# A Recourse-Based Type-2 Fuzzy Programming Method for Water Pollution Control under Uncertainty

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

^{3}$/km

^{2}with additional information as the possibility of “the most possible unit benefit of 499.8 × 10

^{3}$/km

^{2}” is 0.8, and the possibilities of ‘‘there is no possibility that the unit benefit is lower than 424.1 × 10

^{3}$/km

^{2}or higher than 575.6 × 10

^{3}$/km

^{2}” are 0.2 and 0.3, respectively. Under such a situation, unit benefit from planting fruit/vegetable should be described as type-2 fuzzy sets (T2FS). Thus, type-2 fuzzy programming (TFP) can be adopted to tackle such uncertainties, which can be presented as [24]:

- Step 1.
- Formulate the RTFP model.
- Step 2.
- Discrete probability distribution into several values with each corresponds to one probability.
- Step 3.
- Convert the output of T2FS into conventional fuzzy sets.
- Step 4.
- Conduct defuzzification of T2FS according to the critical value (CV)-based reduction method.
- Step 5.
- Run the RTFP model.
- Step 6.
- Obtain the optimal solutions of the objective function ($f$), first-stage decision variable (${x}_{j}$), and second-stage decision variable (${y}_{js}$).

## 3. Case Study

^{2}, and the majority of the county (i.e., approximately 1800 km

^{2}) lies within the middle reaches of the Heshui River Basin [31]. The basin features subtropical monsoon humid climatic conditions (abundant rainfall and sunlight, and long frost-free periods) with an average annual precipitation of 1530.7 mm and an average annual temperature of 18.2 °C. The Heshui River has a total length of 225 km, with 77 km flowing within the borders of Yongxin County from west to east. The county mainly relies on the Heshui River to support its agriculture, industry, livestock husbandry, forestry, and fishery [32].

