*Fasa University, Water Science and Engineering Dept., Daneshjou blvd., Fasa, Fars Province, I.R. of Iran.*

*Shiraz University, Irrigation Dept., Shiraz, Badjgah, Fars Province, I.R. of Iran.*

*Shiraz University, Drought Research Center, Shiraz, Badjgah, Fars Province, I.R. of Iran.*

^{3}
/ha for 0, 60 and 120 mm rainfalls, respectively. The corresponding values for N were 301.1, 299.5 and 295.5 kg/ha, respectively. Optimum amounts of irrigation water and N decreased by increase in rainfall amount.

_{f}(total net income from all irrigated area);

_{l}(net income per unit area);

_{l}(optimum level of N fertilizer under limited land conditions);

_{m}(amount of applied N which results in maximum yield);

_{r}(soil residual mineral N);

_{w}(optimum level of N fertilizer under limited water conditions);

_{c}(crop price);

_{c16}(base price of sugar beet);

_{n}(N fertilizer cost);

_{w}(water price);

_{l}(optimum level of water under limited land conditions);

_{m}(applied water which results in maximum yield);

_{w}(optimum level of water under limited water conditions).

Funding agencies/Institutions |

Shiraz University Research Council, Iran |

Drought National Research Center, Iran |

Center of Excellence for On-Farm Water Management, Shiraz University, Iran |

Iranian National Science Foundation (INSF) |

Application of extra nitrogen (N) fertilizers and irrigation water result in water resources pollution (

Irrigation water and N fertilizers affect the sugar content of sugar beet (

sugar concentration decreased by increasing N.

Optimization problems can be solved by means of mathematical programming or heuristic methods. Mathematical solutions to the problems are more understandable, analyzable and result in exact answers. However, heuristic methods do not necessarily lead to exact answers, and researchers should be consent to obtain the answers with an acceptable accuracy (

In this study, mathematical formulas were derived to determine the optimum amounts of applied water and N at variable crop prices and rainfall conditions for sugar beet under land and water limiting conditions. This theory was applied for sugar beet data obtained in Alborz Province of Iran (

The mathematical formulation was presented by

where A is the irrigated area (ha), W is the applied water (m
^{3}
/ha), R is the rainfall (m
^{3}
/ha), N is the N application rate (kg/ha), Nr is the soil residual mineral N (kg/ha), y (W+R, N+Nr) is the crop yield (kg/ha), c (W, N) is the production costs (Rls
^{1}
/ha), P
_{c}
(W+R, N+Nr) is the crop price (Rls/kg), i
_{l}
(W+R, N+Nr) is the net income per unit area (Rls/ha), and I
_{f}
(W+R, N+Nr) is the total net income (Rls) from all irrigated area. Contrary to W and N that are management parameters and can be selected, R and Nr are not easily available under field conditions; however, R can be predicted. Although the values of R and Nr affect the crop yield and price, they do not influence the crop production costs.

The irrigated area (A) can be a function of used water and is not dependent on N fertilizer because of no limitation for N use. The irrigated area can be determined as follows:

where W
_{T}
is the total available water supply (m
^{3}
).

Under land limited conditions (when the arable land area is limited), the farmer put all available arable land under irrigation and cannot increase the land area. Therefore, net income per unit area gets a maximum value since the partial derivative of farm net income equation [

Under land limiting conditions, all available land is put under irrigation and A is not dependent on applied water and N. In this condition, the derivative of A is zero. Therefore, the optimum levels of water and N fertilizer under limited land conditions,
_{l}
and N
_{l}
can be determined as follows:

Therefore, the partial derivatives of

Therefore, under limited land conditions, optimum applied water and N can be determined by solving:

Under water limiting conditions, the farmer may put more land area under irrigation to use all the water supply. Therefore, when the applied water and N are limited and non-limited, respectively, the partial derivative of A with respect to W and N can be written as follows:

By substituting _{w}
and N
_{w}
, can be calculated as follows:

By simplifying the

and solving _{w}
) and nitrogen (N
_{w}
) under limited water conditions and variable crop price.

