Uncertainty based budget allocation of wastewater infrastructures' flood resiliency considering interdependencies

Flood is one of the most devastating natural disasters that leads to significant economic loss and other casualties. Historical evidence indicates that coastal cities are extremely vulnerable to floods and storms (McGranahan et al. 2007; Kantamaneni 2016; Brown et al. 2018). More importantly, critical infrastructures in these areas are usually a flood target. Of these infrastructures, wastewater treatment plants (WWTPs) are usually very close to coastlines to discharge their effluent into receiving waters. Hence, they are among the most vulnerable infrastructures to coastal flooding (Hummel et al. 2018). On the other hand, any functional failure in these systems results in irreparable environmental damage which, in turn, places more emphasis on their flood resilience improvement. This improvement is not facilitated except through allocating resources and making investment. Resilience resource allocation needs a framework to help decision-makers to optimally decide how to invest in their systems. Creating such a framework requires the assessment of the system's resilience and the consideration of different interventions and related costs (MacKenzie & Zobel 2016).

The concept of resilience was originally introduced by Holling (1973) in the early 1970s. However, this concept has recently found a new application in urban resilience such that Karamouz et al. (2012) consider it the most important risk management indicator. Resilience concept varies in different scientific disciplines (Ayyub 2014). In engineering, resilience is the capacity of a system to adapt to any stressed situations (Alderson et al. 2015). Specifically, flood resilience is defined as the ability of resisting against flood, absorbing the related negative impacts, and returning to normal condition (Simonovic & Peck 2013). With such a definition, many researchers have provided formulations and frameworks for measuring the resilience of urban communities and infrastructures. One widely used framework has been proposed by Bruneau et al. (2003) that is based on quantifying system's resilience dimensions including rapidity, robustness, resourcefulness, and redundancy (four Rs). Rapidity is the ability of a system to achieve the expected level of performance in the shortest possible time while robustness is the system's resistance to maintain its function against stressors. On the other hand, resourcefulness is defined as the ability to employ material and human resources to meet the expected level of service and redundancy is the units that are available for replacement in different parts of the system to ensure its continued operation. On this basis, Kendra & Wachtendorf (2003) investigated the reconstitution of Emergency Operations Centre (EOC) after the destruction in the World Trade Center attack. A conceptual framework based on four Rs was developed by Cutter et al. (2008) for understanding the disaster resilience of communities. Karamouz et al. (2016) utilized four Rs to quantify the flood resilience of WWTPs against coastal flooding.

From the resilience standpoint, various systems should be prioritized to distribute resources among them. In other words, the more a system has the capacity for resilience improvement, the more investment is justified in that system. To accomplish this a priority multi-criterion decision making (MCDM) technique has been extensively used. Ronco et al. (2015) employed the MCDM technique to develop flood risk maps in Switzerland based on ranking the different dimensions of risk. Karamouz et al. (2019) utilized MCDM in two ways, to rank flood mitigation alternatives and to assess the flood resilience of WWTPs. By identifying the flood resilience attributes of WWTPs and taking their relative importance into account, Karamouz et al. (2018) established a resilience evaluation framework based on the MCDM technique. The flood resilience of WWTPs were quantified in terms of four Rs in this framework to examine the priority of these facilities in the allocation of resources.

Recent flood events such as Hurricane Katrina and Superstorm Sandy in the USA have demonstrated the role of interdependencies in cascading failures (Leavitt & Kiefer 2006; Comes & Van de Walle 2014; Sharkey et al. 2015). Interdependency is a linkage connecting two infrastructures in a way that the state of one infrastructure influences the state of the others (Rinaldi et al. 2001). This issue was first reviewed by the US Commission on critical infrastructure protection (Ellis et al. 1997) and then studied in a growing body of investigations. Wastewater treatment plants are very energy demanding and for normal operation they rely on energy infrastructures. During Superstorm Sandy critical energy infrastructures were damaged by flooding. The storm, for example, caused serious damage to generation, transmission, substation and distribution systems (Bloomberg 2013). Subsequently, some treatment plants lost their power for days, forcing operators to partially operate on emergency generators. Therefore, the reliability of utility power and back-up generators needs to be considered for assessing WWTPs' flood resilience (Hyland et al. 2013). If there is no such consideration, resilience tends to be overestimated or underestimated and leads to wrong judgments.

