One of the most important decisions that farmers face is what to produce. By land allocation, it is possible to determine the maximum income from the consumption of a certain number of production factors, or at least the amount of costs for a certain combination of products. The production and activity of agricultural products are always affected by many conditions and factors that are not under the control of farmers; therefore, these two indicators always change with changes in production conditions and affect the stability of farmers income (Di Falco et al., 2007). Land allocation is the division of land between different types of crops to increase the firms total profit. In addition, it shows which product is more beneficial than others and how acreage will be allocated to increase profits. Linear programming was used to determine land distribution to optimize different crop types. Different methods can be used to identify the best land allocation (Sofi et al., 2015). One of the main reasons for the increase in productivity of agricultural products is the correct selection and management of cropping patterns. The best allocation of available production resources to activities that are most useful for farmers and the selection of suitable crops for each region is one of the basic and important issues in agricultural management to prevent excessive consumption of production factors (Rasikh et al., 2024). 

In the current study, the necessary data about the cost of production of the products was collected by completing the 379 questionnaires in Qarghayi district of Laghman province. Wang et al. (2010) developed an uncertain interval multi-objective linear program-ming model to evaluate the optimal land use structure in Pi County, Sichuan Province. The findings of their analysis are as follows: 1. The optimal range for cropland area is identified between 27,976.75 hectares and 31,029.08 hectares; however, the current area falls below the lower threshold. 2. The optimal range for garden land area spans from 4,736.49 hectares to 12,967.11 hectares, with the existing area also being below the minimum limit. 3. For construction land, the optimal adjustment range is between 7,761.95 hectares and 10,393.18 hectares, while the current area exceeds the upper limit. 4. The optimal range for industry and mining land is set between 557.29 hectares and 693.54 hectares, and similarly, the current area surpasses this upper limit.

Besides, Singh and Panda in their 2012 study developed a linear programming model aimed at optimizing land and water resource allocation to maximize net annual returns from irrigated areas in Haryana State, India. The findings of the model indicated a strategic shift in cropping patterns, resulting in a decrease in the cultivation areas for rice, mustard, barley, and gram, while simultaneously increasing the acreage dedicated to cotton, sugarcane, wheat, millet, and sorghum. This optimal allocation of land and water resources led to an increase in groundwater utilization, effectively alleviating issues related to waterlogging and salinity within the region. Consequently, the net annual returns from the study area rose by approximately 26%. Furthermore, the sensitivity analysis conducted on the model parameters revealed that the market price of crops emerged as the most critical factor influencing outcomes, followed closely by crop area and cultivation costs.

Felix et al. (2013) demonstrated that a linear programming model, which incorporates a selection of farm activities, can effectively operate within a defined set of fixed farm constraints to maximize income while also addressing additional objectives such as food security. A comparative analysis of results obtained through linear programming versus traditional methods revealed a significant advantage in favor of the linear programming approach, with a notable difference in gross income of 44.65%. Furthermore, research conducted in India by Shreedhar et al. (2015) highlighted the efficacy of the linear programming method as a valuable tool for optimizing cropping patterns. This approach takes into account critical constraints related to water, labor, fertilizer, and seed consumption, ultimately aiming to enhance overall income. Joolaie et al. (2016) employed linear programming and goal programming methodologies to develop a management model for agricultural products in Mazandaran Province, Iran. The study involved the collection of 493 questionnaires from 19 districts within the province. The findings indicated that implementing linear and goal cropping patterns resulted in an increase in gross margin by 25% and 4%, respectively, compared to the existing cropping pattern in the region.

Ahmadzai et al. (2016) conducted a study aimed at maximizing farm income through the optimization of resource allocation and the development of an optimal farm plan for a cooperative farm in the Ahmad Abad district. A comprehensive field survey was undertaken to gather primary input-output data from 32 cooperative farmers. The findings indicated that an optimal allocation of 179.20 hectares for wheat, 11.80 hectares for maize, 9.97 hectares for rice, and 4.92 hectares for barley would significantly enhance productivity. In contrast, the farmers actual allocation was 72.0 hectares for wheat, 57.9 hectares for maize, 36.9 hectares for rice, and 39.1 hectares for barley. The implementation of the optimal plan resulted in a substantial increase in gross revenue, rising from 3,275,986 AFs under the current plan to 4,518,906 AFs in the optimized scenario - representing a remarkable 37.9% improvement. Additionally, this optimal plan facilitated a 13.5% reduction in labor time. The analysis revealed that land is the primary constraint in resource allocation, while factors such as capital, irrigation water, and tractors do not pose limitations to the production process. Ultimately, the study highlighted that wheat is the most profitable crop for farmers in the region. It is recommended that farmers allocate a greater proportion of arable land to wheat cultivation to enhance the sustainability and commercial viability of their farming operations.

