A GIS-aided two-phase grey fuzzy optimization model for nonpoint source pollution control in a small watershed

Abstract

A method for allocating allowable ranges of total nitrogen (TN) load to nonpoint (diffuse pollution) sources in a watershed has been developed by adopting the two-phase grey fuzzy optimization approach. Competing goals of water quality management authorities and TN load dischargers at nonpoint sources such as paddy field, upland crop field, and residential area are described with linear imprecise membership functions including interval numbers. TN load discharged from each cell of the nonpoint sources is assumed to be transported along with surface, subsurface, and river flow under the conventional first-order kinetic removal with respect to distance. The travel length of the load is estimated with a digital elevation model in a geographic information system (GIS). Uncertainty of river discharge and self-purification coefficients appearing in the TN transport model is also expressed with interval numbers. The GIS-aided grey fuzzy optimization model developed here is applied to the Seimei River watershed, Japan. By solving the optimization model, the allowable load represented by an interval number at each cell is procured, which would be a scientific base for effluent control regarding nonpoint sources in the area.

Appendix

The formulation of eight submodels for the grey fuzzy optimization model [Eqs. (6)–(13)] is shown here.

Case 1

Submodels 1 and 2 are formulated and successively solved in Phases 1 and 2, respectively, to produce the upper limit of $\xi^{ + }$ with optimal $L_{ji}^{ - }$.

<Phase 1>

Submodel 1:

$${\text{Maximize}}\quad \, \xi^{ + }$$

(16)

subject to

$$T^{ - } \le T^{H + } - \xi^{ + } (T^{H - } - T^{D + } )$$

(17)

$$L_{ji}^{ - } \ge L_{ji}^{L - } + \xi^{ + } (L_{ji}^{D - } - L_{ji}^{L + } ),\quad \forall i,j$$

(18)

$$T^{D - } \le T^{ - } \le T^{H + }$$

(19)

$$L_{ji}^{L - } \le L_{ji}^{ - } \le L_{ji}^{D + } ,\quad \forall i,j$$

(20)

$$0 \le \xi^{ + } \le 1$$

(21)

$$T^{ - } = f(L_{ji}^{ - } ).$$

(22)

Submodel 1 [Eqs. (16)–(22)] can be solved by the simplex method to procure $L_{ji}^{ - }$. The obtained objective value of $\xi^{ + }$ is defined as $\xi_{1}^{ + }$. In the second phase, the smallest values of $L_{ji}^{ - }$ are pursued with the overall satisfaction level kept at $\xi_{1}^{ + }$ by solving the following optimization model.

<Phase 2>

Submodel 2:

$${\text{Maximize}}\quad \frac{{T^{H + } - T^{ - } }}{{T^{H - } - T^{D + } }}$$

(23)

subject to

$$\xi^{ + } = \xi_{1}^{ + }$$

(24)

$$T^{ - } \le T^{H + } - \xi^{ + } (T^{H - } - T^{D + } )$$

(25)

$$L_{{j_{i} }}^{ - } \ge L_{{j_{i} }}^{L - } + \xi^{ + } (L_{{j_{i} }}^{D - } - L_{{j_{i} }}^{L + } ),\quad \forall i,j$$

(26)

$$T^{D - } \le T^{ - } \le T^{H + }$$

(27)

$$L_{{j_{i} }}^{L - } \le L_{{j_{i} }}^{ - } \le L_{{j_{i} }}^{D + } ,\quad \forall i,j$$

(28)

$$0 \le \xi^{ + } \le 1$$

(29)

$$T^{ - } = f(L_{{j_{i} }}^{ - } ).$$

(30)

It is noted that the difference between Submodel 1 [Eqs. (16)–(22)] and Submodel 2 [Eqs. (23)–(30)] is appearing in the objective function, Eq. (23), and the additional constraint, Eq. (24). By solving Submodel 2 just after solving Submodel 1, the solution of $L_{ji}^{ - }$ in Case 1 where the goal of water quality management authorities is prioritized is finally procured.

