A Fuzzy Multi-Criteria Decision-Making Approach for Agricultural Land Selection

Abstract

Decision-making involves selecting the best alternative based on evaluation criteria while considering environmental impacts. The translation of environmental factors into quantifiable mathematical expressions is challenging due to the inherent uncertainties. Decision-makers can address the subjective characteristics of alternatives by incorporating fuzzy set theory into decision-making processes where uncertainty and ambiguity exist. Game theory is introduced as another approach to enhance the robustness of decision-making models, leading to more informed and flexible decision outcomes. This approach promotes strategic thinking and aids decision-making by allowing individuals to visualize the potential consequences of different decisions under various conditions. This study proposes a fuzzy multi-criteria decision support system that provides a structured framework to address the complexities of agricultural land selection. The decision support system employs a two-person zero-sum game to identify the optimal land management option, considering the strategic interactions between players. The results from the payoff matrix reveal the equilibrium point, providing an ideal solution for more effective land use planning decisions.

1. Introduction

Climate change has seriously affected agriculture, especially water resources and food production. The increasing demands of the population and the deterioration of natural resources present critical issues for the sustainability of agricultural practices [1,2]. Although new agricultural technology increases productivity, natural resources are depleted over time. Technology deployment in agriculture must be balanced with ecological impact. Precision agriculture can substantially enhance resource efficiency by ensuring that water, nutrients, and other inputs are applied only when and where needed. Policymakers should promote sustainable practices through incentives and regulations, while education programs can empower farmers and communities to adopt new technologies and methods effectively. As a result of climate change, the agricultural production sector faces significant challenges that could increase the risk of food insecurity and famine. A plant requires rain during specific periods, and the soil must maintain a certain moisture content. Climate-induced drought poses a severe threat to humanity. Drought is a problem caused by precipitation that falls substantially below the seasonal normal. The water-related problem also significantly affects production [3]. The agricultural production sector faces increasing demand for food and limited water resources because of climate change and population growth [4,5,6]. Sustainable use of agricultural resources is crucial for addressing current and future challenges such as food security, climate change, and biodiversity loss [7,8]. Furthermore, selecting suitable agricultural land is critical for promoting sustainable farming practices [9]. Conscious producers can assess various factors, such as soil quality, water availability, and local climate conditions, to select land that optimally supports crop growth while minimizing environmental impacts. By utilizing appropriate agricultural land, producers can reduce the need for inputs such as fertilizers and irrigation, thereby decreasing costs and environmental degradation [10].

Türkiye has agricultural potential in terms of climate, soil fertility, and natural resources, particularly for fruit tree cultivation. An intricate balance between ecological conditions and practical considerations must be achieved in establishing orchards. Fruit trees are long-lived plants that can produce products for many years, unlike plants that produce seasonally. Of course, this lifespan can vary greatly depending on the type of tree, soil conditions, and climatic conditions. Therefore, before an orchard project is undertaken, it is crucial to correctly determine the climate and soil conditions required for the type of fruit tree to be produced [11]. Before planting, conducting thorough soil tests can provide valuable information about soil pH, nutrient content, organic matter, and texture. Understanding climatic patterns (temperature ranges, precipitation, and humidity) is essential for predicting fruit tree viability. Geographic location influences various factors, such as sunlight exposure and wind patterns. Given the importance of water for fruit tree cultivation, planning an efficient irrigation system is vital. Water sources, quality, and seasonal availability can help mitigate the risks associated with drought or overwatering. In addition to ecological considerations, economic factors, such as transportation infrastructure, labor availability, market access, and energy sources, should be analyzed. These factors can significantly impact the overall feasibility and sustainability of an orchard project. By taking a holistic approach that encompasses ecological and economic considerations, fruit tree cultivation can become productive and sustainable over the long term [12]. The environmental and financial factors of agricultural production are diverse. It is difficult to evaluate these factors because they are at different levels and contain uncertainties. Therefore, mathematical methods can be used concisely and systematically for more informed decision-making [13]. Access to timely and accurate information enables informed and consistent decision-making, ultimately strengthening the resilience and success of agricultural enterprises.