- (1)
- COD discharge constraints:$$\sum _{i=1}^{{I}_{f}}(T{F}_{ij}^{}-X{F}_{ijh}^{})\cdot CC{O}_{ij}^{}}\le PC{F}_{jh}^{},\text{\hspace{1em}}\forall j,\text{}h,$$$$\sum _{i=1}^{{I}_{i}}(T{L}_{ij}^{}-X{L}_{ijh}^{})\cdot DP{L}_{ij}^{\pm}}\le PC{L}_{jh}^{},\text{\hspace{1em}}\forall j,\text{}h,$$$$\sum _{i=1}^{{I}_{i}}(T{I}_{ij}^{}-X{I}_{ijh}^{})}\cdot DP{I}_{ij}^{\pm}\le PC{I}_{jh}^{},\text{\hspace{1em}}\forall j,\text{}h,$$$$\begin{array}{l}{\displaystyle \sum _{j=1}^{J}[{\displaystyle \sum _{i=1}^{{I}_{f}}(T{F}_{ij}^{}-X{F}_{ijh}^{})\cdot CC{O}_{ij}^{}}+{\displaystyle \sum _{i=1}^{{I}_{i}}(T{L}_{ij}^{}-X{L}_{ijh}^{})\cdot DP{L}_{ij}^{}}}\\ +{\displaystyle \sum _{i=1}^{{I}_{i}}(T{I}_{ij}^{}-X{I}_{ijh}^{})}\cdot DP{I}_{ij}^{}]\le MC{L}_{h}^{},\text{\hspace{1em}}\forall h\end{array},$$
- (2)
- Phosphorus discharge constraints:$$\sum _{i=1}^{{I}_{a}}(T{A}_{ij}^{}-X{A}_{ijh}^{})\cdot (S{L}_{ij}^{}\cdot A{P}_{ij}^{}+R{A}_{ij}^{}\cdot R{P}_{ij}^{})}\le PA{P}_{jh}^{},\text{\hspace{1em}}\forall j,\text{}h,$$$$\sum _{i=1}^{{I}_{f}}(T{F}_{ij}^{}-X{F}_{ijh}^{})\cdot F{P}_{ij}^{\pm}}\le PF{P}_{jh}^{},\text{\hspace{1em}}\forall j,\text{}h,$$$$\begin{array}{l}{\displaystyle \sum _{j=1}^{J}[{\displaystyle \sum _{i=1}^{{I}_{a}}(T{A}_{ij}^{}-X{A}_{ijh}^{})\cdot (S{L}_{ij}^{}\cdot A{P}_{ij}^{}+R{A}_{ij}^{}\cdot R{P}_{ij}^{})}}\\ +{\displaystyle \sum _{i=1}^{{I}_{f}}(T{F}_{ij}^{}-X{F}_{ijh}^{})\cdot F{P}_{ij}^{}}]\le MP{L}_{h}^{},\text{\hspace{1em}}\forall h\end{array},$$
- (3)
- Nitrogen discharge constraints:$$\sum _{i=1}^{{I}_{}}(T{A}_{ij}^{}-X{A}_{ijh}^{})\cdot (S{L}_{ij}^{}\cdot A{N}_{ij}^{}+R{A}_{ij}^{}R{N}_{ij}^{})}\le PA{N}_{jh}^{},\text{\hspace{1em}}\forall j,\text{}h,$$$$\sum _{i=1}^{{I}_{f}}(T{F}_{ij}^{}-X{F}_{ijh}^{})\cdot F{N}_{ij}^{}}\le PF{N}_{jh}^{},\text{\hspace{1em}}\forall j,\text{}h,$$$$\begin{array}{l}{\displaystyle \sum _{i=1}^{{I}_{a}}(T{A}_{ij}^{}-X{A}_{ijh}^{})\cdot (S{L}_{ij}^{}\cdot A{N}_{ij}^{}+R{A}_{ij}^{}\cdot R{N}_{ij}^{})}\\ +{\displaystyle \sum _{i=1}^{{I}_{f}}(T{F}_{ij}^{}-X{F}_{ijh}^{})\cdot F{N}_{ij}^{}}\le MN{L}_{h}^{},\text{\hspace{1em}}\forall h\end{array},$$
- (4)
- Soil loss constraints:$$\sum _{i=1}^{{I}_{i}}(T{A}_{ij}^{}-X{A}_{ijh}^{})\cdot S{L}_{ij}^{}}\le PS{L}_{jh}^{},\text{\hspace{1em}}\forall j,\text{}h,$$$$\sum _{i=1}^{{I}_{w}}(T{W}_{ij}^{}-X{W}_{ijh}^{})\cdot DP{W}_{ij}^{}}\le PW{S}_{jh}^{},\text{\hspace{1em}}\forall j,\text{}h,$$$$\begin{array}{l}{\displaystyle \sum _{j=1}^{J}[{\displaystyle \sum _{i=1}^{{I}_{i}}(T{A}_{ij}^{}-X{A}_{ijh}^{})\cdot S{L}_{ij}^{\pm}}}\\ +{\displaystyle \sum _{i=1}^{{I}_{w}}(T{W}_{ij}^{}-X{W}_{ijh}^{})\cdot DP{W}_{ij}^{}}]\le MS{L}_{h}^{},\text{\hspace{1em}}\forall h\end{array},$$
- (5)
- Water supply balance constraint:$$\begin{array}{l}{\displaystyle \sum _{h=1}^{H}(}{\displaystyle \sum _{i=1}^{{I}_{i}}(T{A}_{ij}^{}-X{A}_{ijh}^{})\cdot W{A}_{i}^{}}+{\displaystyle \sum _{i=1}^{{I}_{f}}(T{F}_{ij}^{}-X{F}_{ijh}^{})\cdot W{F}_{i}^{}}\\ +{\displaystyle \sum _{i=1}^{{I}_{l}}(T{L}_{ij}^{}-X{L}_{ijh}^{})\cdot W{L}_{i}^{}}+{\displaystyle \sum _{i=1}^{{I}_{i}}(T{I}_{ij}^{}-X{I}_{ijh}^{})\cdot W{I}_{i}^{}}\\ +{\displaystyle \sum _{i=1}^{{I}_{w}}(T{W}_{ij}^{}-X{W}_{ijh}^{})\cdot W{W}_{i}^{}}]\le MAX{W}_{j}^{},\text{\hspace{1em}}\forall j\end{array},$$
- (6)
- Product demand constraints:$$T{A}_{i\text{}\mathrm{min}}^{}\le {\displaystyle \sum _{j=1}^{J}T{A}_{ij}^{}}\le T{A}_{i\text{}\mathrm{max}}^{},\text{\hspace{1em}}\forall i,$$$$T{F}_{i\text{}\mathrm{min}}^{}\le {\displaystyle \sum _{j=1}^{J}T{F}_{ij}^{}}\le T{F}_{i\text{}\mathrm{max}}^{},\text{\hspace{1em}}\forall i,$$$$T{L}_{i\text{}\mathrm{min}}^{}\le {\displaystyle \sum _{j=1}^{J}T{L}_{ij}^{-}}\le T{L}_{i\text{}\mathrm{max}}^{},\text{\hspace{1em}}\forall i,$$$$T{I}_{i\text{}\mathrm{min}}^{}\le {\displaystyle \sum _{j=1}^{J}T{I}_{ij}^{-}}\le T{I}_{i\text{}\mathrm{max}}^{},\text{\hspace{1em}}\forall i,$$$$T{W}_{i\text{}\mathrm{min}}^{}\le {\displaystyle \sum _{j=1}^{J}T{W}_{ij}^{-}}\le T{W}_{i\text{}\mathrm{max}}^{},\text{\hspace{1em}}\forall i,$$
- (7)
- Technical constraints:$$X{A}_{ijh}^{},\text{}X{F}_{ijh}^{},\text{}X{L}_{ijh}^{},\text{}X{I}_{ijh}^{},\text{}X{W}_{ijh}^{}\ge 0,\text{\hspace{1em}}\forall i,\text{}j,\text{}h.$$

