Global Advanced Research Journal of Agricultural Science (ISSN: 2315-5094) Vol. 3(8) pp. 223-232, August, 2014. Available online http://garj.org/garjas/index.htm Copyright © 2014 Global Advanced Research Journals

Full Length Research Paper

Optimising irrigation water for field crops to maximise the yield and economic return

M.H.Ali1,*, I. Abustan2, M.H. Zaman3, A.K.M.R. Islam4, A. AlBassam5

1 Agricultural Engineering Division, Bangladesh Institute of Nuclear Agriculture, Mymensingh 2202, Bangladesh.

2 School of Civil Engineering, University Sains Malaysia, Engineering Campus, 14300 Nibong Tebal, Penang, Malaysia.

3 Agricultural Engineering Division, Bangladesh Institute of Nuclear Agriculture, Mymensingh 2202, Bangladesh.

4 Graduate Training Institute, Bangladesh Agricultural University, Bangladesh Agricultural University, Mymensingh 2202,

Bangladesh. 5 A. AlBassam, Department of Geology and Geophysics, King Saud University, Kingdom of Saudi Arabia.

Accepted 10 August, 2014

The optimum use of land and water resources for maximising production and/or profit is a current societal demand. A general procedure has been presented in this paper to optimise irrigation for field crops. The procedure includes determining the irrigation depth for maximising the net financial return under water-limiting conditions and maximising the yield under land-limiting conditions. This approach is based on the general form of the production function (the yield-water relationship), the cost function and the theory of maxima and minima. The derived equations have been applied to two published field crops – wheat and mustard, which are grown in a humid and sub-tropic environment. The procedure and the derived equations can be applied to any field crop to determine the irrigation depth for maximising the yield or economic return. The implications of the derived equations for irrigation management on a larger scale are discussed.

Keywords: Water-limiting, land-limiting, economic return, production function, wheat, mustard.

INTRODUCTION Water supply is an essential aspect of crop production. Irrigated agriculture is the user of approximately 70% of the available water resources worldwide and approximately 80-85% in Southeast Asia. Due to the increase in cropping intensity and coverage (area) and the competition for water in other sectors, such as urban, industrial, and environmental, the availability of water for crop irrigation is becoming limited. Overexploitation of groundwater and the

*Corresponding Author’s Email: hossain.ali.bina@gmail.com; mha_bina@yahoo.com

consequent environmental impact, quality degradation, depletion of reserves, and increased pumping costs are reported for many regions of the world (Sarkar and Ali, 2009; Chawla et al., 2010; Ali et al., 2012).

Water is becoming scarce not only in arid and drought-prone areas but also in humid areas ((Nay-Htoon et al., 2013; Ali and Abustan, 2013). Water scarcity concerns the quantity of resources that are available and the quality of the water because degraded water cannot satisfy the requirements. Sustainable use of water, which includes resource conservation, environmental friendliness, appropriateness of technologies, economic viability, and socially

224. Glo. Adv. Res. J. Agric. Sci. acceptability of development issues, is needed for both regions that are water scarce and regions in which water is abundant (Ali et al., 2011).

The irrigation depth that is required for maximum economic benefit might not be the same as the depth that obtains maximum production. Based on the limitations of the water or land, the situation is referred to as water-limiting and land-limiting conditions (English, 1990; Ali, 2010). Under some circumstances, the available land for cultivation can be limited, but the available water could be abundant. If the land is limited but there is no opportunity to expand the irrigated acreage even if more water is available, then the situation is a land-limiting case. Under such a condition, the optimum irrigation strategy would be to apply the amount of water that would maximise the net income incurred from each unit of land. An area that has limited water and abundant land and in which the farm in question has an opportunity to irrigate additional land if water becomes available is a water-limiting case. In such water-limiting conditions, the water that is saved by a reduction (or deficit) in the irrigation of one piece of land might be used to irrigate additional land, thus increasing the farm income. Thus, under water-limiting conditions, the optimum strategy would be the strategy that generates the maximum net income from each unit of water.

