# Optimization Model for Agricultural Reclaimed Water Allocation Using Mixed-Integer Nonlinear Programming

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Problem Definition and Objective

## 4. Mathematical Formulation of the Optimization Model

#### 4.1. Objective

^{3}); and ${\mathrm{RWC}}_{\mathrm{i}}$ is the cost of RW type i ($/m

^{3}).

#### 4.2. Decision Variables

- (1)
- ${\mathrm{FA}}_{\mathrm{i},\mathrm{x},\mathrm{c}}$: assigned area of farm x to cultivate crop c using RW type i (ha)
- (2)
- ${\mathrm{RW}}_{\mathrm{i},\mathrm{x},\mathrm{c}}$: assigned RW of type i to farm x farming crop c (m
^{3}) - (3)
- ${\mathrm{N}}_{\mathrm{x},\mathrm{i}}$: defines the connectivity of RW type i to farm x (binary variable)
- (4)
- ${\mathrm{M}}_{\mathrm{x},\mathrm{c}}$: defines the connectivity of crop c to farm x (binary variable)

#### 4.3. Constraints

#### 4.3.1. Reclaimed Water Availability Constraints

^{3}) discharged from the WWTP.

^{3}/ha) of each crop c times the cultivated area ${\mathrm{FA}}_{\mathrm{i},\mathrm{x},\mathrm{c}}$ (ha), which is:

^{3}/ha) considering each cultivated crop c is computed as:

^{3}/ha.

#### 4.3.2. Irrigated Farmland Constraints

#### 4.3.3. Connectivity Constraints

#### 4.3.4. Minimum Allowed Net Benefit by the Farm x Constraint

## 5. Baghdad as a Case Study

_{A}) (tertiary treated wastewater), and reclaimed water type B (RW

_{B}) (secondary treated wastewater) are to be allocated on a total of 84 farms with a total area of 5300 ha to the south of Baghdad allowing up to four crops to be cultivated in each farm. Each cultivated farm is based on actual land ownership and is therefore of different land areas starting from a minimum area of 17.5 ha up to a maximum area of 193 ha.

## 6. Data Input for the Model

_{A}, and group B crops are to be irrigated using RW

_{B}. RW

_{A}will be tertiary treated water with both filtration and disinfection to reduce both pathogens and suspended solids. RW

_{B}will be secondary treated water that includes basic disinfection and this water cannot be used on root crops including potatoes and onions. To limit the cultivated area of each crop to ensure a variety in production, the maximum area to be cultivated by each crop is listed in Table 1.

## 7. Results and Discussion

^{5}$ lower than the computed value using ANTIGONE. ANTIGONE was 11.6 times faster than BARON for solving the same optimization model. For example, BARON took about 186 s to solve the problem to find the optimal solution after 109 iterations by exploring 109 nodes. Meanwhile, ANTIGONE took only 17 s to solve the same problem, exploring only one node. Other models such as BONMIN, COUENNE, and DICOPT were also evaluated solving the same MINLP optimization problem, but all these solvers resulted in infeasible solutions.

_{A}and RW

_{B}on the proposed 84 farms for different irrigation efficiencies and different quantities of water are presented in Figure 2 and Figure 3. Results showed that the net benefit of using RW

_{A}and RW

_{B}increases with the increase of the amount of reclaimed water used. The use of 6.0 MCM of RW

_{A}with a 45% irrigation efficiency (IE) has a net benefit of 2.21 × 10

^{6}$ from the cultivation of approximately 384 ha of tomatoes. For the use of 6.0 MCM of RW

_{A}with 85% IE, the model predicts a net benefit of 4.55 × 10

^{6}$ while cultivating a total of 701.2 ha comprised of 500 ha of tomatoes and 201.2 ha of potatoes. The model demonstrates that the use of higher irrigation efficiencies, which means more water availability due to advanced irrigation techniques, can produce a higher net benefit and greater crop diversity. On using the same 6.0 MCM of RW

_{A}with irrigation efficiencies of 55, 65, 75, and 85%, the net benefit increases by 30.7, 57.3, 81.7, and 106.1%, respectively, as compared to the results for a 45% IE. Small increases in irrigation efficiency are clearly beneficial. The use of 6.0 MCM of RW

_{A}with 65% IE has a net benefit increase of 20.4% as compared to a 55% IE, and the 75% IE has a net benefit increase of 15.5% higher as compared to a 65% IE. Finally, the use of 85% IE has a net benefit increase of about 13.4% as compared to a 75% IE. The increase in net benefit will decrease as higher IEs are achieved.

