Many farms in India's agricultural sector are small or medium-sized, struggling to allocate their limited resources among the numerous crops chosen based on agro-climatic conditions. Exploring the potential for optimal resource distribution across varying farm sizes is crucial for boosting agricultural productivity. Population growth increases the demand for agricultural products, necessitating careful planning. One critical phase of agricultural production that significantly impacts farmer income is crop planning. Decisions related to crop distribution, crop mixing, and operational processes are essential for maximizing farm returns. In addition to cultivating various crops, farmers have various methods to choose from. Crop strategy involves several variables, including the prices of supplies such as seeds, labor, fertilizers, and the availability of land, manpower, and tractors. Many researchers have developed goal-programming models to increase crop production and net profit while minimizing land usage. These models consider strategies that account for factors such as seed prices, labor availability, fertilizer costs, and the efficient utilization of available land. Such approaches aim to provide farmers with actionable insights for optimizing their crop plans and achieving sustainable agricultural practices. Goal programming was firstly introduced by Charnes and Cooper in 1967 later Lee and Ignazio implemented to several fields. Fuzzy GP model was applied by Rezayi et. all[1] to various cropping systems in Khoramabad region. Abbas et. all[2] used multi-objective fractional goal programming to managing water supplies on a regional scale present planning challenges for crop patterns. In order to ensure the equitable distribution of the restricted, scarce water resources and farming land among the competing crops, Ahmed M. et. all[3] created a mathematical model to optimize the current year return of the Sudi Arabia farming area. The outcome demonstrates that improving the cropping pattern enhances net return in comparison to current year earnings. In a review article, Alanoud et. all[4] specifically highlights the linear programming model's use in optimising agricultural solutions. Bozena Piech et. all[5] explored the various MCDM approaches to farm planning using the case of a university farm in the UK. Romero and Tahir[6] implemented linear programming and the construction of goal programming to farm planning in their study. This study is accomplished in order to put the potential value of GP and its link to linear programming into context and to inspire different applications of MCDM in farm planning. Singh et. all[7] constructed Linear programming model to different cropping pattern to increase net profit at various water levels. Ishtiaq et. all[8] used LP model to Dera Ghazi Khan division of Punjab province to increase the profit, production at different cropping system. In order to determine the income, production, and best cropping pattern, Arif Raza et. all[9] executed LP in the Faisalabad division. As a result, they were able to increase their income by 2% in comparison to the previous cropping pattern. To maximise net profit, production, and reduce labour and machine working hours, J. Soltani[10] created the LP, GP, and FGP models. After comparing all three models, they concluded that FGP is the best and produces the maximum income. Jernej et. all[11] improved organic agriculture in Slovenia using weight goals and linear programming. The cropping patterns in different districts of Rajasthan were established using linear programming by Mahak B et. all[12] They compared present farm plan to ideal crop plan as a result they achieved maximum net benefit. Heydari et. all[13] developed linear programming to reduce fertiliser use for various agricultural systems. Najafabadia et. all[14] implemented multi objective structural programming model to different cropping pattern. To optimise the cropping pattern for the Barana irrigation project, Viswanathan et. all[15] applied linear and GP models. In the end, they discovered that the GP model is the best optimization model for the Barana irrigation project. At Sekayam District, Burhansyah et. all[16] established a goal programming model (GP) for intercropping of crops. They framed seven objectives and used sampling method to gather data. At the conclusion, they received the highest yearly profit in comparison to their current yearly revenue. Srilatha et. all[17] create an optimal crop plan utilising a linear programming approach to obtain higher returns than the current net average results to farmers. Jeyavanan et.all[18] created goal programming in the field of agriculture to maximise crop yield. To choose the next cropping plant for each field in a farm-wide way, Chono et. all[19] used a mixed zero one goal programming model. Saman et. all[20] created fuzzy linear programming for cropping patterns, which has the result of decreasing the area allotted to crops. Robert Gentleman and Ross Ihaka from New Zealand created R in 1993 as a computer language and analytical tool. Data scientists, statisticians, programmers, and data miners all frequently use it. This is one of the analytics tools most frequently used in business and data analytics. It is employed extensively in a wide range of industries, including medicine, education, consulting, finance, and the media, among many others. There is a higher demand for R language specialists due to the extensive use of the language in statistics, data visualization, and machine learning.