Many studies have focused on water allocation models (Klocke et al., 2006; Gorantiwar and Smout, 2005; Reca et al., 2001; Panda et al., 1996; de Juan et al., 1996; Martin et al., 1989; Bernardo et al., 1988). Most of the studies were intended as a planning tool for crop selection and seasonal allocations of land and water to crop rotations. These choices are not intended for scheduling water applications during the growing season to maximise the return. Single growing-season water allocation among several crops or the irrigation amounts for a specific crop from a limited source of water is based on how each crop responds in terms of its net economic return in response to the water. From a farm-level perspective, a systematic and generalised approach for determining the irrigation depth for maximising the profit is scarce. English (1990) derived equations for different situations but did not include the secondary product(s) in the benefit function, which is important for many field crops. In addition, the cost function is not comprehensive – irrigation-dependent costs and irrigation-independent costs are not separated.

Based on the principle of English (1990) and using the theory of maxima and minima (of mathematics), a detailed analysis has been presented in this paper for selecting the correct irrigation depth under land- and water-liming conditions, with a view towards helping valued utilisation of the available land or water resources. The derived equations for different conditions were tested with field experiment results.

MATERIALS AND METHODS Formulation of mathematical functions Yield and revenue function Let us consider a crop in which x cm water is applied during the crop period. Let the grain (primary yield) and straw yield (secondary yield) of the crop be expressed in functional form as: Yg = a1 + b1x + c1 x

2 ………………..(1)

Ys = a2 + b2x + c2 x2 ………………..(2)

where Yg and Ys are the grain and straw yield (t/ha), respectively; x is the depth of irrigation water applied (cm); a1 is the intercept, and b1, c1 are the coefficients for the grain yield function; and a2 is the intercept, and b2, c2 are the coefficients for the straw yield function. Next, let us consider that: p1 = unit price of grain (or primary yield), $/t p2 = unit price of straw (or secondary yield), $/t Thus, the gross income from one hectare of land (GI, in $), is: GI (x) = (Yg× p1) + (Ys× p2) = (a1 p1+ b1 p1x + c1 p1 x

2) + (a2 p2+ b2 p2x + c2 p2 x

2)

In other words, GI (x) = x

2(c1 p1+ c2 p2) + x (b1 p1+ b2 p2) + (a1 p1+ a2 p2)

……………(3) Cost function Next, let the cost function be expressed as (in $/ha): C(x) = d1 + d2 (x) + d3 (x) …………………(4) where d1 = cost of irrigation-independent items (such as the land lease value, cost of land preparation, the sowing/planting, weeding, and fertiliser cost, and the cost for insecticide/pesticide application), $/ha d2 (x) = irrigation cost, for x cm water application (including pump hire, fuel cost, and labour cost, if any), $/ha d3 (x) = cost of harvest, transportation, threshing, cleaning, and other steps (which is a function of irrigation because the harvestable yield increases due to the irrigation application), $/ha The case of a water-limiting condition Let us assume that an x cm irrigation depth is applied to achieve a targeted yield. For the x cm irrigation depth, the total volume of water per hectare = 1 ha × x cm = x ha-cm Let p3 = cost of the unit water (cost of 1 ha-cm water)

(including the application cost, e.g., the incurred labour, if any), $ p4 = cost of the unit harvest, transportation, threshing, cleaning/processing, and other steps, $/t Then, d2 (x) = p3x d3 (x) = p4 × (Yg + Ys) Thus, equation. 4 can be written as: C(x) = d1 + d2 (x) + d3 (x) = d1 + (p3 × x) + p4 (Yg + Ys) = d1 + (p3 × x) + p4 (a1 + b1x + c1 x

2) + p4 (a2 + b2x + c2 x

2)

Irrigation depth for maximising the net income Here, the net income from one hectare land, NI (x) = GI(x) - C(x) = x

