An integrated optimization method for river water quality management and risk analysis in a rural system

Abstract

In this study, an interval-stochastic-based risk analysis (RSRA) method is developed for supporting river water quality management in a rural system under uncertainty (i.e., uncertainties exist in a number of system components as well as their interrelationships). The RSRA method is effective in risk management and policy analysis, particularly when the inputs (such as allowable pollutant discharge and pollutant discharge rate) are expressed as probability distributions and interval values. Moreover, decision-makers’ attitudes towards system risk can be reflected using a restricted resource measure by controlling the variability of the recourse cost. The RSRA method is then applied to a real case of water quality management in the Heshui River Basin (a rural area of China), where chemical oxygen demand (COD), total nitrogen (TN), total phosphorus (TP), and soil loss are selected as major indicators to identify the water pollution control strategies. Results reveal that uncertainties and risk attitudes have significant effects on both pollutant discharge and system benefit. A high risk measure level can lead to a reduced system benefit; however, this reduction also corresponds to raised system reliability. Results also disclose that (a) agriculture is the dominant contributor to soil loss, TN, and TP loads, and abatement actions should be mainly carried out for paddy and dry farms; (b) livestock husbandry is the main COD discharger, and abatement measures should be mainly conducted for poultry farm; (c) fishery accounts for a high percentage of TN, TP, and COD discharges but a has low percentage of overall net benefit, and it may be beneficial to cease fishery activities in the basin. The findings can facilitate the local authority in identifying desired pollution control strategies with the tradeoff between socioeconomic development and environmental sustainability.

Appendices

Appendix 1

Solution method

In RSRA model, the first-stage decision variables x ^± _j in the model are expressed as intervals, which should be identified before the random variables are disclosed (Huang and Loucks 2000). Consequently, decision variables u _j are introduced to identify an optimized set of the first-stage values for supporting the related policy analyses. Let x ^± _j  = x ⁻ _j  + Δx _j u _j , where Δx _j  = x ⁺ _j  − x ⁻ _j and u _j  ∈ [0, 1]. Then, a two-step process is proposed to transform the model into two deterministic submodels that are associated with the lower and upper bounds of the desired objective. Since the objective is to be maximized, the submodel corresponding to f ⁺ (i.e., most desirable objective) can be firstly formulated:

$$ \max \kern0.5em {f}^{+}={\displaystyle \sum_{j=1}^{n_2}{c}_j^{+}\left({x}_j^{-}+\varDelta {x}_j{u}_j\right)}-{\displaystyle \sum_{j=1}^{k_2}{\displaystyle \sum_{h=1}^v{p}_h{e}_j^{-}{y}_{jh}^{-}}}-{\displaystyle \sum_{j={k}_2+1}^{n_2}{\displaystyle \sum_{h=1}^v{p}_h{e}_j^{-}{y}_{jh}^{+}}} $$

(18)

subject to:

$$ {\displaystyle \sum_{j=1}^{n_1}{\left|{a}_{rj}\right|}^{+}\mathrm{Sign}\left({a}_{rj}^{+}\right)\left({x}_j^{-}+\varDelta {x}_j{u}_j\right)}\le {b}_r^{+},\kern0.5em \forall r $$

(19)

$$ \begin{array}{l}{\displaystyle \sum_{j=1}^{n_1}{\left|{a}_{rj}\right|}^{+}\mathrm{Sign}\left({a}_{rj}^{+}\right)\left({x}_j^{-}+\varDelta {x}_j{u}_j\right)}+{\displaystyle \sum_{j=1}^{k_2}{\left|{a}_{tj}^{\prime}\right|}^{+}\mathrm{Sign}\left({a^{\prime}}_{tj}^{+}\right){y}_{jh}^{-}}\\ {}+{\displaystyle \sum_{j={k}_2+1}^{n_2}{\left|{a}_{tj}^{\prime}\right|}^{-}\mathrm{Sign}\left({a^{\prime}}_{tj}^{-}\right){y}_{jh}^{+}}\ge {\tilde{w}}_h^{-},\kern0.5em \forall t,\ h\end{array} $$