The complexity of environmental factors in decision-making has increased considerably over time, and addressing this complexity often requires a structured, competitive approach [14,15]. Game theory, which emerged during World War II to solve problems in competitive environments using mathematical models, provides a robust framework for this type of decision-making. Today, game theory is applied in various fields, including engineering, economics, and management science [16,17]. The basic game theory structure comprises four fundamental problems. The first problem is to construct a mathematical model that accurately represents a given situation or event. The proposed model simplifies real-life interactions into a format that can be mathematically examined. It outlines the players, strategies, payoffs, and other relevant factors in the competitive scenario. Once the model is built, the next problem is to identify each player’s best strategy. This involves analyzing all available strategies and determining which strategy provides the most favorable outcome. After outlining the potential strategies, the third challenge is to investigate the existence of the best strategy for each player. The final objective is to develop effective analytical and numerical methods to obtain the desired results. This involves creating algorithms or formulae to solve the game and provide insights into the most advantageous strategies for all players [18]. Multi-criteria decision-making (MCDM) methods are essential for evaluating alternatives across multiple criteria [19]. Chakraborty et al. [20] highlighted that these methods help find the best results by assessing qualitative and quantitative variables. Most MCDM problems involve human-driven uncertainties, which align with the principles of strategic game theory, in which independent decision-makers make choices based on their preferences [21]. The criteria, alternatives, and performance of each criterion relative to other options form the basis of MCDM [22,23]. This corresponds to the fundamental elements of a strategic game [24]. A strategic game involves multiple independent decision-makers who make individual choices that determine outcomes, and have individual preferences regarding the possible consequences of the conflict [25]. Agricultural land selection can be framed as a strategic game involving multiple decision-makers. This game represents conflicting interests in which each player has preferences over the possible outcomes. Game theory helps simulate this competitive decision-making environment, allowing for more structured conflict resolution [26,27].

This paper presents a decision support system that integrates game theory with fuzzy TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) to solve the land selection problem when establishing an orchard. We defined the factors for agricultural land selection based on expert interviews, brainstorming, and cross-functional teams, and performed the necessary tests and analyses. Considering the protection of natural resources during production, the most critical stage of this problem is determining which criteria should be used to evaluate agricultural lands. When dealing with multi-criteria selection problems, producers aim to invest in agricultural land that provides profits while contributing positively to humankind and nature. In addition to providing financial gains, the projects are assessed based on their potential to positively contribute to environmental sustainability. The land selection problem is transformed into a structured game in which the strategies of each decision-maker are evaluated according to multiple criteria. This integration ensures that financial and environmental objectives are met, resulting in more sustainable establishment processes.

The remainder of this paper is organized as follows: Section 2 introduces a case study to illustrate the integrated approach using fuzzy TOPSIS and game theory. We present comparison results and emphasize how the integrated approach thoroughly assesses the alternatives. Finally, Section 3 concludes the study and offers potential extensions for future work.

2. Materials and Methods

This study analyzes the problem of selecting land for fruit tree cultivation in the agricultural sector, focusing on the complex and interrelated factors that determine the success and sustainability of orchards. Strategies and criteria are established when selecting among alternatives. The closeness coefficients of the alternatives are derived from the analysis, reflecting how effectively each alternative can be implemented and its potential impact. Once the closeness coefficients are calculated, they can be used to generate payoffs for each player. These payoffs are typically derived by multiplying the effectiveness (or benefit) of a particular strategy by the implementation coefficient of the corresponding alternative. The value of each alternative is determined using the equilibrium point from game theory. MCDM methods are essential for converting qualitative expressions into quantitative data in multi-criteria decision-making problems. When choosing between two alternatives, the advantages held by each player may affect the others. In this case, game theory is an effective tool for determining the optimal option. Several factors need to be considered when establishing an orchard. The primary processes include site selection, species and variety selection, rootstock selection, pollination needs, planting systems and frequency, seedling type and number, planting time, land preparation, seedling planting, and intermediate cultivation. We consider selecting the best land among alternatives for establishing a pear garden. According to the predetermined criteria, two land alternatives were identified in Türkiye. Fuzzy TOPSIS and game theory were integrated to address the decision-making problem involving multiple stakeholders with different objectives. This process is considered from a two-person, zero-sum game perspective. It aims to obtain results by establishing two separate games in which the gains and losses of both players are not necessarily zero-sum. In the first game, the players were the producer and the first land, while in the second game, they were the producer and the second land. Land selection was performed by comparing the results of the two games. The game payoff matrix was created using the closeness coefficient values obtained from the fuzzy TOPSIS method. The decision support system development in this case study includes four main phases.

Preparation Phase (P1)

At this stage of the land selection process, producer targets are defined, and the problem is thoroughly analyzed. This phase is critical for aligning the selection process with the overarching goals of the agricultural project, ensuring that the decision-making process focuses on factors that will contribute to the orchard’s success.

Step 1.1. Analysis of the situation: The producer’s goal is to choose land that meets specific criteria for establishing a pear garden. The agricultural engineer evaluated alternative lands by examining key factors, like soil quality, which includes pH, fertility, drainage, and texture, as well as climate conditions such as temperature, rainfall, and frost risk. This thorough assessment ensured that the selected land would provide an optimal environment for establishing and sustaining a productive pear garden. As a result of preliminary assessments, two alternative land options have been identified for further evaluation. Fuzzy TOPSIS is applied in problem-solving to assess and rank alternatives based on multiple, often conflicting criteria and to deal with uncertainty or imprecise decision-making. Since the aim is to evaluate the characteristics of the lands and the features desired by the producer, game theory is a useful method for obtaining the optimal gain. The properties of the land alternatives were converted from linguistic variables to numerical data to improve the accuracy and precision of decision-making. This conversion is crucial for obtaining objective and reliable results, as it enables qualitative assessments to be quantified and compared in a more systematic manner. The two-person zero-sum game was applied because the interaction between the producer and the land, where one player’s success (the land’s suitability) results in a gain for the producer without causing a direct loss to the other player, aligns with the principles of this game.

