Introduction

Agriculture in Mexico, as in most developing economies, is an important source of jobs and income. Thirteen-point three percent of the population participates in primary activities, who represent 7.1 million economically active people (^{INEGI, 2018}). In 2019 in the agricultural sector, 20 664 554.08 ha were allocated for agricultural production activities, 68% of this area was distributed in production units with supply of irrigation water 4 175 356.45 ha and rainfed 9 833 626.38 ha (^{SIAP, 2019}). As for the value of production, the main crop is grain corn, which generates a value of $106 245 747.07 pesos, in an area of 7 157 586.88 ha.

In contrast to the value of all horticultural production, which generates an income of $89 269 654.01 pesos, in an area of 524 984.14 ha, horticultural production represents 7.3% of the cultivated area and generates 84% of the value of production, with respect to the cultivation of grain corn (^{SIAP, 2019}). Horticultural production under protected agriculture represents 0.5% of all national production, and has a production value of $26 852 081.78 pesos, which means 6% of the national total (^{SIAP, 2019}). The states of Sinaloa, San Luis Potosí, Jalisco, Sonora and Coahuila represent more than 50% of the value of total protected agriculture production. Puebla ranks within the top ten states that contribute the most to the total value of production in this same area, with a participation in its production of $906 857.92 pesos and that represents 4.6% (^{SIAP, 2019}).

Among the main horticultural products under protected agriculture with respect to the value of production, tomato, onion, husk tomato and zucchini stand out; together they add up to 46.7% of the total value of the production of the state of Puebla (^{SIAP, 2019}). In 2010, the horticultural area sown in Puebla was 51 243.8 ha, of which a production of 662 873.88 t was obtained, with a total value of $2 014 126.32 pesos, for 2017 the area increased to 62 159 ha and 1 073 449.62 t were produced, with a total value of $4 157 965.34 pesos (^{SIAP, 2019}).

In the country there are 4 069 938 production units, of which 69% are made up of five or less ha, which contribute 39% of production, also generate 63% of jobs in the agricultural sector considering family labor, as well as hired labor (^{INEGI, 2018}). Credit in the agricultural sector has suffered variations over the years, in 1994 there was a portfolio of $ 154 340.3 million pesos, which fell until 2006 reaching only $17 503.5 million pesos and having an increase to $20 425 million in 2009 and a stabilization that has been maintained since 2009 (^{Reyes and Reyes, 2018}).

At the national level, only 6% of producers have access to institutional credit, about 70% of rural economic production units are of subsistence and self-consumption (^{CONEVAL, 2018}), as access to tools to formulate projects is limited, it makes it impossible for the producer to access sufficient resources and preferential rates that allow them to make the leap to competitive production units (^{Fundar, 2014}). Currently, the prevailing methodology for the formulation and evaluation of projects, established by ECLAC; in the ‘manual on economic development projects’, it contemplates the criteria of engineering or technical studies, investment of budget of income, expenses, financing and organization of the company (^{Pérez, 2000}).

Farmers must make complex production and marketing decisions throughout the year, what crops to produce, how much land to rent, labor to employ and the optimal time to do it, production can be organized with the use of LP (^{Kaiser and Messer, 2011}). ^{Gittinger (1982}); ^{Alvarado (2009}); ^{Render et al. (2012}) mention that LP is a mathematical method that allows analyzing and choosing the best way to distribute limited resources to maximize the profits of the company.

The objective of the present research was to optimize the combination of horticultural crops to improve the financial indicators used in the evaluation and formulation of investment projects, maximizing the net income of production, under a scenario of temporality in land use, compared to the traditional methodology. As a hypothesis, it was established that it is possible to determine in one hectare of soil the succession of horticultural crops that maximizes the net benefits, optimizing the results that would be obtained with the traditional methodology for the formulation of investment projects.

