1 Introduction

Irrigation is essential to increase agricultural production for fulfilling the food requirement of burgeoning global population, which is expected to touch the 9.3 billion mark by 2050 from the current 7.1 billion (United Nations 2010). However, without appropriate planning and management, irrigated agriculture can be detrimental to the environment and may endanger its sustainability (van Dam et al. 2006). For example, during the last few decades, many canal irrigated areas of the world are facing rising groundwater levels, and problems of waterlogging and salinisation have emerged (Boumans et al. 1988; Hoffman and Durnford 1999; McFarlane and Williamson 2002; Konukcu et al. 2006; Guganesharajah et al. 2007; Singh 2012a, 2013). The increase in population and decline in good quality water resources have intensified competition among different water users. This competitiveness can build up the agricultural water shortage and serious problems can thus arise from poorly planned water management systems.

The optimal use of available surface water and groundwater resources is of paramount importance in irrigation management for achieving food security and sustaining livelihoods (Smith 2000). This can be attained by using an optimization technique (Singh 2012b). It helps to find the answer that yields the best result for minimizing or maximizing an objective function subjected to various constraints (Deninger 1970; Anwar and Clarke 2001; Salazar et al. 2005; Joubert et al. 2007; Easa and Hossain 2008; Wei and Hsu 2009; Majone et al. 2010; Zhang and Huang 2011; Stray et al. 2012). The various optimization techniques have been used by researchers from worldwide during the last five decades for irrigation management (Castle and Lindeborg 1960; Yakowitz 1982; Chávez-Morales et al. 1987; Paudyal and Gupta 1990a, b; Chen and Huang 2001; Maqsood et al. 2005; Khare et al. 2006; Luo et al. 2007; Li et al. 2008; Karamouz et al. 2009; Yang et al. 2009; Li et al. 2011; Huang et al. 2012). This study presents a review of different programming techniques used for irrigation planning and management.

2 Linear Programming

Linear programming (LP) optimization models were extensively used by researchers for irrigation management because of its easy formulation and application (Castle and Lindeborg 1960; Morel-Seytoux 1975; Maknoon and Burges 1978; Loucks et al. 1981; Yaron and Dinar 1982; Feinerman and Yaron 1983; O’Mara 1988; Vincent and Dempsey 1991; Mohan and Jothiprakash 2003; Lu et al. 2011; Singh 2012c). Boster and Martin (1979) used an LP model in irrigated farms in Arizona. Male and Mueller (1992) presented a dual-objective LP model, considering the use of groundwater without stream depletion. While, Peralta et al. (1995) developed an LP model to obtain the sustainable groundwater extractions over a period of five decades under conjunctive water use scenario.

An LP model was proposed by Tyagi and Narayana (1981) to allocate available surface water and groundwater for irrigation in a semi-arid area of India. Kumar and Pathak (1989) presented an LP model for optimal crop planning in a canal-aquifer system. Ibáñez-Castillo et al. (1997) used a combination of LP and simulation models for planning the operation of an irrigation system. Barlow et al. (2003) presented an LP-based conjunctive management model to evaluate the tradeoffs between groundwater withdrawal and streamflow depletion for alluvial-valley stream aquifer systems of the northeastern United States. Lu et al. (2011) developed and applied an inexact rough-interval fuzzy LP model to generate conjunctive water allocation strategies. The researchers, i.e., Aron (1969), Suryavanshi and Reddy (1986), Gaur et al. (2011), and Kashyap and Chandra (1982) have also used similar models for management and planning of surface water and groundwater resources.

Matanga and Mariño (1979) developed a stochastic inter-seasonal model to determine irrigation policy. Karmarkar (1984) has established that interior point LP (IPLP) algorithm was quite efficient in solving very large LP problems. Later IPLP approach was used by Ponnambalam et al. (1989) for reservoir operation. Easwaramoorthy et al. (1989) suggested optimal cropping pattern for the lower Bhawani project using an LP model. A multilevel optimization technique was used by Paudyal and Gupta (1990b) to solve the complex problem of irrigation management in a large heterogeneous basin. Tyagi (1986, 1988) formulated and applied decision models for optimal allocation of good quality limited surface water and saline groundwater. Shyam et al. (1994) used LP technique for devising an improved method of water allocation in India.