Step 1.2. Identification of the decision-making group: The decision-making group comprises an agricultural engineer, a producer, an experienced expert, and three landowners. Considering the required features, land alternatives are evaluated based on criteria such as climate, soil structure, groundwater, land slope, transportation, energy access, and land reclamation. By incorporating insights from multiple experts and aggregating their fuzzy evaluations through fuzzy averaging, we achieve a more robust fuzzy representation.

Determining Phase (P2)

At this stage, criteria, strategies, and linguistic variables are systematically established, providing a structured basis for informed and robust decision-making.

Step 2.1. Determining strategies and criteria: Strategies highlight the strengths of alternative land options, emphasizing the unique advantages each land offers based on factors like soil quality, climate, and resource availability. These strategies for the two land options were developed by analyzing soil and landscape properties, which are crucial for maximizing agricultural efficiency and ensuring long-term sustainability. The primary advantage of the first land is its lower cost, which reduces investment expenses, making it more feasible for producers to enter the market. The land’s climate suitability, coupled with convenient irrigation, easy energy access, and excellent road connectivity, strengthens its position as an ideal choice for establishing a pear garden. These factors contribute significantly to operational efficiency, reducing logistical challenges and costs for establishing and maintaining the pear garden. On the other hand, the second land is particularly suitable for pear cultivation because of its soil structure. The soil type and nutrient content of the land are well suited for pear cultivation, leading to better crop yields and quality. Additionally, the land benefits from favorable climate characteristics, including ideal temperature, rainfall, and minimal frost risk. The mild breezes of the Mediterranean climate offer several advantages, including moderating temperatures, enhancing air circulation, and mitigating the risk of excessive wind damage. The land has a gentle slope, which ensures soil stability and a smooth surface, reducing the risk of erosion and facilitating farming. The strategies of the first land include a lower purchasing cost, access to the city water network, mild winters with moderate temperatures, easy energy access, and convenient road transportation. Alternatively, the strategies of the second land focus on soil type and nutrient content, climate and wind conditions, and erosion control.

In this strategic game, the producer and the land are two players, each with distinct strategies and objectives. They work towards optimizing outcomes based on their respective priorities and conditions. The strategies are evaluated based on seven criteria essential for land selection. These criteria assess how effectively each strategy meets the producer’s needs and ensures the land’s suitability.

Table 1. Strategies and evaluation criteria for land selection.

Strategies of the First Land	Strategies of the Second Land	Producer Criteria
(SF1) Low purchasing cost	(SS1) Soil reaction (pH) and salinity	(C1) Climate
(SF2) Availability and quality of water	(SS2) High nutrient levels	(C2) Soil quality
(SF3) Suitable winter climate	(SS3) Moderate temperature	(C3) Water availability
(SF4) Easy access to energy	(SS4) Soil stability	(C4) Topography
(SF5) Easy road transportation	(SS5) Smooth land surface	(C5) Transportation
	(SS6) Low risk of wind damage	(C6) Energy access
		(C7) Biodiversity

outlines the strategies of the two lands and the producer’s evaluation criteria. A strategy of land represents a specific strategic approach or a set of characteristics associated with the land that aligns with the producer’s objectives, such as climate suitability, soil quality, or cost considerations. The producer’s criteria indicate the producer’s choice or preference based on their evaluation of the land’s characteristics, including factors like irrigation potential, slope, and access to transportation.

Step 2.2. Determining Linguistic Variables: The producer defined the linguistic variables and evaluation scores applied in the assessment process. The purpose of using fuzzy numbers is to reduce the uncertainties that arise when translating human expressions into numerical data [28,29,30,31]. The linguistic expressions and their corresponding triangular numbers are presented in

Table 2. Fuzzy linguistic terms and fuzzy numbers for evaluating the criteria and strategies.

The Importance of the Criteria	The Importance of the Strategy	Fuzzy Numbers
Extreme importance (E)	Very good (VG)	(1.00	1.00	0.86)
Very high (VH)	Good (G)	(0.97	0.86	0.68)
High (H)	Medium good (MG)	(0.86	0.68	0.53)
Medium (M)	Medium (M)	(0.68	0.53	0.39)
Low (L)	Medium poor (MP)	(0.58	0.39	0.25)
Very low (VL)	Poor (P)	(0.39	0.25	0.10)
No importance (N)	Very poor (VP)	(0.25	0.00	0.00)

.

Calculating Phase (P3)

The importance of the criteria and their effects on the strategies were determined through a questionnaire administered to the decision-making group.

Step 3.1. Calculating the linguistic variable scale: The linguistic expressions related to the criteria for land selection are presented in

Table 3. Evaluation of criteria based on linguistic variables.

