# A Semi-Infinite Interval-Stochastic Risk Management Model for River Water Pollution Control under Uncertainty

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## Abstract

**:**

## 1. Introduction

^{3}km) is in the worst two categories of water quality classification system (i.e., no longer fishable and of questionable agricultural value) [1]. Effective planning of water pollution control for watersheds plays an important role for national and/or regional sustainable development [2,3,4]. In fact, decisions in water pollution control are often made on the basis of uncertain information (i.e., various uncertainties) existing in system components and their interactions [5,6]. The major sources of uncertainty in water pollution control are the random characteristics of natural processes (e.g., precipitation and climate change) and stream conditions (e.g., stream flow, water supply, and point/nonpoint source pollution), the errors in estimated modeling parameters, and the imprecision of system objectives and constraints [7]. For example, pollutant discharge allowances are often affected by pollutant discharge rates (which influenced by random events such as temperature and precipitation) and decision makers’ estimations; the allowable loads are not measured with certainty but in fact represent a probability distribution. Consequently, a number of mathematical techniques were launched to examine economic, environmental and ecological impacts of alternative pollution control actions under uncertainty, and thus aid the decision makers in generating effective water pollution control plans and policies [8,9,10,11,12,13,14,15,16,17].

## 2. Methodology

_{h}) is used for balancing benefit for scenario h and the target level ($\eta $) [35].

