AbstractThis study aims to introduce a generic solution in the context of a multicriteria decision making (MCDM) platform to (1) facilitate the optimization of hybrid (de)centralized urban drainage infrastructures with many decisions and often conflicting objectives (reliability, resilience, sustainability, and construction costs); (2) investigate the trade-offs between performance indicators and system configuration; and (3) avoid conflicts between optimization analysts and decision makers by involving the latter in different stages of planning. For this purpose, first, all optimum design scenarios of hybrid urban drainage systems (UDSs) are generated through multiobjective optimization (MOO). Then a platform based on MCDM is presented to comprehensively analyze the solutions found by MOO and to rank the solutions. For the sake of demonstration, the proposed framework is applied to a real case study. The results confirm the ability of the proposed framework in handling many decisions, objectives, and indicators for solving a complex optimization problem in a reasonable time by delivering realistic solutions. In addition, the results demonstrate the important role of the degree of (de)centralization (DC) and the layout configuration in obtaining optimal solutions.IntroductionUrban drainage systems (UDSs) are vital and complex infrastructures in cities which directly influence the public economy, health, and welfare (Dong et al. 2017). Wastewater and stormwater require drainage. Wastewater that is not drained properly can endanger the environment and cause health risks. Stormwater that is not drained appropriately can cause additional health risks, public inconvenience, and damage (Butler and Davies 2011). UDSs are traditionally designed using hydraulic reliability–based approaches. These approaches ensure a sufficient hydraulic capacity to convey the runoff of a specific design storm (Butler and Davies 2011; Yazdi 2018). The literature has widely addressed methods that combine mathematical simulation models with optimization/decision-making methods to design, rehabilitate, or retrofit UDSs (Yazdi 2018). Nevertheless, the performance of existing UDSs in various cities is negatively affected by multiple and uncertain threats, such as climate change, rapid and uncontrolled urbanization, and aging infrastructure. Together, these threats cause more frequent and more severe urban flooding with adverse consequences on society, the economy, and the environment (Butler et al. 2014).Therefore, conventional design approaches have been increasingly questioned. According to various publications, UDSs should (1) be reliable during normal loading conditions to minimize failure (flood) frequency, (2) be resilient to extreme loading conditions to lessen the span and extent of floods, and (3) pursue sustainability in the long term to accomplish economic, environmental, and social aims (Butler et al. 2014; Casal-Campos et al. 2018; Mugume et al. 2015b). Additionally, conventional design approaches have increasingly been questioned because of their centralization and their focus on pipe networks only (Eggimann et al. 2015; Poustie et al. 2014). Recent studies in urban water management favor decentralized solutions for UDSs (Damodaram et al. 2010; Eggimann 2016) to decrease life-cycle costs (Bakhshipour et al. 2019a; Tavakol-Davani et al. 2018); to increase system resilience, flexibility, and adaptability (Chelleri et al. 2015; Mugume et al. 2015b; Tavakol-Davani et al. 2018); and to reduce adverse impacts on the environment (Barron et al. 2017; Goncalves et al. 2018).To achieve decentralization, the application of green-blue infrastructures (GBIs), such as green roofs, permeable pavements, infiltration trenches, and rain barrels, as alternatives is receiving increasing attention (Liu et al. 2015; Wang et al. 2017).GBIs are flexible and adaptable measures that provide several benefits in addition to flood risk reduction. These benefits include water quality improvement, groundwater recharge, water harvesting, restoration of the hydrologic characteristics of the site, increased urban amenities, and alleviation of urban heat island effects that are aligned with the environmental and social aims of sustainability (Alves et al. 2019; Cano and Barkdoll 2017; Eckart et al. 2017). However, GBIs are a relatively expensive investment and have poor resilience against extreme loading conditions (Alves et al. 2019; Jia et al. 2015).In addition to GBIs, decentralization can be obtained via conventional gray infrastructures (CGIs) by dividing heavily centralized pipe networks into several parts with multiple outlets or by employing distributed storage tanks (Bakhshipour et al. 2019a; Eggimann et al. 2015). In contrast to GBIs, pure CGIs are proven to have a degrading impact on the environment due to the discharge of polluted stormwater or wastewater to bodies of water. They are also challenging to upgrade and expensive to maintain (not adaptable) (Barron et al. 2017). However, in comparison with GBIs, GCIs need less capital investment and show higher functional resilience against intense rainstorms (Alves et al. 2019; Jia et al. 2015).Combining the advantages of GBIs and CGIs, so-called hybrid green-blue-gray infrastructures (HGBGIs) have been evaluated in recent studies of urban water management (e.g., Dong et al. 2017). The results showed that the components of HGBGIs tend to complement each other (Zhang and Chui 2018). Hence, HGBGIs might be the most promising way to handle the challenges of the modern urban management era to achieve higher reliability, resilience, and sustainability at a lower price (Alves et al. 2019; Casal-Campos et al. 2015; Damodaram et al. 2010; Damodaram and Zechman 2013; Dong et al. 2017).To conclude, future UDSs need to be reliable, resilient, and sustainable. HGBGIs seem to be the most supportive approach to achieving these goals. Notwithstanding, considering all combinations of possible alternatives among CGIs and GBIs and considering the many possible objectives, designing new UDSs is a notably complex optimization problem. To jointly optimize such complex systems, obtaining the optimum layout of the pipe network considering the different degrees of decentralization (DCs), sizing sewers, and selecting the type, size, and location of GBIs are the subproblems that need to be decided simultaneously. Each of these optimization subproblems contains many decisions and technical and hydraulic constraints. In addition, there are different objectives that, in some cases, conflict (Bocchini et al. 2014) and increase problem complexity. As an example, increasing the size of pipes in the network might increase reliability and functional resilience but is not necessarily financially viable and environmentally sustainable.Currently mathematical optimization is a promising approach urban water management to finding a preferred option from feasible solutions, such as a range of designs, planning, operations, management, and policy scenarios (Maier et al. 2014; Nicklow et al. 2010). Environmental models, such as urban drainage models (e.g., SWMM), broadly to aid decision making by assessing the performance of different alternatives (Maier et al. 2019). Our review of existing methodologies and frameworks for MOOs of UDSs shows that the so-called reliability-based approach is the most common in dealing with this problem. In this approach, the performance of either gray alternatives or green-blue scenarios is evaluated and optimized for a single design storm or a limited number of separate design scenarios (Alves et al. 2016; Azari and Tabesh 2018; Cano and Barkdoll 2017; Damodaram and Zechman 2013; Di Matteo et al. 2017; Eckart et al. 2017; Giacomoni and Joseph 2017; Haghighi 2013; Jia et al. 2015; Liu et al. 2016; Oraei Zare et al. 2012; Sebti et al. 2014; Zhang et al. 2013).A few studies have incorporated resilience in their optimization formulation by including functional (Karamouz and Nazif 2013) or