The applied water and N amounts which result in maximum yield,
_{m}
and N
_{m}
, can be determined by taking the partial derivative of the crop yield function with respect to W and N and set those equal to zero as:

Since crop yield and price equations are nonlinear equations as a function of W and N, a nonlinear system of equations should be used for solving

The data used in this investigation were obtained from
^{2}
) with randomized complete blocks arrangement with three replications. Irrigation treatments were the main plots and N fertilizer rates were subplots. Four levels of irrigation water (40%, 80%, 120% and 160% of evaporation from the surface of class A evaporation pan) and four of N fertilizer (urea) (0, 90, 180 and 270 kg N/ha) were applied. Irrigation frequency was the same for all treatments (every 7 days). Sum of the applied water and seasonal rainfall for different irrigation treatments were 9500, 11850, 14500 and 16450 m
^{3}
/ha, respectively. Seasonal rainfall was 207 m
^{3}
/ha. MSC2 cultivar of sugar beet was planted. Seeds were planted on rows with spacing between rows of 0.6 m and distance between seeds on rows of 0.2 m. After harvest, the sugar concentration was determined by standard procedures by the sugar refining factory. The soil residual mineralized nitrogen (NO
_{3}
and NH
_{4}
) of the root zone was considered as 255.15 kg/ha for two soil layers (5% of the soil total N (

Using the above-mentioned analysis to plan the deficit irrigation is contingent on the prediction of seasonal rainfall amount before the start of the growing season or on the amount of optimum applied water that should be determined by using occurrence probability analysis for a given rainfall (

Rainfall amounts during the growing season for 21 years in the study area are available (

where

Based on the sugar beet root yield, sugar content, sum of irrigation water and rainfall and sum of N application rate and residual N, reported by

In

kg/ha; W and R, m
^{3}
/ha; and SC, %.

As mentioned above, crop price (P
_{c}
) depends on root sugar concentration. Therefore, crop price (P
_{c}
) is a function of W+R and N+Nr and its value was determined as follows (

where P
_{c16}
is the base price of sugar beet (Rls/kg) with SC of 16% and its value was 290 Rls/kg at the time of carrying out the experiment. Values of 3 and 13 are a mean waste (%) and mean sugar concentration (%) in previous years, respectively, as suggested by the Iranian Agricultural Ministry (
_{w}
=120 Rls/m
^{3}
) and N fertilizer cost (P
_{n}
=913 Rls/kg) as follows:

Solving _{m}
) and nitrogen (N
_{m}
) that resulted in maximum yield was obtained as follows:

Under land limiting conditions, saved water compared to W
_{m}
cannot be used to irrigate extra land area. This amount of water maximized the benefit for each unit of land. By solving ^{3}
/ha, Nr=255.15 kg/ha, P
_{c16}
=290 Rls/kg, P
_{n}
=913 Rls/kg and P
_{w}
=120 Rls/m
^{3}
,

the calculated optimum water (W
_{l}
) and N (N
_{l}
) under land limiting conditions were 12174.2 m
^{3}
/ha and

262.6 kg/ha, respectively. For more assessments, W
_{l}
and N
_{l}
were calculated at different rainfalls, water costs, N cost, base crop prices and soil residual N. Results are shown in _{l}
decreased, whereas N
_{l}
did not vary (_{l}
. For constant N cost (P
_{n}
=913 Rls/kg) and base crop price (P
_{c16}
=290 Rls/kg), by increasing water cost (P
_{w}
), optimum amounts of water and N decreased and increased, respectively (_{w}
=120 Rls/m
^{3}
) and base crop price (P
_{c16}
=290 Rls/kg),

by increasing N cost (P
_{n}
), the optimum amount of N decreased; however, the optimum amount of water increased (_{l}
variation was very low by a change in base crop price (_{l}
was not affected by soil residual N content, and the value of N
_{l}
decreased by an increase in Nr (

Net income was calculated based on different irrigation water and N applications at different water and N costs and base crop price for constant rainfall and soil residual N. Relationships between net income and irrigation water at different water costs and base crop prices for R=207 m
^{3}
/ha and N+Nr=517 kg/ha are shown in _{w}
=0 (arrow in _{c16}
=290 Rls/kg and W+R=

12381 m
^{3}
/ha are shown in

For an optimum amount of water under water limiting conditions, the saved water with respect to W
_{m}
can be used to increase the A (more planting area). Therefore, net income gained from total A is increased. By solving ^{3}
/ha, Nr=255.15 kg/ha, P
_{c16}
=290 Rls/kg, P
_{n}
=913 Rls/kg and P
_{w}
=120 Rls/m
^{3}
, the calculated optimum water (W
_{w}
) and N (N
_{w}
) under water limiting conditions were 8410.6 m
^{3}
/ha and

300.7 kg/ha, respectively.