(20)

$$ {\displaystyle \sum_{j=1}^{n_1}{c}_j^{+}\left({x}_j^{-}+\varDelta {x}_j{u}_j\right)}-{\displaystyle \sum_{j=1}^{k_2}{p}_h{e}_j^{-}{y}_{jh}^{-}}-{\displaystyle \sum_{j={k}_2+1}^{n_2}{p}_h{e}_j^{-}{y}_{jh}^{+}}\ge {\eta}^{+}-{\mathrm{UB}}_h{z}_h^{-},\kern0.5em \forall j,\ h $$

(21)

$$ {\displaystyle \sum_{j=1}^{n_1}{c}_j^{+}\left({x}_j^{-}+\varDelta {x}_j{u}_j\right)}-{\displaystyle \sum_{j=1}^{k_2}{p}_h{e}_j^{-}{y}_{jh}^{-}}-{\displaystyle \sum_{j={k}_2+1}^{n_2}{p}_h{e}_j^{-}{y}_{jh}^{+}}\le {\eta}^{+}+{\mathrm{UB}}_h\left(1-{z}_h^{-}\right),\ \forall j,\ h $$

(22)

$$ {\displaystyle \sum_{h=1}^v{p}_h{z}_h^{-}}\le {\varepsilon}^{+},\kern0.5em \forall h $$

(23)

$$ {z}_h^{-}=\left\{\begin{array}{l}1\kern0.75em \mathrm{if}\kern0.5em {\displaystyle \sum_{j=1}^{n_1}{c}_j^{+}\left({x}_j^{-}+\varDelta {x}_j{u}_j\right)}-{\displaystyle \sum_{j=1}^{k_2}{p}_h{e}_j^{-}{y}_{jh}^{-}}-{\displaystyle \sum_{j={k}_2+1}^{n_2}{p}_h{e}_j^{-}{y}_{jh}^{+}}\ge {\eta}^{+}\\ {}0\kern0.5em \mathrm{otherwise}\end{array}\right. $$

(24)

$$ {x}_j^{-}+\varDelta {x}_j{u}_j\ge 0,\kern0.5em j=1,\ 2, \dots,\ {n}_1 $$

(25)

$$ {y}_{jh}^{-}\ge 0,\kern0.5em j=1,\ 2, \dots,\ {k}_2;\kern0.5em \forall h $$

(26)

$$ {y}_{jh}^{+}\ge 0,\kern0.5em j={k}_2+1,\ {k}_2+2, \dots,\ {n}_2;\kern0.5em \forall h $$

(27)

where u _j , z ⁻ _h , y ⁻ _jh , and y ⁺ _jh are decision variables; y ⁻ _jh (j = 1, 2, …, k ₂) are decision variables with positive coefficients in the objective function; y ⁺ _jh (j = k ₂ + 1, k ₂ + 2, …, n ₂) are decision variables with negative coefficients. Solutions of y ⁻ _jh (j = 1, 2, …, k ₂), y ⁺ _jh (j = k ₂ + 1, k ₂ + 2, …, n ₂), u _j , and z ⁻ _h can be obtained through solving submodel (18). The optimized first-stage variables are x ^± _j opt  = x ⁻ _j  + Δx _j u _j opt (j = 1, 2, …, n ₁). Then, the submodel corresponding to the lower-bound objective-function value (f ⁻) is as follows:

$$ \max \kern0.5em {f}^{-}={\displaystyle \sum_{j=1}^{n_1}{c}_j^{-}\left({x}_j^{-}+\varDelta {x}_j{u}_{j\ \mathrm{opt}}\right)}-{\displaystyle \sum_{j=1}^{k_2}{\displaystyle \sum_{h=1}^v{p}_h{e}_j^{+}{y}_{jh}^{+}}}-{\displaystyle \sum_{j={k}_2+1}^{n_2}{\displaystyle \sum_{h=1}^v{p}_h{e}_j^{+}{y}_{jh}^{-}}} $$

(28)

subject to:

$$ {\displaystyle \sum_{j=1}^{n_1}{\left|{a}_{rj}\right|}^{-}\mathrm{Sign}\left({a}_{rj}^{-}\right)\left({x}_j^{-}+\varDelta {x}_j{u}_{j\ \mathrm{opt}}\right)}\le {b}_r^{-},\kern0.5em \forall r $$