structural resilience (Haghighi and Bakhshipour 2015b). Many studies have used multicriteria decision analysis (MCDA) instead of optimization in their proposed frameworks to consider various characteristics of resilience and different aspects of sustainability (Tavakol-Davani et al. 2018). MCDA approaches empower decision makers to cover a full range of decision-relevant indicators (Wang et al. 2017) in a reasonable time but only for a small number of predefined scenarios.In addition, a few studies have combined optimization and MCDA techniques. As an example, Sweetapple et al. (2017) proposed a general framework for reliable, robust, and resilient system design. This framework contained three key components: (1) a MOO was applied to the design of the system under standard loading, and a set of Pareto-optimal solutions were obtained; (2) solutions on the Pareto fronts that provide an acceptable level of robustness were subsequently used for resilience analysis; and (3) solutions that reached this step (being reliable and resilient) were ranked based on their performance objectives and the priorities of the decision makers. Although this is a very promising way to include robustness and resilience in system design, the fundamental optimization step is based on reliability only. Therefore, resilience and robustness analysis is performed for only a limited number of solutions that cannot guarantee a global optimum.Our literature review revealed that, at the moment, no UDS optimization tool or framework exists that can simultaneously consider various performance indicators, many decisions, and different technical and practical constraints in its structure. The available literature has solved this problem by isolating either the objectives or the alternatives. Therefore, the attributes and relationships between these operational and strategic system objectives (reliability, functional and structural resilience, and sustainability) and UDS configuration [layout and degree of (de)centralization of CGIs] are still unclear (Casal-Campos et al. 2018; Dong et al. 2017).In addition to the challenges mentioned, the acceptance of solutions obtained from mathematical optimization by decision makers can be challenging (Di Matteo et al. 2019). Decision makers or stakeholders have valuable experience from several years of working with and confronting real-world challenges that might be ignored by optimization analysts. Additionally, there are other engineering and practical considerations and desires that cannot be formulated in mathematical optimization. Some desires imposed by decision makers might even be irrational. However, if decision makers as clients feel undervalued and unheard or if the presentation of the model results and optimization procedure is not transparent, their acceptance of the optimization results might decrease (Di Matteo et al. 2019; Pigmans et al. 2019). Therefore, to develop trusted strategies that are likely to be adopted in practice, decision-maker engagement should be encouraged in all phases of the optimization frameworks applied to water resource problems (Di Matteo et al. 2019; Maier et al. 2014; Voinov and Bousquet 2010; Wu et al. 2016).This study aimed to address the challenges just discussed and fill identified gaps in the literature. Its key contributions are the following: •A combined MOO and multicriteria decision-making (MCDM) platform that aids the sustainable planning of modern hybrid (de)centralized urban drainage infrastructures.•The tools and materials to explore the trade-offs between operational and strategic system indicators (e.g., reliability, resilience, and sustainability) and system configuration [network layout and degree of (de)centralization].•The ability of urban drainage operators and water authorities to participate in decision making.The remainder of this manuscript is structured as follows: (1) reliability, resilience, and sustainability indicators in urban drainage are defined; (2) the proposed framework is presented in detail; (3) the framework is demonstrated and discussed using a case study; and (4) conclusions and recommendations for further investigations are provided.UDS Reliability, Resilience, and SustainabilityHashimoto et al. (1982) proposed three criteria for assessing the performance of water resource systems in supporting the evaluation and selection of alternative designs and operating policies of water resource projects. From their definition, reliability describes how likely a system is to fail, resiliency measures how quickly it recovers from failure, and vulnerability estimates how severe the consequence of failure might be. To reduce the computational burden, especially when complex system-response models are used, Maier et al. (2001) introduced an efficient approach based on first-order reliability for computing reliability, vulnerability, and resilience.Butler et al. (2014) introduced the “Safe & SuRe” framework for water management, based on which reliability is the bedrock of resilience and sustainability. The authors defined reliability as the degree to which a system minimizes the frequency of failure over its design life when subject to standard loadings. In the context of urban drainage, service failure means failure to comply with the levels required by regulations—for example, when sewer flooding or combined sewer overflows (CSOs) violate a given threshold (Casal-Campos et al. 2018). Therefore, current approaches for hydraulic reliability–based design, retrofitting, and rehabilitation concentrate on avoiding hydraulic failures caused by a design storm of a given return period (Butler and Davies 2011).Binesh et al. (2019) formulated three types of reliability for describing different angles of UDS performance: (1) occurrence, (2) temporal, and (3) volumetric. Occurrence reliability is defined by the number of times a satisfactory state (e.g., no surcharging in the system) has occurred in a certain number of time steps. Temporal reliability represents the amount of time the system remains in the satisfactory state divided by the total range of time considered. Volumetric reliability considers the ratio of water volume conveyed safely through the drainage system to total runoff volume generated from rainfall (Binesh et al. 2019).The definitions of reliability do not consider other sources of failure that the system is not designed for (Mugume et al. 2015a), such as structural failure (e.g., pipe blockage, climate change, and urbanization). Here is where the concept of resilience can help. According to Butler et al. (2014, p. 349), resiliency describes the response of the system after failure due to unforeseen loading conditions. They define it as “the degree to which the system minimizes service failure (magnitude and duration) over its design life when subject to exceptional conditions.” For UDSs, two types of failure may occur, functional and structural. The difference is in the threats that endanger the system. Functional failure is induced by threats that alter the load in the system, such as climate change, urbanization, and extra infiltration, while structural failure is triggered by threats that lead to faults of single or multiple components in the system, such as pipe blockages, pump failure, and clogging in infiltration trenches (Mugume et al. 2015b; Yazdi 2018). Therefore, it is crucial to consider different threats and their combinations in building resilience in UDSs. The resilience of UDSs can be increased by enhancing system redundancy and flexibility (Mugume et al. 2015a; Yazdi 2018).Redundancy is defined as the degree of overlapping functionality in a system. It permits the system to continue vital functions while formerly redundant elements break or take on new functions (Hassler and Kohler 2014). Flexibility, on the other hand, is defined as the system’s intrinsic ability to be modified or reconfigured to preserve adequate performance when subject to multiple (varying) loading conditions (Mugume