## 3. Case Study

#### 3.1. Study Area

#### 3.2. Modelling Formulation

- (1)
- Wastewater treatment capacity constraints:$$P{W}_{sth}^{\pm}\cdot G{T}_{st}^{\pm}\le TP{C}_{st}^{\pm},\forall s,\text{}t$$$$AP{C}_{ith}^{\pm}\cdot W{C}_{it}^{\pm}\le TP{D}_{it}^{\pm},\forall i,\text{}t$$
- (2)
- BOD discharge constraints:$$AP{C}_{ith}^{\pm}\cdot W{C}_{it}^{\pm}\cdot I{C}_{it}^{\pm}\cdot \left(1-{\eta}_{BOD,it}^{\pm}\right)\le AB{C}_{ith}^{\pm}(l),\forall i,\text{}t$$$$P{W}_{sth}^{\pm}\cdot G{T}_{st}^{\pm}\cdot B{M}_{st}^{\pm}\cdot \left(1-{\eta}_{BOD,st}^{\pm}\right)\le AB{W}_{sth}^{\pm}(l),\forall s,\text{}t$$
- (3)
- Nitrogen discharge constraints:$$\sum _{k=1}^{9}\left(N{S}_{jk}^{\pm}\cdot S{L}_{jkt}^{\pm}+\text{}R{F}_{jkt}^{\pm}\cdot D{N}_{jkt}^{\pm}\cdot {10}^{-5}\right)\cdot PP{A}_{jkth}^{\pm}}\le MN{L}_{jth}^{\pm}(l),\forall j,\text{}t$$
- (4)
- Phosphorus discharge constraints:$$AP{C}_{ith}^{\pm}\cdot \left[W{C}_{it}^{\pm}\cdot PC{R}_{it}^{\pm}\left(1-{\eta}_{TP,it}^{\pm}\right)+AS{C}_{it}^{\pm}\cdot SL{R}_{it}^{\pm}\cdot PS{C}_{it}^{\pm}\right]\le APC{M}_{ith}^{\pm}(l),\forall i,\text{}t$$$$\sum _{k=1}^{9}\left(P{S}_{jk}^{\pm}\cdot S{L}_{jkt}^{\pm}+\text{}D{P}_{jkt}^{\pm}\cdot R{F}_{jkt}^{\pm}\cdot {10}^{-5}\right)\cdot PP{A}_{jkth}^{\pm}}\le MP{L}_{jth}^{\pm}(l),\forall j,\text{}t$$
- (5)
- Soil loss constraints:$$\sum _{k=1}^{9}S{L}_{jkt}^{\pm}\cdot PP{A}_{jkth}^{\pm}}\le MS{L}_{jth}^{\pm}(l),\forall j,\text{}t$$
- (6)
- Financial risk management:$$\begin{array}{l}{\displaystyle \sum _{i=1}^{5}{\displaystyle \sum _{t=1}^{2}B{C}_{it}^{\pm}\cdot PL{C}_{it}^{\pm}}}+\text{}{\displaystyle \sum _{n=1}^{6}{\displaystyle \sum _{t=1}^{2}B{P}_{pt}^{\pm}\cdot PL{M}_{pt}^{\pm}}}+\text{}{\displaystyle \sum _{s=1}^{4}{\displaystyle \sum _{t=1}^{2}B{W}_{st}^{\pm}\cdot Q{W}_{st}^{\pm}}}\\ +{\displaystyle \sum _{j=1}^{4}{\displaystyle \sum _{k=1}^{9}{\displaystyle \sum _{t=1}^{2}B{A}_{jkt}^{\pm}\cdot P{A}_{jkt}^{\pm}}}}-{\displaystyle \sum _{i=1}^{5}{\displaystyle \sum _{t=1}^{2}PB{C}_{it}^{\pm}\cdot (PL{C}_{it}^{\pm}-AP{C}_{ith}^{\pm})}}\\ -{\displaystyle \sum _{n=1}^{6}{\displaystyle \sum _{t=1}^{2}PB{P}_{nt}^{\pm}\cdot (PL{M}_{nt}^{\pm}-AP{M}_{nth}^{\pm})}}-{\displaystyle \sum _{s=1}^{4}{\displaystyle \sum _{t=1}^{2}PB{W}_{st}^{\pm}\cdot (Q{W}_{st}^{\pm}-P{W}_{sth}^{\pm})}}\\ -{\displaystyle \sum _{j=1}^{4}{\displaystyle \sum _{k=1}^{9}{\displaystyle \sum _{t=1}^{2}PB{A}_{jkt}^{\pm}\cdot (P{A}_{jkt}^{\pm}-PP{A}_{jkth}^{\pm})}}}\ge {\eta}^{\pm}-U{B}_{h}^{\pm}{z}_{h}^{\pm},\forall h\end{array}$$$$\begin{array}{l}{\displaystyle \sum _{i=1}^{5}{\displaystyle \sum _{t=1}^{2}B{C}_{it}^{\pm}\cdot PL{C}_{it}^{\pm}}}+\text{}{\displaystyle \sum _{n=1}^{6}{\displaystyle \sum _{t=1}^{2}B{P}_{pt}^{\pm}\cdot PL{M}_{pt}^{\pm}}}+\text{}{\displaystyle \sum _{s=1}^{4}{\displaystyle \sum _{t=1}^{2}B{W}_{st}^{\pm}\cdot Q{W}_{st}^{\pm}}}\\ +{\displaystyle \sum _{j=1}^{4}{\displaystyle \sum _{k=1}^{9}{\displaystyle \sum _{t=1}^{2}B{A}_{jkt}^{\pm}\cdot P{A}_{jkt}^{\pm}}}}-{\displaystyle \sum _{i=1}^{5}{\displaystyle \sum _{t=1}^{2}PB{C}_{it}^{\pm}\cdot (PL{C}_{it}^{\pm}-AP{C}_{ith}^{\pm})}}\\ -{\displaystyle \sum _{n=1}^{6}{\displaystyle \sum _{t=1}^{2}PB{P}_{nt}^{\pm}\cdot (PL{M}_{nt}^{\pm}-AP{M}_{nth}^{\pm})}}-{\displaystyle \sum _{s=1}^{4}{\displaystyle \sum _{t=1}^{2}PB{W}_{st}^{\pm}\cdot (Q{W}_{st}^{\pm}-P{W}_{sth}^{\pm})}}\\ -{\displaystyle \sum _{j=1}^{4}{\displaystyle \sum _{k=1}^{9}{\displaystyle \sum _{t=1}^{2}PB{A}_{jkt}^{\pm}\cdot (P{A}_{jkt}^{\pm}-PP{A}_{jkth}^{\pm})}}}\le {\eta}^{\pm}+U{B}_{h}^{\pm}\uff081-{z}_{h}^{\pm}\uff09,\forall h\end{array}$$$$\sum _{h=1}^{5}{p}_{h}{z}_{h}^{\pm}}\le {\epsilon}^{\pm},\forall h$$$${z}_{h}^{\pm}=0\text{}\mathrm{or}\text{}1,\forall h$$
- (7)
- Cropland resources constraints:$$\sum _{j=1}^{4}{\displaystyle \sum _{k=1}^{6}PP{A}_{jkth}^{\pm}}}\ge MF{P}_{th}^{\pm},\text{}\forall h,\text{}t=1$$$$\sum _{j=1}^{4}{\displaystyle \sum _{k=1}^{2}PP{A}_{jkth}^{\pm}}}+\text{}{\displaystyle \sum _{j=1}^{4}{\displaystyle \sum _{k=8}^{9}PP{A}_{jkth}^{\pm}}}\ge MF{P}_{th}^{\pm},\text{}\forall h,\text{}t=2$$$$P{A}_{jkt\text{}\mathrm{min}}^{\pm}\le PP{A}_{jkth}^{\pm}\le P{A}_{jkt}^{\pm}\le P{A}_{jkt\text{}\mathrm{max}}^{\pm},\text{}\forall j,\text{}t,\text{}k,h$$
- (8)
- Industrial production scale constraints:$$PL{C}_{it,\text{}\mathrm{min}}\le AP{C}_{ith}^{\pm}\le PL{C}_{it}^{\pm}\le PL{C}_{it,\text{}\mathrm{max}},\text{}\forall i,\text{}t,\text{}h$$$$Q{W}_{st,\text{}\mathrm{min}}\le P{W}_{sth}^{\pm}\le Q{W}_{st}^{\pm}\le Q{W}_{st,\text{}\mathrm{max}},\forall s,\text{}t,\text{}h$$$$PL{M}_{nt,\text{}\mathrm{min}}\le AP{M}_{nth}^{\pm}\le PL{M}_{nt}^{\pm}\le PL{M}_{nt,\text{}\mathrm{max}},\text{}\forall n,\text{}t,\text{}h$$
- (9)
- Non-negative constraints:$$AP{C}_{ith}^{\pm},\text{}PP{A}_{jkth}^{\pm},\text{}P{W}_{sth}^{\pm},\text{}AP{M}_{nth}^{\pm}\text{}\ge 0,\text{}\forall i,\text{}j,\text{}s,\text{}n,\text{}k,\text{}t,\text{}h$$

