Optimal Land Use Management for Soil Erosion Control by Using an Interval-Parameter Fuzzy Two-Stage Stochastic Programming Approach

Abstract

Soil erosion is one of the most serious environmental and public health problems, and such land degradation can be effectively mitigated through performing land use transitions across a watershed. Optimal land use management can thus provide a way to reduce soil erosion while achieving the maximum net benefit. However, optimized land use allocation schemes are not always successful since uncertainties pertaining to soil erosion control are not well presented. This study applied an interval-parameter fuzzy two-stage stochastic programming approach to generate optimal land use planning strategies for soil erosion control based on an inexact optimization framework, in which various uncertainties were reflected. The modeling approach can incorporate predefined soil erosion control policies, and address inherent system uncertainties expressed as discrete intervals, fuzzy sets, and probability distributions. The developed model was demonstrated through a case study in the Xiangxi River watershed, China’s Three Gorges Reservoir region. Land use transformations were employed as decision variables, and based on these, the land use change dynamics were yielded for a 15-year planning horizon. Finally, the maximum net economic benefit with an interval value of [1.197, 6.311] × 10⁹ $ was obtained as well as corresponding land use allocations in the three planning periods. Also, the resulting soil erosion amount was found to be decreased and controlled at a tolerable level over the watershed. Thus, results confirm that the developed model is a useful tool for implementing land use management as not only does it allow local decision makers to optimize land use allocation, but can also help to answer how to accomplish land use changes.

Appendix

Appendix 1

The IFTSP form of land use management model in this study was formulated as follows:

$${\text{Maximize}}\,\varPsi^{ \pm } = \sum\limits_{i = 1}^{I} {\sum\limits_{j = 1}^{J} {\sum\limits_{k = 1}^{K} {X_{i,j,k}^{ \pm } } \cdot {\text{GEB}}_{j,k}^{ \pm } } \cdot {\text{NY}}_{k} } - \sum\limits_{i = 1}^{I} {\sum\limits_{j = 1}^{J} {\sum\limits_{j' = 1}^{J} {\sum\limits_{k = 1}^{K} {{\text{XX}}_{{_{j',i,j,k} }}^{ \pm } } } \cdot {\text{CLD}}_{j',j,k}^{ \pm } } \cdot {\text{NY}}_{k} } + \sum\limits_{i = 1}^{I} {\sum\limits_{k = 1}^{K} {(X_{{_{i,3,k} }}^{ \pm } + X_{{_{i,5,k} }}^{ \pm } - X_{{_{i,5,k - 1} }}^{ \pm } - X_{{_{i,5,k - 1} }}^{ \pm } )} \cdot {\text{SUB}}_{2,k}^{ \pm } } \cdot {\text{NY}}_{k} + \sum\limits_{i = 1}^{I} {\sum\limits_{k = 1}^{K} {(X_{{_{i,1,k} }}^{ \pm } + X_{{_{i,2,k} }}^{ \pm } )} \cdot {\text{SUB}}_{1,k}^{ \pm } } \cdot {\text{NY}}_{k} - C_{k}^{ \pm } \cdot \sum\limits_{i = 1}^{I} {\sum\limits_{k = 1}^{K} {\sum\limits_{l = 1}^{n} {\omega_{i,k,l} \cdot p_{i,k,l} } } } \cdot \mu \cdot {\text{TLA}}_{i}$$

(9a)

Subject to:

$$\sum\limits_{i = 1}^{I} {\sum\limits_{j = 1}^{J} {g_{i,k} (X_{i,j,k}^{ \pm } ) \cdot {\text{TLA}}_{i} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ \le } {\text{SEC}}_{k}^{ \pm } } } \cdot \sum\limits_{i = 1}^{I} {{\text{TLA}}_{i}^{{}} } ,\quad \forall k$$

(9b)

$$\sum\limits_{j = 1}^{J} {g_{i,k} (X_{i,j,k}^{ \pm } )} , - \omega_{i,k,l}^{ \pm } \le \delta_{i,k,l}^{ \pm } ,\quad \forall i,k,l$$

(9c)

$$\sum\limits_{j = 1}^{J} {X_{i,j,k}^{ \pm } } \le {\text{TLA}}_{i} ,\quad \forall i,k$$

(9d)

$$\sum\limits_{j' = 1}^{J} {{\text{XX}}_{j',i,j,k}^{ \pm } = X_{i,j,k}^{ \pm } } ,\quad \forall i,j,k$$

(9e)

$$\sum\limits_{j = 1}^{J} {{\text{XX}}_{{j^{\prime } ,i,j,k}}^{ \pm } \le X_{{i,j^{\prime } ,k - 1}}^{ \pm } } ,\quad \forall i,j^{\prime } ,k$$

(9f)

$$\sum\limits_{i = 1}^{I} {(X_{i,1,k}^{ \pm } + X_{i,2,k}^{ \pm } )} \ge {\text{MFA}}_{k}^{ \pm } ,\quad \forall k$$

