Water Resources and Farmland Management in the Songhua River Watershed under Interval and Fuzzy Uncertainties

Abstract

The Songhua River Watershed (SHRW) in Northeastern China has been challenged by water scarcity, water contamination, and soil erosion for decades. These problems will remain or even worsen in the following decades, threatening regional eco-environmental quality and socio-economic development. Mitigation of these problems through integrated water resources and farmland management (WRFM) is desired but is challenged by multiple system complexities, e.g. interrelations of diverse system components. To fill this gap, an interval fuzzy water resources and farmland programming (IFWRFP) approach is developed in this study for eliminating the potential problems in the SHRW, leading to increased reliability of the decision support process. A series of systematic WRFM measures are proposed for enabling harmonious development of ecological environment and social economy in the SHRW. For instance, planting should always be the priority due to the major contribution of agriculture to the regional economy. As the primary commercial crop, rice cultivation should be allocated the most irrigation water, followed by corn, potato and soybean. Potato yield should be increased to compensate for reduced productivity of the other crops since 2019. It is also revealed that economic benefits are proportional to water environmental pollution in the SHRW. Therefore, decision-makers should adopt the most reasonable suggested schemes after fully balancing the trade-off of environment and economy. Most importantly, a variety of supporting policies are required for enabling sufficient implementation of these measures across the SHRW. For instance, individual farmers can be encouraged to follow the overall crop cultivation plan by the alteration of subsidiaries, taxes, and prices on crop-related activities. The modeling solutions show that the IFWRFP approach can systematically optimize allocations of water resources and cultivation patterns and thus potentially eliminate the problems of water scarcity, water contamination, and soil erosion in the SHRW.

Appendices

IFWRFP model

Based on the aforementioned analyses, the objective function of the IFWRFP model is formulated as follows:

$$ {\displaystyle \begin{array}{c}\operatorname{Max}\ {f}^{\pm}\cong \sum \limits_{i=1}^3\sum \limits_{j=1}^3\sum \limits_{t=1}^3{\mathrm{EP}}_{\mathrm{ijt}}^{\pm}\cdot {\mathrm{YP}}_{\mathrm{ijt}}^{\pm}\cdot {P}_{ijt}^{\pm}\\ {}+\sum \limits_{k=1}^2\sum \limits_{j=1}^3\sum \limits_{t=1}^3{\mathrm{EI}}_{\mathrm{kjt}}^{\pm}\cdot {AI}_{kjt}^{\pm }+\sum \limits_{j=1}^3\sum \limits_{t=1}^3{\mathrm{ET}}_{\mathrm{jt}}^{\pm}\cdot {AT}_{jt}^{\pm }+\sum \limits_{j=1}^3\sum \limits_{t=1}^3{\mathrm{ER}}_{\mathrm{jt}}^{\pm}\cdot {AR}_{jt}^{\pm}\\ {}-\sum \limits_{j=1}^3\sum \limits_{t=1}^3\left({\mathrm{KR}}_{\mathrm{jt}}^{\pm}\cdot {HR}_{jt}^{\pm }+{\mathrm{KG}}_{\mathrm{jt}}^{\pm}\cdot {HG}_{jt}^{\pm}\right)\\ {}-\sum \limits_{k=1}^2\sum \limits_{t=1}^3\left({\mathrm{KWI}}_{\mathrm{kt}}^{\pm}\cdot \sum \limits_{j=1}^3\left({\mathrm{RI}}_{\mathrm{kjt}}^{\pm}\cdot {AI}_{kjt}^{\pm}\right)\right)\\ {}-\sum \limits_{t=1}^3\left({\mathrm{KWT}}_{\mathrm{t}}^{\pm}\cdot \sum \limits_{j=1}^3\left({\mathrm{RT}}_{\mathrm{jt}}^{\pm}\cdot {AT}_{jt}^{\pm}\right)\right)\\ {}-\sum \limits_{t=1}^3\left({\mathrm{KWR}}_{\mathrm{t}}^{\pm}\cdot \sum \limits_{j=1}^3\left({\mathrm{RR}}_{\mathrm{jt}}^{\pm}\cdot {AR}_{jt}^{\pm}\right)\right)\end{array}} $$

(1)

Constraints of the IFWRFP model consist of the following inequalities.

