting solutions. Multiple linked plots of the same data set can assist with identifying and rationalizing trade-offs, such as conflicts and areas of diminishing returns between objectives and emergent behavior caused by the inclusion of particular BMPs within portfolios. Interactive visualization of optimization objectives and decision spaces simultaneously enables decision makers, with the assistance of analysts, to rapidly identify subsets of portfolios that contain preferred projects and compare their performance to that of other portfolios. In this way, browsing through solutions to investigate and learn about the impact of individual project preferences on total catchment benefits can allow decision makers to overcome institutional decision-mak ing biases (Kollat and Reed, 2007; Matrosov et al., 2015). Ultimately, several desirable portfolios are selected for further consideration.


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3 Case Study

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In this study, we demonstrate the many-objective BMP selection approach on a regional stormwater management strategy for a major coastal city in Australia. A catchment manageme nt authority commissioned engineering consultants to identify sites for stormwater BMPs within an integrated catchment with an outlet flowing into a prominent marine body. The integrated catchment covers an area of approximately 700km2 , with average annual rainfall of 400-700mm, and is comprised of highly urbanized and peri-urban regions managed by three local governme nt authorities. A primary objective for the catchment management authority was to reduce the nutrient load from urban stormwater runoff flowing into the marine body. In addition, since the potential sites for BMPs were within public open spaces managed by local government authorities, stormwater harvesting for irrigation of open spaces and increasing vegetation and public amenity value were considered important additional benefits. The consultants identified 70 (N p =70) potential biofiltration, wetland and swale projects at locations distributed in open spaces throughout the three local government authority regions through stakeholder consultatio n. Thirteen of these have a capacity for stormwater harvesting. In addition, the consultants suggested that a portfolio of 20 projects or fewer (N max =20) was practical. The BMPs were considered to be mutually independent for the purposes of demonstrating the optimization approach, as the contributing catchment areas to each BMP did not coincide (i.e. downstream impact of BMPs would not affect the performance of other BMPs within the large regional catchment and as such, interdependencies between BMPs were not considered in the case study application). The number 17 of possible portfolios was ∑20 , which is too large to 𝑘=1(70!/{( 70 − 𝑘) ! 𝑘!) = 2.59898 × 10 fully enumerate using a typical desktop computer.

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The application of the proposed optimization approach was part of a real-world study involving a multi-criteria analysis conducted to identify a portfolio of BMP projects for a regional catchment. This allowed the authors to demonstrate how the proposed approach can consider existing BMP selection practices, which is a study objective. As the case study application was only intended to demonstrate the optimization approach, the results of the study were reviewed by consultants but were not used to inform decision- making. Engagement between decision makers, engineering consultants, and the optimization analysts (who are the authors of this study), was carried out as follows. Firstly, the engineering consultants ran one workshop where the broad stormwater management objectives were established, which was attended by a working group of 16 decision makers from local government authorities and the catchment management authority. The consultants then identified sites, assessed them for quantitative metrics (e.g. required size of BMPs to meet water quality constraints, cost, and stormwater harvesting capacity) and carried out a preliminary scoring each of the qualitative metrics (e.g. vegetation improvement and amenity value) using objective thresholds. Consultants then sent these preliminary scores to local government authorities who were asked to provide a response. These were generally reviewed by landscape, bushland, horticultural and parks and open space staff. The staff involved and level of response varied between the local government authorities. Consultants then had a workshop with each of the individual local government authorities to review the sites, establish a common understanding of the whole catchment management opportunity and confirm the proposed individual project scoring. Then, important objectives were refined into formal optimiza tio n objectives by the consultants and optimization analysts. The analysts used the multi-crite r ia evaluation data to inform the optimization problem formulation including decision variables (projects), and to develop objective functions, objective function values and constraints. The names of the decision makers and catchment regions involved are not disclosed in this study. The


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data used for this study are listed in the references, tables, supplements and repository at Di Matteo et al. (2016)..

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Based on information provided by regulators, a single climate scenario was considered. However, it should be noted that alternative climate scenarios could have been considered. For example, the Pareto optimal solutions could be each assessed against a dry climate scenario, and solutions dominated under that scenario removed from further consideration as in Chichakly et al. (2013). It should also be noted the formulation does not preclude a robustness evaluation (Maier et al., 2016; McPhail et al., 2018; Riddell et al., 2018). However, this would require the analyst to evaluate the independent BMPs and sub-systems of BMPs under different climate scenarios and to use multiple look-up tables to store the a priori evaluation results.

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The portfolio optimization problem formulation developed for the case study, the optimization process used to solve the problem, and the visual analytic approach used to analyze, explore, and select from optimal BMP portfolios, are presented in the following sections.

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3.1 Problem Formulation 3.1.1 Decision Variables

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The 70 potential BMPs (Table 1) were formulated as 70 decision variables with two corresponding decision options; to adopt or not adopt a BMP in a portfolio. Following a preliminary desktop analysis, BMPs were determined by decision makers to have contributing catchments ranging in size from 3 ha to 421.2 ha, with an assumed 50% pervious and 50% impervious area. The functional areas of BMPs were pre-determined by consultants and sized to meet functio na l requirements for total nitrogen, total phosphorous and total suspended solids runoff polluta nt reduction targets (Dr. Dale Browne, personal communication, 2016).

