Coffee is one of the most important agricultural commodities traded internationally, the main arabica and robusta type grain producers are: Brazil, Vietnam, Indonesia, Colombia and Ethiopia. Major variables affecting coffee production are planting density, planting age, variety, growing conditions and crop management, fertilization falls within the latter. According to ^{Sadeghian and González (2012}) fertilization aims to maintain or increase organic matter, soil nutrients and plant resistance to stress conditions such as pests, diseases and drought incidence.

There are several investigations related to the nutrients effect in coffee cultivation, some authors mention that nutrients demand varies with the crop developmental stage (^{Echeverría, 1994}). However, the recommended annual dose is 1 000 kg of a mixture of simple fertilizers for coffee plantations with a year average of 5000 kg ha^-1, regardless of soil analysis (^{Valencia, 1992}).

^{Ribeiro et al. (2003)} in the State of Bahía in Brazil have used nitrogen doses between 600 to 800 kg N ha^-1 yr^-1. Furthermore, ^{Sadeghian and González (2012)} in their research suggested the following amounts for a year: 300 kg ha^-1 of nitrogen (N), 260 kg ha^-1 potassium (K2O) and 50 kg ha^-1 phosphorus (P2O5), magnesium (MgO) and sulfur (S), for technified coffee cultivation with high density (7 500 and 10 000 plants ha^-1) when lacking soil analysis.

Obtaining maximum productivity with a reduction of nitrogen fertilizers is a big challenge (^{Bruno et al., 2011}). Figure 1 shows the high cost of the main nutritional sources used in coffee crops, thus it is of great interest for producers to make a more efficient application of this input while maintaining or increasing productivity.

Figure 1 Monthly cost of major fertilizers in U.S. dollars (USD). Fuente: ^IndexMundi. 

^{Sadeghian (2008)}, states that coffee producers currently face an increasingly competitive global market, it becomes a priority to review factors affecting the crop profitability, including fertilizer costs, whose continuous increases have generated concern in Colombia, since the contribution of fertilization for total production costs has risen from 10 to 20% in the past two years.

The aim of this work is to find the optimal fertilizer dose to minimize costs for this item in coffee production, providing a feasible recommendation to the producers of the Gigante municipality, in the Huila department, Colombia.

Gigante is located in the Huila department at 2 23' 12.01" north latitude and 75 32' 45.96" west longitude. According to the 2012 agriculture statistical yearbook of the Huila department, the Gigante municipality had planted coffee on 5 294 hectares, of which 2 396.9 are in renewal (pruning and planting), and 2 897.1 hectares are in production stage. The latter showed a production of 3 166.8 tons of dry parchment coffee this year.

In order to address the problem of cost minimization, linear programming will be used given its strength in complex problem modelling (^{Alvarado, 2010}), and the possibility for users to solve large-scale models using computer programs based on the simplex method.

Some researchers such as Zambrano et al. (2005) used linear programming in substrate mixtures formulation and found that this method allows up to a 93% reduction in the number of mixtures for analysis compared to the conventionally used method (trial and error).

The cost minimization problem due to fertilizer use, appears in the expression (1).

(1)

Subject to

Ax_i ≥ B

x_i ≥ 0

Where:

Z= Represents fertilizer costs to be minimized;

C_i= Cost coefficients for the 20 nutritional sources used (Table 1);

x_i = nutritional sources (Table 1);

A= Matrix of technical coefficients, which represents the nutritional intake of nitrogen, phosphorus, potassium, calcium, sulfur and magnesium of the 20 nutritional sources;

B=Nutritional requirements established by ^{Sadeghian and González 2012}).

Simple and compound fertilizers used in linear programming are available in Table 1. It is important to mention that the prices of these sources are in U.S. dollars (USD) and were consulted on April 30^th, 2013 in Huila Coffee Growers Central Cooperative Ltd.

Table 1 Sources of simple and compound fertilizers. 

C= fertilizantes simples; S= fertilizantes compuestos; † (N= nitrógeno; P= fosforo; K= potasio; Ca= calcio; S= azufre; Mg= magnesio).

The linear programming model proposed in (1) was solved using the Lindo 6.1 software, the results obtained are summarized in Table 2, by multiplying the quantities of fertilizers obtained (Table 2 column 4) by the price of each food source (Table 1, column 4), the cost/hectare/ year obtained is $ 885.52 USD for fertilization of coffee crops.

Table 2 Quantities of sources used in the fertilization of coffee crops. 

The amounts of nutritional sources minimizing fertilization costs in coffee crops are 8.67 bags of potassium chloride, 2.22 of ammonium phosphate, 0.78 of ammonium sulfate, 11.82 of urea and 6.25 of magnesium sulfate (Table 2, column 4).

In Table 3, an economic comparison is made using the sources and amounts selected by the Lindo 6.1 software (optimal) with two fertilizers commonly used for coffee crops in Colombia.

Table 3 Nutritional sources commonly used in coffee crops. 

1=nitrógeno; 2= fósforo; 3= potasio; 4= azufre; 5= magnesio.

Considering one hectare (1 ha) as reference, a period of 4 years production (productive period after the first harvest), and assuming constant price for nutritional sources in this period, the annual savings were estimated in $ 329.58 USD as the cost difference between Hydranova fertilization and the optimal amounts obtained (Table 3).

Considering the four years coffee production period. The current flow of annual savings is obtained using the equation (2) with an annual interest rate of3.96% set by the Bank of the Republic of Colombia the total savings are $ 1 197.47 USD.

(2)

Where:

VP= Present value;

A= Annual flow of savings ($ 329.58 USD);

i= Annual interest (3.96%);

n= Number of years (4).

Although currently there are more sophisticated tools for optimization, linear programming remains an important technique in solving optimization problems in agriculture. The present study was focused specifically on the Gigante municipality in Colombia with 2 897.1 hectares of coffee production. Results showed a 38.52% reduction in fertilization costs by 2013, compared with the Hydranova compound fertilizer commonly used by coffee growers of this region in Colombia. Furthermore, applying the recommended doses obtained by the linear programming model, producers may obtain savings of $ 1 197.47 USD per hectare in 4 years and considering the coffee production hectares in the Gigante municipality, the economic benefit for the region is considerable.