• 1) Songhua River Watershed farmland availability

• (a) Maximum cultivation areas

$$ \sum \limits_{i=1}^3{P}_{ijt}^{\pm}\underset{\sim }{\le }{\mathrm{MAXA}}_{\mathrm{jt}},\kern1em \forall j,t $$

(2)

• (b) Minimum cultivation areas

$$ \sum \limits_{i=1}^3{P}_{ijt}^{\pm}\underset{\sim }{\ge }{\mathrm{MINA}}_{\mathrm{jt}},\kern1em \forall j,t $$

(3)

• 2) Songhua River Watershed water resources availability

• (a) Surface water availability

$$ \sum \limits_{j=1}^3{HR}_{jt}^{\pm}\underset{\sim }{\le }{\mathrm{MAXR}}_{\mathrm{t}}^{\pm },\kern1em \forall t $$

(4)

• (b) Groundwater availability

$$ \sum \limits_{j=1}^3{HG}_{jt}^{\pm}\underset{\sim }{\le }{\mathrm{MAXG}}_{\mathrm{t}}^{\pm },\kern1em \forall t $$

(5)

• (c) Total water resources availability

$$ \sum \limits_{i=1}^2\sum \limits_{j=1}^3{\mathrm{RDP}}_{\mathrm{ijt}}^{\pm}\cdot {P}_{ijt}^{\pm }+\sum \limits_{k=1}^2\sum \limits_{j=1}^3{AI}_{kjt}^{\pm }+\sum \limits_{j=1}^3{AT}_{jt}^{\pm }+\sum \limits_{j=1}^3{AR}_{jt}^{\pm}\underset{\sim }{\le}\sum \limits_{j=1}^3\left({HR}_{jt}^{\pm }+{HG}_{jt}^{\pm}\right),\kern1em \forall t $$

(6)

• 3) Songhua River Watershed water supply constraints

• (a) Water supply for agriculture

$$ \sum \limits_{i=1}^2\sum \limits_{j=1}^3{\mathrm{RDP}}_{\mathrm{ijt}}^{\pm}\cdot {P}_{ijt}^{\pm}\underset{\sim }{\le }{\mathrm{MAXWP}}_{\mathrm{t}}^{\pm },\kern1em \forall t $$

(7)

• (b) Water supply for industry

$$ \sum \limits_{j=1}^3{AT}_{jt}^{\pm}\underset{\sim }{\le }{\mathrm{MAXWT}}_{\mathrm{t}}^{\pm },\kern1em \forall t $$

(8)

• (c) Water supply for tourism

$$ \sum \limits_{j=1}^3{AR}_{jt}^{\pm}\underset{\sim }{\le }{\mathrm{MAXWR}}_{\mathrm{t}}^{\pm },\kern1em \forall t $$

(9)

• (d) Water supply for household

$$ \sum \limits_{j=1}^3{AR}_{kjt}^{\pm}\underset{\sim }{\le }{\mathrm{MAXWR}}_{\mathrm{t}}^{\pm },\kern1em \forall t $$

(10)

• 4) Songhua River Watershed wastewater treatment capacity constraints

$$ \sum \limits_{k=1}^2\sum \limits_{j=1}^3{\mathrm{RI}}_{\mathrm{kjt}}^{\pm}\cdot {AI}_{kjt}^{\pm }+\sum \limits_{j=1}^3{\mathrm{RT}}_{\mathrm{jt}}^{\pm}\cdot {AT}_{jt}^{\pm }+\sum \limits_{j=1}^3{\mathrm{RR}}_{\mathrm{jt}}^{\pm}\cdot {AR}_{jt}^{\pm}\underset{\sim }{\le }{\mathrm{MAXUT}}_{\mathrm{t}}^{\pm },\kern1em \forall t $$

(11)

• 5) Songhua River Watershed eco-environment constraints

• (a) Soil erosion control

$$ \sum \limits_{i=1}^3\sum \limits_{j=1}^3{\mathrm{CSL}}_{\mathrm{ijt}}^{\pm}\cdot {P}_{ijt}^{\pm}\underset{\sim }{\le }{\mathrm{MAXCS}}_{\mathrm{t}}^{\pm },\kern1em \forall t $$

(12)

• (b) Nitrogen discharge control

$$ \sum \limits_{i=1}^3\sum \limits_{j=1}^3{\mathrm{QN}}_{\mathrm{ijt}}^{\pm}\cdot {P}_{ijt}^{\pm }+\left(\sum \limits_{k=1}^2\sum \limits_{j=1}^3{\mathrm{QN}\mathrm{I}}_{\mathrm{kjt}}^{\pm}\cdot {AI}_{kjt}^{\pm }+\sum \limits_{j=1}^3{\mathrm{QN}\mathrm{T}}_{\mathrm{jt}}^{\pm}\cdot {AT}_{jt}^{\pm }+\sum \limits_{j=1}^3{\mathrm{QN}\mathrm{R}}_{\mathrm{jt}}^{\pm}\cdot {AR}_{jt}^{\pm}\right)\left(1-{\mathrm{NRE}}_{\mathrm{t}}^{\pm}\right)\underset{\sim }{\le }{\mathrm{MAXTN}}_{\mathrm{t}}^{\pm },\kern.5em \forall t $$

