An interval-parameter two-stage stochastic fuzzy program with type-2 membership functions: an application to water resources management

Abstract

This paper presents an interval-parameter two-stage stochastic fuzzy programming with type-2 membership functions (ITSFP–T2MF) approach for supporting water resources management under uncertainty. ITSFP–T2MF is capable not only of dealing with a variety of uncertainties expressed as probability distributions, intervals, and type-2 fuzzy sets, but also of reflecting the complexity of uncertainty presented as the concept of a flexible fuzzy decision. A scenario-based solution method is proposed for solving ITSFP–T2MF, which takes into account different attitudes of decision makers (DMs) towards the objective-function value and constraints. Moreover, the solution method can ensure that no infeasible solutions are included in the results by means of a feasibility test and a constricting algorithm, leading to an enhanced system safety. ITSFP–T2MF is applied to a case study of water resources allocation under uncertainty. The results indicate that interval solutions can be obtained under different scenarios, which enhances the diversity of solutions for supporting the decisions of water resources allocation. Furthermore, a variety of decision alternatives can be generated under different policies for water resources management, which permits an in-depth policy analysis associated with different levels of economic penalties when the promised water-allocation targets are violated, and thus helps DMs identify desired water-allocation plans according to practical situations.

Appendix: Solution method

A scenario-based solution method is proposed for solving the ITSFP–T2MF model, which takes into account six scenarios that represent different attitudes of DMs towards the objective-function value and constraints. Besides, the solution method can guarantee the feasibility of the generated interval solutions by means of a feasibility test and a constricting algorithm. The detailed solution procedures for solving ITSFP–T2MF are shown as follows.

1.1 Scenario 1

Step 1. Solve the following two deterministic submodels, respectively.

Submodel (A1):

$$ {\text{Max}}\,\lambda $$

(A1a)

subject to:

$$ \sum\limits_{i = 1}^{m} {NB_{i}^{ + } } (T_{i}^{ - } + \Updelta T_{i} y_{i} ) - \sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {p_{j} C_{i}^{ - } S_{ij}^{ - } } } \ge \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f}^{ + } + \lambda (\bar{f}^{ + } - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f}^{ - } ), $$

(A1b)

$$ \sum\limits_{i = 1}^{m} {(T_{i}^{ - } + \Updelta T_{i} y_{i} - S_{ij}^{ - } )} \le \bar{q}_{j}^{ + } - \lambda (\bar{q}_{j}^{ - } - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{q}_{j}^{ + } ),\quad \forall j, $$

(A1c)

$$ T_{i}^{ - } + \Updelta T_{i} y_{i} \le T_{i\hbox{max} }^{ + } - \lambda (T_{i\hbox{max} }^{ + } - T_{i\hbox{max} }^{ - } ),\quad \forall i, $$

(A1d)

$$ S_{ij}^{ - } \le T_{i}^{ - } + \Updelta T_{i} y_{i} ,\quad \forall i,j, $$

(A1e)

$$ S_{ij}^{ - } \ge 0, \, \forall i,j, $$

(A1f)

$$ 0 \le y_{i} \le 1,\quad \forall i, $$

(A1g)

$$ 0 \le \lambda \le 1. $$

(A1h)

Submodel (A2):

$$ {\text{Max}}\,\lambda $$

(A2a)

subject to:

$$ \sum\limits_{i = 1}^{m} {NB_{i}^{ - } } (T_{i}^{ - } + \Updelta T_{i} y_{iopt} ) - \sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {p_{j} C_{i}^{ + } S_{ij}^{ + } } } \ge \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f}^{ - } + \lambda (\bar{f}^{ - } - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f}^{ + } ), $$

(A2b)

$$ \sum\limits_{i = 1}^{m} {(T_{i}^{ - } + \Updelta T_{i} y_{{i{\text{opt}}}} - S_{ij}^{ + } )} \le \bar{q}_{j}^{ - } - \lambda (\bar{q}_{j}^{ + } - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{q}_{j}^{ - } ), \, \forall j, $$

(A2c)

$$ S_{ij}^{ + } \le T_{i}^{ - } + \Updelta T_{i} y_{{i{\text{opt}}}} ,\quad \forall i,j, $$

