
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12197-4
10.1016/j.heliyon.2024.e36166
e36166
Research Article
A new bi-level TOPSIS based neutrosophic programming technique for land allocation to medium farm holders
S Angammal
G Hannah Grace hannahgrace.g@vit.ac.in
⁎
School of Advanced Sciences, Department of Mathematics, Vellore Institute of Technology Chennai, Vandalore, Chennai, 600127, India
⁎ Corresponding author. hannahgrace.g@vit.ac.in
17 8 2024
30 8 2024
17 8 2024
10 16 e3616625 11 2023
8 8 2024
9 8 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
Agriculture impacts a country's social and economic growth. Crop allocation, crop combinations, and crop production processes are all necessary to achieve optimal results during various growing seasons. To maximize farm earnings, proper farm planning and resource allocations are necessary. In agriculture, land allocation problems involve several uncertainties and unpredictable variables, includes water supply, labour demands, fertility use, and food requirements. The objective of this study is to propose novel bi-level programming approaches to overcome such issues and obtain optimal land allocation for medium-sized farmers. The current study examines a bi-level, TOPSIS-based neutrosophic programming approaches in two cases, including non-interactive and interactive approaches with linear, exponential, and hyperbolic membership functions to maximize net profit and minimize the expenditure. The proposed methods are compared to other approaches, such as the Torabi & Hassini approach, the Fuzzy Optimization Technique (FOT), and the Intuitionistic Fuzzy Optimization Technique (IFOT) and are found to be more effective than the existing ones.

Highlights

• A framework to optimize production, profit, and expenditure.

• Bilevel neutrosophic techniques have been used for land allocation with the TOPSIS approach.

• Non-interactive and interactive neutrosophic approaches have been used to get an optimal solution.

• A framework's efficacy was evaluated by comparing it with existing approaches.

Keywords

Land allocation
Bi-level programming
TOPSIS approach
Neutrosophic programming approach
Optimal net profit
==== Body
pmc1 Introduction

The Indian economy depends significantly on agriculture. Around 54.6% of the workforce in India is employed in the agricultural sector, which contributes 13.9% of the nation's GDP [1]. Indian agriculture has grown tremendously during the past few decades. In Tamil Nadu, 93% of farmers are small and marginal farmers, and over two-thirds of rural households depend on agriculture as their primary source of income. The Tamil Nadu government is making every attempt to boost farmers income and productivity by embracing frontier agriculture technology to a greater extent for diverse crops cultivated in Tamil Nadu by actively incorporating farmers and extension officials with proper research backing. The present study deals with the medium-sized farmers of Ariyalur district, which is one of the Cauvery delta zone districts. In this district, 70% of the population depends on agriculture and allied enterprises for a living, making it the most significant sector of the economy. The division strategy and targets aid in ensuring the sustainability of agricultural production and in building horticultural production in a sustainable manner to meet the food needs of an expanding population as well as the natural resource needs of industries based on agriculture, providing rural residents with employment opportunities. The geographical area of the Ariyalur District is 1,93,338 hectares. The total cropped area is 1,11,874 hectares. There are around 45,136 hectares of irrigated land and approximately 66,738 hectares of rain-fed land. Agriculture is the principal activity for most of the people in this district, and they grow a variety of crops. In Ariyalur district, 1% of farmers are medium-sized farm holders. The data for the present study was collected from the medium-sized farm holders in the Ariyalur district. Proper crop planning is crucial for increasing farmers profits. For the crop land allocation problem, many researchers have used fuzzy and intuitionistic fuzzy optimization approaches, which contain only truth and falsity membership functions. But there are many indeterminate factors that occur in agriculture, such as seed growth, suitable fertilizer, the growth of the crop, etc. Although fuzzy and intuitionistic fuzzy sets manage all types of uncertainty in crop planning problems, they need a modern framework to manage indeterminate understanding. Based on the neutrosophic set, which was introduced by Smarandache [2] to address acceptance, indeterminacy, and rejection, the present study is a new attempt to overcome such an issue. The primary goal and motivation of this study are as follows:• To improve the profit and production of medium-sized farmers with minimum expenditure.

• Using novel bilevel neutrosophic interactive and non interactive approach with TOPSIS technique.

Multilevel Programming Problems (MLPP) with two-level structures are referred to as bi-level programming problems. When a decision maker (DM) is tasked with maximizing one or more objective functions at each decision-making level, MLPP is employed. MLPP is the primary mathematical optimization problem for representing large-scale decentralized decision problems in a hierarchical organization with multiple interacting decision-makers. They are frequently used in industry, transportation, agriculture, finance, public policy, and supply chain management. The decision-making process is organized from top to bottom. Although the upper level takes precedence over the lower level, lower-level reactions are still necessary for the upper level to function. Additionally, every decision-maker maximizes the objective function as much as possible. For a multi-objective, bi-level optimization problem, Kirti Sharma et al. [3] developed a chance-constrained optimization technique. This method uses triangular intuitionistic fuzzy numbers for objective coefficients and random variables for constraint coefficients. Also, the methodology was implemented in the production planning problem. V.P. Singh et al. [4] proposed a bi-level fuzzy multi-objective optimization technique based on the TOPSIS approach with intuitionistic fuzzy parameters and Murshid Kamal et al. [5] used a bi-level fuzzy goal programming algorithm with multi-choice parameters for production planning problems. For the optimal allocation of available water in Taunsa Barrage in Pakistan to all competing water use sectors, M. Masood et al. [6] used a bi-level programming approach. Reza Lotfi et al. [7] introduced a unique bilevel programming method and game theory to find renewable energy locations in order to meet the government's ideal energy requirement. To address the bi-level multi-objective linear fractional problem, Rizk M. Rizk Allah and Mahmoud A. Abo Sinna [8] developed fuzzy TOPSIS and the Jaya technique. The present TOPSIS-based bilevel neutrosophic program is an extension of fuzzy optimization with the TOPSIS technique. In the present investigation, the first-level decision maker controls the decision variable y2,y4,y5,y6,y7,y9, namely the crop area of kharif season groundnut, rabi season paddy, groundnut, pearl millet, sweet corn, and summer season brinjal. Setting higher and lower bounds for the decision variable under the first-level decision maker's control allows the feasible region to be expanded while proceeding to the next level. A satisfying solution for decision-makers at both levels is obtained by expanding the feasible region.