(13)

• (c) Phosphor discharge control

$$ \sum \limits_{i=1}^3\sum \limits_{j=1}^3{\mathrm{QS}}_{\mathrm{ijt}}^{\pm}\cdot {P}_{ijt}^{\pm }+\left(\sum \limits_{k=1}^2\sum \limits_{j=1}^3{\mathrm{QPI}}_{\mathrm{kjt}}^{\pm}\cdot {AI}_{kjt}^{\pm }+\sum \limits_{j=1}^3{\mathrm{QPT}}_{\mathrm{jt}}^{\pm}\cdot {AT}_{jt}^{\pm }+\sum \limits_{j=1}^3{\mathrm{QPR}}_{\mathrm{jt}}^{\pm}\cdot {AR}_{jt}^{\pm}\right)\left(1-{\mathrm{PRE}}_{\mathrm{t}}^{\pm}\right)\underset{\sim }{\le }{\mathrm{MAXTP}}_{\mathrm{t}}^{\pm },\kern1em \forall t $$

(14)

where f = the expected net system benefit ($); t = time period, t = 1, 2, 3 (where t = 1 for 2014–2018, 2 for 2019 to 2023, 3 for 2024 to 2028); i = the type of crop, i = 1, 2, 3, 4 (where i = 1 for corn, 2 for soybean, 3 for potato, 4 for rice); j = sub-region, j = 1, 2, 3 (where j = 1 for Inner Mongolia, 2 for Jilin, 3 for Heilongjiang); k = the type of industry, k = 1, 2 (where k = 1 for metallurgical industry, 2 for food industry);${\mathrm{EP}}_{\mathrm{ijt}}^{\pm }$ = market price of crop i in sub-region j in period t ($/kg); ${\mathrm{YP}}_{\mathrm{ijt}}^{\pm }$ = yield of crop i in sub-region j in period t (kg/km²); ${\mathrm{EI}}_{\mathrm{kjt}}^{\pm }$ = unit benefit of water allocated to industry k in sub-region j in period t ($/m³); ${\mathrm{ET}}_{\mathrm{jt}}^{\pm }$ = unit benefit of water allocated to tourism in sub-region j in period t ($/m3); ${\mathrm{ER}}_{\mathrm{jt}}^{\pm }$ = unit benefit of water allocated to household in sub-region j in period t ($/m³); ${\mathrm{KR}}_{\mathrm{jt}}^{\pm }$ = cost for pumping and delivering the surface water in sub-region j in period t ($/m³); ${\mathrm{KG}}_{\mathrm{jt}}^{\pm }$ = cost for pumping and delivering the ground water in sub-region j in period t ($/m³); ${\mathrm{KWI}}_{\mathrm{kt}}^{\pm }$ = treatment cost of wastewater from industry k in period t ($/tonne); ${\mathrm{KWT}}_{\mathrm{kt}}^{\pm }$ = treatment cost of wastewater from tourism industry in period t ($/tonne); ${\mathrm{KWR}}_{\mathrm{t}}^{\pm }$ = treatment cost of wastewater from household in period t ($/tonne); ${\mathrm{RI}}_{\mathrm{kjt}}^{\pm }$ = unit wastewater discharge by industry k in sub-region j in period t (tonne/m³); ${\mathrm{RT}}_{\mathrm{jt}}^{\pm }$ = unit wastewater discharge by tourism industry in sub-region j in period t (ton/m³); ${\mathrm{RR}}_{\mathrm{jt}}^{\pm }$ = unit wastewater discharge by household in sub-region j in period t (tonne/m³);MAXA_jt = the maximum area allocated to crop i in sub-region j in period t (km²); MINA_jt = the minimum area allocated to crop i in sub-region j in period t (km²);${\mathrm{MAXR}}_{\mathrm{t}}^{\pm }$ = the maximum allocated surface water amount in sub-region j in period t (m³); ${\mathrm{MAXG}}_{\mathrm{t}}^{\pm }$ = the maximum allocated groundwater amount in sub-region j in period t (m³); ${\mathrm{RDP}}_{\mathrm{ijt}}^{\pm }$ = the unit irrigation demand for crop i in sub-region j in period t (m³/km²);${\mathrm{MAXWP}}_{\mathrm{t}}^{\pm }$ = the maximum water amount allocated to agriculture in period t (m³); ${\mathrm{MAXWI}}_{\mathrm{maxt}}^{\pm }$ = the maximum water amount allocated to industry in period t (m³); ${\mathrm{MAXWT}}_{\mathrm{maxt}}^{\pm }$ = the maximum water amount allocated to tourism in period t (m³); ${\mathrm{MAXWR}}_{\mathrm{maxt}}^{\pm }$ = the maximum water amount allocated to household in period t (m³);${\mathrm{MAXUT}}_{\mathrm{t}}^{\pm }$ = total wastewater treatment capacity in period t (tonne);${\mathrm{CSL}}_{\mathrm{ijt}}^{\pm }$ = amount of soil loss from the land planted with crop i in sub-region j in period t (kg/km²); ${\mathrm{MAXCS}}_{\mathrm{t}}^{\pm }$ = the allowed amount of soil loss in period t (kg);${\mathrm{QN}}_{\mathrm{ijt}}^{\pm }$ = nitrogen percent content of the soil in sub-region j in period t (%); ${\mathrm{QNI}}_{\mathrm{kjt}}^{\pm }$ = unit nitrogen discharge by industry k in sub-region j in period t (tonne/m³); ${\mathrm{QNT}}_{\mathrm{jt}}^{\pm }$= unit nitrogen discharge by tourism industry in sub-region j in period t (tonne/m³); ${\mathrm{QNR}}_{\mathrm{jt}}^{\pm }$= unit nitrogen discharge by household in sub-region j in period t (tonne/m³); ${\mathrm{NRE}}_{\mathrm{t}}^{\pm }$ = nitrogen removal efficiency in period t (%); ${\mathrm{MAXTN}}_{\mathrm{t}}^{\pm }$ = the allowed amount of nitrogen discharge in period t (kg);${\mathrm{QS}}_{\mathrm{ijt}}^{\pm }$ = phosphorus percent content of the soil in sub-region j in period t (%); ${\mathrm{QPI}}_{\mathrm{kjt}}^{\pm }$= unit phosphor discharge by industry k in sub-region j in period t (tonne/m³); ${\mathrm{QPT}}_{\mathrm{jt}}^{\pm }$ = unit phosphor discharge by tourism industry in sub-region j in period t(tonne/m³); ${\mathrm{QPR}}_{\mathrm{jt}}^{\pm }$ = unit phosphor discharge by household in sub-region j in period t (tonne/m³); ${\mathrm{PRE}}_{\mathrm{t}}^{\pm }$ = phosphor removal efficiency in period t (%); ${\mathrm{MAXTP}}_{\mathrm{t}}^{\pm }$ = the allowed amount of phosphor discharge in period t (kg).