Next, Submodels 3 and 4 are provided and successively solved in Phases 1 and 2, respectively, to produce the lower limit of $\xi^{ - }$ with optimal $L_{ji}^{ + }$.

<Phase 1>

Submodel 3:

$${\text{Maximize}}\quad \xi^{ - }$$

(31)

subject to

$$T^{ + } \le T^{H - } - \xi^{ - } (T^{H + } - T^{D - } )$$

(32)

$$L_{ji}^{ + } \ge L_{ji}^{L + } + \xi^{ - } (L_{ji}^{D + } - L_{ji}^{L - } ),\quad \forall i,j$$

(33)

$$T^{D - } \le T^{ + } \le T^{H + }$$

(34)

$$L_{ji}^{L - } \le L_{ji}^{ + } \le L_{ji}^{D + } ,\quad \forall i,j$$

(35)

$$L_{ji}^{ + } \ge \hat{L}_{ji}^{ - } ,\quad \forall i,j$$

(36)

$$0 \le \xi^{ - } \le 1$$

(37)

$$T^{ + } = f(L_{ji}^{ + } ).$$

(38)

It is noted that the solution of Submodel 2, $L_{ji}^{ - } = \hat{L}_{ji}^{ - }$, is set as the lower limit of $L_{ji}^{ + }$ in Eq. (36).

<Phase 2>

Submodel 4:

$${\text{Maximize}}\quad \sum\limits_{j} {\sum\limits_{i} {\frac{{L_{ji}^{ + } - L_{ji}^{L + } }}{{L_{ji}^{D + } - L_{ji}^{L - } }}} }$$

(39)

subject to

$$\xi^{ - } = \xi_{2}^{ - }$$

(40)

$$T^{ + } \le T^{H - } - \xi^{ - } (T^{H + } - T^{D - } )$$

(41)

$$L_{ji}^{ + } \ge L_{ji}^{L + } + \xi^{ - } (L_{ji}^{D + } - L_{ji}^{L - } ),\quad \forall i,j$$

(42)

$$T^{D - } \le T^{ + } \le T^{H + }$$

(43)

$$L_{ji}^{L - } \le L_{ji}^{ + } \le L_{ji}^{D + } ,\quad \forall i,j$$

(44)

$$L_{ji}^{ + } \ge \hat{L}_{ji}^{ - } ,\quad \forall i,j$$

(45)

$$0 \le \xi^{ - } \le 1$$

(46)

$$T^{ + } = f(L_{ji}^{ + } ),$$

(47)

where $\xi_{2}^{ - }$ in Eq. (40) is the overall satisfaction level obtained in Phase 1, i.e., Submodel 3. Solving Submodel 4 [Eqs. (39)–(47)] results in obtaining the final solution of $L_{ji}^{ + }$.

Case 2

Submodels 5 and 6 described below are first formulated and successively solved in Phases 1 and 2, respectively, to produce the upper limit of $\xi^{ + }$ with optimal $L_{ji}^{ + }$.

<Phase 1>

Submodel 5:

$${\text{Maximize}}\quad \xi^{ + }$$

(48)

subject to

$$L_{ji}^{ + } \ge L_{ji}^{L - } + \xi^{ + } (L_{ji}^{D - } - L_{ji}^{L + } ),\quad \forall i,j$$

(49)

$$T^{ + } \le T^{H + } - \xi^{ + } (T^{H - } - T^{D + } )$$

(50)

$$T^{D - } \le T^{ + } \le T^{H + }$$

(51)

$$L_{ji}^{L - } \le L_{ji}^{ + } \le L_{ji}^{D + } ,\quad \forall i,j$$

(52)

$$0 \le \xi^{ + } \le 1$$

(53)

$$T^{ + } = f(L_{ji}^{ + } ).$$

(54)

Submodel 5 [Eqs. (48)–(54)] is solved to procure $L_{ji}^{ + }$. The obtained objective value of $\xi^{ + }$ is specified as $\xi_{3}^{ + }$. In the second phase, the smallest values of $L_{ji}^{ + }$ are pursued with the overall satisfaction level kept at $\xi_{3}^{ + }$ by solving the following optimization model.

