Guided adaptive optimal decision making approach for uncertainty based watershed scale load reduction

w a t e r r e s e a r c h 4 5 ( 2 0 1 1 ) 4 8 8 5 e4 8 9 5

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Guided adaptive optimal decision making approach for uncertainty based watershed scale load reduction Yong Liu a, Rui Zou b,*, John Riverson b, Pingjian Yang a, Huaicheng Guo a a

College of Environmental Science and Engineering, Peking University, The Key Laboratory of Water and Sediment Sciences, Ministry of Education, Beijing 100871, China b Tetra Tech, Inc. 10306 Eaton Place, Ste. 340, Fairfax, VA 22030, USA

article info

abstract

Article history:

Previous optimization-based watershed decision making approaches suffer two major

Received 12 January 2011

limitations. First of all, these approaches generally do not provide a systematic way to

Received in revised form

prioritize the implementation schemes with consideration of uncertainties in the water-

29 April 2011

shed systems and the optimization models. Furthermore, with adaptive management, both

Accepted 26 June 2011

the decision environment and the uncertainty space evolve (1) during the implementation

Available online 3 July 2011

processes and (2) as new data become available. No efficient method exists to guide optimal adaptive decision making, particularly at a watershed scale. This paper presents

Keywords:

a guided adaptive optimal (GAO) decision making approach to overcome the limitations of

Adaptive management

the previous methods for more efficient and reliable decision making at the watershed

Guided adaptive optimal approach

scale. The GAO approach is built upon a modeling framework that explicitly addresses

Risk explicit interval

system optimality and uncertainty in a time variable manner, hence mimicking the real-

linear programming

world decision environment where information availability and uncertainty evolve with

Load reduction

time. The GAO approach consists of multiple components, including the risk explicit

Lake Qionghai Watershed

interval linear programming (REILP) modeling framework, the systematic method for prioritizing implementation schemes, and an iterative process for adapting the core optimization model for updated optimal solutions. The proposed approach was illustrated through a case study dealing with the uncertainty based optimal adaptive environmental management of the Lake Qionghai Watershed in China. The results demonstrated that the proposed GAO approach is able to (1) efficiently incorporate uncertainty into the formulation and solution of the optimization model, and (2) prioritize implementation schemes based on the risk and return tradeoff. As a result the GAO produces more reliable and efficient management outcomes than traditional non-adaptive optimization approaches. ª 2011 Elsevier Ltd. All rights reserved.

1.

Introduction

The need for load reduction to restore aquatic ecosystem health has been well recognized over the past decades (National Research Council, 2001; USEPA, 2001; Diaz and Rosenberg, 2008), and it has been considered necessary to

identify optimal nutrient management strategies both environmentally sound and economically effective (DePinto et al., 2004; Liu et al., 2008). Uncertainty due to data limitation and the stochastic nature of environmental systems presents a difficult challenge for developing and implementing optimal load

* Corresponding author. E-mail address: [email protected] (R. Zou). 0043-1354/$ e see front matter ª 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.watres.2011.06.038

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reduction management schemes (National Research Council, 2001). Various methods have been developed to account for the uncertainty in environmental optimization models (Chang et al., 1997; Freedman and Nemura, 2004; Fiedler et al., 2006; Bazargan, 2007). Although many of these methods appeared to be able to effectively handle uncertainties in the solution process, they all suffer the limitation that the model formulation and optimal solutions are assumed to be essentially static; however, implementation of management schemes is a dynamic process that may influence both the model formulation and the optimal state of a solution. Even though some optimization models such as multi-stage linear programming and dynamic programming appear to be dynamic in their formulation, they are essentially static in their predictive ability because the model formulations (and therefore, the resulting space of possible optimal solutions) remain the same with each successive solution iteration toward an optimal solution. The contradiction between the static nature of decision support models and the dynamic nature of the implementation processes can be illustrated using a hypothetic watershed management problem. When solving a comprehensive optimization model in a watershed that consists of 500 decision variables, the solution leads to optimal/near-optimal solutions for all the 500 decision variables. In other words, there are up to 500 management measures that need be implemented in order to achieve the environmental target. Implementation of these management measures, however, is a dynamic process where some of the management measures would be implemented before others. Furthermore, implementing each management measure would need a certain amount of time and financial resources, and many of these measures cannot be implemented simultaneously. In practice, it is highly possible that after implementing some of the management measures, new information about cost and environmental responses will become available, and might show that certain key coefficients in the original optimization model were inaccurate, making some of the original optimal solutions invalid. In such a case, if decision makers continue to implement management schemes based on the original optimal solutions and assumptions, the culminating management strategy would become ineffective and non-optimal. A preferred alternative to the traditional decision making approach is the adaptive management approach, which provides an effective way to cope with the uncertainties in environmental decision making (Walters, 2007; Harrison, 2007b). Adaptive management was proposed in 1970s as a “learning by doing” process to reduce uncertainty in decision making and correct decision error early in the process (Holling, 1978; Walters, 1986). Adaptive management is a continuous learning process in which decision-making is supported by models while management alternatives are modified when new information becomes available (Marttunen and Vehanen, 2004; Gregory et al., 2006b). Both the traditional environmental decision making and the adaptive management approaches recognize uncertainty. However, traditional methods try to use existing knowledge about the system uncertainty to formulate the best future course of action; while adaptive management explicitly evolves the decision making in the implementation processes by refining