(A2d)

$$ S_{ij}^{ + } \ge S_{{ij{\text{opt}}}}^{ - } ,\quad \forall i,j, $$

(A2e)

$$ 0 \le \lambda \le 1. $$

(A2f)

where $f_{\text{opt}}^{ + }$, $S_{{ij{\text{opt}}}}^{ - }$, and $y_{{i{\text{opt}}}}$ are solutions of submodel (A1); $f_{\text{opt}}^{ - }$ and $S_{{ij{\text{opt}}}}^{ + }$ are solutions of submodel (A2). Interval solutions of $f_{\text{opt}}^{ \pm } = [f_{\text{opt}}^{ - } ,f_{\text{opt}}^{ + } ]$ and $S_{{ij{\text{opt}}}}^{ \pm } = [S_{{ij{\text{opt}}}}^{ - } ,S_{{ij{\text{opt}}}}^{ + } ]$ can thus be obtained through solving submodels (A1) and (A2).

Step 2. Conduct the feasibility test. If pass, then stop. Otherwise, go to step 3.

Step 3. Apply the constricting algorithm to eliminate infeasible solutions.

1.2 Scenario 2

Step 1. Different from scenario 1, constraints (A1c) in submodel (A1) and constraints (A2c) in submodel (A2) can be respectively converted into the following two inequalities:

$$ \sum\limits_{i = 1}^{m} {(T_{i}^{ - } + \Updelta T_{i} y_{i} - S_{ij}^{ - } )} \le \bar{q}_{j}^{ - } - \lambda (\bar{q}_{j}^{ + } - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{q}_{j}^{ - } ),\quad \forall j, $$

(A3)

$$ \sum\limits_{i = 1}^{m} {(T_{i}^{ - } + \Updelta T_{i} y_{iopt} - S_{ij}^{ + } )} \le \bar{q}_{j}^{ + } - \lambda (\bar{q}_{j}^{ - } - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{q}_{j}^{ + } ),\quad \forall j, $$

(A4)

Step 2. Conduct the feasibility test. If pass, then stop. Otherwise, go to step 3.

Step 3. Apply the constricting algorithm to eliminate infeasible solutions.

1.3 Scenario 3

Step 1. Solve the following two deterministic submodels, respectively.

Submodel (A5):

$$ {\text{Max}}\,\lambda $$

(A5a)

subject to:

$$ \sum\limits_{i = 1}^{m} {NB_{i}^{ - } } (T_{i}^{ - } + \Updelta T_{i} y_{i} ) - \sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {p_{j} C_{i}^{ + } S_{ij}^{ + } } } \ge \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f}^{ - } + \lambda (\bar{f}^{ - } - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f}^{ + } ), $$

(A5b)

$$ \sum\limits_{i = 1}^{m} {(T_{i}^{ - } + \Updelta T_{i} y_{i} - S_{ij}^{ + } )} \le \bar{q}_{j}^{ + } - \lambda (\bar{q}_{j}^{ - } - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{q}_{j}^{ + } ),\quad \forall j, $$

(A5c)

$$ T_{i}^{ - } + \Updelta T_{i} y_{i} \le T_{i\hbox{max} }^{ + } - \lambda (T_{i\hbox{max} }^{ + } - T_{i\hbox{max} }^{ - } ),\quad \forall i, $$

(A5d)

$$ S_{ij}^{ + } \le T_{i}^{ - } + \Updelta T_{i} y_{i} ,\quad \forall i,j, $$

(A5e)

$$ S_{ij}^{ + } \ge 0,\quad \forall i,j, $$

(A5f)

$$ 0 \le y_{i} \le 1,\quad \forall i, $$

(A5g)

$$ 0 \le \lambda \le 1. $$

(A5h)

Submodel (A6):

$$ {\text{Max}}\,\lambda $$

(A6a)

subject to:

$$ \sum\limits_{i = 1}^{m} {NB_{i}^{ + } } (T_{i}^{ - } + \Updelta T_{i} y_{iopt} ) - \sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {p_{j} C_{i}^{ - } S_{ij}^{ - } } } \ge \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f}^{ + } + \lambda (\bar{f}^{ + } - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f}^{ - } ), $$

(A6b)

$$ \sum\limits_{i = 1}^{m} {(T_{i}^{ - } + \Updelta T_{i} y_{iopt} - S_{ij}^{ - } )} \le \bar{q}_{j}^{ - } - \lambda (\bar{q}_{j}^{ + } - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{q}_{j}^{ - } ),\quad \forall j, $$

