代写An Integrated Model of Lake Pollution Management based on Robust Decision Making: A Case Study of
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1 Research Background
Lakes are important components of ecosystems, providing ecosystem services such as biodiversity conservation, plant and animal habitats, climate change regulation, drinking water resources, fish- eries, tourism and sightseeing [1]. In particularly, lakes are one of the natural environments on which human beings depend. Natural and artificial lakes contain about 90% of the total surface freshwater on Earth, making them one of the most significant sources of freshwater for human daily activities and socioeconomic development [2]. They also create immeasurable direct and indirect economic value for the basin economy, supporting the socioeconomic development of the basin.
But compared to river ecosystems, lakes are less mobile, oxygenated, and less resistant toecosys- tem stability, while more vulnerable to pollution and degradation problems. It has been a global phenomenon where local government permitted factories built near lakes to achieve rapid economic growth in the basin and accelerate the process of industrialization and urbanization. This, however, has inevitably encouraged the reclamation of farmland, overfished, introduced invasive alien species, as well as serious pollution problems in water bodies such as eutrophication, water quality degra- dation, soil erosion, weakening of biodiversity, etc. In the context of the dramatic global climate change the occurrence of extreme weather events (e.g. persistent droughts and high temperatures), the problems faced by lakes have been exacerbated, with their climate regulation capacity weakened on the one hand and the stability of water ecosystems under threat on the other. It is therefore imperative to explore better management methods for the lake system.
2 Research Objectives
The seriousness and urgency of the water pollution problem in lake basins has drawn considerable policy attention from governments across the globe. Still, the complex nature-social system of the lake basin has made it challenging for policymakers to design effective pollution control strategy and corresponding evaluation metrics with limited time and budget. This research attempts to fill in this knowledge gap by:
1) constructing a lake pollution management model based on the core idea of Dynamic Integrated Model of Climate and the Economy (DICE) proposed by Nordhaus[3];
2) proposing lake pollution management strategies through the robust decision making (RDM) approach [4][5], as an alternative to the Bayesian decision theory[6] [7], to address the issue of non- probabilistic distributions of factors.
In so doing, it could provide theoretical and methodological support for the formulation and evaluation of pollution management decisions, thus contributing to the sustainable development of the basin economy and lake ecological protection.
3 Literature Review
3.1 Lake Complex Systems
Mitsch et al. [8] pointed out that ecosystems are typically complex systems with system dynamics such as self-adaptation, self-growth, self-regulation and evolutionary capacity, which require the use of theories and methods from complexity science to analyze the dynamic behavior. of ecosystem. In following them, more scholars are using the systems science approach to analyze lake basin prob- lems, rather than conducting research from a single engineering or technical perspective. Guneralp et al. [9] constructed a system dynamics model through lake ecosystems, economic subsystems and social subsystems to study and analyze the impact of socioeconomic development on lake ecosystems in relevant watersheds. Liu et al. [10] analyzed sustainable water resources planning and manage- ment processes in the semi-arid southwest states of the USA based on integrated modelling of the interactions between lake basin ecosystems and human society. Similarly, Liu et al.[11] embedded a pollutant load model and a lake water quality model in a socioeconomic system dynamics model to analyze the expected efficiency of water quality restoration in Dian Lake, China, under differ- ent treatment strategies to help support the decisions needed in development and environmental protection.
3.2 Decision Making under uncertainty
Most traditional risk decision theories are based on the Bayesian decision theory, which uses a spe- cific probabilistic distribution to characterize the uncertain factors and identifies the best strategy with the optimal expectation indicator as the objective function[6][7]. However, for water resources planning and management decision problems, natural factors (e.g. climate change) and human factors (e.g. land use change, pollution emission and resource consumption caused by human economic and social activities) are difficult to be reliably described by the relevant probabilistic distributions [12][13]. As a response, scholars proposed the concept of “deep uncertainty”[4] and its operational framework, such as robust decision making (RDM).
According to Lempert et al. [5][14], traditional decision making methods are applicable only when uncertain factors are characterized by probabilistic features. In the context of deep uncer- tainly where there is an absence of some key information, and the level of previous knowledge and experience is difficult to explain the newly occurred problem, they argued that robustness could be a better indicator for decision evaluation. Following this, they proposed theoretical models and analytical methods for RDM, and then developed multi-objective robust decision models.