management alternatives over time (McLain and Lee, 1996; Gregory et al., 2006a; Walters, 2007). More importantly, adaptive management is an incremental approach in which each proposed decision is viewed as an experiment in which the corresponding outcomes will be monitored, evaluated, responded and adapted to better reflect the studied systems (Harrison, 2007a,b; Pahl-Wostl, 2007). Adaptive management has been widely used in climate change studies (Brooks et al., 2005; Maciver and Wheaton, 2005; IPCC, 2007), fisheries management (Marttunen and Vehanen, 2004), water resource management (Pahl-Wostl, 2007), marine reserve evaluation (Grafton et al., 2005), and others. However, no study has reported explicit application of adaptive management in load reduction; USEPA (1991) proposed a ‘phased approach’ for Total Maximum Daily Load (TMDL), which is similar, but not completely the same, to the idea of adaptive management (Freedman and Nemura, 2004). Although there is wide acceptance of the idea of adaptive management in watershed pollution control and load reduction, there lacks a systematic approach for achieving optimal adaptive management in the decision making process. Most of the previous adaptive management research has focused on the “adaptive” process, but have overlooked the “optimization” perspective. Likewise, most of the “optimization” research has overlooked the “adaptive” process of problem formulation. A systematic approach for optimal adaptive management of watershed system should have three major traits. First, it should be able to explicitly handle uncertainty in the optimization model. Second, it should be able to handle relatively large scale problems that are expected for watershed scale analysis. Third, it should be able to provide guidance for prioritizing management measures to achieve reliable and efficient decisions. There is very little literature that addresses optimal adaptive management in environmental systems. For example, Harrison (2007a,b) proposed a two-stage adaptive approach with Bayesian Programming (BP) for optimal adaptive management of a river basin, in which new data were collected after implementation of the first-stage decision to update the priors for the second decision stage using Bayesian analysis. This approach was shown to have merits over earlier approaches in terms of computational efficiency and capability of handling relatively larger problems; however, it still can be computational prohibitive for watershed-scale problems because the BP is essentially a type of stochastic method where a large number of numerical realizations are needed to achieve model solutions. In addition, the BP and other previous researches provide no systematic way to prioritize the implementation schemes given uncertainties in the watershed systems and the optimization models. More importantly, the BP approach is not a full optimization-based adaptive management approach because it does not address cost effectiveness in an uncertainty-based decision making framework. The absence of an effective optimization approach and the lack of a systematic method to prioritize implementation reveal the need for a new methodology for uncertainty-based watershed-scale optimal adaptive management. This paper presents an uncertainty-based adaptive management framework that has been developed for optimal load reduction

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decision making. A recently developed risk explicit interval linear programming (REILP) framework was used as the platform for formulating the uncertainty based optimal load reduction model (Zou et al., 2010; Liu et al., 2010). An adaptive management decision making approach was then proposed as a way to assist with prioritizing the implementation order of optimal solutions obtained at each stage of decision making, while the optimal solution at each stage is adaptively obtained through the process of implementation and data assimilation. This approach was applied to a real-world nutrient load reduction case; the system performance was compared and analyzed with and without applying the proposed approach.

2.

Materials and methodology

s:t: A
X  B


(2)

X0

(3)





where, f is objective function; C
¼ ½c
1 ; c2 ; .; ci ; .; cn  and g ði ¼ 1; 2; .; n; j ¼ 1; 2; .; mÞ are the interval coeffiA
¼ fa
ji cients defined as the lower bound and upper bound estimated from data; X is the unknown decision variables;

T represents the right-hand side B
¼ ½b
1 ; b2 ; .; bm  constraints. Eqs. (1)e(3) can be decomposed into two sub-models corresponding to the lower and upper bounds of the objective function and solved using standard LP algorithms (Tong, 1994). Upon completing the ILP solution for the lower and upper bound of the optimal objective function, the corresponding REILP model can be formulated as (Zou et al., 2010):

2

2.1. Uncertainty based optimization modeling framework

min xi ¼ 4i 4

3     þ  þ  5 lij aij  aij xj þ hi bi  bi

(4)

