*2.3. Fuzzy-Interval Dynamic Programming (FIDP)*

It is noted that multiple uncertainties and dynamic variability exist in the water resources system, which seriously affect effective planning and management of water resources. Although FILP and DP can efficiently address interval uncertainty, coordinate conflicts among different objective functions and characterize systems' dynamics individually, they are unable to deal with those problems at the same time. Therefore, this paper aims to propose a FIDP model by incorporating FILP and DP into one framework to comprehensively reflect both uncertainties and dynamic features in the water resources system. In addition, the function of the users' satisfaction is considered to solve the contradiction caused by uneven distribution of resources. The developed model is shown as follows.

$$\text{Max}\lambda^{\pm} \tag{4a}$$

Subject to:

$$\mathcal{C}\_{\mathcal{S}}^{\pm} X^{\pm} \ge f\_{\mathcal{S}}^{-} + \lambda^{\pm} (f\_{\mathcal{S}}^{+} - f\_{\mathcal{S}}^{-}) \quad \text{g} = 1, \text{ 2, } \dots, \text{ m} \tag{4b}$$

$$\frac{X^{\pm}}{G^{\pm}}a \ge f\_p^{-} + \lambda^{\pm}(f\_p^{+} - f\_p^{-}) \quad p = m+1, \ m+2, \ldots, r \tag{4c}$$

$$\mathcal{C}\_{\hbar}^{\pm}X^{\pm} \le f\_{\hbar}^{+} - \lambda^{\pm}(f\_{\hbar}^{+} - f\_{\hbar}^{-}) \quad \hbar = r + 1, \ r + 2, \dots, n \tag{4d}$$

$$A\_i^{\pm} X^{\pm} \le B\_i^{\pm} \quad i = 1, 2, \dots, k \tag{4e}$$

$$S\_{j}^{\pm} = T(S\_{j-1'}^{\pm}, x\_{j-1}^{\pm}) \quad j = 1, 2, \ldots, l \tag{4f}$$

$$A\_j^{\\\pm} X^{\\\pm} \le \mathbb{S}\_j^{\\\pm} \quad j = 0, 1, \ldots, l \\ \tag{4g}$$

$$0 \le X^{\pm} \le G^{\pm} \tag{4h}$$

$$S\_0^{\pm} = 0\tag{4i}$$

$$0 \le \lambda^{\pm} \le 1 \tag{4}$$

where the symbol *G* ± means the user's ideal demand for resources, and *α* is the weight coefficient of different users. And Equation (4c) can reflect the fairness for different users, Equation (4f) realizes the dynamic transition, and the state constraint after phase transition is achieved by Equation (4g).

The steps of solving the FIDP model can be summarized as: (i) Establish FIDP model. (ii) Divide the model into two submodels through an improved two-step method [45]. In order to maximize *λ* ±, the upper bound submodel should be formulated firstly. (iii) Solve the upper bound submodel and obtain *x* + *opt* and *λ* + *opt*. (iv) Formulate the lower bound submodel for the FIDP model. (v) Solve the lower bound and obtain *x* − *opt* and *λ* − *opt*. (vi) According to the results of the above two models, the objective function values are calculated by formulate (2). (vii) Combining these two submodels, the optimal solution can be expressed as *f* ± *g opt* = [ *f* − *g opt*, *f* + *g opt*] (*g* = 1, 2, . . . , *m*), *f* ± *p opt* = [ *f* − *p opt*, *f* + *p opt*] (*p* = *m* + 1, *m* + 2, . . . , *r*), *f* ± *h opt* = [ *f* − *h opt*, *f* + *h opt*] (*h* = *r* + 1, *r* + 2, . . . , *n*), *λ* ± *opt* = [*λ* − *opt*, *λ* + *opt*], *X* ± *opt* = [*X* − *opt*, *X* + *opt*].

In general, the presented model can be applicable for the following problems: (i) For those problems with uncertain factors, this method can reflect them in model establishment, solution process and results in the form of interval numbers. (ii) For multi-stage decisionmaking problems, this model can provide specific schemes for every stage and global optimal solutions for the whole process. (iii) For multi-objective and multi-user problems, this model can coordinate the conflicts among different objective functions by maximizing satisfaction of the objective functions, and reduce the contradictions among users by considering the principle of fairness.