et al. 2015b). In UDSs, redundancy is enriched by presenting multiple elements providing similar functions, such as additional storage tanks and parallel pipes and higher spare capacity at critical points in the network. Flexibility can also be achieved by designing future-proof options. These include distributed (decentralized) elements, such as GBIs (Mugume et al. 2015a).Considering resilience in the design process of UDSs exponentially increases the complexity of the design problem. Different threats and their combinations need to be taken into account. Additionally, trying to enhance inbuilt system flexibility and redundancy introduces extra decision variables into the optimization problem.Notwithstanding, the complexity added to the problem by introducing resilience is still not enough for future UDSs. Sustainability is another concern that needs to be addressed in the design framework. Sustainability, in general, ought to simultaneously address today’s demands and their impacts on future generations. It requires a holistic view that considers environmental, economic, and social aspects equally (Bocchini et al. 2014). Butler et al. (2014, p. 350) define sustainability as “the degree to which the system maintains levels of service in the longterm while maximizing social, economic, and environmental goals.” This definition can bring new objectives, such as pollution control, rainwater usage, public acceptance, and annual energy savings, to the problem. These additional objectives cannot be accounted for in reliability and resilience indicators (Casal-Campos et al. 2018).Reliability, resilience, and sustainability are three aspects of UDSs that should be pursued simultaneously during decision making. Usually, better decisions lead to enhancement of all aspects, but in some cases the same decision may lead to an improvement in one aspect and a worsening in another (Bocchini et al. 2014). For example, increasing pipe diameters in the system can improve system reliability, resiliency, and public sustainability. However, at the same time it can decrease environmental and economic sustainability by conveying more pollution to water bodies and requiring more capital investment. Alternatively, adding parallel pipes does not necessarily enhance system reliability, but it does improve system resilience and public sustainability.Butler et al. (2014) introduced a pyramid to explain the connection between reliability, resilience, and sustainability. This pyramid demonstrates that resilience should build on reliability and sustainability should build on resilience. Although these indicators are inevitably interlinked, their complicated relationship is still unknown. Therefore, one aim of this paper is to investigate the trade-offs between life-cycle costs, DC, and performance indicators (reliability, resilience, and sustainability).Proposed FrameworkThe proposed framework for sustainable hybrid (de)centralized urban drainage infrastructure planning is shown schematically in Fig. 1. This framework consists of three main steps: (1) system definition, (2) simulation optimization, and (3) final decision making. The details of all steps, subprocesses, and algorithms are provided in the following sections.Step 1: System DefinitionFirst, stakeholders or decision makers introduce all types of urban drainage systems and technologies that they wish to be considered in the design process in addition to all technical and practical considerations. Any screening tool for GBI type or sanitation technology selection, location identification, and narrowing of dimensioning variables without detailed optimization can be employed in this step. Primary screening restricts the search space and enhances optimization efficiency (Zhang and Chui 2018). Examples of screening tools can be found in Eaton (2018), Inamdar et al. (2013), Johnson and Sample (2017), Kuller et al. (2019), Martin-Mikle et al. (2015), Mitchell (2005), and Spuhler et al. (2018).Next, all combinations of the proposed systems and technologies are generated systematically to perform the mathematical optimization. To do that, Step 1a outlines a base graph for the pipe network which includes all drainage feasibilities concerning street alignments, topology, barriers, watercourses, and existing sewers in the area under design. In Steps 1b and 1c, candidate locations of outlets; potential GBI type, size, and location; and, if necessary, treatment solutions such as on-site facilities, constructed wetlands, or traditional wastewater treatment plants are determined. These steps provide the potential decision variables d of the optimization problem [Eqs. (1) and (2)]. Step 1d defines all relevant performance indicators, such as reliability, resilience and sustainability, for MOOs. To prepare for the simulation-optimization step (Step 2), it is crucial to define simple performance indicators that are inexpensive to evaluate. Otherwise, the optimization in Step 2 may take a long time to converge and come close to acceptable solutions. For example, when evaluating hydraulic reliability or functional resilience, one or more design storms (which can also be considered for a more robust design) or multiple extreme events derived from historical rain data should be used instead of recorded time series. The section “Results and Discussion” and the studies mentioned in the previous section can aid selection of appropriate indicators for a specific problem.In Step 1e, a numerical UDS model (EPA SWMM in this study) is constructed to evaluate the predefined performance indicators. Here the model parameters (e.g., impervious area, manning roughness, and soil characteristics) are calibrated or estimated, and design conditions and physical and technical constraints (e.g., design loadings, maximum velocity, minimum slope, maximum buried depth, and water quality standards) are defined. Based on the resulting data, a base model is constructed from which different design schemes are generated during the simulation-optimization step (Step 2).Step 2: Simulation-OptimizationMathematically, the multiobjective optimization of HGBGIs can be formulated as follows: (1) dopt=arg maxd∈D[⟨fIndicator 1⟩,⟨fIndicator 2⟩,…,⟨fIndicator n⟩](2) d=[⟨DC and layout parameters⟩,⟨CGI parameters⟩,⟨GBI parameters⟩]where dopt = optimal choice for decision variables that define the UDS; and d contains elements of at least three subproblems [Eq. (2)]: 1.DC and layout parameters: connectivity between the gray components of the system in each part and the distribution of the system as a whole when multiple outlet candidates are available. Here, DC is defined in Eq. (3) adopted from Bakhshipour et al. (2019a) (3) DC=100×(1−NSO−1NPO−1)(%)where NSO = number of outlets selected from a list of candidates; and NPO = number of possible candidate outlets.2.CGI parameters: size of each conventional gray component, such as pipe diameter, slope, location, and technical details of pump stations, as well as location and size of storage tanks.3.GBI parameters: GBI type, size, and location. Here, D is the feasible space where all structural, technical, hydraulic, environmental, and economic constraints are met. fIndicator i is the ith performance indicator or objective function.To perform a systematic optimization, it is vital to systematically generate various HGBGI schemes that satisfy all technical and physical constraints. Three different modules are employed within the proposed framework to do this: (1) the hanging gardens algorithm to generate feasible layouts with an arbitrary degree of (de)centralization; (2) an adaptive algorithm to hydraulically design the generated layouts; and (3) an algorithm to define GBI type, size, and location. Fig. 2 schematically shows the simulation-optimization procedure proposed in this study and demonstrates the connection between different algorithms.The hanging gardens algorithm, recently introduced by Bakhshipour et al. (2019a), generates a sewer layout based on graph