j¼1

Linear Programming (LP) has been one of the most widely applied mathematical programming techniques for assisting optimal environmental decision making during the past decades (Dantzig, 1955; Huang et al., 1992; Chang et al., 1997; Bazargan, 2007). When considering uncertainty, various types of LP approaches were developed to deal with uncertaintybased decision making problems within a LP frameworkestochastic linear programming (SLP), fuzzy linear programming (FLP) and interval linear programming (ILP) are the most widely researched and applied approaches. Among the three types, the ILP models have been identified by many previous studies, to be the most suitable for handling uncertainty in practical environmental systems analysis because it has the lowest data requirement that is compatible with the data availability in most real cases (Chang et al., 1997; Chinneck and Ramadan, 2000; Fiedler et al., 2006; Ozdemir and Saaty, 2006). Nevertheless, traditional ILP approaches that present optimal solutions in interval numbers have been found to be ineffective in supporting practical decision due to the problems of infeasibility, non-optimality, and the inability to relate decisions to risks (Zou et al., 2010; Liu et al., 2010). As a result, these approaches cannot provide a platform for systematically guiding an optimal adaptive management decision making process. To overcome the limitations of the traditional ILP approaches, Zou et al. (2010) developed a REILP approach to fully explore the uncertainty space defined by an ILP model for optimal solutions that explicitly relates system performance to decision risk. This study shows that the REILP would serve as an effective mathematical framework for optimal adaptive management decision making at a watershed scale. The following text details the formulation of the uncertainty based optimization modeling framework, the solution algorithm, and the methods for prioritizing implementation and guiding adaptive management decision making. A typical ILP model can be formulated as below (Tong, 1994):

Min f
¼ C
X

n X

(1)

s:t:

n X

  þ þ  cþ j xj  m  fopt  l0 fopt  fopt

(5)

j¼1

bþ i 

n X

aij xj  xi ;

ci

(6)

j¼1

l0 ¼ lpre

(7)

0  lij  1

(8)

xj  0;

cj

(9) l0 ðcþ j

c j Þxj ;

 fopt

þ fopt

where, m ¼  and are the lower and upper bound of the optimal solutions for the objective function of the original ILP model; l0 is the system aspiration level with a value between 0 and 1; lij and hi are real numbers between 0 and 1; xi is the risk function of constraint i, which is defined P þ   as xi ¼ nj¼1 lij ðaþ ij  aij Þxj þ hi ðbi  bi Þ; x is the risk function of the entire system; and lpre is the prescribed system aspiration level; 4 is a general arithmetic operator which can be a simple addition, a weighted addition, simple arithmetic mean, weighted arithmetic mean, or a max operator. The risk function forms a platform for risk and system return tradeoff such that when xi takes values greater than 0, some or all constraints would be relaxed for higher system return (Zou et al., 2010). The REILP model (4) to (9) can be solved using a nonlinear programming algorithm at each pre-specified aspiration level to form a decision front for risk-based decision making.

2.2. REILP-based implementation prioritization approach (IPA) The IPA plays a critical role in the optimal adaptive management decision making. As noted earlier, adaptive decision making is preferred over the traditional static approach because of uncertainties that exist in the decision making process. Adaptive management would allow system analysts and decision makers to update model results and

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management schemes based on newly obtained data and refined characterization of uncertainty. Adaptive management is an experimental “learn by doing” approach to reduce uncertainty and improve the cost-effectiveness of management alternatives. In the context of uncertainty and experimental methods, decision makers need to make choices on what decision variables need to be implemented with higher priority either due to consideration of limited resources or interactions between different schemes (Gregory et al., 2006b). During the implementation process, those management schemes that are relatively insensitive to uncertainties should be implemented with higher priority than sensitive ones because they are less likely to be invalidated by newly obtained information. In other words, through implementing at an earlier stage those decision variables with lower risks, it is possible to acquire additional information to reduce the uncertainty and risk associated with those decision variables with higher risks, hence reduce the overall risks of the entire load reduction decision and implementation program. With this consideration, the IPA is formulated on the basis of a sensitivity analysis of the REILP solutions. In the REILP modeling framework, the impact of uncertainty on the optimal solutions is reflected in the aspiration levels (l0), which represents different levels of trade-offs between the cost of management schemes and the risk of potentially violating water quality standards due to uncertainties in the system. A higher aspiration level indicates that the decision makers’ would adopt management schemes that incur lower cost while accepting higher risk of non-compliance in water quality standards, and vice versa. At each aspiration level, a distinct set of optimal solutions can be obtained during decision making. Although it is possible to obtain optimal solutions for any aspiration levels between zero and one, an overly accurate stipulation of aspiration levels is not meaningful to decision makers who usually think in a quasi-qualitative way. In real-world decision processes, evaluations of risk or performance of a system are usually conducted on either a 10-level or 5-level scale to better emulate human brain function. In the present study, we propose to categorize the risk-cost tradeoff on a five-level scale, which are Level I: extremely conservative, Level II: moderately conservative, Level III: intermediate, Level IV: moderately aggressive and Level V: extremely aggressive. These five levels are respectively corresponding to the aspiration levels of [0.0, 0.2], [0.2, 0.4], [0.4, 0.6], [0.6, 0.8] and [0.8, 1.0]. An IPA is then devised based on the fivelevel scale. The proposed IPA consists of two steps, including an overall sensitivity analysis on system return with regard to system risk for achieving the desired tradeoff level (DTL), and an individual sensitivity analysis for each decision variable to decide the implementation priority under DTL.

2.2.1.