	Decision-Makers
Criteria	DM1	DM2	DM3	DM4	DM5	DM6
C1	E	E	E	E	E	E
C2	H	H	H	VH	VH	VH
C3	VH	VH	VH	E	E	H
C4	H	M	L	M	M	H
C5	E	E	E	E	E	E
C6	VH	VH	VH	E	E	H
C7	H	VH	H	E	E	M

. The expressions indicating the importance of the strategies for the first land are listed in

Table 4. Evaluation of the first land based on the first criterion.

Criterion		Linguistic Variables
		SF1	SF2	SF3	SF4	SF5
C1	DM1	P	P	M	M	G
	DM2	VP	MP	MG	MP	VG
	DM3	MP	M	VG	M	VG
	DM4	VG	VG	MG	M	G
	DM5	P	P	M	M	G
	DM6	P	P	M	MP	G

and

Table 5. Producer’s evaluation of the criteria for the first strategy.

Strategy		Linguistic Variables
		${\mathit{C}}_1$	${\mathit{C}}_2$	${\mathit{C}}_3$	${\mathit{C}}_4$	${\mathit{C}}_5$	${\mathit{C}}_6$	${\mathit{C}}_7$
SF1	DM1	G	MG	MG	MP	G	P	M
	DM2	VG	MP	M	MP	MG	P	MG
	DM3	MG	M	M	G	MG	P	MP
	DM4	VP	VP	MP	M	MP	MP	MP
	DM5	VP	VP	MP	M	MP	MP	MP
	DM6	G	MG	M	MG	P	P	P

.

${DM}_k$ = decision-makers $\left(k=1,2,\dots ,6\right)$

$C_i$ = $i$th criteria of the producer $(i=1, 2, \dots , 7)$

${SF}_j$ = $j$th strategy of the first land $(j=1, 2, \dots , 5)$

${SS}_j$ = $j$th strategy of the second land $(j=1, 2, \dots , 5)$

Step 3.2. Calculating criteria and their importance: In this step, the fuzzy triangular numbers corresponding to the linguistic variables, as well as the importance of the criteria and strategies, are calculated [29,30,31,32].

Table 6. Corresponding fuzzy numbers used in the criteria evaluation.

	${\mathit{C}}_1$			${\mathit{C}}_2$			${\mathit{C}}_3$			${\mathit{C}}_4$			${\mathit{C}}_5$			${\mathit{C}}_6$			${\mathit{C}}_7$
DM1	1.00	1.00	0.86	0.86	0.68	0.53	0.97	0.86	0.68	0.86	0.68	0.53	1.00	1.00	0.86	0.97	0.86	0.68	0.86	0.68	0.53
DM2	1.00	1.00	0.86	0.86	0.68	0.53	0.97	0.86	0.68	0.68	0.53	0.39	1.00	1.00	0.86	0.97	0.86	0.68	0.97	0.86	0.68
DM3	1.00	1.00	0.86	0.86	0.68	0.53	0.97	0.86	0.68	0.58	0.39	0.25	1.00	1.00	0.86	0.97	0.86	0.68	0.86	0.68	0.53
DM4	1.00	1.00	0.86	0.97	0.86	0.68	1.00	1.00	0.86	0.68	0.53	0.39	1.00	1.00	0.86	1.00	1.00	0.86	1.00	1.00	0.86
DM5	1.00	1.00	0.86	0.97	0.86	0.68	1.00	1.00	0.86	0.68	0.53	0.39	1.00	1.00	0.86	1.00	1.00	0.86	1.00	1.00	0.86
DM6	1.00	1.00	0.86	0.97	0.86	0.68	0.86	0.68	0.53	0.86	0.68	0.53	1.00	1.00	0.86	0.86	0.68	0.53	0.68	0.53	0.39

and

Table 7. Fuzzy weight matrix for criteria.

Criteria	a	b	c	Weight
(C1)	1.000	1.000	0.860	0.977
(C2)	0.860	0.770	0.680	0.770
(C3)	0.860	0.877	0.860	0.871
(C4)	0.580	0.557	0.530	0.556
(C5)	1.000	1.000	0.860	0.977
(C6)	0.860	0.877	0.860	0.871
(C7)	0.680	0.792	0.860	0.784

show triangular fuzzy numbers based on the linguistic expressions for the criteria and fuzzy weight matrix, respectively. The triangular fuzzy weights of the criteria and aggregated fuzzy weights are calculated as follows.

$$a=\mathit{min}\left\{1.00 1.00 1.00 1.00 1.00 1.00\right\}=1.00$$

(1)

$$b={\displaystyle \frac{1}{6}}\left(1.00+1.00+1.00+1.00+1.00+1.00\right)=1.00$$

(2)

$$c=max\left\{0.86 0.86 0.86 0.86 0.86 0.86\right\}=0.86$$

(3)

$$R\left(C_1\right)={\displaystyle \frac{1.00+\left(4x1.00\right)+0.86}{6}}=0.977$$

(4)

Step 3.3. Analyzing strategies based on evaluation criteria and calculating closeness coefficients (CC): The strategies in the game are compared with those of the competing players [33].