_{B}was 4.46 × 10

^{6}$ with 20.0 MCM of RW

_{B}with an 85% IE while cultivating 2031 ha with 10 different types of crops. As illustrated in Figure 3, optimizing the use of RW

_{B}results in lower net benefit values in comparison to RW

_{A}(Figure 2), due to the difference in the crops allowed to be cultivated using both RW types.

_{B}has followed a different trend to that observed with RW

_{A}. Using 6.0 MCM of RW

_{B}with a 45% IE produces a net benefit of 2.33 × 10

^{6}$. In contrast, the use of 6.0 MCM of RW

_{B}with 55, 65, 75, and 85% IEs results in increases of about 29.1, 46.9, 58.7, and 69.8%, respectively, in comparison to a 45% IE. The increase in net benefit decreases as the quantity of RW

_{B}used increases, and the same is true for the increases in IEs. On using 12.0 MCM of RW

_{B}with 55, 65, 75, and 85% IEs, the net benefit increases by 12.3, 18.9, 24.5, and 30.1%, respectively, as compared to a 45% IE which has a net benefit of about 3.4 × 10

^{6}$. The decreases in the ratio of the net benefit with higher irrigation efficiencies is due to the increase in the practically employed amount of water which tends to irrigate the maximum allowed area of the most economic crops first and later finds crops of lower economic value. The most economic crops identified by the water allocation optimization model using RW

_{B}are tomatoes, eggplant, cucumber, okra, and clover.

_{B}with 45% IE has a computed net benefit of 2.33 × 10

^{6}$, which is higher than the net benefit computed using RW

_{A}, cultivating the same area of 384 ha of tomatoes. RW

_{B}has shown a significant advantage over RW

_{A}when both are used to cultivate the same types of crops on the same areas as with the cultivation of tomatoes using of 6.0 MCM of RW

_{B}with 45 and 55% IE and using 7.0 and 8.0 MCM of RW

_{B}with 45% IE. The advantage of RW

_{B}over RW

_{A}is because the cultivation cost and the selling price of the cultivated crops are the same, although RW

_{B}is less expensive than RW

_{A}.

_{A}had a peak value of 7.6 × 10

^{6}$ when 15.0 MCM of RW

_{A}has been used with 85% IE, as illustrated in Figure 2. Thereafter, the maximized net benefit declined with an increase in the quantity of water used because the model reached the maximum area for the highest economic value crops (Table 1), such as tomatoes, while lower economic value crops are cultivated until crops with negative economic value, such as clover, are the only crops available for cultivation. Optimizing the use of higher water availabilities with RW

_{B}results in a similar decline in the net benefit with higher irrigation efficiencies, as illustrated in Figure 3, due to the previously mentioned reason.

_{A}are presented in Figure 4. Increasing the quantities of RW

_{A}used results in a commensurate increase in the cultivated area. Using 6.0 MCM of RW

_{A}with 45, 55, 65, 75, and 85% IEs results in irrigated areas of 384.8, 470.3, 549.5, 625.3, and 701.2 ha, respectively. The model satisfies the maximum allowed area of the most economic crop then it starts cultivating the crop with the next higher economic value and so on. Therefore, tomatoes were selected first by the model to be cultivated using RW

_{A}followed by potatoes, onion, eggplant, cucumber, and okra. For instance, using 6.0 MCM of RW

_{A}with 45% IE, the model selected tomatoes to be cultivated first and when the quantity of RW

_{A}reached 8.0 MCM with 45% IE, the model cultivated 500 ha of tomatoes, then 11.6 ha of potatoes, which is the second most economical crop in the system.

_{B}with different irrigation efficiencies are presented in Figure 5. The results show that the increase in the reclaimed water quantities used, the served area will increase accordingly depending on the evapotranspiration of the crops cultivated. The model predicts the maximum net benefit by cultivating the optimum area using a variety of crops as a function of the available quantity of water. Using 10.0 MCM of RW

_{B}with 85% IE results in the cultivation of the maximum allowable hectares of tomatoes, eggplant, cucumber, and okra followed by the cultivation of 131.7 ha of clover (Table 1). Meanwhile, using 11.0 MCM of RW

_{B}with 85% IE results in the cultivation of the maximum allowable area of tomatoes, eggplant, and cucumber, followed by 176.3 ha of clover, 93.5 ha of sesame, and 9.3 ha of alfalfa. Instead of cultivating only 209.3 ha of clover, the model maximizes the net benefit by including sesame and alfalfa which provide a similar net benefit to clover (Figure 5). The same trend was predicted by the model using from 13.0 MCM to 19.0 MCM of RW

_{B}with 85% IE. One of the features of the model is to allow for cultivating as many crops as possible which satisfy the maximum net benefit. In addition, the minimum allowed area of crops to be cultivated may be adjusted based on specific conditions to provide constraints in the model consistent with supply and demand.