Step 1: DTL determination

The key information needed by decision makers to determine the DTL is the sensitivity of system return (the value of the objective function in the original ILP model) to the decision risk. For example, let’s assume that a REILP model was solved at the tradeoff level III and IV, and that the system return at

level IV is significantly better than that of level III, while the decision risk (in the form of the value of the risk function in the REILP model) is only slightly higher. In such a case, decision makers would likely choose level IV as the DTL for decision making. On the other hand, if the decision risk of level IV is significantly higher than that of level III, then decision makers might be more cautious and tend to choose level III as the DTL. Let r represents the decision risk level, which is equivalent to the Normalized Risk Level (NRL) in the REILP framework. The overall sensitivity of the system return ( f ) with regard to decision risk r can be expressed as:  df ¼ dr

d

Pn



i¼1 ci xi

dr

¼

n X dðci xi Þ i¼1

dr

(10)

Eq. (10) measures how system return interacts with the decision risk given the uncertainty present in the model formulation. A higher value of the sensitivity means that an increase in risk would be accompanied by large system gains, which reflects a situation where decision makers might consider taking the slightly increased risk for the chance of larger gains. On the other hand, a smaller value means that the same increase in decision risk would be accompanied by only small system gains, reflecting a situation where decision makers might be reluctant to take the risk for only minor gains. In practice, Eq. (10) can be evaluated across the neighboring tradeoff levels to obtain the distribution of system return-risk tradeoff efficiency. A cross-category analysis of the sensitivity can be produced to help decide the DTL reflecting the optimal tradeoff between risk and system performance. Let lt denotes an aspiration level corresponding to the tradeoff level t. The corresponding risk level is denoted as r ¼ rt, and the corresponding coefficient ci ¼ cti ; and the optimal solutions obtained by the REILP at lt are xi ¼ xti , where i ¼ 1, 2, ., n. Eq. (10) can thus be numerically evaluated at tradeoff level t using a finite difference approach. A backward finite difference expression of Eq. (10) is: df t Df t f t  f t1 ¼ ¼ drt Drt rt  rt1

(11)

where t ¼ 2, 3, ., 5 The sign of the value obtained from Eq. (11) depends on whether the original ILP model was minimization or maximization. To measure the absolute sensitivity of system return to risk level for the sake of simplicity, an absolute value of Eq. (11) is taken to define the Combined Sensitivity Coefficient at level t (CSCt) as:       df t  Df t  f t  f t1        CSCt ¼  t  ¼  t  ¼  t  dr  Dr   r  rt1 

(12)

Note that each tradeoff level represents a range of aspiration levels; therefore, the objective function value and risk level of each category t should be calculated before evaluating the value for the CSCt. Here we propose to use the average values for the objective function values and risk levels of each

w a t e r r e s e a r c h 4 5 ( 2 0 1 1 ) 4 8 8 5 e4 8 9 5

category to represent the corresponding system return and risk level values, leading to: Pm

j¼1 fj

ft ¼

m

¼

fLt þ fUt 2

(13)

ZUt

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each decision variable to the system return with regard to the decision risk level, i.e., the absolute individual sensitivity (AIS) can be expressed as:   dðci xi Þ  AISi ¼  dr 

(15)

rdl r ¼ t

Lt

Ut  L t

(14)

where, m is the total number of aspiration levels within tradeoff level t where REILP solutions are available; Lt and Ut represent the lower and upper bound aspiration levels for the tradeoff level; fLt and fUt are the lower and upper bounds of system return within tradeoff level t. Eq. (13) holds because the variation of system return values between aspiration levels follows a linear relationship by definition in REILP formulation, making it straightforward to evaluate the value of ft. However, because decision risk is a nonlinear function of aspiration level, an integration process is needed to evaluate the value for rt. In practice, Eq. (14) can be evaluated using a numerical integration method such as Simpson’s rule (Atkinson, 1989). With Eqs. (13) and (14), the CSC of each tradeoff level can be evaluated using Eq. (12), and the calculated CSC can be provided to decision makers to help decide a DTL upon which the optimal implementation plan will be based. Take a decision process starting from level I as example. Let’s assume that the CSC for level II is very large. It is anticipated that decision makers would likely step up from level I to II for significantly higher system return with relatively smaller risk. The decision makers can then evaluate the CSC for level III, which is assumed to also have a large value. Under such a circumstance, the decision makers might be encouraged to further step up to level III while checking the CSC value for level IV. Let’s further assume that the CSC value for level IV is a considerably smaller number, which means to further enhance system return would mean being subjected to significantly higher risk. Under these circumstances, decision makers would likely decide to stay on level III unless they have special reasons to step up to level IV. Since each tradeoff level is defined by a range of aspiration levels, the representative aspiration level for the DTL needs to be determined for the subsequent analysis. To avoid further complexity, we propose to use the mid-value of the range of aspiration levels for each tradeoff level as the representative aspiration level to reflect a preference for balanced decision within a specific risk tolerance category. As such, the aspiration level corresponding to the five tradeoff levels are 0.1, 0.3, 0.5, 0.7, and 0.9, respectively. In practice, system analysts and decision makers might choose to use different interval range for each category, hence different representative aspiration level for each category. In the latter text, the representative aspiration level corresponding to the DTL is referred to as the desired aspiration level (DAL).

