Crop Planning and Water Management: A Survey

Crop Planning and Water Management: A Survey

Walid Gomaa1,

∗

Nermine Harraz2,

∗

Amr ElTawil2,

∗

1 Department of Computer Science and Engineering

Egypt-Japan University of Science and Technology, Alexandria-Egypt

[email protected] Department of Industrial Engineering and System Management,

Egypt-Japan University of Science and Technology, Alexandria-Egypt

[email protected]

[email protected]

Abstract

Agricultural systems pose many challenges and problems that can be formulated as optimization problems. Amongthese major challenges are crop selection and irrigation planning. That is, to decide on the proper set of crops tobe cultivated and a proper irrigation scheme. Such decisions are made to achieve a certain objectives that typicallyinclude the maximization of net profit and/or the minimization of water waste. The problem is complicated bythe existence of conflicting multiple objectives. In addition, water management represents one of the most criticalproblems that face the national interests in the current and near future, especially in the middle east, where, accordingto UNESCO, the main interstate conflicts over water occur/will occur in that region. Given that agricultural irrigationwater accounts for 80% consumption of the world’s water resources, better agricultural systems management can playa critical role in the peaceful resolution of such crisis. This article presents an initial investigation of the recentwork published in this field involving the mathematical formulation and the computational methods used to solve theresulting models.

1 Introduction

Agricultural systems pose many challenges and problems that can be formulated as optimization problems. Suchproblems include crop selection, country-wide crop planning, irrigation planning, vegetable production, beef production,wildlife and livestock management, and sugarcane transportation. For this article we focus on crop planning and watermanagement [13].

Given a farmland, a water resource, and a list of crops, the objective is to determine the optimal (or near optimal)cropping patterns and irrigation plans. That is to determine the best set of crops to be cultivated over the season(typically one year), the area allocation for each of these crops, the sequencing of crops (if several crops are to besequentially cultivated over the same piece of land in the same season), and the irrigation plans. All determined so as toachieve some set of goals. Typically, these goals involve the maximization of net profit, the minimization of investment,the maximization of total area cultivated, and/or the minimization of irrigation water.

The main entities of the task involve the farmland itself, the labor, the water resource(s), and a list of crops. Themain properties and characteristics of each such entity along with the interactions among them need to be determineda-priori. For example, for the farmland it may be needed to determine its area, its topographical properties, soilcharacteristics such as fertility, and the kind and amount of human labor available. As for the water resource we mayneed to determine such properties as the maximum permissible monthly and annual release, whether it is used for otherpurposes - along with irrigation, such as electricity generation and the associated economic utility. Some comprehensiveset of crops that can be feasibly cultivated in the area under investigation need to be determined. For each such cropsome set of parameters that relate the crop to the given farmland and water resource are to be determined. Theseparameters involve the yield per unit area, water requirement, labor requirement, net return for production of the crop,etc.

Naturally the problem imposes constraints over its parameters and constituent entities. Some constraints are logicalsuch as the fact that the total areas allocated for all crops must not exceed the area of the farmland. Other constraintsare imposed by the different types of decision makers involved in the enterprise. For example, the government authority

∗Currently on-leave from the Faculty of Engineering, Alexandria University

Proceedings of the 41st International Conference on Computers & Industrial Engineering

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may impose an upper limit on the amount of water released for different months of the year, a lower bound on theamount of human labor that must be employed, etc. The farmer, on the other hand, may have economic constraintssuch as limited investment.

The problem is generally a constrained optimization problem, though it is complicated by the fact that there aremultiple, rather than single, goals (objectives) that are to be achieved. Typically, such objectives are conflicting, hence,the notion of ‘optimum solution’ is not very well defined [5] since there will not be an assignment to the decision variablesthat would optimize all the objective functions simultaneously as if the global multi-objective problem is divided intoseveral independent single-objective problems.

Several mathematical programming models have helped in solving special irrigation management problems, particu-larly for a mixed cropping pattern. Surface water and ground water options have been considered by different authorsaccording to the characteristics of agricultural area under study. Classically the models were formulated as single ob-jective optimization problems, but the fact that real life problems incorporate many goals and aspects led recently tothe use of goal programming and multi-criteria decision making.