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Table 1 Details of available stormwater best management practice (BMP) projects

Local Project BMP Contributing Lifecycle TN Total Green government ID Type catchment cost Reduction Supply score authority area (ha) ($NPV) (kg/yr) (ML/yr) (LGA) 1 3 Biofilter 22.5 305,157 72.75 0 4 4 Biofilter 11.6 271,251 37.4 0 4 5 Biofilter 7.7 175,626 24.86 0 5 6 Biofilter 9.3 131,719 30.16 0 5 7 Biofilter 8.2 43,906 26.63 0 5 8 Biofilter 9.4 87,813 30.25 0 5 12 Biofilter 50.3 1,220,630 162.82 0 5 13 Wetland 4.8 169,532 15.49 0 5 23 Wetland 3.0 98,438 9.58 0 5 24 Wetland 13.5 459,379 43.63 0 5 25 Wetland 13.2 459,379 42.79 0 5 35 Wetland 21.5 918,757 69.5 0 5 36 Biofilter 45.2 949,379 146.3 0 5


2

3

45 46 50 55 56 57 1 2 9 16 19 20 21 22 27 29 37 42 47 49 51 52 58 59 60 61 63 66 68 70 10 11 14 15 17 18 26 28 30 31 32 33 34 38

Biofilter Swale Biofilter Biofilter Biofilter Wetland Biofilter Biofilter Wetland Biofilter Wetland Wetland Wetland Wetland Biofilter Wetland Wetland Wetland Biofilter Biofilter Wetland Wetland Biofilter Biofilter Biofilter Biofilter Biofilter Biofilter Wetland Wetland Biofilter Biofilter Wetland Biofilter Biofilter Biofilter Wetland Biofilter Biofilter Biofilter Swale Swale Wetland Biofilter

24.8 271,251 64.5 123,814 9.6 187,282 8.7 305,157 84.9 237,345 29.4 1,206,996 20.4 508,596 25.4 542,502 91.9 1,220,630 28.5 474,689 22.5 787,506 14.8 525,004 59.0 718,815 21.3 406,877 15.3 305,157 6.2 196,877 13.6 590,630 37.5 951,570 57.9 712,034 36.0 610,315 17.4 590,630 21.3 721,881 25.5 592,986 7.8 224,031 50.4 189,135 57.7 381,297 10.4 178,041 88.6 2,027,127 98.4 976,171 22.0 768,630 53.1 1,017,191 32.8 305,157 11.5 295,315 16.0 203,438 43.7 474,689 417.2 474,689 4.5 164,064 10.6 87,813 40.8 542,502 7.2 131,719 10.1 114,970 13.5 88,438 51.4 732,378 97.7 213,213

80.17 208.58 31.08 28.27 274.58 95.12 55.79 69.5 251.32 78.09 61.66 40.55 161.29 58.23 41.89 16.89 37.31 102.47 158.47 98.48 47.56 58.23 69.9 21.21 137.78 157.92 28.59 242.49 269.13 60.29 145.28 89.68 31.52 43.78 119.47 1141.49 12.44 29.13 111.75 19.65 27.73 37.02 140.6 267.23

0 0 11.95 0 0 12.83 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3.0 10.0 2.42 40.0 1.5 5.0 2.0 2.5 0 0 0 0 0 0 0 0 0 0 0 0 0 1.73

7 7 6 8 5 5 4 4 4 6 4 4 4 4 6 6 5 6 7 7 6 6 6 6 6 6 6 6 7 7 5 5 6 6 7 7 6 6 6 6 7 7 5 6


39 40 41 43 44 48 53 54 62 64 65 67 69

15.9 27.2 97.7 18.7 43.7 421.2 15.1 63.1 14.0 18.8 47.4 8.4 47.4

175,626 610,315 97,587 440,783 576,409 915,472 525,004 962,941 576,409 656,255 847,660 95,157 169,532

43.59 74.48 267.23 51.25 119.47 1152.38 41.28 172.58 38.19 51.43 129.69 23.11 129.69

0 0 1.73 0 0 0 0 0 0 0 0 1.29 0

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3.1.2 Objectives

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3.1.2.1 Cost

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Biofilter Biofilter Biofilter Biofilter Biofilter Biofilter Wetland Wetland Wetland Wetland Biofilter Biofilter Biofilter

The objective function for lifecycle cost of each portfolio, LCC [$], was calculated using (Eq. 2-4). The parameters for LCCBMP [$] (Eq. 3) were estimated from cost schedules developed by Melbourne Water Australia (2013) (Table 2). A typical lifecycle period of 25 years, a discount rate of 6.5% per year, an establishment cost factor of 3, and an establishment period of 2 years, were adopted. The parameters for LCCSWH [$] (Eq. 4) were estimated as follows. A cost model for the total net present value (NPV) of stormwater harvesting components was determined using linear regression (r2 = 0.814) between levelized lifecycle cost [$/ML] and estimated annual volume supplied [ML/yr], using detailed costing data for six stormwater harvesting projects derived by Inamdar (2014). Thus, the lifecycle cost of stormwater harvesting components from Eqn. 4 was calculated using the following equation:

559 $

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𝐿𝐶𝐶𝑆𝑊𝐻 = {

∑𝑁 𝑖 =1(−104.49 ⋅ 𝑆𝑢𝑝𝑝𝑙𝑦i + 6622.6)[ 𝑀𝐿] ⋅ Supply𝑖 [𝑀𝐿] , if 𝑆𝑢𝑝𝑝𝑙𝑦 > 0 i 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

(8)

where Supplyi is the average annual supply capacity of the ith BMP in a candidate portfolio of N BMPs. Table 2 Cost variables for stormwater best management practice (BMP) types.

BMP Surface Area (SA) (m2 ) Wetland 0 < SA ≤ 500 500 < SA ≤ 10,000 SA > 10,000

Construction Cost ($/m2 ; year 0)

Establishment Cost ($/m2 /yr; year 1-2)

Maintenance Cost ($/m2 /yr; year 3-25)

150

30

10

100

6

2

75

1.5

0.5


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Biofiltration basin 0 < SA ≤ 100 1,000 15 5 100 < SA ≤ 500 350 15 5 SA > 500 250 15 5 Swale All sizes 35 9 3 Note: Establishment cost = Annual maintenance cost × establishment cost factor. Costs are in Australian Dollars (2013$), based on Melbourne Water Australia (2013). Values were scaled using an inflation adjustment factor of 1.03053 from 2013$ to 2016$ (Dr. Dale Browne, personal communication, 2016). 3.1.2.2 Water Quality Improvement

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Total Nitrogen (TN) was the specific pollutant constituent adopted for the water quality objective. TN load reduction was particularly important since in the urban catchment, it was found by the consultants that maximizing TN reduction through treatment of stormwater also tended to reduce phosphorous, total suspended solids and other pollutants to within target levels (Dr. Dale Browne, personal communication, 2016). The introduction of excess anthropogenically-generated nutrients into coastal systems can cause eutrophication, which has negative impacts. These impacts often include excessive, and sometimes toxic, production of algal biomass, loss of important nearshore habitat, changes in marine biodiversity and species distribution, increased sedimentatio n of organic particles, and depletion of dissolved oxygen.

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The mean annual pollutant mass of TN retained by each candidate portfolio (𝑓𝑞𝑢𝑎𝑙𝑖𝑡𝑦 ; Eqn. 5) was calculated based on the sum of average annual TN mass retained by individual BMPs in a portfolio. The water quality improvement of individual BMPs (i.e. not an integrated system of a portfolio of BMPs) (Eqn. 5) was assessed using the integrated stormwater model MUSIC, version 6.1 (Model for Urban Stormwater Improvement Conceptualizion, eWater (2009)), as suggested by the relevant catchment management authority regulations. MUSIC is an integrated stormwater model that evaluates rainfall/runoff and pollutant generation and transport, as well as the hydraulic and pollutant removal performance of BMPs (Bach et al., 2014). MUSIC algorithms simulate runoff based on models developed by Chiew and McMahon (1999) and urban pollutant load relationships based on analysis by Duncan (1999).

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3.1.2.3 Stormwater Harvesting

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To determine stormwater harvesting capacity of projects, experts on stormwater harvesting from each local government authority were asked to evaluate the stormwater harvesting potential of BMPs within their jurisdiction. They estimated the expected irrigation demand required by open spaces near each BMP, and the average annual potential capacity to supply the demand. The estimates were based on procedures specific to each local government authority, and reflect the stormwater harvesting objective performance values accepted by decision makers.

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3.1.2.4 Urban Vegetation and Amenity Improvement

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The ‘green’ score’ of individual projects (which is a weighted score of several indicators that was developed by the authors and agreed to be used as an optimization objective by the


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consultants), uses scores assigned by experts from each local government authority interviewed in a workshop session by the consultants. The experts were asked to answer the following questions about the BMP projects within their jurisdiction: Answer ‘Yes’ ‘No’ or ‘Maybe’ to the following questions: 1) “will native vegetation increase at the site?”, 2) “will tree cover increase at the site?”, and, 3) “will the quality of recreation spaces in the area increase?”. The total catchment ‘green’ score objective function was:

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𝐺𝑟𝑒𝑒𝑛𝑖 = ∑3𝑗=1 𝑆𝑐𝑜𝑟𝑒𝑗

(9)

3 𝑖𝑓 𝑎𝑛𝑠𝑤𝑒𝑟 𝑖𝑠 ′𝑌𝑒𝑠′ 𝑆𝑐𝑜𝑟𝑒𝑗 = { 2 𝑖𝑓 𝑎𝑛𝑠𝑤𝑒𝑟 𝑖𝑠 ′𝑀𝑎𝑦𝑏𝑒′ 1 𝑖𝑓 𝑎𝑛𝑠𝑤𝑒𝑟 𝑖𝑠 ′𝑁𝑜′

(10)

