In the present study, a linear optimization model has been formulated for maximizing the net annual return of the crops subjected to various land and water resources constraints. The management model has been used to find out the optimal land (cropping pattern) and water resources (groundwater and surface water) allocation policies. Mathematically, the objective function of land and water resources allocation model can be expressed as:

The net incomes of crops were calculated from the potential yield of crops, market price of crops and the cost of cultivation. The yield estimates for individual crops are based on the historical performance of the farm, yield potential of the crops, recorded yields of trial and demonstrations, improved cultural practices and efficient management. The current yields of crops were taken from the agronomy feasibility report of Kobo-Girana pressurized irrigation project (Kobo-Girana pressurized irrigation project crop agronomy feasibility unpublished report, 2007). The cost of cultivation of different crops was estimated mainly in consultation of the local farmers and Woreda and Zonal government institutions including agricultural and other pertinent offices. The cost of cultivation includes seed cost, fertilizer, plant protection, draft power and labor costs. The cost of land was not considered in cost of cultivation since it belongs to farmers own land. The administration costs, such as salary of experts, telephone service costs and so on were not considered as it is covered by the Kobo-Girana pressurized irrigation project and Wereda Agricultural Administration Offices. The unit costs of agricultural crops of the local area were taken from Central Statistical Agency of Ethiopia. Since there is no fixed water tariff rate for irrigation paid by the beneficiaries in the study area, the water tariff rate was considered from the other areas having similar water sources. Thus, the costs of canal water were considered as 3 Ethiopian Birr per 1000 m3 of water which is irrigation canal water tariff of nearby basin (Awash river basin) ([30] and the cost of groundwater were assumed 3.8 Ethiopian Birr per m3 which is the water tariff of Addis Ababa Town for non-domestic purposes. Summary of cost and net return of dry and wet season crops is shown in Table 1.

## 2.1 Model Constraints

Land availability constraints

Land allocated to various crops in different seasons must not exceed the total avialable land. This limit ratio can be expressed as:

$$\sum _{\text{j}=1}^{\text{n}}{\text{a}}_{\text{i}\text{j}}\le {\text{A}}_{\text{i}} \forall \text{i} \left(2\right)$$

Where Ai is the total caltivated command area in season i

Water requirement constraint

The gross irrigation requirement of all the crops grown in the irrigation command area in all the seasons must satisfy the available quantity of groundwater and diverted surface water resources. It can be expressed as:

$$\sum _{j=1}^{n}{a}_{ij}*\left({GIR}_{ij}\right)*10\le \left({E}_{gw}{\left(Gw\right)}_{i}+{E}_{sw}{\left(Sw\right)}_{i}\right) \forall i \left(3\right)$$

Where \({GIR}_{ij}\) = growth irrigation requirement of crop j grown in season i (mm); \({a}_{ij}\) = area

allocated to crop j grown in season i (ha); \({E}_{sw}\)=conveyance efficiency of surface water ; \({E}_{gw}\) = conveyence efficiency of groundwater; \({\left(SW\right)}_{i}\) = allocation of surface water in season i (m3); and \({\left(Gw\right)}_{i}\) = allocation of groundwater in season i (m3). In this study, conveyance efficiency of surface water and groundwater were asummed as 98.2 % and 100 %, respectively (Belay, 2012).

The allocation of groundwater and surface water must not exceed their seasonal

avialablity

$${\left(GW\right)}_{i}\le \sum _{\xi =1}^{43}\sum _{k=1}^{6}{\left({Q}_{gw}\right)}_{\xi ,k}*{\left({\varDelta t}_{p}\right)}_{k}*{5.4*10}^{4} for i=1 \left(4\right)$$

$${\left(Gw\right)}_{i}\le \sum _{\xi =1}^{43}\sum _{k=1}^{5}{\left({Q}_{gw}\right)}_{\xi ,k}*{\left({\varDelta t}_{p}\right)}_{k}*{5.4*10}^{4} for i=2 \left(5\right)$$

$${{\left(Sw\right)}_{i}\le \sum _{k=1}^{5}{\left({Q}_{SD}\right)}_{i ,k}}_{ }* {\left({\varDelta t}_{d}\right)}_{k}*8.64*{10}^{4} for i=1 \left(6\right)$$

$${\left(SW\right)}_{i}\le \sum _{k=6}^{12}{\left({Q}_{SD}\right)}_{i ,k} * {\left({\varDelta t}_{d}\right)}_{k}*8.64*{10}^{4} for i=2 \left(7\right)$$

Where \({\left({\text{Q}}_{\text{g}\text{w}}\right)}_{{\xi },\text{k}}\) is groundwater pumping from well \({\xi }\) in the time period k (m3/s), \({\left({\text{Q}}_{\text{S}\text{D}}\right)}_{\text{l}, \text{k}}\) is

diversion from stream reach \(\text{l}\) in time period k (m3/s), \({\left({\varDelta t}_{p}\right)}_{k}\)is pumping time in the kth time period (for example, number of days of groundwater pumping in the kth month), \({\left({\varDelta t}_{d}\right)}_{k}\) is diversion time in the kth time period (for example, number of days of stream diversion in the kth month) and i is the number of growing seasons in a year. In the present study area, the crop growing seasons is 2. The first crop growing season (dry season) is from December to May and the second crop growing season (wet season) is from July to November. All the wells assumed to be operated for 15 hours per day and the stream flow diversion for 24 hour per day. The maximum seasonal availability limit of groundwater and surface water resources was obtained from the conjunctive water use simulation-optimization model [31].

Minimum and maximum area constraints

There could be some limitations in the area under a particular crop depending upon the past practices, food requirment, land suitablity, water avialablity and agricultural income. It can be expressed as lower and upper bounds.

$${\text{A}}_{\text{j}}^{\text{l}}\le \sum _{\text{i}=1}^{2}{\text{a}}_{\text{i}\text{j}}\le {\text{A}}_{\text{j}}^{\text{u}} \text{j}=1, 2, 3\dots \dots \dots \dots . 14 \left(8\right)$$

Where \({\text{A}}_{\text{j}}^{\text{l}}\)is the minimum area under the jth crop, and \({A}_{j}^{u}\) is the maximum area under the jth crop.

The maximum and minimum areas were set on the consideration of farmer’s knowledge on cultural practices of crops and basic food requirements of the local farmers. Based on these criteria, the maximum area was fixed as 50% more than the target area, and the minimum area was set as one quarter of the target area.