Data collection, Area, Methods, and Model formation:
The study was conducted in Uttara Kannada district Karnataka state, India. The district's primary agricultural products include rice, wheat, cotton, and maize. Crop cultivation is the main occupation of farmers in the study area, with goals oriented towards both consumer satisfaction and marketing. A random sample technique was employed to select thirty farmers for the study. These farmers provided information on land use, fertilizer application, labor hours, seed prices, machine hours, and pesticide usage for four crops. To optimize net returns, a Goal Programming model was developed using the averages of the observed data. The model took into account resource constraints such as land, labor (measured in man-days), machine hours, and expenditures. Goal Programming is adept at handling multiple and conflicting objectives. Each metric has a target value that must be achieved, and deviations from these objectives are measured in both directions from the target. Undesired deviations from the set of desired values are then minimized in an achievement function. The goal programming model assumes a satisficing ideology, where reaching the goal is considered satisfactory for the decision-maker(s). The general goal programming model is outlined below:
Min \(Z=\sum _{i=1}^{n}{P}_{i}\left({g}_{i}^{+}+{g}_{i}^{-}\right)\)
Subjected to the constraints \(\sum _{j=1}^{m}{c}_{ij}{x}_{j}+{g}_{i}^{-}-{g}_{i}^{+}={r}_{i}\) and \({x}_{j}\), \({g}_{i}^{-}\), \({g}_{i}^{+}\ge 0\) for \(i=\text{1,2}\dots n, j=\text{1,2}..m\)
Where \(Z\) is the objective function, \({c}_{ij}\) is the coefficient with respect to the variable j in the ith goal, \({x}_{j}\) is the decision variable, \({r}_{i}\)is the right-hand side value, \({P}_{i}\) is the priorities, \({g}_{i}^{-}\)and \({g}_{i}^{+}\)are the negative and positive deviational variable from the ith goal.
Using the general GP model, we construct the following goals:
i) Land: The most significant limiting element is land, which also serves as the primary resource for production. All land-related operations are measured in terms of acres. 115 acres of land in total are accessible for the current study. The primary crop farmed by farmers in the studied area is Rice.
$${R}_{1}+{W}_{2}+{C}_{3}+{M}_{4}+{g}_{1}^{-}-{g}_{1}^{+}=115$$
ii) Man Power: The most valuable resource for completing the tasks is labour. For the study, the labour is measured in man days.
$${59R}_{1}+30{W}_{2}+309{C}_{3}+25{M}_{4}+{g}_{2}^{-}-{g}_{2}^{+}=5820$$
iii) Machine Hours: In order to prepare their field and harvest their combined crops, farmers use tractor services. Therefore, we considered machine hours constraints.
$${5R}_{1}+6{W}_{2}+{3C}_{3}+5{M}_{4}+{g}_{3}^{-}-{g}_{3}^{+}=526$$
iv) Funds (Expenditure): The cash needed to cover the cost of fertilizers, seeds, manure from farm yards, crop protection chemicals, insurance costs, selling costs, and payment for the labour of people and bullocks as well as tractor power are referred to as funds.
$${38898R}_{1}+32475{W}_{2}+111963{C}_{3}+21714{M}_{4}+{g}_{4}^{-}-{g}_{4}^{+}=4343869$$
v) Net Profit: According to the current plan, farmers might dedicate up to rice on 81 acres, wheat on 21 acres, cotton on 8 acres, and maize on 5 acres, giving returns of Rs. 33,84,345.
$${46188R}_{1}+15436{W}_{2}+160607{C}_{3}+14088{M}_{4}+{g}_{5}^{-}-{g}_{5}^{+}=3384345$$
The goal programming Model of the problem is given below
Minimize \({Z=P}_{1}\left({g}_{1}^{-}\right)+ {P}_{2}\left({g}_{2}^{-}\right)+ {P}_{3}\left({g}_{3}^{-}\right)+ {P}_{4}\left({g}_{4}^{-}\right)+{P}_{5}\left({g}_{5}^{+}\right)\)
Subjected to the constraints
$${R}_{1}+{W}_{2}+{C}_{3}+{M}_{4}+{g}_{1}^{-}-{g}_{1}^{+}=115$$
$${59R}_{1}+30{W}_{2}+309{C}_{3}+25{M}_{4}+{g}_{2}^{-}-{g}_{2}^{+}=5820$$
$${5R}_{1}+6{W}_{2}+{3C}_{3}+5{M}_{4}+{g}_{3}^{-}-{g}_{3}^{+}=526$$
$${38898R}_{1}+32475{W}_{2}+111963{C}_{3}+21714{M}_{4}+{g}_{4}^{-}-{g}_{4}^{+}=4343869$$
$${46188R}_{1}+15436{W}_{2}+160607{C}_{3}+14088{M}_{4}+{g}_{5}^{-}-{g}_{5}^{+}=3384345$$
R Programming Code for the Problem (Fig. 1): The goal programming solver and lpSolve package were used to find out the answer to the goal programming problem.