Asaadi Mehrabani et al. (2018) conducted a comprehensive study on the implementation of fully fuzzy linear programming to address multi-objective optimization of cropping patterns and net benefits under uncertain conditions in the Zarrinehroud Basin. Their findings revealed a 2.53% increase in net benefits compared to traditional crisp optimization methods, and a remarkable 36.34% improvement relative to the existing cropping pattern. This enhancement was achieved by replacing low-income crops with higher-income alternatives. Furthermore, the research demonstrated that utilizing fully fuzzy linear programming, as opposed to crisp linear programming, resulted in a significant reduction in water consumption, amounting to 88.22%. The study also indicated that incorporating uncertainty bands of 10% and 20% for the optimization parameters led to additional water savings and increased net benefits from the optimized cropping pattern.

Zenis et al. (2018) conducted a study to determine the optimal land-use composition for maximizing agricultural income. Utilizing a linear programming model for their optimization analysis, the researchers identified the ideal distribution of land across various crops. Specifically, the recommended land allocation included 135.314 hectares for rice, 11.798 hectares for corn, 2.290 hectares for soybeans, and 2.818 hectares for peanuts. This strategic allocation is projected to yield a total farmer income of IDR 2,682,020,000,000 per year. The findings of this research provide valuable insights that can inform farmers decision-making processes regarding cropping patterns.

Nevertheless, Soltani and Khajehpour (2020) conducted an analysis of optimal cropping patterns in Afghanistan, emphasizing environmental sustainability. Their research focused on key crops- wheat, barley, sesame, cumin, and saffron-that collectively represent over 70% of the cultivated area in Herat province. Utilizing goal programming and linear programming models, the study aimed to identify cropping strategies that not only maximize gross margins but also minimize the reliance on chemical fertilizers and pesticides. The goal programming model specifically targeted the dual objectives of enhancing profitability while promoting environmental sustainability. The findings from the linear programming model revealed that, to achieve an optimal cropping pattern for the region, it would be beneficial to increase the cultivated areas of sesame, barley, and saffron, while reducing the areas dedicated to wheat and cumin. This strategic shift underscores the potential for improved agricultural practices that align economic viability with environmental stewardship in Afghanistans agricultural landscape.

In their 2020 research study, Haq et al. employed a Linear Programming (LP) model to assess the potential for maximizing profits from the three primary crops- wheat, maize, and alfalfa- in the Central Yasin District of Ghizer. This quantitative research aimed to compare the annual profit margins of these major crops and identify the most advantageous option among them. The findings revealed that, on average, the land allocated for the cultivation of wheat, maize, and alfalfa in the study area was 0.924 acres, 0.664 acres, and 0.75 acres, respectively. Notably, the results derived from the LP model indicated that alfalfa emerged as the optimal crop choice, outperforming both wheat and maize. The analysis concluded that cultivating alfalfa yields an average annual net profit of Rs. 195,634.49, highlighting its economic viability in the region.

Moulogianni, (2022) conducted a comprehensive comparison of three mathematical programming models aimed at promoting sustainable land and farm management practices. The study focused on a sample of 219 agricultural holdings that participated as beneficiaries in the Modernization of Agricultural Holdings initiative under the Rural Development Plan for the Central Macedonia region of Greece. Utilizing the crop plans associated with these farms, the research employed mathematical programming models to derive optimal solutions while addressing various, often conflicting, objectives. The analysis compared the outcomes of Linear Programming (LP), Positive Mathematical Programming (PMP), and Weighted Goal Programming (WGP) in relation to proposed changes in agricultural land use. To assess the sustainability of the farms, the study utilized a set of eleven indicators encompassing economic, social, and environmental dimensions. Each mathematical model demonstrated distinct advantages and limitations, highlighting their applicability in different contexts and decision-making scenarios.

Rasikh et al. (2024) employed a linear programming model to optimize cropping patterns, yielding significant results. The study revealed reductions in the consumption of chemical fertilizers, agricultural pesticides, irrigation water, and total area under cultivation by 4.33%, 3.04%, 2.09%, and 2.32%, respectively. Additionally, the gross margin increased by 4.78% under the optimized model. These findings suggest that optimizing cropping patterns not only enhances profitability for farmers in the province but also contributes to environmental sustainability by minimizing input consumption.