2 Literature review

Optimization methods can be categorized into linear and non-linear classes. Since 1950, agriculture has made extensive use of linear programming (LP). The LP problem is commonly applied to the yield planning problem. M.O. Wankhade et al. [9] employed a linear programming approach to figure out the best land allocation for ten key crops in the Stalin track of the red zone. Bhatia et al. [10] used the linear programming technique to increase farm revenue in many districts of Rajastan. To improve the net profit, Meselu Tegenie Mellaku et al. and Sofi et al. [11], [12] used the LP technique for crop land allocation and livestock enterprise arrangement. Elizabeth Gosling et al. [13] studied agroforestry systems by utilizing goal programming to maximize farmers' productivity and returns. For the canal command area land allocation problem, P. Srivastava et al. [14] suggested two-level fuzzy goal programming. Fuzzy optimization techniques are used by K. Ponnalagu et al. and K. Thomas Felix et al. [15], [16] to find the highest production of a variety of crops. Usha Rani Basumatary and Dipak K. Mitra [17] devised a fuzzy optimization approach to maximize profit and production. To design the distribution of land, Babita Mishra et al. [18] developed fuzzy multi-fractional program. Demmelash Mollalign Moges et al. [19] proposed a two-phase weighted intuitionistic fuzzy goal programming technique to solve an intuitionistic fuzzy multi-objective linear fractional optimization problem and applied it to an agricultural production planning problem for efficiency. Under uncertain decision variables, Chung Fu Huang et al. [20] suggested a multi-objective program that integrates non-linear, mixed integer, and compromise fuzzy programs to design a regional sewage system plan. To find the best compromise solution for the imbalanced fixed charge transportation problem, Shivani et al. [21] proposed a novel method in which all decision variables, coefficients of objectives, and limits are represented by a rough interval. K.T. Attanasov [22] developed the Intuitionistic Fuzzy Set, which is an extension of the fuzzy set. P.P. Angelov [23] extends the fuzzy optimization strategy into an intuitionistic fuzzy optimization technique by using an intuitionistic fuzzy set. By using the concept of the Angelov approach, S.K. Bharati et al. [24] proposed an intuitionistic fuzzy optimization strategy to maximize the profit and production of small farm holders. S. Angammal et al. [25] conducted a comparison of fuzzy and intuitionistic fuzzy optimization techniques for allocating agriculture land to each crop. To address the real-life problem with intuitionistic fuzzy parameters, Sujeet Kumaran Singh et al. [26] proposed a methodology based on intuitionistic fuzzy sets with linear and non-linear membership functions. S.V. Pawar et al. [27] applied intuitionistic fuzzy optimization methods to an irrigation system.

In the work of Abdullah Ali H. Ahamadini et al. [28] the linear multi-objective optimization problem with trapezoidal intuitionistic fuzzy parameters was investigated utilizing the neutrosophic programming approach. For the multi-objective transportation problem, novel compromised algorithms based on neutrosophic sets were proposed by Rizk. M. Rizk Allah et al. [29] and the outcomes were confirmed using the TOPSIS ranking method. For the multi-objective non-linear transportation problem, Firoz Ahmad et al. [30] developed the NCPA. For the multi-product production planning problem, Mohammad Faisal Khan et al. [31] suggested an intuitionistic and neutrosophic programming technique. To provide the best possible land allocation to medium-sized farm owners, S. Angammal et al. [32] proposed an INPA with exponential and linear membership functions. Sajida Kousar et al. [33] employed neutrosophic programming with a linear membership function to optimize Pakistan's canal command area's Kharif and Rabi seasons production and profit. Azza H. Amer and Mahmoud A. Abo Sinna [34] proposed a new computational method based on the neutrosophic set to solve a multi-objective non-linear programming problem. For the multi-level, multi-objective linear programming problem with the neutrosophic number parameter, Indrani Maiti et al. [35] suggested a novel goal programming approach. Aakanksha Singh et al. [36] proposed two approaches based on bi-level programming techniques to transport the COVID-19 vaccination from the vaccine manufacturing company to different distribution centres. Suizhi Luo et al. [37] proposed three interactive programming approaches to solve the multi-level programming problem in the neutrosophic environment. In order to minimize the overall cost, back order level, and delivery time for integrated production distribution planning in a two-echelon supply chain scenario, Badhotiya et al. [38] introduced a novel neutrosophic programming technique. Ahteshamul Haq et al. [39] suggested an intuitionistic programming technique for production planning using multi-choice and interval-valued trapezoidal neutrosophic numbers for parameters and resource vectors. Many studies utilized intuitionistic and fuzzy approaches to the crop land allocation problem, whereas some researchers adopted a neutrosophic approach that merely used a linear membership function. However, as Table 1 demonstrates, this work offers a fresh approach that makes use of the TOPSIS bi-level neutrosophic programming technique with linear and nonlinear membership functions such as exponential and hyperbolic.Table 1 Literature review on crop land allocation problem.

Table 1Reference﹨ Methodology	LP	GP	Fuzzy	FGP	IF	Neutrosophic	TOPSIS Neutrosophic	Membership	
M.O. Wankhade, H.S. Lunge (2012) [9]	✓								
S.K. Bharati, S.R. Singh (2014) [24]					✓			Linear	
B. Mishra et al. (2014) [18]			✓					Linear	
P. Srivastava, R.M. Singh (2017) [14]				✓				Linear	
K. Thomas et al. (2017) [16]			✓					Linear	
Mellaku et al. (2018) [11]	✓								
U.R. Basumatary et al. (2020) [17]			✓					Linear	
M. Bhatia, A. Rana (2020) [10]	✓								
E. Gosling et al. (2020) [13]		✓							
K. Ponnalagu, D. Madhumath (2020) [15]			✓					Linear	
Angammal and Grace (2022) [25]			✓		✓			Linear	
Angammal and Grace (2022) [32]						✓		Linear, Exponential	
S.V. Pawar, P.L. Patel (2022) [27]					✓			Linear	
Kousor et al. (2023) [33]						✓		Linear	
D.M. Moges et al. (2023) [19]					✓			Linear	
Proposed Research							✓	Linear, Exponential, Hyperbolic	
Abbreviations: LP-Linear Program, GP-Goal Program, FGP-Fuzzy Goal Program, IF-Intuitionistic Fuzzy.

3 Problem description

A suitable land allocation with a suitable amount of resources is required to maximize the output and profit of a farmer with the minimum expenditure. Typically, a multi-objective optimization model is used to formulate agricultural crop planning problems. Farmers in the agricultural industry are divided into five categories. Marginal farm holders are those who have less than one hectare of cropland; small farm owners have between one and two hectares; semi-medium farmers have between two and four hectares; medium farmers have between four and ten hectares; and large farm owners have more than ten hectares. In this study, medium-sized landowners in the Ariyalur district are focused on. A medium-sized farmer with 15 acres of land in the Ariyalur district contributed the data for this study. The farmer with 15 acres of land grew cotton, paddy, and groundnuts in the kharif; maize, paddy, groundnuts, and pearl millet in the Rabi; sesame, onions, paddy, and brinjal in the summer seasons. The present study deals with six objectives and four types of constraints. Regarding the limitations imposed by land, labour, water, and food requirements, they are maximizing production and profit and minimizing labour, seed, fertilizer, pesticides, and miscellaneous costs. To handle the rising food interests of the nation's quickly developing populations, there is generally a need to further develop the food system. The proposed TOPSIS bi-level non-interactive and interactive neutrosophic programming approach is used to maximize the productivity and profit of a 15-acre farm owner in the Ariyalur district. In this study, the farm owners are the first-level decision-makers, and the well-experienced farming employees are the second-level decision-makers. The Ariyalur district map and land holding pattern are shown in the following Figure 1, Figure 2. Also, the flowchart of the proposed approach is shown in Fig. 3.Figure 1 Ariyalur district map; Source: Google Map.

