Identification of water quality management policy of watershed system with multiple uncertain interactions using a multi-level-factorial risk-inference-based possibilistic-probabilistic programming approach

Abstract

In this study, a multi-level-factorial risk-inference-based possibilistic-probabilistic programming (MRPP) method is proposed for supporting water quality management under multiple uncertainties. The MRPP method can handle uncertainties expressed as fuzzy-random-boundary intervals, probability distributions, and interval numbers, and analyze the effects of uncertainties as well as their interactions on modeling outputs. It is applied to plan water quality management in the Xiangxihe watershed. Results reveal that a lower probability of satisfying the objective function (θ) as well as a higher probability of violating environmental constraints (q _i ) would correspond to a higher system benefit with an increased risk of violating system feasibility. Chemical plants are the major contributors to biological oxygen demand (BOD) and total phosphorus (TP) discharges; total nitrogen (TN) would be mainly discharged by crop farming. It is also discovered that optimistic decision makers should pay more attention to the interactions between chemical plant and water supply, while decision makers who possess a risk-averse attitude would focus on the interactive effect of q _i and benefit of water supply. The findings can help enhance the model’s applicability and identify a suitable water quality management policy for environmental sustainability according to the practical situations.

Appendices

Appendix A: Solution method

A robust two-step method is proposed to convert model (10) into two submodels that correspond to lower and upper bounds of the objective function value. Since the objective is to maximize the system benefit, the submodel corresponding to the upper bound of the objective function value (h ⁺) should be first formulated. Submodel (2) corresponding to h ⁻ is then formulated. In the first step, a set of submodels corresponding to h ⁺ can be reformulated as:

$$ \operatorname{Max}\ {h}^{+} $$

(22)

subject to:

$$ \frac{\sum_{j=1}^{k_1}\left({\lambda}_j^{+}-{\mu}_j^{+}\right){x}_j^{+}+\sum_{j={k}_1+1}^n\left({\lambda}_j^{+}-{\mu}_j^{+}\right){x}_j^{-}-{\Phi}^{-1}\left(1-\theta \right)\sqrt{\sum_{j=1}^{k_1}{\left({\sigma}_j^{+}{x}_j^{+}\right)}^2+\sum_{j={k}_1+1}^n{\left({\sigma}_j^{+}{x}_j^{-}\right)}^2}+{z}_0^{+}}{\sum_{j=1}^{k_1}{\lambda}_j^{+}{x}_j^{+}+\sum_{j=1}^n{\lambda}_j^{+}{x}_j^{-}-{z}_1^{+}+{z}_0^{+}}\ge {h}_{+} $$

(23)

$$ \sum_{j=1}^{j_1}\mid {a}_{i j}\mid {}^{-} Sign\left({a}_{i j}^{-}\right)\ {x}_j^{+}+\sum_{j={k}_1+1}^n\mid {a}_{i j}\mid {}^{+} Sign\left({a}_{i j}^{+}\right)\ {x}_j^{-}\le {b}_i{\left(\omega \right)}^{q_i},\kern0.5em i=1,2,\dots, m $$

(24)

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

(25)

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

(26)

where ${x}_j^{+}$ (j = 1, 2, ..., k ₁) are upper bounds of the decision variables (${x}_j^{\pm }$) with positive coefficients in the objective function, and ${x}_j^{-}$ (j = k ₁ + 1, k ₁ + 2, ..., n) are lower bounds with negative coefficients. Submodels corresponding to h ⁻ can be formulated as:

$$ \operatorname{Max}\ {h}^{-} $$

(27)

subject to:

$$ \frac{\sum_{j=1}^{k_1}\left({\lambda}_j^{-}-{\mu}_j^{-}\right){x}_j^{-}+\sum_{j={k}_1+1}^n\left({\lambda}_j^{-}-{\mu}_j^{-}\right){x}_j^{+}-{\Phi}^{-1}\left(1-\theta \right)\sqrt{\sum_{j=1}^{k_1}{\left({\sigma}_j^{-}{x}_j^{-}\right)}^2+\sum_{j={k}_1+1}^n{\left({\sigma}_j^{-}{x}_j^{+}\right)}^2}+{z}_0^{-}}{\sum_{j=1}^{k_1}{\lambda}_j^{-}{x}_j^{-}+\sum_{j=1}^n{\lambda}_j^{-}{x}_j^{+}-{z}_1^{-}+{z}_0^{-}}\ge {h}^{-} $$