(29)

$$ \begin{array}{l}{\displaystyle \sum_{j=1}^{n_1}{\left|{a}_{rj}\right|}^{-}\mathrm{Sign}\left({a}_{rj}^{-}\right)\left({x}_j^{-}+\varDelta {x}_j{u}_{j\ \mathrm{opt}}\right)}+{\displaystyle \sum_{j=1}^{k_2}{\left|{a}_{tj}^{\prime}\right|}^{-}\mathrm{Sign}\left({a^{\prime}}_{tj}^{-}\right){y}_{jh}^{+}}\\ {}+{\displaystyle \sum_{j={k}_2+1}^{n_2}{\left|{a}_{tj}^{\prime}\right|}^{+}\mathrm{Sign}\left({a^{\prime}}_{tj}^{+}\right){y}_{jh}^{-}}\ge {\tilde{w}}_h^{+},\kern0.5em \forall t,\ h\end{array} $$

(30)

$$ {\displaystyle \sum_{j=1}^{n_1}{c}_j^{-}\left({x}_j^{-}+\varDelta {x}_j{u}_{j\ \mathrm{opt}}\right)}-{\displaystyle \sum_{j=1}^{k_2}{p}_h{e}_j^{+}{y}_{jh}^{+}}-{\displaystyle \sum_{j={k}_2+1}^{n_2}{p}_h{e}_j^{+}{y}_{jh}^{-}}\ge {\eta}^{-}-{\mathrm{UB}}_h{z}_h^{+},\kern0.5em \forall j,\ h $$

(31)

$$ {\displaystyle \sum_{j=1}^{n_1}{c}_j^{-}\left({x}_j^{-}+\varDelta {x}_j{u}_{j\ \mathrm{opt}}\right)}-{\displaystyle \sum_{j=1}^{k_2}{p}_h{e}_j^{+}{y}_{jh}^{+}}-{\displaystyle \sum_{j={k}_2+1}^{n_2}{p}_h{e}_j^{+}{y}_{jh}^{-}}\le {\eta}^{-}+{\mathrm{UB}}_h\left(1-{z}_h^{+}\right),\ \forall j,\ h $$

(32)

$$ {\displaystyle \sum_{h=1}^v{p}_h{z}_h^{+}}\le {\varepsilon}^{-},\kern0.5em \forall h $$

(33)

$$ {z}_h^{-}=\left\{\begin{array}{l}1\kern0.75em \mathrm{if}\kern0.5em {\displaystyle \sum_{j=1}^{n_1}{c}_j^{-}\left({x}_j^{-}+\varDelta {x}_j{u}_{j\ \mathrm{opt}}\right)}-{\displaystyle \sum_{j=1}^{k_2}{p}_h{e}_j^{+}{y}_{jh}^{+}}-{\displaystyle \sum_{j={k}_2+1}^{n_2}{p}_h{e}_j^{+}{y}_{jh}^{-}}\ge {\eta}^{-}\\ {}0\kern0.5em \mathrm{otherwise}\end{array}\right. $$

(34)

$$ {y}_{jh}^{+}\ge {y}_{jh\ \mathrm{opt}}^{-},\kern0.5em j=1,\ 2,\dots,\ {k}_2;\kern0.5em \forall h $$

(35)

$$ 0\le {y}_{jh}^{-}\le {y}_{jh\ \mathrm{opt}}^{+},\kern0.5em j={k}_2+1,\ {k}_2+2,\dots,\ {n}_2;\kern0.5em \forall h $$

(36)

$$ {z}_h^{-}\le {z}_{h\ \mathrm{opt}}^{+},\kern0.5em \forall h $$

(37)

where z ⁻ _h , y ⁺ _jh (j = 1, 2, …, k ₂), and y ⁻ _jh ( j = k ₂ + 1, k ₂ + 2, …, n ₂) are decision variables. Solutions of z ⁻ _h , y ⁺ _jh ( j = 1, 2, …, k ₂), and y ⁻ _jh (j = k ₂ + 1, k ₂ + 2, …, n ₂) can be obtained through solving submodel (19). Through integrating the solutions of the two sets of submodels, the solution for the objective-function value can be obtained. The detailed computational processes for solving the RSRA model can be summarized as follows:

• Step 1. Formulate the RSRA model.