#### 3.3. Data Analysis

## 4. Results and Discussion

#### 4.1. Benefit and Cost

^{6}), the system benefit would be RMB¥ [121.9, 646.3] × 10

^{6}; when ε = 0.64 (with ${\eta}^{\pm}$ = RMB¥ [160,800] × 10

^{6}), the system benefit would be RMB¥ [160.8, 800.6] × 10

^{6}. An increased ε level (a raised risk) corresponds to a decreased strictness for the constraints and thus an increased system benefit; such an increased value, however, would be associated with a raised constraint-violation risk.

#### 4.2. Risk Analysis

^{6}under ε = 0.36 and RMB¥ [173.7, 820.1] × 10

^{6}under ε = 0.88, respectively. The higher risk level corresponds to a higher targeted benefit and thus a higher production level, leading to a higher benefit. A straightforward way of evaluating the tradeoff between risk and benefit is to use the cumulative risk curve. Figure 5 illustrates the lower and upper bound cumulative risk curves under different risk levels, respectively. Under a fixed ε level, the cumulative risk curve indicates the level of incurred risk for the generation target at each benefit level (which is the difference between regular benefit and penalty under a designed pollutant discharge allowance scenario). The cumulative risk curves would monotonically increase because they are cumulative probability functions; therefore, the risk is given by the cumulative probability. The risk area would increase as the ε level is raised. Generally, decisions at a lower ε level (i.e., risk averse) would lead to an increased reliability in fulfilling the system requirements but with a lower system benefit; conversely, a strong desire to obtain high system benefit (i.e., risk neutral or risk prone) would yield a raised risk of violating the constraints. It is revealed that a tradeoff between system benefit and system failure risk exists in the decision processes.

#### 4.3. Proportion of Benefit under Different Scenarios

#### 4.4. Optimal Production Target and Actual Production Level

#### 4.5. Excess Pollutant Discharge

## 5. Conclusions

## Acknowledgements

## Author Contributions

## Conflicts of Interest

## Appendix A. Solution Method

## Appendix B. Nomenclatures of SIRM Model

i | chemical plant, 1 = Gufu (GF), 2 = Baishahe (BSH), 3 = Pingyikou (PYK), 4 = Liucaopo (LCP), 5 = Xiangjinlianying (XJLY) |

j | agricultural zone, and j = 1, 2, 3, 4 |

k | main crop, 1 = citrus, 2 = tea, 3 = wheat, 4 = potato, 5 = rapeseed, 6 = alpine rice, 7 = second rice, 8 = maize, 9 = vegetables |

n | phosphorus mining company; 1 = Xinglong (XL), 2 = Xinghe (XH), 3 = Xingchang (XC), 4 = Geping (GP), 5 = Jiangjiawan (JJW), 6 = Shenjiashan (SJS) |

s | town, 1 = Gufu, 2 = Nanyang, 3 = Gaoyang, 4 = Xiakou |

t | planning time period, 1 = dry season, 2 = wet season |

L_{t} | length of period (day) |

l | independent variable representing time with the range of [0, 180] |

h | allowable pollutant discharge scenario; h = 1, 2, 3 |

${p}_{h}$ | probability of occurrence allowable pollutant discharge level h (%); |

$B{C}_{it}^{\pm}$ | net benefit from chemical plant i in period t (RMB¥/t); |

$PL{C}_{it}^{\pm}$ | production level of chemical plant i in period t (t/day); |

$B{P}_{nt}^{\pm}$ | average benefit for per unit phosphate ore (RMB¥/t); |

$PL{M}_{nt}^{\pm}$ | production level of phosphorus mining company n during period t (t/day); |

$B{W}_{st}^{\pm}$ | net benefit from water supply to municipal uses (RMB¥/m^{3}); |

$Q{W}_{st}^{\pm}$ | quantity of water supply to town s in period t (m^{3}/day); |

$B{A}_{jkt}^{\pm}$ | average benefit for agricultural product (RMB¥/t); |

$P{A}_{jkt}^{\pm}$ | planning area of crop k in agricultural zone j during period t (ha); |

$PB{C}_{it}^{\pm}$ | economic cost when the production targets from plant i in period t are not met (RMB¥/t); |

$AP{C}_{ith}^{\pm}$ | actual production level of plant i under scenario h in period t (t/day); |

$PB{P}_{nt}^{\pm}$ | economic cost when the production targets from company n in period t are not met (RMB¥/t); |

$AP{M}_{nth}^{\pm}$ | actual production level of company n under scenario h in period t (t/day); |