The statistical population of this research was the number of farmers in the Qarghayi district of Laghman Province. According to the report of Qarghayi District, in the last census conducted in the field of agriculture in 2023, the number of farmers are 28,000. The data required for this research were collected by filling 379 questionnaires in (21) villages of Qarghayi District using the linear programming model, which includes the objective, decision variables, and constraints. A linear programming problem with “n” decision variables and “m” constraints can be mathematically modeled as (Higle & Wallence 2003; Sofi et al., 2015).

Profit Maximization      

Subject to (s.t)

This can be written as,

Objective function:

The products produced in Qarghayi district are wheat, corn, barley, paddy, beans, onions, potatoes, eggplant, tomatoes, Cucumber, Carrots, Okra, Cabbage, lentils, pumpkins, garlic, watermelon, pepper, leek, cotton, cabbage, sugarcane, apricot, peach, orange, and lemon (eight crops included wheat, cucumber, tomato, potato, eggplant, okra, round zucchini, and leek). These are the major products of this district and are grown on more than 250 hectares.

In the present study, linear programming models with 11 constraints and 8 decision variables were developed for the Qarghayi district. Specific constraints included area under cultivation, types of fertilizers (DAP, Urea, FYM), types of poisons (herbicides, insecticides, and fungicides), seeds, labor, machinery, and capital. Table 1 list the indicators and variables, respectively, used in the model.

 

Table 1: Crops indices.

Source: Research findings

 

DAP, Urea and FYM are the three most commonly used fertilizers in the Qarghayi district of Laghman Province. In this study, to determine the technical coefficient of chemical fertilizer constraints, the amount of consumption of each of the above fertilizers for each product was checked. In relation (2) to (4), Pi, Ui, and FYMi showed the technical coefficients of DAP, Urea, and FYM fertilizers, respectively. DAPTotal, UTotal, and FYMTotal  showed the availability of DAP, Urea, and FYM fertilizers in the Qarghayi district, respectively.

DAP, Urea and FYM are the three most commonly used fertilizers in the Qarghayi district of Laghman Province. In this study, to determine the technical coefficient of chemical fertilizer constraints, the amount of consumption of each of the above fertilizers for each product was checked. In relation (2) to (4), Pi, Ui, and FYMi showed the technical coefficients of DAP, Urea, and FYM fertilizers, respectively. DAPTotal, UTotal, and FYMTotal  showed the availability of DAP, Urea, and FYM fertilizers in the Qarghayi district, respectively.

Herbicides, insecticides, and fungicides are the three most widely used poisons in the Qarghayi District. In this study, to determine the technical coefficient of the constraints of poisons, the amount of consumption of each poisons for each product was investigated. In equations (5)–(7), Hi, Ii, and Fi represent the technical coefficients of herbicides, insecticides, and fungicides, respectively.

In this study, to determine the technical coefficient of the seed constraint, the amount of consumption of each product was checked. In addition, to determine the right-hand side of this constraint, Equation (8) shows the seed limitation, where Si is the seed technical coefficient and   is the seed stock in this district.

In this study, to determine the technical coefficient of labor constraints, the number of man-days required for the four stages of agricultural operations (preparation, planting, planting-harvesting, and harvesting) was investigated for each product. The values of the technical coefficients were then calculated according to the agricultural calendar. In addition, to determine the right hand side of this constraint, the entire working population in the agricultural sector of Qarghayi District was considered. Equation 10 shows the monthly constraint of labor, where Li and Lj are the technical coefficient and right-hand side value of labor, respectively.

In this study, in order to determine the technical coefficient of machinery constraint, the cost of machinery services was analyzed by separating the four stages of agricultural operations (preparation, planting, planting-harvesting and harvesting) for each product. Considering that the cost of each hour of machinery was different according to the type of operation, in order to homogenize these costs, using the average price of one hour of machine operation in, these costs were converted into hourly units per hectare and using the agricultural calendar, the technical coefficient of this the constraint was calculated. Equation (11) shows the monthly constraint of machinery, where Mi and Mj are the technical coefficient and total working hours of machinery in each month, respectively.

To determine the technical coefficient of capital constraints, the annual variable cost of each crop in this district was considered. The value on the right hand side was calculated using calibration. Equation (12) shows the capital constraint, where Ci and CTotal are the technical coefficient and the capital stock, respectively.

In this study, to control the total cultivated area of crops within the framework of variable land, land constraints were also considered. Equation (13) shows the land constraint, where Xi and XTotal are the decision variables and total cultivated area of this district, respectively.