RDM has been widely applied in the analysis of management strategies for complex systems. It begins with decision framing based on the XLRM (the most important uncertainties, levers, relationships and measures) framework to construct the research problem, where X is the exogenous parameters of the research problem under deep uncertainty, L is the change in the system that the decision maker can adopt strategies, R is the research model linking the decision maker’s strategy
L with the evaluation index M, and M is the evaluation index of the strategy implementation effect.
X : uncertain factors in the socioeconomic and environmental systems
L : policies for water pollution control and improvement
R : lake basin pollution management models
M : total social welfare, management cost and water quality
Weaver et al. [15] analyzed the conflict between Southern California’s rapidly growing population and water allocation in the context of climate change, and the impact of rapid sea level rise on the Port of Los Angeles with RDM. Lampert et al. [14] employed RDM to analyze the vulnerability of the Colorado River Basin to determine water resource management strategies and illustrated how RDM can be used to develop effective investment for the Green Climate Fund. Kim et al.[16] proposed an index-based robust decision framework for watershed management to deal with water quantity and quality under changing climatic conditions. Bhave et al.[17] pointed out the prospects of RDM in applications such as forest resource management and disaster risk management and analyzed some barriers and limitations of RDM in practice.
4 Research Methodology
4.1 Research Object
This study will focus on the Lake Kasumigaura, the second-largest lake in Japan with tremendous environmental and social significance and data availability. Meanwhile, the lake has notoriously un- dergone water bloom, moldy water odor, clogging in water purification system, and other ecological problems with Integrated Measures for Water Quality Conservation being enforced.
The environmental statistics can be obtained from the Lake Kasumigaura Database and Min- istry of the Environment of Japan, relevant studies, local environmental statistical yearbooks and so on. Relevant economic statistics can be obtained from the Statistics Bureau of Japan, FRED database etc.
4.2 Research Steps
STEP 1: Constructing a Lake Pollution Management Model
The lake pollution management model that this research attempts to construct mainly consists of three parts: the objective function, the economic system and the environmental system. After constructing the lake pollution management model, each parameter in the model is estimated and tested by collecting, processing and analyzing the relevant data to obtain a part of relatively certain parameters (i.e. equal to a certain value) and another part of parameters with uncertainty (i.e. satisfying certain distributions to facilitate subsequent sampling operations).
STEP 2: Robust Decision Making Framework
What follows the model construction is the incorporation of the model into a robust decision making framework to analyze the lake management under deep uncertainty. This section will by identify and analyze the weaknesses and vulnerability scenarios of the set of strategies to select the ones that are more robust under different scenarios.
5 Integrated Model of Lake Pollution Management
5.1 Objective Function
The DICE model assumes that the objective of economic and climate policies is based on maximizing the social welfare over the study period. Similarly, the objective function of the lake pollution management model is the social welfare function of the basin during the study period, while the pollution management policies are designed and implemented with the aim of maximizing the total social welfare. Through the evaluation and analysis of different pollution management policies, we are able to identify the most robust decision, and the corresponding cost of the management strategies is the constraint in the optimization model.
The social welfare function is an accumulation of individual utility functions over the study period, and the utility of each member of society as a function of per capita consumption c(t), so we have
(1)
where u(t) is the individual utility function at period t and α is the elasticity of marginal utility of consumption [18].
Because the social welfare function spans multiple time periods, a discount factor needs to be considered which is given by
D(t) = (1 + ρ)−t (2)
where D(t) is the discount factor in period t and ρ is the time preference rate.
If the total population of the study area in period t is N(t), then the social welfare function (i.e. objective function) W(t) can be expressed as
When formulating and evaluating the management policies, it is necessary to consider the eco- nomic cost of policy implementation and the impact caused to the lake ecosystem, such as the government funding budget, the proportion of pollutant reduction, water quality changes, etc. Mcons defines the cost constraint matrix for formulating the strategy, and Mpolicy represents the vector of constraint values faced by each strategy, then the constraints can be expressed as
Mcons ≤ Mpolicy (4)
Thus we can construct the optimization model by
s.t. Mcons ≤ Mpolicy
5.2 Economic System
The main role of the economic system is to estimate the social welfare function. Learning from DICE model,we may estimate the per capita consumption function c(t), the parameters of economic
loss and pollution control cost etc. from the production function of the lake basin, the economic
loss function of lake pollution, and the cost function of lake pollution control.