Table 3 Benefits of water supply for end-users, and costs for pumping and delivering water resources ($/m³)

Fig. 3

Interactive relationships in watershed system

Solution Algorithm

The constructed IFWRFP model can be generalized as an interval fuzzy linear programming (IFLP) problem that is formulated as follows:

$$ \max {f}^{\pm}\cong {C}^{\pm }{X}^{\pm } $$

(15)

subject to:

$$ {A}^{\pm }{X}^{\pm}\underset{\sim }{\le }{B}^{\pm } $$

(16)

$$ {X}^{\pm}\ge 0 $$

(17)

where A^±∈{R^±}^m × n, B^±∈{R^±}^m × l, C^±∈{R^±}^l × n, X^±∈{R^±}^n × l, and {R^±} denotes sets of interval numbers. The IFLP model is equivalent to an interval linear programming model if the fuzziness in the objective function (15) and the constraints (16) is removed. For the latter model, a two-step solution algorithm (Huang et al. 1993, 1996; Cai et al. 2007, 2012) can be employed to solve it, generating interval-form solutions as follows:

$$ {x}_{jopt}=\left[{x}_{jopt}^{-},{x}_{jopt}^{+}\right]\kern0.75em \mathrm{for}\ \mathrm{any}\ j\in \left\{1,2,\dots, n\right\} $$

(18)

$$ {f}_{opt}=\left[{f}_{opt}^{-},{f}_{opt}^{+}\right] $$

(19)