2(c1 p1+ c2 p2) + x (b1 p1+ b2 p2) + (a1 p1+ a2 p2) – [d1 +

(p3 × x) + p4 (a1 + b1x + c1 x2) + p4 (a2 + b2x + c2 x

2) ]

In other words, NI (x)= x

2 (c1 p1+ c2 p2 - c1 p4- c2 p4) + x(b1 p1+ b2 p2 – p3 -

b1 p4 – b2 p4) + (a1 p1+ a2 p2 – d1 - a1 p4 - a2 p4) ………………….(5) Let the total available water in the area = V ha-m For the x cm irrigation depth, the irrigable area, A = �������� = 100 �

� ℎ

Thus, for V ha-m of water, the net income,

NI(x) = 100 �� [�� − ����]

or NI(x) = 100Vx (c1 p1+ c2 p2 - c1 p4- c2 p4) + 100V (b1 p1+ b2

p2 – p3 - b1 p4 – b2 p4) + ����� (a1 p1+ a2 p2 – d1 - a1 p4 - a2

p4) …………………..(6) To find out the value of x, for which the NI(x) will be a maximum or minimum, we must differentiate NI(x) with respect to x and set it equal to zero (from the theory of maxima and minima). Next, differentiating NI(x) [equation (6)] with respect to x, we obtain ��� {�����} = 100V(c1 p1+ c2 p2 - c1 p4- c2 p4) -

������ (a1 p1+ a2

p2 – d1 - a1 p4 - a2 p4)

Equating ��� {�����} to zero (for a maximum or minimum

value), we obtain

100V(c1 p1+ c2 p2 - c1 p4- c2 p4) = ������ (a1 p1+ a2 p2 – d1 -

a1 p4 - a2 p4)

Or � = ��� �!�� ������� "��� "�� �!�� ���� "��� " #�/% ………………(7)

To confirm that the value of NI(x) will be maximum for the above value of x (equation. 7), we must find out the second derivative of NI (x). If the second derivative is negative,

Ali et al 225 then NI (x) will be maximum for the above value of x (Hancock 1917).

Here, ����� {�����} =

%����& (a1 p1+ a2 p2 – d1 - a1 p4 - a2 p4)

………………….(8) The value of the second derivative will be negative if (a1 p1+ a2 p2 ) < (d1 + a1 p4 + a2 p4) ………………..(9) Here, a1, a2 are the intercept of the grain- and straw-yield function (with respect to the water applied); p1, p2 are the price of unit of the grain and straw, respectively; d1 is the cost of the irrigation-independent items ($/ha); and p4 is the cost of the unit harvest, transportation, processing, and other steps.

Hence, equation (9) is the necessary condition for maximising the net benefit.

Under the water-limiting condition (land is not limiting), the aim is to maximise the yield from the available irrigation water, not the maximum yield per unit (say, hectare) of land. Thus, when the net income is maximised, the result is the condition for the maximum total yield. The condition for maximising the yield per unit area does not exist (see Appendix-1). Land-limiting condition Maximising the yield per unit area The grain-yield function is (equation. 1): Yg = a1 + b1x + c1 x

2

Then, ��� �'(� = 0 + b1 + 2c1x

Equating the derivative to zero, we get

x = - )�%�� ………………. (10)

Next, to confirm that the yield will be the maximum for the above value of x, we must check the second derivative: ����� *'(+ = 2-� The second derivative will be negative for all negative values of c1. In practice, the value of c1 is negative in most of the yield functions (polynomial in nature). Thus, the value of Yg can be expected to be maximum for the above value of x (equation. 10). Maximising the net income per unit area The net income from one hectare (01 ha) of land with x cm of irrigation depth is (equation. 5): NI (x)= x

2 (c1 p1+ c2 p2 - c1 p4- c2 p4) + x(b1 p1+ b2 p2 – p3 -

b1 p4 – b2 p4) + (a1 p1+ a2 p2 – d1 - a1 p4 - a2 p4)

226. Glo. Adv. Res. J. Agric. Sci. Table 1. Details of irrigation treatments

* ‘1’ indicates one irrigation at this stage, and ‘0’ indicates no irrigation (deficit).

