Research PaperIrrigation water resources management under uncertainty: An interval nonlinear double-sided fuzzy chance-constrained programming approach
Introduction
Agricultural sector in China plays a decisive role in safeguarding national security and supporting sustainable development of socio-economy (Wang et al., 2019). Irrigation is an effective compensation measure for mitigating water shortages and guaranteeing crop growth under arid and semiarid environments, as a result, agricultural development highly relies on irrigation activities (Elliott et al., 2014). Despite the shortage of water resources in the arid oasis agricultural region of Northwest China, flooding irrigation is commonly applied as the main irrigation pattern due to lower cost, which usually exceeds the actual crop water demands, resulting in over irrigation and low water use efficiency. Under these circumstances, it is desirable that scientific strategies of irrigation water management for optimally allocate irrigation water should be made to enhance agricultural water use efficiency and reduce water resources waste, and finally boost sustainable agricultural development. Moreover, in terms of irrigation water resources planning and management problems, uncertain factors may be ubiquitous rather than exceptional in agricultural system, i.e. e water use volume, crop parameters and decision-making preferences of managers (Zarghami and Hajykazemian, 2013, Liu et al., 2014, Pingale et al., 2014, Cai et al., 2016, Fu et al., 2018, Li et al., 2019). Therefore, it’s meaningful and practical to study on optimization model of irrigation water management in an uncertain environment.
Due to limited water and water scarcity in arid areas, the available water cannot sufficiently meet the water demands for the total irrigated area. The principle of determining irrigation water amount should not be the maximum crop yields based on the crop water production functions (CWPFs), but lie in achieving the maximum agricultural returns resulting from the overall irrigated area (Tong and Guo, 2013). CWPFs, as its name suggests, are quantitative mathematical equations that empirically estimates crop yields at different crop growth stages. Much research on CWPFs including linear and nonlinear forms between obtained crop yield and the total amount of irrigation water have been undertaken (Stewart et al., 1977, Zhang and Oweis, 1999, Brumbelow and Georgakakos, 2007, Geerts and Raes, 2009, Wichelns, 2014). The nonlinear CWPFs with logistic shape are generally acceptable than linear ones when applying more irrigation water than required (Geerts and Raes, 2009). In practice, lower operating and management level, and non-water efficient irrigation pattern make excessive water be used in localized areas, causing over-irrigation and reduction of crop yield. In addition, due to the error of observation and calculation for field experimental data, the fitted deterministic CWPFs may have limitations to fully reflect actual conditions (Li et al., 2016). Therefore, interval-based CWPFs are introduced and employed as the basic optimization framework, thereby resulting in a nonlinear objective function in optimization model for irrigation water management. For example, Tong and Guo (2013) initially estimated interval quadratic CWPFs and developed an inexact quadratic programming for optimal crop water allocation. However, it can only deal with interval uncertainty characterized as extreme conditions with upper and lower limits. Moreover, it doesn't tackle the violation of system constraints (Huang et al., 1993) and it is difficult for manager to make effective decision when the range of interval is relatively large (Zhang and Achari, 2010).
