An interval fuzzy land-use allocation model (IFLAM) for Beijing in association with environmental and ecological consideration under uncertainty

Abstract

In this study, an interval fuzzy land-use allocation method (IFLAM) is developed for the planning of urban land-use allocation system. The developed IFLAM method is based on interval parameter programming technique and fuzzy flexible programming, which could handle uncertainties expressed as interval values and fuzzy sets in the model’s left-and right-hand sides and objective function. A management problem for allocating land resources of Beijing, the capital of China, is studied to illustrate applicability of the proposed IFLAM approach. Results show that the optimized $\lambda$ values (which means the satisfaction degree of all model constraints) are in the range of [0.67, 0.89], and the corresponding system benefit is between $ [534.03, 698.75] $\times$ $10^{12}$ and $ [961.37, 1028.33] $\times ~10^{12}$ while the optimized urban land in the first studying period will be [71905, 77894] ha in the Function Central Zone, [35219, 39711] ha in the Function Extended Zone, [22012, 30547] ha in the New Urban Zone and [17609, 22910] ha in the Eco-Conservation Zone. Results indicate that reasonable solutions have been generated under various environmental, ecological and social conditions. They can help decision makers to identify desired alternatives for making the utmost of land resources of Beijing and for providing services for urban land management decisions; moreover, they are helpful for land managers to gain insight into the tradeoff between economic objective and eco-environment violation risk.

Appendix: The IFP model

According to Zhou et al. (2013, 2014), an interval fuzzy programming (IFP) model can be expressed as follows:

$$\begin{aligned} Max f^{\pm }\;\cong \;C^{\pm }x^{\pm } \end{aligned}$$

(1a)

subject to:

$$\begin{aligned}&\displaystyle C^{\pm }x^{\pm }\;\gtrsim b_{_{opt} }^\pm&\end{aligned}$$

(1b)

$$\begin{aligned}&\displaystyle A_i^\pm x\;\lesssim \;b_{_i}^\pm \quad i = 1, 2, {\ldots }, m.&\end{aligned}$$

(1c)

where $A^\pm \in \{R^\pm \}^{m\times n}, B^\pm \in \{R^\pm \}^{m\times 1}, C^\pm \in \{R^\pm \}^{1\times n}, X^\pm \in \{R^\pm \}^{n\times 1}$, and ${R^\pm }$ denotes a set of interval numbers; Superscripts ‘$-$’ and ‘+’ represent lower and upper bounds of the interval parameters, respectively. symbols ‘$\cong$’, ‘$\gtrsim$’ and ‘$\lesssim$’ represent fuzzy equality and inequality. On the basis of the principle of fuzzy flexible programming (FFP), let $\lambda ^{\pm }$ value correspond to the membership grade of satisfaction for a fuzzy decision. Specifically, the flexibility in the constraints and fuzziness in the system objective, which are represented by fuzzy sets and denoted as “fuzzy constraints” and a “fuzzy goal”, respectively, are expressed as membership grades $\lambda ^{\pm }$ corresponding to the degrees of overall satisfaction for the constraints/objective. Thus, model (1) can be converted to

$$\begin{aligned} Max\;\;\lambda ^{\pm } \end{aligned}$$

(2a)

subject to

$$\begin{aligned}&\displaystyle C^{\pm }X^{\pm }\le f_d^- +(1-\lambda ^{\pm })(f_d^+ -f_d^{-})&\end{aligned}$$

(2b)

$$\begin{aligned}&\displaystyle A^{\pm }X^{\pm }\le B^{-}+(1-\lambda ^{\pm })(B^{+}-B^{-})&\end{aligned}$$

(2c)

$$\begin{aligned}&\displaystyle X^{\pm }\ge 0&\end{aligned}$$

(2d)

$$\begin{aligned}&\displaystyle 0\le \lambda ^{\pm }\le 1&\end{aligned}$$

(2e)

where $x_j^\pm$ denotes interval decision variables and $x_j^\pm \in X^{\pm }; f_d^+$ and $f_d^-$denote the upper and lower bounds of the objective’s aspiration level as designated by decision makers, respectively; $\lambda ^{\pm }$ denotes the control decision variable corresponding to the degree (membership grade) to which $X^{\pm }$ solution fulfils the fuzzy objective or constraints. IFLP model can be transformed into two deterministic sub-models, which corresponding to the upper and lower bounds for the desired objective function value. In detail, the sub-model corresponding to $\lambda ^{-}$ is first formulated and solved. This is based on the fact that the $\lambda ^{-}$ corresponds $f^{-}$ and the system objective is to be minimized. If $b_i^\pm \ge 0$ and $f^{\pm }\ge 0$, the sub-model corresponding to $\lambda ^{-}$ can be formulated as follows:

$$\begin{aligned} \hbox {Max}\;\lambda ^{-} \end{aligned}$$

(3a)

subject to:

$$\begin{aligned}&\displaystyle \sum _{j = 1}^{k_1 } {c_j^+ } x_j^+ +\sum _{j = k_1 + 1}^n {c_j^+ } x_j^- \le f_d^- +(1-\lambda ^{-})(f_d^+ -f_d^- )&\end{aligned}$$