Materials and methods

The research was conducted with information on the state of Puebla. It is in the central region of Mexico, has an area of 34 251 km², composed of 217 municipalities and 6 500 localities. Of the total, 6 200 are rural with a total area of 3 391 900 ha, of these 1 048 999 ha are for agricultural use (898 875 rainfed ha and 150 124 irrigated ha) (^{INEGI, 2019}). In particular, information on the region of the Tecamachalco Valley was used, because producers are gradually adopting horticultural crops of high sowing density, production and value, information on the production cycles 2018-2019, to make a five-year projection.

In 2018, the cultivated area was 18 819.9 ha, with a production of 441 267.58 t and a production value of $1 942 453.74 pesos, compared to the rest of the municipalities with a cultivated area of 40 902.93 ha (^{SIAP, 2019}). A LP model in the form of a network flow, proposed by ^{Bazaraa et al. (2010}), was used. One hectare was established as the land area with potential to be sown, and it was proposed to sow 5 crops: onion (ce), tomato (ji), cucumber (pe), bell pepper (mo) and zucchini (ca).

A planning period of 10 agricultural cycles (spring-summer (ss 01) and autumn-winter (aw 02)) was considered, represented as t= 1, 2, 3,…10. Crop succession needs depend only on the types of crops that are sown and are given in the form of sequences, so that two crops cannot be produced on the same land or in the same cycle. The model was built as follows:

1).

Where:i= horticultural crop, onion (ce), tomato (ji), cucumber (pe), bell pepper (mo) and zucchini (ca); t-1= succession of crops in cycles; TI_i ^t-1= total income from the production of crop i in cycle t-1 (ss, aw); x_i ^t-1= variable of land use decision (1 ha) of crop i, in cycle t-1 (ss, aw); c_i ^t-1, la_i ^t-1 and wa_i ^t-1= costs of production, labor and water extraction (variable costs); N=165: it represents the number of crop income activities and N=150: it represents the number of crop cost activities.

Subject to:

2), restriction of maximum limit of availability of land to be sown for the preceding crop “j” at the beginning of the project in cycle

3), restriction of land requirement for crop i

at the end of the period in cycle

4), restriction of limit that represents the way in which the land available for preceding crop j

at the beginning of cycle t-1is distributed to the successor crop i

in cycle

5).

Restriction of non-negativity

Transfer restrictions provide the means by which the product obtained in one activity can be transferred within the model to other activities.

∑i=1Ncit-1*xit-1≤1

6), it transfers the costs (c) of production of crop i

in cycle t-1for the use of land for cultivation.

∑i=1Nlait-1*xit-1≤1

7), it transfers the labor (la) costs of crop i

in cycle t-1for the use of land for cultivation.

 ∑i=1Nwait-1*xit-1≤1

8), it transfers the costs of kW for water (wa) extraction to crop i

in cycle t-1for the use of land for cultivation. The distribution of available land between the cycles can be seen in (Table 1).

Table 1 Horticultural production schedule for the ss and aw cycles. 

Month/activities	ce01	ji01	mo01	pe01	ca01	ce02	ji02	mo02	pe02	ca02
December	0	0	0	0	1⸋	0	0	0	0	0
January	1	0	0	0	1	0	0	0	0	0
February	1	0	0	0	1	0	0	0	0	0
March	1	0	0	0	1⸙	0	0	0	0	0
April	1	0	1	0	0	0	0	0	0	0
Mayo	1	1	1	1	0	0	0	0	0	0
June	1	1	1	1	0	0	0	0	0	0
July	1§	1	1	1	0	0	0	0	0	0
August	0	1	1¥	1	0	0	0	0	0	1
September	0	1	0	1⸶	0	1	0	0	0	1
October	0	1¶	0	0	0	1	0	0	1◊◊	1⸙⸙
November	0	0	0	0	0	1	1	1	1	0
December	0	0	0	0	0	1	1	1	1	0
January	0	0	0	0	0	1	1	1	1	0
February	0	0	0	0	0	1	1	1	1	0
March	0	0	0	0	0	1§§	1	1¥¥	1⸶⸶	0
April	0	0	0	0	0	0	1¶¶	0	0	0