2 Review

Mathematical models based upon linear programming (LP) have been used extensively in the analysis and planningof water resources systems. In their work [11] M. Mainuddin et al studied how much water should be allocated todifferent cropped areas setting up two strategies for the application of water to the crops. The first was to applyirrigation water at a level which gives maximum net income. This approach would be used when there is no con-straint on the irrigation supplies. The second approach was applied when water shortage existed. The authors considerthree different levels of application of irrigation water (0% deficit, 25% deficit, and 50% deficit). The authors de-veloped a monthly irrigation planning model for determining the optimal cropping pattern and for determining thegroundwater abstraction requirement. Two objectives were considered: the maximization of net economic benefit givenby Max Z =

∑n

i=1

∑l

j=1

∑m

k=1 (EYijkbi − Cijk)Aijk −∑12

t=1 CtGt and the maximization of irrigated area given by

Max A =∑n

i=1

∑l

j=1

∑m

k=1 Aijk. Where Z is annual net return, A is total irrigated area, i is an index for crop type, j

is an index for land type (lowland and upland), k is an index for level of water application (no deficit, 25% deficit, 50%deficit), n is the total number of crops in a year, l is the total number of land types, m is the total number of waterapplication levels, t is an index for time period (month: 1, 2, 3,....12), EYijk is the expected yield per unit area of crop i

in land type j with level of water application k, bi is the unit price of crop i, Cijk is the unit cost of cultivation of crop i

in land j with level of water application k (without considering the cost of irrigation water), Aijk is the area under cropi, in land type j, with level of water application k, Ct is the unit cost of groundwater in month t, and Gt is groundwaterallocated in month t.

J. Adeyemo and F. Otieno in [2] devised a crop planning model and solved it with an evolutionary algorithm. Theysolved the model of a farmland in the Vaalharts irrigation scheme (VIS) in South Africa with total area 771000m2

and supplied with 704694m3 of irrigation water annually. The model is multiobjective aiming at: (1) minimization ofirrigation water, (2) maximization of total net income, and (3) maximization of the total agricultural output. The lastobjective is typically employed to satisfy the livelihood demands of the local community or the nationwide demands.Four crops were targeted: Maize, Groundnut, Lucerne, and Pecannuts. The decision variables are the planting area foreach of these four crops. The constraints of the model were the limitation of total planning area and total irrigationwater. The constraints also include putting minimum and maximum boundaries on the planting area for each crop,this is imposed to allow for diversity and prevent the domination of one crop over the others. The resulting is arelatively simple linear programming model. The main objective of this study is solve this crop planning model usingfour strategies Multiobjective Differential Evolution Algorithm (MDEA). The MDEA algorithm was originally proposedby the same authors in [1]. MDEA1 and MDEA2 employ binomial crossover operator whereas MDEA3 and MDEA4employ exponential crossover operator. On the other hand MDEA1 and MDEA3 use one vector for the perturbationwhereas MDEA2 and MDEA4 use two vectors for the perturbation. All these strategies found the non-dominatedsolutions that converge to the Pareto fronts. The solutions are diverse on the Pareto fronts, though generally MDEA1and MDEA2 generate better solutions than MDEA3 and MDEA4.

B. Feiring et al in [6] constructed an optimization model for water resource sequential planning. The reservoir isassumed to have a dual purpose of generating electricity and supplying irrigation water. The reservoir has a maximumcapacity M , though its level state Ln+1 at time period n + 1 is affected by its previous level Ln, the inflow In at timen due to rainfall, and the outflow at time n due to water released for agricultural irrigation. Hence, the equilibriumequation of the system is Ln+1 = Ln + In − On.

There were two interesting aspects in this work. The first was the linearization of an originally non-linear opti-mization model. For example, the original objective function contained the following term EPn+1 = E min{e, Ln+1} −K max{0, e− Ln+1}, where EPn+1 is the total profit from electrical generation, E is the unit income for electrical use,


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and K is the unit penalty for electrical shortage. This apparently nonlinear objective function can be linearized bydefining the new variables Wn+1 = E min{e, Ln+1} and Xn+1 = K max{0, e − Ln+1}. The new variables satisfy theconstraints

Wn+1 ≤ e, Wn+1 ≤ Ln+1

Xn+1 ≥ 0, Xn+1 ≥ e − Ln+1

(1)

Then the objective function EPn+1 can be rewritten as EPn+1 = Wn+1 − Xn+1 with the inequalities in (1) addedto the constraints of the model. Now the new formulation is apparently linear. The second interesting aspect aboutthis reservoir model is that the stochastic nature of the inflows (rainfall) is taken into consideration. The problem isthen formulated as a chance-constrained stochastic linear program (CCDP), which is a form of the general paradigmof stochastic dynamic programming (SDP). For sufficiently large time steps, the sample mean of the rainfall randomvariable can be assumed to be normally distributed using the central limit theorem. The mean and variance of suchdistribution is unknown and yet to be estimated. Estimation of such parameters are done recursively as new data, atsuccessive time steps, are fed into the system. Using these estimates the model is reformulated as a deterministic linearprogram. So overall from an initially nonlinear stochastic program is eventually transformed into a linear deterministicprogram that is much easier to handle.