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where Greeni is the sum of scores for each project, and Scorej is the number of points assigned to the answer to the j th question. Since there were three questions, each project could achieve a maximum of 9 green points, and each portfolio a theoretical maximum of (20 × 9 = 180) green points. 3.1.2.5 Evaluation of Individual BMPs Before the optimization process was run, the costs and performance values of each BMP were determined (Table 1). Firstly, the stormwater harvesting capacity of individual projects was determined from local government authority expert interviews. Secondly, the individual project lifecycle costs were determined using cost parameters from Eqns. 2, 3, 4 and 8 for each project. Thirdly, the water quality performance of each BMP was determined with the aid of MUSIC. To do this, a stormwater model for a 1 ha catchment area for each local government authority was developed. The model consisted of a 0.5 ha pervious catchment node, a 0.5 ha impervio us catchment node, and an outlet node to estimate the average annual TN load per unit area of catchment with an average 50% impervious surface area (Browne et al., 2012). One year of continuous climate data and pervious surface parameters provided by the catchment manageme nt authority were used for the catchment nodes. To estimate Source [kg] for each BMP, the TN load from a 1 ha unit catchment area for the respective local government authority was multiplied by the contributing catchment area to each BMP in hectares. Each BMP was assumed to remove 45% of the TN load from its contributing catchment (i.e. Residi = (1 - 0.45) × Sourcei), which was suggested as an acceptable performance based on advice from the consultants (Dr. Dale Browne, personal communication, 2016). Finally, Eqns. 7, 9 and 10 were applied to determine the individual project green scores. 3.1.3 Constraints A single constraint was applied to limit portfolios to 20 or fewer projects, since more than 20 projects was determined to be impractical to design and construct by the catchment management authority, as mentioned previously. The projects were assumed be independent in that the inclusion of one project did not influence the expected benefit, cost, or feasibility of another. This assumption was considered acceptable since the catchments contributing to each BMP were mutually exclusive, and customers for stormwater harvesting projects could receive supply from only one project.


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3.2 Pareto-Ant Colony Optimization (P-ACO) Algorithm

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To solve the optimization problem, a variant of the original Pareto-Ant Colony Optimization algorithm (P-ACO; Doerner et al. (2004)) metaheuristic search algorithm was used. P-ACO was selected because it was originally developed to solve portfolio optimization problems (Doerner et al., 2004; Doerner et al., 2006), it has been used successfully and adopted as a benchmark algorithm in recent three-objective portfolio optimization applications (Cruz et al., 2014), and it has been applied to complex multi-objective water resources problems (Nguyen et al., 2016; Szemis et al., 2014; Szemis et al., 2013). The variant adopted here, PACOA, was demonstrated to outperform other multi-objective ant colony optimization algorithms in a recent water resources allocation study (Szemis et al., 2013). The algorithm mimics the cooperative foraging behavior of an ant species that leaves a chemical pheromone on a ground surface. In reallife, since ants traverse short paths to food more frequently, more pheromone is laid on short (efficient) paths. Thus, paths with higher pheromone levels are more likely to be selected by an ant. In the algorithm, artificial ants select between paths, which represent decisions whether or not to adopt a BMP in a portfolio in this instance. An input template and executable for the algorithm are available as Data Set 3 in (Di Matteo et al., 2016).

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A summary of the steps in the PACO algorithm is shown in Figure 4. In the initializa tio n phase, the PACO search control parameters are set. The iterative process commences when b ants are generated, each ant starting with an empty portfolio x = (0), and the objective weights (i.e., the ant’s individual preferences) are determined randomly for each ant. In the construction phase of the algorithm, first the order of BMPs is randomly shuffled, to ensure BMPs are provided an equal chance of being considered first by each ant (see Golding et al. (2017)). Then, the ant decides whether to add each BMP to a portfolio, x, by applying a pseudo-random-proportional rule using pheromone information τi. The pheromone information is stored in one 2xN matrix for each jth objective, representing the binary options for the N possible BMPs. If the ant adds the maximum number of BMPs, Nmax, before all BMPs have been considered, then none of the remaining BMPs are selected. After a portfolio has been constructed, its performance is evaluated using the objective functions (Eqns. 2, 5, 6, and 7). In this case, as individual projects were determined to be independent, the portfolio objective functions were a summation of the constituent individ ua l project objective function values in Table 1.