** in addition to irrigation at each stage, irrigation was given when the total available moisture within the root zone dropped to below 50 %.

For maximising NI (x), differentiating it with respect to x and equating it to zero yields

��� {�����} = 2x(c1 p1+ c2 p2 - c1 p4- c2 p4) +

(b1 p1+ b2 p2 – p3 - b1 p4 – b2 p4) = 0 Thus,

� = � &!)� "!)� "�)� ��)� �%��� �!�� ���� "��� "� # ……………… (11)

The second derivate of NI (x) is

����� {�����} = 2(c1 p1+ c2 p2 - c1 p4- c2 p4)

The second derivative will be negative if c1 p4 + c2 p4) > (c1 p1+ c2 p2). This inequality is the necessary condition for maximising the net profit per unit area. Example application - Application for irrigation optimisation in wheat and mustard The mathematical functions described herein were applied to wheat and mustard production (experimental plot) to demonstrate and evaluate their use. Wheat The experimental details have been described elsewhere (Ali et al. 2007; Ali 2008). Here, a brief description is given. A field experiment was conducted for three consecutive years at the experimental farm of Bangladesh Institute of Nuclear Agriculture (BINA), Ishurdi, Bangladesh (co-

ordinates are: latitude 24 0 06′ N, longitude 89

0 01′ E). The

climate of the region falls under humid sub-tropic and has summer dominant rainfall. The texture of the field soil was silty loam. The field capacity and wilting point of the field soil were 45% and 19% (by volume), respectively. The wheat-growing period, November to March, is characterised by a dry winter. The wheat cultivar was a semi-dwarf variety (the average height is 88 cm). This crop is a 120-130 day cereal crop that suits the prevailing climate of the winter season (November - March). Details of the irrigation treatments are given in Table 1. The experimental design was a randomised complete block (RCB) with four replications.

Soil moisture was measured in one replication by the gravimetric method and/or neutron moisture meter. Access tubes were installed at the centres of the plots. Measurements were taken at the soil depths of 15, 30, 45, 60, 75, and 90 cm at sowing, at every growth stage, and at physiological maturity. The soil moisture was measured by the gravimetric method (weight basis) and was converted into volumetric proportion by multiplying by the bulk density. The irrigation water was applied in the unit plots by using a hose pipe and calibrating the rate with a large bucket of known volume. Evapotranspiration (ET) was calculated by using the general water balance equation:

ET = I + P ± ∆W ………………........... (12) where ET is the crop evapotranspiration, I is the irrigation

water applied, P is the effective rainfall, and ∆W is the change in the soil moisture storage in the soil profile. The rainfall amounts during the wheat growing seasons were 75, 11 and 25 mm during the 1

st, 2

nd, and 3

rd year,

respectively.

Treat-ment

Irrigation at growth phase*

CRI

Jointing to Shooting

Booting to

Heading

Flowering to soft dough

T1 0 0 0 0

T2 1 1 1 1

T3 0 1 1 1

T4 1 0 1 1

T5 1 1 0 1

T6 1 1 1 0

T7 1 0 1 0

T8 0 1 0 1

T9** 1 1 1 1

T10 1 0 0 0

Ali et al 227

Figure 1. Yield function of wheat (combined data, 3 years)

Figure 2. Straw yield vs irrigation plus rainfall (combined data, 3 years)