Table 8. Fuzzy decision matrix for the first land.

	${\mathit{C}}_1$			${\mathit{C}}_2$			${\mathit{C}}_3$			${\mathit{C}}_4$			${\mathit{C}}_5$			${\mathit{C}}_6$			${\mathit{C}}_7$
SF1	0.25	0.36	0.86	0.58	0.56	0.86	0.58	0.51	0.53	0.39	0.54	0.53	0.39	0.47	0.68	0.86	0.83	0.68	0.58	0.74	0.86
SF2	0.39	0.45	0.86	0.58	0.56	0.68	0.86	0.77	0.68	0.58	0.56	0.53	0.58	0.59	0.68	0.68	0.68	0.86	0.68	0.69	0.68
SF3	0.68	0.66	0.86	0.86	0.85	0.86	0.68	0.79	0.86	0.86	0.79	0.86	0.68	0.79	0.86	0.86	0.95	0.86	0.68	0.77	0.86
SF4	0.58	0.69	0.39	0.58	0.56	0.53	0.68	0.61	0.53	0.68	0.66	0.53	0.58	0.54	0.53	0.68	0.77	0.86	0.68	0.66	0.53
SF5	0.97	0.91	0.86	0.86	0.89	0.86	0.68	0.87	0.86	0.68	0.87	0.86	0.86	0.95	0.86	0.68	0.92	0.86	0.86	0.89	0.86

displays the fuzzy decision matrix of the criteria for evaluating the strategies of the first land.

Calculating triangular fuzzy weights for strategies and aggregated fuzzy ratings:

$${\widetilde{SF}}_1(a, b, c), {DM}_k=1, 2, \dots , 6$$

$$a=\min \left\{0.39 0.25 0.58 1.00 0.39 0.39\right\}=0.25$$

(5)

$$b={\displaystyle \frac{1}{6}}\left(0.25+0.00+0.39+1.00+0.25+0.25\right)=0.36$$

(6)

$$c=max\left\{0.10 0.00 0.25 0.86 0.10 0.10\right\}=0.86$$

(7)

Construction of the normalized fuzzy decision matrix: The normalized fuzzy decision matrix is constructed as shown in Equations (8) and (9).

Table 9. Normalized fuzzy decision matrix for the first land.

	${\mathit{C}}_1$			${\mathit{C}}_2$			${\mathit{C}}_3$			${\mathit{C}}_4$			${\mathit{C}}_5$			${\mathit{C}}_6$			${\mathit{C}}_7$
SF1	0.26	0.37	0.89	0.65	0.63	0.96	0.67	0.59	0.61	0.45	0.62	0.61	0.41	0.50	0.72	0.91	0.88	0.72	0.65	0.83	0.76
SF2	0.40	0.46	0.89	0.65	0.63	0.76	0.99	0.89	0.78	0.67	0.64	0.61	0.61	0.62	0.72	0.72	0.72	0.91	0.76	0.77	0.96
SF3	0.70	0.68	0.89	0.96	0.95	0.96	0.78	0.91	0.99	0.99	0.91	0.99	0.72	0.83	0.91	0.91	1,00	0.91	0.76	0.86	0.59
SF4	0.60	0.71	0.40	0.65	0.63	0.59	0.78	0.70	0.61	0.78	0.75	0.61	0.61	0.57	0.56	0.72	0.81	0.91	0.76	0.73	0.96
SF5	1,00	0.93	0.89	0.96	1,00	0.00	0.78	1,00	0.99	0.78	1,00	0.99	0.91	1,00	0.91	0.72	0.97	0.91	0.96	1,00	0.96

presents the corresponding normalized fuzzy decision matrix.

$$d_1^{*}=\underset{i}{\max }\left\{0.25 0.36 0.86 \dots 0.39 0.97 0.91 0.86\right\}=0.97$$

(8)

$$r_{11}=\left({\displaystyle \frac{0.25}{0.97}}, {\displaystyle \frac{0.36}{0.97}}, {\displaystyle \frac{0.86}{0.97}} \right)=(0.26 0.37 0.89)$$

(9)

Construction of the weighted normalized fuzzy decision matrix: The weighted normalized fuzzy decision matrix shown in

Table 10. Weighted normalized fuzzy decision matrix for the first land.