theory, generating all possible sewer layouts and exploring different DCs. The algorithm starts with nominating several outlet candidates in the area under design and generating a centralized layout with an arbitrary outlet from the candidate list. Then it adds other arbitrary outlets from the candidates to the generated layout considering the desired DC. Using graph theory, it assigns parts of the layout to different outlets and generates a decentralized layout. The algorithm needs 2×(NL+NPO) decision variables to generate one feasible layout: NL is the number of loops in the base graph, and NPO is the number of possible outlets. Full details on the hanging gardens algorithm are available in Bakhshipour et al. (2019a).For each generated layout, CGI specifications such as pipe diameters, pump stations, and invert elevation are designed in a way that satisfies all hydraulic and technical constraints. Technical constraints such as telescopic pattern, minimum cover depth, maximum excavation depth, and minimum and maximum slope, are satisfied via the adaptive approach introduced in Haghighi and Bakhshipour (2015a, 2012). The hydraulic constraints of maximum velocity and no flooding are handled by a penalty function during optimization. If NP is the number of pipes in the UDS, this algorithm needs 3NP decision variables to design the system hydraulically: one variable per pipe to assign the size of a pipe, one to assign the slope, and one to determine whether there is a pump station upstream of a pipe. Finally, distributed measures, here GBIs, are added to the designed UDS using an adaptive algorithm to construct a hybrid green-blue-gray UDS, as explained in the following paragraph.Generating a hybrid alternative requires three extra decision variables for each subcatchment (3NS decisions, where NS is the number of subcatchments). For each subcatchment, a binary variable decides whether it is equipped with any GBI. The second variable determines the type of GBI(s) from a list of feasible candidates for each subcatchment using Eq. (4). The third variable defines the size of the GBI(s) considering the minimum and maximum feasible size of each measure using Eq. (5) (4) GBI_Typei=round(1+(NGBIi−1)×gbi_typei)where gbi_typei = uniform random variable generated by the optimization engine; NGBIi = number of GBI candidates in subcatchment i; and GBI_Typei = decoded GBI type in that subcatchment. As an example, suppose the GBI candidate list in one subcatchment is as follows, and gbii randomly generated by the optimization engine is 0.67: GBI_Candidatei={1: Rain barrel2: Infiltration trench3: Green roof4: Rain barrel+Infiltration trench5: Rain barrel+Green roof6: Infiltration trench+Green roof7: Rain barrel+infiltration trench+Green roof}Using Eq. (15), GBIi=round(1+(7−1)×0.67)=5, which means that Rain barrel+Green roof is selected for that catchment.GBI_Sizei is handled as follows: (5) GBI_Sizei=GBI_Sizemin,i+(GBI_Sizemax,i−GBI_Sizemin,i)×gbi_sizeiwhere gbi_sizei = uniform random number generated by the optimization engine; GBI_Sizemin,i and GBI_Sizemax,i = minimum and maximum permissible sizes of GBI_Typei, respectively; and GBI_Sizei = decoded size of GBI_Typei in that subcatchment.Using the algorithms, any arbitrarily generated set of decision variables d constructs a feasible HGBGI scheme. These algorithms, together with performance indicators and a numerical model for evaluating system performance (calculating values of indicators), provide essential tools and materials to perform MOOs. The MOO engine generates a set of Pareto-optimal solutions using different sets of decision variables that satisfy the defined constraints. There are several genetic algorithms (GAs) based MOO engines (e.g., NSGA-II, BORG, GALAXY) in the literature that can be employed for the current optimization (BORG in our study).Step 3: Final Decision MakingThe MOO in Step 2 identifies the solutions that dominate other solutions and constructs the Pareto front. As there is no clear preference between the solutions on the Pareto front according to the optimization objectives from Step 2, a final decision step is required to help decision makers select an appropriate option. This step can account for aspects not covered by the MOO in Step 2.The MOO compares different solutions by evaluating only a limited number of objective functions. However, some criteria cannot be considered for practical reasons such as computational power limitations. For example, applying the global resilience analysis introduced by Mugume et al. (2015b) to evaluate the structural resilience of UDSs requires generating and evaluating many pipe failure scenarios for each solution. This increases the computational burden exponentially. Second (and again due to the computational burden), modeling flooding of specific locations in the urban area has to be done separately using a 2D model based on an accurate detailed digital terrain model (DTM). This is feasible only for a few selected solutions. Third, there are other aspects that cannot be represented mathematically and criteria that cannot be quantified adequately (e.g., risk of human casualties or public nonacceptance).Multicriteria decision analysis (MCDA) techniques [e.g., analytic hierarchy process (AHP) and analytic network process (ANP), technique for order of preference by similarity to ideal solution (TOPSIS)] provide the opportunity to include numerous ranges of indicators and thoroughly analyze the limited number of solutions that exceed this step (Malczewski and Rinner 2015; Wang et al. 2017). Hence, in this step of the proposed framework, a full range of desired indicators is first defined. Then, the performance of all selected solutions is evaluated. Finally, the solutions are ranked employing an MCDA. No specific MCDA technique is prescribed here for this purpose, as the choice must be suitable for the problem. For our application, we use TOPSIS in the next step.Case StudyThis section demonstrates the performance of the proposed framework step by step by applying it in a realistic case study. The case study, featuring a section of Ahvaz a city in southwestern Iran, was is introduced by Bakhshipour et al. (2018). Ahvaz has a semidesert climate with long and scorching summers and short and mild winters. Annually, urban flooding due to the lack of a stormwater management system causes public inconvenience as well as economic and environmental destruction. A few vital public facilities, such as hospitals and schools, are scattered over the area. The area under design is located in a highly urbanized area with flat topography and a relatively high groundwater level. Technically, the issues mentioned previously make constructing a conventional centralized pipe network with large pipe diameters and deep excavations too expensive and almost impossible in practice. Bakhshipour et al. (2019a) partially solved this problem by finding the optimal degree of (de)centralization, which reduced the average pipe size and burial depth. However, we only considered CGIs. In a follow-up study, (Bakhshipour et al. 2019b) we proposed a framework for a hybrid green-blue-gray UDS for. Nevertheless, we optimized only for cost and did not account for other crucial indicators in the current MCDA context.Step 1: System DefinitionSteps 1a to 1c: Drawing the Base Graph, Locating Candidate Outlets, and Selecting GBIsThe case study has an area of approximately 500 ha divided into 181 subcatchments (loops in the base graph), including 530 pipes (approximately 75 km in length) and ten candidate outlets, as shown in Fig. 3. We selected rain barrels and infiltration trenches as GBI options. The reason for this choice, the maximum size of each measure, and their suitable location are discussed in Bakhshipour et al. (2019b). After rainfall, runoff from the roofs is diverted into rain barrels to supply water for toilet flushing and household irrigation. A percentage of the runoff from impervious areas, such as roads and parking lots, as