## 4. Results Analysis and Discussion

^{6}to $195.7 × 10

^{6}. Decisions at a higher allowable pollutant discharge would lead to a higher system benefit, but the reliability in fulfilling the environmental requirements would decrease; on the contrary, decisions at a lower allowable pollutant discharge would lead to a decreasing of risk for violating the pollutant discharge constraints, but with a lower system benefit. It demonstrates a trade-off between environmental requirement violation risk and benefit due to the uncertainties existing in various system components. In practice, when the plan aims to a higher system benefit, the environmental requirements may not be adequately satisfied; contrarily, planning with a lower system benefit may guarantee that the requirements be met. Additionally, the benefit of agriculture activity would take the largest proportion in total benefit and would increase slightly with the raising of pollutant discharge allowances. Moreover, the benefit of fishery would be stable at a low level, approximately occupying 2.1% of the total benefit.

^{6}) would be lower than that from RTFP ($191.1 × 10

^{6}). This is due to the fact that cost/benefit coefficients are handled by RFP, resulting in higher loss of uncertain information than that handled by RTFP. Figure 3 presents the optimal target and standard production scale of each activity under RFP and RTFP. Target of each activity discharge pollutant exceeds standards can be calculated through multiplying excess scale of economic activity (i.e., target—standard) by pollutant discharge rate. Results indicate that the excess planning scale of agricultural activity would be high. This may be attributed to their high crop yields and great selling prices. It is also depicted that the excess feeding size of livestock husbandry activity would also maintain high levels due to its high annual incomes. The excess outputs of industrial activity would also be significant since industry is promoted by the authority to push up the local income. Excess fishery and forestry activities would be low due to their limited planning lands. Furthermore, excess economic activities corresponding to RFP would be higher than that corresponding to RTFP. For instance, target of agricultural activity would be 180 ha in zone 1, while the standard production scale under RTFP and RFP would, respectively, be 156.4 ha and 148.5 ha corresponding to very-low level. Thus, excess agricultural activity would be 23.6 ha and 31.5 ha, respectively. It is revealed that the varied uncertain information would affect the water pollution control plans. Any simplifications may result in unreliable or misleading plans.

^{3}, 152.1, 22.4, and 99.2 t under low pollutant discharge allowance level; in comparison, under high pollutant discharge allowance level, they would respectively decrease to 1.3 × 10

^{3}, 8.2, 7.2, 5.4 t. In general, results discover that a more restrictive pollution control would result in a higher excess pollutant discharge while a looser pollution control would bring on a lower excess pollutant discharge.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclatures

i | index for economic activities; for agricultural activities, i = 1, 2, …, I_{a}; for fishery activities i = 1, 2, …, I_{f} (e.g., fish and prawn farming); for livestock husbandry activities i = 1, 2, …, I_{l}; for industrial activities i = 1, 2, …, I_{i} (e.g., manufacturing, mining, architecture, transportation and others); for forestry activities i = 1, 2, …, I_{w} |

j | index for zones; j = 1, 2, …, J |

k | index for pollutants; k = 1, 2, …, K (e.g., COD discharge, TN loss, TP loss, and soil loss) |

h | allowable pollutant discharge level; h = 1, 2, …, H |

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

$\tilde{B}{A}_{ij}^{}$ | unit benefit from agricultural activity i in zone j (RMB¥/km^{2}) |

$T{A}_{ij}^{}$ | land area target for agricultural activity i in zone j (km^{2}) |

$PE{A}_{ik}^{}$ | reduction of net benefit from agricultural activity i for excess discharge of pollutant k (RMB¥/kg when k = 2, 3; RMB¥/tonne when k = 4) |

$DP{A}_{ijk}^{}$ | discharge rate of pollutant k from agricultural activity i in zone j (kg/km^{2} when k = 2, 3; tonne /km^{2} when k = 4) |

$X{A}_{ijhk}^{}$ | decision variables representing amount by which the target of agricultural activity i discharge pollutant k exceeds standards in zone j when level is h (km^{2}) |