$PB{W}_{st}^{\pm}$ | economic cost when the water supply targets for town s in period t are not met (RMB¥/m^{3}); |

$P{W}_{sth}^{\pm}$ | actual water supply level of town s under scenario h in period t (m^{3}/day); |

$PB{A}_{jkt}^{\pm}$ | economic cost when the plant targets of crop farm j in period t are not met (RMB¥/ha); |

$PB{A}_{jkth}^{\pm}$ | actual plant area of crop farm j under scenario h in period t (ha); |

$W{C}_{it}^{\pm}$ | wastewater generation rate of chemical plant i during period t (m^{3}/t) |

$G{T}_{st}^{\pm}$ | wastewater generation rate at town s during period t (m^{3}/m^{3}) |

$TP{C}_{st}^{\pm}$ | capacity of wastewater treatment capacity (WTPs) (m^{3}/day) |

$TP{D}_{it}^{\pm}$ | capacity of wastewater treatment capacity (chemical plants) (m^{3}/day) |

$I{C}_{it}^{\pm}$ | BOD concentration of raw wastewater from chemical plant i in period t (kg/m^{3}) |

${\eta}_{BOD,it}^{\pm}$ | BOD treatment efficiency in chemical plant i during period t (%) |

$AB{C}_{ith}^{\pm}(l)$ | allowable BOD discharge for chemical plant i under scenario h in period t (kg/day) |

${\mathrm{BM}}_{\mathrm{st}}^{\pm}$ | BOD concentration of municipal wastewater at town s during period t (kg/m^{3}) |

${\eta}_{BOD,st}^{\pm}$ | BOD treatment efficiency of WTPs at town s during period t (%) |

$AB{W}_{sth}^{\pm}(l)$ | allowable BOD discharge for WTPs at town s during period t (kg/day) |

$N{S}_{jk}^{\pm}$ | nitrogen content of soil in agricultural zone j planted with crop k (%) |

$S{L}_{jkt}^{\pm}$ | average soil loss from agricultural zone j planted with crop k in period t (t/ha) |

$R{F}_{jkt}^{\pm}$ | runoff from agricultural zone j with crop k in period t (mm) |

$D{N}_{jkt}^{\pm}$ | dissolved nitrogen concentration in runoff from agricultural zone j planted with crop k in period t (mg/L) |

$MN{L}_{jth}^{\pm}(l)$ | maximum allowable nitrogen loss in agricultural zone j under scenario h during period t (t) |

$PC{R}_{it}^{\pm}$ | phosphorus concentration of raw wastewater from chemical plant i in period t (kg/m^{3}) |

${\eta}_{TP,it}^{\pm}$ | phosphorus treatment efficiency in chemical plant i in period t (%) |

$AS{C}_{it}^{\pm}$ | amount of slag discharged by chemical plant i in period t (kg/t) |

$SL{R}_{it}^{\pm}$ | slag loss rate due to rain wash in chemical plant i during period t (%) |

$PS{C}_{it}^{\pm}$ | phosphorus content in slag generated by chemical plant i in period t (%) |

$AP{C}_{ith}^{\pm}(l)$ | allowable phosphorus discharge for chemical plant i under scenario h in period t (kg/day) |

$PC{M}_{st}^{\pm}$ | phosphorus concentration of municipal wastewater at town s in period t (kg/m^{3}) |

${\eta}_{TP,st}^{\pm}$ | phosphorus treatment efficiency of WTP at town s in period t (%) |

$AP{W}_{sth}^{\pm}(l)$ | allowable phosphorus discharge for WTP at town s under scenario h in period t (kg/day) |

$WP{M}_{nt}^{\pm}$ | wastewater generation from phosphorus mining company n in period t (m^{3}/t) |

$MW{C}_{nt}^{\pm}$ | phosphorus concentration of wastewater from mining company n in period t (kg/ m^{3}) |

${\eta}_{TP,nt}^{\pm}$ | phosphorus treatment efficiency in mining company n (%) |

$AS{M}_{nt}^{\pm}$ | amount of slag discharged by mining company n during period t (kg/t) |

$PC{S}_{nt}^{\pm}$ | phosphorus content in generated slag (%) |

$SL{W}_{nt}^{\pm}$ | slag loss rate due to rain wash (%) |

$APC{M}_{nth}^{\pm}(l)$ | allowable phosphorus discharge for mining company n under scenario h during period t (kg/day) |

$P{S}_{jk}^{\pm}$ | phosphorus content of soil in agricultural zone j planted with crop k (%) |

$S{L}_{jkt}^{\pm}$ | average soil loss from agricultural zone j planted with crop k in period t (t/ha) |

$D{P}_{jkt}^{\pm}$ | dissolved phosphorus concentration in runoff from agricultural zone j with crop k (mg/L) |

$MP{L}_{jth}^{\pm}(l)$ | maximum allowable phosphorus loss in agricultural zone j under scenario h in period t (t/ha) |

$MS{L}_{jth}^{\pm}(l)$ | maximum allowable soil loss agricultural zone j under scenario h in period t (t) |