(28)

$$ \sum_{j=1}^{j_1}\mid {a}_{rj}\mid {}^{+} Sign\left({a}_{rj}^{+}\right)\ {x}_j^{-}+\sum_{j={k}_1+1}^n\mid {a}_{rj}\mid {}^{-} Sign\left({a}_{rj}^{-}\right)\ {x}_j^{+}\le {b}_i{\left(\omega \right)}^{q_i},\kern0.5em i=1,2,\dots, m $$

(29)

$$ 0\le {x}_j^{-}\le {x}_{j\ opt}^{+},\kern0.5em j=1,2,\dots, {k}_1 $$

(30)

$$ {x}_j^{+}\ge {x}_{j\ opt}^{-},\kern0.5em j={k}_1+1,{k}_2+1,\dots, n $$

(31)

where ${x}_{\; j\kern0.37em opt}^{+}$ (j = 1, 2, ..., k ₁) and ${x}_{\; j\kern0.37em opt}^{-}$ (j = k ₁ + 1, k ₁ + 2, ..., n) are solutions corresponding to h ⁻. The model can be similarly solved when NM is adopted.

Appendix B: Nomenclatures and the RIPP model under NM

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

P :

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

r :

livestock, 1 = pig, 2 = ox, 3 = sheep, 4 = domestic fowls

s :

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

t :

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

${\lambda}_{it}^{\pm }$ _, ${\mu}_{it}^{\pm }$ _, ${\sigma}_{it}^{\pm }$ :

mean value, expected value, and standard deviation of benefit from chemical plant (RMB¥/t)

${PLC}_{it}^{\pm }$ :

production level of chemical plant (t)

${\lambda}_{pt}^{\pm }$ _, ${\mu}_{pt}^{\pm }$ _, ${\sigma}_{pt}^{\pm }$ :

mean value, expected value, and standard deviation of benefit for phosphate ore (RMB¥/t)

${PLM}_{pt}^{\pm }$ :

production level of phosphorus mining company p during period t (t)

${\lambda}_{st}^{\pm }$ _, ${\mu}_{st}^{\pm }$ _, ${\sigma}_{st}^{\pm }$ :

mean value, expected value, and standard deviation of benefit from water supply (RMB¥/m³)

${QW}_{st}^{\pm }$ :

quantity of water supply (m³)

${CY}_{jkt}^{\pm }$ :

yield of crop (t/ha);

${\lambda}_{jkt}^{\pm }$ _, ${\mu}_{jkt}^{\pm }$ _, ${\sigma}_{jkt}^{\pm }$ :

mean value, expected value, and standard deviation of benefit for agricultural product (RMB¥/t)

${PA}_{jkt}^{\pm }$ :

planning area of crop k in agricultural zone j during period t (ha)

${\lambda}_r^{\pm }$ _, ${\mu}_r^{\pm }$ _, ${\sigma}_r^{\pm }$ :

mean value, expected value, and standard deviation of benefit from livestock (RMB¥/unit)

${NL}_r^{\pm }$ :

number of livestock r in the study area (unit)

$\lambda {\hbox{'}}_{it}^{\pm }$ _, $\mu {\hbox{'}}_{it}^{\pm }$ _, $\sigma {\hbox{'}}_{it}^{\pm }$ :

mean value, expected value, and standard deviation of wastewater treatment cost of chemical plant (RMB¥/t)

$\lambda {\hbox{'}}_{st}^{\pm }$ _, $\mu {\hbox{'}}_{st}^{\pm }$ _, $\sigma {\hbox{'}}_{st}^{\pm }$ :

mean value, expected value, and standard deviation of wastewater treatment cost at town (RMB¥/m³)

${\lambda}_{jt}^{\pm }$ _, ${\mu}_{jt}^{\pm }$ _, ${\sigma}_{jt}^{\pm }$ :

mean value, expected value, and standard deviation of cost for manure disposal (RMB¥/t)