<Phase 2>

Submodel 6:

$${\text{Maximize}}\quad \sum\limits_{j} {\sum\limits_{i} {\frac{{L_{ji}^{ + } - L_{ji}^{L - } }}{{L_{ji}^{D - } - L_{ji}^{L + } }}} }$$

(55)

subject to

$$\xi^{ + } = \xi_{3}^{ + }$$

(56)

$$L_{ji}^{ + } \ge L_{ji}^{L - } + \xi^{ + } (L_{ji}^{D - } - L_{ji}^{L + } ),\quad \forall i,j$$

(57)

$$T^{ + } \le T^{H + } - \xi^{ + } (T^{H - } - T^{D + } )$$

(58)

$$T^{D - } \le T^{ + } \le T^{H + }$$

(59)

$$L_{ji}^{L - } \le L_{ji}^{ + } \le L_{ji}^{D + } ,\quad \forall i,j$$

(60)

$$0 \le \xi^{ + } \le 1$$

(61)

$$T^{ + } = f(L_{ji}^{ + } ).$$

(62)

By solving Submodel 6 [Eqs. (55)–(62)] just after solving Submodel 5, the final solution of $L_{ji}^{ + }$ in the case that the goals of dischargers are prioritized is procured.

Then, Submodels 7 and 8 are created and successively solved in Phases 1 and 2, respectively, to produce the lower limit of $\xi^{ - }$.

<Phase 1>

Submodel 7:

$${\text{Maximize}}\quad \xi^{ - }$$

(63)

subject to

$$L_{ji}^{ - } \ge L_{ji}^{L + } + \xi^{ - } (L_{ji}^{D + } - L_{ji}^{L - } ),\quad \forall i,j$$

(64)

$$T^{ - } \le T^{H - } - \xi^{ - } (T^{H + } - T^{D - } )$$

(65)

$$T^{D - } \le T^{ - } \le T^{H + }$$

(66)

$$L_{ji}^{L - } \le L_{ji}^{ - } \le L_{ji}^{D + } ,\quad \forall i,j$$

(67)

$$L_{ji}^{ - } \le \hat{L}_{ji}^{ + } ,\quad \forall i,j$$

(68)

$$0 \le \xi^{ - } \le 1$$

(69)

$$T^{ - } = f(L_{ji}^{ - } ).$$

(70)

It is noted that the solution of Submodel 6, $L_{ji}^{ + } = \hat{L}_{ji}^{ + }$, is set as the upper limit of $L_{ji}^{ - }$ in Eq. (68).

<Phase 2>

Submodel 8:

$${\text{Maximize}}\quad \frac{{T^{H - } - T^{ - } }}{{T^{H + } - T^{D - } }}$$

(71)

subject to

$$\xi^{ - } = \xi_{4}^{ - }$$

(72)

$$L_{ji}^{ - } \ge L_{ji}^{L + } + \xi^{ - } (L_{ji}^{D + } - L_{ji}^{L - } ),\quad \forall i,j$$

(73)

$$T^{ - } \le T^{H - } - \xi^{ - } (T^{H + } - T^{D - } )$$

(74)

$$T^{D - } \le T^{ - } \le T^{H + }$$

(75)

$$L_{ji}^{L - } \le L_{ji}^{ - } \le L_{ji}^{D + } ,\quad \forall i,j$$

(76)

$$L_{ji}^{ - } \le \hat{L}_{ji}^{ + } ,\quad \forall i,j$$

(77)

$$0 \le \xi^{ - } \le 1$$

(78)

$$T^{ - } = f(L_{ji}^{ - } ),$$

(79)

where $\xi_{4}^{ - }$ in Eq. (72) is the overall satisfaction level obtained in Phase 1, i.e., Submodel 7. Solving Submodel 8 [Eqs. (71)–(79)] leads to obtaining the final solution of $L_{ji}^{ - }$.