${\eta}^{\pm}$ | targeted benefit level of economic activities (RMB¥) |

${z}_{h}^{\pm}$ | integer variable, which would take a value of zero if the benefit for each economic activity under scenario h is greater than or equal to the target level (${\eta}^{\pm}$) and a value of one otherwise |

$U{B}_{h}$ | upper bound benefit under each scenario h (RMB¥) |

${\epsilon}^{\pm}$ | desired risk exposure level of economic activity |

$MF{P}_{th}^{\pm}$ | the government requirement for minimum area of farmland during period t (ha); |

$PL{C}_{it,\text{}\mathrm{min}}^{\pm}$ | minimum production level of chemical plant i in period t (t/day) |

$PL{C}_{it,\text{}\mathrm{max}}^{\pm}$ | maximum production level of chemical plant i in period t (t/day) |

$Q{W}_{st,\text{}\mathrm{min}}^{\pm}$ | minimum quantity of water supply to town s in period t (m^{3}/day) |

$Q{W}_{st,\text{}\mathrm{max}}^{\pm}$ | maximum quantity of water supply to town s in period t (m^{3}/day) |

$PL{M}_{nt,\text{}\mathrm{min}}^{\pm}$ | minimum production level of phosphorus mining company n during period t (t/day) |

$PL{M}_{nt,\text{}\mathrm{max}}^{\pm}$ | maximum production level of phosphorus mining company n during period t (t/day) |