2.2.2.

Step 2: implementation order determination

After determining the DAL, the corresponding optimal solutions can be used as the basis for formulating management schemes. At the DAL, the sensitivity of the contribution of

The value of AISi at a specific DAL can be evaluated within the corresponding DTL using a centered finite difference scheme as:   clþ1=2h xlþ1=2h  cl1=2h xl1=2h    i i i AISi ¼  i    rlþ1=2h  rl1=2h

(16)

where h is the aspiration level range of the DTL. A large AISi value indicates that the contribution of the corresponding decision variable to the total cost is sensitive to the uncertainty in the system representation. It is apparent that a robust implementation scheme should be associated with the decision variables that are less sensitive to the uncertainty in the model, i.e., those having small values of AIS. Therefore, the implementation process should follow the order that the more robust solutions are to be implemented prior to the less robust ones. To allow direct comparison of the relative sensitivity of each individual decision variable with all others, a relative AIS (RAIS) is defined as: AISi RAISi ¼ Pn i¼1 AISi

(17)

Ordering RAISi from the smallest to the largest, one can easily obtain decision variables which are least sensitive to the uncertainty, which would also tend to be more robust. These solutions should be implemented prior to others.

2.3.

Adaptive management scheme

The basic idea of adaptive management is to recognize the presence of uncertainty and adjust the modeling and practical implementation with time as new information becomes available. We propose a four-step approach for a risk-based optimal adaptive watershed management.

2.3.1.

Step 1: REILP formulation and DTL determination

This step involves formulating and solving the ILP and REILP models for a specific watershed-scale load reduction problem, and then conducting the analysis shown in Eqs. (10)e(14) to determine the DTL.

2.3.2.

Step 2: implementation order determination

This step involves calculating the AISi and RAISi for each decision variable using Eqs (16) and (17) to determine the implementation order. The decision variables with low RAISi values are considered robust and will be placed a higher priority in implementation. At this stage, it is possible that decision makers would incorporate additional stipulations regarding budget and implementation periods for the prioritized decision variables. In practice, multiple implementation

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periods might be needed, considering the availability of financial resources and construction efforts.

2.3.3. Step 3: staged implementation and optimization analysis Throughout each implementation period, new datasets, monitoring data, and other information may become available to better quantify the coefficients in the ILP and REILP models. As modeling-based decisions are implemented, newly collected and monitored data would be available to help increase the accuracy of system representation and reduce uncertainty in the model formulation. Therefore, the optimization model should be updated with the new information to obtain enhanced decision support capability. In this step, the ILP and REILP models developed in Step 1 and used in Step 2 are updated and solved to produce new optimal solutions, after which the analyses in Step 2 are to be repeated to obtain the DTL, implementation order, and new management schemes for the subsequent period.

2.3.4. Step 4: repeating until accomplishment of load reduction goal This step involves repeating Steps 1 to 3 whenever new data become available in the implementation process until the completion of the anticipated management program.

3.

Case study

3.1.

Study area: Lake Qionghai Watershed

Lake Qionghai, the second largest freshwater lake in Sichuan Province, China, is on top of the agenda for water quality protection in the local area (Liu et al., 2008). The watershed is divided into 20 sub-watersheds (Supporting Materials). Lake Qionghai is experiencing rapid eutrophication caused by excessive watershed nutrient loading. A long-term watershed nutrient loading reduction program was initiated to restore the water quality and to protect the aquatic ecosystem. Previously, integrated watershed optimization analysis was conducted in a static decision manner, which is considered insufficient to support the long-term management goal (Liu et al., 2008, 2010). In this study, the proposed optimal adaptive management approach is applied to this system to illustrate the procedure and advantage.

3.2.

3.3.

Adaptive management: initial stage

The ILP and REILP model equations represent the uncertainty based optimization model built upon the currently available information (Supporting Materials). The model can be solved to provide solutions guiding the decision making for the initial stage of the adaptive management. Note that in traditional optimal decision making framework, the solutions obtained in this stage were previously used to formulate long-term implementation plan, which will later be shown to be ineffective and unreliable. Fig. 1 presents the optimal solutions showing the tradeoff between system performance (total cost, f ) and the risk levels (Liu et al., 2010). As shown, for lpre ¼ 0, the risk level is 0 by definition, which corresponds to the upper bound solutions of the ILP model with a total cost of f ¼ $1.546 billion. The upper bound solution reflects the situation that when the decision makers are willing to spend $1.546 billion to treat the pollution sources in the watershed, the risk of violating the water quality target in the lake would be at the minimum. On the other hand, the risk level for lpre ¼ 1.0 reaches the maximum value, suggesting that a most aggressive decision corresponds to the lower-bound solutions of the ILP model with a total cost of f ¼ $0.773 billion. This extreme solution means that if the decision makers are willing to spend the lower amount of $0.773 billion for the load reduction based on an extremely optimistic interpretation of uncertainty in the optimization model, there would be a pretty good chance that the water quality target would still be violated. In Fig. 1, the tradeoff curve shows a sharp decrease of total cost ( f ) when the risk level starts to increase from zero. This decent of cost, however, gradually levels off until reaching the point after which any further increase in risk only result in relatively small reductions in total cost. The optimal adaptive decision making starts with determining the order of implementation for each decision variable. For this purpose, Eqs. (12)e(14) were used to calculate CSCt. The calculated CSCt value for level I to II is 2199.5, for II to III 2022.9, for III to IV 1384.8, and for IV to V 605.0. Since the CSCt values for level I to II and II to III are similar to each other,