_{A}and RW

_{B}is presented in Figure 6 and Figure 7, respectively. With an increase in irrigation efficiency using a specific quantity of water, the computed net benefit per cultivated hectare of crops increased until a limit was reached. The factors that limit the net benefit are the increase in the cultivated area along with the requirement to grow more lower economic value crops. For instance, using 6.0 MCM of RW

_{A}with 45% IE predicted a net benefit of about 5732 $/ha when only tomatoes are cultivated on 384 ha. Meanwhile, the model predicted a net benefit of 6483 $/ha when it cultivated 500 ha of tomatoes, and 201 ha of potatoes using 6.0 MCM of RW

_{A}with 85% IE. In contrast, the model results experienced a significant decline in the predicted net benefit per hectare with the increase in irrigation efficiencies using higher quantities of water due to the increase in the cultivated area, and the decrease of the total maximized net benefit computed from the cultivation of crops with a lower net benefit. Using 20.0 MCM of RW

_{A}with 45% IE has predicted a net benefit of about 4734 $/ha while cultivating 500 ha of tomatoes, 500 ha of potatoes, 15 ha of onion, and 19 ha of eggplant. A net benefit of 3737 $/ha was predicted by cultivating 500 ha of tomatoes, 500 ha of potatoes, 200 ha of eggplant, 150 ha of onion, 150 ha of cucumber, 100 of okra, and 419 ha of clover using 20.0 MCM of RW

_{A}with 85% IE (Figure 6). The net benefit per hectare using different availabilities of RW

_{B}with different irrigation efficiencies, as illustrated in Figure 7, decreases with the increase in the quantities of RW

_{B}with the increase in IEs due to the same reasons mentioned under the use of RW

_{A}.

_{A}with 45, 65, and 85% IEs are presented in Figure 8, Figure 9 and Figure 10, respectively. There are 14 different types of crops available for cultivation using RW

_{A}as listed in group A in Table 1. Each crop has its own evapotranspiration value, selling price, production cost, and yield per hectare. Starting with 6.0 MCM with 45% IE, the model predicted cultivation of 384 ha of tomatoes. Tomato is the crop which satisfied the highest net benefit per hectare as compared to the other competitive crops in Table 1. All of the 84 cultivated farms of the system have the opportunity to cultivate tomatoes depending on the ratio of their areas to the total observed area of farms. Increasing the quantity of RW

_{A}and/or increasing the irrigation efficiency, increases the quantity of water which is allocated on farms cultivating more crops. With 45% IE using different RW

_{A}availabilities, tomatoes, potatoes, onion, and eggplant have been cultivated, respectively, starting from the highest economic value crop then next highest and so on, as illustrated in Figure 8. Increasing the irrigation efficiencies using a certain quantity of reclaimed water provides the opportunity to cultivate more crops after cultivating the maximum allowed area for each crop. For example, at 65% IE the model predicts the cultivation of up to 8 crops (Figure 9). Meanwhile, with 85% IE using certain availabilities of RW

_{A}, the model has predicted the cultivation of up to seven different crops when 20.0 MCM of RW

_{A}was used (Figure 10).

_{B}with 45, 65, and 85% IEs are illustrated in Figure 11, Figure 12 and Figure 13, respectively. The use of RW

_{B}has followed the same trends observed with RW

_{A}by cultivating the highest economic value crop then the next highest and so on while selecting from the 12 crops listed in group B in Table 1. This began with irrigating only 384 ha of tomatoes using 6.0 MCM of RW

_{B}with 45% IE, later reaching the irrigation of 500 ha of tomatoes, 150 ha of eggplant, 200 ha of cucumber, 100 ha of okra, 177 ha of clover, and 1.6 ha of sesame by using 20.0 MCM of RW

_{B}, as illustrated in Figure 11. Figure 12 and Figure 13 illustrate the cultivated crops using different RW

_{B}availabilities with 65% and 85% IEs, respectively. Even though the optimization model allows up to 4 crops to be cultivated simultaneously on the same farm, results showed that most of the farms cultivated up to 2 crops depending on the RW availability and the IE implemented.