(A6c)

$$ S_{ij}^{ - } \le T_{i}^{ - } + \Updelta T_{i} y_{{i{\text{opt}}}} ,\quad \forall i,j, $$

(A6d)

$$ S_{ijopt}^{ + } \ge S_{ij}^{ - } ,\quad \forall i,j, $$

(A6e)

$$ 0 \le \lambda \le 1. $$

(A6f)

where $f_{\text{opt}}^{ - }$, $S_{{ij{\text{opt}}}}^{ + }$, and $y_{{i{\text{opt}}}}$ are solutions of submodel (A5); $f_{\text{opt}}^{ + }$ and $S_{{ij{\text{opt}}}}^{ - }$ are solutions of submodel (A6). Interval solutions of $f_{\text{opt}}^{ \pm } = [f_{\text{opt}}^{ - } ,f_{\text{opt}}^{ + } ]$ and $S_{{ij{\text{opt}}}}^{ \pm } = [S_{{ij{\text{opt}}}}^{ - } ,S_{{ij{\text{opt}}}}^{ + } ]$ can thus be obtained through solving submodels (A5) and (A6).

Step 2. Conduct the feasibility test. If pass, then stop. Otherwise, go to step 3.

Step 3. Apply the constricting algorithm to eliminate infeasible solutions.

1.4 Scenario 4

Step 1. Different from scenario 3, constraints (A5c) in submodel (A5) and constraints (A6c) in submodel (A6) can be respectively converted into the following two inequalities:

$$ \sum\limits_{i = 1}^{m} {(T_{i}^{ - } + \Updelta T_{i} y_{i} - S_{ij}^{ + } )} \le \bar{q}_{j}^{ - } - \lambda (\bar{q}_{j}^{ + } - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{q}_{j}^{ - } ),\quad \forall j, $$

(A7)

$$ \sum\limits_{i = 1}^{m} {(T_{i}^{ - } + \Updelta T_{i} y_{{i{\text{opt}}}} - S_{ij}^{ - } )} \le \bar{q}_{j}^{ + } - \lambda (\bar{q}_{j}^{ - } - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{q}_{j}^{ + } ),\quad \forall j, $$

(A8)

Step 2. Conduct the feasibility test. If pass, then stop. Otherwise, go to step 3.

Step 3. Apply the constricting algorithm to eliminate infeasible solutions.

1.5 Scenario 5

Step 1. Firstly, solve the following submodel (A9) in which all interval parameters can be converted into their mid-values. Then, solve the submodels (A10) and (A11), respectively.

Submodel (A9):

$$ {\text{Max}}\,\lambda $$

(A9a)

subject to:

$$ \sum\limits_{i = 1}^{m} {NB_{i}^{\text{m}} } T_{i}^{\text{m}} - \sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {p_{j} C_{i}^{\text{m}} S_{ij}^{\text{m}} } } \ge \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f}^{\text{m}} + \lambda (\bar{f}^{\text{m}} - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f}^{\text{m}} ), $$

(A9b)

$$ \sum\limits_{i = 1}^{m} {(T_{i}^{\text{m}} - S_{ij}^{\text{m}} )} \le \bar{q}_{j}^{\text{m}} - \lambda (\bar{q}_{j}^{\text{m}} - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{q}_{j}^{\text{m}} ),\quad \forall j, $$

(A9c)

$$ T_{i}^{\text{m}} \le T_{i\hbox{max} }^{\text{m}} ,\quad \forall i, $$

(A9d)

$$ S_{ij}^{\text{m}} \le T_{i}^{\text{m}} ,\quad \forall i,j, $$

(A9e)

$$ S_{ij}^{\text{m}} \ge 0,\quad \forall i,j, $$

(A9f)

$$ 0 \le \lambda \le 1. $$

(A9g)

Submodel (A10):

$$ {\text{Max}}\,\lambda $$

(A10a)

subject to:

$$ \sum\limits_{i = 1}^{m} {NB_{i}^{ + } } (T_{i}^{ - } + \Updelta T_{i} y_{i} ) - \sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {p_{j} C_{i}^{ - } S_{ij}^{ - } } } \ge \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f}^{ + } + \lambda (\bar{f}^{ + } - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f}^{ - } ), $$