The grain yield function and straw yield function for the combined data set (3 years) are depicted in Figure 1 and Figure 2, respectively. Mustard The experimental results have been reported elsewhere (Ali et al., 2013; BINA, 2009). Here, a brief description is presented. A field experiment was conducted at the experimental farm of Bangladesh Institute of Nuclear Agriculture, Magura (23

o30΄ N, 89

o35΄ E), Bangladesh. The

soil type was loam. The climate of the regions falls within humid sub-tropic with summer dominant rainfall. The mustard growing period (November–February) is characterised by dry winter. The experimental design was a randomised complete block (RCB) with split-plot arrangements of the treatments. Irrigation treatments were allocated in the main plots (6 m × 5 m), and the mustard cultivars were in the sub-plots (5 m × 2 m), with three replications. The irrigation treatments were the following: T0 = Control (no irrigation)

228. Glo. Adv. Res. J. Agric. Sci.

Table 2. Mean seed and straw yield and input water of mustard*

* Table 2, Ali et al. 2013

Table 3. Comparison of the English (1990) formula and a new derivation for different conditions

Sl Element English (1990) equations1 Newly derived equations

2

1 Water-limiting condition – irrigation depth for maximising the net financial return

Ww = [(Pca1 –a2)1/2

]/ Pcc1 (equation. 30)

� = . �/� + %/% − 1� − �/2 − %/2-�/� + -%/% − -�/2 − -%/2 3

�/%

(equation. 7)

2 Land-limiting condition – irrigation depth for maximising the net financial return

Wl = (b2 - Pcb1)/2Pcc1

(equation. 29) � = ./4 + 5�/2 + 5%/2 − 5�/� − 5%/%

2�-�/� + -%/% − -�/2 − -%/2� 3 (equation. 11)

3 Land-limiting condition – irrigation depth for maximising the yield per unit area

Wm = - )�%��

(equation. 21)

x = - )�%��

(equation. 10)

1Definition of the elements of the English (1990) equations:

a1 = intercept of the yield function

a2 = intercept of the cost function

b1 = coefficient of w in the yield function

b2 = coefficient of w in the cost function

c1 = coefficient of w2 in the yield function

Pc = crop price ($/t) 2Definition of the elements of the newly derived equations:

a1 = intercept of the primary yield (grain yield) function

a2 = intercept of the secondary yield (straw yield) function

b1 = coefficient of w in the primary yield function

b2 = coefficient of w in the secondary yield function

c1 = coefficient of w2 in the primary yield function

p1 = price of unit grain (or primary yield)

p2 = price of unit straw (or secondary yield)

p3 = price of 1 (one) ha-cm water (including the application cost)

p4 = cost of unit harvest (including transportation and threshing-cleaning) ($/t)

d1 = cost of the irrigation-independent items per unit area ($/ha)

T1 = Irrigation at the vegetative stage (25–30 days after sowing, DAS) T2 = Irrigation at the flowering stage (45–50 DAS) T3 = Irrigation at the vegetative stage (25–30 DAS) and flowering stage (45–50 DAS) where DAS = days after sowing The seed and straw yield and the input water of the mustard are presented in Table 2.

RESULTS AND DISCUSSION Comparison of newly derived equations with the English (1990) equations A comparison of the derived equations for the different conditions with those of English (1990) is given in Table 3. Due to the inclusion of a secondary yield (straw in this case) and a different form of cost function, two derivations

Treatments Seed yield

(t/ha)

Straw yield

(t/ha)

Irrigation + rainfall

(cm)

T0 0.510 1.129 8.88

T1 0.848 1.853 10.28

T2 0.876 2.059 10.68

T3 0.961 2.094 11.58

Ali et al 229 Table 4. Comparison of the ‘water depth’ for maximising the yield or economic return under different conditions in wheat

Situation Condition for irrigation depth

Water depth (cm) calculated using-

Corresponding to the experimental treatment/

Observation*

(IR + Re, cm)

From the yield function

(IR + Re, cm) The derived formula

English (1990)

Land-limiting condition

Maximising the yield per unit area

26.27 26.27 28.5 26.25

Maximising the net profit

20.54 18.5 - -

Water-limiting condition

Maximising the net profit

16.5 25.47 - -

* Table 7, Ali et al. 2007

Note: IR = Irrigation, Re = effective rainfall

[No. 1 and No. 2, i.e., equation (7) and equation (10)] are different from those of English (1990). In the case of yield maximisation (per unit area) based on the irrigation depth under the land-limiting condition, the derivations are similar (No. 3).