## 8. Summary and Conclusions

_{A}results in a higher net benefit as compared to RW

_{B}. With lower quantities of available water, only the most economic crops are grown with both RW

_{A}and RW

_{B}, while the cost of RW

_{B}is less than that of RW

_{A}. For instance, using 6.0 MCM of RW

_{B}with 45% IE has a predicted a net benefit of 2.33 × 10

^{6}$, which is higher than the net benefit of 2.21 × 10

^{6}$ using RW

_{A}while cultivating the same area of 384 ha of tomatoes.

_{A}. Using reclaimed water for irrigation will help in reducing the potential negative environmental impacts of wastewater discharge while increasing the potential uses of RW for agriculture. Since most of Iraq’s built or under construction WWTPs are located in or adjacent to agricultural lands, it is logical and efficient to invest in using their secondary or tertiary treated wastewater for agricultural irrigation to enhance the economy of farmers and the environment while providing a diverse range of crops.

## Author Contributions

## Funding

## Acknowledgment

## Conflicts of Interest

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**Figure 1.**Locations of wastewater treatment plants in Iraq [41].

**Figure 2.**Total net benefit (million $) predicted using reclaimed water type A (RW

_{A}) with five different irrigation efficiencies (IEs).

**Figure 3.**Total net benefit (million $) predicted using reclaimed water type B (RW

_{B}) with five different irrigation efficiencies (IEs).

**Figure 4.**Total cultivated area (ha) predicted using reclaimed water type A (RW

_{A}) with five different irrigation efficiencies (IEs).

**Figure 5.**Total cultivated area (ha) predicted using reclaimed water type B (RW

_{B}) with five different irrigation efficiencies (IEs).

**Figure 6.**Net benefit per hectare ($/ha) predicted using reclaimed water type A (RW

_{A}) with five different irrigation efficiencies (IEs).

**Figure 7.**Net benefit per hectare ($/ha) predicted using reclaimed water type B (RW

_{B}) with five different irrigation efficiencies (IEs).

**Figure 10.**Predicted area (ha) of crops irrigated using reclaimed water type A (RW

_{A}) with 85% IE.

**Figure 11.**Predicted area (ha) of crops irrigated using reclaimed water type B (RW

_{B}) with 45% IE.

**Figure 12.**Predicted area (ha) of crops irrigated using reclaimed water type B (RW

_{B}) with 65% IE.

**Figure 13.**Predicted area (ha) of crops irrigated using reclaimed water type B (RW

_{B}) with 85% IE.

**Table 1.**Maximum allowed areas (ha) to be cultivated by certain types of crops irrigated using two reclaimed water (RW) qualities.

Cotton | Wheat | Maize | Potato | Tomato | Barley | Clover | Cucumber | Alfalfa | Onion | Eggplant | Sunflower | Sesame | Okra | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Group A | 1000 | 1000 | 1000 | 500 | 500 | 1000 | 750 | 200 | 750 | 150 | 150 | 750 | 250 | 100 |

Group B | 1000 | 1000 | 1000 | 0 | 500 | 1000 | 750 | 200 | 750 | 0 | 150 | 750 | 250 | 100 |

Crop | Cotton | Wheat | Maize | Potato | Tomato | Barley | Clover | Cucumber | Alfalfa | Onion | Eggplant | Sunflower | Sesame | Okra |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Cost ($/ha) | 1200 | 820 | 900 | 750 | 1300 | 720 | 320 | 1350 | 500 | 580 | 1250 | 550 | 475 | 1230 |

Crop | Cotton | Wheat | Maize | Potato | Tomato | Barley | Clover | Cucumber | Alfalfa | Onion | Eggplant | Sunflower | Sesame | Okra |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Yield (tons/ha) | 2.0 | 2.6 | 2.26 | 15.7 | 19.0 | 1.2 | 16.25 | 9.2 | 22.4 | 7.9 | 23.0 | 1.32 | 1.0 | 7.8 |

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## Share and Cite

**MDPI and ACS Style**

Aljanabi, A.A.; Mays, L.W.; Fox, P.
Optimization Model for Agricultural Reclaimed Water Allocation Using Mixed-Integer Nonlinear Programming. *Water* **2018**, *10*, 1291.
https://doi.org/10.3390/w10101291

**AMA Style**

Aljanabi AA, Mays LW, Fox P.
Optimization Model for Agricultural Reclaimed Water Allocation Using Mixed-Integer Nonlinear Programming. *Water*. 2018; 10(10):1291.
https://doi.org/10.3390/w10101291

**Chicago/Turabian Style**

Aljanabi, Ahmed A., Larry W. Mays, and Peter Fox.
2018. "Optimization Model for Agricultural Reclaimed Water Allocation Using Mixed-Integer Nonlinear Programming" *Water* 10, no. 10: 1291.
https://doi.org/10.3390/w10101291