Keywords

1 Introduction

Best management practices (BMPs) have been widely used for over two decades globally to manage flood risks, remove pollutants, or reduce sediment in water bodies [1,2,3]. To optimize the placement of BMPs, the present approaches use a range of mathematical optimization techniques such as integer programming, nonlinear programming, and evolutionary algorithms [4]. Behroozi et al. [5] used multiobjective particle swarm optimization (PSO) in District 10, Tehran, Iran, to minimize peak water flow rate and pollutant concentration. Singh et al. [6] describe a case study from Heredia, Costa Rica, where bioretention areas, green roofs, and infiltration trenches are placed in an urban setting to control flood risks. They use a nonlinear programming technique to optimize and formulate the trade-offs between land use and cost. Foomani and Malekmohammadi [7] proposed fuzzy logic and analytic hierarchy process for identifying optimum locations of BMPs in the northern region of Tehran, Iran. Li [8] developed SWMM_FLC, a combination of SWMM, fuzzy logic control, and GA, to reduce downstream flooding volume. Zhang et al. [9] applied Storm Water Management Model (SWMM) and System for Urban Stormwater Treatment and Analysis Integration (SUSTAIN) to conduct watershed-level optimization for Sponge City, China. Annual average runoff volume and total pollutants reduced workout to 87.61% and 85%. Dwivedula et al. [10] employed an ensemble of (1) non-dominated sorting genetic algorithm-III and (2) constrained two-archive evolutionary algorithm for optimizing zone-wise BMP placement in GHMC. Studies presented here, including that of [10] and elsewhere, have not reported any applications of fuzzy optimization in the placement of BMPs.

2 Study Area and Data Source

2.1 Greater Hyderabad Municipal Corporation

In this study, a fuzzy multiobjective approach is used to optimize the placement of BMPs for GHMC as a whole (not zone-wise). This section briefly describes the following:

  • Case study and process(es) used to identify potential BMP sites.

  • Multiobjective optimization problem, i.e., the decision variables, objectives, and constraints.

  • Fuzzy optimization framework and membership functions used.

  • Single Objective Genetic Algorithm (SOGA).

The fuzzy optimization process allows converting a multiobjective problem to a single objective problem, enabling us to use a Single Objective Genetic Algorithm (SOGA). Figure 1 presents the study area.

Fig. 1
A map of Greater Hyderabad has legends for the G H M C boundary and streams. It has a horizontal stream towards the bottom part of the map labeled Musi River. The left end of the river has a shaded area marked Osmansagar. It has 3 shaded areas for Hussain Sagar, Mirajam Tank, and Himayat Sagar.

Study area of GHMC

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2.2 Data Used

The present study examines a historic extreme rainfall event of 237.5 mm during September 20–28, 2016. We attempt to analyze the impact of BMP placement if a similar extreme event was to happen again. Hydrologic Engineering Centre’s-Hydrologic Modeling System (HEC-HMS) was employed to simulate surface runoff [11] and SUSTAIN [12] was used for identifying the potential BMP sites. EPA-SUSTAIN siting tool identified a total of 5,45,895 possible sites. Nine types of BMPs are being considered for placement in the GHMC.

2.2.1 Multiobjective Optimization Problem

The three objectives we wish to optimize are maximizing runoff reduction volume Z1 (in m3) and pollutant load reduction Z2 (in tons) while minimizing the cost of construction Z3 (in Indian Rupees). For an individual BMP for total area Ak, we can select (all the areas/a fractional part of an area/none of an area. This choice is encapsulated as a decision variable, denoted by Xk (0 ≤ Xk ≤ 1). There is a total of K decision variables. The decision variables (X) are related to the objectives (Z) as:

$$\left[ {\begin{array}{*{20}c} {Z_{1} } \\ {Z_{2 } } \\ {Z_{3} } \\ \end{array} } \right] = \mathop \sum \limits_{k} \left\{ {\left[ {\begin{array}{*{20}c} {R_{k } *\rho_{k} } \\ {S_{k} *\eta_{k} } \\ { - 1*d_{k} *c_{k} } \\ \end{array} } \right]*A_{k} *X_{k} } \right\}\forall k \in \left[ { 0,k} \right]$$
(1)

where R is the rainfall, S is the runoff, ρ is the runoff reduction efficiency of the BMP, η is the pollutant reduction efficiency, d is the depth of the BMP, and c is the construction cost of the BMP per unit volume. More details about case studies, modeling, and data requirements are available from [10].

2.2.2 Fuzzy Optimization and Membership Functions

Our problem in optimization is maximizing the objectives (Z). Lower ZL and upper limits ZU for goals are shown in Table 1.

Table 1 Lower and upper limits of the objective functions
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Each objective Z can be represented as a function of the decision variables (X), i.e., Zi = fi(X) (Eq. 1). In this work, we define a membership function denoted by µz(X) for each objective. We studied hyperbolic, exponential, and nonlinear membership functions (refer to Table 2).