If the components/coefficients of the secondary yield (or straw) are omitted from equation (7) (i.e., p2 = 0), it reduces to

� = ��� ������� "��� "�� ���� "��� " #�/% ……………… (13)

which contains additional components from that of English (1990), such as the cost of irrigation-independent items (d1) and the unit cost of harvest (p4) (including transportation and threshing-cleaning, if any), which are functions of the irrigation. These components are logical from a practical point of view. Some costs are compulsory (land preparation, seed, and other factors) whether we apply irrigation or not and whether we vary the frequency of the irrigation or not. Similarly, based on the irrigation application (or the frequency of irrigation), the harvestable yield will eventually increase and, hence, the cost of harvest, transportation and threshing-cleaning will increase to some extent.

In the case of ‘irrigation depth for maximising net financial return’ under land-limiting conditions, the newly derived equation contains the component ‘price of unit water (p3)’ (but the English (1990) equation does not), which is a logical demand in the net return function.

Application of the equations in the wheat yield The calculated irrigation depths under different situations using the newly derived and English (1990) equations are summarised in Table 4. In the case of the land-limiting condition, the equations are similar for ‘maximum yield per unit area’, and thus, the resultant irrigation depths ( d = 26.27 cm) are similar. The depth is very close to the experimental observation (28.5 cm) and also to that from the yield function (26.25 cm). For a maximum net profit, the outputs from the equations are also almost equal (20.54 and 18.5 cm).

In the case of the water-limiting condition, for a maximum net profit, the newly derived equation yielded a logical irrigation depth (16.5 cm), which is less than the yield-maximising level of the ‘land-limiting condition’ (26.27 cm). However, the English (1990) equation yielded a much higher value (25.47 cm). As mentioned earlier, the English (1990) equation does not contain the ‘value of water’ or the ‘coefficient of the cost function’ but only an intercept of the cost function [‘a2’ in equation. Sl.(1)], which indicates the cost of the irrigation-independent items. Application of equations in mustard yield The calculated irrigation depths under different situations using the derived and English (1990) equations are summarised in Table 5. In the case of a land-limiting

230. Glo. Adv. Res. J. Agric. Sci. Table 5. Comparison of the ‘water depth’ for maximising the yield or economic return under different conditions in mustard

Situation Condition for irrigation depth

Water depth (cm) calculated using -

Corresponding to the experimental treatment/ observation*

(IR + Re, cm)

From the yield function

(IR + Re, cm) The derived formula

English (1990)

Land-limiting condition

Maximising the yield per unit area

11.86 11.86 11.58 11.90

Maximising the net profit

11.76 11.12 - -

Water-limiting condition

Maximising the net profit

11.40 1.96 - -

* Table 2, Ali et al., 2013

Note: IR = Irrigation, Re = effective rainfall

condition, the equations are similar for ‘maximum yield per unit area’ and, thereby, the resultant irrigation depths ( d = 11.86 cm). The depth is very close to the experimental observation (11.58 cm) and also to that from the yield function (11.90 cm). For a maximum net profit, the outputs from the equations are also almost equal (11.76 and 11.12 cm).

In the case of a water-limiting condition, for a maximum net profit, the newly derived equation yielded a logical irrigation depth (11.4 cm), which is less than the yield-maximising level of the ‘land-limiting condition’ (11.86 cm). However, the English (1990) equation yielded an un-realistic value (1.96 cm). For both crops, the English (1990) equation (for maximising the net profit) yielded an un-realistic output. It should be mentioned here that, in the yield function and also in comparing the calculated water depth, the total of the irrigation and effective rainfall (which can be called the ‘added water’) is presented. This presentation is logical and appropriate because the ‘added’ water ultimately contributes to the final yield (the graph of the R