Initialize PACOA

For each iteration iter (iter = 1 to w) For each ant (ant = 1 to b) Construct candidate portfolio Calculate fitness functions if ant = b Non-dominant sort

Global update pheromone matrices if its = w Non-dominated portfolios of BMPs 672 673

F igure 4 Portfolio optimization process for Pareto Ant Colony Optimization Algorithm (PACOA)

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After each iteration, of the b portfolios generated by the b ants, the non-dominated portfolios are stored offline in an array. Then, as part of a global update of every element of the j pheromone matrices, the first and second best performing solutions ranked for each j th objective are used to apply the following equation. 𝜏𝑡𝑗 = (1 − 𝜌) ∙ 𝜏𝑡𝑗 + 𝜌 ∙ 𝛥𝜏𝑡𝑗

(11)

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682

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15, 10, 𝛥𝜏𝑡𝑗 = { 5, 0,

𝑡 𝑖𝑛 𝑏𝑜𝑡ℎ 𝑏𝑒𝑠𝑡 𝑎𝑛𝑑 2𝑛𝑑 𝑏𝑒𝑠𝑡 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑡 𝑖𝑛 𝑏𝑒𝑠𝑡 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑡 𝑖𝑛 𝑠𝑒𝑐𝑜𝑛𝑑 𝑏𝑒𝑠𝑡 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

where, for each BMP, the current pheromone value for each t th binary option and j th objective is reduced by pheromone evaporation, ρ, and increased by a pheromone value (𝛥𝜏𝑡𝑗 ). Pheromone is evaporated from decisions that are not in the best solutions for each objective, which makes it less likely these decisions will be selected again in future iterations. In this way, the ant’s decisionmaking landscape is modified to guide ants into regions of the search space that contain nondominated portfolios. Since the single constraint was handled in the construction phase, no penalty function is required for this case study as all constructed portfolios are feasible. The process of developing, assessing and updating the pheromone trails to guide the PACOA to near-optimal trade-offs continues until a specified maximum number of iterations, w, is reached.


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Before the PACOA was applied, a sensitivity analysis was conducted to identify suitable values of parameters that control the searching behavior of the algorithm to maximize the likelihood that the best possible approximation of the Pareto front was generated. The ranges of parameter values tested and the final parameters selected are given in Table 3.

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Table 3 Pareto Ant Colony Optimization Algorithm (PACOA) parameters tested and adopted in sensitivity analysis.

PACOA Parameter Number of ants (b) Initial pheromone (τ o ) Evaporation rate (ρ) Evaluations (b × w)

Range of Values Tested Adopted Value 20, 200, 300,500 0.5, 1.0, 10.0 0.1, 0.15, 0.2, 0.4, 0.5 Up to 2,000,000

500 0.5 0.4 600,000

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In this study, the PACO was run for 1200 iterations of 500 ants, which equates to 600,000 objective function evaluations. This number of evaluations was selected because the progress of the Pareto front ceased to be meaningful (assessed by visually inspecting the Pareto optimal solution set at 5,000 evaluation intervals) after this number of evaluations in a trial run of 2,000,000 evaluations. The optimization results were replicated 50 times using different random starting seeds for the pseudo-random number generator used in the algorithm to minimize the impact of probabilistic effects of some of the operators that influence the search. Each run took approximately 26 minutes on a 3.10GHz computer with 8 GB of RAM, although multiple instances were run on one machine simultaneously. The Pareto optimal solutions shown in this paper are the result of a non-dominated sort of the solutions from the 50 replicate runs. 3.3 Interactive Visual Analytics to Explore Pareto Optimal Solutions

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To visualize and analyze the objective and decision space trade-offs of the Pareto optimal set of portfolios, an interactive visual analytics package was selected. The combined objective space and decision space visualizations were carried out using the approach of Kollat and Reed (2007) using the DiscoveryDV software package (DiscoveryDV Version 0.72; available at https://www.decisionvis.com/discoverydv/). The package features an interactive data plot that allows brushing, linked views of solutions, marking and tracing of solutions of interest, as well as rapid browsing through solution objective, decision and non-objective performance data. The package has been used successfully in several recent many-objective optimization studies (Piscopo et al., 2015; Woodruff et al., 2013). The Pareto optimal solution objective and decision data were uploaded into the interactive visual analytics package. This allowed the analyst to 1) visualize and analyze trade-offs between the four objectives, 2) isolate portfolios from several regions of the trade-off front using interactive brushing and visualization in multiple linked plots, and 3) visua lize the decision and objective space to analyze the impact and prevalence of particular projects on the performance of Pareto optimal solutions. The Pareto optimal solution data file uploaded into the package is available as Data Set 3, and a ‘.ddv’ file for the DiscoveryDV program containing the visualizations is included as Data Set S4 in (Di Matteo et al., 2016).

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4 Results and Discussion

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This section presents the results of the many-objective optimization process for the stormwater management portfolio selection case study outlined in Section 3. The results of the PACOA runs,


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from 50 random starting positions, show the algorithm identified 3654 Pareto optimal (or nearPareto optimal) portfolios as solutions to the optimization problem. 4.1 Identifying Many-Objective Management Portfolios

Trade-Offs Between Pareto Optimal Stormwater

Figure 5 shows the trade-offs between four objectives of the Pareto optimal portfolios in a 4-dimensional coordinate plot. A sharp trade-off exists between TN reduction and cost, and between reuse capacity and cost, indicating small increments in cost can return large increases in both of these objectives. In contrast, green score tends to increase with cost, which is expected as higher cost portfolios have more BMPs distributed in the catchment to enable larger total catchment urban greening and amenity improvement.