For more assessments, the values of W
_{w}
and N
_{w}
were calculated at different rainfall amounts, water costs, N costs, base crop prices and soil residual N contents. The results are shown in _{w}
increased by an increase in the N cost (_{w}
, with exception of base crop price (_{w}
decreased by increase in rainfall (

The net income under conditions of water limiting for seasonal rainfall of 20.7 mm and N+Nr=517 kg/ha at different water costs and base crop prices are shown in

Optimum water, N, grain yield, net income and land increase for Nr=255 kg/ha, P
_{w}
=120 Rls/m
^{3}
, P
_{n}
=

913 Rls/kg and P
_{c16}
=290 Rls/kg at different rainfall depths were calculated and presented in _{w}
. The maximum point was shifted to the left and optimum irrigation water decreased by an increase in the water cost and base crop price (arrows in _{c16}
=290 Rls/kg and W+R=12381 m
^{3}
/ha are shown in

When the crop price is fix (it does not change with crop quality), its value will be determined based on supply and demand system or government pricing. For fixed crop price, term
_{c}
). Anyway, the fixed crop price may be either higher than the variable crop price or lower than it. Therefore, the results of these two situations are similar to those obtained by changing the base crop price in

Nr=255 kg/ha, P
_{w}
=120 Rls/m and P
_{n}
=913 Rls/kg in case of the fixed crop price and equal to the base crop price (P
_{c}
= P
_{c16}
=290 Rls/kg) at different rainfall depths were calculated. In other words, crop price did not change with crop quality and it was assumed that the value of SC in

P
_{c}
= P
_{c16}
=290 Rls/kg. In these conditions, crop price was higher than the varied crop price due to root sugar concentration lower than 16% (mean sugar content is 13% in Iran;
_{c16}
, to higher crop price under constant crop price conditions resulted in higher W
_{l}
, W
_{w}
, N
_{w}
and total net income and lower N
_{l}
compared with the crop price under varied crop prices (

For planning deficit irrigation, the amounts of optimum applied water at different N levels were obtained for 21 years of seasonal rainfall values in the study area. A frequency analysis was applied for the amounts of optimum applied water at different N levels obtained in different years. The occurrence probability for the values of optimum applied water was estimated by the Weibull equation [_{w}
as 50% and 80% that is corresponding to 3001.9 and 3400.3 m
^{3}
/ha of water for N
_{w}
=0.0 kg/ha, 7135.6 and 7430.3 m
^{3}
/ha of water for N
_{w}
=100.0 kg/ha and 7996.0 and 8278.9 m
^{3}
/ha of water for N
_{w}
=200.0 kg/ha, respectively, could be used in irrigation water resources planning, which is in accordance with the probability of occurrence of 50% and 80% for seasonal rainfall.

In the current study, required equations for determining the optimum applied irrigation water depth and N fertilizer under full irrigation, limited land and water conditions for sugar beet were derived when the crop price is a function of the sum of irrigation water and seasonal rainfall and N fertilizer. Results showed that applied water and N to maximize yield decreased by an increase in rainfall and soil residual mineral N, respectively. The higher rainfall supplied, the higher portion of crop water requirements. These results are similar to those obtained by

Under land limiting conditions by an increase in rainfall and water price, optimum water decreased due to partly supplying of the crop water requirement by rainfall. Similar results were reported by
_{l}
as shown by
_{l}
was not affected by Nr. Similarly to the results obtained by
_{l}
increased by an increase in water price and a decrease in base crop price and N cost. Net income decreased by an increase in water and N costs due to the increase in production costs. Therefore, the maximum net income per unit area occurred at higher water and N levels in which yield was higher.

Under limited water conditions, by an increase in rainfall, soil residual N, water cost and base crop price, the value of optimum applied water decreased (
_{w}
increased by an increase in the N cost. With the exception of base crop price, the value of optimum N under water limiting conditions decreased by an increase in rainfall, soil residual N, water cost, and N cost. As discussed by
_{m}
/W
_{w}
. In this condition, as obtained in the current study, the net income is maximized at less applied water in comparison with that obtained at limited land conditions. Under limited water conditions, the net benefit per unit water is maximized, while under limited land conditions, the net benefit per unit land is maximized. According to

Optimum N application under limited land and limited water conditions was higher than its value under full irrigation due to the fact that N fertilizer application was not limited and maximum net income per unit area and water occurred at higher N level in which yield was higher (

For no rainfall in the growing season, deficit irrigation as 27% and 48% compared to full irrigation resulted in 6.4% and 25.4% decrease in yield and 21.4% and 96.2% increase in total net income under land and water limiting conditions, respectively. Under water limiting conditions, for given base crop price, water cost, N cost and soil residual N, cultivated land area increased by 93.7, 108 and 128% for 0, 60 and 120 mm rainfall, respectively.

For fixed crop price, derived equations in this study is similar to the derived equations by
_{l}
, W
_{w}
, N
_{w}
and total net income and decrease in N
_{l}
.

The economic-mathematical analysis presented here can be used for other regions and crops, e.g., sugarcane and other crops for which the crop price is dependent on yield quality. Yield and crop price functions used in the current study were empirical and should be determined for specific climates and cultivars.