Based on the fuzzy flexible programming (Huang et al. 1993), both of the flexibility in constraints as well as the fuzziness in system objective can be assigned membership functions and represented by fuzzy sets. ‘Fuzzy constraints’ and a ‘fuzzy goal’ can then be expressed as a membership grade (λ) corresponding to the overall satisfaction degree of constraints and the objective. By incorporating an interval membership grade (λ^±) into the existing interval linear model (Huang et al. 1996; Chakraborty and Chandra 2005; Lin and Huang 2008; Cheng et al. 2015a), the IFLP model can be equivalently reformulated as follows:

$$ \max {\lambda}^{\pm } $$

(20)

subject to:

$$ {C}^{\pm }{X}^{\pm}\le {f}^{+}-\left(1-{\lambda}^{\pm}\right)\cdot \left({f}^{+}-{f}^{-}\right) $$

(21)

$$ {A}^{\pm }{X}^{\pm}\le {b}_i^{+}-\left(1-{\lambda}^{\pm}\right)\cdot \left({b}_i^{+}-{b}_i^{-}\right) $$

(22)

$$ {X}^{\pm}\ge 0 $$

(23)

$$ 0\le {\lambda}^{\pm}\le 1 $$

(24)

where f ⁻ and f ⁺ are the lower and upper bounds of the objective’s aspiration level, respectively; λ^± is a control variable corresponding to the satisfaction degree for the fuzzy decision (Huang et al. 1993; Huang et al. 1996).

Through employing the two-step solution algorithm (Huang et al. 1993, 1996; Cai et al. 2012) again, this model would be transformed into two sub-models. The first sub-model corresponding to the upper bound of λ^± can be formulated as:

$$ \max {\lambda}^{+} $$

(25)

subject to:

$$ \sum \limits_{j=1}^k{c}_j^{-}{x}_j^{-}+\sum \limits_{j=k+1}{c}_j^{-}{x}_j^{+}\le {f}^{+}-\left(1-{\lambda}^{+}\right)\cdot \left({f}^{+}-{f}^{-}\right) $$

(26)

$$ \sum \limits_{j=1}^{k_1}{\left|{a}_{ij}\right|}^{+} Sign\left({a}_{ij}^{+}\right){x}_j^{-}+\sum \limits_{j={k}_1+1}^n{\left|{a}_{ij}\right|}^{-} Sign\left({a}_{ij}^{-}\right){x}_j^{+}\le {b}_i^{+}-\left(1-{\lambda}^{+}\right)\cdot \left({b}_i^{+}-{b}_i^{-}\right),\forall i,j $$

(27)

$$ {x}_j^{-}\ge 0,j=1,2,\dots, {k}_1 $$

(28)

$$ {x}_j^{+}\ge 0,j={k}_1+1,{k}_1+2,\dots, n $$

(29)

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

(30)

$$ {x}_j\in {X}^{\pm } $$

(31)

Based on the optimal solutions from formulas (25) to (31), the second sub-model corresponding to the lower bound of λ^± can be presented as follows:

$$ \max {\lambda}^{-} $$

(32)

subject to:

$$ \sum \limits_{j=1}^k{c}_j^{+}{x}_j^{+}+\sum \limits_{j=k+1}{c}_j^{+}{x}_j^{-}\le {f}^{+}-\left(1-{\lambda}^{+}\right)\cdot \left({f}^{+}-{f}^{-}\right) $$

(33)

$$ \sum \limits_{j=1}^{k_1}{\left|{a}_{ij}\right|}^{-} Sign\left({a}_{ij}^{-}\right){x}_j^{+}+\sum \limits_{j={k}_1+1}^n{\left|{a}_{ij}\right|}^{+} Sign\left({a}_{ij}^{+}\right){x}_j^{-}\le {b}_i^{+}-\left(1-{\lambda}^{+}\right)\cdot \left({b}_i^{+}-{b}_i^{-}\right),\forall i $$

(34)

$$ {x}_j^{+}\ge {x}_{jopt}^{-}\ge 0,j=1,2,\dots, {k}_1 $$

(35)

$$ {x}_{jopt}^{+}\ge {x}_j^{-}\ge 0,j={k}_1+1,{k}_1+2,\dots, n $$

(36)

$$ 0\le {\lambda}^{-}\le {\lambda}_{opt}^{+} $$

(37)

$$ {x}_j\in {X}^{\pm } $$

(38)

According to this solution algorithm, the IFWRFP model for water resources and farmland management in the Songhua River Watershed can be solved, and the corresponding lower and upper bounds of solutions can be obtained. The generated interval solutions can provide decision makers with multiple decision alternatives according to practical situations and their preferences.