F igure 5 A many-dimensional interactive coordinate plot showing the performance of Pareto optimal solutions in objective space. Each sphere represents a portfolio of stormwater management BMPs. The lifecycle cost, average annual total nitrogen (TN) reduction, and average annual s tormwater reuse capacity performance are represented on the cardinal axes. The green score performance is represented in colour. Arrows i ndicate the preferred performance direction.

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The above inferences are supported and supplemented by the alternate representation of the trade-off surface in parallel coordinates (Inselberg, 1997). In Figure 6, small slopes on some line segments between the adjacent axes of lifecycle cost and stormwater reuse indicate high reuse


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portfolios exist for low costs. However, these low cost-high reuse capacity solutions appear to have lower TN reduction and green score compared to other solutions. As mentioned above, green score appears to be correlated with lifecycle cost, however, some solutions exist that have a high green score and relatively low cost.

Figure 6 Parallel coordinate plot, where each portfolio is represented as a line interval over four vertical axes indicating objective performance values. Lifecycle cost is also represented by colour to identify the cost trade-offs against each objective. The preferred direction for each objective i s upwards.

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In the low cost region, from Figure 5, clusters of solutions form in TN reduction-reuse capacity space. This indicates that individual projects dominate the contribution to total portfolio reuse capacity or total nitrogen reduction in this region. Analysis of the BMPs comprising solutio ns in these clusters shows that these portfolios contain a small number of ‘flagship’ projects with exceptionally large reuse capacity (e.g. project 61, 40 ML/year; project 67, 12.8 ML/year; project 18, 12.0 ML/year) or TN reduction (e.g. project 48, 1152 kg/year; project 18, 1141 kg/year). Portfolios containing only a few of these flagship projects are able to achieve relatively high total reuse capacity or TN capacity at relatively low cost, but also a low green score. This causes the noticeable discontinuity in the objective space in the low-cost region, characterized by clusters of solutions emanating from a small number of portfolios in the low-cost region in Figure 5 and overlapping dark blue (low-cost) line segments joining parallel axes in Figure 6. Moving in the preferred objective direction, adding a flagship project to create a new portfolio on the front can cause a large increase in TN reduction or reuse capacity. Therefore, decision makers desiring lowcost trade-off solutions could consider portfolios of a small number of ‘flagship’ projects, but this


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would considerably compromise the urban greening and amenity performance of the stormwater management strategy. 4.2 Importance of a Many-Objective Problem Formulation for BMP selection The cost and total nitrogen reduction trade-off projections in Figure 7 show trade-offs between water quality and cost objectives, which correspond to the most frequently used formulation in stormwater management optimization studies to date. On the front, a slight ‘knee’ region appears such that when moving along the front away from the knee region, there is a diminishing return in these objectives. This suggests that solutions in this region may represent a desirable trade-off between total nitrogen reduction and cost. The trade-off pattern is consistent with those in other BMP selection studies (Chichakly et al., 2013; Lee et al., 2012; Maringanti et al., 2009). However, only considering trade-offs between water quality and cost objectives neglects the influence of other objectives that may be important to stormwater manageme nt decision makers (Moglia et al., 2012). This could bias decision makers towards selection of solutions that would lie at extremities in objective space should other formal objectives be considered (Kollat et al., 2011).

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Figure 7 Pareto optimal solutions identified by the optimization process projected in water quality-cost objective space, which have been typical objectives in previous stormwater management optimization studies. Non-dominated solutions with respect to the two objectives are shown a s solid, and approximate the best trade-off between total nitrogen (TN) reduction and cost. Other Pareto optimal solutions in 4-objective s pace, but dominated in water quality-cost space, appear transparent.

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The importance of the many-objective representation of the BMP selection problem adopted in this study is demonstrated by tracing a solution from the two-objective knee region in


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Figure 7 through higher dimensional objective space represented in Figure 8. For this purpose, Portfolio 1 (Table 4) is selected and marked for further analysis because it lies at an inflec tio n point (observed by visual inspection) in the knee region of the two-objective trade-off front (Figure 7). Using the visual analytics package, an additional harvesting capacity axis and a green score color axis are added to create a 4-dimensional plot of the objective space (Figure 8). To compare Portfolio 1 with other solutions similar in cost, the analytics package’s data brushing tool is used to highlight solutions with lifecycle costs in the range [$1.90 M, $2.70 M]. In Figure 8, these solutions of interest appear opaque, and the remaining solutions that have been ‘brushed out’ appear transparent. Portfolio 2 (Table 4) is selected for comparison because although it has a 22% greater lifecycle cost and similar TN reduction compared to Portfolio 1, it has a vastly higher reuse capacity and green score. Therefore, although Portfolio 1 appeared in the region of best trade-off (knee region) in the lower-dimensional TN reduction-cost representation of the objective space (Figure 7), it performed poorly in the reuse capacity and green score objectives. Portfolio 2 lies near but not on the non-dominated water quality-cost front in Figure 7. Thus, it would not have been considered by decision makers in a bi-objective Pareto optimization approach, which has been typical in BMP selection optimization studies to date.