well as roof runoff overflowing the rain barrels, is diverted into infiltration trenches. It is assumed that each apartment can be equipped with a 2-m3 rain barrel available on the local market. The infiltration trenches are installed along streetscapes and can cover, on average, up to 5% of the impervious area in each subcatchment. Each infiltration trench unit is assumed to have a width of 2 m, a length of 5 m, and a berm height of 250 mm. Other design parameters are assigned or estimated according to the literature as follows: a vegetation volume fraction of 0.1, a storage (gravel) layer thickness of 1,500 mm, a void ratio of 0.75, a seepage rate of 0.56  mm/h, a drain flow exponent of 0.5, and an offset height of 100 mm (Cano and Barkdoll 2017; Chui et al. 2016; Eckart 2015). These specifications remove the selection of GBI type and size from the optimization, leaving only locations to be determined.Step 1d: Defining Performance IndicatorsTo prepare for the simulation-optimization step (second step), it is crucial to define simple performance indicators that are inexpensive to evaluate. For this reason and to satisfy the technical criteria given in the regional guidance manual, we use the following indicators for the case study.ReliabilityAccording to the local manuals, stormwater collection systems must be designed for two- to five-year design storms (normal loading conditions) in urban areas. The design storms can be found in the Supplemental Materials. No system flooding is allowable for the selected design storm. We quantify the reliability of the system as the following: (6) Rel={0if  HPI2<1HPI5if  HPI2=1(7) HPIT=1−VfloodingVrunoffwhere HPIT = hydraulic performance index of a design storm with return period T; Vflooding = total water that overflows the nodes; and Vrunoff = total runoff volume. To calculate the reliability index Rel according to Eq. (6), each design alternative must be evaluated one or two times for two- and five-year design storms. The reliability is zero if there is any flooding in the system for T=2  years. If the system handles the two-year design storm properly, its reliability is calculated using a five-year design storm. All solutions with reliability greater than zero are acceptable; however, they might have different functional properties and construction costs.ResilienceThe case study considers both functional and structural resilience. Functional resilience accounts for the magnitude and duration of failure when extreme loading conditions occur. Here, a 25-year design storm is used as an extreme loading condition. Eq. (8), introduced by Mugume et al. (2015b), is adopted to calculate functional resilience: (8) ResFunctional=1−VfloodingVrunoff×TfloodingTSimulationwhere ResFunctional = functional resilience; Vflooding = total water that overflows the nodes; Vrunoff = total runoff volume; Tflooding = spatial average flood duration computed for all flooded nodes in the system; and TSimulation = total simulation time.To compute the structural resilience, a simple index that uses the adjacency matrix of the sewer layouts is adopted. The main idea of this index is that when the area affected by pipe failure is low, the structural resilience of the sewer network is high (Haghighi and Bakhshipour 2015b). On this basis, the structural resilience caused by every individual link (pipe) is defined as the following: (9) Resstructrual,i=100 (1−AiAT)%where Ai = area connected to pipei, and AT = total area. To obtain a structural resilience index for the entire layout, we take the average over all pipes as follows: (10) Resstructrual=∑i=1NPResstructural,iNP(%)Haghighi and Bakhshipour (2015b) have shown that using Eq. (10) to evaluate the structural resilience is not sensitive to the layout configuration in some cases. As a remedy and to account for the effect of DC on the resilience of the layout, we suggest (especially for large sewer networks) using only sewers with a structural resiliency index less than a threshold (90% in this study) to calculate Resstructrual. The percentage of sewers with individual structural resiliency greater than 90% is multiplied by Resstructrual to provide a more meaningful index for comparison between layouts with different DC: (11) Resstructrual={NPResstr,i>90%NP(∑i=1NPResstr,i<90%Resstr,i<90%NPResstr,i<90%)(%)100%if NPResstr,i<90%=0where NPResstr,i>90% = number of pipes with a structural resiliency of greater than 90% and NPResstr,i<90% = number of pipes with structural resiliency less than 90%.Theoretically, for a layout with ten outlet candidates, this index could be 100% if each outlet is connected to 10% percent of the total area and zero if all sewers are connected to more than 10% percent of the total area (e.g., in a completely centralized alternative).SustainabilityWe consider here environmental and economic sustainability. As the economic indicator of sustainability, we use life-cycle costs (LCCs). LCC evaluates capital and operation and maintenance (O&M) costs of the pipe network and of implemented GBIs over a typical service period of 30 years (Chui et al. 2016). The LCC of each alternative is calculated by compiling all capital and O&M costs. The cost functions used here can be found in Bakhshipour et al. (2019a, b).As a simple index of environmental sustainability, we use the ratio of storage quantity to precipitation under storm design, so (12) SusEnvironmental=infiltration volume+final GBI storageTotal PrecipitationStep 1e: Constructing the Simulation ModelIn the present study, EPA SWMM version 5.1 software is used (Rossman 2017) for the hydrologic-hydraulic simulation of pipe networks and GBIs. The dynamic wave method is selected as the routing method because of its ability to account for channel storage, backwater effects, flow reversal, and pressurized flow (Rossman 2017). The main parameters for each subcatchment—for example, area, impervious area, width, slope, infiltration parameters, and Manning’s roughness—are estimated using Google Earth and engineering judgment.Step 2: Simulation-OptimizationWe considered two simulation-optimization scenarios for the case study. In the first scenario, the proposed framework is applied to the four fixed layouts with different DCs recommended by Bakhshipour et al. (2019b). Therefore, the problem is simplified to the optimal sizing of the sewers and the location of the GBIs. In the second scenario, the layout and DC variables are also considered. This scenario provides the material to explore the effect of layout configuration on the different performance criteria.In both scenarios, minimizing LCC is used as an optimization objective because LCC is the most determinative parameter for the stakeholders in the area and is different from other performance criteria, thus presenting an independent problem dimension. The second optimization objective is maximizing total sustainability (SusTotal), here defined as the geometric mean of reliability, resilience, and environmental sustainability. The reasons are as follows: (1) considering each performance indicator as a separate objective function increases the computational effort exponentially due to the large scale of the test case (more than 1,000 decision variables); and (2) as discussed earlier, these performance indicators have a pyramidal structure with an unknown relationship that can be obtained through the proposed formulation, as explained in the following paragraphs (13) SusTotal=Rel×Res×SusEnvironmental3(14) Res=ResFunctional×ResStructural2As mentioned before, sustainability has three aspects: public, economic, and environmental. The economic aspect of sustainability is regarded here as a separate objective function (LCC). In addition, increasing reliability and resilience in Eq. (13) results in decreasing urban flood probability, which automatically enhances public sustainability. Therefore, we can guarantee that all aspects of sustainability are acknowledged in our proposed MOO formulation. In this study, for convenience