Science of the Total Environment 515–516 (2015) 39–48
Contents lists available at ScienceDirect
Science of the Total Environment journal homepage: www.elsevier.com/locate/scitotenv
Reliability-oriented multi-objective optimal decision-making approach for uncertainty-based watershed load reduction Feifei Dong a, Yong Liu a,d,⁎, Han Su a, Rui Zou b,c, Huaicheng Guo a a
College of Environmental Science and Engineering, Key Laboratory of Water and Sediment Sciences (MOE), Peking University, Beijing 100871, China Tetra Tech, Inc., 10306 Eaton Place, Ste 340, Fairfax, VA 22030, USA Yunnan Key Laboratory of Pollution Process and Management of Plateau Lake-Watershed, Kunming 650034, China d Institute of Water Sciences, Peking University, Beijing 100871, China b c
H I G H L I G H T S
G R A P H I C A L
A B S T R A C T
• Reliability-oriented multi-objective (ROMO) optimal decision approach was proposed. • The approach can avoid specifying reliability levels prior to optimization modeling. • Multiple reliability objectives can be systematically balanced using Pareto fronts. • Neural network model was used to balance tradeoff between reliability and solutions.
a r t i c l e
i n f o
Article history: Received 7 December 2014 Received in revised form 5 February 2015 Accepted 7 February 2015 Available online xxxx Editor: Eddy Y. Zeng Keywords: Stochastic Multi-objective evolutionary algorithm Pareto fronts Tradeoff analysis Back propagation neural network
a b s t r a c t Water quality management and load reduction are subject to inherent uncertainties in watershed systems and competing decision objectives. Therefore, optimal decision-making modeling in watershed load reduction is suffering due to the following challenges: (a) it is difficult to obtain absolutely “optimal” solutions, and (b) decision schemes may be vulnerable to failure. The probability that solutions are feasible under uncertainties is defined as reliability. A reliability-oriented multi-objective (ROMO) decision-making approach was proposed in this study for optimal decision making with stochastic parameters and multiple decision reliability objectives. Lake Dianchi, one of the three most eutrophic lakes in China, was examined as a case study for optimal watershed nutrient load reduction to restore lake water quality. This study aimed to maximize reliability levels from considerations of cost and load reductions. The Pareto solutions of the ROMO optimization model were generated with the multiobjective evolutionary algorithm, demonstrating schemes representing different biases towards reliability. The Pareto fronts of six maximum allowable emission (MAE) scenarios were obtained, which indicated that decisions may be unreliable under unpractical load reduction requirements. A decision scheme identification process was conducted using the back propagation neural network (BPNN) method to provide a shortcut for identifying schemes at specific reliability levels for decision makers. The model results indicated that the ROMO approach can offer decision makers great insights into reliability tradeoffs and can thus help them to avoid ineffective decisions. © 2015 Elsevier B.V. All rights reserved.
1. Introduction ⁎ Corresponding author at: College of Environmental Science and Engineering, Key Laboratory of Water and Sediment Sciences (MOE), Peking University, Beijing 100871, China. E-mail address: [email protected] (Y. Liu).
http://dx.doi.org/10.1016/j.scitotenv.2015.02.024 0048-9697/© 2015 Elsevier B.V. All rights reserved.
The deterioration of surface water bodies has been a global concern since the 1980s, challenging environmental decision makers and stakeholders. Scientists have demonstrated that nutrient enrichment is one
40
F. Dong et al. / Science of the Total Environment 515–516 (2015) 39–48
of the main causes of lake and estuarine eutrophication (Seehausen et al., 1997; Huisman et al., 2005; Hautier et al., 2009; Shen et al., 2014). It is thereby necessary to implement nutrient load reduction programs to improve water quality and restore aquatic ecological health (Diaz and Rosenberg, 2008; Faulkner, 2008; Conley et al., 2009). Alternative strategies for load reduction should be organized effectively and efficiently at the watershed scale (Kramer et al., 2006), in which the optimization models can assist the decision-making process by proposing economically effective decision schemes and determining the optimal arrangements of alternative measures (Zou et al., 2010; Cao et al., 2010; Sadegh and Kerachian, 2011). The traditional deterministic optimal decision is based on the assumption of fully recognized systems and states. However, the decision-making process on nutrient load reduction is inevitably complicated by uncertainties, including (a) uncertain parameters, (b) competing and ambiguous decision objectives, and (c) deviations in judgment of stakeholders (Beck, 1987; Refsgaard et al., 2007; Tavakoli et al., 2014). Neglecting or incorrectly estimating uncertainties will result in ineffective and misleading decisions. Uncertainty analysis in environmental decision making has triggered much attention, and various methods have been developed to represent and handle uncertainty information in optimization models for reliable decisions (Zhang et al., 2009; Wang et al., 2012; Housh et al., 2013). In modeling, uncertain parameters are usually quantitatively represented as stochastic distributions (Wagner and Gorelick, 1987; He et al., 2006; Qin et al., 2007; Bastin et al., 2013). Stochastic mathematical programming (SMP) is a widely used method for optimal decision making in stochastic systems. It has been applied to handling probabilistic parameters for decades (Stedinger et al., 1984; Shea and Possingham, 2000; Santoso et al., 2005; Pena-Haro et al., 2011), among which chance constrained programming (CCP) developed by Charnes and Cooper (1959) is the most commonly used (Sniedovich and Davis, 1975; Xie et al., 2011; van Ackooij et al., 2014). For optimal decision making on water quality management, some strict constraints for implementing load reduction plans will always exist, such as limited public budget and total environmental capacity (TEC). In traditional deterministic optimization models, the restrictions of available resources of budget and environmental capacity are represented as constraint functions. In CCP models, the probability that the decision scheme is feasible with the limitation of resources is analyzed and defined as the decision reliability (Morgan et al., 1993). CCP models have shown great potential in assisting decision making on practical optimal water quality management under randomness (Huang, 1998; Wang et al., 2004; Gren, 2008). For CCP models, however, there are some limitations for its decision application, including (a) the reliability aspirations of the decision makers that must be prescribed, and (b) schemes at various reliability levels that require multiple runs of the optimal solution algorithm (He et al., 2008; Xie et al., 2011). There will be many difficulties in determining the reliability levels without available information on the relevant decision performances. Decisions should be made based on a tradeoff analysis between reliability levels and system performance, such as cost-risk tradeoffs for the arrangement of load reduction strategies. In previous studies, a tradeoff analysis was usually realized by testing a series of different reliability scenarios (Liu et al., 2008; Roy et al., 2010; Rasekh et al., 2010). For each scenario, the reliability levels of various constraints were assigned with a unified value, and the diversity of the reliability levels of the constraints is not considered. Decision makers may possess diverse reliability preferences towards different constraints. To depict the diversity among constraints, Dong et al. (2014) applied the Taguchi method to design multiple preference scenarios. However, all possible combinations of reliability levels cannot be enumerated exhaustively. The choice space of the decision makers is limited because only a few scenarios were provided in previous studies. Multiple reliability scenarios also require solving CCP models repeatedly, resulting in low decision efficiency.