The motivation behind quantifying resilience is usually to assist decision makers to prepare for and respond to interruptions in the systems (MacKenzie & Zobel 2016). This preparation is typically done by distributing physical and financial resources among infrastructures to mitigate the impacts of disruptive events. However, due to budget limitations, the optimal allocation of these resources is inevitable (Zhang et al. 2018). The optimal allocation of flood resilience resources among different facilities of an agency is difficult for two reasons: (1) because of the nature of resource allocation problems in which the interest of one facility may work against the others (Chakraborti et al. 2015); and (2) because of the uncertainties that exist in measuring and improving their flood resilience (Berkes 2007). The latter is the main reason for investors' unwillingness to invest in systems' resilience enhancement (Juan-García et al. 2017). Therefore, it is important to better understand the uncertainties in order to inform agencies of their investment risk.

In distributing flood resilience resources, uncertainties are due to the stochastic nature of floods and the uncertain resilience attributes of systems. Furthermore, due to the subjective views of experts and decision-makers, another set of uncertainties is entered into the process. Agarwal et al. (2004) categorized the earlier and the latter into stochastic (aleatory) and subjective (epistemic) uncertainties, respectively. The stochastic uncertainty stems from the inherent randomness of events and cannot be reduced while the subjective uncertainty arises due to insufficient knowledge and can be reduced. Stochastic uncertainty is typically analyzed using probability theory. Karamouz et al. (2016) utilized probability theory and the Monte Carlo technique to estimate stochastic uncertainty in measuring the flood resilience of WWTPs.

In respect to policy making, the extent of uncertainty obviously includes subjectivity and different perspectives of various actors involved in the decision making process (Walker et al. 2012). Specifically, in resource allocation usually a limited number of experts offer their opinions based on their limited knowledge and information (Ballent et al. 2019). In addition, in this process, the experts cannot easily express their judgements on the available options with exact and crisp interpretations (Ju & Wang 2012). It is therefore important to combine the views of experts to estimate the subjective uncertainty in the allocation of resources. Conventional probability theory is unsuitable to represent subjective uncertainty (Ali et al. 2012). Instead, the Dempster-Shafer theory (Shafer 1976) that is also called evidence theory makes it possible to deal with such problems and to combine information coming from independent sources (Ju & Wang 2012). This theory offers a degree of belief as a measure of an expert's belief over a proposition (Vick 2002). Many researchers such as Yang & Xu (2002), Yang et al. (2006), and Guo et al. (2007), have utilized the Dempster–Shafer theory for developing an evidential reasoning algorithm for MCDM in decision-making and selection processes. In this paper, the Dempster–Shafer theory is utilized as a post processor for evidential reasoning in the resource allocation process.

Such an approach increases decision confidence in the allocation of resources.

In recent years, the resilience of urban infrastructures to natural hazards has been the focus of many investigations. However, as could be observed in the literature review, the way resilience is measured and resources are allocated among multiple facilities have received limited attention. In many flood disasters, disruption in energy/electrical resources is the key factor in systems shutdown as observed in the case of Superstorm Sandy in NYC. Furthermore, many urban facilities are run by an agency that has the responsibility for funds allocations. It is often done through a non-technical platform with a lack of a performance/growth potential base matric. Some previous studies realized that it is done on technical ground without combining it with many subjective evidences and information and the uncertainties associated with them as well as technical attributes. These shortcomings have been realized in the previous works of the authors (Karamouz et al. 2016, 2018). However, their studies are significantly extended to include energy dependencies in evaluating the coastal flood resilience of waste water infrastructures; and different allocation methods including maximizing overall resilience (MOR), and examining uncertainties in the financial allocation process by having a better way of assigning utility functions to different facilities. The new contribution of this paper is, however, to develop a framework extending the application of Nash product (MNP) and consumer behavior theory (CBT) in order to further close the gap between how agencies allocate funds and how decision making tools combining different analyses and different views could be utilized.

The remainder of the paper is organized as follows. In the next section, methodology is described in detail. In this section, different dimensions of WWTPs' flood resilience and their improvement through investment will be addressed. Also, the allocation of financial resources among these facilities will be examined in different ways. Next, a case study is presented. The results of uncertainty based allocations are illustrated and discussed followed by a summary and conclusion.