$\lambda {\hbox{'}}_{jt}^{\pm }$ _, $\mu {\hbox{'}}_{jt}^{\pm }$ _, $\sigma {\hbox{'}}_{jt}^{\pm }$ :

mean value, expected value, and standard deviation of cost for purchasing fertilizer (RMB¥/t)

${z}_0^{\pm }$ :

minimum total net benefit that decision makers want to obtain (RMB¥)

${z}_1^{\pm }$ :

maximum total net benefit that decision makers want to obtain (RMB¥)

${AM}_{jkt}^{\pm }$ :

amount of manure applied to agricultural zone (t)

${AF}_{jkt}^{\pm }$ :

amount of fertilizer applied to agricultural zone (t)

${TPC}_{st}^{\pm }$ :

capacity of wastewater treatment capacity (WTPs) (m³)

${TPD}_{it}^{\pm }$ :

capacity of wastewater treatment capacity (chemical plants) (m³)

${IC}_{it}^{\pm }$ :

BOD concentration of raw wastewater from chemical plant (kg/m³)

${\eta}_{BOD, it}^{\pm }$ :

BOD treatment efficiency in chemical plant (%)

${ABC}_{it}^{\pm }$ :

allowable BOD discharge for chemical plant (kg)

${BM}_{st}^{\pm }$ :

BOD concentration of municipal wastewater at town (kg/m³)

$\eta {\hbox{'}}_{BOD, st}^{\pm }$ :

BOD treatment efficiency of WTPs at town (%)

${ABW}_{st}^{q_i}$ :

allowable BOD discharge for WTPs at town (kg)

${AML}_{rt}^{\pm }$ :

amount of manure generated by livestock [t/ unit]

${AMH}_t^{\pm }$ :

amount of manure generated by humans [t/ unit]

${RP}_t^{\pm }$ :

total rural population in the study area during period t (unit)

${MS}_t^{\pm }$ :

manure loss rate in period t (%)

${\varepsilon}_{NM}^{\pm }$ :

nitrogen content of manure (%)

${ACW}_t^{\pm }$ :

wastewater generation of per capita water consumption during period t (m³/ unit)

${DNR}_t^{\pm }$ :

dissolved nitrogen concentration of rural wastewater during period t (t/m³)

${ANL}_t^{q_i}$ :

maximum allowable nitrogen loss from rural life section in period t (t)

${NS}_{jk}^{\pm }$ :

nitrogen content of soil in agricultural zone (%)

${SL}_{jkt}^{\pm }$ :

average soil loss from agricultural zone (t/ha)

${RF}_{jkt}^{\pm }$ :

runoff from agricultural zone (mm)

${DN}_{jkt}^{\pm }$ :

dissolved nitrogen concentration in runoff from agricultural zone (mg/L)

${MNL}_{jt}^{\pm }$ :

maximum allowable nitrogen loss in agricultural zone j during period t (t/ha)

${TA}_{jt}^{\pm }$ :

tillable area of agricultural zone (ha)

${PCR}_{it}^{\pm }$ :

phosphorus concentration of raw wastewater from chemical plant (kg/m³)

${\eta}_{TP, it}^{\pm }$ :

phosphorus treatment efficiency in chemical plant (%)

${ASC}_{it}^{\pm }$ :

amount of slag discharged by chemical plant (kg/t)

${SLR}_{it}^{\pm }$ :

slag loss rate due to rain wash in chemical plant (%)

${PSC}_{it}^{\pm }$ :

phosphorus content in slag generated by chemical plant (%)

${APC}_{i t}^{q_i}$ :

allowable phosphorus discharge for chemical plant (kg)

${\varepsilon}_{PM}^{\pm }$ :

phosphorus content of manure (%)

${DPR}_t^{\pm }$ :

dissolved phosphorus concentration of rural wastewater (t/m³)

${APL}_t^{\pm }$ :

maximum allowable phosphorus loss from rural life during period t (t)

${PCM}_{st}^{\pm }$ :

phosphorus concentration of municipal wastewater at town (kg/m³)

${\eta}_{TP, st}^{\pm }$ :

phosphorus treatment efficiency of WTP at town (%)

${APW}_{st}^{\pm }$ :

allowable phosphorus discharge for WTP at town (kg)