R. Sarker and T. Ray in [13] developed a multiobjective crop planning optimization model. They proposed anevolutionary algorithm, called Multiobjective Constrained Algorithm MCA, to solve the planning problem. They alsoconsidered two other optimization approaches that preexists in the literature. The first is the conventional ǫ-constrainedmethod and the second is also evolutionary-based called Non-dominated Sorting Genetic Algorithm NSGAII. Twoobjectives were considered in their work: maximization of total gross income and minimization of total working capitalrequired. The income comes from two sources, namely the cultivation of crops and the import of crops. The workingcapital is mainly the cultivation cost. The constraints are classified into several categories. The first is a demandconstraint dictating that the local production and the imported quantity of each crop must exceed the local requirement.The second is a land constraint where the total area of cultivated land can not exceed the total area available. Thethird is a capital constraint where there is an upper bound on the total working capital that can cover the costs of cropsproduction. The fourth is area/crops constraint where an upper bound is set on the total area of land allocated forcertain crops. This is due to soil characteristics and regional aspects, for example, the unsuitability of certain lands forfruit cultivation makes it necessary to set an area limit for fruit. The fifth is an import constraint where an upper limitis set on the amount of crops that are to be imported. The last couple of constraints are logical ones involving the non-negativity of the decision variables. There are two types of decision variables: the area of land to be cultivated for eachcrop and the amount of import of each crop. The authors classify the cultivated land into three types according to theconsecutive production of crops. The first type is single-cropped land where only one crop is cultivated during the year.The others are double- and triple-cropping lands where two and three crops respectively are cultivated cultivated peryear. Their multiobjective optimization model is essentially linear, however, they also develop a non-linear variation ofthis model. The latter is motivated by two issues: (1) the fact that in double- and triple-cropped lands, the yield is littlehigher than for single-cropped land due to the frequent use of fertilizers, hence a non-linear relationship is establishedto reflect this change and (2) to illustrate the power of using evolutionary algorithms in general, and their own inparticular, for solving multiobjective optimization problems. Pareto optimality is used as the conceptual definition of‘optimum’ with the aim of any solution paradigm is to arrive at a set of Pareto optimal solutions. Their main resultsare the following. For the linear version of their optimization model the evolutionary algorithm NSGAII performed theworst, whereas in the nonlinear model the conventional ǫ-constrained method performed the worst. In both cases theauthors’ evolutionary algorithm MCA had the best performance.

In [14] Sarker et al conducted their work aiming at increasing the contribution of the agricultural sector in Bangladesh.They devised a linear programming model to determine the areas to be used for different crops to gain maximumcontribution. The results reveal that an annual contribution can be increased significantly through proper planning.The linear model accounted for various factors such as land types, alternative crops/crop combinations, crop patterns,input requirement, investment, output, and influence of the crop cultivation. The model was single objective. Theobjective function was to maximize the financial contribution that can be obtained from cropping in a single-crop yearplus imported crops.

Gupta el al [9] in their work followed another approach by accounting for multiple-criteria and using fuzzy rela-tionships. Their study dealt with the real-world problem of irrigation water management of evolving suitable croppingpattern. In The Narmada river basin - India, the irrigation system under study comprised eleven reservoirs either exist-ing, planned, or under construction. The authors used a thirty-years time horizon to simulate the monthly operation ofthe system. Then they develop a multi-objective fuzzy linear programming (MOFLP) area allocation model to cope withthe diverse and/or conflicting interests of different decision makers such as the irrigation authority (government) and the


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individual farmers. Simulation output in the form of optimal monthly releases for irrigation is one of the main inputs tothe MOFLP area allocation model. Variable irrigation demand over the planning time horizon has been incorporatedinto the formulated model considering high variation in precipitation. Besides this, a cropping pattern corresponding to80% dependable releases and rainfall is also analyzed. The analysis was useful in deciding on an appropriate croppingpattern in any command area that minimizes the average crop failure risk in view of the uncertainty about irrigationwater availability, especially in dry years.