(9g)

$$\sum\limits_{i = 1}^{I} {X_{i,5,k}^{ \pm } } \ge \sum\limits_{i = 1}^{I} {X_{i,5,k - 1}^{ \pm } } + {\text{new}}T_{k}^{ \pm } ,\quad \forall k$$

(9h)

$$\sum\limits_{i = 1}^{I} {X_{i,3,k}^{ \pm } } \ge \sum\limits_{i = 1}^{I} {{\text{TLA}}_{i} \cdot {\text{MFC}}_{k}^{ \pm } } ,\quad \forall k$$

(9i)

$$X_{i,4,k}^{ \pm } \ge {\text{TLA}}_{i} \cdot {\text{MGA}}_{i,k}^{ \pm } ,\quad \forall i,k$$

(9j)

$$X_{i,8,k}^{ \pm } \ge {\text{TLA}}_{i} \cdot {\text{MOL}}_{i,k}^{ \pm } ,\quad \forall i,k$$

(9k)

$$\sum\limits_{i = 1}^{I} {X_{i,7,k - 1} } \le \sum\limits_{i = 1}^{I} {X_{i,7,k}^{ \pm } } \le \sum\limits_{i = 1}^{I} {X_{i,7,(k - 1)}^{ \pm } } + {\text{MUE}}_{k}^{ \pm } ,\quad \forall k$$

(9l)

$$\sum\limits_{i = 1}^{I} {X_{3,7,k}^{ \pm } } \ge \sum\limits_{i = 1}^{I} {X_{3,7,k - 1}^{ \pm } } + {\text{U}}3{\text{D}}_{k}^{ \pm } ,\quad \forall k$$

(9m)

$$X_{i,3,k}^{ \pm } \le {\text{SFC}}_{i}^{ \pm } ,\quad \forall i,k$$

(9n)

$$X_{i,1,k}^{ \pm } \le {\text{SPF}}_{i}^{ \pm } ,\quad \forall i,k$$

(9o)

$$X_{i,2,k}^{ \pm } \le {\text{SDL}}_{i}^{ \pm } ,\quad \forall i,k$$

(9p)

$$\omega_{i,k,l}^{ \pm } \ge 0,\quad \forall i,k,l$$

(9q)

$${\text{XX}}_{{j^{\prime } ,i,j,k}}^{ \pm } \ge 0,\quad \forall j^{\prime } ,i,j,k$$

(9r)

$$X_{i,j,k}^{ \pm } \ge 0,\quad \forall i,j,k,$$

(9s)

where symbol “±” denotes an interval set with “+” and “−” as the upper and lower bounds of an interval-parameter, respectively; sign “$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ \le }$” suggests a fuzzy relationship between the left- and right-hand sides; ${\text{XX}}_{{j^{\prime } ,i,j,k}}^{ \pm }$ is decision variables; $X_{i,j,k}^{ \pm }$ represents the decision state of land use allocation.

Appendix 2

The transformed form for solving the IFTSP model was given as follows:

$${\text{Maximize}}\;\lambda^{ \pm }$$

(10a)

Subject to:

$$\begin{aligned} & \sum\limits_{i = 1}^{I} {\sum\limits_{j = 1}^{J} {\sum\limits_{k = 1}^{K} {X_{i,j,k}^{ \pm } } \cdot {\text{GEB}}_{j,k}^{ \pm } } \cdot {\text{NY}}_{k} } - \sum\limits_{i = 1}^{I} {\sum\limits_{j = 1}^{J} {\sum\limits_{j' = 1}^{J} {\sum\limits_{k = 1}^{K} {{\text{XX}}_{{_{j',i,j,k} }}^{ \pm } } } \cdot {\text{CLD}}_{j',j,k}^{ \pm } } \cdot {\text{NY}}_{k} } \\ & \quad + \sum\limits_{i = 1}^{I} {\sum\limits_{k = 1}^{K} {(X_{{_{i,3,k} }}^{ \pm } + X_{{_{i,5,k} }}^{ \pm } - X_{{_{i,5,k - 1} }}^{ \pm } - X_{{_{i,5,k - 1} }}^{ \pm } )} \cdot {\text{SUB}}_{2,k}^{ \pm } } \cdot {\text{NY}}_{k} \\ & \quad + \sum\limits_{i = 1}^{I} {\sum\limits_{k = 1}^{K} {(X_{{_{i,1,k} }}^{ \pm } + X_{{_{i,2,k} }}^{ \pm } )} \cdot {\text{SUB}}_{1,k}^{ \pm } } \cdot {\text{NY}}_{k} \\ & \quad - C_{k}^{ \pm } \cdot \sum\limits_{i = 1}^{I} {\sum\limits_{k = 1}^{K} {\sum\limits_{l = 1}^{n} {\omega_{i,k,l} \cdot p_{i,k,l} } } } \cdot \mu \cdot {\text{TLA}}_{i} \ge \psi^{ + } \cdot \lambda^{ \pm } + \psi^{ - } \cdot \left( {1 - \lambda^{ \pm } } \right) \\ \end{aligned}$$