To overcome the above limitations of CCP, a reliability-oriented multi-objective (ROMO) optimal decision-making approach was proposed in this study. The probability of achieving particular goals under a random environment (i.e., decision reliability) was established as an optimization objective. It was achieved in two steps through (a) ROMO optimization modeling and (b) scheme identification. Because various criteria may be considered to judge decision performance in practice, ROMO optimization models involve objectives for maximizing different aspects of decision reliability. In this study, the models were solved by the controlled elitist non-dominated sorting genetic algorithm (CE-NSGA-II), an advanced multi-objective evolutionary algorithm (MOEA). Scheme identification, which was achieved by artificial neural networks (ANNs), was implemented to reveal the quantitative relationship between decision reliability and implementation schemes. The ROMO approach provides a shortcut for accessing decision variables at expected reliability levels and contributes to deducing other alternative schemes in addition to a limited number of CE-NSGA-II solutions. Using the reliability-oriented approach developed in this study, all of the available schemes, reflecting diverse reliability preferences, can be assessed by one single run of the optimal solution algorithm. This reflects the intention of decision makers to maximize the feasibility of the decision and thus avoid predefinition of the aspirations for reliability levels. The ROMO approach was applied for decision making on watershed nutrient load reduction in Lake Dianchi, one of the three most eutrophic lakes in China, to provide effective support for decision making under uncertainties and two competing reliability objectives. 2. Materials and methods 2.1. Reliability-oriented optimization model A typical CCP model can be formulated as follows (Charnes and Cooper, 1963): Max f ðw; xÞ
ð1Þ
subject to: n o e j ; x ≤ b j ≥ β j ; j ¼ 1; 2; …; k ej w Pr g
ð2Þ
x∈D
ð3Þ
where x = (x1, x2, …, xn)T is an n-dimensional vector of decision variables; D is a deterministic convex set, restricting the feasible region of x. f(w, x) is the objective function, which is aimed to be maximized under the constraint formulated as Eqs. (2) and (3); w = (w1, w2 , …, wn ) is the deterministic parameter vector e j ; x is the constraint functions with stofor the objective function; e gj w ej¼ w e 1; j ; w e 2; j ; …; w e n; j is the stochastic coefficient chastic parameters; w vector for the jth constraint function; bj is the allowable maximum value n o e j ; x ≤ b j denotes the of the corresponding constraint function; Pr e gj w e j ; x ≤ b j inside the braces; probability of respecting the inequality e gj w and βj(0 ≤ βj ≤ 1) is the reliability level, which limits the minimum n o e j ; x ≤b j . In this paper, stochastic coefficients or value of Pr e gj w functions are labeled with tildes to indicate their involvement with uncertainties. CCP models are limited by prescribed reliability levels. Decision makers are required to determine the reliability levels βj before solving by CCP. It is a common practice to assign βj with a unified value R(0 ≤ R ≤ 1), and the constraints of Eq. (2) were then transformed into: n o e j ; x ≤ b j ≥ R; j ¼ 1; 2; …; k: ej w Pr g
ð4Þ
F. Dong et al. / Science of the Total Environment 515–516 (2015) 39–48
The limitations of CCP include the following: (a) diverse reliability aspirations towards different constraints are not considered, and (b) CCP models with multiple values of βj must be solved repeatedly. To address these problems, a linear ROMO optimization model to maximize decision reliability and improve the feasibility of the schemes was developed as follows in this study: n o c 1 ; x ; ef 2 e c2 ; x ; …; ef k e ck ; x Max e F ¼ ef 1 e
ð5Þ
subject to: x∈D
ð6Þ
where e F is the objective vector with stochastic parameters, including k obn o jectives ef e c ¼ ec ; e is c ; x ¼ Pr e c x ≤ u ; j ¼ 1; 2; …; k. e c ; …; e c j
j
j
j
j
1; j
2; j
k; j
the vector of uncertain coefficients of the jth objective function with specific probability distributions. Gaussian distribution is considered as an adequate assumption for many uncertain variables (Li et al., 2008). Inequality e c j x ≤ u j constrains e c j x to not exceed the threshold value uj; The stochastic functions ef e c ; x for maximizing the probability with rej
j
spect to e c j x ≤ u j are named as the reliability objective functions (ROFs). Definition of reliability levels ahead of decision making can be avoided by the introduction of ROFs instead of chance constraints in CCP. Assuming that the uncertain parameters e c j follow Gaussian distribu e e tions, the reliability objective functions f c ; x can be transferred into j
j
deterministic equivalent forms as below (Wets, 1974; Bes and Sethi, 1989; Vishnaykov and Kibzun, 2006): 8 9 > > : x0 V j x ; x0 V j x> 8 9 ! > > σj > σj : x0 V x ; j
ð7Þ c j and Vj is the covariance where c j denotes the expected value of e matrix of e c j . The expected values of objectives ef j ¼ e c j x can be denoted qffiffiffiffiffiffiffiffiffiffiffiffi 0 as f j ¼ c j x with standard deviation σ j ¼ x V j x. The symbol ϕ, which represents the cumulative distribution function of the standard normal distribution. The following equivalence relation can be deduced according to the progressive increase characteristic of ϕ: ! u j− f j max ef j x; e ; j ¼ 1; …; k: c j ⇔ max ϕ σj
ð8Þ
The linear ROMO optimization model formulated as Eqs. (5) and (6) can therefore be transformed into: ( Max F ¼
! ! !) u1 − f 1 u2 − f 2 uk − f k ;ϕ ; …; ϕ ϕ σ1 σ2 σk
ð9Þ
41
