Here, to apply this proposed framework in real life, sustainable agriculture in India is taken for the study. As the population of India is gradually increasing on one side [

1,

3], it is challenging for the Government to match the agricultural needs of the people [

20]. Additionally, Indian farmers are also facing many issues in making the produces due to challenges of environment and scarcity of resources [

2]. Furthermore, due to the uncertainty of the agricultural market in India [

37,

38], farmers are not able to make the minimum profit [

2,

19,

39]. So, it is an important issue to address for changing the dreadful situation of the Indian farmers and for making the economic background of the nation healthier.

As per the findings of past researchers, agricultural sustainability is the solution to this problem. So, we decided to do the study on the selection of crop pattern for Rabi season crops, by considering all the challenges in accordance with sustainable agricultural practices. The proposed rough AHP-TOPSIS is implemented to find the perfect crop pattern, which yields all the desirable outputs. The data were collected in the form of a matrix, in which the elements of the matrix compare the criteria and the alternatives with the range of scale values representing the magnitude of influence. Four experts were chosen, who have immense experience in the field of agriculture over a long tenure. Among the four participants, three are academic experts, and the other one is a farmer. At last, the expert’s opinions were compiled and examined using the developed rough AHP-TOPSIS framework, and the results were analyzed. The procedure of the developed model is illustrated below.

#### Implementation of Rough AHP-TOPSIS Methods

**Step 1: Formation of hierarchal structure**

Firstly, all the criteria, associated sub-criteria, and the alternatives are chosen for the selection of ideal crop pattern of Rabi crop in accordance with sustainable agricultural practices. Then the selected criteria, sub-criteria, and alternatives are compiled and organized to form a hierarchal structure. In that structure, the goal is at level 1, alternatives are in level 4, and the criteria and sub-criteria are in level 3 and level 4, respectively. The hierarchal framework for the selected problem is presented in

Table 5.

**Step 2: Construction of judgement matrices**

The decision-makers were informed of all the necessary details of this study at the beginning of the survey (

Table 3 and

Table 6). Then the experts were allowed to fill the decision matrices that are necessary to perform rough AHP-TOPSIS calculations for selecting the ideal crop pattern. A sample of judgments made by the decision-makers is presented in

Appendix A (

Table A1,

Table A2,

Table A3,

Table A4,

Table A5 and

Table A6). The pairwise comparison table values from

Table 3 are used to fill the matrices which compare one criterion with the other criteria, which are employed in finding the weight of each criteria using a rough AHP method. Another judgment matrix was also formed by the experts, which compares criteria with the alternatives for performing the rough TOPSIS calculations. For forming that matrix, a performance rating was given based on the score values, which are presented in

Table 6.

**Step 3: Finding weights of criteria by rough AHP**

The decisions made by the four decision-makers are aggregated by combining the values of particular criteria over others in the form of sets. The integrated decision matrix is converted into rough set values by using the Equations (1)–(6). The integrated rough decision matrices of all the criteria and sub-criteria are presented in

Appendix B (

Table A7,

Table A8,

Table A9,

Table A10 and

Table A11). The converted rough numbers are used in the Analytical Hierarchy Process, and the weights of each criterion and the sub-criteria are found in the form of a rough set using Equation (21). Then, with the help of Equation (22), weights are normalized to the scale of 0 to 1. Then, the normalized weights of the main criteria were multiplied with the weights of the sub-criteria to form the global weight. They are shown in

Table 7 below.

**Step 4: Finding the ideal crop pattern by rough TOPSIS**

As similar to the rough-AHP method, the judgment matrix of decision-makers is integrated, and the aggregated decision matrix was converted into the matrix of rough values, and the normalization was done by using the formula (30). In

Appendix C, tables of integrated judgment matrix (

Table A12) and the normalized matrix (

Table A13) are presented. The weighted normalized matrix (

Table A14) was formed by multiplying the global weights of each sub-criteria with corresponding normalized performance ratings according to the Equations (31) and (32).

The positive and negative ideal solutions of each sub-criteria correlated with the alternatives are found using the Equation (33). Then, the distance from PIS and NIS of each sub-criteria in relation to the respective alternatives were found using formula (34) and (35), respectively. These values are presented in

Table 8.

According to Equation (36), the closeness coefficients (C

_{i}) of each alternative were found, and the values are ranked in descending order and are illustrated in

Table 9. The alternative which ranked number 1 (Rabi Pulses) was chosen as an ideal crop pattern for the sustainable agricultural practice considering all the influencing factors in sustainable farming.

According to the results obtained from the rough AHP-TOPSIS method, the ideal crop pattern for the Rabi cropping season for sustainable agricultural practices is Rabi pulses, as it got the highest closeness coefficient (Ci) value at about 0.5701, and is ranked number 1 among all alternatives that are considered. The least preferred crop pattern would be barley, and it has the minimum Ci value, only 0.3759 by considering all the criteria that influence agricultural sustainability. The ranking of all the alternative crop patterns are in the order of Rabi pulses > Rabi fruits > Rabi cereals > Rabi spices > Rabi vegetables > gram > wheat > barley.

Figure 3 presents the graphical representation of a comparison of weights of all the criteria obtained by the Rough AHP-TOPSIS method to the conventional crisp AHP-TOPSIS method. It is interesting to note that the order of ranking of weights of all the criteria by rough AHP-TOPSIS is more or less the same with the rank order of weights by the AHP-TOPSIS method. C8 > C7 > C6 > C5 > C9 > C1 > C2 > C10 > C3 > C4 > C11 is the rank order obtained by rough AHP-TOPSIS method, and the sequence of rank by AHP-TOPSIS method is C8 > C7 > C6 > C5 > C9 > C10 > C1 > C2 > C3 > C11 > C4. In the aforementioned sequence of rank orders, only a few criteria (C1, C2, C4, C10, and C11) have a different position of rank by both methods. In the rough method, the uncertainties of decision-makers can be understood, as it fits the values of decision-makers in the form of upper and lower limits [

17,

18,

28]. Here, in this graph, the spread of judgment by experts is represented in the form of the bar for the rough AHP process as opposed to a line by the crisp AHP method. The more the length of the bar indicates, the more the uncertainties of decisions by the decision-makers. The less the length of the spread represents, the higher the accuracy of the decisions. When it comes to the weights by the AHP method, they are represented in the form of lines, even though multiple decision-makers are involved in this problem. It takes only the mean value of decisions by the experts, and so the vagueness and uncertainties of the judgment values cannot be found in the conventional AHP method.

In

Figure 4, the rank of the alternatives found by rough AHP-TOPSIS is compared with the rank of alternative by the crisp AHP-TOPSIS method. It can be seen from the graph that the rank of alternatives of Rabi fruits, Rabi spices, wheat, and barley are the same by both methods; they are 2, 4, 7, and 8, respectively. Rabi pulses are the most suggested crop pattern for the sustainable agricultural practices by the rough AHP-TOPSIS method, whereas Rabi cereals by the AHP-TOPSIS method. The least preferred crop pattern for Rabi season in terms of agricultural sustainability is barely, as it ranked eight among the alternatives by both methods.