Resilience assessment is generally classified into one of two categories: attribute based (qualitative) and performance based (quantitative) assessments (Hosseini et al. 2016; Vugrin et al. 2017). Performance based measuring of resilience requires system's simulation during hazards. With the increasing number of systems and their complexity, the simulation has become more and more difficult. In particular, consideration of systems' interdependencies increases the simulation time burdens. However, it provides a detailed description of systems' performance during hazards. In comparison, in the evaluation of systems' resilience based on their attributes, there is no need to simulate systems. This approach is a low-cost technique to compare the systems' resilience with a minimum degree of complexity in resource allocation problems. Thus, this paper extends the work of Karamouz et al. (2016, 2018) at using an attribute based approach to assess the resilience of WWTP systems through four criteria including rapidity, robustness, resourcefulness and redundancy (four Rs) and sub-criteria that include different data types, such as hydrological, environmental, economic, and technical data. Also, given the importance of energy for the operation of sewage treatment systems and the possible failure of electrical infrastructures during floods, the relevant sub-criteria are identified and incorporated into the framework. The proposed framework is illustrated in Figure 1. According to this figure, the framework consists of four steps:

• 1.

  Identifying the attributes of systems that are effective in flood resilience.

• 2.

  Comparing the flood resilience of systems using MCDM.

• 3.

  Investing in systems and examining their flood resilience improvement.

• 4.

  Allocating a fixed budget among systems considering uncertainties.

Each of these steps will be described in detail.

Among different WWTPs' flood resilience attributes, 23 attributes, 18 of which were identified by Karamouz et al. (2019), are utilized for evaluating the system's flood resilience. These sub-criteria are grouped into four Rs according to Table 1. Selection of these sub-criteria has been done to cover a wide range of different situations of WWTP in normal (such as the design capacity of plant), emergency (such as the number and capacity of backup generators) and recovery conditions (such as recovery time). The expert assessment and weighting methodology are also involved based on these physical processes and site specific characteristics as explained in the case study section. Some more details are presented in the supplementary material of this paper. Steps 1–3 of the flow chart are extension of previous works by adding new energy sub-criteria and new financial utility function. Step 4 is the main contribution of this paper in this paper.

One important aspect which is often neglected in examining systems' resilience is interdependencies among them. Sewage treatment systems need electricity to operate. This necessity becomes more acute in flood situations when high flow is entered into the systems. However, there is always a high probability of experiencing power outages at the time of severe storms. Wastewater treatment plants are usually equipped with back-up power to support their performance in emergency situations. However, due to the high cost of back-up generators, the total electrical power capacity required is not provided. Therefore, the number and capacity of generators in critical situations play a significant role in the rapid recovery of them. Furthermore, the safety of substations inside the plants and electrical feeders against floods should also be considered. Because of the high inundation depth at the unit substations, main plant switchgear, and transformers may also result in complete shutdown of the plants. Based on historical observation of disruption to electricity connected to wastewater treatment plants, their flood resilience has also been assessed. This number is set to zero and one, where one indicates disruption during a storm. All the above-mentioned factors are specified into five sub-criteria in Table 1 and are assumed to represent water–energy interdependencies. These factors are bolded in this table besides 18 other sub-criteria defined by Karamouz et al. (2019).

Using the values of sub-criteria and their respective weights, the attribute-based resilience for any of the WWTPs can be calculated. For this purpose, different approaches based on MCDM can be used. One of the recent approaches that has been successfully used is the Promethee and Gaya method (Behzadian et al. 2010). The relationships of this method, that is explained in detail in Karamouz et al. (2016), are presented in the supplementary material.

Here, the optimal improvement of system's resilience for a given budget is of interest. In doing so, economic factors are identified among the sub-criteria. These factors include items that potentially can be improved by investment. These factors are listed in Table 1A in the supplementary material. In this table, all the costs are presented in 2013 US dollars and are averaged over the range of available costs. In the next step, allocation constraints are determined for each economic sub-criterion (ESC) in an optimization problem. Then, the resilience improvement of each plant is evaluated for different budgets. Accordingly, a financial allocation-resilience improvement curve (hereafter utility function) is developed for each plant that is further used as their utilities of the received budget in the financial allocation process.

