Multiple scenarios-based on a hybrid economy–environment–ecology model for land-use structural and spatial optimization under uncertainty: a case study in Wuhan, China

Abstract

The comprehensive optimization analysis of the quantitative structure and spatial allocation of land use is the key research direction of current and future land use planning. There are a lot of uncertain factors in the land use system. However, few studies couple the uncertain factors in the land use system with the spatial layout model. Based on this, the uncertain mathematical model and the spatial allocation model (GeoSOS-FLUS) are coupled to simulate the land use optimal allocation in Wuhan in 2030. The quantitative structure and spatial allocation optimization model of land use under three scenarios of economic development priority, ecological protection priority and low carbon emission priority were predicted. The coupling model solves the quantitative problem of land system uncertainty and applies it to spatial layout. The results show that the coupling model can help decision-makers to make future land use planning from the perspective of economy, ecology and low carbon emissions, and realize the sustainable development of future land use patterns.

Appendix

According to Zhou (2015) and Zhou et al. (2013), the model (1) can be transformed into:

$$Max {\omega _1}\lambda _{1}^{ \pm }+ {\omega _2}\lambda _{2}^{ \pm }$$

(2a)

subject to:

$$f_{k}^{ \pm }({X^ \pm }) \leqslant f_{k}^{ \pm } - \lambda _{1}^{ \pm }(f_{k}^{+} - f_{k}^{ - }),\quad k = 1,2,\ldots, p,\quad p \ne q$$

(2b)

$$f_{l}^{ \pm }({X^ \pm }) \leqslant f_{l}^{ - } - \lambda _{2}^{ \pm }(f_{l}^{+} - f_{l}^{ - }),\quad l = 1,2,\ldots,q,\quad q \ne p$$

(2c)

$$A_{i}^{ \pm }{X^ \pm } \gtrsim b_{i}^{ - }{\text{+(1-}}{\lambda ^ \pm }{\text{)(}}b_{i}^{+} - b_{i}^{ - })\quad i = 1,2,\ldots,m, \quad i \ne s$$

(2d)

$$A_{s}^{ \pm }{X^ \pm } \leqslant b_{s}^{{({p_s})}}\quad s = 1,2,\ldots,n,\quad s \ne i$$

(2e)

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

(2f)

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

(2g)

where ω₁ and ω₂ are weight coefficients; λ^± denotes the control decision variable corresponding to the degree (membership grade) to which X^± solution fulfills the fuzzy objective or constraints. Model (2) can be divided and solved by a two-step method, and if ${\text{b}}_{\text{i}}^{\pm }$ ≥ 0 and f^± ≥ 0, the sub-model corresponding to λ⁻ can be formulated as follows:

$$Max\,{\omega _1}\lambda _{1}^{ - }{\text{+}}{\omega _2}\lambda _{2}^{+}$$

(3a)

subject to:

$$\sum\limits_{{s = 1}}^{t} {c_{{k's}}^{+}x_{s}^{+} \leqslant f_{{k'}}^{+} - } \lambda _{1}^{ - }(f_{{k'}}^{+} - f_{{k'}}^{ - }),\quad k'=1,2,\ldots,q$$

(3b)

$$\sum\limits_{{s = 1}}^{t} {c_{{l's}}^{+}x_{s}^{+} \leqslant f_{{l'}}^{ - }+} \lambda _{2}^{+}(f_{{l'}}^{+} - f_{{l'}}^{ - }),\quad l' =q + 1,q + 2,\ldots,2q$$

(3c)

$$\sum\limits_{{s = 1}}^{t} {{{\left| {{a_{is}}} \right|}^ - }sign(a_{{is}}^{ \pm })x_{s}^{+} \leqslant b_{i}^{+} - \lambda _{1}^{ - }} (b_{i}^{+} - b_{i}^{ - }),\quad \forall i$$

(3d)

$$\sum\limits_{{s = 1}}^{t} {{{\left| {{a_{js}}} \right|}^ - }sign(a_{{js}}^{ \pm })x_{s}^{+} \leqslant b_{s}^{{({p_s})}}} ,\quad \forall s,\quad s \ne i$$

