Introduction

The main activities in Irrigation District 028 are irrigated agriculture and intensive livestock farming (^{Sánchez et al., 2016}), highlighting the dairy production chain (^{Sanchez and Terrones, 2015}). As the production of fodder crops is an indispensable factor in the development of livestock, in the District 97% of its surface is assigned to the sowing of these (^{CONAGUA, 2016}; ^{SIAP, 2018}), historically supplied through the network of channels from the Tulancingo river and the volume of concession for extraction for agricultural use.

As is known, the water resources that are used beyond their capacity for renewal modify the equilibrium conditions for their long-term sustainability and necessary supply (^{Hanasaki et al., 2010}), this being the case of the Tulancingo Valley aquifer (key 1317), within whose official limits the district under study is located, since the average annual volume of groundwater available for the year 2013 presented a deficit of 6 842.2 thousand m³ per year, extracted from non-renewable storage, which is why 87% of its surface is subject to the provisions of the decree establishing a closure for the illumination of groundwater (^{DOF, 2013}).

For the year 2018, the balance sheet update showed a considerable increase, now of 20 117.7 thousand m³ of deficit (^{DOF, 2018}). Given the above, the planning and programming of the crop pattern established year after year in the district, considers as volume available for irrigation only that which comes from bypass and from the La Esperanza dam, both driven by gravity.

Finally, in order to seek greater benefits for producers in the region, given the resources available, a decrease in unit costs; however, seen from another perspective, the maximization of benefits from the optimization of resources, through linear programming with the Simplex method, has proven to be more efficient than other algorithms.

Thus, with the help of said method, this research aims to provide a proposal for a crop pattern to be established in the irrigation district 028 Tulancingo, in which the available resources are efficiently managed, the requirements of each crop are supplied , and the economic benefit that is currently had is increased.

Materials and methods

The irrigation district 028 Tulancingo is located in the southeast of the state of Hidalgo, between the extreme coordinates 98° 24” west longitude 20° 10” north latitude and 98° 20” west longitude 20° 3” north latitude (Figure 1), at a height of 2 150 meters above sea level. The predominant climate in the area according to the Köppen classification, modified by ^{Garcia (2004)}, is BS₁kw corresponding to semi-arid temperate dry with long cool summer, rains in summer and winter between 5 and 10% of the annual total, average temperature annual between 12 and 18 °C.

Figure 1 Location of the irrigation district 028 Tulancingo in the state of Hidalgo. 

Selection of crops and historical areas

The total surface area dominated is 987 ha, of which 980 are irrigable, distributed in two modules: Module 01 ‘Aguas Blancas’ with 408 ha and Module 02 ‘Aguas Negras’ with 572 ha. This information was contrasted with that reported by the agricultural statistics of the irrigation districts published by the National Water Commission from 1997 to 2016 (^{CONAGUA, 1997-2016}) and was complemented by the 2016-2017 and 2017-2018 cycles, registered with the district headquarters, to have a total of 21 complete agricultural cycles.

In the total agricultural cycles that were recorded in the district, perennial forage crops show dominance in planted area (87% on average), Figure 2. Of these, the crops to consider in the linear programming model (LPM) They were green lucerne (Medicago sativa L.), green clover (Trifolum repens L.) and associated forages, in force for the last 10 years. Associated forages are: Ovillo (Dactylis glomerata L.), perennial ryegrass (Lolium perenne L.), red clover (Trifolum pratense L.) and fescue (Festuca arundinacea L.). In addition to spring summer crops: green forage oats (Avena sativa L.), green forage maize (Zea mays L.) and other grasses, such as annual ryegrass (Lolium multiflorum L.) and wheat (Triticum aestivum L.). The information regarding yield, rural average price, cost and net benefit per crop, presented in Table 1, corresponds to the 2017-2018 cycle and shows the maximum and minimum historical area that has been planted (^{CONAGUA, 1997-2016}).

Figure 2 Percentage historical distribution of the area allocated to each agricultural sub-cycle in Irrigation District 028. Elaboration, with agricultural statistics of the irrigation districts (^{CONAGUA, 1997-2016}) and information provided by the district. 

Table 1 Limits of historically planted area and value of agricultural production in Irrigation District 028 Tulancingo. 

Crop Spring-Summer	Surface (ha)		Yield (t ha-1)	Rural average price ($ t-1)	Production value ($ ha-1)	Cost ($ ha-1)	Net profit ($ ha-1)
	maximum	minimal
Green forage oats	21	6	22.5	510	11 475	10229.7	1 245.3
Green fodder corn	68	2	42.3	500	21 150	9 792.7	11 357.3
Other grasses Perennes	25	0	5.3	6 500	34 450	27043.8	7 406.2
Green lucerne	170	0	101	335	33 835	24652.9	9 182.1
Associated forages	602	0	100	350	35 000	27268.6	7 731.4
Green clover	596	0	95.5	331.1	31 621.5	24118	7 503.5

$= Mexican pesos (MXN), to the year 2017. Source: own elaboration, with the agricultural statistics of the irrigation districts (^{CONAGUA, 1997-2016}) and information provided by the district.