2.2. Multiple objective evolutionary algorithm The ROMO models are formulated as multiple objective programming (MOP) to simultaneously minimize or maximize two or more than two reliability objectives. None of the MOP solutions are absolutely optimal because competing objectives cannot reach their optimal values simultaneously. Therefore, the solutions of MOP are not unique. Near-optimal solutions are usually exploited and known as Pareto optimal solutions (Coello, 2011), which are dominated by none of the other solutions. This means that a solution which is superior to the Pareto optimal solutions with regard to each objective does not exist. For the ROMO model formulated as Eqs. (5) and (6), solution x1 is defined to dominate x2 if the following conditions are satisfied (Coello, 1999): e ef x ; e e i 1 c i ≥ f i x2 ; c i ; ∀i ∈ f1; 2; …; kg
ð11Þ
ef x ; e e e j 1 c j N f j x2 ; c j ; ∃ j ∈ f1; 2; …; kg:
ð12Þ
The solutions that are non-dominated by any other comprise the Pareto optimal front (Zitzler and Thiele, 1998). There are diverse methods for handling the conflicting objectives of MOP. The traditional method is to convert multiple objectives into a new objective, which is an overall fitness metric by a weighted summation of the original objectives (Coello, 1999). However, determination of the weights is difficult because it requires tradeoffs among objectives and negotiations among stakeholders. As a consequence, it may result in subjective and inefficient decision making. The multi-objective evolutionary algorithm (MOEA) is proven to be an effective alternative approach for solving MOP problems (Bekele and Nicklow, 2007; Chen et al., 2007; Zhang et al., 2010). MOEA is a population-based algorithm that analyzes a set of potential effective solutions instead of identifying the single best solution. Various MOEAs have been developed, such as the nondominated sorting genetic algorithm (NSGA) (Srinivas and Deb, 1995), non-dominated sorting genetic algorithm-II (NSGA-II) (Deb et al., 2000), multi-objective particle swarm approach (Reddy and Kumar, 2007), and strength Pareto evolutionary algorithm2 (SPEA2) (Kim et al., 2004). These algorithms are based on the idea of population evolution and the ranking and selection of non-dominated solutions. They can capture a number of Pareto optimal solutions simultaneously rather than just one. NSGA-II is the most popular MOEA as a modified version of NSGA, using a fast non-dominated sorting and elitism approach. The approach contributes to reducing the computation complexity and accelerates the capture of the Pareto front effectively. NSGA-II exhibits high performance and has been widely applied in various disciplines, i.e., water quality management (Fu et al., 2008; Singh and Chakrabarty, 2010; Rodriguez et al., 2011). To solve the ROMO model and generate the optimal Pareto front, an improved version of NSGA-II, named the controlled elitist nondominated sorting genetic algorithm (CE-NSGA-II) (Deb and Goel, 2001), was used in this study. In contrast to traditional NSGA-II (Deb et al., 2000), CE-NSGA-II is able to maintain diversity of the population by reserving individuals from many Pareto fronts during the iterative progress of the algorithm. As a variant of NSGA-II, CE-NSGA-II showed a better convergence property than the original NSGA-II with uncontrolled elitism (Deb and Goel, 2001). 2.3. Scheme identification
subject to: x ∈D
ð10Þ
where F is the objective vector including k reliability objectives ! u j −f j ϕ ; j ¼ 1; 2; …; k. σj
Scheme identification is a shortcut for detecting schemes with prescribed reliability levels. The process can be realized by identifying the quantitative relationship x = g(F) between the expected reliability objectives F and decision variables x by analyzing the MOEA solutions. Any of the other alternative solutions can be deduced based on the identified relationship, given specific reliability levels F'. To reduce the dimension of x, preprocessing is made to evaluate variability of decision variables x
42
F. Dong et al. / Science of the Total Environment 515–516 (2015) 39–48
among the MOEA solutions. Decision variables with low variability hold stable values and thus can be ignored in the relationship analysis process. Variability of each decision variable xi ∈ x is judged by the relative standard deviation (RSD): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N uX s 2 u xi −xi =ðN−1Þ u sdi t s¼1 ; i ¼ 1; 2…; k ¼ RSDi ¼ xi xi
ð13Þ
where sdi and xi are the standard deviation and the mean value of xis(s = 1, 2, …, N) respectively; N is the sample number. The ANNs are powerful techniques for modeling complex multivariate nonlinear relationships, among which the back propagation neural network (BPNN) is the most commonly applied. BPNN is comprised of three parts: (a) an input layer where the data are input into the network, (b) a hidden layer where data are intermediately processed, and (c) an output layer where the results are generated (Sahoo Goloka et al., 2006). Training of the BPNN involves two steps: (a) a feedforward process, where input data at the input layer is propagated forward to compute the results at the output layer, and (b) a backward process, where connection weights of layers are adjusted based on the differences between the computed and actual results at the output layer. 2.4. General procedure A ROMO decision-making procedure was proposed to handle uncertainty and multiple decision reliability objectives for watershed nutrient load reduction. It consists of the following steps (Fig. 1): Step 1: Model formulation. A ROMO optimization model with uncertain objectives to maximize multiple decision reliability was formulated as Eqs. (5) and (6). It is based on basic information on the watershed studied, such as watershed geographic information, MAE and alternative pollution control measures. Uncertain parameters can be demonstrated with probability distributions. Step 2: Model solution. Objectives with uncertain parameters of Gaussian distributions are transferred into their deterministic equivalent forms based on Eqs. (7) and (8). The converted ROMO model with substituted deterministic equivalent forms is solved by the