(3b)

$$\begin{aligned}&\displaystyle \sum _{j = 1}^{k_1 } {\left| {a_{ij} } \right| ^{-}} Sign(a_{ij}^- )x_j^+ +\sum _{j = k_1 + 1}^n {\left| {a_{ij} } \right| ^{+}} Sign(a_{ij}^+ )x_j^- \le b_i^- +(1-\lambda ^{-})(b_i^+ -b_i^- ),\quad \forall i&\nonumber \\ \end{aligned}$$

(3c)

$$\begin{aligned}&\displaystyle 0\le \lambda ^{-}\le 1&\end{aligned}$$

(3d)

$$\begin{aligned}&\displaystyle x_j^- \ge 0,\quad j=1,\;2,\ldots ,k_1&\end{aligned}$$

(3e)

$$\begin{aligned}&\displaystyle x_j^+ \ge 0,\quad j=k_1 +1,\;k_1 +2,\ldots ,n&\end{aligned}$$

(3f)

where Sign is a signal function, which is defined as:$Sign(x^{\pm }) \hbox {=} \left\{ {{\begin{array}{l} {1\quad \hbox {if }x^{\pm }\ge 0} \\ {\hbox {-}1\quad \hbox {if }x^{\pm }\le 0} \\ \end{array} }} \right.$. Let $x_{j opt}^{+}\; (j = 1, 2,{\ldots }, k_{1})$ and $x_{j opt}^{-} \; (j=k_{1} + 1, k_{1} + 2,{\ldots }, n)$ be solutions of sub-model (3). Then, the second sub-model corresponding to $\lambda ^{+}$ can be formulated supported by the solution of sub-model (3):

$$\begin{aligned} \hbox {Max}\;\lambda ^{+} \end{aligned}$$

(4a)

subject to:

$$\begin{aligned}&\displaystyle \sum _{j = 1}^{k_1 } {c_j^- } x_j^- +\sum _{j = k_1 + 1}^n {c_j^- } x_j^+ \le f_d^- +(1-\lambda ^{-})(f_d^+ -f_d^- )\end{aligned}$$

(4b)

$$\begin{aligned}&\displaystyle \quad \sum _{j = 1}^{k_1 } {\left| {a_{ij} } \right| ^{+}} Sign(a_{ij}^+ )x_j^- +\sum _{j = k_1 + 1}^n {\left| {a_{ij} } \right| ^{-}} Sign(a_{ij}^- )x_j^+ \le b_i^- +(1-\lambda ^{-})(b_i^+ -b_i^- ),\quad \forall i\end{aligned}$$

(4c)

$$\begin{aligned}&\displaystyle \quad 0\le \lambda ^{+}\le 1\end{aligned}$$

(4d)

$$\begin{aligned}&\displaystyle \quad x_{j\;opt}^+ \ge x_j^- \ge 0,\quad j=1,\;2,\ldots ,k_1\end{aligned}$$

(4e)

$$\begin{aligned}&\displaystyle \quad x_j^+ \ge x_{j\;opt}^- ,\quad j=k_1 +1,\;k_1 +2,\ldots ,n \end{aligned}$$

(4f)

Let $x_{j opt}^{-} (j = 1, 2,{\ldots }, k_{1})$ and $x_{j opt}^+ \; (j=k_{1} + 1, k_{1} + 2,{\ldots }, n)$ be solutions of sub-model (4). Thus, we can obtain the interval solutions as follows:

$$\begin{aligned} \lambda _{opt}^\pm&= [\lambda _{opt}^{-} ,\lambda _{opt}^+]\end{aligned}$$

(5a)

$$\begin{aligned} x_{j\;opt}^\pm&= [x_{j\;opt}^- ,x_{j\;opt}^+ ],\quad \forall j \end{aligned}$$

(5b)

Then, $f_{opt}^{-}$ and $f_{opt}^+$ can be calculated as follows:

$$\begin{aligned} f_{opt}^-&= \sum _{j = 1}^{k_1 } {c_j^- } x_j^- +\sum _{j = k_1 + 1}^n {c_j^- } x_j^+\end{aligned}$$

(6a)

$$\begin{aligned} f_{opt}^+&= \sum _{j = 1}^{k_1 } {c_j^+ } x_j^+ +\sum _{j = k_1 + 1}^n {c_j^+ } x_j^- \end{aligned}$$

(6b)

Thus, we have

$$\begin{aligned} f_{j\;opt}^\pm =[f_{opt}^- ,f_{opt}^+ ],\quad \forall j \end{aligned}$$

(7)