Study Area

Qarghayi District is the largest district in Laghman Province, and is located 30 Km from the provincial centre of Mihtarlam. It borders Mihtarlam District to the north, Alingar District to the northeast, Nangarhar Province to the south and Kabul Province to the west. The district center is the village of Lalkhanabad, located between the Kabul River and its tributary the Alingar River. Longitude of this district 70° 18 18" E and latitude is 34° 34 14".

Fig. 1: Geographical Location of Qarghayi District.

Sampling method

The number of farmers in this district is 28,000 according to information on the extension administration of the Qarghayi district, the number of farmers in this district. If the number of farmers (population) is known, Cochrans formula or Morgans table can be used to select the sample size or number of questionnaires. 

(379) questionnaires were filled by a simple random sampling method in (21) villages of this district.

Table 2, considering the type of product, average yield, and value of the product, total cost, and net profit are shown. Wheat has the highest and lowest profits.

Table 2: Gross Profit of Major Competitive Crops.

Objective function (Profit Maximization)

High consumption of DAP (Leek) and low consumption (wheat), high consumption of urea (Leek) and low consumption (wheat), herbicide overuse (Potatoes) and underuse (Round zucchini), excessive use of insecticide (eggplant) and less use (Leek), over-consumption (Cucumber) and under-consumption (Leek) of fungicide, high consumption of labor (Okra) and low consumption (wheat), high consumption of machinery (wheat) and low consumption (potatoes), and high consumption of capital (Leek) and low consumption (wheat), as shown in Table 3 below:

Table 3: Average Coefficient for Constraints.

 Source: Research findings

Fertilizer Constraints

Current cropping pattern

Fig. 2: Software Result for LP Lingo.11.

Source: Research findings

According to the table above, no land has been allocated for tomatoes, potatoes, eggplants, and round zucchini, that is, the linear model has already been deleted. (889.45) and (442.6) ha of land were added to Cucumber and Leek, respectively. Wheat and okra decreased (176) and (272.25) hectares, respectively.

 

Table 5: Changes in production factors by percentage.

Source: Research findings 

Table 5 Shows the additional consumption of limited resources in the models of current cropping patterns and linear cropping patterns. Using the best linear model, DAP was reduced by 0.23%, and herbicide and insecticide by 1.32% and 26.02%, respectively. Besides, machinery 5.68%, capital 1.03%, land 8.13% and seeds 49.76% reduce compared to the current cropping pattern. (In the current cropping model, more production factors are consumed than the percentage mentioned above). The current research model was estimated using Lingo 19 software, and the results are shown in Table 3. In the spring season, the Qarghayi district, Wheat, and Cucumber had the largest cultivated areas of 3700 and 700 ha, respectively. In 2023, the total cultivated area of the eight products under investigation in this district was approximately 6350 hectares and the annual profit was 629.4 million Afghanis. With a decrease of 516.19 hectares of the total cultivated area of the crops, the amount of Gross profit has increased by 56.5 million Afghanis, which indicates a change in the gross profit (9%), therefore, without the interference of external factors in this district. It is possible to increase gross margin by only allocating land.

Increasing farmers incomes and improving their living conditions are always among the priorities of planners and policymakers in the agricultural sector. The linear programming method determines the optimal combination of products by selecting products with the highest gross margin. The results of the current research showed that if the best agricultural model is implemented in the Qarghayi district, the gross margin of the crop can be increased by about 56.5 million Afghanis. The results of the linear model, aimed at maximizing gross margin, showed that in the optimal cropping pattern of the region, the cultivated area of Leek and Cucumber should be increased, and the cultivated area of Wheat and Okra should be decreased. This study suggests a suitable policy for the Agricultural Directorate of Laghman province, so steps should be taken to implement it.

The dynamic nature of parameters such as production cost, transportation cost, material cost, and selling price can pose a challenge in maintaining the relevance and accuracy of the Liner programming model for cropping pattern design over time. 

1. In the present study, obtained profit maximiz-ation for spring season crops; therefore, in future research, the fall season crops should be considered.
2. This research is based on linear programming, and other methods will be considered in future research.
3. This study has only profit maximization; therefore, future studies should consider environmental conditions in addition to the economic aspect.

The paper has five authors and their contribution to the paper is described below: The first author analysed the data and wrote a complete report, the second and third authors reviewed the paper and provided there valu-able feedbacks, the remaining authors fill ques-tionnaires from respondents and collected data for the paper.

We would like to thank the directorate of agriculture in Laghman province for their constant support in providing the data, we also appreciate the extension administration as well for providing the secondary data and overall supported our work.

The datasets generated during the current study are available from the corresponding author upon reasonable request.

Conflict of Interest the authors declare no competing interests.