## References

- Ministry of Water Resources, People’s Republic of China. China Water Resources Bulletin; China Water Power Press: Beijing, China, 2017.
- Cederkvist, K.; Jensen, M.B.; Ingvertsen, S.T.; Holm, P.E. Controlling stormwater quality with filter soil—Event and dry weather testing. Water
**2016**, 8, 349. [Google Scholar] [CrossRef] - He, J. Probabilistic evaluation of causal relationship between variables for water pollution control. J. Environ. Inform.
**2016**, 28, 110–119. [Google Scholar] [CrossRef] - Toyosada, K.; Otani, T.; Shimizu, Y.; Managi, S. Water quality study on the hot and cold water supply systems at Vietnamese Hotels. Water
**2017**, 9, 251. [Google Scholar] [CrossRef] - Li, Y.P.; Zhang, N.; Huang, G.H.; Liu, J. Coupling fuzzy-chance constrained program with minimax regret analysis for water pollution control. Stoch. Environ. Res. Risk Assess.
**2014**, 28, 1769–1784. [Google Scholar] [CrossRef] - Li, Y.P.; Huang, G.H. Fuzzy-stochastic-based violation analysis method for planning water resources management systems with uncertain information. Inf. Sci.
**2009**, 179, 4261–4276. [Google Scholar] [CrossRef] - Li, Y.P.; Huang, G.H. Two-stage planning for sustainable water-quality management under uncertainty. J. Environ. Manag.
**2009**, 90, 2402–2413. [Google Scholar] [CrossRef] [PubMed] - Liu, J.; Li, Y.P.; Huang, G.H.; Nie, S. Development of a fuzzy-boundary interval programming method for water pollution control under uncertainty. Water Resour. Manag.
**2014**, 29, 1169–1191. [Google Scholar] [CrossRef] - Kerachian, R.; Karamouz, M. A stochastic conflict resolution model for water pollution control in reservoir-river systems. Adv. Water Resour.
**2007**, 30, 866–882. [Google Scholar] [CrossRef] - Li, Y.P.; Huang, G.H.; Yang, Z.F.; Nie, S.L. IFMP: Interval-fuzzy multistage programming for water resources management under uncertainty. Resour. Conserv. Recycl.
**2008**, 52, 800–812. [Google Scholar] [CrossRef] - Ocampo-Duque, W.; Osorio, C.; Piamba, C.; Schuhmacher, M.; Domingo, J.L. Water quality analysis in rivers with non-parametric probability distributions and fuzzy inference systems: Application to the Cauca River, Colombia. Environ. Int.
**2013**, 52, 17–18. [Google Scholar] [CrossRef] [PubMed] - Mohammad, R.N.; Najmeh, M. Water quality zoning using probabilistic support vector machines and self-organizing maps. Water Resour. Manag.
**2013**, 27, 2577–2594. [Google Scholar] [CrossRef] - Chen., C.F.; Tsai, L.Y.; Fan, C.H.; Lin, J.Y. Using exceedance probability to determine total maximum daily loads for reservoir water pollution control. Water
**2016**, 8, 541. [Google Scholar] [CrossRef] - Hwang, S.A.; Hwang, S.J.; Park, S.R.; Lee, S.W. Examining the relationships between water shed urban land use and stream water quality using linear and generalized additive models. Water
**2016**, 8, 155. [Google Scholar] [CrossRef] - Ji., Y.; Huang, G.H.; Sun, W.; Li, Y.F. Water pollution control in a wetland system using an inexact left-hand-side chance-constrained fuzzy multi-objective approach. Stoch. Environ. Res. Risk Assess
**2016**, 30, 621–633. [Google Scholar] [CrossRef] - Zolfagharipoor, M.A.; Ahmadi, A. A decision-making framework for river water pollution control under uncertainty: Application of social choice rules. J. Environ. Manag.
**2016**, 183, 152–163. [Google Scholar] [CrossRef] [PubMed] - Chen, Y.; Zou, R.; Han, S.; Bai, S.; Faizullabhoy, M.; Wu, Y.Y.; Guo, H.C. Development of an integrated water quality and macroalgae simulation model for Tidal Marsh eutrophication control decision support. Water
**2017**, 9, 277. [Google Scholar] [CrossRef] - Li, Y.P.; Huang, G.H.; Huang, Y.F.; Zhou, H.D. A multistage fuzzy-stochastic programming model for supporting sustainable water resources allocation and management. Environ. Model. Softw.
**2009**, 24, 786–797. [Google Scholar] [CrossRef] - Nematian, J. An extended two-stage stochastic programming approach for water resources management under uncertainty. J. Environ. Inform.
**2016**, 27, 72–84. [Google Scholar] [CrossRef] - Huang, G.H.; Loucks, D.P. An inexact two-stage stochastic programming model for water resources management under uncertainty. Civ. Eng. Environ. Syst.
**2000**, 17, 95–118. [Google Scholar] [CrossRef] - Zhang, N.; Li, Y.P.; Huang, W.W.; Liu, J. An inexact two-stage water pollution control model for supporting sustainable development in a rural system. J. Environ. Inform.
**2014**, 24, 52–64. [Google Scholar] [CrossRef] - Luo, B.; Li, J.B.; Huang, G.H.; Li, H.L. A simulation-based interval two-stage stochastic model for agricultural non-point source pollution control through land retirement. Sci. Total Environ.
**2006**, 361, 38–56. [Google Scholar] [CrossRef] [PubMed] - Harrison, K.W. Two-stage decision-making under uncertainty and stochasticity: Bayesian Programming. Adv. Water Resour.
**2007**, 30, 641–664. [Google Scholar] [CrossRef] - Tavakoli, A.; Nikoo, M.R.; Kerachian, R.; Soltani, M. River water pollution control considering agricultural return flows: Application of a nonlinear two-stage stochastic fuzzy programming. Environ. Monit. Assess.
**2015**, 187, 158. [Google Scholar] [CrossRef] [PubMed] - Hu, X.H.; Li, Y.P.; Huang, G.H.; Zhuang, X.W.; Ding, X.W. A Bayesian-based two-stage inexact optimization method for supporting stream water pollution control in the three gorges reservoir region. Environ. Sci. Pollut. Res.
**2016**, 23, 9164–9182. [Google Scholar] [CrossRef] [PubMed] - Ahmed, S.; Sahinidis, N.V. Robust process planning under uncertainty. Ind. Eng. Chem. Res.
**1998**, 37, 1883–1892. [Google Scholar] [CrossRef] - Li, Y.P.; Huang, G.H. Interval-parameter robust optimization for environmental management under uncertainty. Can. J. Civ. Eng.
**2009**, 36, 592–606. [Google Scholar] [CrossRef] - Khan, U.T.; Valeo, C. Short-term peak flow rate prediction and flood risk assessment using fuzzy linear regression. J. Environ. Inform.
**2016**, 28, 71–89. [Google Scholar] [CrossRef] - Koppol, A.P.R.; Bagajewicz, M.J. Financial risk management in the design of water utilization systems in process plants. Ind. Eng. Chem. Res.
**2004**, 42, 5249–5255. [Google Scholar] [CrossRef] - Barbaro, A.; Bagajewicz, M.J. Managing financial risk in planning under uncertainty. Process Syst. Eng.
**2004**, 50, 963–989. [Google Scholar] [CrossRef] - Liu, Z.F.; Huang, G.H. Dual-interval two-stage optimization for flood management and risk analyses. Water Resour. Manag.
**2009**, 23, 2141–2162. [Google Scholar] [CrossRef] - Ahmed, S.; Elsholkami, M.; Elkamel, A.; Du, J.; Ydstie, E.B.; Douglas, P.L. Financial risk management for new technology integration in energy planning under uncertainty. Appl. Energy
**2014**, 128, 75–81. [Google Scholar] [CrossRef] - Zhu, Y.; Li, Y.P.; Huang, G.H. Planning carbon emission trading for Beijing’s electric power systems under dual uncertainties. Renew. Sustain. Energy Rev.
**2013**, 23, 113–128. [Google Scholar] [CrossRef] - Felfel, H.; Ayadi, O.; Masmoudi, F. Multi-objective stochastic multi-site supply chain planning under demand uncertainty considering downside risk. Comput. Ind. Eng.
**2016**, 102, 268–279. [Google Scholar] [CrossRef] - McCray, A.W. Petroleum Evaluations and Economic Decisions; Prentice Hall: Englewood Cliffs, NJ, USA, 1975; ISBN 0136622136. [Google Scholar]
- Chen, H.W.; Chang, N.B. Decision support for allocation of watershed pollution load using grey fuzzy multobjective programming. J. Am. Water Resour. Assoc.
**2006**, 42, 725–745. [Google Scholar] [CrossRef]

**Figure 2.**Study area. Note: GF, Gufu chemical plant; XL, Xinglong phosphorus mining company; XH, Xinghe phosphorus mining company; XC, Xingchang phosphorus mining company; BSH, Baishahe chemical plant; PYK, Pingyikou chemical plant; LCP, Liucaopo chemical plant; GP, Geping phosphorus mining company; JJW, Jiangjiawan phosphorus mining company; SJS, Shenjiashan phosphorus mining company; XJLY, Xiangjinlianying chemical plant.