	${\mathit{C}}_1$			${\mathit{C}}_2$			${\mathit{C}}_3$			${\mathit{C}}_4$			${\mathit{C}}_5$			${\mathit{C}}_6$			${\mathit{C}}_7$
SF1	0.26	0.37	0.76	0.56	0.49	0.65	0.57	0.51	0.52	0.26	0.34	0.32	0.41	0.50	0.62	0.78	0.77	0.62	0.44	0.65	0.65
SF2	0.40	0.46	0.76	0.56	0.48	0.52	0.85	0.78	0.67	0.39	0.36	0.32	0.61	0.62	0.62	0.62	0.63	0.78	0.52	0.61	0.83
SF3	0.70	0.68	0.76	0.83	0.73	0.65	0.67	0.80	0.85	0.57	0.50	0.52	0.72	0.83	0.78	0.78	0.88	0.78	0.52	0.68	0.51
SF4	0.60	0.71	0.35	0.56	0.48	0.40	0.67	0.61	0.52	0.45	0.42	0.32	0.61	0.57	0.48	0.62	0.71	0.78	0.52	0.58	0.83
SF5	1,00	0.93	0.76	0.83	0.77	0.00	0.67	0.88	0.85	0.45	0.56	0.52	0.91	1,00	0.78	0.62	0.85	0.78	0.65	0.79	0.83

was obtained by multiplying the normalized fuzzy decision matrix by the weights of the evaluation criteria.

$${\tilde{v}}_{11}={\tilde{r}}_{11}\left(.\right){\tilde{w}}_1=\left(0.26 0.37 0.89\right)x \left(1.00 1.00 0.86\right)=(0.26 0.37 0.76)$$

(10)

The Fuzzy Positive Ideal Solution (FPIS) and Fuzzy Negative Ideal Solution (FNIS) for the alternatives are determined as follows:

$$v_1^{+}=\underset{j}{\mathit{max}}\left\{0.26 0.37 0.76 \dots 1.00 0.93 0.76\right\}=0.76$$

(11)

$$v_1^{-}=\underset{j}{\mathit{min}}\left\{0.26 0.37 0.76 \dots 1.00 0.93 0.76\right\}=0.26$$

(12)

Calculating the distance for each alternative from the FPIS and FNIS: The FPIS and FNIS were calculated from the weighted normalized fuzzy values, as presented in

Table 11. Ideal solutions for the first land.

	${\mathit{C}}_1$			${\mathit{C}}_2$			${\mathit{C}}_3$			${\mathit{C}}_4$			${\mathit{C}}_5$			${\mathit{C}}_6$			${\mathit{C}}_7$
Av−	0.26	0.26	0.26	0.40	0.40	0.40	0.51	0.51	0.51	0.26	0.26	0.26	0.41	0.41	0.41	0.62	0.62	0.62	0.44	0.44	0.44
Av+	0.76	0.76	0.76	0.83	0.83	0.83	0.85	0.85	0.85	0.57	0.57	0.57	0.83	0.83	0.83	0.88	0.88	0.88	0.83	0.83	0.83

,

Table 12. Distance from the FNIS.

	${\mathit{C}}_1$	${\mathit{C}}_2$	${\mathit{C}}_3$	${\mathit{C}}_4$	${\mathit{C}}_5$	${\mathit{C}}_6$	${\mathit{C}}_7$
SF1	0.298	0.570	0.036	0.060	0.128	0.128	0.174
SF2	0.325	0.521	0.265	0.099	0.205	0.095	0.248
SF3	0.458	0.741	0.272	0.276	0.369	0.201	0.150
SF4	0.329	0.485	0.109	0.149	0.151	0.109	0.241
SF5	0.649	0.653	0.301	0.255	0.493	0.166	0.325

and

Table 13. Distance from the FPIS.

	${\mathit{C}}_1$	${\mathit{C}}_2$	${\mathit{C}}_3$	${\mathit{C}}_4$	${\mathit{C}}_5$	${\mathit{C}}_6$	${\mathit{C}}_7$
SF1	0.579	0.271	0.340	0.268	0.498	0.172	0.264
SF2	0.485	0.309	0.131	0.220	0.383	0.214	0.218
SF3	0.288	0.115	0.127	0.050	0.227	0.079	0.270
SF4	0.474	0.353	0.280	0.184	0.450	0.187	0.229
SF5	0.142	0.479	0.118	0.076	0.137	0.161	0.102

.

$$d_1^{-}=\sqrt{{\displaystyle \frac{1}{3}}[{\left(0.26-0.26\right)}^2+{\left(0.37-0.26\right)}^2+{\left(0.76-0.26\right)}^2]}=0.298$$

(13)

$$d_1^{+}=\sqrt{{\displaystyle \frac{1}{3}}[{\left(0.26-1.00\right)}^2+{\left(0.37-1.00\right)}^2+{\left(0.76-1.00\right)}^2]}=0.579$$

(14)

Calculating the closeness coefficients (CC_i) of alternatives: Closeness coefficients indicate the effectiveness of strategies based on given criteria. They represent the relative closeness to the ideal solution by evaluating each strategy’s performance in relation to the opposing player’s strategies [34].

Table 14. CCs of the first land strategies for the producer.

	${\mathit{C}}_1$	${\mathit{C}}_2$	${\mathit{C}}_3$	${\mathit{C}}_4$	${\mathit{C}}_5$	${\mathit{C}}_6$	${\mathit{C}}_7$
SF1	0.340	0.678	0.096	0.183	0.205	0.427	0.397
SF2	0.401	0.627	0.668	0.310	0.349	0.307	0.531
SF3	0.614	0.866	0.682	0.848	0.619	0.718	0.356
SF4	0.410	0.579	0.279	0.448	0.251	0.368	0.513
SF5	0.820	0.577	0.718	0.771	0.783	0.508	0.761

and

Table 15. CCs of the producer criteria for the first land.