MOEA. A series of Pareto optimal solutions representing nearoptimal decision schemes are generated. Step 3: Reliability tradeoffs. The Pareto front is delineated based on the MOEA solutions, which reveals tradeoffs among reliability levels. Different areas of the Pareto front demonstrate distinctions in reliability preferences. Decision makers can select a preferable scenario as the implementation scheme from solutions along the Pareto front, and they can proceed to the next step if none of the output solutions are satisfactory and some adjustments are required. Step 4: Scheme identification. After preprocessing and filtration of decision variables with Eq. (13), the relationship between reliability levels and decision variables is analyzed by training the BPNN. The detected quantitative relationship is presented to decision makers as a shortcut for identifying schemes under specific reliability objectives, which is helpful for the adjustment of expected reliability levels and the determination of the final implementation scheme. 3. Case study 3.1. Study area Lake Dianchi is the sixth-largest freshwater lake in China and one of its most eutrophic. It is located in Yunnan province at E 102°29′– 103°01′ and N 24°29′–25°28′, with a watershed area of 2920 km2, a capacity of 9.92 × 108 m3 and a normal average water depth of 5.2 m (Fig. 2). Lake Dianchi is in the high priority decision agenda of the central and local governments for water quality restoration because it is now threatened by eutrophication, which is induced by nutrient loading from point and nonpoint sources (Liu et al., 2004; Kunming Municipal Government, 2011). An optimal plan was conducted for effective arrangement of measures on nutrient load reduction for the restoration of water quality in Lake Dianchi. Eight sub-watersheds were divided (C1 and W1, W2,…,W7) (Fig. 2) according to the hydrology and nutrient transport processes (Liu et al., 2012). The entire nutrient load in subwatershed C1 will be discharged into Songhua Dam, the drinking water source for sub-watersheds W1, W2 and W3; therefore, there is no direct nutrient loading from C1 to Lake Dianchi. Thus, sub-watershed C1 was excluded from the nutrient load reduction decision in this study. Previous studies have shown that phosphorus (P) is the limiting nutrient for eutrophication in Lake Dianchi (Liu et al., 2006), so the reduction of the total
ROMO Optimization Modeling
Scheme Identification
ROMO model formulation
Variability analysis of DVs Select a MOEA solution˛
Deterministic conversion of reliability objectives
No
Relationship detection between RLs and DVs
Yes
Solving ROMO models with MOEA
Reliability tradeoffs
Determine expected RLs Scheme Determination Detect DVs under expected RLs
Fig. 1. The ROMO decision procedure. Notes: DVs represent decision variables; RLs represent reliability levels.
F. Dong et al. / Science of the Total Environment 515–516 (2015) 39–48
43
Fig. 2. Location of the Lake Dianchi watershed and eight sub-watersheds.
phosphorus (TP) load was considered in the optimal model. The basic geological, load and water quality data were drawn from the Twelfth Five-year Plan for Water Pollution Control in Lake Dianchi (12·5 PWPCDC), which was compiled by the Kunming Municipal Government and approved by the China State Council (Kunming Municipal Government, 2011). In the 12·5 PWPC-DC, alternative categories of load reduction strategies and unit costs were proposed for sub-watersheds W1 to W7, but without optimal arrangements. Proper arrangements of the strategies were then explored in this study based on a stochastic programming model with two reliability-oriented objectives.
3.2. ROMO model development for Lake Dianchi watershed A ROMO optimization model was developed for supporting the arrangement of nutrient load reduction strategies in the Lake Dianchi watershed. The model is formulated to obtain reliable schemes for achieving the environmental target under a strict public budget. Various uncertainties of the system may result in lower-than-expected load reduction performance or extra cost. There was an assumption that costs and phosphorus load abatement increased with control strategies linearly. In this model, the unit capital cost and TP abatement of each
Fig. 3. Pareto fronts of six MAE scenarios.
44
F. Dong et al. / Science of the Total Environment 515–516 (2015) 39–48
distributions. The distributions of IICj and APRj are demonstrated in the Supporting materials. ai,j is the suitability factor; ai,j = 0 indicates that strategy j is unsuitable in sub-watershed i, whereas ai,j = 1 indicates that strategy j is suitable in sub-watershed i. Some constraints were taken into consideration for decision making, including government restrictions for farmland area, forest area and treatment rate of domestic wastewater. Lake riparian vegetation buffer area is also constrained in the optimization model because it is effective to intercept pollutants before they enter the lake, and the local government will make great efforts to restore the vegetation buffers. Therefore, the optimization model is subject to the following constraints: Farmland constraint: I X
X i;3 ai;3 ≤ RML TLAQ
ð16Þ
i¼1
Fig. 4. Extreme and compromise solutions in scenario 4.