Information from INIFAP’s technology development and transfer centers. ce= onion, ji= tomato; mo= bell pepper; pe= cucumber; ca= zucchini; 01= period 1 ss cycle, 02= period 2 aw cycle, ^§1= end of production of ce01 July 4; ^¶1= end of production of ji01 October 12; ^¥1= end of production of mo01 August 23; ^⸶1= end of production of pe01 September 27; ^⸋1= start of production December 16 and ^⸙1= end of production March 15 of ca01; ^§§1= end of production of ce02 March 4; ^¶¶1= end of production of ji02 April 14; ^¥¥1= end of production of mo02 March 25; ^◊◊1= start of production October 16 and ^⸶⸶1= end of production March 15 of pe02; ^⸙⸙1= end of production of ca02 October 29.

To avoid overlaps between crops, a time space of 15 days was established to give continuity to the production of a second crop. In the case of available agricultural labor, the information from the report on the National Survey of Agricultural Day Laborers (^{SEDESOL, 2011}) was taken as the basis. The minimum wage of $123.22 pesos per day for workers in Mexico, excluding the northern zone bordering the United States of America, was considered as the salary of workers (^{CONASAMI, 2020}). To determine the coefficient of water costs, the cost of extraction by pumping was calculated, for that the power required by the equipment was obtained, in watts (W):

Pe=Q*H*γηe 

9).

Where:

(ηe)

is the technical information of the various pump manufacturers in the country, the efficiency has a value of 82%, (^{CONUEE, 2011}).

In the Tecamachalco Valley, the wells have an average instantaneous expenditure of 23 L s^-1 (Q), a specific water weight of 9.81

N m-3 (γ)

and a total pumping equipment load of 91.4 m (H), (^{CRETEALC-Mexico, 2011}). As a result, the pumping equipment delivers a power (P_e) of 25 150 watts (w)= 25.15 kW. Table 2 shows the calculations of kWh (kilowatt-hour) necessary for the pumping of water required by crop and the cost of electrical energy, starting from:

kWh= kW*t

10).

Table 2 Water extraction costs based on the information on water requirement 2018. 

			Onion	Tomato	Bell pepper	Cucumber	Zucchini
Volume of water required (l)			7 600 000	5 908 000	4 432 000	6 000 000	3 650 000
Extraction time (h)			91.78	71.35	53.52	72.46	44.08
Kwh			8 310.27	6 460.14	4 846.20	6 560.74	3 991.11
RABT fee		$2.50	$20 813.00	$16 189.53	$12 156.29	$16 440.92	$10 019.4

Preparation with data from CFE, RABT (agricultural irrigation in low voltage) fee 2018.

The objective function consisted of 315 decision variables, 165 of sales and 150 of production costs. Two hundred two restrictions are included: 2 of land resource, 50 of alternation of crops in ten cycles, and 150 of resource transfer (50 of yields, 50 of number of daily wages and 50 of energy required for water extraction). The succession of possible crops to be sown, according to the production schedule, is expressed in a network flow model b_t-1. The b represents the amount of land allowed to produce in a previous cycle t-1.

The crops ce, ji, pe, mo and ca represent the nodes of supply and demand of land, 01 and 02 represent the production cycles ss and aw, respectively, the lines that connect the nodes represent the possible direction of production from the beginning to the end of the production horizon (Figure 1).

Figure 1 Graphical representation of the network flow for the proposed horticultural crops. Preparation based on the models of LP in network (^{Bazaraa et al., 2010}). 

Results

The cash flow projection of the sequencing of optimal crops of the obtained model was analyzed, as well as the financial indicators of the project. Below is the LP model of network flow, which was run with Solver^® Lindo^®.