${WPM}_{pt}^{\pm }$ :

wastewater generation from phosphorus mining company (m³/t)

${MWC}_{pt}^{\pm }$ :

phosphorus concentration of wastewater from mining company (kg/ m³)

${\eta}_{TP, pt}^{\pm }$ :

phosphorus treatment efficiency in mining company (%)

${ASM}_{pt}^{\pm }$ :

amount of slag discharged by mining company (kg/t)

${PCS}_{pt}^{\pm }$ :

phosphorus content in generated slag (%)

${SLW}_{pt}^{\pm }$ :

slag loss rate due to rain wash (%)

${APM}_{pt}^{q_i}$ :

allowable phosphorus discharge for mining company (kg)

${PS}_{jk}^{\pm }$ :

phosphorus content of soil in agricultural zone (%)

${SL}_{jkt}^{\pm }$ :

average soil loss from agricultural zone (t/ha)

${DP}_{jkt}^{\pm }$ :

dissolved phosphorus concentration in runoff from agricultural zone (mg/L)

${MPL}_{jt}^{\pm }$ :

maximum allowable phosphorus loss in agricultural zone (t/ha)

${MSL}_{jt}^{\pm }$ :

maximum allowable soil loss agricultural zone (t/ha)

${NVF}_t^{\pm }$ :

nitrogen volatilization/denitrification rate of fertilizer (%)

${NVM}_t^{\pm }$ :

nitrogen volatilization/denitrification rate of manure (%)

${\varepsilon}_{NF}^{\pm }$ :

nitrogen content of fertilizer (%)

${\varepsilon}_{PF}^{\pm }$ :

phosphorus content of fertilizer (%)

${\varepsilon}_{NM}^{\pm }$ :

nitrogen content of manure (%)

${\varepsilon}_{PM}^{\pm }$ :

phosphorus content of manure (%)

${NR}_{jkt}^{\pm }$ :

nitrogen requirement of agricultural zone (t/ha)

${PR}_{jkt}^{\pm }$ :

phosphorus requirement of crop k in agricultural zone (t/ha)

${TAH}_{jt}^{\pm }$ :

dry farmland of agricultural zone (ha)

${TAS}_{jt}^{\pm }$ :

paddy farmland of agricultural zone (ha)

${MFP}_t^{\pm }$ :

the government requirement for minimum area of farmland (ha);

${PLC}_{it, \min}^{\pm }$ :

minimum production level of chemical plant (t/day)

${PLC}_{it, \max}^{\pm }$ :

maximum production level of chemical plant (t/day)

${NL}_{r, \min}^{\pm }$ :

minimum number of livestock (unit)

${NL}_{r, \max}^{\pm }$ :

maximum number of livestock (unit)

${QW}_{st, \min}^{\pm }$ :

minimum quantity of water supply to town (m³/day)

${QW}_{st, \max}^{\pm }$ :

maximum quantity of water supply to town (m³/day)

${PLM}_{pt, \min}^{\pm }$ :

minimum production level of phosphorus mining company (t/day)

${PLM}_{pt, \max}^{\pm }$ :

maximum production level of phosphorus mining company (t/day)

The RIPP model under NM for supporting support water quality management in the Xiangxihe watershed can be represented as follows:

$$ \operatorname{Max}\kern0.5em {h}^{\pm } $$

(32)

subject to:

• (1) Risk inference of decision maker:

$$ \frac{\begin{array}{c}\Big\{\sum_{i=1}^5\sum_{t=1}^2\left[{\mu}_{i t}^{\hbox{'}\pm }-{\mu}_{i t}^{\pm}\right]{PLC}_{i t}^{\pm }-\sum_{p=1}^6\sum_{t=1}^2{\mu}_{p t}^{\pm }{PLM}_{p t}^{\pm}\\ {}+\sum_{s=1}^4\sum_{t=1}^2\left[{\mu}_{s t}^{\hbox{'}\pm }-{\mu}_{s t}^{\pm}\right]{QW}_{s t}^{\pm }-\sum_{j=1}^4\sum_{k=1}^9\sum_{t=1}^2{\mu}_{j kt}^{\pm }{CY}_{j kt}^{\pm }{PA}_{j kt}^{\pm}\\ {}+\sum_{r=1}^4{\mu}_r^{\pm }{NL}_r^{\pm }+\sum_{j=1}^4\sum_{k=1}^9\sum_{t=1}^2{\mu}_{j t}^{\pm }{AM}_{j kt}^{\pm }+\sum_{j=1}^4\sum_{k=1}^9\sum_{t=1}^2{\mu}_{j t}^{\hbox{'}\pm }{AF}_{j kt}^{\pm}\Big\}\\ {}-{\Phi}^{-1}\left(1-\theta \right)\sqrt{\begin{array}{l}\sum_{i=1}^5\sum_{t=1}^2\left[{\left({\sigma}_{i t}^{\pm }{PLC}_{i t}^{\pm}\right)}^2-{\left({\sigma}_{i t}^{\hbox{'}\pm }{PLC}_{i t}^{\pm}\right)}^2\right]+\sum_{p=1}^6\sum_{t=1}^2{\left({\sigma}_{p t}^{\pm }{PLM}_{p t}^{\pm}\right)}^2\\ {}+\sum_{s=1}^4\sum_{t=1}^2\left[{\left({\sigma}_{s t}^{\pm }{QW}_{s t}^{\pm}\right)}^2-{\left({\sigma}_{s t}^{\hbox{'}\pm }{QW}_{s t}^{\pm}\right)}^2\right]+\sum_{j=1}^4\sum_{k=1}^9\sum_{t=1}^2{\left({\sigma}_{j kt}^{\pm }{CY}_{j kt}^{\pm }{PA}_{j kt}^{\pm}\right)}^2\\ {}+\sum_{r=1}^4{\left({\sigma}_r^{\pm }{NL}_r^{\pm}\right)}^2-\sum_{j=1}^4\sum_{k=1}^9\sum_{t=1}^2{\left({\sigma}_{j t}^{\pm }{AM}_{j kt}^{\pm}\right)}^2-\sum_{j=1}^4\sum_{k=1}^9\sum_{t=1}^2{\left({\sigma}_{j t}^{\hbox{'}\pm }{AF}_{j kt}^{\pm}\right)}^2\end{array}}+{z}_0^{\pm}\end{array}}{\begin{array}{c}\sum_{i=1}^5\sum_{t=1}^2\left({\gamma}_{i t}^{\pm }-{\gamma}_{i t}^{\hbox{'}\pm}\right){PLC}_{i t}^{\pm }+\sum_{p=1}^6\sum_{t=1}^2{\gamma}_{p t}^{\pm }{PLM}_{p t}^{\pm }+\sum_{s=1}^4\sum_{t=1}^2\left({\gamma}_{s t}^{\pm }-{\gamma}_{s t}^{\hbox{'}\pm}\right){QW}_{s t}^{\pm }+\sum_{j=1}^4\sum_{k=1}^9\sum_{t=1}^2{\gamma}_{j kt}^{\pm }{CY}_{j kt}^{\pm }{PA}_{j kt}^{\pm}\\ {}+\sum_{r=1}^4{\gamma}_r^{\pm }{NL}_r^{\pm }-\sum_{j=1}^4\sum_{k=1}^9\sum_{t=1}^2{\gamma}_{j t}^{\pm }{AM}_{j kt}^{\pm }-\sum_{j=1}^4\sum_{k=1}^9\sum_{t=1}^2{\gamma}_{j t}^{\hbox{'}\pm }{AF}_{j kt}^{\pm }-{z}_1^{\pm }+{z}_0^{\pm}\end{array}}\ge {h}^{\pm } $$

(33)

• (2) Constraints of water supply:

$$ {QW}_{st}^{\pm}\cdot {GT}_{st}^{\pm}\le {TPC}_{st}^{\pm },\kern0.5em \forall s, t $$

(34)

$$ {QW}_{st}^{\pm}\cdot {GT}_{st}^{\pm}\cdot {BM}_{st}^{\pm}\cdot \left(1-{\eta}_{BOD, st}^{\pm}\right)\le {ABW}_{st}^{q_i},\kern0.5em \forall s, t $$