strategy were regarded as uncertain factors that interfere with decision reliability. The objectives of the model were thus set to (a) maximize the probability that actual spending is under the government budget and (b) maximize the probability that TP load discharge will not exceed the maximum allowable emissions (MAEs). The objective functions regarding cost and load reliability were formulated as below: Objective 1: Maximize cost reliability 9 8 J =
Contents lists available at ScienceDirect
Science of the Total Environment journal homepage: www.elsevier.com/locate/scitotenv
Reliability-oriented multi-objective optimal decision-making approach for uncertainty-based watershed load reduction Feifei Dong a, Yong Liu a,d,⁎, Han Su a, Rui Zou b,c, Huaicheng Guo a a
College of Environmental Science and Engineering, Key Laboratory of Water and Sediment Sciences (MOE), Peking University, Beijing 100871, China Tetra Tech, Inc., 10306 Eaton Place, Ste 340, Fairfax, VA 22030, USA Yunnan Key Laboratory of Pollution Process and Management of Plateau Lake-Watershed, Kunming 650034, China d Institute of Water Sciences, Peking University, Beijing 100871, China b c
H I G H L I G H T S
G R A P H I C A L
A B S T R A C T
• Reliability-oriented multi-objective (ROMO) optimal decision approach was proposed. • The approach can avoid specifying reliability levels prior to optimization modeling. • Multiple reliability objectives can be systematically balanced using Pareto fronts. • Neural network model was used to balance tradeoff between reliability and solutions.
a r t i c l e
i n f o
Article history: Received 7 December 2014 Received in revised form 5 February 2015 Accepted 7 February 2015 Available online xxxx Editor: Eddy Y. Zeng Keywords: Stochastic Multi-objective evolutionary algorithm Pareto fronts Tradeoff analysis Back propagation neural network
a b s t r a c t Water quality management and load reduction are subject to inherent uncertainties in watershed systems and competing decision objectives. Therefore, optimal decision-making modeling in watershed load reduction is suffering due to the following challenges: (a) it is difficult to obtain absolutely “optimal” solutions, and (b) decision schemes may be vulnerable to failure. The probability that solutions are feasible under uncertainties is defined as reliability. A reliability-oriented multi-objective (ROMO) decision-making approach was proposed in this study for optimal decision making with stochastic parameters and multiple decision reliability objectives. Lake Dianchi, one of the three most eutrophic lakes in China, was examined as a case study for optimal watershed nutrient load reduction to restore lake water quality. This study aimed to maximize reliability levels from considerations of cost and load reductions. The Pareto solutions of the ROMO optimization model were generated with the multiobjective evolutionary algorithm, demonstrating schemes representing different biases towards reliability. The Pareto fronts of six maximum allowable emission (MAE) scenarios were obtained, which indicated that decisions may be unreliable under unpractical load reduction requirements. A decision scheme identification process was conducted using the back propagation neural network (BPNN) method to provide a shortcut for identifying schemes at specific reliability levels for decision makers. The model results indicated that the ROMO approach can offer decision makers great insights into reliability tradeoffs and can thus help them to avoid ineffective decisions. © 2015 Elsevier B.V. All rights reserved.
1. Introduction ⁎ Corresponding author at: College of Environmental Science and Engineering, Key Laboratory of Water and Sediment Sciences (MOE), Peking University, Beijing 100871, China. E-mail address: [email protected] (Y. Liu).
http://dx.doi.org/10.1016/j.scitotenv.2015.02.024 0048-9697/© 2015 Elsevier B.V. All rights reserved.
The deterioration of surface water bodies has been a global concern since the 1980s, challenging environmental decision makers and stakeholders. Scientists have demonstrated that nutrient enrichment is one
40
F. Dong et al. / Science of the Total Environment 515–516 (2015) 39–48
of the main causes of lake and estuarine eutrophication (Seehausen et al., 1997; Huisman et al., 2005; Hautier et al., 2009; Shen et al., 2014). It is thereby necessary to implement nutrient load reduction programs to improve water quality and restore aquatic ecological health (Diaz and Rosenberg, 2008; Faulkner, 2008; Conley et al., 2009). Alternative strategies for load reduction should be organized effectively and efficiently at the watershed scale (Kramer et al., 2006), in which the optimization models can assist the decision-making process by proposing economically effective decision schemes and determining the optimal arrangements of alternative measures (Zou et al., 2010; Cao et al., 2010; Sadegh and Kerachian, 2011). The traditional deterministic optimal decision is based on the assumption of fully recognized systems and states. However, the decision-making process on nutrient load reduction is inevitably complicated by uncertainties, including (a) uncertain parameters, (b) competing and ambiguous decision objectives, and (c) deviations in judgment of stakeholders (Beck, 1987; Refsgaard et al., 2007; Tavakoli et al., 2014). Neglecting or incorrectly estimating uncertainties will result in ineffective and misleading decisions. Uncertainty analysis in environmental decision making has triggered much attention, and various methods have been developed to represent and handle uncertainty information in optimization models for reliable decisions (Zhang et al., 2009; Wang et al., 2012; Housh et al., 2013). In modeling, uncertain parameters are usually quantitatively represented as stochastic distributions (Wagner and Gorelick, 1987; He et al., 2006; Qin et al., 2007; Bastin et al., 2013). Stochastic mathematical programming (SMP) is a widely used method for optimal decision making in stochastic systems. It has been applied to handling probabilistic parameters for decades (Stedinger et al., 1984; Shea and Possingham, 2000; Santoso et al., 2005; Pena-Haro et al., 2011), among which chance constrained programming (CCP) developed by Charnes and Cooper (1959) is the most commonly used (Sniedovich and Davis, 1975; Xie et al., 2011; van Ackooij et al., 2014). For optimal decision making on water quality management, some strict constraints for implementing load reduction plans will always exist, such as limited public budget and total environmental capacity (TEC). In traditional deterministic optimization models, the restrictions of available resources of budget and environmental capacity are represented as constraint functions. In CCP models, the probability that the decision scheme is feasible with the limitation of resources is analyzed and defined as the decision reliability (Morgan et al., 1993). CCP models have shown great potential in assisting decision making on practical optimal water quality management under randomness (Huang, 1998; Wang et al., 2004; Gren, 2008). For CCP models, however, there are some limitations for its decision application, including (a) the reliability aspirations of the decision makers that must be prescribed, and (b) schemes at various reliability levels that require multiple runs of the optimal solution algorithm (He et al., 2008; Xie et al., 2011). There will be many difficulties in determining the reliability levels without available information on the relevant decision performances. Decisions should be made based on a tradeoff analysis between reliability levels and system performance, such as cost-risk tradeoffs for the arrangement of load reduction strategies. In previous studies, a tradeoff analysis was usually realized by testing a series of different reliability scenarios (Liu et al., 2008; Roy et al., 2010; Rasekh et al., 2010). For each scenario, the reliability levels of various constraints were assigned with a unified value, and the diversity of the reliability levels of the constraints is not considered. Decision makers may possess diverse reliability preferences towards different constraints. To depict the diversity among constraints, Dong et al. (2014) applied the Taguchi method to design multiple preference scenarios. However, all possible combinations of reliability levels cannot be enumerated exhaustively. The choice space of the decision makers is limited because only a few scenarios were provided in previous studies. Multiple reliability scenarios also require solving CCP models repeatedly, resulting in low decision efficiency.