Objective function MAX Z= 718447.60ca0111 + 1000645.18pe0111 + 1000645.18pe0112 + 1000645.18pe0113 + 1000645.18pe0114 + 1648876.11mo0111 + 1648876.11mo0112 + 1648876.11mo0113 + 1648876.11mo0114 + 1648876.11mo0115 + 1225449.92ji0111 + 1225449.92ji0112 + 1225449.92ji0113 + 1225449.92ji0114 + 1225449.92ji0115 + 729757.00ce0111 + 598803.56ca0211 + 598803.56ca0212 + 564685.51pe0211 + 564685.51pe0212 + 564685.51pe0213 + 564685.51pe0214 + 786622.69mo0211 + 786622.69mo0212 + 786622.69mo0213 + 786622.69mo0214 + 786622.69mo0215 + 1148579.52ji0211 + 1148579.52ji0212 + 1148579.52ji0213 + 1148579.52ji0214 + 1148579.52ji0215 + 63,307.42ce0211 + 631307.42ce0212 + 718447.60ca0121 + 1000645.18pe0121 + 1000645.18pe0122 + 1000645.18pe0123 + 1000645.18pe0124 + 1648876.11mo0121 + 1648876.11mo0122 + 1648876.11mo0123 + 1648876.11mo0124 + 1648876.11mo0125 + 1225449.92ji0121 + 1225449.92ji0122 + 1225449.92ji0123 + 1225449.92ji0124 + 1225449.92ji0125 + 729757.00ce0121 + 598803.56ca0221 + 598803.56ca0222 + 564685.51pe0221 + 564685.51pe0222 + 564685.51pe0223 + 564685.51pe0224 + 786622.69mo0221 + 786622.69mo0222 + 786622.69mo0223 + 786622.69mo0224 + 786622.69mo0225 + 1148579.52ji0221 + 1148579.52ji0222 + 1148579.52ji0223 + 1148579.52ji0224 + 1148579.52ji0225 + 63,307.42ce0221 + 631307.42ce0222 + … + 598803.56ca0251 + 598803.56ca0252 + 564685.51pe0251 + 564685.51pe0252 + 564685.51pe0253 + 564685.51pe0254 + 786622.69mo0251 + 786622.69mo0252 + 786622.69mo0253 + 786622.69mo0254 + 786622.69mo0255 + 1148579.52ji0251 + 1148579.52ji0252 + 1148579.52ji0253 + 1148579.52ji0254 + 1148579.52ji0255 + 63307.42ce0251 + 631307.42ce0252 - 254525.85C0111 - 273148.60C0112 - 607723.83C113 - 620681.02C0114 - 334164.54 C0115 - … - 254525.85C0151 - 273148.60C0152 - 607723.83C153 - 620681.02C0154 - 334164.54 C0155 - 123.22MO0112 - 123.22MO0113 - 123.22MO0114 - 123.22MO0115 - … - 123.22MO0251 - 123.22MO0152 - 123.22MO0153 - 123.22MO0154 - 123.22MO0155 - 2.5AG0111 - 2.5AG0112 - 2.5AG113 -2.5AG0114 - 2.5AG0115 - … - 2.5AG0251 - 2.5AG0152 - 2.5AG0153 - 2.5AG0154 - 2.5AG0155.

Conclusions

Based on the information from the area of the Tecamachalco Valley, Puebla, the LP of the network flow type allowed optimizing the crop pattern. Also formulating a feasible investment project with better results when comparing it with the traditional methodology. In a period of 10 agricultural cycles, it was possible to determine an optimal production dynamic that generated a total net income of $7 422 367 pesos, growing bell pepper in the s-s cycles and tomato in the a-w cycles.

Compared to a monoculture production, as traditionally established in the formulation and evaluation of projects for the crops selected in this study, profits of $6 254 668.49 and $6 339 146.18 pesos were calculated for tomato and bell pepper, respectively. Compared to the $7 422 637 pesos of the combination of tomato and bell pepper resulting from the LP model.