The interval copula-measure Me based multi-objective multi-stage stochastic chance-constrained programming for seasonal water resources allocation under uncertainty

Abstract

A copula-measure Me based interval multi-objective multi-stage stochastic chance-constrained programming (CMIMOMSP) model is proposed for water consumption optimization. It can conduct water allocation amid multiple users and multiple stages, and deal with the uncertainties presented as interval numbers, random fuzzy interval numbers, and stochastic variables. It improves upon multi-stage stochastic chance-constrained programming by introducing the multi-objective programming, and it can tradeoff the relationships amid economic benefit, full usage of water resources, and economic loss. It enhances the accuracy of copula function and conditional distribution function through proposing the interval functions. Besides, it can deal with the impact of the decision attitudes of managers on water allocation by formulating the function equation between water demand and the optimistic-pessimistic factor. The CMIMOMSP model is applied to a case study of the Heihe River Basin to verify its application. The results indicate that: (1) the optimistic-pessimistic factors have different degrees of positive influences on water allocation for industrial, domestic and ecological sectors; (2) the joint violated probability and optimistic-pessimistic factor have various range of impacts on agricultural water allocation; (3) tthe objective function values have different variation tendencies with the rise of joint violated probabilities and optimistic-pessimistic factors. Its robustness is enhanced by comparing it with the three single-objective programming models. The CMIMOMSP model can provide various water allocation schemes for managers with different risk attitudes in semi-arid and arid districts.

Appendix

The measure Me can be defined as follows:

$$Me\left\{ {\xi \leq r} \right\}=Nec\left\{ {\xi \leq r} \right\}+\lambda (Pos\left\{ {\xi \leq r} \right\} - Nec\left\{ {\xi \leq r} \right\}),$$

where $Pos\left\{ {\xi \leq r} \right\}{\text{=}}\mathop {\sup }\nolimits_{{u \leq r}} \mu (u)$ and $Nec\left\{ {\xi \leq r} \right\}{\text{=1-}}\mathop {\sup }\nolimits_{{u>r}} \mu (u)$; Pos is the possibility measure; Nec is the necessary measure; λ is the optimistic-pessimistic parameter to determine the combined attitude of a manager.

The probability distribution of triangular fuzzy numbers $\widetilde {\xi }=({\xi _1},{\xi _2},{\xi _3})$ based on the measure Me is show as follows:

$$Me\left\{ {\widetilde {\xi } \geq r} \right\}{\text{=}}\left\{ \begin{array}{ll} 1& \quad r \leq {\xi _1} \hfill \\ \lambda +(1 - \lambda )\frac{{{\xi _2} - r}}{{{\xi _2} - {\xi _1}}}& \quad {\xi _1} \leq r<{\xi _2} \hfill \\ \lambda \frac{{{\xi _3} - r}}{{{\xi _3} - {\xi _2}}}& \quad {\xi _2} \leq r<{\xi _3} \hfill \\ 0& \quad r>{\xi _3} \hfill \\ \end{array} \right.$$

where ${\xi _1},{\xi _2},{\xi _3}$ are the minimum possible value, possible value and maximum possible value, respectively.

The expected value of $\widetilde {\xi }$ based on the measure Me.

$$E[\xi ]={E^{Nec}}[\xi ]+\lambda ({E^{Pos}}[\xi ] - {E^{Nec}}[\xi ])=\lambda {E^{Pos}}[\xi ]+(1 - \lambda ){E^{Nec}}[\xi ]$$

Finally, the expected value of $\widetilde {\xi }$ is expressed as ${E^{Me}}[\xi ]=\frac{1}{2}({\xi _2}+{\xi _1})+\frac{\lambda }{2}({\xi _3} - {\xi _1})$