Figure 1

Figure 2 Ariyalur district's land holding pattern; Source: tn.data.gov.in.

Figure 2

Figure 3 Flow chart of the proposed model.

Figure 3

The preliminary findings, notations, and methodology used in this study are stated in the following sections.

3.1 Preliminaries

This section discusses the fundamental ideas of Multi objective optimization problem, distance measure, and Neutrosophic set.

3.1.1 Definition

The general structure of the r objective, m constraint, and n decision variable multi-objective optimization problem is as follows [40]:(1) Max/Min{Z1,Z2,...,Zr}Subject tohj(y)≤ or ≥ or =bj,j=1,2,…,myi≥0,i=1,2,…,n

3.1.2 Distance measure

If the objective function vector Z(y)={Z1(y),Z2(y),…,Zr(y)} is one that should be maximized. The distance between two points Z(y) and Z⁎ is then defined by the Lp metric asdp={∑i=1rwip(Zi⁎−Zi(y))p}1p

where wi represents the relative importance of the objective Zi, i=1,2,…,r, Zi⁎=Z1⁎,Z2⁎,…,Zr⁎ and Zi⁎=max(Zi(y)). Due to the incommensurability of the objective functions, a scaling function in the range [0, 1] should be used for each one. As a result, the next metric is used [41].dp={∑i=1rwip(Zi⁎−Zi(y)Zi⁎−Zi′)p}1p

where Zi′=min(Zi(y)).

3.1.3 Single valued neutrosophic set

A single valued neutrosophic set (SVNS) A, over the universe of discourse Y is defined asA={(y,TA(y),IA(y),FA(y))/y∈Y}

where TA(y),IA(y),FA(y)∈[0,1] and 0≤TA(y)+IA(y)+FA(y)≤3. [2]

3.1.4 Definition

If P and Q are two different single valued neutrosophic sets then the union of P and Q is also a single valued neutrosophic set which is written as S=P∪Q and its truth, indeterminacy and falsity membership functions are described as follows [2]:

Type 1 TS(y)=max(TP(y),TQ(y))IS(y)=max(IP(y),IQ(y))FS(y)=min(FP(y),FQ(y))

Type 2 TS(y)=max(TP(y),TQ(y))IS(y)=min(IP(y),IQ(y))FS(y)=min(FP(y),FQ(y))

3.1.5 Definition

If P and Q are two different single valued neutrosophic sets then the intersection of P and Q is also a single valued neutrosophic set which is written as S=P∩Q and its truth, indeterminacy and falsity membership functions are described as follows [2]:

Type 1 TS(y)=min(TP(y),TQ(y))IS(y)=min(IP(y),IQ(y))FS(y)=max(FP(y),FQ(y))

Type 2 TS(y)=min(TP(y),TQ(y))IS(y)=max(IP(y),IQ(y))FS(y)=max(FP(y),FQ(y))

3.2 Notation

Oi(y) Objective function

bj Resource vector

G Decision set

αk,βk Scaling factor

Uiμ,Uiσ,Uiγ Upper bound of truth, indeterminacy, falsity

Liμ,Liσ,Liγ Lower bound of truth, indeterminacy, falsity

μiL,σiL,γiL Linear membership function of truth, indeterminacy, falsity

μiE,σiE,γiE Exponential membership function of truth, indeterminacy, falsity

μiH,σiH,γiH Hyperbolic membership function of truth, indeterminacy, falsity

yu Decision variable controlled by leader

yl Decision variable controlled by follower

dpPISu Distance of positive ideal solution at upper level

dpNISu Distance of negative ideal solution at upper level

dpPISl Distance of positive ideal solution at lower level

dpNISl Distance of negative ideal solution at lower level

wi Weight of each objective function

Ti(yk),Ii(yk) Linear truth, indeterminacy, falsity

Fi(yk) membership function of the ith level decision variable

λ2 Minimum acceptance's degree

ϑ2 Maximum indeterminacy's degree

δ2 Maximum rejection's degree

3.3 Neutrosophic programming technique for MOOP

Real-life complexity frequently leads to an indeterminacy problem when making the best decisions. The degree of indeterminacy in the decision-making process is crucial, in addition to the acceptance and rejection rates. As a result, the indeterminacy degree is covered in the feasible solution set. Smarandache investigated a neutrosophic set that consists of an indeterminacy membership function with truth and falsity membership functions. NS is distinguished from all other uncertain decision sets, including fuzzy and intuitionistic fuzzy sets, by the idea of an independent indeterminacy degree.

Bellman and Zadeh [42] developed fuzzy objectives (O), fuzzy decisions (G), and fuzzy constraints (C), which are now commonly used in many decision-making scenarios incorporating ambiguity and hesitation. The intersection of integrated fuzzy objectives and constraints forms decision set G.G=O∩C

As a result, the neutrosophic decision set GN, which considers neutrosophic objectives and constraints, is depicted as follows:GN=(∩i=1rOi)(∩j=1mCj)=(y,TG(y),IG(y),FG(y))

HereTG(y)=Min{μO1(y),μO2(y),…,μOr(y)μC1(y),μC2(y),…,μCm(y)}IG(y)=Max{σO1(y),σO2(y),…,σOr(y)σC1(y),σC2(y),…,σCm(y)}FG(y)=Max{γO1(y),γO2(y),…,γOr(y)γC1(y),γC2(y),…,γCm(y)},∀y∈Y

TG(y),IG(y),FG(y) represent the membership functions of truth, indeterminacy and falsity of decision set GN. The present study deals with linear, hyperbolic & exponential membership functions and is described by using upper and lower limits as discussed in Section 3.4. According to Bellman and Zadeh, the neutrosophic optimization technique is as follows:Max mini=1,2,…,rμi(Oi(y))Min maxi=1,2,…,rσi(Oi(y))Min maxi=1,2,…,rγi(Oi(y))Subject tohj(y)≤ or ≥or=bj,j=1,2,…,myi≤0,i=1,2,…,n.

The preceding problem is transformed into the single objective programming problem below by utilizing the auxiliary parameters λ, ϑ, and δ.Max(λ−ϑ−δ)Subject toμi(Oi(y))≥λσi(Oi(y))≤ϑγi(Oi(y))≤δλ≥ϑ,λ≥δλ+ϑ+δ≤3hj(y)≤or≥or=bj,j=1,2,…,myi≥0,i=1,2,…,n.

An optimal land allocation for medium-sized farmers in the Ariyalur district is achieved by using the above concept with the TOPSIS bi-level programming approach, as explained in the following section.