of interpretation all introduced indicators are a real number between zero and one; zero indicates the lowest performance of each indicator; one, the highest performance. Generally, the geometric mean is used when considering several criteria that cannot compensate for each other; that is, one aspect rated zero sets the overall performance to zero.By using the geometric mean, total sustainability is zero when one or more of the indicators are equal to zero, and one if and only if all indicators are equal to one. Therefore, the suggested pyramidal structure of the performance indicators can be obtained. The remaining technical and social aspects and indices are treated in the decision-making step. It is worth mentioning that the choice of performance indicators and MOO problem formulation depend on the problem at hand. We cannot prescribe any general formulation here. However, any MOO formulation can be handled using our proposed framework.For the second scenario, maximizing DC is also used as an extra objective function. This means that, for a certain amount of LCC and total sustainability, solutions with greater DC (lower number of parts or outlets) are selected. This allows the optimization engine to fully explore the feasible space and determine the influence of DC on the performance indices. Therefore, the general optimization problem for the test case is reformulated and simplified as follows: (15) dopt=arg maxd∈D[⟨−LCC⟩,⟨−DC⟩,⟨SusTotal⟩](16) d=[⟨DC and layout parameters⟩,⟨pipe diameter⟩, ⟨GBI location⟩]Table 1 summarizes the number, type, and function of decision variable vector d for the test case.Table 1. Decision variables and the algorithms that use themTable 1. Decision variables and the algorithms that use themAlgorithmNo. of decision variablesFunctionTypeHanging gardens algorithm=2(NL+NPO)=2(181+10)=382Determine layout configuration and DCBinary and realSewer sizing adaptive algorithmNP=530Determine pipe sizeIntegerGBI locator algorithmNS=530Determine GBI locationsBinaryTotalIn this study, the Borg multiobjective evolutionary algorithm (Hadka and Reed 2013) is employed for the MOO based on its successful application to water resource problems (Eckart et al. 2018). More detail about this algorithm is given in Appendix.Step 3: Final Decision MakingAs described earlier, some solutions from the Pareto front obtained in the previous step are selected in this step for more comprehensive assessments and then for the final decision. The SWMM simulations for the selected solutions calculate all additional indicators and rank the solutions. In addition to the simple indicators used in the simulation-optimization step, 15 additional indicators (Table 2) are used here for final decision making. These indicators are evaluated under continuous simulation, which uses six months of rainfall data from October 2018 to March 2019, a period recognized for its extreme events in recent years that caused many problems in the area, such as flooded streets, infectious diseases, traffic jams, and public protests. The total rainfall depth during this period was 268.4 mm, which was 45% more than the 30-year average. The maximum precipitation in 24 h during this period was 45.1 mm.Table 2. Additional indicators, their category, symbols and unitsTable 2. Additional indicators, their category, symbols and unitsIndicator categorysymbol (units)NoteTechnical and construction concerns (structural resilience)AvgD (m)Average diameter of all pipes in the networkmaxD (m)Maximum diameter of all pipes in the networkAvgE (m)Average buried depth of all pipes in the networkmaxE (m)Maximum buried depth of all pipes in the networkLD>1 (%)Percent of the length of pipes with diameter>1  m to the total length of pipesLE>2.5 (%)Percent of the length of pipes with buried depth>2.5  m to the total length of pipesEnvironmental sustainabilitySt (mm)Percent of the rainfall that is infiltrated and stored during the continues simulation to the total precipitationmaxflow,t (L/s)Maximum flow between all outlets during the continues simulationFunctional resiliencemaxvel,t (m/s)Maximum velocity in the pipes during the continues simulationavgvel,t (m/s)Mean value of velocity in the pipes during the continues simulationNf,tNumber of flooding manholes during the continues simulationFt (m3)Accumulated flood volume from sewer manholes during the continues simulationAvgh_f,t (h)Mean value of flood duration of all flooded manholes during the continues simulationmaxh_f,t (h)Maximum value of flood duration between all flooded manholes during the continues simulationStakeholders acceptanceStkRankRank of solutions considering public acceptanceThe proposed indicators are divided into four categories: •Technical and construction concerns (structural resilience): primarily UDS pipe diameter and burial depth. Installing pipes larger than 1 m is problematic in the existing narrow streets in our case study area. Moreover, as the average groundwater level in the area is 2.5 m below the ground surface, installing pipes deeper than this depth is costly and increases the risk of inflow. In extreme flood conditions, such additional loads can cause pipe blockages, reducing structural resiliency.•Environmental sustainability: the environmental sustainability of each scenario, assessed by measuring the percent of the rainfall infiltrated and stored during the continuous simulation and by measuring the maximum outflow to the river during this period. Additional water quality assessments are disregarded due to a lack of data.•Functional resilience: the hydraulic performance of each scenario, assessed during the continuous simulation using indicators such as maximum velocity in the pipes, number of flooded manholes, accumulated flood volume, and flood duration (Table 2).•Public acceptance: stakeholder or decision maker buy-in, based on which solutions are ranked. We assumed that traditional centralized alternatives that rely mostly on a pipe network have a greater acceptance.Here TOPSIS is used to aid in prioritizing solutions, and the entropy method is adopted to assign indicator weights (Li et al. 2014). TOPSIS, first proposed by Hwang and Yoon (1981) and later developed by Yoon (1987) and Hwang et al. (1993), ranks alternatives based on relative similarity to ideal solutions (Roszkowska 2011). It is a practical technique with an intuitive and clear logic that represents the rationale of human choice. It can be straightforwardly applied and enjoys high computational efficiency (Hung and Chen 2009).As the determination of a specific weight for each index in TOPSIS is usually subjective, the entropy method was utilized to calculate the weights and reduce the subjectivity (Li et al. 2014). Entropy is a term in information theory introduced by Shannon (1948). The calculation steps of the so-called entropy method for weight determination are as follows, supposing a decision matrix xij with m alternatives (rows) and n indicators (columns) (Wang et al. 2017): 1.Compute the normalized pij of the ith alternatives to the jth indicator in the decision matrix (17) pij=xij∑i=1mxij(1≤i≤m,1≤j≤n)2.Compute the output entropy ej of the jth factor (18) ej=−k∑i=1mpijln(pij)(k=1ln(m),1≤j≤n)3.Compute the weight of entropy wj(19) wj=(1−ej)∑j=1n(1−ej)(1≤j≤n)Results and DiscussionIn the first scenario, for each layout, 100,000 simulation evaluations are defined as the termination criterion in the BORG optimization engine. Considering 2 s for each simulation, the whole optimization procedure for each layout took approximately 2.5 days (10 days for all layouts) using a personal laptop (Intel Core i7 with a 2.8 GHz dual-core CPU and 16 GB RAM). For the second scenario, all layout and DC variables are optimized. In this case 800,000 simulation evaluations are defined as the termination criterion, leading to a computational time of approximately 18 days. The Pareto-optimal solutions identified by the proposed framework for all scenarios are shown in Fig. 4, which accurately