The resilience improvement of a plant is maximized using genetic algorithm. This algorithm is repeated for different budgets (i.e. $10, $50, $100, $200, and $300 million) until the resiliency improvement rate is insignificant) Also, there is no need to extend the budgets to draw allocation-resilience improvement curves, given the $187 million budget that is distributed among the plants to compare with the NYCDEP's (the organization responsible for the operation of sewage treatment systems serving NYC) allocation. Although repeating the algorithm for more budgets in this interval provides more points to derive the functions of these curves, these five budgets seem sufficient to also compare the results with Karamouz et al. (2018). Equation (1b) shows the incremental improvement of sub-criteria values after the allocation of budgets. The budget allocated to increase a plant's treatment capacity and on-site storage is defined at a maximum of (Equation (1e)). When the generator capacity is less than the power required by the plant, the capacity will be increased depending on the allocated budget to finally supply the power needed (Equation (1f)). WWTPs are typically equipped with 1,000–2,000 KW generators (NYCDEP 2007), thus, based on Table 2A in the supplementary material, it is assumed that for every 2,000 KW addition, one unit is added to the number of generators, which is equivalent to 1.6 million dollars. This amount of money has already been included in the allocation to the generators' capacity. The allocation limit on sand bagging, water proofing, and elevating critical equipment is assumed based on Equation (1g) up to ⁠. The upper bound of increasing the number of technical workers is γTB (Equation (1h)). Equation (1i) indicates that the budget allocated to a plant on dewatering facilities is $7 million (M₅ = 7 from Table 2A in the supplementary material) provided that the plant does not have these facilities. Otherwise, no fund will be allocated to this section. The allocated budget limit to increase the on-site storage of plants is defined up to (Equation (1j)). The coefficients of ⁠, ⁠, ⁠, and represent the percentage of TB, which are defined by the experts of each plant. It should be noted that these constraints are defined based on preliminary assessment. In practice, more rigorous analysis can be carried out by the experts of each system to better define these constraints.

In this study, three methods are investigated to allocate financial resources among systems. The first method is to maximize overall resilience improvement and the allocations to the plants is geared toward this goal. In the second method, considering systems' utilities of the received budgets, the Nash bargaining solution is used and the financial allocation is performed based on maximizing the Nash product. Finally, the allocation problem is solved using consumer behavior theory. In the following sections, each of these methods will be discussed.

Uncertainties in the financial allocation process to improve the flood resilience of wastewater treatment plants are divided into two categories:

• 1.

  Uncertainties due to the random nature of flood and the uncertain flood resilience attributes of plants (stochastic uncertainty).

• 2.

  Uncertainties due to subjective views of experts and decision-makers (subjective uncertainty).

Therefore, the financial allocation to each plant has some degree of uncertainty. These uncertainties are discussed in the following sections, respectively.

In the process of allocating funds, an agency should consider different views of its experts. Based on their information and knowledge, these individuals will reach a degree of agreement on the level of allocation to each system. Such a degree of belief can be achieved in different ways including resilience analysis, post-storm damage assessment, building and infrastructure-level vulnerability assessment, etc. Combining the degree of expert beliefs is inevitable to avoid unilateral decisions and judgments. For this purpose, the financial allocation ranges are categorized into the types of low, medium and high (Figure 3).

Based on the values of belief and plausibility, one can interpret the uncertainty over each proposition of allocation. If the degree of belief over a proposition is 1, the evidence absolutely supports the proposition, and the proposition is quite true. In case of zeroing belief and plausibility, it is ensured that the intended statement about the allocation is quite false. If the degrees of belief and plausibility become 0 and 1, respectively, the evidence about the proposition is absolutely unknown. By increasing the degree of belief, the correctness of the proposition increases and vice versa. The above interpretations of Figure 4 are summarized in Table 2.

New York City's wastewater treatment system is at high risk of inundation caused by hurricanes and tropical storms and is selected to illustrate the methodology. The city's Department of Environmental Protection (NYCDEP) is in charge of the entire operation and maintenance of the sewage treatment system. The NYCDEP wastewater treatment system includes 14 plants that discharge their effluent into the receiving water surrounding the city. These facilities together process over 1.3 billion US gallons (nearly five million cubic meters) of wastewater daily. After Superstorm Sandy, the insufficient flood resilience of the plants was perceived by the NYCDEP. Accordingly, the agency's study acknowledged that all fourteen wastewater treatment plants are at risk with a consequence of over $900 million against a 100-year flood. In this respect, the recommended budget to improve the flood resilience of these facilities was estimated at $187 million (NYCDEP 2013). This budget mainly includes the cost of elevating critical equipment, flood proofing, sand bagging, and constructing barriers.

Figure 5 illustrates the plants' names, locations, capacities, and their service area boundaries on the NYC map. According to this figure, some of these treatment plants are directly exposed to surges in the Atlantic Ocean, such as Coney Island and Rockaway. Some of them are exposed after passing through Long Island Sound, such as Hunts Point, and therefore part of the storm surge is dissipated. Others are exposed by the propagation of surges in the rivers, such as the North River plant. These site-specific exposures to hazards are considered in assigning values to the sub-criteria in Table 3.

Table 4 shows the proposed coefficients of financial constraints for these plants based on expert advice. Given the considerable budget required for increasing the plants' resilience against coastal flooding, efforts to optimally distribute this budget among them are of great importance. In the next section, the results of such efforts are presented.