$\tilde{B}{F}_{ij}^{}$ | unit benefit from fishery activity i in zone j (RMB¥/km^{2}) |

$T{F}_{ij}^{}$ | land area target for fishery farming activity i in zone j (km^{2}) |

$PE{F}_{ik}^{}$ | reduction of net benefit from fishery activity i for excess discharge of pollutant k (RMB¥/kg) |

$DP{F}_{ijk}^{}$ | discharge rate of pollutant k from fishery activity i in zone j (kg/km^{2}) |

$X{F}_{ijhk}^{}$ | decision variables representing amount by which target of fishery activity i discharge pollutant k exceeds standards in zone j when level is h (km^{2}) |

$\tilde{B}{L}_{ij}^{\pm}$ | unit benefit from livestock husbandry activity i in zone j (RMB¥/head) |

$T{L}_{ij}^{}$ | target for livestock husbandry activity i in zone j (head) |

$PE{L}_{i}^{}$ | reduction of net benefit from livestock husbandry activity i for excess discharge of pollutant (i.e., COD) (RMB¥/kg) |

$DP{L}_{ij}^{}$ | discharge rate of pollutant (i.e., COD) from livestock husbandry activity i in zone j (kg/head) |

$X{L}_{ijh}^{}$ | decision variables representing amount by which target of livestock husbandry activity i discharge pollutant (i.e., COD) exceeds standards in zone j when level is h (head) |

$T{I}_{ij}^{}$ | output target for industrial activity i in zone j (RMB¥) |

$PE{I}_{i}^{}$ | reduction of net benefit from industrial activity i for excess discharge of pollutant (i.e., COD) (RMB¥/kg) |

$DP{I}_{ij}^{}$ | discharge rate of pollutant (i.e., COD) from industrial activity i in zone j (kg/RMB¥) |

$X{I}_{ijh}^{}$ | decision variables representing amount by which target of industrial activity i discharge pollutant (i.e., COD) exceeds standards in zone j when level is h (RMB¥) |

$\tilde{B}{W}_{ij}^{}$ | unit benefit from forestry activity i in zone j (RMB¥/head) |

$T{W}_{ij}^{}$ | land area target for forestry activity i in zone j (unit) |

$PE{W}_{i}^{}$ | reduction of net benefit from forestry activity i for excess discharge of pollutant (i.e., soil loss) (RMB¥/tonne) |

$DP{W}_{ij}^{}$ | discharge rate of pollutant (i.e., soil loss) from forestry activity i in zone j (tonne/km^{2}) |

$X{W}_{ijh}^{}$ | decision variables representing amount by which target of forestry activity i discharge pollutant (i.e., soil loss) exceeds standards in zone j when level is h (unit) |

$CO{F}_{ij}^{}$ | COD discharge from fishery farming activity i in zone j (kg/km^{2}) |

$PC{F}_{jh}^{}$ | maximum allowable COD discharge for fishery farming activities in zone j with probability ${p}_{h}$ of occurrence under level h (kg) |

$PC{L}_{jh}^{}$ | maximum allowable COD discharge for livestock husbandry activities in zone j with probability ${p}_{h}$ of occurrence under level h (kg) |

$PC{I}_{jh}^{}$ | maximum allowable COD discharge for industrial activity i in zone j with probability ${p}_{h}$ of occurrence under level h (kg) |

$MC{L}_{h}^{}$ | maximum allowable COD discharge from economic activities with probability ${p}_{h}$ of occurrence under level h (kg) |

$S{L}_{ij}^{}$ | soil loss from agricultural activity i in zone j (tonne/km^{2}) |

$A{P}_{ij}^{}$ | phosphorous content of soil corresponding to agricultural activity i in zone j (kg/tonne) |

$R{A}_{ij}^{}$ | runoff from agricultural activity i in zone j (kg/km^{2}) |

$R{P}_{ij}^{}$ | dissolved phosphorous content of runoff corresponding to agricultural activity i in zone j (%) |

$PA{P}_{jh}^{}$ | maximum allowable phosphorous loss from agricultural activities in zone j with probability ${p}_{h}$ of occurrence under level h (kg) |

$F{P}_{ij}^{}$ | dissolved phosphorous loss from fishery farming activity i in zone j (kg/km^{2}) |

$PF{P}_{jh}^{}$ | maximum allowable phosphorous loss from fishery farming activities in zone j with probability ${p}_{h}$ of occurrence under level h (kg) |

$MP{L}_{h}^{}$ | maximum allowable phosphorous loss from economic activities with probability ${p}_{h}$ of occurrence under level h (kg) |

$A{N}_{ij}^{}$ | nitrogen content of soil corresponding to agricultural activity i in zone j (kg/tonne) |