Sarker and Quaddus [15] developed first an LP model then turned it into a goal program as they found this type ofmathematical programs more appropriate to the problem. The three goals were: (1) the minimization of import whichputs restriction on the quantity imported from certain crop such as the cereals, (2) the minimization of investments, and(3) the maximization of the return from cultivated land. The authors also defined three types of land: single, double, andtriple cropped. Three goals were formulated: (1) import goal restriction

∑i∈L2

Ii +d−2 −d+1 = b, where Ii is the amount

of crop I that should be imported, (2) capital goal restriction∑

i

∑j

∑k CijkXijk +d−2 −d+

2 = Ca, where Xijk is the areaof land cultivated for crop i of crop combination j in land type k and Cijk is the cost of production per acre of crop i ofcrop combination j in land type k, and (3) contribution goal restriction

∑i

∑j

∑k BijkXijk +

∑i ICiIi +d−3 −d+

3 = EC,where Bijk is the benefit that can be obtained per acre of land from crop i of crop combination j in land type k andICi is the cost per unit of import of crop i.

The three goals led to the objective function of the Goal Program GP minZ1 = w1d+1 + w2d

+2 + w3d

−

3 , wherew1, w2, w3 are the weights assigned to the deviational variables. The authors accounted for the following constrinats.

1. Food demand constraint implying that the sum of local production and the import quantity of crop i in a yearmust be greater than or equal to the total requirements in the country.

2. Land constraint, the authors used the classical land constraint associated with a land type coefficient that was setto 1 for single cropped area, 1

2 and 13 for double and triple cropped areas respectively.

3. The capital (budget) constraint.

4. The contingent constraint: because the model accounts for single and multiple cropped areas, the area used forany crop under a crop combination for double- or triple-cropped land must be equal for every crop. Two casesarise:

(a) Double cropped land: the area used for two crops belonging to any combination of double cropped land isequal: X(i1∈j)j2 − X(i2∈j)j2 = 0, where i1 is the first crop of combination j and i2 is the second crop for thesame the combination j for type of land k = 2.

(b) Triple cropped land: the area used for three crops belonging to any combination of triple cropped land isequal X(i1∈j)j3 − X(i2∈j)j3 = 0 and X(i2∈j)j3 − X(i3∈j)j3 = 0, where i1, i2, i3 are the first, second, and thirdcrop of combination j for type of land k = 3.

5. Area bound constraint.

6. Import bound constraint.

7. And finally the non-negativity constraints.

X. Cai et al [4] incorporated the sustainability dimensions. They considered two objectives: sustaining irrigatedagriculture for food security and preserving the associated natural environment. The study was conducted in theAral Sea region in Central Asia, a region that is most famous for its conflict between sustaining irrigated agricultureand preserving the environment. The paper presented an integrated modeling framework for sustainable irrigationmanagement analysis and then applied it to analyze the irrigation water management in the Aral Sea region. Based onthe modeling outputs, alternative future plans for the irrigation practices in the region are explored.

The objective of the work conducted by M.N. Azaiez et al [3] is to maximize the total expected profit of an entire basinirrigated by both surface and ground water. The model considered a multi-period framework. The stochastic nature ofthe inflows to the main reservoir as well as the irrigation demand are incorporated in the study. The uncertainties in theinflow were treated through chance constraints and penalties of failure to release the planned amounts of surface waterfrom the main reservoir. Although the authors accepted the deficit irrigation and used adequate production functionsto estimate the expected crop yields in order to reflect uncertainties in irrigation, they tried in their model to avoidlarge deficits except for periods where crop yields are relatively insensitive to water shortage. Data used to illustratethe model are partially hypothetical. The authors also conducted some sensitivity analysis that provides guidelines forselecting an appropriate confidence level for not violating the chance constraints and showed how the optimal policiesmay notably change with this level.


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The sustainability dimension considered previously in the work done in [4] in the Aral sea region was also consideredby K. Rajua et al [12] in India. In this work, three linear objectives were developed. The first is to maximize theeconomic benefit. The second is to maximize the agricultural production. And the third objective is to maximize thelabor employment. Compared to the model developed in [4], which is more biased towards environmental aspects, themodel of Raju et al [12] considers three objectives that are explicitly related to economic performance and implicitlytargeting the other sustainability pillars. To suggest an acceptable irrigation plan and cropping pattern the authors usedfour phases. In phase one, separate LP models are formulated for the three above mentioned objectives. In phase two,non-dominated (compromise) irrigation planning strategies are generated using the constraint method of multi-objectiveoptimization. In phase three a neural network based classification algorithm is employed to sort the non-dominatedirrigation planning strategies into smaller groups. In phase four, a Multi-Criterion Analysis (MCA) technique, namely,Compromise Programming is applied to rank the strategies obtained from phase three. It was concluded that the aboveintegrated methodology was effective for modeling multi-objective irrigation planning problems and the approach canbe extended to situations where the number of irrigation planning strategies are large in number. Such work and othersstress on the inherent need to design the agricultural systems sustainably in developing regions where local culture havelittle awareness of this practice.