• Step 2. Transform the RSRA model into two submodels, based on an interactive algorithm.

• Step 3. Formulate and solve the upper bound submodel corresponding to f ⁺, obtain x ^± _j opt and y ⁺ _jh opt , and calculate f ⁺_opt .

• Step 4. Formulate and solve the f ⁻ submodel base on the solutions obtained through step 3, obtain y ⁻ _jh opt , and calculate f ⁻_opt .

• Step 5. Construct the corresponding risk curve. If the decision-maker is satisfied with the current level of risk then stop; otherwise, go to step 6.

• Step 6. Let the allowable risk exposure level from the decision-makers be ε ^± and the corresponding cost aspiration level be η ^±.

• Step 7. Formulate the RSRA problem with the objective being maximized under chosen ε ^± and η ^±.

• Step 8. If the decision-maker is satisfied with the solutions, then stop. Otherwise, go to step 6 for the next level.

Appendix 2

Nomenclatures for variables and parameters

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, and transportation); 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 scenario; h = 1, 2,…, H

p _h :

probability of occurrence allowable pollutant discharge level h (%)

f ^±_AGR :

sub-total benefit of agriculture (RMB¥)

f ^±_FIS :

sub-total benefit of fishery (RMB¥)

f ^±_LIV :

sub-total benefit of livestock husbandry (RMB¥)

f ^±_IND :

sub-total benefit of industry (RMB¥)

f ^±_FOR :

sub-total benefit of forestry (RMB¥)

BA ^± _ij :

unit benefit from agricultural activity i in zone j (RMB¥/km²)

TA ^± _ij :

land area target for agricultural activity i in zone j (km²)

PEA ^± _ik :

reduction of net benefit from agricultural activity i for excess discharge of pollutant k (RMB¥/kg when k = 2, 3; RMB¥/t when k = 4)

DPA ^± _ijk :

discharge rate of pollutant k from agricultural activity i in zone j (kg/km² when k = 2, 3; t/km² when k = 4)

XA ^± _ijhk :

decision variables representing amount by which the target of agricultural activity i discharge pollutant k exceeds standards in zone j when scenario is h (km²)

BF ^± _ij :

unit benefit from fishery activity i in zone j (RMB¥/km²)

TF ^± _ij :

land area target for fishery farming activity i in zone j (km²)

PEF ^± _ik :

reduction of net benefit from fishery activity i for excess discharge of pollutant k (RMB¥/kg)

DPF ^± _ijk :

discharge rate of pollutant k from fishery activity i in zone j (kg/km²)

XF ^± _ijhk :

decision variables representing amount by which target of fishery activity i discharge pollutant k exceeds standards in zone j when scenario is h (km²)

BL ^± _ij :

unit benefit from livestock husbandry activity i in zone j (RMB¥/head)

TL ^± _ij :

target for livestock husbandry activity i in zone j (head)

PEL ^± _i :

reduction of net benefit from livestock husbandry activity i for excess discharge of pollutant (i.e., COD) (RMB¥/kg)

DPL ^± _ij :

discharge rate of pollutant (i.e., COD) from livestock husbandry activity i in zone j (kg/head)

XL ^± _ijh :

decision variables representing amount by which target of livestock husbandry activity i discharge pollutant (i.e., COD) exceeds standards in zone j when scenario is h (head)

TI ^± _ij :

output target for industrial activity i in zone j (RMB¥)

PEI ^± _i :

reduction of net benefit from industrial activity i for excess discharge of pollutant (i.e., COD) (RMB¥/kg)

DPI ^± _ij :

discharge rate of pollutant (i.e., COD) from industrial activity i in zone j (kg/ RMB¥)

XI ^± _ijh :

decision variables representing amount by which target of industrial activity i discharge pollutant (i.e., COD) exceeds standards in zone j when scenario is h (RMB¥)

BW ^± _ij :

unit benefit from forestry activity i in zone j (RMB¥/head)

TW ^± _ij :

land area target for forestry activity i in zone j (unit)

PEW ^± _i :

reduction of net benefit from forestry activity i for excess discharge of pollutant (i.e., soil loss) (RMB¥/t)