	SF1	SF2	SF3	SF4	SF5
C1	0.595	0.599	0.386	0.683	0.180
C2	0.606	0.373	0.134	0.421	0.423
C3	0.904	0.332	0.318	0.721	0.282
C4	0.734	0.841	0.189	0.662	0.282
C5	0.759	0.651	0.381	0.749	0.217
C6	0.869	0.243	0.202	0.243	0.174
C7	0.665	0.366	0.358	0.382	0.360

present the closeness coefficients of the first land strategies and the producer criteria, respectively.

$${CC}_1={\displaystyle \frac{0.298}{0.298+0.579}}=0.340$$

(15)

Decision Phase (P4)

Game theory enables the quantification of potential outcomes based on various combinations of land characteristics and producer priorities. The payoff matrix for the second land was also developed by following the calculation steps outlined in the third phase. After constructing the game payoff matrix in strategic form, the equilibrium point can be identified, enabling decision analysis.

Step 4.1. Constructing the decision payoff matrix: The corresponding payoff matrices for the two-player non-zero-sum game are presented in

Table 16. The payoff matrix for the first land and producer.

	${\mathit{C}}_1$		${\mathit{C}}_2$		${\mathit{C}}_3$		${\mathit{C}}_4$		${\mathit{C}}_5$		${\mathit{C}}_6$		${\mathit{C}}_7$
SF1	0.643	0.188	0.606	0.620	0.904	0.096	0.389	0.183	0.759	0.265	0.869	0.665	0.665	0.397
SF2	0.845	0.269	0.373	0.627	0.332	0.668	0.841	0.310	0.651	0.262	0.446	0.307	0.442	0.531
SF3	0.712	0.722	0.134	0.866	0.657	0.682	0.189	0.848	0.613	0.570	0.403	0.718	0.499	0.356
SF4	0.555	0.428	0.283	0.579	0.721	0.279	0.662	0.448	0.749	0.166	0.243	0.368	0.382	0.513
SF5	0.313	0.739	0.423	0.521	0.282	0.718	0.282	0.601	0.217	0.756	0.174	0.508	0.360	0.761

and

Table 17. The payoff matrix for the second land and producer.

	${\mathit{C}}_1$		${\mathit{C}}_2$		${\mathit{C}}_3$		${\mathit{C}}_4$		${\mathit{C}}_5$		${\mathit{C}}_6$		${\mathit{C}}_7$
SS1	0.651	0.322	0.756	0.332	0.464	0.445	0.616	0.524	0.616	0.575	0.599	0.340	0.291	0.742
SS2	0.607	0.138	0.645	0.182	0.606	0.233	0.651	0.606	0.709	0.545	0.612	0.243	0.652	0.493
SS3	0.786	0.079	0.672	0.335	0.975	0.216	0.455	0.692	0.771	0.590	0.555	0.296	0.316	0.679
SS4	0.137	0.558	0.196	0.720	0.440	0.285	0.534	0.521	0.748	0.461	0.483	0.459	0.390	0.554
SS5	0.116	0.566	0.186	0.924	0.606	0.230	0.385	0.606	0.469	0.469	0.469	0.368	0.390	0.554
SS6	0.304	0.630	0.793	0.285	0.511	0.505	0.425	0.386	0.280	0.642	0.457	0.377	0.545	0.469

.

Step 4.2. Finding equilibrium point and optimal strategies: The Nash equilibrium can be considered a strategy pair in which all players simultaneously choose their best responses to the strategies of the other players. The equilibrium point of the game between the producer and the first land (first game) is identified as the strategy pair SF₁ and C₂, highlighted in bold in Table 16. The values (0.606 and 0.620) represent the outcomes of the best strategy. The resulting payoff matrix for the game between the producer and the second land, along with the equilibrium points, is given in Table 17. The equilibrium point representing the strategy pair SS₆ and C₃ is the balance point in the payoff matrix, with values of (0.511 and 0.505).

Step 4.3. Decision analysis: The values obtained by determining the equilibrium point represent the gains of the players. Considering the two games separately, the equilibrium point for the first game is achieved when the producer and the first land play their first strategy. In the second game, the optimal gain for the producer occurs when he plays his third strategy, assuming the second land plays its sixth strategy. While choosing between the two lands, the lands’ gains are compared to determine the optimal choice. The equilibrium points of the games are as follows.

$$\mathrm{For} \mathrm{the} \mathrm{first} \mathrm{land} \left(0.606 0.620\right) \Rightarrow {\displaystyle \frac{0.606}{0.620}}=1.01$$

(16)

$$\mathrm{For} \mathrm{the} \mathrm{second} \mathrm{land} (0.511 0.505) \Rightarrow {\displaystyle \frac{0.511}{0.505}}=0.98$$