$R{N}_{ij}^{}$ | dissolved nitrogen content of runoff corresponding to agricultural activity i in zone j (%) |

$PA{N}_{jh}^{}$ | maximum allowable nitrogen loss from agricultural activities in zone j with probability ${p}_{h}$ of occurrence under level h (kg) |

$F{N}_{ij}^{}$ | dissolved nitrogen loss from fishery activity i in zone j (kg/km^{2}) |

$PF{N}_{jh}^{}$ | maximum allowable nitrogen loss from fishery farming activities in zone j with probability ${p}_{h}$ of occurrence under level h (kg) |

$MS{L}_{jh}^{}$ | maximum allowable soil loss from economic activities with probability ${p}_{h}$ of occurrence under level h (tonne) |

$MN{L}_{h}^{}$ | maximum allowable nitrogen loss from economic activities with probability ${p}_{h}$ of occurrence under level h (kg) |

$PS{L}_{jh}^{}$ | maximum allowable soil loss from agricultural activities in zone j with probability ${p}_{h}$ of occurrence under level h (tonne) |

$PW{S}_{jh}^{}$ | maximum allowable soil loss from forestry activities in zone j with probability ${p}_{h}$ of occurrence under level h (tonne) |

$W{A}_{i}^{}$ | water demand for agricultural activity i (m^{3}/km^{2}) |

$W{F}_{i}^{}$ | water demand for fishery activity i (m^{3}/km^{2}) |

$W{L}_{i}^{}$ | water demand for livestock husbandry activity i (m^{3}/head) |

$W{I}_{i}^{}$ | water demand for industrial activity i (m^{3}/RMB¥) |

$W{W}_{i}^{}$ | water demand for forestry activity i (m^{3}/km^{2}) |

$MAX{W}_{j}^{}$ | maximum allowable water resources supply amount in zone j (m^{3}) |

$T{A}_{i\text{}\mathrm{min}}^{}$ | minimum demand for agricultural activity i (km^{2}) |

$T{A}_{i\text{}\mathrm{max}}^{}$ | maximum demand for agricultural activity i (km^{2}) |

$T{F}_{i\text{}\mathrm{min}}^{}$ | minimum demand for fishery activity i (km^{2}) |

$T{F}_{i\text{}\mathrm{max}}^{}$ | maximum demand for fishery activity i (km^{2}) |

$T{L}_{i\text{}\mathrm{min}}^{}$ | minimum demand for livestock husbandry activity i (head) |

$T{L}_{i\text{}\mathrm{max}}^{}$ | maximum demand for livestock husbandry activity i (head) |

$T{W}_{i\text{}\mathrm{min}}^{}$ | minimum demand for industrial activity i (RMB¥) |

$T{W}_{i\text{}\mathrm{max}}^{}$ | maximum demand for industrial activity i (RMB¥) |