(A10b)

$$ \sum\limits_{i = 1}^{m} {(T_{i}^{ - } + \Updelta T_{i} y_{i} - S_{ij}^{ - } )} \le \bar{q}_{j}^{ + } - \lambda (\bar{q}_{j}^{ - } - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{q}_{j}^{ + } ),\quad \forall j, $$

(A10c)

$$ T_{i}^{ - } + \Updelta T_{i} y_{i} \le T_{i\hbox{max} }^{ + } - \lambda (T_{i\hbox{max} }^{ + } - T_{i\hbox{max} }^{ - } ),\quad \forall i, $$

(A10d)

$$ S_{ij}^{ - } \le T_{i}^{ - } + \Updelta T_{i} y_{i} ,\quad \forall i,j, $$

(A10e)

$$ S_{ij}^{ - } \ge 0,\quad \forall i,j, $$

(A10f)

$$ S_{ij}^{\text{m}} \ge S_{ij}^{ - } ,\quad \forall i,j, $$

(A10g)

$$ 0 \le y_{i} \le 1,\quad \forall i, $$

(A10h)

$$ 0 \le \lambda \le 1. $$

(A10i)

Submodel (A11):

$$ {\text{Max}}\,\lambda $$

(A11a)

subject to:

$$ \sum\limits_{i = 1}^{m} {NB_{i}^{ - } } (T_{i}^{ - } + \Updelta T_{i} y_{iopt} ) - \sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {p_{j} C_{i}^{ + } S_{ij}^{ + } } } \ge \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f}^{ - } + \lambda (\bar{f}^{ - } - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f}^{ + } ), $$

(A11b)

$$ \sum\limits_{i = 1}^{m} {(T_{i}^{ - } + \Updelta T_{i} y_{iopt} - S_{ij}^{ + } )} \le \bar{q}_{j}^{ - } - \lambda (\bar{q}_{j}^{ + } - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{q}_{j}^{ - } ), \, \forall j, $$

(A11c)

$$ S_{ij}^{ + } \le T_{i}^{ - } + \Updelta T_{i} y_{{i{\text{opt}}}} , \, \forall i,j, $$

(A11d)

$$ S_{ij}^{ + } \ge S_{ij}^{\text{m}} , \, \forall i,j, $$

(A11e)

$$ 0 \le \lambda \le 1. $$

(A11f)

where $S_{ij}^{\text{m}}$ are solutions of submodel (A9); $f_{\text{opt}}^{ + }$, $S_{{ij{\text{opt}}}}^{ - }$, and $y_{{i{\text{opt}}}}$ are solutions of submodel (A10); $f_{\text{opt}}^{ - }$ and $S_{{ij{\text{opt}}}}^{ + }$ are solutions of submodel (A11). Interval solutions of $f_{\text{opt}}^{ \pm } = [f_{\text{opt}}^{ - } ,f_{\text{opt}}^{ + } ]$ and $S_{{ij{\text{opt}}}}^{ \pm } = [S_{{ij{\text{opt}}}}^{ - } ,S_{{ij{\text{opt}}}}^{ + } ]$ can thus be obtained through solving submodels (A9), (A10), and (A11).

Step 2. Conduct the feasibility test. If pass, then stop. Otherwise, go to step 3.

Step 3. Apply the constricting algorithm to eliminate infeasible solutions.

1.6 Scenario 6

Step 1. Different from scenario 5, constraints (A10c) in submodel (A10) and constraints (A11c) in submodel (A11) can be respectively converted into the following two inequalities:

$$ \sum\limits_{i = 1}^{m} {(T_{i}^{ - } + \Updelta T_{i} y_{i} - S_{ij}^{ - } )} \le \bar{q}_{j}^{ - } - \lambda (\bar{q}_{j}^{ + } - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{q}_{j}^{ - } ), \, \forall j, $$

(A12)

$$ \sum\limits_{i = 1}^{m} {(T_{i}^{ - } + \Updelta T_{i} y_{iopt} - S_{ij}^{ + } )} \le \bar{q}_{j}^{ + } - \lambda (\bar{q}_{j}^{ - } - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{q}_{j}^{ + } ), \, \forall j, $$

(A13)

Step 2. Conduct the feasibility test. If pass, then stop. Otherwise, go to step 3.

Step 3. Apply the constricting algorithm to eliminate infeasible solutions.