3.4 Proposed bilevel TOPSIS neutrosophic programming approach

A classic form of a multi-objective, bi-level linear programming problem is given by:maxyu⁡O1(yu,yl)=O11(yu,yl),O12(yu,yl),…,O1k1(yu,yl)maxyl⁡O2(yu,yl)=O21(yu,yl),O22(yu,yl),…,O2k2(yu,yl)Subject toA(yu)+B(yl)≤b

(2) yu,yl≥0

where Oij=Cij(yu)+Dij(yl) for i=1,2 and j=1,2,…,ki, yu=(yu1,yu2,…,yun1)∈Rn1, yl=(yl1,yl2,…,yln2)∈Rn2, n1+n2=n. Also A∈RmXn1, B∈RmXn2, b∈Rm, Cij∈R1Xn1 and Dij∈R1Xn2.

Finding an efficient solution for an upper-level decision maker is the first step of solving the problem. The following TOPSIS model used to find an optimal solution for upper-level multi-objective optimization problem to the above Eq. (2).(3) MinimizedpPISu(yu,yl)MaximizedpNISu(yu,yl)Subject toA(yu)+B(yl)≤byu,yl≥0

where(4) dpPISu={∑i=1k1wip(O1i⁎−O1iO1i⁎−O1i′)p}1p

(5) dpNISu={∑i=1k1wip(O1i−O1i′O1i⁎−O1i′)p}1p

Also O1i⁎=Uacc(O1i), O1i′=Lacc(O1i) and wi are each objective function's relative importance such that wi≥0 and ∑i=1k1wi=1. Using the lower and upper acceptance of positive and negative ideal solutions, the goal of Eq. (3) is fixed as follows:(6) Find(yu,yl)Subject todpPISu≈dpPISu⁎dpNISu≈dpNISu⁎A(yu)+B(yl)≤byu,yl≥0

Here ≈ is an intuitionistic fuzzy goal, meaning that some departures from the strict goal are permitted. Also, dpPISu⁎=Min(dpPISu), and dpNISu⁎=Max(dpNISu). In the neutrosophic programming approach, several linear and non-linear truth, falsity and indeterminacy membership functions are used for fuzzification. The present study deals with linear, exponential &, hyperbolic truth, falsity and indeterminacy membership functions and their upper and lower bounds are defined as follows:

MaximizationUiμ=Uacc(Oi(y)),Liμ=Lacc(Oi(y))Uiσ=Uiμ−αk(Uiμ−Liμ),Liσ=LiμUiγ=Uiμ−βk(Uiμ−Liμ),Liγ=Liμ

MinimizationUiμ=Uacc(Oi(y)),Liμ=Lacc(Oi(y))Uiσ=Uiμ,Liσ=Liμ−αk(Uiμ−Liμ)Uiγ=Uiμ,Liγ=Liμ−βk(Uiμ−Liμ)

Where αK & βK are the scaling factors also αK, βK∈[01].

For maximization and minimization objectives the linear membership functions are defined as follows:

MaximizationμiL(Oi(y))={1,Oi(y)≥UiμOi(y)−LiμUiμ−Liμ,Liμ<Oi(y)<Uiμ0,Oi(y)≤Liμ

σiL(Oi(y))={0,Oi(y)≥UiσUiσ−Oi(y)Uiσ−Liσ,Liσ<Oi(y)<Uiσ1,Oi(y)≤Liσ

(7) γiL(Oi(y))={0,Oi(y)≥UiγUiγ−Oi(y)Uiγ−Liγ,Liγ<Oi(y)<Uiγ1,Oi(y)≤Liγ

MinimizationμiL(Oi(y))={0,Oi(y)≥UiμUiμ−Oi(y)Uiμ−Liμ,Liμ<Oi(y)<Uiμ1,Oi(y)≤Liγ

σiL(Oi(y))={1,Oi(y)≥UiσOi(y)−LiσUiσ−Liσ,Liσ<Oi(y)<Uiσ0,Oi(y)≤Liσ

(8) γiL(Oi(y))={1,Oi(y)≥UiγOi(y)−LiγUiγ−Liγ,Liγ<Oi(y)<Uiγ0,Oi(y)≤Liγ

For maximization and minimization objectives the exponential membership functions are defined as follows:

MaximizationμiE(Oi(y))={1,Oi(y)≥Uiμe−d(Uiμ−Oi(y)Uiμ−Liμ)−e−d1−e−d,Liμ<Oi(y)<Uiμ0,Oi(y)≤Liμ

σiE(Oi(y))={0,Oi(y)≥Uiσe−d(Oi(y)−LiσUiσ−Liσ)−e−d1−e−d,Liσ<Oi(y)<Uiσ1,Oi(y)≤Liσ

(9) γiE(Oi(y))={0,Oi(y)≥Uiγe−d(Oi(y)−LiγUiγ−Liγ)−e−d1−e−d,Liγ<Oi(y)<Uiγ1,Oi(y)≤Liγ

MinimizationμiE(Oi(y))={0,Oi(y)≥Uiμe−d(Oi(y)−LiμUiμ−Liμ)−e−d1−e−d,Liμ<Oi(y)<Uiμ1,Oi(y)≤Liμ

σiE(Oi(y))={1,Oi(y)≥Uiσe−d(Uiσ−Oi(y)Uiσ−Liσ)−e−d1−e−d,Liσ<Oi(y)<Uiσ0,Oi(y)≤Liσ

(10) γiE(Oi(y))={1,Oi(y)≥Uiγe−d(Uiγ−Oi(y)Uiγ−Liγ)−e−d1−e−d,Liγ<Oi(y)<Uiγ0,Oi(y)≤Liγ

where d is the shape parameter chosen by the decision-maker.

For maximization and minimization objectives the hyperbolic membership functions are defined as follows:

Maximization(11) μiH(Oi(y))={1,Oi(y)≥Uiμ12[1+tanh⁡(θi(Oi(y)−Uiμ+Liμ2))],Liμ<Oi(y)<Uiμ0,Oi(y)≤LiμσiH(Oi(y))={0,Oi(y)≥Uiσ12[1+tanh⁡(θi(Uiσ+Liσ2−Oi(y)))],Liσ<Oi(y)<Uiσ1,Oi(y)≤LiσγiH(Oi(y))={0,Oi(y)≥Uiγ12[1+tanh⁡(θi(Uiγ+Liγ2−Oi(y)))],Liγ<Oi(y)<Uiγ1,Oi(y)≤Liγ

Minimization(12) μiH(Oi(y))={0,Oi(y)≥Uiμ12[1+tanh⁡(θi(Uiμ+Liμ2−Oi(y)))],Liμ<Oi(y)<Uiμ1,Oi(y)≤LiμσiH(Oi(y))={1,Oi(y)≥Uiσ12[1+tanh⁡(θi(Oi(y)−Uiσ+Liσ2))],Liσ<Oi(y)<Uiσ0,Oi(y)≤LiσγiH(Oi(y))={1,Oi(y)≥Uiγ12[1+tanh⁡(θi(Oi(y)−Uiγ+Liγ2))],Liγ<Oi(y)<Uiγ0,Oi(y)≤Liγ

where θi=6Ui−Li,i=1,2,…r.