Narayanan Kannan et al conducted a study in Rwanda [10] and S. Geerts et al [7] conducted a study in Bolivia.Both works targeted increasing yield in dry seasons. The objective of the first study was to assess the possibility foradditional crop production during the long dry season, while regular crop production is carried out during wet seasonswith low yielding short duration varieties of crop. In the article the authors suggested straightforward guidelines forDeficit Irrigation (DI) to help in increasing crop water productivity in agriculture in Bolivia.

3 Conclusion

The survey for the recently published work in the area of crop planning and water management revealed that themajority of studies are conducted in developing countries due to scarcity of water, bad practice, and lack of sustainabilityawareness in such regions. Generally studies can be categorized into experimental work and modeling using mathematicalprogramming or simulation. However the current paper focuses on the mathematical modeling approach as experimentalsetups require long time span, may be costly, and require the availability of large amount of data which might not bealways available. Most of the developed models were based on linear programming with deterministic, stochastic orfuzzy parameters according to the depth of the study.

In the following subsections the different components of the mathematical models for crop planning and watermanagement optimization problems are presented.

3.1 parameters

The parameters included in the mathematical models are either deterministic, stochastic, and/or fuzzy. In [11] theparameters dealt with are deterministic, but to account for the uncertainty the authors solved the model for threelevels of reliability for rainfall and ground water resources. Also in [14] the authors dealt with deterministic parameters,however, they conducted a sensitivity analysis. The analysis showed that the contribution of the binding constraintswith positive shadow prices can be increased by increasing the single-, double- and triple-cropped land, the area forfruit, juice, drugs, and narcotics, and cereal import quantities.

Fuzzy parameters have been considered in [9]. The authors used the hyperbolic membership function. They justifiedtheir particular choice of this membership function by appealing to real-world problems where the marginal satisfactionof the decision maker decreases as the level of satisfaction (grade of membership) with respect to attainment of objectivesincreases.

Stochastic parameters are used in [6] to model the uncertainties about the rainfall. The rainfall is modeled by arandom variable with an arbitrary distribution with mean µ and variance σ2. The observations (the random sample)are independent. The problem is then formulated as a chance-constrained stochastic program. Both parameters, themean and variance, are estimated using the available observations. The mean is estimated using the recursive leastsquares approach [8]. Whereas the variance is estimated using the maximum likelihood method. Given the most recentestimates of the parameters, the information is used to reformulate the problem by an equivalent deterministic model.

3.2 Constraints

Among the classes of constraints spotted in the crop planning and water management problems one can generallyenumerate the following.

1. Water allocation constraint: Ensuring that the total water requirements for the crops at any level of waterapplication in any time period should not exceed the total water supplied in that period.


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2. Land area constraints: The sum of the cropped area in land type j in time period t can not be greater than thetotal available land of that particular type.

3. Constraints for maximum allowable area: These constraints are introduced to limit the amount of production ofcertain crops for the sake of maintaining market price and to allow for cultivating variety of other crops.

4. Constraints for minimum required area: These constraints are needed to fulfill social obligations such as theproduction of minimum requirements of certain crops and the maximization of calories.

5. Constraints due to pumping capacity: The total water supplied from the tube wells is limited by the pumpingcapacity. This is affected by the pumping rate per hour, the total pumping hours per day, and the number of daysof pumping per month.

6. Non-negativity constraints: All decision variables have to be non-negative values.

Acknowledgement

Many thanks and appreciation are due to Professor Samir Ismail, Vice Dean for Community Service and EnvironmentDevelopment, Faculty of Agriculture, Alexandria University, for his valuable comments and fruitful discussions.

References

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[14] R. Sarker, S. Talukdar, and A. Haque. Determination of Optimum Crop Mix for Crop Cultivation in Bangladesh.Applied Mathematical Modelling, 21(10):621–632, 1997.

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