The weights assigned by the experts are combined and the relative importance of each factor is determined using an analytical hierarchy process (Saaty 1990). The weights of criteria and sub-criteria are presented in Table 5. As illustrated, the weights of four Rs are not the same. Rapidity and robustness are assigned higher weights with 0.395 and 0.349, respectively. This is because the resilience of a system mainly results from the inherent strength and characteristics to withstand flood and the speed of the system to recover (Bruneau et al. 2003; Karamouz et al. 2016; Ouyang 2017). However, resilience can also be improved by means of resourcefulness and adding redundant elements to the system. Hence, all the elements of the four Rs are important and need to be counted in quantifying resilience.

The flood resilience of NYC's wastewater treatment plants based on MCDM (Equation (4a)) is illustrated in Figure 6. The Red Hook plant has the highest amount of resilience among others. This plant did not experience major flooding and preserved its basic performance during Superstorm Sandy. In contrast, the Rockaway and Coney Island plants have the least amount of resilience. Both of these plants experienced major flooding during Sandy and many of their electrical equipment were submerged by the flood (NYCDEP 2013). In the Rockaway plant, for example, both utility feeders serving the plant were interrupted during the storm. Although this plant was equipped with two 900 KW diesel generators, they were not sufficient to support all treatment processes. Besides the Coney Island and Rockaway plants, North River, Oakwood Beach, Port Richmond, 26th Ward, and Owls Head are the plants which experienced major flooding during Sandy. However, their different flood resilience attributes result in different resiliency values. The resilience of the Red Hook plant has increased significantly while the resilience of Coney Island has considerably decreased, which is more consistent with NYCDEP reports.

Considering that the implementation of resilience interventions needs investment in the systems, the resilience improvement of each system versus investment is illustrated in Figure 7. A logarithmic function (⁠⁠) is fitted to the behavior of each of these systems as their utilities of the received budgets. The derivative of these functions (a/x), which defines the marginal utility of NYCDEP (in CBT), is strictly decreasing and represents that the additional utility of NYCDEP reduces with spending more dollars on the plants and the axiom of law of diminishing marginal utility is satisfied.

The coefficient a indicates the rate of resilience improvement. Here, the Rockaway and Coney Island plants with coefficients of 2.92 and 2.48 have the highest resilience improvement rate, respectively. In other words, these plants have the highest potential for resilience improvement and the highest investment priorities. Interestingly, these plants are those which have the least amount of resilience among others. In contrast, the Wards Island plant with the coefficient of 0.39 has the lowest potential to improve resilience. In comparison to the results of Karamouz et al. (2018), the priority of the plants in terms of investment is relatively stable; however, the resilient improvement rate of the plants has changed.

Because of the uncertainties previously discussed, the utility of the plants or the improvement of their resilience cannot be represented merely in the form of a fixed curve (solid lines in Figure 7). Rather, a set of curves could offer the utility of the systems in the process of distributing flood resilience resources. These curves are based on statistical distribution of random sub-criteria (Table 2A). The upper and the lower envelope of these curves are illustrated for each of these plants. These envelopes represent the maximum and minimum resilience improvement for a given budget, respectively. Accordingly, the plants can be classified into three categories. For some of the plants the fixed curve is very close to the lower envelope. The fixed utility of the Wards Island plant, for example, is almost touching the lower envelope. For some other plants, such as Hunts Point, the fixed curve is very close to the upper envelope. The fixed curve of others, such as Oakwood Beach, is almost in the middle of the upper and the lower envelope. The gap between the upper and the lower envelope of a plant implies the uncertainty of allocation to that plant. For example, due to the wide gap between the upper and the lower envelope of Wards Island, the financial allocation to this plant is expected to be more uncertain.

In the following subsections, the fixed curve of the systems is used for deterministic financial allocation and the set of curves defined between the upper and the lower envelope of each system is utilized for stochastic allocation among them.

Assuming that the resilience improvement of the plants is deterministic, the $187 million budget of NYCDEP is distributed among them according to Figure 8. This budget is allocated to the plants based on maximizing overall resilience (MOR) and Nash product (MNP). For some of the plants, such as 26th Ward and Coney Island, the allocation based on MOR is very close to the NYCDEP's recommended values. The Coney Island plant, for example, receives $15.5 million from the NYCDEP compared to $15.3 million based on MOR. The Wards Island plant, which has the lowest potential for resilience enhancement, almost receives no allocation while the smaller plant, Rockaway, with low resilience and high potential of improvement, receives the maximum allocation and resilience benefit. This is because many factors, including the depth of flood, contribute to the utility of each plant and the Wards Island plant is less vulnerable to coastal flooding compared to the Rockaway plant which suffers direct surge impact. For some other plants, such as Hunts Point, the allocation based on MNP is closer to the recommended allocation of NYCDEP. Based on the MNP approach, the Hunts Point plant receives the highest allocation of $26 million which is close to the NYCDEP's $24.2 million while the North River, with $4.6 million, receives the lowest.