Using the above truth, indeterminacy, and falsity membership functions, Eq. (6) is converted into the crisp programming problem in two cases, such as the non-interactive and interactive neutrosophic approaches, and it is defined as follows:

Case 1 Neutrosophic Programming Approach (NPA)Max(λ1−ϑ1−δ1)Subject toμ(dpPISu)≥λ1;μ(dpNISu)≥λ1σ(dpPISu)≤ϑ1;σ(dpNISu)≤ϑ1;γ(dpPISu)≤δ1;γ(dpNISu)≤δ1λ1+ϑ1+δ1≤3λ1≥ϑ1;λ1≥δ1A(yu)+B(yl)≤b

(13) yu,yl≥0

Case 2 Interactive Neutrosophic Programming Approach (INPA)Maxη1(λ1−ϑ1−δ1)+(1−η1){w1u[μ(dpPISu)−σ(dpPISu)−γ(dpPISu)]+w2u[μ(dpNISu)−σ(dpNISu)−γ(dpNISu)]}Subject toμ(dpPISu)≥λ1;μ(dpNISu)≥λ1σ(dpPISu)≤ϑ1;σ(dpNISu)≤ϑ1;γ(dpPISu)≤δ1;γ(dpNISu)≤δ1λ1+ϑ1+δ1≤3λ1≥ϑ1;λ1≥δ1A(yu)+B(yl)≤b

(14) yu,yl≥0

where λ1 is the minimum satisfactory level, (w1u,w2u,η1) is the upper-level relative importance of the positive and negative ideal solutions and compensation coefficient between the overall satisfaction level respectively. If the solution vector of Eqs. (13) & (14) are (yu⁎,yl⁎) then (yu1⁎,yu2⁎,…,yun1⁎,yl1⁎,yl2⁎,…,yln2⁎) be the best solution of upper-level optimization problem. The leader establishes the tolerance limits of the decision variables that he controls at this stage in accordance with the theory behind a bi-level programming problem. Let TkL & TkU represents the lower and upper-level tolerance limits, respectively, of the leader-controlled decision variable. Additionally, these limits are not necessarily equal. The first-level decision-maker's decision vector's linear truth, indeterminacy, and falsity membership functions can be expressed as follows:Ti(yk)={yk−(yk⁎−TkLT)TkLT,yk⁎−TkLT≤yk≤yk⁎(yk⁎+TkUT−yk)TkUT,yk⁎≤yk≤yk⁎+TkUT0,otherwise

Ii(yk)={(yk⁎−TkLI)−ykTkLT−TkLI,yk⁎−TkLT≤yk≤yk⁎−TkLIyk−(yk⁎+TkUI)TkUT−TkUI,yk⁎+TkUI≤yk≤yk⁎+TkUT0,otherwise

(15) Fi(yk)={(yk⁎−TkLF)−ykTkLT−TkLF,yk⁎−TkLT≤yk≤yk⁎−TkLFyk−(yk⁎+TkUF)TkUT−TkUF,yk⁎+TkUF≤yk≤yk⁎+TkUT0,otherwise

Where TkLT, TkLF, & TkLI are the lower tolerance limit of truth, falsity and indeterminacy membership function, TkUT, TkUF, & TkUI are the upper tolerance limit of truth, falsity and indeterminacy membership function respectively. The limits are determined as follows:TkLI=αkTkLT;TkLF=βkTkLTTkUI=αkTkUT;TkUF=βkTkUT,αk,βk∈[01].

By decreasing the objective function's distance from the positive ideal solution and maximizing the distance from the negative ideal solution, TOPSIS is one way of locating the best answer to an optimization problem. The distances of the positive and negative ideal solutions at the second level are categorised as follows:(16) dpPISl={∑i=1k1wip(O1i⁎−O1iO1i⁎−O1i′)p+∑i=k1+1k2wip(O2i⁎−O2iO2i⁎−O2i′)p}1p

(17) dpNISl={∑i=1k1wip(O1i−O1i′O1i⁎−O1i′)p+∑i=k1+1k2wip(O2i−O2i′O2i⁎−O2i′)p}1p

The bi-level conflicting multi-objective optimization problem given in Eq. (2) is reduced to the following Eq. (18) in order to obtain an efficient solution.MinimizedpPISl(yu,yl)MaximizedpNISl(yu,yl)Subject toA(yu)+B(yl)≤b

(18) yu,yl≥0

At this stage, the truth, indeterminacy & falsity membership functions of second-level positive and negative ideal solutions are created by using Eqs. (7) to (12), and the truth, indeterminacy, and falsity membership functions for decision variables controlled by the first-level decision maker are created by using Eq. (15). After creating these membership functions, the optimal solution of Eq. (2) will be found by solving the following two cases of a single objective optimization problem:

Case 1 NPAMax(λ2−ϑ2−δ2)Subject toμ(dpPISl)≥λ2;μ(dpNISl)≥λ2σ(dpPISl)≤ϑ2;σ(dpNISl)≤ϑ2;γ(dpPISl)≤δ2;γ(dpNISl)≤δ2yk−(yk⁎−TkLT)TkLT≥λ2;(yk⁎+TkUT−yk)TkUT≥λ2(yk⁎−TkLI)−ykTkLT−TkLI≤ϑ2;yk−(yk⁎+TkUI)TkUT−TkUI≤ϑ2(yk⁎−TkLF)−ykTkLT−TkLF≤δ2;yk−(yk⁎+TkUF)TkUT−TkUF≤δ2λ2+ϑ2+δ2≤3λ2≥ϑ2;λ2≥δ2A(yu)+B(yl)≤b

(19) yu,yl≥0

Case 2 INPAMaxη2(λ2−ϑ2−δ2)+(1−η2){w1l[μ(dpPISl)−σ(dpPISl)−γ(dpPISl)]+w2l[μ(dpNISl)−σ(dpNISl)−γ(dpNISl)]}Subject toμ(dpPISl)≥λ2;μ(dpNISl)≥λ2σ(dpPISl)≤ϑ2;σ(dpNISl)≤ϑ2;γ(dpPISl)≤δ2;γ(dpNISl)≤δ2yk−(yk⁎−TkLT)TkLT≥λ2;(yk⁎+TkUT−yk)TkUT≥λ2(yk⁎−TkLI)−ykTkLT−TkLI≤ϑ2;yk−(yk⁎+TkUI)TkUT−TkUI≤ϑ2(yk⁎−TkLF)−ykTkLT−TkLF≤δ2;yk−(yk⁎+TkUF)TkUT−TkUF≤δ2λ2+ϑ2+δ2≤3λ2≥ϑ2;λ2≥δ2A(yu)+B(yl)≤b

(20) yu,yl≥0

where λ2 is the minimum satisfactory level, (w1l,w2l,η2) is the lower-level relative importance of the positive and negative ideal solutions and compensation coefficient between the overall satisfaction level respectively.