ILP and REILP model formulation

The goal of optimal nutrient load reduction analysis is to minimize the implementation cost while meeting the water quality targets. Since significant uncertainty exists in the system characterization and no data were available to specify the probabilistic or possibilistic distribution of model coefficients, interval numbers were used to represent the uncertainty in a LP framework, resulting in an ILP model for the watershed. In our previous studies, a multi-stage ILP-based and REILP-based optimization model were developed to deal with the nutrient load reduction issues in Lake Qionghai Watershed in three 5-year periods (Liu et al., 2008, 2010; Supporting Materials).

Fig. 1 e Trade-off curve of total cost versus normalized risk level (NRL) of decision.

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either of them can be chosen as the DTL in practice depending on the decision makers’ preference. In this study, the highest CSCt value of 2199.5 was chosen to determine the DTL for illustrative purposes to suggest that it is most effective to formulate a management scheme based on conservative interpretation (tradeoff level II) of the uncertainty in the ILP model (Supporting Materials). After determining the DTL, the AIS and RAIS were calculated and presented in Table 1. As shown, solutions for all the decision variables except those corresponding to j ¼ 6, 8 and 10 have very small RAIS. Therefore, all decision variables with small RAIS are considered robust with regard to uncertainty in the ILP model. Based on this analysis, the solutions for these robust decision variables would be set at a higher priority for implementation than the solutions for the decision variables corresponding to j ¼ 6, 8, and 10. It is noted that to fully implement all these robust solutions would require approximately $1.3 billion, which is far beyond the funds available at the current stage. In contrast, the total funds approved for the load reduction projects in the first three years are approximately $200 million, which also cannot meet the investment requirements for implementing all of the robust solutions; therefore, decision makers need to further prioritize the implementation order based on the available budget, RAIS of each decision variable, and the associated construction times and maintenance costs. This analysis led to the selection of the four decision variables corresponding to j ¼ 1, 2, 4, and 7 to be implemented in the first three years.

3.4.

Adaptive decision making

In real-world practice, decision making is always a dynamic process as the system representation continually changes and new information becomes available. In some cases the new data and information might provide justification for the validity of the parameter values used in the initial optimization model, indicating that the original optimal solutions can still be valid for the next stage of implementation. However, in cases where the new data and information indicates that the parameter values in the original optimization model are inaccurate, the original optimal solutions for decision variables not yet being implemented might become invalid or suboptimal for guiding the next stage of implementation. In such cases, it is desirable to obtain updated optimal solutions that better characterize the changing system. For watershed management, examples of new data and information include: (1) water quality monitoring data characterizing the watershed and lake system, (2) an updated water quality model linking watershed loadings to in-lake water quality response, or (3) a more accurate cost function

acquired through the construction and maintenance of the treatment facility built during the initial stage of implementation. In the case of Lake Qionghai, the original optimization model was configured using a water quality response matrix derived from a simple box model of the lake. If a sophisticated eutrophication model is developed after the initial stage, which provides a more accurate response matrix, the optimization model would need to be updated using the newly available information. For illustrative purposes, this paper presents a hypothetical condition where new information becomes available that convinced the decision makers to find updated optimal decisionsdthe new information will help to reduce the previous uncertainties and therefore shrink the parameter intervals. The hypothetical decision conditions are as follows: (1) after the first implementation period, more information becomes available to update the cost functions

ðIIC
j and ASCj Þ for decision variables; (2) TEC was refined using new water quality monitoring and load data; (3) after evaluating the performances of the implemented load reduction schemes in the initial stage, the decision makers decide to refine their anticipated goals, leading to the updates

of APR
j , RFLi and RLR , etc. (please refer to the Supporting Materials). To support the second stage of adaptive decision making, the original ILP/REILP system was updated using the aforementioned new information, and the updated model was then solved for new solutions (Supporting Materials). Note that for those decision variables already implemented during the first stage, no updated solution were generated because they are now part of the physical reality in the system. Therefore, they are labeled as “NaN”. The corresponding AIS, RAIS and cost for the second stage REILP was calculated and shown in Table 2. The results revealed that j ¼ 3, 9 and 8 should be on top of the list for being implemented in the following years since they have relatively low RAIS values. The decision making will follow a similar adaptive process as in the first implementation period, which we will not further elaborate here. In subsequent implementation periods, the ILP/REILP model will be updated whenever new data and information become available to guide the optimal decision making process over time.

4.

Results and discussions

Fig. 2 compares the total cost and load reduction that resulted from the original non-adaptive REILP optimization model (NAO) and the guided adaptive REILP optimization model. It suggests that by applying the guided adaptive management approach, the total cost for load reduction can be reduced

Table 1 e AIS, RAIS and cost for each decision variable.