Morel-Seytoux (1975) solved an LP model by using the discrete kernel generator (Morel-Seytoux and Daly 1975). Bowen and Young (1985) used an LP model to derive estimates of financial and economic benefits of irrigation in Egypt. An LP model was developed by Afzal et al. (1992) to optimize the use of different quality waters by alternative irrigations in an area of Pakistan. Malek-Mohammadi (1998) presented a mixed-integer LP model for planning an irrigation system. Latif and James (1991) applied an LP model in the Indus basin in Pakistan to maximize the net income of irrigators through cycles of wet and dry years over the long period. Similarly, Yamout and El-Fadel (2005) developed a regional LP model to assist decision makers in the planning and setting policies for optimal water resources allocation.

An LP model was applied by Feinerman and Yaron (1983) for guiding annual decision making with regard to saline irrigation water mixing from various sources. Panda et al. (1983) developed and applied an LP model for conjunctive use of surface water and groundwater in a canal command area of Indian Punjab. Tyagi and Narayana (1984) developed a deterministic LP model for allocation of surface water and groundwater for irrigation. Kaushal et al. (1985) developed a deterministic LP model to find out the optimal cropping pattern and optimum use of saline groundwater in a canal command area in India. The results of a study on a forecasting system of irrigation water requirements using the fuzzy theory was presented by Saruwatari and Yomota (1995). Khare et al. (2007) used an LP model to investigate the scope of conjunctive use of surface water and groundwater for a link canal command in Andhra Pradesh, India.

Khanjani and Busch (1983) developed a procedure to specify optimal plans for an irrigation system with temporary internal storage. An LP model for planning the management of an irrigation district in Mexico was developed by Chávez-Morales et al. (1987). Panda et al. (1996) developed and linked three non-structural management models for the canal command area of a semi-arid region of Indian Punjab. Teixeira and Mariño (2002) presented an optimization methodology for reservoir operation coupled with an irrigation scheduling scheme that maximized the net income to an irrigation district. Sun et al. (2011) concluded that irrigation water productivity can be improved for the double cropping system under optimized water management.

Smout and Gorantiwar (2005) presented a water allocation model, which incorporates deficit irrigation for optimizing the use of irrigation water. Later, Gorantiwar and Smout (2005) applied the model to a medium irrigation scheme in India to obtain the land and water allocation plans. Moradi-Jalal et al. (2007) developed an LP model for the optimal multi-crop irrigation areas associated with reservoir operation policies in a reservoir-irrigation system in Iran. Recently, Singh (2012c) developed and applied an LP model for the optimal allocation of land and water resources in a semi-arid region of India, which is underlain by poor quality groundwater.

Chance constrained linear programming (CCLP) is one of the approaches of LP under risk wherein some or all parameters are random variables. Nieswand and Granstrom (1971) used a set of CCLP models for the conjunctive use of surface water and groundwater. Smith (1973) formulated a chance-constrained stochastic model for an irrigation project in Bangladesh for studying the complex hydrologic and economic interactions on conjunctive use of surface water and groundwater. The researchers, i.e., Mishra (1975), Lakshminarayana and Rajagopalan (1977), Maji and Heady (1978), and Panda et al. (1985) have also used CCLP approach for irrigation planning and management. While, Li et al. (2010) utilized an inexact two-stage water management model for irrigation planning in the Zhangweinan River Basin, China.