To overcome the above limitations of CCP, a reliability-oriented multi-objective (ROMO) optimal decision-making approach was proposed in this study. The probability of achieving particular goals under a random environment (i.e., decision reliability) was established as an optimization objective. It was achieved in two steps through (a) ROMO optimization modeling and (b) scheme identification. Because various criteria may be considered to judge decision performance in practice, ROMO optimization models involve objectives for maximizing different aspects of decision reliability. In this study, the models were solved by the controlled elitist non-dominated sorting genetic algorithm (CE-NSGA-II), an advanced multi-objective evolutionary algorithm (MOEA). Scheme identification, which was achieved by artificial neural networks (ANNs), was implemented to reveal the quantitative relationship between decision reliability and implementation schemes. The ROMO approach provides a shortcut for accessing decision variables at expected reliability levels and contributes to deducing other alternative schemes in addition to a limited number of CE-NSGA-II solutions. Using the reliability-oriented approach developed in this study, all of the available schemes, reflecting diverse reliability preferences, can be assessed by one single run of the optimal solution algorithm. This reflects the intention of decision makers to maximize the feasibility of the decision and thus avoid predefinition of the aspirations for reliability levels. The ROMO approach was applied for decision making on watershed nutrient load reduction in Lake Dianchi, one of the three most eutrophic lakes in China, to provide effective support for decision making under uncertainties and two competing reliability objectives. 2. Materials and methods 2.1. Reliability-oriented optimization model A typical CCP model can be formulated as follows (Charnes and Cooper, 1963): Max f ðw; xÞ
ð1Þ
subject to: n o e j ; x ≤ b j ≥ β j ; j ¼ 1; 2; …; k ej w Pr g
ð2Þ
x∈D
ð3Þ
where x = (x1, x2, …, xn)T is an n-dimensional vector of decision variables; D is a deterministic convex set, restricting the feasible region of x. f(w, x) is the objective function, which is aimed to be maximized under the constraint formulated as Eqs. (2) and (3); w = (w1, w2 , …, wn ) is the deterministic parameter vector e j ; x is the constraint functions with stofor the objective function; e gj w ej¼ w e 1; j ; w e 2; j ; …; w e n; j is the stochastic coefficient chastic parameters; w vector for the jth constraint function; bj is the allowable maximum value n o e j ; x ≤ b j denotes the of the corresponding constraint function; Pr e gj w e j ; x ≤ b j inside the braces; probability of respecting the inequality e gj w and βj(0 ≤ βj ≤ 1) is the reliability level, which limits the minimum n o e j ; x ≤b j . In this paper, stochastic coefficients or value of Pr e gj w functions are labeled with tildes to indicate their involvement with uncertainties. CCP models are limited by prescribed reliability levels. Decision makers are required to determine the reliability levels βj before solving by CCP. It is a common practice to assign βj with a unified value R(0 ≤ R ≤ 1), and the constraints of Eq. (2) were then transformed into: n o e j ; x ≤ b j ≥ R; j ¼ 1; 2; …; k: ej w Pr g
ð4Þ
F. Dong et al. / Science of the Total Environment 515–516 (2015) 39–48
The limitations of CCP include the following: (a) diverse reliability aspirations towards different constraints are not considered, and (b) CCP models with multiple values of βj must be solved repeatedly. To address these problems, a linear ROMO optimization model to maximize decision reliability and improve the feasibility of the schemes was developed as follows in this study: n o c 1 ; x ; ef 2 e c2 ; x ; …; ef k e ck ; x Max e F ¼ ef 1 e
ð5Þ
subject to: x∈D
ð6Þ
where e F is the objective vector with stochastic parameters, including k obn o jectives ef e c ¼ ec ; e is c ; x ¼ Pr e c x ≤ u ; j ¼ 1; 2; …; k. e c ; …; e c j
j
j
j
j
1; j
2; j
k; j
the vector of uncertain coefficients of the jth objective function with specific probability distributions. Gaussian distribution is considered as an adequate assumption for many uncertain variables (Li et al., 2008). Inequality e c j x ≤ u j constrains e c j x to not exceed the threshold value uj; The stochastic functions ef e c ; x for maximizing the probability with rej
j
spect to e c j x ≤ u j are named as the reliability objective functions (ROFs). Definition of reliability levels ahead of decision making can be avoided by the introduction of ROFs instead of chance constraints in CCP. Assuming that the uncertain parameters e c j follow Gaussian distribu e e tions, the reliability objective functions f c ; x can be transferred into j
j
deterministic equivalent forms as below (Wets, 1974; Bes and Sethi, 1989; Vishnaykov and Kibzun, 2006): 8 9 > > : x0 V j x ; x0 V j x> 8 9 ! > > σj > σj : x0 V x ; j
ð7Þ c j and Vj is the covariance where c j denotes the expected value of e matrix of e c j . The expected values of objectives ef j ¼ e c j x can be denoted qffiffiffiffiffiffiffiffiffiffiffiffi 0 as f j ¼ c j x with standard deviation σ j ¼ x V j x. The symbol ϕ, which represents the cumulative distribution function of the standard normal distribution. The following equivalence relation can be deduced according to the progressive increase characteristic of ϕ: ! u j− f j max ef j x; e ; j ¼ 1; …; k: c j ⇔ max ϕ σj
ð8Þ
The linear ROMO optimization model formulated as Eqs. (5) and (6) can therefore be transformed into: ( Max F ¼
! ! !) u1 − f 1 u2 − f 2 uk − f k ;ϕ ; …; ϕ ϕ σ1 σ2 σk
ð9Þ
41