In contrast, fuzzy numbers can gradually adjust the range of choice based on different levels of ambiguity according to the human-induced judgements of decision-makers. In practice, the water use volume cannot be accurately determined in the case of water resources management problems, which may be vaguely expressed as fuzzy linguistic terms like “approximately 200 × 106 m3 or 220 × 106 m3”. Clearly, fuzzy mathematical programming (FMP) method is capable of addressing such a fuzzy problem by defuzzification into its crisp deterministic one. Among these FMP methods, fuzzy chance-constrained programming, namely FCCP, was developed to solve violation of constraints by incorporating predetermined confidence levels (Liu and Iwamura, 1998). Based on the FCCP, fuzzy constraints can be presented as deterministic ones corresponding to different confidence levels and satisfaction degree levels of constraints (Rong and Lahdelma, 2008). In additional, FCCP has advantages in (1) allowing to involve fuzzy variables into an optimization model where possibilistic distributions are easily defined than probabilistic distributions; (2) generating a series of stable solutions at several confidence levels and the related allowable violation risk levels while the constraints are not required to be totally satisfied and (3) having a lower computational requirement and a higher computational efficiency. Therefore, FCCP methods have been widely employed in many real-world case studies (Rong and Lahdelma, 2008, Cao et al., 2009, Xu et al., 2010, Zhou, 2015, Chen et al., 2017, Cai et al., 2018). However, the above-mentioned FCCP method can merely address fuzzy uncertainty existing in the right-hand side of constraints (Liu and Iwamura, 1998). Generally, both right-hand and left-hand sides of constraints are likely related to fuzzy uncertainties (e.g. variables and parameters). A double-sided FCCP method can be employed to improve upon applicability of the conventional FCCP. The basic concept of DFCCP method is to generate results of satisfying constraints at different confidence levels. For example, Xu and Qin (2010) proposed an inexact double-sided fuzzy chance-constrained programming model for agricultural effluent control. Liu et al. (2016) developed a fuzzy fractional chance-constrained programming model for air quality management under uncertainty. Cheng et al. (2017) proposed an interval double-sided fuzzy chance-constrained programming model for water resources management. Moreover, previous studies integrating double-sided FCCP method into optimization model of irrigation water allocation have been undertaken. For instance, Zhang et al., 2018a, Zhang and Guo, 2018 developed fuzzy linear fractional programming and interval fuzzy chance-constrained programming with double-sided fuzziness for optimal allocation of irrigation water, respectively. Nevertheless, the above two studies focused on the linear shape CWPFs, which cannot reflect nonlinear relationships between crop yield and applied irrigation water or rainfall (Geerts and Raes, 2009).
Therefore, in response to above concerns, an interval nonlinear double-sided fuzzy chance-constrained programming (INDFCCP) approach is formulated for optimal allocation of irrigation water resources. It incorporates double-sided fuzzy chance-constrained programming method and inexact quadratic programming within an optimization framework, formulating an enhanced optimization model of irrigation water allocation. Uncertainties of intervals and double-sided fuzziness in the model, and nonlinearity in the objective function can be handled simultaneously. Then, a case study on irrigation water resources allocation in the Yingke Irrigation District (YID) in the Heihe River Basin, northwest China will be provided for demonstrating its applicability. Interval quadratic CWPFs are estimated by interval regression analysis method using field experimental data, which can overcome the limitation of linear ones. Two reliability scenarios including the maximum reliability scenario and minimum reliability scenario are derived from the DFCCP method. A range of optimal solutions under several confidence levels are generated through solving the INDFCCP approach, which is useful for supporting irrigation water management. These results are expected to give insight into interactions among objective function values, confidence levels and their arisen constraints-violation risks.
Section snippets
Methodology
This section entails four subsections on the formulation of the methodology: (1) interval quadratic CWPFs; (2) double-sided fuzzy chance-constrained programming and (3) interval nonlinear double-sided fuzzy chance-constrained programming approach; (4) solving process.
Problem statement
The study area, Yingke Irrigation District (YID), is the third largest agricultural irrigation area in the Heihe River Basin (100°17′–100°34′ E, 38°50′–38°58′ N) (see Fig. 2). The east-west span of the YID is 17.4 km the north-south distance is 14.2 km, covering a total area of 192.2 km2. The study area is relatively flat and its altitude ranges from 1456 m to 1600 m. The topographic trend is that the altitude gradually decreases from southwest to northeast. Moreover, the study area has a
Results analysis and discussion
Because the INDFCCP approach is formulated by integrating techniques of IQP and DFCCP method, thus it has the advantages arising from the above two methods. Therefore, it is particularly critical for managers to determine the confidence level. Eleven scenarios by confidence levels are considered in this study, that is α = 0, 0.1, 0.2, , 1.0. Through solving the INDFCCP approach by transforming into the minimum and maximum reliability scenarios, twenty-two sets of feasible solutions can be
Conclusions
An interval nonlinear double-sided fuzzy chance-constrained programming (INDFCCP) approach is formulated for supporting optimal irrigation water allocation under complexity and uncertainty. In this study, an extension regarding the form of CWPFs from linear, interval linear to interval quadratic has been done to present more valuable information associated with system uncertainties and complexities in practical conditions. It can deal with discrete intervals and fuzzy uncertainties. Allowable
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This research was supported by National Natural Science Foundation of China (51679236 and 52009133). We gratefully acknowledge the Project funded by China Postdoctoral Science Foundation (2019M660871 and BX20190373). We are deeply grateful to the reviewer for his/her insight and careful review. The provided comments and suggestions have greatly improved the manuscript.
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