## References

- Pastori, M.; Udías, A.; Bouraoui, F.; Bidoglio, G. A multi-objective approach to evaluate the economic and environmental impacts of alternative water and nutrient management strategies in Africa. J. Environ. Inform.
**2017**, 29. [Google Scholar] [CrossRef] - Steenbergen, R.D.J.M.; Gelder van, P.H.A.J.M.; Miraglia, S.; Vrouwenvelder, A.C.W.M. Safety, Reliability and Risk Analysis: Beyond the Horizon. In Proceedings of the 22nd Annual Conference on European Safety and Reliability (ESREL), Wroclaw, Poland, 14–18 September 2014; CRC Press-Taylor & Francis Group: Boca Raton, FL, USA, 2014; pp. 1115–1120. [Google Scholar]
- David, F.; Sandra, P. On the complementary nature of CGC-MS, CGC-FTIR, and CGC-AED for water pollution control. J. Sep. Sci.
**2015**, 14, 554–557. [Google Scholar] [CrossRef] - Pietrucha-Urbanik, K.; Tchorzewska-Cieslak, B. Water Supply System Operation Regarding Consumer Safety Using Kohonen Neural Network; Steenbergen, R.D.J.M., van Gelder, P.H.A.J.M., Miraglia, S., Vrouwenvelder, A.C.W.M.T., Eds.; CRC Press: Boca Raton, FL, USA, 2013. [Google Scholar]
- Li, Y.P.; Zhang, N.; Huang, G.H.; Liu, J. Coupling fuzzy-chance constrained program with minimax regret analysis for water quality management. Stoch. Environ. Res. Risk Assess.
**2014**, 28, 1769–1784. [Google Scholar] [CrossRef] - Jorba, L.; Adillon, R. A generalization of trapezoidal fuzzy numbers based on modal interval theory. Symmetry
**2017**, 9, 198. [Google Scholar] [CrossRef] - Hu, B.; Bi, L.Q.; Dai, S.S. The orthogonality between complex fuzzy sets and its application to signal detection. Symmetry
**2017**, 9, 175. [Google Scholar] [CrossRef] - Dimitrov, V.; Driankov, D.; Petrov, A. Fuzzy equations and their applications to water pollution control. IFAC Proc.
**1977**, 10, 369–371. [Google Scholar] [CrossRef] - Chang, N.B.; Chen, H.W.; Shaw, D.G. Water pollution control in river basin by interactive fuzzy interval multiobjective programming. J. Environ. Eng.
**1997**, 123, 1208–1216. [Google Scholar] [CrossRef] - Karmakar, S.; Mujumdar, P.P. Grey fuzzy optimization model for water quality management of a river system. Adv. Water Resour.
**2006**, 29, 1088–1105. [Google Scholar] [CrossRef] - Singh, A.P.; Ghosh, S.K.; Sharma, P. Water quality management of a stretch of river Yamuna: An interactive fuzzy multi-objective approach. Water Resour. Manag.
**2007**, 21, 515–532. [Google Scholar] [CrossRef] - Sakawa, M. Interactive fuzzy goal programming for multiobjective nonlinear problems and its application to water quality management. Electron. Commun. Jpn.
**2010**, 68, 49–56. [Google Scholar] [CrossRef] - Chmielowski, W.Z. Fuzzy Control in Environmental Engineering; Springer International Publishing: Basel, Switzerland, 2016. [Google Scholar]
- Forio, M.A.E.; Mouton, A.; Lock, K. Fuzzy modelling to identify key drivers of ecological water quality to support decision and policy making. Environ. Sci. Policy
**2017**, 68, 58–68. [Google Scholar] [CrossRef] - Maeda, S.; Kuroda, H.; Yoshida, K. A GIS-aided two-phase grey fuzzy optimization model for nonpoint source pollution control in a small watershed. Paddy Water Environ.
**2016**, 1–14. [Google Scholar] [CrossRef] - Liu, M.; Nie, G.; Hu, M. An interval-parameter fuzzy robust nonlinear programming model for water quality management. J. Water Resour. Prot.
**2013**, 5, 12–16. [Google Scholar] [CrossRef] - Tavakoli, A.; Nikoo, M.R.; Kerachian, R. River water quality management considering agricultural return flows: application of a nonlinear two-stage stochastic fuzzy programming. Environ. Monit. Assess.
**2015**, 187, 158. [Google Scholar] [CrossRef] [PubMed] - Ji, Y.; Huang, G.; Sun, W. Water quality management 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] - Wu, D.R.; Tan, W.W. Computationally Efficient Type-reduction Strategies for a Type-2 Fuzzy Logic Controller. In Proceedings of the 14th IEEE International Conference on Fuzzy System, Reno, NV, USA, 25–25 May 2005; pp. 353–358. [Google Scholar]
- Cervantes, L.; Castillo, O. Type-2 fuzzy logic aggregation of multiple fuzzy controllers for airplane flight control. Inf. Sci.
**2015**, 324, 247–256. [Google Scholar] [CrossRef] - Castillo, O.; Melin, P. A review on interval type-2 fuzzy logic applications in intelligent control. Inf. Sci.
**2014**, 279, 615–631. [Google Scholar] [CrossRef] - Precup, R.E.; Angelov, P.; Costa, B.S.J.; Sayed-Mouchaweh, M. An overview on fault diagnosis and nature-inspired optimal control of industrial process applications. Comput. Ind.
**2015**, 74, 75–94. [Google Scholar] [CrossRef] - Li, Y.P.; Huang, G.H.; Li, H.Z. A recourse-based interval fuzzy programming model for point-nonpoint source effluent trading under uncertainty. J. Am. Water Resour. Assoc.
**2015**, 50, 1191–1207. [Google Scholar] [CrossRef] - Suo, C.; Li, Y.P.; Wang, C.X. A type-2 fuzzy chance-constrained programming method for planning Shanghai’s energy system. Int. J. Electr. Power Energy Syst.
**2017**, 90, 37–53. [Google Scholar] [CrossRef] - Alcantud, J.C.R.; García-Sanz, M.D. Evaluations of infinite utility streams: Pareto Efficient and Egalitarian Axiomatics. Metroeconomica
**2013**, 64, 432–447. [Google Scholar] [CrossRef][Green Version] - Precup, R.E.; Sabau, M.C.; Petriu, E.M. Nature-inspired optimal tuning of input membership functions of Takagi-Sugeno-Kang fuzzy models for Anti-lock Braking Systems. Appl. Soft Comput. J.
**2015**, 27, 575–589. [Google Scholar] [CrossRef] - Vrkalovic, S.; Teban, T.-A.; Borlea, I.-D. Stable Takagi-Sugeno fuzzy control designed by optimization. Int. J. Artif. Intell.
**2017**, 15, 17–29. [Google Scholar] - Chen, Z.; Zhou, S.; Luo, J. A robust ant colony optimization for continuous functions. Expert Syst. Appl.
**2017**, 81, 309–320. [Google Scholar] [CrossRef] - Tchórzewska-Cieślak, B.; Pietrucha-Urbanik, K.; Urbanik, M. Analysis of the gas network failure and failure prediction using the Monte Carlo simulation method. Eksploatacja i Niezawodnosc Maint. Reliab.
**2016**, 18, 254–259. [Google Scholar] - Qin, R.; Liu, Y.K.; Liu, Z.Q. Methods of critical value reduction for type-2 fuzzy variables and their applications. J. Comput. Appl. Math.
**2011**, 235, 1454–1481. [Google Scholar] [CrossRef] - Yongxin Bureau of Statistics Statistical Communiqué on the 2012 Economic and Social Development in Yongxin. Available online: http://old.yongxin.gov.cn/tjj/a/detail-7021.html (accessed on 28 October 2013).
- Huang, G.H.; Chen, B.; Qin, X.S.; Mance, E. An Integrated Decision Support System for Developing Rural Eco-Environmental Sustainability in the Mountain-River-Lake Region of Jiangxi Province, China; Final Report; United Nations Development Program: New York, NY, USA, 2006. [Google Scholar]
- Huang, G.H.; Qin, X.S.; Sun, W.; Nie, X.H.; Li, Y.P. An optimisation-based environmental decision support system for sustainable development in a rural area in China. Civ. Eng. Environ. Syst.
**2009**, 26, 65–83. [Google Scholar] [CrossRef] - Statistics Bulletin of the National Economic and Social Development in Yongxin County. 2016. Available online: http://www.yongxin.gov.cn/doc/2017/04/25/52136.shtml (accessed on 25 April 2017).