2.2. Multiple objective evolutionary algorithm The ROMO models are formulated as multiple objective programming (MOP) to simultaneously minimize or maximize two or more than two reliability objectives. None of the MOP solutions are absolutely optimal because competing objectives cannot reach their optimal values simultaneously. Therefore, the solutions of MOP are not unique. Near-optimal solutions are usually exploited and known as Pareto optimal solutions (Coello, 2011), which are dominated by none of the other solutions. This means that a solution which is superior to the Pareto optimal solutions with regard to each objective does not exist. For the ROMO model formulated as Eqs. (5) and (6), solution x1 is defined to dominate x2 if the following conditions are satisfied (Coello, 1999): e ef x ; e e i 1 c i ≥ f i x2 ; c i ; ∀i ∈ f1; 2; …; kg
ð11Þ
ef x ; e e e j 1 c j N f j x2 ; c j ; ∃ j ∈ f1; 2; …; kg:
ð12Þ
The solutions that are non-dominated by any other comprise the Pareto optimal front (Zitzler and Thiele, 1998). There are diverse methods for handling the conflicting objectives of MOP. The traditional method is to convert multiple objectives into a new objective, which is an overall fitness metric by a weighted summation of the original objectives (Coello, 1999). However, determination of the weights is difficult because it requires tradeoffs among objectives and negotiations among stakeholders. As a consequence, it may result in subjective and inefficient decision making. The multi-objective evolutionary algorithm (MOEA) is proven to be an effective alternative approach for solving MOP problems (Bekele and Nicklow, 2007; Chen et al., 2007; Zhang et al., 2010). MOEA is a population-based algorithm that analyzes a set of potential effective solutions instead of identifying the single best solution. Various MOEAs have been developed, such as the nondominated sorting genetic algorithm (NSGA) (Srinivas and Deb, 1995), non-dominated sorting genetic algorithm-II (NSGA-II) (Deb et al., 2000), multi-objective particle swarm approach (Reddy and Kumar, 2007), and strength Pareto evolutionary algorithm2 (SPEA2) (Kim et al., 2004). These algorithms are based on the idea of population evolution and the ranking and selection of non-dominated solutions. They can capture a number of Pareto optimal solutions simultaneously rather than just one. NSGA-II is the most popular MOEA as a modified version of NSGA, using a fast non-dominated sorting and elitism approach. The approach contributes to reducing the computation complexity and accelerates the capture of the Pareto front effectively. NSGA-II exhibits high performance and has been widely applied in various disciplines, i.e., water quality management (Fu et al., 2008; Singh and Chakrabarty, 2010; Rodriguez et al., 2011). To solve the ROMO model and generate the optimal Pareto front, an improved version of NSGA-II, named the controlled elitist nondominated sorting genetic algorithm (CE-NSGA-II) (Deb and Goel, 2001), was used in this study. In contrast to traditional NSGA-II (Deb et al., 2000), CE-NSGA-II is able to maintain diversity of the population by reserving individuals from many Pareto fronts during the iterative progress of the algorithm. As a variant of NSGA-II, CE-NSGA-II showed a better convergence property than the original NSGA-II with uncontrolled elitism (Deb and Goel, 2001). 2.3. Scheme identification
subject to: x ∈D
ð10Þ
where F is the objective vector including k reliability objectives ! u j −f j ϕ ; j ¼ 1; 2; …; k. σj
Scheme identification is a shortcut for detecting schemes with prescribed reliability levels. The process can be realized by identifying the quantitative relationship x = g(F) between the expected reliability objectives F and decision variables x by analyzing the MOEA solutions. Any of the other alternative solutions can be deduced based on the identified relationship, given specific reliability levels F'. To reduce the dimension of x, preprocessing is made to evaluate variability of decision variables x
42
F. Dong et al. / Science of the Total Environment 515–516 (2015) 39–48
among the MOEA solutions. Decision variables with low variability hold stable values and thus can be ignored in the relationship analysis process. Variability of each decision variable xi ∈ x is judged by the relative standard deviation (RSD): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N uX s 2 u xi −xi =ðN−1Þ u sdi t s¼1 ; i ¼ 1; 2…; k ¼ RSDi ¼ xi xi
ð13Þ
where sdi and xi are the standard deviation and the mean value of xis(s = 1, 2, …, N) respectively; N is the sample number. The ANNs are powerful techniques for modeling complex multivariate nonlinear relationships, among which the back propagation neural network (BPNN) is the most commonly applied. BPNN is comprised of three parts: (a) an input layer where the data are input into the network, (b) a hidden layer where data are intermediately processed, and (c) an output layer where the results are generated (Sahoo Goloka et al., 2006). Training of the BPNN involves two steps: (a) a feedforward process, where input data at the input layer is propagated forward to compute the results at the output layer, and (b) a backward process, where connection weights of layers are adjusted based on the differences between the computed and actual results at the output layer. 2.4. General procedure A ROMO decision-making procedure was proposed to handle uncertainty and multiple decision reliability objectives for watershed nutrient load reduction. It consists of the following steps (Fig. 1): Step 1: Model formulation. A ROMO optimization model with uncertain objectives to maximize multiple decision reliability was formulated as Eqs. (5) and (6). It is based on basic information on the watershed studied, such as watershed geographic information, MAE and alternative pollution control measures. Uncertain parameters can be demonstrated with probability distributions. Step 2: Model solution. Objectives with uncertain parameters of Gaussian distributions are transferred into their deterministic equivalent forms based on Eqs. (7) and (8). The converted ROMO model with substituted deterministic equivalent forms is solved by the