2 value of

the added-water versus the yield is higher than that of the only irrigation-water versus the yield). For determining the irrigation need, the effective rainfall should be subtracted from the calculated water depth. Discussion and implications Water plays a vital role in modern agriculture. The optimum use of land and water resources for maximising the productivity and/or profit is essential. A step-by-step detailed methodology is presented in this paper to determine the irrigation amount (depth) for a specific crop to maximise the income and/or yield under land- and water-limiting conditions. Because the production and cost

functions depend on a number of local factors, a specific form of the production and cost function might not exist in real-world situations; hence, deviations in the optimum levels could occur. Thus, following the procedure described herein (under the prevailing cost elements and expected price of the product), individuals and farms can optimise the farm income, and the decision makers can make an exact prescription for the amount of water to apply. The managers can also evaluate and compare alternative crop(s) and optimise the use of the available irrigation water to maximise the farm income. The procedure will help to find a utilisation strategy for the available water resources under varying conditions of the cost of the production elements and the prices of the products (both primary and secondary products). This approach will help to make decisions under both land- and water-limiting conditions. In addition, it can be used to evaluate any cropping pattern (and be applied individually for each crop), to maximise the farm return with the available water resources, by using the following expression:

6789 =:;<=

<>�× 6<

where PTCj = the total profit for the whole year for a specific cropping pattern (j) Ai = the area (ha) occupied by a specific crop in season (i) with the optimal irrigation depth, under the cropping pattern (j) Pi = the profit ($/ha) for the specific crop for season (i), under the cropping pattern (j) n = the number of crop seasons per year j = the cropping pattern, j = 1 to N The cropping pattern that produces the maximum profit with the available water should be chosen.

SUMMARY AND CONCLUSIONS This paper is concerned with yield and economic optimisation of irrigation water. A general economic optimisation procedure is presented for analysing the irrigation depth under both land- and water-limiting conditions. Optimisation here means maximising the net benefit/profit that can be achieved with the available irrigation water / water-resources (in water-limiting conditions) and with available land resources (in land-limiting conditions). The procedure is based on the maxima and minima theory of mathematics and the yield and cost functions. A step-by-step detailed methodology is presented to find the optimum solution. The applicability of this approach for estimating the irrigation depth that is required to maximise the net economic return (under the water-limiting condition) and to maximise the yield (under the land-limiting condition) is presented and verified for winter wheat and mustard, which are cultivated in a humid, sub-tropic environment.

The methodology that is presented here is a precise technique for choosing the irrigation strategy on a farm to maximise the profit. The methodology can be used to analyse the impact of changes in the irrigation water price, other inputs and the output prices. This technique can also be applied in larger areas such as basins or regional plans to analyse the water usage and other factors. REFERENCES Ali MH (2008). Deficit irrigation for wheat cultivation under limited water

supply condition. PhD thesis, Dept. of Irrigation and water management, Bangladesh Agricultural University, p.183.

Ali MH (2010). Economics in Irrigation Management and Project Evaluation. In: Fundamentals of Irrigation and On-farm Water Management, Volume 1. Springer, NY, USA, pp.511-555.

Ali MH, Abustan I (2013). Irrigation management strategies for winter wheat using AquaCrop model. J. of Natural Resour. and Dev. 03: 106-113.

Ali MH, Abustan I, Rahman MA, Haque AAM (2011). Sustainability of Groundwater Resources in the North-Eastern Region of Bangladesh. Water Resour. Manage. 26:623–641.

Ali MH, Hoque MR, Hassan AA, Khair MA (2007). Effects of deficit irrigation on wheat yield, water productivity and economic return. Agric. Water Manage. 92: 151- 161.

Ali et al 231 Ali MH, Sarkar AA, Rahman MA (2012). Analysis on groundwater-table

declination and quest for sustainable water use in the North-western region (Barind area) of Bangladesh. J. Agril. Sci. and Appli. 1(1):26-32.