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When considering the project options selected in the two portfolios (Table 4), it is apparent Portfolio 2 is similar to Portfolio 1 except for one small project (Project 38 instead of Project 33) and, importantly, two additional projects located in local government authority 1 (projects 60 and 61). Consequently, decision makers may consider that Portfolio 2 provides a better compromise between objectives compared with Portfolio 1, due to the reuse capacity and green score benefit that the two additional projects provide.

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Importantly, for stormwater managers, the results of this study show that when they use water quality and cost as the only optimization objectives, they may not identify solutions that represent good trade-offs between water quality, cost, and other important objectives, includ ing stormwater harvesting capacity and green score. The above results are consistent with findings in several other studies including (1) findings by Kollat et al. (2011), Kasprzyk et al. (2015), and Woodruff et al. (2013) that, generally in optimization studies, lower dimensional problem formulations may bias selection of solutions that would otherwise exist at low-performing extremes if additional performance criteria were considered as formal optimization objectives; (2) a finding by Chichakly et al. (2013) that, for BMP selection optimization, desirable solutions lie near but away from the two-objective non-dominated Pareto front for water quality improve me nt and cost objectives; and (3) trade-offs for a stormwater harvesting system design determined by Di Matteo et al. (2017a), which showed slight increases in system costs, could provide large increases in both water quality improvement and harvesting capacity.


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F igure 8 4-dimesional coordinate plot showing the trade-off space with solutions in a defined low-cost range as solid, with all other solutions brushed out and appearing transparent. Portfolio 2 may provide a more desirable alternative to Portfolio 1 in 4-objective space.

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Table 4 Objective values and decision options of selected solutions.

Solution TN- cost compro mise (2D) Portfolio 1

Low cost compro mise (4D) Portfolio 2

Objectives Lifecycle Cost ($M) Total Nitrogen Reduction (kg/yr.)

2.06

2.51

3312

3377


Stormwater reuse capacity (ML/yr.) Green Score (no units) Portfolio project decisions Projects in Municipality 1 Projects in Municipality 2 Projects in Municipality 3 Total No. projects

3.46

44.15

38

51

-

60, 61

46, 56

46, 56

18, 38, 41, 48

18, 33, 41, 48

6

8

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4.3 Identifying Impacts of Project Options on Pareto Optimal Portfolio Performance

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Figure 9 shows combined objective performance and decision characteristics of the Pareto optimal portfolios, which helps the analyst to overcome biases arising from artificial distinctio ns between objective performance and other characteristics of the problem (Matrosov et al., 2015). For example, the visual interactive plot allows the analyst to inspect which area of the trade-off front each project features in Pareto approximate portfolios. In this way, an analyst can infer the impact of particular projects on portfolio performance.

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In Figure 9 (a) the opaque spheres represent portfolios containing Project 61 (lifecycle cost $381,297; TN reduction 157.92 kg/year; reuse capacity 40 ML/year; green score 6), which was the project with the highest reuse capacity. Importantly, all portfolios with 40 ML/year or greater reuse capacity include Project 61, and these portfolios occur in nearly the full range of cost, TN reduction and green score of Pareto solutions. Therefore, this indicates decision makers should probably consider Project 61 in their final portfolio. In Figure 9 (b), the opaque spheres represent portfolios containing Project 48 (lifecycle cost $915,472; TN reduction 1152 kg/year; reuse capacity 0 ML/year; green score 7), which was the project with the highest TN reduction. Importantly, in the lower cost region, Pareto optimal portfolios with a number of smaller solutions dominated infer ior portfolios containing project 48. This was because although the green score of Project 48 was high (7 out of 9), the cumulative green score and/or reuse capacity of low-cost portfolios with more projects dominated portfolios containing a small number of larger projects, including Project 48. This indicates multiple additional benefits can be achieved for a similar cost by using a portfolio of projects rather than one ‘flagship’ project. In addition, decision makers can view and assess additional (non-objective) characteristics that may influence decision-making, for example the


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percentage of the catchment treated, spatial distribution of projects throughout the catchment, or socio-political preferences for particular projects.

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Figure 9 Coordinate plot showing the combined Pareto optimal objectives, decisions, and alternate data spaces. Portfolios that include, in part a) Project 61, and in part b) Project 48, are shown as opaque spheres, other portfolios are brushed out and appear transparent. The size of spheres i s proportional to the number of projects in a portfolio. DCIA = directly connected impervious area.

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5 Summary and Conclusion

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A general multi-objective optimization formulation was developed for the selection of a portfolio of BMPs for stormwater management. This study advances the BMP selection optimization field of research (Chen et al., 2015; Chichakly et al., 2013; Di Matteo et al., 2017a; Lee et al., 2012; Maringanti et al., 2009; Zare et al., 2012) as it addresses the need for a manyobjective optimization formulation for the selection of stormwater BMPs that a) can cater to a large number of performance criteria, b) can handle a large number of decision options and potential strategies, c) can identify solutions that represent the best possible trade-offs between performance criteria, d) enables trade-off information to be communicated in an easy-tounderstand fashion, and e) enables the development of solutions that are trusted by decision makers.