6

AIS (10 ) RAIS Cost ($ million)

j¼1

j¼2

j¼3

j¼4

j¼5

j¼6

j¼7

j¼8

j¼9

j ¼ 10

97.64 0.05 132.02

40.03 0.02 55.38

24.26 0.01 37.79

0.31 0 0.79

164.58 0.08 79.89

1015.83 0.48 321.05

0 0 0.02

315.09 0.15 422.66

17.39 0.01 34.6

435.17 0.21 230.09

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w a t e r r e s e a r c h 4 5 ( 2 0 1 1 ) 4 8 8 5 e4 8 9 5

Table 2 e AIS, RAIS and cost for each decision variable based on updated REILP model.

6

AIS (10 ) RAIS Cost ($ million)

j¼3

j¼5

j¼6

j¼8

j¼9

j ¼ 10

21.67 0.01 37.79

365.81 0.14 79.89

1526.10 0.57 321.05

268.49 0.10 422.66

17.19 0.01 34.60

479.93 0.17 230.09

from $1.31 billion to $1.18 billion (a savings of about 10%) due to the incorporation of new information throughout the implementation period. Correspondingly, the total load reduction amount between the two scenarios changes from the 222.4 tonnes/year to 221.12 tonnes/year. Compared with the original NAO model results, the costs and load reduction for j ¼ 5, 6, 8 and 10 and i ¼ 3, 9, 10, 11 and 12 varied noticeably, suggesting that the optimal solutions of the NAO model that had a higher range of uncertainty, were no longer valid given the updated, more accurate characterization of the system. The reason why the guided adaptive optimal approach would produce a lower long-term cost than the static approach is that with the new information incorporated, the original optimal solutions might become obsolete and no longer optimal in the updated model framework. To further demonstrate the advantage of the adaptive approach, the original REILP solutions at various aspiration levels were entered into the updated REILP framework to determine the maximum achievable aspiration levels (MAAL). The MAAL is defined as the maximum aspiration level, which when exceeded, would cause the original REILP solutions to become infeasible in the updated model. The introduction of the MAAL into the analytic framework provides a consistent way of evaluating the performance of different REILP solutions (Liu et al., 2010). After determining the MAAL, the corresponding risk and system performance ( f ) at lpre ¼ MAAL was then

calculated by incorporating the original REILP solutions into the modified REILP modeling framework, allowing for direct comparison of the solutions of the adaptive REILP model at various risk levels (Fig. 3). As shown, at each specific decision risk (any given point along the X-axis), the decision based on the original static REILP model solutions incurs a higher total cost than the corresponding one based on the adaptive approach. Similarly, at each specific total cost (any given point along the Y-axis), the decision based on the original REILP suffers a higher risk of violating water quality standards than the one based on the adaptive approach. The results of this analysis demonstrate one of the main points of this paper: the traditional optimization-based watershed management approaches which use static optimal solutions to guide longterm management plan can be ineffective at achieving cost reduction and risk control. In contrast, the proposed adaptive optimal approach was shown to be able to provide improved decision support for cost effectiveness and risk control simultaneously. Another major intention of this paper is to fill in the knowledge gap of traditional adaptive management by developing a systematical way to conduct adaptive management that is optimal with regard to evolving uncertainties in the system characterization. To distinguish our approach to the conventional adaptive management concept, our approach is hence referred to as the “guided adaptive optimal” (GAO) approach, which features the capability of conducting guided prioritization of implementation order. Here we will illustrate the advantage of the GAO approach by contrasting it against the unguided adaptive optimization approach (UGAO) that is adaptive, but without guidance for prioritizing the implementation order at each adaptive stage. While the GAO was able to perform an informed decision by selecting j ¼ 1, 2, 4, and 7 to be implemented in the first three years based on the RAIS calculation, the UGAO could only proceed no better than randomly picking values, for example j ¼ 2, 3, 4 and 5, to be

Fig. 2 e Total cost and load reduction summarized by strategy and subwatershed for the non-adaptive REILP versus guided adaptive optimal approach.

w a t e r r e s e a r c h 4 5 ( 2 0 1 1 ) 4 8 8 5 e4 8 9 5

Fig. 3 e The total cost versus normalized risk level of decision for non-adaptive optimal REILP versus the guided adaptive optimal solutions.

implemented in the first three years due to lack of systematic guidance. The resulting solutions for the GAO and UGAO comparison are shown in Fig. 4. With the systematic guidance for adaptive management, the GAO results in optimal solutions for the subsequent implementation stage that is different from those of UGAO. With the GAO approach, the total cost for load reduction would be $1.18 billion, which is lower than the $1.22 billion of the UGAO approach (a savings of only about 3%). However, the risk level of the decisions obtained by the GAO approach is only 0.099 (versus 0.106 for the UGAO approach), which represents lower risk level. By using a systematic process to prioritize the implementation order, the GAO approach produced a more cost-effective and safer (more reliable) management decision than the more