3 Nonlinear Programming

The inability of LP models to handle nonlinear problems and difficulty in attaining global optimal solution of other algorithms (Sedki and Ouazar 2011) thrusts the use of nonlinear programming (NLP) models in irrigation management (Rydzewski and Rashid 1981). Khan (1982) presented an NLP model for managing irrigated agriculture with different quality waters. Danskin and Gorelick (1985) developed a groundwater-surface water management model using NLP technique. Ghahraman and Sepaskhah (2004) used LP and NLP models for exploring the irrigation optimization. A conjunctive use planning model was formulated by Chiu et al. (2010), considering optimal pumping and recharge strategy.

An NLP model was formulated by Gupta et al. (1987) for conjunctive use through blending of poor quality groundwater and good quality canal water. The potential of a quadratic programming-based optimization approach was evaluated by Wardlaw and Barnes (1999) for improving the real time irrigation water management in systems with complex distribution networks. Benli and Kodal (2003) formulated a crop water benefit function-based NLP model for the determination of irrigation water needs and farm income under adequate and limited water supply conditions in southeast Anatolian Region of Turkey. The similar approach was adopted by Mainuddin et al. (1997), Alaya et al. (2003), Raju and Kumar (2004), and Shang and Mao (2006).

A methodology was developed by Haimes and Dreizin (1977) for solving the problems of conjunctive use of a large scale complex groundwater system. Rastogi (1989) used NLP for the simulation of a groundwater management model in the Blue Lake aquifer in Northern California. Carvallo et al. (1998) developed an NLP model for determining the optimal cropping pattern with cultivated area in each soil as decision variable. Similarly, Ghahraman and Sepaskhah (1997) also used NLP for maximizing farm income, while, Paudyal and Gupta (1990a) proposed an NLP model for irrigation management. Takahashi and Peralta (1995) computed optimal perennial groundwater yield pumping strategies for a complex multilayer aquifer of the Great Salt Lake in Utah.

For the efficient utilization of water resources in a coastal groundwater basin of Orissa in India, NLP and LP models were developed and applied by Rejani et al. (2009). Montazar et al. (2010) developed an integrated soil water balance algorithm and coupled to an NLP model for carrying out water allocation planning in complex deficit agricultural water resources systems. Similarly, Huang et al. (2012) developed an integrated two-stage interval-quadratic programming model for water resources planning and management in China. Austin et al. (1998) analysed different programming models and concluded that NLP models do not give appreciably better performance compared with LP models.

4 Dynamic Programming

Because of its inherent advantages, the use of dynamic programming (DP) technique is very common in irrigation planning and management (Yakowitz 1982) and it has been extensively used by various researchers worldwide (e.g., Flinn and Musgrave 1967; Hall et al. 1968; Hall and Butcher 1968; Burt 1970; Yaron et al. 1987; Rao et al. 1988, 1992; Sunantara and Ramirez 1997; Naadimuthu et al. 1999; Shangguan et al. 2002; Abdallah et al. 2003; Tran et al. 2011). The important developments in the area of the DP application in irrigation management has been reported by many researchers, i.e., Yeh (1985), Kennedy (1986), Bosch et al., (1987), and Benedini (1988). Usually, the DP approach has been limited to two or three state variables. However, Philbrick and Kitanidis (1998) applied a second order gradient DP method which had five state variables. Panda (1992) applied DP for irrigation planning and management of a semi-arid region through conjunctive use of surface water and gypsum treated sodic groundwater.

Yaron and Dinar (1982) presented a system analysis approach for water allocation during peak seasons to alternative crops. A differential DP algorithm was developed by Jones et al. (1987) to reduce dimensionality. Tsakiris and Kiountouzis (1984) developed a DP model to optimize the intra-seasonal distribution of irrigation water under the constraint of predetermined irrigation timing. An integrated LP-DP model for intra-seasonal irrigation allocation was developed by Vedula and Kumar (1996). Rao et al. (1990) developed an optimization model for optimal weekly irrigation scheduling by considering both seasonal as well as intra-seasonal competition of water.