4 Results & discussion

The percentage of medium-sized farmers in Tamil Nadu is 2%, whereas in the district under investigation for this article, Ariyalur, it is 1%. This study aims to provide medium-sized farmers with the best framework for allocating crop land by employing the suggested method, which will boost their net profit margin and meet the world's food needs. In this section, the suggested interactive and non-interactive neutrosophic strategies are examined using the problem from Section 3. Sensitivity and comparative analyses are used to discuss the results. The mathematical formulation of the problem mentioned in Section 3 is created by using Table 2, which shows the production, profit, four types of expenditure, and resources of each crop per acre, respectively. Using Table 2, the multi-objective crop planning problem is described mathematically in the following way, as mentioned in Eq. (1):MaxO1=3430y1+1555y2+1275y3+3760y4+1455y5+1365y6+2730y7+2830y8+11600y9+9650y10+355y11MaxO2=73745y1+98742.5y2+74268.75y3+80840y4+92392.5y5+29347.5y6+50505y7+60845y8+232000y9+154400y10+24335.25y11MinO3=9150y1+5550y2+19650y3+9150y4+5550y5+4650y6+6150y7+9150y8+8850y9+10650y10+3900y11MinO4=3175y1+4500y2+1500y3+3175y4+4500y5+350y6+2000y7+3175y8+1600y9+6000y10+650y11MinO5=3310y1+2500y2+5000y3+3310y4+2500y5+1180y6+2800y7+3310y8+5500y9+5000y10+1380y11MinO6=3915y1+4000y2+3400y3+3915y4+4000y5+1150y6+1100y7+3915y8+3600y9+1900y10+2620y11

Subject toy1+y2+y3≤15y3+y4+y5+y6+y7≤15y8+y9+y10+y11≤15y1+y2+y3≤25825y3+y4+y5+y6+y7≤24875y8+y9+y10+y11≤2487530.5y1+18.5y2+65.5y3≤26030.5y4+18.5y5+15.5y6+20.5y7≤26030.5y8+29.5y9+35.5y10+13y11≤2603430y1+3760y4+2830y8≥37601365y6≥230

(21) y1,y2,…,y11≥0

The multi-objective linear optimization problem described above is transformed into the following bi-level programming problem (22) using Eq. (2).maxyu⁡O1(yu,yl)=O11(yu,yl),O12(yu,yl)minyl⁡O2(yu,yl)=O21(yu,yl),O22(yu,yl),O23(yu,yl),O24(yu,yl)Subject to

(22) All the constraints of Eq. (21)

In this study, production and profit are at the upper level, while labour costs, seed costs, pesticides, fertilizer, and miscellaneous costs are at the lower level. That is yu=(y2,y4,y5,y6,y7,y9); yl=(y1,y3,y8,y10,y11).Table 2 Production, Profit, Expenditure and Labour per acre.

Table 2Crop	Production	Profit	Labour cost	Seed cost	Pest & Fert	Misc	Labour	
Paddy (y1)	3430	73745	9150	3175	3310	3915	30.5	
Ground nut (y2)	1555	98742.5	5550	4500	2500	4000	18.5	
Cotton (y3)	1275	74268.75	19650	1500	5000	3400	65.5	
Paddy (y4)	3760	80840	9150	3175	3310	3915	30.5	
Ground nut (y5)	1455	92392.5	5550	4500	2500	4000	18.5	
Pearl millet (y6)	1365	29347.5	4650	350	1180	1150	15.5	
Sweet Corn (y7)	2730	50505	6150	2000	2800	1100	20.5	
Paddy (y8)	2830	60845	9150	3175	3310	3915	30.5	
Brinjal (y9)	11600	232000	8850	1600	5500	3600	29.5	
Onion (y10)	9650	154400	10650	6000	5000	1900	35.5	
Sesame (y11)	355	24335.25	3900	650	1380	2620	13	

The individual maximum and minimum values of each objective function are (165983.2, 4651386, 234000, 167223.3, 118108.2, 156171.5) and (3990, 85785, 9933.516, 3233.974, 3508.829, and 4108.773), respectively. Also, the positive and negative ideal solutions of first-level decision-makers are (165983.2, 4651386) and (3990, 85785), respectively. Using the equation of positive and negative ideal solution's distance mentioned in Eqs. (4) and (5), dpPISu and dpNISu are formulated as follows:dpPISu={w1p(165983.2−O11(yu,yl)161993.2)p+w2p(4651386−O12(yu,yl)4565601)p}1pdpNISu={w1p(O11(yu,yl)−3990161993.2)p+w2p(O12(yu,yl)−857854565601)p}1p

The decision maker may give various combinations of w1 and w2 to get the most satisfied solution in the first level. In this work, at w1=0.3 and w2=0.7, the first level decision maker reaches the most satisfactory solution. The maximum and minimum values of dpPISu and dpNISu are (0.7616, 0.0503) and (0.7314, 0), respectively. Thus, (dpPISu⁎, dpNISu⁎)=(0.0503,0.7314). Therefore, the problem (22) is converted into the following goal programming problem:Find(yu,yl)Subject todpPISu≈0.0503dpNISu≈0.7314

(23) All the constraints of Eq. (21)

Using the linear truth, indeterminacy, and falsity membership functions mentioned in Eqs. (7) & (8) for dpPISu & dpNISu, the above problem (23) is converted into the following problems (24) and (25).

Case 1Max(λ1−ϑ1−δ1)Subject toUiμ−dpPISuUiμ−Liμ≥λ1;dpNISu−LiμUiμ−Liμ≥λ1dpPISu−LiσUiσ−Liσ≤ϑ1;Uiσ−dpNISuUiσ−Liσ≤ϑ1dpPISu−LiγUiγ−Liγ≤δ1;Uiγ−dpNISuUiγ−Liγ≤δ1λ1+ϑ1+δ1≤3λ1≥ϑ1;λ1≥δ1

(24) All the constraints of Eq. (21)

Case 2Maxη1(λ1−ϑ1−δ1)+(1−η1){w1u[μ(dpPISu)−σ(dpPISu)−γ(dpPISu)]+w2u[μ(dpNISu)−σ(dpNISu)−γ(dpNISu)]}Subject to

(25) All the constraints of Eq. (24)

Solving the above problems, the first-level decision maker gets (λ1,ϑ1,δ1,y1,y2,y3,y4,y5,y6,y7,y8,y9,y10,y11)=(0.9987,0.0921,0.0921,0,14.0541,0,1,12.1705,0.1685,0.1414,0,8.8136,0,0) in case 1, using linear membership functions. In case 2, after verifying various compensation coefficients η1∈[0,1], the decision maker gets the most satisfactory solution (λ1,ϑ1,δ1,y1,y2,y3,y4,y5,y6,y7,y8,y9,y10,y11)=(1,0.0741,0.0741,0,14.0541,0,1,12.2642,0.1685,0,0,8.8136,0,0) at η1=(0.1,0.2,…,0.7). The first-level decision maker decides 10 cents is the upper and lower tolerance limit of the truth membership function of (y2,y4,y5,y6,y7,y9). Indeterminacy and falsity's tolerance limits are obtained by using the tolerance limits of truth membership functions, as mentioned earlier. After finding the tolerance limits, the truth, indeterminacy, and falsity membership functions for the decision variables controlled by the first-level decision makers are created by using Eq. (15).