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traditional unguided adaptive optimal management approach. Finally, when evaluating solutions associated with the NAO, UGAO and GAO, it becomes evident how the selected strategies for different sub-watersheds determine the predicted costs and load reductions. Fig. 5 shows that although the average relative distribution over the study area for selected strategies is similar, there are some big differences between selected solutions among some of the sub-watersheds. Subwatersheds i ¼ 1, 3, 5, 8, 12, and 16 had some of the most notable differences between scenarios. For example, the GAO solution completely forwent lake riparian vegetation covers ( j ¼ 10) in subwatershed 12 in favor of increased extreme erosion restoration ( j ¼ 3) elsewhere in subwatershed 3 and wastewater treatment plant enhancements ( j ¼ 5) in subwatershed 8. Within subwatershed 8, wastewater treatment plant enhancement was also prioritized over NPS controlling measures in the GAO solution. On the other hand, the UGAO solution invested significantly more money on NPS controlling measures than wastewater treatment in subwatershed 8. As part of a wider planning effort for the entire study area, the GAO-selected solutions resulted in a net cost savings for comparable load reduction at a lower risk level. By minimizing the risk of selecting ineffective solutions, the proposed guided adaptive optimization approach has a greater potential to deliver optimal solutions in the long run than both non-adaptive and unguided optimization approaches. For larger systems with more management options or even subsequent implementation rounds of the same system, the advantages of the guided approach would likely become even more pronounced in terms of cost savings, load reduction, and risk management. The major limitation of the presented approach is that it is not based on a full simulation-optimization framework, therefore, it is best fit for systems with insufficient data

Fig. 4 e Total cost and load reduction summarized by strategy and subwatershed for unguided adaptive versus guided adaptive optimal approaches.

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w a t e r r e s e a r c h 4 5 ( 2 0 1 1 ) 4 8 8 5 e4 8 9 5

Fig. 5 e Cost distribution by strategy (J) for non-adaptive, unguided, and guided adaptive optimal approaches for entire study area and selected sub-watersheds (I).

available to accurately quantity the nonlinear cause-andeffect responses between management measures and water quality. In cases where data are sufficient to quantify the nonlinear cause-and-effect responses, and where the nonlinearity is strong and non-negligible, it would be more desired to apply a direct simulation-optimization framework to support the adaptive management decision-making process. Such methodological framework will be the focus of follow-up researches.

5.

Conclusions

This study developed a GAO approach for risk-based decision making for nutrient reduction at the watershed scale. The methodological procedure of the GAO approach was demonstrated through a case study application of the Lake Qionghai Watershed in China. The advantages of the GAO approach were further demonstrated by comparing the resulting solutions using: (1) the non-adaptive model versus the guided adaptive model, and (2) the unguided adaptive model versus the guided adaptive model. The conclusions obtained in this study are as follows: 1) Traditional non-adaptive optimization-based decision support approaches are not effective in a dynamic management environment; therefore, an adaptive optimal approach offers a better way to dynamically take into account new information for updating the system characterization and quantifying uncertainty. In an adaptive management process, the order of implementation can have a significant impact on the system performance. Traditional unguided adaptive management approaches

lack a systematic way that simultaneously addresses optimization while prioritizing the implementation order. The proposed GAO approach provides a solid analytical framework and a systematic way to guide optimal decision making in a changing decision environment by addressing the evolution of uncertainty with newly available information during the implementation process, and providing timely guidance to subsequent implementation phases for a more reliable outcome. 2) The case study shows that with the GAO approach, the resulted decision making is (a) more cost-effective and (b) has lower risk liability than the case when a static optimization-based decision making was used. The case study also demonstrates that the adaptive process alone is not guaranteed to produce optimal management results; in contrast, the systematic prioritization component of the GAO provides a sound basis for informed decision making with regard to the implementation order, resulting in higher cost-effectiveness and lower risk-liability than a more conventional, unguided adaptive management approach. The model results shows that the total GAO cost for load reduction would be $1.18 billion, which is lower than the $1.22 billion of the UGAO approach. 3) While the case study example demonstrated an improving cost-effectiveness and risk reduction trend between the non-adaptive, unguided, and guided adaptive optimization approaches in the Lake Qionghai Watershed, only two rounds of adaptive management were performed. It is expected that the differences between these approaches would become even more pronounced (1) after subsequent management rounds, or (2) in a larger watershed with more management strategy options or more compliance constraints.

w a t e r r e s e a r c h 4 5 ( 2 0 1 1 ) 4 8 8 5 e4 8 9 5

4) The ILP and REILP mathematical framework appears to be an effective analytic basis for handling the adaptive decision support problem given their direct consideration of uncertainty in both the formulation and the solution, coupled with the capability of generating a risk-based decision front upon which to conduct in-depth tradeoff analysis for guiding the adaptive management process.

Acknowledgments This paper was supported by the “China National Water Pollution Control Program” (2008ZX07102-001), Los Angeles County Department of Public Works, and Research Fund for the Doctoral Program of Higher Education of China (20100001120020).

Appendix. Supplementary material The supplementary data associated with this article can be found in the on-line version at doi:10.1016/j.watres.2011.06.038.

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