5.2.1 Production Function
Learning from the construction methods of economic model in DICE model[3], here we use Cobb-
Douglas production function to depict the economic growth of the lake basin, which is given by
Q(t) = A(t)KαK (t)LαL (t) (5)
where Q(t) is the gross product of the lake basin at stage t; A(t) is the total factor productivity at stage t and could be portrayed using a logistic type curve, whose growth rate decreases with time; K(t) is the capital stock at stage t; L(t) is the number of labor force population at stage t; αK and αL are the elasticity coefficients of capital and labor output, respectively, satisfying αK + αL = 1.
More specifically, for the number of labor force population L(t), we have
L(t) = γ1 N(t) (6)
where γ1 is the coefficient of the labor force share of the total population; N(t) is the total popu- lation.
The Malthusian model is very simple and applicable for small population, and for large popu- lation it is preferable to use logistic ODE population model[19]. Therefore, N(t) can be expressed as
where Nm is the limit value of population number; N0 is the initial population number; r is the natural population growth rate; and t0 is the initial stage.
5.2.2 Economic Loss Function
Eutrophication (overabundance of nutrients in lakes, which promotes excessive plant and algae growth) occurs in the lake basin, and the resulting water bloom causes such problems as foul smelling drinking water, fisheries damage, and loss of landscape beauty etc. It is highly relevant to the concentration of total nitrogen, so the economic loss function is given by
Economic Loss Function = Γ(t) = δ1 ϕ(t) + δ2 ϕ2 (t) (8)
where Γ(t) is the loss function caused by pollution; δi is the coefficient of loss function; and ϕ(t) is the concentration of total nitrogen at stage t.
In addition, Γ(t) denotes the proportion of the loss due to pollution to the difference between the output and the loss, which is set to ensure that the pollution loss does not exceed the output. The loss from total nitrogen pollution of the lake as a percentage of output is .
The form. of loss function assumes that:
• There is no critical value or turning point for the loss of pollution;
• The loss due to pollution is marginal increasing, i.e., the second order derivative of Γ(t) with
respect to ϕ(t) is positive.
5.2.3 Pollution Control Cost Function
There are three main sources of nitrogen pollution: rural, industrial, and urban. With the formula of carbon dioxide abatement cost in the DICE model, we can construct a function of total nitrogen abatement cost for each source into the lake, which is set as a high-dimensional convex function and given by
where Ω(t) is the ratio of abatement cost at stage t to the total output Q(t); γ2 is the coefficient of capital required to be invested per ton of total nitrogen treated; σi (t) is the intensity of total nitrogen emissions into the lake per unit GDP at stage t for rural, industrial and urban sources respectively; µi (t) represents the reduction rate of total nitrogen emissions into the lake at stage t of each source; θi is the abatement cost coefficient of each source.
The form. of cost function assumes that:
• The marginal cost of the emission reduction rate µi (t) is incremental;
• The marginal cost of the emission reduction rate increases in a non-linear proportion.
5.2.4 Net Output Function
Based on equations (5)-(9), we can get the net output function Y (t) at stage t given by
(10)
5.2.5 Capital Accumulation Function
Since the net output of the economy consists mainly of social consumption and gross investment, there is
Y (t) = C(t) + I(t) (11)
where C(t) is the value of total consumption at stage t; I(t) is the total investment at stage t.
Besides, due to capital depreciation, under the perpetual inventory method, the capital stock
has
K(t) = (1 - δ)K(t - 1) + I(t) (12)
where δ is the capital depreciation rate. Thus, we have
(13)
5.3 Environmental System
The main role of the environmental system is to estimate each environmental parameters involved in Mcons and economic system, such as total nitrogen concentration ϕ(t) of the lake, reduction rate µi (t), emission intensity σi (t) and so on by constructing water quality model such as the Vollenweider-OECD model, the Dillon model [20] and inflow model of total nitrogen.