Figure 9 shows the resilience improvement of the plants associated with the financial allocation performed above. The Wards Island plant is not improved based on MOR. The maximum resilience improvement is also related to the Rockaway plant with 9.4%. Interestingly, these two plants had received the highest and the lowest allocations, respectively. Similar to MOR, the lowest resilience improvement based on MNP is associated with the Wards Island plant, but in this case the resilience of this plant is slightly improved. Rockaway is still the most improved plant; however, this improvement has decreased to 7.5%. In contrast to MOR, the maximum and minimum resilience improvement is not related to the plants that had received the highest and lowest allocations (Hunts Point and North River in Figure 8), respectively. In general, plants with lower resilience improvement than the MOR average resilience improvement of 4.6%, including Bowery Bay, Hunts Point, Tallman Island, Wards Island, Oakwood Beach, and 26th Ward, have higher resilience improvement in MNP compared to MOR.

Among all the plants, the budgets allocated to the Wards Island plant are significantly different. This plant almost receives no fund based on MOR but receives a significant allocation of $15.1 million based on MNP. This difference arises because in MOR as long as the overall resilience improvement of systems is increased, any budget distribution is acceptable even if a system receives nothing from the budget. However, MNP has a strong tendency to distribute the budget to minimize dissatisfaction. In other words, this allocation scheme tends to increase the resilience improvement of the system with minimal improvement. Since the Wards Island plant has the lowest improvement of resilience among other plants (see Figure 9), the maximum difference of the two allocation schemes belongs to this plant.

Depending on organizational strategic goals, any of these allocations can be acceptable. The result of consumer behavior theory (CBT) is identical to those of MOR and for the sake of brevity we skipped presenting CBT results. The difference between MNP and CBT is that in MNP it is assumed that each plant has its own utility in the bargaining process. In other words, 14 players participate in the bargaining process in order to maximize their benefits, while in CBT, utilities are defined for one player (NYCDEP agency) that wants to maximize its benefit from financial allocation to its facilities.

Here, distributing the budget of $187 million among the plants is investigated in an uncertain environment. The probability distribution of allocation for each plant based on MOR and MNP is illustrated in Figure 10. Using Monte Carlo sampling from random sub-criteria's distributions (see Table 2A) and examining null hypothesis in the Chi Square test, it was observed that a random sample of allocation to each plant accepts the null hypothesis and therefore normal distribution was fitted to random allocations to each plant. The standard deviation (SD) of distribution for the plants shows the degree of allocation uncertainty. In this figure, the cumulative distribution represents the probability that the optimal allocation will take an amount less than or equal to a certain amount of budget. On this basis, in the case of Bowery Bay, the probability that the optimal allocation is less than or equal to $10 million is almost 70 and 40% based on MOR and MNP, respectively. A similar observation can be extended to other plants.

According to MOR, investment in the Hunts Point and 26th Ward plants are associated with the highest risk and uncertainty. Here the risk of investment means getting lower resilience improvement than expected for a given allocation (see Figure 10). Therefore, optimal investment in these plants is associated with more risk. On this basis, the Bowery Bay plant has the lowest investment risk.

In allocation based on MNP, the standard deviation of allocation for the plants becomes smaller. In this case, the SD ranges are between 1.0 and 2.9 and the Wards Island plant has the highest SD or allocation uncertainty. The 95% confidence interval of allocation for each of these plants is illustrated in Figure 11 based on MOR and MNP.

The difference at the 95% confidence intervals indicates that the use of different allocation schemes influences the degree of uncertainty in the allocation process.

Regarding different investment risks of allocations, it is important to know which allocation option is associated with the lowest risk. For this purpose, the financial allocation with the lowest investment risk is determined in Figure 12 based on maximizing the expected value of MOR and MNP.

Based on maximizing the expected value of overall resilience, the Rockaway plant still receives the highest share of the total budget with $19.2 million, and the Wards Island plant receives almost no fund similar to the deterministic allocation. In this case, the maximum change compared to the deterministic allocation is related to the Hunts Point plant, which is a $4.1 million decrease.

By maximizing the expected value of the Nash product, the highest budget is received by the Hunts Point plant with $23.1 million and the lowest by the North River with $4.1 million.