DPW ^± _ij :

discharge rate of pollutant (i.e., soil loss) from forestry activity i in zone j (t/km²)

XW ^± _ijh :

decision variables representing amount by which target of forestry activity i discharge pollutant (i.e., soil loss) exceeds standards in zone j when scenario is h (unit)

COF ^± _ij :

COD discharge from fishery farming activity i in zone j (kg/km²)

PCF ^± _jh :

maximum allowable COD discharge for fishery farming activities in zone j with probability p _h of occurrence under scenario h (kg)

PCL ^± _jh :

maximum allowable COD discharge for livestock husbandry activities in zone j with probability p _h of occurrence under scenario h (kg)

PCI ^± _jh :

maximum allowable COD discharge for industrial activity i in zone j with probability p _h of occurrence under scenario h (kg)

MCL ^± _h :

maximum allowable COD discharge from economic activities with probability p _h of occurrence under scenario h (kg)

SL ^± _ij :

soil loss from agricultural activity i in zone j (t/km²)

AP ^± _ij :

phosphorous content of soil corresponding to agricultural activity i in zone j (kg/t)

RA ^± _ij :

runoff from agricultural activity i in zone j (kg/km²)

RP ^± _ij :

dissolved phosphorous content of runoff corresponding to agricultural activity i in zone j (%)

PAP ^± _jh :

maximum allowable phosphorous loss from agricultural activities in zone j with probability p _h of occurrence under scenario h (kg)

FP ^± _ij :

dissolved phosphorous loss from fishery farming activity i in zone j (kg/km²)

PFP ^± _jh :

maximum allowable phosphorous loss from fishery farming activities in zone j with probability p _h of occurrence under scenario h (kg)

MPL ^± _h :

maximum allowable phosphorous loss from economic activities with probability p _h of occurrence under scenario h (kg)

AN ^± _ij :

nitrogen content of soil corresponding to agricultural activity i in zone j (kg/t)

RN ^± _ij :

dissolved nitrogen content of runoff corresponding to agricultural activity i in zone j (%)

PAN ^± _jh :

maximum allowable nitrogen loss from agricultural activities in zone j with probability p _h of occurrence under scenario h (kg)

FN ^± _ij :

dissolved nitrogen loss from fishery activity i in zone j (kg/km²)

PFN ^± _jh :

maximum allowable nitrogen loss from fishery farming activities in zone j with probability p _h of occurrence under scenario h (kg)

η ^± :

targeted benefit level of economic activities (RMB¥)

z ^± _h :

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 (η ^±) and a value of 1 otherwise

UB _h :

upper bound benefit under each scenario h (RMB¥)

ε ^± :

desired risk exposure level of economic activity

MSL ^± _jh :

maximum allowable soil loss from economic activities with probability p _h of occurrence under scenario h (t)

MNL ^± _h :

maximum allowable nitrogen loss from economic activities with probability p _h of occurrence under scenario h (kg)

PSL ^± _jh :

maximum allowable soil loss from agricultural activities in zone j with probability p _h of occurrence under scenario h (t)

PWS ^± _jh :

maximum allowable soil loss from forestry activities in zone j with probability p _h of occurrence under scenario h (t)

WA ^± _i :

water demand for agricultural activity i (m³/km²)

WF ^± _i :

water demand for fishery activity i (m³/km²)

WL ^± _i :

water demand for livestock husbandry activity i (m³/head)

WI ^± _i :

water demand for industrial activity i (m³/RMB¥)

WW ^± _i :

water demand for forestry activity i (m³/km²)

MAXW ^± _j :

maximum allowable water resources supply amount in zone j (m³)

TA ^± _i min :

minimum demand for agricultural activity i (km²)

TA ^± _i max :

maximum demand for agricultural activity i (km²)

TF ^± _i min :

minimum demand for fishery activity i (km²)

TF ^± _i max :

maximum demand for fishery activity i (km²)

TL ^± _i min :

minimum demand for livestock husbandry activity i (head)

TL ^± _i max :

maximum demand for livestock husbandry activity i (head)

TW ^± _i min :

minimum demand for industrial activity i (RMB¥)

TW ^± _i max :

maximum demand for industrial activity i (RMB¥)