(35)

$$ {QW}_{st}^{\pm}\cdot {GT}_{st}^{\pm}\cdot {PCM}_{st}^{\pm}\left(1-{\eta}_{TP, st}^{\pm}\right)\le {APW}_{st}^{\pm },\forall s,\kern0.5em t $$

(36)

$$ {QW}_{st, \min}\le {QW}_{st}^{\pm}\le {QW}_{st, \max },\kern0.5em \forall s, t $$

(37)

• (3) Constraints of chemical plant production:

$$ {PLC}_{it}^{\pm}\cdot {WC}_{it}^{\pm}\le {TPD}_{it}^{\pm },\kern0.5em \forall i, t $$

(38)

$$ {PLC}_{it}^{\pm}\cdot {WC}_{it}^{\pm}\cdot {IC}_{it}^{\pm}\cdot \left(1-{\eta}_{BOD, it}^{\pm}\right)\le {ABC}_{it}^{\pm },\kern0.5em \forall i, t $$

(39)

$$ {PLC}_{i t}^{\pm}\cdot \left[{WC}_{i t}^{\pm}\cdot {PCR}_{i t}^{\pm}\left(1-{\eta}_{TP, it}^{\pm}\right)+{ASC}_{i t}^{\pm}\cdot {SLR}_{i t}^{\pm}\cdot {PSC}_{i t}^{\pm}\right]\le {APC}_{i t}^{q_i},\kern0.5em \forall i, t $$

(40)

$$ {PLC}_{it, \min}\le {PLC}_{it}^{\pm}\le {PLC}_{it, \max },\kern0.5em \forall i, t $$

(41)

• (4) Constraints of phosphorus mining company production:

$$ {PLM}_{pt}^{\pm}\cdot \left[{WPM}_{pt}^{\pm}\cdot {MWC}_{pt}^{\pm}\left(1-{\eta}_{TP, pt}^{\pm}\right)+{ASM}_{pt}^{\pm}\cdot {PCS}_{pt}^{\pm}\cdot {SLW}_{pt}^{\pm}\right]\le {APM}_{pt}^{q_i},\kern0.5em \forall p, t $$

(42)

$$ {PLM}_{pt, \min}\le {PLM}_{pt}^{\pm}\le {PLM}_{pt, \max },\kern0.5em \forall p, t $$

(43)

• (5) Constraints of crop farming:

$$ \sum_{k=1}^9\left({NS}_{jk}^{\pm}\cdot {SL}_{jk t}^{\pm }+{RF}_{jk t}^{\pm}\cdot {DN}_{jk t}^{\pm}\cdot {10}^{-5}\right)\cdot {PA}_{jk t}^{\pm}\le {MNL}_{jt}^{\pm}\cdot {TA}_{jt}^{\pm },\kern0.5em \forall j, t $$

(44)

$$ \sum_{k=1}^9\left({PS}_{jk}^{\pm}\cdot {SL}_{jk t}^{\pm }+{DP}_{jk t}^{\pm}\cdot {RF}_{jk t}^{\pm}\cdot {10}^{-5}\right){PA}_{jk t}^{\pm}\le {MPL}_{jt}^{\pm}\cdot {TA}_{jt}^{\pm },\forall j, t $$

(55)

$$ \sum_{k=1}^9{SL}_{jkt}^{\pm}\cdot {PAQ}_{jkt}^{\pm}\le {MSL}_{jt}^{\pm}\cdot {TA}_{jt}^{\pm },\forall j, t $$

(56)

$$ \left(1-{NVF}_t^{\pm}\right)\cdot {\varepsilon}_{NF}^{\pm}\cdot {AF}_{jkt}^{\pm }+\left(1-{NVM}_t^{\pm}\right)\cdot {\varepsilon}_{NM}^{\pm}\cdot {AM}_{jkt}^{\pm}\ge {NR}_{jkt}^{\pm}\cdot {PA}_{jkt}^{\pm },\kern0.5em \forall j, k, t $$

(57)

$$ {\varepsilon}_{PF}^{\pm}\cdot {AF}_{jkt}^{\pm }+{\varepsilon}_{PM}^{\pm}\cdot {AM}_{jkt}^{\pm}\ge {PR}_{jkt}^{\pm}\cdot {PA}_{jkt}^{\pm },\kern0.5em \forall j, k, t $$