demonstrates the relation between LCC, DC, and total sustainability for the presented test case.Influence of (De)CentralizationAs a first analysis, we look at the influence of DC on the Pareto front. For the fixed layout scenarios, DC has two main effects. First, all solutions with a certain DC are dominated by solutions with a lower DC, so the lower DC values are beneficial. This is more obvious for the most centralized alternative (e.g., compare DC=100% and DC=55%). However, the more decentralized scenarios (DC=33% and DC=0%) are close to each other, indicating that more structural decentralization does not cause a recognizable additional benefit. The same trend is observed for the second scenario, where the layouts are not fixed. The distance between Pareto fronts is high for more centralized solutions and decreases as DC decreases. Second, the range of solutions discovered by the MOO engine is highly dependent on the DC. As can be observed in Fig. 4, the broadest range of solutions belongs to the layout with DC=66% (total sustainability ranging from 40 to 95), followed by DC=33%, DC=0%, and DC=100%. As a result, a very centralized or very decentralized layout might restrict the feasible search space, so more investigation is needed to determine whether there is a meaningful and insightful relation between the feasible search space or number of solutions on the Pareto fronts and system flexibility.FlexibilityOur next analysis examines the effect of layout configuration flexibility on the quality of solutions for the optimum HGBGI. As seen in Fig. 4, for a fixed DC all optimal solutions of the second scenario (layout optimized) dominate solutions of the first scenario (fixed layout). The solutions of the second scenario with DC=88% (two outlets) practically overlap with the solutions of the first scenarios with DC=66% (four outlets). Particularly for DC=100%, solutions from the second scenario are observed to explore a broader range of objectives than the solutions from the first scenario. Only for DC=33% are the results of the two scenarios close. This similarity might be explained by the well-designed layout of the first scenario for this DC. Overall, we conclude that joint global optimization, as is possible with our formulation, is advisable. Here, we omit the fixed layout cases (Scenario 1) from further analysis.Final Decision MakingFrom the Pareto front of the second scenario, seven solutions are selected to proceed to the final decision-making stage. Five solutions with different DC and LCC lower than 300,000 million rials and a total sustainability greater than 70%—also the solution with the lowest LCC and the greatest total sustainability—are chosen from the Pareto fronts. Table 3 presents the indicator values, entropy weights, TOPSIS scores, and final solution rankings. The results show that the solution with DC=33% has the greatest score (Rank 1) among all selected solutions. Fig. 5 shows this design and its specifications. Other selected solutions (DC=100%, 88%, 55%, 0%) are shown in Figs. 6–9. The TOPSIS scores for solutions with Ranks 1–5 are relatively close. This suggests that most of the solutions on the Pareto front can provide satisfactory quality. It must be noted that all of these conclusions are based on a specific case study and no general conclusion can therefore be made. However, the proposed framework can be applied to different case studies with completely different situations without restriction.Table 3. Results of indicators, entropy weights, and the final rankingTable 3. Results of indicators, entropy weights, and the final rankingIndicator (unit)DC=100%DC=88%DC=55%DC=33%DC=0%DC=0%DC=0%Entropy weight(a)(b)(c)LCC (million rials)293,490283,590280,230263,530259,320224,200306,0300.0012Rel (%)98.9999.0598.4998.9499.188.9110.0002ResFunctional (%)80.1983.6386.7485.2886.1977.499.760.0008ResStructural (%)61.7168.9374.5579.9681.7681.7482.060.0013SusEnvironmental (%)62.459.0556.3258.759.1832.1784.190.0080DC (%)1008855330000.0506AvgD (m)0.420.420.430.410.410.410.430.0001maxD (m)1.51.521. (m)1.951.941.911.861.861.861.890.0000maxE (m)6.956.956.>1 (%)5.795.626.745.074.514.626.690.0031LE>2.5 (%)14.0414.5515.5814.8314.8215.1316.070.0002St (%)43.6643.0640.8344.1844.0236.0757.490.0024maxflow,t (L/s)6,9965,7365,8512,5782,1213,2751,7310.0319maxvel,t (m/s)4.043.393.082.652.723.692.250.0048avgvel,t (m/s),t1361132314100.2671Ft (m3)1,2286423233,42000.3436Avgh_f,t (h)0.750.490.,t (h)1.670.770.730.330.34.3100.1601StkRank62134750.0371TOPSIS score0.4960.9000.9120.9430.9310.0240.939—Rank6541372—Stakeholder ParticipationAs mentioned before, the proposed framework enables stakeholder involvement in the complex decision-making processes. In the problem definition stage, these stakeholders can suggest their desired technologies, such as different types of GBIs, treatment facilities, or conventional systems. In the simulation-optimization stage, all possible combinations of these technologies are systematically generated and evaluated to provide a Pareto front of nondominated solutions. Finally, in the final decision-making stage, stakeholders participate in selecting and ranking solutions. They can do so either by determining the weights of the indicators or by ranking the solutions based on their preference as an extra indicator. This approach might reduce the unfavorable procedural outcomes, resistance, and conflicts that stakeholders often cause when they feel undervalued and unheard (Pigmans et al. 2019). However, this framework restricts the domain of their contribution to only feasible, previously optimized, and plausible solutions. We also calculate solution ranks without considering the stakeholders’ opinions (without including StkRank in Table 3 to calculate TOPSIS scores). The only change in comparison with the ranking in Table 3 is that the order of the first and second solutions is reversed.Analyzing Structural ResilienceFinally, the role of structural resilience in the layout configuration of the final design is investigated. Structural resilience is often neglected in the design of UDSs. However, it can play a significant role in system performance during extreme events. For instance, the fluvial flood that occurred in the study area in the winter of 2020 dramatically increased the groundwater level in the riversides, resulting in the choking and blockage of several main pipes of the existing centralized wastewater collection network. As a consequence, several parts of the network, even far away from the riversides, were out of service, causing wastewater to spill from manholes and overflow in the streets and resulting in serious disturbances and health problems for citizens.In Fig. 5, the structural resilience of each outlet [using Eq. (8)] is presented. These values can be interpreted as the percentage of total area not affected by any failure in that outlet. As an example, the minimum structural resilience among all outlets is 79%, which belongs to Outlet 4. This means that only 21% percent of the total area is connected to that outlet. The critical pipes, which are the first pipe in each part of the system with a structural resilience less than 90%, are also shown in Fig. 5. The pipes downstream of the critical pipes have less structural resilience. However, increasing the system’s redundancy in weak points can resourcefully increase network resilience by for example, adding some loops or pipes that divert the flow direction to other parts of the network in emergency conditions. In this way, the effects of single pipe failures can be significantly restricted. Of the 530 pipes in the case study, 4 are recognized as critical, as seen in Fig. 5. Therefore, only by introducing 4 additional elements can a minimum (90% here) structural resilience be achieved for all pipes in the system. Interestingly, the structural resilience of Outlet 9 is 84%; however, none of its upstream pipes has a structural resilience of less than 90%. The reason is that, in this part of the network, the