Table 2 Types of membership functions and corresponding equations for \(Z_{L} < Z < Z_{U}\)
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For all three membership functions, \(Z \le Z_{L}\) is 0 and \(Z \ge Z_{U}\) is 1.

We notice that the hyperbolic membership function does not have any parameters that the decision-maker must choose, unlike nonlinear or exponential membership functions. S is a non-zero parameter 0 < S ≤ 1 [13]. β determines the shape of the membership function.

The fuzzy optimization problem (with N objectives) is as follows:

Maximize λ, subject to constraints:

  • µz(X) ≥ λ ∀i ∈ {1, 2,...N}

  • 0 ≤ λ ≤ 1

  • Z1 ≥ 3.5 × 106 m3 and Z2 ≥ 25 tons

along with other existing constraints and bounds.

2.2.3 Single Objective Genetic Algorithms

SOGA with a population size of 1000, simulated binary cross-over probability of 0.9 [14], polynomial mutation probability of 0.1 [15], and tournament selection are used for optimization. The PyMoo library [16] is employed for implementing the optimization functions.

3 Results and Discussions

The results of optimization with three membership functions are as follows. All source codes used to run these experiments have been open sourced under the MIT license and are available online.Footnote 1

3.1 Hyperbolic Membership Function

Performing the optimization procedure with hyperbolic membership function for all three objectives yielded a solution of satisfaction λ = 0.8796, corresponding to a real-world configuration of BMPs spanning 61.98 km2 of area, which reduce surface runoff by 1.02 × 107m3, while removing 73.87 tons of pollutant at a monetary cost of Rs. 1.16 × 1010. The progress of SOGA can be visualized by plotting the best-discovered value of λ against the number of function evaluations as depicted in Fig. 2.

Fig. 2
A map of lambda versus function evaluations for optimization convergence. It has an increasing trend from (0, 0.65) to (100000, 0.88). Values are approximated.

Optimization process with hyperbolic membership function for all three objectives

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3.2 Exponential Membership Function

Next, we present the results of the exponential membership function. Optimization procedure was run with 125 different configurations of the parameter s such that: s1, s2, s3 ∈ {0.2, 0.4, 0.6, 0.8, 1}.

Here, s1, s2, s3 are the parameters for runoff reduction, pollutant load reduction, and cost (Z3). Use of exponential membership function with these 125 different sets of parameters yielded solutions with λ ranging from 0.5479 to 0.6432. The average value of λ is 0.5950. Optimization convergence of all these 125 different sets of parameters can be visualized in Fig. 3. Each line in Fig. 3 represents a different set of parameters. It is noticed that all the lines follow similar trends, suggesting that the optimization process is not very sensitive to changes in parameters s1, s2, s3. We also notice that the value of λ begins to plateau for most lines after the 60th generation (or 60,000 function evaluations), indicating that the optimization approach has converged.

Fig. 3
A map of lambda versus function evaluations for optimization convergence exponential. It has multiple increasing curves between (0, 0.41) to (100000, 0.54) and (0, 0.525) to (100000, 0.64). Values are approximated.

Optimization process with exponential membership function for all three objectives and various values of the parameter

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3.3 Nonlinear Membership Function

Use of nonlinear membership function with 125 different sets of parameters yielded solutions with λ ranging from 0.1307 to 0.9601 with a moderate satisfiability of λ = 0.5454. The parameters of βi used were: β1, β2, β3 ∈ {0.1, 0.4, 1.0, 3.0, 5.0}.

Here, β1, β2, and β3 are the parameters for Z1, Z2, and Z3, respectively. Optimization convergence for nonlinear membership functions is plotted in Fig. 4, similar to the plots in previous sections. One key difference noticed is that solution is susceptible to changes in the parameter β. For example, using (β1, β2, β3) = (5, 5, 5) yields the least value of λ = 0.1307, while (β1, β2, β3) = (0.1, 0.1, 0.1) yields the highest value of λ = 0.9601. Values of βi will have to be decided based on the relative importance of each objective; that is, objectives that are relatively more important must have a larger β relative to others.

Fig. 4
A map of lambda versus function evaluations for optimization convergence non-linear. It has multiple increasing curves that move linearly between (0, 0.07) to (100000, 0.12) and (0, 0.98) to (100000, 0.98). Values are approximated.

Optimization process with nonlinear membership function with varying values of membership function parameter \(\beta\)

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4 Conclusions

A fuzzy optimization approach was applied to optimize BMPs in the GHMC. Experimentation was done with a wide range of parameters to analyze the sensitivity of each membership function with its parameters. It is observed that the nonlinear membership function is relatively more sensitive to changes in parameters when compared to the exponential membership function. Future work could include experimentation with more optimization algorithms, extending the analysis for potential future rainfall events, and applying this framework to other case studies.