**Figure 3.**Optimal target and standard production scale of each activity. (

**a**) Agriculture; (

**b**) Livestock husbandry; (

**c**) Industry; (

**d**) Forestry; (

**e**) Fishery.

**Figure 4.**Excess pollutant discharge under different pollutant allowances (unit: t). (

**a**) soil loss; (

**b**) COD discharge; (

**c**) TP discharge; (

**d**) TN discharge.

**Figure 5.**Proportion of pollutant discharges under medium-high level. (

**a**) soil loss; (

**b**) TN discharge; (

**c**) TP discharge; (

**d**) COD discharge.

Zone | Agriculture | ||

Paddy Farm | Dry Farm | Fruit/Vegetable | |

Unit Net Benefit (10^{3} $/km^{2}) | |||

Zone 1 | (201.4, 272.6, 292.3; 0.2, 0.6) | (104.5, 148.4, 163.5; 0.2, 0.6) | (345.3, 457.4, 486.2; 0.2, 0.6) |

Zone 2 | (174.2, 219.6, 269.6; 0.3, 0.5) | (96.9, 130.3, 140.9; 0.3, 0.5) | (152.9, 205.9, 215.1; 0.3, 0.5) |

Zone 3 | (169.6, 231.7, 254.4; 0.4, 0.7) | (87.8, 112.1, 127.2; 0.4, 0.7) | (337.7, 472.6, 569.4; 0.4, 0.7) |

Zone 4 | (187.8, 265.1, 290.8; 0.4, 0.7) | (92.4, 118.1, 124.2; 0.4, 0.7) | (254.5, 343.4, 384.7; 0.4, 0.7) |

Zone 5 | (184.8, 245.4, 275.7; 0.5, 0.7) | (92.4, 122.7, 133.3; 0.5, 0.7) | (289.3, 390.8, 402.9; 0.5, 0.7) |

Zone | Livestock Husbandry | ||

Pig | Cattle | Poultry | |

Unit Net Benefit ($/head) | |||

Zone 1 | (152.5, 166.3, 170.1; 0.4, 0.7) | (785.4, 892.4, 998.4; 0.4, 0.7) | (5.3, 6.2, 7.1; 0.4, 0.7) |

Zone 2 | (139.6, 151.2, 156.6; 0.5, 0.8) | (833.9, 933.6, 974.2; 0.5, 0.8) | (6.2, 6.8, 7.4; 0.5, 0.8) |

Zone 3 | (139.6, 159.6, 161.2, 0.4, 0.7) | (732.9, 831.1, 868.1; 0.4, 0.7) | (4.7, 6.9, 7.1; 0.4, 0.7) |