MOEA. A series of Pareto optimal solutions representing nearoptimal decision schemes are generated. Step 3: Reliability tradeoffs. The Pareto front is delineated based on the MOEA solutions, which reveals tradeoffs among reliability levels. Different areas of the Pareto front demonstrate distinctions in reliability preferences. Decision makers can select a preferable scenario as the implementation scheme from solutions along the Pareto front, and they can proceed to the next step if none of the output solutions are satisfactory and some adjustments are required. Step 4: Scheme identification. After preprocessing and filtration of decision variables with Eq. (13), the relationship between reliability levels and decision variables is analyzed by training the BPNN. The detected quantitative relationship is presented to decision makers as a shortcut for identifying schemes under specific reliability objectives, which is helpful for the adjustment of expected reliability levels and the determination of the final implementation scheme. 3. Case study 3.1. Study area Lake Dianchi is the sixth-largest freshwater lake in China and one of its most eutrophic. It is located in Yunnan province at E 102°29′– 103°01′ and N 24°29′–25°28′, with a watershed area of 2920 km2, a capacity of 9.92 × 108 m3 and a normal average water depth of 5.2 m (Fig. 2). Lake Dianchi is in the high priority decision agenda of the central and local governments for water quality restoration because it is now threatened by eutrophication, which is induced by nutrient loading from point and nonpoint sources (Liu et al., 2004; Kunming Municipal Government, 2011). An optimal plan was conducted for effective arrangement of measures on nutrient load reduction for the restoration of water quality in Lake Dianchi. Eight sub-watersheds were divided (C1 and W1, W2,…,W7) (Fig. 2) according to the hydrology and nutrient transport processes (Liu et al., 2012). The entire nutrient load in subwatershed C1 will be discharged into Songhua Dam, the drinking water source for sub-watersheds W1, W2 and W3; therefore, there is no direct nutrient loading from C1 to Lake Dianchi. Thus, sub-watershed C1 was excluded from the nutrient load reduction decision in this study. Previous studies have shown that phosphorus (P) is the limiting nutrient for eutrophication in Lake Dianchi (Liu et al., 2006), so the reduction of the total
ROMO Optimization Modeling
Scheme Identification
ROMO model formulation
Variability analysis of DVs Select a MOEA solution˛
Deterministic conversion of reliability objectives
No
Relationship detection between RLs and DVs
Yes
Solving ROMO models with MOEA
Reliability tradeoffs
Determine expected RLs Scheme Determination Detect DVs under expected RLs
Fig. 1. The ROMO decision procedure. Notes: DVs represent decision variables; RLs represent reliability levels.
F. Dong et al. / Science of the Total Environment 515–516 (2015) 39–48
43
Fig. 2. Location of the Lake Dianchi watershed and eight sub-watersheds.
phosphorus (TP) load was considered in the optimal model. The basic geological, load and water quality data were drawn from the Twelfth Five-year Plan for Water Pollution Control in Lake Dianchi (12·5 PWPCDC), which was compiled by the Kunming Municipal Government and approved by the China State Council (Kunming Municipal Government, 2011). In the 12·5 PWPC-DC, alternative categories of load reduction strategies and unit costs were proposed for sub-watersheds W1 to W7, but without optimal arrangements. Proper arrangements of the strategies were then explored in this study based on a stochastic programming model with two reliability-oriented objectives.
3.2. ROMO model development for Lake Dianchi watershed A ROMO optimization model was developed for supporting the arrangement of nutrient load reduction strategies in the Lake Dianchi watershed. The model is formulated to obtain reliable schemes for achieving the environmental target under a strict public budget. Various uncertainties of the system may result in lower-than-expected load reduction performance or extra cost. There was an assumption that costs and phosphorus load abatement increased with control strategies linearly. In this model, the unit capital cost and TP abatement of each
Fig. 3. Pareto fronts of six MAE scenarios.
44
F. Dong et al. / Science of the Total Environment 515–516 (2015) 39–48
distributions. The distributions of IICj and APRj are demonstrated in the Supporting materials. ai,j is the suitability factor; ai,j = 0 indicates that strategy j is unsuitable in sub-watershed i, whereas ai,j = 1 indicates that strategy j is suitable in sub-watershed i. Some constraints were taken into consideration for decision making, including government restrictions for farmland area, forest area and treatment rate of domestic wastewater. Lake riparian vegetation buffer area is also constrained in the optimization model because it is effective to intercept pollutants before they enter the lake, and the local government will make great efforts to restore the vegetation buffers. Therefore, the optimization model is subject to the following constraints: Farmland constraint: I X
X i;3 ai;3 ≤ RML TLAQ
ð16Þ
i¼1
Fig. 4. Extreme and compromise solutions in scenario 4.
strategy were regarded as uncertain factors that interfere with decision reliability. The objectives of the model were thus set to (a) maximize the probability that actual spending is under the government budget and (b) maximize the probability that TP load discharge will not exceed the maximum allowable emissions (MAEs). The objective functions regarding cost and load reliability were formulated as below: Objective 1: Maximize cost reliability 9 8 J =
