## 1. Introduction

[2] *Deficit irrigation* (DI), an irrigation practice whereby water supply is reduced below maximum level and mild stress is allowed with minimal effect on yield [*Kirda*, 2002], is known as a demand management measure toward dealing with the water scarcity and improving water productivity. Under conditions of droughts and scarcity of water supply, DI may lead to greater economic gains than maximizing the yield per unit water volume applied.

[3] Several efforts have been made in using mathematical optimization models of irrigation management including two categories of models, that is, single-crop (SC) and multicrop (MC; multifield). In a SC model, the problem is either to determine an optimal applied water (intensive irrigation), an optimal cultivated area (extensive irrigation) [*English et al*., 2002], or the intraseasonal water application for deficit (extensive) irrigation [*Bras and Cordova*, 1981]. Mathematical programming techniques have been also used in MC problems to optimize the performance of an entire irrigation field as a single planning unit. The problems have addressed determining at least one of the aspects of cropping pattern, irrigation scheduling, and resource (surface reservoir and/or groundwater aquifer) operation policies.

[4] For intensive irrigation, an economic objective function of the net benefit resulting from irrigation has been considered [*Tsakiris and Spiliotis*, 2006]; but for DI, an objective function of either maximization of net benefit or maximization of crop yield has been used [e.g., *Vedula et al*., 2005]. More comprehensive models of irrigation scheduling include soil moisture dynamics equations [e.g., *Ghahraman and Sepaskhah*, 2002]. To determine the allocation policies from a surface or groundwater reservoir, the reservoir's water balance equations combined with soil water dynamics equations may also be considered [e.g., *Teixeira and Mariño*, 2002].

[5] Irrigation planning and management becomes more complicated due to uncertainties on agrohydrological variables (e.g., rainfall, potential evapotranspiration, and crop and soil properties), and socioeconomic factors (e.g., crop prices, production costs, farmers behavior, and political forces). From another point of view, the sources of uncertainty may be classified as temporal variability, spatial heterogeneity, measurement errors, human's behavior complexity, etc. Some socioeconomic factors are mainly under long-term temporal variation. Also, some of the factors caused by spatial heterogeneity, which have been considered in a few irrigation management studies [e.g., *Russo*, 1986; *Gates and Grismer*, 1989], are only influential in small-scale problems; so they progressively become less important as the spatial scale of the problem increases [*Rodriguez-Iturbe et al*., 2006]. *Rhenals and Bras* [1981] have shown that the effect of the uncertainty of potential evapotranspiration on irrigation performance measures could be apparently minimal in a semiarid climate such as Lower South Platte, Colorado.

[6] Rainfall, a relevant part in supplying the agricultural water requirement, is under uncertainty. By assessing different hydroclimatic conditions, *Daly and Porporato* [2006] have shown that the impact of rainfall temporal variability on soil moisture dynamics is much more significant than that of potential evapotranspiration, whose fluctuations do not considerably affect the soil moisture statistical properties. Therefore, attempts have been taken to consider rainfall uncertainty in both long-term and short-term irrigation optimization models. Stochastic dynamic programming (SDP) models [e.g., *Dudley and Burt*, 1973] and the detailed soil moisture dynamics simulation models [e.g., *Ahmed et al*., 1976; *Ines et al*., 2006] have been developed to precisely survey SC farm problems. However, these models are not applicable in MC problems due to being computationally intensive.

[7] Most of the MC models used for long-term irrigation planning consist of a long-term stochastic optimization constituent, for example, SDP, and a short-term deterministic optimization module, for example, linear programming, nonlinear programming, or dynamic programming [e.g., *Dudley et al*., 1976; *Ghahraman and Sepaskha*, 2002; *Vedula and Kumar*, 1996]. The long-term part seeks for the interseasonal water allocation policies, while the short-term model is about intercrop and intraseason optimal water allocations. In this regard, *Alizadeh and Mousavi* [2013a] developed a model coupling Particle Swarm Optimization (PSO) and Monte Carlo simulation for stochastic-order-based design of a reservoir-irrigation district system. Moreover, Positive Mathematical Programming has been used for calibration of the agricultural-production and resource-use models with parameters that are implicit in the observed land allocation decisions [e.g., *Howitt*, 1995; *Paris and Howitt*, 1998].

[8] The computational burden of Monte Carlo-based approaches has led to a class of optimization models in which an explicit stochastic simulation component of soil moisture dynamics is used. *Bras and Cordova* [1981] presented a short-term SDP model for determining optimal intraseasonal DI policies for a SC farm based on a stochastic physically based soil moisture model considering rainfall uncertainty as a Poisson process. *Ganji et al*. [2006] developed a stochastic model of short-term DI optimization based on an explicit stochastic simulation model using indicator functions.

[9] Recently, more realistic explicit stochastic models of soil moisture dynamics have been introduced [*Rodriguez-Iturbe and Porporato*, 2004], which have not yet been used in an optimization model of irrigation management. This paper presents a new explicit stochastic optimization model of short-term MC DI management in the form of a simulation-optimization model with an economic objective function. The agrohydrological simulation model will estimate the actual crop yield and irrigation water requirement based on soil water and salt balance of root zone. It also takes the effects of shallow water table and different irrigation methods into account. Then, the PSO algorithm, linked to the agrohydrological simulation model, will solve the resulting nonlinear nonconvex program for different SC and MC problems.