Zone 4 | (154.7, 172.2, 175.4; 0.3, 0.8) | (662.1, 734.9, 760.7; 0.3, 0.8) | (4.9, 6.7, 7.2; 0.3, 0.8) |

Zone 5 | (158.5, 168.7, 185.2; 0.4, 0.8) | (904.8, 984.2, 999.2; 0.4, 0.8) | (6.2, 7.1, 7.8; 0.4, 0.8) |

Pollutant | Level | Probability | Zone | ||||
---|---|---|---|---|---|---|---|

Zone 1 | Zone 2 | Zone 3 | Zone 4 | Zone 5 | |||

COD (10^{3} kg) | Very-low | 0.05 | 2453.5 | 496.4 | 3768.4 | 3284.2 | 2307.5 |

Low | 0.10 | 2469.8 | 499.3 | 3780.8 | 3298.5 | 2213.3 | |

Low-medium | 0.20 | 2498.3 | 500.3 | 3792.4 | 3390.7 | 2309.8 | |

Medium | 0.30 | 2523.5 | 502.5 | 3897.9 | 3469.3 | 2468.3 | |

Medium-high | 0.20 | 2760.7 | 515.6 | 4506.2 | 3607.8 | 2647.8 | |

High | 0.10 | 2984.4 | 535.5 | 4512.5 | 3812.9 | 2851.2 | |

Very-high | 0.05 | 3286.2 | 578.7 | 4714.7 | 4214.2 | 3154.9 | |

TN (10^{3} kg) | Very-low | 0.05 | 166.5 | 193.2 | 235.7 | 286.3 | 223.6 |

Low | 0.10 | 216.4 | 297.3 | 307.3 | 369.4 | 374.3 | |

Low-medium | 0.20 | 266.5 | 401.4 | 379.5 | 451.8 | 526.8 | |

Medium | 0.30 | 324.3 | 506.7 | 458.8 | 561.2 | 679.2 | |

Medium-high | 0.20 | 381.4 | 614.4 | 544.3 | 683.3 | 836.2 | |

High | 0.10 | 457.2 | 747.5 | 653.9 | 815.5 | 994.1 | |

Very-high | 0.05 | 545.5 | 883.9 | 756.3 | 935.8 | 1168.7 | |

TP (10^{3} kg) | Very-low | 0.05 | 24.2 | 31.1 | 35.6 | 43.2 | 39.9 |

Low | 0.10 | 34.4 | 52.0 | 50.0 | 59.5 | 70.5 | |

Low-medium | 0.20 | 44.5 | 73.5 | 64.5 | 76.6 | 101.7 | |

Medium | 0.30 | 55.0 | 94.6 | 81.0 | 93.6 | 132.2 | |

Medium-high | 0.20 | 65.1 | 118.1 | 100.2 | 113.9 | 165.4 | |

High | 0.10 | 79.5 | 142.6 | 114.8 | 134.3 | 200.3 | |

Very-high | 0.05 | 93.2 | 166.1 | 130.4 | 152.6 | 231.6 | |

Soil loss (10^{3} t) | Very-low | 0.05 | 97.3 | 92.5 | 133.4 | 168.5 | 61.7 |

Low | 0.10 | 92.4 | 93.6 | 134.2 | 169.5 | 63.0 | |

Low-medium | 0.20 | 99.6 | 94.7 | 135.3 | 170.7 | 64.2 | |

Medium | 0.30 | 100.3 | 98.9 | 140.6 | 176.0 | 71.5 | |

Medium-high | 0.20 | 118.0 | 133.1 | 155.7 | 186.1 | 82.7 | |

High | 0.10 | 133.1 | 140.4 | 165.9 | 201.3 | 89.9 | |

Very-high | 0.05 | 133.2 | 140.5 | 171.6 | 274.4 | 91.9 |

© 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.; Chen, L.
A Recourse-Based Type-2 Fuzzy Programming Method for Water Pollution Control under Uncertainty. *Symmetry* **2017**, *9*, 265.
https://doi.org/10.3390/sym9110265

**AMA Style**

Liu J, Li Y, Huang G, Chen L.
A Recourse-Based Type-2 Fuzzy Programming Method for Water Pollution Control under Uncertainty. *Symmetry*. 2017; 9(11):265.
https://doi.org/10.3390/sym9110265

**Chicago/Turabian Style**

Liu, Jing, Yongping Li, Guohe Huang, and Lianrong Chen.
2017. "A Recourse-Based Type-2 Fuzzy Programming Method for Water Pollution Control under Uncertainty" *Symmetry* 9, no. 11: 265.
https://doi.org/10.3390/sym9110265