(17)

The non-zero-sum game model indicates that the producer’s benefits vary based on the agricultural land selected, reflecting the differing potential outcomes of each option. Since the decision is not purely adversarial, the producer’s gains or losses are influenced by how well the criteria (e.g., climate, soil quality, water availability, and topography) align with their strategic goals. The interaction between the producer (decision-maker) and the agricultural lands (alternatives) affects the outcome, as their interests are not entirely opposed. MCDM integration enables the precise evaluation of trade-offs between competing objectives in game theory. The first land is preferred because it promises better economic outcomes and offers a higher potential return. The purchasing cost and soil structure are key factors influencing the selection of the first land. Although the second land offers similar water access, the superior soil structure of the first land provides a competitive advantage.

Evaluating all critical factors related to agricultural development and future planning can help determine the most profitable strategies for the players. The result from the payoff matrix of the game between the producer and the first land illustrates the Nash equilibrium (SF₁, C₂), with values of (0.606 and 0.620). The second game between the producer and the second land has the Nash equilibrium (SS₆, C₃), with values of (0.511 and 0.505). The key contribution of the proposed strategies is the efficient utilization of agricultural resources and enhanced sustainability. This result indicates that the producer’s land selection decisions influence the overall land strategy. Integrating fuzzy TOPSIS with game theory enables robust analysis in multi-criteria settings by addressing uncertainty, conflict resolution, and establishing resilient decision-making rules. Thus, we derive more reliable weights that reflect both importance and variability, while also considering strategic interactions. Land is a finite resource that will remain the primary foundation for human sustenance. Improving land management and optimizing soil resource use are crucial for sustainable agriculture. Agricultural practices and poor management lead to significant environmental degradation. The environment maintains its ecological balance sheet if its ecosystems are properly managed. An effective way to achieve a balance between the producer and the land is by using agricultural resources efficiently while ensuring environmental sustainability. If the proper analysis is conducted and decision-makers effectively manage decision-making processes, this reduces the investments required to improve soil fertility, water availability, drainage, irrigation, and erosion control. This also contributes to sustainable agricultural systems and can help mitigate the impact of the increasing climate change crisis to some extent. Moreover, it introduces effective management practices in agricultural regions that not only enhance environmental sustainability but also improve economic performance. Combining these two methods can be considered an eco-friendly approach to harnessing the Earth’s biological capacity. The proposed methodology contributes to the systematic evaluation of factors for the specific use of land in agricultural applications. However, there are still several limitations to consider. Future land selection policies will focus on multi-objective optimization to enhance both socioeconomic and ecological impacts by involving multiple stakeholders and broadening the scope of consideration.

3. Conclusions

The integration of fuzzy TOPSIS and game theory combines two powerful decision-making tools to improve the evaluation of competing alternatives in complex scenarios. This study proposes a fuzzy TOPSIS approach to derive a comprehensive game payoff matrix, which can then be analyzed using game theory to identify optimal strategies for agricultural land selection. The evaluation criteria and strategies were determined, and the importance weights were derived through the decision-making group evaluations. Finding the game’s equilibrium point ensures that players select the optimal alternatives. With the TOPSIS method, solution accuracy improves as uncertainties in real-life expressions are translated into linguistic and numerical values. This advantage is further enhanced by using fuzzy triangular numbers. Using the sensitive solution of the fuzzy TOPSIS method, the land properties were evaluated based on the producer’s demands, and proximity coefficients were derived. The relative closeness coefficients are derived to compare alternatives through the features of game theory, where players aim to maximize profits by considering each other’s actions. A broad knowledge base is essential for making informed choices among different options in decision-making processes. MCDM methods provide a framework to evaluate strategies based on multiple criteria. Converting linguistic variables from the knowledge base into mathematical expressions is crucial for effective decision-making. Game theory is a key decision-making method primarily used for addressing problems involving competitive situations. In competitive environments, determining player strategies and constructing the payoff matrix are the most critical stages of game theory. Integrating fuzzy TOPSIS with game theory offers a structured and practical approach to agricultural land selection, especially in environments with competing interests. This methodology enhances decision accuracy by accounting for multiple criteria and addressing uncertainties inherent in the decision-making process. The fuzzy TOPSIS method aids in evaluating and ranking alternatives under uncertainty, while game theory provides a framework to identify stable and robust solutions in both competitive and cooperative environments. This hybrid methodology enables decision-makers to explicitly evaluate trade-offs between competing objectives and strategic interactions within the agricultural sector. Furthermore, MCDM frameworks offer the flexibility to adapt to shifting priorities or changing external conditions, ensuring that the equilibrium remains robust and relevant over time. Because agricultural processes are affected by unpredictable climate and market conditions, future research could focus on developing adaptive payoff matrices that dynamically adjust based on real-time environmental and economic data. Exploring game theory models with adaptive strategies could assist stakeholders in re-evaluating and updating land selection criteria as environmental conditions evolve.