**Figure 3.**Benefits and costs under different risk levels. Note: (

**a**) lower-bound targeted benefit and system benefit, (

**b**) upper-bound targeted benefit and system benefit, (

**c**) lower-bound regular benefit and penalty, (

**d**) upper-bound regular benefit and penalty; unit: 10

^{6}RMB¥.

**Figure 4.**Benefit corresponding to each scenario and risk level. (

**a**) Lower bound and (

**b**) Upper bound.

**Figure 6.**Proportion of benefit under different scenarios. (

**a**) Lower bound under low scenario; (

**b**) Upper bound under low scenario; (

**c**) Lower bound under low-medium scenario; (

**d**) Upper bound under low-medium scenario; (

**e**) Lower bound under medium scenario; (

**f**) Upper bound under medium scenario; (

**g**) Lower bound under medium-high scenario; (

**h**) Upper bound under medium-high scenario; (

**i**) Lower bound under high scenario and (

**j**) Upper bound under high scenario.

**Figure 7.**Actual production levels of industrial activities. Note: (

**a**) lower-bound production level of chemical plant; (

**b**) upper-bound production level of chemical plant; (

**c**) lower-bound production level of phosphorus mining company; (

**d**) upper-bound production level of phosphorus mining company; (

**e**) lower-bound water supply and (

**f**) upper-bound water supply.

Industrial Activities | Net Benefit | Penalty | ||
---|---|---|---|---|

Period 1 | Period 2 | |||

Chemical plant (RMB¥/t) | ||||

Gufu (GF) | [791.3, 832.4] | [803.2, 864.7] | [823.6, 891.5] | [833.1, 910.4] |

Baishahe (BSH) | [1217.4, 1493.8] | [1532.3, 1682.4] | [1357.4, 1634.1] | [1723.7, 1841.6] |

Pingyikou (PYK) | [832.8, 912.4] | [943.5, 1032.4] | [861.5, 943.8] | [993.5, 1145.7] |

Liucaopo (LCP) | [1532.6, 1843.5] | [1682.2, 1942.5] | [1735.7, 2011.1] | [1835.3, 2113.5] |

Xiangjinlianying (XJLY) | [1843.3, 2174.3] | [1932.4, 2243.5] | [1992.3, 2294.5] | [2014.5, 2385.3] |

Water supply (RMB¥/m^{3}) | ||||

Gufu | [45.4, 52.7] | [58.8, 61.4] | [52.4, 58.9] | [63.7, 68.2] |

Nanyang | [32.4, 38.5] | [41.4, 46.7] | [39.5, 43.1] | [47.2, 53.4] |

Gaoyang | [48.3, 52.4] | [56.4, 59.4] | [55.5, 58.7] | [63.7, 70.1] |

Xiakou | [42.1, 46.2] | [48.3, 53.4] | [49.2, 54.5] | [53.8, 59.1] |

Phosphorus mining company (RMB¥/t) | ||||

Xinglong (XL) | [167.3, 185.3] | [167.3, 194.2] | [196.7, 213.5] | [196.7, 221.3] |

Xinghe (XH) | [154.5, 164.9] | [163.4, 173.4] | [178.4, 197.3] | [182.1, 207.3] |

Xingchang (XC) | [161.5, 178.4] | [161.5, 178.4] | [188.4, 204.5] | [188.4, 204.5] |

Geping (GP) | [166.4, 174.7] | [168.3, 179.4] | [189.2, 201.3] | [191.2, 207.3] |

Jiangjiawan (JJW) | [161.4, 178.6] | [166.4, 181.4] | [188.3, 204.6] | [189.2, 211.3] |

Shenjiashan (SJS) | [161.4, 183.4] | [171.9, 191.5] | [188.3, 211.3] | [198.2, 201.4] |

Industrial Activities | Period 1 | Period 2 |
---|---|---|

Allowable phosphorus discharge for each chemical plant (kg/day) | ||

Gufu (GF) | [0.60 + 0.002l, 1.05 + 0.002l] | [0.70 + 0.001l, 1.15 + 0.001l] |

Baishahe (BSH) | [195.63 + 0.089l, 432.56 + 0.089l] | [162.64 + 0.086l, 407.87+ 0.086l] |

Pingyikou (PYK) | [30.49 + 0.09l, 70.71+ 0.09l] | [50.03 + 0.013l, 90.30 + 0.013l] |

Liucaopo (LCP) | [124.7 + 0.046l, 294.7 + 0.046l] | [213.40 + 0.043l, 409.5 + 0.043l] |