The positive and negative ideal solutions of a second-level decision maker are (9933.516, 3233.974, 3508.829, 4108.773) and (234000, 167223.3, 118108.2, 156171.5), respectively. Using Eqs. (16) & (17), the maximum and minimum dpPISl & dpNISl with w1=0.2,w2=0.6,w3=w4=w5=w6=0.05 are (0.6325, 0.0947) & (0.6252, 0.0922) respectively. Thus, (dpPISl⁎,dpNISl⁎)=(0.0947,0.6252). Then, using Eqs. (19) and (20), the problem is transformed into the following crisp linear programming problems.

Case 1Max(λ2−ϑ2−δ2)Subject toUiμ−dpPISlUiμ−Liμ≥λ2;dpNISl−LiμUiμ−Liμ≥λ2dpPISl−LiσUiσ−Liσ≤ϑ2;Uiσ−dpNISlUiσ−Liσ≤ϑ2dpPISl−LiγUiγ−Liγ≤δ2;Uiγ−dpNISlUiγ−Liγ≤δ2yk−(yk⁎−TkLT)TkLT≥λ2;(yk⁎+TkUT−yk)TkUT≥λ2(yk⁎−TkLI)−ykTkLT−TkLI≤ϑ2;yk−(yk⁎+TkUI)TkUT−TkUI≤ϑ2(yk⁎−TkLF)−ykTkLT−TkLF≤δ2;yk−(yk⁎+TkUF)TkUT−TkUF≤δ2λ2+ϑ2+δ2≤3λ2≥ϑ2;λ2≥δ2

(26) All the constraints of Eq. (21)

Case 2Maxη2(λ2−ϑ2−δ2)+(1−η2){w1l[μ(dpPISl)−σ(dpPISl)−γ(dpPISl)]+w2l[μ(dpNISl)−σ(dpNISl)−γ(dpNISl)]}Subject to

(27) All the constraints of Eq. (26)

Solving the above problems (26) and (27) by using exponential, hyperbolic, and linear membership functions, the second-level decision-maker gets the optimal area of crops to cultivate to get the maximum profit with minimum expenditure, as mentioned in Table 3, Table 4.Table 3 Non-Interactive Neutrosophic Solutions with existing approaches.

Table 3Crop area	FOT	IFOT	Torabi approach	NLMPA	NEMPA	NHMPA	
y1	0	0	0	0	0	0	
y2	14.0541	14.0541	14.0541	14.0538	14.0537	14.0541	
y3	0	0	0	0	0	0	
y4	2.3129	2.3129	2.3129	1	1	1	
y5	2.0998	2.0998	2.0998	12.1072	12.1072	11.7591	
y6	1.2649	1.2649	1.2649	0.1685	0.1685	0.1685	
y7	0	0	0	0.1411	0.1410	0.4559	
y8	0	0	0	0	0	0	
y9	8.8136	8.8136	8.8136	8.8136	8.8136	8.8136	
y10	0	0	0	0	0	0	
y11	0	0	0	0	0	0	
λ2	0.8246	0.8246	0.8246	0.9975	0.996	0.9976	
ϑ2	-	-	-	0.0932	0.0568	0.0075	
δ2	-	0.0838-	-	0.0932	0.0568	0.0075	
MaxO1	137570	137570	137570	146080	146080	146440	
MaxO2	3850600	3850600	3850600	4644000	4644000	4627800	
MinO3	194700	194700	194700	234000	233990	234000	
MinO4	94580	94580	94580	135340	135340	134410	
MinO5	98008	98008	98008	117780	117780	117790	
MinO6	106850	106850	106850	140640	140640	139590	
Abbreviations: NLMPA- Neutrosophic Linear Membership Programming Approach, NEMPA- Neutrosophic Exponential Membership Programming Approach, NHMPA- Neutrosophic Hyperbolic Membership Programming Approach.

Table 4 Interactive Neutrosophic solutions.

Table 4Crop area	INLMPA	INEMPA	INHMPA	
η2 = 0.1,…,0.7	η2 = 0.1,…,0.6	η2 = 0.1	
y1	0	0	0	
y2	14.0541	14.0541	14.0541	
y3	0	0	0	
y4	1	1	1	
y5	12.2642	12.2642	12.2642	
y6	0.1685	0.1685	0.1685	
y7	0	0	0	
y8	0	0	0	
y9	8.8136	8.8136	8.8136	
y10	0	0	0	
y11	0	0	0	
λ2	0.9971	0.9955	0.9974	
ϑ2	0.09353	0.057	0.0075	
δ2	0.09353	0.057	0.0075	
MaxO1	145926	145926	145926	
MaxO2	4651397	4651397	4651397	
MinO3	234000	234000	234000	
MinO4	135768	135768	135768	
MinO5	117779	117779	117779	
MinO6	141110	141110	141110	
Abbreviations: INLMPA- Interactive Neutrosophic Linear Membership Programming Approach, INEMPA- Interactive Neutrosophic Exponential Membership Programming Approach, INHMPA- Interactive Neutrosophic Hyperbolic Membership Programming Approach.

4.1 Comparative analysis

The bi-level TOPSIS non-interactive and interactive neutrosophic programming methods have been used in this study to resolve the multi-objective linear agricultural land allocation problem for medium-sized farm owners in the Ariyalur district. The comparative analyses of cases 1 and 2 are as follows:

Case 1

In this case, the first-level decision maker's optimal results for productivity, profit, labour cost, seed cost, pesticides and fertilizer cost, and miscellaneous cost using linear, exponential, and hyperbolic membership functions are (146084.3, 4644061, 234000.4, 135345.7, 117783.6, 140639.7), (146084.5, 4644070, 234000.9, 135346.2, 117783.8, 140640.1), and (146436.1, 4627737, 234000.5, 134406, 117793, 139591); the degree of acceptances in these three cases are 0.9987, 0.998, and 0.9974, respectively. The second level optimal solutions, using these three types of membership functions are tabulated in Table 3. Comparing profit and net profit, the linear and exponential membership programming approaches provided almost the same results, and the results obtained from these approaches are superior to the hyperbolic membership programming approach. Comparing the degrees of acceptance, indeterminacy, and falsity, the hyperbolic membership programming approach provided the most satisfactory solution.

Case 2

In this case, the compromise solution is found by varying the compensation coefficient at both levels between 0 and 1. After analysis, the decision maker receives the best optimal solution (MaxO1,MaxO2,MinO3,MinO4,MinO5,MinO6)=(145926.299,4651397,234000.5,135768.1,117779.4,141110.9) at both levels with the same compensation coefficients η1 & η2 as (0.1, 0.2, 0.3, 0.4, 0.5, 0.6,0.7) in linear, η1 & η2 as (0.1,0.2, 0.3, 0.4, 0.5, 0.6) in exponential and η1 & η2 as 0.1 in hyperbolic membership function.