stormwater is smartly collected from three different main collectors, as depicted in Fig. 5.Conclusion and OutlookThis study introduced a multicriteria decision-making platform for sustainable planning of urban drainage infrastructures considering centralized or decentralized strategies. Our platform encourages decision-maker engagement in all phases of the optimization procedure to increase the buy-in of the optimization results. The proposed framework is divided into three main steps Step 1 is system definition, in which the area under design is characterized, all desired types of urban drainage systems and technologies are introduced by decision makers, and the performance indicators are determined. Step 2 is simulation-optimization, in which a multiobjective optimization problem is formulated based on simple reliability, resilience, and sustainability indices. Many hybrid design schemes [gray, green, and blue elements with different sewer layouts and different degrees of (de)centralization] are generated and evaluated. This results in a Pareto front of nondominated solutions. Step 3 is final decision making, in which the selected solutions undergo a comprehensive assessment using a full range of indicators, including decision-maker preferences. Finally, a multicriteria decision-support technique is employed to rank the solutions.The proposed framework was applied to the design of a stormwater management system for a section of Ahvaz, a city in southwestern Iran. Four simple indices to assess reliability, resilience (structural and functional), and sustainability were defined for the simulation-optimization problem. Additionally, 15 mostly technical indices were proposed to rank the solutions. Two optimization scenarios were considered: (1) fixed layouts; and (2) (de)centralized layouts and the degree of (de)centralization. The second scenario constructs an extremely complex nonlinear, mixed integer-real, highly constrained optimization problem with 1,093 decisions, including layout configuration, pipe diameters, and green-blue infrastructure locations, in addition to three objectives: (1) life-cycle costs, (2) total sustainability, and (3) degree of (de)centralization. During the simulation-optimization procedure, approximately 800,000 HGBGI schemes were generated and evaluated.The results confirm the ability of the proposed framework to handle many decisions, objectives, and indicators to solve a complex optimization problem in a reasonable time by delivering realistic solutions. The comparison of the results of the first and second scenarios demonstrates the significant role of layout configuration and degree of (de)centralization in the optimum HGBGI. For a fixed level of sustainability, a significant reduction in life-cycle costs may be obtained through a well-designed layout. The layout configuration can also determine the system’s structural resilience. The green-blue infrastructures used in this study can theoretically increase system flexibility. However, future studies might investigate tactics to increase system redundancy in the design of layouts or sewers, such as by introducing additional storage tanks and parallel pipes, or by allowing loops in critical zones.The proposed framework facilitates stakeholder (decision maker) involvement in decision making as a way to decrease conflicts between them and optimization analysts. In the first step, alternative technologies are suggested; in the final step, solutions are prioritized or the indicator weights are assigned. In future work, the proposed framework could be extended to resolve the conflict between multiple stakeholders with contradictory objectives (Di Matteo et al. 2019; Raei et al. 2019).In the present study, simulations were limited to selected design storms. The choice of storms influenced the optimization results. We are planning to evaluate the robustness of our results against changes in rainfall scenarios in future studies considering rainfall patterns and variations in spatial distribution. We also plan to include multievent simulations (rainfall series) or synthetic rainfall series considering climate change scenarios in the postprocessing step for final decision making.Before applying the proposed framework to larger-scale problems, the computation time needs to be reduced. To do this, the use of metamodels instead of simulation software might be practical (Carbajal et al. 2017; Mahmoodian et al. 2018). Another alternative is to propose indices based on layout characteristics as a prescanning step in simulation-optimization so that only layouts with a predefined acceptable condition for hydraulic design are considered. Parallel processing can be used in all alternatives to accelerate the process. Although the frameworks are proposed for urban drainage system design, the general framework is relevant to the design of other kinds of water resource systems.Appendix. Borg MOEAThe Borg MOEA employs what is called ε-box dominance archive fitness assignment, which improves convergence and diversity throughout the search as defined here. Most of this section has been obtained from Hadka and Reed (2013).For a given ε>0, a vector (objective values) u=(u1,u2,…,um)ε-box dominates another vector v=(v1,v2,…,vm) if and only if one of the following occurs: 1.⌊uε⌋≺⌊vε⌋, Or2.⌊uε⌋=⌊vε⌋and∥u−ε⌊uε⌋∥<∥v−ε⌊vε⌋∥Borg MOEA executes an update archive procedure for every generated solution to add the solutions that ε-box dominate all solutions in the archive to the archive. Abstractly, the ε box dominance archive provides a minimum search resolution by dividing the objective space into hyper-boxes with side length ε. This is useful when decision makers can define their precision goals (e.g., 1 million rials for LCC in this study) or computational limits.Moreover, Borg includes different design principles and several novel components that improve its efficiency in handling many-objective, extremely multimodal problems. The main components are as follow: 1.ε-Progress, which evaluates search progress and prevents stagnation in a search. ε-Progress happens when a solution d is accepted into the archive such that there is no member of the archive with the same ε-box index vector.2.Restart mechanisms, which revive the search after stagnation is detected using ε-progress. These mechanisms include (1) adaptive population size, (2) adaptive tournament selection size, and (3) emptying the population and repopulating using solutions from the archive.3.Autoadaptive multioperator recombination, which enhances a search in a wide assortment of problem domains to systematically recognize operators that produce more thriving offspring, rewarding them by increasing the number of offspring they generate. Borg includes six recombination operators: simulated binary crossover, differential evolution, parent-centric crossover, simplex crossover, unimodal normal distribution crossover, and uniform mutation.The Borg MOEA combines the components just listed in its main exploration algorithm as follows: 1.An initial population is randomly generated in the feasible search space.2.For all chromosomes, the corresponding objective functions are evaluated. The ε-box dominance condition is checked for all generated solutions, and the archive is updated.3.Using autoadaptive multioperator recombination, one of the recombination operators is selected.4.For a recombination operator requiring k parents, one parent is selected randomly from the archive. The remaining k−1 parents are selected from the population using tournament selection.5.The generated offspring solutions are evaluated and considered for inclusion in the population and archive.6.After a user-defined number of iterations, ε-Progress and the population-to-archive ratio are checked. If a restart is needed, the main loop pauses and restart is executed. Once restart has been completed, the algorithm, now with a new population, is repeated from Step 2. 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