A DP model was developed and applied by Karamouz et al. (2004) for fulfilling the objective of meeting agricultural water demands in the Tehran metropolitan area. Prasad et al. (2006) developed a deterministic DP model for obtaining optimal irrigation planning in a multi-crop and multi-season environment in Nagarjuna Sagar right canal command of India. Similarly, Shangguan et al. (2002) developed and used a DP model for regional optimal allocation of irrigation. Recently, Li et al. (2011) developed and used a robust multistage interval-stochastic programming method and applied it to the planning of regional water management systems.

The stochastic dynamic programming (SDP) demonstrated to be a potential tool in solving irrigation management problems (Rhenals and Bras 1981; Duldley 1988; Rajput and Michael 1989; Protopapas and Georgakakos 1990). Bras and Cordova (1981) proposed a SDP algorithm for optimal temporal allocation of irrigation water. The researchers i.e., Stedinger et al. (1984), Rao et al. (1988), Sunantara and Ramirez (1997), and Paul et al. (2000) also used SDP approach for addressing the problems of optimal allocation.

A SDP model was used by Dudley et al. (1971) for integrating the systems for determining the optimal irrigation timing. Later, Dudley and Burt (1973) and Duldley (1988) have made improvement in the model. Gupta and Chauhan (1986) modeled stochastic irrigation needs of paddy. Reca et al. (2001a) proposed an optimization model for planning of deficit irrigation in the Bembezar river irrigation district, Spain (Reca et al. 2001b). Datta and Dhiman (1996) developed a model for designing a groundwater quality monitoring network. Azaiez et al. (2005) developed a chance constrained model for optimal multi-period operation of a multi-reservoir system. An optimization model using fuzzy theory within the SDP was proposed by Bertsekas and Tsitsiklis (1996) to deal with highly nonlinear problems.

A SDP technique was developed by Alaya et al. (2003) to satisfy the irrigation water demand and ensure minimal water storage in the Nebhana dam. Vedula and Mujumdar (1992) developed a three state variable SDP model to obtain an optimal reservoir operating policy. Saad et al. (1996) developed a fuzzy learning disaggregation method to decompose monthly optimal policies of an aggregated reservoir using DP. Ghahraman and Sepaskhah (2002) developed an NLP-SDP model for optimal allocation of water in Ardak reservoir dam in an arid region.

5 Genetic Algorithms

The inability of conventional LP and NLP models in handling nonlinear non-convex problems and difficulty in attaining global optima generate demand for other types of algorithms. The genetic algorithms (GA) (Holland 1975; Fogel 1994) has been established as a valuable tool for solving complex optimization problems (Wang 1991; Oliveira and Loucks 1997; Wardlaw and Sharif 1999; Wu et al. 2007; Liu et al. 2008; Rana et al. 2008; Sedki et al. 2009; Nicklow et al. 2010). This technique has been used by many researchers for dealing with a wide range of optimization problems (Davis 1991; McKinney and Lin 1994; Mckinney and Lin 1995; Huang and Mayer 1997; Haupt and Haupt 1998).

Although GA has been extensively used for many optimization problems, its application for irrigation planning is relatively new (Aly and Peralta 1999; Hilton and Culver 2000, 2005; Morshed and Kaluarachchi (2000); Nixon et al. 2001; Wu and Simpson 2001; Ahmed and Sarma 2005; Espinoza et al. 2005; Vedula et al. 2005; Kumar et al. 2006). Sharif and Wardlaw (2000) presented a GA algorithm to the optimization of a multi-reservoir system in Brantas Basin in Indonesia. Wardlaw and Bhaktikul (2004) utilized the time block approach and GA for solving the irrigation scheduling problem. However, the approach used by Wardlaw and Bhaktikul (2004) was criticized by Haq et al. (2008). An irrigation allocation model was presented by Kumar et al. (2006) by using a GA approach. Kuo et al. (2003) performed a comparative study on optimization techniques for irrigation project planning. Maskey et al. (2002) used the groundwater flow and particle tracking models and a global optimization tool, GLOBE for the groundwater remediation.