Therefore, the plants that received the highest and the lowest funding from the deterministic allocation do not change. In this case, the most notable allocation change is for the Bowery Bay plant but with a $3.8 million decrease compared to the deterministic allocation based on MNP.

Due to different analysis and expert perspectives on allocation to each infrastructure, assessing the uncertainty of allocation to each plant can help managers of these facilities make the right decisions and manage their finances. Therefore different beliefs are combined for each plant, and the range of belief, disbelief, and uncertainty are measured over each proposition of allocation. An expert based on the resilience analysis of this study and another based on the building and infrastructure-level vulnerability assessment performed by the NYCDEP, arrive at the degrees of belief in each proposition. The belief mass distribution of the latter expert based on the interval of NYCDEP's allocation is assumed trapezoidal. Accordingly, the mass of belief in the allocation interval of this agency is 50%, and in the adjacent intervals of 25% each (Figure 13(a)). These values for the first allocation interval are 67 and 33%, respectively (Figure 13(b)).

The belief and disbelief intervals and the range of uncertainty over each proposition of allocation are illustrated in Figure 14 based on the evidence theory and Dempster–Shafer rule of combination (Equations (10)–(16)). The black bars in this figure show the degree of inconsistency between the two expert beliefs for each plant. According to these bar charts, the most inconsistency corresponds to the Bowery Bay, Hunts Point, and Jamaica plants with almost 68%.

The Dempster–Shafer rule of combination is inappropriate for inference from highly inconsistent beliefs because it does not take into account the inconsistency between beliefs (Agarwal et al. 2004). Therefore, no inference is made about the allocation to the Bowery Bay, Hunts Point, and Jamaica plants. For the other plants, the inconsistency between the beliefs is far lower. In the cases of the Tallman Island, Wards Island, Port Richmond and Owls Head plants, this is less than 5%.

In the cases of the Newtown Creek, North River and Oakwood Beach plants, evidence rejects the allocation types of low and high, and the degrees of plausibility and belief are almost zero. While the evidence almost entirely supports the allocation type of medium, and these plants are placed in the category of plants for which the allocation type can be defined as medium.

The inconsistency between the expert beliefs for the Tallman Island plant is negligible and about 3%. The low-type of allocation to this plant is refuted by nearly all the evidence and the degree of belief in the high-type of allocation is negligible. Therefore, the allocation to this plant can be interpreted as medium.

In the case of the Wards Island plant, the inconsistency between the expert beliefs is almost 4%. Given the high-type allocation is refuted by the evidence and the degree of belief in the allocation type of medium is negligible, the allocation to the plant could be considered the type of low.

The conflict between experts' beliefs for the Port Richmond plant is only 1%. The evidence denies the allocation type of low for this plant. Furthermore, the agree of belief in the allocation type of high is negligible. Hence, the allocation type of medium could be considered for this plant.

The inconsistency between the expert beliefs for the Red Hook plant is almost 12%. The evidence refutes the allocation type of low for this plant. The degree of belief in the allocation types of medium and high are 53 and 17%, respectively. Given that most evidence supports the medium-type of allocation and the upper limit of belief in the allocation type of high is less than 50% (Pl = 45%), the allocation to the plant could be defined as medium.

In the case of the 26th Wards Plant, the conflict between the expert beliefs is 8%. The degrees of belief in the high and low-type of allocations to this plant are insignificant. The degree of belief in the medium-type of allocation is 84%, which can increase to 97% with uncertainty (Pl = 97%). Therefore, the allocation to this plant could be interpreted as medium-type.

The inconsistency between the expert beliefs for the Coney Island plant is almost 8%. The evidence denies the low-type of allocation to this plant. The degrees of belief in the allocation types of medium and high are 44 and 20%, respectively. the plausible interval of these two types of allocations are 80 and 55%, which are greater than 50%. Therefore, the allocation type for the plant could be defined as either medium or high.

In the case of the Owls Head plant, there is only 5% inconsistency between the beliefs of the experts. The evidence refutes the low-type of allocation to this plant. The degree of belief in the high-type of allocation is also insignificant. The allocation to this plant, therefore, could be considered as medium-type.

Finally, the inconsistency between the expert beliefs for the Rockaway plant is almost 12%. The evidence denies the low-type of allocation to this plant. The degree of belief in the high-type of allocation is 52%. Although the belief's degree in the medium-type of allocation is 19%, the plausibility of this belief is less than 50%. Therefore, the allocation to this plant could be considered as high-type. The type of allocation to each plant is illustrated in Table 6.