(3e)

$$0 \leqslant \lambda _{1}^{ - } \leqslant 1$$

(3f)

$$0 \leqslant \lambda _{2}^{+} \leqslant 1$$

(3g)

$$c_{{k's}}^{{\text{+}}} \geqslant 0$$

(3h)

$$c_{{l's}}^{+} \leqslant 0$$

(3i)

$$x_{s}^{+} \geqslant 0,\quad s = 1,2,\ldots,k$$

(3j)

$$Sign({x^ \pm })=\left\{ \begin{array}{ll} 1,&\,if\,{x^ \pm } \geqslant 0 \hfill \\ - 1,&\,if\,{x^ \pm } \leqslant 0 \hfill \\ \end{array} \right.$$

(3k)

Then, let $x_{{s{\text{ opt}}}}^{+}$ be solution of sub-model (3). where opt⁻ and opt⁺ denote the upper and lower bounds of the objective's aspiration level as designated by policy makers. The second sub-model corresponding to λ^±can be formulated supported by the solution of sub-model (4):

$$Max\,{\omega _1}\lambda _{1}^{+}{\text{+}}{\omega _2}\lambda _{2}^{ - }$$

(4a)

subject to:

$$\sum\limits_{{s = 1}}^{t} {c_{{k's}}^{ - }x_{s}^{ - } \leqslant f_{{k'}}^{+} - } \lambda _{1}^{+}(f_{{k'}}^{+} - f_{{k'}}^{ - }),\quad k'=1,2,\ldots,q$$

(4b)

$$\sum\limits_{{s = 1}}^{t} {c_{{l's}}^{ - }x_{s}^{ - } \leqslant f_{{l'}}^{ - }+} \lambda _{2}^{ - }(f_{{l'}}^{+} - f_{{l'}}^{ - }),\quad l'=q + 1,q + 2,\ldots,2q$$

(4c)

$$\sum\limits_{{s = 1}}^{t} {{{\left| {{a_{is}}} \right|}^+}sign(a_{{is}}^{ \pm })x_{s}^{ - } \leqslant b_{i}^{+} - \lambda _{1}^{+}} (b_{i}^{+} - b_{i}^{ - }),\quad \forall i$$

(4d)

$$\sum\limits_{{s = 1}}^{t} {{{\left| {{a_{js}}} \right|}^+}sign(a_{{js}}^{ \pm })x_{s}^{ - } \leqslant b_{s}^{{({p_s})}}},\quad \forall s,s \ne i$$

(4e)

$$0 \leqslant \lambda _{1}^{+} \leqslant 1$$

(4f)

$$0 \leqslant \lambda _{2}^{ - } \leqslant 1$$

(4g)

$$c_{{k's}}^{ - } \geqslant 0$$

(4h)

$$c_{{l's}}^{ - } \leqslant 0$$

(4i)

$$x_{s}^{ - } \geqslant 0,\quad s = 1,2,\ldots,k$$

(4j)

$$Sign({x^ \pm })=\left\{ \begin{gathered} 1,\,if\,{x^ \pm } \geqslant 0 \hfill \\ - 1,\,if\,{x^ \pm } \leqslant 0 \hfill \\ \end{gathered} \right.$$

(4k)

Then, let $x_{{s{\text{ opt}}}}^{ - }$ be the solution of sub-model (4). Thus, we can get the interval solutions as follows:

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

(4l)

$$x_{{s\,opt}}^{ \pm }=\left[ {x_{{s\,opt}}^{ - },x_{{s\,opt}}^{+}} \right]$$

(4m)

Finally, we can obtain the solution of the optimized objective by substituting (4m) into (2a). The optimized objective $f_{{{\text{opt}}}}^{ - }$ and $f_{{{\text{opt}}}}^{+}$ can be calculated as follows:

$$f_{{s\,opt}}^{ \pm }=\left[ {f_{{s\,opt}}^{ - },f_{{s\,opt}}^{+}} \right],\,\forall s$$

(5)