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The approach was applied to a case study catchment plan for a catchment authority in a major coastal city in Australia. The benefits of the proposed formulation specifically with regard to stakeholder acceptance of solutions were evident for the case study. By using the proposed formulation, the only options considered were solutions that stakeholders had proposed and were familiar with. Consequently, the final set of solutions obtained after the optimization process consisted of combinations of trusted solutions, rather than solutions that were obtained via processes and models end-users were not familiar with. Stakeholders would have been able to identify portfolios that contain particular projects they may have a preference for (or bias towards selecting) and compare these with other optimal portfolios. In this way, when analyzing optimal solutions stakeholders could identify other projects that when packaged with preferred projects maximise the total portfolio benefits, or experience a shift in preferences for particular projects by visualizing the prevalence of other projects within optimal solutions.

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The results demonstrate the benefits of exploring full portfolio solution trade-offs in a many-dimensional Pareto optimal front, which can reduce the prevalence of decision making biases due to the optimization formulation. A comparison between the trade-off spaces of the lower dimensional water quality-cost problem formulation, and the many-objective formulatio n,

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demonstrated that low-objective formulations can result in Pareto optimal portfolios with low performance in non-objective performance criteria. In this study, when stormwater harvesting and vegetation and amenity improvement scores were included as objective functions, solutions that were in a region of best trade-off in water quality-cost space performed poorly in these additiona l objectives. The many-objective optimization results show that sharp trade-offs exist between TN reduction and cost, and between reuse capacity and cost, indicating small increments in cost can return large increases in both of these objectives. Portfolios in the low-cost regions typically featured a small number of projects including cost-efficient ‘flagship’ projects that provide high TN reduction or reuse capacity. However, in order to maximize the vegetation improvement and amenity benefits, portfolios with a larger number of lower cost BMPs distributed throughout the catchment were preferred. Notably, the optimization formulation in the case study does not consider that interaction between having a higher harvest capacity might allow for more irrigatio n of green spaces. Using the visual analytics approach to explore combined optimization and decision spaces, the impact of individual projects that may be preferred by decision makers was rapidly visualized. This approach could assist in overcoming institutionally influenced biases to include particular projects or BMP technologies to demonstrate alternative similar cost options to decision makers.

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Future studies applying the framework could account for differences in preferences between multiple decision makers that may be responsible for funding over different periods of the project lifecycle. For example, in some funding schemes, catchment management authorities fund the capital expenses, whereas local government authorities fund the maintenance and ongoing expenses. The many-objective problem formulation could be adapted to include specific objectives important to local government authorities, which might include the operating expenses for each individual local government authority, in addition to total catchment benefits, thus identifying the tradeoffs between CAPEX and OPEX. In addition, the Pareto optimal solutions could be explored taking into account individual formal objective and informal non-objective preferences of multip le decision makers. In this way, decision makers can visualize their preferences on a trade-off curve and compare and, through an iterative approach, visualize and negotiate acceptable outcomes and solutions. This may be preferable to other approaches where weightings are set a priori, which do not account for decision maker preferences in the decision space, nor allow a visual comparison of the regions of interest preferred by several decision makers. For the green index and stormwater harvesting objectives, the objective formulations do not consider how diminishing returns due to interdependencies between projects in these objectives may be represented. Finally, the constraint for the number of projects should consider the difficulty of constructing individual BMP types (e.g., 20 swales might be easier to construct than 20 wetlands).

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Data and Software availability

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Name of software: Pareto Ant Colony Optimization Algorithm, and problem data set

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Developers: Michael Di Matteo, Holger R. Maier, Graeme C. Dandy

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Year first available: 2017

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Hardware required: PC/Mac

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Program language: FORTRAN

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Program size: 10.0 MB


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Contact address: School of Civil, Environmental and Mining Engineering, University of Adelaide, Adelaide, South Australia.

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Telephone: +61 8 8303 4313 12

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Fax: +61 8 8313 4359 13

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E-mail: [email protected]

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Url: https://figshare.com/articles/A_MANYOBJECTIVE_OPTIMIZATION_AND_VISUAL_ANALYTICS_APPROACH_TO_PROJECT_ SELECTION_FOR_INTEGRATED_CATCHMENT_MANAGEMENT/4233119

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Data set software required: The ddv. files require DiscoveyDV visualization software; DiscoveryDV is available for academic use under a Beta license (as of April 2017) at https://www.decisionvis.com/discoverydv/ . Some data files are as .csv or .xlsx, which can be opened using Microsoft Excel other spreadsheet packages. Matlab scripts and data files, which were used to convert the raw optimization run data into a useful form, are available and require Matlab software package,

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Availability: Software and data are available via figshare public and online repository, and submitted to the journal editor for publication.

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Cost: Free for non-commercial use.

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Acknowledgements

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Research funding was provided by the Australian Postgraduate Award, the University of Adelaide, and the Goyder Institute for Water Research. The authors thank Dr. Dale Browne and e2designlab, Australia for assistance with the case study data and the anonymous reviewers for their comments, which helped improve the quality of the paper. The data used are listed in the references, tables, supplements and repository at 10.6084/m9.figshare.4233119.

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