Ali MH, Sarkar AA, Zaman MH, Rahman MA (2013). Impact of irrigation schedules on seed yield, water use and water productivity of Mustard mutants. Bangladesh J. Nuclear Agric.

Bernardo DJ, Whittlesey NK, Saxton KE, Bassett DL (1988). Irrigation optimization under limited water supply. Trans. ASAE 31(3): 712-719.

BINA (Bangladesh Institute of Nuclear Agriculture) (2009). Irrigation scheduling of Mustard cultivars at different locations. In: Annual Report for 2007-08 (Agricultural Engineering Division), Bangladesh Institute of Nuclear Agriculture, Mymensingh 2202, Bangladesh.

Chawla JK, Khepar SD, Sondhi SK, Yadav AK (2010). Assessment of long-term groundwater behavior in Punjab, India. Water Int. 35(1): 63-77.

de Juan JA, Tarjuelo JM, Valiente M, Garcia P (1996). Model for optimal cropping patterns within the farm based on crop water production functions and irrigation uniformity. I: Development of a decision model. Agric. Water Manage. 31: 115-143.

English M (1990). Deficit irrigation. I. Analytical framework. J. Am. Soc. Civil Eng. 108(IR2): 91-106.

Gorantiwar SD, Smout IK (2005). Multilevel approach for optimizing land and water resources and irrigation deliveries for tertiary units in large irrigation schemes. II: Application. J. Irr. and Drain. Engg. 131(3): 264-272.

Hancock H (1917). Theory of maxima and minima. The Athenxum Press (GINN and Company), USA, pp.1-8.

Klocke NL, Stone LR, Clark GA, Dumler TJ, Briggeman S (2006). Water allocation model for limited irrigation. Applied Engg. and Agric. 22(3): 381-389.

Martin DL, van Brocklin J, Wilmes G (1989). Operating rules for deficit irrigation management. Trans. ASAE 32(4): 1204-1215.

Nay-Htoon B, Phong NT, Schlüter S, Janaiah A (2013). A water productive and economically profitable paddy rice production method to adapt water scarcity in the Vu Gia-Thu Bon river basin, Vietnam. J. Natural Resour. and Dev. 03: 58-65.

Panda SN, Khepar SD, Kaushal MP (1996). Interseasonal irrigation system planning for waterlogged sodic soils. J. Irri. and Drain. Engg. 122(3): 135-144.

Reca J, Roldan J, Alcaide M, Lopez R, Camacho E (2001). Optimisation model for water allocation in deficit irrigation systems. I. Description of the model. Agric. Water Manage. 48: 103-116.

Sarkar AA, Ali MH (2009). Water-table dynamics of Dhaka city and its long-term trend analysis using the “MAKESENS” model. Water Int. 34(3): 373-382.

232. Glo. Adv. Res. J. Agric. Sci. APPENDEIX-1 The condition for maximising the yield per unit area under the water-limiting condition Let y be the irrigation depth for maximising the yield per unit area (Y t/ha) under the water-limiting condition. Then, from V ha-m of available water, the total irrigable area can be expressed as follows: A (ha) = (V ha-m)/ (x cm)

= 100 �� ha ..................................................... (1.1)

The total grain yield

YT = (a1 + b1x + c1 x2)× 100 �

� = (100a1V/x) + 100b1V + (100c1Vx) ...................(1.2) ��� �'7� = -(200a1V/x

2) + 0 + 100c1V

Equating the first derivative of YT to zero, we obtain x = @%���� ........................(1.3)

Next, to examine where the YT will be maximum or minimum for the above value of x, we must find the second derivative of YT. If the second derivative is negative, then YT will be maximum for the above value of x.

����� �'7� = %������&

The value of the second derivative will be negative for negative values of a1 (As ‘V’ and ‘x’ cannot be negative). However, in reality, the value of ‘a1’ (the intercept of the grain yield curve) cannot be negative. Thus, the irrigation depth for the maximum yield under the water-limiting condition does not exist.