(58)

$$ \sum_{k=1}^9\left({\varepsilon}_{NF}^{\pm}\cdot {AF}_{jkt}^{\pm }+{\varepsilon}_{NM}^{\pm}\cdot {AM}_{jkt}^{\pm }-{NR}_{jkt}^{\pm}\cdot {PA}_{jkt}^{\pm}\right)\le {MNL}_{jt}^{\pm}\cdot {TA}_{jt}^{\pm },\kern0.5em \forall j, t $$

(59)

$$ \sum_{k=1}^9\left({\varepsilon}_{PF}^{\pm}\cdot {AF}_{jkt}^{\pm }+{\varepsilon}_{PM}^{\pm}\cdot {AM}_{jkt}^{\pm }-{PR}_{jkt}^{\pm}\cdot {PA}_{jkt}^{\pm}\right)\le {MPL}_{jt}^{\pm}\cdot {TA}_{jt}^{\pm },\kern0.5em \forall j, t $$

(60)

$$ \sum_{j=1}^4\sum_{k=1}^6{PA}_{j kt}^{\pm}\ge {MFP}_t^{\pm },\kern0.5em t=1 $$

(61)

$$ \sum_{j=1}^4\sum_{k=1}^2{PA}_{j kt}^{\pm }+\sum_{j=1}^4\sum_{k=8}^9{PA}_{j kt}^{\pm}\ge {MFP}_t^{\pm },\kern0.5em t=2 $$

(62)

$$ {PA}_{jkt}^{\pm}\le {TAS}_{jt}^{\pm },\kern0.5em k=6, t=1 $$

(63)

$$ \sum_{k=1}^5{PA}_{jkt}^{\pm}\le {TAH}_{jt}^{\pm },\kern0.5em t=1 $$

(64)

$$ {PA}_{jkt}^{\pm}\le {TAS}_{jt}^{\pm },\kern0.5em k=7, t=2 $$

(65)

$$ \sum_{k=1}^2{PA}_{jkt}^{\pm }+\sum_{k=8}^9{PA}_{jkt}^{\pm}\le {TAH}_{jt}^{\pm },\kern0.5em t=2 $$

(66)

$$ { TA S}_{jt}^{\pm }+{ TA H}_{jt}^{\pm }={TA}_{jt}^{\pm },\kern0.5em \forall j, t $$

(67)

• (6) Constraints of livestock husbandry:

$$ \begin{array}{l}\left(\sum_{r=1}^4{ AM L}_{r t}^{\pm}\cdot {NL}_r^{\pm }+{ AM H}_t^{\pm}\cdot {RP}_t^{\pm }-\sum_{j=1}^4\sum_{k=1}^9{AM}_{j kt}^{\pm}\right)\cdot {MS}_t^{\pm}\cdot {\varepsilon}_{NM}^{\pm}\\ {}+{RP}_t^{\pm}\cdot {ACW}_t^{\pm}\cdot {DNR}_t^{\pm}\le {ANL}_t^{q_i},\kern0.5em \forall t\end{array} $$

(68)

$$ \sum_{r=1}^4{ AM L}_{r t}^{\pm}\cdot {NL}_r^{\pm }+{ AM H}_t^{\pm}\cdot {RP}_t^{\pm}\ge \sum_{j=1}^4\sum_{k=1}^9{AM}_{j kt}^{\pm },\kern0.5em \forall t $$

(69)

$$ {NL}_{r, \min}\le {NL}_r^{\pm}\le {NL}_{r, \max },\kern0.5em \forall r $$

(70)

• (7) Non-negative constraints:

$$ {PLC}_{it}^{\pm },\kern0.5em {PA}_{jkt}^{\pm },\kern0.5em {NL}_r^{\pm },\kern0.5em {QW}_{st}^{\pm },\kern0.5em {PLM}_{pt}^{\pm },\kern0.5em {AM}_{jkt}^{\pm },\kern0.5em {AF}_{jkt}^{\pm}\ge 0 $$

(71)

Solutions for the MRPP model under a given risk level can be obtained through integration of solutions of the lower and upper submodels. A set of interval solutions associated with possiblistic and probabilistic information for the objective and decision variables can be obtained by solving the submodels under the other risk levels.