Xiangjinlianying (XJLY) | [126.22 + 0.042l, 321.76 + 0.042l] | [174.4 + 0.039l, 364.8 + 0.039l] |

Allowable phosphorus discharge for each mine company (kg/m^{3}) | ||

Xinglong (XL) | [47.41 + 0.043l, 123.49 + 0.043l] | [78.31 + 0.41l, 191.59 + 0.041l] |

Xinghe (XH) | [23.68 + 0.044l, 86.87 + 0.044l] | [39.19 + 0.42l, 86.87 + 0.042l] |

Xingchang (XC) | [30.54 + 0.038l], 62.75, + 0.038l] | [52.74 + 0.034l, 115.76 + 0.034l] |

Geping (GP) | [48.39 + 0.012l, 109.36 + 0.012l] | [70.03 + 0.011l, 173.95 + 0.011l] |

Jiangjiawan (JJW) | [25.91 + 0.014l, 40.13 + 0.014l] | [38.49 + 0.013l, [87.51 + 0.013l] |

Shenjiashan (SJS) | [20.23 + 0.011l, 42.3l + 0.011l] | [28.19 + 0.009l, [82.6l + 0.009l] |

Allowable BOD discharge for each WTP (kg/day) | ||

Gufu | [70 + 0.003l, 110 + 0.003l] | [75 + 0.002l, 115 + 0.002l] |

Nanyang | [3 + 0.0001l, 8 + 0.0001l] | [4 + 0.0009l, 9 + 0.0009l] |

Gaoyang | [10 + 0.0003l, 25 + 0.0003l] | [12 + 0.0002l, 27 + 0.0002l] |

Xiakou | [15 + 0.0002l, 28 + 0.0002l] | [17 + 0.0001l, 30 + 0.0001l] |

Activity | Designed Target | ε = 0 | ε = 1 | |||
---|---|---|---|---|---|---|

t = 1 | t = 2 | t = 1 | t = 2 | t = 1 | t = 2 | |

Production level of chemical plants (t/day) | ||||||

i = 1 | [21.8, 28.6] | [17.6, 28.3] | 28.6 | 28.3 | 28.6 | 28.3 |

i = 2 | [457.5, 573.7] | [461.6, 625.6] | 573.7 | 461.6 | 573.7 | 625.6 |

i = 3 | [69.5, 112.6] | [62.8, 110.5] | 69.5 | 110.5 | 112.6 | 110.5 |

i = 4 | [347.2, 526.8] | [327.6, 552.4] | 347.2 | 552.4 | 526.8 | 552.4 |

i = 5 | [260.4, 376.8] | [256.7, 373.7] | 260.4 | 256.7 | 376.8 | 373.7 |

Production level of phosphorus mining companies (t/day) | ||||||

n = 1 | [679.4, 1435.6] | [736.5, 1324.5] | 679.4 | 736.5 | 1435.6 | 1324.5 |

n = 2 | [246.7, 424.5] | [194.6, 412.5] | 424.5 | 412.5 | 424.5 | 412.5 |

n = 3 | [257.4, 436.4] | [227.5, 415.3] | 436.4 | 415.3 | 436.4 | 415.3 |

n = 4 | [385.3, 783.5] | [325.6, 704.5] | 783.5 | 704.5 | 783.5 | 704.5 |

n = 5 | [348.6, 568.4] | [287.4, 503.5] | 568.4 | 503.5 | 568.4 | 503.5 |

n = 6 | [495.6, 689.4] | [437.8,612.7] | 689.4 | 437.8 | 689.4 | 612.7 |

Quantity of water supply to towns (m^{3}/day) | ||||||

s = 1 | [9921.5, 17533.5] | [10385.7, 21756.4] | 9921.5 | 10385.7 | 17533.5 | 21756.4 |

s = 2 | [729.5, 1232.7] | [798.5, 1463.6] | 1232.7 | 1463.6 | 1232.7 | 1463.6 |

s = 3 | [2804.5, 3424.5] | [2964.5, 3853.7] | 3424.5 | 3853.7 | 3424.5 | 3853.7 |

s = 4 | [1428.7, 2104.7] | [1564.6, 2746.4] | 2104.7 | 2746.4 | 2104.7 | 2746.4 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Liu, J.; Li, Y.; Huang, G.; Fan, Y.
A Semi-Infinite Interval-Stochastic Risk Management Model for River Water Pollution Control under Uncertainty. *Water* **2017**, *9*, 351.
https://doi.org/10.3390/w9050351

**AMA Style**

Liu J, Li Y, Huang G, Fan Y.
A Semi-Infinite Interval-Stochastic Risk Management Model for River Water Pollution Control under Uncertainty. *Water*. 2017; 9(5):351.
https://doi.org/10.3390/w9050351

**Chicago/Turabian Style**

Liu, Jing, Yongping Li, Guohe Huang, and Yurui Fan.
2017. "A Semi-Infinite Interval-Stochastic Risk Management Model for River Water Pollution Control under Uncertainty" *Water* 9, no. 5: 351.
https://doi.org/10.3390/w9050351