Comparing the outcomes of the two cases, Case 2 offered the most effective solutions. Also, for validation, the results obtained in these two cases are compared with some existing techniques such as FOT [43], IFOT [23], and the Torabi approach [44]. The optimum production, profit, labour cost, seed cost, pesticides and fertilizer cost, miscellaneous cost, and net profit obtained by using the above existing methods are the same, and they are (137570kg, Rs. 3850600, Rs. 194700, Rs. 94580, Rs. 98008, Rs. 106850, Rs. 3356462) with a 0.8246 degree of acceptance, which is less than the degree of acceptance of the proposed approach. The comparison of case 1 with existing approaches is tabulated in Table 3, and the case 2 findings are tabulated in Table 4. Additionally, Figure 4, Figure 5 compare the production and expenditure as well as the profit and net profit in non-interactive neutrosophic and existing approaches, and Fig. 6 shows the comparative analysis of the net profit of cases 1 and 2.Figure 4 Comparative Analysis of Production and Expenditure in Bi-level TOPSIS Non-Interactive Neutronosophic and Existing Approaches.

Figure 4

Figure 5 Comparative Analysis of Profit and Net Profit in Bi-level TOPSIS Non-Interactive Neutrosophic and Existing Approach.

Figure 5

Figure 6 Comparative Analysis of Net Profit in Bi-level TOPSIS Non-Interactive and Interactive Neutrosophic Programming Approaches.

Figure 6

4.2 Sensitivity analysis

In the interactive neutrosophic programming approach, the compensation coefficients η1 and η2 implicitly control the minimum satisfaction level of objectives and the degree of compromise between objectives. That is, by varying the value of parameters η1 and η2, the suggested formulation of case 2 can produce compromised solutions that are both unbalanced and balanced for a particular problem instance, depending on the decision maker's preferences. In case 2, a sensitivity analysis is performed to find potential effective solutions to the problem by changing the compensation coefficient values η1 and η2 from 0 to 1 at levels 1 and 2. This study shows that when a linear membership function is used, INPA offers balanced solutions when the compensation coefficient is 0.1≤η1,η2≤0.7 and unbalanced solutions when the compensation coefficients are η1,η2≥0.8 in terms of production, profit, labour cost, seed cost, fertilizer & pesticide costs and miscellaneous cost. It generates a balanced solution when the compensation coefficients are 0.1≤η1,η2≤0.6 and an unbalanced solution when the compensation coefficients are η1,η2≥0.7 using exponential membership functions. The INPA is highly sensitive to the parameter η1,η2 when utilising the hyperbolic membership function. Therefore, the imbalanced solution is given for values of η1,η2 between 0.1 and 0.9. The level 2 solution set for profit from this analysis utilising linear, exponential, and hyperbolic membership functions is shown in Table 5. Additionally, this study shows that in linear, exponential, and hyperbolic membership functions, profit and net profit decrease as the compensation coefficient increases from 0.1 to 0.9.Table 5 Sensitivity analysis for profit.

Table 5η2	Profit	
Linear	Exponential	Hyperbolic	
0.1	4651397	4651397	4651397	
0.2	4651397	4651397	4649623	
0.3	4651397	4651397	4644091	
0.4	4651397	4651397	4644070	
0.5	4651397	4651397	4644050	
0.6	4651397	4651397	4641048	
0.7	4651397	4644087	4637032	
0.8	4643690	4644048	4633508	
0.9	4616089	4643973	4630441	

5 Managerial insights and practical implications

The food demand of the people of India, which has reached the first place in population, is increasing day by day. The government of India has come up with several new schemes to meet the food demand of India's growing population. To manage food demand, crop planning is crucial to agriculture. Successful crop planning helps to manage soil fertility, seasonality, and productivity variation. The present study, using a bilevel TOPSIS-based interactive and non-interactive neutrosophic programming technique, is a new approach for crop land allocation to a medium farmer in Ariyalur district, and it has given a more satisfactory solution than the existing approaches. Therefore, it helps to make a crop plan for state and national-level agriculture.

6 Conclusion

Neutrosophic set theory is an effective technique for addressing data uncertainty and inaccuracy in a variety of real-world issues. In this study to discover the best solution to the multi-objective crop land allocation problem for the medium farm owner in the Ariyalur district, which has a 15-acre plot of land, a new bi-level TOPSIS interactive and non-interactive neutrosophic programming technique with linear, hyperbolic, and exponential membership functions is utilized, whereas the other researchers used linear membership functions only. Comparing the three membership functions, linear and exponential membership functions offered a more satisfactory solution than the hyperbolic membership function approach in terms of net profit. However, when comparing the degree of acceptance, the hyperbolic membership function offered the most satisfactory solution than the linear and exponential membership functions in case 1. In case 2, all the membership functions provided the same net profit, and comparing the acceptance's degree, the hyperbolic membership function is superior to the linear and exponential membership functions. The problem is also resolved by utilizing three other existing approaches, namely fuzzy and intuitionistic fuzzy optimization approaches and Torabi interactive fuzzy optimization strategies, to show the effectiveness of the bi-level TOPSIS-based neutrosophic programming approach. The outcomes of the suggested methodology are tabulated in Table 3, Table 4. Also, Figure 4, Figure 5 show the comparative analysis of case 1's production, expenditure, and net profit with existing approaches, and Fig. 6 shows the comparative analysis of net profit in two cases. From these tables and figures, it is clear that the proposed strategy outperforms the current approaches. The proposed method is employed only for efficient land allocation on medium-sized farms. However, this methodology can also be applied to improve irrigation systems and crop combinations at the state and national levels using bi-level or multi-level optimization techniques. Also, by using the proposed methodology, the crop planning model can be designed for rain-fed areas to meet the food requirements of the growing population. Numerous real-world issues, including those in business, transportation, agriculture, finance, public policy, and supply chain management, have been tackled using fuzzy and intuitionistic fuzzy optimization techniques. As a result, the suggested strategy is utilized to identify the most effective, gratifying remedies to the aforementioned issues. Nowadays, a wide variety of bio-inspired algorithms are employed to find the optimal solution for many complex real-world problems. The most effective solutions will be discovered by fusing the suggested methodology with bio-inspired algorithms for real-world issues where uncertainty and indeterminacy factors occur. Future studies can also focus on creating a generic neutrosophic framework with bio-inspired algorithms that is appropriate for a variety of real-world issues, especially to optimize agricultural benefits.

CRediT authorship contribution statement

Angammal S: Writing – original draft, Validation, Methodology, Investigation, Data curation. Hannah Grace G: Writing – review & editing, Validation, Supervision, Investigation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

All data analysed during this study are included in the submitted article.

Acknowledgments

Authors are extremely grateful to Mr. A. Sivakumar and family for providing the farm data necessary to use the model. The authors appreciate the efforts of the anonymous reviewers toward this publication. The support from Vellore Institute of Technology University, Chennai is also deeply appreciated.
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References

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