5.4 Construction of Strategies Set
Theoretically, the potential strategies may consist of an infinite set, and it is impossible to get any analytical solution in this case. By summarizing and analyzing a series of schemes issued by the Chinese and Japanese governments to control eutrophication pollution in lakes, this study would restrict the set to five strategies (scenarios):
• with no control;
• Weak abatement strategy: to reduce the total nitrogen concentration in the lake by 3% per year (just an example value, specific values will be adjusted according to local policy documents);
• Strong abatement strategy: to reduce the total nitrogen concentration in the lake by 9% per year;
• Maintenance target strategy: to keep annual total nitrogen concentrations in the lake below a certain value (e.g. 2mg/L);
• Direct funding strategy: Investment funds of lake basin pollution management account for 0.5% of the annual GDP in the basin economy.
Therefore, the Mpolicy can be expressed as a set of Target Reduction Rate (TRR,e.g. 3% or 9%), Target Maintenance Value (TMV, e.g. 2mg/L) and Government Funding Budget (GFB, e.g. 0.5% of the annual GDP in the basin economy). And the Mcons represents the reduction rate of total nitrogen (µi (t)), the total concentration of nitrogen in the lake (ϕ(t)) and the cost of pollution control (Q(t) · Ω(t)) in the model. Then the constraint condition Mcons ≤ Mpolicy is given by
('' Σ µi { ϕ(t)
'(Q(t) · Ω(t)
5.5 Connection between Subsystems
The Integrated Model of Lake Pollution Management consists of three parts: objective function, economic system and environmental system.
The objective function includes the utility function and the strategy constraint, and the opti- mization objective is to maximize the value of the social welfare function during the study period, in which the utility function involves the consumption and population at each stage in the economic model, and the strategy constraint governs the variation of the total nitrogen emissions into the lake from each source in the environmental model, and affects the calculation of the cost function in the economic model.
The economic model consists of production function, population model, economic loss function, pollution control cost function, and capital accumulation equation, where the population and con- sumption at each stage affect the utility function in the objective function, and the output solved by the production function at each stage and the policy constraint in the objective function affect the calculation of total nitrogen emissions from each source into the lake in the environmental model.
The environmental model mainly consists of two parts: the exogenous input model of total nitrogen to the lake and the water quality model, in which the solved concentration of total nitrogen in the water quality model influences the calculation of the loss function in the economic model.
6 Robust Decision Making
6.1 Evaluate Strategies Across Futures
A collection of candidate strategies to tackle lake pollution problems will be derived from sources such as relevant public debates, expert think tank, existing case experiences and so on. Next, numerous scenarios will be generated and the candidate strategies are tested against these possible scenarios representing future uncertainties, In this sense, the initial robust strategies are identified.
Specifically, in generating a large number of different scenarios, the method of Latin Hypercube Sampling will be used to sample the uncertain parameters and construct different combinations of parameters to represent different scenarios. In the evaluation of candidate strategies, the robustness of different strategies is evaluated in terms of the relative regret of different metrics (M) calculated by the research model (R) for each strategy (L) under numerous scenarios.
6.2 Vulnerability Analysis
Simon argued that the ideal robust strategy is free of vulnerability, but strategies in real situations often have difficulty in achieving such results[7]. Therefore, it is necessary to analyze the vulnerabil- ity of the initial robust strategy using the Scenario Discovery (SD) method, which can be employed to construct the vulnerable scenarios of the initial robust strategy by the Patient Rule Induction Method (PRIM) [21].
6.3 Trade-off Analysis
After the vulnerability analysis of the initial robust strategy, the large number of scenarios generated in step 2 can be classified into two categories: vulnerable and non-vulnerable scenarios. Next, we search for strategies whose evaluation metrics do not differ significantly from the initial robust strategy in all scenarios, but perform. better than the initial robust strategy in the vulnerable scenarios, and thus overcome the vulnerability problem of the initial robust strategy.
6.4 New Futures and Strategies
If the implementation effects of the initial set of robust strategies are unacceptable, a new robust de- sign can be carried out, i.e., the candidate strategies are modified and improved by the vulnerability information perceived in the scenario discovery, and then a new round of robust decision iteration is started. After the set of candidate robust strategies is analyzed and the decision maker is satisfied with the implementation results, the candidate robust strategies in the set can be evaluated and analyzed to choose the final strategy.