(10b)

$$\frac{{\sum\limits_{i = 1}^{I} {\sum\limits_{j = 1}^{J} {g_{i,k} (X_{i,j,k}^{ \pm } )} } \cdot {\text{TLA}}_{i} }}{{\sum\limits_{i = 1}^{I} {{\text{TLA}}_{i} } }} \le \lambda^{ \pm } \cdot {\text{SEC}}_{k}^{ + } + (1 - \lambda^{ \pm } ) \cdot {\text{SEC}}_{k}^{ - } ,\quad \forall k$$

(10c)

$$\sum\limits_{j = 1}^{J} {g_{i,k} (X_{i,j,k}^{ \pm } )} - \omega_{i,k,l}^{ \pm } \le \delta_{i,k,l}^{ \pm } ,\quad \forall i,k,l$$

(10d)

$$\sum\limits_{j = 1}^{J} {X_{i,j,k}^{ \pm } } \le {\text{TLA}}_{i} ,_{i} \forall i,k$$

(10e)

$$\sum\limits_{{j^{\prime } = 1}}^{J} {{\text{XX}}_{{j^{\prime } ,i,j,k}}^{ \pm } = X_{i,j,k}^{ \pm } } ,\quad \forall i,j,k$$

(10f)

$$\sum\limits_{j = 1}^{J} {{\text{XX}}_{{j^{\prime } ,i,j,k}}^{ \pm } \le X_{{i,j^{\prime } ,k - 1}}^{ \pm } } ,\quad \forall i,j^{\prime } ,k$$

(10g)

$$\sum\limits_{i = 1}^{I} {(X_{i,1,k}^{ \pm } + X_{i,2,k}^{ \pm } )} \ge {\text{MFA}}_{k}^{ \pm } ,\quad \forall k$$

(10h)

$$\sum\limits_{i = 1}^{I} {X_{i,5,k}^{ \pm } } \ge \sum\limits_{i = 1}^{I} {X_{i,5,k - 1}^{ \pm } } + {\text{new}}\;T_{k}^{ \pm } ,\quad \forall k$$

(10i)

$$\sum\limits_{i = 1}^{I} {X_{i,3,k}^{ \pm } } \ge \sum\limits_{i = 1}^{I} {{\text{TLA}}_{i} } \cdot {\text{MFC}}_{k}^{ \pm } ,\quad \forall k$$

(10j)

$$X_{i,4,k}^{ \pm } \ge {\text{TLA}}_{i} \cdot {\text{MGA}}_{i,k}^{ \pm } ,\quad \forall i,k$$

(10k)

$$X_{i,8,k}^{ \pm } \ge {\text{TLA}}_{i} \cdot {\text{MOL}}_{i,k}^{ \pm } ,\quad \forall i,k$$

(10l)

$$\sum\limits_{i = 1}^{I} {X_{i,7,k - 1} } \le \sum\limits_{i = 1}^{I} {X_{i,7,k}^{ \pm } } \le \sum\limits_{i = 1}^{I} {X_{i,7,(k - 1)}^{ \pm } } + {\text{MUE}}_{k}^{ \pm } ,\quad \forall k$$

(10m)

$$\sum\limits_{i = 1}^{I} {X_{3,7,k}^{ \pm } } \ge \sum\limits_{i = 1}^{I} {X_{3,7,k - 1}^{ \pm } } + {\text{U}}3{\text{D}}_{k}^{ \pm } ,\quad \forall k$$

(10n)

$$X_{i,3,k}^{ \pm } \le {\text{SFC}}_{i}^{ \pm } ,\quad \forall i,k$$

(10o)

$$X_{i,1,k}^{ \pm } \le {\text{SPF}}_{i}^{ \pm } ,\quad \forall i,k$$

(10p)

$$X_{i,2,k}^{ \pm } \le {\text{SDL}}_{i}^{ \pm } ,\quad \forall i,k$$

(10q)

$$\omega_{i,k,l}^{ \pm } \ge 0,\quad \forall i,k,l$$

(10r)

$$XX_{{j^{\prime } ,i,j,k}}^{ \pm } \ge 0,\quad \forall j^{\prime } ,i,j,k$$

(10s)

$$X_{i,j,k}^{ \pm } \ge 0,\quad \forall i,j,k$$

(10t)

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

(10u)

where λ ^± is the control decision variable corresponding to the degree (membership grade) to which $X_{i,j,k}^{ \pm }$ solution fulfills the fuzzy objective or constraints; Ψ ⁺ and Ψ ⁻ represent the most and least desirable system objective values, corresponding to upper and lower submodels of model (9), respectively.