Wastewater treatment plants are vulnerable infrastructures against coastal storms. The failure of these facilities and the interruption of their recovery will result in irreparable environmental consequences. Improving the flood resilience of these systems is essential and requires considerable funds. However, due to budget limitations it is inevitable to optimize the allocation of financial resources among them. The optimal allocation among these systems needs an assessment of their performance indicators such as resiliency. For such an assessment the characteristics of these systems in terms of hydrological, environmental, economic, technical and operational aspects have been taken into account. Also, given the potential power outages during floods and the urgent need of these systems for power, the dependence of these facilities on energy has been included in the assessment of their flood resilience. All flood resilience attributes of these systems are categorized in terms of rapidity, robustness, resourcefulness and redundancy (four Rs).

The allocation of resources among the facilities of an agency is typically performed based on non-technical negotiations and default allocation routine. This study has provided a robust metric based on expert views for reinforcing the resource allocation mechanism of water/wastewater agencies. Using multiple expert views, the relative importance of four Rs has been determined. Since more weights were assigned to rapidity and robustness, the importance of these criteria is more dominating to evaluate the resilience of these facilities. Assessing the resilience of NYC's wastewater treatment plants, with regard to interdependencies, shows better agreement with NYCDEP reports after Superstorm Sandy. In other words, the flood resilience of the plants such as Coney Island that were seriously damaged in Superstorm Sandy have been far less than the plants that were not exposed to major damage. Considering the dependence of sewage treatment facilities on energy and the related flood resilience attributes such as the number of backup generators and their capacity, a better comparison of their resiliency has been made in Figure 6.

Taking into account different resilience interventions and related costs, the resilience improvement of each plant was determined for a wide range of allocations and is assumed as their utilities of the allocations. Assuming that these utilities are deterministic, the allocation of resources has been implemented based on maximizing overall resilience (MOR), Nash product (MNP), and consumer behavior theory (CBT). Allocation based on consumer behavior theory confirms the result of MOR. The Rockaway and Wards Island plants have the highest and the lowest resilience improvement, respectively in both MOR and MNP. Based on MOR, these two plants have also received the highest and the lowest allocations, respectively. The Wards Island plant has received insignificant allocation in MOR. However, in MNP, the highest and the lowest allocations are not attributed to these plants. In contrast to MOR, Wards Island has received significant allocation in MNP. In other words, the resilience improvement of Wards Island with minimal improvement increases in MNP, which demonstrates that this allocation scheme tends to increase the resilience improvement of the system with minimal improvement and to distribute funds more uniformly.

Due to the uncertainties in flood characteristics and some of the resilience attributes of these facilities, they were considered in the utility of each plant. Accordingly, allocations were made among the plants using MOR and MNP. Based on the standard deviation and the 95% confidence interval of allocations, it was demonstrated that the level of uncertainty is much lower in MNP compared to MOR. Therefore, the level of uncertainty is influenced by how resources are distributed.

Considering the different views of experts in the allocation process, the subjective uncertainties of each plant have been examined by the theory of evidence. Accordingly, the degree of inconsistency between experts' beliefs was measured and the belief, plausibility and uncertainty were estimated for each proposition of allocation. The results showed that the inconsistency is high in the cases of Bowery Bay, Hunts Point and Jamaica plants and is far less for the other plants. For plants where the degree of inconsistency is high on the allocation intervals, the Dumpster rule of combination cannot be applied and the uncertainty intervals of these plants cannot be determined. Subsequently, for other plants, according to the degree of belief in each proposition, the type of allocation has been determined. The results show that the medium-type of allocation could be considered for most of the plants.

The result of this study demonstrates the significant value of resilience based funds considering interdependencies and uncertainties related to different views of entities and analyses. The methodology of the paper is applicable to other geographic settings in coastal areas. One limitation of this framework is the sensitivity of its results to model input subjective values. A sensitivity analysis performed by Karamouz et al. (2018) on the input of the model indicates that plants with high investment potential are more sensitive to the model's subjective inputs. Global sensitivity analysis on the dependence of the results on the model inputs is suggested as it is outside the scope of this study.

The authors would like to thank Dr M. A. Olyaie and P. Khalili, PhD student from the University of Alberta, for their valuable remarks in the preparation of this paper. All input data used in this research are based on NYCDEP reports on the performance of New York City's wastewater treatment plants during Superstorm Sandy in October of 2012. These reports are publicly available at www1.nyc.gov/html/dep/html/home/home.shtml.

The Supplementary Material for this paper is available online at https://dx.doi.org/10.2166/hydro.2020.145.