Results and discussion

As a first result, the proposed LPM for the availability of the 2017-2018 cycle was obtained, in order to compare with the cultivation pattern that was actually established for that period in the district. Both results are presented in Table 3, area and net benefit, by crop and total; the incorporation of 26 ha of green clover and increases in total area (15 ha) and net profit ($150.64 thousand). From the results obtained, it follows that the model performs its function of optimizing resources well, by complying with the imposed restrictions and proposing a crop pattern that increases not only the area to be sown, but also the net profit. Similar to works such as that of ^{Osama et al. (2017)} and that of ^{Daghighi et al. (2017)} where the increase in BN that they achieve with their proposed crop pattern was 6.4 and 8% on average, respectively.

Subsequently, for each LPM that by scenario in combination with availability variant was elaborated, the patterns present in Figure 3 were obtained and in it is observed how the cultivation of associated forages is suggested to dominate a larger area in all cases, despite not being the one that provides the greatest net benefit. The foregoing is explained since in the model its water requirement was distributed throughout the year, as it is a perennial crop, not so with green fodder maize, which, being the most profitable, requires water at the time of greatest demand (from March to June).

Figure 3 Crop pattern obtained by LPM for the different scenarios, grouped in charts by availability. 

With the available volume of 13 866 thousand m³ (agricultural cycle 2017-2018), the areas for green forage maize and green alfalfa remain constant in the 4 scenarios, with 68 and 170 ha, respectively. The option of sowing ‘other grasses’ is suggested only in case of covering and the second crops are distributed in 14 and 2 ha for grain corn and other crops.

Given a volume with a probability of 50%, scenario 2 further distributed the areas between crops, increasing the area of green clover (null in SA) to 155 ha and that of other pastures (6.7 ha in SA) to 90 ha. The second crops suggest, in case of scenario 4, the planting of 14 ha only for grain corn.

With the volumes of 75 and 95% probability, the trend was similar to that of 50%, reducing all areas, with the exception of green forage oats, which presented maximum areas in both cases in scenario 2. In this case, water availability it affects not only in yield, but also in the feasible surface of being sown (^{Bredel et al., 2017}), therefore it is not surprising that when the volume offered (with a greater probability of exceedance) decreases, the maximum surfaces that the LPM suggest for each crop they were also reduced, a result that coincides with those obtained in investigations such as those of ^{Singh et al. (2001)}, ^{Ortega et al. (2009)} and ^{Mazengo et al. (2018)}, in the latter they found that with 10% more water available, the LPM proposes the sowing of an additional 133 ha.

Likewise, the actions proposed by scenario, in all cases, increased the feasible area to a greater or lesser extent. In Table 4 of total surfaces, it can be seen how with a lower availability and covering the conduction network, the surface recommended by the model is comparable to that in the current situation with higher availability (CRPexc95 = 746.1 ha vs SAPexc50 = 757.9 ha) and even higher in some cases, if you compare CRPexc75> SAPexc50, for example.

The response to total net profit was remarkably similar to that of surfaces. In this regard, the values presented in Table 5 show that: compared to the current situation, the profitability of scenarios 2, 3 and 4 was higher, with less water availability, the gain in a scenario with coating is comparable to the one with the highest volume of supply in the current situation and for the same scenario, the reduction in volume directly impacts the decrease in profit.

The aforementioned indicates that the LPM efficiently carried out the optimization, since the maximum benefit is not obtained with any pattern that suggests planting a larger area, but with an adequate distribution by crop, as occurred with the LPM proposed in the ^{Sofi et al. (2015)}; ^{Upadhyaya (2017)}, where with the help of the same method, they propose the modification to cultivation patterns in India.

In Figure 4, the differences in total surface area results and net profit of scenarios in relation to the current situation are exposed, in order to verify under which situation, the impact of the actions proposed by the scenarios will be greater. In the research carried out by ^{Al-Nassr (2018)}, where he determines the optimal crop and surface combination for the Rasheed/Hamorabi farm in Baghdad, the proposed crop rotation offers a net profit of 63.9% higher than his actual plan for the 2015-2016 cycle, in this case, although without significant changes in the crops to be considered, the addition of second crops presents an increase in net profit between 61 and 105.3 thousand pesos, with a variation in the area of between 4.7 and 5.6 ha, above the current situation.

Figure 4 Difference between total results by scenario in relation to the current situation. 

The effect of the coating (CR-SA) on the total surface is greater as availability is reduced, however, the maximum difference in net profit is reached with the availability of Pexc50 (812.2 thousand pesos). With both options, incorporating second crops and performing the conduction network coating (CRSC) have the highest differences, however, the greatest impact on the surface (126.7 ha) does not coincide with that of BN (907.2 thousand pesos) in a situation of availability: the first is for a Pexc95 and the second for a Pexc75.

Conclusions

The elaborated LPM efficiently optimized resources, by complying with the imposed restrictions and proposing a crop pattern that increases not only the area to be sown, but also the net profit. In them, when decreasing the volume offered (with a greater probability of exceedance), the maximum areas that the LPM suggest for each crop are also reduced. The proposed actions to cover the conduction network (scenario 2), incorporation of second crops (scenario 3) or both (scenario 4), in all cases increased to a greater or lesser extent the feasible area, as well as its net profit, compared to the current situation, even in cases where it had greater water availability.

An important recommendation, in favor of the improvement of the LPM elaborated, is the one to incorporate new restrictions that have to do with the human resource (jornales), machinery and available supplies; important aspects of production, with which you can further enrich the models.