An inexact two-stage water management model for planning agricultural irrigation under uncertainty

Abstract

In this study, an inexact two-stage water management (ITWM) model is developed for planning agricultural irrigation in the Zhangweinan River Basin, China. The ITWM model is derived from the incorporation of interval-parameter programming (IPP) within a two-stage stochastic programming (TSP) framework. It can reflect not only uncertainties expressed as probability distributions but also interval numbers. Moreover, it can provide an effective linkage between conflicting economic benefits and the associated penalties attributed to the violation of the predefined policies. Four decision scenarios associated with different water-resources management policies are examined. Targeted incomes, recourse costs, and net system benefits under different scenarios are analyzed, which indicates that different policies for agricultural irrigation targets correspond to different water shortages and surplus, and thus lead to varied system benefit and system-failure risk. The results are valuable for supporting the adjustment or justification of the existing irrigation patterns and identify a desired water-allocation plan for agricultural irrigation under uncertainty.

Introduction

Water-resources management is at the forefront in formulating sustainable development policies for many countries (Loukas et al., 2007). Over the past decades, controversial and conflict-laden water-allocation issues among competing municipal, industrial and agricultural interests have raised increasing concerns (Li et al., 2007a, Li et al., 2007b, Bashir et al., 2009, Menezes and Inyang, 2009). Particularly, growing population, varying natural conditions, and shrinking water availabilities have exacerbated such competitions. Shrinking water availabilities can result in reduced water supplies, while growing population can lead to increased water demands; these two facts can further intensify the water shortage. Among all of the water consumers, agricultural sector is a major consumer for many regions throughout the world. For example, in China, agricultural water consumption accounts for approximately 70% of the total water uses (MWRPRC, 2008); however, the available water resources is less than 2200 m³ per capita, only one quarter of the world average level (Shan et al., 2000). Losses can hardly be avoided when the resources are insufficient to satisfy essential demands (Li and Huang, 2008). Thus, a sound strategy for agriculture water-resources allocation is desired for reduction of such conflicts and losses.

Previously, a number of systems analysis methods were developed for agricultural water management and planning (Smith, 1973, Abrishamchi et al., 1991, Javaid and David, 1992, Manocchi and Mecarelli, 1994, Kodal, 1996, Mainuddin et al., 1997, Kumar et al., 1998, Carvallo et al., 1998, Raju and Kumar, 1999, Amir and Fisher, 1999, Benli et al., 2001, Singh et al., 2001, Reca et al., 2001). For example, Smith (1973) proposed a linear programming model for the optimizing agricultural water-resources allocation in an irrigation district. Raju and Kumar (1999) developed a multicriterion decision making model for irrigation planning in Sri Ram Sagar Project in India, which dealt with three conflicting objectives such as net benefits, agricultural production, and labor employment. Reca et al. (2001) developed an economic optimization model for planning water resources allocation in an insufficient irrigation system. In summary, the above conventional optimization methods were useful for planning agriculture water-resources systems with considerations of a number of impact factors (e.g. economic objective, environmental requirement, and policy regulation); however, they had difficulties in tackling various uncertainties existing in the agricultural water management problems. In fact, in agricultural water management problems, many system components and their interrelationships might be uncertain. For example, spatial and temporal variations exist in many system components, such as available amount of water, irrigation targets and irrigation quota; moreover, fluctuations would be associated with crop prices that are functions of many stochastic factors. These complexities could become further compounded by not only interactions among the uncertain parameters but also their economic implications (Li et al., 2006).

In response to the above complexities and uncertainties, a number of stochastic mathematical programming (SMP) methods were proposed for supporting water-resources system planning (Loucks et al., 1981, Eiger and Shamir, 1991, Luo et al., 2003, Maqsood et al., 2005, Mujumdar and Nirmala, 2007, Ganji et al., 2008, Qin et al., 2008, Li et al., 2009a, Li et al., 2009b, Cui et al., 2009, Cui et al., 2010). Among these methods, two-stage stochastic programming (TSP) is effective for problems where an analysis of policy scenarios is desired and the related data are random in nature (Li and Huang, 2008). In TSP, a decision is first undertaken before values of random variables are known; then, after the random events have happened and their values are known, a second-stage decision can be made in order to minimize “penalties” that may appear due to any infeasibility (Loucks et al., 1981, Birge and Louveaux, 1988, Birge and Louveaux, 1997, Li et al., 2008a). The initial decision is called the first-stage decision, and the corrective action is named the second-stage decision. TSP methods were widely applied to water-resources management over the past decades (Mobasheri and Harboe, 1970, Pereira and Pinto, 1985, Kovacs et al., 1986, Wang and Adams, 1986, Trezos and Yeh, 1987, Ferrero et al., 1998, Huang and Loucks, 2000, Kibzun and Nikulin, 2001, Seifi and Hipel, 2001, Luo et al., 2003, Maqsood et al., 2005, Li et al., 2007a, Li et al., 2007b, Li et al., 2008b, Li and Huang, 2008). For example, Pereira and Pinto (1985) proposed a stochastic optimization approach for the planning of a multi-reservoir hydroelectric system under uncertainty, through associating a given probability to each of a range of inputs that occurred at different stages of an optimization horizon. Ferrero et al. (1998) examined hydrothermal scheduling of multi-reservoir systems using a two-stage dynamic programming approach. However, the conventional TSP methods had difficulties in solving large-scale practical problems with all uncertain parameters being expressed as probability distributions, while non-probability-distribution information could not be directly acquired. Moreover, the quality of information on uncertainties in many practical problems is often not good enough to be presented as probability distributions. In fact, these uncertainties may exist as ambiguous intervals because planners and engineers typically find it more difficult to specify distributions than to define fluctuation ranges. Aiming to reflect such uncertainties within the TSP framework, Huang and Loucks (2000) developed an inexact two-stage stochastic programming method and applied it to a hypothetic water-resources management system; the developed method could tackle uncertainties expressed as both probability distributions and intervals and account for economic penalties due to infeasibility. Maqsood et al. (2005) developed an inexact two-stage integer-stochastic programming for regional water-resources management under uncertainty. Li and Huang (2008) proposed an interval-parameter two-stage stochastic nonlinear programming (ITNP) method for supporting decisions of water-resources allocation within a multi-reservoir system. However, few studies are found in developing inexact TSP for agricultural water management.

Therefore, this study aims to develop an inexact two-stage water management (ITWM) model for agricultural water management and planning in the Zhangweinan River Basin, China. The developed model will incorporate approaches of two-stage stochastic programming (TSP) and interval-parameter programming (IPP) within a general optimization framework, such that uncertainties expressed as both interval values and probability distributions can be addressed. Moreover, it can reflect tradeoffs between conflicting economic benefits and the associated penalties attributed to the violation of irrigation targets. When irrigation targets approach their upper bounds, high economic benefits could be obtained if the targets are satisfied, but high penalties might have to be paid when the promised water is not delivered. However, when irrigation targets reach their lower bounds, we might have lower economic benefits when the promised water is delivered but, at the same time, a lower risk of violating the promised targets (i.e. lower penalties) if the water demands are not satisfied. The developed ITWM will be applied to planning agricultural water-resources allocation in the Zhangweinan River Basin. This basin is one of the main food and cotton producing regions of the North China; moreover, it faces serious water shortage (e.g. the average water resource per capita is only 212 m³ per year) due to increasing demand and decreasing availability (EBCZR, 2003, HRCC, 2003). The results obtained can help decision makers to identify a desired water-resources allocation plan for agricultural irrigation under uncertainty.