A multiobjective planning model was developed (Yang et al. 2009) by integrating a multiobjective GA, constrained differential DP (CDDP), and a groundwater simulation model. Sarker and Ray (2009) provided useful insights about solutions that are generated using population-based approaches. Raju and Kumar (2004) applied GA for evolving an optimal cropping pattern utilizing surface water resources. A GA model was used by Karamouz et al. (2009) to optimize a water resources allocation scheme considering the conjunctive use of surface water and groundwater resources. Zeng et al. (2010) developed a fuzzy multiobjective LP model for crop area planning of Liang Zhou region, Gansu province, China.

An integrated LP-GA model was formulated by Md. Azamathulla et al. (2008). Hilton and Culver (2000) compared an additive penalty method with a multiplicative penalty method in a GA. Reed et al. (2000) presented a review of the existing tools from literature to ensure that a GA converges to an optimal or near-optimal solution. Wardlaw and Sharif (1999) performed the evaluation of GA for a four-reservoir problem. A GA model was formulated by Karamouz et al. (2008) to optimize the cropping pattern of irrigation networks. Wu and Simpson (2001) concluded that messy GA requires fewer design trials than for other GAs. Archibald et al. (2006) proposed a new approach that can be applied to more general reservoir systems, allowing arbitrary partitions. Moghaddasi et al. (2010) developed a regional optimal allocation model which allocates water samong different crops and irrigation units.

An integrated GA-CDDP model was used by Hsiao and Chang (2002) for obtaining the optimal solutions of the groundwater resources planning problem. A hybrid two-stage GA model was developed by Afshar et al. (2010) for optimizing the design and operation of a nonlinear and semi-distributed cyclic reservoir system in an irrigated area. The researchers, i.e., Dariane and Hughes (1991) and Mujumdar and Ramesh (1997) addressed the problem of real time reservoir operation for irrigation. Nixon et al. (2001) applied a GA for optimizing off-farm irrigation scheduling. Huang et al. (2002) developed a GA-SDP model to cope with the dimensionality problem of a multiple-reservoir system. The researchers, i.e., Rao et al. (2004), Bhattacharjya and Datta (2005), Rao et al. (2006), and Safavi et al. (2010) have used artificial neural networks (ANN) (Rogers and Dowla 1994) to solve the optimization problems of water resources.

6 Conclusions

The literature review revealed that the management models used in the past for irrigation planning and management mainly considered the objectives of maximization of net farm income, minimization of waterlogging, and minimization of groundwater depletion. The LP technique was extensively used because of its easy formulation and application. However its inability to handle nonlinear problems propels the use of NLP. CCLPs were used to accommodate random variables in decision making. The use of DP techniques was very common in irrigation planning and management because of its ability to model sequential decision-making processes and ability to incorporate stochasticity of hydrological processes. The complex nonlinear non-convex optimization problems were solved by using GA as it could yield better results as compared to the traditional optimization techniques. The GA-based optimization approach is mainly suitable for externally linking the numerical simulation model within the optimization model. Some past studies developed and used multiobjective planning models by integrating GA and DP. The ANNs were used because of their simple structure and ability to approximate even highly complex systems. Many researchers concluded that simulated annealing with an ANN can be useful for real problems of modest size. Combined use of optimization and simulation models have been preferred recently to explore the unique advantages of the technique.

The reviews on the different programming techniques used for the planning and management of irrigation was done and presented in this paper. This review provides the basis for the selection of appropriate methodology. Although the study has examined all possible literature sources, it seems to be practically impossible to include in a review all publications. This review paper highlights an overall approach for optimization modelling applications for the planning and management of irrigation. It is likely that some aspects of some of the subjects have either been overlooked or only briefly referred to. Some of the subjects may deserve a more comprehensive, special review. It is expected that these gaps could be filled by subsequent contributions and that there is scope for further discussion about the subject covered in this review.